Geophysical Modelling of the Polar Motion 9783110298048, 9783110298093, 9783110389135

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Christian Bizouard Geophysical Modelling of the Polar Motion

De Gruyter Studies in Mathematical Physics

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Edited by Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA Dmitry Gitman, São Paulo, Brazil Alexander Lazarian, Madison, Wisconsin, USA Boris Smirnov, Moscow, Russia

Volume 31

Christian Bizouard

Geophysical Modelling of the Polar Motion |

Physics and Astronomy Classification 2010 91.10.Nj Author Dr. Christian Bizouard Observatoire de Paris Syrte-UMR 8630, PSL 61 avenue de l’Observatoire 75014 Paris France [email protected]

ISBN 978-3-11-029804-8 e-ISBN (PDF) 978-3-11-029809-3 e-ISBN (EPUB) 978-3-11-038913-5 ISSN 2194-3532 Library of Congress Control Number: 2020935901 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Contents Acronyms, abbreviations and common notation | XIII Forewords | XVII Introduction | XIX

Part I: Astrometric estimation 1 1.1 1.2 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 1.4 1.5

Earth rotation and space-time reference systems | 3 Earth rotation | 3 Terrestrial reference system | 4 Celestial reference system | 6 End of the absolute system | 6 The ether as a reminiscence of the absolute system | 8 Kinematic celestial reference systems | 9 Dynamical celestial reference system | 10 Drift of the Geocentric Dynamical Reference System with respect to the Kinematic Barycentric Reference System | 11 Reference time scale | 12 Space-time reference systems | 13

2 2.1 2.2 2.3 2.4 2.5

Geophysical irregularities of Earth’s rotation: an overview | 17 Earth’s rotation is irregular | 17 Precession–nutation | 18 Polar motion | 20 Variations of the Earth angular velocity | 24 Synthesis | 27

3 3.1 3.2 3.2.1

Astro-geodetic observations | 30 Introduction | 30 The Earth Orientation Parameters (EOP) | 30 Description of the transformation between terrestrial and celestial systems | 30 EOP n°1, n°2: Celestial Pole Offsets | 32 EOP n°3: UT1-UTC | 32 EOP n°4, n°5: pole coordinates (x, y) | 34 Full coordinate transformation from celestial reference system to terrestrial reference system | 35

3.2.2 3.2.3 3.2.4 3.2.5

VI | Contents 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.4 3.4.1 3.4.2 3.5 3.5.1 3.5.2 3.6 3.6.1 3.6.2 3.6.3 3.7

Relation between EOP and components of the instantaneous rotation vector | 36 Kinematic relations | 36 Earth orientation parameters formulated as Euler angle perturbations | 37 Relation between the CIP and the instantaneous rotation pole | 39 Relation between LOD and UT1 | 39 Terrestrial motion of the instantaneous rotation pole | 40 Principles of the Earth Orientation Parameters determination | 42 Modeling of the astro-geodetic observations | 42 Linearization and least-square inversion | 43 Observations techniques | 47 Modern astro-geodetic techniques | 47 Evolution of the observation techniques | 50 Pole coordinates accuracy | 53 Need for a combined reference EOP solution | 53 Accuracy of the combined C04 solution | 54 Uncertainty versus stability for intra-technique solutions | 54 Conclusion | 59

Part II: Polar motion theory 4 4.1 4.2 4.3 4.3.1 4.3.2 4.3.3 4.4 4.4.1 4.4.2 4.4.3 4.5 4.6

Liouville equations | 63 General introduction of the second part | 63 Newtonian framework of the Earth rotation | 63 Liouville equations | 65 Dynamical Liouville equations | 65 Linearization at sub-secular scale | 70 Linearized Liouville equation in the Terrestrial Reference System | 72 Free and forced polar motion for a rigid Earth model | 73 Rigid Earth model: definition | 73 Biaxial Earth | 73 Triaxial Earth | 74 Celestial and terrestrial motion of the CIP | 76 Incompleteness of this formulation | 78

5 5.1 5.2 5.3 5.4

Dissipative rotational excitation function | 79 Centrifugal deformation | 79 Centrifugal potential | 81 Mean figure of the Earth | 82 Variable centrifugal effect on the inertia moments | 85

Contents | VII

5.5 5.6 5.6.1 5.6.2 5.7 6 6.1 6.2 6.3 6.4 6.5 6.5.1 6.5.2 6.6 6.7 7 7.1 7.2 7.2.1 7.3 7.3.1 7.3.2 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.5 7.5.1 7.5.2 7.5.3 7.6 7.6.1 7.6.2 7.6.3 7.6.4

Influence of the Earth’s non-rigidity on polar motion | 87 Damping | 88 Complex Chandler frequency | 88 Damping estimation | 91 Effect of a mass redistribution on polar motion | 93 Ocean pole tide | 95 Introduction | 95 Ocean pole tide | 95 Oceanic rotational excitation | 99 Liouville equation for a triaxial, anelastic Earth covered by the oceans | 103 Observational consequence | 105 Symmetric and asymmetric response to a circular excitation of fixed frequency | 105 Intrinsic ellipticity | 107 Dynamical ocean pole tide in the diurnal band | 109 Conclusion | 111 Influence of the fluid core | 113 Introduction | 113 Liouville equation for a biaxial fluid core | 114 Fluid core angular momentum | 114 Polar motion differential equation for a biaxial mantle | 119 General case | 119 Polar motion periods longer than 2 days | 120 Coupled core-mantle system in the retrograde diurnal band | 121 Differential equation system for core and extended mantle | 121 Free core nutation | 124 Solution in frequency domain | 126 Free core nutation | 127 Polar motion equation for a superficial fluid layer excitation in the retrograde diurnal band | 129 Effect of the matter term | 129 Effect of the motion term | 130 Total effect | 130 Frequency dependence of the Love number and consequences | 131 Convolution in time domain | 131 Theoretical frequency dependence of polar motion resonance | 132 Confirmation from nutation analysis | 133 Influence of the frequency dependent rheology on the geophysical transfer function | 134

VIII | Contents 7.7 7.8

Influence of the solid inner core | 136 Conclusion | 137

Part III: Geophysical forcing 8 8.1 8.2 8.2.1 8.2.2 8.2.3 8.3 8.4 8.4.1 8.4.2 8.4.3 8.4.4 8.5 8.5.1 8.5.2 8.5.3 8.5.4 8.6 8.7 8.8

Hydro-atmospheric excitation | 141 Introduction | 141 Hydro-atmospheric excitation | 142 A brief historical account | 142 Matter and motion term of the angular momentum functions | 143 Gravimetric matter term | 145 Nature of the coupling between a fluid layer and the solid Earth | 146 Atmospheric excitation | 151 Pressure term | 151 Wind term | 153 Inverted barometer oceans | 153 Atmospheric Angular Momentum functions and uncertainty | 155 Ocean excitation | 160 Dynamical ocean model | 160 Sea water level variations | 160 Currents | 161 Ocean Angular Momentum function and uncertainties | 161 Hydrological excitation | 165 Total excitation | 169 Conclusion | 169

9

Equatorial angular momentum balance from two days to decadal time scale | 171 Liouville equation from two days to some decades | 171 Amplification of the prograde seasonal band | 173 Methodology | 174 Overall budget | 180 Accuracy of geodetic and angular momentum excitations | 180 Overall comparison | 180 Coherence over the time span 2000–2019 | 182 Allan deviation analysis over the period 2000–2019 | 184 Elliptical polarization of the excitation function | 185 Are prograde and retrograde terms interrelated independently from frequency? | 185 Prograde and retrograde terms and elliptical motion at a given frequency | 185

9.1 9.2 9.3 9.4 9.4.1 9.4.2 9.4.3 9.4.4 9.5 9.5.1 9.5.2

Contents | IX

9.5.3 9.5.4 9.5.5 9.6

Example of the seasonal excitation | 186 Linear dependence of prograde and retrograde parts | 189 Statistical distribution in spectral domain | 192 Concluding remarks | 195

10 Rapid, seasonal, inter-annual, and decadal excitations | 198 10.1 Introduction | 198 10.2 Rapid fluctuations | 198 10.2.1 Overview | 198 10.2.2 Spectral analysis of excitation functions | 199 10.2.3 Irregularities of the rapid fluctuations | 204 10.3 Seasonal cycle | 205 10.4 Inter-annual and infra-decadal variations | 207 10.5 Decadal and secular variations | 211 10.6 Conclusion | 212 11 Chandler’s wobble | 214 11.1 An excited mode | 214 11.2 Theoretical lineaments | 215 11.3 Estimation of the Chandler wobble parameters | 217 11.4 Hydro-atmospheric reconstruction of the Chandler wobble | 220 11.5 Multi-decadal modulation of the Chandler excitation | 225 11.6 Asymmetric excitation of the Chandler wobble | 227 11.6.1 Theoretical reminder | 227 11.6.2 Analysis | 229 11.7 Conclusion and discussion | 229 12 Diurnal and sub-diurnal hydro-atmospheric effect | 232 12.1 Geophysical diurnal and sub-diurnal effects | 232 12.2 Liouville equation in the retrograde diurnal band | 233 12.3 Diurnal and sub-diurnal atmospheric effect | 234 12.3.1 Four-daily AAM series | 234 12.3.2 Retrograde diurnal band | 235 12.3.3 Prograde diurnal and semi-diurnal bands | 237 12.3.4 Comparison of the results associated with NCEP, ECMWF-TUW, JMA models | 237 12.4 Diurnal and sub-diurnal non tidal oceanic effect on polar motion | 238 12.5 Combined effect of the atmosphere and the oceans | 240 12.6 Conclusion | 241 13 13.1

Fluid layer effect on nutation | 242 Formalism for fluid layer effect on Earth’s nutation | 242

X | Contents 13.2 13.2.1 13.2.2 13.2.3 13.3 13.3.1 13.3.2 13.3.3 13.3.4 13.3.5 13.4 13.4.1 13.4.2 13.5 13.6 14 14.1 14.2 14.3 14.4 14.5 14.6

Overview of the celestial angular momentum series | 245 Data | 245 Spectral analysis of the atmospheric CEAM | 246 Spectral analysis of the oceanic CEAM | 246 The 2–30 day band of the atmospheric CEAM | 249 Proportionality between wind and pressure terms | 249 Tidally coherent wave at 13.6 days | 251 The quasi-weekly band | 253 Hydrostatic model of the tidal O1 oscillation of the pressure term | 254 Oceanic CEAM in the 2–30 day band (25–48 hours) | 256 Contribution on lunisolar nutation | 256 Fluid layer contribution to lunisolar nutations | 256 Comparison with observed lunisolar nutations | 257 Excitation of the free core nutation | 262 Conclusion | 266 Seismic effect | 267 Introduction | 267 Theoretical bases | 268 Modeling of the co-seismic excitation | 270 Excitation of the Chandler wobble | 275 Discussion | 276 Conclusion | 278

15 Epilogue: Geological polar motion | 279 15.1 Geological mass redistribution | 279 15.2 Nonlinear Liouville equations | 279 15.3 Visco-elastic variation of the moments of inertia | 280 15.4 Linear approach | 284 15.4.1 General formulation | 284 15.4.2 Qualitative analysis for an increasing surface load | 286 15.4.3 Transient mass redistribution like glaciation followed by melting | 288 15.5 Nonlinear treatment | 291 15.5.1 General formulation | 291 15.5.2 Permanent mass redistribution like mountain upheaval | 294 15.6 Readjustment of the main pole of inertia | 294 15.7 Conclusion | 297 Synthesis | 299 Appendix | 303

Contents | XI

A A.1 A.2 A.2.1 A.2.2 A.2.3 A.3

Generalized Liouville equations | 303 Definition | 303 Solution of the generalized linearized Liouville equations | 303 Solution in frequency domain and eigenfrequencies | 303 Solution in time domain | 305 Effect on a forced oscillation at the angular frequency σ0 | 306 Generalized geodetic excitation and digitization | 306

B

Spherical harmonic coefficients of the geopotential and moments of inertia | 309 Spherical harmonic functions | 309 Spherical harmonic development of the geopotential | 311 Stokes coefficients of degree 2 and moment of inertia of the Earth | 312

B.1 B.2 B.3

C C.1 C.2 C.3

Earth figure | 315 Spherical harmonic expansion of the radius of the Earth | 315 Effect of a infinitesimal rotation on a spherical harmonic development of degree 2 | 315 Inertia matrix transformation from principal axis frame to TRF | 317

D D.1 D.2 D.3

Tidal perturbation and Love number resonance in the diurnal band | 321 Tesseral tidal potential | 321 Resonance of the geopotential Love number k2 | 322 Resonance of the loading Love number k2󸀠 | 322

E E.1

Matter term, pressure torque and loading effect | 325 Matter term expressed as a function of the tesseral component of the surface pressure | 325 Spherical harmonic expansion of the pressure torque | 325 Equatorial torque | 326 Axial pressure torque | 328 Loading Love number | 329 Relation between atmospheric torques and angular momentum in a non rotating frame | 331

E.2 E.2.1 E.2.2 E.3 E.4

F F.1 F.2

Modeling the fluid core motion | 333 Poincaré flow | 333 Order of magnitude of the residual torque on the fluid core | 335

G G.1

Statistics | 341 Correlation coefficient | 341

XII | Contents G.2 G.3 G.4 G.5 H

Explained variance rate | 341 Coherence function | 341 Panteleev pass-band filter | 342 Allan variance | 343 Usual constants | 347

Bibliography | 351 Index | 363

Acronyms, abbreviations and common notation Scientific abbreviations AAM(F) AMF CIO CIP CRF CRS GCRS cpy cpd DORIS EAMF EEAMF ECCO EOP FCN FWHM GCM GLONASS GNSS GPS HAM(F) IB ICRF ITRF IVS LOD mas NIB OAM(F) SLR TAI TIO TPW TRF TRS ATRS UTC

Atmospheric Angular Momentum (Function) Angular Momentum Function Two meanings: Celestial Intermediate Origin / Conventional International Origin Celestial Intermediate Pole Celestial Reference Frame Celestial Reference System Geocentric Celestial Reference System cycle per year cycle per day Doppler Orbitography by Radiopositioning Integrated on Satellite Equatorial Angular Momentum Function Effective Equatorial Angular Momentum Function Estimating the Circulation and Climate of the Oceans Earth Orientation Parameters Free Core Nutation Full Width at Half Maximum Global Circulation Model GLObalniy NAvigationniy Sputnikoviy System Global Navigation Satellite System Global Positioning System Hydrological Angular Momentum (Function) Inverted Barometer (response of the oceans) International Celestial Reference Frame International Terrestrial Reference Frame International VLBI Service Length Of Day (offset of the) milliarcsecond Non Inverted Barometer (response of the oceans) Oceanic Angular Momentum (Function) Satellite Laser Ranging Temps Atomique International Terrestrial Intermediate Origin True Polar Wander Terrestrial Reference Frame Terrestrial Reference System Auxiliary Terrestrial Reference System Universal Time Coordinated

https://doi.org/10.1515/9783110298093-201

XIV | Acronyms, abbreviations and common notation UT1 VLBI

Universal Time 1 (Earth rotation time scale) Very Long Baseline Interferometry

Organisations CNES ECMWF GFZ IDS IERS IGN IGS ILRS MIT NCEP

Centre National d’Etudes Spatiales European Centre for Meteorological Weather Forecasts GeoForschungZentrum International DORIS Service International Earth Rotation and reference system Service Institut National de l’Information Géographique et Forestière International GNSS Service International Laser Ranging Service Massachusetts Institute of Technology National Centre for Environmental Prospect

Notations A B C A = (A + B)/2 e = (C − A)/A λA Am Cm C −A em = mA m m Af Cf ef = Rf G GM⊕ M⊕ J2 = g Re r0 f Ω

Cf −Af Af

C−A M⊕ R2e

Small equatorial principal moment of inertia Large equatorial principal moment of inertia Axial principal moment of inertia Mean equatorial principal moment of inertia Dynamical ellipticity Longitude of the principal axis of inertia A Equatorial principal inertia moment of the mantle Axial principal inertia moment of the mantle Dynamical ellipticity of the mantle Equatorial principal inertia moment of the fluid core Axial principal inertia moment of the fluid core Dynamical ellipticity of the core Core equatorial radius Gravitational constant Geocentric constant of gravitation Earth’s mass

Coefficient of the geopotential degree 2 spherical harmonic Gravity field at Earth’s surface Earth’s mean equatorial radius Earth’s mean radius Geometrical flattening Mean Earth’s angular velocity

Notations | XV

m1 , m2 m1 + mf1 , m2 + mf2 m = m1 + im2 p = x − iy Tc fc σc (σ̃ c ) σf (σ̃ f ) Q k2 ko k2󸀠 kf󸀠

ks = 3G(C−A) Ω2 R5e h2 ρo ρ⊕ c = c13 + ic23 h = h1 + ih2 χG χA/O/H e χA/O/H

Equatorial cosine direction of the Earth’s instantaneous rotation vector in the TRS Equatorial cosine direction of the inner core rotation vector in the TRS Complex rotation pole coordinate Complex coordinate of the CIP Chandler wobble’s period Chandler wobble’s frequency Chandler wobble’s angular frequency (complex) Free core nutation angular frequency (complex) in the TRS Quality factor of the Chandler wobble Solid Earth Love number Oceanic Love number Loading Love number for the mantle Loading Love number for the core Secular Love number Displacement Love number for the solid Earth Ocean density Earth’s density Off-diagonal inertia moment Equatorial relative angular momentum Geodetic excitation Angular momentum function of the atmosphere/oceans/hydrosphere Effective angular momentum function of the atmosphere/oceans/ hydrosphere

Forewords Fruit of long years spent in the astrogeodetic team of the Paris Observatory, this book reflects a collective spirit, combining science and service, and thus anchoring our theoretical projections in various applications such as orbitography, geopositioning by satellite and space navigation. Indeed, the research presented in this book has been developed in the frame of the Earth Rotation Service which pursues the astrometric tradition of the Paris Observatory in terms of modern astrogeodetic techniques and is entrusted of producing reference Earth rotation parameters time series by the International Earth Rotation and reference system Service (IERS). Our quasi-daily monitoring of the Earth rotation variations is printed in this synthesis through many metrological considerations. So, my gratitude goes to all the staff, first of all to my closest collaborators: Daniel Gambis, former director of the service, Olivier Becker, JeanYves Richard, Sébastien Lambert, Lucia Seoane, Ibnu Nurul Huda, Teddy Carlucci, Jean Souchay, Maria Karbon, Yann Ziegler, and Nicole Capitaine with whom I take my first steps in research. In this book is found, in various ways, the footprint of discussions, collaborations, exchanges of information that I had with many colleagues or scholars, for most of them located in a East-West strip of land stretching from the Ural mountains to the Californian coast, and to whom I address the expression of my friendship or my recognition. My scientific reflection was particularly nourished by the work carried out with Alexandre Couhert (CNES, Toulouse), Leonid Zotov (Moscow State University), and, in my first publications, with Véronique Dehant (Royal Belgium Observatory) and her former PhD student Olivier de Viron (La Rochelle University), Aleksander Brzeziński (Space Research Center, Warsaw) and his former doctoral student Sergei Petrov (Saint Petersburg University). I am very grateful to Alberto Escapa (Alicante University) for its valuable corrections pertaining to intricacies of precession–nutation theory and Celestial Mechanics. Finally I am indebted to Michael Efroimsky (US Naval Observatory) for giving me the opportunity to publish this book at de Gruyter publishing house.

https://doi.org/10.1515/9783110298093-202

Introduction Science is the belief in the ignorance of experts. Richard Feynman, What is Science? Fifteenth annual meeting of the National Science Teachers Association in 1966

The Earth rotation variations The rotation of the Earth not only rules social and biological life, but it is also at the crossroads of many scientific disciplines encompassing biology, geophysics and astronomy. Indeed, the rotation of the Earth determines biological cycles at quasidaily periods (circadian cycles), our perception of the sky, duration of the ocean tides, and many geophysical processes like cyclone formation, oceanic currents, and the magnetic field. Owing to its tiny variability, almost imperceptible to our senses, concerning both the angular rate and the direction of the rotation axis, rotation of the Earth arouses great interest. First, for practical reasons: variations of the rotation of the Earth day after day modify astrometric pointing at a given sidereal instant and then influence measurements done by space geodetic techniques; processing these measurements, for instance for deriving the orbits of the implied satellite or for doing ground positioning, needs accurate estimates of these variations. More fundamentally, the Earth’s rotation changes reflect global geophysical properties and processes within the Earth. Thus, by analyzing the observed fluctuations, we can better come to know our planet. The progressive discovery of these fluctuations has a long history. In terms of observation techniques, three epochs can be distinguished. The first, from Antiquity to the early classical science in the seventeenth century, is that of astrometric pointing to the naked eye, using instruments made of wood or metal (quarter circle for example). In the seventeenth century begins the telescopic age, benefiting a double technological breakthrough: angular measurements are not only much more precise, but they are dated more accurately thanks to Huygens invention of pendulum clocks, regulated by a stable pendulum period. This second era ended around 1960 with the advent of space and atomic clocks technology: the astrometric pointing were abandoned in favor of ultra-precise measurements of flight time or frequencies of electromagnetic signals propagating over Earth scale distances.1 These technological advances, combined with the development of Newtonian mechanics first revealed the astronomical nature of the fluctuations of the Earth rotation, and at the end of nineteenth century interference of geophysical causes.

1 Or difference of flight time for the VLBI technique, yielding the time delay of a given radio wavefront at two remote antennas. https://doi.org/10.1515/9783110298093-203

XX | Introduction The Earth’s rotation: an astronomical theme until the nineteenth century By the end of the nineteenth century rotation of the Earth was still an astronomical discipline. As the precision of the angular measurements verged on 0.1󸀠󸀠 (about 1000 times the resolution power of human eye), the unique variation was the precession– nutation of the rotation axis with respect to the starry sky. The precession—this gradual shift in the orientation of the Earth axis of rotation, which, similar to a precessing top, traces out a cone in a cycle of approximately 25800 years with a declination of 23°26󸀠 with respect to the ecliptic pole axis—had been discovered from the Antiquity (around 200 B.-C. by the Greek Hipparchus); the nutation—a composition of periodic oscillations with amplitude smaller than 20󸀠󸀠 and superimposed to the precession— was unveiled thanks to telescopic astrometry; Bradley discovered its main component of 18.6 years in 1748. In the light of the new mechanics of Newton, precession and nutation result from the gravitational lunisolar forces on the Earth’s equatorial bulge. Even if the amplitude of each term is proportional to the Earth’s dynamical ellipticity, its cause is nonetheless astronomical. Geophysical breakthrough By the beginning of the nineteenth century Laplace investigated the influence of various terrestrial causes on the Earth’s angular velocity. However, the order of magnitude of the estimated effects was out of reach of the observation of his epoch.2 The geophysical breakthrough came where it was least expected. In 1750 Euler enlarged Newtonian mechanics to extended solid bodies, and from its new theorems could prove that the rotation pole freely moves with respect to the Earth’s surface with a period equal to the sidereal period divided by the Earth dynamical ellipticity [84].3 During 150 years, this polar motion was actively looked for in astronomical latitudes (angle between the true equator, perpendicular to rotation axis, and the local vertical). Finally, in 1891, Chandler [34, 35] related an oscillation of about 0.2󸀠󸀠 with a period of 430 days, not of 305 days, as expected. One year later, Newcomb [162] explained how Earth’s non rigidity can lengthen the Euler period of 130 days. This allowed him to state that the solid Earth had an elasticity comparable to the steel one. From that moment Earth’s rotation became a means for inferring geophysical properties. In the same time an annual oscillation was discovered, with the half amplitude of the Chandler wobble [36]. So, after Lord Kelvin [119], Newcomb [163] and other scientists naturally assumed the role of seasonal air and water circulation in polar motion. 2 “J’ai discuté dans le cinquième livre de la Mécanique céleste l’influence des causes intérieures telles que les volcans, les tremblements de terre, les vents, les courants de la mer, etc., sur la durée de rotation de la terre; et j’ai fait voir au moyen du principe des aires que cette influence est insensible (…)” P. S. de Laplace, Exposition du système du monde, Sixth ed. (1827) p. 344. 3 Euler determined a flattening of about 1/230 from the available measurement of the epoch [83], so he found a period of about 230 sidereal days against 305 for the contemporaneous value of the so-called Euler period.

Introduction

| XXI

In the vein of a pioneering study by Hopkins in 1839 [76], Hough, Sloudsky and Henri Poincaré theorized the effect of a fluid core, still hypothetical, on the Earth rotation at the dawn of the twentieth century. Assuming incompressible, homogeneous and non-viscous fluid contained in an ellipsoidal rigid cavity, Hough [116] and Sloudsky [202] independently predicted a second free oscillation of the rotation pole, quasidiurnal and retrograde in the terrestrial frame, resulting in a celestial nutation, of which the period is inversely proportional to the flattening of the fluid core. In 1910, Poincaré [172] showed how the terms of the nutation resonate at this period and therefore differ from those of a rigid Earth. Discovery of variations in the rotation angular rate In the early twentieth century, it was proved that for the Earth the axis of rotation wobbles, but the diurnal rotation constituted a flawless clock until the 1920s: the succession of seconds of the mean solar day (representing conventionally 1.00273 times the stellar rotation period assumed to be constant) realized the Universal Time (UT) with the convention that the mean sun passes the meridian of Greenwich at 12 hour UT. But, following the work of Newcomb and Spencer Jones (1926) [208], there was recognized in the Length Of Day (LOD) a secular growth of about +1.6 ms per century and decadal fluctuations of a few milliseconds, which are interpreted by the dissipation accompanying tidal deformations and core–mantle coupling, respectively. Then, in the 1930s, comparing UT with the time of quartz clocks which had just been developed, Stoyko found seasonal variations of about 20 ms, namely 0.5 ms in LOD, and it was immediately thought of in terms of the impact of seasonal atmospheric circulation. Advent of the Space Age and consequences In the period following the discovery of polar motion, instrumentation did not make substantial progress, and the quality of optical surveys was still affected by atmospheric turbulence. But regular observations of the latitude were then carried out over several observatories around the world to determine the terrestrial oscillations of the pole. And from 1900 to 1960, the temporal resolution going from months to weeks, the accuracy from grew from 0.03󸀠󸀠 to 0.01󸀠󸀠 . In the 1960s began the Space Age. Satellites, providing measurements of the whole Earth’s surface, enabled the determination of overall physical properties of the solid Earth and its fluid envelope, in addition to balloon or local surface measurements. Processing an increasing quantity of information was then made possible thanks to growing computer capabilities. This resulted in considerable progress in the measurement and modeling of meteorological, oceanographic and hydrological hazards. Analysis of satellite orbital perturbations also revolutionized our knowledge of the Earth’s gravity field on a large scale. From this epoch, “planetary” technology, such as Very Long Baseline Interferometry (VLBI), was developed to determine the shape of the Earth, its gravity field and its motions, and gradually replaced optical astrometry for ground positioning as well as

XXII | Introduction for geodynamic studies. Space geodesy was born. In the 1980s, it addressed global deformations like tectonic drift. In 1985, it permitted to determine the Earth’s rotation irregularities 10 times more precisely than the optical astrometry does; this technique, downgraded, was abandoned in geodynamic studies and geodetic applications. This concomitant progress in the knowledge of Earth’s surface movements and mass redistributions operating at its surface, or within it, confirmed the hypotheses made at the end of the nineteenth century. First, polar motion is indeed influenced by the atmosphere, a little less by the oceans and inland waters (ice, snow, soil moisture and vegetation included). Second, from the 1980s the VLBI observations clearly have unveiled the resonance of the lunisolar terms of the nutation at a 430 day period, in 1986 a non-tidal nutation was discovered at this period and immediately attributed to the corresponding free mode. Decoupling the rotational speed from the polar motion Over periods smaller than one century, changes in the rotation of the Earth are small enough to be treated as perturbations. The law of angular momentum written in the terrestrial reference system then provides a set of first order linear differential equations, the Liouville linear equations decoupling angular rotation velocity (axial perturbation) from rotation axis direction (equatorial perturbation involving polar motion and nutation). In these equations the various geophysical and astronomical effects can be calculated separately. Summing them, we obtain the total effect which can be compared with the observations. Often this summation is not necessary because the studied effect has such a particular signature that it is easily recognizable in the observed variation and cannot be confused with the imprint of another phenomenon. Nature of axial and equatorial excitations For periods of less than 10 years, changes in LOD (1 ms amplitude) primarily result from tidal deformations and atmospheric winds. Beyond 10 years, the variations are larger, and they are commonly attributed to the interaction between the core and the mantle. However, the modeling of this internal mechanism is widely speculative and cannot be verified by direct observations. In contrast, polar motion originates from the surface fluid layer or the lithosphere, that is to say, superficially with regard to the diameter of the Earth.4 They are observable, at least estimable, in contrast to internal processes for which there is only indirect observations like magnetic field traducing mass transport within the core as far as dynamo model is sound. So, looking at their causes, polar motion and length of day variation differ substantially. 4 With the exception of the Markowitz term (20–30 years), which may result from the core-mantle interaction.

Introduction

| XXIII

Purpose of this book These considerations lead us to treat polar motion, and in a broader sense nutation, irrespective of the LOD variation. Polar motion remains better measured than it is understood. The compilation and analysis of historical observations can trace it since the middle of the nineteenth century. Over longer periods, the secular drift at the rate of about 0.4󸀠󸀠 /century, is larger than the Chandler and seasonal wobble with a total amplitude not exceeding 0.4󸀠󸀠 . This secular trend not only comes from mass redistribution but it can eventually result from the continental drift. If all stations, owing to their lithospheric drift, rotate on the mantle, the pole will shift accordingly. So in the drift will overlap both tectonic effect and changes of the moment of inertia, such as those caused by the mountain uplift or ice melting. Some observations, such as fossilized tropical plants in now temperate areas, demonstrate a shift in the pole of rotation relative to the continents, but it is difficult to distinguish the tectonic shift from the effects of mass transports. But we only evoke this distant past in the last chapter, for we shall focus on the contemporaneous or astrometric polar motion, started about 150 years ago. This time window allows us to embrace any oscillation from a few hours to hundred years. The purpose of this book is to model them in the light of global geophysical processes, as we modeled them today. Sometimes the theory is so complex that we cannot see its relevance to interpret the observation; conversely the usual theoretical formalism cannot be fully consistent with the present accuracy of Earth rotation determination. So, we endeavor to tie the theoretical modeling to the uncertainty that spoils the geophysical excitation. By stressing this “metrological” aspect, we hope to complement the books of Sidorenkov [200, 201], throwing meteorological light on the instabilities of the Earth rotation. Besides seasonal, daily and half-daily variations, polar motion has a strong stochastic part, making it hardly predictable in contrast to the lunisolar precession–nutation. In this view, this work is a logical extension of the recent book of Dehant and Mathews [60], mostly devoted to the regular wobble of the rotation axis,5 namely the lunisolar precession–nutation. The description of the Earth response to an external potential through the Love numbers formalism is a cornerstone of the polar motion theory. At contemporaneous time scale, this question is treated in the book by Dehant and Mathews, but more comprehensively in the book by Symlie (2013) Earth dynamics: deformations and oscillations of the Rotating Earth [204], which also brings precious insight for investigating the possible seismic effect on rotation of the Earth. Whereas it is mostly focused on the sub-secular polar motion, this synthesis finally touches on the polar motion at secular and geological time scales. Then, the quasi-elastic solid Earth has to be given up for a visco-elastic Earth. For completing our short review of 5 In a general sense, encompassing the observed rotation axis or celestial intermediate axis, the instantaneous rotation axis, the figure axis or the angular momentum axis. All these notions will be precised later.

XXIV | Introduction

Figure 1: Study of polar motion.

this problem, we recommend the more comprehensive book of Sabadini, Vermeersen and Cambiotti [178]. In tracking and modelling polar motion, the space-time reference systems are cornerstones, of which the reader will find an updated and comprehensive presentation in the book of Soffel and Langhans [205]. The other references go back to the 1980s. Munk and MacDonald (1960) [153], Lambeck (1980) [126], and Moritz and Mueller (1987) [152] contain much valuable material, but these references have not not benefited from the advances of space geodesy and global circulation models of the hydro-atmosphere. Contents While this book presents the polar motion studies with its most recent developments, it is addressed not only to specialists but also to a wide public not having prerequisite knowledge in geophysics or astrogeodesy. The reader will discover or deepen the intricacies of the polar motion, studied in light of a dynamical system: i) the input, namely the geophysical excitation, ii) the transfer operated by the Earth system and iii) the output, namely the polar motion (see Figure 1). First we give an overview of the system output, that is to say, changes in the rotation of the Earth. After exposing the basic aspects (space time reference systems, Chapter 1), we give a synthetic outlook of the Earth rotation variations (Chapter 2), and we describe the way they are parametrized and determined by processing astrogeodetic measurements (Chapter 3). In the second part we formulate how the Earth system transmits the mechanical excitation to polar motion. After giving the dynamical foundations (Chapter 4), we consider the effect of the rotational deformation of the mantle (Chapter 5) and of the ocean surface (Chapter 6) accompanying the polar

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motion, and model the influence of the fluid core (Chapter 7). In the light of the current observation accuracy, a biaxial Earth model and other theoretical approximations are insufficient. Accordingly triaxiality and the asymmetric effect of the oceans are considered and lead us to introduce the formalism of generalized Liouville equation in the equatorial plane (Appendix A). Finally, the third part is devoted to the study of geophysical excitation (mainly produced by the mass redistribution) and its effect on polar motion within the theoretical framework of the second part and for time scales ranging from 12 hours to a few decades. Atmospheric, ocean, hydro-continental excitations are described (Chapter 8), they are compared globally to polar motion (Chapter 9), and then in more detail for characteristic frequency bands of the contemporaneous polar motion: rapid, annual, inter-annual, and decadal (Chapter 10), Chandler band (Chapter 11), diurnal and sub-diurnal (Chapter 12), and retrograde diurnal band (Chapter 13). This allows us to clarify some outstanding issues such as the origin of the Chandler term (Chapter 11), the impact of inland fresh waters (Chapter 9), diurnal and subdiurnal variations (Chapter 12), and their effect on nutation (Chapter 13). Finally, extending our considerations even further, we deal with transient or jerk-like excitation by modeling the seismic contributions (Chapter 14), and treat the polar drift produced by the largest known mass redistribution, occurring at prehistoric or geological time scales (Chapter 15), and then determined by a visco-elastic mantle.

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Part I: Astrometric estimation

1 Earth rotation and space-time reference systems To determine the relationship between physical quantities, i. e., the measures of these quantities— we have express term of comparison units and arbitrary states, without knowing the natural units and preferred states (…); for studying the movements of the body we are forced to adopt reference systems related to some of these bodies, so moving reference systems, unable to reach the absolutely fixed system; and to date the events we have to choose our starting time in the temporal limits of human experience, without knowing the first moment of world history. Augustin Sesmat in Le système absolu classique et les mouvements réels, translated from French (Ed. Hermann 1936).

1.1 Earth rotation It is probably useful to remember what the rotation of the Earth and the basic concepts related to it are, even before coming to its irregularities like polar motion. At first glance it is the rotation of the lithosphere1 relative to the stars. But the astrogeodetic observations, combining Earth bound quantities and those associated with celestial bodies including artificial satellites,2 have revealed that the crust undergoes deformation (several cm per day) and the stars are not fixed beacons. As rotation can only be defined between two rigid axis systems, the determination of the Earth’s rotation requires the prior realization of a terrestrial coordinate system and of a celestial reference coordinate system, endowed with orthonormal axes for easing the processing of the astro-geodetic observations. Given real “objects” as ground stations and observed directions of stars, we have to extract a network of fiducial points that do not move with respect to one another, a kind of ideal rigid body, against which we shall coordinate all the astro-geodetic observations. Our knowledge of the Earth’s rotation changes is all the broader as the celestial and terrestrial references are precise. For example, if the axes of the celestial reference frame are given with an accuracy of 0.1󸀠󸀠 , the non-rigidity of the Earth will remain veiled in the nutation, and the rapid polar motion, with an amplitude 0.001󸀠󸀠 will be out of scope. Measuring the Earth’s rotation also requires a time reference that is more stable than the one given by the diurnal cycle. With the advent of relativity in astrometry since the 1960s, time and space intertwine to form a space-time reference system. All these reasons lead us to devote this introductory chapter to space-time reference systems.

1 From the Greek language lithos, stone, specifically the lithosphere is the mineral portion of the crust, excepting all liquid or gaseous elements. 2 The field devoted to the corresponding techniques and observation processing is called astrogeodesy or space geodesy. https://doi.org/10.1515/9783110298093-001

4 | 1 Earth rotation and space-time reference systems

1.2 Terrestrial reference system The Terrestrial Reference System, abbreviated TRS, is an orthonormal Cartesian coordinate system Gxyz attached to the crust, and which center is located on the center of mass of the Earth, and in which the relative angular momentum3 of the lithosphere is zero: this is the so-called Tisserand condition, of which the realization through a set of tectonic rigid plate model is termed no-net rotation condition. Consequently, referred to TRS, the lithospheric displacement field contains pure strains only. Such a system is set to a constant rotation transformation, so that it has to be specified by stating both the terrestrial position of the geographic north pole and the prime meridian. Until the end of the nineteenth century the rotation pole defined the geographic pole, but as soon as its wobble was discovered with the latitudinal variations, the rotation pole was substituted with its mean position of the observation epoch as the geographic origin. Then the prime meridian was taken as the semi-major circle intersecting the north pole and Greenwich Observatory. But the mean pole drifts, as was first observed in the 1920s by the American astronomer Lambert [131], and confirmed in the 1960s. As attested by Figure 1.1, the rotational pole has shifted by several meters since 1900—about 300 milliarcseconds (mas) for the corresponding direction of the rotation axis. In this respect, the mean pole of reference was conventionally defined by fixing in 1967 the geographic latitudes of five stations of the International Latitude Service, corresponding to the position of the mean rotation pole for the epoch 1900–1905 according to astrometric reduction at 10 mas level accuracy. This reference pole has been called the Conventional International Origin (CIO) [218]. The advent of space technique in the 1960s, and their increasing weight in determining the Earth’s rotation parameters until the discontinuation of optical data in the 1990s made both CIO and Greenwich meridian obsolescent. In the current implementation of the TRS, based upon observations carried through by the Doppler Radio Integrated System (DORIS), the Satellite Laser Ranging (SLR), the Very Long Baseline Interferometry (VLBI), and the Global Navigation Satellite System (GNSS), the prime meridian is no longer fixed relative to some Earth ground station and passes about a few hundred meters of the Greenwich geodetic point, whereas CIO is replaced by the IERS reference pole, of which the relative position to observation stations is determined with an accuracy of a few millimeters. The first version of the modern TRS goes back to 1984 with the BIH Terrestrial System 1984 (BTS 84). The geographic axis Gz is close to the figure axis, that is, the axis of symmetry of the Earth’s reference ellipsoid. This ellipsoid is an idealized figure of Earth, composed of homogeneous layers and not affected by any variable deformation. At time scales longer than the period of convection in the mantle (millions of years), it is 3 See Chapter 4 for the definition of the relative angular momentum.

1.2 Terrestrial reference system

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Figure 1.1: Mean polar path from 1900 to 2020 obtained by Gaussian smoothing from IERS C01 pole coordinates. This plot also displays the full polar motion corresponding to the periods 2018–2020 and 1903–1905. In 1900 the pole position in the terrestrial frame was determined with an uncertainty of about 50 mas, which is lower than 0.1 mas nowadays.

conceivable that regional deformations fade so that the Earth takes the form of the reference ellipsoid imprinted by the centrifugal force. The figure’s axis, mean rotation axis, and mean principal axis of largest inertia moment (and corresponding poles) then become collinear. The practical realization of the TRS, the so-called Terrestrial Reference Frame (TRF), according to the distinction fashioned by Jean Kovalesky [123] and recommended by IAU is to choose a set of geodetic points (where geodetic techniques are located) and specify their Cartesian coordinates. Until the 1960s, points anchored to the ground with invariant coordinates were sufficient. Indeed, tectonic drifts of a few cm per year and tidal displacements were not detectable by optical astrometry (excluding changes in the vertical). But with centimeter accuracy (sub-centimeter after the advent of GNSS), space geodetic techniques have made this approximation invalid, forcing us to separate the geodetic points from the ground. These points, idealizing the location of receiving or emitting antenna, are freed from the well-modeled solid tidal movement (40 cm) and eventually from displacement resulting from the ocean’s tidal loading (a few cm in the coastal area). This provides a network of immaterial points (abusively called Station Coordinates) matching the tectonic movements. In practice, the coordinates of these points are composed of initial coordinates at a given epoch t0 and a linear function of time representing the tectonic drifts. By abuse of language these points are confused with the TRF whereas they implicitly define the direction of the axes of the TRF at a given time with respect to the ground. Insofar as observation refines the tectonic shift models, this leads one

6 | 1 Earth rotation and space-time reference systems to redefine regularly the TRF by imposing the requirement that between two consecutive realizations there is no overall rotation. The achievement and maintenance of the TRF, namely the International Terrestrial Reference Frame (ITRF) is attributed by the IERS to the LAboratoire de Recherche en Géodésie (LAREG) de l’Institut National d’information géographique et forestière (IGN). It is difficult to quantify accurately the residual angular momentum of the lithosphere in the ITRF, and thus to find to what extent the ITRF satisfies the Tisserand condition. In fact, if the coordinates of the TRF stations are determined across one day or one week, they highlight additional fluctuations at centimeter level, reflecting the deformations produced by the hydrological and atmospheric loading (then mostly at seasonal periods), the post-seismic ground relaxation, or mis-modeling of the tidal displacements. The origin’s G of the ITRF is implicitly defined with respect to the Cartesian coordinates of the stations. Its does not coincide with the Earth center of mass (the origin of the ITRS!), which position fluctuates because of the permanent mass transport within the Earth and at its surface. In this respect observations done by satellite techniques—SLR and DORIS, GNSS to a lesser extent—allow one to determine displacements of the center of mass at centimeter level with respect to the network of observing stations [212].

1.3 Celestial reference system The celestial reference frame is the foundation of astronomy: it is both a means to identify and coordinate the positions of the stars, and a framework to theorize as regards their motion. It is also the purpose, because there is no more a burning issue such as that of our location in the universe. Undoubtedly, the pursuit of a heavenly absolute reference, was one of the main motivations of astronomy and physics until the twentieth century, before the mainstream rallies of the Copernican principle and its rejection of any center.4 Since antiquity the question has arisen of the spatial reference to which the planetary motions have to be assigned. Long under a bushel by the Aristotelian geocentrism, it would reappear in the sixteenth century, and crystallize in the passionate arguments of the seventeenth century that opposed heliocentrists and geocentrists. 1.3.1 End of the absolute system The most natural way is to take as a reference the relative directions of some electromagnetic signals (visible or not): they form a “mesh” allowing for the identification 4 Citing a fragment attributed to the mythic Ancient philosopher Hermes Trismegistus, theologian Nicolas of Cusa wrote in the fifteenth century: God is an infinite sphere, the center of which is everywhere, the circumference nowhere.

1.3 Celestial reference system

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of the neighboring stars. Until the seventeenth century it was commonly believed that the stars were stowed in a sphere, the center of which was the Earth. With measurements made with the naked eye, stars kept the same relative positions regardless of the position of the observer, and it made clear that they were placed on a single sphere very far in terms of the size of the Earth. A metaphysical sphere was still the gold standard in response to the quest of a harmonic cosmos, desired by Greek philosophy and the Christian faith. The emerging heliocentric vision of the sixteenth century had not shaken the spherical cosmos. Copernicus essentially shifted the traditional center of the Cosmos from the Earth to the Sun. Most astronomers of the time admitted the traditional dichotomy between the earthly and the heavenly worlds that began with the orb of the Moon.5 But with the discovery of sunspots, the variable brightness of certain stars, etc., celestial bodies lost their status of incorruptibility, and one gradually gave up on the idea that they were made of a material other than the Earth.6 In the same vein the discovery of lunar mountains, of the elliptical nature of planetary orbits revealed a celestial world where sphere, circle and uniform circular motion—guaranties of perfection—were no more exclusive.7 The sphere of the fixed stars was losing its heavenly splendor. The coup de grace was firmly administered with the recognition of the proper motion of the stars in the eighteenth century. The absolute reference was disappearing. However, measurements of the star positions were always more accurate. During the seventeenth century, the introduction of optical instruments revolutionized astrometry, since the angular resolution, only visual up to that time, went from 2󸀠 to 1󸀠󸀠 . It was expected that one could solve some puzzles such as the distance of stars. Thus, as the reference of the “fixed star” lost its absolute character, its practical interest grew. To measure the movement of the planets, look for parallax, and detect changes in the Earth’s ro5 If the four elements—earth, water, air and fire—constituted the changing and corruptible terrestrial substance, the ether gave to the celestial world perfection and incorruptibility, so that one should meet there only the perfect figures of the circle and the sphere. But note that, in his book La corruption des cieux par le péché (1672), the French cleric Francois Placet argued that also most ancient philosophers believed that the heavens were corrupted. 6 Like the backwash, ideas outweigh an epoch or break it to reappear on tiptoe. And some astrophysical data suggest that the universe is full of dark matter and energy, of which the nature remains unknown and it would constitute up to 80 % of the total mass of the universe (assuming the matter– energy equivalence)! Moreover, the cosmic black body radiation, akin to that of a spherical shell in thermal equilibrium on the edge of observable universe, is not without evoking the cosmic sphere of the Ancients. Some would even suggest that the cosmic microwave background is an absolute reference in the sense that any movement can be reported by Doppler effect. The redshift of the galaxies in the Hubble law involves the characteristic distance R ≈ c/H, H denoting the Hubble constant. In cosmological models based on general relativity, R is close to the radius of the observable universe or horizon radius. 7 This ancient obsession brought certainly some fruits: by breaking down planetary motion into a sum of uniform circular motions, it was a prefiguration of the Fourier transform. Dynamic theory of the polar motion, by exhibiting a free circular mode, requires this Ancient practice.

8 | 1 Earth rotation and space-time reference systems tation, catalogues giving the star declination and right ascension with an increasing accuracy had to be provided. The star’s proper motion in itself is not a fundamental obstacle provided that it is known. Knowledge of the spherical coordinates of the stars at a given time, and their proper motion if necessary, is sufficient to provide a grid of reference at the date of observation. In 1740 Bradley discovered a curious annual movement that systematically affects the star positions, which he could explain by the composition of the speed of light along the sight-line with the orbital velocity of the Earth. As it was not expected, it was called annual aberration. In 1830 or a little bit earlier, by observing Vega at Pulkovo observatory Struve discovered the first annual parallax (0.125󸀠󸀠 ), and a few months later Bessel made a more sound estimation for 61 Cygn (0.314󸀠󸀠 ). It thus appeared that the geocentric stellar directions undergo parallactic oscillation accompanying the annual orbit in the geocentric reference frame. This was not the appropriate framework to coordinate the stars. It became clear that the apparent positions of the stars were to be coordinated in a heliocentric system where parallax and annual aberration vanish, provided that these perturbations are calculated and taken into account at the time of the observation.

1.3.2 The ether as a reminiscence of the absolute system In the early nineteenth century, whereas one gave up any illusion as to the absolute nature of the reference provided by the stars, the wave nature of light was highlighted, and in the same time, as it was believed to be a matter of fact, the existence of its propagation medium, the ether. This invisible and hypothetical substance is a new avatar of absolute reference. As the light is assumed to propagate with an uniform speed in the ether, its speed is changed with respect to any object moving with respect to this medium. One of the problems of physics becomes the detection of motion of the Earth in the ether. Many optical experiments were realized to detect the famous ether wind that should sweep the surface. The most famous experiment is by Michelson and Morley (1887) [148]. But none of it conclusively demonstrates an effect of the orbital motion or cosmic velocity of the Earth.8 The ether, in the unified theory of Maxwell, becomes 8 Experimental controversy raged: in 1920–1930 the American physicist Miller repeated the Michelson experiment and reported an ether wind, which was attributed later to a thermal effect but where we can observe curious astronomical patterns [2]. The most amazing lie in the Sagnac effect (1913) [179]. This French physicist designed a new type of interferometer in which two light beams travel a closed path circumscribing a surface in opposite direction (in the Michelson interferometer beams circumscribe a null area). The device having been deposited on a turntable, Sagnac observed displacement of the interference fringes corresponding precisely to the Galilean composition of the light speed with the speed of rotation, according to classical electromagnetic theory. Following a suggestion by Fitzgerald, Michelson realized a giant Sagnac interferometer secured to the ground, which allowed him to observe the diurnal rotation in accordance to the expected ether wind [149]. Special relativity

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the substrate of all electromagnetic and electrodynamic phenomena. In particular, the emergence of a magnetic field is dependent on the movement of electric charges with respect to the ether. But the experience of Trouton and Noble (1903) shook the certainty [216]. The discussion of the ether was closed with the theory of relativity, according to which the speed of light is the same in all reference frames and the magnetic field depends only on the speed of the electric charge relative to the observer. 1.3.3 Kinematic celestial reference systems Thus, the stars remained the best Celestial Reference System (CRS). By removing their annual aberration and, if necessary, their annual parallax, and their proper motion, from their apparent positions, we obtain a network of fictitious points absolutely immobile between them. The elimination of aberration and parallax is equivalent to placing the origin of the CRS at the center of mass of the solar system (Barycentric Celestial Reference System, BCRS). As for the TRS, the choice of the pole or the equator is conventional (the pole of the BCRS is chosen at proximity of the rotational pole without nutation on the 1st of January 2000, 12 hours UTC). For the study of the Earth’s rotation, we prefer the geocentric celestial system (GCRS), of which the axes keep the same directions as those of the BCRS, but whose center is coincident with the center of mass of the Earth. Since the eighteenth century, the star catalogs were permanently completed with more and more accurate positions. Through surveys of the Hipparcos satellite in the 1990s, star positions reach an accuracy of a few mas. With the GAIA satellite (launched from French Guyana on December 19 2013) the catalog extends to about 1.7 billions stars with an uncertainty smaller than 1 mas. Meanwhile, in 1970, the VLBI technique put an end to the unchallenged reign of the stars: while their optical pointing is limited by atmospheric turbulence, radio interferometry avoids such pitfalls and extragalactic radio sources photo-centers (quasars, QSOs) are positioned with unparalleled precision, confining 0.5 mas. In 1997 the International Astronomical Union (IAU) has raised the CRS of extragalactic radio sources to the rank of International Celestial Reference System (ICRS). Its concrete realization, the International Celestial Reference Frame (ICRF), is constituted of the Cartesian coordinates of a few hundred extragalactic radio sources whose relative angular movements do not exceed 10−6 󸀠󸀠 /year. In 2018 the IAU adopted the ICRF of third generation (ICRF3). All these systems or frame have in common that they are based on celestial objects. As they do not involve any physical theory, and are entirely determined by the positions, possibly the relative motion of these objects, they are called kinematic systems or frames. and general relativity describes very well the Sagnac effect in predicting additional terms in (Ω/c)3 but these are too small to be measurable.

10 | 1 Earth rotation and space-time reference systems 1.3.4 Dynamical celestial reference system The exposure of celestial reference systems would be quite incomplete if we would ignore the developments of mechanics from the seventeenth century. The scholastic philosophers had long wondered about the cause of the conservation of the motion. Following the work of Galileo and Descartes, physical science became experimental and operative, and distanced himself from metaphysics and theology. Focusing more on “how” than on “why”, on the mathematization and quantification of the phenomena, scientists were able to identify descriptive laws of the motion, whose synthesis was made by Newton. The first of these laws is the principle of inertia, formulated by Galileo and Descartes: “any moving body tends to continue its motion at straight and uniform speed”. Newton will then specify: “provided that it is not subjected to any force”, and thereby defines force as “the cause of the change in the intensity of the motion”. Experience showed him a body X times heavier accelerates X times less under the same effort, and he defined force as the product of mass and acceleration. This is the famous fundamental equation of dynamics. The question then arises of in which reference system such laws must be formulated. The empirical answer is that such a system is not accessible, but can be determined approximatively. First this is the lab, but if the considered motion is on a larger scale, involving increasingly low acceleration, we shall have to consider successively the geocentric non-rotating system, the heliocentric system, the galactocentric system, etc. This Newtonian absolute reference, somewhat mythical as it is, seems to challenge the observable universe because all celestial bodies present relative acceleration. So how could one imagine that there is a body in a state of perfectly uniform motion or of absolute rest? The discovery of the Sun’s galactic rotation at the velocity of 300 km/s has dethroned the center of mass of the solar system as a possible origin of the inertial system. So should one position the origin at the center of the galaxy? But does it not accelerate towards the galactic Virgo cluster? The absolute reference body is therefore out of the scope of the physical realm. In general relativity this Gordian knot is untied by stating there is no preferred or absolute spatio-temporal system for describing physical laws. Physical time is local, the inertial character is local and applies to all bodies subject to gravity alone, that is to say, free fall is the natural state, reflecting the surrounding properties of space-time (the curvature). The global inertial frame of Newton is abandoned in favor of a local inertial frame. Despite the admirable mathematical synthesis of Newtonian mechanics and general relativity, one and the other cannot explain the gravitational motion from the perspective of the classical philosophy: if it is considered that any movement proceeds from a cause, that is to say, requires the action of a motor as in the example of a moving car, it must be recognized that the gravitational force or the curvature of the space-time in Einstein is not the cause but the effect (i. e. the acceleration as the inverse square of the distance). Whether it is the mysterious action-at-distance force, of which Newton was denying reality, or the space-time curvature, the engine of the

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orbital movements alias gravitation remains occult. And under these conditions the gravitation theory, whether Newtonian or relativistic, saves appearances as well as the Ptolemaic model. The inaccessibility of the Newtonian inertial system must not hide its approximate achievements, or dynamical reference system characterizing a local inertial frame. The most common example is the Dynamical Barycentric Reference Frame whose origin coincides with the center of mass of the solar system, and axes are implicitly determined with respect to the planets position at a given instant. Indeed the application of the law of universal gravitation permits one to accurately reconstruct the trajectory of the planets and Sun in this frame, despite some minor defects solved by general relativity. These trajectories realize a celestial canvas for any other body in the sky. For instance, if we observe a particular comet near Jupiter, we are able, through the coordinated path of the center of mass of this planet, to locate it in the barycentric frame. A second kind of Dynamical Reference Frame is the geocentric one realized by the orbits of artificial satellites for which orbits are accurately determined (GNSS, DORIS, satellites equipped with laser reflectors). Another way to find the orientation of a body relative to a local inertial frame, without the intermediary of orbital mechanics, is possible by placing a gyro-unit measuring its instantaneous rotation vector. Then, by numerical integration of the kinematic equations one can reconstruct the rotation matrix of the body with respect to the local inertial frame. Pocket optical gyroscopes, based upon the Sagnac effect, successfully applied to aircraft guidance, are sensitive to the Earth’s rotation. But only a large laser gyro (a few meters size), such as those from Christchurch (New Zealand) and Wettzell (Germany), can track its irregularities with increasing performances: having first detected the diurnal and semi-diurnal oscillations of the instantaneous rotation vector (mixed with ground displacements of the same period) [192], they are now able to record seasonal variations [191]. With larger stability and much smaller surface (∼ 10 cm2 ), matter wave gyros represent a promising alternative (at best their stability is 1 nrad/s at 104 s of integration time [72], that is, about 10−5 of the Earth’s angular velocity).

1.3.5 Drift of the Geocentric Dynamical Reference System with respect to the Kinematic Barycentric Reference System According to general relativity, the Dynamical Geocentric Reference System derives from the kinematic Barycentric Reference System reference: it slightly rotates at the rate of 20 mas per year around the axis of the ecliptic poles. This is the so-called geodetic precession, included in the global precession of the equatorial plane with respect to the ecliptic. Nevertheless, the common Newtonian precession of the Earth cannot be even modeled with an accuracy of 20 mas/year—because of our ignorance of the mass distribution in the Earth, so that the relativistic precession cannot be isolated in

12 | 1 Earth rotation and space-time reference systems the global effect. The geodetic precession is accompanied by an eponymous annual nutation, of the order of 30 µas per year. In principle, it should be considered in the VLBI measurements of the annual wobble. So far it could not be clearly separated from geophysical effects [19].

1.4 Reference time scale Motion is inseparable from time. As wrote Aristotle, “Time is not motion, whereas it does not exist without motion”.9 And, later in the same opus, he stated that “time is the number of motion”. Relativity reaffirmed this nesting, refusing the metaphysical, so unprovable absolute time. Berkeley and Mach had already denied the absolute space as objective data. The purpose of spatial reference frames is not to coordinate stationary objects, but moving ones, that is to say, to determine their movement. In the measuring process, the time coordinate had to be added to measured positions for distinguishing and ordering them along time. For this purpose, it is necessary to build a reference time scale. Until 1972 the succession of mean solar days was realizing the Universal Time (UT) used worldwide as the basis of the various legal time scales. In 1937 Nicolas Stoyko while officiating at Observatoire de Paris10 found seasonal variations in the UT compared to quartz clocks, invented in the late 1920s [211]. Some observations of the planets and the moon had already highlighted that the UT was not as uniform as the time associated with their motion, the so-called Ephemeris Time (ET). The time scale ET is neither more nor less than the practical realization of “Newtonian absolute time” involved in the differential equations ruling planetary motion. But it is inappropriate to describe a number of phenomena such as atomic clocks drift with altitude and with the speed of the satellite on board of which they are embedded, etc. General relativity provides a consistent description of these anomalies provided that the time scale is specific to a spatio-temporal frame where the curvature of spacetime is constant and uniform. Regarding to the current precision of the atomic clocks (at best 10−16 s), the most natural choice is the time referred to the geoid 11 , the Terrestrial Time (TT) (tidal effects on curvature are ignored). Terrestrial Time is the proper 9 Aristotle, Physics, IV. 10 Nicolas Stoyko (Odessa, 1894–Menton, 1976). Russian-born, trained at the University of Odessa by Prof. Alexander Orlov, a leading specialist in polhody, Stoyko was driven from his country by the Bolshevik Revolution and settled in France in 1923. The following year the Paris Observatory offered him a position in the Bureau International de l’Heure, created in 1912 and he became its director in 1945. 11 A given equipotential surface of the geopotential (composed of the gravitational and centrifugal potentials), approximating the mean sea level at equilibrium. The value W0 of the equipotential is fixed conventionally by the International Association of Geodesy.

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time of an atomic clock located on the geoid. The practical realization of TT is achieved through a network of atomic clocks distributed throughout the globe, and is called the International Atomic Time (Temps Atomique International, TAI). Its time unit, the SI second, is defined as an integer number of periods of transition between two energy levels in the cesium 137 atom. The constant difference TT − TAI = 32.184 s results from the fact that TT extends in a continuous way the ephemeris time whereas TAI extends the UT, so at the moment of their creation on 1 January 1958, TT − TAI was taken as the difference ET − UT = 32.184 s of this epoch. Standard unit time, the TAI second accumulated since 1 January 1958. Due to the slowdown of the Earth rotation, the mean solar day increases on average by about 1.4 ms/century according to ancient records of lunisolar eclipses. Equivalently the UT second increases at the rate of 2 10−5 ms/century. This is confirmed by 150 years of astrometric observations, but over decadal time scale fluctuations of several ms dominate the length of day offset (see Figure 2.5). The TAI second was chosen as the 1/86400th part of the mean solar day in the 1820s. As a result, the TAI beats slightly faster than the UT, and presents an increasing excess with respect to UT, which is accelerating due to the secular increase of the UT second mentioned earlier. For applications of Celestial Mechanics or processing of observations of objects outside the Earth and its close environment (stars, quasars, space probe), the most convenient system is the BCRS, in association with the Temps Coordonnée Barycentrique (TCB), namely the barycentric time coordinate. The time reference is all the more important as modern astro-geodetic measurements are not, strictly speaking, measures of angles or distances but frequency or time measurements, like the time travel of a radio pulse from a GNSS satellite to a terrestrial receiver. Building a physical model linking these time measurements to geodetic parameters, it is possible to determine the TRF orientation with respect to the CRF with an uncertainty of 0.1 mas, thanks to the incredible accuracy of the electromagnetic pulses dating, about 10 picoseconds. With traditional optical astrometry, the uncertainty did not drop below the 10 mas level.

1.5 Space-time reference systems The reference systems of time and space are the basis of any rational knowledge of our environment, like Earth’s rotation, and it permits one to master Nature and use it. The manner we coordinate objects in space and time takes its roots in the most ancient civilizations. Their definition and implementation have been refined over the ages to match potentialities of the observation techniques. Thus a deep understanding of present space-time reference systems cannot be restricted to the present technical achievement brought about by atomic clocks and space geodesy, but has also to consider the long term maturation of time-space measurements in conjunction with the discoveries and physical theories they led to.

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Figure 1.2: Two-thousand years of space-time sciences: from Antiquity to the eighteenth century.

1.5 Space-time reference systems | 15

Figure 1.3: Two-thousand years of space-time sciences: from the eighteenth century to nowadays.

16 | 1 Earth rotation and space-time reference systems Of the very rich history underlying the development of space-time references, we offer the synthetic representation in chronological order in Figures 1.2 and 1.3. It can be seen how the conceptual representation of space and time references, as well as our knowledge of increasingly small celestial and ground motions, is associated with the development of clocks and astrometry. If many discoveries and physical theories have resulted from technical advances or have been confirmed by them, the inverse is also true: the search for tiny celestial motions (parallaxes, annual and secular aberrations, proper motion, …), Earth shape (flattening, geoid, topography), and ground displacement (tides, continental drift, …) suggested by theoretical considerations has motivated much technical progress, especially in optics and time keeping. The tremendous advances of the last 50 years are mostly due to the invention of atomic clocks and space exploration, their improvement, and their introduction in space geodesy for determining angles and distances, so that in their practical realization time and space references become totally entangled, as described by relativity. With the advent of modern science and technology, space-time references have not only diverged from the first reality attributed by our five senses (directions of the stars for the CRF, points on the ground for the TRF, succession of solar or lunar days for time scale) but also lost the absolute or metaphysical meaning that mankind conceived. Notice, however, that the conception of reference frame as a realization of the ideal reference system [123], leads one to believe that reference systems exist independently from the material world and the way the observation is carried out. Indeed reference systems are not really concerned with emitters (quasar and stars), receivers (antennas at station observations), and transmitters (electromagnetic waves), and look like a reminiscence of platonic philosophy. Although their development is an increasing abstraction, their concrete realizations are nonetheless, and paradoxically enough, precise tags of the space-time convolutions. Space-time reference frames are a matrix concerning the surrounding space-time enabling one to coordinate and archive consistently all our measurements, and the geometric and physical parameters we are drawing from these measurements.

2 Geophysical irregularities of Earth’s rotation: an overview A physical theory is by so much the more true, as it puts in evidence more true relations. (…) So, the “The earth turns round” has a richer content expressed by the flattening of the Earth, the rotation of Foucault’s pendulum, the gyration of cyclones, the trade-winds, and who knows what else? Poincaré in La valeur de la Science (1905)

2.1 Earth’s rotation is irregular In 10 µs a meridian sweeps about 7.292115 10−10 radians or 0.15041 mas, that is, an arc of 4.646 mm at the distance r0 = 6371.0083 km (the Earth mean radius) from the geocenter.1 Yet, this tiny angle is larger than the uncertainty on the rotation transformation between a terrestrial reference frame Gxyz and a celestial reference frame GXYZ, about 0.1 mas or 3 mm, as currently determined. This outstanding precision, as reached nowadays, has resulted from the concomitant development of atomic clocks (maser clocks in particular) and space techniques after the 1960s, leading to the invention of the Very Long Baseline radio Interferometry (VLBI), the Satellite Laser Ranging (SLR), the Doppler Orbitography Radio Integrated System (DORIS) and of the Global Navigation System Satellite (GNSS) from the 1990s with the American Global Positioning Systems (GPS), the Russian GLObalnaya NAvigatsionnaya Sputnikovaya Systema (GLONASS), and the European GALILEO and Chinese BEIDOU. For comparison, ground techniques, based on the chronometric records of the stars position, yield the Earth rotation with an uncertainty above 10 mas. Astronomical and geophysical variations In the absence of any internal or external disturbance, any meridian line would sweep invariably the starry sky over 23 h 56 min 4.10 s around a fixed axis in space, making an angle of 23°26󸀠 with the pole axis of the ecliptic, so that the matter would not present a strong physical interest. But in our changing world, the Earth rotation has fluctuations, which can be analyzed and classified in the light of their causes of astronomical and terrestrial nature. An astronomical effect is one caused by the external environment, mostly by the action of lunisolar gravitational tides on the equatorial bulge of the Earth, responsible for 99.9 % of the precession–nutation. Conversely a geophysical effect is that resulting from terrestrial processes, mainly mass redistribution. This distinction is not without ambiguity, since many geophysical fluctuations, 1 One mas is the angle subtended by an arc of 3.01 cm on the surface of the Earth seen from the geocenter at a distance of 6371.0083 km. https://doi.org/10.1515/9783110298093-002

18 | 2 Geophysical irregularities of Earth’s rotation: an overview such as those prevailing in the Earth fluid layers, either would not exist without the astronomical thermal heating of the Sun, or are influenced by lunisolar tides. But that categorization has a more fundamental reason: astronomical effects are by nature regular (lunisolar tides),2 except rare catastrophic events, and are therefore modeled analytically with great precision, making them predictable; geophysical forcing results from aleatory processes, and therefore gives the non-predictable part of the Earth rotation, which has to be monitored permanently for the needs of space techniques. Hereafter we briefly present the Earth rotation variation—precession-nutation, length-of-day change, and polar motion—from a historical perspective. For each of these components, we unveil the geophysical irregularities in contrast to regular astronomical variations. The chronology of their discovery reflects both astrometric progress and model sophistication involving increasingly tenuous effects.

2.2 Precession–nutation As early as 200 BC the Greek astronomer Hipparchus discovered that the vernal point or the direction of the Sun at spring equinox3 drifts clockwise along the ecliptic relative to the stars at the rate of 50󸀠󸀠 per year. From an Earthbound perspective, the vernal point travels the sky slightly faster than the stars; it “precesses” the star positions, doing a full turn on the zodiac in 25800 years (Figure 2.1), and the motion is therefore called precession. Currently it is pointing in the constellation Pisces and is approaching the Aquarius (Figure 2.1). Equivalently the equatorial plane, at the constant inclination of 23°26󸀠 to the ecliptic, precesses with the same rate; the rotation axis, perpendicular to the equatorial plane, describes a cone about the axis of the poles of the ecliptic in the clockwise direction, and this angular motion is also called precession. Indicating nowadays the polar star, the axis of rotation will point to Vega in 13000 years. To precession are superimposed periodic oscillations composing the nutation of the rotational axis. Having a magnitude smaller than 1󸀠 , that is to say, below the resolving power of the human eye, the nutation was discovered thanks to the advent of astrometric telescopic observations. In 1748 the English astronomer Bradley discovered its principal term in 18.6 years, with an amplitude of about 10󸀠󸀠 and associated with the precession of the lunar orbital plane on the ecliptic. Additional terms, a few thousand in the current conventional model, are at least 10 times smaller. 2 This regularity does not exist anymore at geological time scales as the deterministic chaos impedes making any sound analytical model. 3 The descending node of the equator on the ecliptic plane when the equator circle is described counter-clockwise.

2.2 Precession–nutation

| 19

Figure 2.1: Left: precession–nutation. Right: precession of the rotation pole (P) round the ecliptic pole and precession of the vernal point on the zodiac.

As early as the 18th century, it was established that precession and the 18.6 year nutation have an astronomical cause in the sense that they are described by the action of the lunisolar gravitational forces on the equatorial bulge of the Earth, considered as a pure rigid body. A semi-annual term of about 1󸀠󸀠 was discovered in the nineteenth century, and in the twentieth century many other terms of smaller amplitude were observed in correspondence with the rigid Earth nutation model, for instance the Woolard one [232]. However, since the 1970s, the astrometric observations show that nutation terms do not fit perfectly the ones that are calculated for a rigid Earth model: they can deviate by about 50 mas from the rigid case, as illustrated in Figure 2.2, thus unveiling the footprints of non-rigidity. A perturbation of 50 mas represents about 5 thousandths of the amplitude of the principal nutation (10󸀠󸀠 ), that is, less than 1 % of the “rigid” effect. Nonetheless, this deviation is evidence for the fluid core effect on nutation as predicted by Poincaré in 1910 [172] and fully confirmed from the middle of the 1980s [110, 108]. The perturbation of lunisolar precession–nutation brought about by the internal structure of the Earth and its rheological properties is presently modeled at 0.5 mas accuracy level. Residuals are monitored by VLBI since the mid-1980s at 5–7 day interval, and exhibit irregular variations, of which the cause cannot be attributed to gravitational tides, but to some terrestrial processes, probably lying in the hydro–atmospheric layer. In this respect, nutation, in addition to the lunisolar effect modulated by non-rigidity, contains a minor geophysical effect, with an order of magnitude equal to 1/10000 the astronomical component. As precession reflects mostly the average effect of the lunisolar tide on the average mass distribution of the Earth, it is much less affected by the non-rigidity of the Earth than the nutation terms, nevertheless coupling of the mass redistribution and the lunisolar potential leads to second order effects up to 0.4 mas/year [8].

20 | 2 Geophysical irregularities of Earth’s rotation: an overview

Figure 2.2: Celestial oscillation of the rotation axis during the year 2017, projected on the equator of the epoch J2000. Mostly because of the fluid core, observed nutation shifts from the modeled nutation for a rigid Earth up to 50 mas (theoretical model of Kinoshita–Souchay, 1990 [120]). Geophysical irregularities less than 1 mas cannot be seen at the scale of the figure.

2.3 Polar motion In the middle of the eighteenth century, the famous Swiss–Russian scientist Leonard Euler generalized Newton’s laws to an extended body, and derived the law of the angular momentum balance [86]. Whereas the Earth’s flattening at the poles was still a hypothesis, giving the Earth at least two uneven moments of inertia, Euler explored theoretical consequences [84]. He showed that, if the pole of rotation is not aligned with one of the principal axes of inertia, then there follows spontaneously on the Earth surface an uniform counter-clockwise circular motion whose frequency increases with the relative difference in moments of inertia, also called dynamical ellipticity. He calculated that the axis of rotation can oscillate freely at a period of 303.6 days with an indeterminate amplitude. Consequently, the angle between the zenith of a given place and the axis of rotation, that is to say, the astronomical co-latitude, fluctuates with the same frequency. Seeking latitude changes was undertaken throughout the nineteenth century. These were actually suspected and eventually detected at Pulkovo Observatory (Russia) from 1843 to 1873 by Peters, Nyrén and Gyldén, then Künstner in Berlin

2.3 Polar motion

| 21

in the 1880’s.4 In 1891 the American astronomer Chandler performs the decisive step [34] by exhibiting an oscillation of about 0.3󸀠󸀠 in about 430 days, not the expected 304 days of Euler.5 Shortly after he realized that the Chandler wobble was accompanied by an annual oscillation of about 0.1󸀠󸀠 . The proximity of these cycles produces a beating over 6.4 years in the direction cosines xp and yp of the rotation pole along axes Gx and Gy of the terrestrial frame. This is shown in Figure 2.3 where the so-called pole coordinates x = xp and y = −yp are displayed over the period 1962–2019, along with the Chandler and seasonal components extracted by singular spectral analysis. In the process, Newcomb interpreted the observed 430 day period6 as the Euler period lengthened by the Earth non-rigidity. As wisely advocated by Chandler and Newcomb [163], the annual oscillation comes from seasonal mass redistributions in the hydro–atmospheric layer. With the observation of the polar motion, the Earth rotation is emancipated from celestial mechanics and had to deal with geophysics. In the spectral domain (see Figure 2.4), the contrast between an “astronomical Earth” and a “geophysical Earth” is strengthened: the isolated “Dirac-delta” at 0.83 year (304 days) for a rigid Earth becomes a broad band peak in 1.185 year (433 days), accompanied by a seasonal term. Until the 1960s no considerable progress was made, despite instrumental improvement, like the introduction of Photographic Zenith Tube (PZT) and reinforcement of the observational network: pole coordinate uncertainty remains above 30 mas. But, as time series became longer, a secular trend at the rate of about 4 mas/century towards Greenland appeared in the 1920s [131], confirmed later in the 1960s along with a pseudo-cyclic term in about 25 years, the Markowitz wobble [141]. The smoothed path of the pole of Figure 1.1 or Singular Spectrum Analysis of Figure 2.3 puts forward this secular trend as well as multi-annual oscillations, encompassing the Markowitz wobble. From the 1960s the advent of space geodesy revolutionized our knowledge. Doppler tracking of terrestrial beacons from satellites lead to first space geodetic determination of the pole coordinates, with uncertainty dropping down to 15 mas in 1965. This unveils in the 1970s the inter-annual polar motion (periods between 1.5 and 10 years, 10 mas amplitude), and sub-annual fluctuations in the range 90–365 days, dominated by semi-annual and ter-annual harmonics of the seasonal polar motion. The development of VLBI (Very Long Baseline Radio Interferometry) and SLR (Satellite 4 The story of the discovery of polar motion is reported in the Proceedings of IAU Symposium 178. Polar Motion, historical and scientific problems Ed. Steven Dick, Dennis McCarthy and Brian Luzum, ASP Conf Ser, Vol. 208. 5 According to S. Debarbat [54] observations conducted at the Observatoire de Paris by Yvon Villarceau (1813–1883) could have led to this discovery. 6 Now more than 100 years of observations favor a period of 432 ± 1 day.

22 | 2 Geophysical irregularities of Earth’s rotation: an overview

Figure 2.3: Pole coordinates x = and y from 1962 up to nowadays (IERS C04 combined series), corresponding to the direction cosines xp = x and yp = −y of the rotation axis in the TRF, and decomposed into secular/decadal, Chandler, seasonal, rapid and inter-annual components by Singular Spectral Analysis.

2.3 Polar motion

| 23

Figure 2.4: Complex Fourier spectrum of the complex pole coordinate x − iy (period 1962–2019) compared to the analogous theoretical spectrum for a rigid flattened Earth in the band from −2 to +2 cycle/year.

Laser Ranging) permitted one to observe rapid fluctuations of about 1 mas (period between 2 and 100 days) in the late 1980s, and diurnal/semi-diurnal oscillations (1 mas) from the 1990s. Whereas all the variations discovered after 1950 are at least 6 times smaller than the annual and Chandler terms, they highlight geophysical processes of miscellaneous natures: – The secular drift is probably caused by the viscoelastic rebound of the ground that begun with the melting of the glaciers of the last glaciation era: in subarctic areas such as Scandinavia, the soil that had been depressed by the ice layer, uplifts, causing a secular variation of the equatorial moments of inertia of the Earth and resulting in a pole secular drift (see Chapter 15); this post-glacial rebound is certainly completed or inflected by the present ice sheet variation [40]. – The Markowitz wobble resists any interpretation (see Chapter 10). – The inter-annual and sub-annual oscillations are due to the action of hydro– meteorological layer (see Chapters 9, 10, 11). – The diurnal and subdiurnal oscillations are caused by the oceanic tides, and, to a much lesser extent, result from thermodynamic mass transports in the hydro–atmosphere and from the coupling of the lunisolar tides with the Earth triaxiality (see Chapter 12).

24 | 2 Geophysical irregularities of Earth’s rotation: an overview

Figure 2.5: Long term variation of the length of day from 1832 to 2019. The reference value is 86400 s SI (Paris Observatory C02 series, available on http://hpiers.obspm.fr/eop-pc).

2.4 Variations of the Earth angular velocity As far as the triaxiality is neglected, the lunisolar torque does not cause an axial torque on the rigid Earth, so that the variation of the angular velocity should rather be seen as the footprint of non-rigidity or geophysical forcing. As early as the eighteenth century, deceleration of the Earth rotation was expected because of the tidal friction. Indeed, the tidal bulge is not perfectly aligned along the gravitational forces, but occurs with some delay during which the Earth has turned by a small angle; so the tidal bulge is slightly rotated forward with respect to the Moon direction, causing a braking torque and deceleration of the spin rate. The braking produced by the Sun according to the same mechanism is too small for being noticed in UT or LOD. However, the diurnal rotation ensured the function of a perfect clock until Newcomb, Spencer Jones [208], and de Sitter (1927) [50] put forward in UT−TE (see Section 1.4) the expected secular deceleration of the rotation speed of about ω̇ T = −4 10−22 rad/s−2 , or, equivalently an increase of the Length of Day (LOD) of 1.6 ms/cy.7 The same scholars discovered 7 In virtue of the angular momentum balance (see Chapter 4), the associated braking torque is ΓT = C ω̇ T where C ≈ 8 1037 kg m2 is the axial moment of inertia, that is, ΓT ≈ −3 1016 kg m2 s−2 .

2.4 Variations of the Earth angular velocity | 25

Figure 2.6: Zonal tide effect on the length of day over 2008. Defraigne and Smits model [55] recommended by IERS conventions 2003 (from WEB tool on http://hpiers.obspm.fr/eop-pc).

decadal fluctuations of a few ms and whose periods are spread from 10 to 70 years (see Figure 2.5), paving the way to a fluid core effect on the mantle, of which modeling remains quite elusive. A few years later, in 1937, Stoyko [211] discovered seasonal variations of the order of 0.5 ms, which is nowadays very well explained by the action of winds on Earth surface, especially on the mountain range. Pure mass redistributions are mixed with zonal tidal deformations changing the axial moment of inertia. In turn the LOD presents a tidal oscillation of up to 0.5 ms, mostly at the lunar periods of 13.6 and 27.3 days. This tidal effect, reconstructed over one year in Figure 2.6, is modeled very precisely, and has to be removed from the observed length of day for isolating its geophysical forcing. Definition of the Length of Day change Processing space geodetic observations (SLR, GPS) permits one to estimate the length of day offset ΔLOD with respect to its nominal value of LOD = 86400 s SI, also called the mean solar day for it coincides with the mean solar day found in the nineteenth century. Let ω = Ω(1 + m3 ) be the angular frequency, where Ω is the reference or nominal angular velocity, and m3 means its relative variation. According to the IERS 2010 conventions [168], the nominal angular frequency Ω/2π, expressed in cycle per nominal day LOD, is k = 1.002 737 811 911 354 48 .

(2.1)

26 | 2 Geophysical irregularities of Earth’s rotation: an overview Equivalently Ω k = 2π LOD

or LOD = k

2π , Ω

(2.2)

meaning that k is also the ratio of the SI day LOD (equivalently the nominal length of day or the mean solar day) to the mean rotation period 2π/Ω, called the mean stellar day. From (2.2) we have Ω = 7.292 115 146 706 980 10−5 rad/s .

(2.3)

As the Earth’s angular velocity is determined with a relative uncertainty of 10−10 , IERS recommends the significant figures Ω = 7.292 115 10−5 rad/s (IERS Conventions 2010, Table 1.2 [168]). Sometimes the constant k is wrongly confused with k̃ = 1.002 737 909 350 795 giving the slightly larger angular velocity referred to the equinox point Ω̃ = 7.292 115 855 306 587 10−5 rad/s, or equivalently the mean sidereal day [4]. Equation (2.2) applies to the total angular velocity ω and length of day LOD: LOD = k

2π 2π =k , ω Ω(1 + m3 )

(2.4)

appears, and neglecting second order where the nominal length of day LOD = k 2π Ω terms in O(m23 )LOD ∼ 10−16 LOD, the length of day offset8 is ΔLOD = LOD − LOD = −LOD m3 .

(2.5)

The angle of rotation between dates t0 and t is defined by the summation of the infinitesimal angles ω dt: t

t

θ(t) − θ(t0 ) = ∫ ω dt = ∫ Ω(1 + m3 )dt , t0

(2.6)

t0

that is, t

θ(t) − θ(t0 ) = Ω(t − t0 ) + ∫ Ωm3 dt

(2.7)

t0

or t

θ(t) − θ(t0 ) = Ω(t − t0 ) − Ω ∫ t0

ΔLOD LOD

dt .

8 Wrongly called “excess” of the length of day since it happens to be negative.

(2.8)

2.5 Synthesis | 27

Here, we distinguish the term varying linearly with time from an irregular component expressing the time integration of length of day changes. The Earth rotation time scale UT1 is merely defined from the angle of rotation through a linear transformation exposed in the next chapter (see Eq. (3.3)). When LOD increases, UT1 decreases, and conversely. Until the 1980’s the UT1 fluctuations were recorded through optical observations made in particular by meridian circles, nowadays by VLBI time recordings of quasar front waves.

2.5 Synthesis Our current understanding of the geophysical effects on Earth rotation is summed up in Table 2.1. The nutation is the “most” astronomical component, and for this reason is very well modeled, with an accuracy below 1 mas. The resulting predictability allows nutation corrections to be determined from the VLBI observation processing with an average latency of two weeks, without compromising the operability of other astro-geodetic techniques. Conversely polar motion is highly dependent on internal and surface mass redistributions, so that after a few days the prediction error of the pole coordinates, becoming larger than 1 mas, can spoil applications of space techniques. It has to be estimated continuously, from daily to even four-daily processing of the GNSS measurements. Although polar motion cannot be reduced to a predictive model, it is relatively well interpreted afterwards by the hydro–atmospheric mass redistribution, determined from the relevant global circulation models. Length of the day or rotation angle presents astronomical regularities and geophysical irregularities: annual and sub-annual variations are very well explained by the effect of zonal winds superimposed on the zonal tides (≈ 1 ms on LOD); the secular increase of LOD or parabolic decrease of UT1 is a mixed effect of dissipation and lunar torque; but the decadal fluctuations—the largest (≈ 5 ms)—are rebellious to any accurate modeling because of lack of data describing their cause: if it lies in fluid core motions, as commonly admitted, the only signature is the surface magnetic field generated in the fluid core but shielded by the mantle. Study of the Earth rotation is not limited to modeling the impact of geophysical processes, it also leads to determining global geophysical parameters or properties, such as the elasticity of the mantle, the mantle conductivity, the ellipticity of the fluid core, that local geophysical observations cannot specify [121]. The mass redistributions are observed partially, in turn geophysical modeling struggles to compete with the extraordinary precision of modern determinations of the Earth rotation. However, space missions dedicated to the continuous observation of Earth’s gravity field could change that.9 These gravimetric measurements allow one to better monitor changes in 9 There are four methods to determine the gravity field: i) orbit determination by ground laser telemetry on geodetic satellites (LAGEOS), ii) satellite orbit determination by the GPS receiver on board, cou-

ΔLOD < 0.5 ms ΔUT 1 < 0.1 ms 10 mas

Diurnal/semi-diurnal terms [109]

Sub-decadal polar motion

Modeling: – Enigmatic; * Mediocre; ** Average; *** Good; **** Excellent.

2010

2 mas 1 mas

Rapid polar motion (≤ 100 d) Diurnal/semi-diurnal terms [109]

1985 1994

0.1 mas

10 mas 50 mas

Inter-annual terms (1.5–10 yr) Defects of the rigid Earth nutation models [110]

1970 1980

1 mas

Markowitz wobble (25 years) [141]

Fortnightly and monthly terms

1960

Seasonal variations [211]

100 mas 4 mas/cy ΔLOD < 2 ms/cy ΔUT 1 < 70 s/cy ΔLOD < 2 ms ΔUT 1 < 1 s ΔLOD < 1 ms ΔUT 1 < 0.2 s ΔLOD < 0.3 ms ΔUT 1 < 3 ms 20 mas

200 mas

Order of magnitude

15 mas

1937

Annual term [36] Secular drift [131] Secular increase [208, 50]

1892 1922 1926

30 mas

Nutation

Multi-decadal variations [208, 50]

Chandler term at 430 d [34]

1891

Observed effect

Angular velocity / UT1 / LOD

50 mas

error

polar motion

*** *** * *** **

Inland freshwater transports [1]

** ***

–

***

****

*

** ** ** ** **

Core–mantle coupling? [112] Ice cap fluctuations? Hydro–atmosph. transports [126] Fluid core effect Mantle elasticity [172, 150] Hydro–atmosph. transports [9, 20] Oceanic tides [39] Hydro–atmosph. transports [97] Oceanic tides [21]

Atmosph. transports [127] + Seasonal zonal tides Zonal tides

Core–mantle coupling

Euler period lengthened by Earth non-rigidity [162] Hydro–atmosph. transports [97] Hydro–atmosph. transports [119, 117, 97] Post-glacial rebound [233] Tidal friction

Cause and modeling

Table 2.1: Overview of the geophysical variation of the Earth rotation: effect, discovery, cause and model (at left evolution of the astrometric accuracy).

28 | 2 Geophysical irregularities of Earth’s rotation: an overview

2.5 Synthesis | 29

ice sheet thickness of geographic zones, like arctic or antarctic, where in-situ data are lacking. At the same time, they allow one to reconstruct the geopotential variations, which the degree 2 spherical harmonics give the Earth moments of inertia changes regardless of any geophysical model [197].

pled to the measurement of the non-gravitational acceleration by an on-board inertial system (satellite CHAMP 2000–2010), iii) two satellites with GPS orbit determination, and measurement of their relative distance by micrometer telemetry (GRACE mission began in 2002), iv) satellite localized by GPS with three on-board accelerometers for measuring gradient gravity field (GOCE satellite launched in 2009). While GOCE is designed to measure the static part of the gravity field with a spatial resolution of at least 100 km, GRACE gives the variable part with a spatial resolution up to 300 km.

3 Astro-geodetic observations Quand il s’agit du mouvement de la terre, il faut observer que l’axe de la terre est différent de l’axe de rotation; car, puisque l’axe de la terre se trouve dans un mouvement continuel à cause de sa nutation et de la précession des équinoxes, il ne convient jamais avec l’axe de rotation, qui à chaque instant est absolument immobile, faisant abstraction du mouvement annuel. Leonard Euler in Recherches sur le mouvement de rotation des corps célestes (1759) [85]

3.1 Introduction Determined by astro-geodetic techniques with a 0.1 mas accuracy, the observed pole coordinates are not the ones of the instantaneous rotation pole. They give the terrestrial position of a Celestial Intermediate Pole (CIP), permitting one to describe and adjust the transformation of rotation between Geocentric Celestial Reference System (GCRS) and the International Terrestrial Reference System (ITRS). The relative difference between both kinds of coordinates increases with frequency: less than 1 % below 1/100 cycle per day (cpd), it reaches 100 % for 1 cpd. Since the dynamical interpretation is based upon the instantaneous pole coordinates, we have first to specify how they can be derived from the observed coordinates of the CIP. For doing this, we shall sketch the larger picture of the astrometric modeling of Earth’s rotation (Section 3.2). Then using the kinematic relations, we shall establish the relationship between rotational pole and CIP (Section 3.3). Finally we give the principle of the Earth rotation monitoring from astro-geodetic observation, and briefly qualify the uncertainty of the pole coordinates (Section 3.4).

3.2 The Earth Orientation Parameters (EOP) 3.2.1 Description of the transformation between terrestrial and celestial systems The instantaneous rotation vector is not directly connected to the raw observations. These—namely height, azimuth, time in the case of historic astrometric pointing, and phase, dated time delay or frequency for contemporaneous space geodetic observations—can be modeled as a function of the parameters describing the rotation of the ITRS with respect to the GCRS, but they are not functions of the components of the instantaneous rotation vector. Thus, the orientation of the ITRS with respect to the GCRS can be represented by the product of three successive rotations: Rl (αl )

Rm (βm )

Rn (γn )

(Xi ) 󳨀→ (xi(1) ) 󳨀→ (xi(2) ) 󳨀→ (xi ) ,

(3.1)

where Rl (αl ) means a rotation of angle αl around the axis Xl of the GCRS, transforming the GCRS to the coordinate system (xi(1) ), Rl (βm ) means a rotation of angle βm around https://doi.org/10.1515/9783110298093-003

3.2 The Earth Orientation Parameters (EOP)

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(1) the transformed axis xm with m ≠ l, and Rn (γn ) means a rotation of angle γn around (2) the axis xn resulting from the second transformation and with n ≠ m. This rotation sequence can be coded by the subscripts of the Cartesian axes around which the rotations take place, that is, (lmn). It can be realized in 12 ways. The one most common in astronomy, (313), is called Eulerian and is associated with the Euler angles, which are α3 = ψ (precession), β1 = θ (nutation) and γ3 = Φ (proper rotation). The angle (1) ? θ = (OX 3 , Ox3 ) is oriented by the vector x⃗1 . Commonly this sequence starts from an ecliptic celestial frame (the fundamental plane is the ecliptic of a given epoch of reference like J2000, and the origin is the mean equinox of this epoch). In principle, these three angles readily determine the Earth’s rotation. Because of the diurnal rotation, at least one of them has a rate having the order of magnitude of Ω ≈ 7.292 115 10−5 rad/s ≈ 15.041 mas/ms according to (2.3). In turn, for describing a rotation transformation with the current uncertainty of 0.1 mas, we would need to provide this angle every 0.1/15.041 = 0.007 ms. So, after one day the Earth rotation would be described by a temporal series of 86400 103 /0.007 ≈ 1010 angles! This consideration shows that taking three angles is not appropriate for monitoring the Earth’s rotation irregularities. As the main part the Earth rotation is very regular—diurnal rotation at nominal angular rate Ω and precession–nutation—, the Earth orientation is parametrized by separating the unpredictable perturbations from the modeled variations. These angular perturbations define the Earth Orientation Parameters (EOP) and are then estimated from space geodesy with a temporal resolution between a few hours in the best case and a few days. The Euler angles Ψ and θ, which determine the direction of the geographic axis Gz (see Section 1.2) in the GCRS, can be modeled for their variations above two days, according to the action of tidal force on Earth’s bulge. We obtain a precession–nutation model with an accuracy of about 0.5 mas, and defining an axis whose intersection with the northern hemisphere of the celestial sphere is called the Celestial Intermediate Pole (CIP).1 One can show that the celestial intermediate axis is within 20 mas of the instantaneous rotation axis. That is why the diurnal rotation can be reckoned around this axis. The non-modeled part of the angles Ψ and θ is then divided into two parts: (i) a correction of the precession–nutation, that is to say, the spatial position of the CIP, and (ii) an “Earth-bound” or “geophysical” term varying mostly with quasi-diurnal periodicities. This term is not coordinated in the celestial system but in the ITRS by pole coordinates x, y after applying the diurnal rotation matrix. So the non-modeled part of the Earth rotation is parametrized by five Earth Orientation Parameters (EOPs): (i) two parameters correcting the celestial position of the CIP in the celestial frame or celestial pole offsets (dX, dY), (ii) a correction of the ro-

1 Before the IAU General Assembly in 2000, the definition of this pole, then called the Celestial Ephemeris Pole (CEP), did not state if it contained diurnal and sub-diurnal terms.

32 | 3 Astro-geodetic observations tation angle Ω(UT1 − UTC), giving its non uniform variation, and (iii) the terrestrial coordinates of the CIP or pole coordinates (x, y). Additional parameters are commonly considered, strictly these are not EOP, i. e., angles and direction cosines (or a quantity proportional to an angle like UT1), but the time derivative of the EOP, especially the pole rate x,̇ ẏ and the length of day offset ΔLOD with respect to LOD0 = 86400 s TAI, which can be derived from UT1 according to (3.29). The EOP and their time derivatives can be termed more generally Earth Rotation Parameters (ERPs). In what follows, we make precise the definition of EOP.

3.2.2 EOP n°1, n°2: Celestial Pole Offsets The precession–nutation transformation brings the GCRS in coincidence with the true equatorial frame or Celestial Intermediate System (CIS) GXi Yi Zi , of which the fundamental plane GXi Yi has for pole the CIP. By definition of the CIP, its wobble is composed of circular oscillations whose periods are longer than two days. The first axis of this system GX0 is called the Celestial Intermediate Origin (CIO).2 The CIO is chosen as a non-rotating origin in the intermediate equatorial plane, that is, an angular origin that does not present any instantaneous axial rotation around the CIP with respect to the GCRS. More precisely the precession–nutation is described by the coordinate transformation illustrated in Figure 3.1a and given by [CIS] = PN[GCRS] with PN = R3 (−s)R3 (−E)R2 (d)R3 (E) ,

(3.2)

where E and d are the longitude and colatitude of the CIP in the GCRS, respectively, and s allows one to locate the CIO on the intermediate equatorial frame from the kinematic condition stipulated above. Actually, IAU and IERS favor the equatorial direction cosines of the CIP, namely X = cos E sin d, Y = cos E sin d for modeling precession– nutation. The defects of the conventional precession–nutation model are estimated as corrections dX, dY or celestial pole offsets. As such, the CIP has for equatorial direction cosines (X = Xmod + dX, Y = Ymod + dY) in the GCRS. So the precession–nutation matrix will be expressed by PN(Xmod + dX, Ymod + dY, s). 3.2.3 EOP n°3: UT1-UTC As the CIP is near the instantaneous pole of rotation within 20 mas, the diurnal rotation and its fluctuations are counted around the axis of the CIP, giving the third EOP, 2 The abbreviation also CIO holds for Conventional International Origin; see Section 1.2.

3.2 The Earth Orientation Parameters (EOP) | 33

namely the angle of rotation of the Earth Θ. It specifies the angle between the Terrestrial Intermediate Origin (TIO) and the Celestial Intermediate Origin (see above), as shown in Figure 3.1b. As for CIO, TIO is a non-rotating origin in the equatorial plane, but now with respect to the ITRS. The concept of non-rotating origins in GCRS and ITRS was introduced by Bernard Guinot in the 1980s, in order to define the Earth angle of rotation independently from the ecliptic plane as it is done with the traditional representation using the Greenwich sidereal time referred to the vernal point [31]. Later on, in the 1990s Nicole Capitaine and her collaborators specified CIO and TIO positions and their evolution for astrometric modeling of the Earth’s rotation [30, 129]. The Earth rotation time scale or Universal Time 1 (UT1) is defined from the Earth rotation angle Θ through the scaling law Θ = Θ0 + Ω(UT1 − UT10 ) ,

(3.3)

where Ω is the conventional angular velocity Ω = 2π 1.00273781191135448 rad/day and UT10 is taken as the epoch 1 January 2000 at 12 h UT1 (when the TIO crosses the mean Sun). Here the time unit day is the SI day of 86400 s TAI3 yielding a stellar day of 2π/Ω = 86164.098 903 691 s. At the epoch UT10 , the angle Θ between CIO and TIO of this date is then Θ0 = 2π 0.779 057 273 264 ≈ −79.53°. The angle of rotation is composed of a part that varies linearly with the atomic time TAI with the angular rate Ω, Ω(TAI − TAI0 ), and of an irregular component Ω(UT1 − TAI): Θ = Θ0 + Ω(TAI − UT10 ) + Ω(UT1 − TAI)

= Θ0 + Ω(TAI − TAI0 ) − Ω(UT10 − TAI0 ) + Ω(UT1 − TAI) ,

(3.4)

where TAI0 is the 1 January 2000 at 12 h TAI. This expression exhibits the correction UT1 − TAI, considered as well as the EOP fully specifying the angle of rotation. Since the 1940s seasonal change of UT had been noticed thanks to the quartz clock [211]. The UT time scale was made regular by removing a rough model composed of annual and semi-annual harmonics at the level of 0.03 s. This regularized time scale was called UT2. In this respect UT was renamed UT1. With the beginning of a continuous atomic time scale in 1958, named in 1971 TAI, the Earth’s rotation time UT1 could be also monitored with respect to TAI. The first January of 1958 at 0 h UT2 TAI was chosen equal to UT2.4 Since this epoch, UT1 delayed by about 37 s (in 2018); this effect is not because of the secular Earth rotation deceleration but because the TAI second has been chosen 2 10−5 ms shorter than the averaged UT second of 1958 (shift amplified by the decadal deceleration of the 1970s). 3 Corresponding to Mean solar day of the mid nineteen century. 4 The time scale UT2 became totally obsolete when UT1 variations were fully measured by VLBI.

34 | 3 Astro-geodetic observations In fact, the correction that is disseminated is not UT1 − TAI but UT1 − UTC where the so-called Universal Time Coordinated or UTC is a discontinuous atomic time, irregularly delayed from TAI by one second at fixed dates of the year (31 December, 30 June), so that the difference |UT1 − UTC| is kept less than 0.9 s. Since the date of its creation in 1972, UTC ensures a worldwide time synchronization replacing the Earth rotation time, and constitutes the basis of any national or international legal time. The introduction of a leap second allows one to reflect Earth’s rotation time with one second accuracy, and to perform longitude determination to 15󸀠󸀠 from star or Sun observations. The interest of UTC and leap seconds has strongly diminished since the advent of GNSS techniques, allowing much better positioning independently from UTC. Whereas the difference TAI − UT1 grows grossly as a parabolic time function, the introduction of the leap second is modulated by Earth’s rotation irregularities, and therefore cannot be predicted long in advance. For instance, in the period 1999–2006, the Earth’s rotation speeded up so much that the UT1 second becomes closer even smaller than TAI second, avoiding any leap second for 6 years. On December 31, 2016 a leap second was introduced at 23 h 59 min 59 s UTC, and this day lasted 86401 s TAI, whereas TAI − UTC increased from 36 s to 37 s. Finally, the axial rotation of angle θ rotates the Celestial Intermediate System (CIS) to the Terrestrial Intermediate System (TIS), having for equatorial origin the TIO: [TIS] = R3 (Θ0 + Ω(TAI − UT10 ) + Ω(UT1 − TAI))[CIS] .

(3.5)

3.2.4 EOP n°4, n°5: pole coordinates (x, y) After having applied the geometric transformations of precession–nutation and diurnal rotation, the resulting frame is the Terrestrial Intermediate System (TIS) Gxi yi Zi . For reaching the ITRS, we apply an axial rotation R3 (s󸀠 ) associated with the parameter s󸀠 locating the TIO in the equatorial frame with respect to the ITRS, and two successive rotations around equatorial axes, which rotate the intermediate equator to the geographic equator according to [ITRS] = W[TIS] with W = R1 (−y)R2 (−x)R3 (s󸀠 ) ,

(3.6)

both angles x and y not exceeding 1󸀠󸀠 (see Figure 3.1c and d). It is easy to show that the equatorial direction cosines of the CIP in the ITRS, or pole coordinates, are given at first order by xp = x, yp = −y (zp ≈ 1). The corresponding path of the CIP in Figure 3.3 appears as a quasi-diurnal average of the motion of the instantaneous rotation pole.

3.2 The Earth Orientation Parameters (EOP)

| 35

Figure 3.1: Decomposition of the rotation transformation from the GCRS GXYZ to the ITRS through the celestial intermediate equator, Celestial Intermediate Origin GXi (CIO), and Terrestrial Intermediate Origin Gxi (TIO).

3.2.5 Full coordinate transformation from celestial reference system to terrestrial reference system To transform the ICRS into the ITRS, the origin of the ICRS, namely the solar system barycenter, has to be translated to the origin of the ITRS (the center of masses) by an appropriate relativistic transformation, thus forming the geocentric CRS or GCRS. This space-time transformation also involves the time coordinates TCB (Temps Coordonné Barycentrique) and TCG (Temps Coordonné Géocentrique). Grouping precession– nutation PN, diurnal rotation R and polar motion W, the total transformation from GCRS to ITRS reads [ITRS] = Q [GCRS] ,

(3.7)

with the rotation matrix Q = R1 (−y)R2 (−x)R3 (s󸀠 )

R3 (Θ0 + Ω(TAI − UT10 ) + Ω(UT1 − TAI)) NP(Xmod + dX, Ymod + dY, s) .

(3.8)

36 | 3 Astro-geodetic observations

3.3 Relation between EOP and components of the instantaneous rotation vector 3.3.1 Kinematic relations The theory of the rotation of the Earth is based upon the components of instantaneous rotation vector ω⃗ in the ITRS, that is, ωx = Ωm1 ,

ωy = Ωm2 ,

(3.9)

ωz = Ω(1 + m3 ) ,

where Ω is the reference angular velocity defined in Section 2.4. In contrast to EOP, they are not the parameters fitted from measurements, except ΔLOD (proportional to m3 according to Eq. (2.5)), which is determined by GNSS and SLR processing.5 Therefore, to interpret EOP in terms of the extensive meaning (common EOP, as those defined above, Euler angles, elements of the rotation matrix, components of the quaternion of rotation), it is imperative to state how they are connected to the instantaneous rotation vector. The vector ω⃗ is composed of the instantaneous rotation around each of the three axes of the Euler sequence (lmn) following the notation of (3.1): (1) + γṅ x⃗n(2) . ω⃗ = αl̇ X⃗ l + βṁ x⃗m

(3.10)

Expressing the terrestrial components of ω⃗ according to the angles αl , βm and γn , we obtain the kinematic relations combining the mi , the Euler angles and their time derivative. For classical Euler angles α3 = Ψ, β1 = θ and γ3 = Φ, defined in Section 3.2.1, we obtain the so-called Euler kinematic relations θ̇ + iΨ̇ sin θ = Ωm eiΦ , Φ̇ + Ψ̇ cos θ = Ω(1 + m3 ) ,

(3.11)

where we have introduced the complex equatorial coordinate m = m1 + im2 . Doing the same for the celestial components (ωX , ωY , ωZ ) of ω,⃗ we obtain ̇ iΨ + iΦ̇ sin θ eiΨ , ωX + iωY = θe ωZ = Ψ̇ + Φ̇ cos θ .

(3.12) (3.13)

5 Note that the ωi can be directly measured by in situ laser or matter wave gyroscopes, as the corresponding Sagnac effect is a linear function of the ωi , but at such a slow angular velocity inertial captors are still far from competing with space techniques.

3.3 Relation between EOP and components of the instantaneous rotation vector

| 37

Those angular kinematic relations have the matrix equivalence 0 [ QQ̇ −1 = 𝒲ITRS = [ ωz [ −ωy

−ωz 0 ωx

ωy ] −ωx ] , 0 ]

(3.14)

linking the rotation matrix Q and its time derivative to the terrestrial components of ω.⃗ 3.3.2 Earth orientation parameters formulated as Euler angle perturbations First we distinguish the modeled part of the Earth orientation, given by the a priori Euler angles Ψ0 , θ0 , Φ0 as analytical functions of time (precession–nutation model, diurnal uniform rotation, possibly tidal modeled effects), and an unknown part, to be determined by observations and parameterized by the angular corrections δΨ, δθ and δΦ: θ + iΨ sin θ0 = (θ0 + δθ) + i(Ψ0 + δΨ) sin θ0 .

(3.15)

Rotating an ecliptic GCRS to the ITRS is achieved by applying the Eulerian sequence 313 specified by [ITRS] = R3 (Φ0 + δΦ)R1 (θ0 + δθ)R3 (Ψ0 + δΨ)[GCRSec ] .

(3.16)

The a priori angles θ0 and Ψ0 , given by a precession–nutation model, define the direction of the modeled CIP as well as the a priori Celestial Intermediate System CIS0 , also noted GXi0 Yi0 Zi0 . The CIS0 can be deduced from the ecliptic frame GXYZec by the rotation sequence R1 (θ0 )R3 (Ψ0 ) (see Figure 3.2): [CIS]0 = R1 (θ0 )R3 (Ψ0 )[GCRSec ] .

(3.17)

Combining (3.16) and (3.17), there results [CIS]0 = R1 (θ0 )R3 (Ψ0 ) {R3 (Φ0 + δΦ)R1 (θ0 + δθ)R3 (Ψ0 + δΨ)}

−1

[ITRS]

= R1 (θ0 )R3 (Ψ0 ) R3 (−Ψ0 − δΨ)R1 (−θ0 − δθ)R3 (−Φ0 − δΦ)[ITRS] = R1 (θ0 )R3 (−δΨ)R1 (−θ0 − δθ)R3 (−Φ0 − δΦ)[ITRS] .

(3.18)

Thus, in CIS0 the unit vector ẑ pointing towards the geographic pole admits the components Xẑ 0 ( Yẑ ) = R1 (θ0 )R3 (−δΨ)R1 (−θ0 − δθ) ( 0 ) . Zẑ 1

(3.19)

38 | 3 Astro-geodetic observations

Figure 3.2: Celestial Intermediate System GXi Yi Zi and a priori Celestial Intermediate System GXi0 Yi0 Zi0 .

After suppressing of the second order terms (smaller than 5 10−9 mas with celestial pole offsets δψ and δθ not exceeding 1 mas), we obtain Xz ̂ δΨ sin θ0 ( Yz ̂ ) = ( ) . −δθ Zz ̂ 1

(3.20)

So in CIS0 the equatorial direction cosines of the Gz axis are given by the complex coordinate 𝒫 = Xẑ + iYẑ = −i(δθ + iδΨ sin θ0 ) .

(3.21)

Now, the a priori Terrestrial Reference System TIS0 or Gxi0 yi0 zi0 is defined as the system deduced from the a priori Celestial Intermediate System CIS0 by applying the axial rotation R3 (Φ0 ). In the complex plane Gxi0 yi0 the offset from GZ0 to Gz is represented by the complex coordinate 𝒫̃ = 𝒫 e−iΦ0 . As the a priori geographic axis Gz0 = GZ0 is close to the rotation axis (offset below 1󸀠󸀠 ), Φ0 has almost the same rate as the Earth rotation angle Θ, that is, Φ0 = Ωt + δΦ0 (t) where the phase δΦ0 (t) slightly depends on time t because of the equinox motion and of the Earth rotation variations. The perturbation 𝒫 is commonly split in a “slow” precession–nutation component P of period larger than 2 days and a rapid component P 󸀠 = −peiΦ0 grouping the whole spectral band below 2 days. In the ITRS P 󸀠 is determined by the complex coordinate p = x − i y = −P 󸀠 e−iΦ0 (of which the largest part has a spectral content above 2 days). In summary we have 𝒫 = P + P = P − pe 󸀠

iΦ0

.

(3.22)

This splitting of the angle between the modeled CIP and the geographic pole allows one to specify the position of the observed CIP both in ICRS and ITRS. Before IAU

3.3 Relation between EOP and components of the instantaneous rotation vector

| 39

adopted the definition of the CIP in 2000, the spectral border between P and p was not clearly stated as 𝒫 could include diurnal or sub-diurnal components overlapping the spectral content of p. The corresponding intermediate pole was then called the Celestial Ephemeris Pole (CEP). 3.3.3 Relation between the CIP and the instantaneous rotation pole Let us introduce in the first Euler kinematic equation (3.11) the a priori Euler angles and their perturbations, as defined in the former subsection: ̇ sin θ = ΩmeiΦ , θ̇0 + δθ̇ + i(Ψ̇ 0 + δΨ)

(3.23)

where δθ + iδΨ sin θ ≈ i𝒫 from (3.21) (sin θ ≈ sin θ0 ), with 𝒫 specifying the direction of the Gz axis in the a priori CIS. On the other hand the complex equatorial coordinate m of the instantaneous rotation pole is composed of two parts: the first one, m0 , is associated with the rotation of the a priori ITRS, that is, with the Euler angles Ψ0 , θ0 , Φ0 ; the second one, δm, results from Euler angle perturbations. As θ̇0 + iΨ̇ 0 sin θ = Ωm0 eiΦ0 , Eq. (3.23) is reduced to ΩδmeiΦ0 ≈ i𝒫̇ , (3.24) that is, δm ≈ or, by replacing 𝒫 by Eq. (3.22),

i ̇ −iΦ0 𝒫e , Ω

(3.25)

i ̇ −iΦ0 i ṗ + Pe . (3.26) Ω Ω This expression gives the contribution of the observed pole coordinates p and of Celestial Pole Offsets P on the terrestrial coordinates of the instantaneous rotation pole m. It corresponds to the kinematic Euler relation restricted to equatorial oscillations of the CIP. Equation (3.26), originally derived by Brzezinski and Capitaine (1993) [25] and independently by Gross [96], is of great importance for interpreting observed pole coordinates of the CIP p = x − iy: the dynamic Liouville equations, expressed as a function of m, are transcribed as a function of p by applying (3.26), as will be done considering Chapter 4. δm ≈ p −

3.3.4 Relation between LOD and UT1 Replacing in (2.8) the stellar angle Θ by its UT1 equivalent, expressed by (3.3), we obtain UT1 as an uniform time scale t perturbed by integrated length-of-day offsets: t

UT1(t) − UT1(t0 ) = t − t0 − ∫ t0

ΔLOD LOD

dt .

(3.27)

40 | 3 Astro-geodetic observations For Earth rotation studies we take t as realized by TAI: t

UT1(t) − UT1(t0 ) = TAI − TAI0 − ∫

ΔLOD

t0

LOD

dt .

(3.28)

Then, taking the time derivative we obtain the practical relation ΔLOD d(UT1 − TAI) =− = m3 , dt LOD

(3.29)

found in the literature in the confusing form ΔLOD = −d(UT1 − TAI)/dt, which is only valid if the time unit of t is LOD = 1 day.6 3.3.5 Terrestrial motion of the instantaneous rotation pole The precession–nutation of the CIP in GCRS—overlapping the frequency band ℬGCRS = ]−0.5 Ω, +0.5 Ω[—can be likened on sufficiently short time scales (below 50 years) to an angular perturbation P. Then, in the terrestrial frame, it is accompanied with a composition of retrograde quasi-diurnal loops of m, termed diurnal nutation, belonging to the frequency band ℬITRS = ]−1.5 Ω, −0.5 Ω[. Indeed, according to (3.26) a circular 󸀠 nutation at the frequency σ 󸀠 ≪ Ω, that is, P = P0 eiσ t , is associated in the ITRS with the instantaneous rotation pole oscillation δm = −

󸀠 σ󸀠 P0 eiσ t e−i(Ωt+ϕ0 ) Ω

(3.30)

of angular frequency σ = −Ω + σ 󸀠 close to −Ω. The diurnal nutation reaches an amplitude of 20 mas, modulated with periods of 13.66 and 182.6 days, because of the superposition of near frequency circular oscillations, chiefly at tidal periods −Ω(1 − 1/13.7) (O1 ), −Ω (K1 ) and −Ω(1 − 1/183.1) (P1 ) (see Figure 3.3). Processing of the geodetic observations brings corrections to the precession– nutation (in ℬGCRS ) or celestial pole offsets, not exceeding 1 mas. Complementary frequency band ℬ󸀠 GCRS = ]−∞, −0.5 Ω[ ⋃ ]0.5 Ω, ∞[ is also determined, but in the ITRS by the polar oscillation p = x − iy, covering the frequency band ℬITRS = ]−∞, −1.5 Ω[ ⋃ ] − 0.5 Ω, ∞[. In contrast to the small amplitude of the celestial pole offsets P, p can reach 300 mas over several months and remains widely unpredictable (10 mas error after a month, 30–50 mas after one year) and has to be monitored continuously. 6 Sometimes this expression is wrongly taken as a definition of LOD, so that LOD becomes dimensionless, appearing as the “UT1 rate” expressed in ms/day!

3.3 Relation between EOP and components of the instantaneous rotation vector

| 41

Figure 3.3: Upper plot: path of the instantaneous rotation pole from 1 January 2015 to 31 December 2015, determined from the matrix product Q−1 Q̇ according to (3.14), where the rotation matrix Q includes the UAI 2000 precession–nutation model and EOP of the IERS combined series C04. Turning clockwise around the Celestial Intermediate Pole (CIP) (bold blue line) with a diurnal period, the rotation pole does not shift from CIP more than 20 mas. This so-called diurnal nutation reflects the lunisolar precession–nutation P, viewed from a terrestrial frame according to Eq. (3.26). Bottom plot: m1 component over January 2015.

Equation (3.26) allows us to draw a sketch of the terrestrial oscillations of instantaneous rotation pole in correspondence with those of the CIP in the same system. After having cast aside the celestial contribution P in (3.26), we apply a Fourier transform: m(σ) = (1 +

σ )p(σ) . Ω

Thus, differences between p and m grow with frequency.

(3.31)

42 | 3 Astro-geodetic observations –

– – –

Long periods (|σ| ≪ Ω): paths of the CIP and rotation pole R do not deviate more than 1 mas (considering the main component at 433 days with a maximum amplitude of 0.4󸀠󸀠 ). Prograde diurnal band (σ = Ω + ε, ε ≪ Ω): m(σ) ≈ 2 p(σ). Prograde semi-diurnal band (σ = 2 Ω + ε, ε ≪ Ω): m(σ) ≈ 3 p(σ). Retrograde semi-diurnal band (σ = −2 Ω + ε ; ε ≪ Ω): m(σ) ≈ − p(σ).

Note that any retrograde diurnal oscillation (σ = −Ω + σ 󸀠 , σ 󸀠 ≪ Ω) is by convention excluded from p and has to be considered as a Celestial Pole offset. Actually subdiurnal EOP estimates produce “leakage” of the celestial pole offsets towards pole coordinates. The corresponding oscillation for m, m(σ) = σ 󸀠 /Ω p(σ), has a much lower amplitude.

3.4 Principles of the Earth Orientation Parameters determination 3.4.1 Modeling of the astro-geodetic observations Strictly speaking, the EOP are not measured but estimated from astro-geodetic measurements. The latter ones partly depend on the Earth’s orientation, more generally on the geometry of the receptor network (GNSS receptor, VLBI antenna) with respect to celestial radio sources (quasars, GNSS satellites) or reflector (SLR satellites). So one can recover the hidden information on EOP by numerical inversion. Nowadays astro-geodetic measurements consist of very accurate time delays (10 picoseconds) and Doppler frequency shifts, dated with microsecond accuracy. The most important part of the delay is the geometric one (∼ 70 ms for GNSS), depending on the terrestrial positions of the stations, on Earth’s rotation and EOP and celestial positions of the emitting sources (orbital elements for GNSS satellites). But the geometric delay is not sufficient for describing the measurement: we have to account for atmospheric propagation (tropospheric correction and ionospheric corrections), relativistic correction, clocks biases and instrumental delay. For instance, GNSS observation of the light time τ is modeled by station sat τ = τgeo + τiono + τtropo + τrel + τclock + τclock + τinstru .

(3.32)

Fortunately, the ionospheric delay is either negligible (for laser observation) or can be eliminated by operating two frequency observation modes as it is done for VLBI and GNSS: expressed theoretically as S/f where f is the frequency, and S depends on the electronic content along the sight line, it is given by S/fi for the delay τi observed with the two frequencies f1 , f2 , so that the time travel difference τ2 −τ1 = S(1/f22 −1/f12 ) gives S and in turn the ionospheric delay itself. For GNSS the geometric component τgeo reads τgeo =

⃗ 1 )‖ ⃗ 2 ) − X(t ‖x(t , c

(3.33)

3.4 Principles of the Earth Orientation Parameters determination

| 43

Figure 3.4: Trip time from satellite to terrestrial receptor.

⃗ 1 ) is the observed satel⃗ 2 ) is the station position at reception time t2 and X(t where x(t lite position at emission time t1 , as illustrated in Figure 3.4. The vector x⃗ is naturally referred to the ITRS by column matrix x and X⃗ to GCRS by column matrix X. In ITRS X⃗ has for component QX where Q is the matrix transforming celestial coordinates into terrestrial coordinates and expressed by (3.8). So, the geometric delay reads τgeo =

‖x(t2 ) − Q(x, y, UT1 − UTC, dX, dY)X(t1 )‖ , c

(3.34)

giving the dependence of measurement τ with respect to EOP. These as well as station coordinates can be considered common to a lot of measurements gathered over some period, chosen between one hour (high temporal resolution) and a few days. Considering 100 stations, three EOP, we obtain a reduced set of about 300 common geometric parameters per day, whereas we have collected during the same period up to three million GNSS observations, representing 1 gigabyte. Meanwhile, a bunch of other parameters have to be estimated: orbital elements, tropospheric delay, clock biases, so that the total number of parameters can be up to 10000. The procedure is comparable to reducing a whole library of 10000 volumes to a single book.

3.4.2 Linearization and least-square inversion As illustrated from the above example, the measures oi depend on the influence parameters xj by the non-linear function fi (x1 , . . . , xp ). But linearity of n measures oi with respect to p parameters xi can be achieved if we know approximate values of these parameters, namely x̃i . Then a Taylor development of first order gives p

oi (pj ) = oi (p̃ j ) + ∑ j=1

𝜕fi (x − x̃j ) + O(xj − x̃j )2 . 𝜕xj j

(3.35)

44 | 3 Astro-geodetic observations Let x be the p×1 column matrix of the parameters xj − x̃j referred to their a priori values, let z = O − Õ be the n vector of the measures oi minus their calculated part from the model fi and a priori parameters (the so-called O–C), the former system is equivalent to the matrix form z = Ax + v ,

(3.36)

where A is the n×p matrix of observation partial derivatives, with elements aij = 𝜕fi /𝜕xi and v is the n vector of the residuals. The better the model and a priori values, the smaller is the norm of v. Even though the model was perfect, letting no systematic effect in z, the latter ones would remain noisy, affected by different random errors. In what follows, residuals are supposed to reflect only the random error on z, so that the expected matrix of the residuals v is E(v) = 0. The least square solution x̂ is the one that minimizes the variance of the residuals J(x) = vt v = (z − Ax)t (z − Ax). As the quadratic form J(x) is positive, a sufficient and necessary condition for its minimum is dJ(x) = 0

(3.37)

whatever be the arbitrary infinitesimal variation dx around x. As we have dJ(x) = δ(z − Ax)t (z − Ax) + (z − Ax)t δ(z − Ax) = −δx t At (z − Ax) − (z t − x t At ) Aδx

t

= −δx t At (z − Ax) − (δxt At (z − Ax)) ,

(3.38)

the condition dJ(x) = 0 is equivalent to the normal equations At (z − Ax) = 0

⇐⇒

At Ax = At z .

(3.39)

If the model is well conditioned in the sense that z = Ax contains at least p non-linearly dependent equations for x (A has at least rank p), then it can be shown that the p × p square matrix At A has also rank p, and thus can be inverted, giving the least square solution x̂ ⏟⏟ ⏟⏟⏟⏟⏟ p×1

=

(At A) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ p×p −1

⏟⏟⏟A⏟⏟t⏟⏟

p×n

z ⏟⏟ . ⏟⏟⏟⏟⏟ n×1

(3.40)

First step of the least square procedure consists in building the normal equation (3.39), which reduces or makes compact the measures through the model A. According to the Gauss–Markov theorem, of all linear estimators, x̂ is not only unbiased7 but has minimal variance. If the measure oi is considered as the realization of a random variable 7 Indeed the mean of the estimator is E(x)̂ = E[(At A)−1 At z] = (At A)−1 At E(z). As E(z) = AE(x) + E(v) with E(v) = 0 (no systematic effect in residuals), and we obtain E(x)̂ = (At A)−1 At AE(x) = E(x).

3.4 Principles of the Earth Orientation Parameters determination

| 45

y, the probability to find it in the interval [y, y + dy] is given by Φ(y)dy where Φ(y) is the probability density function. Mostly Φ(y) is the normal law Φi (y) =

y−μ 2 1 ) −( e 2σi , σi √2π

(3.41)

where μ is the mean of the aleatory variable oi . In this case 68 % of the possible values lie within 1 standard deviation of the mean; 95 % lie within 2 standard deviations; and 99.7 % lie within 3 standard deviations. If, moreover, the measures are independent, as happens mostly, the covariance matrix of the observations vector z is reduced to the variances appearing on the diagonal: σ12 0 C = ( ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

0 σ22 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

0 0 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

0 0 0 ) . 0 σn2

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ 0

(3.42)

Measures with high standard deviation should less influence the estimated parameters than measures with lower standard deviation. Therefore, in the least square procedure, the estimated vector residual v̂ = z − Ax̂ is weighted by C −1/2 . The quadratic form to be minimized becomes t

J(x) = (C −1/2 z − C −1/2 Ax) (C −1/2 z − C −1/2 Ax) = (z − Ax)t C −1 (z − Ax) .

(3.43)

In a similar way as given above, we derive the normal equation (At C −1 A)x = At C −1 z ,

(3.44)

and the associated least square solution x̂ = (At C −1 A) At C −1 z . −1

(3.45)

The uncertainties of the measures propagate in the least square solution. It can be shown that the covariance matrix of the least square solution is given by8 D = (At C −1 A)

−1

.

(3.46)

8 The error on the least square estimates x is given by the vector e = x̂ − x where x represents the real but unknown values of the parameters. Let W be the matrix At C −1 A, from the least square solution (3.45) we have e = W −1 W(x̂ − x) = W −1 At C −1 (Ax̂ − Ax) ;

̂ On the other hand As z = Ax + v and z = Ax̂ + v,̂ we obtain e = W −1 At C −1 (v − v). v̂ = z − AW −1 At C −1 z = (I − AW −1 At C −1 )z = (I − U)z

46 | 3 Astro-geodetic observations The variance of the estimated parameter xi is Dii , and its covariance with other parameters xj are given by Dij , j ≠ i. Both quantities are linear combinations of the observation variances. Besides the measurement errors giving the formal uncertainties x

σii = (Dii )1/2

(3.47)

of the least square estimates, the other factor limiting our knowledge of the EOP is the astro-geodetic modeling. We are frequently confronted with over- or under-valued uncertainties σi of the observations. So the variance–covariance matrix (3.42) is known to a unknown factor fσ2 , called unitary variance factor, and should be replaced by C 󸀠 = fσ2 C .

(3.48)

Then it can be shown that an unbiased estimate of fσ2 is fσ 2 =

vt C −1 v . n−p

(3.49)

In turn variance–covariance matrix of the estimated parameter is D󸀠 = (At C 󸀠−1 A)

−1

= fσ2 D ,

and the formal uncertainty of the least square fit is x

σii = √

vt C −1 v D . n − p ii

(3.50)

with the n × n matrix U = AW −1 At C −1 . The expression z = Ax + v yields v̂ = (I − U)(Ax + v) But UA = AW −1 At C −1 A = AW −1 W = A, so that v̂ = (I − U)v or v − v̂ = Uv. Putting this last expression in e, we obtain e = W −1 At C −1 Uv = W −1 At C −1 AW −1 At C −1 v = W −1 WW −1 At C −1 v = W −1 At C −1 v . This expression allows us to derive the covariance matrix of the error, t

t

D = E(eet ) = E(W −1 At C −1 v vt (C −1 ) A(W −1 ) ) , that is,

t

t

t

t

D = W −1 At C −1 E(v vt )(C −1 ) A(W −1 ) = W −1 At C −1 C(C −1 ) A(W −1 ) .

As the covariance matrix is symmetric as well as its inverse, we obtain t

t

t

D = W −1 At C −1 A(W −1 ) = W −1 (W −1 At C −1 A) = W −1 (W −1 W) = W −1 . This demonstration is taken from [105].

3.5 Observations techniques | 47

3.5 Observations techniques 3.5.1 Modern astro-geodetic techniques Very Long Baseline Interferometry (VLBI) For VLBI the basic observable is the time arrival difference of a radio wave front in two telescopes that are a few thousand km apart.9 The number of collected delays has grown from a few thousand delays in the 1980s to about 1 million in 2015. The VLBI delay is modeled in the same way as the GNSS light time (3.32). The astrô where B⃗ is the vector geodetic information is contained in the geometric delay, B⃗ ⋅ k/c, ̂ joining the two telescopes and k the unit vector pointing towards the observed radio source. When considering a network of radio telescopes observing several extragalactic radio sources of the ICRF, the corresponding VLBI geometric delays are expressed in the ITRS by τij = [x i (ti ) − xj (tj )] ⋅ Qk/c

(3.51)

where the x i (ti ) are the terrestrial coordinates of radio telescope i at the reception time ti of the wave front. These delays depend on the geometric transformation Q from the GCRS to ITRS, and thus contain EOP. Other sources of delay are reported in Table 3.1 with their order of magnitude. Observations are scheduled for some radio-telescope networks (like R1, R4, IRIS A, IRIS S, NEOS), observing over 24 hours a set of radio sources constituting the ICRS. The observation sessions are either spaced 5 to 7 days, or 1–3 days for east–west bases dedicated to intensive UT1 determination. The arrival of the wave fronts are recorded independently at each site and stored on hard disk (in the past on magnetic tapes). This data is then sent to a center where a correlator ensures the determination of the delays. Thanks to fiber optic networks, the treatment tends to be more and more near real time (electronic or e-VLBI). Satellite Laser Ranging (SLR) A global network of stations measures the round trip time of ultrashort laser pulses to satellites equipped with retro-reflectors. The round trip time is modeled in a sim9 First developed in the USSR from 1950–1960 for positioning radio-lunar probes, it consisted of two radio telescopes spaced by 500 m near Evpatoria (Crimea). Matveenko then proposed to increase the baseline up to several thousand kilometers and thus invented the VLBI or rather the Radiointerferometriya s Bol’shoy Basoy. The first experiment, in the framework of a Soviet–American cooperation, took place in 1969 between the radio telescope of Simeiz (Balnear station of Crimea, 40 km east from Yalta) and Green-Bank (Maryland, USA). This technique had encountered a great soar in the USA from the 1970s with the “Crustal Dynamics Project”, which allowed the first measurements of continental drift. The history of the technique is reported by one of its inventors in [146].

48 | 3 Astro-geodetic observations Table 3.1: Modeling of the space geodetic delay and orders of magnitude.

frequency (GHz) reference altitude one way/echo measurement error delay in 10−9 s geometric delay ionospheric delay tropospheric delay instrumental delay receptor clock bias relativistic delay

VLBI

GNSS

SLR

8.4 (X), 2.3 (S) 2.2 GHz (K) quasars ∞ quasar → ground see below

1.2, 1.5

optical band

satellites 25000 km sat. → ground see below

satellites 1000–25000 km ground → sat. → ground 50 ps

108 3–300 bi-frequency correction 1–10

107 –108 negligible

0–108 3–300 bi-frequency correction 1–10 100–1000 100–1000 0–100 gravitational deflection

3000 3 satellite velocity

ilar way to GNSS and VLBI. At present the measurement error reaches 50 ps or less, corresponding to a distance of a few mm. In 2018 the system was operating up to 41 satellites. The ones with low orbits (below 6000 km) are much more sensitive to the low degree harmonics of the geopotential, so that their orbital tracking brings about precious determination of the Earth gravitational field and of the center of mass position in the International Terrestrial Reference Frame (ITRF). With a pseudodistance10 dependent on EOP, station coordinates and reflector positions in the CRF, SLR significantly contributes to pole coordinates and LOD with accuracies of 100 µas and 30 µs, respectively. In contrast to GNSS, SLR measurements yield accurate determination of the pseudo-distance between station and satellites, and after appropriate corrections the geometric distance corresponding to the conventional value of the light velocity in vacuum. Therefore the SLR observations have a major impact on the scale of length of the ITRF, also evaluated though VLBI analysis. In the same vein SLR is of primary importance for the calibration of radar altimeters and separation of instrumentation drift from secular changes in ocean surface topography. Lunar Laser Ranging (LLR) is done with the retro-reflector placed on the Moon by human-powered (Apollo) or automated mission (Lunakhod). Because of the distance of the Moon, the echo is much longer (2 s), and the pulse has to be much more energetic, about 1 mJ or more instead of some tens of µJ for SLR. 10 The flight time multiplied by the light velocity c.

3.5 Observations techniques | 49

Global Navigation Satellite System (GNSS) This technique operates on ground by a set of satellites emitting radio pulses, which enclose their emission time tie provided by on board synchronized clocks, where i is index to the satellite. A terrestrial receiver measures simultaneously the travel time τi = ti − tie (ti is the arrival time) and phases Φi of the received signals emitted by the GNSS satellites i = 1, . . . , N. Generally the number N of satellites observed above the site is at least N = 8. The rough trip times ti − tie give with 1 km accuracy the geometric distances between satellites and receiver, so that this one can be positioned at the intersection of the N spheres centered on satellites and spheres of which the radius is equal to (ti −tie )/c. As far as the positions of the satellites are supposed to be well known in the geocentric inertial frame, they can be accurately computed in ITRF by applying the Earth rotation matrix Q. Then the receiver can be located in the ITRF. Conversely, if the positions of a set of receivers are well known in the ITRF, the GNSS measurements allow us to refine our knowledge of the rotation of ITRF with respect to the non-rotating frame. However, for reaching cm accuracy, we have to determine the pure geometric effect. Receiver clock biases are the main source of disturbance (1 km in pseudodistance), but they can be eliminated by doing the simple difference of light time for two different satellites observed simultaneously. Indeed, as the clock bias τclock does i i not depend on the satellite, we have τi = τgeo + τclock + τtrop + ⋅ ⋅ ⋅ for satellites i = 1, 2, 2 1 2 1 and the simple difference given by Δτ12 (t) = τ2 (t) − τ1 (t) = τgeo − τgeo + τtrop − τtrop +⋅⋅⋅ does not depend anymore on clock bias. The instrumental errors of the receiver are eliminated by the same procedure. Forming and collecting these simple differences cleanse the measures from these undesirable effects, and allow one to reach a few meters accuracy satisfactory for many civil purposes and corresponding to the remaining perturbing effects, first of all tropospheric ones (1 m). Now let us gather the simultaneous simple difference for two receivers α and β observing satellites 1 and 2, close enough for considering that tropospheric delays are common: α 2α 1α 2 1 Δτ12 (t) = τgeo − τgeo + τtrop − τtrop ,

(3.52)

β

2β 1β 2 1 Δτ12 (t) = τgeo − τgeo + τtrop − τtrop . β

2β

1β

α 2α 1α We see immediately that the double difference Δτ12 (t)−Δτ12 (t) = τgeo −τgeo −(τgeo −τgeo ) permits one to eliminate tropospheric delay, and isolate the f . For instance, suppose that 8 satellites are simultaneous observed by a network of 5 close GNSS antenna. For each receiver we can built 8(8 − 1)/2 = 28 simple differences. Now, forming the double difference for two different receivers, we build 28(28 − 1)/2 = 378 double differences. As there are 5(5 − 1)/2 = 10 pairs of receivers, we finally get 3780 double differences, which provide an overdetermined system with respect to the 5 × 8 = 40 geometric delays to be estimated. This kind of procedure can be applied to phase carriers, which are more precisely measured. Today, with 31 satellites, the Global Positioning System (GPS) of the U. S. Army is the dominant GNSS. Its most serious rival with 24 satellites is the Russian GLOblaniy

50 | 3 Astro-geodetic observations NAvigatsionny Sputnikoviy System (GLONASS) meaning exactly GNSS, and recent years have seen the emergence of the European GALILEO system11 and the Chinese BEIDOU (15 satellites in 2018). Doppler Orbitography Radio positioning Integrated by Satellite (DORIS) Taking over the US transit system, DORIS was created by CNES and IGN in the 1980s and made operational in 1990. It is composed of nearly 60 ground stations well spread across the globe and about 12 low-altitude satellites (h ≈ 1000 km). In contrast to the other space geodetic techniques, the radio-frequency signals (at ≈ 0.4 GHz and ≈ 2 GHz) are emitted from the ground stations to the satellites, where their Doppler– Fizeau shift is measured by DORIS receivers with a present relative precision of 10−12 (3 10−4 m/s on relative velocity between ground and satellite). By integration of the corresponding velocity, it is possible to adjust the orbit and the position of the ground station with a sub-centimeter precision. This unrivaled accuracy for orbit determination is exploited in satellite altimetric missions (JASON 1/2, TOPEX/POSEIDON), all equipped with DORIS receivers. As ground positioning is limited to fixed emitting beacons, DORIS does not compete with the portability of the GNSS. But as accurate as GNSS, most of the DORIS beacons are integrated in the network constituting the ITRF and thus contribute to its maintenance. On the other hand, DORIS processing yields a pole coordinate with an accuracy of 0.2 mas with errors mostly pertaining to rapid polar motion (below 30 days).

3.5.2 Evolution of the observation techniques Following the Chandler wobble discovery, the International Latitude Service (ILS) was created in 1895 for coordinating accurate observations of latitude of some stations with the aim to determine polar motion. Because of the advent of the quartz clock in the 1930s, variations of the Universal Time became measurable, so that UT was considered as another EOP like x and y: this was the UT0, the terrestrial prime meridian time determined by the Bureau International de l’Heure (BIH). As the UT0 monitoring also contributed to polar motion determination, both time and latitude observations were merged from the 1950s. This practice was recognized by restructuring the ILS, becoming in 1962 the International Polar Motion Service (IPMS). After 1965, the emergence of space geodetic techniques, first the Doppler satellite tracking12 and LLR, and from 1975, SLR and VLBI on extragalactic radio sources were 11 The first satellite was sent December 28, 2005, from the Baikonur Cosmodrome, and the operational phase begun in December 2016 with 22 satellites. 12 With the U. S. TRANSIT system surpassed by the french DORIS system in the 1980s; at the time the Doppler tracking gave the pole coordinates with an unsurpassed precision of about 50 cm (15 mas)

3.5 Observations techniques | 51

Figure 3.5: Weighting of the techniques in EOP determination from 1970 to nowadays.

accompanied by a drop of the EOP uncertainty by a factor 10 (formal uncertainties of about 1 mas). Taking advantage of this progress, the respective contribution of the techniques to EOP determination has evolved as shown by Figure 3.5. Since the middle of the 1980s, the optical astrometry is almost no longer used for Earth rotation monitoring. The last development in this area was the reanalysis of an exhaustive set of astrometric observations from 1900 to 1992 by a team of the Czech Academy of Science led by with respect to optical astrometry, but today DORIS gives the pole with an uncertainty of 0.2 mas, which is four times the one reached by GNSS.

52 | 3 Astro-geodetic observations Jan Vondràk. As it benefited from much better positions of the observed star thanks to the results obtained by the Hipparcos missions in the 1990s, especially those pertaining to proper motions, this reanalysis has provided one of the most consistent pole coordinate time series from 1900 to the 1970s [222, 226]. In this respect it is included in the long term IERS C01 solution covering the period of 1900 to the present. From 1993 the GNSS has become gradually the best technique for determining the pole coordinates and length of day variations. However, VLBI remains the only technique capable of providing UT1 and Celestial Pole Offsets. With the adoption of atomic time scale TAI as legal time in 1972, BIH had lost its raison d’être. It was redesigned with the IPMS within a larger structure coordinating both astro-geodetic techniques and analysis centers: the International Earth Rotation Service (IERS). Until 1997 the IERS activities were mainly concentrated at the Observatoire de Paris (Central Office) and the U. S. Naval Observatory (“rapid series” for use in real time). As the activities coordinated by the IERS were rapidly growing, a restructuring was imposed in 1997. Space geodetic techniques that were previously overseen by the IERS took their autonomy within the framework of specific international services. The Central Office is now in BKG (Bundesamt fuer Kartographie und Geodesie) in Frankfurt, but the Paris Observatory kept and extended its scientific task; first of all it provides the EOP time series constituting the international reference (monthly Bulletin B, bi-weekly combined series C04). Maintenance of the ITRF has been given to IGN, while the Paris Observatory and USNO are jointly responsible for maintaining the ICRF. In regard of the fundamental role of the geophysical fluids (atmosphere, oceans, fluid core, inland waters, etc.) in global geodynamics (Earth rotation and large scale deformation), and taking advantage of the progress realized by global circulation model, IERS coordinates the production and dissemination of the related data (especially angular momentum) in the framework of its Global Geophysical Fluid Center (GGFC). Ability of the techniques to determine the EOP The VLBI technique is certainly the most complete, but also the heaviest and most expensive. Only VLBI permits one to adjust long term (over several days) Earth rotation irregularities in the ICRF, namely the precession–nutation and UT1. The reason is that VLBI measurements are attached to a stable celestial source, the quasars. This is not the case of satellite measurements, since they are referred to orbital paths in the geocentric dynamical CRF (for the satellite center of mass) which cannot be predicted accurately enough in the CRF after a few days, mainly because of the atmospheric drag. Satellites do not provide stable celestial “anchors” for periods exceeding a week. At one time or another, the orbit must be re-determined by the most recent observations. If satellite techniques are unsuitable to the precession–nutation and UT1 for periods exceeding seven days, they are very effective in determining the pole coordinates and

3.6 Pole coordinates accuracy | 53

the length of day offset. Indeed, the polar motion in the inertial frame appears as a counter clockwise quasi-diurnal circular motion of the geographic pole around of the CIP, and the radius and the direction of this circular oscillation can be determined over one day or a few hours. The determination of the pole coordinates (x, y) therefore does not require the stability of the satellite frame for several days, and they are well estimated by GNSS and SLR. Despite the high accuracy of SLR and DORIS, these techniques are restricted by the low number of satellites (DORIS) or laser stations (SLR), and are hampered by their heaviness. In contrast, the GNSS, by the huge amount of data they provide and their spatial coverage, has emerged since 2000 as the best technique to track the polar motion.

3.6 Pole coordinates accuracy 3.6.1 Need for a combined reference EOP solution As we have seen, various centers produce EOP from the processing of a given technique, but the corresponding time series do not provide the full set of EOP. For instance, the international GNSS service (IGS) provides only pole coordinates and LOD, VLBI analysis centers provide all EOP, but the associated pole coordinates are not as accurate and as frequently sampled as the GNSS ones, and so on. Moreover various intra-techniques as regards the EOP series are derived according to a different processing pertaining to algorithm, a priori models, station network or selected observations. In other words these EOP data are not determined consistently, and this deteriorates their accuracy. Because of the multiples, strategic, societal applications of EOP, complete and ITRF/ICRF consistent EOP values have to be regularly updated and provided to the various users. In the framework of the International Earth Rotation and Reference System Service (IERS), production of this reference is entrusted to Paris Observatory for the final version, ending 30 days back from the current date and to USNO for the rapid or real-time solution (last 30 days and 6 month prediction). The Paris Observatory solution or the C04 combined series, starting in 1962, is obtained from the combination of “operational” EOP series derived from the various astro-geodetic techniques: Doppler tracking, LLR or SLR, VLBI and more recently GNSS. The combination performed twice a week consists in series given at one-day intervals for each of those parameters [16]. For overcoming the shortcomings of the intra-technique solutions, multi-technique combination at normal equation level is now applied by several analysis centers for estimating the astro-geodetic parameters. Even if the normal equations disseminated by technique centers are constructed on the basis of different a priori data or reference systems, the inconsistencies can be accounted for when combining the SINEX

54 | 3 Astro-geodetic observations files containing both normal equations and a-priori data. At Paris Observatory, such a combination and the subsequent inversion is performed through the DYNAMO software for determining the five EOPs, station coordinates of the observation networks, and possibly radio-source coordinates. As this approach is not yet operational, the present procedure for producing the C04 solution remains an a posteriori combination of various intra-techniques EOP series.

3.6.2 Accuracy of the combined C04 solution The evolution of the pole coordinates determination is summarized by the combined C04 series, showing a historical continuity since 1962, while reflecting the performance of the observation techniques. Based on optical astrometry in the early 1960s, a C04 series is built from Doppler measurements from 1965, then SLR and VLBI analyses were introduced from the 1980s with the abandonment of the optical data. From the 2000s GNSS solutions stand out as the best, and contribute for more than 90 % to C04 pole coordinates. In about 40 years the accuracy of the pole coordinates has been improved by a factor 600 (Table 3.2). Table 3.2: Evolution of the mean uncertainty in mas of the pole coordinates (C04 series). Period 1962–1967 1968–1971 1972–1979 1980–1983 1984–1995 1996–2000 2000–2019 30

20

15

2

0.7

0.2

0.05

3.6.3 Uncertainty versus stability for intra-technique solutions As seen earlier, formal errors of the EOP estimates do not give the real EOP accuracy, but rather quantify the EOP ability to fit the measurements together with the other parameters. In this sense they also reflect the shortcoming and inconsistency of the measurement modeling. In particular EOP can present systematic effects: bias, spurious periodic terms resulting from correlation with orbital parameters, and so on. Looking at GNSS solutions, considered as the most reliable, we notice that their differences are mostly larger than the formal errors affecting each point. Whereas the formal error oscillates between 5 and 30 µas, the mean standard deviation of the paired differences amounts to 50 µas, with biases up to 100 µas. Formal uncertainties of pole coordinates x and y can be rescaled in the light of these paired differences. The same is true for the various VLBI and SLR solutions.

3.6 Pole coordinates accuracy | 55

Figure 3.6: Paired differences of four GNSS pole coordinate solutions (x and y) for the period 2000– 2016: Allan Deviation (AD) (top plots) and spectral amplitude (bottom plots). In log–log scale, for a period T between 10 and 1000 days, ADs have a slope s󸀠 = 0 revealing a frequency modulation flicker noise (PSD in 1/f ).

Given n solutions of the pole coordinates for a given technique over the time interval (2000–2019 for GNSS, 2004–2019 for SLR, 1990–2019 for VLBI), we build n(n − 1)/2 paired differences. Systematic errors at a given period or frequency are detected by the discrete Fourier spectrum, whereas the random uncertainty can be assessed by analyzing the Allan deviation (or Allan variance) (Appendix G.5). GNSS Let us start with GNSS time series, commonly considered as the most reliable. We take five of them (IGS, ESOC, GFZ, JPL, CODE). Reported in the log–log plots of Figure 3.6, Allan deviations of the paired differences show more or less horizontal lines between 10 and 1000 days, with a floor of c󸀠 = 50 µas. This reflects a frequency modulation flicker noise or a pink noise with power spectral density and spectral amplitude, respectively, given by s(f ) =

c󸀠2 , 4 ln(2)f

ε(f ) = √

30 µas s(f ) 1 = c󸀠 √ = , P 4 ln 2fP √fP

(3.53)

56 | 3 Astro-geodetic observations

Figure 3.7: Paired differences of three SLR pole coordinate solutions (x and y) for the period 2004– 2019: Allan Deviation (AD) (top plots) and spectral amplitude (bottom plots).

where f is the frequency, and P the sampling period (see Eq. (G.24)). The quantity ε(f ) will be considered as the random error affecting the spectral component at frequency f for a given sampling of length P. On the other hand the amplitude spectra of the paired differences are evidence (here in log–linear scale; see Figure 3.6) of systematic differences of about 50 µas at 1 year and inter-annual periods,13 much larger than the background noise ε(f ). SLR We select three SLR solutions (MCC, IAA, ILRS).14 According to Figure 3.7, the Allan deviation of the corresponding paired differences have slopes between 0 and −1/3 from 10 days, with a noise level at least 2 times larger than for GNSS. Annual and interannual systematic differences are up to 400 µas. 13 Inter-annual periods correspond to the range between 365 days and 36525 days (10 years). Notice that the x component presents larger inconsistencies in this band. 14 MCC: Russian Mission Control Center; IAA: Institute for Applied Astronomy (Saint-Petersburg); ILRS: International Laser Ranging Service.

3.6 Pole coordinates accuracy | 57

Figure 3.8: Paired differences of four VLBI pole coordinate solutions (x and y) for the period 1990– 2019: Allan Deviation (AD) (top plots) and spectral amplitude (bottom plots).

VLBI We select five VLBI series: USNO (US Naval Observatory) and GSFC (Goddard Space Flight Center) obtained by the SOLVE software, IAA (Institute for Applied Astronomy) obtained by the QUASAR software, and BKG. From 10 days to inter-annual periods Allan deviations of the corresponding paired differences have slopes near −1/2 in log– log scale (see Figure 3.8), revealing a white noise. Its flat power spectral density is s(f ) = AV(T = 1 d) = K0 ≲ 4002 µas2 d. In turn this noise has a spectral amplitude ε(f ) ≲

400 µas cpd−1/2 . √P [d]

(3.54)

We notice that VLBI pole coordinates become as stable as GNSS at seasonal scales. Inconsistency is striking at annual and inter-annual period, reaching 50 µas, but does not increase with the period as in the case of GNSS and SLR. For all these reasons, VLBI series are certainly more adequate for investigating Markowitz wobble and secular drift (actually GNSS, covering only 20 years, cannot provide any valuable information for the last two components).

58 | 3 Astro-geodetic observations

Figure 3.9: Paired differences of four VLBI solutions of the nutation offsets dX and dY for the period 1990–2019: Allan Deviation (AD) (top plots) and spectral amplitude (bottom plots).

VLBI celestial pole offsets The former analysis is extended to the nutation offsets of the same series, and results are plotted in Figure 3.9. As for VLBI pole coordinates, random differences can be considered as white noise till 1000 days with AV(T = 1 d) ≲ 4002 µas2 d corresponding to the spectral amplitude ε(f ) ≲

400 µas cpd−1/2 . √P [d]

(3.55)

Discussion Each technique distinguishes itself by the stochastic nature of the paired difference. Is it associated with the nature of the measurements, the model, and algorithmic part of the processing? The underlying software, for which algorithm and a priori models are the same, has a strong influence. For example, USNO and GSFC VLBI series, both being obtained by the software SOLVE, yield the lowest Allan deviation in Figures 3.8 (polar motion) and 3.9 (nutation). Yet, the stochastic nature of the noise remains a feature of the technique.

3.7 Conclusion

| 59

3.7 Conclusion The pseudo-geometric nature of astro-geodetic observations led to a description of the rotation of the ITRS relative to the GCRS by introducing a Celestial Intermediate Pole (CIP), whose spatial position is determined by the superposition of precession– nutation terms with periods over two days. Its allows one to clearly split the rotation transformation from ITRS to GCRS into a regular part (precession–nutation model and uniform diurnal rotation) and an irregular part (nutation corrections, non-uniformity of diurnal rotation, polar motion) described by 5 EOPs. The terrestrial path of the CIP, or polar motion is estimated with a formal uncertainty of 50 µas, but accounting for systematic effects the accuracy is not below 100 µas. The common time resolution is 1 day. Throughout the world many analysis centers process astro-geodetic observations, and extend day after day the pole coordinate time series. Nowadays the reference time series provided by the IERS (C04) mostly reflects the GNSS solution. But, as noticed from Allan variance analysis, the VLBI solution seems to have higher stability after 1000 days and is of particular interest for estimating the polar drift. So, in reason of their complementary performances, observations of the astro-geodetic techniques have to be combined. Until the 2000s the multi-technique combination was only applied to astro-geodetic parameters obtained independently for each technique. In order to increase the consistency of the results, especially station coordinates and Earth rotation parameters, IERS favors now the combination at the level of the normal equations. This allows to estimate and suppress systematic effects coming from observing networks or from the processing (for instance length scale bias or drift, residual rotation between TRS, difference between a-priori parameters), and reduce the EOP uncertainty.

|

Part II: Polar motion theory

4 Liouville equations Mais si un corps céleste n’est pas sphérique, ou que ses moments d’inertie par rapport à ses trois axes principaux ne sont pas égaux, et qu’il ait commencé à tourner autour d’un axe différent de ses axes principaux, alors quand même il n’y aurait point de forces sollicitantes, son mouvement de rotation serait troublé, et l’axe de rotation changerait de direction. Leonard Euler in Recherches sur le mouvement de rotation des corps célestes (1759) [85]

4.1 General introduction of the second part In the second part we aim at establishing the fundamental differential equations ruling the polar motion. The problem cannot be reduced to the one of a rigid spinning top, for the Earth non-rigidity determines the polar motion features. As an introductory chapter we present the fundamental Liouville equations, valid for any rotating body, and their linearization for rotational perturbations, permitting one to study the geophysical causes independently from one another. The modeling of the non-rigidity influence is done in the next chapters: first by modeling the Earth as a quasi-elastic body (Chapter 5), then by adding the anisotropic ocean layer considered at equilibrium, and finally by considering the fluid core (Chapter 7).

4.2 Newtonian framework of the Earth rotation First we recall the theoretical foundations that are used to understand the variations of the Earth rotation. This is the law of balance of angular momentum expressed in the Terrestrial Reference System (TRS) and made more malleable by linearization. No relativistic effect has ever been observed on the Earth rotation, the framework of Newtonian mechanics is ample.1 To write the Newtonian equations of a rotating celestial body, the origin of the inertial frame is not involved, and the Earth’s rotation can be studied in the dynamical GCRS defined in Chapter 1, as we recall below. Consider the fundamental relationship dynamics, expressed in inertial frame whose origin is not specified: dm

d2 R⃗ = dF⃗ , dt 2

(4.1)

where R⃗ sets a mass element dm of the body in any inertial system, and dF⃗ is the sum of the forces applied to dm. The elemental force dF⃗ consists of an external force dF⃗ ext 1 The drift of the dynamical celestial reference system relative to quasars (geodetic precession– nutation) is ghostly (see Chapter 1), the post-Newtonian nutation (≤ 1 µas) even more so [19]. https://doi.org/10.1515/9783110298093-004

64 | 4 Liouville equations system and an internal force dF⃗ int resulting from the interaction of dm with the other components of the body. Time in Newtonian mechanics is supposed to flow in the same way everywhere, so it is possible to coordinate the whole time measurements by a single parameter t. Let R⃗ 0 be the radius vector of the mass barycenter in the inertial frame, and consider r ⃗ setting the dm position with respect to the center of mass. We have dm

d2 R⃗ 0 d2 r ⃗ + dm = dF⃗ . dt 2 dt 2

(4.2)

By integrating this relation over all matter composing the celestial body, we obtain M

d2 R0⃗ = F⃗ ext + ∫ dF⃗ int , dt 2

(4.3)

⊕

where F⃗ ext is the result of external forces acting on the body, reduced almost exclusively to the lunisolar gravitation, except for small bodies like artificial satellites or asteroids, of which the orbital motion is strikingly influenced by the radiation pressure. If the internal forces obey the principle of action and reaction (gravitational force, electric force), it is easy to see that ∫⊕ dF⃗ int = 0,⃗ and the above relation gives the center of mass theorem, M

d2 R0⃗ = F⃗ ext . dt 2

(4.4)

Within the planets, circulating electric charges develop electromagnetic forces, which are known to violate the principle of action and reaction. Therefore there is no reason why ∫⊕ dF⃗ int should be equal to zero, and the center of mass of the system is expected to accelerate even in the absence of external force. But, as demonstrated by Poincaré in 1898 in a lecture taught at the Sorbonne [171], the radiation emitted by the interacting electric charges is endowed with a momentum canceling the effect of internal forces, so that the center of mass theorem still holds [12].2 The combination of (4.2) and (4.4) gives dm

d2 r ⃗ dm ⃗ ext = dF⃗ − F . M dt 2

(4.5)

This equation characterizes the movement of the mass element dm in a barycentric system maintaining a fixed orientation in an inertial indefinite reference system. In such a barycentric dynamically non-rotating reference system, a dynamical GCRS in the case of the Earth, the motion of the mass elements is described by the difference ⃗ between the mass element acceleration dF/dm and the acceleration of the center of 2 More precisely any volume element containing the amount of electromagnetic energy dE is endowed with the fictitious mass m = dE/c2 (first form of mass–energy equivalence) and carries the momentum p = dE in the direction of the local radiation. c

4.3 Liouville equations | 65

mass, mostly resulting from the planetary orbital motion. This is the tidal acceleration γ⃗T or tidal force dmγ⃗T given by dF⃗ F⃗ ext γ⃗T = − . (4.6) dm M All developments of Newtonian mechanics can be applied in such a dynamically nonrotating reference system as far the external forces are reduced to the tidal forces dmγ⃗T . To address the problem of the Earth rotation, we can then consider two approaches. The first method is based on the direct application of the fundamental relation of dynamics to all mass elements which compose the Earth. In addition its accounts for the boundary conditions and includes certain simplifying assumptions for the mutual interactions between these elements. By involving certain symmetries (radial density, etc.), the number of equations can be reduced considerably, thereby avoiding a purely finite element approach as it is done in industry to simulate mechanical deformations, and reducing the number of variables of the problem to the essential degrees of freedom of the system. Integration yields the displacement field of these elements, from which we can extract the purely rotational component [203, 227]. The method hardly permits one to treat any kind of geophysical processes. Therefore we favor the more global approach based upon the angular momentum balance of the body in the dynamically non-rotating system. Directly related to the rotational change of the system, it links the rotation parameters of the Earth with global quantities characterizing geophysical process, like mass transports, or external excitation. In a complementary sense, the balance of the angular momentum of the body can be formulated in variational terms, which have been very fruitful for building the non-rigid Earth nutation theory (see e. g. [79]), and which some benefits [151] have not been fully exploited in modeling the geophysical fluctuations of the Earth rotation.

4.3 Liouville equations 4.3.1 Dynamical Liouville equations The mechanical system includes at least the solid Earth to which we can add the core, the atmosphere and the oceans. However, we are free to reject the fluid layers in an external system interacting with the solid Earth. In the dynamical GCRS, the law of the angular momentum or second Euler law [86] tells us that the variation of the angular momentum H⃗ of the Earth is equal to the moment L⃗ of external forces (in the sense given in the previous section) that are applied:3 dH⃗ = L⃗ . (4.7) dt 3 This law is actually a theorem, which is easily derived when the internal forces are central and satisfy the principle of action and reaction. When electromagnetic forces are involved, the electromagnetic angular momentum of the emitted radiation has to be considered.

66 | 4 Liouville equations We shall develop this equation in a tied-crust system for two reasons. On the one hand our point of view is geophysical, dealing with earthbound quantities (gravimetric tide, pressure, wind, etc.). On the other hand, in such a system, we can adopt a perturbation approach by separating moment inertia matrix and instantaneous rotation vector into a mean part, which is known, and small disturbances which remain to be determined. For treating triaxiality, it is advantageous to express this equation with respect to an Auxiliary Terrestrial Reference System (ATRS), diverging from the TRS (see Chapter 1) by any constant rotation around the center of system. Let ω⃗ be the instantaneous rotation vector of the ATRS with respect to the CRS, Eq. (4.7) transcribed in ATRS reads dH +ω∧H=L, dt

(4.8)

where H, ω, L are column matrices of the ATRS components of the corresponding vectors. The theoretical origin of the ATRS is at the center of mass of the Earth and a priori differs from the ITRF one by a translation d of about one centimeter [212], so that the confusion of the two systems neglects differences in moments of inertia of the order of Md2 ≈ 10−18 A (A is a mean principal moment of inertia). This is fully justified in the light of the precision wherewith the inertia moments and their variations, are determined, as discussed below. The Earth’s angular momentum is the sum of elementary angular momentum associated with mass dm, positioned by the vector r ⃗ from the center of mass of the Earth. If v⃗ is the velocity of dm in the CRS, we have H⃗ = ∫ r ⃗ ∧ v⃗ dm .

(4.9)

⊕

At this point we introduce the instantaneous rotation vector of the Earth (or more precisely of the ATRS) with respect to the celestial system. Indeed, the velocity vector v⃗ consists of v⃗ = ω⃗ ∧ r ⃗ + v⃗r ,

(4.10)

where v⃗r is the velocity vector relative to the crust. Recall that, for the crust, by definition of the terrestrial system, v⃗r constitutes an irrotational field. This partition is not arbitrary: it reflects the fact that the velocity of a mass element is mainly due to the rotation of the Earth (460 m/s at the equator) and to a much lesser extent to relative displacements in ATRS. Indeed, for the time scales that we will explore, more than a few hours, the only significant velocities are associated with the oceanic currents (1 m/s) and the winds reaching up to 30 m/s (jet streams in the stratosphere at 10–12 km altitude); other global mass motions, like lunisolar tides in the mantle and oceans, atmospheric thermal tides activated by solar heating, are extremely slow, hav-

4.3 Liouville equations | 67

ing an order of magnitude of one meter per day. By putting Eq. (4.10) in (4.9), we obtain ⃗ H⃗ = ∫[r 2 ω⃗ − (r ⃗ ⋅ ω)⃗ r]dm + ∫ r ⃗ ∧ v⃗r dm . ⊕

(4.11)

⊕

The first integral, associated with the overall rotation of the system, is the matter term. The second integral, associated with the velocity field v⃗r , is the relative momentum or motion term. The previous equation can be expressed in any coordinate systems by the matrix relation H = I(t)ω + h ,

(4.12)

where I(t) is the moment inertia matrix expressed by Iij = ∫(r 2 δij − xi xj )dm ,

(4.13)

⊕

with the Kronecker symbol δij . So Eq. (4.8) can be extended to a non-rigid body by substituting H with (4.12): d(I(t)ω + h) + ω ∧ (I(t)ω + h) = L . dt

(4.14)

Euler first formulated this vectorial equation for a rigid Earth (constant inertia moments, no relative angular momentum) [86]. In this case their projections on the x, y, and z axes of the terrestrial system are called dynamical Euler equations. Then the French mathematician Joseph Liouville (1858) [135] extended it by considering changes of the moment of inertia. At the eve of twentieth century the relative angular momentum was conceived and led to the modern form, the so-called Liouville equations [153, 126]. Some authors, like Moritz and Müller [152] favor the term “Euler– Liouville equation”, probably to avoid a possible confusion with the equations associated with Liouville’s theorem. Mean moments of inertia the Earth and inertia moment variations On average, for periods longer than a few years, the principal moments of inertia of the Earth A < B < C have values reported in Table 4.1.4 Note that (B − A)/A ≈ 2.5 10−5 , while (C−A)/A ≈ (B−C)/B ≈ 3 10−3 . Over the period 1700–2100, the axes of the two first moments A and B are located within 1󸀠󸀠 of the equatorial geographic plane. In other words, the mean principal axis of inertia IC is close to the axis of the geographical 4 See Appendix H for details pertaining to their determination.

68 | 4 Liouville equations poles—less than 1󸀠󸀠 away.5 But the IA and IB axes are not close to the Gx and Gy axes of TRS. Indeed the IA axis has a longitude of λA = −14.9291 ± 0.0010°.6 This leads us to specify the ATRS as the Earth coordinate system Gx󸀠 y󸀠 z 󸀠 deduced from Gxyz by an axial rotation of angle λA . As Gx󸀠 y󸀠 z 󸀠 neighbors the mean principal axis frame (IA , IB , IC ), the instantaneous inertia matrix is expressed in the ATRS in a quasi-diagonal form (symmetric by definition): [ I(t) = [ [

󸀠 A + c11 󸀠 c21

󸀠 c13

󸀠 c12

󸀠 B + c22 󸀠 c23

󸀠 c13

󸀠 c23

] ] ,

(4.15)

󸀠 C + c33 ]

where the cij󸀠 are assumed to have small increments of inertia moment due to mass redistributions or permanent misalignment of the reference system Gx󸀠 y󸀠 z 󸀠 relative to the mean principal axes. Global mass redistributions within the Earth (tides and thermal processes in the fluid layers) produce changes in moments of inertia at most of the order of one hundred millionth of the principal moments of inertia (cij󸀠 ≤ 10−8 A), and a misalignment smaller than 1󸀠󸀠 (over the period 1700–2100) causes constant increments not exceeding 10−8 A.7 Although their magnitude is at least 1000 times smaller than the uncertainty of the total moment of inertia (10−5 A), they are very finely modeled, at least for some parts of the Earth. This illustrates the rule that it is always easier to determine the variations of a physical quantity than its absolute value. Fortunately systematic uncertainties on A, B and C do not affect the analysis of the terrestrial rotation instabilities, which are determined by the relative differences between A, B, and 󸀠 󸀠 󸀠 C and changes c13 , c23 , and c33 . The involved time scales are at least a few hours, so 󸀠 −8 that ċij ≤ 10 AΩ. The strongest earthquakes cause mass redistributions over a few minutes, accompanied by changes cij󸀠 ≈ 10−11 A and ċij󸀠 ≤ 10−7 AΩ if we give credit to estimates of Chapter 14, Table 14.2. The Gx󸀠 y󸀠 z 󸀠 system is treated as a Tisserand system, where the contribution of the lithosphere to h is zero, so that the h components must be sought in the mantle, core, hydro and atmospheric layers if they are included in the mechanical system. The relative angular momentum is mainly produced by atmospheric winds, and reaches a value of about 1026 kg m2 s−1 for the axial component. So, h ≤ 10−8 AΩ (AΩ ≈ CΩ ≈ 6 1033 kg m2 s−1 ), or one hundred millionth of the total Earth angular momentum. Given that changes in global circulation occur at least on one day, we have the upper bound ḣ ≤ 10−8 AΩ2 . 5 This order of magnitude can be easily derived: the axis IC , coinciding with the mean rotation pole, also drifts from the geographic axis at the rate of about 0.4󸀠󸀠 /cy towards 80° west in the terrestrial system. Both IC and geographic axes were confounded over the period 1900–1906; see Chapter 1. 6 According to the Chen and Shen estimates (2010) [42] founded on recent models of the gravity field as in EGM 2008. 7 This order of magnitude can be easily derived from (C.17) in Appendix C.2.

4.3 Liouville equations | 69 Table 4.1: Constants pertaining to mean principal moments of inertia. Values Equat. principal inertia moment Equat. principal inertia moment Mean equat. principal inertia mom. Axial principal inertia moment Dynamical ellipticity Triaxiality coefficient Longitude of the principal inertia axis A

Unit

A B A

8.010083(9) 10 8.010260(9) 1037 8.010171(9) 1037

kg m2 kg m2 kg m2

C&S C&S C&S + Eq. (4.21)

C e

8.036481(9) 1037 3.284 51(1) 10−3 ≈ 1/304.46 3.284 547 9(1) 10−3 1.10485 10−5 −14.92851(8)

kg m2

C&S C&S + Eq. (4.22) MHB 2000 C&S + Eq. (4.23) C&S

e󸀠 λA

37

°

C&S: Chen and Shen (2010) [42], Table 2a, zero-tides value associated with the gravity model EGM2008. IERS: IERS Conventions (2010), Table 2.1 [168]. MHB 2000: Mathews Herring Buffet precession–nutation model [145].

The Gx󸀠 y󸀠 z 󸀠 system is driven by a rotation around an axis remaining close to the axis Gz = Gz 󸀠 (deviation less than 1󸀠󸀠 ), with almost uniform speed. Consequently, the instantaneous rotation vector ω⃗ is expressed in the coordinate system Gx󸀠 y󸀠 z 󸀠 by m󸀠1 ω⃗ = Ω ( m󸀠2 ) , 1 + m󸀠3

(4.16)

where Ω = 7.292 115 10−5 rad/s is the nominal mean Earth angular velocity (see Eq. (2.3)), and the off-dimensional offsets mi are small compared to 1 (observation shows that m󸀠1 ∼ m󸀠2 ∼ 10−6 , m󸀠3 ∼ 10−8 and ṁ 󸀠1 ∼ ṁ 󸀠2 ∼ 10−6 Ω, ṁ 󸀠3 ∼ 10−8 Ω). The indices 1, 2, 3 stand for the x󸀠 , y󸀠 , z 󸀠 components, respectively. From Eqs. (4.8), putting together the equatorial components by defining complex coordinates H 󸀠 = H1󸀠 + iH2󸀠 , L󸀠 = L󸀠1 + iL󸀠2 , and m󸀠 = m󸀠1 + im󸀠2 , we get Ḣ 󸀠 + iΩ[(1 + m󸀠3 )H 󸀠 − m󸀠 H3󸀠 ] = L󸀠 , Ḣ 3󸀠 + Ω[m󸀠1 H2󸀠 − m󸀠2 H1󸀠 ] = L󸀠3 .

(4.17)

Then we express the angular momentum (4.12) as a function of the inertia moments from (4.15) and a motion term: 󸀠 󸀠 󸀠 H1󸀠 = (A + c11 )Ωm󸀠1 + c12 Ωm󸀠2 + c13 Ω(1 + m󸀠3 ) + h󸀠1 ,

󸀠 󸀠 󸀠 H2󸀠 = (B + c22 )Ωm󸀠2 + c12 Ωm󸀠1 + c23 Ω(1 + m󸀠3 ) + h󸀠2 ,

H3󸀠

= (C +

󸀠 c33 )Ω(1

+

m󸀠3 )

+

󸀠 c13 Ωm󸀠1

+

󸀠 c23 Ωm󸀠2

+

h󸀠3

(4.18)

.

Let h = h1 + ih2 be the equatorial relative angular momentum and c = c13 + ic23 , we have

70 | 4 Liouville equations H 󸀠 = (Am󸀠1 + iBm󸀠2 + c󸀠 )Ω + h󸀠 + 𝒪(cij󸀠 m󸀠k )Ω ,

(4.19)

H3󸀠

(4.20)

= [C(1 +

m󸀠3 )

+

󸀠 c33 ]Ω

+

h󸀠3

+

󸀠 󸀠 𝒪(cij mk )Ω

.

Let A be the mean equatorial moment of inertia reading A=

A+B , 2

(4.21)

then the mean dynamical ellipticity resulting from the Earth biaxility is defined by e=

C−A A

(4.22)

.

Its value determines the precession rate and the luni-solar components of the nutation, of which observation gives back the value e = 0.003 284 547 9(1) ≈ 1/304.46 [145]. More recent studies, based upon a new geopotential model, yield slight different values (see Table 4.1). On the other hand triaxility is quantified by the coefficient e󸀠 =

B−A A

=

A−A A

=

B−A . A+B

(4.23)

Following the formal approach given in [143], Eq. (52), the complex equatorial angular momentum (4.19) is also expressed by H 󸀠 = (Am󸀠 − e󸀠 Am󸀠 ∗ + c󸀠 )Ω + h󸀠 + 𝒪(cij󸀠 m󸀠k )Ω ,

(4.24)

where m󸀠 ∗ is the complex conjugate of m󸀠 . By putting the angular momentum expression (4.24) and (4.20) in Eqs. (4.17), we fully expand the Liouville equations. When cij = 0 and hi = 0, the Liouville equations take the form of the dynamical Euler equations, as already mentioned. 4.3.2 Linearization at sub-secular scale At sub-secular scale, the mass redistributions are chiefly located in the fluid part, and yield inertia moment variations of about 10−8 A and relative angular momentum smaller than 10−6 AΩ. This order of magnitude is exceeded by the cumulated effect of mantle convection, tectonic displacements and post-glacial rebound over prehistoric or geological periods. Except the last chapter, our considerations are restricted to the sub-secular scale; in turn the development of (4.17) makes terms appear having the dimension of a moment of force with the order of magnitude 10−6 –10−8 AΩ2 (10−6 AΩ2 = 1024 kg m2 s−2 ): Ωċij󸀠 , AΩṁ 󸀠i , ḣ 󸀠i , AΩ2 m󸀠i , Ωh󸀠i , and terms of much smaller amplitude:

4.3 Liouville equations |

71

ċij󸀠 Ωm󸀠k ≤ 10−13 AΩ2 ,

(C − A)Ω2 m󸀠3 m󸀠1/2 , (B − A)Ω2 m󸀠1 m󸀠2 ≤ 10−16 AΩ2 ,

cij󸀠 m󸀠k Ω2 ≤ 10−14 AΩ2 ,

cij󸀠 m󸀠k m󸀠l Ω2 ≤ 10−19 AΩ2 ,

h󸀠i m󸀠j Ω ≤ 10−14 AΩ2 ,

reaching no more than 10−14 AΩ2 (1016 kg m2 s−2 , estimates from the upper bounds on the quantities m󸀠i , cij󸀠 , hi and their time derivative, given earlier). So, we retain only first order terms, and obtain the linearized Liouville equations − iΩ2 (C − A)m󸀠 + ΩAṁ 󸀠 − ie󸀠 Ω2 Am󸀠 ∗ − e󸀠 ΩAṁ 󸀠 ∗ + iΩ2 (c󸀠 +

h󸀠 ) + ċ󸀠 Ω + ḣ 󸀠 = L󸀠 , Ω (4.25a)

󸀠 CΩṁ 󸀠3 + ċ33 Ω + ḣ 󸀠3 = L󸀠3 .

χ3󸀠

(4.25b)

Let us define the equatorial and axial Angular Momentum Function (AMF) χ 󸀠 and by χ󸀠 =

c󸀠

+

h󸀠

C − A (C − A)Ω c󸀠 h󸀠 χ3󸀠 = 33 + 3 . C CΩ

,

(4.26)

This denomination stems from the fact that, for a part of the Earth (such as the atmosphere, oceans, …) representing a small proportion of the land mass and having the equatorial and axial angular momentum H 󸀠 and H3󸀠 , respectively, the functions χ 󸀠 and χ3󸀠 are the dimensionless angular momentum χ 󸀠 ≈ H 󸀠 /[(C − A)Ω] and χ3󸀠 ≈ H3󸀠 /(CΩ), respectively (justification in Section 8.2.2). Then, dividing (4.25a) by −iΩ2 (C − A) and (4.25b) by CΩ, the linearized Liouville equations become m󸀠 +

i 󸀠 e󸀠 󸀠 ∗ e󸀠 i L󸀠 ṁ + m − i ṁ 󸀠 ∗ = χ 󸀠 − χ̇ 󸀠 + i , σe e σe Ω (C − A)Ω2 ṁ 󸀠3 = −χ3̇ 󸀠 +

L󸀠3 , CΩ

(4.27)

with the so-called Euler frequency, σe =

C−A A

Ω = eΩ ,

(4.28)

corresponding to a period of 304.5 sidereal days or 303.6 mean solar days. This linearization was introduced by Jeffreys (1916) [117] and notations are those of the book of Munk and MacDonald (1960) [153]. The error introduced by the deletion

72 | 4 Liouville equations of second order terms is equivalent to neglecting in m󸀠1 , m󸀠2 and m󸀠3 variations of up to 3 10−12 rad (0.6 µas), which is justifiable under the current precision of the Earth rotation parameters (at best 50 µas or 2.5 10−10 radians). However, it is not excluded that in the future, increasing accuracy obliges us to consider the coupling terms. The right hand sides of these equations, called excitation functions and scored (Ψ󸀠 , Ψ󸀠3 ) group external torque (L󸀠i ) and angular momentum resulting from a mass redistribution (with relative angular momentum hi and inertia moment variation cij󸀠 ). We have Ψ󸀠 = χ 󸀠 − Ψ󸀠3

=

−χ3󸀠

i 󸀠 L󸀠 χ̇ + i , Ω Ω2 (C − A)

L󸀠 + ∫ 3 dt . CΩ

(4.29)

4.3.3 Linearized Liouville equation in the Terrestrial Reference System As observed pole coordinates are commonly referred to the TRS Gxyz as well as modeled excitation, for practical purpose we have to go back to this reference system by applying in (4.27) the complex coordinate change m = m󸀠 eiλA (corresponding to the axial rotation of angle −λA which brings the equatorial inertia axis Gx󸀠 in coincidence with Gx). In the TRS, we also have c = c󸀠 eiλA and h = h󸀠 eiλA , and thus a similar transformation for equatorial excitation function. Axial components are the same. So we finally obtain m+

e󸀠 e2iλA ∗ e󸀠 e2iλA ∗ i L i ṁ + ṁ = χ − χ̇ + i m −i , σe e σe Ω Ω2 (C − A) L ṁ 3 = −χ3̇ + 3 . CΩ

(4.30a) (4.30b)

If we neglect the triaxial coefficient e󸀠 , we get back the well-known form of the Liouville equation for a biaxial rigid Earth (Munk and MacDonald 1960 [153], Section 6.1). If we remove the relative angular momentum in χ, (4.30a) is equivalent to Eq. (54a) in [143] by making the substitution e󸀠 e2iλA → Z, χ → c̃3 /(eA), and by casting aside the fluid core (Af = 0, m̃ f = 0). Matter term and motion term of the angular momentum function In the AMF the term proportional to the increment of the moment of inertia c, resulting from the rotation of the considered mass re-distribution, is the matter term, denoted ma. The term proportional to the relative angular momentum is the motion term, denoted mo. Subsequently, we use the notations c c χma = , χ3ma = 33 , C−A C (4.31) h3 h mo χmo = , χ3 = . (C − A)Ω CΩ

4.4 Free and forced polar motion for a rigid Earth model | 73

Meaning of the linearized Liouville equations While governing the temporal evolution of the terrestrial components of the instantaneous rotation vector, the system of linear equations (4.27) in ATRS or (4.30) in TRS operates a decoupling of the processes affecting the Earth rotation. Therefore they can be analyzed separately. Often a process is so characteristic that its signature is printed directly in the Earth rotation, so that there is no need to combine them all before comparing the EOP. For example, we can calculate the atmospheric effect, without considering the influence of lunisolar tides and vice versa, and recognize it directly in the Earth’s rotation parameters. With respect to the present accuracy of the observations and geophysical models, there is no evidence to challenge the linearization at subsecular time scale. A more fundamental limitation stems from the very principles of Newtonian mechanics, in the sense that the Liouville equations mask the cause and effect relationship by proposing a non-causal equality between excitation and irregularities in the rotation of the Earth. If the atmospheric circulation is partly the cause of the seasonal polar motion and the Chandler term, it would be rash to assert that the polar motion produces climatic variations!

4.4 Free and forced polar motion for a rigid Earth model 4.4.1 Rigid Earth model: definition What is usually meant by a rigid Earth model is a stiffened Earth including its fluid layers, in the sense that the material particles of this ideal body keep their mutual distances constant. In such a context the excitation functions are not influenced by changes of the instantaneous rotation vector (because of the centrifugal effect), so that the rough linearized Liouville equations enable a perfect decoupling between instantaneous rotation and excitation. 4.4.2 Biaxial Earth The equatorial moments of inertia A and B have a relative deviation of 0.002 %, almost 100 times lower than they have with the axial moment of inertia C. As a first approximation this difference can be neglected. Then e󸀠 = 0 and the Earth model is termed biaxial. The equatorial part of (4.30) is then reduced to the well-known equation m+

i i iL ṁ = Ψ = χ − χ̇ + . σe Ω (C − A)Ω2

(4.32)

Equatorial Liouville equation for a given Earth layer In the next chapters, the Liouville equations will be restricted to any layer of the earth, whose main axes of inertia are in the vicinity of those of the global Earth (the case of

74 | 4 Liouville equations the mantle, the fluid core, the lithosphere). For this it is sufficient to replace the global relative angular momentum and moments of inertia by those of this layer, indexed by the letter p. In the biaxial case (Ap = Bp ) Eq. (4.25a) reads iΩ2 (Ap − Cp )m + ΩAp ṁ + iΩ2 (cp +

hp Ω

) + ċp Ω + ḣ p = Lp ,

(4.33)

where the variations of the moment of inertia cp , relative angular momentum hp and moment of force L⃗ p only pertain to the considered layer. General solution The free solution is the circular term m = m0 ei(σe t+ϕ0 ) ,

(4.34)

where m0 and ϕ0 are the amplitude and the phase, respectively, which we let undefined. This means that the rotation pole can freely wobble on the Earth crust, completing an anticlockwise circle in a period of 303.6 days, known as Euler’s free wobble. The general solution of (4.32) can be obtained by applying the variation of the constant method, which gives t

t

m(t) = −iσe eiσe t ∫ Ψ(τ)e−iσe τ dτ = m(t0 )eiσe (t−t0 ) − iσe eiσe t ∫ Ψ(τ)e−iσe τ dτ .

(4.35)

t0

−∞

It shows up as the convolution of the free wobble (the impulse response) by the equatorial excitation. 4.4.3 Triaxial Earth For a triaxial Earth the most suitable reference system is the ATRS Gx󸀠 y󸀠 z 󸀠 , in which the polar motion obeys Eq. (4.27): m󸀠 +

i 󸀠 e󸀠 󸀠 ∗ e󸀠 ṁ + m − i ṁ 󸀠 ∗ = Ψ . σe e σe

(4.36)

It can be assimilated to the generalized Liouville equation (A.1) as defined in Appendix A: (1 − U) m󸀠 +

i i (1 + eU)ṁ 󸀠 − Vm󸀠 ∗ + eV ṁ 󸀠 ∗ = Ψ , σe σe

(4.37a)

with U=0,

V =−

e󸀠 . e

(4.37b)

4.4 Free and forced polar motion for a rigid Earth model | 75

According to the solution given in Appendix A, the free solution is composed of two circular uniform motions of frequencies given by (A.13) and (A.14), that is, ±σe (1 + O(e2 )): m󸀠 (t) = m+0 eiσe t + m−0 e−iσe t .

(4.38)

Putting this expression in (4.36) with Ψ = 0, we obtain the condition m−0 = −(m+0 )

∗

e󸀠 . 2e

(4.39)

So, considering m+0 = |m+0 |eiΦ0 , the free solution can be written e󸀠 ) cos(σe t + Φ0 ) , 2e e󸀠 m󸀠2 = |m+0 |(1 + ) sin(σe t + Φ0 ) . 2e m󸀠1 = |m+0 |(1 −

(4.40)

This corresponds to an elliptical motion, of which the semi-major axis is directed towards 0y󸀠 = IB , being about e󸀠 /e = 0.34 % larger than the semi-minor axis aligned towards 0x󸀠 = IA . If we consider the typical magnitude |m+0 | = 200 mas, the difference, about 0.7 mas, is significant in the light of the pole coordinate accuracy (less than 0.1 mas). In the TRS this elliptical polarization points towards the longitude λA + 90° = 75°. A circular excitation Ψ0 eiσ0 t of frequency σ0 produces the forced polar motion according to (A.24), where we take σc = σe , σc− = −σe , and U = U1 = 0: m󸀠σ0 (t) = m+0 eiσ0 t + m−0 e−iσ0 t ,

(4.41a)

with m+0 = −Ψ0 m−0 = Ψ∗0

σe , σ0 − σe

e󸀠 σe (σe + eσ0 ) . e (σ0 − σe )(σ0 + σe )

(4.41b)

Added to the circular term at angular frequency σ0 there appears a small term circling e󸀠 (σ +eσ )

at the opposite frequency and in the amplitude ratio | e(σe +σ 0) |. This additional term e 0 makes elliptical an oscillation which would be purely circular for a biaxial Earth. If the perturbation has an order of magnitude less than 1 % for any frequency of the same sign as σe , this is no longer the case for a retrograde pulsation, when σ0 approaches −σe . In Chapter 6 the ocean pole tide will make the polar motion more elliptic, but in another direction.

76 | 4 Liouville equations

4.5 Celestial and terrestrial motion of the CIP As we saw in Chapter 3, what is monitored is not the rotation pole but the Celestial Intermediate Pole (CIP), determined in the CRS by a precession–nutation model and celestial pole offsets (dX, dY) and in the TRS by pole coordinates (x−iy). We now describe how these parameters are related to the dynamic theory which has been outlined. Precession–nutation The rotation of the TRS with respect to the CRS is specified by three Euler angles, that are linked to the terrestrial components of the instantaneous rotation vector by the Euler kinematic relations (3.11). Insofar as θ deviates slightly from the average θ0 value, the integration of these relations gives as a first approximation [θ]tt0

+

i[Ψ]tt0

t

sin θ0 = Ω ∫ meiΦ dτ , t0

t

(4.42)

Φ = Φ(t0 ) + [Ψ]tt0 cos θ0 + Ω ∫(1 + m3 )dτ . t0

The rotation angle Φ essentially reflects the average diurnal rotation at the angular rate Ω: Φ(t) = Ωt + Φ0 (t) ,

(4.43)

where the phase Φ0 (t), nearly constant, is slightly affected by the irregularities of the speed of rotation and the precession–nutation of the nodes of the equator on the ecliptic. The angles θ and Ψ determine the spatial direction of the axis of the geographic poles. The CIP is a kind of geographic pole, of which the celestial wobble is restricted to variations beyond two days, chiefly caused by the lunisolar tidal torque L. By restricting our consideration to the biaxial case governed by (4.32), this torque rules m according to m+

i iL ṁ = , σe (C − A)Ω2

(4.44)

L(σ) . AΩ(σ − σe )

(4.45)

or in the frequency domain m(σ) = −i

Considering that the tidal torque with respect to the ground is chiefly composed of retrograde quasi-diurnal component of frequency σ = −Ω+σ 󸀠 with σ 󸀠 ≤ Ω2 , the induced polar motion at frequency σ reads m(t) = m0 ei((−Ω+σ )t+ζ0 ) . 󸀠

(4.46)

4.5 Celestial and terrestrial motion of the CIP | 77

Then the corresponding precession–nutation component of the CIP is derived from (4.42): [θ]tt0

+

i[Ψ]tt0

t

sin θ0 = Ω ∫ m0 ei(σ τ+Φ0 (τ)+ζ0 ) dτ . 󸀠

(4.47)

t0

It appears as a variation with a period larger than two days. Considering all spectral components of the tidal torque, taking its origin from the Sun, Moon and planets (see e. g. [60]), and taking into account the non-rigidity of the Earth (tides induce an offdiagonal moment of inertia c with the same frequency as L), a precession–nutation theory can be built for giving the position of the CIP in the CRS at any time. Despite their deterministic character, the current luni-solar precession–nutation models are not fully consistent (precession and nutation constructed separately, nonconsistent formulation of second order effects). VLBI observations are used to adjust the imperfections of the conventional theoretical model (UAI 2000) and the geophysical effects not included in it8 as deviations (dX, dY) from the modeled celestial pole coordinates (X,Y). Polar motion of the CIP for a biaxial Earth Pole coordinates p = x − iy quantify the offsets between CIP and geographic pole, and correspond to any motion m of the instantaneous rotation pole apart from the retrograde diurnal. From (3.26) this variation m is related to p by m=p−

i ṗ . Ω

(4.48)

Substituting m with this expression in (4.32) allows one to obtain the second order differential equation ruling the CIP coordinates p+

i i d i c h i ċ ḣ iL ̇ = ṗ − (p + p) + − ( + )+ . σe Ω dt σe C − A (C − A)Ω Ω C − A (C − A)Ω (C − A)Ω2 (4.49)

As the linearity permits permits us to get rid of the external torque, this equation establishes a direct comparison between the geophysical Angular Momentum Function (AMF) χ geoph =

c

C−A

+

h

(C − A)Ω

(4.50)

8 Especially the free core nutation, probably forced by the diurnal hydro-atmospheric transports or geomagnetic jerks (see Chapter 13).

78 | 4 Liouville equations and the observed or geodetic excitation, derived from the observed pole coordinates by χG = p +

i ṗ . σe

(4.51)

Indeed (4.49) reads i i χ̇ = χ geoph − χ̇ geoph , Ω G Ω

(4.52)

i c h ṗ = + = χma + χmo . σe C − A (C − A)Ω

(4.53)

χG − so that χG = χ geoph or p+

This equation is modified by the Earth’s non-rigidity but keeps the present form, which is very appropriate for studying the influence of a mass redistribution on polar motion outside the nutation frequency band [−1.5, −0.5] cpd. Actually some mass wandering (due to thermal cycling and diurnal ocean tides) also result in retrograde diurnal components in the excitation function, which has the effect of competing with the tidal torque. In this case, Eq. (4.53) is not sound, and the Liouville equations involve an additional resonance, associated with the free core nutation that will be characterized in Chapter 7.

4.6 Incompleteness of this formulation A priori all the equations are posed, and the problem is completely acknowledged: now, it would be sufficient to characterize the mass redistributions like the atmospheric, oceanic, hydrologic ones, or those caused by lunisolar tides to obtain the path of the instantaneous rotation pole. But this formulation ignores the fact that the solid Earth is not rigid: by the centrifugal effect, the fluctuations in the rotation of the Earth create a deformation or a mass redistribution, subsequently variations of the inertia tensor cij . Moreover the superficial fluid loading causes a deformation of the solid Earth, responsible for additive cij . In the excitation function, these terms are as important as the one produced by the direct load. This is why it is important to model as accurately as possible the “interference” operated by the rheology of the Earth and its internal structure, and to derive the corresponding equations governing the polar motion. This is the object of Chapters 5, 6, and 7.

5 Dissipative rotational excitation function Mr. Chandler discovery gives rise to the question whether there can be any defect in the theory which assigns 306 days as the time of rotation. The object of this paper is to point out that there is such a defect—namely, the failure to take account of the elasticity of the Earth itself, and of the mobility of the ocean. Newcomb, Astronomical Journal (1892).

5.1 Centrifugal deformation As a non-rigid body, the Earth undergoes a deformation produced by: (i) the lunisolar tides (∼ 1 m), (ii) the load of the ocean and the atmosphere (∼ 10 cm), (iii) the tectonic processes with the continental drift (a few cm/year) and seismic displacements (up to 10 m), and (iv) the centrifugal force associated with the Earth rotation (∼ 1 cm). These deformations induce inertia moment variations and in turn affect the rotation vector. Of all the considered strains, the rotational distortion depends on the change of the rotation of the Earth, leading to a feed-back effect. We must distinguish the constant part of the rotational deformation, caused by the uniform rotation, that is, the equatorial bulge, and the variable part induced by the disturbances m and m3 of the rotation vector. Regarding the constant centrifugal bulge, the process happens in several hundred thousands of years. Once the deformation is completed, any mass element of the Earth is in equilibrium under the combined action of internal gravitational forces, centrifugal and pressure forces. Then the Earth behaves like a fluid in hydrostatic equilibrium, of which we have to determine the shape. This question kept scholars of the seventeenth and eighteenth centuries in suspense. In advancing the idea that the Earth at the equator is oblong and flattened at the poles, supporters of Newtonian mechanics opposed Jean-Dominique Cassini and his school, who supported the reciprocal proposal. The equatorial bulge was endorsed by the scholars of the time when, after scientific expeditions funded by the King of France, it was observed that the same meridian arc of the order of 1 degree (difference of astronomical latitude) is longer at the equator than at north or temperate latitudes (the equator is farthest from the center of the Earth, rather than the poles). Meridian arcs of one degree were measured jointly in France, in Lapland (expedition organized by de Maupertuis in 1736–1737), and in Peru (expedition led by La Condamine 1735–1739) and then in the Cape (1751). When the rotation varies at sub-secular time scale, centrifugal forces acting on the Earth’s mass elements in the TRS are perturbed. Then the solid mass elements undergo quasi-elastic strain, whereas the ocean surface follows hydro-statically the centrifugal potential change, at least for periods longer than 10 days. Shortly after Chandler (1891) [34, 35] revealed the oscillation of the pole in about 430 days, Newcomb explained how the variable centrifugal effect generated by the https://doi.org/10.1515/9783110298093-005

80 | 5 Dissipative rotational excitation function

Figure 5.1: Newcomb interpretation of the Chandler period. While pole R rotates, instantaneous inertia pole I moving with Euler’s angular frequency σe , this one moves towards R, slowing down the same oscillation with respect to the ground (xy frame) which appears to have the angular velocity σc ≤ σe .

polar motion can lengthen the period of the free Euler motion from 304 days to 433 days. Newcomb’s interpretation [162], because of its simplicity, is worth remembering. Suppose that the rotation pole R initially coincides with the mean inertia pole I.̄ At time t the pole shifts from I ̄ to R(t). Then the figure of the Earth tends to balance around the axis of rotation under the action of centrifugal forces: if the Earth is elastic or partially fluid, it instantly distorts so that the instantaneous pole of inertia I approaches the rotation pole R by segment IR, as shown in Figure 5.1. This readjustment of the equatorial bulge around the rotation axis is accompanied by a deformation of the order of cm, called pole tide.1 Then Euler’s motion is not around the mean pole of inertia I,̄ but around the displaced pole of inertia with a shorter amplitude, ̄ In other words, R while is turning at angular frequency that is, IR rather than IR. σc around I ̄ (relative to the Earth’s crust) it is still running with Euler’s angular frequency σe around I. During the period Δt, the pole of rotation R then traverses the arc ̄ ̄ by an amount depending on the σe IRΔt = σc IRΔt. As the IR segment is shorter than IR ̄ In turn, overall non-rigidity of the Earth, σc is smaller than σe within the ratio IR/IR. ̄ the observed period of the free motion, the Chandler period, is IR/IR times larger than Euler’s one. According to the observed excess of about 130 days, while considering the hydrostatic ocean pole tide, Newcomb concluded that the solid Earth has an elasticity comparable to that of steel. Notice that, if the Earth would be an inviscid fluid, 1 This phenomenon is difficult to measure because it is at least one order of magnitude smaller than the lunisolar tide, and mixes the effects of atmospheric and oceanic loading.

5.2 Centrifugal potential | 81

points I and R would overlap, σc would be equal to zero, and the free mode would vanish. In this chapter, adopting the frame of a quasi-elastic solid Earth model (slightly anelastic), we aim at characterizing the feed-back effect of the polar tide. We shall see that Earth’s angular velocity change causes a much smaller effect, negligible in line with the present geophysical reconstruction. Accordingly Liouville equation will be modified, and will exhibit the presence of Chandler’s angular frequency.

5.2 Centrifugal potential At the point located by the vector r,⃗ the gradient of the centrifugal or rotational potential (marked by “r” from the first letter of rotational) U (r) is given by the centrifugal acceleration according to ⃗ (r) = −ω⃗ ∧ (ω⃗ ∧ r)⃗ = ω2 r ⃗ − (ω.⃗ r)⃗ ω⃗ . ∇U

(5.1)

By introducing the components ω1 = Ωm1 , ω2 = Ωm2 , ω3 = Ω(1 + m3 ) of the rotation vector and Cartesian coordinates (x, y, z) of r,⃗ it reads also

⃗ (r) ∇U

(ω22 + ω23 )x − ω2 ω1 y − ω3 ω1 z [ 2 2 =[ [ (ω1 + ω3 )y − ω1 ω2 x − ω3 ω2 z 2 2 [ (ω1 + ω2 )z − ω1 ω3 x − ω2 ω3 y

] ] . ]

(5.2)

]

Thereby U (r) is expressed by U (r) =

ω22 + ω23 2 ω21 + ω23 2 ω21 + ω22 2 x + y + z − (ω1 ω2 xy + ω1 ω3 xz + ω2 ω3 yz) . 2 2 2

(5.3)

The centrifugal potential is composed of a constant part resulting from the nonperturbed Earth angular velocity vector (Ω, 0, 0), that is, U

(r)

=

Ω2 (x2 + y2 ) , 2

(5.4)

and of a variable part associated with the perturbations mi of ω:⃗ ΔU (r) = Ω2 [m21

y2 + z 2 x2 + z 2 x2 + y2 + m22 + (m23 + 2m3 ) 2 2 2

− m1 m2 xy − m1 (1 + m3 )xz − m2 (1 + m3 )yz] .

(5.5)

82 | 5 Dissipative rotational excitation function By introducing the Legendre polynomial expressions (where ϕ means the latitude and λ the longitude) 2r 2 P20 (sin ϕ) = 3z 2 − r 2 = −x 2 − y2 + 2z 2 , x2 + y2 = r 2

P22 , 3

P P22 + cos(2λ) 22 ) , 3 6 P P y2 + z 2 = r 2 (P20 + 22 − cos(2λ) 22 ) , 3 6

x2 + z 2 = r 2 (P20 +

(5.6)

r 2 cos λP21 (sin ϕ) = 3 xz , r 2 sin λP21 (sin ϕ) = 3 yz ,

xy = r 2 sin(2λ)

P22 , 6

we obtain after some algebra m21 + m22 + m23 + 2m3 m2 + m22 m23 + 2m3 +( 1 − )P20 3 6 3 m (1 + m3 ) cos λ + m2 (1 + m3 ) sin λ − 1 P21 3 m2 − m21 mm cos λ − 1 2 sin(2λ))P22 ] . +( 2 12 6

ΔU (r) = r 2 Ω2 [

(5.7)

This expression is useful when considering non-linear effects in the Earth rotation in the last chapter. At sub-secular time scales, we know that m3 ≤ 10−8 and m1 , m2 ≤ 10−6 , and in turn the second order terms of the perturbations can be neglected in the centrifugal potential: ΔU (r) ≈

Ω2 r 2 [2m3 (1 − P20 (sin ϕ)) − (m1 cos λ + m2 cos λ)P21 (sin ϕ)] . 3

(5.8)

Here the terms produced by polar motion predominates in a ratio of 100.

5.3 Mean figure of the Earth Over secular time scale, skipping the secular deceleration, the Earth has an uniform rotation rate Ω. Then, under the action of constant centrifugal force, gravity and pressure of surrounding masses, any mass element reaches a state of equilibrium, corresponding to a constant mass distribution. This hydrostatic balance is reflected in the local equation ⃗ + ρ∇W ⃗ =0, − ∇P

(5.9)

5.3 Mean figure of the Earth

| 83

where ρ is the density, P the pressure, and W the geopotential, that is, the sum of the gravitational potential U and centrifugal potential U (r) . Any equipotential sur⃗ and to ∇P. ⃗ Therefore isobaric surfaces are also face of W is perpendicular to ∇W equi-geopotential. On the terrestrial surface the pressure is grossly PS = 1.013 105 Pa, with variable or constant anomalies up to 10 % of this value. These anomalies cause deformation of the land surface up to 1 cm, and ocean surface up to 10 cm. So, averaging these anomalies, the terrestrial surface is isobaric. If the atmospheric layer is included, the isobaric surface becomes P = 0. So, whatsoever the case (surface or upper bound of the atmosphere), the Earth figure is an equipotential taking the value W0 . This conclusion permits one to establish theoretically the shape of the Earth. The mean gravitational potential U mostly shifts from the one of an homogeneous sphere by a spherical harmonic of degree 2 given by (B.27). Thus, given M⊕ , the Earth mass, and Re , the equatorial radius, the mean gravitational geopotential is given by U=

2

GM⊕ R 3 sin2 ϕ − 1 [1 + ( e ) C 20 ] r r 2

(5.10)

in the system of the mean principal axes. Considering (5.4), the geopotential W =

U +U

(r)

associated with an uniform angular velocity Ω is expressed by W=

2

GM⊕ R 3 sin2 ϕ − 1 Ω2 2 [1 + ( e ) C 20 ]+ r cos2 ϕ . r r 2 2

(5.11)

As this potential has the same value at equator (radius Re ) and at north pole (radius Rp , U (r) = 0), we obtain 2

GM⊕ R C Ω2 2 GM⊕ (1 − 20 ) + Re = [1 + ( e ) C 20 ] . Re 2 2 Rp Rp

(5.12)

Introducing geometric flattening f = (Re − Rp )/Re , that is, Rp = Re (1 − f ), and neglecting second order terms in f 2 or f C 20 (with an order of magnitude of (1/300)2 ), we have 3 q f = − C 20 + , 2 2

(5.13)

where q is defined as the ratio of the centrifugal acceleration to the gravitational acceleration at the equator: q=

Ω2 R3e . GM⊕

From the values of Ω, Re , GM⊕ in Appendix H, we have q = 3.461 10−3 ≈ 1/289.

(5.14)

84 | 5 Dissipative rotational excitation function The constant centrifugal potential (5.4) can be developed in spherical harmonics of degree 0 and 2: U

(r)

=

(r) (r) Ω2 r 2 (1 − P20 (sin ϕ)) = U 0 + U 20 . 3

(5.15)

Now, we notice that the mean gravity potential of degree 2 (B.27) and the degree 2 part of the rotational potential, namely U 20 =

2

GM⊕ Re ( ) C 20 P20 (sin ϕ) and r r

(r)

U 20 = −

Ω2 r 2 P (sin ϕ) , 3 20

(5.16)

are proportional. Considering Eq. (B.24) of C 20 for a biaxial Earth with the mean equatorial inertia moment A, we get U 20 =

3G(C − A) (r) U 20 . Ω2 r 5

(5.17)

At the Earth’s surface, taking r = Re , we have (r)

U 20 (Re ) = ks U 20 (Re )

(5.18)

with ks =

3G(C − A) ≈ 0.938 . Ω2 R5e

(5.19)

The so-called secular Love number ks expresses that, at the surface of the Earth, the additional degree 2 gravitational potential resulting from the equatorial bulge is almost equal to the constant centrifugal potential of degree 2, as expected from hydrostatic equilibrium. Substituting Eq. (B.24) of C 20 in (5.13), we obtain the well-known Clairaut equation, f =

1 q (1 + ks ) ≈ , 2 297.5

(5.20)

expressing geometrical flattening as a function of the acceleration ratio q = γc /g and secular Love number. The geometrical shape of the Earth, given by the radius vector r, is derived from the surface equi-geopotential W, equated to W(Re ): 2

GM⊕ GM⊕ R C 3 sin2 ϕ − 1 Ω2 2 Ω2 2 [1 + ( e ) C 20 ]+ r cos2 ϕ = (1 − 20 ) + R . r r 2 2 Re 2 2 e

(5.21)

It is seen that r does not depend on longitude, so that the Earth takes the shape of a spheroid of revolution around the polar axis. Considering the form r = Re (1 + δr ), the former equation can be easily solved with respect to δr by neglecting second order terms in f and δr , leading to r = Re (1 − f sin2 ϕ) .

(5.22)

5.4 Variable centrifugal effect on the inertia moments | 85

5.4 Variable centrifugal effect on the inertia moments Under the influence of the variable part of the rotational potential given by (5.8), considering a sub-secular time scale, the solid part of the Earth no longer deforms like a fluid but like a quasi-elastic solid body. In ΔU (r) there is spherical harmonic of degree 0, proportional to m3 , and a degree 2 spherical harmonic. The degree 0 spherical harmonic is associated with a radial force and produces a radial deformation, and thus only influences degree 0 harmonic of the geopotential. Variation of the geopotential of degree 2, hence a modification of the moment of inertia c13 and c23 (see Appendix B.3), can only result from the degree 2 of ΔU (r) . At the surface of the Earth (r ≈ Re ) this spherical harmonic is ΔU2(r) = −

Ω2 2 R [2m3 P20 (sin ϕ) + (m1 cos λ + m2 sin λ)P21 (sin ϕ)] . 3 e

(5.23)

According to the static linear theory of bodily tides (see e. g. [75]), introduced by Love (1909) [136], a time variable degree 2 potential variable W2 , like ΔU2(r) , produces a proportional variation of the gravitational potential at the surface of the Earth, according to ΔU = kW2 ,

(5.24)

where k takes intermediate values between the rigid case (k = 0, no deformation) and the hydrostatic case (k = ks ) ruled by (5.18). The parameter k is called the degree 2 Love number. When it pertains to the geopotential perturbation at the solid Earth surface, it is commonly denoted k2 . For periods stretching from a few hours to some decades, for which the solid Earth deforms almost elastically, k2 equals about 0.3. Such a value can be empirically derived from the determination of the tilt factor 1 + k2 − h2 2 and of the gravimetric factor 1 + h2 − 3/2k2 3 for components of the gravimetric tide (deriving from a degree 2 potential). At higher frequency, the solid Earth tends to behave like a rigid body and k2 goes down: k2 = 0.23 for 100 Ω, k2 = 0.04 for 1000 Ω [213]. On the lower frequency side, the smaller the frequency is, the more manifest is the viscoelastic behavior: k2 increases up to 0.42 for Ω/105 [213]. For a permanent perturbation W2 , an hydrostatic equilibrium is established over geological periods, and k2 becomes the secular Love number ks . By the way, whatever the frequency band the Earth cannot be reduced to a solid body, and k2 is modified by the ocean layer, of which the influence can also be incorporated into the Love number formalism according to Chapter 6. In any case, the proportionality (5.24) is only true in limited frequency bands. This leads one to assume that ΔU(σ) = k(σ)W2 (σ), translating in the time domain by the 2 A coefficient characterizing the vertical deviation produced by the gravimetric tide. 3 The ratio of the observed gravimetric tide to the theoretical one corresponding to a rigid Earth.

86 | 5 Dissipative rotational excitation function convolution product ΔU(t) = k(t) ∗ W2 (t) ,

(5.25)

where W2 eventually contains a constant part, like the permanent centrifugal potential. As far as we are dealing with rotational changes m̃ 3 , m̃ of the solid Earth, corresponding to the intermediate frequencies in the sense specified above, k2 is assumed to be the constant ∼ 0.3, and the pole tide potential of degree 2 causes a change of the gravitational potential at the surface of the Earth according to (5.24), that is, ΔU = k2 ΔU2(r) = −k2

Ω2 2 R [2m̃ 3 P20 (sin ϕ) + (m̃ 1 cos λ + m̃ 2 sin λ)P21 (sin ϕ)] , 3 e

(5.26)

which can be expressed as a spherical harmonic development, ΔU =

GM⊕ (r) (r) (r) (C20 P20 (sin ϕ) + (C21 cos λ + S21 sin λ)P21 (sin ϕ)) , Re

(5.27)

with the “rotational Stokes coefficients” (r) C20 = −k2

2R3e 2 Ω m̃ 3 , 3GM⊕

(r) C21 = −k2 (r) S21 = −k2

R3e Ω2 m̃ 1 , 3GM⊕

R3e Ω2 m̃ 2 . 3GM⊕

(5.28) (5.29) (5.30)

These increments account for the effect of the variable centrifugal deformation. From Appendix B.3, the Earth’s moments of inertia are determined by the degree 2 terms of the gravitational potential taken at the surface of the Earth. In particular the offdiagonal moments of inertia c13 and c23 are given by (B.17) and (B.18): c13 = −M⊕ R2e C21 ,

c23 = −M⊕ R2e S21 .

(5.31)

Thus, the additional potential ΔU caused by the polar motion yields the inertia moment changes R5e 2 Ω m̃ 1 , 3G R5 = k2 e Ω2 m̃ 2 . 3G

(r) c13 = k2 (r) c23

(5.32)

By considering the secular Love number ks of Eq. (5.19), these rotational variations, (r) (r) put in the complex form c(r) = c13 + ic23 , can be written c(r) =

k2 (C − A)m̃ . ks

(5.33)

5.5 Influence of the Earth’s non-rigidity on polar motion

| 87

5.5 Influence of the Earth’s non-rigidity on polar motion In the polar motion differential equation for a biaxial Earth (4.32) or a triaxial Earth (4.36) the excitation function includes the term (5.33), yielding the rotational excitation or centrifugal excitation: χ (r) =

c(r)

C−A

=

k2 m̃ . ks

(5.34)

This quantity is restricted to a sub-secular spectral band, corresponding to the complex pole coordinates m.̃ Let m be the secular counterpart, mostly encompassing a linear drift, possibly reflecting multi-secular cycles. Then we have m = m̃ + m ,

(5.35)

as depicted by Figure 5.2. Isolating in the Liouville equation (4.32) the secular part m, the time derivative can be neglected and we obtain m=χ.

(5.36)

As it can be reasonably assumed that the relative angular momentum has no trend, we have χ = c/(C − A). According to (C.26) this means that the mean pole m and the mean inertia pole coincide. In contrast, at the sub-secular scale, the Liouville equation (4.32) reads m̃ + i

k ṁ̃ i k2 ̇ = ψpure + 2 m̃ − m̃ , σe ks Ω ks

(5.37)

where ψpure represent the “pure” geophysical excitation, free from any centrifugal effect. Then m̃ can be interpreted as the wobble of the rotation pole R round the mean inertia pole I. The estimated pole coordinate m is the sum of m = PI and m̃ = IR, where P means the geographic north pole. From the former equation we see that, in the absence of excitation, during Δt the rotation pole R moves the arc ̃ − k2 /ks )Δt Δm̃ ≈ iσe m(1

(5.38)

around I, with the angular frequency σc = σe (1 − k2 /ks ). On the other hand, if we consider that under the rotational effect, the instantaneous inertia pole I is shifted ̃ − k2 /ks ) = IR − II = IR from I by the arc II = c(r) /(C − A) = k2 /ks m̃ = 0.3 m,̃ then m(1 and Δm̃ = iσe IR, meaning that R moves round the instantaneous inertia pole with the angular frequency σe . This is the Newcomb interpretation illustrated by Figure 5.1. From now until Chapter 14, we shall treat sub-secular polar motion only, and the associated flag ̃above m will be removed. In turn, Eq. (5.37) can be expressed by m+

ψpure i ṁ = σc 1 − kk2 s

(5.39)

88 | 5 Dissipative rotational excitation function where σc = σe

k2 ks e kk2 s

1− 1+

(5.40)

.

Euler’s frequency is reduced by the factor (1 − kk2 )/(1 + e kk2 ) ≈ 1 − kk2 ≈ 0.7, and its s s s period is lengthened by the ratio 1.45. The free oscillation is thus expected with the period 440 days, close to the observed Chandler period of 433 ± 2 days. Meanwhile the excitation is amplified by 1/(1 − kk2 ) = 1.43. s

5.6 Damping 5.6.1 Complex Chandler frequency Converting (5.39) in the frequency domain, m(σ) = −

σc Ψpure (σ) , σ − σc 1 − k2 k

(5.41)

s

it is easy to see that polar motion should present an infinite resonance at the Chandler wobble, even in the case of any tiny aleatory excitation, like white noise. But the Chandler wobble keeps a moderate amplitude, below 0.5󸀠󸀠 , thus hinting that the resonance is damped by some dissipative processes. Moreover the Chandler oscillation is very variable, showing a decrease from 1900 to 1925, its quasi-disappearance from 1925 to 1930, its recovery from the year 1940 [47], and other strong fluctuations since this epoch (see Figure 11.7). Thus we can reasonably assume that the damping is compensated for by some variable exciting processes. The damping can result from the non-perfect elasticity of the solid Earth. As the Earth presents heterogeneous rheological properties, the pole tide generates friction. This results in both a loss of mechanical energy and delay of the centrifugal deformation with respect to its cause. In turn the rotational excitation (5.34) presents a delay τ with respect to the polar displacement m (at sub-secular scale): it lies towards the direction of m(t − τ), as illustrated in Figure 5.2. In other words, we have χ (r) (t) =

k k2 ṁ m(t − τ) ≈ 2 (1 − τ )m(t) , ks ks m

(5.42a)

where τ is assumed to be lower than 1 hour. Equivalently, this expression can be put in the form χ (r) (t) =

k2 k(t) δ(t − τ) ∗ m(t) = ∗ m(t) , ks ks

(5.42b)

5.6 Damping

| 89

where δ(t) is the Dirac function. With k(t) = k2 δ(t − τ), this rotational excitation is consistent with the general form (5.25) of the geopotential perturbation. Now, considering the mean angular frequency σm of the polar motion (σm has a value between the Chandler and the annual frequency), we have χ (r) (t) ≈

k2 (1 − iσm τ)m(t) . ks

(5.42c)

Let ε be the angle −σm τ. As τ > 0 and σm > 0, ε is a priori negative. We have the order of magnitude |ε| ≤ 2π/365 × 1/24 rad ≈ 0.5°, and χ (r) (t) =

k2 (1 + i ε)m(t) . ks

(5.42d)

By introducing the complex Love number k̃2 = k2 + i k2 = k2 (1 + iε) ≈ k2 eiε ,

(5.42e)

the rotation excitation has a form similar to (5.34) holding for a non-dissipative Earth. So, the derivation of Section 5.5 for an elastic Earth can be maintained as far as the

Figure 5.2: Phase shift ε and delay τ between sub-secular polar motion m(t) and rotational excitation function χ (r) (t) caused by dissipative processes (xy is the equatorial plane of the TRF).

real Love number k2 is replaced by the complex k̃2 . The polar motion thus obeys the differential equation m+

ψpure i , ṁ = ̃ σ̃ c 1 − kk2

(5.43)

s

where σ̃ c is the complex pulsation σ̃ c = σe

1− 1+

k̃2 ks

̃ e kk2 s

= σc + iα ≈ σe (1 −

k̃2 ), ks

(5.44)

90 | 5 Dissipative rotational excitation function with the damping coefficient α ≈ −σe

k2 ε . ks

(5.45)

As the free motion, expressed by mc (t) = m0 eiσc (t−t0 ) = m0 eiσc (t−t0 ) e−α(t−t0 ) , ̃

(5.46)

is damped, we have α ≥ 0, implying ε ≤ 0, as in Figure 5.2. Assuming ε ∼ −1° (this order of magnitude is justified below), we have α = 10−4 rad/d, so that the amplitude of the free mode would be divided by e after the relaxation time4 τ = 1/α ∼ 30 years in the absence of excitation. On the other hand the effective geophysical excitation function defined by Ψgeoph . =

ψpure 1−

k̃2 ks

≈

ψpure 1−

k2 ks

(1 + i

k2 ε ) ks − k2

(5.47)

can be derived from the “non-dissipative” one, namely ψpure /(1 − k2 /ks ), by a rotation of angle k2 ε/(ks − k2 ) ∼ ε/2 ∼ −1°. Transfer function Taking the Fourier transform of (5.43) we get the relation m(σ) = T(σ)Ψgeoph . (σ)

with T(σ) = −

σ̃ c , σ − σ̃ c

(5.48)

determining the frequency transfer function T(σ) from effective geophysical excitation Ψgeoph . (σ) to polar motion m(σ). So, the spectral power of the polar motion is σc2 + α2 󵄨󵄨 geoph . 󵄨󵄨2 󵄨󵄨 󵄨2 (σ)󵄨󵄨 . 󵄨Ψ 󵄨󵄨m(σ)󵄨󵄨󵄨 = (σ − σc )2 + α2 󵄨

(5.49)

yielding the amplitude of the transfer function: σc2 + α2 󵄨󵄨 󵄨 . 󵄨󵄨T(σ)󵄨󵄨󵄨 = √ (σ − σ )2 + α2 c

(5.50)

This one is shown in Figure 5.3 for lower and upper limits of the damping coefficient (α = 0.013 rad/yr and α = 0.066 rad/yr, respectively). The transfer function is symmetric with respect to the free mode frequency. 4 The duration after which the amplitude is divided by e.

5.6 Damping

| 91

Figure 5.3: Transfer function amplitude: |T (σ)| = |m(σ)|/|Ψgeoph . (σ)| around the Chandler frequency for high damping and low damping coefficients. The corresponding quality factors Qc = σc /(2α) are also indicated.

Time domain response In frequency domain the impulse response i(σ) (when the forcing is δ(t)) is precisely the transfer function, for we have i(σ) = T(σ) ℱ σ (δ(t)) = T(σ) .

(5.51)

So, in time domain the polar motion forced by Ψgeoph. (t) is

m(t) = i(t) ∗ Ψgeoph. (t) = −iσc̃ H(t)eiσc t ∗ Ψgeoph. (t) ̃

t

= −iσc̃ ∫ eiσc (t−τ) Ψgeoph. (τ) dτ , ̃

(5.52)

−∞

where H(t) is the Heaviside function. 5.6.2 Damping estimation Whereas the free mode period can be directly determined from the pole coordinate time series with a relative uncertainty below 1 % [217], the damping coefficient relies on the modeling of the underlying excitation in the Chandler band.

92 | 5 Dissipative rotational excitation function White noise excitation In Chandler frequency band σc (not pertaining to a seasonal band), the geophysical excitation can be approximated as white noise, characterized by a flat spectral power density |Ψgeoph . (σ)|2 = K, as will be justified in Chapter 8. Therefore, in the vicinity of the Chandler frequency, the power spectrum of the polar motion reflects the transfer function. In particular, it is symmetric with respect to σc , presenting at this frequency the maximal value Kσc2 /α2 . So, the resonance frequency is determined by the maximum of the spectral power. Moreover, considering the symmetric angular frequencies σ − = σc − Δσ and σ + = σc + Δσ , for which the spectral power is only half of the central 2 2 power, we find the Full Width at Half Maximum (FWHM), that is, Δσ = 2α. The larger the damping is, the larger is the FWHM. This leads to the practical definition of the quality factor Qc as the ratio of the central frequency over the FWHM5 : Qc =

σc σ = c . Δσ 2α

(5.53)

If the white noise approximation allows one to estimate the quality factor from the spectral power of a polar motion (Figure 5.4), it is noteworthy that the spectral width is blurred by the limited length T of the sampling of m(t), equivalent to multiplying ̂ ) m(t) by a boxcar of length T. In the frequency domain this is a convolution of m(f by T sinc (πfT) with f = σ/(2π), having a FWHM of 0.88/T (spectral resolution is 1/T). For T = 40 years, this introduces a numerical broadening of 0.02 cpy, comparable to the value of the physical width between 0.02 cpy (Qc = 40) and 0.004 cpy (Qc = 200). Thus a rough spectral estimation tends to provide underestimated Qc values. The power spectrum of Figure 5.4, associated with the polar motion over the 120 year period 1900–2019, leads to Qc ∼ 47. Real excitation As demonstrated in Section 9.4.4 by an Allan deviation analysis, the Chandler wobble excitation is not strictly white noise, so that a mere spectrum of the pole coordinates series for determining the resonance frequency as well as the damping is not fully sound. A refined approach is to adjust (Tc , Qc ) by fitting the modeled counter̂ part of the polar motion T(σ, Tc , Qc )Ψ(σ) to the observed spectrum m(σ), where Ψ is the modelled geophysical excitation, as the hydro-atmospheric angular momentum function. An equivalent approach can be applied in the time domain, where σ̃ c = 2π/Tc (1 + i/2Qc ) characterizing the dynamical system (5.43) corresponding to the best fit between geodetic and geophysical excitations. Whereas Tc can be estimated within 1 ΔE 5 The genuine definition of the quality factor of a damped oscillation is Q1 = 2π where ΔE the E E c relative energy dissipation during one cycle (energy is proportional to the spectral power). When the σ excitation has vanished, the damped oscillation follows (5.46), and we have Qc = 2αc . The larger Qc is, the smaller is the energy leak.

5.7 Effect of a mass redistribution on polar motion

| 93

Figure 5.4: Spectral power of the polar motion around Chandler frequency (0.845 cpa) over the period 1900–2019 (combined series IERS C01).

an error bar of 2 days, Qc value remains loosely determined because of the error affecting geophysical excitation, which, on the other hand, covers just a little bit more than the relaxation time τ = QTc /π (from 15 to 76 years). This method is revisited in Chapter 11 by considering updated time series of the fluid layer excitation, and then reducing the interval of Qc values to (40, 120). σ From α = 2Qc = −σe kk2 ε we get c

s

ε=−

ks σc k − k2 ≈− s . 2Qc k2 σe 2Qc k2

(5.54)

With Qc in the interval (40, 120), considering a sub-secular frequency band, the rotational excitation precedes the polar motion as in Figure 5.2 by an angle ε between −0.5 and −1.5 degree.

5.7 Effect of a mass redistribution on polar motion In the case where excitation is reduced to matter and motion terms reflecting the association with the given mass redistribution, the polar motion is ruled by an equation similar to (4.53): p+

χ +χ i ṗ = ma ̃ mo . σ̃ c 1 − kk2

(5.55)

s

If a mass redistribution, associated with inertia moment variation cs = c13 + ic23 is located in the surface fluid layer (atmosphere, oceans, or hydric content of the continents), the supplementary loading produces a deformation of the crust, and a com-

94 | 5 Dissipative rotational excitation function pensation Δcs of the inertia moment increment cs . Within the quasi-elastic approximation Δcs is given by (see Appendix E) Δcs = k̃2󸀠 cs ,

(5.56)

where k̃2󸀠 ≈ −0.3075 is called the loading Love number. Thus the effective variation of the moment of inertia is cstot = (1 + k̃2󸀠 )cs .

(5.57)

The elastic deformation induced by a surface load compensates for almost a third of the equatorial moment inertia associated with the load itself. By introducing the Effective Angular Momentum Function (EAMF), e χma = e χmo =

1 + k̃2󸀠 1−

1

1−

k̃2 ks

k̃2 ks

χma ≈ 1.027χma , χmo ≈ 1.47χmo ,

(5.58)

we obtain a differential equation similar to the one derived for a rigid Earth: p+

i e e ṗ = χma + χmo . σ̃ c

(5.59)

6 Ocean pole tide It is easily intelligible that, as the axis of rotation shifts in the earth, the oceans will tend to swash about, and that a sort of tide will be generated. If the displacement of the axis were considerable, whole continents would be drowned by a gigantic wave, but the movement is so small that the swaying of the ocean is very feeble. George Darwin, The tides and kindred phenomena in the solar system (1897).

6.1 Introduction The solid Earth is deformed by the polar motion, and the oceanic surface as well. However, the ocean layer does not cover the Earth uniformly. So, in contrast to the quasielastic solid Earth, this ocean pole tide is only effective over pelagic zones. The generated variation of the moment of inertia is thus larger when the pole goes towards oceanic zones or goes away from them. As the oceans extend further around the great circle of Greenwich (Atlantic and Pacific), such an asymmetry favors the influence of rotation pole coordinate m1 . The Liouville equations become more complex and are processed in the form of the generalized Liouville equation introduced in Chapter 4, defined and solved in Appendix A. Having been addressed from the late nineteenth century by Kelvin, the ocean pole tide raised a large scientific interest soon after it was advocated by Newcomb [162] as one of the causes of the lengthening of the Euler period. Assuming equilibrium, an effect of the order of 1 cm is obtained. Modern models account for gravitational selfattraction of the oceans, and the consequent additional yielding of the solid Earth [153, 126, 46, 66]. But the determination of the ocean rotational response using tidal gauges and satellite altimetry remains delicate because of the blurring of tidal and atmospheric-induced variations brought about. In this chapter, the rotational change of the sea level at equilibrium is presented through a simplified approach. Then we derive the rotational excitation accompanying the polar motion and develop the consequences for the theory of polar motion.

6.2 Ocean pole tide Because of the variable centrifugal potential U (r) the ground rises or goes down along the vertical by the length ξs according to U (r) (θ, λ) ξs (θ, λ) = h̃ 2 , g

(6.1)

where h̃ 2 = h2 + ih2 ∼ 0.6 is the complex deformation Love number, the imaginary part representing the dissipative process accompanying the deformation. This anelastic displacement is accompanied with the geopotential variation k̃2 U (r) and rotational https://doi.org/10.1515/9783110298093-006

96 | 6 Ocean pole tide excitation χ (r) = k̃2 /ks m introduced in the former chapter. Meanwhile, as a first approximation, the oceanic masses redistribute hydrostatically under the action of centrifugal potential and of the geopotential change k̃2 U (r) , causing the sea level variation U ξe = (1 + k̃2 ) . g (r)

(6.2)

The effective sea level change should be estimated from the seafloor, which has moved by ξs (θ, λ): U (r) (θ, λ) . ξ (θ, λ) = ξe − ξs = (1 + k̃2 − h̃ 2 ) g

(6.3)

Taking for U (r) Eq. (5.23) where the angular velocity change m3 is neglected, the ocean pole tide is expressed by 2 2

Ω r0 (m1 cos λ + m2 sin λ)P21 (cos(θ)) , ξ (θ, λ) = −(1 + k̃2 − h̃ 2 ) 3g

(6.4)

where r0 is the mean equatorial radius.1 Let λm be the longitude of the polar displacement: m = m1 + im2 = |m|eiλm , and the former expression reads also 2 2

Ω r0 |m| cos(λ − λm )P21 (cos(θ)) . ξ (θ, λ) = −(1 + k̃2 − h̃ 2 ) 3g

(6.5)

With a typical polar displacement |m| of 200 mas, the expected ocean pole tide (6.5) has an order of magnitude of 1 cm. It is displayed on a latitude–longitude grid (Figure 6.1), where the longitude λ − λm is referred to the longitude of the rotation pole. In the region of the northern hemisphere having the longitude of the polar displacement, the centrifugal effect decreases as well as the sea level, in the southern hemisphere the process is opposite. Now, we can figure out how the pattern of Figure 6.1 propagates towards the east as the pole moves with increasing longitude λm .2 Yet, observing the ocean pole tide remains a challenge, as it is hard to distinguish it among all processes that affect sea level: lunisolar tides, barometric changes, thermal expansion, and so one. Some studies report its possible observation in North Sea tidal gauges [234]. Altimetry observations (vertical radar echo), operated by the Topex/Poseidon mission (1992–2001) and covering almost the entire world ocean, are more reliable. After removing all modeled fluctuations, Desai (2002) [62] isolated a field of degree 2 and order 1, consistent with the static response to the pole tide potential. Adding the observations of Jason-1 and Jason-2/OSTM missions, more recently 1 Starting from Eq. (C.3) of r0 as a function of Re , we neglect the second order terms fm1/2 where f is the Earth’s flattening. 2 Notice that when annual and Chandler wobbles cancel each other, the pole can present a westwards motion during a few days, as during the winter of 2005–2006. But this phenomenon scarcely ever occurs. See Section 10.2.

6.2 Ocean pole tide

|

97

Figure 6.1: Ocean pole tide in cm associated with a polar displacement of 200 mas in the direction λm .

Desai and his collaborators fitted the ocean centrifugal tide over the period 1992–2014 [61], and slightly improved the earlier result. In what follows, the complex spherical harmonic function Ylm = Nlm eimλ Pl|m| (see Appendix B.1) will be very useful because of the simplifications brought about by their orthonormal properties. So the ocean pole tide will be written ξ (θ, λ) = −(1 + k̃2 − h̃ 2 )

Ω2 r02 (m∗ Y21 + mY2−1 ) , 6gN21

(6.6)

with the normalization coefficient N21 = √5/(24π) according to Eq. (B.4b). Correction of mass conservation The part of the ocean mass pertaining to pole tide is ΔM = ∮ ρo ξ (θ, λ)dS ≈ r02 ρ0 ∮ ξ (θ, λ) sin θdλdθ , oceans

(6.7)

oceans

where ρo ≈ 1025 kg/m3 is the surface water density. If the oceans would cover the whole Earth surface, this mass would be equal zero, for the integrals of Y21 and Y2−1 appearing in the pole tide expression (6.6) would vanish. But, as the ocean layer is interrupted by continents, ΔM is not equal to zero, and the ocean mass conservation is not granted. The resulting mass default or excess ΔM can be canceled mathematically

98 | 6 Ocean pole tide Table 6.1: Normalised spherical harmonic development of the ocean function up to degree 2. l

m

alm

ãlm

0 1 1 2 2 2

0 0 1 0 1 2

2.51664 −0.41032 −0.54913 −0.21349 −0.20585 0.18763

0.00000 0.00000 −0.29852 0.00000 −0.31460 −0.01419

if the sea level is raised or lowered by the uniform height ξ 󸀠 such as ΔM + r02 ρo ξ 󸀠 ∮ sin θdλdθ = 0 .

(6.8)

oceans

This correction3 can be written ∮ ξ (θ, λ) sin θdλdθ + ξ 󸀠 ∮ sin θdλdθ = 0 ,

oceans

(6.9)

oceans

that is, after substitution of ξ (θ, λ) by (6.6) π

−(1+k2 −h2 )

2π

π

2π

Ω2 r02 ∫ ∫ 𝒪(m∗ Y21 +mY2−1 ) sin θdλdθ+ξ 󸀠 ∫ ∫ 𝒪 sin θdλdθ = 0 . (6.10) 6gN21 θ=0 λ=0

θ=0 λ=0

Here 𝒪 is the “ocean function” equal to 1 over the oceans and 0 over the continents and developed in spherical harmonics according to m=+l

𝒪 = ∑ ∑ (alm cos(mλ) + ã lm sin(mλ))P̄ lm , l=0 m=0

(6.11)

where the P̄ lm are the normalized Legendre polynomials given by (B.4). The coefficients of this development are estimated from a latitude–longitude grid representing the ocean distribution with a resolution of 0.25° × 0.25°4 by a trapezoidal integration of Eqs. (B.9b), and are reported on Table 6.1 up to degree 2. The expansion in complex spherical harmonic functions Ylm reads m=+l

m

m

𝒪 = ∑ ∑ 𝒪l Yl , l=0 m=−l

(6.12a)

3 Introduced by the famous English geophysicist George Darwin (son of Charles) [48], who improved the tidal theory and proposed the symbols S1 , S2 , M1 , . . . of its components. 4 Published by Hans-Peter Plag on the WEB site WEB http://geodesy.unr.edu/hanspeterplag/tools/ ocean_function/

6.3 Oceanic rotational excitation

| 99

with m

𝒪l =

alm − iã lm 2

for m > 0 ;

0

𝒪l = al0 ;

m

𝒪l =

al,−m + iã l,−m 2

for m < 0 . (6.12b)

Substituting (6.12a) in (6.10) and applying the orthonormal relation (B.5), we obtain 2 2

Ω r0 ∗ −1 (m 𝒪2 + m𝒪21 ) + ξ 󸀠 𝒪00 √4π = 0 . − (1 + k̃2 − h̃ 2 ) 6gN21

(6.13)

Hence ξ 󸀠 = (1 + k̃2 − h̃ 2 )

Ω2 r02 (m∗ 𝒪2−1 + m𝒪21 ) , 0 1 √ 6gN2 𝒪0 4π

(6.14)

amounting to 1 mm, which is about 10 times less than the rotational variation.

6.3 Oceanic rotational excitation The pole tide of the sea level ξ (θ, λ) produces a rotational excitation, which can be expressed like the water height term of the ocean angular momentum according to (8.37): χo(r) = −

=−

ρo r04

C−A

π

2π

∫ ∫ 𝒪 ξ (θ, λ) cos θ sin2 θ eiλ dλdθ θ=0 λ=0

ρo r04

3N21 (C − A)

π

2π

∫ ∫ 𝒪 ξ (θ, λ)Y21 sin θ dλdθ .

(6.15)

θ=0 λ=0

Pole tide effect without mass correction The pole tide (6.6), without mass correction, causes the rotational excitation χo(r)

=

(1 + k̃2 − h̃ 2 )Ω2 ρo r06 18g(N21 )2 (C − A)

π

2π

∫ ∫ 𝒪 (m∗ Y21 + mY2−1 )Y21 sin θ dλdθ ,

(6.16)

θ=0 λ=0

in which the spherical harmonic 𝒪Y21 = 𝒪N21 cos λP21 + i 𝒪N21 sin λP21 appears. Here, the real functions 𝒪c = 𝒪N21 cos λP21 and 𝒪s = 𝒪N21 sin λP21 admit the spherical harmonic expansions c

m=+l

𝒪 =∑ ∑( l=0 m=0

aclm − iã clm m aclm + iã clm −m Yl + Yl ) , 2 2

as − iã slm m aslm + iã slm −m 𝒪 = ∑ ∑ ( lm Yl + Yl ) . 2 2 l=0 m=0 s

m=+l

(6.17)

100 | 6 Ocean pole tide Hence 𝒪Y21 admits the development m=+l

1

𝒪Y2 = ∑ ∑

l=0 m=0

aclm + iaslm − i(ã clm + iã slm ) m aclm + iaslm + i(ã clm + iã slm ) −m Yl + Yl . 2 2

(6.18)

Thus m=+l

1

𝒪 Y2 = ∑ ∑ 𝒪 l=0 m=−l

󸀠m m l Yl

(6.19)

,

with 𝒪

󸀠m l

=

aclm + iaslm − i(ã clm + iã slm ) , 2

𝒪

󸀠 −m l

=

aclm + iaslm + i(ã clm + iã slm ) . 2

(6.20)

Then, applying the orthonormal relations (B.5), Eq. (6.16) is reduced to χo(r) = (1 + k2 − h2 )

Ω2 ρo r06

18g(N21 )2 (C

−1

− A)

1

(m∗ 𝒪󸀠 2 + m𝒪󸀠 2 ) .

(6.21)

There results 1

−1

ac + ias21 − i(ã c21 + iã s21 ) ac21 + ias21 + i(ã c21 + iã s21 ) + m 21 2 2 s c s c (6.22) = m1 a21 + m2 ã 21 + i(m1 a21 + m2 ã 21 ) .

m∗ 𝒪󸀠 2 + m𝒪󸀠 2 = m∗

The coefficients ac21 , ã c21 , as21 , ã s21 are estimated from the integrals (B.9b): ac21

π

2π

π

θ=0 λ=0

θ=0 λ=0

ã c21

π

2π

π

c

2π

= 2 ∫ ∫ 𝒪 P̄ 21 sin λ sin θdθdλ = 2 ∫ ∫ 𝒪 cos λ sin λ (P̄ 21 )2 sin θdθdλ , θ=0 λ=0

as21

2π

= 2 ∫ ∫ 𝒪 P̄ 21 cos λ sin θdθdλ = 2 ∫ ∫ 𝒪 cos2 λ (P̄ 21 )2 sin θdθdλ , c

π

2π

θ=0 λ=0 s

= 2 ∫ ∫ 𝒪 P̄ 21 cos λ sin θdθdλ =

ã c21

θ=0 λ=0 π

2π

(6.23)

,

π

2π

ã s21 = 2 ∫ ∫ 𝒪s P̄ 21 sin λ sin θdθdλ = 2 ∫ ∫ 𝒪 sin2 λ (P̄ 21 )2 sin θdθdλ . θ=0 λ=0

θ=0 λ=0

Their numerical evaluation is done from the digitalization of the oceanic function 𝒪 given above4 , leading to A1 = ac21 = 0.784,

A2 = as21 = B1 ,

B1 = ã c21 = −0.0179 ,

B2 = ã s21 = 0.630 .

(6.24)

6.3 Oceanic rotational excitation |

101

1

−1

Substituting m∗ 𝒪󸀠 2 + m𝒪󸀠 2 with (6.22) in (6.21), the oceanic rotational excitation becomes Ω2 ρo r06 4π χo(r) = (1 + k̃2 − h̃ 2 ) 3g(C − A) 5

󵄨󵄨 󵄨󵄨 A1 m1 + B1 m2 , 󵄨󵄨 󵄨󵄨 B m + B m . 󵄨󵄨 1 1 2 2

(6.25)

In this expression g is approximated by GM/r02 = 4π/3 Gρ⊕ r0 (ρ⊕ ≈ 5.513 kg/m3 is the mean density of the Earth), and we use the secular Love number ks ≈ 3G(C − A)/(Ω2 r05 ) (Eq. 5.19): χo(r) =

󵄨 3 1 + k̃2 − h̃ 2 ρo 󵄨󵄨󵄨󵄨 A1 m1 + B1 m2 , 󵄨 5 ks ρ⊕ 󵄨󵄨󵄨󵄨 B1 m1 + B2 m2 .

(6.26)

A rapid evaluation gives χo(r) ∼ 0.05 m, meaning that the contribution of the ocean pole tide amounts to about 16 % of the solid Earth rotational excitation. Effect of the ocean mass correction In virtue of (6.15) the corrective term gives the rotational excitation χo󸀠 (r) = −

ρo r04

3N21 (C − A)

π

2π

ξ 󸀠 ∫ ∫ 𝒪 Y21 sin θdλdθ .

(6.27)

θ=0 λ=0

After substituting ξ 󸀠 with (6.14) we get χo󸀠 (r)

π

2π

m=l (1 + k̃2 − h̃ 2 )(m∗ 𝒪2−1 + m𝒪21 )Ω2 ρo r06 =− ∑ ∑ 𝒪lm ∫ ∫ Ylm Y21 sin θdλdθ . (6.28) 18(C − A)g(N21 )2 𝒪00 √4π l=0 m=−l θ=0 λ=0

Application of the orthonormal relations reduces the expansion to χo󸀠 (r) = −

(1 + k̃2 − h̃ 2 )Ω2 ρo r06 4π −1 ∗ −1 1 𝒪2 (m 𝒪2 + m𝒪2 ) . 3(C − A)g 𝒪00 √4π

(6.29)

From (6.12b) m∗ 𝒪2−1 + m𝒪21 = a21 m1 + ã 21 m2 , a + i ã 21 −1 𝒪2 = 21 , 2 0 𝒪0 = a00 .

(6.30)

By considering as above that g ≈ 4π/3 Gρ⊕ r0 and accounting for the secular Love number, we obtain a similar expression to (6.26): χo󸀠 (r) =

󵄨 3 1 + k̃2 − h̃ 2 ρo 󵄨󵄨󵄨󵄨 A󸀠1 m1 + B󸀠1 m2 , 󵄨 5 ks ρ⊕ 󵄨󵄨󵄨󵄨 B󸀠1 m1 + B󸀠2 m2 ,

(6.31a)

102 | 6 Ocean pole tide with A󸀠1 = −

(a21 )2 = −0.006, a00 √4π

B󸀠1 = − B󸀠2

a21 ã 21 = −0.007 , a00 √4π

(ã 21 )2 =− = −0.011 . a00 √4π

(6.31b)

So, the total rotational excitation, including the mass correction, keeps the form (6.26) with slightly modified coefficients A1 , B1 , and B2 : A1 = 0.778, A2 = B1 ,

B1 = −0.025 ,

B2 = 0.619 .

(6.32)

Moreover, the load of the ocean pole tide generates a deformation of the solid Earth that has to be accounted for. According to the formulation of the effective matter term produced by a thin fluid layer, as given by (5.57), the rotational excitation (6.31a) should be multiplied by the factor 1 + k2󸀠 : χo(r) =

ρ 3 1 + k̃2 − h̃ 2 (1 + k2󸀠 ) o [A1 m1 + B1 m2 + i(B1 m1 + B2 m2 )] . 5 ks ρ⊕

(6.33)

Comparison with Desai’s model (Section 4 of [62]) recommended by IERS conventions 2010 [168] By using our formalism, this pole tide model is given by the coefficients Ad1 = 0.942 ,

Bd1 = Ad2 = −0.021 ,

Bd2 = 0.746 .

(6.34)

Whereas A1 and B2 are slightly larger, the relative offset remains the same. These values will be taken for the numerical applications. Oceanic Love number By writing m1 = (m + m∗ )/2 and m2 = −i(m − m∗ )/2, the ocean pole tide excitation is put in the form χo(r) =

k̃o k 󸀠̃ m + o m∗ , ks ks

(6.35)

with ρ A + B2 3 k̃o = (1 + k̃2 − h̃ 2 )(1 + k2󸀠 ) o 1 ≈ 0.0477 + i 0.0002 , 5 ρ⊕ 2 ρ A − B2 + 2 i A2 3 k̃o󸀠 = (1 + k̃2 − h̃ 2 )(1 + k2󸀠 ) o 1 ≈ 5.4 10−3 − i 1.1 10−3 . 5 ρ⊕ 2

(6.36)

6.4 Liouville equation for a triaxial, anelastic Earth covered by the oceans | 103

The coefficient k̃o is at the ocean pole tide what the Love number k̃2 is to the solid Earth pole tide (5.42), and it is called the oceanic Love number. A rough estimate of the real part of k̃o is ko = 0.048. The dissipation associated with the ocean pole tide is restricted to processes affecting solid Earth (complex part of the Love number k̃2 and h̃ 2 ). So the complex part of k̃o = ko + iko is given by ko ≈ 3/5(k2 − h2 )

ρo A + B2 (1 + k2󸀠 ) 1 ≈ 0.13(k2 − h2 ) . ρ⊕ 2

(6.37)

Under the action of an external potential of degree 2, radial deformation and geopotential perturbation undergo a phase lag ε of comparable magnitude, about 1 degree (see Chapter 5), so that k2 ≈ h2 and ko should be much smaller than k2 . However, our simple equilibrium model does not take into consideration the friction over the coasts and the shallow seas, and in turn ignores this contribution on ko . From the anelastic Love numbers k̃2 and h̃ 2 (see Table H.1), k̃o ≈ 0.0477 + i 0.0002. With a typical polar motion of 10−6 rad at quasi-seasonal scale, the oceanic rotational excitation has the order of magnitude 5 10−8 rad or 10 mas, which is about the magnitude of the hydro–atmospheric excitation at seasonal scale.

6.4 Liouville equation for a triaxial, anelastic Earth covered by the oceans As well as triaxiality, the ocean pole tide excitation makes the Liouville equations asymmetric with respect to components m1 and m2 . So, in order to have a consistent approach, we have to start from the Liouville equation for a triaxial Earth (4.30a), in which the non-rigid effects have not been expressed yet: m+

i e󸀠 e2iλA ∗ e󸀠 e2iλA ∗ i i ṁ + m −i ṁ = χ − χ̇ + χ (r) − χ̇ (r) , σe e σe Ω Ω

(6.38)

where we discriminate the rotational excitation χ (r) from the purely geophysical one. In this equation the influence of the triaxiality on the geophysical excitation has been neglected (a few thousandths of the excitation) with regard to the present accuracy of the geophysical forcing (≥ 1 % relative uncertainty). The ocean rotational excitation (6.35) is merged with the one pertaining to the solid Earth and provided by (5.42): χ (r) =

k̃ 󸀠 k̃2 + k̃o m + o m∗ = Um + U 󸀠 m∗ , ks ks

(6.39a)

with U=

k̃2 + k̃o , ks

U󸀠 =

k̃o󸀠 . ks

(6.39b)

104 | 6 Ocean pole tide The numerical values of the coefficients U and U 󸀠 are reported in Table 6.2. Transferring (6.39) from the right hand side of (6.38) to the left hand side, the right hand side is reduced to the pure excitation free from the rotational effect: (1 − U) m +

i i i (1 + eU)ṁ − Vm∗ + eV ṁ ∗ = χ − χ̇ , σe σe Ω

(6.40a)

with the total asymmetric coefficient V = U󸀠 −

e󸀠 2iλA e . e

(6.40b)

It can be seen that the asymmetry introduced by the ocean pole tide (coefficient U 󸀠 ) is partly balanced by triaxiality (see numerical values in Table 6.2). The form (6.40a) is the one of the generalized equatorial linearized Liouville equation, formulated by (A.1) in Appendix A, and it can be brought closer to Eq. (19) of Okamoto and Sasao (1977) [165] associated with a more complicated derivation. In the case considered above, the coefficients of these equations respect the following orders of magnitude: U = U1 + i U2 |V| ≈ e

with U1 ≈ 0.36, U2 = O(e) ,

(e = flattening) .

(6.41)

Symmetric approximation Neglecting asymmetric coefficients, we recognize the common Liouville equation associated with a biaxial Earth and symmetric pole tide: m(1 − U) +

i (1 + eU)ṁ = Ψ(pure) . σe

(6.42)

The effective complex Love number k̃ accounting for both mantle anelasticity, oceans and dissipation is defined by k̃ = k̃2 + k̃o = ks U ≈ 0.355 ,

(6.43)

̃ where the rheological coefficient U = k/k s is given by (6.39b). The complex free mode frequency of system (6.42) is σc̃ = σe

1−U k̃ i ≈ σe (1 − ) ≈ σc (1 + ) 1 + eU ks 2Q

(6.44)

where Q is the quality factor reflecting the imaginary part of k.̃ Selecting U = k̃2 /ks , we go back to an anelastic Earth, and the Chandler frequency is expressed by (5.44) associated with a period of about 433 days. But, considering the slightly larger Love number k = 0.355, we see that So the oceans lengthen the Chandler period up to Tc = 2π/σc ≈ 482 days. In the next chapter it will be exposed how the core passivity at periods larger than 2 days brings back Tc to the observed period of 433 days.

6.5 Observational consequence |

105

Complete analysis From Appendix A the generalized Liouville equations represent the prograde Chandler mode with complex frequency given by (A.16), that is, σ̃ c = σe [1 − U − e U1 (1 − U1 ) + O(e2 )] ,

(6.45)

and the retrograde mode with frequency σ̃ c− = −σ̃ c∗ (Eq. (A.15)). It can be easily checked that the form (6.45) is consistent with Eq. (6.44) found in the symmetric case. Dissipation and imaginary values of coefficients U and V Dissipative processes can be globally quantified by the imaginary part of k̃ = k(1 + iε). The imaginary part of (6.44) yields σe

k −kε k 1 1 −0.86 −0.29 = σe (1 − ) ⇐⇒ ε = (1 − s ) ≈ ⇐⇒ k = ℑ(k)̃ ≈ . (6.46) ks ks 2Q k 2Q Q Q

Taking the median value Q = 100 (40 < Q < 200, see Chapter 11), the phase lag is ε ≈ −0.01 rad (−0.5°). Numerical values of coefficients U, U 󸀠 , V, given in Table 6.2, are estimated according to the coefficients (6.34) of the Desai model and Love numbers of Table H.1. Table 6.2: Numerical values of the coefficients k,̃ U, U 󸀠 and V consistent with the Desai model and Love numbers h2 and k2 of Table H.1, ρo = 1025 kg/m3 , ρ⊕ = 5513 kg/m3 . The imaginary parts of k̃ and U are determined by the Chandler wobble quality factor. Constant

Symbol

Values

Oceanic Love number Effective Love number Rheological coefficient Pole tide asymmetric coefficient Total asymmetric coefficient

k̃

0.0477 + i 0.0002 0.355 − i 0.29/Q 0.378 − i 0.309/Q 0.0058 − i 0.0012 0.0028 + i 0.00046

o

k̃ = k2̃ + kõ ̃ U = k/k s U󸀠 V

6.5 Observational consequence 6.5.1 Symmetric and asymmetric response to a circular excitation of fixed frequency From (A.24) a circular excitation with angular frequency σ0 , that is, Ψ0 eiσ0 t causes the elliptic oscillation: mσ0 (t) = m+0 eiσ0 t + m−0 e−iσ0 t

(6.47a)

106 | 6 Ocean pole tide

Figure 6.2: Ratio |m−0 /Ψ0 | in function of the excitation frequency for three asymmetric cases: asymmetric pole tide alone (red), triaxality alone (blue), combined effect (dashed). Notice the double resonance at Chandler frequency and its opposite.

with m+0 = −Ψ0 m−0 = Ψ∗0

σe (1 − e U1 ) , σ0 − σ̃ c

σe V σ + eσc σe − eσc σe + σ0 e (− e + ) = −Ψ∗0 σe V . 2σc σ0 − σ̃ c σ0 − σ̃ c− (σ0 − σ̃ c )(σ0 − σ̃ c− )

(6.47b)

We recognize the common term m+0 (t) circling with the same angular frequency, and exhibiting the unique resonance at Chandler angular frequency σ̃ c . But also the additional contribution m−0 (t) of opposite frequency −σ0 appears: its double resonance, both at Chandler frequency and its opposite, is strongly reduced by the smallness of the numerator V in (6.47). In Figure 6.2, we plot the amplitude and the phase of m−0 /Ψ0 in three asymmetric cases: i) asymmetric pole tide without triaxiality (by canceling e󸀠 in V), ii) triaxiality

6.5 Observational consequence |

107

but symmetric pole tide (by cancelling U 󸀠 in V), and iii) the combined effect (accounting for e󸀠 and U 󸀠 ). Whereas the asymmetric ocean pole tide contribution is the largest one (ratio above 1.5 at −σc or σc ), it is compensated for by the out-of-phase effect of triaxiality, so that the combined effect reaches about 0.6 at −σc or σc . As far as the free mode excitation can be reduced to a single harmonic of about 1 mas (see Chapter 11), we expect an asymmetric or a retrograde Chandler term at 1 mas level. It has to be pointed out that with larger quality factor, like Q = 180, favored by analysis of the pole coordinate series covering about one century and including optical data [97], the ratio |m−0 /Ψ0 | is up to 2 for σc . At proximity, the annual excitation, about 10 mas both in retrograde and prograde bands (see Figure 10.7), produces asymmetric terms of about 0.01 ∗ 10 = 0.1 mas. The same conclusion can be drawn for multi-annual excitation. At the other edge of the spectrum, below 100 days, the rapid fluctuations of the excitation with an amplitude up to 10 mas (see Chapter 9) present a negligible and asymmetric counter-part in the light of the pole coordinate accuracy. 6.5.2 Intrinsic ellipticity The relative impact of m−0 with respect to the common effect m+0 is quantified by the ratio 󵄨󵄨󵄨 V(σe + σ0 e) 󵄨󵄨󵄨 󵄨󵄨 (6.48) |m−0 /m+0 | = 󵄨󵄨󵄨 󵄨 󵄨󵄨 σ0 + σ̃ c 󵄨󵄨

represented in Figure 6.3a for the three cases considered above. This ratio also determines the intrinsic ellipticity of the polar motion at frequency σ0 , given by the relative difference between great and small axes, that is, E=

|m+0 | + |m−0 | − (||m+0 | − |m−0 ||) , |m+0 | + |m−0 |

(6.49)

plotted in Figure 6.3b. Assuming a quality factor Q = 100 for complex Chandler frequency, the ratio |m−0 /m+0 | reaches a maximum of 0.74 at −σc , but drops to 0.002 around σc (for the total asymmetric effect). Generally, the ratio is lower than 0.05. Correspondingly the intrinsic elliptic polarization is larger in the retrograde band, with a maximum around −σc (almost 1), but it is less than 1 % for the main polar motion harmonics (prograde 365 and 433 day terms) as shown by Figure 6.3b. Actually the total polar motion at frequency σ0 also comes from circular excitation 󸀠 Ψ󸀠0 eiσ0 t at an opposite frequency σ0󸀠 = −σ0 and with complex amplitude Ψ󸀠0 . From (6.47) it causes the elliptic term −iσ0 t iσ0 t m󸀠σ0 (t) = m+󸀠 + m−󸀠 , 0 e 0 e

m+󸀠 0 = Ψ0󸀠

σe , σ0 + σ̃ c

∗󸀠 m−󸀠 0 = −Ψ0

σe V(σe − σ0 e) , (σ0 + σ̃ c )(σ0 + σ̃ c− )

(6.50)

108 | 6 Ocean pole tide

Figure 6.3: Amplitude of the ratio m−0 /m+0 as a function of the frequency (a) and polar motion intrinsic ellipticity E as a function of the frequency (b) for the three cases: asymmetric ocean pole tide alone (red line), triaxality alone (blue line), combined effect (dashed line).

so that the total elliptic polar motion at frequency σ0 is expressed by 󸀠 mtot σ0 (t) = mσ0 (t) + mσ0 (t)

≈ −(Ψ0

σe σe V(σe − σ0 e) + Ψ∗󸀠 )eiσ0 t 0 σ0 − σ̃ c (σ0 + σ̃ c )(σ0 + σ̃ c− )

+ (Ψ0󸀠

σe σe V(σe + σ0 e) )e−iσ0 t . − Ψ∗0 σ0 + σ̃ c (σ0 − σ̃ c )(σ0 − σ̃ c− )

(6.51)

This expression shows that the asymmetric effects associated with coefficient V cannot be extracted from the polar motion harmonics without any information or hypothesis on geophysical excitation (circular terms of complex amplitudes Ψ0 and Ψ󸀠0 ). In the words of Okamoto and Sasao [165], intrinsic elliptic polarization is hardly distin-

6.6 Dynamical ocean pole tide in the diurnal band | 109

guishable from the polarization associated with geophysical excitation. Outside the Chandler frequency band, the effect of the damping can be neglected and the elliptic polar motion resulting from geophysical forcing is given by mtot σ0 (t) ≈ −(Ψ0

σe σe V(σe − σ0 e) + Ψ∗󸀠 )eiσ0 t 0 σ0 − σc (σ0 + σc )(σ0 − σc )

+ (Ψ0 󸀠

σe σe V(σe + σ0 e) − Ψ∗0 )e−iσ0 t . σ0 + σc (σ0 − σc )(σ0 + σc )

(6.52)

At seasonal scale, the polar motion chiefly results from the hydro–atmospheric forcing and, in turn, can be modeled with 1 mas precision (see Chapter 9). As seen above, the asymmetric perturbation brought about by the coefficient V is about 10 times less than this order of magnitude, so that its consideration cannot improve the present geophysical budget of the observed annual wobble.

6.6 Dynamical ocean pole tide in the diurnal band Can we totally exclude dynamical effects in the ocean pole tide? In the late 1980s Dickman [64, 63] concluded that the dynamical effects at quasi-seasonal scales lengthens the Chandler period by one day. Equivalently the oceanic Love number is increased by about 0.0014. Moreover the dynamical processes slightly delay the ocean tide response: with a time damping of about 500–700 years, much longer than the Chandler relaxation time (30 years), the damping introduced an imaginary part in the Love number of about −2 10−4 . So, in respect to the equilibrium value of 0.0477, the hydrostatic pole tide remains an excellent approximation when considering common polar motion. However, below 10 days, many studies have shown that the ocean response to an atmospheric pressure variation strongly departs from equilibrium, the so-called inverted barometer response, so the hydrostatic pole tide is not sound. In the diurnal band, this issue can be solved in the light of the diurnal ocean tides. For, as the pole tide potential has the same form as the lunisolar tesseral potential and concerns the same frequency band, the Earth response should be formally the same. It is well known that the diurnal ocean tides are strongly affected by dynamical processes. Currents are generated, and in turn a relative angular momentum. Meanwhile, the observed diurnal ocean tide height is smaller than the theoretical equilibrium tide produced by the lunisolar tesseral potential, and is strongly out-of-phase with respect to it. The tidal component at frequency σ causes the equatorial oceanic angular momentum H(t) = H1 cos(θ(σ) + χ − Φ1 ) + i H2 cos(θ(σ) + χ − Φ2 ) , h(t) = h1 cos(θ(σ) + χ − ϕ1 ) + i h2 cos(θ(σ) + χ − ϕ2 ) ,

(6.53)

where θ(σ) is the tidal argument, H(t) and the associated coefficients H1 , H2 , Φ1 , Φ2 hold for the matter term; h(t) and the associated coefficients h1 , h2 , ϕ1 , ϕ2 describe

110 | 6 Ocean pole tide Table 6.3: Main terms of the oceanic angular momentum generated by tesseral diurnal gravitational tides according to FES 2012, as reported in [137]. The reported coefficient corresponds to Eq. (6.53). The amplitudes H1 , H2 , h1 , h2 are expressed in units of 1025 kg m2 /s, and γ = GMST + π.

Q1 O1 P1 K1 J1

Tidal argument θ

χ (°)

H1

Φ1 (°)

H2

Φ2 (°)

h1

ϕ1 (°)

h2

ϕ2 (°)

γ − l − 2F − 2? γ − 2F − 2? γ − 2F + 2D − 2? γ γ+l

−90 −90 −90 +90 +90

0.116 0.476 0.169 0.462 0.026

340.4 330.1 310.6 308.3 294.0

0.264 1.178 0.450 1.377 0.076

215.4 221.9 223.2 224.2 228.8

0.058 0.291 0.183 0.557 0.036

307.8 299.7 287.4 288.8 292.0

0.075 0.442 0.255 0.774 0.055

217.1 206.1 192.8 192.1 186.7

the current term. According to the FES 2012 ocean tidal model, the main diurnal constituents are for tesseral tides J1 , K1 , P1 , O1 , Q1 . The corresponding coefficients calculated in [137] are provided in Table 6.3. We have close estimates for an ancient model going back to 1996, as reported in [39]. For a given tidal constituent, the retrograde term is −

−

−

ℋ (t) = (H + h )e

−i(θ+χ)

(6.54)

with H1 cos(Φ1 ) − H2 sin(Φ2 ) H sin(Φ1 ) + H2 cos(Φ2 ) +i 1 ), 2 2 h sin(ϕ1 ) + h2 cos(ϕ2 ) h cos(ϕ1 ) − h2 sin(ϕ2 ) +i 1 ). h− = ( 1 2 2

H− = (

(6.55)

̃ 𝒴 −1 ), with From (D.5), the corresponding tesseral lunisolar potential is −Ω2 R2e /3Re(Φ(t) 2 1 ̃ = 3gN2 ξ e−i(θσ −π/2) , ϕ(t) Ω2 R2e σ

(6.56)

where ξσ is the equilibrium tidal height. Accounting for the deformation effect of the tidal loading, the retrograde effective angular momentum function caused by the tidal ̃ is pontential ϕ(t) χo (t) =

H − (t)(1 + k2󸀠 ) + h− (t) (C − A)Ω

=

H − (1 + k2󸀠 ) + h− (C − A)Ω

e−i(θ+χ) .

(6.57)

̃ is formally equivalent to m(t) In the tidal potential W, expressed through (D.4), ϕ(t) ̃ in the pole tide potential. So, χo is proportional to ϕ(t), as the rotation excitation is proportional to m(t): χo =

k̃o ̃ Φ, ks

(6.58)

6.7 Conclusion

| 111

Table 6.4: Oceanic Love number in the diurnal band.. Q1 O1 P1 K1 J1

−0.037 + i 0.039 −0.031 + i 0.038 −0.023 + i 0.042 −0.023 + i 0.042 −0.023 + i 0.047

where ko is the oceanic Love number. Then we obtain H − (1 + k2󸀠 ) + h− ΩR2e H − (t)(1 + k2󸀠 ) + h− (t) = −ks k̃o = ks . 3gN21 ξσ (C − A)ΩΦ̃ C−A

(6.59)

Then we can estimate k̃o for the tidal components above by considering the ξσ values reported in Table D.1. For the loading Love number k2󸀠 = −0.3075, the obtained values are given in Table 6.4. The frequency dependence of k2󸀠 , especially resulting from the free core nutation resonance, as described by (D.8), introduces negligible changes. The Love number values in the diurnal band strikingly differ from ko = 0.048 estimated for an equilibrium pole tide (see Eq. (6.36)): the real part becomes negative and is superseded by a positive imaginary part. The values of Table 6.4 allow one to model ko (σ) through a degree 2 polynomial of the frequency: kod (f ) = (−0.716 + i 0.721)f 2 + (−1.483 + i 1.337)f + (−0.791 + i 0.658) ,

(6.60)

where f is in cpd, and the “d” superscript means that this expression of ko is limited to the diurnal domain. In Chapter 7, when modeling the Earth’s rotation response at diurnal periods, we shall see how this will modify the resonance frequency of the polar motion.

6.7 Conclusion The ocean pole tide resulting from the common polar motion (with periods larger than 2 days) is estimated in the approximation of hydrostatic equilibrium, with a method that differs notably from that proposed in the monographs of Munk and MacDonald [153] and Lambeck [126]. The results confirm the Desai model (2002) [62]. However, dynamic effects are manifest at periods smaller than 10 days. In particular, at diurnal periods, polar motion produces a dynamic ocean tide as the Moon and the Sun do. The resulting proportionality coefficient ko between the oceanic rotational excitation and pole displacement m is strongly phase-shifted with respect to the one computed at equilibrium. In consequence the polar motion frequency resonance should be modified as confirmed by the analysis of the nutation terms (corresponding to retrograde nearly-diurnal polar motion). This issue will be investigated in more detail in Chapter 7.

112 | 6 Ocean pole tide On the other hand anisotropic ocean pole tide and to a lesser extent the triaxiality of the Earth break the symmetry of the equatorial linearized Liouville equations. Casting aside the decoupling from the fluid core, which is analyzed in the next chapter, their consistent handling leads to an extended form of the linearized Liouville equation, providing an elegant separation between rotational symmetry and asymmetry, with a formalism simpler than the one found in [165]. We derive a general solution of these equations, putting forward a secondary resonance of the circular excitation at the negative Chandler frequency. We strive to characterize the observational consequences. A given circular excitation gives an elliptical polar motion, the ellipticity reaching 1 in the vicinity of the negative Chandler frequency. But the elliptic polarization for the main circular components remains small (1 %), and therefore can hardly be distinguished from the one brought about by the forcing itself. Whereas various authors show how the triaxiality couples the amplitude of any circular wobble to that of the opposite frequency (see e. g. [143]), we emphasize that the ocean pole tide is the main contributor to this coupling and that triaxiality counterbalances half of it.

7 Influence of the fluid core Il y a donc double résonance; donc l’amplitude des nutations différera notablement de ce qu’elle serait pour un corps rigide. Poincaré (1910) [172]

7.1 Introduction In the wake of the previous chapter the polar motion theory is deepened by taking into account the influence of the fluid core.1 Because of its fluid nature, the fluid core tends to decouple from the mantle and develop its own rotational changes. However, even in the absence of viscous or electromagnetic processes, we could not erase a moment of force of the core on the mantle: the simple ellipsoidal shape of the core–mantle interface ensures the existence of a pressure coupling opposing the rotation of the core relative to the mantle. By hitting the mantle, the core cannot develop a large equatorial tilt, so that its rotation axis may wobble with respect to the mantle, but only on the short-term scale, that is, for diurnal and sub-diurnal periods as we shall see later.2 Early theories of the rotation of a rigid shell containing a fluid core were developed by Hopkins in 1839, Kelvin, Hough [116], Sloudsky [202] in the late nineteenth century and Poincaré in 1910 [172]. According to him the core induces a second free mode for the polar motion, with a retrograde diurnal frequency, and thus introduces a resonance for the lunisolar nutation terms. This phenomenon, of which the theoretical modeling was refined by considering the mantle elasticity (especially by Molodenskii [150]), was confirmed in the years 1970–1980 [147]. From the 1980s, the starting VLBI observations of the nutation at 1 mas level implied that the fluid core is slightly more flattened than it was thought, in violation of the hydrostatic equilibrium [110, 108]. The unprecedented precision of the nutation then motivated many refinements of the nutation theory of a two layered Earth, either forced by lunisolar tides [227, 184, 29, 94] or by surface loading or friction [114, 115, 185, 113, 132]. 1 Located between 1215 km and 3485 km depth, composed of molten nickel and iron, and including an iron solid core, its existence is attested by seismology. 2 By contrast, the axial coupling fades, and the differential rotation of the core relative to the mantle can develop over several years or even a few centuries. Therefore the influence of the fluid core on UT1 or the Earth’s angular velocity may happen in the longer term. To that extent it is not surprising to note decadal fluctuations in LOD (a few ms) or UT1, first observed in the 1930s (see Figure 2.5). As changes in the Gauss coefficients of the geomagnetic field are partially correlated with LOD decadal variations, one suspected one and the other result from the same motions in the core. According to magneto-hydrodynamic models, the Earth’s magnetic field at its surface can reconstruct fairly well the axial angular momentum variations of the core and its effect of the LOD variations [111]. The multiannual climate variations affecting both the zonal winds and the thickness of the glaciers may provide an alternative or a complementary explanation [201]. In support of this hypothesis, the zonal winds contribute up to 10 % of the LOD changes observed between 10 and 20 years [53, 101]. https://doi.org/10.1515/9783110298093-007

114 | 7 Influence of the fluid core From the 1990s, theoretical studies focused on the possible effects of the solid inner core of 1200 km radius, detected in the 1930s from seismic analysis [144, 59]. This has moments of inertia not exceeding 10−4 of the whole Earth versus 10−1 for the fluid core, and it theoretically induces effects in the Earth’s rotation, which are not clearly evidenced for now. For this reason the theoretical developments presented here neglect its presence. We summarize the modeling in the most simple manner, but sufficiently advanced to interpret polar motion in terms of hydro-meteorological excitation. The mechanical system is now subdivided in the mantle (in the broad sense, that is to say, including lithosphere and fluid surface layers) and the fluid core. Since there is no observational evidence of the possible effects of the triaxiality of the fluid core on the Earth’s rotation, especially on normal mode frequencies [236], the core will be considered biaxial. For each of these sub-systems we need to develop two Liouville equations. Viscous and electromagnetic torque will be evoked, but their potential effects on the polar motion will not be explored.

7.2 Liouville equation for a biaxial fluid core 7.2.1 Fluid core angular momentum As a first approximation, the fluid core is rotating like a rigid body with respect to the mantle with the angular velocity vector ω⃗ f , admitting the components [Ωmf1 , Ωmf2 , Ωmf3 ] in the TRF. So the velocity field relative to the mantle should be v⃗f = ω⃗ f ∧ r.⃗ But, as the fluid boundary is an ellipsoid of revolution around ∼ Oz, the meridian velocity is not perfectly tangential to this boundary, and it can produce an outer flow. To satisfy the impermeability condition at the mantle-fluid boundary, it is necessary that the meridian velocity field contains an additional part v⃗fe . More generally the fluid core velocity with respect to the TRF is composed of a rotational part and a irrotational ⃗ residual velocity v⃗fe (∇⃗ ∧ v⃗fe = 0): v⃗f = ω⃗ f ∧ r ⃗ + v⃗fe .

(7.1)

The corresponding velocity field with respect to the CRF is thus V⃗ f = Ω⃗ f ∧ r ⃗ + v⃗fe ,

(7.2)

where we have introduced the angular velocity vector Ω⃗ f = ω⃗ + ω⃗ f of the core with respect to the CRF. If we assume that the fluid core is incompressible and has a velocity field, of which the components are fully linear functions of the terrestrial Cartesian coordinates, and which is tangential to the core–mantle boundary, namely if we deal with a Poincaré flow, then it can be shown that its components in TRF are v⃗fe = (

−ef ωf2 z ef ωf1 z

ef (−ωf2 x

+

ωf1 y)

) ∼ ef ωf r ,

(7.3)

7.2 Liouville equation for a biaxial fluid core | 115

where ef is the dynamical flattening of the core (demonstration in Appendix F.1; see Eq. (F.16)). Experiments of rotating ellipsoidal shell enclosing a fluid confirm the existence of the Poincaré flow. Whereas the condition of a Poincaré flow can be violated, we shall assume that the order of magnitude (7.3) is always satisfied [60, 144, 133]. The fluid core angular momentum H⃗ f can be expressed by H⃗ f = ∫ ρr ⃗ ∧ V⃗ f dτ

(7.4)

𝒱f

where 𝒱f is the volume of the fluid core, and ρ the density of any volume element dτ in the core. Let L⃗ f be the torque on the fluid core. In the TRF the angular momentum balance reads dHf dt

+ ω ∧ Hf = Lf ,

(7.5)

where the notation A means that we consider the TRF components of the vector A.⃗ Any volume element of the core dτ is submitted to gravitational forces deriving from ⃗ dτ and to a non-conservative force the potential Φg , to a pressure gradient force −∇P ⃗ ρ dτ t, like the viscous force at the core–mantle boundary. Thus, the elementary force per volume unit is ⃗ + ρ∇Φ ⃗ g + ρt ⃗ , f ⃗ = −∇P

(7.6)

and the total torque acting on the core is ⃗ + ρ∇Φ ⃗ g + ρt)⃗ dτ . L⃗ f = ∫ r ⃗ ∧ f ⃗ dτ = ∫ r ⃗ ∧ (−∇P 𝒱f

(7.7)

𝒱f

We introduce the centrifugal potential Φc with respect to the axes of the fluid core, rotating with the angular velocity vector Ωf⃗ (it will be shown that this system almost obeys the Tisserand condition). As for the centrifugal potential (5.1) expressed in the TRF, ∇Φc = −Ωf ∧ (Ωf ∧ r) .

(7.8)

Let Φgc = Φg + Φc be the sum of the gravitational and centrifugal potential. Then considering fluid core and mantle rotating with the uniform angular velocity vector ω⃗ (0) (ω⃗ f = 0), that is, the state of equilibrium indexed by (0) , we have ⃗ (0) = 0⃗ , ⃗ (0) + ρ(0) ∇Φ − ∇P gc

(7.9)

where the only forces at stake are self-gravitation, invariable centrifugal force and pressure gradient. In the non-equilibrium state, the pressure, the density and the total

116 | 7 Influence of the fluid core potential read P = P (0) + P (1) , (7.10)

ρ = ρ(0) + ρ(1) , Φgc =

Φ(0) gc

+

Φ(1) gc

,

where the terms indexed by (1) can be considered as perturbations brought about by tidal forces, angular velocity changes, and so on. Then the volume force (7.6) can be expressed by ⃗ (0) + P (1) ) + (ρ(0) + ρ(1) )∇(Φ ⃗ (0) + Φ(1) ) + ρt ⃗ − ρ∇Φ ⃗ c. f ⃗ = −∇(P gc gc

(7.11)

Neglecting the second order terms with respect to the perturbations, and accounting for (7.9) the former expression reduces to ⃗ (1) + ρ(0) ∇Φ ⃗ (1) + ρ(1) ∇Φ ⃗ (0) + ρt ⃗ − ρ∇Φ ⃗ c. f ⃗ = −∇P gc gc

(7.12)

⃗ (1) + ρ(0) ∇Φ ⃗ (1) + ρ(1) ∇Φ ⃗ (0) , the torque exerted on the core (7.7) Let us define f ⃗(1) = −∇P gc gc reads ⃗ c + ρt ⃗ ) dτ . L⃗ f = ∫ r ⃗ ∧ (f ⃗(1) − ρ∇Φ

(7.13)

𝒱f

Order of magnitude The moment of inertia of the core Af is approximatively the one of a homogeneous sphere of radius af : 2 8π 5 Af = Mf a2f = ρa . 5 15 f

(7.14)

The contribution of the centrifugal forces on the torque (7.13) is ⃗ c dτ = O( 4π ρ a3 a2 Ω2 ) = O(5/2Af Ω2 ) . ∫ −r ⃗ ∧ ρ∇Φ f f 3

(7.15)

𝒱f

In Appendix F.2 the contribution of f ⃗(1) to the torque is shown to have the order of magnitude 4πa5f ρef2 Ωωf = ef2 15/2Af Ωωf , and is thus negligible with respect to the centrifugal contribution (7.15). It is also negligible in (7.5) with respect to ω⃗ ∧ H⃗ f = O(ΩAf Ωf ) =

O(Af Ω2 ). So, the angular momentum balance reduces to dH f dt

+ ω ∧ H f = ∫ ρr ∧ (−∇Φc + t) dτ . 𝒱f

(7.16)

7.2 Liouville equation for a biaxial fluid core | 117

The fluid core angular momentum reads H f = ∫ ρr ∧ (Ωf ∧ r + vef ) dτ .

(7.17)

𝒱f

The condition of Poincaré flow ∇⃗ ∧ v⃗fe = 0⃗ implies that v⃗fe derives from a potential ϕ = O(ef ωf r 2 ). So, neglecting the gradient of the density, the contribution of v⃗fe to the fluid core angular momentum is ⃗ dτ = ∫ r ⃗ ∧ ∇(ρϕ) ⃗ H⃗ fe = ∫ ρr ⃗ ∧ ∇ϕ dτ = − ∫ ρϕ r ⃗ ∧ n̂ dS = O(4πa5f ρef2 ωf ) , 𝒱f

𝒱f

(7.18)

𝒮f

where n̂ is the outer normal to the core–mantle boundary 𝒮f (r ⃗ ∧ n̂ = O(ef r)). So, this relative angular momentum is negligible, and the system of axes rotating with angular velocity Ω⃗ f approximates a Tisserand frame for the fluid core; the TRF is a Tisserand axis for the mantle (see Section 4.3.1). Moreover, considering Eq. (7.8) in the angular momentum balances (7.16), we obtain dH f dt

= −ω ∧ ∫ ρr ∧ (Ωf ∧ r) dτ + ∫ ρr ∧ (Ωf ∧ (Ωf ∧ r) + t) dτ . 𝒱f

(7.19)

𝒱f

Noting that −ω ∧ (r ∧ (Ωf ∧ r)) = r 2 Ωf ∧ ω + (Ωf ⋅ r)ω ∧ r

= r 2 Ωf ∧ (Ωf − ωf ) + (Ωf ⋅ r)(Ωf − ωf ) ∧ r = −r 2 Ωf ∧ ωf + (Ωf ⋅ r)(Ωf − ωf ) ∧ r

and that r ∧ (Ωf ∧ (Ωf ∧ r)) = (Ωf ⋅ r)r ∧ Ωf , the integrand of (7.19), casting aside the non-conservative forces, is proportional to −r 2 Ωf ∧ ωf + (Ωf ⋅ r)(Ωf − ωf ) ∧ r + (Ωf ⋅ r)r ∧ Ωf = −r 2 Ωf ∧ ωf + (Ωf ⋅ r)(−ωf ) ∧ r = ωf ∧ (r ∧ (Ωf ∧ r)) . Finally we obtain dH f dt

= ωf ∧ ∫ ρr ∧ (Ωf ∧ r) dτ + ∫ ρr ∧ t dτ 𝒱f

= ωf ∧ H f + T ,

𝒱f

(7.20)

118 | 7 Influence of the fluid core where T⃗ is the torque exerted on the mantle by the non-conservative force. This is the equation found by Sasao et al. (1980) (Eq. (31)) [184], which the above demonstration is inspired by [60]. The terrestrial components of the fluid core angular momentum are

H f = I f Ωf = (

f Af + c11 f c21

f c31

f c12

f c13

f Af + c22

Ω(m1 + mf1 )

f c23

f c32

f Cf + c33

)(

Ω(m2 + mf2 )

Ω(1 + m3 + mf3 )

(7.21)

) .

After neglecting the second order terms O(cij mk ), we obtain

Hf = Ω (

f Af (m1 + mf1 ) + c13

Cf mf2

f Af (m2 + mf2 ) + c23

f Cf (m3 + mf3 ) + Cf + c33

),

ωf ∧ H f ≈ Ω2 ( −Cf mf1 ) ,

(7.22)

0

so that (7.20) yields the set of linear Liouville equations for the core: ΩAf (ṁ + ṁ f ) + Ωċf = −iΩ2 Cf mf + T , ΩCf (ṁ 3 +

ṁ f3 )

+

f Ωċ33

= T3 ,

(7.23) (7.24)

f f where mf = mf1 + i mf2 , cf = c13 + i c23 , and T = T1 + i T2 . The equatorial part reads

ΩAf (ṁ + ṁ f ) + iΩ2 (1 + ef )Af mf + Ωċf = T .

(7.25)

Noticeable is the absence of the external gravitational torque. Substituting in (7.5) dH f /dt with Eq. (7.20), the total torque acting on the core L⃗ f is Lf = Ḣ f + ω ∧ Hf = (ω + ωf ) ∧ H f + T .

(7.26)

Then this torque can be rearranged with (7.22): f Af (m1 + mf1 ) + c13 Cf mf2 m1 2 f f Lf = Ω ( m2 ) ∧ Ω ( ) + Ω ( −Cf mf ) + T . (7.27) Af (m2 + m2 ) + c23 1 f f 1 + m3 0 Cf (m3 + m3 ) + Cf + c33

Getting rid of second order terms, the equatorial component is Lf = iΩ2 [cf − (Cf − Af )(m + mf )] + T .

(7.28)

7.3 Polar motion differential equation for a biaxial mantle

| 119

7.3 Polar motion differential equation for a biaxial mantle 7.3.1 General case Let us consider the mechanical system composed of the mantle, of the lithosphere and of the surface fluids. Being mostly an extension of the mantle, it will be called extended mantle and any quantity relative to it will be indexed by the letter “m”. Applying the Liouville equation (4.33) to that part of the Earth, we obtain (m) − iΩ2 (Cm − Am )m + ΩAm ṁ = −iΩ2 c − iΩh − Ωċ − ḣ + Lext m − Lf

(7.29)

where the quantities c and h pertain to the extended mantle, Lext m is the tidal torque on this system, and L(m) is the equatorial torque impressed by the mantle on the core. f In the former section the full torque acting on the core Lf was evaluated, and it was shown that the contribution of the gravitational perturbation, including tides, could be neglected, reducing Lf to the interaction torque L(m) with Lf given by (7.28). So, with f

A = Am + Af , and C = Cm + Cf , (7.29) is rewritten

− iΩ2 (C − A)m + ΩAm ṁ − iΩ2 ef Af mf = −(iΩ2 cf + iΩ2 c + iΩh + Ωċ + h)̇ + Lext m − T , (7.30) that is, Lext − T i ḣ ċ + )+i m . ( (C − A)Ω eA C − A (C − A)Ω Ω C − A (C − A)Ω (C − A)Ω2 (7.31) Like a quasi-elastic Earth model covered by the oceans of Chapter 6, Eq. (6.39), the rotational excitation function of the extended mantle, is expressed by m+i

Am

ṁ +

ef Af

mf =

χ (r) =

c(r)

cf + c

C−A

=

+

h

−

k̃ 󸀠 k̃ k̃2 m + o m + o m∗ = Um + U 󸀠 m∗ , ks ks ks

(7.32)

with U = (k̃2 + k̃o )/ks and U 󸀠 = k̃o󸀠 /ks . Moving rotational contribution c(r) = (C − A)(Um + U 󸀠 m∗ ) to the left hand side of (7.31), the effective equation driving the polar motion becomes ef Af A U󸀠 i ( m + U)ṁ − U 󸀠 m∗ + i ṁ ∗ + mf Ω C−A Ω eA cf Lext − T i = + χ − χ̇ + i m , Ω C−A (C − A)Ω2

(1 − U)m +

where χ =

c C−A

+

h (C−A)Ω

(7.33)

is the equatorial angular momentum function of the extended

mantle, in the sense defined before (in particular it contains a hydro-atmospheric excitation).

120 | 7 Influence of the fluid core 7.3.2 Polar motion periods longer than 2 days As will be shown later in this chapter, it seems reasonable to assume that the equatorial rotation of the core competes with the one of the mantle only for retrograde quasidiurnal periods. Over common periods of the polar motion, larger than a few days, the core does not present significant local or rotational motion with respect to the mantle, so that the core figure axis remains stable with respect to the mantle. Consequently mf ≈ 0, cf is constant, and the friction torque included in T vanishes, reducing this one to a pure electromagnetic torque. With mf canceled, cf playing no role in (7.33), this equation reduces to (1 − U)m +

Lext − T A i U󸀠 i . ( m + U)ṁ − U 󸀠 m∗ + i ṁ ∗ = χ − i χ̇ + i m Ω C−A Ω Ω (C − A)Ω2

(7.34)

Neglecting the asymmetric part, and not considering torque effect but only mass redistribution, this equation takes the common form m+

1 i i ̇ , ṁ = (χ − χ) σ̃ c 1−U Ω

(7.35)

where σ̃ c is the Chandler angular frequency given by σ̃ c = Ω

1−U

Am C−A

+U

= σe

A A (1 − U)(1 − eU + O(e2 )) . Am Am

(7.36)

For a near-elastic Earth including the oceans the Chandler angular frequency was found to be σc ≈ σe (1 − R(U)) according to (6.45), corresponding to a period of 480 days. We see that core passivity and reduction of the mechanical system to the enlarged mantle increases this angular frequency by the ratio A/Am = 1.128. In other words, the Chandler period Tc is brought back from 480 days to 433.6 days, fitting the observed value. As the involved inertia moments are really determined with a relative accuracy smaller than 0.1 %, the main source of uncertainty comes from U, more precisely from the ocean pole tide coefficients A1 and B2 according to (6.39b) and (6.36). Then, by taking the errors ΔA1 ∼ ΔA2 ∼ 0.1, we have R(U) ∼ 0.378 ± 0.005. So, the uncertainty on U is ten times larger than the term eU 2 A/Am ≈ 5 10−4 , which can be dropped in (7.36). The numerical application shows that the theoretical value of Tc lies in the interval (430.1,437.7). Inserting the effective Love number k̃ = ks U = k̃2 + k̃o in accordance with (6.43), the Chandler angular frequency reads also k̃ A σ̃ c ≈ σe (1 − ) , (7.37a) Am ks where

k̃ = 0.355 − i 0.29/Q ,

(7.37b)

according to the value proposed in Table 6.2 and resulting from the quasi-hydrostatic ocean response. Taking Q ≈ 100, we recover the empirical value 0.353 − i 0.003 determined in [194]. The dissipation is also determined by a phase lag ε such as k̃ = k(1+iε).

7.4 Coupled core-mantle system in the retrograde diurnal band | 121

As for a quasi-elastic Earth covered by the oceans of the former chapter, this phase lag is deduced from the Chandler quality factor Q like (6.46): ε = (1 −

ks 1 −0.82 ) ≈ . k 2Q Q

(7.38)

Similarly to Section 5.7, the differential equation (7.35) for instantaneous rotation pole m is converted into complex pole coordinates p = x − iy of the CIP: p+

i e e ṗ = χma + χmo , σ̃ c

(7.39a)

with the effective angular momentum functions e χma =

1 + k2󸀠 χma , ̃ 1 − k/k s

e χmo =

1 χmo . ̃ 1 − k/k s

(7.39b)

Here the loading deformation has been taken into account through the Love number k2󸀠 . The complex part of k̃ of about 0.003 shifts the angular momentum function by ∼ 0.3°, or delays it by less than one day at seasonal period, which is negligible. For this reason, the complex part of k̃ is ignored in the final definition of the effective angular momentum functions: e χma =

1 + k2󸀠 χ = 1.112χma , 1 − k/ks ma

e χmo =

1 χ = 1.606χma . 1 − k/ks mo

(7.39c)

7.4 Coupled core-mantle system in the retrograde diurnal band 7.4.1 Differential equation system for core and extended mantle For periods close to 1 day, the core reacts to the polar oscillation of the mantle and brings about an additional resonance. This is what we shall show in the way of Sasao and Wahr (1981) [185]. The asymmetric effects are still ignored. The core–mantle coupling is treated by grouping the Liouville equations (7.30) and (7.25) for the extended mantle and the core, respectively: −iΩ2 (C − A)m + ΩAm ṁ − iΩ2 ef Af mf + iΩ2 cf + iΩ2 c + iΩh + Ωċ + ḣ = Lext m −T , ΩAf (ṁ + ṁ f ) + iΩ2 (1 + ef )Af mf + Ωċf = T .

(7.40)

Variations of c and cf both result from deformations induced by the pole tide (indexed by “r”) and by surface loading (indexed by “l”): c = c(r) + c(l) , cf = cf(r) + cf(l) .

(7.41)

Considering a mono-layered quasi-elastic Earth, the inertia moment produced by the solid pole tide is expressed by (5.33), and adding the contribution of the hydrostatic

122 | 7 Influence of the fluid core Table 7.1: Compliances from PREM & 1066 A model (Mathews et al. 1991, Table 1 [144]) and estimated from VLBI observations (Table 2 of Mathews et al. 2002 [145]). parameter a

R(κ) R(κ)b ξ γ β a b

PREM

1066 A

Observations

1.039 10 1.212 10−3 2.222 10−4 1.965 10−3 6.16 10−4

1.046 10 1.220 10−3 2.250 10−4 1.980 10−3 6.228 10−4

1.0340(92) 10−3 1.230(3) 10−3 − 1.9662(14) 10−3 −

−3

−3

Values corresponding to k = 0.297(2) and to the resonance frequency σcd . Values corresponding k = 0.354(1) and to Chandler wobble frequency σc .

ocean pole tide it takes the form (7.3.1). Anyway, it is a linear function of the rotation pole coordinates m1 and m2 at the same instant. Sasao et al. (1980) [184] have shown that this linearity can be extended to the rotational perturbation brought about by the fluid core, and to the fluid core itself. According to the formalism introduced in [144, 145, 60] (r) ctot = A(κm + ξmf ) ,

cf(r) = Af (γm + βmf ) ,

(7.42)

(r) where ctot stands for the whole Earth system; the so-called compliances κ, ξ , γ, β describing the rotational response, possibly include small complex part accounting for dissipative processes. Their real parts are given in Table 7.1. The coefficient κ determines the rotational change of the extended mantle, and thus contains the contribution of the oceans. For now, the compliances are constant throughout the retrograde diurnal band. Frequency dependence will be addressed in Section 7.6. The aforemeñ (C − A)m, is related to the comtioned effective Love number k,̃ defined by c(r) = k/k s pliance κ according to k̃ κ=e . (7.43) ks

Now, we exclude the core and focus on the extended mantle system, for which the rotation response yields (r) c(r) = ctot − cf(r) = A(κm + ξmf ) − Af (γm + βmf )

= A(κ − ξ )m + Af (γ − β)mf ,

(7.44)

where we have used the property Aξ = Af γ [184]. To these rotational changes we have to add the effect of the variable loading of the hydro-atmospheric layer. According to [185], the induced moment of inertia changes on the global Earth and on the fluid core are (l) ctot = −A(τ − χ) Φ(l) ,

cf(l) = Af η Φ(l) ,

(7.45)

7.4 Coupled core-mantle system in the retrograde diurnal band | 123

respectively. Here τ = e/ks , χ = 1.073 10−3 , η = 1.940 10−3 , and Φ(l) represents the tesseral potential of the load, linked with the equatorial moment of inertia of the load c(l) by Φ(l) = −

k 3G (l) c = − s c(l) . Ω2 R5e eA

(7.46)

In turn ks χ)c(l) = (1 + k2󸀠 )c(l) , e k = −Af η s c(l) = kf󸀠 c(l) , eA

(l) ctot = (1 −

cf(l)

(7.47)

with k2󸀠 = −

ks χ = −0.307 e

and

kf󸀠 = −

ηks Af eA

= −0.063 .

(7.48)

The loading effect on the extended mantle system alone is (l) c(l) = ctot − cf(l) = (1 + k2󸀠 − kf󸀠 )c(l) = (1 + k 󸀠 )c(l) ,

(7.49)

with k 󸀠 = k2󸀠 − kf󸀠 = −0.244. Finally, cumulating the rotational and loading responses, the total variations of the equatorial moment of inertia are c = A(κ − ξ )m + Af (γ − β)mf + (1 + k 󸀠 )c(l) , cf = Af (γm + βmf ) + kf󸀠 c(l) .

(7.50)

Including these expressions in the system (7.40), this becomes (e − κ)m +

Af i Am i Af ( + κ − ξ )ṁ + (ef − γ)mf + (γ − β)ṁ f = eΨm , Ω A Ω A A Af i Af i Af (1 + γ)ṁ − (1 + ef ) mf + (1 + β)ṁ f = eΨf , Ω A Ω A A

(7.51)

where Ψm and Ψf are the excitation function pertaining to the mantle and the core, respectively, given by Ψm = (1 + k2󸀠 )χma + χmo −

Lext − T i ̇ ]+i m [(1 + k 󸀠 )χ̇ ma + χmo , Ω (C − A)Ω2

i T ̇ +i Ψf = − kf󸀠 χma , Ω (C − A)Ω2

(7.52)

and expressed as a function of the matter term χma and the motion term χmo defined by (4.31), that is, χma =

c(l)

C−A

,

χmo =

h

Ω(C − A)

.

124 | 7 Influence of the fluid core Let us recall that the matter term does not contain any pole tide effect, formulated separately by (7.44), and results from the loading anomaly over the mantle. Whereas the external tidal forces are included in the excitation, their full treatment for a real Earth requires additional developments that are out of scope of this book, focusing on geophysical causes. First, the lunisolar tide-generating potential (chiefly of degree 2) causes a retrograde quasi-diurnal torque on the Earth mass distribution, in turn the precession–nutation of the CIP in the celestial system. Second, like the pole tide potential, it is accompanied by global deformations, and in turn by additional off-diagonal inertia moments c(t) and cf(t) proportional to this potential according to the rheology of each of this layer. From now on, the external tidal torque is not considered. The system (7.51) has the form E1 m + E2 ṁ + E3 mf + E4 ṁ f = eΨm , F2 ṁ + F3 mf + F4 ṁ f = eΨf ,

(7.53a)

with E1 = e − κ ,

i Am ( + κ − ξ) , Ω A Af (ef − γ) , E3 = A i Af E4 = (γ − β) , Ω A E2 =

i Af (1 + γ) , Ω A Af F3 = − (1 + ef ) , A i Af F4 = (1 + β) . Ω A F2 =

(7.53b)

7.4.2 Free core nutation We shall solve this system in the frequency domain. By Fourier transforming, we obtain [

E1 + iσE2 iσF2

E3 + iσE4 m(σ) eΨm (σ) ][ ]=[ ] . F3 + iσF4 mf (σ) eΨf (σ)

(7.54)

Inverting this system, the solution reads [

m(σ) F + iσF4 1 ]= [ 3 mf (σ) −iσF2 Det

−(E3 + iσE4 ) eΨm (σ) ][ ] , E1 + iσE2 eΨf (σ)

(7.55)

where Det is the determinant of the system, Det = (F2 E4 − E2 F4 )σ 2 + i(E2 F3 + E1 F4 − F2 E3 )σ + E1 F3 .

(7.56)

This degree 2 polynomial in σ can be factorized: Det = (F2 E4 − E2 F4 )(σ − σ+ )(σ − σ− ) ,

(7.57)

7.4 Coupled core-mantle system in the retrograde diurnal band | 125

where σ+ and σ− are the roots, σ± =

−i(E2 F3 + E1 F4 − E3 F2 ) ± √Δ . 2(E4 F2 − E2 F4 )

(7.58)

The angular frequencies σ+ and σ− appear as resonant frequencies in the solution m(σ) and mf (σ). Neglecting the second order term with respect to the Earth’s ellipticity, we have √Δ = √−(E2 F3 + E1 F4 − E3 F2 )2 − 4(E4 F2 − E2 F4 )E1 F3 ≈ i(E2 F3 − E1 F4 − E3 F2 ) ,

(7.59)

that is, σ+ =

−iE1 F4 , E4 F2 − E2 F4

σ− = i

E3 F2 − E2 F3 . E4 F2 − E2 F4

(7.60)

Keeping only first order term of the ellipticity, the roots are expressed by σ+ = Ω

A (e − κ) , Am

σ− = −Ω[1 +

A (e − β)] . Am f

(7.61)

Expressed in terms of the effective Love number k thanks to (7.43), σ+ can be assimilated to the polar motion resonance frequency (7.37a), which had been obtained by assuming the passivity of the core: σ+ = σ̃ cd = Ω

A A k̃ (e − κ) = σe (1 − ) . Am Am ks

(7.62)

Actually, the Love number k in the diurnal band cannot be confused with the one reigning at multi-day scales.3 One reason for this has already been put forward in Section 6.6: the dynamical oceanic processes introduces a strong phase shift of the oceanic part of the Love number, which varies from 0.048 + i 0.0002 at seasonal period to −0.02+i0.04 in the diurnal retrograde band. So, σ+ cannot be confused with the Chandler frequency, but should be treated as an analog frequency that will be symbolized by σ̃ cd (“d” for “diurnal”), and which only pertains to the diurnal frequency band. This question will be investigated in more detail in Section 7.6. Moreover, the fluid core brings about a new resonance, with the retrograde diurnal angular frequency σ− = σ̃ f = −Ω[1 +

1 A ]. (e − β)] ≈ −Ω[1 + Am f 436

(7.63)

In the celestial frame the angular frequency becomes σ̃ f󸀠 = σ̃ f + Ω ,

(7.64)

corresponding to a period of about 435 mean solar days. ̃ 3 In the resonance frequency, the compliance κ = ek/k s quantifies how the pole tide compensates for the Earth’s ellipticity e caused by its permanent rotational response.

126 | 7 Influence of the fluid core 7.4.3 Solution in frequency domain From (7.57) we have 1 1 1 1 1 = ). ( − Det E4 F2 − E2 F4 σ̃ cd − σ̃ f σ − σ̃ cd σ − σ̃ f

(7.65)

As σ̃ cd − σ̃ f = √Δ/(E4 F2 − E2 F4 ) the result is 1 1 1 1 ). = − ( Det √Δ σ − σ̃ cd σ − σ̃ f

(7.66)

Taking into account Eqs. (7.53b), we obtain, after elimination of the terms in O(e2 ), √Δ =

Af Am 2

AΩ

(7.67)

.

Then, expressing (7.66) accordingly, and using it in (7.55), we obtain the frequency domain solutions 2

1 1 eA Ω ( − )[(F3 + iσF4 )Ψm (σ) − (E3 + iσE4 )Ψf (σ)] , m(σ) = Af Am σ − σ̃ cd σ − σ̃ f

(7.68)

2

1 1 eA Ω ( )[−iσF2 Ψm (σ) + (E1 + iσE2 )Ψf (σ)] . − mf (σ) = Af Am σ − σ̃ cd σ − σ̃ f After development in simple elements,4 we obtain m(σ) = mf (σ) =

2 d iE4 σ̃ cd + E3 iE4 σ̃ f + E3 eA Ω iF4 σ̃ c + F3 iF4 σ̃ f + F3 [( )Ψ (σ) − ( )Ψf (σ)] , − − m Af Am σ − σ̃ f σ − σ̃ f σ − σ̃ cd σ − σ̃ cd 2

E1 + iE2 σ̃ cd E1 + iE2 σ̃ f −iF2 iF2 eA Ω [( )Ψ (σ) + ( )Ψf (σ)] , + − m Af Am σ − σ̃ cd σ − σ̃ f σ − σ̃ f σ − σ̃ cd

which gives, after substitution of the terms Ei and Fi by their expressions (7.53b)

(7.69)

A ef − γ ef − β eAΩ 1 + ef + Am (e − κ) Af ef − β m(σ) = − [( )Ψm (σ) + ( )Ψf (σ)] , + − d d ̃ ̃ ̃ Am A σ − σ σ − σ̃ f σ − σc σ − σc m f

mf (σ) =

A

d

σ̃ f σ̃ d eA eA Am (e − κ)Ω − σ̃ c ( c d − )Ψm (σ) + ( − Am σ − σ̃ c σ − σ̃ f Af σ − σ̃ cd

1 4 By using the rule (a + bσ)( σ−σ − 1

1 ) σ−σ2

=

a+bσ1 σ−σ1

−

a+bσ2 . σ−σ2

A (e Am

− κ)Ω − σ̃ f

σ − σ̃ f

)Ψf (σ) . (7.70)

7.4 Coupled core-mantle system in the retrograde diurnal band | 127

According to (7.62) the former system simplifies to m(σ) = − mf (σ) =

d Af (ef − β)Ω (ef − γ)Ω (ef − β)Ω eA (1 + ef )Ω + σ̃ c + − [( )Ψm (σ) + ( )Ψf (σ)] , d ̃ ̃ Am A σ − σ σ − σ̃ f σ − σc σ − σ̃ cd m f

d σ̃ f σ̃ d eA eA σ̃ c − σ̃ f ( c d − )Ψm (σ) − ( )Ψf (σ) . Am σ − σ̃ c σ − σ̃ f Af σ − σ̃ f

(7.71) If we consider low frequencies, much lower than Ω ∼ |σ̃ f |, then we have the approximation m(σ ≪ Ω) ≈ −

(ef − γ)Ω Ω eA [ Ψ (σ) + Ψf (σ)] , Am σ − σ̃ c m σ − σ̃ c

eA σ eA mf (σ ≪ Ω) ≈ ( )Ψm (σ) − Ψ (σ) , Am σ − σ̃ c Af f

(7.72)

where the resonance angular frequency σ̃ cd , only found in the retrograde diurnal band, has been replaced by the common value σ̃ c . Considering the order of magnitude, it is easy to see that mf (σ) ∼ σ/Ω m(σ) for periods much larger than 1 day: the larger the period is, the smaller is mf and the more the core follows the equatorial wobble of the extended mantle. This somewhat justifies the assumption made in Section 7.3.2 for common polar motion.

7.4.4 Free core nutation The retrograde diurnal frequency σf , given by the real part of (7.63), entails a second normal mode of the polar motion. According to (7.64) this yields a retrograde nutation at the frequency σf󸀠 ∼ −Ω/430 in a celestial reference frame. From (3.30), the corresponding oscillation P(σf󸀠 ) of the CIP admits the amplitude 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󸀠 󵄨 󸀠 󵄨󵄨P(σf )󵄨󵄨󵄨 = 󵄨󵄨󵄨Ω/σ m(σf )󵄨󵄨󵄨 ∼ 430󵄨󵄨󵄨m(σf )󵄨󵄨󵄨 .

(7.73)

Though it pertains to the mantle, this nutation is termed Free Core Nutation, FCN. The true FCN is the one given by mf (σf ). As the core is viscous, the compliance β has a small complex part, of which the value is difficult to model. So the mantle diurnal nutation at σf is a damped oscillation presenting the quality factor Qf ≈ 20000 according to [145]. The FCN very likely corresponds to the 430 day pseudo-harmonic oscillation up to 0.5 mas observed in the celestial pole offsets—obtained by VLBI from 1985 [110] and bringing about corrections to the conventional IERS precession–nutation model. Its excitation is probably caused by the hydro-atmospheric transport at diurnal scale (see Chapter 13). Moreover, the lunisolar terms resonate at the frequency σf󸀠 , so that the discrepancy with a rigid Earth nutation term can reach several mas, depending on

128 | 7 Influence of the fluid core frequency (see Figure 2.2 for the cumulated discrepancy). Resonance parameters can be estimated by fitting the theoretical nutation terms to the VLBI observed lunisolar nutation, thus yielding −430.1 ≤ Tf ≤ −429.3 days and 15392 ≤ Qf ≤ 16866 [177].5 In the retrograde diurnal band, encompassing the FCN period, the frequency solution (7.71) is approximated by m(σ ∼ −Ω) ≈ − mf (σ ∼ −Ω) ≈ −

Af (ef − β)Ω (ef − β)Ω eA [(−1 + )Ψm (σ) − (ef − γ + )Ψf (σ)] , ̃ Am Am σ − σf σ − σ̃ f eA σ̃ f eA Ω Ψ (σ) − Ψ (σ) . Am σ − σ̃ f m Af σ − σ̃ f f

(7.74) So mf (σ) can exceed m(σ), especially near σf . Considering this frequency solution (7.74) at σf , after introducing the expression σ̃ f = σf (1 −

1 i i )≈− (1 − ) cpd , 2Qf 430 32000

(7.75)

and taking into account (7.64), we obtain m(σf ) ≈ ≈

Ω Af Ω eA [(1 + i2Qf (e − β))Ψm (σf ) − i2Qf (ef − β)Ψf (σf )] Am σf Am f σf 󸀠 Af σf󸀠 A σf eA [(1 − i2Qf )Ψm (σf ) + i2Qf m Ψf (σf )] , Am A σf A σf

mf (σf ) ≈ i2Qf

(7.76)

eA Ω eA Ψ (σ ) + i2Qf Ψ (σ ) . Am m f σf Af f f

For a mechanical excitation applied to the extended mantle, we have mf (σf )/m(σf ) ≈ i2Qf /(1 − i2Qf

Af σf󸀠 A σf

) ∼ −430

A ∼ −3800 . Af

(7.77)

In the celestial reference frame the complex FCN angular frequency reads σ̃ f󸀠 = σ̃ f + Ω = σf + Ω − σf

i i = σf󸀠 (1 − 󸀠 ) 2Qf 2Qf

(7.78)

with the quality factor Q󸀠f

= Qf

σf󸀠 σf

≈

Qf

430

in the interval (35, 39) ,

corresponding to the lower limit of the Chandler wobble quality factor. 5 That is the relaxation time 2Qf /σf in the interval of (13, 15) years.

(7.79)

7.5 Polar motion equation for a superficial fluid layer excitation in the retrograde diurnal band |

According to (7.77) the core instantaneous rotation pole, given by mf (σf ), oscillates in opposition of phase with an amplitude equal to |mf (σf )| ∼ 3800 |m(σf )|, that is, |mf (σf )| = 3800|σf󸀠 /Ω P(σf󸀠 )| from (7.73). Consequently, with |P(σf󸀠 )| ≤ 0.5 mas, |mf (σf )| reaches an amplitude of 4 mas. In the celestial frame, the nutation of the core figure can be deduced from 󵄨󵄨󵄨Pf (σ 󸀠 )󵄨󵄨󵄨 = 󵄨󵄨󵄨Ω/σ 󸀠 mf (σf )󵄨󵄨󵄨 = 3800 󵄨󵄨󵄨P(σ 󸀠 )󵄨󵄨󵄨 . (7.80) f 󵄨 f 󵄨 󵄨 󵄨 󵄨 󵄨 The upper value is |Pf (σf󸀠 )| ∼ 2󸀠󸀠 , amounting to the order of magnitude of the largest lunisolar nutations.

7.5 Polar motion equation for a superficial fluid layer excitation in the retrograde diurnal band Now we restrict our consideration to the excitation of a surface fluid layer, determined by the matter term χma and the motion term χmo . By contrast to the mantle, the fluid core reacts only to the matter term. 7.5.1 Effect of the matter term In virtue of (7.52) the matter excitation reads, in the frequency domain, 󸀠 Ψm ma (σ) = (1 + k2 +

σ (1 + k 󸀠 ))χma (σ) , Ω

σ 󸀠 k χ (σ) . Ω f ma These expressions are reported in (7.70): Ψfma (σ) =

mma (σ) = −

(7.81)

A eAΩ 1 + ef + Am (e − κ) Af ef − β σ {[ + ](1 + k2󸀠 + (1 + k 󸀠 )) Am Am σ − σ̃ f Ω σ − σ̃ cd

+[

ef − γ

σ − σ̃ cd

−

ef − β σ 󸀠 ] k }χ (σ) . σ − σ̃ f Ω f ma

(7.82)

But what is determined from geodetic observation is not m(t) but the pole coordinate of the CIP p(t), related to m(t) by (4.48). In the frequency domain we have Ω m (σ) . σ + Ω ma After development and decomposition in simple elements, we have pma (σ) =

pma (σ) =

kf 1 + k2󸀠 eAΩ + [− ]χma (σ) . d Am σ − σ̃ c σ − σ̃ f 󸀠

(7.83)

(7.84)

In addition to the Chandler mode and that of the free core nutation, we note a resonance at the diurnal angular frequency Ω.

129

130 | 7 Influence of the fluid core 7.5.2 Effect of the motion term In virtue of (7.52) the motion term has the Fourier transform σ+Ω χ (σ) , Ω mo Ψfmo (σ) = 0 . Ψm mo (σ) =

(7.85)

From (7.70), the corresponding perturbation on polar motion is 1 + ef + AA (e − κ) Af ef − β eA m + (σ + Ω)[ ]χ (σ) . mmo (σ) = − Am Am σ − σ̃ f mo σ − σ̃ cd

(7.86)

The corresponding oscillation of the CIP is pmo (σ) = −

Af ef − β 1 eAΩ [ ]χ (σ) . + Am σ − σ̃ cd Am σ − σ̃ f mo

(7.87)

7.5.3 Total effect Grouping the effect of the matter and motion terms, (7.84) and (7.87), respectively, we have p(σ) =

kf 1 + k2󸀠 eAΩ [− ]χma (σ) + d Am σ − σ̃ c σ − σ̃ f 󸀠

+

Af ef − β eAΩ 1 − [− ]χmo (σ) . Am σ − σ̃ cd Am σ − σ̃ f

(7.88)

Now, we inject the polar resonance frequency at multi-day scales, namely the Chandler frequency σc = eΩA/Am (1 − k/ks ) with k = 0.354. So, the former expression is rewritten p(σ) = σc [−

kf󸀠 1 + k2󸀠 1 1 ] + χma (σ) σ − σ̃ cd 1 + k2󸀠 σ − σ̃ f 1 − k

Af ef − β 1 1 − + σc [− ] d σ − σ̃ c Am σ − σ̃ f 1 −

ks

k ks

χmo (σ) .

(7.89)

This equation shows Eq. (7.39c) of the effective angular momentum function. In turn the frequency response (7.89) is similar to the common form popularized by Brzezinski [23] from the theoretical expressions of Sasao and Wahr (1981) [185], that is, p(σ) = −(

σc σc σc σc e + ama )χma (σ) − ( + amo )χ e (σ) , d d ̃ ̃ ̃ σ − σ σ − σ̃ f mo σ − σc σ − σc f

(7.90a)

7.6 Frequency dependence of the Love number and consequences | 131

with

amo

kf󸀠

= 9.1 10−2 , 1 + k2󸀠 Af = (e − β) = 2.6 10−4 . Am f

ama = −

(7.90b)

Here we have the values of k2󸀠 = −0.3075, kf󸀠 = −0.063, Af /Am = 0.128, ef = 0.002545 (see Table H.1), and β = 6.16 10−4 (see Table 7.1). The coefficients of Brzezinski or Sasao and Wahr have close values: ama = 9.2 10−2 and amo = 2.510−4 . But, when the traction of the ground by the relative motion of the fluid is included, the coefficient amo is magnified, becoming amo = 5 10−4 [185]. Anyway this estimation may be not so sound, and we shall privilege the value of amo that we obtain, keeping in mind that the amo value can present a large uncertainty. Then the resonance at the angular frequency σ̃ f is about ama /amo ∼ 290 larger for the matter term than for the motion term. Geodetic excitation After multiplication of (7.90a) by (σ − σ̃ f )(σ − σ̃ cd ), we go back to the time domain by the inverse Fourier transform: σ̃ cd + σ̃ f

σc d d e e e e [(i + σ̃ f )(χma + χmo ) + (i + σ̃ cd )(ama χma + amo χmo )] . d dt dt σ̃ c σ̃ f (7.91) The left hand side exhibits the geodetic excitation to be compared to the effective AMF in the right hand side.

p+i

σ̃ cd σ̃ f

ṗ −

p̈

σ̃ cd σ̃ f

=

7.6 Frequency dependence of the Love number and consequences 7.6.1 Convolution in time domain Until now, Love number or compliances had been assumed as independent from frequency. But the MHB nutation theory makes use of a compliance value κ consistent with the Love number k = 0.3 (see Table 7.1), diverging from the one corresponding to the Chandler frequency (0.354). This mere fact means that the Love number is frequency dependent. According to [145] this frequency dependence only pertains to κ or k,̃ the loading Love number k 󸀠 = k2󸀠 − kf󸀠 , and γ. Then the off-diagonal inertia moment of the mantle in (7.50) shows convolution products: c(σ) = A(κ(σ) − ξ )m(σ) + Af (γ(σ) − β)mf (σ) + (1 + k 󸀠 (σ))c(l) (σ) .

(7.92)

In the time domain this expression becomes c(t) = A(κ(t) − ξδ(t)) ∗ m(t) + Af (γ(t) − βδ(t)) ∗ mf (t) + (δ(t) + k 󸀠 (t)) ∗ c(l) (t) . (7.93)

132 | 7 Influence of the fluid core Actually, most of the development exposed in Sections 7.4 and 7.5 remains unchanged, for in the frequency domain all expressions keep their form. But the ordinary differential equation (7.91), obtained by the inverse Fourier transform, is only valid in bounded frequency bands for which κ and k2󸀠 or k 󸀠 = k2󸀠 − kf󸀠 can be considered as constant. On the other hand γ, and subsequently any frequency dependence, does not impact the transfer function because it is eliminated by limiting the development to O(e) or O(ef ). 7.6.2 Theoretical frequency dependence of polar motion resonance The mere fact that κ or k̃ = k̃2 + k̃o depends on the ocean response through ko implies its frequency dependence (as seen in Section 6.6), for ko switches from the quasi-real value 0.048 at seasonal period to about k̃od ∼ −0.03 + i 0.04 at daily time scale. Then the strong complex part reflects the dynamical behavior of the oceans. In the diurnal band, the parameter k̃od also presents a slow frequency dependence, modeled by (6.60). Moreover, in the diurnal band, close to the free core nutation diurnal period, the solid Earth tide departs from the one of a quasi-elastic Earth. Indeed, the induced tilt of the core with respect to the mantle modifies the Earth mass distribution and in turn the surface gravity. Modeled in the 1960s, this phenomenon has been confirmed from the 1990s through the supra-conducting gravimeter measurements [43]. Other perturbations, of much lesser amplitude (100 times less), occur because of the Free Inner Core Nutation (FICN) mode at σi ∼ 1.0017 cpd in the TRF, and because of the polar motion resonance appearing at a period of about 380 days, as justified below. From IERS Conventions 2010 [168], Table 6.4, Eqs. (6.9) and (6.10), the perturbation of the diurnal tide on the geopotential can be described through the “diurnal” body Love number k̃2d given by (D.7). Replacing in (7.62) the pure anelastic value of k2 by its resonant version (D.7) and k̃o by k̃od (σ), we conclude that the eigenfrequency, to which nutation or diurnal retrograde polar motion resonates, is σcd (σ) = σe

d d A k̃2 (σ) + k̃o (σ) . Am ks

(7.94)

The associated period and quality factor, namely Tcd (σ) =

2π , Re(σcd )

Qdc (σ) =

Re(σcd )

2 Im(σcd )

,

(7.95)

are displayed in Figure 7.1 over the frequency band [−1.15 cpd, −0.85 cpd] (denoted band I) fully covering the VLBI observed nutation band, of which the periods in a nonrotating frame stretch from 7 days to 18.6 years at least. Outside a narrow band around the free core nutation frequency, the effect to the FCN fades, k̃2 (σ) tends to the constant value 0.3, and the resonance parameters join

7.6 Frequency dependence of the Love number and consequences | 133

Figure 7.1: Resonance parameters of the polar motion in the diurnal retrograde band for an anelastic Earth covered by oceans and containing a fluid core. Green crosses specify the values obtained from nutation inversion over the restricted frequency bands II1 (ν1 , OO1 ), III2 (ϕ1 , ψ1 ), III3 (K1 ), III4 (S1 , P1 , O1 , Q1 ): the horizontal bar extension gives the frequency band, and the vertical bar the uncertainty of the estimated value.

the values of an anelastic Earth covered by oceans. Ignoring the small frequency variation of the oceanic Love number by taking the mean value k̃o = −0.027 + i 0.042, the resonance parameters are Tcd ∼ 380 days and Qdc ∼ −10. 7.6.3 Confirmation from nutation analysis This theoretical model can be confronted to lunisolar nutation components, resulting from the tesseral lunisolar potential. For a rigid Earth model (flagged by “R”), a given component of the tesseral tidal potential at diurnal frequency σ causes a retrograde quasi-diurnal polar motion pR (σ), equivalently a nutation term of the CIP P R (σ 󸀠 ) at low frequency σ 󸀠 = σ + Ω (period longer than 2 days) as explained in Section 3.3.5. For an axially symmetric two-layered non-rigid Earth model this nutation is modified according to Nf σ̃ d /e σ −σ N0 (1 + (1 + σ)(− c d + ))pR (σ) . (7.96) p(σ) = e Ω + σe σ − σ̃ c σ − σ̃ f

More refined expressions have been developed for a three layered Earth model, introducing two supplementary resonances (see next section). In this expression, the

134 | 7 Influence of the fluid core rigid Earth term pR (σ) reflects the tidal excitation, whereas p(σ) yields the modified retrograde diurnal oscillation for a non-rigid two-layered Earth. Until the 1990s, the non-rigid Earth nutation theories considered an ocean-less Earth, corresponding to k̃o = 0 and thus the theoretical value Tcd ≈ 401 days [185, 144]. From 2000, the time interval covered by VLBI observations has become large enough to fit the resonance parameters of the transfer function, in particular σ̃ cd from a set of observed nutation terms and corresponding theoretical values for a rigid Earth, given for instance by the Kinoshita and Souchay model [120]. This yielded a frequency of 0.96 cpy or a period of 383.5 days and a quality factor of −9.5 [145]. A recent adjustment was based on a set of 42 dominant circular nutation terms covering the whole band I in the TRF (retrograde and prograde terms from 7 days to 18.6 years in the CRF), and determined through 36 years of VLBI observation (1984–2018) [164]. Taking the upper bounds of the estimates, it gives Tcd = 382 ± 2 d and Qdc = −11 ± 1 (see Table 7.2). Table 7.2: Parameters of the polar motion resonance as determined from observed nutation inversion over different frequency bands specified in the TRF [164]. Band

frequency (cpd) in the TRF

I II1 II2 III1 III2 III3 III4

(−Ω − 1/6.86 ≤ σ ≤ −Ω + 1/6.86) (−Ω − 1/6.86 ≤ σ ≤ −Ω − 1/386) (−Ω − 1/1095.18 ≤ σ ≤ −Ω + 1/6.86) (−Ω − 1/6.86 ≤ σ ≤ −Ω − 1/31.81) (−Ω − 1/121.75 ≤ σ ≤ −Ω − 1/386) (−Ω − 1/1095.18 ≤ σ ≤ −Ω + 1/1095.18) (−Ω + 1/386 ≤ σ ≤ −Ω + 1/6.86)

Tcd

Qcd

382.0 ± 1.3 418.5 ± 7.2 381.8 ± 1.2 415.1 ± 3.3 486.8 ± 58.4 381.7 ± 7.6 381.8 ± 1.3

−10.4 ± 0.5 −8.24 ± 1.7 −10.4 ± 0.5 −7.7 ± 0.7 13.4 ± 30.7 −10.2 ± 2.9 −10.4 ± 0.5

7.6.4 Influence of the frequency dependent rheology on the geophysical transfer function The resonance produced by the free core nutation strongly affects the tidal lines surrounding K1 —pertaining to the precession and the long period nutation terms (6798 d, 1095 d, 3399 d) in the CRF (band III3 )—and tidal lines close to ψ1 —retrograde nutation terms in 365.25 d (ψ1 ), 182.6 d (ϕ1 ), 386 d in the CRF (band III2 ). Can we detect this modification? The estimated value of Tcd from the nutation terms in the band III2 (Tcd = 486.8 ± 58.4 d) confirms the enhancement of the resonance period around Ψ1 (theoretical value of 470 d at Ψ1 ). Also, the nutation inversion in the band III3 supports the theoretical decrease around K1 (modeled value Tcd ∼ 360 d versus estimated value Tcd = 382 d). Meanwhile, the nutation inversion allows one to get the modeled quality factor

7.6 Frequency dependence of the Love number and consequences | 135

of the band K1 (observed −10 versus modeled −8.5). For the band Ψ1 the interval of the estimated value (QPM = 13 ± 31) is too loose for confirming the modeled value (−5), but it can include the modeled quality factor at the side frequency σFCN = 1.005 cpd (∼ 0). Outside the narrow band of the free core nutation frequency, far from K1 and ψ1 , the resonance parameters rejoin the curves obtained for an anelastic Earth covered by oceans. At the right part of the spectrum corresponding to band III4 , covering tidal lines S1 and O1 , the estimates (Tcd = 381.8 ± 1.3 d, Qdc = −10.4 ± 0.5) slightly differ from the modeled parameters. For the opposite band (II1 ), the estimated period increased up to 418 days, as expected from the resonance. Whereas Tcd strikingly varies in the interval [−1.006, −1.004] cpd presenting values between 1000 days and −10 days, even going through retrograde diurnal band at σFCN , this does not have any striking resonant effect. Indeed at σf the quality factor is below 0.5, and strongly mitigating the resonance at such a peculiar eigenfrequency. Thus, the polar motion is determined by an equation similar to (7.88), where k2󸀠 is substituted with k2󸀠 d (σ) given by (D.8), and σ̃ cd with σ̃ cd (σ): p(σ) =

kf 1 + k2󸀠 (σ) eAΩ [− ]χma (σ) + d Am σ − σ̃ c (σ) σ − σ̃ f 󸀠

+

Af ef − β eAΩ 1 − [− ]χmo (σ) . Am σ − σ̃ cd (σ) Am σ − σ̃ f

(7.97)

In a first approximation the Love number can be fixed in the retrograde diurnal band by ignoring their resonant part, and taking k̃2d = 0.29954 − i 0.1412 10−2 according to (D.7), k̃od = −0.027 + i 0.042 (see above), k2󸀠 d = −0.3075 according to (D.8). Then all coefficients of the transfer function become constant. In particular σ̃ cd = 2π/Tcd (1 + i/(2Qdc )) with Tcd ≈ 380 days and Qdc ≈ −8. We check that the frequency dependence of the Love numbers does not impact much the transfer function ensuring the passage from the AMF to polar motion in the diurnal band. To this aim we compare the transfer function corresponding to (7.97), 1 Tma (σ) = 1 Tmo (σ)

kf 1 + k2󸀠 (σ) eAΩ + [− ], d Am σ − σ̃ c (σ) σ − σ̃ f 󸀠

Af ef − β 1 eAΩ [− − ], = d ̃ Am A σ − σc (σ) m σ − σ̃ f

(7.98)

with their counterparts associated with the constant Love numbers (as given above), Tma (σ) =

kf 1 + k2󸀠 eAΩ [− ], + d ̃ Am σ − σ̃ f σ − σc 󸀠

Af ef − β eAΩ 1 Tmo (σ) = − [− ], d Am σ − σ̃ c Am σ − σ̃ f

(7.99)

by plotting T 1 (σ)/T(σ) in Figure 7.2. The differences which are smaller than 1 % are not significant in the light of the uncertainties affecting the AMF (see Chapter 8).

136 | 7 Influence of the fluid core

Figure 7.2: The transfer functions accounting for the Love number dependence in the diurnal band 1 1 (σ)) are compared to the transfer function for which the Love numbers are fixed (Tma (σ), (σ), Tmo (Tma Tmo (σ)).

7.7 Influence of the solid inner core For developing a comprehensive theory of the influence of the Earth internal structure on the polar motion, we have to refine the mechanical system by considering the solid inner core, which, in the heart of the core extends from 0 to 1215 km. Initiated in the 1970s, the theory of a three layer Earth model has been deepened in the 1990s (see e. g. [144, 59, 78]), and has introduced two new free modes: the Inner Core Wobble (ICW) whose period ranges between 6 and 7 years [71] and the Free Inner Core Nutation (FICN) with the retrograde diurnal frequency of ∼ −1 + 1/1000 cpd in the TRF [145]. These free modes result in a resonance of the polar motion of about 7 years and resonant prograde nutations near 1000 days. Whereas the ICW remains speculative [107, 58], it seems that the consideration of FICN better accounts for the residuals of the observed nutation, in particular the 18.6 year term (order of magnitude of 200 µas). The mere fact that one can adjust a prograde frequency near 900 days and a quality factor of 700 favors the existence of the FICN [145]. However, this mode has no observable impact on nutation caused by geophysical excitation. So, at present the consideration of two layers, core and mantle, is sufficient to model the polar motion in the light of its geophysical forcing, as we will specify in the next chapter.

7.8 Conclusion

| 137

7.8 Conclusion By considering an axi-symmetric two-layered Earth model, we established the fundamental differential equation ruling the polar motion under the action of mass transport occurring in the outer layers of the Earth. In contrast to a quasi-elastic Earth covered by oceans, we get a second resonance mode associated with the FCN. Whereas this resonance modifies the lunisolar nutation of a rigid Earth up to 50 mas, it is also of primary importance for accounting for the hydro-atmospheric effects in the retrograde diurnal band. Moreover, at retrograde diurnal frequencies the dynamical response of the ocean strongly modifies the common resonance parameters of the polar motion, becoming Tcd ∼ 380 days and Qdc ∼ −10. The resonance of the solid Earth deformation at the FCN frequency even makes these parameters frequency dependent in the retrograde diurnal band, as confirmed by fitting Tcd days and Qdc over succesive sections of the retrograde diurnal band.

|

Part III: Geophysical forcing

8 Hydro-atmospheric excitation Donc toute la masse entière de la sphère de l’air qui est au monde pèse ce même poids de 8 283 889 440 000 000 000 livres [≈ 4 1018 kg]. (...) La saison où le mercure est le plus haut pour l’ordinaire est l’Hiver. Celle où d’ordinaire il est le plus bas est l’Esté. Où il est moins variable est aux Solstices; Et où il est le plus variable est aux Equinoxes. Blaise Pascal, Traités de l’Equilibre des Liqueurs et de la Pesanteur de la Masse de l’air (1648).

8.1 Introduction In the second part we have shown how the Earth’s non-rigidity and its two layer structure influence the polar motion at a subsecular scale. This has been achieved by accounting for both the rotational deformation and decoupling of the core from the mantle. The basic linear Liouville equations were modified accordingly, as well as the frequency of the free mode. Armed with these equations, we can investigate the effect of various geophysical excitations for periods stretching from one hour to several decades. The purpose of this chapter is to precisely describe the principal geophysical excitation at these periods, namely the one proceeding from the mass redistributions within the superficial fluid layers or hydro-atmosphere composed of the atmosphere, the oceans and the inland freshwater (including snow, ice, soil moisture and vegetation if any). According to the Liouville equation only a mass redistribution inside the mechanical system1 or an external coupling on this system may alter the course of the diurnal rotation of the Earth crust. Although some processes are very energetic (such as an earthquake or a geomagnetic storm), their mechanical efficiency on the polar motion is not ensured. In Table 8.1, we list the most significant moments of force acting on the Earth. Equatorial coupling between the solid Earth and its fluid layers is about 1020 kg m2 s−2 with periods between 12 hours and some years.2 This is about 100 times smaller than the lunisolar torque at retrograde diurnal frequency. As was noted in the introduction of Chapter 2, a perfect separation between geophysical and astronomical causes is not possible. Mass displacements occurring in the hydro-atmospheric layer result both from: – the lunisolar tide, causing the ocean tide (with a global order of magnitude of 1 m) and barometric tide, about 10 % of the surface atmospheric pressure fluctuations, and – solar heating; thermal differences activate the thermodynamic “Earth machine” generating air and water displacements, superimposed to the tidal effects. 1 The Earth as a whole or the extended mantle or the solid Earth. 2 The main part resulting form the equatorial bulge is given by Ω2 (C − A)χma according to (8.29) with χma ∼ 10 mas ∼ 5 10−8 rad; hence the proposed order of magnitude. https://doi.org/10.1515/9783110298093-008

142 | 8 Hydro-atmospheric excitation Table 8.1: Order of magnitude of the most important moment of force acting on the Earth. equatorial moment kg m2 s−2

axial moment kg m2 s−2

1020 0.3 1020 0.2 1020

1019 1017

?

5 1018

atmosphere oceans inland freshwater Core–mantle coupling electromagnetic–viscous gravitation–pressure lunisolar tidal torque

21

10 1023

–

typical time scale seasonal seasonal seasonal 10–100 years diurnal retrograde diurnal retrograde

As the ocean tides have an astronomical cause, and are modeled apart independently from the thermohaline redistribution,3 they are out of the scope of this book. Occurring mostly at diurnal and semi-diurnal frequencies, they cause: – diurnal retrograde polar motion, that is, perturbation on the nutation of the order of 1 mas (associated with the diurnal retrograde part of the oceanic angular momentum function); – diurnal and semi-diurnal oscillations on the polar motion: ∼ 0.2 mas (10 µs on the length of day). In contrast to oceans, the effect of the lunisolar tide on the atmosphere—the barometric tide—is not well modeled, and is blurred by the dominant role of the thermal processes. Therefore the surface atmospheric pressure and wind measurements, subsequently the atmospheric angular momentum, contain tidal and thermal effects jointly.

8.2 Hydro-atmospheric excitation 8.2.1 A brief historical account Shortly after the discovery of the Chandler and annual terms, various scientists endeavored to determine their causes. The seasonal air mass redistribution was an ideal candidate. The first attempt to estimate the seasonal atmospheric influence on the offdiagonal moments of inertia and polar motion was made by the Austrian astronomer and meteorologist Spitaler at the eve of the twentieth century [210, 209]. A more refined estimate is found in Jeffreys (1916) [117]. Although the equivalent Atmospheric Angular Momentum Function (AAMF) is overestimated (almost 100 mas instead of 3 Large-scale ocean circulation, driven by global density gradients created by surface heat and freshwater fluxes.

8.2 Hydro-atmospheric excitation |

143

10 mas), the counter-clockwise direction of the annual term of the AAMF is correctly established. In the same paper, the Oceanic Angular Momentum Function (OAMF), exceeding a few tens of mas, is given as the second factor of the polar motion; then come the rainfalls, the vegetation and the level of the polar ice cap (a few mas) grouped in what is called nowadays the hydrological excitation. Despite overestimated values, this hierarchy is still valid today. After the Second World War, when the meteorological observations were extended to the whole atmosphere (launch of balloon probe in the upper atmosphere), the determination of the global atmospheric circulation tremendously improved. So, the 1960s saw more realist estimates of the atmospheric excitation, partly accounting for seasonal oscillation of the pole [153]. The first time series of the atmospheric angular momentum go back to the decade 1970–1980. Whereas the axial term caused by the zonal winds explained the annual fluctuation of the length of day [127], the equatorial term could not enable a satisfactory account of the annual path of the pole [9] and a large part of the spectral continuum [128]. Meanwhile, it indicated rapid fluctuations (below 100 days) at mas level, observed one decade after thanks to the progress of the astro-geodetic techniques. Following the endeavors of Jeffreys, Munk and Groves (1952) [154] estimated the annual excitation of the ocean currents to about 1 mas. But, because of the difficulty to measure the currents and the sea level, the OAMF was largely undetermined until the years 1980. With Ocean Global Circulation Model (OGCM) improvements of the 1990s, it was possible to constitute time series of the OAMF, which, superimposed to AAMF, give a better account of the observed excitation at subdecadal periods. To find the distribution of freshwater content of the continents remains a challenge. Indeed inland fresh waters present a wide variety of forms at the surface (glaciers, water content of vegetation, snow, water levels in rivers and lakes), but are also hidden in the bowels of the earth (soil moisture, underground waters). This is why realistic estimates of freshwater transport across the planet came after those of the air and ocean masses, and only begins with the twenty-first century. They were boosted by the additional information provided by the space observation of the global gravitational field (GRACE missions, LAGEOS, CHAMP, GOCE). When the effects of modeled atmospheric and oceanic loads are removed, gravitational field variations provide residual fluctuations that are mainly attributed to the fresh water transports, although deeper localized processes (tectonic and seismic motions, magma motion in the fluid core for instance) are not ruled out.

8.2.2 Matter and motion term of the angular momentum functions Despite the great variety of these mass distributions, they all induce off-diagonal inerF F tia moment changes cF = c13 + ic23 no larger than 10−8 A and relative angular momenF F −8 tum h1 + ih2 smaller than 10 AΩ at sub-secular scale. So their effect on polar motion

144 | 8 Hydro-atmospheric excitation can be described in a standardized way by using the linear Liouville equations of an anelastic Earth (Chapter 5), covered by the oceans (Chapter 6) and including a fluid core (Chapter 7). More precisely, we deal with the angular momentum function, composed of the matter term χma = c/(C − A) and of the motion term χmo = h/(C − A)/Ω in accordance with (4.31). The angular momentum of a fluid layer is expressed similarly to the one for the whole Earth in compliance with (4.19) and (4.20): HF = (AF m1 + iBF m2 + cF )Ω + hF + 𝒪(cijF mk )Ω ,

F H3F = [CF (1 + m3 ) + c33 ]Ω + hF3 + 𝒪(cijF mk )Ω ,

(8.1)

where AF ≈ BF ≈ CF are the mean principal moments of inertia of this fluid layer, maximized by 10−5 A (order of magnitude for the most important surface fluid layer, namely the oceans). Thus, neglecting terms like AF mΩ ≤ 10−11 AΩ, we get H F = cF Ω + hF ,

F H3F = [CF + c33 ]Ω + hF3 Ω ,

(8.2)

so that cF and hF directly yield the angular momentum of the fluid layer. In spherical coordinate r, λ (longitude) and θ (colatitude), cF and hF read rs +h π

2π

cF = − ∫ ∫ ∫ ρF r 4 cos θ sin2 θ eiλ dr dλ dθ ,

(8.3)

r=rs θ=0 λ=0

rs +h π

2π

hF = ∫ ∫ ∫ ρF r 3 sin θ eiλ (−v cos θ + iu) dr dλ dθ ,

(8.4)

r=rs θ=0 λ=0

where the radial integration is done from some bottom surface rs (λ, θ) (the ground, the sea floor, the mean sea level) to some upper surface rs (λ, θ) + h(λ, θ) (the upper atmospheric layer where pressure is nul, the instantaneous sea level, . . . ), u is the southward meridian velocity and v the eastward velocity.4 In the case of the atmosphere, assuming hydrostatic equilibrium and exchanging the vertical with the radius, we introduce in (8.3) the variable change dP = −ρF gdr. 4 Intermediate steps for deriving these expressions are cF = − ∫(xz + iyz)ρF dτ = − ∫ r 2 (cos λ + i sin λ) cos θ sin θρF r 2 sin θ dr dθ dλ , h⃗ F = ∫ r ⃗ ∧ v⃗ ρF dτ = ∫ r 4 û r ∧ (uû θ + vû λ + wû r )ρF sin θ dλdθdr = ∫ r 4 (uû λ − vû θ )ρF sin θ dλ dθ dr . Expressing Cartesian components of the unit vectors û λ (along meridian, eastward) and û θ (along parallel, southward), we obtain the desired expression for hF .

8.2 Hydro-atmospheric excitation

| 145

Then, neglecting the variation of r 4 in the integrand (thin layer approximation) and taking r = r0 (mean radius of the Earth), we obtain π

2π

0

r4 cF = 0 ∫ ∫ ∫ cos θ sin2 θ eiλ dP dλ dθ g θ=0 λ=0 P=PS π

=−

2π

r04 ∫ ∫ PS cos θ sin2 θ eiλ dλ dθ , g

(8.5)

θ=0 λ=0

where PS is the ground or surface pressure. The estimation of angular momentum functions χma = cF /(C−A) and χmo = hF /(C− A)Ω from these integrals requires the knowledge of the pressure field, the velocity field and density of the fluid layer at any point of a sufficiently dense two-dimensional or three-dimensional mesh. As geophysical observations are partial, it is necessary to implement Global Circulation Models (GCMs) for interpolating the state of the layer where it is not observed. Without going into the complexity of the models, we shall characterize the global redistribution of mass operating in each layer, and try to give the magnitude of the corresponding effects on cF and hF . Although their angular momentum can be calculated separately, oceans, atmosphere and hydrological layer evolve in a coupled manner. Firstly the ocean surface is deformed dynamically under the action of surface atmospheric pressure variations and frictional effect of the wind. On the other hand hydrological layer acquires its mass mainly by precipitation of atmospheric water vapor and mainly lost it by flowing into the oceans.

8.2.3 Gravimetric matter term Since two decades the total matter term cF Ω is estimated independently through the variations of the degree 2 order 1 Stokes coefficients of the geopotential, as determined by precise laser orbitography of low altitude satellites like LAGEOS 1/2 (from 1976), or gradiometric measurements of the GRACE mission (2002-2017). From (B.17) and (B.18) the complex equatorial inertia moment associated with the variations ΔC21 and ΔS21 is cF (1 + k2󸀠 ) = −(ΔC21 + iΔS21 )M⊕ R2e .

(8.6)

Here, the quasi-elastic deformation produced by the fluid load is a fortiori included in the geopotential coefficients reflecting the whole Earth’s mass distribution, and is accounted for by the factor 1 + k2󸀠 , by which the pure load inertia moment cF is multiplied, as explained in Section 5.7.

146 | 8 Hydro-atmospheric excitation

8.3 Nature of the coupling between a fluid layer and the solid Earth The hydro-atmospheric excitation could be treated as an external torque on the Earth without its hydro-atmosphere. The angular momentum approach is preferred in the sense that it eludes the complicated modeling of the interaction forces between the solid Earth and its surface fluid layer. Nevertheless, for a better understanding of the physical processes at stake, this section gives a brief description of the nature of this interaction, and explains how the exchange of angular momentum operates. Compared to the lunisolar tidal torque on the equatorial bulge, with an order of magnitude of 1023 N m, the moment of forces produced by the hydro-atmospheric layer, is 100 times smaller, and presents four major differences: i) the lunisolar torque is regular and modeled with a great accuracy, but the coupling with a fluid layer is not subject to analytic predictive model and is reconstructed only after the fact; ii) lunisolar torque is mostly effective in the retrograde diurnal band, causing the precession– nutation, and the fluid layer torque is responsible for a larger range of frequencies, from 12 hours to several years; iii) the lunisolar torque is mostly equatorial, whereas the fluid layer torque has a strong axial component; iv) the tidal torque results from pure volume gravitational forces, but the interaction between the fluid layer and the solid Earth is also driven by surface forces. Because of the ellipticity of the Earth surface, the pressure forces do not pass generally through the center of gravity, causing in consequence at each point of the surface a moment of force. As the pressure field is not uniform, all these elementary contributions result in a non-zero pressure torque. Increasing the pressure in a given point on the surface (or the sea floor) corresponds to an accumulation of air (or water) mass, which in turn exerts a modified gravitation force on the solid Earth, counterbalancing the increment of pressure force. There results a gravitational coupling between the solid Earth and the fluid layer offsetting the pressure effects. Moreover, owing to fluid motion on the Earth surface, there appears a frictional coupling, much more difficult to model than the gravity–pressure one. In summary, the coupling is composed of three terms (see Figure 8.1): – A pressure term resulting from the contributions of the vertical columns of air or water on the whole Earth surface. – A gravitation term caused by the attraction of the hydro-atmospheric masses on the solid Earth masses. – A friction term associated with the friction produced by the wind or the ocean currents on the sea floor. Pressure and gravitational torques are computed from surface integrals, only requiring knowledge of the surface or bottom pressure Ps .

8.3 Nature of the coupling between a fluid layer and the solid Earth

| 147

Figure 8.1: Atmospheric forces on the solid Earth: pressure, gravitation, and friction (courtesy of O. de Viron).

Pressure torque It is expressed by Γ⃗ p = − ∫ r ⃗ × Ps n̂ dS ,

(8.7)

S

where S is the bottom surface of the considered fluid (ground or see floor), r ⃗ is the radius from the Earth center, n̂ the outward unit normal vector to the bottom surface, which admits the spherical harmonic development lmax

l

r = r0 [1 + ∑ ∑ (ulm cos mλ + ũ lm sin mλ)Plm (cos θ)] , l=1 m=0

(8.8)

where θ is the colatitude, λ the longitude, and r0 is the mean radius of the Earth (r0 = 6371000 m [see Appendix C.1 for its definition]). The second degree term u20 quantifies the flattening,5 the terms of larger degree are associated with the topography. Surface pressure admits a similar development: lmax

l

Ps = P0 + ∑ ∑ (plm cos mλ + p̃ lm sin mλ)Plm (cos θ) . l=1 m=0

(8.9)

5 In the reference ellipsoid frame the other coefficients of degree 2 are equal to zero, but in the TRF appear non-null coefficients u21 and ũ 21 . See Appendix C for more details.

148 | 8 Hydro-atmospheric excitation The unit normal vector n̂ to any Earth surface element dS is given by 𝜕r ⃗ 𝜕r ⃗ ∧ dθ dλ , 𝜕θ 𝜕λ where r ⃗ is the radius vector expressed by n̂ dS =

(8.10)

r sin θ sin λ r ⃗ = ( r sin θ sin λ ) . r cos θ

(8.11)

After differentiating r ⃗ with respect to θ and λ, we obtain cos λ cos θ − sin λ sin θ ] [ 2 𝜕r 2 𝜕r ( sin λ cos θ ) − r ( cos λ sin θ )] dλ dθ . r ⃗ × n̂ dS = [r 𝜕λ 𝜕θ − sin θ 0 ] [

(8.12)

Hence, the equatorial component of the pressure torque (8.7) is equivalent to the complex coordinate Γp = − ∫ Ps r 2 (cos θ S

𝜕r 𝜕r − i sin θ )eiλ dλ dθ , 𝜕λ 𝜕θ

(8.13)

whereas the axial component is expressed by Γz = ∫ Ps r 2 sin θ S

𝜕r dλ dθ . 𝜕λ

(8.14)

Substituting the spherical harmonic developments (8.8) and (8.9) of r and Ps , respectively, into (8.13) and (8.14), and neglecting second order and higher terms in ulm , we obtain Eqs. (E.23) and (E.25) of Appendix E.2, that is, l

1 + δm1 (p − ip̃ l,m−1 )(ulm + iũ lm ) (2Nlm )2 l,m−1 l=1 m=1

Γp = ir03 (∑ ∑ l

1 + δm0 (p + ip̃ l,m+1 )(ulm − iũ lm )) , m+1 )2 l,m+1 (2N l=1 m≥0 l

−∑ ∑

(8.15)

l

m (ulm − iũ lm ) (plm + ip̃ lm )(1 + δm0 )2 , m )2 (2N l l=1 m=1

Γz = ir03 ∑ ∑

. with (Nlm )2 = (2l+1)(l−m)! 4π(l+m)! The equatorial coupling is mostly associated with the equatorial bulge, namely the term u20 in the Earth radius expansion; then we see it only involves the (2,1) tesseral component of the surface pressure according to Γ20 p =−

12π 3 8π ir u (p + ip̃ 21 ) = i f r03 (p21 + ip̃ 21 ) , 5 0 20 21 5

(8.16)

since u20 = −2/3f from (C.4). This is the principal term of the pressure torque, since any departure from an ellipsoid (that is, topography) is represented by coefficients ulm at least 105 times smaller.

8.3 Nature of the coupling between a fluid layer and the solid Earth

| 149

Compensation by the gravitational torque In virtue of the action and reaction principle, the gravitational torque Γ⃗ g that the fluid layer exerts on the solid Earth is the opposite of the gravitational torque that the solid Earth exerts on this layer, so that ⃗ dV , Γ⃗ g = − ∫ ρr ⃗ ∧ ∇U

(8.17)

V

where V is the volume of the fluid, ρ its density, and U the geopotential. Assuming ⃗ + ρ∇(U ⃗ + U (r) ) = 0⃗ where U (r) is the centrifugal hydrostatic equilibrium, we have −∇P potential (5.9). Hence ⃗ − ρ∇U ⃗ (r) ) dV . Γ⃗ g = − ∫ r ⃗ ∧ (∇P

(8.18)

V

⃗ ∧ r ⃗ = ∇P ⃗ ∧ r,⃗ the gravitational torque Considering the formula ∇⃗ ∧ (P r)⃗ = P ∇⃗ ∧ r ⃗ + ∇P reads ⃗ (r) dV . Γ⃗ g = ∫ ∇⃗ ∧ (P r)⃗ dV + ∫ r ⃗ ∧ ρ∇U V

(8.19)

V

Then applying the rotational formula, ⃗ (r) dV , Γ⃗ g = − ∫ P r ⃗ ∧ n̂ V dS + ∫ r ⃗ ∧ ρ∇U

(8.20)

V

SV

where SV is the surface enclosing the volume V and n̂ V the outward normal to this surface. For a fluid layer like atmosphere, only the bottom surface—where P = PS and ̂ n̂ V = −n—contributes to the first integral, yielding ⃗ (r) dV , Γ⃗ g = ∫ PS r ⃗ ∧ n̂ dS + ∫ r ⃗ ∧ ρ∇U

(8.21)

V

S

where the first integral is precisely the opposite of the pressure torque (8.7). In turn we have ⃗ (r) dV , Γ⃗ p + Γ⃗ g = ∫ r ⃗ ∧ ρ∇U

(8.22)

V

showing that the sum of the pressure and gravitational torque is the torque of the centrifugal forces acting on the fluid layer in the TRF. This torque will be termed centrifugal torque. Neglecting the Earth’s rotation variation in the TRF by taking ω⃗ = [0, 0, Ω], and ⃗ (r) with Eq. (5.1), an elementary contribution to the centrifugal torque substituting ∇U is expressed by − sin λ ⃗ (r) = −ρ(ω⃗ ⋅ r)⃗ r ⃗ ∧ ω⃗ = ρΩ2 r 2 sin θ cos θ ( cos λ ) . ρr ⃗ ∧ ∇U 0

(8.23)

150 | 8 Hydro-atmospheric excitation So, as far as polar motion is neglected, the centrifugal torque is purely equatorial, given by the complex Γr = ∫ ρΩ2 r 2 sin θ cos θ i eiλ r 2 sin θ dr dθ dλ .

(8.24)

V

Then doing the variable change dP = −ρgdr (hydrostatic equilibrium by equating the vertical with the radius), and making the thin layer approximation, r = r0 , we obtain π

Γr =

=

2π

P=0

θ=0 λ=0

P=PS

iΩ2 r04 ∫ ∫ sin θ cos θ eiλ ( ∫ −dP) sin θ dθ dλ g iΩ2 r04 3 g N21

π

2π

∫ ∫ PS (θ, λ)Y21 sin θ dθ dλ .

(8.25)

θ=0 λ=0 l

max Considering the expansion Ps = ∑l=0 ∑lm=−l p󸀠lm Ylm (θ, λ) in terms of the complex spherical harmonics defined by (E.2), and applying the orthonormality relation (B.5) between Ylm and Y21 , we reduce the centrifugal torque to

Γr =

iΩ2 r04 p21 + ip̃ 21 iΩ2 r04 󸀠 p = 2,−1 2 3 g N21 3 g (N21 )2

(8.26)

with N21 = √5/(24π). Considering the quantity q ≈ Ω2 r03 /(GM⊕ ) ≈ Ω2 r0 /g according to (5.14), we have Γr = Γp + Γg =

4π 3 iqr0 (p21 + ip̃ 21 ) . 5

(8.27)

By comparison with the pressure torque on the equatorial bulge (8.16), as f ≈ q, we see that the gravitational interaction compensates about one half of the pressure effect. As shown in Appendix E, Eq. (E.3), the pressure term of the angular momentum can be expressed similarly: χp = −

r04 4π (p + ip̃ 21 ) , 5 (C − A)g 21

(8.28)

so that the total bulge torque or centrifugal torque Γb = Γr = Γp + Γg reads also Γb = iΩ2

4 4π r0 (p + ip̃ 21 ) = −i(C − A)Ω2 χp = −iΩHp , 5 g 21

(8.29)

where Hp is the matter or pressure component of the angular momentum and χp the associated angular momentum function. This relation was first derived in [10] (see Eq. (16) of this paper).

8.4 Atmospheric excitation

| 151

Friction torque Considering viscous force in the Navier–Stokes equation, the moment of friction forces acting on the Earth surface is expressed by (see, e. g., Eq. (19) of [51]) Γ⃗ f = − ∫ η r ⃗ ∧, (n̂ . ∇)⃗ v⃗r dS ,

(8.30)

S

where η describes is the fluid viscosity and v⃗r is the fluid velocity with respect to the bottom surface. Budget In what pertains to the atmosphere, the equatorial pressure and gravitation torques associated with the Earth’s bulge account for 90 % of the equatorial torque. The residual is caused by pressure on topography (10 %), friction (0.05 %) and remaining gravitational terms of the geopotential. In the case of axial coupling, the pressure torque caused by topography—ellipsoidal term is equal to zero according to (8.15)— contributes to 65 % of the total torque, the friction torque to 30 %, and the gravitation torque to 2–5 % [52]. Computation of atmospheric torque time series was developed in the 1990s [51, 52] and revisited more recently in [190]. As the frictional force remains poorly determined, this method is occasionally used to identify the nature of the coupling and its geographical location. For routine analysis of the fluid layer excitation, it is easier to evaluate in the TRF off-diagonal moment of inertia cij and relative angular momentum hi from global circulation models assimilating measures of pressure, wind, temperature, humidity, sea level, etc.

8.4 Atmospheric excitation The atmosphere is conditioned by two thermal cycles that shape our existence: the alternation of day and night, and the seasons. It is therefore not surprising that these two time scales characterize the fluctuations of the atmospheric angular momentum. 8.4.1 Pressure term With a mass MA ≈ 5.2 1018 kg,6 equal to one part per million of the Earth mass, the principal moments of inertia of the atmosphere have an order of magnitude of 10−6 A (the 6 As did Blaise Pascal in his famous Traités de l’Equilibre des Liqueurs et de la Pesanteur de la Masse de l’air (1648) but considering other units, this value is easily obtained from the mean surface atmospheric pressure of 105 Pa giving the mass of a vertical air column per meter square, that is, dm = 105 /g ≈ 104 kg/m2 .

152 | 8 Hydro-atmospheric excitation

Figure 8.2: Surface pressure difference between January and July. Extremal values are reached over the continents, corresponding to a less than 1 % of the mean atmospheric pressure. Isolines in g/cm2 (equivalent to 10 kg/m2 or 100 Pa): solid the air-mass inflow; broken the air-mass outflow (courtesy of Sidorenkov, 2002, p. 161 [200]).

atmospheric layer can be considered as a hollow sphere of inertia moment 2/3MA r02 ≈ 8.44 1031 kg m2 where r0 ≈ 6371 km is the mean radius of the Earth). Atmospheric surface pressure shows a strong seasonal variation. From the world map in Figure 8.2, surface pressure changes between July and January, reported in g/cm2 (or 10 kg/m2 , equivalent to about 100 Pa), reach 20 g/cm2 ≈ 2000 Pa on the Eurasian continent, but are much smaller over the oceans because of their regulatory role on the outside temperature. The accumulation of air mass in the northern hemisphere winter is accompanied by depressions in the southern hemisphere. The phenomenon is reversed in summer. So, one would expect a dominant semi-annual oscillation in atmospheric angular momentum. If a semi-annual term is observed, the annual harmonic is dominant due to the fact that the northern hemisphere regroups 70 % of land areas where seasonal pressure variations are the most important. The moment of inertia change (8.5) integrates the surface pressure field fluctuation, of which the largest component, at the seasonal period, represents no more than 103 Pa, that is, 2 % of the total pressure. From (8.28) cijA has an upper bound of |cpmax | = 4π/5 r04 /g|p21 | ≈ 4 1029 kg m2 , that is, 5 10−9 A. The matter term χp is thus bounded by 5 10−9 A/(C − A) = 1.5 10−7 rad or 300 mas.7 Precise meteorological esti7 In the following the angular momentum functions are converted in mas by dividing them by 1 mas = 4.848 10−9 rad, making them commensurable with the pole coordinates expressed in this unit.

8.4 Atmospheric excitation

| 153

mates of χp are deduced from (8.5), namely π 2π

r04 cA χp = =− ∫ ∫ Ps sin2 θ cos θ eiλ dθ dλ , C−A (C − A)g

(8.31)

0 0

and give an order of magnitude 30 times smaller at the annual period. By performing a global reanalysis of the atmospheric data, the NCEP together with NCAR8 provides a four daily time series of χp (0hTU, 6hTU, 12hTU, 18hTU) beginning in 1948. In the evolution of χp illustrated over the period 2006–2009 in Figure 8.3, the seasonal cycle dominates up to 50 mas on component χy and 10 mas on χx . This asymmetry reflects the fact that the x component sums up the contributions of the moderate oceanic zones (Pacific and Atlantic), whereas the y component results from continental zones (Eurasia and the Americas). This temporal overview is supplemented by a spectral decomposition of the AMF (bottom of Figure 8.3), showing that the seasonal term is composed of two opposite circular terms of comparable magnitudes. Decreasing with growing frequency, the spectral amplitude rises abruptly to diurnal frequency at the level of 1 mas.

8.4.2 Wind term The winds are mainly governed by the eastward circulation, and above all have an impact on the length of the day: 95 % of the variations between a few days and six years are explained by the wind term of the axial angular momentum. Conversely, the winds much less impact the common polar motion above 2 days than the pressure fluctuations. The equatorial wind term χmo = χv has a seasonal fluctuation of the order of 1 mas (Figure 8.3), about a tenth of the χp one. This notwithstanding, χv is significant in the diurnal and sub-diurnal bands with fluctuations up to 10 mas. The analysis of those bands is postponed to Chapters 12 and 13.

8.4.3 Inverted barometer oceans Over the oceans the pressure variation dP induces an opposite sea level change. If the time scale is sufficiently long, beyond 10 days [182], the mass anomaly dm in any vertical air column from the surface to the upper atmospheric layer—reflected by surface pressure change dP = g dm/S (S is the section of the column)—is compensated by a variation −dm in the underlying column of water. Thus the sea level changes by dh such that dm = −ρo dh S, where ρo is the water density, that is, dh = −dP/(gρo ) (+5 cm 8 National Center for Environmental Prospect and National Center for Atmospheric Research, USA.

154 | 8 Hydro-atmospheric excitation

Figure 8.3: Equatorial atmospheric angular momentum function: pressure term χp in totality/with inverted barometer correction (“IB”), and wind term χv according to NCEP/NCAR reanalysis. The seasonal variation is dominant in the pressure term but is mitigated in the wind term where the diurnal oscillation is the most important.

8.4 Atmospheric excitation

| 155

for a typical cyclonic pressure drop dP = −500 Pa). In other words, the mass of the whole column from the ocean bottom to the upper atmosphere does not change. As sea level change reflects an opposite pressure variation, the oceans are said to react as Inverted Barometer (IB). If the oceans do not deform, they respond as a Non-Inverted Barometer (NIB). Below 10 days, the ocean response is neither IB or NIB, but becomes dynamical, as seen for the diurnal band in Section 6.6. Considering the fluid layer composed of the atmosphere and the oceans, the ocean bottom pressure is invariable over the oceans zones, and integral (8.31) associated with the moment of inertia variation (8.5) can be restricted to the continents: π 2π

χp(IB)

R4e =− ∫ ∫ 𝒞 (λ, θ)Ps sin2 θ cos θeiλ dθ dλ , g(C − A)

(8.32)

0 0

where 𝒞 (λ, θ) is the continent function, equal to 1 over the continents and 0 over the world ocean. This numerical procedure is the inverted barometer correction, and models the hydrostatic response of the oceans in front of the atmospheric circulation discarding any dynamical effects. As two-thirds of the Earth surface are cast aside, this correction tapers the pressure term, especially seasonal variation (see Figure 8.3). It permits one to better account for polar motion excitation at period larger than 2 days (Chapter 9), but is not sound for modeling angular momentum exchanges with the solid earth at diurnal and sub-diurnal time scales (see Chapters 12 and 13).

8.4.4 Atmospheric Angular Momentum functions and uncertainty Atmospheric global circulation models and AAM series (AGCM) For studying change of the rotation of the Earth the most appropriate AAM time series are the longest and the ones which are regularly updated. Four AGCMs allow one to estimate such AAM: i) the NCEP model from the National Center for Environmental Prediction Prospect, ii) the ECMWF model from the European Center for Meteorological Forecasts, iii) the JMA model operated by the Japan Meteorological Agency, and iv) the MERRA model of the NASA. The AAM series are estimated from the output of these models either by the organizations having developed those models or by independent teams throughout the world. So the ECMWF AAM time series are computed both by GeoForschungZentrum in Postdamm (GFZ) and TU Wien. These series have a subdaily sampling (6 hours for NCEP and MERRA, 3 hours for ECMWF estimated at GFZ). For a given model, they either result from a reanalysis of the past meteorological data or are computed in an operational way with a latency not exceeding of one day. The longest of them (NCEP reanalysis, ECMWF ERA) began as early as the 1950s. Table 8.2 groups the atmospheric series available in the public domain, and the ones that have been used for the analyses presented in this book.

[69]

OMCT8 [215, 214]

MPIOM6 [68] NEMO7

ECCO5 [188]

Oceanic time series

JMA2 MERRA3 NCEP4 [181]

ECMWF1

Atmospheric time series

model

ERA40 Interim Operational-old

50 yr kf079 kf080i v4r3 [91] Operational GFZ

Operational Reanalysis [118]

Operational TUW [189] Operational GFZ [68] ERA40 [69] Interim [69] Operational-old [69]

version/center

1958–2001 1989–2010 2001–2017

1949–2002 1993– 1993– 1992–2016 1976–now 1979.0–2012.0

1980–now 1976–now 1958–2001 1989–2010 2001–2017 1993– 1980– 1948–

time interval

Table 8.2: AAMF, OAMF and HAMF time series available on internet (except NEMO).

6 h from 0hUT 6 h from 0hUT 6 h from 0hUT

10 days–0hTU 1 day–0hTU 1 day–0hTU 1 h from 0hUT 3 h from 0hUT 1 day

6 h from 0hUT 3 h from 0hUT 6 h from 0hUT 6 h from 0hUT 6 h from 0hUT 6 h from 0hUT 6 h from 0hUT 6 h from 0hUT

sampling

id id

SBO [188]♭ id id id GFZ♠ MERCATOR-OCEAN GFZ♯

TUW♦ GFZ♠ GFZ♣ id id SBA [186]♥ NASA♦ SBA [186]♥

origin WEB/FTP site

no altim. assimilation altim. assimilation altim. assimilation

NIB/IB IB only IB only IB only IB only 10 hPa 10 hPa 10 hPa

other features

156 | 8 Hydro-atmospheric excitation

Operational GFZ

version/center

1948–2007 1979–2007 1976–now

time interval

♥ Special Bureau for the Hydrology: http://www2.csr.utexas.edu/research/ggfc/dataresources.html.

2

European Centre for Medium-Range Weather Forecasts. Japan Meteorological Agency. 3 Modern-Era Retrospective analysis for Research and Applications – NASA. 4 National Centre for Environmental Prospect. 5 Estimating the Circulation and the Climate of the Oceans, model developed at MIT. 6 Max Planck Institute Ocean Model. 7 Nucleus for European Modelling of the Ocean, http://www.nemo-ocean.eu. 8 global Ocean Model for Circulation and Tides. 9 Climate Prediction Centre. 10 Global Land Data Assimilation System. 11 Land Surface Discharge Model. ♥ Special Bureau for the Atmosphere: http://files.aer.com/aerweb/AAM/. ♦ https://www.iers.org/IERS/EN/DataProducts/GeophysicalFluidsData. ♠ ftp://rz-vm115.gfz-potsdam.de/EAM. ♣ ftp://ftp.iers.org/products/geofluids/atmosphere/eamf/GGFC2010/GFZ/old_series. ♦ http://aam.earthrotation.net. ♭ Special Bureau for the Oceans: https://euler.jpl.nasa.gov/sbo/. ♯ ftp://ftp.iers.org/products/geofluids/oceans/eamf/GGFC2010/GFZ/old_series/.

1

CPC9 [87] GLDAS10 [176] LSDM11 [69]

Hydrological time series

model

Table 8.2: (continued)

30 days 30 days 1 day

sampling

SBH [187]♥ SBH [187]♥ GFZ♠

origin WEB/FTP site

other features

8.4 Atmospheric excitation | 157

158 | 8 Hydro-atmospheric excitation

Figure 8.4: Pressure term: spectrum and Allan deviation of the paired differences NCEP–ECMWF and NCEP–MERRA over the period 1980–2019. The pressure term is IB and all AMF are taken as noneffective.

Time series comparison As the error of the angular momentum values is not provided, the uncertainties that affect them are assessed from the differences between the AAM series of various centers. The pressure term time series of NCEP, MERRA and ECMWF are consistent, the mutual explained variances have rates of 99 % and more (over the period 1990–2019). But the associated wind terms do not fit so well, with mutual explained variance of about 76–79 % (NCEP/MERRA, 1990–2019) or 23–27 % (NCEP/ECMWF, 1990–2019). Estimated from the abundant surface pressure measures, the pressure term is much more precise than the wind term. For the latter relies on a circulation model at different altitudes and on the variable choice of the upper altitude for computing the integral (8.4) (generally taken as an isobaric surface). Paired differences between the time series are composed of both systematic offsets for characteristic periods (1 year, 24 h, 12 h) and stochastic noises, as shown in more detail now. Pressure term uncertainty We build the paired difference NCEP–ECMWF (IB-pressure) over the period 1980–2019. The amplitude spectrum—obtained by DFT—affords one to have an overall estimate of the inconsistency in each frequency band. It mixes both systematic differences, independent from the sampling duration P and random uncertainty (or random error) decreasing as 1/√P when P increases. The random uncertainty is characterized by an Allan deviation analysis. Both spectrum and Allan deviation are plotted in Figure 8.4. The Allan deviation AD is modeled against the time scale T as a piece-wise linear func-

8.4 Atmospheric excitation

| 159

Figure 8.5: Wind term: spectrum and Allan deviation of the paired differences NCEP–ECMWF and NCEP–MERRA over the period 1980–2019.

tion in log–log scale, namely ln(AD(T)) = s󸀠 ln(T) + log(c󸀠 ) with slope s󸀠 and constant c󸀠 given at T = 1 time unit (see Appendix G.5). Such a linear function corresponds to a colored noise of power spectral density Kα f α with α = −2s󸀠 − 1 and Kα = c󸀠 2 /Jα where the coefficient Jα is given in Table G.1 for some particular values of α. Neglecting the drop accompanying the semi-annual period, we grossly approximate AD(T) by the constant function c󸀠 = 0.5 mas (s󸀠 = 0). According to (G.24), this corresponds to a frequency modulation flicker or pink noise (α = −1) of the spectral amplitude: εAp (f ) =

c󸀠 −s󸀠 −1/2 0.5 mas −1/2 f = f , √4 ln 2P √J−1 P

(8.33)

with P is the considered time interval. Then, taking at least P = 7 years in order to split the annual and Chandler oscillations, the spectral uncertainty in the Chandler band associated with the global circulation modeling is about 0.1 mas. Notice that the y component is slightly noisier than the component x. Yet, at annual periods the model’s inconsistency reaches 1 mas, and this is larger than the random error, maximized by 0.4 mas for the minimal duration P = 1 year. Noise of the wind term A similar Allan deviation analysis is performed for paired differences pertaining to the wind terms and displayed in Figure 8.5. The instability is larger, as expected. From 10 days to 2000 days, it can be grossly modeled by a flat function of about 1 mas. This

160 | 8 Hydro-atmospheric excitation indicates a pink noise of spectral amplitude εAw (f ) =

c󸀠 −1/2 1 mas −1/2 f . f = √4 ln 2P √J−1 P

(8.34)

In the seasonal/Chandler band, this yields an uncertainty of about 0.6 mas for P = 1 year, and 0.2 for P = 7 years. Yet, systematic uncertainty of the modeling reaches 3 mas at annual periods, and this is much larger than the random error.

8.5 Ocean excitation 8.5.1 Dynamical ocean model Under the action of atmospheric pressure, the ocean masses redistribute. For time scales longer than a few days, this can be modeled as a hydrostatic sea level change (“inverted barometer” oceans) of a few cm, reducing the integration of atmospheric matter term to land areas. But this adjustment is far from covering all transport taking place in the seas, especially currents. They have to be modeled by local dynamical equations (Navier–Stokes equations, conservation of mass, etc.) possibly accounting for wind stress. Assimilating different observational data (temperature, salinity, etc.) the equations are integrated, then giving currents at each point of a three-dimensional grid covering the oceans, and also a 2-D chart of additional corrections of the sea level, permitting one to evaluate time series of the ocean angular momentum. 8.5.2 Sea water level variations From (8.3) the matter term of the ocean reads χh = −

rs +h π

2π

1 ∫ ∫ ∫ 𝒪 ρo r 4 cos θ sin2 θ eiλ dr dλ dθ , C−A

(8.35)

r=rs θ=0 λ=0

where h is the water height reckoned from the ocean surface at equilibrium parametrized by rs (λ, θ), 𝒪(λ, θ) is the ocean function introduced in (6.10), and ρo ≈ 1000 kg/m3 is the ocean water density. Assuming an uniform density we obtain π

2π

r 5 − (rs + h)5 1 χh = − ρo ∫ ∫ 𝒪 s cos θ sin2 θ eiλ dλ dθ . C−A 5

(8.36)

θ=0 λ=0

Considering (rs + h)5 ≈ rs5 (1 + 5h/rs ) we have π

2π

1 χh = − ρ ∫ ∫ 𝒪 h rs4 cos θ sin2 θ eiλ dλ dθ , C−A o θ=0 λ=0

(8.37)

8.5 Ocean excitation

| 161

showing that variations of the oceanic matter term are caused by sea level change h with respect to the geoid. In addition to the ocean tide (1 m) and inverted barometer effect (up to 5–10 cm), this sea level change results from the following processes [126]: – Volume fluctuations associated with salinity and temperature (1–2 cm). – Dragging by the surface wind (1 cm). – Hydrological transports: evaporation/precipitation and runoff (1 cm). In Figure 8.6 the water height term estimated from the Max Planck Institute Ocean model (MPIOM) shows a seasonal variation of a few mas, 5 times smaller than the one of the atmosphere. 8.5.3 Currents Although Coriolis forces caused by Earth rotation produce strong ocean currents, these do not contribute to the relative angular momentum [153]. From (8.4) the current term is estimated by the integral rs +h π

2π

1 χc = ∫ ∫ ∫ ρo r 3 sin θ eiλ (−v cos θ + iu) dr dλ dθ . (C − A)Ω

(8.38)

r=rs θ=0 λ=0

Looking at the spectrum of Figure 8.6, we see that the seasonal change has an amplitude of about 1 mas and is less prominent than for the sea level term χh . As in the case of the atmosphere, the rapid retrograde band from 0.01 cpd (100 days) to 0.5 cpd (2 days) shows a larger power (∼ 0.5 mas) than the prograde one (∼ 0.1 mas). In Chapter 9 this will be related to the retrograde atmospheric normal mode in about 10 days. 8.5.4 Ocean Angular Momentum function and uncertainties Three time series of the OAMF are considered. The first are estimated at a daily rate from the OGCM ECCO (Estimating the Circulation and Climate of the Oceans) developed at MIT, the second are also produced from the ECCO model with hourly sampling rate, and the third result from the MPIOM model (Max Planck Institute Ocean Model) processed at the GeoForschung Zentrum. ECCO daily time series (1949–) It merges two distinct series produced from the ECCO model. The first one, covering the period 1949–1993 (but actually spreading over 1949–2002), has a temporal resolution of about 10 days, and does not include any assimilation of the altimetric data.9 The complementary series, covering the period 1993–2019, are sampled at daily rate 9 See Gross et al. (2005), Atmospheric and Oceanic Excitation of Decadal-Scale Earth: Orientation Variations, J. Geophys. Res. 110, B09405.

162 | 8 Hydro-atmospheric excitation

Figure 8.6: Equatorial oceanic angular momentum function according to the MPIOM model: water height term χh and current term χc .

8.5 Ocean excitation

| 163

and benefit from altimetric observations (time series kf080i). The latitude domain excludes the 10∘ band surrounding the poles, the horizontal resolution is of 1/3 degree at equator to 1 degree at high latitudes by 1 degree longitude, there are 46 vertical levels ranging in thickness from 10 m at surface to 400 m at depth. One makes use of the Boussinesq approximation and the ocean is forced by the similar surface wind stress, surface heat flux, surface freshwater flux obtained by the NCEP/NCAR model. ECCO hourly time series (1992–2016) These series are derived from ECCO Version 4 Release 3 with the following features: horizontal resolution with a variable spacing ranging from 22 km to 111 km, vertical resolution with 50 levels ranging in thickness from 10 m at surface to 460 m at depth, Boussinesq model, inclusion of steric effects and of mass effects from net real freshwater flux, forcing by 6-hourly ERA-Interim values, assimilation of sea level altimetry, in situ temperature/salinity, satellite sea surface temperature and salinity, bottom pressure, sea-ice concentration.10 MPIOM time series The MPIOM series produced at GFZ since 2018 are sampled four times daily. Its spatial resolution is 1∘ × 1∘ . Arguing the poor modeling of the ocean circulation at the diurnal and subdiurnal tidal frequencies (underestimation of the atmospheric forcing in many areas of the ocean surface, no assimilated observations), Henryk Dobslaw and Robert Dill at GFZ [68] have preferred to disseminate OAM series from which the 12 largest diurnal and sub-diurnal lunisolar harmonics are removed according to an adjustment made over the period 2007–2014. Those removed harmonics are even not provided separately. This is in contrast to the outdated OAM series produced at GFZ using the output of the OMCT model (Ocean Model for Circulation and Tides), merging both reanalysis series (forced by ERA and ERAInterim atmospheric pressure) over 1958–2011 and an operational version made available until 2017. Time series comparison Uncertainty affecting OAMF can be assessed from the paired differences ECCO– MPIOM and ECCO–ECCO1h11 over their common period 1993–2016 (that of ECCO1h). The paired differences ECCO–MPIOM have a standard deviation of about 9–10 mas for the matter term, and 12–15 mas for the current term, which is about the standard deviation of these terms considered alone. The correlation coefficients are up to 0.8 for the matter term, but do not exceed 0.45 for the current term. Stemming from the same circulation model, the current terms of ECCO and ECCO1h are in better agreement with 10 See Fukumori et al. 2018, ftp://ecco.jpl.nasa.gov/Version4/Release3/doc/v4r3_data.pdf 11 After removing the diurnal and sub-diurnal variations of MPIOM and ECCO1h, for they are not present in ECCO.

164 | 8 Hydro-atmospheric excitation

Figure 8.7: Ocean water height term: spectrum and Allan deviation of the paired difference ECCO– MPIOM and ECCO–ECCO1h over the period 1993–2016.

the correlation coefficients slightly above 0.7, but the corresponding matter terms present discrepancies at the same level from ECCO and MPIOM. So the oceanic time series are far from reaching the consistency observed for the atmospheric AMF. Indeed oceanographic observations are much more sparse and models are less constrained. Noise of the water height term An Allan deviation analysis of the paired differences permits one to characterize the instability associated with the modeling at a given time scale, and to infer the random spectral uncertainty. In Figure 8.7, the Allan deviations AD(T) are plotted in log–log scales for the offsets ECCO–MPIOM and ECCO–ECCO1h. For T ∈ [10, 1000] days, they are roughly modeled by log(AD(T)) = s󸀠 log(T) + log(c󸀠 ) with s󸀠 = −1/4 , c󸀠 = 10 mas cpds , 󸀠

(8.39)

associated with a noise index α = −2s󸀠 − 1 = −1/2 having the spectral amplitude εOh (f ) =

c󸀠 √J−1/2 P

󸀠

f −s −1/2 =

10 mas cpd−1/4 −1/4 f , √1.562P

(8.40)

where P is the period of study. Taking P = 7 years, this yields an uncertainty of about 0.16 mas in Chandler or seasonal band. Meanwhile, the models exhibit much larger systematic offsets at annual period, up to 2 mas, as evidenced by the spectra of Figure 8.7.

8.6 Hydrological excitation

| 165

Figure 8.8: Currents terms: spectrum and Allan deviation of the paired difference ECCO–MPIOM and ECCO–ECCO1h over the period 1993–2016.

Noise of the current term Both paired difference Allan deviations displayed in Figure 8.8 are approximatively modelled through the piece-wise linear function log(AD(T)) = s󸀠 log(T) + log(c󸀠 ) s󸀠 = −1/3, c󸀠 = 8 mas cpds

󸀠

={

s󸀠 = −0.5, c󸀠 = 10 mas cpds

󸀠

for T ∈ [10, 200] days , for T > 200 days ,

(8.41)

associated with the spectral amplitude εOc (f )

{ ={ {

8 mas cpd−1/3 −1/6 f √P [d] 15 mas cpd−1/2 √P [d]

for T = 1/f ∈ [10, 200] days , for T = 1/f > 200 days (white noise) .

(8.42)

In the Chandler or seasonal band, as for the water height term, the uncertainty associated with the ocean circulation modeling can be assimilated to a white noise. Taking P = 7 years, this yields an uncertainty of about 0.2 mas. Systematic offsets are observed at annual and semi-annual periods up to 2 mas.

8.6 Hydrological excitation Although fresh waters are scattered throughout the crust or on its surface, they can be likened to a water shell of variable thickness. Freshwaters represent about 4 % of

166 | 8 Hydro-atmospheric excitation

Figure 8.9: Hydrological reservoirs and fluxes [33].

the total water content of the Earth. More than half is in the form of ice, snow or permafrost. The rest consists of lakes, glaciers, groundwater, soil moisture and vegetation [67] (see Figure 8.9). Paradoxically, land storage and retrieval of freshwater affect the polar motion almost as much as the oceans. From a given instant t0 , the stored water mass per unit area q(θ, λ, t) at time t results from the balance between precipitation P, evaporated water E and discharged water into oceans D according to q(θ, λ, t) = P(θ, λ, t) − E(θ, λ, t) − D(θ, λ, t) .

(8.43)

The terrestrial water storage can be also seen as the sum of soil moisture (SM) over a typical depth (∼ 2 m) and snow water equivalent (SNW), and computed accordingly. To achieve a hydrological model means to determine the evolution of P, E and D or SM and SNW over all the Earth’s continents. Because of the various forms of the inland fresh waters, often hidden in direct measurements, it is more difficult to model its global distribution than for air and oceans masses. This explains why quality time series of the HAM came after AAM and OAM only on the eve of the twenty-first century, after the partial modeling of the 1990s. Thus the past decade has seen the emergence of models offering an almost complete coverage of the Earth: 1. The Land Data Assimilation System of Climate Prediction Center (LDAS-CPC) depending on National Oceanic and Atmospheric Administration (NOAA) [87] assimilates solar radiation, soil moisture and precipitation as well as various atmospheric data produced by NCEP: temperature, ground pressure and winds. Over the period 1948–2008 this model gives one time per month mass of stored freshwater Δq per unit surface, except over Antarctica.

8.6 Hydrological excitation

2.

3.

| 167

The Global Data Assimilation System (GLDAS) [176] was jointly developed by NOAA and Goddard Space Flight Center (GSFC). This model incorporates the ground features like vegetation and urban zones, as well a large number of space and ground observations. The surface mass density Δq is estimated from 1979 to 2008 with a spatial resolution of 1 degree and sampled monthly or every three hours. The Land Surface Discharge Model (LSDM) combines modeling of the land surface (Simplified Land surface Scheme) and hydrological fluxes (Hydrological Discharge Model) [67], compatible with the ocean model ECCO and the atmospheric model ECMWF. The corresponding angular momentum output, as computed by GFZ, covers the period from 1976 to present.

As the level water change h and the mass change per unit area are related by Δq = ρh (ρ is the water density), the mass term of the Hydrological Angular Momentum (HAM) function obeys an expression similar to (8.37) for the OAM function, that is, π

χTWS = −

2π

1 ∫ ∫ (1 − 𝒪) Δq(θ, λ, t) rs4 cos θ sin2 θ eiλ dλ dθ . C−A

(8.44)

θ=0 λ=0

The associated relative angular momentum is negligible in confrontation with the mass term and will not be considered. Inland water mass redistribution mostly happens with the annual period and in the northern hemisphere where more than 70 % of the continents are grouped. So, examining LSDM time series over the period 2005– 2015 (Figure 8.10), it is not surprising to see a prominent seasonal term of about 5 mas (three times less than the corresponding variation of the atmospheric pressure term). In addition, hydrological excitation has staggering inter-annual and decadal changes (∼ 4 mas), clearly evidenced by the Fourier spectrum done over the period 1976–2019 (Figure 8.10). This is related to the extent of the ice caps, modulated by the climate change observed in air surface temperature. Notice that decadal variations of such an amplitude are not observed in atmospheric and oceanic AMF. Time series differences The uncertainty of the hydrological series is assessed from a statistical study of their offsets. The inconsistencies are striking: the standard deviation of LSDM-GLDAS, LSDM-CPC, GLDAS-CPC differences (4 mas for x, up to 15 mas for y) can be larger than the standard deviation of the series themselves (4 mas for x, 10 mas for y). From the Allan deviation of the paired differences (Figure 8.11), we can see that instability level increases with time scale and exceeds the one of the atmospheric and the oceanic angular momentum functions above 100 days. From 10 days to 100 days the slope s󸀠 in log–log scale is between 0.5 (α = −2, red noise) and 1 (α = −3); after 200 days s󸀠 ≈ 0, revealing a pink noise like behavior. So, in the Chandler or seasonal band, as for the

168 | 8 Hydro-atmospheric excitation

Figure 8.10: Equatorial hydrological angular momentum function according to LSDM.

Figure 8.11: Spectrum and Allan deviation of the paired difference LSDM–CPC, LSDM–GLDAS and GLDAS–CPC over the period 1990–2008.

8.7 Total excitation

| 169

Figure 8.12: Random uncertainty affecting the hydro-atmospheric layer for sampling duration P = 1 year at time scales ranging from 10 to 2000 days. For other values of P in years, the plotted random error has to be divided by √P.

atmospheric pressure term, the uncertainty brought about by the modeling is a pink noise, of which the spectral amplitude is approximated by εH (f ) =

c󸀠

√J−2s󸀠 −1 P

f

−s󸀠 −1/2

{ ={ {

3 mas −1/2 f √4 ln 2P 6 mas −1/2 f √4 ln 2P

for x component , for y component ,

(8.45)

amounting at 1 cpy to about 0.5 mas for the x component and 2 mas for the y component with P = 7 years. Yet, systematic offsets at annual periods, as shown by the spectra of various HAM series, are much larger up to 7 mas for the y component.

8.7 Total excitation For establishing the budget of the polar motion excitation, we favor atmospheric, oceanic, and hydrological angular momentum time series that are produced with consistent models. In the ECCO model, oceans are affected by surface winds of the NCEP model, as well as the CPC hydrological model is built from NCEP water evaporation data. The ocean model MPIOM and the hydrological model LSDM operated at GFZ depend on the output of the atmospheric model ECMWF (“ERA40” and “ERA Interim” re-analyses). Thus, in the following chapter, a consistent analysis of the polar motion shall be performed by using either the combination NCEP(Reanalyses)+ECCO+CPC or ECMWF+MPIOM+LSDM.

8.8 Conclusion At sub-secular time scale polar motion modeling relies on the estimates of the angular momentum variations of the hydro-atmospheric layer. The atmospheric component is

170 | 8 Hydro-atmospheric excitation both the most important and the one best determined. The oceanic contribution, with two times smaller variations, is more uncertain. Fluctuations of the hydrological AMF, dominated by a seasonal component, represent about 20 % of the atmospheric AMF, but compared to the case of AAMF and OAMF, their various models are much less consistent. The whole set of “uncertainty functions”, derived from an Allan deviation analysis for each term of the excitation function, are plotted in Figure 8.12. Then we see that in the near-seasonal domain, the sources of uncertainty are by decreasing order of magnitude: the land waters (∼ 3/√P mas), the sea level (∼ 2/√P mas), the oceanic currents (∼ 1/√P mas), the winds (∼ 0.6/√P mas), and the surface pressure (∼ 0.3/√P mas), where P is the sampling duration in years. For the rapid fluctuation (below 100 days), in the absence of a fresh water contribution, the main source of error stems from the oceans, introducing an uncertainty of about ∼ 0.6/√P mas between 10 and 100 days.

9 Equatorial angular momentum balance from two days to decadal time scale Thus, we have decisive demonstration that the motion, relatively to the earth, of the Earth’s instantaneous axis of rotation is the cause of variations of latitude which had been observed in Berlin, Greenwich, and other great observatories, and which could not wholly be attributed to errors of observation. This [...] gives observational proof of a dynamical conclusion [...] to the effect that irregular movements of the Earth’s axis to the extent of half a second may be produced by the temporary changes of sea-level due to meteorological causes. Kelvin, On Variations of Short Period in the Latitude, Astro. Soc. Pacific: 33–34 (1891).

9.1 Liouville equation from two days to some decades As far as hydro-atmospheric AMF is the main source of equatorial excitation below 100 years, we are able to initiate an interpretation of the polar motion. Other excitations are either minor or unknown, and among them only a seismic mass redistribution will be considered in Chapter 14. In this chapter our interest is limited to the “common” polar motion, whose fluctuations extend from two days to some decades. In the second part of this book we have seen how the structure of the Earth and its rheology modify the differential equations governing the instantaneous rotation pole coordinates (m = m1 + i m2 ) or the ones of the celestial intermediate pole (p = x − iy). Having first considered a quasi-elastic Earth (Chapter 5), we introduced the modification brought about by the oceans (Chapter 6), and the fluid core (Chapter 7). Because of ocean pole tide and triaxiality, these differential equations are slightly asymmetric as regards m1 and m2 , making any polar motion oscillation weakly elliptical that is caused by an uniform circular excitation, mostly near the Chandler frequency and its opposite. In the light of the uncertainty of the hydro-atmospheric excitation, to consider asymmetric effects outside the Chandler band is probably unnecessary. They will be investigated only in the case of the Chandler band excitation in Chapter 11. On the other hand the influence of the core is only effective in the retrograde diurnal band, because of the resonance at the free core nutation frequency. So pole oscillations from two days to several decades will be described in the framework of an extended mantle system (including the hydro-atmospheric layer) decoupled from the core, of which the polar motion was described in Chapter 7. In the case of a surface fluid layer excitation the complex pole coordinate p = x − iy should obey the first-order differential equation (7.39a) characterized by the Chandler wobble resonance: p+

i e e ṗ = χma + χmo , σ̃ c e e p(σ) = T(σ)[χma (σ) + χmo (σ)]

https://doi.org/10.1515/9783110298093-009

with T(σ) = −

σ̃ c σ − σ̃ c

(spectral version) ,

(9.1a)

172 | 9 Equatorial angular momentum balance from two days to decadal time scale with the effective AMF defined by 1 + k 󸀠̃ 2 = 1.112 , 1 − k/ks 1 = = 1.606 . 1 − k/ks

e χma = αma χma ,

αma =

e χmo

αmo

= αmo χmo ,

(9.1b)

Here the estimates of αma and αmo correspond to the numerical values k2󸀠 = −0.3075, k = 0.354, ks = 0.938. Because of the uncertainties affecting these Love numbers, αma and αmo present miscellaneous values in the literature. Collecting the currently used values in Table 9.1, we see that relative offsets of αma do not exceed 2 %, and likewise for αmo . Table 9.1: Various determinations of the coefficients defining the effective AMF for uncoupled core/extended mantle. Model Wahr (1982) [228] Eubanks (1993) [82] Dobslaw and Dill (2018) [68] This work

αma

αmo

1.118 1.098 1.101 1.112

1.61 1.5913 1.610 1.606

Because of the deformation induced by the fluid load, the matter term is partially compensated and is almost reduced to the one for a rigid Earth. But the motion term is amplified by more than 50 % of its “rigid Earth” value. Constant term of the angular momentum functions Matter terms have generally a non-zero mean, associated to the mean distribution of the considered fluid layer in the terrestrial frame. These biases reflect the fact that the principal axes of the fluid layer do not lie on the axes of the TRF. For the oceans, a non-zero mean of about 5 mas is noticeable in the current term (see Figure 8.6). It can either reflect a secular cycle or result from model errors. Actually the most important non-zero mean pertains to the axial wind term reaching 1028 kg m2 s−1 (that is onehundred thousandth of the Earth’s angular momentum) because of the super-rotation of the atmosphere. As large as they are, these mean components, because of their invariability, do not induce any observable effect on the polar motion and Earth’s rotation speed. Assuming their existence over decades, they impact definitively the polar motion, by introducing a constant shift with respect to the geographic pole. Only the variable part, keeping in mind that we deal with a sub-secular time scale, can cause an observable displacement of the pole, and is thus of interest in analyzing polar motion excitation.

9.2 Amplification of the prograde seasonal band | 173

Figure 9.1: Amplitude and argument of the transfer function T (σ) = p(σ)/χ e (σ) for the whole frequency range (left, in cycle/day) and seasonal band (right, in cycle/per).

9.2 Amplification of the prograde seasonal band Dominated by the seasonal cycle at the level of 15 mas and multi-annual processes at the level of 5 mas, the forcing is also present at weekly and monthly periods, to judge by the spectrum of Figure 9.2: these are rapid oscillations reaching an amplitude of a few mas. At a given frequency, the equatorial excitation describes an ellipse, composed of prograde and retrograde circular terms of more or less comparable amplitudes. Due to the resonance at Chandler frequency (+0.85 cpa), this equality disappears in polar motion, essentially composed of prograde terms, as evidenced by the complex Fourier spectrum of the C04 series (Figure 9.2). Indeed, according to the frequency transfer function drawn in Figure 9.1, the excitation is multiplied by a factor 5 at 1 cpa, but it is halved at −1 cpa. Therefore the polar motion is dominated by an annual prograde term (≈ 100 mas) and the Chandler mode itself (≈ 200 mas). This is the result of an excitation from 100 to 400 times smaller, of the order of 1 mas, which cannot be reduced to a particular frequency, but has to be considered as a random noise spanning a wide frequency range, of which the effective width depends on the nature of the noise. This problem is addressed in Chapter 11. At the moment, we want to know

174 | 9 Equatorial angular momentum balance from two days to decadal time scale

Figure 9.2: Complex Fourier spectra of the pole coordinates (bottom) and of the corresponding excitation (top) (1990–2018, C04 series). Due to the resonance at Chandler frequency, the polar motion is dominated by the terms at +0.85 cpy and +1 cpy. The plots at right present a zoom of the low frequency range between 0.1 and 1.5 cycle/year.

to which extent the hydro-atmospheric AMF globally reproduces the reconstructed excitation from observed pole coordinates in the range of time scales stretching from two days to some decades.

9.3 Methodology The geophysical or modeled polar motion can be obtained by integrating the geophysical excitation through (9.1); then it can be compared with the observed polar motion. However, this approach emphasizes the prograde terms of the excitation around the Chandler frequency, and it mitigates the other. Moreover, it requires the knowledge of initial conditions, causing a mixture of the free mode with forced part of the polhody. Therefore the integration will not be used, except in Chapter 11 when the Chandler wobble will be analyzed. The preferred approach is to compare the effective modeled excitation with the geodetic or observed excitation χG defined by the LHS of the Liouville equation, and thus to establish a kind of geophysical budget of the excitation. The modeled excitations or effective EAMF are deduced from the atmospheric, oceanic and hydrological AMF (χAe , χOe and χHe , respectively) from (9.1b). The atmo-

9.3 Methodology | 175

spheric series include the IB correction, which quite well accounts for the atmospheric–ocean coupling beyond 10 days. Anyway the ocean series take account of dynamical effects produced by the atmosphere and thermal heating. Eventually, we consider the additional impact of the continental water. The geodetic excitation is estimated from the pole coordinates p = x − iy by the digital filtering described below.1 Digitization of the observed excitation function Given the pole coordinates series with constant sampling time h, the digital version of the continuous excitation function χG = p + i/σ̃ c ṗ has to be built. This is the inverse problem. This operation introduces a distortion, which has to be known before one proceeds. We could approach ṗ by a two point difference, and thus compute χt = pt + i/σ̃ c (pt+h − pt−h )/2h, but this causes a loss of amplitude at short periods. So it is necessary to develop a more refined formula, as was done by Wilson and its collaborators from the 1970s [231, 229, 230]. The simplest way is to build a digital linear filter admitting the generic expression j=m

̂ χt(+h/2) = ∑ aj+n+1 pt+jh j=−n

(9.2)

from the n + m + 1 sampled values pt+jh . Here the estimation at t + h/2 instead of t is optional, and should be put between parentheses. So, the inverse transfer function transforming sampled pole coordinates in digital excitation reads in the frequency domain j=m

T̂ −1 (σ) = (e−iσh/2 ) ∑ aj+n+1 eiσjh . j=−n

(9.3)

For determining the coefficients ak with k = 1, . . . , n + m + 1, T̂ −1 (σ) is constrained to be equal to the inverse of the analog transfer function T(σ) appearing in (9.1a), namely T −1 (σ) =

σ̃ c − σ σ̃ c

(9.4)

for n+m+1 appropriate frequencies σk . This yields a system of n+m+1 linear equations to be solved for the n + m + 1 unknowns ak . For instance, considering ̂ χt+h/2 = a1 pt + a2 pt+h ,

(9.5)

1 In the following, the letter A means the atmospheric series, O the oceanic series, H the hydrological series and any other combination gives the corresponding combination of these series.

176 | 9 Equatorial angular momentum balance from two days to decadal time scale the transfer function of the digital filter is T̂ −1 (σ) = a1 e−ihσ/2 + a2 eihσ/2 ,

(9.6)

and its identification with (9.4) for angular frequencies σ = σc and σ = 0 gives the coefficients a1 =

σ̃ c − σc − σ̃ c eiσc h/2 , 2iσ̃ c sin(σc h/2)

σ̃ − σc − σ̃ c e−iσc h/2 a2 = c . −2iσ̃ c sin(σc h/2)

(9.7)

After appropriate approximations, these coefficients get in close agreement with the those of the Wilson filter [229], given by χt+h/2 =

ie−iπfc h ̃ (pt+h − pt eiσc h ) . σ̃ c h

(9.8)

The ratio T̂ −1 (σ)/T −1 (σ) is plotted in Figure 9.3 (yellow line) for h = 0.5 and in the frequency range [−182 cpy, +182 cpy] associated with the common polar motion (σc = 0.843 cycle/year and quality factor Q = 100). It can be numerically checked that T̂ −1 (σ) tends to T −1 (σ) when h tends to zero. The ability of the Wilson filter to reproduce the inverse transfer function is spoilt for periods approaching the time sampling (of course frequencies larger than the Nyquist frequency 1/(2h) are not meaningful). So, taking h = 0.5 day, Wilson’s digitization produces an amplitude ratio T̂ −1 (σ)/T −1 (σ) stretching from 0.97 for 100 cpy (or 3.6 days) to 0.9 for two days (see Figure 9.3). This defect can be overcome by taking the Wilson and Chen filter [230] designed for high frequency range, or a more simple filter, as shown (Bizouard 2019) in Figure 9.3, which we designed according to the seven point formula j=3

χt̂ = ∑ aj+4 pt+jh . j=−3

(9.9a)

Its transfer function is equated to the continuous one for angular frequencies σ1 = −0.5Ω, σ2 = σc − Ω/50, σ3 = σc − Ω/400, σ4 = σc , σ5 = σc + Ω/400, σ6 = σc + Ω/50, σ7 = 0.5Ω, giving for h = 0.5 day: a1 = 0.03602364 − i 4.10880996 ,

a2 = −0.06139002 + i 27.91398968 ,

a3 = −0.34789453 − i 112.41304908 ,

a4 = 0.48072169 + i 0.00370900 ,

a5 = 1.25795968 + i 112.40654887 ,

a6 = −0.46013470 − i 27.91026465 , a7 = 0.09471425 + i 4.10787616 .

(9.9b)

9.3 Methodology | 177

Figure 9.3: Ratio of the digital inverse transfer function to the analytical one (T ̂ −1 (σ)/T −1 (σ)) for time sampling h = 0.5 and considering three digitizations of the geodetic excitation: three point formula, Wilson (1985) and Bizouard 2019.

The three point formula yields not only too large a loss of high frequencies range, but also distorts the phase close to the Chandler frequency. Spectral uncertainty of the geodetic excitation At a given frequency, the uncertainty on geodetic excitation εG (f ) and the one on polar motion εp (f ) are related by 󵄨󵄨 f ̃ − f 󵄨󵄨 󵄨 󵄨󵄨 εG (f ) = εp (f )󵄨󵄨󵄨 c 󵄨. 󵄨󵄨 f ̃ 󵄨󵄨󵄨 c

(9.10)

In Section 3.6.3, we have seen that the spectral uncertainty stems from two kinds of error: one is random and decreases with the duration P of the time series, the other is systematic and its level does not depend from P. Considering the best pole coordinate time series resulting from contemporaneous GNSS data processing (like C04 after 2000), the random uncertainty is given by (3.53), that is, 30 µas/√fP where P is the time span of the data, and the systematic error can be maximized by 40 µas at

178 | 9 Equatorial angular momentum balance from two days to decadal time scale seasonal and inter-annual periods (see the spectrum of Figure 3.6). As a result 󵄨󵄨 f ̃ − f 󵄨󵄨 30 µas 󵄨 󵄨󵄨 εG (f ) = 󵄨󵄨󵄨 c + ∑ 40 µas δ(f − fi )) , 󵄨( 󵄨󵄨 f ̃ 󵄨󵄨󵄨 √fP c i

(9.11)

where the fi correspond to annual and some specific inter-annual specific frequencies. Data used e The modeled series χMOD are atmospheric AMF series, possibly combined with consistent OAMF and HAMF, leading to two data sets: i) NCEP+ECCO+CPC, and ii) ECMWF+MPIOM+LDSM (see Table 8.2). In order to estimate the geodetic excitation χG , pole coordinates are taken from the combined series C01 and C04, for they benefit from three essential qualities for carrying out geophysical interpretation: sampling at constant rate, maximal accuracy, and long time span over several decades, all of these criteria being not simultaneously fulfilled for series associated with a unique technique. The C04 pole coordinates do not have only the precision of the best GNSS series (since 1995), but they are provided without discontinuity since 1962 in consistent terrestrial frames, now the ITRF 2014 (see Section 3.6.3). Intricacies of the overall comparison After digitizing of χG according to the Wilson filter studied in the former section, these e series are compared with χMOD over the longest available span, that is, from the beginning of AMF determinations (1950/1960), to nowadays. Most often three successive e e comparisons will be done: χG /χAe , χG /χAO , χG /χAOH . Indeed, as the atmospheric excitation is the most important, and also the one better estimated, this permits one to know to what extent the oceanic forcing, and then hydrological forcing, refines the geophysical budget of χG . A global comparison over the whole span, using for instance correlation coefficients, is not appropriate, for it mingles heterogeneous geodetic data. With the advent of space geodesy, the temporal resolution and precision of the pole coordinates have strongly increased since the 1950s. From the 1970s, processing of the Doppler measurements showed the existence of inter-annual oscillations (10 mas). Then, in the 1980s, the processing of SLR and VLBI observations clearly reveal short term fluctuations (below 100 days).2 Hydro-meteorological sciences did not present e such a “technical breakthrough”: the series χMOD continuously improve along time, benefiting from the intensification and spatial extension of the meteorological observations, partly thanks to the satellite observations. Thus, as evidenced by wavelet transforms in Figure 9.4, the patterns of the modeled excitation do not present any rupture, whereas geodetic excitation is clearly defective before 1985 for periods bee low 100 days. Before 1985, only the seasonal prograde band of χG agrees with χAOH , 2 Rapid variations in the LOD were detected a little earlier by Feissel and Gambis [88].

9.3 Methodology | 179

Figure 9.4: Wavelet decomposition of geodetic excitation (G, left) and effective hydro-atmospheric excitation (NCEP+ECCO+CPC, right) for periods between 0 and 500 days over the span 1962–2008. Amplitude in mas. Notice the stable prograde seasonal band, the irregular “rapid” band (below 100 days), and the striking discrepancy between modeled and geodetic excitation before 1985.

and seems to merit consideration. Indeed, associated with polar motion variations of 100 mas and more, this is the best determined range in the 1960–1980 when the uncertainty was above 10 mas. With the advent of the GNSS in 1993, the uncertainty of χG decreases, becoming smaller than the one of the fluid excitation. e The level compliance of χG with χMOD will be estimated over appropriate periods by using various statistical or spectral methods described in Appendix G, namely: – Estimation of sample correlation coefficient. – Estimation of rate of the variance of the geodetic excitation explained by the modeled one, expressed by [var(G) − var(G − MOD)]/var(G). e – Spectral comparison, reading χG (σ) ≡ χMOD (σ), in order to identify the physical processes at stake. e – Spectral coherence between χG and χMOD , giving a kind of complex correlation for each frequency band. – Allan deviation analysis permitting one to characterize the stochastic processes enclosed in these series. As shown by Figure 9.4, polar motion cycles are irregular, even transient, so that spectral analysis has to be completed by numerical tools isolating irregularities in time: wavelet transform, sliding window fit, Singular Spectrum Analysis (SSA). Data pre-treatment As already pointed out, over the considered period the geophysical series present nonzero means, which do not produce any observable effect on the polar motion. Therefore they are removed for AMF series. The corresponding biases of the geodetic excitation are eliminated as well.

180 | 9 Equatorial angular momentum balance from two days to decadal time scale As modeled AMF are sampled differently, sometimes at a higher rate than the observed excitation, the series have to be homogenized by re-sampling them at the dates of the geodetic excitation given at equal time interval h. First a low-pass filter is applied with cut-off period equal to the Nyquist period 2h; then the filtered series are interpolated at the chosen dates. Because of the progressing quality of the astro-geodetic solution, path h is chosen according to the real temporal resolution of the pole coordinates over the considered period of investigation. So, whereas h cannot be smaller than 20 days from 1962 to 1984, we can take h ≤ 1 day from 1984 to present.

9.4 Overall budget In this section the geodetic excitation is derived from the C04 series sampled at 0hUTC and 12hUTC according to the seven point digital filter (9.9). The geophysical data are sampled at the corresponding dates, after the removing of possible diurnal or subdiurnal fluctuations.

9.4.1 Accuracy of geodetic and angular momentum excitations From the epoch 2000 the spectral random uncertainty of the geodetic excitation is given by the first member of Eq. (9.11). It is displayed in Figure 9.5 along with the random uncertainties of the effective hydro-atmospheric AMF for a sampling duration of P = 1 year. These are provided in Chapter 8 for the rough AMF (of which the matter term has to be multiplied by the coefficient ≈ 1.1 and the motion term by the coefficient ≈ 1.6): by Eqs. (8.33) and (8.34) for the atmosphere, (8.40) and (8.42) for the oceans, and (8.45) for the continental fresh water. The uncertainty of the geodetic excitation is much smaller than the modeled one except for periods below one month.

9.4.2 Overall comparison The global agreement is quantified by correlation coefficients and the rate of variance of χG explained by the fluid layer excitation. As justified earlier, the analysis is not carried out over the whole time span (1950–2019), but over successive periods of about 10 years, representative of the pole coordinate uncertainty and its evolution. Correlations and rate of explained variance given in Table 9.2 show both the convergence of the polar motion and its modeling along time, and to which extent the combined effect of the atmospheric and oceanic mass transports matches the observed excitation: for our epoch correlations are larger than 70 % (x component) and 80 % (y component), explained variance rate larger than 50 % (x component) and 70 % (y component). Inland fresh waters do not significantly improve the budget. It can be noticed that the

9.4 Overall budget | 181

Figure 9.5: Spectral uncertainty of geodetic excitation compared with the random spectral uncertainty of the angular momentum functions for sampling duration P = 1 year. For other values of P in years, the plotted random error has to be divided by √P.

Table 9.2: Compliance of geodetic excitation (G) with effective angular momentum functions of the fluid layer composed of the atmosphere alone (A), atmosphere and oceans (AO), atmosphere, oceans and continental fresh water (AOH): correlation coefficient, rate of explained variance of G by fluid layer excitation, and standard deviation of the differences for two coupled hydro-atmospheric models. Model 1: A-NCEP, O-ECCO and H-CPC. Model 2: A-ECMWF, O-MPIOM and H-LSDM. Correlation coef. A AO AOH

Explained var. % A AO AOH

Standard dev. G G-A G-AO

Model 1 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2

0.12 0.31 0.31 0.35 0.35 0.57 0.62 0.55 0.60

0.17 0.42 0.49 0.42 0.44 0.71 0.75 0.68 0.72

0.30 0.50 0.56 0.44 0.44 0.74 0.79 – 0.68

−3 5 6 11 11 31 37 29 35

−2 15 23 17 19 50 56 45 51

7 24 30 19 18 55 62 – 44

26 21 21 46 46 27 27 29 29

26 21 21 43 43 22 21 24 23

26 20 19 41 41 19 18 21 20

25 18 18 41 41 18 16 – 21

Model 1 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2

0.56 0.69 0.61 0.49 0.51 0.71 0.70 0.75 0.75

0.60 0.75 0.67 0.56 0.62 0.84 0.86 0.86 0.89

0.54 0.72 0.58 0.56 0.61 0.83 0.88 – 0.89

29 48 34 23 25 50 48 56 56

35 55 41 31 37 70 72 73 78

27 51 22 31 36 68 77 – 78

35 36 36 60 60 44 44 46 46

30 26 29 53 52 31 31 30 30

28 24 28 50 48 24 23 23 21

30 25 32 50 48 24 20 – 21

G-AOH

x component (mas)

1966.0–1976.01 1976.0–1986.01 1986.0–1997.0 1997.0–2007.7 2008.0–2018

y component (mas) 1966.0–1976.01 1976.0–1986.01 1986.0–1997.0 1997.0–2007.7 2008.0–2018 1

Elimination of the spectral band below 20 days.

182 | 9 Equatorial angular momentum balance from two days to decadal time scale coupled ECMWF–MPIOM–LSDM model better matches the geodetic excitation than NCEP–ECCO–CPC. According to the integral (8.5), the component y of the pressure term results in greater measure from the continental regions (λ ∼ π/2: Central Asia or λ ∼ 3π/2: America) than from the oceanic zone surrounding the great circle of Greenwich (Atlantic and Pacific oceans). As the strongest variations of the atmospheric pressure are located on the continents, the y component is more dominated by the atmospheric pressure than the x component. In contrast to the y component, x is more influenced by the oceanic circulation. The ocean angular momentum being more uncertain than the atmospheric one, the AO excitation better matches the geodetic excitation for the y component than for x (correlation of 0.89 for y instead of 0.72 for x over the period 2008–2019 for coupled model 2).

9.4.3 Coherence over the time span 2000–2019 This comparison is deepened by estimating the spectral coherence between χG on the e e one hand and χAe , χAO , χAOH on the other hand. With the exception of ice sheets, transport of hydro-atmospheric mass spans mainly from 12 hours to a few years. Beyond 10 years, the fluid layer effect amounts to a few mas, and partly accounts for decadal fluctuations (Markowitz wobble; see Section 10.5). Here the coherence function is examined from 2000 to 2019, where data and models are the most accurate. Judging by Figure 9.6, the ocean contribution improves the coherence with χG in the whole subdecadal range. The effect is spectacular for the seasonal band (see plot from −4 to 4 cpy) where the inconsistency of the atmospheric excitation disappears by adding χOe . The role of hydrological processes, only shown for the triplet ECMWF–MPIOM– LSDM (for the CPC model ends in 2007), is evident in the seasonal and low frequency bands, where the phase coherence is improved by adding χHe . For rapid fluctuations ranging from 0.01 cpd (100 days) to 0.5 cpd (2 days) the hydrological effect fades. In this respect the coherence function G/AOH is only displayed on the band [−5, +5] cpy. Complementary plots for G/AO over the whole band [−0.5, +0.5] cpd favor the representation of high frequency edges. Then the coherence function G/AO shows the key role of the oceans at a time scale smaller than 100 days. Meanwhile, at frequencies near 0.5 cpd, the combination NCEP+ECCO is less coherent with χG than the pair ECMWF+MPIOM, and the coherence amplitude drops for two possible reasons: firstly the sampled polar motion is too smoothed (in principle our digital filtering does not produce attenuation on excitation), secondly the AMF uncertainty, mostly resulting from the oceans, is about 1/√P ≈ 0.2 mas for duration sampling of P = 19 years (see Figure 9.5), which is of the order of magnitude of the geodetic excitation in this frequency range. These first results must be tempered in view of the disagreement between the AMF series. In addition to the differences noticed in the rapid band, the hydrological CPC

9.4 Overall budget | 183

Figure 9.6: Coherence (amplitude and phase) of the observed excitation (G) with the modeled one (A/AO/AOH) over the period 2000–2019 for two sets of hydro-atmospheric coupled models.

184 | 9 Equatorial angular momentum balance from two days to decadal time scale

Figure 9.7: Comparative Allan deviation analysis of equatorial excitations over the period 2000– 2019. Left panels: the geodetic excitation G is compared with fluid layer excitations A + O (ECMWF+MPIOM) and A + O + H (ECMWF+MPIOM+LSDM). Right panels: the difference G − A − O is compared with H.

and LSDM models present notable discrepancies, of up to a few mas in the seasonal band, the ECCO and MPIOM ocean models to a lesser extent (see the Fourier spectrum of Figure 10.7). 9.4.4 Allan deviation analysis over the period 2000–2019 Except for the seasonal, diurnal and semi-diurnal components which have a quasiharmonic nature, the variations of the geodetic and geophysical excitations look like stochastic processes, which can be characterized by carrying out an Allan deviation analysis (Appendix G.5). This one is reported in Figure 9.7, left panel, for the series χG , e e χAO and χAOH (after removal of the mean seasonal oscillation by least square fit over the whole period) considering the geophysical A−ECMWF, O−MPIOM, H−LSDM models over 2000–2019. The trend and noise level are consistent, and the improvement brought about by the hydrological series is noticeable for seasonal and biennial time scales. Notice that from 100 to 500 days the log–log scale slopes s󸀠 of the geodetic and geophysical signals lie in the interval [−0.5, −0.25]. From Table G.1, the corresponding power spectral density is f α with α ∈ [−0.5, 0] (α = −2s󸀠 − 1), and does not exactly

9.5 Elliptical polarization of the excitation function

| 185

comply with the white noise assumption for the Chandler band (α = 0) proposed in Chapter 5. As shown by Figure 9.7 (right panel), the residuals G − AO do not reproduce so well the pure hydrological series, especially in the rapid band (2–100 days), where G − AO shows much higher spectral noise.

9.5 Elliptical polarization of the excitation function 9.5.1 Are prograde and retrograde terms interrelated independently from frequency? Considering (9.1a), the interest of splitting prograde and retrograde components is evident, for prograde terms having periods close to 433 days are amplified, while the transfer function T(σ) of the retrograde terms has a modulus smaller than 1, tending towards 0 with increasing absolute value of the frequency. Within this frequency domain approach, prograde and retrograde terms of the excitation are considered as specific to the frequency, and are estimated in the light of their effect on the polar motion. To the best of our knowledge, no study has ever raised the question whether the prograde and retrograde terms could be interrelated independently from the frequency. Could it be that, whatever the frequency is, the elliptical path resulting from the addition of prograde and retrograde terms presents the same privileged direction? It had been noted in studies going back to the early twentieth century [201, 153, 126] that the annual geodetic excitation is an ellipse, of which the semi-major axis has a stable ellipticity and longitude (80–90° east). The same characteristics were found later in the atmospheric excitation [126]. We can extend the computation of ellipticity (or retrograde to prograde term ratio) and longitude of the semi-major axis to the whole set of frequencies representing the polar motion and its excitation. We aim at showing that the obtained values tend to privilege the 80–90° east polarization, and a mean ellipticity of about 0.8. 9.5.2 Prograde and retrograde terms and elliptical motion at a given frequency Over a given time interval complex coordinates (in the TRF) of the equatorial excitation can be decomposed into a complex Fourier series reading χ(t) = ∑ a+σ eiσt + a−σ e−iσt + χ0 ,

(9.12)

σ>0

where a+σ is the complex amplitude of the prograde (counter-clockwise) term of the angular frequency σ, a−σ is the analogue quantity specifying the retrograde term of the same frequency; χ0 is a constant term. In the time domain, prograde and retrograde terms at a given frequency are determined by χσ+ (t) = a+σ eiσt = A+σ eiΦσ eiσt ; +

χσ− (t) = a−σ e−iσt = A−σ eiΦσ e−iσt , −

(9.13)

186 | 9 Equatorial angular momentum balance from two days to decadal time scale where A+σ , A−σ and Φ+σ , Φ−σ are for the amplitude and phase, respectively. The total prograde and retrograde components in the time domain are determined by adding the individual frequency terms of the Fourier decomposition: χ + (t) = ∑ a+σ eiσt ;

χ − (t) = ∑ a−σ e−iσt .

σ>0

σ>0

(9.14)

Considering Eqs. (9.13), we see easily that prograde and retrograde terms at a given frequency are related by χσ− (t) =

a−σ A−σ i(Φ+σ +Φ−σ ) + ∗ ∗ + (χ (t)) = e (χσ (t)) . σ (a+σ )∗ A+σ

(9.15)

On the other hand, the total contribution at frequency σ is χσ (t) = χσ+ (t) + χσ− (t), that is, χσ (t) = ei(Φσ +Φσ )/2 (A+σ ei(Φσ −Φσ )/2 eiσt + A−σ ei(−Φσ +Φσ )/2 e−iσt ) . +

−

+

−

+

−

(9.16)

So, by doing the coordinate transformation (x, y) → (x 󸀠 , y󸀠 ) associated with the axial angle of rotation ασ = (Φ+σ + Φ−σ )/2 , +

(9.17)

−

the complex coordinate is multiplied by e−i(Φσ +Φσ )/2 and reads in the new system χσ󸀠 (t) = A+σ ei(Φσ −Φσ )/2 eiσt + A−σ e−i(Φσ −Φσ )/2 e−iσt . +

−

+

−

(9.18)

Consider θσ = (Φ+σ − Φ−σ )/2; then splitting real and imaginary parts of the former expression, we obtain χσ󸀠 (t) = (A+σ + A−σ ) cos(θσ + σt) + i(A+σ − A−σ ) sin(θσ + σt) ,

(9.19)

showing that the equatorial component of the frequency σ describes an elliptical path with the semi-major axis A+σ + A−σ directed along x 󸀠 and the semi-minor axis |A+σ − A−σ | directed along y󸀠 (see Figure 9.8). The orientation of the ellipse is determined with respect to the x axis by the angle ασ = (Φ+σ + Φ−σ )/2. 9.5.3 Example of the seasonal excitation Practically the prograde and retrograde components at a given frequency are determined by fitting the model χσ (t) = (a+r + ia+i )eiσt + (a−r + ia−i )e−iσt

(9.20)

9.5 Elliptical polarization of the excitation function

| 187

Figure 9.8: Ellipsoidal path of the excitation at a given frequency σ.

to the observations, where ar and ai are for the real and imaginary parts, respectively, with the time origin taken as the epoch J2000. The estimated parameters a+r , a+i , a−r , a−i yield 2

2

√(a− )2 r

2 (a−i )

A+ = √(a+r ) + (a+i ) , A = −

+

,

Φ+ = arctan

a+i + π (1 − sign(a+r ))/2 , a+r

a− Φ = arctan −i + π (1 − sign(a−r ))/2 . ar

(9.21)

−

Besides the diurnal and semi-diurnal oscillations—their study is reported in Chapter 12—the most prominent cycle of the excitation is seasonal. From year to year its amplitude instability does not exceed 5 mas around a mean of 15 mas and its phase presents a variation smaller than 30 degree (see Section 10.3). Therefore, the annual, semi-annual and terannual elliptical paths averaged over several years will be taken as a starting example. The model (9.20) composed of the harmonics at 1, 1/2, 1/3 year and of linear trend is adjusted over the period 1997–2007. This seasonal–linear adjustment is redone for the corresponding angular momentum function of the atmosphere (ECMWF) and the oceans (MPIOM) and of the fresh water layer (LSDM). The seasonal components of the χ(t) vectors in the equatorial plane are reconstituted in Figure 9.9; in Table 9.3 we report the amplitude and phase of the annual prograde and retrograde terms. Apart from the terannual term, the atmospheric contribution is strongly polarized towards ∼ 70°–90° in the x, y equatorial plane, as shown by studies of the 1970s (results are

188 | 9 Equatorial angular momentum balance from two days to decadal time scale

Figure 9.9: Mean annual, semi-annual and terannual elliptical path of the polar motion excitation (period 1997−2007): deduced from pole coordinates (G) and associated with atmospheric A−ECMWF, oceanic O−MPIOM, hydrological H−LSDM angular momentum series. We also display the total fluid layer contribution AOH and its matter term AOH.ma. Plain arrows represent amplitude and phase (at January 1) of the prograde terms, dot arrows those of the retrograde terms. The numbers 1, 2, 3 . . . along the annual ellipses correspond to the number of the elapsed month from the first of January.

9.5 Elliptical polarization of the excitation function

| 189

Table 9.3: Annual components of the geodetic excitation and of the atmospheric, oceanic and hydrological AMF estimated over the period 1997−2007. Series A−ECMWF, O−MPIOM and H−LSDM.

χG χO χA χAO χAOH χH

A+ (mas)

Φ+

A− (mas)

Φ−

α = (Φ+ + Φ− )/2

16.74 ± 0.72 5.65 ± 0.42 17.18 ± 0.40 20.00 ± 0.64 19.64 ± 0.69 4.51 ± 0.10

−59.3 ± 2.5° −15.8 ± 4.3° −83.8 ± 1.3° −68.6 ± 1.8° −55.6 ± 2.0° 32.5 ± 0.7°

9.98 ± 0.75 2.99 ± 0.47 14.54 ± 0.41 11.82 ± 0.54 10.66 ± 0.70 8.17 ± 0.10

227.0 ± 4.3° 56.7 ± 9.1° 258.8 ± 1.7° 264.2 ± 2.6° 222.0 ± 3.8° 145.4 ± 0.7°

84° 20° 87° 100° 83° 89°

gathered in Figure 6.4 of [201]). The fresh water AMF has almost a linear polarization towards ∼ 90° East (annual term, the other harmonics are below 1 mas). The ocean seasonal contribution is elliptically polarized towards 0°–20°; but being three times smaller than the atmospheric effect, it is not surprising that the geodetic excitation is also polarized like the atmospheric one. Notice also the good agreement between the total fluid layer contribution and the geodetic elliptical path at the annual period, but the disagreement at semi-annual and terannual periods.

9.5.4 Linear dependence of prograde and retrograde parts According to (9.15) and (9.17), we have χσ− (t) =

A−σ i2ασ + ∗ e (χσ (t)) . A+σ

(9.22)

If the polarization angle of the annual excitation, namely α ∼ 80°, tends to be the same for the other frequency components, then, by adding all individual frequency components corresponding to Eq. (9.22), we will find as a result χ − (t) = e2iα ∑ σ

A−σ + ∗ (χ (t)) , A+σ σ

(9.23)

where χ − (t) is the total retrograde part of the excitation. Moreover, assuming that the ratio Rσ = A−σ /A+σ slightly depends on the frequency, we have the relation χ − (t) ≈ (χ + (t)) R e2iα , ∗

R>0,

(9.24)

where χ + (t) is the total prograde part of the excitation. The former relation means that prograde and retrograde parts are correlated in the time domain. To determine this possible correlation, χ − (t) and χ + (t) are first reconstructed from the complex direct Fourier transform of χ(t) over a given period: χ − (t) is obtained by adding up the negative frequency terms and χ + (t) by adding up the positives ones according to Eqs. (9.14).

190 | 9 Equatorial angular momentum balance from two days to decadal time scale

Figure 9.10: Retrograde part χx− (t) of the excitation against time compared with −χx+ (t). Geodetic (bottom plot) and atmospheric (upper plot) excitations split in two frequency bands: 2–182 days (a), and above 182 days (b). In both frequency bands a negative prograde variation along x axis reflects the retrograde ones. Time in modified Julian date.

Then, considering α ∼ 80°, χ − (t) is compared with (χ + (t))∗ R ei160° ∼ −R (χ + (t))∗ . So, as far as (9.24) is valid, we should have χx− (t) ∼ −Rχx+ (t). Equivalently the y components χy− (t) and χy+ (t), which are nothing more than the x components out-of-phased by π/2 and −π/2, respectively, are related by χy− (t) ∼ Rχy+ (t). The comparison of χx− (t) and −χx+ (t) is shown in Figure 9.10 for geodetic and the ECMWF-atmospheric excitations, split in two spectral bands by a Vondrak low-pass filter [221]: seasonal and inter-annual band above 100 days (admittance ≥ 95 % for period larger than 182 days), and rapid band 2–100 days (admittance ≤ 5 % for periods larger than 182 days). The “rapid” band is shown on the whole year 2006, and the low frequency band from 1997 to 2007. The match between prograde and retrograde signals can be noticed visually for each case, confirmed by correlation coefficients between χx− (t) and −χx+ (t) of ∼ 0.5 below 182 days and at least equal to 0.7 above 182 days. The angle α giving the maximum correlation between χ − (t) and (χ + (t))∗ e2iα can be derived by computing the complex correlation between χ − (t) and (χ + (t))∗ . Indeed, the phase of the complex correlation is precisely 2α. The correlation amplitude CR and phase 2α are reported Table 9.4 for geodetic excitations for both triads of hydroatmospheric series: A-ECMWF/O-MPIOM/H-LSDM on the one hand, and A-NCEP/OECCO/H-CPC on the other hand. Given the high sample size (n ≈ 3650, corresponding to 10 years of data sampled at 1 day interval), a correlation coefficient above 0.3 has a p value lower than 0.01, meaning that such a correlation is significant at 99 % level and more. So, a significant correlation pertains not only to geodetic (α = 75°, CR = 0.50) and atmospheric excitations (α = 85°, CR = 0.61 for ECMWF), but also to a hydrological one (α = 90°, CR = 0.81 for LSDM). The oceanic excitation is slightly less polarized (α = 50°, CR = 0.33 for MPIOM but α = 65°, CR = 0.48 for ECCO). The observed correlations partly result from the prominent annual oscillation, of which the elliptical motion presents the favored orientation of ∼ 80° according to the

9.5 Elliptical polarization of the excitation function

| 191

Table 9.4: Complex correlation (amplitude CR and phase 2α) over the period 1997–2007 between conjugate prograde and retrograde excitations for geodetic excitation (G) and geophysical angular momentum functions: atmospheric (A), oceanic (O), hydrological (H), their sum (AO, AOH), corresponding matter (ma) and motion term (mo). For fluid layer excitations, the first line corresponds to the A-ECMWF/O-MPIOM/H-LSDM circulation models, the second line to the A-NCEP/O-ECCO/H-CPC circulation models. The phase of the complex correlation can be interpreted as the double of the elliptical polarization angle α.

G A O H AO AOH A ma O ma AO ma AOH ma A mo O mo AO mo AOH mo

Total excitation CR 2 α (°)

Seasonal harmonics removed CR 2 α (°)

0.50 0.61 0.61 0.34 0.48 0.80 0.91 0.42 0.50 0.43 0.54 0.77 0.75 0.33 0.48 0.57 0.64 0.55 0.67 0.09 0.08 0.35 0.27 0.17 0.04 0.17 0.04

0.46 0.42 0.41 0.37 0.48 0.63 0.75 0.34 0.42 0.33 0.44 0.48 0.46 0.32 0.48 0.37 0.48 0.34 0.51 0.11 0.07 0.36 0.21 0.10 0.03 0.10 0.03

2 × 75 2 × 85 2 × 75 2 × 60 2 × 70 2 × 90 2 × 80 2 × 85 2 × 80 2 × 80 2 × 80 2 × 75 2 × 75 2 × 49 2 × 70 2 × 75 2 × 75 2 × 70 2 × 75 2 × 40 2 × 60 2 × 34 2 × 34 2 × 16 2 × 55 2 × 16 2 × 55

2 × 75 2 × 80 2 × 75 2 × 65 2 × 75 2 × 95 2 × 75 2 × 75 2 × 80 2 × 75 2 × 80 2 × 75 2 × 75 2 × 50 2 × 70 2 × 65 2 × 75 2 × 65 2 × 75 2 × 65 2 × 60 2 × 38 2 × 38 2 × 34 2 × 80 2 × 34 2 × 80

former section. Yet, as the seasonal harmonic model of the former section is removed from the geodetic and fluid layer excitations, the correlations remain important (see Table 9.4). So, the polarization is not only a feature of the seasonal part of the signal, but also of its irregular variations. The separate analysis of matter and motion terms shows that the maximum correlation is obtained for the atmospheric (CR = 0.8) and hydrological (CR = 0.9) matter terms (hydrological excitation itself is a matter term), whereas prograde and retrograde parts of motion term are not significantly correlated (or slightly correlated for

192 | 9 Equatorial angular momentum balance from two days to decadal time scale the ocean motion term). So, the elliptical polarization towards ∼ 80° of the geodetic excitation can be attributed to the atmospheric and hydrological matter terms. The geophysical A-ECMWF/O-MPIOM/H-LSDM and A-NCEP/O-ECCO/H-CPC trio yield globally consistent results. However, we notice significant differences pertaining to oceanic and hydrological excitation (correlation amplitudes diverging by more than 0.1, and phases up to 40°). 9.5.5 Statistical distribution in spectral domain As pointed out above, the elliptical polarization of the excitation implies that prograde and retrograde circular oscillations at any frequency σ smaller than 0.5 cpd act to obey the condition χσ− (t) ≈ Rσ (χσ+ (t)) e2iασ ∗

(9.25)

with privileged values of ασ and Rσ . Considering spectral complex components a+σ = + − A+σ eiΦσ and a−σ = A−σ eiΦσ obtained by a complex direct Fourier transform, the former relation reads also rσ =

a−σ ≈ Rσ e2iασ . (a+σ )∗

(9.26)

Considering the geodetic excitation and the atmospheric ECMWF model, the Rσ and ασ values were computed for the period 2000−2014, corresponding to more precise pole coordinates than prior to 2000 [16]. The initial sample size of 2557 points sweeps the frequencies from 2×10−4 to 0.5 cycle/day. But some of these ratios are not significant in the light of the uncertainty which affects the spectral coefficients. They are eliminated according to the uncertainty level of the geodetic excitation, which is taken as the random part of (9.11): εG (f ) =

0.03 mas f − fc̃ , √Pf fc̃

(9.27)

where f means the frequency, P the period of study, and fc̃ = σ̃ c /2π is the complex Chandler frequency. The ratio rσ = a−σ /(a+σ )∗ is considered as far as the amplitude A+σ is larger than εG (f ), and frequencies not satisfying this criterion are rejected. The same criterion is applied to fluid layer excitations, for which we do not have a sound assessment of the errors. For the selected frequencies, the complex ratio rσ = Rσ e2iασ is estimated. In Figure 9.11 we plot Rσ and ασ along the frequency σ for geodetic (G), ECMWF-atmospheric (A) and combined ECMWF-atmospheric/MPIOMoceanic (AO) excitations. This manifests privileged distributions with 1 ≲ Rσ ≲ 2 and 50° ≲ ασ ≲ 150° whatsoever the frequency band is, except for frequencies approaching the Nyquist frequency of 0.5 cpd of the polar motion series. Because of

9.5 Elliptical polarization of the excitation function

| 193

Figure 9.11: Plots of Rσ (retrograde to prograde amplitude ratio) and ασ in degrees (elliptical orientation with respect to the x axis) as a function of the frequency σ for the geodetic, ECMWFatmospheric, and ECMWF-atmospheric/MPIOM-oceanic excitations. Period 2000–2014.

this proximity, the shift of ασ towards 180° noticed in geodetic excitation could be considered as doubtful, but it is accounted for the combined AO excitation, of which the initial sampling is 4-daily (before low-pass filtering and interpolation at the daily time step). Complementary insight is brought about by the histograms of Rσ and ασ distributions, displayed in Figure 9.12 for the geodetic and atmospheric excitations. The histograms corresponding to the combined fluid layer excitations (AO or AOH) are very similar to the one of χG , therefore are not displayed. The statistic distribution is envisaged in two cases: i) for the whole spectrum encompassing periods from 2 days to 14 years; ii) for periods beyond 10 days (frequency smaller than 0.1 cpd). With the mean value μ ∼ 80° and standard deviation σ ∼ 40°, the histogram of ασ has a bell form, which looks like the corresponding normal distribution 𝒩 (μ; σ). The agreement with the normal law seems to be reinforced by selecting frequencies below 0.1 cpd. On the other hand we found empirically that the amplitude ratio Rσ can be approximated by the gamma probability law fΓ (x, k, θ) = e−x/θ

xk−1 , θk Γ(k)

(9.28)

194 | 9 Equatorial angular momentum balance from two days to decadal time scale

Figure 9.12: Histograms of the ratio Rσ = |a−σ /(a+σ )∗ | and of ασ for the geodetic and atmospheric excitation. Left panels: Rσ between 0 and 4 with 0.04 bin-width and gamma distribution law corresponding to the mean μ ∼ 1.5 and standard deviation σ ∼ 1 of the geodetic excitation (maximum for 1). Right panel: ασ between 0 and 180° with 5° bin-width, and normal law corresponding to the mean μ ∼ 80° and standard deviation σ ∼ 40° of the geodetic excitation. Upper plots correspond to the frequencies lower than 0.1 cpd. Initial sample size of 2557 points (before elimination of not significant terms).

where Γ(k) = ∫0 t k−1 e−t dt is the Euler Gamma function, θ is the scale parameter and k the shape parameter, related to the mean value μ and the standard deviation σ of the distribution by μ = kθ, σ 2 = kθ2 . In the case studied, the parameters k and θ are determined by μ ∼ 1.5 and σ 2 ∼ 12 , giving k = (σ/μ)2 ∼ 0.44 and θ = σ 2 /μ ∼ 0.66. The maximum is reached for equal amplitude ratio. The Pearson chi-squared test allows us to test the proposed probability laws. For both ασ and Rσ distributions, we estimated the reduced chi-squared, that is, ∞

χ2e =

(O − Ei )2 1 , ∑ i N −3 i Ei

where N is the number of bins (N = 36 for ασ , N = 25 for Rσ ), Oi is the observed distribution (expected outcome frequencies), Ei is the expected theoretical frequency for bin i, asserted by the null hypothesis. The number of degrees of freedom is the number of bins N minus 3 constraints: same mean, standard deviation and number of bins for sample and tested distributions. Both χ2e and the probability that χ2 is larger

9.6 Concluding remarks | 195 Table 9.5: Goodness of fit of Rσ and ασ histograms by gamma and normal law, respectively (having the mean and standard deviation of the corresponding samples). The lower is the probability P(χ2 ≥ χ2e ), the poorer is the fit. Below 0.1 % it can be rejected.

G A G f < 0.1 cpd A f < 0.1 cpd

Rσ (against gamma law) χ2e P(χ2 ≥ χ2e ) %

ασ (against normal law) χ2e P(χ2 ≥ χ2e ) %

4.23 3.60 1.26 1.44

6.12 6.49 2.93 1.58

0.00 0.00 18.25 8.29

0.00 0.00 0.00 1.80

than the estimated value are reported in Table 9.5. The null hypothesis is accepted if χ2e ∼ 1, the cases with 2 ≤ χ2e ≤ 3 are questionable, and those with χ2e ≥ 3 are rejected. So the ratio Rσ for f < 0.1 cpd clearly obeys the gamma law, but the slight skewness and sharpness of the ασ distributions, except in the case A, f ≤ 0.1 cpd, make them diverge from the normal law. Considering the geodetic excitation for f ≤ 0.1 cpd, the mean of ασ becomes 75° with the number of occurrences (∼ 44) significantly higher than the value expected from the normal distribution (∼ 28), given that the expected value obeys a Poisson probability law with a standard deviation of ∼ √28 ≈ 5. This makes the elliptical orientation of 75° even more probable than in the case of a normal distribution. The corresponding statistic distributions for the hydrological AMF, not represented here, are at odds with the atmospheric and geodetic ones: to Rσ corresponds a gamma probabilistic distribution centered on 1; the angular distribution ασ , centered on ∼ 95° follows a much sharper and asymmetric bell. This sharpness results from the fact that the hydrological excitation results from seasonal and inter-annual processes located on continental regions, making it much more polarized than the rapid components observed in geodetic and atmospheric/oceanic excitations. The gamma distribution Rσ centered on 1.5 and the bell form centered on ∼ 75° were also tracked down as regards the oceanic excitation. By selecting a frequency of f < 0.1 cpd, the influence of the matter term (resulting from bottom oceanic pressure variations) is highlighted; this shifts the maximum of the oceanic ασ towards ∼ 50°, as expected from Table 9.4, and it causes a strong skewness. However, as the fluid layer excitation is dominated by the atmospheric contribution, geodetic statistical distributions mostly reflect the atmospheric ones.

9.6 Concluding remarks Beyond two days to some decades, the mass redistributions taking place in the surface fluid layer explain up to 80 % of the variations observed in the equatorial excitation. The joint contribution of the atmosphere and oceans dominates the seasonal and sub-

196 | 9 Equatorial angular momentum balance from two days to decadal time scale seasonal time scale, where the impact of the fresh waters is restricted to the seasonal and longer periods. These ones tremendously ameliorate the budget at inter-annual and decadal periods. With the years this agreement improves and extends to rapid processes (from 2 to 100 days). This reflects concomitant progress of global circulation models and space geodetic techniques. We show that the retrograde and prograde parts of the polar motion excitation are related. This reflects not only an elliptical polarization towards 80° east (India, Tibet, Central Siberia), equivalently 180° + 80° = 260° east or 100° west (Central Northern America), but also a privileged amplitude ratio of about 1.5 between retrograde and prograde terms (not the case of the annual terms) with predominance of the retrograde terms. These characteristics clearly result from the air and continental water mass redistributions, and they are explained by the matter term of their associated angular momentum. Atmospheric Angular Momentum Function (AMF) polarization towards 80° east can be explained as far as continental zones of maximal hydro-atmospheric mass redistribution, whatsoever the frequency band is, are close to a great meridian circle, namely the one of longitude ∼ 80 ± 10° east. Actually, the largest continental contribution to the pressure term are located over Eurasia and North-America, as has been documented in [161] for several spectral bands namely, inter-annual (730–1825 days), annual (230−450 days), semiannual (150–230 days), terannual (90–150 days), and subseasonal/rapid (30–90 days). In the Chandler band the pressure term mostly results from longitudinal band 0°–145° (Eurasia) [240]. Moreover, a major contribution to the hydrological AMF in the seasonal band comes form basins located in South-Eastern Asia, South-America and NorthernAustralia, and southern Africa [160]; apart from the last one, these regions are near the great circle of longitude ∼ 80 ± 10°, so that the hydrological AMF tends to be maximum by crossing the great circle of longitude ∼ 80 ± 10°. Though the result is less significant, the oceanic excitation also presents an overall elliptical polarization towards a more western longitude, namely ∼ 65–75 ± 10° east for a reason which has not become elucidated, but this does not influence much the polarization of the geodetic excitation. The gamma law of the A− /A+ distribution is more difficult to account for. The predominance of the retrograde terms could partly result from the atmospheric normal mode Ψ13 , propagating towards the west (thus being retrograde) with a period of about 10 days, thus enhancing retrograde components of pressure term around 0.1 cycle/day. This effect will be explored in more detail in the next chapter. As elliptical fixed-frequency components tend to adopt the ratio A− /A+ ∼ 1.5, their ellipticity, defined as the relative difference between semi-major and semi-minor axis, that is, A+ + A− − |A+ − A− | |1 − A− /A+ | = 1 − A+ + A− 1 + A− /A+ tends to take the value 0.8.

9.6 Concluding remarks | 197

So, we detected a linear statistical dependence between retrograde and prograde parts of the polar motion excitation in time domain; equivalently elliptical fixedfrequency components act to take a favored orientation (80° east) and ellipticity (0.8), and thus conspire towards an overall elliptical polarization. This results from the air and continental water mass redistributions according to a process which has not fully been elucidated.

10 Rapid, seasonal, inter-annual, and decadal excitations The word, force, is merely an abstraction which we use for linguistic convenience. For mechanics, force is the relation of a movement to its cause. For physicists, chemists and physiologists, it is fundamentally the same. As the essence of things must always remain unknown, we can learn only relations, and phenomena are merely the results of relations. Claude Bernard in Introduction à la médecine expérimentale (1865)

10.1 Introduction After the overall diagnostic of the geodetic excitation function from two days to decades in the light of the hydro-atmospheric angular momentum function, this chapter will focus on each of the characteristic frequency ranges: i) rapid band between 2 and 100 days, ii) the quasi-seasonal band from 100 to 400 days, iii) inter-annual range from 500 days to 10 years, iv) decadal band and secular changes (over 10 years). Despite its large effect, the Chandler wobble excitation is just above the uncertainty level of the hydro-atmospheric AMF. So the equatorial angular momentum balance around Chandler frequency deserves a specific treatment, which will be presented in Chapter 11.

10.2 Rapid fluctuations 10.2.1 Overview Below seasonal period and its harmonics at 182 and 121 days, the hydro-atmospheric angular momentum shows fluctuations reflecting somehow weather changes at weekly and monthly time scale. As evidenced by the wavelet transform in Figure 9.4, these variations, with amplitude up to 30 mas, are sometimes stronger than the seasonal fluctuation, but much more irregular. As the Chandler period is remote, resonance fades, so that the corresponding polar motion or rapid polar motion does not exceed a few mas, that is, one hundredth of the main term, as shown in Figure 10.1 for the y pole coordinate. It was predicted in the 1970–1980s from refined estimates of the atmospheric angular momentum, making weekly variations appear [9]. As the precision of the pole coordinates was then only 2 mas, it was observed later, from 1985, thanks to the progress of VLBI and SLR [80, 22, 81, 156]. So, as often happened, the geophysical modeling had preceded the astrometric determination of EOP or boosted its progress. The rapid fluctuations were confirmed from the 1990s thanks to the contribution of GPS data processing [102, 158], which permitted a drop of uncertainty down to 0.2 mas and an increased temporal resolution of 24 hours. On the eve of the twenty-first century, improved circulation models have shown that oceans https://doi.org/10.1515/9783110298093-010

10.2 Rapid fluctuations | 199

Figure 10.1: An extract of the y pole coordinate (combined series C04) over 200 days of the year 2013 after filtering out the variations below 100 days by a Vondrak filter.

significantly contribute to rapid polar motion [159, 173], as already demonstrated by the coherence function in Figure 9.6. For data prior to 2000, atmospheric and oceanic excitations account for the rapid excitation with a coherence drop below 5 days [122]. But, for data after 2000, this lack of coherence fades, whereas the role of inland fresh waters is imperceptible (see Figure 9.6). During the winter 2005–2006, because of a rare destructive interference between Chandler and annual wobbles, the rapid fluctuations were naturally highlighted in the polar motion (Figure 10.2), appearing as consecutive loops with time span from 7 to 14 days [130]. At first sight, the transient formation of these loops seems quite irregular, but their periods are reminiscent of the fortnightly tide Mf (13.7 days). We can wonder weather the Moon would not trigger these fluctuations. Actually, the zonal lunar tide causes cyclic fluctuations of the sea level, and thus of the oceanic angular momentum, which is reflected in the polar motion by a term of about 200 µas at the period of 13.7 days. Whereas this fortnightly effect is modeled [98], its identification in pole coordinate series is not easy, the more it is mixed with the effect of the corresponding atmospheric tide. 10.2.2 Spectral analysis of excitation functions Atmospheric mode Ψ13 In the range of 2–60 days, the excitation exhibits peaks at 10, 13, 20, 45 and 52 days up to 10 mas as evidenced by the amplitude Fourier spectrum of Figure 10.3 (pair NCEP+ECCO). Below 20 days, the retrograde band is globally more powerful than the prograde one. Contrast would be even more tremendous if the non-IB pressure term would have been considered. This results from the atmospheric normal mode Ψ13 ,

200 | 10 Rapid, seasonal, inter-annual, and decadal excitations

Figure 10.2: Rapid fluctuations of the polar motion in the winter 2005–2006, of the size of a cellular phone [130].

propagating towards the west (thus retrograde) with a period of about 10 days. Its footprint in the atmospheric angular momentum had been reported by Brzeziński (1981) [22]; its effect on polar motion, which is a retrograde circular term of about 100 µas,1 was confirmed in the observed pole coordinates determined from VLBI and GNSS observations [24]. The prograde band is dominated by a term at 50 days, with an amplitude of about 10 mas (1.3 mas on polar motion). Note that the elliptical oscillation at 20 days is a sub-harmonic of Ψ13 , and could be associated with this mode. Matter term versus motion term and atmosphere versus oceans For both AAM and OAM excitations, matter term (surface pressure/sea water height) and motion term (winds/currents) are quite comparable in the rapid band, as evidenced by the spectra of Figures 8.3 and 8.6. For the atmosphere, this is in contrast to the seasonal band dominated by a pressure term. Except the retrograde band between −9 and −10 days, dominated by the atmospheric mode Ψ13 , the atmospheric and oceanic contributions are quite comparable, as shown by the spectrum of Figure 10.3. Fortnightly ocean tide At 13.6 days, the prograde and retrograde circular terms (of about 5 mas) of χG are poorly explained by the fluid layer excitation. Indeed they are strongly influenced by the lunar oceanic tide, not included in χOe . From the Dickman model (1993) [65], reformulated by Gross (1998) [100], the oceanic tide contributes to the excitation function 1 Transfer function |T(0.1 cpj)| ≈ 0.025 according to (9.1a).

10.2 Rapid fluctuations | 201

Figure 10.3: Rapid band spectrum of the equatorial excitation: geodetic (G), geodetic minus oceanic tidal effect (G-T) and modelled (A: atmosphere / O: Oceans / T: Ocean tides). Atmospheric NCEP models and oceanic ECCO model. Period 2006–2010.

202 | 10 Rapid, seasonal, inter-annual, and decadal excitations

Figure 10.4: ‘‘Hyper–rapid’’ fluctuations (from 2 to 10 days): geodetic (G), geodetic minus oceanic tidal effect (G-T) and modelled (A: atmosphere / O: Oceans / T: Ocean tides). Atmospheric NCEP model, oceanic ECCO model and empirical model of Gross (2009) [98]. Period 2006–2010.

at periods of 9, 13.6 and 27.3 days with a maximal amplitude of about 2 mas at 13.6 days. In order to find evidence for them, the corresponding empirical model of Gross [98] is removed from χG . The resulting excitation, denoted ‘‘G-T’’ (T for Tides), is plotted in Figures 10.3 and 10.4 and shows a much closer agreement with A+O at 13.6 and at 9.1 days in a lesser extent, thus confirming the influence of the fortnightly ocean tide in polar motion, as advanced by Gross [100] after removal of the atmospheric excitation. However, the tidal correction at −27.3 days spoils the agreement of G with A+O (see Figure 10.3, upper plot). Spectral noise We ensure that the spectral peaks that we discussed are well above the random spectral error of χG , evaluated according to (9.11). With P = 4 years, we get εG (f ) ≈

15 µas 󵄨󵄨󵄨󵄨 fc̃ − f 󵄨󵄨󵄨󵄨 󵄨 󵄨. √f [cpa] 󵄨󵄨󵄨 fc̃ 󵄨󵄨󵄨

(10.1)

As seen in Figure 10.3, the peaks of χG widely exceed the spectral noise εG (f ) superimposed to the spectrum. In the rapid band the spectral error of the fluid layer contribution is up to 0.5 mas according to the abacus of Figure 9.5. Anyway the observed peaks of χAO are at least three times larger than this spectral uncertainty.

10.2 Rapid fluctuations | 203

Figure 10.5: Rapid fluctuation decomposed of a Morlet wavelet: geodetic excitation (G, left) and ocean–atmospheric (A-NCEP, O-ECCO, right) contribution from 2000 to 2008. Amplitude in mas.

204 | 10 Rapid, seasonal, inter-annual, and decadal excitations

Figure 10.6: Amplitude and phase variability of the −13.6/−10 day term in geodetic minus tidal excitation (G-T) and modeled AO (NCEP+ECCO) excitation.

10.2.3 Irregularities of the rapid fluctuations The tremendous irregularities of the spectral components are caught by the wavelet decomposition of Figure 10.5. The most important variability is concentrated in the retrograde band from 9 to 15 days. Reaching sometimes 30 mas, they can largely exceed the seasonal variation. At a given period, the rapid fluctuation can persist over some months; then it suddenly attenuates. The corresponding ocean–atmospheric excitation accounts quite well for the patterns observed in χG . Fortnightly barometric tide in the geodetic excitation free from ocean tide Our interest is now focused on the band surrounding the −13.6 day period. A least square fit over 110 day sliding window reveals that the effect mostly stems from the atmosphere (Figure 10.6). The variability of the ocean tide free geodetic excitation χG-T

10.3 Seasonal cycle

| 205

e at 10 and 13.6 days is well reproduced by the one of χAe (∼ 80 % correlation) and χAO e (∼ 90 % correlation), but the modeled contribution χAO presents a slight defect in amplitude variation of a few mas with respect to the geodetic excitation. The Allan deviae tion analysis of Figure 9.7 had already shown evidence that χAO is less noisy than χG in the rapid band, especially for the y component. As χG is very precise (error of 0.1 mas at 13.6 days from Eq. (10.1)), this either hints at some defect in global circulation models at 2–30 day periods or some shortcoming in the differential equation (9.1) describing the polar motion. In particular, we have neglected the frequency dependence of the resonance frequency. As this tends to larger values in the rapid band (Ω/400) due to the dynamical effect that mitigates the amplitude of the pole tide (see Section 6.6), the transfer function T(σ) ≈ σc /(σc − σ) ≈ −σc /σ (for σc ≪ σ in the rapid band) should be slightly larger than it is for a resonance period of 433 days. So, when computing a geodetic excitation χG (σ) = p(σ)/T(σ) from polar motion by using the period of 433 days, one obtains in principle overestimated χG values. The transient atmospheric power at 13.6 days could be caused by an episodic shift of the normal mode period towards the 13.6 day period; then the lunar barometric tide in 13.6 days would be amplified. This is poorly documented, but it has been observed both in surface barometric records [7, 6] and in zonal winds [134]. From measurements done in the suburbs of Thessalonica, the barometric tide in 13.6 days rises up to 25 Pa [6], that is, about one-fourth of the tesseral diurnal barometric tide [57], which causes a retrograde diurnal pressure term χA of about 1 mas (see Figure 8.3). If the variation, as measured in Thessalonica, has a tidal origin and yields at planetary scale a tesseral component p21 ≈ 25 Pa, the produced pressure term of the excitation should have the order of magnitude 0.3 mas according to (E.3), that is, a significant part of the magnitude observed in χAe at 13.6 days. As the atmospheric response to the lunar tide is complex, it is somewhat more natural to observe a second harmonic at 6.8 days, as shown by Figure 10.4.

10.3 Seasonal cycle Sometimes dominated by rapid fluctuations, the seasonal cycle is nonetheless the most important fluctuation affecting the hydro-atmosphere along with the diurnal and semi-diurnal ones. In the seasonal band, the influence of the atmosphere is the largest, as evidenced by the spectra of Figure 10.7 over the period 1990–2008. The seasonal variation is not reduced to the 1-year harmonic term, but is also composed of semi-annual and ter-annual terms with amplitudes at least five times smaller. For the 1-year harmonic, the atmospheric excitation, of about 20 mas, would be too large compared to geodetic excitation, and it is balanced by the oceanic terms, which amounts to about 5 mas. The inland fresh waters also excite the seasonal polar motion at almost the same level, but do not permit to close the budget of χG . Most probably we can blame the imperfections of the hydrological models.

206 | 10 Rapid, seasonal, inter-annual, and decadal excitations

Figure 10.7: Complex Fourier spectra between 0 and 2 years of the equatorial excitations: geodetic (G) and modeled (A/O/H). Models NCEP+ECCO+CPC and ECMWF+MPIOM+LSDM. Period 1990–2008.

10.4 Inter-annual and infra-decadal variations | 207

Numerous works [197, 69] report the values of the seasonal terms for each excitation estimated over some years. This has already been done for the annual term in Table 9.3 for determining its mean elliptical path in the equatorial plane. These averaged estimates should not hide the strong amplitude and phase variability, which can e e be grasped in Figure 10.8 for excitations χG , χAO , and χAOH . There we plotted the amplitudes A±j and phases ϕ±j of the seasonal harmonics, obtained from a least-square fit of the model 3

χ = ∑ Aj eiΦj ei2πfj (t−t0 ) + A−j eiΦ−j e−i2πfj (t−t0 ) , j=1

(10.2)

where f1 = 1 cpy, f2 = 2 cpy, f3 = 3 cpy and t0 is the J2000 epoch, over 2 year sliding window. The variability mostly results from the combined effect of the atmosphere and the oceans (NCEP+ECCO). The hydrological excitation (CPC) is not dominant but significantly improves the agreement with χG (according to the correlation coefficient and the explained variance rate). The harmonics at 2 cpy and 3 cpy have larger relative variations. Globally the fluid layer excitation yields a satisfactory explanation of the observed variability at seasonal scale, but this does not fully explain the mean values: in particular at −1 cpy the mean of the modeled AOH term exceeds by 5 mas the mean of the geodetic one. Probably we should challenge the hydrological model. It should be recalled here that the geodetic excitation has been computed according to symmetric Liouville equations, neglecting an asymmetric effect of about 0.5 mas at −1 cpy (see Section 11.6) that should be taken into account as well. Note that the ECMWF+MPIOM+LSDM series yield comparable results.

10.4 Inter-annual and infra-decadal variations Beyond one year, we enter into the ‘‘no-man’s land’’ of the stochastic Chandler excitation (of about 1 mas). It requires the special treatment postponed in Chapter 11. So we come up to periods ranging from 500 days to 10 years. In this band we perform the e e wavelet decompositions of χG and of angular momentum functions χAO and χAOH associated with coupled A-ECMWF, O-MPIOM and H-LSDM models, processed at GFZ over the period 1976–2019. The results are reported in Figure 10.9. The influence of inland fresh waters is noticeable, as shown in numerous studies like [41, 89, 196, 198]. This analysis is completed by a time comparison of the excitation series restricted to the inter-annual domain (applying a band pass filter) over the period 2002–2015 covering of GRACE mission. Along this line, we append the gravimetric matter term defined in Section 8.2.3 and determined from the processing of GRACE data (release 06). Inter-annual oscillations covering the periods from 500 days to 2 years, have an amplitude of 1–2 mas (see also the spectra of Figure 10.7). As shown by Figure 10.10, the fresh waters compete with the atmospheric–oceanic mass transports. Because of the vicinity of the Chandler period, the prograde part of these fluctuations is amplified

208 | 10 Rapid, seasonal, inter-annual, and decadal excitations

Figure 10.8: Variability of the annual, semi-annual, and ter-annual terms in geodetic (G) and modeled excitation AO (NCEP+ECCO), AOH (NCEP+ECCO+CPC).

10.4 Inter-annual and infra-decadal variations | 209

Figure 10.9: Wavelet amplitudes of the inter-annual fluctuations of the geodetic and fluid layer excitations. Coupled models A-ECMWF, O-MPIOM and H-LSDM. The zone outside back inclined lines is spoilt by edge effects.

in the polar motion by a factor 7.5 at 500 days and 3.5 at 600 days. The pole coordinate analysis confirms the presence of a prograde term around 500–510 days, presenting a variable amplitude up to 20 mas. The amplitude of χG rises again near 3–4 years especially in the retrograde band. Here the dominant influence of the atmosphere and of the ocean is strikingly reinforced by the land water transport, as also reflected in the terrestrial components of

210 | 10 Rapid, seasonal, inter-annual, and decadal excitations

Figure 10.10: Inter-annual variation of the geodetic excitation (G) and corresponding ones of H-LSDM model, A-ECMWF, and O-MPIOM (left panels), and full mass term as determined by GRACE (right panels). Here H-GRACE is the equivalent hydrological mass term after removing from GRACE excitation the A-ECMWF and O-MPIOM matter terms. The rate of the explained variance (%) between G and the reconstructed excitation are reported on the plots.

the hydrological excitation given by GRACE and LSDM model (respectively curves “HGRACE” and “H” in Figure 10.10). Furthermore, the wavelet analysis unveils a zone of oscillations from 5 to 10 years. Again, global circulation in the hydro-atmosphere is clearly at stake, except in the prograde band around 6.5 years, which evokes the Inner Core Wobble (see Section 7.7). At this time scale, the GRACE matter term takes precedence, as clearly shown in the right panel of Figure 10.10. However, the modeled matter terms of the hydro-atmospheric excitation and GRACE excitation show differences up to 5 mas, especially for the x component (respectively curves “AOH matter” and “GRACE” on the right panel). In the inter-annual band we can see the footprint of the El-Nino South Oscillation (ENSO), taking place in South Pacific. The corresponding index or Southern Oscillation Index (SOI) has transient components with periods of 2.4, 3.6, 4.8 and 6 years, as shown by the wavelet transform of Figure 10.11.2 The SOI signature has been also identified in the equatorial atmospheric torque [140]. 2 Oddly these periods are integer multiples of 1.2 years, that is, the Chandler period [201].

10.5 Decadal and secular variations | 211

Figure 10.11: Wavelet amplitudes of the South Oscillation Index (SOI). The zone outside the back inclined lines is spoilt by edge effects.

Figure 10.12: Wavelet decomposition in the decadal rage of the geodetic excitation (IERS C01 series). Left panel: prograde component; right panel: retrograde component letting the Markowitz wobble appear at −30 years.

10.5 Decadal and secular variations Beyond 10 years, the resonance is strongly mitigated, and geodetic excitation becomes equal to polar motion. It is dominated by a 25–35 year oscillation with a mean amplitude close to 10 mas, as evidenced by the wavelet decomposition of Figure 10.12. This is the Markowitz term [141, 142]. The retrograde component is more stable than the prograde component, which has faded nowadays. In this frequency range the contemporaneous uncertainty of the geodetic excitation is 30/√fP µas (from Eq. (9.11)) with f ∼ 0.03 cycle/year and P ∼ 60 years (window of the Morlet wavelet transform), that is, about 20 µas. Before 1960, the error of the pole coordinates was 300 times larger (see Table 3.2), so that the Markowitz wobble had an uncertainty of about 6 mas, which is almost the level of its wavelet amplitude.

212 | 10 Rapid, seasonal, inter-annual, and decadal excitations If the core mantle coupling is invoked to explain the decadal variations of the length of day, there is no proof that it produces an equatorial torque and explains the Markowitz wobble [112, 70]. On the other hand, ocean–atmospheric excitation presents an excitation of about 1 mas at decadal periods [101], and land water impact is even stronger, at the level of a few mas as can be noticed in the spectrum of Figure 8.11. In the longer term polar motion has a drift of 3 ± 0.003 mas/year in the direction 76–79° west [103]. It was firstly noticed in the 1920s thanks to the lengthening of the pole coordinate series [131]. Since the 1980s, it has commonly been interpreted as the Earth visco-elastic response echoing the last deglaciation, mixing both post-glacial rebound of the soil and rotational readjustment [167], as addressed in Chapter 15. But the inflection from 2000 to 2010 towards Eastern Europe (see Figure 1.1) can be explained through the quasi-elastic response of the solid Earth to the ice melting observed on Antarctic and Greenland over the same period [40, 1]. Whereas land water models hardly account for this inflection, the hydrological excitation reconstructed from GRACE data—after eliminating atmospheric and oceanic mass contribution, here associated with ECMWF and MPIOM models—affords one to well model the non usual trend of the polar motion (or of its excitation) from 2002 to 2016. This can be seen in Figure 10.13, where the last GRACE data release quite well describes the drift along the prime meridian (x component), with a positive slope of about 4 mas/year between 2002 and 2014 in contrast with the usual secular slope of about 0.7 mas/year along the prime meridian. It seems that this process is mitigated since 2014, and the polar motion is recovering the trend imparted by the glacial isostatic adjustment.

10.6 Conclusion Dominated by the seasonal cycle, the hydro-atmospheric excitation has a substantial influence within the whole range of periods from 2 days to 100 years. The GRACE mission has revealed its role at decadal periods, mostly resulting from land water transports. As the land water effect, as determined through GRACE from 2002 to 2016, has bent the polar motion in the counter-clock wise direction, it does not seem to be related to the clockwise Markowitz wobble. Only the longer time series of the gravimetric matter term will allow one to quantify the effect of the mass term on the Markowitz wobble, and to find to which extent an internal motion term could be its cause (for the surface motion term is negligible at this period). If the inland fresh waters are crucial to explain the seasonal, inter-annual, and decadal fluctuations, they leave the atmosphere and oceans to settle the rapid oscillations. This budget is still affected by the defects of the hydro-atmosphere models: partial knowledge of the ocean–atmosphere coupling and hydrological processes, discrepancies between models for the wind terms and the ocean current terms. But looking at the progress made in recent decades, we do not doubt that advances in global circulation models will dispel these unknowns in the next years.

10.6 Conclusion | 213

Figure 10.13: Decadal variation of the geodetic excitation (G) and corresponding ones of H-LSDM model, A-ECMWF, and O-MPIOM (left panels), and full mass term as determined by GRACE (right panels). Here H-GRACE is the equivalent hydrological mass term after removing from GRACE excitation the A-ECMWF and O-MPIOM contributions. The rate of the explained variance (%) between G and the reconstructed excitation are reported on the plots.

11 Chandler’s wobble (...) it is convenient to say that the general result of a preliminary discussion is to show a revolution of the Earth’s pole in a period of 427 days, from west to east, with a radius of 30 feet [∼ 10 m], measured at the Earth’s surface. Chandler (1891) [35]

11.1 An excited mode Because of the Earth’s dynamical ellipticity e, the rotation pole presents a free oscillation with respect to its surface. If the Earth would be a rigid ellipsoid, the angular frequency of this motion would be eΩ, where Ω is the mean Earth angular velocity. The corresponding period would be about 304 days. As shown in the second part, polar motion generates a variation of the centrifugal potential, and because of the Earth’s non-rigidity, it induces a deformation tending to compensate the Earth ellipticity. In turn the free mode period is lengthened by about 180 days. At the same time it is shortened by about 40 days because of the passive behavior of the fluid core. This leads to a period of about 433 days (0.843 cycle/year). This notwithstanding, as any resonant phenomenon met in physics, the Chandler wobble is damped because of dissipation accompanying the pole tide, and various studies suggest that the Chandler wobble should lose more than half of its amplitude after 40 years if it would not be excited (see Chapter 5). Conservation of its average amplitude of 200 mas, as observed since the 1950s, requires an average excitation of about 1 mas. This grossly corresponds to the background spectral noise of the atmospheric and oceanic excitation [97, 26] in the full width at half maximum (about 0.06 cycle/year) of the Chandler wobble spectral peak. Because of the chaotic nature of the excitation, the Chandler wobble is expected to be irregular. Indeed, as shown in Figure 11.7, over some decades we observe amplitude variations up to 100 mas and phase changes up to 50°, the Chandler period being fixed [106, 47, 223, 95]. Equivalently the apparent period of the Chandler wobble shifts up to 20 days from the theoretical period of the free mode [220, 217]. Actually the observed Chandler wobble is the free theoretical wobble forced by geophysical excitation, of which the effective frequency band is between 405 and 460 days (see for the justification below). So, amplitude and frequency/phase variabilities stem from the fluctuations of the mechanical excitation in the Chandler band. In this respect pressure and wind fluctuations in the northern hemisphere can be responsible for the observed frequency variations [169]. Before the twenty-first century, the forcing of the Chandler wobble was also sought in the Earthquake mass redistribution. Now, since 2000, the improvement of hydroatmospheric circulation model has led to the conclusion that the Chandler effect is mostly forced by the mass transports within the hydro-atmosphere [97, 26]. This https://doi.org/10.1515/9783110298093-011

11.2 Theoretical lineaments | 215

notwithstanding, some bones of contention remain. So, some papers affirm that the matter and wind terms have a comparable and intermittent influence [5, 49], but other studies, like Seitz & Schmidt (2005) [195] had privileged the matter term as the main factor of variability in the Chandler wobble. Actually this geophysical analysis faces a number of challenges. First, in the Chandler band, the uncertainty of the modeled excitation is almost on the level of the spectral amplitude of the geodetic excitation: for the oceanic AMF the random spectral uncertainty at the Chandler frequency can reach 0.7 mas over a 7 year sampling interval (Chapter 8). Therefore the common spectral comparison χG /χMOD , as carried out in Chapter 9, is delicate. If, on the other hand, we reconstruct the Chandler wobble from geophysical excitation time series, we face another problem: the period covered by hydro-atmospheric series is just over the Chandler relaxation time (τ = 2Qc /σc ∼ 40 years for Qc ∼ 100), so that the free damped part resulting from past processes and the contemporaneous forced part remain mixed together. Finally, we mention the asymmetry of the geodetic excitation with respect to cosine directions m1 and m2 . If this asymmetry, resulting from the anisotropy of the ocean pole tide and from the triaxiality, modifies the amplitude of Chandler’s wobble at the level of a few % (see the end of Chapter 6), the relative change is much more important in the geodetic excitation function. In this chapter, we show how the Chandler wobble variability informs about its geophysical cause, especially confirming the prevailing role of the hydro-atmospheric transports. First, we re-estimate the Chandler parameters (period Tc and quality factor Qc ).

11.2 Theoretical lineaments The Chandler wobble is commonly studied by means of the symmetric equation (9.1), corresponding to a biaxial mantle uncoupled from the core, permitting to investigate the polar motion excitation from 2 days to several decades (Chapters 9 and 10): ṗ p+i ̃c σ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ χG

=

e e χ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ma + χmo ,

χMOD

(11.1)

where one recognizes the complex Chandler angular frequency σ̃ c = σc (1 + i/2Qc ), and the observed excitation χG is equated to the reconstructed excitation χMOD resulting from hydro-atmospheric circulation. In Chapters 5, 6, and 7 it has been established how non-rigidity, namely solid Earth elasticity, oceans, fluid core and dissipation, account for the period and the damping of the free mode. This theoretical development yields Tc = 434 ± 4 days (see Eq. (7.36) or (7.37a) and its numerical application) consistent with the observed period of 433 ± 2 days [217]. The quality factor Qc is not so well determined: its modeling is a very difficult task, and its estimates from geodetic excitation fitted to the hydro-atmospheric

216 | 11 Chandler’s wobble Table 11.1: Estimated period Tc and quality factor Qc of the Chandler wobble from atmospheric (A), oceanic (O), and land water (H) excitations. 95 % confidence intervals.

Furuya and Chao (1996) [92] Kuehne et al. (1996) [125] Aoyama et al. (2003) [5] Seitz et al. (2012) [194] Nastula and Gross (2015) [157]

Tc

Qc

period

433.7 ± 1.8 439.5 ± 2 431.2 432.98 430.9 ± 0.7

49 (35, 100) 72 (30, 500) 179 97 127 (56, 255)

1983–1994 1984–1993 1980–1994 60 1985–2010

excitation A A A AO AOH

AMF offer a large interval of possible values depending on the chosen period and hydro-atmospheric model, as evidenced by the results compiled in Table 11.1. Both integration and excitation approaches, as discussed in Section 9.3, will be applied. The polar motion is obtained by integrating (11.1). Then the reconstructed Chandler wobble is extracted by a band-pass filter and compared to the observed Chandler wobble extracted by the same filter. According to (5.52) the solution of (11.1) can be written pMOD (t) = p(t0 ) e

iσ̃ c (t−t0 )

t

̃ ̃ − iσ̃ c eiσc t ∫ χMOD (t 󸀠 ) e−iσc t dt 󸀠 󸀠

(11.2)

t0

where t0 is the initial time. The first term yields the damped free wobble, of which the amplitude is divided by e ≈ 2.718 after the relaxation time τ = 2Qc /σc (≈ 26 years for Qc = 70). This progressive damping is compensated for by the second term, which is the forced part. In the frequency band around σc , the excitation looks like a stochastic noise, of which the spectral density varies as f α with α between 0 (white noise) and −1 (see Section 9.4.4). This noise, considered in a given frequency band around σc , resonates and contributes significantly to the Chandler wobble. The effective band, defined as yielding 95 % of the observed power, depends on the power law characterizing the noise. Considering the white noise approximation Kf 0 , it can easily be shown that the spectral power included in the window [σc − Δσ, σc + Δσ] is given by PΔσ = σc K4Qc Arctan(

2Qc Δσ ). σc

(11.3)

Then the total power associated with Δσ = ∞ is given by Ptot = 4Qc Kσc π2 , and 95 % of σ this power is obtained for Δσ = 2Qc tan(0.95 π2 ). For the value Qc = 70, justified below, c Δf = Δσ/(2π) ≈ 0.076 cpy, determining the effective frequency band [0.767, 0.919] cpy, which does not interfere with the annual excitation. Thus, the white noise approximation and a quality factor Qc ∼ 70 allow us to clearly split annual and Chandler excitation.

11.3 Estimation of the Chandler wobble parameters | 217

11.3 Estimation of the Chandler wobble parameters The roughest approach for determining the Chandler parameters is a spectral approach. Assuming a white noise excitation, the Chandler frequency corresponds to the maximum of the spectrum, and the quality factor can be derived from the full width at half maximum (FWHM) by Qc = σc /FHWM (see Chapter 5). As the FWHM associated with the damping cannot be determined accurately, statistical procedures were designed on the basis of digitized excitation. They assumed that, in the Chandler band, excitation and pole coordinates are uncorrelated at monthly intervals, after removal of the seasonal signal [231]. From the 1990s, the production of atmospheric angular momentum series led one to give up this statistic approach, and to determine Tc and Qc as the polar motion resonance parameters in the dynamical equation linking the observed excitation with the modeled excitation. Considering the polar motion from monthly to inter-annual time scales, it is reasonable to assume a quasi-elastic regime of the mantle and oceans at equilibrium, entailing constant values of the solid Earth Love number k̃2 and the oceanic Love number k̃o . Then, according to (7.37a), the complex resonance frequency σ̃ c ≈ σe

k̃ + k̃o A (1 − 2 ) = 2π/Tc (1 + i/2Qc ) , Am ks

(11.4)

can be considered as fixed, and pole coordinates and angular momentum series, injected in (11.1), provide an overdetermined system for the unknowns Tc and Qc , which can be inverted. Pioneering studies were restricted to the atmospheric excitation [125]. It had been noted that the bad modeling of seasonal terms strongly affect the estimation Tc due to their proximity with the resonance period, therefore it is preferable to remove them from the input series for mitigating errors [92]. More recent studies include the oceanic excitation [157] and benefit from longer data series, better describing the excitation at stake. Here, the Chandler parameters are derived by improving the approach of [125], and considering the full hydro-atmospheric excitation as known today. According to the Wilson digital filter (9.8), the observed excitation is sampled according to G χt+h/2 = apt + bpt+h ,

(11.5)

with a = −beiσc̃ h and b = ie−iπfc h /σc̃ h. In the original procedure [125], the complex coefficients a and b are fitted to the effective angular momentum of the fluid layers by inverting the set of equations ...

F a pt + b pt+h = χt+h/2 ,

F a pt+h + b pt+2h = χt+h+h/2 ,

⋅⋅⋅

(11.6)

218 | 11 Chandler’s wobble expressed for n available dates t. Then the Chandler parameters can be easily derived. But according to the above relation the coefficients a and b are linearly correlated. To avoid this defect, Eqs. (11.6) are converted into ... pt+h + r pt =

F χt+h/2

b

(11.7)

... with r = a/b = r1 + i r2 = −eiσ̃c h . In turn, what is now estimated is the complex coefficient r. Then, the Chandler parameters are deduced from the expressions fc =

arg(−r) , 2πh

Qc =

−hπfc . log |r|

(11.8)

Here χ F = ama χma +amo χmo is the effective angular momentum function, where ama and amo depend on the effective Love number k according to (7.39c), and thus on σc from ̃ consistent (7.37a). The latter equation yields a value of k,̃ subsequently of k = R(k), with the Chandler wobble frequency: A σ̃ k̃ = ks (1 − m c ) . A σe

(11.9)

The inversion of the system (11.7) requires an initial value of b that is given by a priori values for the Chandler parameters (we choose fc = 0.843 cpy and Qc = 100). These values, including k, ama , and amo , are updated after the first inversion, and the system is reshaped accordingly and inverted again. The procedure is reiterated until the Chandler parameters reach convergent values. Three versions of the modeled excitation are considered: i) atmospheric-NCEP + oceanic-ECCO, ii) atmospheric-ECMWF + oceanic-MPIOM, and iii) the last one supplemented by land water-LSDM. Note that before the inversion, all series are made spectrally consistent by low pass frequency filter (period below 5 days eliminated) and removal of their linear and decadal trends. Indeed, at this time scale, visco-elasticity of the mantle increases and strikingly modify the resonance parameters. For the reason noticed above, we also build the series free from their seasonal components. Here, we favor the C04 pole coordinate series. In contrast to the C01 time series, C04 fully includes the rapid polar motion, which is sensible to the resonance, and is reasonably well modeled by the atmospheric and oceanic excitation. First, we look at the estimates obtained over the period 1980–2019 (see the two last columns of Table 11.2), for which the pole coordinates have an uncertainty below than 1 mas. When the inconsistent seasonal terms are present, the Chandler period strongly varies from one modeled excitation to another, lying in the interval (426.8, 437.2) days. Their suppression tights this interval to (431.8, 435.5) days. Meanwhile, the removal of

11.3 Estimation of the Chandler wobble parameters | 219

Figure 11.1: Chandler Wobble parameters for three sets of hydro-atmospheric excitation as a function of the data range Δ = tf − ti where the final epoch tf is fixed to 2019.0, and ti varies from 2002 back to 1976. In the left part seasonal terms are included, in the right part seasonal terms are removed.

the seasonal terms yields a less constrained quality factor: (33, 83) instead of (29, 68). Adding or not the fresh water contribution does not significantly impact these estimates. The robustness of these results can be better appreciated by iterating the Chandler parameters fit over an increasing data range Δ = tf −ti where the final epoch tf is fixed to 2019.0, and ti starts in 1976 and increases by steps of 1 year until 2004. The obtained estimates are reported in Figure 11.1, where the abscissa gives the time interval Δ = tf −ti . It can be seen that the most accurate estimates are for Δ = 39 years, corresponding to the period 1980–2019. Before 1980, the uncertainty of the data mitigates the results and reintroduce the instability observed for windows of length smaller than 30 years. For the time interval Δ ≤ 25 years (ti ≥ 1994), the quality factor becomes unstable. These results are consistent with a time relaxation of (12, 31) years, favored by Qc lying in the interval (33, 83). For a time interval larger than Δ = 25 years, the removal of the seasonal term ensures the convergence of the different estimates towards the values mentioned earlier. In Table 11.2 we also provide the average estimates ⟨Tc ⟩ and ⟨Qc ⟩ of the Chandler parameters obtained from ti = 1980 to ti = 1994 (Δ = 39 years down to Δ = 25 years).

220 | 11 Chandler’s wobble Table 11.2: Chandler parameters fitted over three hydro-atmospheric angular momentum series. First two columns provide the mean of the values obtained over the decreasing time intervals [ti , 2019], with ti = (1980, 1981, . . . , 1994). The last two columns yield parameters for the interval 1980–2019. ⟨Tc ⟩ (days)

⟨Qc ⟩

Tc (days)

Qc

(426.69, 428.71) (432.60, 434.80) (435.49, 437.71)

(36, 59) (32, 49) (26, 38)

(426.78, 429.33) (432.26, 434.95) (434.53, 437.24)

(37, 68) (29, 45) (24, 36)

(431.88, 434.12) (432.61, 434.99) (432.31, 434.69)

(49, 102) (45, 88) (43, 81)

(431.76, 434.64) (432.52, 435.48) (432.14, 435.10) (437.4, 441.6) (430.20, 431.60)

(39, 83) (33, 63) (35, 68) (30, 500) (56, 255)

with seasonal term AO(NCEP+ECCO) AO(ECMWF+MPIOM) AOH(ECMWF+MPIOM+LSDM) without seasonal term AO(NCEP+ECCO) AO(ECMWF+MPIOM) AOH(ECMWF+MPIOM+LSDM) Kuehne et al. (1996) Nastula & Gross (2015)

11.4 Hydro-atmospheric reconstruction of the Chandler wobble Data The Chandler wobble is reconstructed from the longest data set of atmospheric and oceanic AMF, which we have at hand: by considering first the pair NCEP–ECCO spanning from 1950 to present, then the pair ECMWF–MPIOM covering the period from 1976 to nowadays (see Chapter 8). The geodetic excitation is deduced from the IERS pole coordinates according to the digitization exposed in Section 9.3. Beginning in 1830, only the IERS C01 time series covers a sufficiently longer span.1 From 1950 to 1962 the C01 pole coordinates mostly result from optical observations, and are thus affected with an uncertainty of 10 mas at least. After 1962, uncertainty is reduced, dropping below 1 mas from 1980. But this error is not crucial in the light of the Chandler wobble amplitude, above 100 mas in the second middle of the twentieth century, and of its variations reaching the same level. Any other optical series, for the early years, would yield the same results. Pole coordinates and AMF time series are made spectrally consistent by Vondrák smoothing [221]—equivalent to a low frequency pass filter (transfer coefficient ≥ 99 % for period above 10 days and ≤ 1 % below 4 days)—and by re-sampling at 0hUTC every 10 days. Method Injecting the AMF time series in (11.2), the modeled trajectory pMOD (t) is obtained by trapezoidal integration. The initial condition p0 = p(t0 ) is given by the observed pole 1 Sampling of 0.05 year, available from http://iers.obspm.fr/eop-pc

11.4 Hydro-atmospheric reconstruction of the Chandler wobble

| 221

coordinates at time t0 . Doing so, we assume that the surface fluid layers are the only sources of the observed Chandler wobble. Considering the reconstructed polar motion pMOD (t) and the observed polar motion pG (t), the corresponding Chandler wobbles pcMOD (t) and pcG (t) are extracted by a Panteleev band-pass filter centered on the Chandler frequency with the bandwidth parameter f0 = 0.08 yr−1 . It allows one to select the efficient Chandler band [0.767, 0.919] cpy determined here above with a frequency admittance of 55 % at the edges of window, while the annual term is mitigated (6 % of the signal remains) according to Appendix G.4, Eq. (G.9). In order to minimize the contamination by the annual band, the mean annual harmonic is estimated by a least square fit and removed from both the observed and the modeled pole coordinate time series before applying the band-pass filter; thus, the leakage at the annual frequency concerns only a residual change of about 20 mas, so that the footprint of the annual signal reaches at most 1 mas after Panteleev filtering. The initial wobble damps after t0 according to |p(t0 )| e−σc /(2Qc )(t−t0 ) . So lower Qc values shift the reconstructed amplitude downwards and higher Qc upwards. Sensitivity to Qc The sensitivity to Qc values is investigated by computing pcMOD (t) for Qc sweeping the interval [30, 200] and computing for each run both the correlation coefficient with pcG (t) and the explained variance rate of pcG (t). From Figure 11.2 corresponding to the sets NCEP–ECCO over the period 1950–2019 and ECMWF–MPIOM–LSDM over the period 1976–2019, it can be concluded that low Qc values, 40–120, yield the best explained variance rate (≥ 90 %) and correlation (≥ 0.95), in agreement with the results obtained in the former section. Reconstruction of the Chandler variability by NCEP–ECCO models The filtered time series pc (t) correspond to counter-clockwise circular oscillations at Tc = 433 day period with variable amplitude and phase. Thus, in Figure 11.3 we display their envelope amplitude and their phase shift with respect to 2π/Tc (t − t0 ) where t0 is the epoch J2000 (January 1, 2000) for the NCEP–ECCO data and the value Qc ∼ 70 favored by the former consideration. The 2D signal pcG (t) and pcAO (t) are highly correlated (complex correlation 0.99 e−i 14° and explained variance of 91 %). The oceanic contribution accounts for 78 % of the geodetic signal pcG (t) (complex correlation of 0.96 e−i 17° ). So, it is as important as the atmospheric excitation, as shown by Figure 11.3. Using also the ECCO model, Gross (2000) [97] had already proposed the key role of the ocean bottom pressure in the Chandler wobble excitation. The total matter term even better matches the observed variability (complex correlation 0.99 e−i 11° and an explained variance of 94 %), as shown in Figures 11.2 and 11.3.

222 | 11 Chandler’s wobble

Figure 11.2: Complex correlation coefficient (amplitude and phase) between the observed and reconstructed Chandler modes and explained variance of the observed Chandler wobble as a function of Qc according to the two sets of coupled hydro-atmospheric circulation models.

Contribution of the land water transports according to LSDM model What about the effect of the land water on Chandler wobble? The climatic simulation reported in [32] yields a small contribution compared to the combined effect of the atmosphere and the oceans. This issue can be revisited through the AMF provided by GFZ, derived from the coupled atmospheric-ECMWF–oceanic-MPIOM–hydrologicalLSDM models covering the shorter interval 1976–2019. For the pair ECMWF–MPIOM, Qc ∼ 60 appears as an optimal value. Looking at Figure 11.4, we see that the match of observed and reconstructed Chandler wobbles is even more striking than for the NCEP–ECCO models (complex correlation 1 e−i 0.6° and explained variance of 99 %). Whereas the hydrological LSDM series significantly impact the Chandler wobble amplitude up to 30 mas, it slightly downgrades the budget of the Chandler variability, as reflected through the phase of the AOH reconstruction on the right panel of Figure 11.4.

11.4 Hydro-atmospheric reconstruction of the Chandler wobble

| 223

Figure 11.3: Amplitude and phase variations of the observed and modeled Chandler oscillations (for Tc = 433 days and Qc = 70) according to NCEP–ECCO angular momentum series. Separate contributions of the atmospheric pressure, wind, currents and water height (upper plots), the total effect (AO), and the one restricted to the matter term (AO Matter) (bottom plot). The edge effect of the Panteleev filtering (with central frequency fc = 0.843 cycle/year and bandwidth f0 = 0.08 cycle/year) limits the meaningful period to 1954–2015.

Instantaneous Chandler period versus resonance period In the observed Chandler wobble, phase changes can be grossly assimilated to a piecewise linear function with typical linear change Δϕ ∼ 30° over a period of Δt ∼ 10 years. If we consider the Chandler wobble phase is given against time by ϕ = 2π/Tc t where the Chandler period Tc varies, we have Δϕ = −2πΔTc /Tc2 Δt. Hence the Chandler period change is |ΔTc | = 30°/360°(1.22 /10) year, that is, about 4 days. Thus, the spectrally determined Chandler period should not be considered as the resonance period, of which the stability is physically granted by the Earth properties: this is the period for which we have a maximal forcing in the Chandler band for the time span under consideration. Sensitivity to the initial condition The sensitivity to the initial condition t0 is not crucial. Indeed, when the integration is redone several times by sweeping the initial epoch t0 over the years 1950–1970, the correlation between the reconstructed and the observed Chandler wobble is not significantly changed, as shown on Figure 11.5.

224 | 11 Chandler’s wobble

Figure 11.4: Amplitude and phase variations of the observed and modeled Chandler oscillations (for Tc = 433 days and Qc = 70) according to ECMWF–MPIOM angular momentum series. Separate contributions of the atmospheric pressure and wind, oceanic water height and currents (upper plots), the total effect (AO), the one restricted to the matter term (AO Matter) (bottom plot). We have an additional contribution of the LSDM land waters (H) and total effect (AOH). The edge effect of the Panteleev filtering (with central frequency fc = 0.843 cycle/year and bandwidth f0 = 0.08 cycle/year) limits the meaningful period to 1980–2015.

Figure 11.5: Dependence on initial condition: complex correlation coefficient (amplitude and phase) between the observed and reconstructed Chandler modes (NCEP–ECCO model) and explained variance of the observed Chandler wobble as a function of the initial epoch t0 of integration.

11.5 Multi-decadal modulation of the Chandler excitation

| 225

Figure 11.6: Observed excitation of the Chandler wobble and its reconstruction from NCEP and ECCO models. Time domain – x component, envelope amplitude and phase variation.

11.5 Multi-decadal modulation of the Chandler excitation Geophysical excitation versus geodetic excitation in the Chandler band It has been shown that the variability of the Chandler wobble can be accounted by the influence of atmospheric and ocean transports. In principle, one can say something analogous for the corresponding excitation. In order to check this point, we apply the digital Wilson filter (9.8) to the series pcG (t) and pcAO / AOH (t). In Figure 11.6 we display the derived excitations in the case of the NCEP–ECCO set and with the geodetic excitation computed for Qc = 70. The amplitude and phase variability of the geodetic excitation are well explained by the AO effect, as confirmed by complex correlations over the restricted time span 1956–2013 for mitigating edge effects: 0.87 e−i 8° and explained variance of 75 % (maximal for the selected Qc value). Considering the atmosphere only, the correlation is reduced to 0.60 e−i 23° . For ECMWF–MPIOM the best fit to the geodetic excitation is maximal for Qc = 60 (complex correlation of 0.96 ei0° , explained variance of 90 %), but the hydrological LSDM slightly series spoil this agreement as already noticed for the corresponding wobble. The significance of this striking agreement is compromised by the random uncertainty spoiling the hydro-atmospheric excitation. In time domain it can be grossly evaluated from the divergence between various models of the AMF functions in Allan deviation diagrams considered at the Chandler period and displayed in Figures 8.4,

226 | 11 Chandler’s wobble

Figure 11.7: Chandler wobble and underlying excitation from 1840, x component (left plots). Spectra of their envelope (right plots).

8.5, 8.8 (we discard the land water excitation, that does not seem relevant for Chandler wobble). The instability between modeled AMF reach 1 mas, in particular for ocean water height term. So, it is not surprising to notice that the observed excitation in Chandler band is better reconstructed by the AO excitation when this one has an amplitude exceeding 1 mas. Multi-decadal modulation of the Chandler excitation Now, we go back in time to before the 1950s in the absence of available data for the fluid layer excitation. The analysis encompasses the whole C01 pole coordinates from 1830 to 2019—the C01 is the longest available series. Repeating the same analysis, the Chandler wobble is filtered out of the polar motion and the corresponding excitation is derived by the Wilson digital filter. For the x component, amplitude and phase are displayed in Figure 11.7. The Chandler wobble is more stable than its excitation, except in the 1930s. The wobble envelope (Figure 11.8, left panel) presents a modulation exhibiting oscillations in about 82 yr, 41 yr, 22 yr, 18.6 yr (the analysis is limited to data after 1900 because of their better accuracy). The corresponding harmonic model is fitted to the Chandler amplitude, and represented against the amplitude (bottom plot). The middle left plot shows that the 18.6 yr wave is shifted by 180° with respect to the correspond-

11.6 Asymmetric excitation of the Chandler wobble

| 227

Figure 11.8: Amplitude of the Chandler wobble and of its geodetic excitation: time series and harmonic model (bottom plots), individual harmonic waves of the model (middle plots), and spectra (upper plots). The unit is mas.

ing length of day variation caused the 18.6 year lunar zonal tide. A similar analysis is performed for the envelope of the excitation, where prominent spectral peaks appear at 37.2, 22, 18.6, and 11 years. Whereas Ponte et al. (2002) [174] had already pointed out that the oceans cause a strong multi-annual modulation in the Chandler band, the amplitude modulation in about 18–20 years has first been noticed by Leonid Zotov [239]. There was an attempt to relate these multi-annual changes to the combined effect of the 18.6 year lunar tide and decadal climatic cycle [238]. However, the phase of the associated harmonic strongly depends on the period chosen for the fit, and do not allow one to draw any conclusion on their possible coherence with lunar tide and solar activity cycle.

11.6 Asymmetric excitation of the Chandler wobble 11.6.1 Theoretical reminder Until now, the geophysical interpretation of the polar motion was only produced for symmetric Liouville equations. But it has been shown in Chapter 6 that, owing to the asymmetric effects, the counter-clockwise Chandler wobble can result from the clockwise excitation around −σc . Thus, assuming a typical excitation of 1 mas near −σc (the same order of magnitude as for +σc ), the asymmetry of the Liouville equation converts it into supplementary Chandler oscillation of about 1 mas according to the transfer function of Figure 6.2. Inversely, a prograde excitation at +σc not only maintains the Chandler wobble but also contributes to a retrograde circular term of frequency −σc of about 1 mas. Owing to the uncertainty of the underlying geophysical excitation, it is a hard task to find such an asymmetric perturbation in polar motion. We are going to quantify the asymmetric contribution on the geodetic excitation. At the Chandler frequency the instantaneous rotation pole will be confounded with

228 | 11 Chandler’s wobble the CIP, thus considering m = p. For, according to (4.48) expressed in the frequency domain, the relative difference between their complex coordinate is no higher than 󵄨󵄨 p(σ ) − m(σ ) 󵄨󵄨 󵄨󵄨 σ 󵄨󵄨 󵄨󵄨 󵄨 󵄨 c c 󵄨󵄨 (11.10) 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨 c 󵄨󵄨󵄨 ≈ 1/430 , 󵄨󵄨 󵄨󵄨 󵄨󵄨 Ω 󵄨󵄨 p(σc )

which is below the order of magnitude of the asymmetric perturbation we are looking for. A realistic theoretical value of the Chandler period is obtained in the case where the fluid core is passive; then polar motion is described by the generalized Liouville equations (7.34): (1 − U)m +

Lext A i i U󸀠 m ( m + U)ṁ − U 󸀠 m∗ + i ṁ ∗ = χ − χ̇ + i . Ω C−A Ω Ω (C − A)Ω2

The corresponding geodetic excitation is the first member of this equation divided by 1 − U: A U󸀠 i i ( m + U)ṁ + (−m∗ + ṁ ∗ ) . (11.11) ΨG = m + Ω(1 − U) C − A 1−U Ω

This is an extension of the common definition for symmetric Liouville equations, given by (11.1). Indeed we have i i U󸀠 (−m∗ + ṁ ∗ ) , ṁ + 1−U Ω σ̃ c

ΨG = m + with σ̃ c = Ω

1−U

Am C−A

≈ σe

+U

A (1 − U) Am

(11.12)

(cf. Eq. 7.36) .

The function ΨG is modeled through AMF χ according to ΨMOD =

χ̇ χ 1 (χ − i ) ≈ , 1−U Ω 1−U

(11.13)

where the time derivative of the AMF has been neglected, as it has a relative contribution of about 1/430 in the Chandler band. Here χ = χ1 + i χ2 is the non-effective AMF. For an AMF embodying a surface load term χma and a motion term χmo , the effective AMF reads (1 + k2󸀠 )χma χ + mo . (11.14) 1−U 1−U The geodetic excitation is the sum of a symmetric part depending on m and of the asymmetric part depending on m∗ : χe =

sym

ΨG

=m+

i ṁ , σc̃

asym

ΨG

=

U󸀠 i (−m∗ + ṁ ∗ ) . 1−U Ω

(11.15)

In the frequency domain the asymmetric term reads asym

ΨG

(σ) = −

U 󸀠 (1 + Ωσ ) 1−U

m∗ (−σ) .

(11.16)

11.7 Conclusion and discussion

| 229

Figure 11.9: Symmetric (common) and asymmetric parts of the geodetic excitation from 2000 to 2012, as derived from C04 pole coordinates. Complex Fourier spectrum done over the period 1980– 2012 reveals retrograde terms at 1 year (1 mas) and 433 days (2 mas), of which the sum accounts for a 6.4 year modulation in time domain.

11.6.2 Analysis As the prograde terms at 433 and 365 days dominate the polar motion with amplitudes of about 100–200 mas, Eq. (11.16) shows that the asymmetric part of the excitation is mostly composed of retrograde terms at these periods, at the level of 1 mas. A more accurate estimation is done from the pole coordinate time series by the mean of the digitization derived in Appendix A.3. In Figure 11.9, the asymmetric excitation exhibits variations of about 2 mas, that is, almost 10 % of the common retrograde annual harmonic, dominating the retrograde band. This order of magnitude is significant in the light of the uncertainty of the geodetic excitation (a few tens of µas according to Eq. (9.11)).

11.7 Conclusion and discussion Nowadays, the comparison of polar motion to various sets of hydro-atmospheric excitation through Liouville equations affords one to better constrain the Chandler parameters: according to the mean values of Table 11.2 with removal of the seasonal term, 431.9 d ≤ Pc ≤ 435.0 d and 43 ≤ Qc ≤ 102. From (11.9) the corresponding effective

230 | 11 Chandler’s wobble Love number presents the values (R(k) = 0.3557 ± 0.0021, I(k) = −0.0048 ± 0.0020). If we assume that the hydrostatic contribution of the ocean is given through the ocean Love number ko = 0.0477 + i0.0002 (see Chapter 6 and Table 6.2), its removal from R(k) leads to the quasi-elastic solid Earth Love number (R(k2 ) = 0.3080 ± 0.0021, I(k2 ) = −0.0050 ± 0.0020), consistent with 0.3074 − i0.0036, namely the value of the anelastic Love number at the Chandler wobble frequency according to Eq. (6.12) of the IERS Conventions 2010 [168]. These parameters well account for the resonance in the seasonal band, but when entering the rapid band below 20 days and approaching the diurnal band, the dynamical response of the ocean reshuffles the cards in a way that has not been fully clear until now, except in the nutation band as exposed in Sections 6.6 and 7.6. The modification of the resonance parameters should also concern the whole band stretching from 1 day to fortnightly periods, where the response of the ocean is still dynamical. In particular, we can expect a resonance period between 380 and 430 days. However, our attempt to obtain robust estimates of the resonance parameters between 2 days and 2 weeks has failed, probably because of the error affecting the hydro-atmospheric AMF. This lack of knowledge also affects the resonance parameters at decadal time scale, where the viscous mantle reaction is becoming more prominent and lengthens the Love number k2 by an amount that is undetermined (the resonance period should also lengthen). In 2011 the variability of the Chandler wobble was reconstructed from the mass transports in the oceans and the atmosphere [17]. This has been confirmed by 8 complementary years of hydro-atmospheric and geodetic series. For explaining the phase variation, there is no need to investigate speculative processes, like geomagnetic triggering [11]. Moreover, the mere fact that two independent sets of the coupled circulation model yield similar results cannot be a coincidence. The tiny fluctuations of the excitation (∼ 1 mas) cannot abruptly change the Chandler wobble. After some years, the initial wobble in (11.2) remains preponderant, for it persists over decades depending on relaxation time τ = Qc Tc /π. This study leads to τ ∼ 15–40 years or Qc ∼ 40–100. The Chandler band excitation is gradually integrated and can significantly modify the Chandler wobble after several years. Actually, at multi-decadal time scale the excitation presents some regular patterns of variability, in particular an amplitude modulation of about 18–20 years, corresponding to a 37–40 year modulation of the Chandler wobble amplitude. The interval 1830–2020 reveals a 80–90 year variation of the Chandler wobble envelope (see Figure 11.7). We support the idea that this seemingly regular variability is triggered by climatic cycles [238], which could result from a synchronisation with the long term luni-solar tide or solar activity. Yet, these modulations can be represented by the sum of close harmonics around 433 days, and presenting subsequent multi-decadal beatings. So, some authors advocate a degeneracy of the Chandler normal mode into multiple frequencies (see e. g. [166]).

11.7 Conclusion and discussion

| 231

An asymmetric effect modifies the equatorial excitation by introducing retrograde terms at 433 days and 365 days, up to 2 mas. This is more than the uncertainty whereby the excitation is modeled through mass redistributions within the surface fluid layers. To this extent, asymmetry of Liouville equations should not be dismissed in the geophysical interpretation. But the question of whether asymmetry improves the agreement with atmospheric and oceanic AMF has not received a clear answer because of the error affecting these ones. By estimating the pole solution (A.21), the asymmetric retrograde perturbation produces an effect of a few mas on polar motion, which is minor with regard to the Chandler wobble amplitude. So, the symmetric approximation is sufficient until now.

12 Diurnal and sub-diurnal hydro-atmospheric effect We see the tides of the water; we do not see the tides of the air. Atmosphere, as well as oceans, has its flux and reflux, even more gigantic, rising, huge tumor, to the moon. Victor Hugo, The toilers of sea, free translation

12.1 Geophysical diurnal and sub-diurnal effects The diurnal variation of the temperature completes the seasonal cycle around the world. Its footprint in the Equatorial Atmospheric Angular Momentum Function (EAAMF) is expected and has been already noticed in the spectrum of Figure 8.3. The amplitude of the diurnal peak of the wind term (10 mas) is even larger than the annual variation. The semi-diurnal harmonic is at the border of the temporal resolution permitted by EAAMF series. At daily and half-daily periods the ocean surface also reacts to the solar heating and atmospheric changes, yielding see level variations and currents, which add to the ocean tides. Judging by Figure 8.6, this “high-frequency” ocean contribution is smaller than the atmospheric ones but still significant. The high-frequency fluctuations of the continental waters are not determined. Except for high diurnal rainfall in the equatorial region, hydrological process probably do not bring about a significant contribution to the diurnal and sub-diurnal hydrological equatorial excitation. So, oceanic and atmospheric circulation also impact the high-frequency polar motion. Because of the resonance at the free core nutation frequency, the effect is maximal in the retrograde diurnal band, yielding diurnal terms up to 100 µas [14, 93]. The complementary high-frequency band contributes to the polar motion at the level of 10 µas only [24, 27]. Recall that, according to the IERS convention (Chapter 3), the retrograde quasi-diurnal effect has to be formulated as a spatial nutation of the CIP. However, the diurnal and sub-diurnal hydro-atmospheric polar motion interferes with the tidal oceanic effect, of which the total amplitude reaches about 500 µas and is thus dominant. As ocean tides are regular, this effect is modeled precisely. In this respect, IERS recommends a model composed of 71 diurnal or semi-diurnal harmonic waves [168]. Meanwhile, the triaxial Earth figure or inertia ellipsoid passes every 12 hours in front of the Sun and every 12.9 hours in front of the Moon, so that the Moon and Sun exert a quasi-semi-diurnal equatorial moment of force on the solid Earth, as seen from a celestial reference frame. As a result one has a quasi-semi-diurnal oscillation of the celestial pole (∼ +2Ω), which has to be mapped in the TRF as a prograde diurnal polar motion (∼ +Ω), up to 20 µas [90]. So, the hydro-atmospheric terms at diurnal and sub-diurnal frequencies perturb the dominant lunisolar effect conveyed by ocean tide and triaxiality, but it is considerable, enough for taking it into account. The retrograde diurnal oscillations of the fluid angular momentum contribute to nutation, which has well been monitored by VLBI (see Chapter 13) since 1984. The https://doi.org/10.1515/9783110298093-012

12.2 Liouville equation in the retrograde diurnal band | 233

complementary part of the spectrum below 2 days contributes to the high-frequency polar motion, of which the observation remains an open issue. First, hourly or subhourly GNSS determinations may be ambiguous, probably due to the aftermath of Earth tides on the satellite orbits. On the other hand, discontinuous VLBI 24-h sessions do not afford estimations at a sub-diurnal rate, but diurnal and semi-diurnal tidal components can be fitted as global parameters over a multi-year interval. Such VLBI and GNSS analyses indicate corrections to IERS tidal model no larger than a few tens of µas. The authors of a recent GNSS analysis [199] assess a spectral noise of about 5 µas for the diurnal/sub-diurnal components. But, as further validation is required, especially comparative studies between GNSS and VLBI estimates, an astrometric confrontation will be eluded. And assuming that the uncertainty of modeled terms of the tidal oceanic effect is about 30 µas for the largest ones, the values of hydro-atmospheric or other geophysical contributions cannot be corroborated with more precision. We take this issue in the light of the latest series of atmospheric and oceanic AMF. We perform an overall spectral evaluation of the diurnal and sub-diurnal contributions to polar motion, considering consecutively the atmosphere and the oceans. The retrograde diurnal excitation will be more carefully investigated in Chapter 13.

12.2 Liouville equation in the retrograde diurnal band Investigating a polar motion excitation near −Ω necessitates one to consider the coupled system composed of the extended mantle and of the fluid core. Then the polar motion is governed by (7.91): p+i =

σ̃ cd + σ̃ f σ̃ cd σ̃ f

ṗ −

p̈ d ̃ σc σ̃ f

σc d d e e e e [(i + σ̃ f )(χma + χmo ) + (i + σ̃ cd )(ama χma + amo χmo )] , d dt dt σ̃ c σ̃ f

or in the frequency domain by (7.90): σ σc e e e p(σ) = d c [χma (σ) + χmo (σ)] + [a χ e (σ) + amo χmo (σ)] , σ̃ f − σ ma ma σ̃ c − σ

(12.1a)

(12.1b)

with ama ≈ 9.1 × 10−2 , amo ≈ 2.6 × 10−4 and the effective AMF given by (7.39c): 1 + k2󸀠 ma χ ≈ 1.112χ ma , 1 − k/ks 1 = χ mo ≈ 1.606χ mo . 1 − k/ks

e χma = e χmo

(12.1c)

In contrast to the common frequency band below 0.5 cpd, polar motion reacts differently to the matter term and the motion term of the AMF. Each of them has its own frequency transfer function Tma (σ) or Tmo (σ), plotted in Figure 12.1.

234 | 12 Diurnal and sub-diurnal hydro-atmospheric effect

e (σ) and Tmo = Figure 12.1: Amplitude and argument of the transfer function Tma = p(σ)/χma e p(σ)/χmo (σ) around the frequency of the free core nutation (left) and from −2 cpd to +2 cpd (right). Outside the retrograde diurnal band Tma and Tmo can be confused with the common transfer function T , only affected by the Chandler resonance.

Method To have a global view of the atmospheric and oceanic effects, we estimate its spectral e e content. Considering the time series of the effective AMF χma (t) and χmo (t) determined by (12.1c), we perform their discrete Fourier transform, and multiply it by the transfer function (12.1b) for obtaining the resulting polar motion |p(σ)| in the frequency domain.

12.3 Diurnal and sub-diurnal atmospheric effect 12.3.1 Four-daily AAM series The series NCEP–reanalysis,1 ECMWF or ECMWF–TUW2 and JMA3 (see Table 8.2) are appropriate for investigating the diurnal atmospheric polar motion, for they have a time resolution of 6 hours at least. 1 http://files.aer.com/aerweb/AAM/ 2 Derived at Technische Universität of Vienna, http://ggosatm.hg.tuwien.ac.at/ROTATION/AAM/ VERSION1/ 3 http://files.aer.com/aerweb/AAM/

12.3 Diurnal and sub-diurnal atmospheric effect | 235

Figure 12.2: Atmospheric effect on polar motion in the frequency range 0.5–2 cpd (NCEP model). Period 2004–2016. Upper plots: zoom around 1 cpd for prograde and retrograde components.

The frequency domain procedure is applied to the NCEP series (with NIB correction) from 2004 to 2016. Its 6-hourly regular sampling allows us to catch any circular component circling with periods between 12 and 48 hours. Doing so, we obtain the spectrum of Figure 12.2, where prograde and retrograde frequency bands are shown side by side.

12.3.2 Retrograde diurnal band The largest peaks, up to 100 µas, are noticed in the retrograde diurnal band around −1 cpd (tidal frequency S1 ), and their predominance results from the resonance at the free core nutation frequency σf = −1.0050 cpd. As the pressure term resonates about 100 times more than the wind term, it yields the dominant effect in the vicinity of −1 cpd. But, going away from −1 cpd, the resonance fades and the wind term reveals its strongest power. At the right of the main peak S1 there appears an isolated harmonic of 5 µas at frequency −0.93 cpd (25.8 hours), exactly corresponding to the lunar tide O1 , as well as a broad band peak at −0.85 cpd of the same amplitude. Zooming in around −1 cpd, a “fine structure” emerges unveiling the tidal periodicities π1 , P1 , S1 , K1 , ψ1 , ϕ1 (see the summary of Table 12.1). The peak Ψ1 at frequency

236 | 12 Diurnal and sub-diurnal hydro-atmospheric effect Table 12.1: The retrograde diurnal frequency σ = −Ω + σ 󸀠 with σ 󸀠 ≪ Ω is mapped to a long periodic celestial component of frequency σ 󸀠 in a non-rotating frame. Notation in TRF Ψ11 O1 π1 P1 S1 K1 σf ψ1 ϕ1

Terrestrial frequency σ (cpd)

Celestial frequency σ 󸀠 = σ + Ω (cpd)

−0.85 −0.92953 −0.99452 −0.99726 −1.00000 −1.00274 −1.00507 −1.00547 −1.00822

+1/6.55 +1/13.66 +1/121.747 +1/182.621 +1/365.242 ∞ −1/430 −1/365.242 −1/182.621

fΨ1 = −1.002733 − 1/365.25 = 1.0055 cpd is accompanied just at the right by a peak at frequency σf = −1.002733 − 1/430 = −1.0050 cpd. A least-square adjustment shows that all these terms, except O1 , are strongly dephased with respect to the corresponding diurnal gravitational tides, and thus result from thermal effects to a large extent. First, the S1 wave is associated with the 24-hour heating cycle following the motion of the Sun in the TRF. The peaks P1 and K1 are symmetric with respect to S1 , and they can result from the seasonal modulation of the 24 hour cycle as proposed in [14, 237], since we observe that 1 cpd + 1/365.242 cpd = 1.0027 cpd = σK1 ,

1 cpd − 1/365.242 cpd = 0.9973 cpd = σP1 .

(12.2)

Moreover, the annual modulation develops with a 6 month lag in the southern and the northern hemispheres, so that the diurnal EAAMF can reach a maximum two times per year, suggesting a semi-annual modulation of the S1 cycle, which generates the periodicities 1 cpd + 2/365.242 cpd = 1.0055 cpd = σΨ1 ,

1 cpd − 2/365.242 cpd = 0.9945 cpd = σπ1 .

(12.3)

Finally, a ter-annual modulation is also at stake, explaining the presence of the tidal cycle Φ1 (1.0082 cpd): 1 cpd + 3/365.242 cpd = 1.0082 cpd = σΦ1 .

(12.4)

On the other hand the spectral component O1 , mostly observed in the wind term, cannot result from a thermal flux, and it hints to the direct effect of the lunar tide, as will be demonstrated in the next chapter. The peak at −0.85 cpd probably stems from the atmospheric normal mode Ψ11 , also discussed in the next chapter.

12.3 Diurnal and sub-diurnal atmospheric effect | 237

12.3.3 Prograde diurnal and semi-diurnal bands In the prograde diurnal band and semi-diurnal band (prograde and retrograde), the atmospheric excitation also exceeds 1 mas, but it is strongly attenuated by the transfer function (by a factor 370 at 1 cpd and 720 at 2 cpd), which is no more enhanced by the free core nutation resonance. So, whereas the prograde diurnal band presents the same frequencies as the retrograde diurnal ones, except O1 , the corresponding amplitudes do not exceed 5 µas, which is one order of magnitude smaller. The semi-diurnal excitation, at the level of 1 mas, is reduced to the term S2 (12 h), which can be merely interpreted as the second harmonic of the 24-hour thermal cycle. The corresponding polar motion drops down to 1 µas. Thus, it can be concluded that the atmospheric contribution to the high-frequency polar motion of the CIP, combining semi-diurnal band and prograde diurnal band, does not present any observational interest with regard to the tidal oceanic counterpart, amounting to at least 100 µas.

12.3.4 Comparison of the results associated with NCEP, ECMWF-TUW, JMA models For the AAM time series associated with NCEP, ECMWF-TUW and JMA we adjust the model composed of the above-mentioned diurnal harmonics: N

p(t) = ∑(aj + ibj )eiθ(j) j=1

(12.5)

where j is the index of the wave and the argument θ(j) = ωj t + ϕj linearly varies at the angular rate ωj . If the frequency is also the one of a tidal wave, ϕj is taken as the conventional lunisolar argument of this wave. Over the period 2004–2016, we fit in total 12 harmonic terms, 5 of them being prograde (π1+ , P+1 , S+1 , K+1 , ϕ+1 ) and the other ones retrograde (Ψ11 , O−1 , π1− , P−1 , S−1 , K−1 , ψ−1 ). For the three centers the pressure terms yield consistent fits within the formal error. This was not the case ten years ago, especially for the retrograde diurnal band [235]. So, at a diurnal time scale the atmospheric global circulation models have converged with time. But the wind term harmonics do not reach such a coherency level: from one model to another, the coefficients aj and bj can differ by a ratio ranging from 1 to 3. If ECMWF-TUW and NCEP present similarities for some frequencies (like P−1 ), JMA is not consistent: we have the absence of the O1 retrograde wave in the wind term, and striking discrepancies in the diurnal prograde band. Figure 12.3 allows one to appreciate the differences between ECMWF-TUW and NCEP models. Significant discrepancies affect the wind term contribution at +1 cpd (S1 ), and the pressure term effect in the retrograde Ψ1 band touched by the FCN resonance.

238 | 12 Diurnal and sub-diurnal hydro-atmospheric effect

Figure 12.3: Spectral difference in the high-frequency polar motion estimated from NCEP and ECMWF TU Wien models. Pressure and wind contributions are separated.

12.4 Diurnal and sub-diurnal non tidal oceanic effect on polar motion Four-daily OAM series In 2018 a team of the MIT derived hourly OAM from ECCO model over the period 1992– 2016,4 denoted ECCO-1h. Geoforschung Zentrum of Postdam (see Table 8.2) derives 3 hourly OAM series from the MPIOM model; regularly updated for operational purpose, they cover the period from 1976 to the current day with a latency not exceeding 24 hours.5 For both the ECCO-1h and the MPIOM operational series, the terms at the lunisolar tidal frequencies are excluded or strongly mitigated. However, the ECCO-1h series still contain a power around the FCN frequency (−1.0055 cpd). The fluctuations at tidal frequencies are present in the OAM output of the ocean Model for Circulation and Tides (OMCT), which was produced at GFZ until 2017.6 In the following these series will be denoted OMCT. As for the atmosphere, a spectral overview of the oceanic diurnal/sub-diurnal effects is obtained by multiplying the OAMF spectrum by the frequency transfer function (12.1b). In Figure 12.4 we report the spectra obtained for 4 Series to download from ftp://euler.jpl.nasa.gov/sbo/oam_global/ECCO_v4r3_yesFWF.oam, documented “Data sets used in ECCO Version 4 Release 3” https://dspace.mit.edu/handle/1721.1/120472 5 ftp://rz-vm115.gfz-potsdam.de/EAM 6 ftp://ftp.iers.org/products/geofluids/oceans/eamf/GGFC2010/GFZ/old_series/

12.4 Diurnal and sub-diurnal non tidal oceanic effect on polar motion

| 239

Figure 12.4: Oceanic effect on polar motion in the frequency range 0.5–2 cpd (ECCO-1h and OMCT series). Period 2004–2016. Upper plots: zoom around 1 cpd for prograde and retrograde components.

240 | 12 Diurnal and sub-diurnal hydro-atmospheric effect

Figure 12.5: Atmospheric (ECMWF) and oceanic (MPIOM) effects on polar motion in the frequency range 0.5–2 cpd. Period 2004–2016. Upper plots: zoom around 1 cpd for prograde and retrograde components.

ECCO-1h and OMCT series. The contribution of ECCO-1h OAM is only significant in the free core nutation band. Whereas MPIOM yields even smaller effects and is not represented here, the OMCT series may lead to the conclusion that the oceans contribute to high-frequency polar motion as much as the atmosphere. Prograde quasi-diurnal oscillations reach 1–2 µas. The semi-diurnal variation exceeds 1 µas in the case of the OMCT model only. Whatever the oceanic series would be, as for the atmosphere, the oceans only impact the retrograde diurnal band at an observational level.

12.5 Combined effect of the atmosphere and the oceans In Figure 12.5 the ECMWF atmospheric and MPIOM oceanic contributions are represented side-by-side, as well as their combination. Close to −2 days, the combined effect presents fluctuations up to 3 µas. They are already in the realm of the hyper-rapid polar motion, for which the geophysical transfer function does not discriminate matter and motion terms: |T(−0.5 cpd)| ≈ 0.05. So the effective excitation at −0.5 cpd is of the order of magnitude of 3/0.05 = 60 µas, confirmed by the spectrum of Figure 10.4. But close to +0.5 cpd, the prograde excitation is three times smaller.

12.6 Conclusion

| 241

12.6 Conclusion The atmospheric circulation and the non tidal oceanic transports perturb the diurnal and sub-diurnal oscillations of the rotation pole at an observational level. The corresponding effects are not precisely estimated. In the conventional polar motion, excluding the retrograde quasi-diurnal wobble, they are up to 10 µas. This is about the order of magnitude of the differences between VLBI or GNSS determinations. So, the detection of these effects is an open issue, in particular a better geodetic precision would not ensure an unambiguous separation between the hydro-atmospheric contributions and their corresponding tidal oceanic effects.

13 Fluid layer effect on nutation The action of the sun and moon [...] produces tides in the atmosphere, which delicate observations have been able to render sensitive and measurable. John F. W. Herschel in Astronomy (1834)

13.1 Formalism for fluid layer effect on Earth’s nutation Of all the hydro-atmospheric equatorial oscillations found in the range of diurnal and sub-diurnal frequencies, only the retrograde diurnal ones produce an effect which can be looked for in common EOP determinations with a time resolution larger than 1 day, namely the observed nutation residuals or celestial pole offsets dX and dY. Indeed, according to Section 3.3.5, the retrograde diurnal oscillations, which are conventionally excluded from the pole coordinates p, are determined in the non-rotating frame as celestial pole offsets, P = −p eiΘ ,

(13.1)

with periods larger than 2 days. Here Θ is the Earth angle of rotation expressed by (3.4). As Θ is mostly an increasing angle at the constant rate Ω, a retrograde quasidiurnal oscillation p in the TRF at frequency σ = −Ω + σ 󸀠 with σ 󸀠 ≪ Ω is mapped to a long periodic celestial component P of frequency σ 󸀠 . The accurate determination of the celestial pole offsets (dX, dY) has started in 1984. They are plotted in Figure 13.1 until 2019, along with the complex Fourier spectrum of dX + i dY. The retrograde diurnal band could be handled by determining p from (12.1a) and then estimating nutation correction by (13.1). A more direct and elegant treatment, proposed by Brzezinski in 1994 [23], is to form the Liouville equation in terms of the nutation correction P. This is achieved in two steps. First we replace p with −P e−iΘ and its time derivatives with ṗ = e−iΘ (−Ṗ + iΩP) ,

p̈ = e−iΘ (Ω2 P + 2iΩṖ − P)̈ ,

(13.2)

in the Left Hand Side (LHS) of (12.1a) (Θ̇ ≈ Ω). We obtain LHS = −

e−iΘ {(σ̃ cd σ̃ f + Ω(σ̃ cd + σ̃ f ) + Ω2 )P + (σ̃ cd + σ̃ f + 2Ω)iṖ − P}̈ . σ̃ cd σ̃ f

(13.3)

Now, we introduce the space-referred angular frequency of the free polar motion resonance in the retrograde diurnal band σ̃ c󸀠 d = σ̃ cd + Ω, where σ̃ cd ≈ Ω/380 (1 − i/10) has been modeled in Section 7.6. Considering as well the Free Core Nutation (FCN) angular frequency σf󸀠 = σf + Ω with σf ≈ −Ω(1 + 1/430) (1 − i/32000) (for Qf = 16000, see https://doi.org/10.1515/9783110298093-013

13.1 Formalism for fluid layer effect on Earth’s nutation | 243

Figure 13.1: Celestial pole offsets dX and dY (C04 solution) over the period 1984–2019 and complex spectrum.

Eq. (7.75)), we have σ̃ cd σ̃ f + Ω(σ̃ cd + σ̃ f ) + Ω2 = σ̃ cd (Ω + σ̃ f ) + Ω(Ω + σ̃ f ) = σ̃ f󸀠 (σ̃ cd + Ω) = σ̃ c󸀠 d σ̃ f󸀠 . Hence (13.3) can be reduced to LHS = −

eiΘ {σ̃ 󸀠 d σ̃ 󸀠 P + i(σ̃ c󸀠 d + σ̃ f󸀠 )Ṗ − P}̈ . σ̃ cd σ̃ f c f

(13.4)

Then, the effective Equatorial Angular Momentum (EAM) determining the Right Hand Side (RHS) are replaced with the Celestial Equatorial Angular Momentum (CEAM), denoted χ 󸀠 e , and related to the EAM as P is related to p: 󸀠e e χma = −χma eiΘ ,

󸀠e e χmo = −χmo eiΘ .

(13.5)

This form of CEAM slightly diverges from the original definition [23] stating that 󸀠e e χma/mo = −χma/mo ei GST where GST signifies the Greenwich Sidereal Time. In order to comply with the recommendations issued by IERS for describing rotation of the Earth, GST is replaced by the angle of rotation of the Earth, Θ [168], hence we have Eq. (13.5).1 1 The CEAM correspond to the EAM components χ 󸀠󸀠 of the angular momentum function in the celestial intermediate system GX0 Y0 Z0 – determined by the equatorial plane of the Celestial Intermediate Pole

244 | 13 Fluid layer effect on nutation Noting that χ̇ e = e−iΘ (−χ̇ 󸀠 e + iΩχ 󸀠 e ), the RHS of (12.1a) reads RHS = −

σc e−iΘ 󸀠e 󸀠e 󸀠e 󸀠e ̇ 󸀠 e + χmo ̇ 󸀠 e ) + Ω(χma {i(χma + χmo ) + σ̃ f (χma + χmo ) σ̃ cd σ̃ f

󸀠e 󸀠e 󸀠e 󸀠e ̇ 󸀠 e + Ωχma ̇ 󸀠 e + Ωχmo + ama (iχma ) + amo (iχmo ) + σ̃ cd (ama χma + amo χmo )} .

(13.6)

We recall that the dimensionless coefficients ama ≈ 9.1 × 10−2 and amo ≈ 2.6 × 10−4 express the response of the FCN mode to matter and motion terms, respectively. By setting LHS = RHS, we obtain a kind of Liouville equation in terms of nutation perturbations, but now of second order: 󸀠e 󸀠e 󸀠e 󸀠e σ̃ c󸀠 d σ̃ f󸀠 P + i(σ̃ c󸀠 d + σ̃ f󸀠 )Ṗ − P̈ = σc {σ̃ f󸀠 (χma + χmo ) + σ̃ c󸀠 d (ama χma + amo χmo )

̇ 󸀠 e + χmo ̇ 󸀠 e ) + i(ama χma ̇ 󸀠 e + amo χmo ̇ 󸀠 e )} + i(χma

(13.7)

or P+i

σ̃ c󸀠 d + σ̃ f󸀠 σ̃ c󸀠 d σ̃ f󸀠

Ṗ −

P̈

σ̃ c󸀠 d σ̃ f󸀠

= σc { +

1 1 󸀠e 󸀠e 󸀠e 󸀠e + amo χmo ) (χma + χmo ) + 󸀠 (ama χma 󸀠 d σ̃ f σ̃ c

i i ̇ 󸀠 e + χmo ̇󸀠 e ) + ̇ 󸀠 e + amo χmo ̇ 󸀠 e )} . (13.8) (χma (ama χma 󸀠 󸀠 d 󸀠 d ̃ ̃ ̃ σc σf σc σ̃ f󸀠

In the frequency domain this differential equation reads 󸀠 󸀠e 󸀠 󸀠e P(σ 󸀠 ) = Tma (σ 󸀠 )χma (σ 󸀠 ) + Tmo (σ 󸀠 )χmo (σ 󸀠 ) ,

(13.9)

in which 󸀠 Tma (σ 󸀠 ) = σc [

ama 1 ], + σ̃ c󸀠 d − σ 󸀠 σ̃ f󸀠 − σ 󸀠

(13.10)

and the Conventional Intermediate Origin in this plane (see Section 3.3.2). Indeed, the components χ 󸀠󸀠 can be derived from their terrestrial counterparts χ by applying the matrix transformation χ 󸀠󸀠 = R3 Wχ , where R3 represents the diurnal rotation matrix and W the polar motion matrix. The major part of this transformation is caused by the uniform rotation in R3 with the mean angular velocity Ω. We argue that the residual rotation can be considered as a perturbation. At sub-secular time scales, polar motion produces oscillations of the geographic pole below 1󸀠󸀠 , and the non-uniform part of the rotation angle in R3 is smaller than 20󸀠󸀠 . As the χi are small quantities of the order of 10−7 radian, the maximum effect of the variable rotation is 20 × 10−7 = 2 10−6󸀠󸀠 over 100 years, which can be neglected in the light of the AAM accuracy (> 10−4󸀠󸀠 ). Thus, for sub-secular phenomena, the above transformation can be restricted to an axial rotation, reading χ 󸀠󸀠 = R3 [Θ(t)]χ . Hence the complex equatorial component reads χ 󸀠󸀠 = (cos Θ χ1 − sin Θ χ2 ) + i(sin Θ χ1 + cos Θ χ2 ) = eiΘ χ = −χ 󸀠 .

13.2 Overview of the celestial angular momentum series | 245

󸀠 󸀠 around the frequency of and Tmo Figure 13.2: Amplitude and argument of the transfer function Tma the Free Core Nutation in cycle/year (left) and for the whole nutation band [−0.5, +0.5] cycle/day (right).

󸀠 and a similar expression can be phrased for Tmo using the coefficient amo . The transfer 󸀠 󸀠 functions Tma and Tmo , respectively, for the matter term and of the motion term of the CEAM, are plotted in Figure 13.2.

13.2 Overview of the celestial angular momentum series 13.2.1 Data In the TRF retrograde diurnal terms of the EAM are squeezed in a frequency band around 24 h. By applying (13.5) we remove the diurnal carrier in the TRF-based EAM, and therefore produce the Celestial Angular Momentum (CEAM) function that represents retrograde diurnal oscillations as slow periodic variations from 2 days to several years, reflecting the celestial motions with which they originate. Such “demodulated” time series form the basis for a characterization of the effects of the major retrograde diurnal components (S1 , P1 , K1 ) on Earth’s precession–nutation [14, 235, 24]. Considering the atmospheric and oceanic data on their common time interval 1958–present, described in Table 8.2 and used in the previous chapter, namely the – 6-hourly atmospheric EAMF estimates from the reanalysis model of NCEP/NCAR (1950–2019) [186];

246 | 13 Fluid layer effect on nutation – –

6-hourly atmospheric EAMF estimates from the ECMWF−TUW model from TU Wien (1980–2019) [189]; 3-hourly oceanic EAMF estimates from the MPIOM model (1976–2019) forced by ECMWF wind and surface pressure;

we computed the associated CEAM function according to (13.5) for matter and motion terms. Prior to demodulation, we removed the long term components (periods larger than 2 days) of the EAMF. After application of (13.5), a low band pass filter was used to eliminate residual diurnal/sub-diurnal signal content and to obtain the celestial excitation limited to periods larger than 2 days, corresponding to the precession–nutation frequency band. Moreover, with regard to the angular momentum budget done in the next section, we consider the non-effective estimates of the CEAM, in which the multiplication by (C − A)Ω directly yields the matter and motion terms of the fluid layer angular momentum. 13.2.2 Spectral analysis of the atmospheric CEAM The complex Fourier spectrum of the obtained NCEP-atmospheric CEAM is displayed in Figure 13.3 for both pressure and wind terms (ECMWF series yield similar results) over the period 1984–2019 starting with VLBI monitoring of the precession–nutation. It is dominated by a prograde annual oscillation at 365 days (S1 in the TRF) and its subharmonics at 182 (P1 ), 121 (π1 ), and 91 days (1.0083 cpd in the TRF). The retrograde part of the spectrum only exhibits power at −365 days (ψ1 ) and −182 days (ϕ1 ). As already pointed out, these seasonal oscillations have been extensively analyzed by various authors with regard to their observable effects on Earth’s nutation [237, 14, 235]. Less investigated is the band below 30 days; its spectral zoom clearly unveils a double peak at +13.66 days (25.8 h or O1 in the TRF) and +13.63 days, and a broad band peak around +7 days (28 h in the TRF). 13.2.3 Spectral analysis of the oceanic CEAM The complex Fourier spectrum of the oceanic MPIOM–CEAM over the period 1985– 2019 is displayed in Figure 13.4 for both water height and current terms. In comparison to atmospheric excitation the seasonal and sub-seasonal peaks are much smaller. Indeed, for the reasons mentioned in Chapter 8 (p. 163), the harmonics corresponding to S1 and P1 lunisolar tides have been removed from MPIOM OAM series. Atmospheric and oceanic CEAM are more similar at periods below 30 days. If the 13.6 day term is present in water height and pressure terms with comparable order of magnitude, it is missing in the current term. Moreover, the water height term shows a supplementary broad band signal around 4 days. The current term does not contain any weekly signal but exhibits broad band signals at 4 days and 2.7 days.

13.2 Overview of the celestial angular momentum series | 247

Figure 13.3: Amplitude spectrum of the atmospheric Celestial Angular Momentum from the NCEP/NCAR model, wind (top) and pressure (bottom) components over the period 1984–2019.

248 | 13 Fluid layer effect on nutation

Figure 13.4: Amplitude spectrum of the oceanic Celestial Angular Momentum derived from the MPIOM model, current (top) and water height (bottom) components over the period 1984–2019.

13.3 The 2–30 day band of the atmospheric CEAM

| 249

Figure 13.5: Equatorial components X and Y of the Celestial Atmospheric Angular Momentum for the wind term χw and the NIB pressure term χp multiplied by 2.2 (linear regression coefficient from 1958 to 2015) computed from NCEP AAM series after eliminating long term periods above 1 month. Extract over 130 days, date in Modified Julian Day (MJD).

13.3 The 2–30 day band of the atmospheric CEAM 13.3.1 Proportionality between wind and pressure terms The analysis of seasonal and sub-seasonal band with regard to its effect on nutation is postponed to Section 13.4. We now focus on the rapid band of the atmospheric NCEP−CEAM between 2 days and 30 days, isolating it in the time domain by using an appropriate high band pass filter, having an admittance of 99 % at 30 days. An extract over 130 days of the obtained time series is depicted in Figure 13.5 (from MJD 50000 to MJD 50130), showing both wind terms in the X and Y components and the full Non-Inverted Barometer (NIB) pressure terms, which are void of any approximative correction for the oceanic response to air pressure variations. It is remarkable that, for the X and Y coordinates, wind and pressure terms are evidently proportional. This finding is not limited to the considered time interval, as correlations throughout the period 1958–2015 amount to 0.57 for both X components and Y components. Linear regression over the entire analysis time gives χw󸀠 ∼ 2.2χp󸀠 . For the case of a hydrostatic re-

250 | 13 Fluid layer effect on nutation

Figure 13.6: Contributions of the Southern Hemisphere (SH) and Northern Hemisphere (NH) to the wind term in χX󸀠 (bottom) and χY󸀠 (top) for the 2−30 day band. Time series of 130 days, commencing at modified Julian day 50000 (October 10, 1995).

sponse of the oceans (Inverted Barometer or IB approximation) imposed on the CEAM, the ratio χw󸀠 /χp󸀠 increases up to 5.4, whereas the correlation drops to 0.45. The detected proportionality appears to be a feature of the short term CEAM from 2 days to 1 month; but it does not extend to the complementary spectral bands, ranging from 1 month to several years, where correlations between pressure and wind terms drop to 0.1. Another striking feature distinguishing the 2−30 day band from other parts of the spectrum is the fact that the contributions of northern and southern hemisphere to the wind terms have synchronous variations with similar phases and amplitudes, as shown by Figure 13.6. Such a behavior is not observed for the S1 thermal wave for which the southern and northern hemispheres contribute asymmetrically to χw󸀠 with a phase lag of 6 months. All these results are confirmed by an analogous analysis of the ECMWF AAMF time series. Meaning of the proportionality between pressure and wind term Let Γ⃗ be the torque that the atmosphere exerts on the solid Earth. It is composed of the bulge torque Γ⃗ b acting on the equatorial bulge because of pressure and gravitational forces, and of a local torque Γ⃗ l caused by pressure on the local topography as well as

13.3 The 2–30 day band of the atmospheric CEAM

| 251

the friction drag on Earth’s surface as seen in Section 8.3. In the non-rotating frame we 󸀠 have the following complex quantities: Hw/p for the equatorial wind/pressure term, Γ󸀠l 󸀠 for the local torque, Γb for the bulge torque, Γ󸀠ext for the external gravitational torque on the atmosphere. Following the derivations in Appendix E.4, it can be established in the frequency domain that (Eq. (E.47)) 1−

󸀠 󸀠 󸀠 σ 󸀠 σ 󸀠 Ĥ w −Γ̂ l + Γ̂ ext = , − Ω Ω Ĥ p󸀠 Γ̂ 󸀠b

(13.11)

where the hat ĉ orresponds to the Fourier transform. If the residual torque −Γ̂ 󸀠l + Γ̂ 󸀠ext is much smaller than the bulge torque Γ󸀠b then Ω − σ󸀠 ̂ 󸀠 Ĥ w󸀠 ≈ Hp . σ󸀠

(13.12)

For positive angular frequencies σ 󸀠 of the filtered CEAM, with periods from 2 days to 1 month, we have 1/30 Ω ≤ σ 󸀠 ≤ 1/2 Ω. The retrograde part of the band 2–30 days is much smaller and can be neglected for the following considerations. So, according to (13.12) pressure and wind terms become almost proportional. This is in contrast with the seasonal band (S1 in the TRF) where the smallness of the local torque with respect to the bulge torque is not satisfied [139]. Considering for the lunar tidal band a typical magnitude of |χp󸀠 | ∼ 0.2 mas (see the spectrum of Figure 13.3), the bulge torque magnitude, given by |Γ󸀠b | = Ω|Hp󸀠 | = Ω2 (C − A)|χp󸀠 | according to (8.29), amounts to ∼ 1.5 1018 Nm. In [15] it is shown that the external torque Γ󸀠ext is mostly composed of a 13.6-day component with an amplitude of ∼ 1017 Nm, which is at least 10 times smaller than the equatorial bulge torque Ω|Hp󸀠 |. Hence, as far as the local torque does not exceed the order of magnitude of the external torque, the above condition holds and explains the quasi-proportionality of Hw󸀠 and Hp󸀠 . 13.3.2 Tidally coherent wave at 13.6 days As emphasized by the spectral zoom of Figure 13.3, the main peak is at 13.66 days. The most natural hypothesis for its origin is the diurnal lunar tidal wave O1 determined by the Delaunay argument 2(F + ?) in the non-rotating frame, with F = ω + l being the sum of the perigee argument ω and the mean anomaly l, and ? being the longitude of the ascending node of the Moon on the ecliptic plane. Note that s = F + ? is the mean tropic longitude of the Moon. As expected from tidal theory, the main peak is accompanied by a side-lobe at 13.63 days having the argument 2F + ?. These two components differ by the frequency ? of the displacement of the ascending node of the Moon, that is, 1/18.6 cpy. The fact that we observe both of these peaks in CEAM substantiates the quasi-diurnal tidal lunar influence on CEAM, in particular on the wind component.

252 | 13 Fluid layer effect on nutation The celestial oscillations of arguments Φ1 = 2F + 2? = 2s (13.66 days) and Φ2 = 2F + ? (13.63 days) are fitted by a least-squares method to the model 2

χ 󸀠 = ∑(mjc + imjs )ei(Φj +π/2) .

(13.13)

j=1

For the period 1958–2015 we obtain χp󸀠 IB [mas] = (0.05 − i0.03) ei(Φ1 +π/2) + (0.01 − i0.00) ei(Φ2 +π/2) ,

χp󸀠 NIB [mas] = (0.19 − i0.07) ei(Φ1 +π/2) + (0.05 − i0.02) ei(Φ2 +π/2) , χw󸀠 [mas]

= (0.83 − i0.04) e

i(Φ1 +π/2)

+ (0.27 − i0.05) e

i(Φ2 +π/2)

(13.14)

.

The ms terms are small relatively to mc , except for the IB term where about two third of the regional contribution of the pressure field—the oceanic one—has been replaced by a time-variable mean value computed from all pelagic points. Disregarding the IB solution, the harmonic coefficients are therefore almost in phase with the tidal wave of argument Φi + π/2, confirming the proportionality of wind and pressure terms at this period and supporting their common tidal gravitational cause. The ratio χw󸀠 /χp󸀠 NIB = Hw󸀠 /Hp󸀠 ∼ 4 for the two tidal frequencies does not match the numerical value of the condition (13.12), namely Hw󸀠 /Hp󸀠 = 13.6 − 1 ∼ 13. To explain this difference, it does seem unreasonable to assume that the wind term is underestimated, e. g. as a consequence of the 10 mbar vertical boundary of the NCEP/NCAR model that neglects only about 1 % of the total atmospheric mass. On the other hand the ratio χw󸀠 /χp󸀠 IB ≈ 14 much better fits the expected ratio of 12.6, as if the effective pressure term around the O1 frequency was the one restricted to continents and a static IB ocean. This is quite peculiar, since an IB response of the oceans is generally observed above 10 days but not at diurnal periods in the TRF [182]. Temporal variability By reference to the high frequency CEAM model χ 󸀠 = (m1c + i m1s ) ei(Φ1 (t)+π/2) + (m3c + i m3s ) ei(Φ3 (t)+π/2) with Φ1 = 2(F + ?) (13.66 days, O1 tidal wave in the TRF) and Φ3 = Φ1 + 2lm (lm is the mean anomaly of the Moon) representing the weekly signal (6.86 days, 2Q1 tidal wave in the TRF), the parameters mc and ms were adjusted repeatedly in a sliding window of 6 years. As the 13.66 day and 13.63 day components cannot be split anymore over such a limited time span, they are grouped in the first term, m1c +i m1s . The results are reported in Figure 13.7 for the pressure and wind terms. The IB pressure term at 13.6 days (which gives the best theoretical ratio χw󸀠 /χp󸀠 at 13.6 days), and the NIB pressure term at 6.8 days are multiplied by the corresponding theoretical ratio from (13.12) for χ 󸀠 w /χp󸀠 . For these harmonics, the coherency in the variations of χp󸀠 and χw󸀠 are overwhelming. We notice

13.3 The 2–30 day band of the atmospheric CEAM

| 253

Figure 13.7: Variability of 13.66 d and 6.8 d components of the winds and pressure terms. 6-year sliding window fit of mc (left panel) and ms (right panel) parameters in the model (mc + ims ) eiΦ where Φ is the tidal argument. The NIB or IB pressure terms are multiplied by the theoretical ratio (13.12) corresponding to the considered period.

a 18–20 year modulation for the 13.66-day term, reflecting the beating produced by the 13.63-day term.

13.3.3 The quasi-weekly band As shown by the Morley wavelet decomposition in Figure 13.8, the broad band peak between 5 and 8 days is much more powerful than the thin peaks around 13.6 days, showing episodes with an amplitude exceeding 5 mas. Some studies like [24] attribute this weekly signal to the retrograde Rossby–Haurwitz atmospheric normal mode Ψ11 having in the TRF the geometry of a spherical harmonic cos(ϕ)eiλ (ϕ is the latitude, λ is the longitude) propagating to the west [219]. On the other hand, in the non-rotating frame, this resonant mode propagates from the west to the east as the Moon, and with an averaged period of about 7 days it could be amplified at planetary scale by the minor lunar tides 2Q1 (6.86 days) and σ1 (7.09 days) (at least 100 times smaller than O1 ). As shown by the associated sliding window least-squares fit plotted in Figure 13.7, the linear correlation between variable parts of the wind and NIB pressure term is even more striking than at 13.6 days (correlation coefficient above 0.7). Moreover, at quasiweekly periods (σ 󸀠 /Ω ≈ 1/7) the ratio χw󸀠 /χp󸀠 NIB ≈ 5.8 fits the condition (13.12) reading Hw󸀠 /Hp󸀠 ≈ 7 − 1 = 6, and is thus valid for the full pressure term in contrast to what is observed at 13.6 days. Our validation against complementary series from the ECMWF model reproduced quite well the results associated with NCEP data over the period 1985–2013, except for the wind component at 13.6 days, which contained a relatively large out-of-phase term ms (0.5 instead of 0.07 mas).

254 | 13 Fluid layer effect on nutation

Figure 13.8: Morley wavelet decomposition of the CEAM wind term from 2000 to 2010 in the lunar tidal band of 2–20 days.

13.3.4 Hydrostatic model of the tidal O1 oscillation of the pressure term Why is the phase of the O1 oscillation of the CEAM close to Φ1 = 2(F + ?) + π/2? Let us investigate this question by assuming a simple hydrostatic redistribution of the air masses under the action of the lunar tidal force. Because of the lunar tidal potential U1 , the surface pressure undergoes a variation from P0 to P0 + P1 , satisfying ⃗ 0 + ∇P ⃗ 1 = −(ρ0 + ρ1 )∇(U ⃗ 0 + U1 ) , ∇P

(13.15)

where U0 = W is the non-perturbed equipotential of the gravity field, ρ0 ≈ 1.3 kg/m3 is the non-perturbed uniform air density and ρ1 denotes the variation of the air density caused by the tide. Neglecting second order terms, and accounting for equilibrium in ⃗ 0 = −ρ0 ∇(U ⃗ 0 ), we get the initial state ∇P ⃗ 1 = −ρ0 ∇(U ⃗ 1 ) − ρ1 ∇(U ⃗ 0) . ∇P

(13.16)

Here U0 only depends on height. Thus, projecting this relation along the axes tangent to the local parallel and meridian, and defining a plane that is locally confused with the equipotential U0 , we get 𝜕U 𝜕P1 = −ρ0 1 , 𝜕θ 𝜕θ

𝜕P1 𝜕U = −ρ0 1 . 𝜕λ 𝜕λ

(13.17)

Integrating with respect to the colatitude θ and longitude λ lead to P1 (θ, λ, t) = −ρ0 U1 (θ, λ) + K(λ) ,

P1 (θ, λ) = −ρ0 U1 (θ, λ) + K 󸀠 (θ) .

(13.18)

13.3 The 2–30 day band of the atmospheric CEAM

| 255

Hence the perturbation of the surface pressure (over the equipotential) is P1 (θ, λ, t) = −ρ0 U1 (θ, λ) + K ,

(13.19)

where K is a constant, which cannot lead to any matter term change and will be dropped in what follows. According to (D.5) and Table D.1, gravimetric O1 tide caused by the Moon at the Earth surface is expressed by U1̃ (t) = −gN21 ξO1 Re[e−i(GMST+π−2F−2?−π/2) 𝒴2−1 ]

(13.20)

5 with ξO1 = −0.26223 m and N21 = √ 24π . Hence

P1 (t) = ρ0 gN21 ξO1 Re[e−i(GMST+π−2F−2?−π/2) 𝒴2−1 ] ,

(13.21)

or P1 (θ, λ, t) = (p21 (t) cos λ + p̃ 21 (t) sin λ)P21 (cos ϕ) with p21 = ρ0 gN21 ξO1 cos(GMST − 2F − 2? + π/2) ,

p̃ 21 = −ρ0 gN21 ξO1 sin(GMST − 2F − 2? + π/2) .

(13.22)

It turns out that these tesseral pressure components (of about 1 Pa) determine the AAM pressure term according to (E.3): χp ≈ −

r04 4π (p + ip̃ 21 ) , 5 (C − A)g 21

(13.23)

where r0 ≈ 6371 km is the mean equatorial radius of the Earth and g ≈ 9.81 m/s2 represents the mean gravity acceleration in the vicinity of Earth’s surface. So, the pressure term variation engendered by the Moon is χp = −√

4

2π r0 ρ ξ e−i(GMST−2F−2?+π/2) , 15 (C − A) 0 O1

(13.24)

and the corresponding CEAM reads χp󸀠 ≈ −χp ei GMST ≈ √

4 4 2π r0 ρ0 ξO1 i(2F+2?−π/2) 2π r0 ρ0 |ξO1 | i(2F+2?+π/2) e =√ e . 15 (C − A) 15 (C − A)

(13.25)

Numerical application gives the expression χp󸀠 = 0.28 [mas] ei(2F+2?+π/2) , which not only accounts for the order of magnitude of the observed O1 pressure term in (13.14) (0.2 mas) but also reproduces its phase. Can we also explain the amplitude and the phase of the O1 wind term? If the hydrostatic assumption is true for surface pressure, tidal winds do not blow at Earth’s surface, but at high altitudes. Tidal lunar variations of the horizontal wind have been indeed reported in the mesosphere (above 50 km) [183], in the stratosphere [155, 180], in the upper troposphere [124] but never in the low troposphere. So tidal winds do not

256 | 13 Fluid layer effect on nutation produce any notable friction torque. On the other hand, a tesseral tidal variation of the pressure field exerts a torque on the bulge but cannot contribute to the topographic torque, which results from spherical harmonics of degree higher than 3 according to (8.15). We thus conclude that the tidal atmospheric circulation does not contribute significantly to the local torque. So, in accordance with the condition (13.12), this accounts for the proportionality of χw󸀠 and χp󸀠 . As noted earlier, we obtain the expected ratio for 7-day oscillations, but for 13.6 days the theoretical constraint is satisfied only if we resort to the IB pressure term and not to the total matter term of the atmospheric angular momentum. 13.3.5 Oceanic CEAM in the 2–30 day band (25–48 hours) As shown by Figure 13.4, the 13.6 day oscillation in the water height term (0.2 mas) is as large as the one observed in the atmospheric pressure term, but the current term is about 0.1 mas against almost 1 mas for the wind term. Oddly, the 7 day broad band peak is only noticeable in the water height component, with an amplitude comparable with the atmospheric pressure term (∼ 0.15 mas). This can be related with the fact, highlighted earlier, that winds do not blow at Earth’s surface and thus cannot trigger currents at this period. This 7-day water height term is accompanied by a broad band peak around 4 days of comparable amplitude. In the terrestrial frame such an oscillation appears as a retrograde circular term with the period of about 1.33 day (32 h). This possibly reflects the fundamental Kelvin wave for the southern ocean computed in [170]. The power of the current term is mostly concentrated around 2.5 days (∼ 0.15 mas). From a terrestrial point of view, such an oscillation is retrograde with a period of about 1.66 day or 39.8 hours.

13.4 Contribution on lunisolar nutation 13.4.1 Fluid layer contribution to lunisolar nutations Now, we investigate the effect of the atmospheric and non-tidal oceanic circulation on the Earth’s lunisolar nutation. As before the period of study is 1984–2019. The main harmonic components of the effective CEAM should impact the nutation terms of periods +365 (S1 ), +182 (P1 ), +121 (π1 ), +13.6 (O1 ) days as well as some of their retrograde counterparts at periods −365 days (ψ1 ) and −182 days (ϕ1 ). They are estimated by fitting the model χ 󸀠 = ∑(mjc + imjs )ei(Φj +π/2) j=1

(13.26)

to CEAM, where the Φj are the corresponding lunisolar tidal arguments. To a given harmonic component of frequency σj󸀠 (matter/motion term) corresponds the nutation

13.4 Contribution on lunisolar nutation

perturbation

P(σj󸀠 ) = T(σj󸀠 )ma/mo (mjc + imjs )ei(Φj +π/2) = (pjc + ipjs )ei(Φj +π/2) ,

| 257

(13.27)

in which T(σj󸀠 ) holds for the transfer function (13.10) associated with the matter term or the motion term. Least-square estimates and the corresponding effect on nutation above 4 µas are reported in Table 13.1 for the atmosphere. The largest contributions are observed at the annual and semi-annual periods. The dispersion between ECMWF and NCEP results is less than 10 µas, which is smaller than the difference between the pressure effects for the NIB and the IB ocean responses (for NCEP or ECMWF series). The oceanic perturbations pertaining to the same terms are derived from MPIOM series are reported in Table 13.2. The prograde annual and semi-annual corrections are negligible: as noticed earlier, the corresponding S1 and P1 tidal harmonics were almost completely eliminated from O-MPIOM series [68]. Only the retrograde annual contribution (ψ1 in the TRS) is significant, but is affected by a large uncertainty. In contrast, the outdated OMCT model yields prograde annual and prograde semiannual contributions up to 50 µas, as evidenced by the retrograde S1 and P1 peaks in the spectrum of Figure 12.4. Notice that those oceanic perturbations chiefly result from the mass term, as first shown by Brzezinski et al. (2004) [27] from the Ponte model over the period 1993–2000. Finally we report the total atmospheric-oceanic effects in Table 13.3. 13.4.2 Comparison with observed lunisolar nutations The calculated perturbations on the lunisolar terms are compared with the corrections derived from the celestial pole offsets. In the ideal case, the tidally-induced precession–nutation would be given by the IERS conventional model [145]. This results from the multiplication of the rigid Earth lunisolar nutation terms by a frequency transfer function accounting for mantle anelasticity, core–mantle coupling, resonance effects, and ocean tides (1 mas at 18.6 year period). The coefficients of the transfer function (electromagnetic constants describing the core–mantle electromagnetic coupling, fluid core ellipticity, frequency resonance parameters, …) are fitted to the observed lunisolar terms. Therefore they can wrongly absorb the influence of the hydro-atmospheric layer or a mis-modeling of the ocean tides. Mathews and its coauthors [145] attempt to circumvent this problem for the prograde annual term: prior to making the adjustment, they remove from the observed prograde annual term an optimal estimate of the “Sun-synchronous” effect , namely pc = 10 µas and ps = −108 µas (signs consistent with Eq. (13.27), opposite to those of Table 7 in [145]). These values are consistent with our estimates of the atmospheric perturbation at +365 days to about 40 µas (see Table 13.1). Recently Nurul Huda and coauthors [164] have shown that the removal of these fluid layer perturbations not only at +365 days, but also at −365 and +182 days modify significantly the estimates of the frequency resonance parameters, in particular the quality factor of the FCN and FICN modes.

258 | 13 Fluid layer effect on nutation Table 13.1: Atmospheric effect on in-phase pc and out-of-phase ps components of lunisolar nutation terms above 4 µas: pressure term with inverted barometer response of the oceans (P), pressure term with the non-inverted barometer response (P(nib)), wind term (W) and total contributions (P+W). Estimates of the CEAM lunisolar components mc and ms over the period 1984–2019. period (days) 365.26

182.62

−365.26

13.66

CEAM (mas) P-ECMWF P-NCEP P(nib)-ECMWF P(nib)-NCEP W-ECMWF W-NCEP P+W-ECMWF P+W-NCEP P(nib)+W-ECMWF P(nib)+W-NCEP P-ECMWF P-NCEP P(nib)-ECMWF P(nib)-NCEP W-ECMWF W-NCEP P+W-ECMWF P+W-NCEP P(nib)+W-ECMWF P(nib)+W-NCEP P-ECMWF P-NCEP P(nib)-ECMWF P(nib)-NCEP W-ECMWF W-NCEP P+W-ECMWF P+W-NCEP P(nib)+W-ECMWF P(nib)+W-NCEP P-ECMWF P-NCEP P(nib)-ECMWF P(nib)-NCEP W-ECMWF W-NCEP P+W-ECMWF P+W-NCEP P(nib)+W-ECMWF P(nib)+W-NCEP

mc

ms

−0.745 ± 0.004 −0.892 ± 0.004 −0.912 ± 0.012 −0.568 ± 0.014 −0.234 ± 0.060 −0.364 ± 0.060

1.008 ± 0.004 1.023 ± 0.004 0.822 ± 0.012 0.789 ± 0.014 −6.166 ± 0.060 −6.762 ± 0.060

−0.237 ± 0.004 −0.177 ± 0.004 −0.645 ± 0.012 −0.535 ± 0.014 −2.683 ± 0.060 −1.885 ± 0.060

0.600 ± 0.004 0.515 ± 0.004 0.104 ± 0.012 0.113 ± 0.014 −11.523 ± 0.060 −11.396 ± 0.060

0.099 ± 0.004 0.086 ± 0.004 0.145 ± 0.012 0.118 ± 0.014 0.658 ± 0.060 0.809 ± 0.060

−0.068 ± 0.004 −0.049 ± 0.004 −0.077 ± 0.012 −0.005 ± 0.014 0.150 ± 0.060 1.073 ± 0.060

0.046 ± 0.004 0.055 ± 0.004 0.185 ± 0.012 0.199 ± 0.014 1.137 ± 0.060 0.999 ± 0.060

−0.036 ± 0.004 −0.032 ± 0.004 −0.101 ± 0.012 −0.093 ± 0.014 −0.445 ± 0.060 −0.075 ± 0.060

Nutation (µas) pc 32.8 ± 0.3 39.2 ± 0.3 40.1 ± 0.7 25.0 ± 0.8 −0.8 ± 0.3 −1.3 ± 0.3 32.0 ± 0.6 37.9 ± 0.6 39.2 ± 1.0 23.7 ± 1.1 6.6 ± 0.2 4.9 ± 0.2 17.7 ± 0.5 14.7 ± 0.5 −9.6 ± 0.3 −6.8 ± 0.3 −3.1 ± 0.5 −1.9 ± 0.5 8.1 ± 0.8 7.9 ± 0.8 53.0 ± 3.4 46.6 ± 3.5 78.8 ± 9.5 66.5 ± 10.9 4.0 ± 0.5 5.1 ± 0.5 56.9 ± 3.9 51.7 ± 4.1 82.7 ± 10.0 71.5 ± 11.4 0.0 ± 0.0 0.0 ± 0.0 −0.1 ± 0.0 −0.1 ± 0.0 4.5 ± 0.3 4.0 ± 0.3 4.5 ± 0.3 4.0 ± 0.3 4.5 ± 0.3 3.9 ± 0.3

ps −43.8 ± 0.3 −44.4 ± 0.3 −35.6 ± 0.7 −34.3 ± 0.8 −21.7 ± 0.3 −23.8 ± 0.3 −65.5 ± 0.6 −68.2 ± 0.6 −57.3 ± 1.0 −58.1 ± 1.1 −16.4 ± 0.2 −14.1 ± 0.2 −2.8 ± 0.5 −3.0 ± 0.5 −41.4 ± 0.3 −40.9 ± 0.3 −57.8 ± 0.5 −55.1 ± 0.5 −44.2 ± 0.8 −44.0 ± 0.8 −43.0 ± 3.4 −31.7 ± 3.5 −50.1 ± 9.5 −8.1 ± 10.9 0.8 ± 0.5 6.3 ± 0.5 −42.2 ± 3.9 −25.4 ± 4.1 −49.3 ± 10.0 −1.7 ± 11.4 0.0 ± 0.0 0.0 ± 0.0 0.0 ± 0.0 0.0 ± 0.0 −1.8 ± 0.3 −0.3 ± 0.3 −1.8 ± 0.3 −0.3 ± 0.3 −1.7 ± 0.3 −0.3 ± 0.3

13.4 Contribution on lunisolar nutation

| 259

Table 13.2: MPIOM-oceanic effect on in-phase pc and out-of-phase ps components of the lunisolar nutation terms perturbed by atmosphere: Water height term (H), current term (C) and total contributions (H+C). Estimates of the CEAM lunisolar components mc and ms over the period 1984–2019. period (days) 365.26

182.62

−365.26 13.66

CEAM (mas) H C H+C H C H+C H C H+C H C H+C

mc

ms

0.067 ± 0.022 −0.014 ± 0.041

0.011 ± 0.022 −0.090 ± 0.041

0.015 ± 0.022 −0.039 ± 0.041

0.009 ± 0.022 0.000 ± 0.041

−0.004 ± 0.022 0.131 ± 0.041

−0.012 ± 0.022 0.148 ± 0.041

0.088 ± 0.022 −0.011 ± 0.041

−0.218 ± 0.022 0.043 ± 0.041

Nutation (µas) pc −2.9 ± 1.3 0.0 ± 0.2 −3.0 ± 1.6 −0.4 ± 0.8 −0.1 ± 0.2 −0.5 ± 1.1 −2.8 ± 17.5 0.8 ± 0.4 −2.0 ± 17.9 0.0 ± 0.0 0.0 ± 0.2 −0.1 ± 0.2

ps −0.5 ± 1.3 −0.3 ± 0.2 −0.8 ± 1.6 −0.2 ± 0.8 0.0 ± 0.2 −0.2 ± 1.1 −6.8 ± 17.5 0.9 ± 0.4 −6.0 ± 17.9 0.1 ± 0.0 0.2 ± 0.2 0.2 ± 0.2

So, the terms of the lunisolar nutation model have the same level of uncertainty than the corresponding perturbations brought by the atmosphere and tide free ocean circulation. For that reason, comparing the observed nutation residuals to the CEAM contribution is a delicate task. Variability of the seasonal lunisolar nutations The variability of the atmospheric–oceanic perturbation at +365, −365, 182 days is investigated by fitting (pc , ps ) over a 7-year sliding window. As shown in Figure 13.9, the perturbations associated with A-ECMWF/O-MPIOM series present fluctuations up to 100 µas. Such a variability might be sought in the corresponding estimates for the observed celestial pole offsets (IVS data with respect to the IERS conventional model), which are superimposed on the same graphs. For the prograde annual term, the oceanic contribution is not significant, and the pc and ps coefficients caused by the atmosphere present a tiny variability of about 20 µas that does not match the corresponding variations in observed nutation residuals. Whereas the observed residuals Δη = η − ηmod are close to 0 to about 10 µas except before 1990 where VLBI data are of less quality (η is the total observed term, ηmod its modelled part), the atmospheric effect presents systematic offsets, especially for the out-of-phase term ps (∼ −60 µas). If the reference nutation model ηmod had incorporated the atmospheric correction p1s ∼ −60 µas that we derived from NCEP or ECMWF models instead of p2s = −108 µas (the Mathews et al. correction mentioned above), the mean value of the out-of-phase observed residual would have been Δη = η − (ηmod − p2s + p1s ) ≈ −p1s + p2s = 60 − 108 ∼ −50 µas, and would have much better fitted our atmospheric correction.

260 | 13 Fluid layer effect on nutation Table 13.3: Sum of the ECMWF-atmospheric and MPIOM-oceanic effects on in-phase pc and outof-phase ps components of the lunisolar nutation terms: matter term (Matter), with NIB response (Matter NIB), motion term (Motion) and total contributions (Total / Total NIB). Estimates of the CEAM lunisolar mc and ms terms over the period 1984–2019. period (days) 365.26

182.62

−365.26

13.66

CEAM (mas) Matter Matter NIB Motion Motion Total Total NIB Matter Matter NIB Motion Motion Total Total NIB Matter Matter NIB Motion Motion Total Total NIB Matter Matter NIB Motion Motion Total Total NIB

mc

ms

−0.678 ± 0.023 −0.846 ± 0.028 −0.248 ± 0.069 −0.248 ± 0.069

1.019 ± 0.023 0.833 ± 0.028 −6.256 ± 0.069 −6.256 ± 0.069

−0.222 ± 0.023 −0.630 ± 0.028 −2.722 ± 0.069 −2.722 ± 0.069

0.608 ± 0.023 0.113 ± 0.028 −11.529 ± 0.069 −11.529 ± 0.069

0.095 ± 0.023 0.141 ± 0.028 0.789 ± 0.069 0.789 ± 0.069

−0.081 ± 0.023 −0.090 ± 0.028 0.297 ± 0.069 0.297 ± 0.069

0.134 ± 0.023 0.273 ± 0.028 1.126 ± 0.069 1.126 ± 0.069

−0.254 ± 0.023 −0.319 ± 0.028 −0.402 ± 0.069 −0.402 ± 0.069

Nutation (µas) pc 29.9 ± 1.4 37.2 ± 1.7 −0.9 ± 0.3 −0.9 ± 0.3 29.0 ± 1.8 36.3 ± 2.0 6.2 ± 0.9 17.3 ± 1.1 −9.8 ± 0.4 −9.8 ± 0.4 −3.6 ± 1.2 7.5 ± 1.4 50.2 ± 18.5 76.0 ± 22.1 4.8 ± 0.6 4.8 ± 0.6 55.0 ± 19.1 80.8 ± 22.7 0.0 ± 0.0 −0.1 ± 0.0 4.5 ± 0.4 4.5 ± 0.4 4.4 ± 0.4 4.4 ± 0.4

ps −44.3 ± 1.4 −36.2 ± 1.7 −22.0 ± 0.3 −22.0 ± 0.3 −66.3 ± 1.8 −58.1 ± 2.0 −16.7 ± 0.9 −3.0 ± 1.1 −41.4 ± 0.4 −41.4 ± 0.4 −58.1 ± 1.2 −44.5 ± 1.4 −49.8 ± 18.5 −56.9 ± 22.1 1.6 ± 0.6 1.6 ± 0.6 −48.2 ± 19.1 −55.3 ± 22.7 0.1 ± 0.0 0.1 ± 0.0 −1.6 ± 0.4 −1.6 ± 0.4 −1.5 ± 0.4 −1.5 ± 0.4

At retrograde annual period, the oceanic variability has an order of magnitude comparable to the observed one, and seems to partly reproduce its near 6 year modulation of about 50 µas. At the semi-annual prograde period, observed and reconstructed variabilities are not consistent. It is worth to notice a constant atmospheric contribution of about −60 µas for the out-of-phase term ps (see also Table 13.1) whereas the mean correction to the conventional nutation model does not exceed 10 µas. As this model does not include any atmospheric correction at the semi-annual period (see Table 7 in [145]), this inconsistency hints at a defect in the tidal part of the model, namely in the transfer function converting the lunisolar nutation of a rigid Earth into that of a real Earth (see Eq. (13.27)). These results can be reproduced for the coupled NCEP-ECCO1h model. It can be concluded that, for the lunisolar terms, the hydro-atmospheric effect does not well reproduce the variability of the observed residuals. This lack of con-

13.4 Contribution on lunisolar nutation

| 261

sistency cannot be attributed to the uncertainty affecting the celestial pole offsets. Indeed, the VLBI random error is a white noise of about 400/√(7 ∗ 365.25) ∼ 8 µas for a 7 year window (see Eq. (3.55)), much smaller than the observed variability. On the other hand, the systematic discrepancies between VLBI series at seasonal periods are at the level of 20 µas (see the spectra in Figure 3.9), and remain smaller than the offsets noticed at −365 and +182 day periods between nutation residuals and corresponding atmospheric-oceanic perturbations. Probably the observed disagreements result from both shortcomings in the lunisolar nutation modeling and OAM series at diurnal tidal frequencies. Short term nutation between 2 and 30 days According to the spectrum of Figure 13.1, the largest nutation residuals below 30 days are mostly composed of retrograde oscillations at 13.6 and 27.3 days at the level of 󸀠 󸀠 10 µas. With transfer functions Tma (−1/27.3 cpd) ≈ 0.01 and Tmo (−1/27.3 cpd) ≈ 0.003 according to Eq. (13.10) or Figure 13.2, and smaller values at −1/13.6 cpd, the required power in atmospheric or oceanic CEAM is at least 0.01 mas /0.01 = 1 mas for the pressure term and 0.01/0.003 ∼ 3 mas for the wind term. As this amplitude level is not reached by the atmospheric and oceanic CEAM in the retrograde band between 2 and 30 days, the cause of the nutation residuals at −13.6 and −27.3 days does not seem to originate in the hydro-atmospheric layer. However, the prograde nutation below 30 days can be excited by the atmosphere and the oceans as justified now. It can be estimated that, for prograde circular excita󸀠 tions of periods spanning from 2 days to 1 month, Tma (σ 󸀠 ) is globally smaller than 󸀠 󸀠 󸀠 Tmo (σ 󸀠 ), with Tmo presenting a rather constant value: Tmo (1/7 cpd) ≈ 2.6 10−3 and 󸀠 −3 Tmo (1/13.6 cpd) ≈ 2.3 10 . Given that the pressure term is about 5 times smaller than the wind term, its effect can be neglected. With amplitudes in χw󸀠 reaching 1 mas, broad band oscillation of the wind term around 7 days can produce a nutation of about 1 × 2.6 10−3 ∼ 0.003 mas, an order of magnitude consistent with the one found in [189]. With spectral components smaller than 0.2 mas in the 2–30 day prograde band, the oceanic CEAM has a much smaller contribution, and therefore will not be discussed. Until now, this 2–10 day band is poorly covered by space geodetic techniques, since nutation is routinely monitored by aid of VLBI observations with a mean temporal resolution of about 5 days. Various attempts to densify the nutation determination by jointly processing GNSS (Global Navigation Satellite System) data seem to confirm the broad band nutation around 7 days with the estimated order of magnitude [18]. The tidal atmospheric effect on the 13.6-day nutation has an average magnitude of 1 × 2.3 10−3 = 0.003 mas. But the same term fitted to the celestial pole offsets is uncertain to make any sound conclusion: for the series obtained by the analysis centers of the International VLBI Service, selecting the period 1990–2019, we have the discrepancies −10 ≤ pc ≤ 10 µas, 0 ≤ ps ≤ 30 µas. This level of inconsistency clearly is manifested by the Allan diagram of the paired differences in Figure 3.9 (100 µas at this time scale).

262 | 13 Fluid layer effect on nutation

Figure 13.9: Hydro-atmospheric corrections to the in-phase pc and out-of-phase ps components of the seasonal lunisolar nutation terms estimated by 7-year sliding windows: atmosphere (A–ECMWF, IB case), oceans (O–MPIOM) and comparison with the corresponding quantities fitted to observed Celestial Pole Offsets (OBS).

13.5 Excitation of the free core nutation The free core nutation is the most important component of the celestial pole offsets. It can be isolated by a band pass filter similar to the one used for extracting Chandler wobble. Indeed, both modes, although referred to different reference systems, have close periods and quality factors (see Eq. (7.79)). So we adopt a Panteleev band

13.5 Excitation of the free core nutation

| 263

Figure 13.10: Free core nutation amplitude and phase variation determined by Panteleev band pass filter.

pass filter of central frequency fc = −1/430 cpd with a bandwidth of 2f0 = 0.16 cpy = 4 10−4 cpd, not extending over the retrograde annual nutation (see Appendix G.4 for the description of the filter). As shown in Figure 13.10, the envelope amplitude of the filtered signal varies between 50 and 250 µas with a 180° phase shift taking place in the years 1997–2007 (here the phase shift is the offset with respect to the reference phase σf󸀠 (t − t0 ) where t0 is an arbitrary starting epoch, there J2000). In contrast to Chandler wobble, the free core nutation seems to be excited by a more regular process (see Figure 11.3 for Chandler wobble). A rough estimate of the atmospheric and oceanic effect can be assessed if we mul󸀠 tiply the matter term transfer function of Figure 13.2 at −1/430 cpd, that is, Tma = 10, by the corresponding amplitude of the excitation shown on the spectra of Figures 13.3 and 13.4 (∼ 0.03 mas for the pressure term or the water height term), noting 󸀠 that with Tmo ∼ 0.025 the motion term of about 0.25 mas is not effective. We obtain an order of magnitude of 0.3 mas, consistent with the observed FCN amplitude. However, the free core nutation excitation cannot be reduced to a single frequency. As for the Chandler wobble, it is a broad band process. To find to which extent atmosphere and ocean excite it, the simplest approach is to compare both sides of the differential equation (13.8) in the time domain, that is, to address the celestial geodetic excitation,

264 | 13 Fluid layer effect on nutation namely χG󸀠 = P + i

σ̃ c󸀠 d + σ̃ f󸀠 σ̃ c󸀠 d σ̃ f󸀠

Ṗ −

P̈

σ̃ c󸀠 d σ̃ f󸀠

(13.28)

,

with the celestial fluid layer excitation, that is, χF󸀠 = σc { +

1 1 󸀠e 󸀠e 󸀠e (χ 󸀠 e + χmo ) + 󸀠 (ama χma + amo χmo ) σ̃ f σ̃ c󸀠 d ma

i i ̇󸀠 e ) + ̇ 󸀠 e + amo χmo ̇ 󸀠 e )} . (χ̇ 󸀠 e + χmo (ama χma σ̃ c󸀠 d σ̃ f󸀠 ma σ̃ c󸀠 d σ̃ f󸀠

(13.29)

Celestial geodetic excitation is estimated from equally spaced values of the nutation offsets (sampled period h) by a three point digital filter, designed in the same way as the polar motion excitation filter (Section 9.3): χĜ 󸀠 (t) = aP(t) + bP(t − h) + cP(t + h) .

(13.30a)

Identifying the corresponding transfer function (from nutation offsets to excitation) with the analogous one at the frequency σ = 0, σc󸀠 d and σf󸀠 , we obtain 󸀠d

a=

(1 − eiσc

h

− ασc󸀠 d − β(σc󸀠 d )2 ) sin(σf󸀠 h) − (1 − eiσf h − ασf󸀠 − βσf󸀠 2 ) sin(σc󸀠 d h) 󸀠

(1 − eiσc h ) sin(σf󸀠 h) − (1 − eiσf h ) sin(σc󸀠 d h) 󸀠d

󸀠

󸀠d

󸀠d

α(σf󸀠 − σc󸀠 d ) + β(σf󸀠 2 − (σc󸀠 d )2 ) + α(σc󸀠 d eiσf h − σf󸀠 eiσc h ) + β((σc󸀠 d )2 eiσf h − σf󸀠 2 eiσc h ) 󸀠

b=

,

󸀠

2i((1 − eiσc h ) sin(σf󸀠 h) − (1 − eiσf h ) sin(σc󸀠 d h)) 󸀠d

󸀠

c =1−a−b,

,

(13.30b)

with α=

1 1 + σ̃ c󸀠 d σ̃ f󸀠

and

β=−

1 . σ̃ c󸀠 d σ̃ f󸀠

(13.30c)

By the same token, the matter or the motion celestial geophysical function χF󸀠 = σc {(

ama/mo 󸀠 e iama/mo 󸀠 e 1 i ̇ + )χma/mo + ( 󸀠 d 󸀠 + 󸀠 d 󸀠 )χma/mo } 󸀠 󸀠 d σ̃ f σ̃ c σ̃ c σ̃ f σ̃ c σ̃ f

(13.31)

is digitalized according to χF̂ 󸀠 (t) = uχ 󸀠 e (t) + vχ 󸀠 e (t − h) .

(13.32a)

13.5 Excitation of the free core nutation

| 265

Figure 13.11: Celestial geodetic excitation (G) and fluid layer excitation in the free core nutation band isolated by Panteleev band pass filter. Analysis based upon atmospheric series A–ECMWF (NIB pressure term) and oceanic series O–MPIOM; see Table 8.2.

Identifying the Fourier transform of (13.31) and (13.32a) at σf󸀠 and σ = 0, this gives a linear system with the unknown u, v, of which the solution is u = σc (

σc σf󸀠 1 + ama/mo ama/mo 1 , + ) − 󸀠 σ̃ f󸀠 σ̃ c󸀠 d σ̃ c󸀠 d σ̃ f󸀠 1 − e−iσf h

σc σf󸀠 1 + ama/mo v = 󸀠d 󸀠 . 󸀠 σ̃ c σ̃ f 1 − e−iσf h

(13.32b)

Applying to the free core nutation isolated earlier, the digital filter described by Eqs. (13.30), we derive χG󸀠 in a 2f0 = 0.004 cpd band centered on 1/430 cpd. A similar processing is used to determine the geophysical counterpart χF󸀠 : first the FCN band is selected in effective CEAM, then the geophysical celestial excitation χF󸀠 is derived by the digital filter described by Eqs. (13.32). The filtering is done over the period 1984–2019 for ECMWF AAM series and MPIOM OAM series. The results, plotted in Figure 13.11 for the X component in the celestial intermediate system (Y component, is the same as X out-of-phased of Tf /2), confirm that the hydro-atmosphere, mostly the oceans, excites the FCN with the required power. However, the modeled excitation remains most often out-of-phase with respect to χG󸀠 and does not reproduce its

266 | 13 Fluid layer effect on nutation amplitude modulation. Over the restricted period 1992–2016 coupled models A-NCEP and O-ECCO yield similar results, and for now no study has been able to fully explain the observed FCN by the atmospheric and oceanic celestial angular momentum.

13.6 Conclusion The atmosphere and the oceans perturb the seasonal lunisolar nutation, bringing about in-phase and out-of-phase contributions up to 100 µas at the annual period. However, the oceanic perturbations remain poorly determined. On the other hand, despite the fact that the lunisolar terms of the nutation have an observational uncertainty of about 20 µas, it is not yet possible to isolate the pure footprint of the hydro-atmosphere, mixed with other influences like the ocean tides and the core– mantle electromagnetic coupling. In the nutation residuals with respect to the IERS conventional model, the most important variation is a pseudo-harmonic retrograde oscillation at 430 days attributed to the free core nutation. Its mean amplitude of about 150 µas (∼ 10 µas on the corresponding celestial geodetic excitation) can be accounted by oceanic–atmospheric excitation. But its phase is not reproduced, in contrast to the result obtained in [224] by applying the integration approach and adjusting the initial conditions. This suggests defects in the global circulation models, an other kind of excitation involved [225], such as geomagnetic jerks resetting the phase of the FCN, or a pitfall in Earth rotation theory. Below periods of 1 month, the Celestial Equatorial Atmospheric Angular Momentum is mostly composed of prograde oscillations, dominated by a harmonic at 13.6 days and a broad band peak around 7 days, initially detected in the terrestrial frame [24]. We aimed at showing that this band has a lunar tidal origin, suggested by two specific features that contrast with what is observed for seasonal CEAM variations of thermal origin: i) northern and southern hemispheres contribute equally and synchronously to the globally-integrated signal; ii) pressure and wind terms are almost proportional to each other. The 13.6-day pressure term clearly results from the main lunar tide, explained by a hydrostatic response to the tesseral O1 tidal forcing. The origin of the broad band peak around 7 days in the non-rotating frame is less evident, but seems to be related to the Moon because of some features not observed in the case of the thermal seasonal wave (in particular proportionality between pressure and wind terms). It possibly results from the lunar tide, of which the effect is amplified by the Ψ11 atmospheric resonance. In order to observe a possible effect on nutation, improvements of the VLBI/GNSS data processing as well as optimized observation schedules are required. The characterization of the regional atmospheric contributions is likely to yield further insights into the associated mass transport mechanism.

14 Seismic effect (…) it appears that although large earthquakes (such as the 1960 Chilean earthquake and the 1964 Alaskan earthquake) do produce polar shifts which might someday, given improved data, be visible, the cumulative seismic activity of the Earth is not sufficient to serve as the primary source of excitation of the Chandler wobble. Dahlen, 1973 [45]

14.1 Introduction Whereas the influence of earthquakes on Earth rotation has been a recurrent theme of research since the 1960s, no effect has been observed yet. Theoretically megaearthquakes can modify polar motion at a level observable by modern space geodetic techniques. The gigantic earthquake (magnitude 9) that took place on 2004 December 26 at 01 h UTC, about 200 km from the western coast of northern Sumatra (denoted SUM 2004) was the first mega-earthquake during the Space Geodesy era, and has thus constituted an opportunity for recording a possible effect. It came in third or fourth position in the list of the most powerful earthquakes ever recorded by seismometers, namely Chili (1960, m = 9.5), Alaska (1964, m = 9.2) and Kamchatka (1959, m = 9). It can be described as a thrust-faulting on the interface of the India plate and the Burma microplate. In a period of minutes, the faulting released elastic strains that had accumulated for centuries from ongoing subduction of the India plate beneath the overriding Burma microplate. The ground over 1000 km fault was displaced in average by about 11 m (see Figure 14.1) along the fault plane inclined of 8° with respect to the horizon.1 In the same time the mean pole of figure (that is the mean pole of inertia) would have shifted by about 3 cm according to our own estimation [13] or the one published by Gross and Chao (2006) [99]. But, with this order of magnitude, the effect is not distinguishable from the usual atmospheric and oceanic contributions. Interest for this question was again highlighted by mass media on February 27 2010 when the Chili coast was struck by an earthquake of magnitude 8.8 (denoted CHI 2010). More recently on 2011 March 11 the eastern coast of Japan was devastated by a tsunami caused by a seismic event of magnitude 9 (denominated JAP 2011). Our estimates gave a jump of 15 cm in the inertia pole, which polar motion analysis could not confirm. In addition to the co-seismic jump of the inertia pole, earthquakes could yield a source of excitation for the free mode [153, 126]. In the light of the former chapter, hydro-atmospheric processes are the most important at sub-decadal scale, so that the possible seismic excitation should have a minor role to play, except at a longer time scale. This chapter is an actualization of the problem, based upon elastic dislocation 1 A dip–slip earthquake. https://doi.org/10.1515/9783110298093-014

268 | 14 Seismic effect

Figure 14.1: A graphical representation of the thrust-faulting Sumatran earthquake of 2004 December 26.

model of Dahlen [44, 45]. We shall focus on the recent cases of Chili, Sumatra and Japan, mentioned here-above. Besides the slight push of the pole towards 140° [207], we attempt to uncover other systematic phenomena stemming from the apparent random nature of the seismic effects. The detectability of the co-seismic contributions in the geodetic excitation is also discussed.

14.2 Theoretical bases An earthquake corresponds to a mass redistribution taking place during a few minutes, so that, neglecting post-seismic effects, the resulting co-seismic c = c13 + ic23 is permanent and can be modeled as a step-wise function c(t) = c1 H(t − t1 ) where t1 is the instant of the earthquake [126]. Seismic motions yield also a relative angular momentum h = h1 + ih2 , of about 1025 kg m2 s−1 for the mega-quakes. This is far too negligible compared to the relative angular momentum of the atmosphere, but as this effect is transient, existing only during the seismic event, its influence is not observable because of the too large temporal resolution of the pole coordinates (no less than 1 hour). Therefore we deal only with the seismic matter term, not accounting for viscoelastic relaxation of the ground, which would slightly amplify the corresponding effect along with time [206]. This problem is postponed to Chapter 15 for addressing mass redistributions at geological time scales. Theoretically earthquakes can also excite retrograde diurnal polar motion, namely nutation, because of the resonance at Free Core Nutation period. But with an order of magnitude of 1 µas [56], this effect is negligible, and we shall focus on periods larger than 2 days. We also take the framework of an anelastic Earth model, of which the mantle and lithosphere are uncoupled from the fluid core. Considering the imprecision of the seismic excitation, we apply a symmetric Liouville equation (7.35) for an instantaneous rotation pole. Introducing the observed pole coordinates of the CIP (p = x − iy),

14.2 Theoretical bases | 269

the time derivative of the seismic excitation disappears, giving (see Section 4.5) p+i

ṗ 1 χ = χS , = σ̃ c 1 − k̃ k

(14.1)

s

where σ̃ c = σc (1 + i/2Q) is the complex Chandler pulsation (σc = 2π/433 cycle/day), including damping with a Q value in the interval [40, 200]. Generally, the co-seismic excitation function reads χS = K

c1

C−A

H(t − t1 ) .

(14.2)

As there is no surface loading, K is expressed by K = 1/(1 − k/ks ) ≈ 1.606. The quantity c1 /(C − A) represents the shift of the pole of figure according to Appendix C (Eq. (C.26) for A = B = A), which occurs just after the event. The solution of (14.1) can be deduced from (11.2): p(t) = p(t1 )e

iσ̃ c (t−t1 )

− iσ̃ c e

iσ̃ c t

t

∫ χS (τ)e−iσc τ dτ . ̃

(14.3)

t1

Then integration shows that the effect of a step-wise solution χS (t) = Ψ1 H(t − t1 ), beginning at instant t1 , yields both a polar shift and a modification of the free mode following the expression Δp(t) = Ψ1 H(t − t1 )(1 − eiσc (t−t1 ) ) , ̃

(14.4)

where Ψ1 = Kc1 /(C − A). In the instants, even the days, following the earthquake, t − t1 is small in comparison with the Chandler period, and the co-seismic polar motion can be approximated by Δp(t) = −iΨ1 σ̃ c (t − t1 )

Tc ≫ t ≥ t1 .

(14.5)

Thus the effect is perpendicular to the equatorial excitation function, and grows linearly with time. If many days have passed, the co-seismic shift can be approximated by Δp(t) = −Ψ1 2iei

σc (t−t1 ) 2

sin

σc (t − t1 ) 2

t ≥ t1 ,

(14.6)

showing a maximal displacement (what ever be the direction of Ψ1 ) after half period of Chandler (7 months).2 2 In the sinus term σc (t − t1 )/2 = π/2.

270 | 14 Seismic effect Now, let us consider a sequence of N successive seismic events occurring at instants t1 , . . . , tn , . . . , tN , and causing the variations c1 , . . . , cn , . . . , cN for off-diagonal moments of inertia. At any instant t following tN , the total co-seismic excitation results from the contribution of each isolated contribution, that is , N

χS(n) = K ∑

n=1

cn

C−A

H(t − tn ) .

(14.7)

A corresponding effect on the pole is obtained by cumulating each individual effect obeying the same form as (14.4): N

Δp(t) = K ∑

cn

n=1 C − A

H(t − tn )[1 − eiσc (t−tn ) ] . ̃

(14.8)

14.3 Modeling of the co-seismic excitation From elastic dislocation model proposed by Dahlen in the 1970s [45], the step-wise increment c1 on the off-diagonal inertia moment can be estimated from the seismic parameters, namely: – The area of the fault plane S. – The location of the epicenter: depth h, longitude ϕ, colatitude θ. – The orientation of the fault plane: (i) the strike angle α is measured clockwise from north to the strike direction with the fault dipping down to the right of the strike direction such that 0 ≤ α ≤ 2π, (ii) the dip angle δ with respect to the horizon plane, measured down from the horizontal such that 0 ≤ δ ≤ π/2 (see Figure 14.1). – The mean seismic displacement or slip D in the fault plane at angle λ (the slip angle or rake) reckoned counter-clockwise from the strike direction (see Figure 14.1). However, these definitions given by Aki and Richard (Quantitative Seismology, 1980, p. 106) are not the ones of Dahlen: Dahlen measures the strike counter-clockwise, and allows his dip angle to range between 0° and 180°, whereas Aki and Richards’ dip angle only ranges between 0° and 90°. Furthermore, Dahlen does not assign the direction strike with respect to the fault dipping, so that his strike angle only ranges from 0 to 180° (this is why Dahlen’s dip angle must extend beyond 90° to 180°).3 The surface S and slip D determine the seismic moment M0 = μSD, where μ is the shear modulus of 3 Richard Gross noticed it in a personal communication; see the last sentence on page 178 of Dahlen (1971) [44] for a discussion of why his dip angle must be allowed to be greater than 90°. Also, note the typographical error in Table 1 of Dahlen (1973) where the slip angle for the 1960 Chile event should be 90° [as given in Table 1 of Dahlen (1971)], not 270°. In the last sentence on page 178, Dahlen (1971) gives the dip angle in the usual geological terminology as being 10° in the strike direction N 65° W (65° west from the north). For the 1964 Alaskan earthquake, in Aki and Richard’s convention, the strike angle

14.3 Modeling of the co-seismic excitation | 271

the lithosphere (≈ 75 GPa). The seismic parameters, illustrated in Figure 14.1, are determined for the most significant earthquakes (of magnitude larger than 4); they can be extracted from worldwide solutions of the Centroid Moment Tensor (CMT). One of the most complete is the CMT Harvard University catalogue [73, 77],4 stretching from March 1975 to the current month, including about ∼ 44600 earthquakes of magnitude larger than 4. It becomes much denser from 1977. For some specific mega-earthquakes the solution of the US Geological Survey will be also considered. In Table 14.1 we report the seismic parameters of SUM 2004, CHI 2010 and JAP 2011, associated with the CMT Harvard and USGS solutions. Table 14.1: Seismic parameters of SUM 2004, CHI 2010 and JAP 2011. Aki and Richard’s conventions for α, δ, and λ. Organisation Azimuth Dip angle Slip angle α δ λ

Slip D

Area Seismic moment S M0

Depth h

Sumatra 9.3 m: December 26, 2004 01 h UTC

Epicenter: about 200 km from the north-western coast of Sumatra (latitude 3.3°, longitude 95.8°)

Harvard USGS

329° 274°

8° 13°

110° 11 m ≈ 105 000 km2 55°

4 1022 Nm 2.6 1021 Nm

10 km 30 km

1.8 1022 Nm 1.8 1022 Nm

35 km 30 km

Chile 8.8 m: February 27, 2010 06 h UTC

Epicenter: Pacific coast (latitude −35.85°, longitude −72.72°)

Harvard USGS

18° 166°

18° 19°

112° 104°

5m 5m

≈ 50 000 km2

Japan 9.0 m: March 11, 2011 05h46 UTC

Epicenter: 130 km from north-east of Honshu (latitude 38.32°, longitude 142.37°)

Harvard USGS

203° 193°

10° 14°

88° 81°

5.3 1022 Nm 24.4 km 3.9 1022 Nm 10 km

Based upon an elastic seismic dislocation model, Dahlen [45] derived an expression of the co-seismic moment of inertia c as a function of the following seismic parameters: M0 , strike angle α, dip angle δ, slip angle λ (with its own definition of α, δ and λ is 205°, the dip angle is 10°, and the slip angle is 90° (pure thrust). But in Dahlen’s convention, the strike angle is 155°, the dip angle is 170° and the slip angle is 270° [see Table 1 of Dahlen (1971)]. The conversion algorithm to Dahlen’s parameters is αD = 360° − α α > 180° ⇒ δD = 180° − δ λD = 360° − λ

αD = 180° − α α ≤ 180° ⇒ δD = δ λD = λ

4 Available http://www.globalcmt.org/CMTcite.html

272 | 14 Seismic effect

Figure 14.2: Functions Γ1 , Γ2 and Γ3 .

reported above), colatitude θ, and longitude ϕ: c13 = M0 {Γ1 (h)[(sin 2α sin δ cos λ +

1 cos 2α sin 2δ sin λ) sin 2θ cos ϕ 2

1 − 2( sin 2α sin 2δ sin λ − cos 2α sin δ cos λ) sin θ sin ϕ] 2 + Γ2 (h)(− sin 2δ sin λ sin 2θ cos ϕ)

+ Γ3 (h)[(sin α cos 2δ sin λ − cos α cos δ cos λ) cos 2θ cos ϕ + (sin α cos δ cos λ + cos α cos 2δ sin λ) cos θ sin ϕ]} , c23

1 = M0 {Γ1 (h)[(sin 2α sin δ cos λ + cos 2α sin 2δ sin λ) sin 2θ 2 1 + 2( sin 2α sin 2δ sin λ − cos 2α sin δ cos λ) sin θ cos ϕ] 2

(14.9)

+ Γ2 (h)(− sin 2δ sin λ sin 2θ sin ϕ)

+ Γ3 (h)[(sin α cos 2δ sin λ − cos α cos δ cos λ) cos 2θ sin ϕ

− (sin α cos δ cos λ + cos α cos 2δ sin λ) cos θ) cos ϕ]} , where Γ1 (h), Γ2 (h), Γ3 (h) are functions of the depth h derived from the Earth model SNREI 8073AW [45]. They are plotted in Figure 14.2. For a better understanding, the problem can be simplified by the displacement of a point-like mass Δm of Cartesian coordinates (x,y,z), of which the position changed by (Δx, Δy, Δz). The corresponding

14.3 Modeling of the co-seismic excitation | 273

Figure 14.3: Equatorial co-seismic excitation Ψx , Ψy and its module |Ψ| estimated from 1975 to nowadays.

moment of inertia variation reads Δc13 + iΔc23 ≈ −[xΔz + zΔx + i(yΔz + zΔy)]Δm ,

(14.10)

that is, transcribed in spherical coordinates (r, ϕ, θ), Δc sin 2θ iϕ ≈ r sin 2θe−iϕ Δr − ir 2 e Δϕ − r 2 cos 2θeiϕ Δθ , Δm 2

(14.11)

where r can be taken as the mean radius of the Earth. For SUM 2004, ϕ ≈ π/2, θ ≈ π/2, and the displacement is rather horizontal (Δr = 0) and towards west (Δθ ≈ 0, D = rΔϕ with Δϕ ≤ 0), leading to |Δc|/Δm ≈ rD sin 2θ/2 = 0.11 rD. Despite the huge mass that has been displaced, this is far from an optimal configuration for c. In contrast, JAP 2011, which can also be assimilated to a meridian horizontal displacement, yields |Δc|/Δm ≈ rD sin 2θ/2 = 0.5 rD, and appears as 5 times more effective than SUM 2004 because of its mid-latitude. Using the seismic parameters of the Harvard catalogue in (14.9), the cumulated co-seismic excitation is estimated from 1975 to 2019. Its components Ψx and Ψy are plotted at left of Figure 14.3 from 1975. These data confirm and extend the co-seismic excitation obtained by Gross and Chao (2006) [99] from another model specified in [38]. Both components present a drift over the decade 1990–2000, reinforced by the jumps of the “mega-seismic trilogy” SUM 2004, CHI 2010 and JAP 2011. As shown by the representation in the tangent plane to the north pole, reported in Figure 14.5, the drift is directed towards 145° east (Ψx ∼ −Ψy on left plot of Figure 14.3), as already noticed by former estimates [37, 207]. Plotting the logarithm of the module (log(|Ψ|)) at right, we see that it fits a linear function, meaning that over the period 1975–2019 the drift oddly increases at an exponential rate. According to Spada (1997) [207] this tendency of earthquakes to nudge the pole towards 150° east results from “the combined effects of the geographical distribution of hypocenters and the prevailing dip–slip nature of large earthquakes”. Indeed according to Figure 14.4 this accelerating drift is mostly caused by earthquakes of the Asian–Pacific zone (longitude from 90° to 180° and positive latitude), representing

274 | 14 Seismic effect

Figure 14.4: Amplitude of the total and regional co-seismic excitations from 1975 to 2019 (Harvard catalogue): Asia-Pacific zone (longitude sector 90°/180° limited to Northern Hemisphere), EuropaAfrica (0°/90°), America-Atlantic (−90°/0°), Pacific (−180°/ − 90°).

Table 14.2: Excitation function shifts caused by mega-earthquakes SUM 2004 and CHI 2010. Seismic parameters

Sumatra Chili Japan

Harvard CMT USGS Harvard CMT USGS Harvard USGS

Inertia moments 1026 kg m2 c13 c23

x (mas)

−y (mas)

Ampli./direction

0.81 0.07 19.6 21.0 27.5 29.11

−0.76 −0.06 −0.96 −1.00 −3.20 −3.71

0.10 0.01 2.44 2.61 3.41 3.62

2.3 cm 173° East 0.16 cm 170° East 7.9 cm 111° East 8.5 cm 111° East 14.5 cm 133° East 16 cm 136° East

−6.1 −0.45 −7.7 −8.0 −25.8 −29.86

Excitation function shift

less than one third of the total number of earthquakes reported in the Harvard catalogue. With no regard to the sudden seismic shifts of 2010 and 2011, this drift (from 0.1 to 0.4 mas/year) remains 10 times lower than the one observed in polar motion (∼ 4 mas/an) towards Greenland and its direction is almost opposite. Now we focus on SUM 2004, CHI 2010 and JAP 2011, of which the effects are summed up in Table 14.2. The co-seismic shift on the pole of inertia is up 8 cm for CHI 2010, 15 cm for JAP 2011, but it does not exceed 3 cm for SUM 2004, for the corresponding seismic parameters are less effective. With directions between 110° and 170°, these three jumps reinforce the global drift towards 145° east. These estimations are confirmed by the model developed by Chao and Gross [37, 99].

14.4 Excitation of the Chandler wobble

| 275

Figure 14.5: Co-seismic equatorial excitation (left) and resulting polar motion (right) from 1975 to July 2015.

14.4 Excitation of the Chandler wobble In Figure 14.5 co-seismic polar motion is reconstituted according to the Harvard catalogue. The Chandler mode has its amplitude increasing up to 8 mas over 30 years. This is not much with regard to the observed variability (up to 100 mas over some decades; see Chapter 11). We can wonder whether this increase is hazardous or is evidence of a possible link between Chandler wobbles and earthquake occurrences through the mechanism of the solid pole tide. First, the Allan deviation of the co-seismic excitation, shown in Figure 14.7, reveals that it behaves as a random walk (see Appendix G.5). This means a spectral power in 1/f 2 , as it can be expected theoretically from the Fourier transform of a stepwise function (14.7): N

χS (σ) = K ∑

n=1

cn e−σc tn C−A

(πδ(σ) −

i ), σ

(14.12)

of which the amplitude square gives a term in σ −2 (to be added to the Dirac corresponding to a constant term in the time domain). On the other hand, the Chandler term caused by a sequence of N earthquake at instants t1 , . . . , tN reads at an instant t ≥ tN N

Δpσc (t) = −K ∑

n=1

cn

C−A

eiσc (t−tn ) ̃

(14.13)

276 | 14 Seismic effect

Figure 14.6: Seismic influence on the Chandler amplitude from 1976 to 2019. We superpose the maximal effect, associated with a constant phase θn and the effect resulting from an aleatory distribution of θn .

according to (14.8). Denoting by ϕn the longitude of the co-seismic shift of the excitation function, we have N

Δpσc (t) = −Keiσc t ∑ ̃

n=1

|cn |

C−A

ei(ϕn −σc tn ) . ̃

(14.14)

So the chandler amplitude tends to increase if the phase distribution θn = ϕn − σc tn , n = 1, . . . , N, between 0 and 2π favor a given value θ: the closer the values of θ will be, the more effective will be the summation in pc (σ). Without changing the seismic parameters except tn , we simulate the effects of i) a random phase distribution θn , and ii) a distribution restricted to an unique value of θn , yielding the maximal effect according to (14.14). As shown in Figure 14.6, the increase of the modeled amplitude (real phase ϕn ) can be reconstructed with a random phase distribution θn as well, and this is far from reaching the maximal effect. So the global seismic occurrence does not present any correlation with the Chandler wobble.

14.5 Discussion According to the former estimates, mega-earthquakes perturb the Chandler wobble and polar trend at an observational level. But can we really detect this perturbation? Until now, even for JAP 2011 or SUM 2004, the detection was not successful [99]. Trying to isolate a co-seismic offset of about a few mas in geodetic excitation χG —the left-hand side of (14.1)—is a difficult task, for the main variation of χG , about 10 mas over some

14.5 Discussion

| 277

Figure 14.7: Comparative Allan deviation analysis of equatorial seismic excitation (S), combined atmospheric–oceanic excitation (AO), geodetic excitation (G) and residuals G-AO over the period 1985–2019.

days (see Chapter 9), is brought about by the hydro-atmosphere and modeled with an uncertainty larger than 1 mas. A supplementary difficulty is associated with the fact that the temporal resolution of χG , about 24 h, is not sufficient for localizing a sudden offset of a few mas. The question is also to know whether SUM 2004, CHI 2010, JAP 2011 would have deflected the rotation pole, introducing a shift of 2–15 cm after 7 months. This continuous shift is small with regard to the few meter path that the pole achieves during this period owing to the fluid layer excitation. By integrating hydro-atmospheric AMF, this path is modeled with an error of 20 mas or 60 cm (see Chapter 11), so that the coseismic perturbation remains hardly detectable in polar motion. However, if a mega-earthquake shifts χS above the threshold of the geophysical noise of 20 mas, its effect would be observed. For instance, the Chilean megaearthquake of 1960 theoretically induced an offset of 23 mas in χG [38],5 and it would have been readily detected in χG if the space geodetic techniques already operated at this epoch. But at that time the accuracy of the pole coordinates was above 30 mas. 5 For this event Dahlen (1973) [45] found a tilting of the inertia pole of 10 mas towards 110° east.

278 | 14 Seismic effect Random walk behavior inconsistent with hydro-atmospheric excitation In Figure 14.7 we also display an Allan deviation analysis of the geodetic and hydroatmospheric excitation (Atmospheric model ECCO, Oceanic model ECCO, no fresh water contribution) over the period 1985–2019. From 100 days to the seasonal scale, the observed and hydro-atmospheric excitations as well as their difference G − (A + O) present similar behaviors in terms of Allan deviation, with a negative slope between ∼ −1/2 and 0. In contrast, the corresponding co-seismic forcing S has an Allan deviation with the slope ∼ +1/2 in log–log scale (indicating a random walk noise justified in the former section) and much lower than the G − (A + O) Allan deviation below the threshold of 1000 days. So, from the Allan variance analysis, S does not fit geodetic residuals G − (A + O) in the sub-decadal band; in particular it is excluded as a source of excitation for the Chandler wobble. Yet, the co-seismic excitation and the residuals G −(A+O) meet each other beyond 3000 days. Thus, decadal and secular polar motions should be significantly excited by an earthquake. In particular, the co-seismic drift towards ∼ 145° east, accelerating at an exponential rate from 1976, has a present velocity of 0.4 mas/year (over 2005–2019): this is smaller than the secular polar motion (∼ 4 mas/year) but exceeds its uncertainty (0.07 mas/year over the period 1980–2000). Looking at Figure 14.4, we also notice decadal changes at 0.5 mas level.

14.6 Conclusion In this beginning of the 21st century estimates of the co-seismic excitation from the catalog of seismic parameters fully confirm the mere conclusion of Dahlen in 1973, quoted in the forehead of this chapter. Over 40 years the cumulative effect increases the Chandler wobble by about 6 mas. Meanwhile, such a tiny gain is not enough for sustaining a Chandler wobble, of which the free part in (11.2) is damped by a factor 2.72 after 30 years (Q = 80). Atmosphere and oceans are clearly the main players. This notwithstanding, the co-seismic effect on polar motion is still puzzling, especially in regard of the accelerating drift towards Siberia.

15 Epilogue: Geological polar motion It is a common opinion in Sweden that the sea level is decreasing and that people ford or go on dry land in many places where that was not possible earlier. Some very knowledgeable men have shared this popular view; and de Buch adopts it so far as to suppose that the entire land mass of Sweden is rising little by little. Cuvier (1825) in Discourse on the revolutionary upheavals on the surface of the globe.

15.1 Geological mass redistribution Before closing this book we go back to a very remote past encompassing a mass redistribution much larger than the one taking place within the hydrosphere: mantle upwelling or downwelling, associated mountain formation, glaciation followed by deglaciation. As surprising as it may seem, the current polar drift witnesses these gigantic processes because of the delayed visco-elastic readjustment of the mass of the Earth distribution around the rotation axis. For modeling the polar motion a viscoelastic mantle has to be substituted for a quasi-elastic one. On the other hand, over 1 million years, the relative plate motion can reach 50 km, and thus can blur the polar motion at 0.5 degree level. So over such time scale the polar motion has to be referred to the mantle, and is then called True Polar Wander (TPW) in contrast to the apparent polar motion, which would then present striking differences from one plate to another. In the following we shall sum up the modeling of the TPW, considering what is commonly considered as its two main causes: glaciation cycles and tectonic processes. The comparison of this modeling to observational evidence of the TPW will be limited to the contemporaneous polar drift. So, we shall not discuss the reconstitution of TPW over million years from the analysis of paleomagnetic records. While the moment of inertia perturbation associated with seasonal change is about 10−10 A (10 mas for the corresponding angular momentum function), the geological variation can be 104 times more important, reaching 10−6 A [138], which is almost of the order of magnitude of the difference B − A between the mean equatorial moments of inertia. Moreover, by extrapolating the secular polar motion (4 mas/year), we notice that the pole displacement m = m1 + im2 reaches one degree (0.017 rad) after one million years, and can no longer be considered as a perturbation. So a pure linear approach has to be questioned.

15.2 Nonlinear Liouville equations In the Auxiliary Terrestrial Reference System (ATRS), confused with the mean principal axes frame of a given epoch, the non-linearized Liouville equations (4.17) read Ḣ + i[ω3 H − (ω1 + iω2 )H3 ] = L , Ḣ 3 + ω1 H2 − ω2 H1 = L3 , https://doi.org/10.1515/9783110298093-015

(15.1a)

280 | 15 Epilogue: Geological polar motion where the prime symbols, meaning that the corresponding quantities are referred to the ATRS, have been removed. The whole Earth angular momentum is given by H = I ω + h, that is, H1 = I11 ω1 + I12 ω2 + I13 ω3 + h1 ,

H2 = I22 ω2 + I12 ω1 + I23 ω3 + h2 ,

(15.1b)

H3 = I33 ω3 + I13 ω1 + I23 ω2 + h3 . With respect to the proper values A, B, C of a given epoch, the inertia matrix can be decomposed into A [ I=[ 0 [ 0

0 B 0

0 c11 ] [ 0 ] + [ c21 C ] [ c31

c12 c22 c32

c13 ] c23 ] . c33 ]

(15.1c)

In (15.1b) the cross-terms of the variations, namely cij mk Ω2 , can reach 10−8 AΩ2 after one million year, which is the order of magnitude of the contemporaneous moment of force responsible for the polar motion. So, the linearization by eliminating the second order terms of the perturbations should be applied with caution, depending on the studied mass redistribution.

15.3 Visco-elastic variation of the moments of inertia The moment of inertia change cij in the Earth angular momentum (15.1c) results from

two distinct effects: i) the moment of inertia cij(l,ve) produced by a given load mass and the induced visco-elastic deformation of the solid Earth, ii) the visco-elastic centrifugal deformation, yielding cij(r,ve) . What does the term visco-elastic mean? When the stress (centrifugal or loading potential) varies over thousand years, the asthenosphere, that is, the upper mantle, does not follow instantaneously the stress like an elastic body because of shearing of macro-molecules against each other. So, after applying a constant stress, the material continues to deform. After removing the stress, as happens at the end of a glaciation cycle, the material relaxes until a plastic deformation is retained. The longer is the time scale of the excitation process, the largest will be this plastic deformation and time-lagged deformation. Until now, in order to handle contemporaneous geophysical mass redistribution, visco-elastic effect was reduced to a constant phase lag between excitation process and its response, and thus was considered as independent from frequency. In Chapter 5, the quasi-elasticity of the solid Earth was described through a constant complex Love number k̃2 in the frequency domain, entailing the convolution of k̃2 δ(t) by m(t) in the time domain, namely Eq. (5.42). Then the induced perturbation on the geopotential was proportional to the degree 2 part of the centrifugal potential.

15.3 Visco-elastic variation of the moments of inertia

| 281

Equivalently the induced off-diagonal moment of inertia c = c13 + ic23 was modeled by c(r) (t) =

k̃2 k̃ (C − A)m(t) = 2 (C − A)δ(t) ∗ m(t) , ks ks

and thus was proportional to the pole displacement in the complex plane. However, for secular and longer variation of m(t) this complex proportionality in the time domain breaks down. Fortunately, the visco-elastic rheology ensures the Hooke law in Laplace domain, where the Laplace transforms c(r) (s) and m(s) remain proportional. Reversely, in the time domain the linearity is kept, but in the form of a convolution product integrating the past polar motion until the relevant date. In the most general case, the visco-elastic perturbation of the moments of inertia can be estimated from the degree 2 centrifugal potential U2(r) convoluted with the tidal Love number k(t) according to (5.25): ΔU = k(t) ∗ U2(r) ,

(15.2)

where the total centrifugal potential of degree 2, U2(r) , is obtained by gathering the terms of degree 2 in Eqs. (5.7) (the variable part of the centrifugal potential) and (5.16) (the constant part of the centrifugal potential): U2(r) = r 2 Ω2 [( +(

m21 + m22 (1 + m3 )2 m (1 + m3 ) cos λ + m2 (1 + m3 ) sin λ − )P20 − 1 P21 6 3 3

m22 − m21 mm cos λ − 1 2 sin(2λ))P22 ] . 12 6

(15.3)

Now, the second order terms mi mj of the instantaneous rotation vector are not neglected anymore. From (15.2), the resulting degree 2 Stokes coefficients of the geopotential variations at the Earth surface (r = Re ) read ΔC20 =

Ω2 R3e m2 + m22 (1 + m3 )2 k(t) ∗ ( 1 − ), GM⊕ 6 3

ΔC21 = − ΔS21 = − ΔC22 =

Ω2 R3e m (1 + m3 ) k(t) ∗ 1 , GM⊕ 3

Ω2 R3e m (1 + m3 ) k(t) ∗ 2 , GM⊕ 3

Ω2 R3e m2 − m21 k(t) ∗ 2 , GM⊕ 12

ΔS22 = −

Ω2 R3e mm k(t) ∗ 1 2 . GM⊕ 6

(15.4)

282 | 15 Epilogue: Geological polar motion Then Eqs. (B.17), (B.18), (B.19), (B.21), (B.22), and (B.23) allow us to write the moment of inertia changes produced by the visco-elastic pole tide: Ω2 R5e k(t) ∗ m1 (1 + m3 ) , 3G Ω2 R5e (r,ve) I23 (t) = k(t) ∗ m2 (1 + m3 ) , 3G Ω2 R5e (r,ve) I12 (t) = k(t) ∗ m1 m2 , 3G Ω2 R5e m2 + m22 m23 + 2m3 m22 − m21 (r,ve) k(t) ∗ ( 1 − − ), I11 (t) = 3G 6 3 2 (r,ve) I13 (t) =

(r,ve) I22 (t) =

(15.5)

Ω2 R5e m2 + m22 m23 + 2m3 m22 − m21 k(t) ∗ ( 1 − + ), 3G 6 3 2

(r,ve) I33 (t) = −

2 5 m2 + m22 2 Ω Re k(t) ∗ ( 1 − (1 + m3 )2 ) . 3 3G 2

All those expressions can be gathered in a unique one: Iij(r,ve) (t) =

Ω2 R5e k(t) ∗ Mij (t) , 3G

(15.6a)

with 1 Mij (t) = (δi3 + mi )(δj3 + mj ) − (m21 + m22 + (1 + m3 )2 )δij . 3

(15.6b)

Note that these expressions concern the total moments of inertia Iij , for they include the whole centrifugal effect. The convolution product is expressed by Ω2 R5e Ω2 R5e ∫ k(t − τ)Mij (τ) dτ = ∫ k(t − τ)Mij (τ) dτ , 3G 3G +∞

Iij(r,ve) (t) =

+∞

(15.7)

t0

−∞

because the mi are considered equal to zero for τ ≤ t0 . In the following we take t0 = 0. According to visco-elasticity theory, the Love number k(t) takes the form J

k(t) = k2 δ(t) + H(t) ∑ tj e−sj t , j=1

(15.8)

where H(t) is the Heaviside function. This entails t

Iij(r,ve) (t)

Ω2 R5e = ∫ k(t − τ)Mij (τ) dτ . 3G

(15.9)

0

The number of coefficients tj , sj grows with the number of discontinuities J in the model of the Earth. For a visco-elastic homogeneous sphere, J = 1. For the stratified

15.3 Visco-elastic variation of the moments of inertia

| 283

Table 15.1: Inverse load relaxation times sj , tidal-effective Love numbers tj , and load Love numbers lj for the nine relaxation modes of a 5-layer Earth model A according to [WP, Tab 4] [233]. j

sj (kyr−1 )

tj (kyr−1 )

lj (kyr−1 )

1 2 3 4 5 6 7 8 9

4.15 3.95 3.58 3.48 1.92 3.88 10−1 9.79 10−2 4.08 10−4 9.01 10−6

8.77 10−3 1.45 10−2 9.35 10−3 1.67 10−2 9.35 10−1 5.86 10−2 2.82 10−3 1.13 10−6 8.62 10−10

−5.93 10−3 −1.25 10−2 −6.73 10−3 −1.45 10−2 −7.61 10−1 −8.16 10−2 −5.81 10−3 −8.37 10−6 −1.38 10−8

models used in Wu and Peltier (1984) [233] [WP] and in the recent book of Sabadini, Vermeersen and Cambiotti [SVC] [178], J = 9. We shall use the coefficients sj , tj , lj of [WP]1 reported in Table 15.1. The purely elastic pole tide accompanying the contemporaneous polar displacement, not depending on the frequency, yields the moment of inertia changes t

Iij(r,e) (t) =

Ω2 R5e Ω2 R5e k M (t) . ∫ k2 δ(t − τ)Mij (τ) dt = 3G 3G 2 ij

(15.10)

0

But the visco-elastic response brings about the total contribution t

Iij(r,ve) (t) =

Ω2 R5e Ω2 R5e k2 Mij (t) + ∑ t ∫ e−sm (t−τ) Mij (τ) dτ . 3G 3G m=1 m

(15.11)

0

For the main mode (m = 5), the relaxation time 1/sm is about 0.5 kyr. Visco-elastic loading As for the centrifugal excitation, the moment of inertia response to a surface load is determined by the visco-elastic loading Love number k 󸀠 (t) = k2󸀠 δ(t) + H(t) ∑ lm e−sm t , m=1

(15.12)

where k2󸀠 ≈ −0.3 quantifies the contemporaneous quasi-elastic deformation caused by the surface load, and the summation describes the visco-elastic response. If I (l) (t) is 1 Associated with the stratified model A of this paper, parametrized by an upper mantle viscosity of 1021 Pa s, a lower mantle viscosity of 1021 Pa s, and lithospheric thickness of 120 km.

284 | 15 Epilogue: Geological polar motion the moment of inertia associated with the load, the total visco-elastic variation of the off-diagonal moment of inertia of the solid Earth is Iij(l,ve) (t) = (1 + k 󸀠 (t)) ∗ Iij(l) (t)

t

= (1 + k2󸀠 )Iij(l) (t) + ∑ lm ∫ e−sm (t−τ) Iij(l) (τ) dτ . m=1

(15.13)

0

The total change of the moment of inertia is obtained by summing (15.6) and (15.13): Iij (t) = Iij(r,ve) (t) + Iij(l,ve) (t) =

Ω2 R5e k(t) ∗ Mij (t) + (1 + k 󸀠 (t)) ∗ Iij(l) (t) . 3G

Having at hand a time variable model Iij(l) (t) and the rotational contribution Mij (t), the Liouville equations (15.1a) embodying the angular momentum (15.1b) yield a system of nonlinear differential equations with respect to the variables ωi . The handling of this system is complicated by the fact that the rotational visco-elastic contribution Iij(r,ve) non-linearly convolves the tidal Love number k(t) with the unknown mi (t) from instant t0 = 0 until t. So these equations cannot be solved analytically. Further simplification will have to be brought about, as in the following sections.

15.4 Linear approach 15.4.1 General formulation We investigate the effect of a transient gigantic mass load, of which the corresponding moment of inertia reaches the triaxiality threshold B − A. But, as the load is nonpermanent, the pole displacement remains smaller than 1 degree after 1 My, and a linear approach is still valid. In particular, it is employed for estimating the secular pole drift resulting from the glaciation–deglaciation cycle. Then the basic Liouville equation describing the polar motion is given for instance by (4.32), where the time derivative of the excitation function can be neglected at secular periods: m(t) +

i I (r,ve) (t) + I (l,ve) (t) ṁ = , σe C−A

(15.14)

(r,ve) (r,ve) (l,ve) (l,ve) with I (r,ve) = I13 + iI23 and I (l,ve) = I13 + iI23 . By neglecting a second order term in (15.6) and (15.13) we have

Iij(r,ve) (t) = Iij(l,ve) (t) =

k(t) ∗ m(t) (C − A) , ks (1 + k 󸀠 (t)) ∗ I (l) (t) , ks

(15.15)

15.4 Linear approach

| 285

Table 15.2: Coefficients σj and ρj of polynomial (15.20) corresponding to the model of Table 15.1. j

σj (kyr−1 )

ρj (kyr−1 )

0 1 2 3 4 5 6 7 8 9

0 2.08 10−5 − i 1.84 10−10 3.40 10−3 − i 4.01 10−7 1.32 10−1 − i 4.73 10−6 7.80 10−1 − i 9.07 10−5 3.47 − i 5.28 10−6 3.57 − i 2.20 10−6 3.94 − i 5.23 10−6 4.14 − i 3.06 10−6 1.53 − i 5.27 103

−1.95 10−4 + i 1.38 10−9 −1.38 10−4 + i 4.89 10−9 1.47 10−2 − i 1.31 10−6 2.98 10−2 + i 1.98 10−6 2.03 10−1 + i 5.52 10−5 1.09 10−3 + i 5.28 10−6 1.36 10−3 + i 2.44 10−6 9.83 10−4 + i 5.24 10−6 1.80 10−3 + i 3.58 10−6 −2.52 10−1 − i 5.25 103

with ks = 3G(C − A)/(Ω2 R5e ) according to (5.19). In turn, (15.14) has the linear form m(t) +

i k(t) (1 + k 󸀠 (t)) ∗ I (l) (t) ṁ = ∗ m(t) + . σe ks C−A

(15.16)

Here, we assume that C − A is not influenced significantly by the equatorial viscoelastic readjustment, and is considered to be constant. Equation (5.19) for the secular Love number ks also appears. Then applying the Laplace transform allows one to eliminate the convolution product: m(s)(1 −

s I (l) (s) k(s) , + i ) = (1 + k 󸀠 (s)) ks σe C−A

(15.17)

with J

k(s) = k2 + ∑ j=1

J

tj

lj

and k 󸀠 (s) = k2󸀠 + ∑

sj + s

sj + s

j=1

.

(15.18)

Hence m(s) =

1 + k2󸀠 + ∑Jj=1 1−

k2 ks

− ∑Jj=1

lj sj +s

tj sj +s

I (l) (s)

+ i σs C − A

.

(15.19)

e

The function P(s) =

1 + k2󸀠 + ∑Jj=1 1−

k2 ks

lj sj +s

+ i σs − ∑Jj=1 e

tj sj +s

can be decomposed in simple fractions with J non-zero roots −σj :

(15.20)

286 | 15 Epilogue: Geological polar motion

P(s) =

J ρj ρ0 +∑ . s j=1 s + σj

(15.21)

These coefficients can be easily obtained by Mathematica, and the results for the WP rheological model are reported in Table 15.2. This polynomial is composed of Laplace transforms of the step function H(t) for the first term ρ0 /s and of the function H(t)e−σj t when the root σj is not null. So, forming the inverse Laplace transform of P(s)I (l) (s)/(C− A), we obtain the convolution product J

m(t) = H(t)(ρ0 + ∑ ρj e−σj t ) ∗ 1

I (l) (t)

C−A

.

(15.22)

In Table 15.2, the last mode (complex frequency σ9 ) models the forced damped Chandler wobble, and the contribution of the visco-elasticity to the secular polar motion is given by all the other modes.

15.4.2 Qualitative analysis for an increasing surface load Assume a surface mass redistribution producing a monotonously varying inertia increment c(t) = at. For a rigid or an elastic Earth, the polar displacement is also proportional to time. But a consequence of visco-elasticity is to accelerate it. This is what we are going to show in the light of the basic equation (15.14). As we consider a time scale much longer than the Euler period, the time derivative drops out and we rewrite it m(t) =

I(t) , C−A

(15.23)

where I(t) encompasses the load I (l) (t), and other non-rigid effects. Let us consider successive equally spaced instants, t0 = 0, t1 , t2 , tn , . . . . During each time step τ = [ti , ti+1 ], the off-diagonal moment of inertia of the load increases by ΔI (l) . To illustrate our demonstration, let us take a positive increase ΔI (l) = ΔI23 = − ∫ yz dm in the plane yz, corresponding to a positive mass anomaly symbolized by a trapezoidal form in Figure 15.1. For the sake of simplicity we assume that at time t0 the inertia pole, the geographic pole (the z axis of the terrestrial frame) and the Earth rotation pole coincide at the point R0 . At time t1 = τ, the Earth axis moves to the position Rr1 by Δm =

ΔI (l)

C−A

(15.24)

for a rigid Earth. In the case of a solid elastic Earth, partially covered by hydrostatic oceans, the pole tide is instantaneous. According to (15.15) it yields the pole tide inertia increment

15.4 Linear approach

| 287

Figure 15.1: Schematic representation of the polar shift mechanism in the body-fixed reference frame when off-diagonal moment of inertia c23 linearly increases with time. At time t1 = t0 + τ a positive mass load is added, pictured by a trapezoidal form and producing an increase of c23 . At time t2 = t1 + τ this moment inertia variation is multiplied by 2, and so one. The subsequent displacement of the rotation pole R are represented (magnified) for three Earth models: R r for a rigid Earth, R e for an elastic/fluid Earth and R ve for a visco-elastic Earth.

I (r,e) = k/ks (C − A)m with k ≈ 0.35, thus (15.23) becomes m(t) =

I (l) (t)

1 , C − A 1 − k/ks

(15.25)

where I (l) (t) possibly includes the partial compensation produced by the visco-elastic deformation. We see that the shift is amplified, and rotation pole is now in the position Re1 . For the next interval [t1 , t2 ] the same increment happens for I (l) , the pole shift is doubled for a rigid or an elastic/fluid Earth. The new pole positions are, respectively, Rr2 and Re2 . So the poles of a rigid Earth and of an elastic Earth move linearly with time.

288 | 15 Epilogue: Geological polar motion But, for a visco-elastic rheology, the first step [0, t1 ] did not present a full adjustment of the equatorial bulge around the new direction of the Earth axis. A delayed pole tide begins during the period [t1 , t2 ], function of the displacement R0 Re1 , and it causes a supplementary pole tide with the same sign as I (l) (t), of which the effect on the off-diagonal moment of inertia can be formulated by I (r,v) (t2 ) = ϵ

k (C − A)m(t1 ) ks

with ε ≪ 1 .

(15.26)

Again, this reinforces the purely elastic or hydrostatic effect, and at time t2 by rotation the pole lies in the more shifted position Rve 2 . Iterating this argument at inve stant t3 , we see that R3 integrates the delayed pole tide effect during period [t1 , t2 ] and [t0 , t1 ], so that, at time tn , Rve n results not only from the pure rigid polar shift and elastic pole tide but also from the delayed pole tide over the time intervals [t0 , t1 ], [t1 , t2 ], . . . , [tn−2 , tn−1 ], all adding to the elastic effect of cl . This means that the displacement of Rve is accelerated with time, as far as I (l) (t) increases. 15.4.3 Transient mass redistribution like glaciation followed by melting The former modeling is applied to a transient, or quasi-periodic mass loading, such as the one produced by consecutive glaciations. First, we consider the last glaciation, which can be approximated by a gigantic ice cap mostly spreading over North America. According to [233], the corresponding off-diagonal moments of inertia Ixy and Iyz at the peak of the glaciation reach the values c0 = Ixy + Iyz = −5.880 1031 + i 3.222 1032 kg m2 .

(15.27)

This load is assumed to build up linearly with time over the period τ1 ∼ 90 kyr, then to disappear in τ2 ∼ 10 kyr also at a linearly temporal rate. The predominance of the Northern America cap (the “Laurentia” zone) for which x > 0, y < 0, z > 0 accounts for the signs of Ixy and Iyz . Geological or radio-dating of the ice core showed several consecutive glaciation periods, lasting about 100 kyr. Their cause could be the global insolation variation accompanying the 100 kyr variation of the eccentricity of the Earth’s orbit, according to the Milankovich theory. The moment inertia change produced by N consecutive glaciation cycle of period τ = τ1 + τ2 can be formulated by the saw-like function c0 f (t) with a(t − (n − 1)τ) { { f (t) = { b(t − (n − 1)τ − τ1 ) + 1 { { 0

t ∈ [(n − 1)τ, (n − 1)τ + τ1 ] t ∈ [(n − 1)τ + τ1 , nτ] t ≥ Nτ ,

a = 1/τ1 , b = −1/τ2 ,

(15.28)

where n = 1, 2, . . . , N. This assumes that, once a glaciation cycle is over, the position of the pole has not changed much with respect to the continents, so that the ice sheets

15.4 Linear approach

| 289

of the coming glaciation form on the same location and produces about the same moment of inertia. The convolution product (15.22) then gives t

m(t) =

t

J c0 (ρ0 ∫ f (u) du + ∑ ρj e−σj t ∫ eσj u f (u) du) . C−A j=1 0

(15.29)

0

First, we evaluate the integrals Ln1 (t)

t≤(n−1)τ+τ1

=

a[u − (n − 1)τ] du ,

∫ (n−1)τ

(15.30)

t≤nτ

Ln2 (t) =

[b(u − (n − 1)τ − τ1 ) + 1] du ,

∫ (n−1)τ+τ1

where n is the number of the cycle. We easily obtain Ln1 (t) = a[ Ln2 (t)

t2 (n − 1)2 τ2 − (n − 1)τt + ], 2 2

b((n − 1)τ + τ1 )2 t2 = b[ − ((n − 1)τ + τ1 )t] + t + − (n − 1)τ − τ1 . 2 2

(15.31)

In particular, we have τ12 τ1 τ b = = L01 , Ln2 (nτ) = (τ − τ1 )2 + τ − τ1 = 2 = L02 . (15.32) 2 2 2 2 For the terms depending on σj , we have to evaluate the integrals Ln1 [(n − 1)τ + τ1 ] = a

n J1,j (t) =

t≤(n−1)τ+τ1

∫

eσj u a[u − (n − 1)τ] du ,

(n−1)τ n J2,j (t) =

t≤nτ

(15.33) eσj u [b(u − (n − 1)τ − τ1 ) + 1] du .

∫ (n−1)τ+τ1

Integrating by parts allows us to obtain n J1,j (t) = n J2,j (t)

aeσj t 1 a [t − (n − 1)τ − ] + 2 eσj (n−1)τ , σj σj σj

eσj t b eσj [(n−1)τ+τ1 ] b = [b(t − (n − 1)τ − τ1 ) + 1 − ] − (1 − ) . σj σj σj σj

(15.34)

In particular, n J1,j [(n − 1)τ + τ1 ] = n J2,j (nτ)

eσj ((n−1)τ+τ1 ) 1 1 (τ1 − ) + 2 eσj (n−1)τ , σj τ1 σj σj τ1

1 1 eσj nτ =[ − (1 + )e−σj τ2 ] . σj τ2 σj τ2 σj

(15.35)

290 | 15 Epilogue: Geological polar motion Defining the integer functions t + τ2 ) τ t n2 (t) = int( ) τ n1 (t) = int(

n1 (t) = n2 (t) = N

for t < Nτ , (15.36)

for t < Nτ , for t ≥ Nτ ,

Eq. (15.29) is reduced to m(t) =

c0 n +1 (n +1)τ n +1 1 {ρ [n (t)L01 + n2 (t)L02 + H(Nτ − t)(Πnn11 τ+τ (t) L1 1 (t) + Πn22τ+τ1 (t)L2 2 (t))] τ C−A 0 1 J

n1 (t)

n2 (t)

j=1

n=1

n=1

n n × ∑ ρj e−σj t [H(t − τ1 ) ∑ J1,j [(n − 1)τ + τ1 ] + H(t − τ) ∑ J2,j (nτ) n +1

n +1

1 + H(Nτ − t)(Πnn11 τ+τ (t) J1,j1 (t) + Πn22τ+τ1 (t)J2,j2 (t))]} , τ

(n +1)τ

(15.37) where Πba (t) is the rectangular function equal to 1 in the interval [a, b] and zero elsen n where; H(t) is the Heaviside function. As the J1,j [(n − 1)τ + τ1 ] and J2,j (nτ) are geometric series, they can be summed: n1 (t)

n [(n − 1)τ + τ1 ] = ∑ J1,j

n=1

n2 (t)

1 σj τ1 1 1 1 − eσj n1 (t)τ [e (1 − )+ ] , σj σj τ1 σj τ1 1 − eσj τ

n (nτ) =[ ∑ J2,j

n=1

1 1 eσj τ 1 − eσj n2 (t)τ − (1 + )e−σj τ2 ] . σj τ2 σj τ2 σj 1 − eσj τ

(15.38)

Substituting in (15.37) Eqs. (15.31), (15.32), (15.34), and (15.38), we have a quick way to numerically evaluate the effect of N glaciation cycles on the polar motion. The computation is done according to the coefficients of Table 15.2, as derived from rheological model A proposed in [WP]. First, restricting the forcing to one cycle of 100 kyr, we obtain the polar shift of Figure 15.2. In the displayed time scale our epoch is near t = 6–8 kyr. For a rigid Earth, the pole displacement would be exactly the saw-like moment of inertia divided by the (C − A), reaching a maximum of 250󸀠󸀠 for m2 . In the case of a mono-layer homogeneous visco-elastic Earth, a constant polar displacement remains after t > 0. For a multi-layered viscous elastic Earth, the polar motion is not only amplified, but the polar drift continues after the glacial process has ended. The rate corresponding to our epoch (6–8 kyr) is about 2 mas/year in the direction of 80°, comparable with the contemporaneous value of 3.3 mas/year towards 75° ± 5° [193]. Considering now that seven consecutive glaciation/melting cycles precede our epoch, as evidenced by geological records, we obtain the polar drift of Figure 15.3. The polar displacement increased, almost reaching 800󸀠󸀠 towards Siberia during the

15.5 Nonlinear treatment | 291

Figure 15.2: Pole drift resulting from one glaciation cycle lasting 100 kyr.

seven cycles (24 km at the Earth surface). Post-glacial drift is also strengthened; with a rate of 3 mas/year 6–8 kyr after the end of the last cycle, it grows in better agreement with the observed value. This notwithstanding, such an evaluation remains very indicative, for the modeled rate strongly depends on the Earth model, especially on the lower mantle viscosity, which is poorly constrained by geophysical observations. For instance, a very strong viscosity of the lower mantle (1023 Pa s instead of 1021 Pa s for the WP model) gives a 10 times lower rate. In this case, there would be room for an alternative forcing of the secular pole drift. Anyhow the mantle convection cannot explain the secular pole drift [178]. For a deeper comprehension of the post-glacial polar motion and its sensitivity to the Earth’s model, we refer the reader to Chapter 4 of the book of Sabadini, Vermeersen and Cambiotti [178].

15.5 Nonlinear treatment 15.5.1 General formulation For a glaciation process, after a typical duration of 100 kyr the load is suppressed, and the pole partly goes back to its initial position. If, conversely, the mass redistribu-

292 | 15 Epilogue: Geological polar motion

Figure 15.3: Pole drift resulting from seven glaciation cycles lasting 100 kyr.

tion asymptotically tends towards some threshold, the visco-elastic adjustment of the mass of the Earth around the shifting pole is not stopped. After some million years, it exceeds one degree and a nonlinear approach is required. This problem can be simplified for a time scale much longer than the largest relaxation time 1/sj of the viscoelastic modes (about 100 kyr). First, let us consider that a state of hydrostatic equilibrium is reached for t → ∞. Then the rotation vector components mi are stabilized as well as the quantities Mij (t), becoming the constant Mij . The rotational change of the moment of inertia (15.6) is given by Ω2 R5e k(t) ∗ Mij (t) . t→∞ 3G

lim Iij(r,ve) (t) = lim

t→∞

(15.39)

According to the final value theorem stating that limt→∞ f (t) = lims→0 sf (s), we have lim Iij(r,ve) (t) = lim[sk(s)

t→∞

s→0

Ω2 R5e M (s)] . 3G ij

(15.40)

But lims→0 sMij (s) = limt→∞ Mij (t) = Mij . Hence lim Iij(r,ve) (t) =

t→∞

Ω2 R5e k(s = 0)Mij . 3G

(15.41)

15.5 Nonlinear treatment | 293

In agreement with (15.6), k(s = 0) is the secular Love number ks , which is equal, on the other hand, to 3G(C − A)/Ω2 R5e . From (15.18) we have J

k0 = k(s = 0) = k2 + ∑ j=1

tj

sj

(15.42)

.

A numerical estimation from the coefficients of Table 15.1 yields a value of k0 close to 0.94, as expected. According to (15.6b) the rotational readjustment resulting from the non-perturbed angular velocity associated with mi = 0 is given by Iij(r,ve) (t = ∞) =

Ω2 R5e k M0 3G 0 ij

1 with Mij0 = δi3 δj3 − δij . 3

(15.43)

So, the variable part of the centrifugal change is cij(r,ve) (t) = Iij(r,ve) (t) − Iij(r,ve) (t = ∞) =

Ω2 R5e (k(t) ∗ Mij (t) − k0 Mij0 ) . 3G

(15.44)

Under the condition t ≫ 1/sj , a first order Taylor development of k(s) holds. Starting from (15.18), we have J

k(s) = k2 + ∑ j=1

tj

sj

(1 −

s + O(s2 )) = k0 (1 − T1 s + O(s2 )) , sj

(15.45)

with T1 =

1 J tj . ∑ k0 j=1 s2j

(15.46)

According to Table 15.1 T1 ∼ 20 kyr. Now, the Laplace transform of (15.44) reads cij(r,ve) (s) =

k0 Mij0 Mij0 Ω2 R5e Ω2 R5e (k(s)Mij (s) − )= k0 [(1 − T1 s) Mij (s) − ]. 3G s 3G s

(15.47)

Using the property ℒs {f ̇(t)} = sf (s) − f (t = 0), we obtain cij(r,ve) (s) =

Mij0 Ω2 R5e k0 [Mij (s) − T1 (ℒs {Ṁ ij (t)} + Mij (t = 0)) − ]. 3G s

(15.48)

Thus, inverse transform yields cij(r,ve) (t) =

Ω2 R5e k [M (t) − Mij0 − T1 ( ṁ i mj + mi ṁ j 3G 0 ij 2 − (ṁ 1 m1 + ṁ 2 m2 + ṁ 3 m3 + ṁ 3 ) δij + δi3 ṁ j + δj3 ṁ i + Mij0 δ(t))] . 3

(15.49)

Then this expression and (15.13) for visco-elastic loading contribution cij(l,ve) (t) are put in the angular momentum equation (15.1), yielding a system of second order nonlinear differential equations for mi (t). We see that the visco-elastic response is reduced to the parameter T1 .

294 | 15 Epilogue: Geological polar motion Table 15.3: Parameters of the model.

Upper mantle viscosity (Pa s) Lower mantle viscosity (Pa s) Lithospheric thickness (km)

Model WPA

Model WP B

Model WP C

mono-layered

21

21

21

1021 1021 –

10 1021 120

10 3 1021 120

10 3 1021 30

15.5.2 Permanent mass redistribution like mountain upheaval If the glaciation cycles explain TPW on the 100–100 000 time scale, the longer term TPW probably results from mantle convection happening over million years. We will not to provide a precise model of this effect as in [28] or [178]. But we want to evaluate the order of magnitude of the TPW caused by a permanent mass redistribution, described by the function f (t)M with f (t) = {

at 1

t ∈ [0, τ] t≥τ.

a = 1/τ ,

(15.50)

The mass M = 10−6 M⊕ —M⊕ is the mass of the Earth—is supposed to be point-like and located at the latitude π/4 and longitude π/2, and the moment of inertia changes are computed accordingly. After τ ∼ 5000, the distribution is achieved, and its moment of inertia then grossly represent the one of the Himalayan massif. As τ is much larger than T1 = 20 kyr, we fall in the nonlinear framework of the last section. The viscoelastic rotational response is given by (15.49), allowing one to confer the form of ordinary differential equations of second order to the Liouville equations involving convolution products of the variables mi . These are integrated numerically by Matlab considering four rheological Earth models. Three are stratified (the Wu and Peltier models WP A, WP B, and WP C) and one is for a mono-layered Earth with a uniform viscosity. Characteristic parameters of these models are reported in Table 15.3. The corresponding solutions, displayed in Figure 15.4, demonstrate striking polar displacements of a few tens of degrees after 5 millions years. During the same period, the continental plate can slip by about 250 km (or ∼ 2°) assuming a constant average tectonic drift of 5 cm/year. This is much less than the effect produced by the polar drift, switching high latitude regions to low ones and inversely.

15.6 Readjustment of the main pole of inertia After the permanent loading is achieved (here after 5 millions years), the process of visco-elastic readjustment still goes on, and the main pole of inertia “runs after” the rotation pole. Once the rotational bulge is fully relaxed, the main pole of inertia should

15.6 Readjustment of the main pole of inertia |

295

Figure 15.4: Pole drift caused by moment of inertia change amounting to triaxiality starting at t = 0 and lasting 5000 kyr, modeled as a surface point mas located at λ = 90°, Φ = 45° according to four rheological models.

get to the rotation pole. To check this, we determine the cosine directors of the main of pole of inertia from the eigenvectors of the inertia matrix in the terrestrial frame, at each time step of the integration. The rotational part of the inertia matrix is computed according to (15.49) and loading part from (15.13). Then we compute the angular distance between the rotation pole and the main pole of inertia, and display it in Figure 15.5a. This distance increases up to 10󸀠󸀠 when the mass load ends up increasing. After this maximum, it decreases, and after 10 million years it becomes smaller than 1 mas, whatever the rheological model. This study is repeated in the case of a transitory mass load brought about by a glaciation cycle as treated in Section 15.4.3. Then the angular distance between rotation and inertia poles is depicted in Figure 15.5b. According to this simulation, ∼ 6 kyr after the end of the 100 kyr glaciation, the difference between the two poles can reach 10 mas and higher. As the mean inertia pole is now determined from the degree 2 geopotential coefficients (LAGEOS 1/2 and GRACE satellites) with a few mas accuracy, we have an independent estimate of the contemporaneous difference (with respect to the mean rotation pole), which brings about a new constraint on the Earth’s rheology.

296 | 15 Epilogue: Geological polar motion

Figure 15.5: Distance between the rotation pole and the mean pole of inertia according to four rheological models for a surface mass load of about 10−6 M⊕ . Time unit: kyr.

15.7 Conclusion | 297

15.7 Conclusion These brief considerations on the secular polar motion or true polar wander cannot replace much more comprehensive analyses, especially the synthesis produced by SVC [178]. The primary aim of this chapter was to show how the approach developed at sub-secular scale can be extended to geological scale by replacing the quasi-elastic mantle by a visco-elastic one. Secular pole drift seems to be the unique observable consequence of the geological mass redistribution. But its causes, glacial isostatic adjustment or mantle convention, highly depending on unknown parameters like low mantle viscosity, cannot be discriminated with certainty. Over a million years a polar wander of several degrees takes place, probably driven by mantle convection. Elusive evidence is found in the magnetic grains of the ocean floor basalt, possibly indicating the direction of the magnetic field when the basalt was forming. According to these paleomagnetic data, as far as rotation axis and geomagnetic dipole remain close within 15°, the rotation pole would have shifted by several tens of degrees with respect to the mantle during the last 100 million years.

Synthesis En un mot: observer ce qui est patent; découvrir ce qui est caché; établir les lois qui résultent de la comparaison des faits observés et de toutes les modifications qu’ils éprouvent suivant les temps et les lieux; enfin procéder à la recherche d’une inconnue plus cachée encore que celle dont nous venons de parler, c’est-à-dire remonter aux causes des effets connus, ou prévoir les effets à venir d’après la connaissance des causes. A. M. Ampère, Essai sur la Philosophie des Sciences

The Earth is the only astronomical body of which the rotation is determined with the extraordinary relative accuracy of 10−11 thanks to developments in space geodesy since the 1980s. Astrogeodetic techniques have permitted to refine the already known Earth rotation changes, and lead to the discovery of sub-milliarcsecond variations presenting various causes. Depending on the global properties of the Earth and the geophysical processes taking place there, these changes of the rotation of the Earth are a source of valuable information for Earth rheology, geodynamics and global circulation models in fluid layers. Our attention was focused on the equatorial effect, namely the polar motion. Its study was limited to the most evident geophysical causes. Now, we need to venture into less familiar terrain. If the effect of the mass transport surface has been dissected, the viscous and electromagnetic coupling at the core–mantle interface has not been treated. It is not excluded that this dissipative coupling produces observable effects, like the Markowitz wobble, but in the absence of measurements of magnetic field, pressure, conductivity at the core–mantle interface, its effects on polar motion remain even more speculative than the seismic excitation. Whereas we are far from having exhausted the subject and all geophysical excitations, we can conclude that the fragile and tiny layer of ice, water and air that covers the Earth is the major cause of the polar displacements at time scales from a few hours to one century. It is fortunate, for other sources of excitation, especially internal ones, are not subject to direct observations and cannot be reconstructed precisely. The hydro-atmospheric excitation reflects thermal fluctuations or climatic cycles that we commonly observe: diurnal and seasonal cycle, weather hazards throughout weeks or months. Even the tiny excitation (at a few mas level) describing the Chandler wobble stems from the oceanic and atmospheric mass redistribution, that well reproduces its amplitude and phase variability. The pitfalls encountered in the geophysical budget of the polar motion are largely due to imperfections in hydrological models. In the coming years, the expansion of hydrological observations and the improvement of the ocean model will make the reconstitution of the fluid later excitation more faithful. On the other hand, with the continuation of the satellite mission dedicated in particular to the low degree coefficients of the geopotential, the full matter term time series, determined independently from any circulation models, will be refined and expanded. Usually the pole geophysical excitation is treated by considering that the observed excitation does not depend on longitude (symmetric approximation). On the one hand

300 | Synthesis triaxiality introduces an asymmetry between the equatorial components of the instantaneous rotation vector and makes any uniform circular component of the polar motion slightly elliptical; on the other hand the main asymmetry is introduced by the ocean pole tide. These asymmetric effects are investigated by introducing the concept of the generalized Liouville equation. The asymmetry results in another free mode, having an opposite frequency to that of Chandler. Consequently geodetic excitation is disturbed at the level of 2 mas, which is remarkable in the light of geophysical excitation changes and cannot be neglected in interpreting annual and 433 day terms. But, even at these frequencies, the intrinsic ellipticity of the polar motion cannot be separated from the one associated with the ellipticity of the geophysical excitation. Amazingly, the elliptical path of the geodetic excitation at a given frequency tends to polarize towards 70–80 degree east. This behavior is found in the reconstructed hydro-atmospheric excitation, but it has not been explained until now. Throughout this study, we have seen how the pole tide determines the resonance of the polar motion. Yet of all mass redistributions we have described, the polar tide is one we approach the less readily by observation although it has been modeled profusely. Neither the ocean pole tide nor the solid earth pole tide are subject to certain determinations. In the commonly accepted theory treating polar motion at time scales from one hour to one century, developed in the second part of this book, the viscoelastic response of the solid Earth is approached by a quasi-elastic approximation, and the oceans react hydrostatically to the pole tide potential. Then the compliance or Love number does not depend on the frequency, and the moments of inertia induced by the pole tide are linear combinations of the instantaneous pole coordinates. However, this approach is only valid for the polar motion, of which the time scale spreads out from 10 days to some years. For polar motion below 10 days the dynamical readjustment of the oceans strongly modifies the effective Love number, and in turn the complex resonance frequency of the polar motion. As the ocean pole tide is less effective, the equatorial bulge is less compensated for, thus causing an increase of the resonance frequency, which moves towards the Euler frequency. In the retrograde diurnal band, from the knowledge of the ocean tides, it can be predicted that the resonance period amounts to about 380 days, and that the equivalent quality factor has a much lower value and even becomes negative (−10 against 70 in the seasonal band) because of the phase shift accompanying the ocean tide. Moreover, the resonance of geopotential around the free core nutation frequency, −1.0051 cpd, also affects the effective Love number, entailing a frequency dependence of the polar motion resonance parameters in the retrograde diurnal band, with the resonance period sweeping the interval of 370–580 days for characteristic diurnal tidal frequencies. Actually, the retrograde diurnal band is observed in the celestial system as nutation. The extraordinary precision of the nutation observation and accurate modeling of its lunar–solar excitation, through the rigid Earth nutation model, allows one to confirm the modeled polar motion parameters, accounting for both the dynamical effect of the ocean and the free core nutation resonance.

Synthesis | 301

In the opposite part of the spectrum, from some years to one century, the approximation of a quasi-elastic response of the mantle can be questioned. For now, poor knowledge of the geophysical excitation does not allow one to fit consistently the visco-elastic parameters in this frequency range. In the future an improved time series of the full matter term, as determined from dedicated satellite missions, may better yield the difference between the mean rotation pole and the mean inertia pole, from which we can infer the visco-elastic parameters. At time scales beyond 100 years, the pole tide of the solid Earth is expressed as a convolution product of the polar motion, so that the Liouville equations stop being ordinary differential equations. When considering the effect of glaciation, which spreads over 100 000 years, the Liouville equations keep their linearity, and they can be solved in the Laplace domain. But, when investigating the tectonic redistribution over millions of years, the rotational moment of inertia becomes too large for keeping a linear approximation; fortunately the problem can be approximated by an ordinary differential equation of second order. Supposedly banished from the geophysical excitation, lunar–solar tides have surreptitiously crept up in the analysis by touching both rapid fluctuations of the pole, diurnal fluctuations, and even the Chandler term. Indeed, many of the excitation processes we studied, including atmospheric transports, are strongly affected by tidal cycles either directly or indirectly through mechanisms of synchronization, that have to be clarified yet. The Earth is an open system constantly exchanging gravitational energy, electromagnetic and heat with the celestial bodies, in a way that still eludes us. We have to keep in mind that the objective knowledge of the polar motion is limited by the temporal window of the observations. Determined by optical astrometric measurements from 1840 and by space geodesy from 1980, the longest pole coordinate time series spans over 180 years. This allows one to catch the polar drift, attributed to the visco-elastic readjustment following the last ice melting, which possibly competes with tectonic mass transports on an even longer time scale. On the other edge of the spectrum, the current resolution hardly falls under 12 hours. Although astro-geodetic observations potentially contain the hourly variations of the pole, hourly determinations are still controversial. So, since its discovery, the Earth polar motion still represents an observational challenge, and at the same time it is at the crossroad of a bunch of issues of primary importance for astronomy and geophysics, funding the astrogeophysics.

A Generalized Liouville equations A.1 Definition Accounting for the asymmetric effect on Earth’s polar motion like triaxiality and the ocean pole tide leads to generalized linearized Liouville equations having the generic form (1 − U)m +

i i (1 + eU)ṁ − Vm∗ + eV ṁ ∗ = Ψ(pure) , σe σe

(A.1)

where Ψ(pure) is the geophysical excitation free from the rotational effect and V is the complex asymmetric coefficient. We have the following orders of magnitude: U = U1 + i U2 |V| ≈ e

with U1 ≈ 0.36, U2 = O(e) ,

(e = flattening) .

(A.2)

Neglecting the asymmetric coefficient V, we recognize the common Liouville equations associated with a biaxial Earth and the symmetric pole tide is given by m(1 − U) +

i (1 + eU)ṁ = Ψ(pure) , σe

(A.3)

with the complex free mode frequency σc̃ = σe

1−U . 1 + eU

(A.4)

A.2 Solution of the generalized linearized Liouville equations A.2.1 Solution in frequency domain and eigenfrequencies Let m(σ) be the Fourier transform of m(t), then the Fourier transform of m∗ (t) and ṁ ∗ (t) are m∗ (−σ) and iσm∗ (−σ), respectively. In turn (A.1) is transformed into [1 − U −

σ σ (1 + eU)]m(σ) − V[1 + e ]m∗ (−σ) = Ψ(σ) , σe σe

(A.5)

where Ψ(σ) denotes the Fourier transform of Ψ(t) (hereafter the superscript “pure” of a geophysical function is ignored). Now, by replacing σ by −σ in (A.5), and taking the complex conjugate of this equation, we obtain the linear system with the unknowns m(σ) and m∗ (−σ): [1 − U − [1 − U ∗ +

σ σ (1 + eU)]m(σ) − V[1 + e ]m∗ (−σ) = Ψ(σ) , σe σe

σ σ (1 + eU ∗ )]m∗ (−σ) − V ∗ [1 − e ]m(σ) = Ψ∗ (−σ) . σe σe

https://doi.org/10.1515/9783110298093-016

(A.6)

304 | A Generalized Liouville equations We easily extract the solution m(σ) = −

[1 − U ∗ +

σ (1 σe

+ eU ∗ )]Ψ(σ) + V(1 + e σσ )Ψ∗ (−σ) e

P(σ)

,

(A.7)

with the second order polynomial P(σ) = {

[(1 + eU)(1 + eU ∗ ) − e2 VV ∗ ]( σσ )2 + ⋅ ⋅ ⋅ e

⋅ ⋅ ⋅ + [(1 + e)(U − U ∗ )] σσ − (1 − U)(1 − U ∗ ) + VV ∗ .

(A.8)

e

The discriminant of P(x = σ/σe ) is Δ={

[(1 + e)(U − U ∗ )]2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − 4[(1 + eU)(1 + eU ∗ ) − e2 VV ∗ ][VV ∗ − (1 − U)(1 − U ∗ )] .

(A.9)

From (A.2) U − U ∗ = 2 U2 = O(e) and we have Δ = 4(1 − U)(1 − U ∗ )(1 + eU + eU ∗ ) + O(e2 ) ,

(A.10)

so that the roots of P(x) are U ∗ − U + O(e2 ) ± √4(1 − U)(1 − U ∗ )(1 + eU + eU ∗ ) + O(e2 ) . 2[1 + eU + eU ∗ + O(e2 )]

(A.11)

The corresponding roots of P(σ) are σe (±√

(1 − U)(1 − U ∗ ) U ∗ − U + + O(e2 )) . 1 + eU + eU ∗ 2

(A.12)

One of them, associated with the positive square root, leads to the eigenfrequency σ̃ c = σe (√

(1 − U)(1 − U ∗ ) U ∗ − U + + O(e2 )) , 1 + e(U + U ∗ ) 2

(A.13)

of which the real part is the Chandler angular frequency (the square root) and the imaginary part the corresponding damping. The second root is σ̃ c− = σe (−√

(1 − U)(1 − U ∗ ) U ∗ − U + + O(e2 )) , 1 + e(U + U ∗ ) 2

(A.14)

which turns out to be equal to the negative complex conjugate of the first one: σ̃ c− = −σc∗ .

(A.15)

By expressing the real part and imaginary part of U = U1 + iU2 in (A.13), it is easy to show that σ̃ c = σe [1 − U − e U1 (1 − U1 ) + O(e2 )] .

(A.16)

A.2 Solution of the generalized linearized Liouville equations | 305

Notice that this form is consistent with the eigenfrequency of the symmetric part of (A.5), that is, σe

1−U . 1 + eU

(A.17)

Finally, by considering the factorized polynomial P(σ) =

1 + e(U + U ∗ ) + O(e2 ) (σ − σ̃ c )(σ − σ̃ c− ) , σe2

(A.18)

we obtain m(σ) = −σe

[σe (1 − U ∗ ) + σ(1 + eU ∗ )]Ψ(σ) + V(σe + eσ)Ψ∗ (−σ) , (1 + eU + eU ∗ )(σ − σ̃ c )(σ − σ̃ c− )

(A.19)

which can be decomposed into simple fractions. Then neglecting terms of second order in ellipticity, we have m(σ) = −

σ V(σe + eσc ) ∗ σ V(σe − eσc ) ∗ σe (1 − e U1 ) Ψ(σ) − e Ψ (−σ) + e Ψ (−σ) . (A.20) σ − σ̃ c 2σc (σ − σ̃ c ) 2σc (σ − σ̃ c− )

We easily check that the part in Ψ(σ) is the solution of the symmetric Liouville equations (A.3). It is followed by an asymmetric contribution in Ψ(−σ)∗ , presenting the double resonance at σc and −σc , and the ratio V/2 with respect to the symmetric solution. A.2.2 Solution in time domain By the inverse Fourier transform of (A.20), we obtain the solution in the time domain:1 m = m+ (t) + m− (t)

with

t

m+ (t) ≈

−iσe iσ̃c t V(σe + eσc ) ∗ ̃ e Ψ (τ)]dτ ∫ e−iσc τ [Ψ(τ) + 1 + e U1 2σc −∞

t

iσ ̃− ̃− m (t) ≈ e V(σe − eσc )eiσc t ∫ e−iσc τ Ψ∗ (τ) dτ . 2σc −

−∞

The common solution associated with the symmetric part of (A.1), that is, t

−iσe iσ̃c t ̃ e ∫ e−iσc τ Ψ(τ) dτ, 1 + eU ∞

1 From the inverse Fourier transform ℱσ−1 1/(σ − σa ) = iH(t)eiσa t .

(A.21)

306 | A Generalized Liouville equations is included in the term m+ (t). Other contributions reflect asymmetric effects. The derivation could have been done directly without specifying any special property different from integrability about Ψ(t): by solving the first order differential equation system, which couples (A.1) and its complex conjugate. Put in the matrix form Ẋ = AX + B(t) with X = (m, m∗ ) and B(t) associated with the excitation, we would have obtained the free modes exhibited earlier and the general solution (A.21) by applying the variation of constants method. A.2.3 Effect on a forced oscillation at the angular frequency σ0 Let Ψ(t) = Ψ0 eiσ0 t be a circular excitation at angular frequency σ0 . Its Fourier transform is Ψ(σ) = 2πδ(σ − σ0 )Ψ0 ,

(A.22)

where δ(σ − σ0 ) is the Dirac function. By putting this expression in (A.20), we obtain mσ0 (σ) = m+0 (σ) + m−0 (σ) ,

(A.23a)

with m+0 (σ) = −2πΨ0 m−0 (σ) = 2πΨ∗0

σe (1 − e U1 ) δ(σ − σ0 ) , σ − σ̃ c

σ + eσc σe − eσc σe V (− e + )δ(−σ − σ0 ) . 2σc σ − σ̃ c σ − σ̃ c−

(A.23b)

From the inverse Fourier transform, we conclude that the induced polar motion in the time domain is mσ0 (t) = m+0 eiσ0 t + m−0 e−iσ0 t ,

(A.24a)

with m+0 = −Ψ0 m−0 = Ψ∗0

σe (1 − e U1 ) , σ0 − σ̃ c

σe V σ + eσc σe − eσc (− e + ). 2σc σ0 − σ̃ c σ0 − σ̃ c−

(A.24b)

A.3 Generalized geodetic excitation and digitization Widening the usual definition, the generalized geodetic excitation is the first member of (A.1) divided by 1 − U, namely ΨG = m +

i V i (1 + eU)ṁ + (−m∗ + e ṁ ∗ ) . (1 − U)σe 1−U σe

(A.25)

A.3 Generalized geodetic excitation and digitization

| 307

If m is sampled with sampling time h, ΨG can be estimated by the two point digital linear filter Ψ̂ t+h/2 = a mt + b mt+h + c m∗t + d m∗t+h .

(A.26)

The coefficients a, b, c and d can be determined by equating the Fourier transform of the digitized function Ψ̂ to the one of ΨG : ΨG (σ) = [1 −

σ V σ (1 + eU)]m(σ) − (1 + e )m∗ (−σ) , (1 − U)σe 1−U σe

iσh/2 ̂ Ψ(σ)e = m(σ)(a + beiσh ) + m∗ (−σ) (c + deiσh ) .

(A.27)

̂ Equating Ψ(σ) to Ψ(σ) yields (a + beiσh )e−iσh/2 = 1 − (c + deiσh )e−iσh/2 = −

σ , σ̃ c

σ V (1 + e ) , 1−U σe

(A.28)

with σ̃ c = σe (1 − U)/(1 + eU), consistent with (A.16) at first order of the ellipticity. The coefficients a and b reflect the symmetric part of the excitation. Expressing the first equation of the system (A.28) at frequencies 0 and σc , we obtain a=i

1 − σc /σ̃ c − eihσc /2 , 2 sin(hσc /2)

b=1−a.

(A.29)

These expressions look like the coefficient (9.8) of the Wilson digital filter. By expressing the second equation of (A.28) at frequencies 0 and σe , we obtain the coefficients of the asymmetric part: c=i

V eihσe /2 − (1 + e) , 1 − U 2 sin(hσe /2)

d=−

V −c . 1−U

(A.30)

B Spherical harmonic coefficients of the geopotential and moments of inertia B.1 Spherical harmonic functions Given the longitude λ and the colatitude θ, the associated Legendre polynomials of degree l and order m Plm are defined by m/2 d

m

Pl (cos θ) , dt m where the Pl (x) are the Legendre polynomials given by Plm (cos θ) = (1 − cos θ2 )

(B.1)

1 dl (x2 − 1)l . (B.2) 2l l! dxl These polynomials are reported in Table B.1 up to degree 3. Notice that Pl (x) = Pl0 (x). Now, we define the spherical harmonic functions by Pl (x) =

m

|m] imλ

𝒴l (λ, θ) = Pl e

m = −l, . . . , 0, . . . , +l .

(B.3)

It can be shown that these functions constitute a basis of orthogonal functions. We introduce the normalized harmonic functions Ylm (cos θ) = Nl|m] Pl|m] (cos θ)eimλ

m = −l, . . . , 0, . . . , +l ,

(B.4a)

with Nlm = √

2l + 1 (l − m)! , 4π (l + m)!

(B.4b)

yielding a set of orthonormal functions: 2π

π

∫ ∫ (Ylm ) Ylm󸀠 sin θ dθ dλ = δll󸀠 δmm󸀠 . ∗

󸀠

(B.5)

λ=0 θ=0

Note that another normalization is used in space geodesy for the spherical harmonic coefficients of the geopotential, namely Klm = √(2 − δ0m )(2l + 1)

(l − m)! . (l + m)!

Table B.1: Legendre functions Plm (cos θ) of degree l and order m up to degree 3. Degree l=0 l=1

l=2 l=3

m=0 1 cos θ

3 cos2 θ−1 2 3 5 cos θ−3 cos θ 2

m=1 sin θ 3 sin θ cos θ 2

sin θ 15 cos2

θ−3

https://doi.org/10.1515/9783110298093-017

m=2

3(sin2 θ)

15 sin2 θ cos θ

m=3

15 sin3 θ

(B.6)

310 | B Spherical harmonic coefficients of the geopotential and moments of inertia Table B.2: Normalization coefficients Nlm (Eq. (B.4b)) and Klm (Eq. (B.6)) of degree 2. Order m

N2m

K2m

0

5 √ 4π

√5

1 2

5 √ 24π 5 √ 96π

√ 53

5 √ 12

The normalized Legendre polynomials are defined in the same vein: P̄ lm (cos θ) = Nlm Plm (cos θ) m = 0, . . . , l .

(B.7)

The coefficients Nl|m] of degree 2 are reported in Table B.2. The spherical harmonics functions Ylm form an orthonormal basis of the Hilbert space of square-integrable functions. On a sphere, any square-integrable function f (λ, θ) can thus be expanded as a linear combination of these: +∞

l

f (λ, θ) = ∑ ∑ a󸀠lm Ylm (λ, θ) ,

(B.8a)

l=0 m=−l

with

2π

a󸀠lm

π

= ∫ ∫ f (λ, θ)(Ylm ) sin θ dθ dλ .

(B.8b)

∗

λ=0 θ=0

Equivalently l

+∞

f (λ, θ) = ∑ ∑ (alm cos(mλ) + ã lm sin(mλ))P̄ lm (cos θ) , l=0 m=0

with

2π

(B.9a)

π

alm = 2 ∫ ∫ f (λ, θ)P̄ lm cos(mλ) sin θ dθ dλ

m>0,

λ=0 θ=0 2π

π

ã lm = 2 ∫ ∫ f (λ, θ)P̄ lm sin(mλ) sin θ dθ dλ

m>0,

(B.9b)

λ=0 θ=0

2π

π

al0 = ∫ ∫ f (λ, θ)P̄ l0 sin θ dθ dλ

m=0.

λ=0 θ=0

If alm and ã lm are referred to the non-normalized Plm in (B.9a), then the former expressions have to be multiplied by Nlm . We have the relations a󸀠lm =

alm − iã lm 2

m>0;

a󸀠l0 = al0

m=0;

a󸀠lm =

al,−m + iã l,−m 2

m m and m ≥ 1. The low degree Stokes coefficients of the geopotential are reported in Table B.3. The expansion (B.11) can be expressed in terms of spherical harmonic functions: U(r, λ, θ) =

l C̄ − iS̄lm m C̄ + iS̄lm −m GM +∞ l Re Yl (λ, θ) + lm Yl (λ, θ)) , ∑ ∑ ( ) ( lm r l=0 m=0 r 2 2

(B.13)

with the normalized coefficients C̄ lm = Clm /Nlm and S̄lm = Slm /Nlm . Equivalently Table B.3: Mean Stokes coefficients Clm and Slm up to degree 3. From (Sidorenkov, 2002, p. 73, table 2.4, [200]). Degree n

Order m

Clm

Slm

0 1 1 2 2 2 3 3

0 0 1 0 1 2 0 3

1 0 0

0 0 0 0

−108263 10−8 0.134 10−8 157 10−8 254 10−8 162 10−8

−0.314 10−8 −90 10−8 0 0

312 | B Spherical harmonic coefficients of the geopotential and moments of inertia

U(r, λ, θ) =

l

GM +∞ l Re ∑ ∑ ( ) Dlm Ylm (λ, θ) , r l=0 m=−l r

(B.14a)

with C̄ lm + iS̄lm 2 ̄ Dl0 = Clm C̄ − iS̄lm Dlm = lm 2 Dlm =

for m < 0 , (B.14b)

for m = 0 , for m > 0 .

B.3 Stokes coefficients of degree 2 and moment of inertia of the Earth In a terrestrial frame Oxyz the Earth’s inertia matrix can be expressed by I11 [ I = [ I12 [ I13

∫(y2 + z 2 ) dm I13 [ ] I23 ] = [ [ − ∫ xy dm I33 ] [ − ∫ xz dm

I12 I22 I23

− ∫ xy dm 2

2

∫(x + z ) dm − ∫ yz dm

− ∫ xz dm − ∫ yz dm

] ] . (B.15) ]

∫(x 2 + y2 ) dm ]

The degree 2 Stokes coefficients are given by Eqs. (B.12), where we have the combination of the moments of inertia of the Earth: C20 =

S21

C22

S22

2 M⊕ Re 2

∫ r 2 (3 cos2 θ − 1) dm

I +I 1 [−I33 + 11 22 ], 2 2 2 M⊕ Re M⊕ Re 1 1 2 = ∫ r cos θ sin θ cos λ dm = ∫ xz dm M⊕ R2e M⊕ R2e I = − 13 2 , M⊕ Re 1 1 = ∫ r 2 cos θ sin θ sin λ dm = ∫ yz dm 2 M⊕ Re M⊕ Re 2 I = − 23 2 , M⊕ Re 1 1 = ∫ r 2 sin2 θ cos(2λ) dm = ∫(x 2 − y2 ) dm 4 M⊕ Re 2 4 MRe 2 I −I = 22 112 , 4 MRe 1 1 = ∫ r 2 sin2 θ sin(2λ) dm = ∫ xy dm 2 4 MRe 2 MRe 2 I = − 12 2 . 2 MRe =

C21

1

1

2

∫(2z 2 − x2 − y2 ) dm =

(B.16)

B.3 Stokes coefficients of degree 2 and moment of inertia of the Earth

| 313

Conversely, to proceed to some theoretical development requires one to derive the change of the moment of inertia from variation of the degree 2 Stokes coefficients resulting from a given mass redistribution, like the pole tide. Evidently we have I13 = −M⊕ R2e C21 ,

(B.17)

I23 = −M⊕ R2e S21 , I12 =

−2M⊕ R2e S22

(B.18) .

(B.19)

The derivation of I11 , I22 and I33 requires a supplementary equation based upon the conservation of the trace of the inertia matrix. The trace, given by ∫ 2(x2 + y2 + z 2 ) dm = ∫ 2r 2 dm , is not only a frame invariant, but is constant for a mass redistribution produced by a degree 2 exciting potential [46, 175]. Under this condition, we have I11 + I22 + I33 = 0 .

(B.20)

Then, the system constituted by (B.20) and the terms C20 and C22 of Eq. (B.16) permit one to deduce the expressions I11 = M⊕ R2e (

C20 − 2 C22 ) , 3

(B.21)

I22 = M⊕ R2e (

C20 + 2 C22 ) , 3

(B.22)

2 I33 = − M⊕ R2e C20 . 3

(B.23)

Equations (C.17), as shown later, relate the moments of inertia Iij in the terrestrial frame with the principal moments of inertia A󸀠 , B󸀠 , C 󸀠 . Now, we denote by A, B, and C the mean values of the principal moments of inertia I11 , I22 , and I33 , respectively (the period of average, at least larger than 7 years is left undefined). According to this notation we have I 33 −

I 11 + I 22 A+B =C− . 2 2

So, from (B.16), the corresponding mean value of C20 is C 20 = −

A+B 2 M⊕ Re 2

C−

.

(B.24)

314 | B Spherical harmonic coefficients of the geopotential and moments of inertia Geopotential of degree 2 The geopotential of degree 2 is given by U2 =

2

GM⊕ Re 3 cos2 θ − 1 ) [C + (C21 cos λ + S21 sin λ)3 cos θ sin θ ( 20 r 2 r3 + (C22 cos 2λ + S22 sin 2λ)3 sin2 θ] .

(B.25)

Replacing spherical coordinates by Cartesian coordinates, U2 reads U2 =

2

GM⊕ Re 2z 2 − x2 − y2 ( ) [C20 + 3 (C21 xz + S21 yz) + 3C22 (x2 − y2 ) + 6 S22 xy] . 3 r 2 r (B.26)

Neglecting the Earth’s triaxiality, the mean contribution of the Earth’s bulge is U2 =

2

2

GM⊕ Re GM⊕ Re 2z 2 − x2 − y2 3 cos2 θ − 1 ( ) [C ] = ( ) [C ]. 20 20 r 2 r r 2 r3

(B.27)

C Earth figure C.1 Spherical harmonic expansion of the radius of the Earth According to (5.22), under the action of a constant centrifugal potential, the Earth acts to take the shape of an axi-symmetric flattened spheroid, of which the radius is defined by r = Re (1 − f cos2 θ)

(C.1)

in any point of colatitude θ with respect to the axis of symmetry. Here f is the geometric flattening, that is, the relative difference of equatorial and polar radius, (Re − Rp )/Re . The radius r can be decomposed into the spherical harmonic development of degree 2, 2 f r = Re (1 − ) − Re fP20 (cos θ) . 3 3

(C.2)

The mean radius of the Earth is defined by the constant term of this development, namely f r0 = Re (1 − ) = 6371.0083(3) km , 3

(C.3)

where the reference values of Re and f can be found in Table H.1. Actually r0 is also the geometric mean of the three semi-axes of the spheroid: r03 ≈ R2e Rp . By neglecting the second order term in f 2 , the figure of the terrestrial spheroid reads as a function of r0 , f , and colatitude θ: 2 r = r0 [1 − fP20 (cos θ)] . 3

(C.4)

C.2 Effect of a infinitesimal rotation on a spherical harmonic development of degree 2 Let us consider an axi-symmetric spheroid. In the frame of its axes of symmetry, namely Gx󸀠 y󸀠 z 󸀠 , its surface is expressed by the radius (C.4) r(θ) = r0 [1 + u20 P20 (cos θ󸀠 )],

2 u20 = − f . 3

(C.5)

This spheroidal shape is illustrated by the core–mantle boundary. Yet the practical reference frame Gxyz, like the TRF (associated with spherical coordinates λ, θ) https://doi.org/10.1515/9783110298093-018

316 | C Earth figure

Figure C.1: The system Gx 󸀠 y 󸀠 z 󸀠 constituted by the axes of symmetry of the spheroid is close to the terrestrial reference frame Gxyz.

is not perfectly aligned with Gx󸀠 y󸀠 z 󸀠 (spherical coordinates λ󸀠 , θ󸀠 ) (Figure C.1). The Gz axis is no more the figure axis Gz 󸀠 of the spheroid, conferring Eq. (C.5) to its surface. Then, what is the spherical harmonic development in Gxyz? The systems Gx󸀠 y󸀠 z 󸀠 and Gxyz differ by an infinitesimal rotation R = R(α1 )R(α2 )R(α3 ), associated with the coordinate transformation x󸀠 1 󸀠 󸀠 X = RX ⇐⇒ ( y ) = ( −α3 z󸀠 α2 2

x −α2 α1 ) ( y ) . 1 z

α3 1 −α1

(C.6)

The Legendre polynomial P20 (cos θ󸀠 ) = 3 cos2 θ −1 = 32 ( zr )2 − 21 can be expressed in Gxyz by taking z 󸀠 = α2 x − α1 y + z deduced from (C.6): P20 (cos θ󸀠 ) =

󸀠

󸀠

3 2α2 xz − 2α1 yz + z 2 1 ( )− 2 2 r2

= P20 (cos θ) + P21 (cos θ)[α2 cos λ − α1 sin λ] .

(C.7)

Finally the spherical harmonic decomposition in Gxyz is r = r0 [1 + u20 P20 (cos θ) + (u21 cos λ + ũ 21 sin λ)P21 (cos θ)] ,

u21 = α2 u20 ,

(C.8)

ũ 21 = −α1 u20 . By applying the opposite rotation, namely t R, to the unit vector ẑ󸀠 = (0, 0, 1) of the figure axis, we deduce that its coordinates in Gxyz are (α2 , −α1 , 1).

C.3 Inertia matrix transformation from principal axis frame to TRF | 317

C.3 Inertia matrix transformation from principal axis frame to TRF Now Gx󸀠 y󸀠 z 󸀠 means the system of principal inertia axes, in which the inertia matrix is diagonal, taking the form A󸀠 I =( 0 0 󸀠

0 B󸀠 0

0 0 ) . C󸀠

(C.9)

In the terrestrial frame Gxyz the inertia matrix can be derived from this diagonal matrix by using the formula I = P −1 I P

(C.10)

󸀠

where P is the transformation matrix from Gx󸀠 y󸀠 z 󸀠 to Gxyz, of which the columns are composed of the components of the unit vector x,̂ y,̂ and ẑ expressed in Gx󸀠 y󸀠 z 󸀠 . Corresponding coordinate transformation is X = P −1 X 󸀠 or X 󸀠 = PX. For a triaxial Earth, P is given by the rotation transformation P = R1 (α1 )R2 (α2 )R3 (λA )

(C.11)

where λA ≈ −15° and α1 and α2 are small angles smaller than 1󸀠󸀠 , that allow one to bring the geographic pole in coincidence with the pole of figure, associated with the principal inertia moment C 󸀠 . We obtain 1 P=( 0 α2

0 1 −α1

−α2 cos λA α1 ) ( − sin λA 1 0

cos λA =( − sin λA α1 sin λA + α2 cos λA

sin λA cos λA 0

0 0 ) 1

sin λA cos λA −α1 cos λA + α2 sin λA

−α2 α1 ) . 1

(C.12)

In Gxyz the coordinates of the inertia pole corresponding to the inertia axis endowed with the largest moment (C 󸀠 ) are given by xI 0 −1 ( yI ) = P ( 0 ) zI 1

(C.13)

where P −1 can be easily deduced from the opposite transformation of (C.11) or directly from the transpose of matrix (C.12): cos λA P −1 = ( sin λA −α2

− sin λA cos λA α1

α1 sin λA + α2 cos λA −α1 cos λA + α2 sin λA ) . 1

(C.14)

318 | C Earth figure So the coordinates of the inertia pole in Gxyz are xI = α1 sin λA + α2 cos λA ,

(C.15)

yI = −α1 cos λA + α2 sin λA , zI = 1 .

Accounting for Eqs. (C.12) and (C.14) in the matrix product (C.10) and casting aside second order terms with respect to αi (on the order of 10−11 A󸀠 ), we find as a result cos λA I = ( sin λA −α2

− sin λA cos λA α1

α1 sin λA + α2 cos λA −α1 cos λA + α2 sin λA ) 1

cos λA A󸀠 ×( − sin λA B󸀠 (α1 sin λA + α2 cos λA )C 󸀠

sin λA A󸀠 cos λA B󸀠 (−α1 cos λA + α2 sin λA )C 󸀠

−α2 A󸀠 α1 B󸀠 ) C󸀠

and finally I=(

A󸀠 cos2 λA + B󸀠 sin2 λA

I21

cos λA sin λA (A󸀠 − B󸀠 )

A󸀠 sin2 λA + B󸀠 cos2 λA

α1 sin λA (C 󸀠 − B󸀠 ) + α2 cos λA (C 󸀠 − A󸀠 )

−α1 cos λA (C 󸀠 − B󸀠 ) + α2 sin λA (C 󸀠 − A󸀠 )

I31 I32 ) , C󸀠

(C.16) where we recover the symmetric property I13 = I31 , I23 = I32 , I12 = I21 of any moment of inertia. From the matrix above we have I13 = α1 sin λA (C 󸀠 − B󸀠 ) + α2 cos λA (C 󸀠 − A󸀠 ) ,

I23 = −α1 cos λA (C 󸀠 − B󸀠 ) + α2 sin λA (C 󸀠 − A󸀠 ) ,

I11 + I22 = A󸀠 + B󸀠 ,

I11 − I22 = (A󸀠 − B󸀠 )(cos2 λA − sin2 λA ) = (A󸀠 − B󸀠 ) cos 2λA ,

(C.17)

I12 = cos λA sin λA (A󸀠 − B󸀠 ) ,

I33 = C 󸀠 . We find

I13 + iI23 = −i(C 󸀠 − B󸀠 )α1 eiλA + (C 󸀠 − A󸀠 )α2 eiλA

A󸀠 + B󸀠 A󸀠 + B󸀠 )α1 eiλA + (C 󸀠 − )α2 eiλA 2 2 A󸀠 − B󸀠 A󸀠 − B󸀠 −i α1 eiλA − α2 eiλA 2 2 A󸀠 + B󸀠 A󸀠 − B󸀠 = (C 󸀠 − )(α2 − iα1 )eiλA − (α2 + iα1 )eiλA . 2 2

= −i(C 󸀠 −

(C.18)

But, according to (C.15) xI + iyI = (α2 − iα1 )eiλA ,

(C.19)

C.3 Inertia matrix transformation from principal axis frame to TRF | 319

and from Eqs. (C.17) I11 − I22 + 2iI12 = (A󸀠 − B󸀠 )(cos(2λA ) + i sin(2λA )) = (A󸀠 − B󸀠 )e2iλA , C󸀠 −

I +I A󸀠 + B󸀠 = I33 − 11 22 . 2 2

(C.20)

Hence (C.18) reads

I11 + I22 I − I + 2iI12 )(xI + iyI ) − 11 22 (α2 + iα1 )e−iλA 2 2 I − I + 2iI12 I +I (xI + iyI )∗ . = (I33 − 11 22 ) (xI + iyI ) − 11 22 2 2

I13 + iI23 = (I33 −

(C.21)

According to (B.16) degree 2 Stokes coefficients are linearly related to the moments of inertia by I13 = −M R2e C21 ,

I23 = −M R2e S21 ,

I12 = −2M R2e S22 ,

I11 − I22 = −4 MRe 2 C22 ,

I33 −

(C.22)

I11 + I22 = −M R2e C20 . 2

Thus (C.21) can be expressed in term of the Stokes coefficients: C21 + iS21 = C20 (xI + iyI ) − 2(C22 + iS22 )(xI − iyI )

= (C20 − 2C22 )xI − 2S22 yI + i[(C20 + 2C22 )yI − 2S22 xI ] .

(C.23)

This is a very useful expression for reconstituting the degree 2 Stokes coefficients at multi-annual periods by assimilating the pole of figure to the mean rotation pole. In IERS conventions one makes use of the normalized coefficients C̄ lm = Clm /Nlm with Nlm = √

(l − m)!(2l + 1)(2 − δ0m ) . (l + m)!

We have N20 = √5, N21 = √ 35 , N22 = 21 √ 35 , leading to C̄ 21 + iS̄21 = √3C̄ 20 (xI + iyI ) − (C̄ 22 + iS̄22 ) (xI − iyI ) .

(C.24)

The off-diagonal terms read also c13 + ic23 = I13 + iI23 = (I33 − I11 − iI12 )xI + i(I33 − I22 + iI12 )yI .

(C.25)

According to (C.16) I12 is a small fraction of the triaxiality, thus it can be neglected, and the variation of diagonal term as well. This leads to c13 + ic23 = (C − A)xI + i(C − B)yI

(C.26)

where A, B, C are the mean diagonal moments of inertia in the terrestrial frame.

320 | C Earth figure Case of a biaxial homogeneous Earth Taking the approximation of a biaxial and homogeneous Earth, the axes of G󸀠 x󸀠 y󸀠 z 󸀠 can be assimilated to the symmetry axes of the Earth, where the radius of the surface is given by (C.5). Especially Gz 󸀠 coincides with the direction of the inertia pole. Thus, in virtue of (C.8), the coordinates of the inertia pole are (α2 = u21 /u20 , −α1 = ũ 21 /u20 , 1), and Eq. (C.26), where triaxiality had been neglected, becomes c13 + ic23 = (C − A)

u21 + iũ 21 3(C − A) =− (u21 + iũ 21 ) . u20 2f

(C.27)

Confusing geometric flattening f with the dynamical one, that is (C − A)/A, we obtain 3 c13 + ic23 = − A(u21 + iũ 21 ) . 2

(C.28)

D Tidal perturbation and Love number resonance in the diurnal band D.1 Tesseral tidal potential The material of this appendix is inspired by [60], Section 5.5. Consider a point of the Earth at distance r from the geocenter of latitude ϕ and longitude λ. At this place the tesseral part of the tidal potential generated by a celestial body of mass M—located in the true equatorial frame by its right ascension α, declination δ, and distance d from the geocenter—is given by W=

GM 1 2 1 d P2 (sin δ)r 2 P21 (sin ϕ) cos(λ − α) . d5 3

(D.1)

Here astronomical and geographic latitudes are merged, as well the node of prime meridian and the Terrestrial Intermediate Origin (TIO, see Chapter 3). Introducing the terrestrial Cartesian coordinates (x, y, z) and (dx , dy , dz ) of the location and of the celestial body, respectively, we can easily derive W=

3GM zdz Re[(dx + idy )(x − iy)] . d5

(D.2)

Then, noting that r 2 𝒴2−1 = 3(xz − iyz), where 𝒴2−1 is the degree 2 non-normalized spherical harmonic function defined by (B.3), we obtain W=

GM d r 2 Re[(dx + idy )𝒴2−1 ] . d5 z

(D.3)

For describing the Earth’s rotation the gravimetric tidal potential W is put into the 2 2 form of the pole tide potential ΔU (r) = − Ω3r Re[m(t)𝒴2−1 ]: W =−

Ω2 r 2 ̃ 𝒴 −1 ] Re[ϕ(t) 2 3

̃ =− with ϕ(t)

3GM d (d + idy ) . Ω2 d5 z x

(D.4)

Then, tidal theory allows one to establish the expansion 1 ̃ = 3gN2 ∑ ξ e−i(θσ (t)−π/2) , ϕ(t) 2 2 Ω Re σ≥0 σ

(D.5)

where θσ is the tidal argument and σ the corresponding frequency and N21 = √5/(24π) according to (B.4b). In Table D.1 we report the coefficients of the tesseral lunar–solar tides that are considered in this book. For a given tidal component, the numerical application yields ϕ̃ σ (t) = 3.51 10−5 [m−1 ] ξσ [m]e−i(θσ (t)−π/2) .

https://doi.org/10.1515/9783110298093-019

(D.6)

322 | D Tidal perturbation and Love number resonance in the diurnal band Table D.1: Coefficients of the lunar–solar tides used in this book. σ (cpd) Q1 O1 P1 K1 J1

θσ (t)

ξσ (m)

GMST + π − l − 2F − 2? GMST + π − 2F − 2? GMST + π − 2F + 2D − 2? GMST + π GMST + π + l

0.9295 1.00273

−0.05021 −0.26223 −0.12199 0.36864 0.02062

D.2 Resonance of the geopotential Love number k2 In the diurnal band the tilt of the fluid core modifies the mass distribution of the Earth, and in turn the surface gravity field. As for nutation, it can be established theoretically that the gravity field or the geopotential presents a resonance at the frequency of the free core nutation σf in the TRF. Another perturbation, of much lesser amplitude (100 times less), also occurs because of the free inner core resonance at σi in the TRF. For diurnal frequencies a supplementary effect comes from the polar motion resonance at the period of about 380 days. From the IERS Conventions 2010 [168, Table 6.4, Eq. 6.9 and 6.10], the perturbation of the diurnal tide on the geopotential can be described through the “diurnal” Love number k2d (σ) = 0.29954 − i 0.1412 10−2 −

Lf Lc Li − , − d σ − σ̃ c σ − σ̃ f σ − σ̃ i

(D.7a)

with the following quantities in cycles per sidereal day Lc =

−0.77896 10−3 − i 0.3711 10−4

σ̃ c =

0.0026010 + i 0.0001361 ,

Lf =

0.90963 10

−5

σ̃ f =

−1.023181 + i 2.5 10−5 ,

Li =

−0.11416 10−5 + i 0.5325 10−7

σ̃ i =

−0.999026 + i 0.000780 .

−4

− i 0.2963 10

(D.7b)

Here σ̃ cd ≈ 1/383(1−i/20) cpd corresponds to the complex frequency of the polar motion resonance in the diurnal band, as modeled in Chapter 7.

D.3 Resonance of the loading Love number k2󸀠 As for k2 , the loading Love number is frequency dependent, and it exhibits the triple resonance (IERS Conventions [168], Table 6.4) k2d (σ) = −0.30808 − i 0.1412 10−2 − 󸀠

L󸀠c

σ − σ̃ cd

−

L󸀠f

σ − σ̃ f

−

L󸀠i , σ − σ̃ i

(D.8a)

D.3 Resonance of the loading Love number k2󸀠

| 323

with the numerators L󸀠c = 8.1874 10−4 , L󸀠f = 1.4116 10−4 , L󸀠i = 3.4618 10−7 , in cycles per sidereal day.

(D.8b)

E Matter term, pressure torque and loading effect E.1 Matter term expressed as a function of the tesseral component of the surface pressure The integrand of the pressure or matter term (8.31) can be expressed as a function of the complex spherical harmonic Y21 defined by (B.4a): χp = −

π

r04

3N21 (C − A)g

2π

∫ ∫ Ps Y21 sin θ dθ dλ

(E.1)

θ=0 λ=0

5 . The surface pressure Ps of the fluid layer is given by the spherical with N21 = √ 24π harmonics decomposition (8.9) or l

Ps = ∑ ∑ p󸀠lm Ylm (λ, θ) , l=0 m=−l

(E.2a)

with p󸀠lm = p󸀠lm =

plm − ip̃ lm 2Nl|m|

plm + ip̃ lm 2Nl|m|

m≥0, m≤0.

(E.2b)

Substituting this development in (E.1) and applying the orthonormality relations (B.5) between Y21 and Ylm , this yields χp = −

r04

3N21 (C − A)g

p󸀠2,−1 = −

r04 4π (p + ip̃ 21 ) . 5 (C − A)g 21

(E.3)

This treatment can also encompass the case of inverted barometer oceans by considering PS = 0 over the oceans and computing the corresponding spherical harmonics development.

E.2 Spherical harmonic expansion of the pressure torque Computations are easier by considering the expansion in spherical harmonic functions Ylm (λ, θ) = Nl|m| Pl|m| (cos θ)eimλ . For the surface of the Earth we have lmax

l

r = r0 [1 + ∑ ∑ u󸀠lm Ylm (λ, θ)] , l=1 m=−l

https://doi.org/10.1515/9783110298093-020

(E.4)

326 | E Matter term, pressure torque and loading effect where the coefficients u󸀠lm are deduced from the ulm and ũ lm of Eq. (8.8) according to u󸀠lm =

ulm − iũ lm 2Nl|m|

for m > 0 ;

u󸀠l0 = ul0 /Nl0 ;

ulm + iũ lm

u󸀠lm =

2Nl|m|

for m < 0 ,

or ulm − sign(m) i ũ lm

u󸀠lm = (1 + δm0 )

2Nl|m|

.

(E.5) (E.6)

Similarly the surface pressure reads qmax

q

Ps = ∑ ∑ p󸀠qr Yqr (λ, θ) ,

(E.7a)

q=0 r=−q

where the p󸀠qr are deduced from (8.9) according to p󸀠qr = (1 + δr0 )

pqr − sign(r) ip̃ qr 2Nq|r|

.

(E.7b)

E.2.1 Equatorial torque According to (8.13) the equatorial pressure torque reads Γp = − ∫ Ps r 2 (cos θ S

𝜕r 𝜕r − i sin θ )eiλ dλ dθ . 𝜕λ 𝜕θ

(E.8)

Derivation of (E.4) with respect to λ and θ gives l 𝜕r = r0 ∑ ∑ u󸀠lm imYlm (λ, θ), 𝜕λ l=1 m=−l l 𝜕r cos θ m = r0 ∑ ∑ u󸀠lm (|m| Y − Nl|m| Pl|m|+1 eimλ ) , 𝜕θ sin θ l l=1 m=−l

(E.9) (E.10)

where the time derivation of the Plm (cos θ) is done by using Eq. (B.1), giving dPl|m| dθ

= |m|

cos θ |m| P − Pl|m|+1 . sin θ l

(E.11)

Putting Eqs. (E.9) and (E.10) in (E.8), the pressure torque receives the development l

Γp = − ∑ ∑ ir0 u󸀠lm ∫ Ps r 2 (mYlm cos θ − |m| cos θYlm + sin θNl|m| Pl|m|+1 eimλ )eiλ dλ dθ . l=1 m=−l

S

(E.12)

E.2 Spherical harmonic expansion of the pressure torque

| 327

As r 2 = r02 (1+O(ulm )), neglecting second and higher order terms in ulm , and introducing the expansion of the pressure (E.7a), we have qmax

q

l

Γp = − ∑ ∑ ∑ ∑ ir03 p󸀠qr u󸀠lm ∫ Yqr (cos θYlm (m − |m|) + sin θNl|m| Pl|m|+1 eimλ )eiλ dλ dθ . q=0 r=−q l=1 m=−l

S

The case m ≥ 0 is trivial. Applying the orthonormality relations (B.5) we obtain Nlm

l

Γ(m≥0) = −ir03 ∑ ∑ p

m+1 l=1 m≥0 Nl

u󸀠lm p󸀠l,−(m+1) .

(E.13)

(E.14)

The case m < 0 = −n requires more algebra: qmax

q

l

Γ(m