Advanced Studies in Ocean Physics (Springer Oceanography) [1st ed. 2021] 3030722686, 9783030722685

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Springer Oceanography

Anatoly Kistovich Konstantin Pokazeev Tatiana Chaplina

Advanced Studies in Ocean Physics

Springer Oceanography

The Springer Oceanography series seeks to publish a broad portfolio of scientific books, aiming at researchers, students, and everyone interested in marine sciences. The series includes peer-reviewed monographs, edited volumes, textbooks, and conference proceedings. It covers the entire area of oceanography including, but not limited to, Coastal Sciences, Biological/Chemical/Geological/Physical Oceanography, Paleoceanography, and related subjects.

More information about this series at http://www.springer.com/series/10175

Anatoly Kistovich · Konstantin Pokazeev · Tatiana Chaplina

Advanced Studies in Ocean Physics

Anatoly Kistovich Institute for Problems in Mechanics Russian Academy of Sciences (RAS) Moscow, Russia

Konstantin Pokazeev Faculty of Physics M. V. Lomonosov Moscow State University Moscow, Russia

Tatiana Chaplina Institute for Problems in Mechanics Russian Academy of Sciences (RAS) Moscow, Russia

ISSN 2365-7677 ISSN 2365-7685 (electronic) Springer Oceanography ISBN 978-3-030-72268-5 ISBN 978-3-030-72269-2 (eBook) https://doi.org/10.1007/978-3-030-72269-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Introduction

There are a huge number of water reservoirs on the Earth. Their diversity extends from small rivers and ponds to vast seas and boundless oceans. These water spaces are never at rest, something always happens in them: either the movement of water masses in the depths, or the undulation of the surface, or the mixing of different water flows, etc. Despite all the variety of currents and differences in the scale of their manifestation, there is something that unites them. These are the physical laws that govern the movement of water. The mathematical formulation of the basic laws governing the evolution of flows forms a system of fundamental hydrodynamic equations. The explicit appearance of the system depends on the selected model and the approximations used in the analysis. It is usually assumed that a continuous medium in a non-equilibrium state can be conditionally divided into a collection of small particles, each of which is in an equilibrium state. The local equilibrium approximation makes it possible to characterize a small particle with a set of thermodynamic values, which include specific entropy and internal energy, pressure, temperature, density, and impurity concentrations. Estimating the error of such approximation is an extremely difficult task. For this reason, in the case of small gradients of the thermodynamic parameters, the approximation of the local thermodynamics equilibrium is used on an intuitive base. The rejection of the mentioned approximation leads to a complex problem of describing the medium based on a single thermodynamic function of locally nonequilibrium processes, namely specific entropy. With this approach, it is impossible to move from a statistical description of phenomena to a hydrodynamic one. At the same time, long-term studies of the World Ocean indicate a good correspondence of the local equilibrium model to the observed physical processes, which makes it possible to analyze flows at the hydrodynamic level. The basic laws of mechanics of a moving continuous medium are described by the equations of mass transfer, impurity diffusion, and momentum and energy transfer.

v

vi

Introduction

∂ρ ∂Sn + ∇ · (ρ v) = m , + ∇ · (Sn v + In ) = an ∂t ∂t ∂(ρ vi ) . + ∇j ij = ρ g i + 2ρεijk vj k + f i ∂t   ∂E ∂w i + ∇i (Evi ) + ∇i qi + pvi − σij vj + I =e ∂t ∂Sn n

(1)

Here ρ is the medium density, p is the pressure, v is the velocity field, Sn and In are the mass concentration and the density of the diffusion flux of the n-th impurity ij i j ij ij flux density tensor, correspondingly, ρvi v +pδ −ij σ kisthe  tensor of impulse   i j  = ij j σ = η ∂ v x + ∂ v x − (2 3) δ ∂ v xk +ζ δ ij ∂ vk xk is the symmetric tensor of viscous stresses, δ ij is the fundamental metric tensor, η, ζ are the first and second dynamic correspondingly,  is the angular speed of the Earth rotation,    viscosities E = ρ v2 2+ ε + U is the total energy density, ε is the specific internal energy, w = ε + p ρ is the specific enthalpy, U is the specific gravitational potential, g = −∇U is the gravitational acceleration, q is the heat flux density, m, an , f, e are the densities of the mass, n-th impurity, force, and energy correspondingly. Thermodynamic relations allow us to express density, specific internal energy, specific entropy and other process parameters through temperature T , pressure p, mass concentrations of impurities Sn and velocity field v . In generally accepted models of liquid media, the variables T , p, Sn and v play the role of basic parameters. The combination of equations of the system (1) allows to replace the equation of energy transfer for the equation of evolution of the temperature field.  ρ cp

dT dp − dt dt



∂w 1 = −∇i qi + σij eij + Ini ∇i + Q. 2 ∂Sn

(2)

  Here d d t = ∂ ∂ t + v · ∇, cp is the pressure,  is the  capacityat constant   heat adiabatic temperature gradient, eij = ∂ vi ∂xj + ∂ vj ∂xi 2 is the strain velocity tensor, Q is the density of heat sources. In the system (1), the number of variables is one more than the number of equations. In order for this system to be fully defined, it is necessary to add the equation of state describing the dependence of density on the main parameters of the medium. ρ = ρ(T , p, Sn ).

(3)

Since the medium in most cases is bounded by either solid boundaries or a free surface, the equations of motion (1) must also be supplemented with boundary conditions. On the rigid impenetrable boundaries (bottom of the ocean, coasts, laboratory basins walls), the conditions of liquid non-penetration and absence of the impurities fluxes are valid

Introduction

vii

v · n| = In · n| = 0.

(4)

Here n is the external normal to the boundary surface . In viscous liquid, the condition for the velocity filed is reformed into the adhesion condition v| = 0.

(5)

If the boundary is thermally isolated, then the condition for the heat flux q is valid q · n| = 0.

(6)

In general case for thermally conductive and impurity penetrative rigid boundaries, the boundary conditions have the forms ∂T ∂Sn + γ T T = ϕT , κ n + γSn Sn = ϕSn , (7) ∂n ∂n where χ and κn are the diffusive coefficients of temperature and nth impurity correspondingly, γT , γSn are the coefficients of temperature transfer and mass concentrations transfers, ϕT and ϕSn are the known values of the temperature flux density and impurities concentrations fluxes densities on the boundary. The boundary conditions on the free surface ζ(r, t) follow from the integral formulations of the mass, impurities, momentum, and energy conservation laws and may be partitioned into three groups, namely kinematic, dynamic, and energetic. The kinematic boundary conditions describe the motion of the free surface and the transfer processes on it χ

∂ζ (8) + v · ∇ζ = b|∇ζ| , In · ∇ζ = −bSn |∇ζ|, ∂t where b is the flux of pure water due to evaporation, precipitation, ice formation, and its melting when the external impurity fluxes are absent. The dynamic boundary conditions formulate the condition of the mutual compensation of the all forces acting on the surface        ij ij p1 − p2 + α 1 R1 + 1 R2 ∇i ζ − σ1 − σ2 ∇j ζ − ∇i α = 0,

ij

ij

(9)

where p1 , p2 and σ1 , σ2 are pressures and viscous tension tensors in the media separated by the free surface, α is surface tension coefficient, R1 , R2 are the principal radii of curvatures of the free surface.

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The energetic boundary condition describes the external heat influx into medium q · ∇ζ = Q|∇ζ|,

(10)

where Q is the total heat flux density on the boundary due to different heat sources. It is evident that the concrete set of the boundary conditions used depends on the problem under consideration. For this reason in the consequent sections of the presented guide, only the necessary part of the mentioned boundary conditions will be used, which is connected to the studied physical phenomenon. Presented here the system of the governing equations and boundary conditions is very cumbersome, because it describes the entire complex of processes occurring in the ocean environment, both significant and insignificant under certain conditions. For this reason, when analyzing equations of motion, approximate models are widely used, in which movements with low energy are not taken into account, which leads to simplification of the fundamental equations. The explicit form of an approximate mathematical model of a physical process depends significantly on the specific problem under study. Therefore, the wording of the necessary approximation used in describing a particular phenomenon is given in the corresponding section of this guide. Nevertheless, there are frequently used approximations that it is convenient to describe here to avoid superfluous repetitions in the sequel. The first approximation is the negligibility of the density variations in the inertial and Coriolis forces that are the parts of momentum transfer equation in the system (1). Second, the spatial variations of the velocity field are so small that in the energy transfer equation does not take into account the viscous transfer of mechanical work to heat. In the laboratory and natural conditions in the liquid particles displacement even  on 1000 m distances, the contribution of the term d p d t in Eq. (2) is in many times small than the influence of the term d T d t. In the most physical processes, the kinematic coefficients of the medium will be considered as constant values with the exception of the specific phenomena concerning to convective motions. In common, the announced approximations will be called further for brevity as Oberbeck–Boussinesq approximation [1, 2]. Despite the fact that the described approximations simplify the model of motion of a non-uniform fluid, the differential equations of multicomponent media remain nonstationary, nonlinear, and singularly excited. In most of the problems discussed below, the methods of perturbation theory are not applicable to them, and the significant contribution of different types of transfer phenomena to the dynamics of the described processes does not allow their preliminary regularization by standard methods. In this regard, when studying the nonlinear dynamics of elements of homogeneous or stratified flows, it is necessary to supplement the traditional fluid mechanics with more general mathematical methods. In most problems of physics, it is extremely rare to find an exact solution. This can be caused by nonlinear effects affecting the system behavior, various spatial

Introduction

ix

heterogeneities of the processes occurring, or complex boundary conditions. As a result, we have to resort to various approximate research methods, the essence of which depends on the degree of nonlinearity, inhomogeneity, etc., manifested in the behavior of complex dynamical systems. When creating a mathematical model of the process, evolutionary equations are used that describe various changes that occur with the system under study in space and time. In this case, the physical characteristics and variables of the mathematical model are included in the equations in dimensional form. If it is possible to get an exact solution to the equations, then the dimension of the variables does not complicate the analysis of the system in any way. Otherwise, it is difficult to determine the relative value of the contribution of a particular mechanism of the physical process, in order to use developed applied methods of solution. Sometimes, even in cases where the solution of the system is achievable in an explicit form, it is useful to get rid of dimensional variables, while achieving a better understanding of the behavior of the dynamic system. As a formal mathematical object, a dimensional variable is an extremely inconvenient characteristic, since it allows comparison only with those quantities that have the same dimension. If there are values of different dimensions, it is impossible to make its quantitative comparison. For this reason, it makes sense to transform the dimensional evolutionary equations to a dimensionless form, renormalizing by the adequate manner the variables of the mathematical model under study. Since there is no unified approach to the problem of reducing equations to a dimensionless form in a general situation, this problem will be considered every time taking into account the specifics of a physical phenomenon under consideration. The work was supported by the Ministry of Education and Science, project No. 2020-1902-01-258.

References 1.

2.

Oberbeck A (1879) Über die Wärmeleitung der Flüssigkeiten bei der Berücksichtigung der Strömungen infolge von Temperaturdifferenzen. Annalen der Physik und Chemie B.7:271– 292 Boussinesq J (1903) Théorie analytique de la chaleur. Gauthier-Villars, T. 2, Paris, p 670

Contents

1 Linear and Nonlinear Surface Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Mathematical Models of Wave Phenomena on the Surface . . . . . . . . 1.2 Infinitesimal Waves and Linear Models . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Transfer By Waves of Energy and Momentum . . . . . . . . . . . . . . . . . . 1.4 Weakly Nonlinear Waves of Finite Amplitude . . . . . . . . . . . . . . . . . . 1.5 Soliton and Soliton-Like Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Alternative Approach to the Description of Surface Waves . . . . . . . . 1.7 The Problem of Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 6 13 21 25 33 40 63 76

2 Applications of the Surface Wave’s Theory to Description of Some Natural Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Statement of the Problem and its Mathematical Model . . . . . . . 2.2 The Dispersion Equation of Internal Waves at the Interface . . . . . . . 2.3 Characteristic Features of the Waves at the Interface . . . . . . . . . . . . . 2.4 The Statement of the Problem of Waves on the Flow . . . . . . . . . . . . . 2.5 The Dispersion Equation of Waves on the Flow . . . . . . . . . . . . . . . . . 2.6 Reflection of Waves from Flow Inhomogeneities . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 82 85 87 90 92 94

3 Surface Contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Basic Equations and Boundary Conditions in the Presence of Surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Dispersion Equation and Features of the Wave Motion in the Presence of Surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Effect of the Surfactants on the Surface Waves . . . . . . . . . . . . . . 3.4 Oil Spills in Nature and Methods of Their Elimination . . . . . . . . . . . 3.5 Experimental Studies of the Physical Properties of Various Sorbents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 96 97 104 108 112 127

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Contents

4 Stratification of the Ocean Environment . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Model for Stable Stratification of the Marine Environment . . . . . . . . 4.2 Thermodynamic Characteristics of the Medium . . . . . . . . . . . . . . . . . 4.3 Models of Natural Stratified Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Simplified Models of Natural Stratified Media Used in Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129 129 132 136

5 Internal Gravity Waves in a Stratified Medium . . . . . . . . . . . . . . . . . . . . 5.1 Mathematical Model of Oscillations of a Stratified Medium . . . . . . 5.2 The Dispersion Equation of Internal Gravity Waves . . . . . . . . . . . . . 5.3 Transfer of Momentum and Energy by Internal Gravity Waves and Their Specific Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Some Features of the Generation of Internal Gravity Waves . . . . . . . 5.5 Separation of Internal Gravity and Surface Waves . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 146 151

6 Convective Phenomena in the Sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Model of Heat and Mass Transfer in the Marine Environment . . . . . 6.2 Convection Criteria from Localized Sources . . . . . . . . . . . . . . . . . . . . 6.3 The Problem of Convective Instability of the Marine Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Physical Approach to the Problem of Convection Onset . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 143

153 159 163 167 169 170 175 178 181 193

7 Interaction of Surface Waves With Regions of Near-surface Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Statement of the Problem of the Propagation of Waves on a Surface with Varying Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Spatial Structure of Gravitational-capillary Waves in the Presence of Convective Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Experimental Characteristics of Scattering of Surface Waves by Convective Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

224 228

8 Anomalous Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The Concept of Abnormal Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 History of Observation of Anomalous Waves . . . . . . . . . . . . . . . . . . . 8.3 Researches and Various Models of Anomalous Waves . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229 230 231 235 255

195 196 207

Chapter 1

Linear and Nonlinear Surface Waves

This Chapter is devoted to the problem of describing waves observed on the surface of liquid media (in the vast majority of cases—on the surface of water), located in an external gravitational field. Let’s define the issues that will be discussed here. The object of research is waves propagating on the surface of a liquid of constant density. In the majority of the physical situations used an inviscid model of the environment with (or without) regard of effects of surface tension, the value of which, in most cases, determined by the constant tension coefficient. The characteristic features of the influence of viscosity and variations in surface tension (due to uneven heating of the liquid surface) are discussed separately in the relevant sections. Since this Chapter does not address the problems of the geostrophic flows, the action of the gravitational field is characterized by a constant in space gravitational acceleration vector g. The liquid is considered incompressible, which leads to the exclusion of acoustic phenomena from the analysis. The half-space above the liquid is filled with air, the pressure in which is considered constant. An attempt to describe a surface disturbance based on a single regular approach to the problem, suitable for any type of disturbance, usually fails. This is because the nature of wave processes developing on the sea surface depends on a large number of factors, the degree of influence of which changes during the transition from one natural situation to another. For this reason, it is necessary to study different types of sea waves separately, based on different approaches and mathematical tools used. When studying surface phenomena, it is natural to move from simple surface perturbations to more complex types of disturbance [1, 2]. In order not to complicate the problem of describing surface waves and not to introduce additional types of motion that are not directly related to the studied phenomenon (such as boundary layers, internal gravitational waves, etc.), we will consider sea water an inviscid medium of constant density. The description of the characteristics of the studied phenomenon will be carried out in a Cartesian coordinate system (x, y, z), the plane (x, y) of which is compatible with the undisturbed surface of the sea, and the z axis will be directed to the atmosphere opposite to

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Kistovich et al., Advanced Studies in Ocean Physics, Springer Oceanography, https://doi.org/10.1007/978-3-030-72269-2_1

1

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1 Linear and Nonlinear Surface Waves

acceleration vector g. The deviation of the sea surface from its undisturbed state is described by the function ζ(x, y, t). To describe surface waves, it is necessary to select hydrodynamic equations and boundary conditions that correspond to the problem. Hydrodynamic equations are necessary for the reason that they describe the dynamics of the physical characteristics of sea waves—the velocity field of liquid elements of the medium and the pressure in the water. Boundary conditions are necessary for us because, first, the actual surface waves at the level of visual perception are presented to an outside observer as fluctuations of the water-air interface, and secondly, real natural reservoirs have a bottom that separates seawater from the underlying solid medium. At these borders, it is necessary to link the characteristics of water movement with the characteristics of environments that restrict the marine environment. The system of equations describing the motion of the water medium primarily includes the Euler equation 1 v t + (v∇)v = − ∇ p + g, ρ where v(r, t) is the velocity field, p(r, t) and ρ are pressure in the liquid and its density correspondingly. The used approximation of the constancy of the density of seawater entails the following simplifications of the model of the actual water environment. The absence of impurities, isothermality of the medium, disregard for the effects of compressibility (there are initially no sound waves in the medium, and they are not generated by surface waves during their propagation) and the absence of mass sources simplify the state equation of water to the form ρ = const. Substitution of this relation into mass conservation low ρt σ + ∇ · (ρv) = 0 permits to get its reduced form ∇ · v = 0. Now it is necessary to formulate boundary conditions on the free surface of the sea and on its bottom. These conditions should describe the kinematics and dynamics of liquid elements on the interface surfaces with air and solid bottom. Let’s first consider the free surface, that is, the water—air interface. The dynamic boundary condition consists in the fulfillment of Newton’s third law, that is, in balancing all the forces acting on the liquid element located on the boundary. Let’s select a small element d S of the free surface around the point with coordinates (x, y, ζ(x, y, t)). It is affected by a force pa d S from the atmosphere, where pa is the atmospheric pressure, which in the accepted model is considered constant in space and time. From the liquid side, a force pd S acts on it, where p is the value of the pressure in the liquid near the free surface. Here the question arises: where to get this value p? If we knew the pressure distribution p(x, y, z, t) corresponding to the propagation of the surface wave throughout space and time, we could calculate the necessary value by putting the coordinate z equal to the deflection of the free surface

1 Linear and Nonlinear Surface Waves

3

ζ(x, y, t). But the point is that we do not know in advance either the pressure distribution p(x, y, z, t) or the shape of the free surface ζ(x, y, t). In this case, we are forced to formally describe the force pd S as a quantity p(x, y, z, t)d S|z=ζ(x,y,t) . If the totality of all forces acting on the surface element were limited only to the above-mentioned effects, it would be sufficient to equate them to each other by writing the dynamic boundary condition in the form p(x, y, z, t) d S|z=ζ(x,y,t) = pa d S|z=ζ(x,y,t) ≡ pa d S, from where, taking into account the randomness of the element d S selection, it would follow p(x, y, z, t) |z=ζ(x,y,t) = pa . In the model we have adopted, this means that on the surface of the waves, the pressure from the liquid side takes a constant value in time and space, equal to atmospheric pressure. But limiting all forces acting on a liquid element near the surface to only two forces considered is too serious a simplification, since it does not take into account the force that occurs as a result of the effects of surface tension. The origin of the surface tension force is described at the molecular rather than hydrodynamic level, and is related to the isolation of molecules in contact with air molecules. Without being able to be distracted here by a detailed conclusion of macroscopic relations for surface tension forces, we give the final result, which is that the value of the additional force generated by capillary effects is given by the expression − α(1/R1 + 1/R2 ) d S, where α the surface tension coefficient, R1 and R2 are the principal radii of the surface curvature at the point of application of this force. Thus, taking into account the effects of surface tension, the dynamic boundary condition on a free surface takes the form p(x, y, z, t) − α(1/R1 + 1/R2 )|z=ζ(x,y,t) = pa . The choice of the sign for the surface tension force means that if the free surface has a bulge towards the atmosphere (wave crests), the pressure under the surface is greater than atmospheric pressure, and, conversely, in places where the free surface deflects down (wave troughs), the pressure under the surface is less than atmospheric. So, a dynamic boundary condition on a free surface is formed. Let us now turn to the kinematic condition, which expresses the fact that the velocity of a liquid element located on a free surface can only have a tangential component to this surface. Otherwise, the particle would break away from the liquid and go into the atmosphere. If F(x, y, z, t) = z − ζ(x, y, t) = 0 is the free surface equation, then the mathematical expression of the kinematic condition given above has the form d F/d t| F = 0 = 0.

4

1 Linear and Nonlinear Surface Waves

In the Eulerian description of fluid motion this expression takes the form d F/d t = ∂ F/∂ t + v · ∇ F = −ζt + w − uζx − vζy = 0 when z = ζ(x, y, t). Here u = vx , v = v y , w = vz are the corresponding components of the velocity field v. This decomposition will always be implied in the future, unless a different form is explicitly represented, which is more convenient in special coordinate systems. In the final form, the kinematic boundary condition on a free surface is defined by the relation  w − uζx − vζy z=ζ(x,y,t) = ζt . Let’s turn to the boundary conditions on the solid bottom. In our model, a solid bottom is an absolutely solid body. This means that any finite deformation of the bottom results in an infinite reaction force near the deformation point. Therefore, any final pressure in the liquid near the bottom is compensated by zero deformations of the bottom material and there is no need to formulate a dynamic boundary condition at the bottom. In addition, near the bottom, the normal velocity of liquid particles to its surface should be equal to the velocity of displacement of the bottom. Since the bottom, set by the ratio B = z − η(x, y) = 0, is stationary, and the normal to it is co-directed with the vector ∇ B, a kinematic condition on the bottom naturally occurs v · ∇ B| B=0 = 0. This result can be obtained similarly to the kinematic condition on a free surface. Since the velocities of liquid particles at the bottom must have only a tangential component, the condition must be met d B/d t| B=0 = 0. Since the bottom is stationary and is given by the equation B(x, y, z) = z − η(x, y) = 0, the relations are valid d B/d t = ∂ B/∂ t + v · ∇ B = −ηt + w − uηx − vηy = w − uηx − vηy = 0. In its final form, the kinematic boundary condition on a solid bottom is defined by the expression  w − uηx − vηy = 0z=η(x,y) = 0. Thus, based on the assumptions made the classical system of equations and boundary conditions of the mathematical model of wave propagation over the surface

1 Linear and Nonlinear Surface Waves

5

of a homogeneous inviscid liquid is formed [1–3] 1 v t + (v∇)v = − ∇ p + g, ∇ · v = 0 ρ   w − uηx − vηy z = η = 0, w − uζx − vζy z=ζ = ζt ,    2 p − α K |z = ζ = pa , K = −∇ · ∇ζ/ 1 + (∇ζ)

(1.0)

which is the basis for most further studies of surface waves. Here K = 1/R1 + 1/R2 is the doubled mean curvature of the free surface is expressed, the output of the expression for which can be found in the manuals on differential geometry [4]. The system (1.0) has been an object of active research aimed at obtaining its solutions for more than two hundred years. Like other problems of hydrodynamics, the problem of describing surface waves is complicated by the non-linearity of the Euler equation and boundary conditions, as well as by the fact that the conditions at the liquid-air interface are set on an unknown mobile surface, the shape and evolution of which must be determined! The consequence of this is the fact that at the moment we know the only exact solution [5] of the problem (1.0) obtained in the class of vorticity flows, which significantly reduces its value. Solutions describing potential surface waves are of primary interest, since the main part of the energy of waves observed in nature is contained in potential motion. Another principle of dividing solutions by types is to assign them to the classes of stationary and non-stationary waves. Sections 1.2–1.4 focus on stationary waves of both infinitesimal and finite amplitudes, as well as soliton solutions. They present the main methods used in obtaining approximate solutions, as well as the results of studies that are currently considered classic. Here also a correct way to prove the existence of a “Stokes flow” [6] that occurs when weakly nonlinear waves propagate is presented. Section 1.5 describes the known solutions from the class of stationary waves up to the class of non-stationary waves—wave zugs and the so-called “Schrodinger soliton” [10–14]. Almost all the information contained in the first paragraphs is well known in the scientific literature and is given here because the results of the rest of the Chapter, based on alternative approaches to the study of the system (1.0), are tested by comparison with known classical solutions. Various alternative methods for investigating the problem, their results and comparisons with known conclusions of classical theory are presented in Sect. 1.6. This section is also devoted to finding exact solutions in the class of potential flows. An exact, but complex-valued, solution of the problem (1.0) is explicitly given in the absence of surface tension. It is shown how the results given in Sects. 1.2 and 1.4 may be simply obtained based on this solution. Here we also present a method for reducing the boundary value problem under study to a single functional-differential equation for a single unknown value, namely the shape of the undulated surface. It

6

1 Linear and Nonlinear Surface Waves

is shown how all known approximate solutions for stationary surface waves may be obtained using this equation.

1.1 Mathematical Models of Wave Phenomena on the Surface In common case the surface waves can be divided by two types, namely potential and vortical waves. Such decomposition is inherent to all solutions of hydrodynamic problems in which the Euler equation is used. The velocity field always permits the presentation v = ∇ϕ + ∇ × ψ,

(1.1)

where ϕ and ψ are scalar and vector potentials correspondingly. Because the vorticity ω of the velocity field is defined by relation ω = ∇ × v = ∇ × (∇ϕ + ∇ × ψ) = ∇ × ∇ × ψ,

(1.2)

then it is considered that the value ∇ϕ defines the potential part of velocity and ∇ ×ψ concerned to its vortical part. It is necessary to note that the decomposition (1.1) is not uniquely defined. This is due to the fact that the equation ∇ A = ∇ × B has the infinite set of nontrivial solutions. For example, the functions A = x 2 − y 2 and B = 2x y ez satisfy the above equation, where ez is the basis vector in z direction. Thus always it is possible to present the arbitrary vector potential in the form ψ = ψ1 + ψ2 , where the relations ∇ × ∇ × ψ1 = 0 and ∇ × ∇ × ψ2 = 0 are valid. The first expression is equivalent to the equation ∇ϕ1 = ∇ × ψ1 that means the part of velocity field which is described by the value ∇ × ψ1 is of the potential nature and permits its presentation by scalar potential. For this reason with regard to expression (1.1) the true criterion of the velocity field potentiality is the condition ω = 0 but not ∇ × ψ = 0. As it follows from (1.2), the satisfaction to the criterion ω = 0 means that the relation ∇ ×ψ = ∇ϕ has place, where ϕ is some scalar function which for the given vector potential ψ is defined up to the additive function of time. Since the gradient of an arbitrary function that depends only on time is zero, the velocity field is set uniquely. This result allows to write the relation (1.1) in a form v = ∇(ϕ+ϕ ) that determines the potentiality of the velocity field. For this reason, here and in the further, potential motion is understood as a motion whose velocity field is represented in the form v = ∇ϕ. It is impossible to create a criterion that distinguishes a purely vortical motion, since the condition ω = 0 is generally non-constructive and because any vector potential is represented in the form ψ = ψ1 + ψ2 , where ∇ × ∇ × ψ1 = 0 and

1.1 Mathematical Models of Wave Phenomena on the Surface

7

∇ × ∇ × ψ2 = 0, does not allow excluding the potential part of the motion, if it is present in the physical process under study. So let the velocity field is potential, i.e. v = ∇ϕ. The substitution of this relation into Euler equation of the system (0.1) transforms it to 1 ∇(ϕt + (∇ϕ)2 /2) = − ∇ p − ∇(gz), ρ which integration leads to the result   p = −ρ gz + ϕt + (∇ϕ)2 /2 + f (t) + p0 , where f (t) is some arbitrary function of time and p0 is a constant. The incompressibility condition and boundary conditions on the bottom and free surface take the forms ϕ = 0,

 ϕz − ϕx ηx − ϕy ηy z=η = 0,

 ϕz − ϕx ζx − ϕy ζy z=ζ = ζt ,

which include only the first and second degree derivatives of potential ϕ with respect to spatial coordinates. This fact permits to exclude the function f (t) from the expression for pressure because it isalways possible to present the potential in the form ϕ(x, y, z, t) = ϕ(x, ˜ y, z, t) − f (t) dt so no one value depending on f (t) will not appear in the equations for surface waves. As a value p0 the atmospheric pressure pa is chosen that permits to exclude it from the dynamical boundary condition of the system (0.1). As a result the original system (1.0) for potential surface waves is transformed to the system   ϕz − ϕx ηx − ϕy ηy z=η = 0, ϕz − ϕx ζx − ϕy ζy z=ζ = ζt  .

 ϕt + (∇ϕ)2 /2z=ζ = −gζ − α K , K = −∇ · ∇ζ/ 1 + (∇ζ)2

ϕ = 0,

(1.3)

Here and further, the surface tension coefficient α is normalized to the density of the liquid ρ, so that its dimension is equal to [α] = sm3 /s2 . The value K is taken out from under the sign of the boundary condition, since it does not depend on the variable z. It is the system (1.3) or its simplified forms that will be mainly used in the future in the study of potential surface waves. Let us now turn to the vortical waves. As it was said in introduction to this chapter the only known exact solution of (1.0) describing the waves of such type was constructed by Gerstner [5]. Gerstner have got his solution by the integration of equations presented in the Lagrange coordinates. The expressions for Lagrange coordinates (x, z) of the liquid particle have the form

8

1 Linear and Nonlinear Surface Waves

Fig. 1.1 The form of the vortical Gerstner wave

x =a+

ekb ekb sin(k(a + ct)), z = b − cos(k(a + ct)), k k

(1.4)

where parameters a√and b define the separated particle in the liquid continuum, k is wavenumber, c = g/k is the wave speed. The trajectory of the particle is the circle of the radius ekb /k and the vorticity 2kb and is rapidly vanished with the depth. at the particle place equals to ω = − 2kce 1−e2kb The form of the Gerstner wave is shown on the Fig. 1.1, where the horizontal line represents undisturbed liquid’s level. In accordance to the vortical waves problem [5] the flow on the surface of the infinitely deep water is considered. No physical fields depend on y- coordinate and surface tension effects are absent. In this case the system (1.0) transforms into system ut + uux + wuz = − ρ1 px , wt + uwx + wwz = − ρ1 pz − g, ux + wz = 0  . p|z=ζ = pa , w − uζx z=ζ = ζt (1.5) Because the vortical motion is under investigation it is impossible to introduce the velocity potential. At the same time the form of incompressibility equation ux +wz = 0 permits to introduce the stream function ψ so u = −ψz and w = ψx . For the reason ω = ∇ × v = 0 the relation ψ = wx − uz = 0 is valid. The pressure p in the liquid is presented as sum of atmospheric pressure pa , water hydrostatic pressure and pressure ρ q due to surface wave propagation, i.e. p = pa − ρ gz + ρ q. It is assumed that the wave portrait propagates with constant speed c in positive direction of x- axis. This fact permits to consider the physical fields ψ, q and ζ as the functions of two variables, namely ξ = x − ct and z. Thus the system (1.5) transforms into system cψzξ + ψz ψzξ − ψξ ψzz = −qξ , −cψξξ − ψz ψξξ +ψξ ψzξ = −qz    . q|z=ζ = gζ, ψξ + c + ψz ζξ  = 0

(1.6)

z=ζ

When the surface wave is absent the liquid is in the rest and its undisturbed level is defined by condition z = 0. Then the introduction of the modified stream function  by means of expression  = cz + ψ

(1.7)

1.1 Mathematical Models of Wave Phenomena on the Surface

9

permits to transform (1.6) into z zξ − ξ zz = −qξ , ξ zξ − z ξξ = −qz . q|z=ζ = gζ, |z=ζ = Φ0 = const

(1.8)

The last relation, namely the kinematic boundary condition on the free surface,   is the result of the following transformations. From (1.7)   it follows z = c + ψz     and ξ = ψξ so ψξ + c + ψz ζξ  = ξ + z ζξ  = 0. From the other z=ζ z=ζ       hand ∂∂ξ |z=ζ = ξ + z ζξ  so the relation has place ∂∂ξ |z=ζ = 0, which z=ζ

integration leads to the kinematic boundary condition |z=ζ = 0 = const of the (1.8). The exclusion of the variable q from two first equations of (1.8) forms the equation for modified stream function z ξ − ξ z = 0,  =

∂2 ∂2 + , ∂ ξ2 ∂ z2

which integration leads to result  = F().

(1.9)

Here F() is some arbitrary function of modified stream function which should be vanish when the wave is absent but when the wave is present it should be F() = 0 because the relation  = ψ = 0 has place. Using the Eq. (1.9) the integration of (1.8) give the expression for the function q  1 q = − ξ2 + z2 − 2

F(Φ) d + q0 ,

(1.10)

where q0 is the constant which value should be defined. Substitution of (1.10) into the dynamic boundary condition of (1.8) creates the relation    2  2 (1.11) ξ + z z=ζ + 2 F() d − 2q0 = −2gζ. z=ζ

Expressions (1.9–1.11) have a general character, which makes it possible to define constant integrations 0 and q0 . For this goal the situation of the wave absence q = 0, ζ = 0,  = cz is considered. In this case ξ = 0, z = c and from (1.9) and (1.11) it follows F() = 0 and q0 = c2 /2. The boundary condition |z=ζ = 0 of (1.8) gives the result 0 = 0. Because F() = F(cz) = 0 for all values of its argument then F ≡ 0 and there is no way to achieve a difference from zero of F() in general case. This result is in

10

1 Linear and Nonlinear Surface Waves

contradiction with the condition of the flow vorticity. It means that it is impossible to generate the vortical wave on the surface of liquid that is in the rest. For the vortical surface wave existence it is necessary to assume that initially before the wave generation the liquid is moved in x- direction and the velocity field is defined by the stream function ψ0 (z) and the relations u0 = −ψ0 (z) and w0 = 0 have place. The vorticity of the original flow equals to the value ω0 = ψ0 (z)e y . Let the Gerstner’s wave is specified by the function ψG (ξ, z). Hence the common stream function is defined by expression ψ = ψ0 + ψG . The velocity field as in the previous case is calculated by the formulae u = −ψz and w = ψx . Since ψ0 = ψ0 (z) and ψG = ψG (ξ, z) then ψ = ψ(ξ, z). The introduction of the modified stream function in accordance to (1.7) forms the same relations (1.8–1.11) but with one significant distinction. When the surface wave is absent (q = 0, ζ = 0 and ψG = 0) then  = ψ0 (z) + cz and Eq. (1.9) has the form ∂ 2 ψ0 = F(ψ0 (z) + cz). ∂ z2

(1.12)

The original flow described by the stream function ψ0 (z) is characterized by nonzero vorticity in the whole space. In the contrary case the expression ψ0 (z) = c0 z + const will be valid, which corresponds to horizontal flow of the liquid as a whole with the speed c0 and returns us to the previous situation. Thus, the introduction of the original vortical horizontal flow eliminates the contradiction mentioned above. Now from the boundary condition |z=ζ = 0 = const follows that in the wave absence the result 0 = ψ0 (z) + cz|z=0 = ψ0 (z)|z=0

(1.13)

has place, which define the value 0 .   The boundary condition q|z=ζ = gζ may be presented in the form q|z=ζ ξ = ζξ . The left part of this expression by means of two first equations of the system (1.8) transforms into relation 

q|z=ζ

 ξ

   = qξ + ζξ qz z=ζ = ξ zz − z zξ + ζξ z ξξ − ξ zξ z=ζ . (1.14)

The using  of (1.9) and boundary condition |z=ζ  ξ + ζξ z  = 0 allows to get

= 0 in its form

z=ζ



   1 ∂    2  ∂ 2 2 2 −  + ζ  = F()  + ζ  +  +   z z z ξ ξ ξ ξ ξ 2 ∂ ξ ∂ z ξ   z=ζ .

    = − 21 ∂∂ξ ξ2 + z2 + ζξ ∂∂z ξ2 + z2  = − 21 ∂∂ξ ξ2 + z2  

q|z=ζ



z=ζ

z=ζ

1.1 Mathematical Models of Wave Phenomena on the Surface

11

Thus the boundary condition   ∂ 2 ξ + z2 z=ζ = −2gζξ ∂ξ has place, which integration gives its final form  ξ2 + z2 z=ζ = −2gζ+A,

A = const.1

(1.15)

When the wave is absent (ζ = 0, ψG = 0, ξ = 0, z = ψ0 + c) then A =    = 0 follows the relation (ψ0 + c)2 z=0 . Since from the condition ξ + ζξ z  z=ζ     ξ2  = ζξ2 z2  then the fully transformed problem has the form z=ζ

z=ζ

  = F(), ψ0 = F(ψ0 + cz), |z=ζ = ψ0 (z)z=0 ,     .  , z z=ζ = A−2gζ , A = (ψ0 + c)2 z=0 ξ  = − ζξ A−2gζ 1+ζ 2 1+ζ 2 z=ζ

ξ

(1.16)

ξ

Unfortunately, the Gerstner wave description in the Euler coordinates is intractable problem. This is one of the few examples when the using of Lagrange coordinates is more convenient than Euler’s. To understand the reasons for it let’s represent the solution (1.4) in more rigorous form x = a + Aekb sin(k(a + ct)), z = b − Aekb cos(k(a + ct)),

(1.17)

where the constant A is defined by the wave’s amplitude. The vorticity of velocity field is defined by the relation ω=

∂ x˙ ∂ z˙ ∂w ∂u − = − , ∂x ∂z ∂x ∂z

(1.18)

where the upper dot is the symbol of the partial derivative with respect to time. Differentiation of (1.17) give the expressions x˙ = ck Aekb cos(k(a + ct)), z˙ = ck Aekb sin(k(a + ct)),

(1.19)

which substitution into (1.18) demands the calculation of values ax , az , bx and bz . For this purpose the both relations of (1.17) are differentiated with respect to x and z creating the system      kb kb  kb  kb  1 + β k Aekb  ax + α k Aekb bx = 1, α k Aekb ax + 1 − β k Aekb  bx = 0 , (1.20) 1 + β k Ae az + α k Ae bz = 0, α k Ae az + 1 − β k Ae bz = 1 where for brevity the designations are used

12

1 Linear and Nonlinear Surface Waves

α = sin(k(a + ct)), β = cos(k(a + ct)). The solution of (1.20) define the necessary expressions for ax , az , bx and bz , which substitution into (1.18) gives the expression for vorticity ω=−

2ck 3 A2 e2kb . 1 − k 2 A2 e2kb

(1.21)

To present (1.21) in Euler’s variables the relation x˙ 2 + z˙ 2 = c2 k 2 A2 e2kb following from (1.19) is used, so ω = −2ck

x˙ 2 + z˙ 2 u2 + w2 = −2ck , c2 − x˙ 2 − z˙ 2 c2 − u2 − w2

(1.22)

where u and w are field velocity components in the Euler description. Since the form of the unknown function F is united both in the presence and in the absence of the wave then the relation (1.22) may be used for its form determination. For this purpose the case of the undisturbed situation is used when ω = ψ0 and u2 + w2 = ψ02 . Because the Eq. (1.22) is reckoned among the class of nonlinear equation then after preliminary normalizing of the stream function to c/k and zcoordinate to k it transforms into equation ψ0 = −2

ψ02 . 1 − ψ02

(1.23)

Under action of the same normalization the equation ψ0 = F(ψ0 + cz) gets the form ψ0 = F(ψ0 + z).

(1.24)

The combination of (1.23, 1.24) allows to exclude the stream function ψ0 and to receive the equation for unknown function F  F 1 ±



F F −2

 ± F 5/2 (F − 2)3/2 = 0,

(1.25)

where upper accent means the differentiation with respect to its argument, i.e. with respect to the value y= ψ0 + z. The presence of the symbol ± in (1.25) is due to

F the fact that ψ0 = ± F−2 . The integration of the (1.25) creates the functional equation

F 1 1 y + y0 + ln − ∓ 4 F − 2 2(F − 2)



1 F ±√ = 0, F −2 F(F − 2)

y0 = const., (1.26)

1.1 Mathematical Models of Wave Phenomena on the Surface

13

the solution of that with respect to F is not available at the current level of mathematics development. Thus the required function F is specified in implicit form as a solution of (1.26). This fact does not allow to integrate the system (1.16) an to get the solution expressed in Euler’s coordinates. The Eq. (1.23) allows the integration in explicit form and defines the undisturbed flow stream function by means the relation   

1 z + z0

  2 2 2 (z + z 0 ) − 1 − ln z + z 0 + (z + z 0 ) − 1 , ψ0 (z) = (z + z 0 ) ± 2 2 (1.27) where z 0 is a constant. For the reason of (1.27) the velocity field and vorticity of the origin liquid flow also are defined in explicit form u = −z − z 0 ∓



(z + z 0 )2 − 1, w = 0, ω = 1 ±

z + z0 (z + z 0 )2 − 1

.

(1.28)

Here the velocity field and the vorticity are normalized to c and ck correspondingly. The condition of flow kinetic energy finiteness enclosed in the spatial region of liquid restricted by the vertical band of the finite horizontal size defines the necessity of the vorticity vanishing when the depth of the observation point tends to infinity, i.e. lim z → − ∞ ω = 0. From the other hand, the requirement that the vorticity of the flow on the undisturbed surface is constant value, for example lim z → − 0 ω = 1, defines the sign in (1.28) and the value z 0 . Finally, the expressions in (1.28) transform to the form u = −z + 1 −



(z − 1)2 − 1, w = 0, ω = 1 +

z−1 (z − 1)2 − 1

.

(1.29)

From the result received it follows that the speed of the flux undisturbed should be equal and opposite directed to the propagation speed of the Gerstner’s wave. Vorticity and flux velocity should vanish with the depth in accordance to (1.29). If these demands will not be satisfied then the Gerstner’s wave will not be exist. The artificial nature of the applied conditions negates the value of the solution for a vortical wave, so in the further investigations only potential waves will be considered.

1.2 Infinitesimal Waves and Linear Models As mentioned earlier, the solution of the system (1.0) in its most general form is practically an unsolvable problem. The reason for this is the non-linearity of the

14

1 Linear and Nonlinear Surface Waves

Euler equation and the boundary conditions, as well as the fact that the boundary conditions are set on surfaces of complex shape. For a free surface, there is an additional difficulty because the surface itself changes over time and its shape is unknown in advance. One of the sections of the theory of surface waves, in which it is still possible to obtain constructive results that are in good agreement with experimental and natural observations, is devoted to the study of the propagation of so-called infinitesimal waves. The term “infinitesimal waves” is widely used in all textbooks, monographs and articles devoted to the problems of describing surface waves. From a physical point of view, this term means that the deviation of the free surface of the liquid from its undisturbed state is much smaller than any characteristic physical scale inherent in the phenomenon under consideration. But in the problem of surface waves, it is generally impossible to specify the natural scale of the surface deflection in the vertical direction. For example, the problem of wave propagation over the surface of an infinitely deep liquid does not have a predetermined vertical scale at all. The size of the liquid element underlying the hydrodynamic description cannot be chosen as such scale, since all scales that are smaller than this size are assumed to be identically zero. Thus, the physical concept of an “infinitesimal wave” on the surface of a liquid simply does not exist. If we turn to the mathematical methods used in the study of “infinitesimal waves”, it becomes clear that this term entails the linearization of the system (1.0) with the subsequent solution of the resulting simplified relations. Thus, in order to understand the mathematical meaning of the concept of “infinitesimal waves”, it is necessary to understand what follows such a standard operation as linearization of equations and boundary conditions. In order for the linearized system (1.0) to adequately describe the observed phenomena, the following conditions must be met ((v∇)v)2 v 2t

1,

 2 uηx + vηy z=η

1,  2 w|z=η



((v∇)v)2 (∇ p/ρ)2

1,

((v∇)vz )2

1, g2

 2 uζx + vζy z=ζ

1,  2 w|z=ζ

|∇ζ| 1,

(∇ζ · (∇ζ∇)∇ζ)2 (ζ)2



 2 uζx + vζy z=ζ ζt 2

1,

1,

which ensure the validity of the use of linearized relationships. The first group of conditions provides linearization of the Euler equation, the second—kinematic boundary conditions on the solid bottom and the free surface of the liquid, the third—dynamic boundary conditions on the free surface. A bar over the corresponding relation term implies averaging this term either over the time of observation of the process, or over the spatial region in which the process occurs. At

1.2 Infinitesimal Waves and Linear Models

15

the same time, the above conditions must be met when averaging both over time and space separately. Space averaging in the first group of conditions is carried out by the volume occupied by the liquid, and in the second and third groups—by the areas of the bottom and the free surface of the liquid. Since ζ = ζ(x, y, t), the condition |∇ζ| 1 means that the surface waves have a small steepness and the linearization of the system (1.0) restricts the described surface waves to a class of so-called gentle waves. It is clear that it is impossible to answer the question about the feasibility of the set conditions before obtaining a solution to the linearized problem, and the verification of their validity must be carried out a posteriori. A priori assumption about the linearization conditions’ feasibility reduces the system (1.0) to the form v t = − ρ1 ∇ p + g, ∇ · v = 0 , w|z=η = 0, w|z=ζ = ζt , p|z=ζ + αζ = pa however, this simplified form of the boundary value problem does not yet allow us to obtain constructive results, because the shape of the bottom surface η(x, y) can generally be described by a rather complex function, and the boundary conditions on the free surface of the liquid are set on an unknown surface ζ(x, y, t). The problem of boundary conditions at the bottom in the vast majority of manuals and articles on surface waves is solved by volitional effort: it is assumed that the shape of the bottom is a known function, and the relative local deviations of the depth of the undisturbed liquid at any point of the bottom from a known a priori value is significantly less than one. In particular, when introducing the theory of surface waves, the approximation of a plane horizontal bottom is usually used, that is, it is assumed that η(x, y) = −h where h is the depth of the liquid. When studying waves on the shelf, the depth of the sea is set as a linear function of the distance from the coastline. In the future, the study of the fundamental characteristics of infinitesimal waves uses the approximation of a plane horizontal bottom. It remains to consider how the conditions on a free surface are transformed when using a linear approximation. A commonly used technique is to carry the boundary conditions to the level z = 0 of the free surface of an undisturbed liquid. To find out the conditions that allow such a mapping, we consider an abstract mathematical expression that defines the value of a certain function f (x, y, z, t) on the boundary z = ζ(x, y, t). Because f (x, y, z, t)|z=ζ(x,y,t) = f (x, y, ζ(x, y, t), t), then the formal decomposition of the right side of this expression into a Taylor series near the level z = 0 has the form   1 f (x, y, ζ(x, y, t), t) = f (x, y, 0, t) + ζ(x, y, t) f z z=0 + ζ 2 (x, y, t) f zz z=0 + · · · 2   = f (x, y, z, t) + ζ(x, y, t) f z (x, y, z, t) + . . . z=0 .

16

1 Linear and Nonlinear Surface Waves

Thus, reducing the boundary conditions to the level z = 0 with their subsequent linearization requires the validity of following relations     n (n) (x, y, z, t)z=0   ζ (x, y, t) f z..z  

1, n = 1, 2, . . . , n!  f (x, y, z, t)|z=0  which, in relation to the problem under study, mean that the relative variations of physical fields (velocity and pressure) near a free surface with a vertical shift of the observation point of these fields by an order of magnitude of the wave amplitude are small compared to one. As a result, if the physical fields of the problem (1.0), the bottom shape, and the surface wave satisfy all the above conditions, then the simplified linearized problem of surface wave propagation has the form v t = − ρ1 ∇ p + g, ∇ · v = 0 . w|z=−h = 0, w|z=0 = ζt , p|z=0 + αζ = pa

(1.30)

From the course of obtaining the linearized system (1.30), it follows that no restrictions are imposed on the values of |v|, | p| and |ζ|. This means that the waves under study are not infinitesimal at all. This fact allows us to explain the good coincidence of the results of the theory of surface waves based on the system (1.30) with natural and experimental observations of gentle, but not infinitesimal waves. At the same time, because the term “infinitesimal waves” is widely used in the scientific literature and is well established, in the future this term is reserved for wave solutions of the problem (1.30). Before proceeding to the study of the problem (1.30), it makes sense to consider the equilibrium state of the liquid in the absence of undulation. In this case v = 0, ζ = 0 and the system (1.30) is reduced to the form ∇ p = ρ g,

p|z=0 = pa

The solution to the equilibrium state problem is obvious p = p0 (z) = pa − ρ gz and represents a stationary distribution of hydrostatic pressure in a rest liquid. The occurrence of a wave leads to disturbances of the hydrostatic pressure field p0 (z). To single out these disturbances, it is convenient to set the total pressure p in the form p = p0 (z) + q(x, y, z, t) = pa − ρ gz + q(x, y, z, t),

(1.31)

where the symbol q indicates the pressure deviation from its equilibrium value.

1.2 Infinitesimal Waves and Linear Models

17

In order to simplify the process of obtaining the main characteristics of infinitesimal waves, it is convenient, without losing the generality of the final results, to consider the case of propagation of “one-dimensional” waves traveling along the x axis, that is, such waves whose physical fields do not depend on the second horizontal coordinate y. Under using this simplification and representation (1.31), the system (1.30) takes the form in which all relations are linear and boundary conditions are set on simple surfaces ρ ut = −qx , ρ wt = −qz , ux + wz = 0 w|z=−h = 0, q|z=0 − ρ (g ζ − α ζx x ) = 0,

w|z=0 = ζt

.

(1.32)

The first two equations of the system (1.32) allow us to enter the potential of the velocity field, which determines the components u, w of the velocity field and the pressure perturbation q by means of the relations u = ϕx , w = ϕz , q = −ρ ϕt ,

(1.33)

moreover, the potential itself, as follows from the third equation and boundary conditions (1.32), is the solution of the boundary value problem ϕx x + ϕzz = 0,

 ϕz z=−h = 0,

 ρϕt z=0 + ρ (g ζ − α ζx x ) = 0,

 ϕz z=0 = ζt . (1.34)

First, we will look for the solution (1.34) in the class of harmonic waves traveling in the positive direction of the axis x. In this case the shape of the undulated surface is given by the expression ζ(x, t) = a cos(kx − ω t),

(1.35)

where a is the wave amplitude. It follows from (1.35) and the last two boundary conditions (1.34) that the velocity potential has the form ϕ(x, z, t) = f (z)sin(kx − ω t).

(1.36)

Substituting (1.36) into the system (1.34) generates a problem for the function f (z) f zz − k 2 f = 0,

f |z=0 =

a (g + α k 2 ), ω

 f z z=0 = ω a,

 f z z=−h = 0. (1.37)

Substituting the solution of the problem (1.37) in the expression (1.36) determines the speed potential

18

1 Linear and Nonlinear Surface Waves

ϕ(x, z, t) =

ωa ch(k(z + h))sin(kx − ω t) k sh(kh)

(1.38)

and the dispersion equation of capillary-gravitational waves ω2 = th(kh) (gk + α k 3 ).

(1.39)

As follows from the dispersion equation, the properties of surface waves are determined by the combined action of gravity and surface tension. Their influence is equal in the case when g = α k 2 , that corresponds to the wavelength

λ∗ = 2π α/g.

(1.40)

In the case of short waves, for which λ λ∗ , capillary effects predominate and the dispersion Eq. (1.39) takes the approximate form ω2 ≈ α k 3 th(kh), and the short waves themselves are called capillary. For long waves, when λ λ∗ has place, the gravitational effects significantly exceed the capillary ones, the dispersion equation takes the form of ω2 ≈ gk th(kh), and the waves are called gravitational. In the intermediate region of λ ∼ λ∗ , the Eq. (1.39) comes into force, and the waves are called capillary-gravitational. For water at a temperature of T ∼ 20 C ◦ , the value of α ≈ 72 sm3 /s2 and λ∗ ≈ 1.7 sm. In the case of a large depth (h ≥ λ/π,th(kh) → 1), the velocity of capillarygravitational waves cannot be less than a certain limit value determined by the phase velocity extremum ∂ ω2 (k) ≈ α − g/k 2 = 0 ∂ k k2

⇒ k=



g/α.

(1.41)

This extremum corresponds to the minimum of the phase velocity because ∂ 2 ω2 (k) 2g ≈ 3 > 0. ∂ k2 k2 k It follows from (1.41) that the minimum is reached at the wavelength λ = λ∗ ,

√ and the minimum value of the wave velocity is c ph∗ = 2 α g ≈ 23 sm/s at T ∼ 20 C◦ . Substitution (1.38) in the third equation of the system (1.33) determines the perturbation of the pressure field q. As a result, the complete set of relations that determine the physical fields of capillary-gravitational waves has the form ζ(x, t) = a cos(kx − ωt) ωa ch(k(z + h)) sin(kx − ωt) . ϕ(x, z, t) = kch(kh) 2 ch(k(z+h)) q = ρa(g + αk ) ch(kh) cos(kx − ωt)

(1.42)

1.2 Infinitesimal Waves and Linear Models

19

Checking the requirements used in obtaining the approximate system (1.34) shows that they are equivalent to the condition ka 1, which is a condition of gently sloping waves and is fulfilled for all waves of infinitesimal amplitude at a finite wavelength. Expressions (1.42) allow us to determine the velocity field and total pressure in the marine environment when a capillary-gravitational wave propagates along its surface (g + α k 2 ) ch(k(z+h)) cos(kx − ω t) u = ak ω ch(kh) ak 2 sh(k(z+h)) w = ω (g + α k ) ch(kh) sin(kx − ω t) . cos(kx − ω t) p = pa − ρ gz + ρ a(g + α k 2 ) ch(k(z+h)) ch(kh)

(1.43)

To simplify the analysis of the characteristics of surface waves of this type, it is convenient to go to the limit of an infinitely deep liquid, when kh → ∞ the relations ch(k(z + h))/ch(kh) and sh(k(z + h))/ch(kh) are replaced by a simple limit expression exp(kz). In this case (g + α k 2 ) exp(kz) cos(kx − ω t) u = ak ω ak . w = ω (g + α k 2 ) exp(kz) sin(kx − ω t) p = pa − ρ gz + ρ a(g + α k 2 ) exp(kz) cos(kx − ω t)

(1.44)

It can be seen from (1.44) that perturbations of the initial equilibrium state caused by the propagation of the surface wave decrease exponentially with the depth of the liquid. Thus, at a depth equal to one wavelength, the amplitude of the velocity field decreases by more than 500 times, which justifies the name “surface wave”, since all significant disturbances of the equilibrium physical fields are concentrated near the free surface of the liquid. To determine the trajectories of liquid particles that participate in oscillations of the form (1.44) caused by a surface wave, it is necessary to formulate equations for liquid element’s Lagrange coordinates (r, s) corresponding to displacements along the axes x and z dr dt ds dt

= u(x + r, z + s) = ak (g + α k 2 ) exp(k(z + s)) cos(k(x + r ) − ω t) ω . (1.45) ak = w(x + r, z + s) = ω (g + α k 2 ) exp(k(z + s)) sin(k(x + r ) − ω t)

Taking into account the infinite smallness of the wave amplitude (ka 1) allows us to present the result of integrating the system (1.45) as r = ωa (g + α k 2 ) exp(kz)cos(kx − ω t) + r0 , s = − ωa (g + α k 2 ) exp(kz)cos(kx − ω t) + s0

(1.46)

where r0 , s0 are some characteristics that identify a separate liquid particle at the initial time of observation (for example, its Lagrange coordinates at some time t0 ). From (1.46) follows the ratio

20

1 Linear and Nonlinear Surface Waves

(r − r0 )2 + (s − s0 )2 = R 2 (z),

R(z) =

a (g + α k 2 ) exp(kz), ω

(1.47)

from which it follows that liquid elements move along circles whose radius decreases exponentially with the depth of the liquid. In the case of a liquid of finite depth, similar calculations based on the expressions (1.43) show that liquid particles move along ellipses, in which the ratio of the vertical axis to the horizontal axis tends to zero when approaching the bottom. It is these circular (ellipsoidal) movements that form a sinusoidal wave on the surface. Since the liquid elements of the medium make finite movements, periodically returning to the starting point, infinitesimal surface waves do not transfer mass. As will be shown later, weakly nonlinear waves of finite amplitude create a mass transfer called “Stokes wind” or “Stokes stream”. The linearity of the system (1.32) allows us to represent its solution as a linear combination of solutions of the form (1.42), namely as a packet of waves ζ(x, t) =

∞

a(ω) cos(k(ω)x − ω t) dω

0

ϕ(x, z, t) = q=ρ

∞ 0

∞ 0

ω a(ω) ch(k(ω)(z k(ω) sh(k(ω)h)

+ h))sin(k(ω)x − ω t) dω ,

a(ω)(g + α k 2 (ω)) ch(k(ω)(z+h)) cos(k(ω)x − ω t) dω ch(k(ω)h)

the initial form of which may differ significantly from the sine wave, and a(ω) is the spectral amplitude, k(ω) is the solution of the dispersion Eq. (1.39) with respect to the wave number at a given frequency. Since the phase velocities of individual spectral harmonics are different, such a packet of surface waves will be subject to dispersion and will lose its original shape after passing a certain distance. The exception concerns a packet of long waves running on the surface of a shallow sea. This packet consists of waves for which the relations λ λ∗ , kh 1 are valid. In this case, the dispersion Eq. (1.39) takes the form ω2 = gh,

(1.48)

so the phase velocity all of these partial waves is determined by the expression c ph = ω/k =



gh,

(1.49)

and it is determined only by the depth of the sea and the amount of gravitational acceleration, that is, it does not depend on the frequency. This means that there is no dispersion and, as a result, the shape of the packet of long waves on a shallow sea remains unchanged as it propagates. The solution of the system (1.32) can also be found in the class of waves of constant shape running at a constant speed. In this case, the solutions of the system

1.2 Infinitesimal Waves and Linear Models

21

under study are given as ζ = ζ(ξ), ϕ = A(z)B(ξ), q = A(z)C(ξ), ξ = x − ct,

(1.50)

where c is the speed of the wave propagation. Substituting (1.50) into (1.32) and then solving the boundary value problem gives the result ζ = a cos(kξ), ϕ =

a(g + α k 2 ) ch(k(z + h)) sin(kξ), kc ch(kh)

(1.51)

and the speed of the wave is determined by the ratio c2 = th(kh)(g/k + α k).

(1.52)

A comparison of (1.51, 1.52) with (1.39, 1.42) shows that they pass into each other when replaced ω ↔ kc, but the solutions (1.51, 1.52) cannot form a wave packet, since at a fixed speed it follows from (1.52) that the wave number is also fixed. This result means that in the case of a general position, at arbitrary ratios between the wavelength and the depth of the liquid, infinitesimal waves of constant shape appear only as sinusoidal waves. Here it should be noted at once that this result loses validity for waves of small but finite amplitude.

1.3 Transfer By Waves of Energy and Momentum As has been shown previously to infinitesimal waves do not carry the liquid’s mass. Let us now consider the question of the transfer of energy and momentum by such waves. The specific energy of a fixed volume of liquid in the field of gravity is determined by the ratio E = T + + U,

(1.53)

where T = ρv2 /2, = ρ G, U = ρ u are the kinetic, potential, and internal energy, respectively; G is the gravitational potential and u is the internal energy of a unit of mass of a liquid.

22

1 Linear and Nonlinear Surface Waves

The partial derivative of the total energy with respect to time is given by the expression ∂E v2 ∂ ρ ∂v ∂ ρ ∂ (ρ u) = + ρv · +G + . ∂t 2 ∂t ∂t ∂t ∂t

(1.54)

To convert (1.54) to the required form, you need to refer to the Euler and mass transfer equations, which are given here in general form   ∂v + (v∇)v = −∇ p − ρ ∇G, ρ ∂t

∂ρ + ∇ · (ρ v) = 0. ∂t

(1.55)

Replacing in (1.54) terms ∂ v/∂ t and ∂ ρ/∂ t using Eq. (1.55) converts the total energy derivative to the form  2  v ∂ ρ ∂ (ρ u) ∂E =− + G ∇ · (ρv) − v · (∇ p + ρ∇G + ρ(v∇)v) + . ∂t 2 ∂t ∂t Since for potential waves the relation v·((v∇)v) = v·∇v2 /2 is valid, the resulting result can be given the form   2  v ∂ (ρ u) ∂E = −∇ · ρv +G −v·∇p + . ∂t 2 ∂t

(1.56)

To calculate the last two terms in (1.56) it is necessary to use the basic thermodynamic identity and auxiliary thermodynamic relations d u = T ds +

p 1 p dρ, d w = T ds + dp, w = u + . ρ2 ρ ρ

(1.57)

Here T is the temperature of the medium, s and w are the specific entropy and enthalpy. Using (1.57) generates the following relations d(ρ u) = u dρ + ρ du = u +

p ρ2



dρ + ρ T ds = w dρ + ρ T ds

dp = ρ dw − ρ T ds

.

(1.58)

From (1.58) follow expressions for the necessary quantities ∂ (ρ u) ∂ρ ∂s =w +ρT , ∇ p = ρ∇w − ρ T ∇s. ∂t ∂t ∂t

(1.59)

Substituting (1.59) in (1.56) leads to the ratio   2    ∂E v ∂s = −∇ · ρv +G+w +ρT + v · ∇s . ∂t 2 ∂t

(1.60)

1.3 Transfer By Waves of Energy and Momentum

23

Since adiabatic processes are considered, the condition of adiabaticity, i.e. constancy of specific entropy, is fulfilled ∂s ds = + v · ∇s, dt ∂t by using of which Eq. (1.60) takes the form   2  ∂E v + ∇ · ρv +G+w = 0. ∂t 2

(1.61)

The expression 2 (1.61) iswritten in2 the form of thelaw of conservation, therefore, the value ρv v2 + G + w = ρv v2 + G + ρp + u represents the density of the energy flow. The result obtained is used in relation to the surface wave, for which G = G 0 + gz (G 0 is some constant), and the pressure is determined by the expression p = pa − ρ gz + q. Substituting these values converts the expression for the wave energy flux density to the form 

p v2 +G+ +u ρv 2 ρ





 v2 pa + q = ρv + +u . 2 ρ

In our model, fluid environment temperature is constant, density is constant, no chemical transformations are present, the internal energy u of the mass unit of any liquid element of the medium is also constant. Atmospheric pressure pa is also constant, and the liquid is considered incompressible, that is ∇ · v = 0. In that case    2    2 pa + q v q v + +u = ∇ · ρv + . ∇ · ρv 2 ρ 2 ρ Since this section deals with infinitesimal waves, the term ρv v2 /2 must be ignored, since it is a third-order quantity of smallness. As a result, the law of conservation of energy for infinitesimal surface waves takes the form ∂E + ∇ · (qv) = 0, ∂t

(1.62)

and the interesting density of the energy flow carried by the wave is determined by the value qv. Consider the question of the transfer of momentum by a wave. By definition, the pulse density is given by the value ρv. As in the case of energy transfer, it is necessary to calculate the derivative ∂ (ρv)/∂ t of the pulse density over time. Using the above relations, one can write this value in the form

24

1 Linear and Nonlinear Surface Waves

∂v ∂ρ ∂ (ρv) =ρ +v = −ρ(v∇)v − ∇ p − ρ∇G − v∇ · (ρv). ∂t ∂t ∂t The component-by-component record of the obtained ratio has the form ∂ (ρvi ) ∂t

= −ρv · ∇vi − vi ∇ · (ρv) − ∂∂ xpi − ρ ∂∂ xGi   , = − ∂∂x j ρvi v j + pδi j − ρ ∂∂ xGi

(1.63)

where δi j is the Kronecker symbol, and for repeated indexes, summation is assumed. Since the density of the medium is constant in the problem under consideration, it is possible to introduce a tensor of the momentum flow density defined by the expression i j = ρvi v j + ( p + ρ G)δi j ,

(1.64)

as a result, (1.63) takes the form of the law of conservation ∂ (ρvi ) ∂ i j + = 0. ∂t ∂ xj

(1.65)

By definition (1.64), i j is the i -th component of the amount of momentum transferred per unit of time through a unit area perpendicular to the j -th direction of the coordinate system. Substitution in (1.64) of expressions for the pressure and gravitational potential of the model under consideration leads to the relation i j = ρvi v j + ( pa + q + ρ G 0 )δi j . Since neither the atmospheric pressure pa nor the reference value G 0 of the gravitational potential have any relation to the surface wave, they should be excluded from the above relation when calculating the momentum flow carried by the wave. Thus, the transfer of momentum by a surface wave is defined by the expression i j = ρvi v j + qδi j .

(1.66)

Let’s apply the results obtained (1.62, 1.66) to the surface wave. Since the wave process has an oscillatory character, only the values averaged over the oscillation period have a physical meaning. The average density of the wave energy flow, whose physical fields are determined by the obtained expressions (1.42), has the form 1 T

t 0 +T

qv dt = ex ρ t0

a 2 k 2 c3ph ch2 (k(z + h)) , 2 sh2 (kh)

(1.67)

1.3 Transfer By Waves of Energy and Momentum

25

where T is the oscillation period. The ratio (1.67) is obtained using the dispersion ratio and the phase velocity determination given earlier. This result shows that the energy transfer by the wave occurs in a horizontal direction along the axis x. In the case of momentum transfer, the elements of the momentum density tensor averaged over the period are given by the expressions  x z  =  zx  = 0,  x x  = ( )0 ch2 (k(z + h)), a 2 k 2 c2

ph  zz  = 0 sh2 (k(z + h)), 0 = ρ sh2 (kh)

.

(1.68)

The above calculations did not take into account the surface energy of the waves. The amount of stored linear surface energy on one spatial period of the capillarygravitational wave is determined by the value E α = ρα

λ  0

  λ  1 + ζx2 − 1 d x = ρα 1 + (ka)2 sin2 (kx − ω t) − 1 d x. 0

Using the smallness of the value ka in comparison with the unity allows us to estimate the reserve of surface energy over the spatial period of the wave by the value 2

ka , Eα ∼ = ρα 4

(1.69)

so to the horizontal energy flow (1.67), the value c ph E α must be added.

1.4 Weakly Nonlinear Waves of Finite Amplitude The problem of propagation of waves of small but finite amplitude is considered in the model (1.0) without taking into account surface tension. This approximation allows us to identify the main characteristic features of the studied flows and, since the studied type of waves belongs to the class of gently sloping waves, that is, they have a sufficiently long wave length in comparison with the capillary-gravitational boundary λ∗ , the effects of surface tension appear only as small corrections to the main solution. Also for simplicity, the liquid is considered infinitely deep, and the problem is considered in a two-dimensional setting. The same way as was done earlier, the representation of pressure in the form of p = pa − ρ gz + ρ q (q describes the disturbances that occur when a surface wave propagates), the introduction of a stream function ψ (in this case, u = ψz , w = −ψx ) and the imposition of a requirement uz − wx = 0 (a condition for the potentiality of motion) reduces the system (1.0) to the form

26

1 Linear and Nonlinear Surface Waves    ψzt + ψz ψzx − ψx ψzz = −qx , −ψxt − ψz ψx x +ψ  x ψzx = −qz , ψ = 0 .       q|z=ζ = gζ, ψx + ζx ψz z=ζ = −ζt , q, ψx , ψz z=−∞ = 0 (1.70)

Here  = ∂ 2 /∂ x 2 + ∂ 2 /∂ z 2 is a two-dimensional Laplace operator, and the last condition in (1.70) is a requirement for the absence of perturbations of physical fields at an infinite distance from the free surface. Since the first two equations of the system (1.70) cannot be integrated and cannot represent the pressure perturbation q explicitly, the following mathematical technique is used. The dynamic boundary condition q|z=ζ = g ζ is differentiated by the variable x, that leads to the relation  qx + ζx qz z=ζ = g ζx .

(1.71)

Substituting in (1.71) expressions for qx , qz , whose values are determined by the first two equations of the system (1.70), allows to get rid of the variable q in the dynamic boundary condition, and cross-differentiation of the mentioned equations eliminates the pressure perturbation from the equations of motion, so that the system (1.70) is converted to the form   ψ = 0, ψx + ζx ψz z=ζ = −ζt , ψx , ψz z=−∞ = 0,  . ψzt + ψz ψzx − ψx ψzz + ζx (g − ψxt − ψz ψx x +ψx ψzx )z=ζ = 0

(1.72)

In contrast to the case of infinitesimal waves, it is impossible to simply carry the boundary conditions of the system (1.72) to the equilibrium unperturbed surface z = 0. It is necessary to take into account the deviation of the undulated surface from the level of the stationary liquid. If this deviation were arbitrary in magnitude, then achieving a solution would be an almost impossible task. But at the beginning of the Chapter, it was stipulated that the deviations of the liquid surface are, although finite, but small in comparison with the characteristic longitudinal scale of the wave. The mathematical record of this condition has the form ε = ka 1, where a is the characteristic scale of the deviation of the free surface and k is the value inversely proportional to the longitudinal scale of the wave. The solution of the system (1.72) is sought in the class of waves of constant shape traveling along the surface of a liquid at a constant speed. In this case, the excitement is set by the expression ζ = a H (ξ), ξ = k(x − ct),

H ∼ O(1),

(1.73)

where c is the propagation speed of the wave pattern. The type of boundary conditions and the need to satisfy the stream function to the Laplace equation allows us to use the representation ψ(x, z, t) = ac F(ξ, η), η = kz,

F ∼ O(1).

(1.74)

1.4 Weakly Nonlinear Waves of Finite Amplitude

27

Substitution of (1.73, 1.74) in (1.72) reduces the problem of small-amplitude waves to a form in which all the values are reduced to a dimensionless form, which is necessary for the analysis and solution of nonlinear equations   + Fηη = 0, Fξξ

  Fξ + ε Hξ Fη 

η=ε H

= Hξ ,

  Fξ , Fη 

     + ε Fη Fξη − ε Fξ Fηη ε Hξ (g˜ + Fξξ − ε Fη Fξξ + −Fξη

=0

η=−∞    ε Fξ Fξη )

η=ε H

=0

, (1.75)

where the notation g˜ = g/kc2 is entered. The presence of a small parameter allows us to apply the multi-scale method to the system (1.75) [7], according to which sets of variables are introduced     {ξi } = εi ξ , {ηi } = εi η ,

(1.76)

so the differential operators included in (1.75) are defined by expressions of the form ∞

 ∂ ∂ = εi , etc. ∂ξ ∂ ξi i=0

(1.77)

and the solutions to the problem (1.75) are given by expansions H=

∞ 

εk Hk ({ξi }),

k=1

F=

∞ 

εk Fk ({ξi }).

(1.78)

k=1

Substitution (1.76–1.78) in (1.75) with the subsequent solution, according to the method of many scales [7], up to the terms of the third order of smallness in ε, determines the stream function and the shape of the excited surface. Then, using the system (1.70) determines the pressure perturbation q. As a result, the desired solution to the problem has the form in which, for convenience, the reverse transition to dimensional physical variables is made

ζ = a cos ϕ(x, t) + 2ε cos 2ϕ(x, t) +

3ε2 8

 cos 3ϕ(x, t) + o(ε3 ) ,

. ψ = ac (1 − ε2 /8)ex p(kz) cos ϕ(x, t) + o(ε3 ), ϕ(x, t) = k(x − ct (1 + ε2 /2)) (1.79) The pressure perturbation is determined by the expression   q = ag (1 + 3ε2 /8)exp(kz)cosϕ(x, t) − ε (1 − ε2 /4)exp(kz)/2 .

(1.80)

In the presented expressions (1.79, 1.80) c2 = g/k. The resulting solution is called the Stokes wave. The speed of propagation of this wave, as follows from the expression √ for the phase function ϕ(x, t) in (1.79), is determined by the value (1 + ε2 /2) g/k and depends on the amplitude of the

28

1 Linear and Nonlinear Surface Waves

Fig. 1.2 The form of a weakly nonlinear Stokes wave

wave-the greater the amplitude, the greater the speed. Figure 1.2 shows the shape of the undulated surface for ε = 0.3. The crests of this weakly nonlinear wave are sharpened, and the depressions are more slick than those of the sinusoidal wave. An important property of the Stokes wave, which is absent in infinitesimal waves, is that in addition to momentum and energy, this wave also carries mass. To show this, based on the solution (1.79) for surface waves of small but finite amplitude, the equations for the Lagrange coordinates (α, β) of liquid particles are written dα dt dβ dt

= u = kacexp(k(z + η)) cos(kx − kct + kα) . = v = kacexp(k(z + η)) sin(kx − kct + kα)

(1.81)

Calculating the time derivative of the first Eq. (1.81) using the system itself (1.81) leads to a sequence of relations = ε c dd βt exp(k(z + η)) cos(kx − kct + kα)− ,   (3.12) −k ε dd αt − c exp(k(z + η)) sin(kx − kct + kα) = kc dd βt d2 α d t2

(1.82)

where ε = ka 1. Integration (1.82) gives rise to the relation β=

1 dα + δ(x, z). kc d t

(1.83)

Substituting (1.83) into the first Eq. (1.81) converts it to the form   dα 1dα = ε cexp(k(z + δ(x, z)))exp cos(kx − kct + kα). dt c dt The writing of the Eq. (1.84) in the form −

  1dα 1dα exp − = ε exp(k(z + δ(x, z))) cos(kx − kct + kα) c dt c dt

(1.84)

1.4 Weakly Nonlinear Waves of Finite Amplitude

29

allows to express the derivative d α/d t by the ratio dα = −c W(−ε exp(k(z + δ(x, z))) cos(kx − kct + kα)), dt

(1.85)

where W(x) is the Lambert function, which is a solution of the equation W(x)exp(W(x)) = x whose properties are well studied [8, 9]. As follows from (1.81), the horizontal d α/d t and vertical d β/d t velocities of liquid particles are small compared to the propagation speed of the wave pattern. But there is no reason to talk about small displacements α and β in comparison with the wavelength. For this reason, the decompositions of trigonometric functions used by Stokes in his classical work [6] are at least incorrect. At the same time, it can be shown that the results themselves [6] remain valid. First, a semi-qualitative study of the Eq. (1.85) is performed. For this purpose the displacement is represented as an integral of the horizontal velocity d α/d t t α(x, z, t) = 0

d α(x, z, θ) d θ. dθ

(1.86)

Substituting (1.86) into (1.85) converts this equation to the form ⎛ ⎛ ⎞⎞  t  dα d α(x, z, θ) c− = −c W⎝−ε exp(k(z + δ(x, z))) cos⎝kx − k d θ⎠⎠. dt dθ 0

Since, according to (1.81), the ratio |d α(x, z, θ)/d θ| c is valid, the phase of the cosine function is mainly determined by the value kct, and for qualitative estimates in the zero approximation, we can write dα ≈ −c W(−ε exp(k(z + δ(x, z))) cos(kx − kct)). dt

(1.87)

Since z takes negative values, and the positive offsets of the function δ(x, z) cannot exceed the value of −z +ζ(x, t), where ζ(x, t) is the displacement of the free surface, then, given that k|ζ(x, t)| 1, there is an estimate of exp(k(z + δ(x, z)) < 1. Thus, the value of ε exp(k(z + δ(x, z))) has at least the first order of smallness over ε. This means that Eq. (1.87) can be represented as dα ≈ −c W(− p cos(kx − kct)), dt

p 1.

(1.88)

Figure 1.3 shows the behavior of the right side of the Eq. (1.88) as a function of time at p = 0.1. The time is calculated in conventional units proportional to the value 1/kc. The horizontal velocity is calculated as a fraction of the wave pattern transfer rate c.

30

1 Linear and Nonlinear Surface Waves

Fig. 1.3 Qualitative dependence of the horizontal velocity of liquid particles on time

Fig. 1.4 Horizontal displacement of liquid particles

The dotted horizontal line corresponds to the average speed value d α/d t for this graph. Its value 0.005 c in the estimated approximation corresponds to the Stokes flow velocity. The presence of a non-zero average velocity level indicates the existence of infinite transport of particles horizontally. This is confirmed by integrating the dependency (1.88) (and, consequently, the graph in Fig. 1.3). Figure 1.4 shows the horizontal displacement of liquid particles calculated according to the ratio t W(− p cos(kx − kc θ)) d θ,

α ≈ −c

p 1

0

at the same value of the parameter p. As you can see from Fig. 1.4, there is a constant trend, on which the periodic fluctuations are overlaid. That is, the horizontal movement of liquid particles is infinite. The obtained semi-qualitative results can be confirmed by the following calculations. Differentiating the second equation of the system (1.81) in time allows us to obtain, using the system itself, the equation dα d 2β −k + ck 2 dt dt



dα dt

2

 +

dβ dt

2  = 0.

(1.89)

1.4 Weakly Nonlinear Waves of Finite Amplitude

31

The use of Eqs. (1.82) converts (1.89) to the form 2   2 1 d 2α d 3α 2 2d α 2 dα − + c k − ck = 0. d t3 dt c d t2 dt

(1.90)

The introduction of a new variable A = k ξ and dimensionless time τ = ck t gives the Eq. (1.90) the form d 3A d A − + d τ3 dτ



d 2A d τ2

2

 −

dA dτ

2 = 0.

(1.91)

Since dd Aτ ≡ 1c dd αt and d α/d t can be set in the form d α/d t = ε cF(x, z, t), replacing dd Aτ = ε F(x, z, t) gives Eq. (1.91) the form d2 F + F −ε d τ2



dF dτ



2 +F

2

= 0.

(1.92)

Equation (1.92) is essentially an equation for horizontal velocity obtained exactly from the system (1.81). A regular solution of this equation with accuracy up to second-order terms by ε has the form         F = r cos 1 − ε2 r 2 + s 2 τ − s sin 1 − ε2 r 2 + s 2 τ + ε r 2 + s 2 + o(ε), where r = r (x, z) and s = s(x, z)are some coordinate functions. Thus, there is an equation for the velocity of a liquid particle dα dt

         = ε c r cos 1 − ε2 r 2 + s 2 kc t − s sin 1 − ε2 r 2 + s 2 kc t   , +ε2 c r 2 + s 2 + o(ε2 )

whose integration determines its displacement          2 2 2 2 2 2 r kc t + s cos 1 − ε r kc t + s + s α = a r sin 1 − ε   . +ε2 c r 2 + s 2 t + o(ε2 ) If now, in accordance with the system (1.81), put r = exp(kz) cos(kx), s = exp(kz) sin(kx), then the final result of calculations will look like     dα = ε c exp(kz) cos kx − 1 − ε2 c2 exp(2kz) kc t + ε2 cexp(2kz) + o(ε2 ) dt     . α = −a exp(kz) sin kx − 1 − ε2 c2 exp(2kz) kc t + ε2 cexp(2kz) t + o(ε2 ) (1.93)

32

1 Linear and Nonlinear Surface Waves

The ratio (1.83) defines vertical displacements of liquid particles by the expression     β = a exp(kz) cos kx − 1 − ε2 c2 exp(2kz) kc t + . +ε aexp(2kz) + δ(x, z) + o(ε2 )

(1.94)

The constant vertical displacement that occurred in (1.94), independent of the choice of the initial moment of time, must be eliminated, which is achieved by selecting the function δ(x, z) = −ε aexp(2kz). This result confirms the validity of the estimated ratios (1.87, 1.88). Finally, the vertical displacements of liquid particles are determined by the formula     β = a exp(kz) cos kx − 1 − ε2 c2 exp(2kz) kc t + o(ε2 ).

(1.95)

Expressions (1.93, 1.95) confirm the validity of Stokes’ results (but not the method of obtaining them). According to (1.93) for ε = 0.1, which for p = 0.1 corresponds to the zero approximation in the estimation calculations, the Stokes flow velocity is equal to 0.01 c. This result is the same as the estimate shown in Fig. 1.3. The form of a weakly nonlinear Stokes wave presented in Fig. 1.2 is obtained for the value ε = 0.3, which allows to clearly show its characteristic features. At the same time, this value of parameter ε violates the method of successive approximations applied to the source system (1.0). In order to estimate the limits of acceptable values of this parameter, it is necessary to set the degree of approximation of the obtained solutions, and then, based on the value of the desired error, calculate the restrictions imposed on ε. Let the expected relative error of the solution should not exceed 0.01. In this case, the calculation of the velocity and pressure fields based on the formulas (1.79, 1.80) with subsequent adjustment of the results by means of the Euler equation and the boundary conditions of the system (1.0) show that the √ necessary error value for describing surface perturbations is achieved at ε ≤ 0.1 · 2 ≈ 0.14. Figure 1.5 shows for comparison the Stokes wave calculated for the specified limit value of parameter ε (solid curve) and the normal sine wave shown here for comparison (dotted line).

Fig. 1.5 The Stokes wave for ε = 0.14

1.4 Weakly Nonlinear Waves of Finite Amplitude

33

It can be seen that given a sufficiently high-quality description of the surface wave (the relative error does not exceed the value 0.01), the difference between the Stokes wave and a simple sine wave (which is a solution in the class of infinitesimal waves) is extremely small and is not determined by the naked eye in nature. However, this fact does not detract from the value of the work of Stokes, who proposed a method for describing nonlinear phenomena, modifications of which were the basis for many further studies.

1.5 Soliton and Soliton-Like Solutions Another line of research on finite-amplitude waves was developed in [10, 11], which investigated the problems of propagation of surface perturbations with characteristic longitudinal scales significantly exceeding the depth of the liquid. In particular, special attention was paid to the description of the experimentally observed Russell wave, which is a single “hump” running along the channel with water flowing in it. Rayleigh’s approach to this problem is that the wave is described simultaneously using both the velocity potential and the stream function. The flow is considered stationary, that is, it depends on two variables ξ = x − ct and z, and c is the constant velocity of those liquid particles that are located at a great distance from the surface perturbation both in the direction of wave propagation and in the opposite direction. Since, as follows from (1.0), in this formulation, both the velocity potential and the stream function satisfy the Laplace equation ϕ = 0, ψ = 0,  =

∂2 ∂2 + ∂ ξ2 ∂ z2

and boundary conditions  ϕz z=0 = 0,

 ψξ z=0 = 0,

where (according to Rayleigh) z = 0 is the level of the flat bottom, and the level of the undisturbed surface is given by the relation z = h, these values allow to use representations of the forms ϕ=F−

z3 z5 z 2  z 4 (4) F + F − . . . , ψ = z F  − F  + F (5) , 2! 4! 3! 5!

F = F(ξ). (1.96)

Superscript strokes and indexes for the function F denote the order of the derivative. In this case, it is assumed that the function F and its derivatives experience small changes when its argument ξ changes by values of the order of the depth of the liquid h.

34

1 Linear and Nonlinear Surface Waves

Rayleigh assumed that at the bottom ψ = 0, and on the agitated surface ψ = −ch. This assumption together with the dynamic boundary condition   ϕξ 2 + ϕz2 z=h+ζ = ψξ2 + ψz2 z=h+ζ = c2 − 2gζ,

(1.97)

allows to convert (1.96) to the form F  2 − (h + ζ)2 F  F  + (h + ζ)2 F  2 + . . . = c2 − 2gζ . 3 (h + ζ)F  − (h+ζ) F  + . . . = −ch 3!

(1.98)

Successive exclusion of the function F from (1.98), taking into account the slow nature of its changes, allows to obtain an approximate equation for the deviation of the free surface   g(h + ζ) ζ2 2 . (1.99) ζ =3 2 1− h c2 Analysis (1.99) shows that ζ = 0 only in the cases ζ ≡ 0 and ζ = c2 /g − h. The first case is not of interest, since it describes the undisturbed state of the liquid. To implement the second case, the wave must necessarily consist of either one elevation or one cavity. The wave of a cavity is absolutely unstable and therefore is not considered in the future. Let the elevation wave have a maximum deviation ζ = a (for certainty, it is assumed that this maximum is located at the point ξ = 0). Then it follows from (1.99) that the rate of propagation of such a “hump” is determined by the expression c2 = g(h + a).

(1.100)

The resulting ratio indicates that the higher the wave, the greater the speed of its propagation. Equation (1.99) with (1.100) is divided into two equations ζ

ζ =± 1 − ζ/a, b = h b





h+a , 3a

(1.101)

the “plus” sign refers to the area ξ < 0, and the “minus” sign refers to the area ξ > 0. As a result, the solution (1.101) has the form ζ(ξ) =

a , ch (ξ/2b) 2

(1.102)

the form of which is shown in Fig. 1.6. In Fig. 1.6, the dotted line shows the level of the undisturbed surface, and the arrow indicates the direction of movement of the liquid and wave. Here it is necessary to make a number of comments on the received solution. First, this solution is approximate, since the derivation of equation (1.99) leaves only

1.5 Soliton and Soliton-Like Solutions

35

Fig. 1.6 The form of a solitary Russell-Rayleigh wave

terms up to and including the second order of smallness with respect to the small parameter ε = h/L ξ , where L ξ is the characteristic horizontal scale of the solitary wave.  , and then the Second, as follows from the form (1.101) L ξ = 2b = 2h h+a 3a condition must be met  3a 1

1, (1.103) ε= 2 h+a from which it follows that a h, and this, in turn, negates the result (1.100), which is often used to explain the increased speed of solitary waves with a significant increase in their amplitude compared to the depth of the liquid. The fact of such an increase has been experimentally proved by numerous observations, but it is not supported by the theory presented here, since the restriction (1.103) does not allow applying the results obtained when a ∼ h. Third, the shape of the surface perturbation (1.102) cannot be achieved by creating a wave of arbitrary height. This is due to the fact that in advance, before generating a wave of height a, the undisturbed liquid must move at a speed determined by the expression (1.100), which contains the value of the expected deviation. Thus, you can only create such a solitary wave by accident. And finally, fourth, generation of waves described by displacement of solid boundary (and Russell observed the formation of such waves after the sudden stop in the canal long boats) are prohibited by law by the conservation of mass, which  +∞ requires the fulfilment of condition −∞ ζ(ξ) d ξ = 0 that for waves of the form (1.102) is obviously violated. Despite the mentioned unsatisfactory properties of this solution, it will be used in the future when comparing with the results of describing surface perturbations of the liquid obtained in the framework of alternative approaches. Further analysis of soliton surface waves was carried out by Korteweg and de Vries [12], who studied non-stationary waves traveling along the surface of a constantvelocity liquid of a finite depth, using a modification of the Rayleigh method [11], so a detailed presentation of the method [12] does not make sense. The result of the

36

1 Linear and Nonlinear Surface Waves

research was the famous KdV equation describing the evolution of the deviation of the free surface from its unperturbed position ∂ζ 3c ∂ = ∂t 2h ∂ x



 ζ2 σ ∂ 2ζ 2 , + ζ0 ζ+ 2 3 3 ∂ x2

(1.104)

√ where h is the depth of the liquid, c = gh is the velocity of its flow, σ = h 3 /3 − α h/g and ζ0 is some small (indefinite) constant. The value α included in the definition of parameter σ is the surface tension coefficient normalized on the density of the liquid. The solution of the Eq. (1.104) is one-dimensional cnoidal waves, which in the stationary limit, with the appropriate choice of the parameter ζ0 , pass into a solitary Rayleigh wave (1.102). The development of the approach [13] in the direction of studying twodimensional soliton solutions was carried out in the work of Kadomtsev and Petviashvili [13], which considered the perturbation of solutions (1.104) when the onedimensionality of this equation is violated. Taking into account the transverse dispersion of waves on a two-dimensional surface led to the appearance of the CP equation of the form    3c ∂ ζ2 2 σ ∂ 2ζ ∂ 2ζ ∂ ∂ζ = λ2 σ 2 , − + ζ0 ζ+ (1.105) 2 ∂x ∂t 2h ∂ x 2 3 3∂x ∂y where λ = L x /L y 1 is the ratio of the characteristic spatial scales of the wave in the directions x and y, respectively. Thus, the KP Eq. (1.105) describes such soliton solutions whose transverse dispersion is small. Equations (1.104, 1.105) are necessary for further comparison with the results of alternative approaches to describing surface waves. In the previous sections, we mainly considered stationary surface waves. Practical implementation of such waves is more likely to be achieved in laboratory experiments, while numerous field observations show that the wave patterns observed in nature are rather unsteady or, under certain conditions, weakly unsteady. Research on the problem of weakly unsteady waves led to the discovery of the envelope soliton (sometimes, in relation to waves on water, called the Schrödinger soliton) [14]. The research [14] is based on a weakly nonlinear Stokes wave (1.79), whose amplitude and phase characteristics are considered to be slowly changing functions of distance and time, so that the free surface perturbation is described by the relation 3k 2 a 3 ka 2 cos(2θ)+ cos(3θ)+ . . . , 2 8 ∂θ ∂θ a = a(ε x, ε t), ≈ ω, ≈k ∂t ∂x ζ = a cos(θ)+

(1.106)

1.5 Soliton and Soliton-Like Solutions

37

where ε is a small parameter whose physical nature is not clear, but its smallness is postulated. The decomposition corresponding to this model for the velocity potential has the form   a ω a ϕ = ωka sin(θ)ekz + kt cos(θ) + k 2 x (1 − kz)cos(θ) ekz + , (1.107) 2 + ω2a sin(2θ)e2kz + . . . moreover, (1.107) provides satisfaction of the potential of the Laplace equation with the desired degree of accuracy. The relations (1.106, 1.107) by direct substitution allow us to calculate the average Witham’s Lagrangian, defined by the expression L=

1 2π

2π 0

 ζ   1   2   2 ϕt + 2 ϕx + ϕz + gz dz ≈ dθ 

−h

≈ − ω4ka + aa  + 2ktt + 2 2

ω a a a 2 ω2 a  2 ga 2 + t + 4kx2 z + 8k 3x + 4  4k 2 4 3ω aaxt 3ω2 aa  + 8k 3 x x + gk8a 4k 2

.

(1.108)

The application of the variational principle to (1.108) gives, when the phase function θ varies, the law of conservation of energy 1

∂ a2 ∂  2 ca = 0, c = g/k, + ∂t ∂x 2

(1.109)

where c is the group velocity of the wave in the linear approximation, and when varying by the amplitude function a, a nonlinear dispersion equation is formed ω=



  k2a2 a  gk 1 + + x x2 . 2 8k

(1.110)

The phase velocity determined on the basis of (1.110) differs from the phase velocity of an ordinary weakly nonlinear Stokes wave (1.79) by the presence of a term aaxx /8k 2 responsible for the spatial evolution of the amplitude of a weakly unsteady wave. The mentioned term is omitted in the Lighthill’s theory [3], which is the reason for the lack of solutions of the envelope soliton type. In addition to the relations (1.109, 1.110), the incompressibility equation must be fulfilled, which, using the representations (1.106, 1.107), takes the form kt + ωx = 0.

(1.111)

The set of relations (1.109–1.111) forms a closed system of equations describing the evolution in space and time of instantaneous values of the frequency ω, the wave number k, and the amplitude of the wave a. It is assumed that the frequency and wavenumber variations are significantly less than the amplitude variations. If this

38

1 Linear and Nonlinear Surface Waves

assumption is rejected, additional terms containing the values k xx and k x ax appear in the expression for the Lagrangian. Since the frequency and wavenumber variations are considered small, representations of the form are acceptable ˜ ˜ k = k0 + k, ω = ω0 + ω,

(1.112)

where ω0 and k0 , are constant in space and time, and ω and k˜ describe small deviations from the constant values of the corresponding quantities. Accordingly, the phase function is represented in the form θ = θ0 + θ˜ ,

(1.113)

˜ ˜ θ˜x = k. θ0 = k0 x − ω0 t, θ˜t = −ω,

(1.114)

and the relations are valid

The decomposition (1.114) implies the existence of a constant carrier wave with characteristics denoted by zero indices. Substitution (1.12–1.114) in (1.109–1.111) with subsequent exclusion from the equations of quantities ω˜ and k˜ forms a system of paired nonlinear Schrödinger equations at + θ˜t +

ω0  a 2k0 x ω0 ˜  θ 2k0 x

− −



ω0 ˜  θ 8k02 x x ω0 ˜  2 θx 8k02

 + 2θ˜x ax = 0  . ω k2 a  − ax x + 02 0 a 2 = 0

(1.115)

The system (1.15) is called the pair Schrödinger equations because the introduc˜ tion of a complex variable A = aei θ transforms it into a single equation  i

At +

ω0  A 2k0 x

 −

ω0  ω0 k02 2 |A| A = 0, A − 2 8k02 x x

(1.116)

which is the well-known nonlinear Schrödinger equation. As shown by Zakharov and Shabat [15], Eq. (1.116) allows us to obtain exact solutions under the condition A → 0 when |x| → ∞. For surface waves on water, the solution (1.116) or (1.115) has the following properties. The initial wave packet of any shape splits into several solitons and an oscillatory “tail” as it propagates. The number and structure of these solitons and the structure of the “tail” are completely determined by the initial conditions. The oscillatory “tail” is relatively small compared√to the soliton part and its amplitude decreases over time according to the law 1/ t. Each soliton represents a solution (1.116) describing a progressive wave of the form

1.5 Soliton and Soliton-Like Solutions

39

−i n (x,t)

e ζn = an ch(ψ , ψn (x, t) = n (x,t))

n (x, t) =

k02 an2 2

ω0 t +

4k02 ω0

√

2k02 an ϕn (x, t)

vn (ϕn (x, t) + θn )

,

(1.117)

 ω0 t, and an and vn are the amplitude and velocity + v where ϕn (x, t) = x − X n + 2k n 0 (relative to the speed c0 = ω0 /2k0 ) of the n -th soliton, and the values X n and θn represent its initial positions and phase. In contrast to the soliton solution of the KdV Eq. (1.104), the amplitude an and velocity vn are not related by any explicit relation. These solitons are stable in the sense that they do not interact with each other, except for changes in their initial characteristics X n and θn . The time scale of soliton formation (1.117) from some initial surface perturbation is directly proportional to the length of the perturbation area and inversely proportional to the amplitude of the initial impact. For an initial perturbation with small initial variations ω˜ and k˜ a rough estimation of the number of solitons into which the wave profile decays, is defined by the expression 1√ 2 2k0 a0 n= π

+∞ f (x) d x,

(1.118)

−∞

where the function a0 f (x), 0 ≤ f (x) ≤ 1, sets the shape of the initial perturbation profile of the free surface of the liquid. As follows from (1.117), a separate soliton is a wave consisting of an envelope described by a function ch(ψan n(x,t)) and its frequency filling cos n (x, t). For this reason, in the theory of surface waves, these solitons are called envelope solitons. The envelope and its frequency filling move in the positive direction of the x axis at different speeds! Figure 1.7 shows the evolution over time of a certain soliton from possible solutions (1.117). One can see that the frequency fill (solid lines) shifts over time relative to the envelope (dotted lines). The physics of ocean waves, presented here, are generally called wave zugs. Careful field observations show that even small sinusoidal (at first glance) waves running across the surface of the seas and oceans are wave zugs. The number of carrier wave maxima ranges from 7 to 21. The average” length “ of the observed zugs is seventeen maxima. The highest maximum is approximately in the middle of the envelope. This fact can be applied to explain the “ninth wave” phenomenon during a storm, assuming that storm waves are also described by solutions of the resulting form (1.117), while the highest maximum is on average the ninth. Using solutions (1.117) to explain storm events is not entirely correct. The problem is that the Eqs. (1.115, 1.116) are obtained using a model based on a Stokes weakly nonlinear wave. The weak nonlinearity of this wave means that the ratio ka 1 used in the study of the results presented here is correct. At the same time, it should be remembered that this condition indicates the gently sloping of surface waves, and therefore the solutions presented here describe gently sloping zug waves, the use of which to explain some characteristics of storm events

40

1 Linear and Nonlinear Surface Waves

Fig. 1.7 Dynamics of the envelope soliton

is more than doubtful. On the other hand, it is obvious that envelope solitons have an important independent significance in the theory of surface waves, since they are an intermediate element of the wave pattern, located between the classes of stationary and non-stationary waves.

1.6 Alternative Approach to the Description of Surface Waves Various approximate methods for describing surface waves, developed mainly in the XIX century and described in the previous paragraphs, affect a fairly narrow class of physical problems under study and, consequently, possible types of wave perturbations. Alternative approaches to describing waves on water that have emerged over the past 50 years are primarily aimed at obtaining accurate and comprehensive ratios. One of the first is the work [16], in which an equation was obtained describing steady waves of finite amplitude on the surface of an infinitely deep liquid flowing at a constant speed c. In this case, the class of waves is limited to waves that are mirror-symmetrical with respect to verticals passing through any ridge. In the space occupied by the liquid, an infinitely deep region is distinguished, bounded by a free surface and two vertical lines passing through neighboring depressions of the wave. At first, this region is displayed on a half-band in the plane of a complex variable χ = ϕ + i ψ, with the potential ϕ changing within [−c λ/2, +c λ/2] (λ is the wavelength), and the stream function ψ within (−∞, 0]. When the free surface is displayed in this way, the segment responds ψ = 0 and

1.6 Alternative Approach to the Description of Surface Waves

41

ϕ ∈ [−c λ/2, +c λ/2] then the half-band is mapped to the plane of the auxiliary complex variable σ = ξ + i η using the transformation χ=

cλ ln σ, 2π i

dχ cλ = , dσ 2π i σ

(1.119)

and is converted to a circle of unit radius with a section along the segment ξ = [−1, 0]. On the other hand, the introduction of a complex variable y = x + i z, where x and z are the coordinates of the natural physical space in which the surface wave exists, allows us to determine the transformation of the form dy λ f (σ) =− , dσ 2π i σ

(1.120)

where the function f (σ) is holomorphic in the circle |σ| ≤ 1 and does not vanish in its center. In other words, this function has the form f (σ) = 1 + a1 σ + a2 σ2 + a3 σ3 + . . . ,

(1.121)

so a turn by turn division (1.119) by (1.120) is acceptable, which gives the result c dχ =− . dy f (σ)

(1.122)

If σ = 0, which in physical space corresponds to an infinite depth, it follows from (1.122) that the velocity of the liquid is c, in full accordance with the original statement of the problem. Using boundary conditions in terms of a variable σ means that |σ| = 1 or σ = ei θ . Substitution of this ratio in (1.120) leads to the expression    λ dζ = − Im f ei θ , dθ 2π

(1.123)

where ζ is the shape of the undulated surface. Using (1.123) and a dynamic boundary condition on a free surface generates the relation   c2 d g λ   i θ   =    Im f e . (1.124) d θ f ei θ f e−i θ π  i θ The function is divided into real and imaginary parts by means of an  i θ f e expression f e = A(θ) + i B(θ), as a result of which (1.124) takes the form   1 d gλ = κ2 B, κ2 = . 2 2 dθ A +B π c2

(1.125)

42

1 Linear and Nonlinear Surface Waves

If we represent the functions A(θ) and B(θ) in terms of the angle ω(θ) of inclination of the particle velocity on the free surface to the horizontal axis and the ratio of the liquid velocity at infinity and on the free surface, then from (1.125) follows   1 d = κ2 R sin ω. d θ R2

(1.126)

From the theory of functions of a complex variable, it is known that on the circle of a circle of unit radius, there is a connection between the real g(θ) and imaginary h(θ) parts of any holomorphic function in the circle 1 g(θ) = 2π

2π 0

  sin  h ( p) ln  sin 

p+θ 2 p−θ 2

    dp + const, 

(1.127)

provided that h(2π − θ) = h(θ). If now inside the circle |σ| = 1 consider the function W = −ω(θ)+i ln R(θ) ≡ i ln f (σ), one can see that this function is holomorphic, and if we put g(θ) = −ω(θ) and h(θ) = ln R(θ) it becomes clear that the relation (1.127) applies to it, the use of which and the Eq. (1.126) leads to the relation 1 ω(θ) = 6π

2π 0

  sin μ sin ω( p)  ln  p  sin 1 + μ sin ω(q) dq

p+θ 2 p−θ 2

  3 3   dp, μ = κ2 R 3 (0) = κ2 f 3 (1).  2 2

0

(1.128) The expression (1.128) is a Nekrasov’s Equation [16] for steady-state periodic waves. The unknown desired function in this equation is the angle of inclination ω(θ). For waves of the specified class, this is an exact equation, but no one has yet been able to get an exact solution to it. Approximate solutions of this equation allow us to obtain known results for infinitesimal waves and waves of small but finite amplitude. The representation of wave characteristics by means of non-local relations (this is the type of Eq. (1.128)) was developed, after a certain period of time, in other methods of describing surface waves that are alternative to the standard ones. As has been repeatedly emphasized earlier, one of the most difficult problems in the problem of describing surface waves is the problem of setting boundary conditions on an unknown free surface. The desire to simplify this problem led to the work [17], the main idea of which is to transform the area occupied by a liquid changing in space and time into a simple area with fixed boundaries. If we denote the distance from the bottom to the surface with the symbol h(x, t), then the first mathematical

1.6 Alternative Approach to the Description of Surface Waves

43

problem is to transform the area −∞ < x < +∞, 0 ≤ z ≤ h(x, t) in the band −∞ < ξ < +∞, 0 ≤ ζ ≤ 1. This means that you need to find the x(ξ, ζ, t) and z(ξ, ζ, t) functions that would provide this transformation. Using the result of mathematical work [18], you can write the required transformations explicitly sin(πζ) z(ξ, ζ, t) = 2

+∞ −∞

h(ξ − ξ , t) d ξ , x(ξ, ζ, t) = cos(πζ) + ch(πξ )

ξ 0

∂z  (ξ , ζ, t) dξ . ∂ζ (1.129)

If we describe a surface wave based on the velocity potential ϕ(x, z, t), then the corresponding boundary value problem   ϕ = 0, ϕz z=0 = 0, ϕz − h x ϕx z=h = h t ,  = ∂ 2 /∂ x 2 + ∂ 2 /∂ z 2   , ϕt + 21 ϕx2 + ϕz2 z=h = g(h 0 − h) (1.130) converted in terms of new variables (ξ, ζ, t) to the form ϕ = 0,

 ϕζ ζ=0 = 0,

 ∂(x, z) ,  = ∂ 2 /∂ ξ2 + ∂ 2 /∂ ζ2 , ϕζ ζ=1 = xζ z t − xt z ζ = ∂(ζ, t) (1.131)

and the deviation of the free surface h − h 0 from the undisturbed level is determined using the formula  h0 − h =

ϕt +

   ∂(x, z) 1   2 ∂(ξ, ζ)  ∂(x, z) ϕξ − ϕξ + ϕξ + ϕζ 2 ∂(x, z) ∂(ζ, t) ∂(ξ, t) 2

   

ζ=1

,

(1.132) in which you need to substitute the solution of the problem (1.131). This solution, according to [18], has the form 1 ϕ(ξ, ζ, t) = − 2π

+∞ −∞

∂(x, z) (ξ − ξ , t) ln(cos(πζ) + ch(πξ )) dξ . ∂(ξ, t)

(1.133)

It is the set of relations (1.132, 1.133) that represents the result obtained in [17]. Since, in accordance with (1.129), the values of Jacobians ∂(x,z) and so on ∂(ξ,t) are expressed by integral relations including the unknown function h(x, z, t), the system (1.132, 1.133) belongs to the class of differential-integral systems. The resulting system (1.132, 1.133) is so complex and cumbersome that it can only be presented explicitly on a few pages, and its practical application for complex

44

1 Linear and Nonlinear Surface Waves

nonlinear problems is even more difficult than the original system (1.0). Numerical implementation of the solution of such a system is also almost unattainable. In [19], a different method of nonlocal description of surface waves was proposed. Based on the description of the velocity field through the potential, since the introduction of the stream function in true three-dimensional movements is impossible, the authors of the cited work reduced the system (1.0) to the form   ϕ = 0, ϕz z=−h = 0, ϕz − ζx ϕx − ζy ϕy z=−h = ζt  . ϕt + 21 (∇ϕ)2 z=ζ = α∇ · √ ∇ζ 2 − gζ

(1.134)

1+(∇ζ)

If one enter the notation q(x, y, t) = ϕ|z=ζ , then differentiating this definition of the value q by x and y provides the ratio   qx = ϕx + ζx ϕz z=ζ , q y = ϕy + ζy ϕz z=ζ , using which and the kinematic condition on the free boundary of the fluid, the values on the free surface of all components of the gradient of potential ϕ are determined, which is equivalent to the definition of all velocity components, so that ϕx =

(1+ζy2 )qx −ζx ζy q y −ζx ζt 1+(∇ζ)2

ϕz

=

, ϕy =

(1+ζx2 )q y −ζx ζy qx −ζy ζt

ζt +ζx qx +ζy q y 1+(∇ζ)2

1+(∇ζ)2

.

(1.135)

Substituting (1.135) into the dynamic boundary condition of the system (1.134) generates the equation qt

 2 ζt + ζx qx + ζy q y ∇ζ 1 α 2   + (∇q) + gζ − , = ∇·

2 2 ρ 2 1 + (∇ζ) 1 + (∇ζ)2

(1.136)

which is one of the two final equations that form the result achieved by the authors. The second equation is obtained using the following mathematical method. Let A(x, y, z, t) and B(x, y, z, t) be two distinct functions satisfying the Laplace equation. Based on this fact, the ratio is fair Az B + Bz A = 0, which allows writing the form ∂ ∂ ∂ (Az Bx + Bz Ax ) + (Az B y + Bz Ay ) + (A B  − Az Bx − Ay B y ) = 0. ∂x ∂y ∂z z z (1.137) Substitution in (1.137) A = ϕ, B = exp(i (k x x + k y y + k z z)) and integrating over the entire space occupied by the liquid, taking into account the kinematic boundary

1.6 Alternative Approach to the Description of Surface Waves

45

conditions at the bottom and the free surface, forms a non-local equation   +∞ +∞ sh(|k|(ζ + h)) d x d y exp(i(k x x + k y y)) i ζt ch(|k|(ζ + h)) + k · ∇q = 0, |k|

−∞ −∞

(1.138) which plays the role of the second equation we are looking for. We should pay attention to the fact that in the system (1.136, 1.138) the first equation is defined in the usual configuration space, and the second is defined in the phase space of wave vectors, which significantly complicates their analysis. In addition, the mentioned system has other negative properties, which will be discussed in more detail later. A different approach to obtaining non-local equations was developed in [20–22], which allowed to obtain one equation for one unknown shape of an undulated surface in the case of stationary waves, and to write down all the relations in the configuration space for non-stationary waves. At first, we consider a two-dimensional problem of steady potential waves on the surface of a homogeneous ideal incompressible heavy liquid of finite depth h. The effects of surface tension are not taken into account. The shape of the free surface of the medium is set by the function ζ (x, t) of the horizontal coordinate and time. The vortex-free velocity field v in the medium at z < ζ (x, t) is described by two components vx = u and vz = w. The system of equations of motion of the medium and boundary conditions, following from the system (1.0), in this statement has the form ut + u ux + w uz = − px − g ζx , wt + u wx + w wz = − pz ux + wz = 0, uz − wx = 0 , p |z=ζ = 0, w − u ζt z=ζ = ζt , w|z=−h = 0

(1.139)

where p is the pressure perturbation due to the surface wave normalized to density of the liquid. Since we consider a steady wave pattern propagating at the speed of c, the velocity and pressure fields, as well as the elevation of the free surface, are functions of a variable ξ = x − c t, i.e. ζ = ζ(ξ), u = u(ξ, z), w = w(ξ, z), p = p(ξ, z). The introduction of a stream pseudofunction defined by the relations u = c + ψz and w = −ψx = −ψξ allows to integrate the Euler equation of the system (1.139) p = p0 −

 1 2 ψ + ψz2 − gζ, 2 ξ

and reduce the remaining ratios to the form

p0 = const.,

46

1 Linear and Nonlinear Surface Waves

  ψξξ + ψzz = 0, ψξ + ψz ζξ  = 0,   z=ζ .   = 0, ψξ2 + ψz2  = 2( p0 − gζ) ψξ  z=−h

(1.140)

z=ζ

Since the main part of the surface wave energy is concentrated in the near-surface layer, we study the behavior of the ψ function at small values of z − ζ, when the decomposition is true ψ = 0 (ξ) + 1 (ξ)(z − ζ) + 2 (ξ)(z − ζ)2 + 3 (ξ)(z − ζ)3 + . . . .

(1.141)

Substituting (1.141) into the equation of motion and boundary conditions on the free surface of the system (1.140) defines the i (ξ) functions by the relations n = c Fn , Fn =

F0 = 0,

 gζ /c2 F1 = − 1−2 1+ζ 2

    1 Fn−1 ζ +2Fn−1 ζ −Fn−2 /(n−1) , n 1+ζ 2

n≥2

,

(1.142)

through constants (g, c) of the problem and the shape ζ of the free surface, while p0 = c2 /2 (a consequence of the condition u = p = 0 at ζ ≡ 0) and take the positive values of the square root. Thus, the problem (1.140) is reduced to a problem of the form ψξξ + ψzz = 0,

 ψξ z=−h = 0,

(1.143)

and moreover, solutions (1.143) near the surface z = ζ must meet the conditions (1.141–1.142). The solution (1.143), which satisfies the conditions of the velocity field limitation with unlimited growth of |ξ| and the absence of stationary flows, has the form ∞ ψ = −c z +

sh(k(z + h)) (A(k) cos(kξ) + B(k) sin(kξ)) dk, ch(kh)

(1.144)

0

in which the spectral functions A(k) and B(k) are unknown. The matching condition of the decomposition (1.144) near the free surface and the representation (1.141–1.142) generates a set of relations  ∞ sh(k(ζ + h)) k2 c an + [A(k) cos(kξ) + B(k) sin(kξ)] dk = c Fn , n !ch(kh) , ch(k(ζ + h)) 0 a0 = −ζ, a1 = −1, an = 0, n ≥ 2 (1.145) where the upper expression in curly brackets is selected for even values n, and the lower expression for odd values.

1.6 Alternative Approach to the Description of Surface Waves

47

Summing the ratios (1.145) with the weight coefficients (−ζ)n forms the equation ∞ th(kh)[A(k) cos(kξ) + B(k) sin(kξ)] dk = c F(ξ) = c

∞ 

(−ζ)n Fn , (1.146)

n=1

0

the solution of which has the form c A(k) = cth(kh) π

+∞ F(ξ) cos(kξ) dξ,

−∞

c B(k) = cth(kh) π

+∞ F(ξ) sin(kξ) dξ.

−∞

(1.147) Substituting in (1.144) at z = ζ of expressions (1.147) generates a nonlinear integral equation ζ(ξ) −

1 π

∞

⎤ ⎡ +∞ sh(k(ζ(ξ) + h)) ⎣ cos(k(ξ − η)) F(η) d η⎦ dk = 0, sh(kh)

(1.148)

−∞

0

in which the only unknown function is elevation ζ. It should be particularly noted that when obtaining this equation, no assumptions were made about the smallness of the elevation ζ compared to its characteristic horizontal size (for example, compared to the wavelength). The only limitation is that negative deviations of the free surface cannot exceed the depth of the liquid h by absolute value. A characteristic feature of equation (1.148) is its non-local character: the contribution of the integral term is determined by the entire surface of the liquid. For further research, equation (1.148) is reduced to a dimensionless form, for which a longitudinal scale L and a new longitudinal coordinate y = ξ/L are introduced. The deflection of the free surface is normalized to the depth of the liquid so that ζ = ε h Z (y), where ε = max |ζ|/ h, Z (y) is a new variable describing the surface wave. According to the definition of the parameter ε the expression max |Z | = 1 takes place. The ratio of vertical and horizontal scales is indicated by the symbol δ = h/L. Substituting new variables into the integral equation (1.148) converts it to the form ⎤ ⎡ +∞ ∞ sh(δ k(1 + ε Z(y))) ⎣ 1 cos( k(y − y  )) F(y  ) dy  ⎦ dk = 0. Z(y) − π sh(δ k) 0

−∞

(1.149)

48

1 Linear and Nonlinear Surface Waves

Here k is normalized to L, and the function F is defined by expressions F=

∞ %

(−1)n Zn Fn ,

F1 =



1−ε σ Z , 1+ε2 δ2 Z  2

n=1  2 εF Z  +2ε Fn−1 Z  −Fn−2 /(n−1) Fn = δn n−1 , 1+ε2 δ2 Z  2

σ=

2gh c2

n≥2

.

(1.150)

The validity of Eq. (1.149) is investigated for a number of known cases. The first object of research is infinitesimally small waves, for which the relations are fulfilled   ε σ |Z | 1, ε δ  Z   1. Using the expansions F ≈ −Z F1 + . . . ≈ Z − sh(δ k(1+ε Z )) sh(δ k)

εσ 2 Z 2

−

Z 2

  ε2 δ2 Z  2 + Z Z  +

ε 2 σ2 4

 Z 2 + o(Z 3 )

= ch(k εδ Z ) + sh(k εδ Z )cth(δ k) ≈ 1 + k εδ Z cth(δ k) + o(Z )

reduces (1.149), up to third-order terms of smallness, to the form ∞ +∞ Z (y) cos( k(y − y  ))( δ k cth(δ k) − σ/2) Z (y  ) dy  dk = 0,

(1.151)

0 −∞

which expresses the condition of zeroing the second-order terms of smallness in the original integral equation. In the case where the elevation of the free surface is a harmonic function, namely Z (y) = cos(κ y), it follows from (1.151) that there are no second-order corrections when performing the relation c2 κ = g th(κ h) ∼ ω2 = gκ th(κ h), ω = κ c,

(1.152)

which is exactly the dispersion equation for small-amplitude harmonic waves on the surface of a liquid of finite depth. In (1.152), the reverse transition to dimensional physical variables has already been made. If Z (y) is a superposition of oscillations Z (y) =

N 

an cos(nκ y),

(1.153)

n=1

the wavelengths of which significantly exceed the depth of the liquid h, i.e. n δ κ 1, the condition for the absence of second-order corrections takes the form N  2  c −gh an = 0. n=1

1.6 Alternative Approach to the Description of Surface Waves

49

%N Since in the general case n=1 an  = 0, the last equality is fulfilled only if the condition c2 = gh is met, that is, “the propagation speed does not depend on the wavelength, …, the wave profile remains unchanged during its forward movement“ [1]. Thus, using the proposed integral Eq. (1.148) gives all known results for the case of infinitesimal waves. In the case of waves of finite but small amplitude, when the n κ an 1 conditions are met for the representation (1.153), Eq. (1.149), up to and including the third order terms of smallness, takes the form    4ε σ Z + ε2 σ2 Z 2 + 4δ2 Z  2 + Z Z    , 3 + ε6σ (3σ2 + 16 δ2 κ2 )Z 3 − 36 δ2 Z Z  2 = (y, κ, ε, δ, σ)

(1.154)

where the somewhat cumbersome explicit form of the right side of the equation is not presented. The decomposition (1.154) is assumed to be regular, which makes it possible to assign a n-th degree of smallness to all coefficients an , with the exception of the value a0 . This value determines the value of the average surface level, that is a0 = ζ¯ , where the line above the symbol means averaging over the space. Arbitrariness in the choice of a0 , which is considered acceptable in [23], in fact leads to a violation of the law of conservation of mass for reservoirs of large but finite horizontal dimensions. To avoid this in the further it is assumed a0 = ζ¯ = 0. With this choice of a0 and performing integration operations, a system of algebraic equations is formed with respect to quantities ai , c and κ, the solution of which in physical variables has the form  2 a1 = a, a2 = κ 2a cth(κ h) 1 + 2sh21(κ h)

2 3 (κ h) − , a3 = κ 8a 3cth2 (κ h) + 2sh21(κ h) + 43 1+cth 2 sh (κ h)    h) − cth(κ 1 + 2sh21(κ h) 4 + 14th2 (κ h) − 18cth2 (κ h) 8

(1.155)

and the speed of propagation of surface waves is determined by the ratio c2 =

 g th(κ h) + κ2 a 2 cth(κ h) . κ

(1.156)

In the approximation of an infinitely deep liquid (h → ∞) from (1.156), it follows κ a2 3κ a 2 that a2 = 2 1 , a3 = 8 1 , so that the expression for the surface perturbation ζ(ξ) = a cos(κξ) +

3κ2 a 3 κ a2 cos( 2κξ) + cos( 3κξ) + o(κ2 a 2 ), 2 8

(1.157)

exactly coincides with the Stokes formula [6] in the ζ¯ = 0 representation, and the ratio (1.156) is reduced to a known dependence of the wave velocity on their

50

1 Linear and Nonlinear Surface Waves

amplitude c2 =

 g 1 + κ2 a 2 . κ

(1.158)

The above mentioned concerns those situations when the wave lengths of the components forming the wave were smaller or comparable to the depth of the liquid, and the amplitudes were quite small compared to the longitudinal scale. If the characteristic longitudinal size of the wave significantly exceeds the depth of the liquid, and the elevation is small compared to the depth, the relations ε, δ 1 are fulfilled. In this case, the waves are characterized by a large longitudinal scale, their spatial spectrum is concentrated near small and limited values of the wavenumber k, which, using the smallness of the parameters ε and δ, allows us to decompose the integrand into a double power series and reduce Eq. (1.149) to the form ε + Z δ3ε Z (σ − 2) 2ε + Z2 σ(σ + 4) ε8 + Z3 σ2 (σ + 2) 16 22  2 ε4 4 3 2 +Z σ (5σ + 16) 128 + Z + (3 − 2σ) Z + (6 − 2σ)ZZ ε 6δ . 4 +Z I V δ45ε + O(δ2 ε3 ) = 0 2

3

2

(1.159)

The solution of Eq. (1.159) is related to the study of the possible degree of smallness of the parameter δ in comparison with the parameter ε. In order for the decomposition (1.159) to be a series of natural degrees ε, the parameter δ must be set by a relation of the form   δ = ε p/2 δ0 + εδ1 + ε2 δ2 + . . . ,

p = 0, 1, 2, . . . , δn = O(1).

(1.160)

The σ = 2gh/c2 parameter is also decomposed in a series by ε σ = σ0 + εσ1 + ε2 σ2 + . . . , σn = O(1).

(1.161)

Substituting (1.160, 1.161) in (1.159) shows that for p = 0 there is a single solution Z = 0 that describes the absence of a surface wave. For p = 1, Eq. (1.159) takes the form  2 δ2 Z (σ0 − 2) 2ε + Z 8σ0 (σ0 + 4) + Zσ2 1 + 30 Z ε2  4     δ2 δ + 450 Z I V + Z δ20 1 + σ30 Z + 2δ30 δ1 + 20 + 13 Z 2 .  2   σ02  δ + 16 2 + σ20 Z3 + 60 + σ04σ1 + σ21 Z2 + σ22 Z ε3 + o(ε3 ) = 0

(1.162)

Equating the term to zero for ε gives σ0 = 2. As a result, the term for ε2 turns to zero if Z satisfies the equation Zσ1 δ20  3Z2 Z + + = 0. 3 2 2

(1.163)

1.6 Alternative Approach to the Description of Surface Waves

51

Equation (1.163) is a doubly integrated Boussinesq equation for waves traveling at a constant speed without changing their shape. The solution of Eq. (1.163), for which max |Z| = 1, has the form −1

, Z (y) = 1 − 3α m 2 sn2 (y, m), α = 1 + m 2 ± 1 + m 4 − m 2

(1.164)

where sn(y, m) is an elliptic Jacobi function defined by the relations ϕ sn(y, m) = sin ϕ,



y= 0

dθ 1 − m 2 sin2 θ

, m ∈ [0, 1],

and the coefficients δ0 , σ1 take values δ0 =

√

3 α , σ1 = ∓6α 1 + m 4 − m 2 . 2

(1.165)

Solution (1.165) √ describes a family of cnoidal waves whose longitudinal scales L(m) ≈ h/δ0 (m) ε and propagation velocity c2 (m) = 2gh/(2 + ε σ1 ) depend on the parameter m. In the limiting case m = 1 the shape of the surface wave and the speed of its propagation in physical variables are determined by the relations ζ(ξ) =

ch 2

εh  ξ 2

3ε 1+2ε/7

    , c2 = gh 1 + ε + o ε2 .

(1.166)

The resulting expressions describe a solitary Russell wave. The difference between the formula (1.166) for the wave velocity and the expression obtained by Russell, namely c2 = gh(1 + ε), is very small, and secondly, the Russell formula itself is approximate [1]. The difference between the ratio for  (1.166) 

3ε is also surface elevation and the ratio obtained by Rayleigh ζ(ξ) = ε h/ch 2 2ξ 1+ε insignificant. In addition, Rayleigh used an inaccurate Russell ratio and neglected members of the third order of smallness [1]. This concludes the study of Eq. (1.159), since when p > 1 it has no solutions, and in the case where there is a dependency  δ = εβ δ0 + εδ1 + ε2 δ2 + · · · , where β = p only a trivial solution Z = 0 is possible . In conclusion, the spatial spectrum of the Russell wave is shown in Fig. 1.8 below. The graph shows that the spectral components of the solution (1.166) are concentrated in a bounded region near zero, which ensures the convergence of the decomposition of the integrand function (and, consequently, Eq. (1.159)) and confirms the validity of the results obtained. A distinctive feature of the Eq. (1.148) is that the characteristic parameters of its solutions depend on the square of the velocity of propagation of surface waves. This

52

1 Linear and Nonlinear Surface Waves

Fig. 1.8 Spatial spectrum of the Russell wave

means that the solutions (1.164) remain valid even when the direction of travel is changed, in contrast to the solutions of the KdV equation, which can only propagate in one direction. Another consequence of the Eq. (1.148) is the absence of restrictions on the amount of decrease in the level of the free surface in the wave. As follows from (1.142), at a given wave velocity c, the condition 2gζ/c2 < 1 must be met (otherwise, the functional F(ξ) (1.146) becomes an imaginary quantity and Eq. (1.148) has no solutions). This means that elevations should be less than the value c2 /2g, while depressions are limited only to the bottom. This conclusion suggests that in nature, it is permissible to spread giant waves of lowering ocean levels, which can lead to catastrophic consequences in natural conditions. An important property of Eq. (1.148) is the fact that for waves whose longitudinal scales significantly exceed their vertical ones, the ratio is true F(ξ) ≈ ζ(ξ) + o(ζ(ξ)),

(1.167)

that is, in this case, the nonlinear functional F(ζ(ξ)) acts as a linear transfer function. In particular, the property (1.167) holds for all solutions given in this paragraph. As follows from the analysis of Eq. (1.148), this result means that in nature, the propagation of surface waves of any form is acceptable, if only their spatial spectrum is concentrated near small (compared to the reverse wave amplitude) values of wave numbers. The quasi-linear properties of the F(ζ(ξ)) functional also allow for the simultaneous existence of many different solitons, and their interaction with each other will be weak.

1.6 Alternative Approach to the Description of Surface Waves

53

The system of equations describing non-stationary waves has the same form (1.139), but it is impossible to introduce a variable ξ = x − ct that is so convenient for studying stationary perturbations. For this reason, the problem is investigated by introducing the potential ϕ (u = ϕx , w = ϕz ), which makes it possible to integrate the Euler equations   p = −ϕt − ϕx2 + ϕz2 /2 − gζ

(1.168)

and reduce the original problem (1.0) to the form ϕx x +ϕzz = 0   ϕt + ϕx2 + ϕz2 /2z=ζ = −gζ,

 ϕz − ϕx ζx z=ζ = −ζt ,

 . (1.169) ϕz z=−h = 0

In the ratio (1.168), the integration constant is assumed to be zero, since in the absence of a wave (ζ = 0, ϕ = 0), the pressure perturbations due to the surface wave must also be zero. Following the ideology developed in the previous paragraph, the potential ϕ near the surface z = ζ (x, t) is represented as a decomposition ϕ = 0 (x, t) + Φ1 (x, t)(z − ζ) + 2 (x, t)(z − ζ)2 + 3 (x, t)(z − ζ)3 + . . . , (1.170) substituting which into the Laplace equation of the system (1.69) leads to a sequence of relations n =

 /(n − 1) 1 n−1 ζx x + 2n−1x ζx − Φn−2 xx , n ≥ 2.  2 n 1 + ζx

(1.171)

Substitution of (1.170) into the kinematic boundary condition of the system (1.169) on a free surface generates a connection 1 =

0 x ζx + ζt , 1 + ζx2

(1.172)

and from the dynamic boundary condition, taking into account (1.172), we get the equation for the function 0     1 ∂ 0 2 ∂ 0   1  2  ∂ 0  1 + ζx2 + − ζx ζt − ζt + gζ 1 + ζx2 = 0. ∂t 2 ∂x ∂x 2

(1.173)

Thus, the system (1.169) is reduced to the problem of determining such a potential ϕ that satisfies the equations ϕx x + ϕzz = 0,

 ϕz z=−h = 0,

(1.174)

54

1 Linear and Nonlinear Surface Waves

and near the surface z = ζ(x, t) obeys the ratios (1.172–1.173). As follows from (1.174), the desired potential is represented as ∞

ch(k(z + h)) (A(k, t) cos(kx) + B(k, t) sin(kx)) dk. ch(kh)

ϕ(x, z, t) =

(1.175)

0

Decomposition (1.175) near the surface and comparison of the result with decomposition (1.170) leads to the following relation 1 n!

∞& 0

' kn ch(k(ζ + h)) (A(k, t) cos(kx) + B(k, t) sin(kx)) dk = n , n ≥ 0, sh(k(ζ + h)) ch(kh)

(1.176) in which, for even n, the upper expression is taken in curly brackets, and for odd n, the lower expression is taken. Multiplying each ratio (1.176) by and then summing the results in the ratio ∞ (A(k, t) cos(kx) + B(k, t) sin(kx)) dk = F(x, t) =

∞ 

(−ζ)n n .

(1.177)

n=0

0

As a result, from (1.177) follows 1 A(k, t) = π

+∞







F(x , t) cos(kx ) d x , −∞

1 B(k, t) = π

+∞

F(x  , t) sin(kx  ) d x  .

−∞

(1.178) Substituting (1.178) in (1.175) for results in an equation of the form 1 0 (x, t) = π

∞ 0

ch(k(ζ(x, t) + h)) ch(kh)

+∞

F(x  , t) cos(k(x − x  )) d x  dk.

(1.179)

−∞

Thus, the set of relations (1.171–1.173, 1.179) is closed and determines the dynamics of potential surface waves. It should also be noted here that no additional assumptions are made about the small elevation of ζ in comparison with the characteristic spatial mass (for example, with the wavelength). Again, the only limitation is that negative deviations of the free surface cannot exceed the depth of the liquid by an absolute value h .

1.6 Alternative Approach to the Description of Surface Waves

55

The efficiency of the obtained relations is checked on classical limit problems. Limit of infinitesimal waves. According to the definition of infinitesimal waves, the relation |kζ| 1 holds for all k = ∞, so that the decomposition is allowed ch(k(ζ + h)) ≈ 1 + k ζ(x, t) th(kh) + o(k ζ(x, t)), ch(kh)

(1.180)

substituting which in Eq. (1.179), with the retention of terms up to the first order of smallness in the parameter k ζ, reduces it to the form 0 (x, t) ∞  +∞    . ≈ π1 (1 + k ζ(x, t) th(kh)) 0 (x  , t) − ζ(x  , t)1 (x  , t) cos(k(x − x  )) d x  dk 0 −∞

(1.181) Since the ratios are valid f (x, t) = +∞  −∞

1 π

∞ +∞  0 −∞



f (x  , t) cos(k(x − x  )) d x  dk 



f (x , t) cos(k(x − x )) d x =

− k12n

+∞  −∞

∂ 2n f (x  ,t) ∂ x  2n



cos(k(x − x )) d x



,

Eq. (1.181) allows an equivalent entry ∞ +∞ 0 −∞

 th(kh) ∂ 2    (x , t) +  (x , t) cos(k(x − x  )) d x  dk ≈ 0. (1.182) 0 1 k ∂ x 2

In the infinitesimal limit, the Eqs. (1.172, 1.173) are converted to the form 1 ≈ ζt ,

∂ 0 + gζ ≈ 0. ∂t

(1.183)

Time differentiation (1.182) and usage (1.183) lead to the equation  ∞ +∞ th(kh) ∂ 2 ζ  ζtt − g cos(k(x − x  )) d x  dk ≈ 0. k ∂ x 2

(1.184)

0 −∞

Substitution in (1.184) of the spectral representation for the surface wave 

ζ(x , t) =

+∞  −∞

 A(λ, t) cos(λ x  ) + B(λ, t) sin(λ x  ) dλ

(1.185)

56

1 Linear and Nonlinear Surface Waves

shows that the spectral amplitudes (1.185) satisfy the equation Ftt + gλ th(λ h)F = 0,

(1.186)

where F is A or B. The corresponding (1.186) dispersion equation coincides with the known equation for small harmonic waves on water. When applying the condition of stationarity of a surface wave (ζtt = c2 ζx x , where c is the propagation velocity), the following result follows from (1.184, 1.185) [1]: the wave has a sinusoidal shape ζ(x, t) = a sin(μ(x −c t)+b) (a, b, μ are constants), and its velocity is determined by the expression c2 = g th(μ h)/μ. In the case when the surface elevation is a long-wave packet (i.e. carriers of spectral amplitudes A and B are limited by a region |λ h| 1), the ratio (1.184) takes the form of  ∞ +∞ ∂ 2ζ ζtt − gh cos(k(x − x  )) d x  dk ≈ 0, ∂ x 2 0 −∞

from which it follows that the elevation of the surface satisfies the equation ζtt − gh ζx x = 0.

(1.187)

√ According to (1.187), a long-wave packet propagates with the speed c = gh with almost no dispersion (Stokes’s result). Cnoidal waves, the Russell soliton, and the KdV equation. In the case of waves of finite amplitude a with a characteristic longitudinal scale L, for which the relations are valid a h L, two small parameters of the problem ε = a/ h and δ = h/L arise. Further research is convenient to carry out in dimensionless variables x  and t  , defined by the relations x = L x , t =

L  t , ζ = a Z (x  , t  ), max|Z | = 1, c

where c is the characteristic scale of the propagation speed of surface perturbations. Normalization of the wave number k in the Integral Equation (1.179) on the longitudinal scale L (k = k  /L) and the use of small parameters ε and δ allows for k  δ ≤ 1 the use of decomposition ch(k(ζ+h)) ch(kh) 2 ≈ 1 + εδ2 Z (ε

Z + 2) k 2 +

εδ4 Z 24

   , (ε Z )3 + 4(ε Z )2 − 8 k 4 + o k 6 δ6

(1.188)

where in the right part, as well as in the following, the strokes for dimensionless variables are omitted for brevity.

1.6 Alternative Approach to the Description of Surface Waves

57

Substituting (1.188) into (1.179) results in 0 (x, t) ≈ F(x, t) −

 εδ2 Z εδ4 Z  (ε Z )3 + 4(ε Z )2 − 8 FxIxVx x , (ε Z + 2) Fxx + 2 24 (1.189)

and the function F is defined in (1.177). The introduction of a function S(x, t) such that Z = St allows us to integrate the Eqs. (1.172, 1.173) within the accepted approximation)      ε  2  2  , 0 ≈ ε Lc −σ S + δ Stt − σ2 Sx 2 dt , 1 ≈ ε δ c Stt − ε σ Sx Sxt 2 (1.190) where σ = gh/c2 . In addition, from (1.171) follows 2 ≈

ε σ c  S . 2L x x

(1.191)

Equation (1.189) when substituting in it the relations (1.190, 1.191) is reduced, up to the terms of the third order of smallness by ε and δ, to the equation σ

Sxx

−

Stt

  ε σ2 δ2 σ I V  Sx x x x + ε σ Sx Sxt + + St Sxx + 3 2





Sx 2

 xx

dt ≈ 0. (1.192)

In the case of stationary waves, when S(x, t) = S(x − t), ζ(x, t) = ζ(x − t), the Eq. (1.192) is converted to the form δ2 σ  Z + (σ − 1) Z + εσ(1 + σ/2) Z 2 = 0, 3

(1.193)

from which follow the same results as from Eq. (1.163), confirming the operability of the relations (1.171–1.173, 1.179). In general, differentiating the Eq. (1.174) over time allows you to write it in the form σ Z xx − Z tt + δ3σ Z xI xVx x +     2    , ≈0 Z x dt +ε σ Z x Z x dt + Z Z xx dt t + σ2 2

(1.194)

xx

the structure of which suggests how this result can be achieved without using an integral description. According to the method [24], the original system (1.139) is reduced to a system of two equations of the form

58

1 Linear and Nonlinear Surface Waves

σ Z + δ Rt + ε 2δ Rx 2 − δ2 (1 + ε Z )2 Rtx x + . . . = 0  , 3 Z t + ε δ Rx Z x + δ(1 + ε Z ) Rxx − δ6 (1 + ε Z )3 Rxx x x + . . . = 0 2

3

(1.195)

where the function R is included in the ratio that determines the potential of the % 2 j 1 ∂ 2 j R(x,t) velocity field ϕ = ∞ j=0 (−δ (1 + z) ) (2 j)! ∂ x 2 j . The exception from the system (1.195) variable method of the perturbation theory proposed in [24] (in this method the value Z attributed to some vague order of smallness, contrary to its definition as the principle of building Z ∼ O(1)) leads to the famous Korteweg-de Vries where one saved the terms of the order ε and δ2 . At the same time, the combination of the form Ax x − Bt + δ2 A xI Vx x x /3, where A and B are the first and second equations of the system (1.195), respectively, while preserving the terms of the orders ε and δ2 leads to the Eq. (1.194). In contrast to the KdV equation, (1.194) has the property that if there is a solution to it Z (x − t), the function Z (x + t) also satisfies this equation. As a result, waves can propagate in any direction, and there is no artificial anisotropy of space that contradicts the original system (1.139). A similar anisotropy (for the types of waves studied in this section) is observed in the properties of approximate systems of equations proposed in [17, 19] and given in the preamble to this Chapter. At the same time, this anisotropy is not peculiar to the original exact systems [17, 19]. It occurs as a result of applying the perturbation theory developed in [24] to them. It should be emphasized that the anisotropy of the KDV equation obtained in the original paper [12] is artificial, since the authors initially considered the problem of small slow-scale long-scale perturbations of the fluid flow flowing in the positive direction of the horizontal axis at a speed of u = u0 . If we reject the smallness of perturbations, the use of the approach presented in this paper in relation to the statement [12] leads to the emergence of a pair of KDV equations     σ  2  σ εσ δ2 σ  Z Z x ∓ 1+ Z ∓ = 0, Z t ± μ − x μ μ 2μ2 3μ x x x

(1.196)

where μ = u0 /c, c is the characteristic scale of the velocity of propagation of surface perturbations. In (1.196) for a wave running in the positive direction of the axis, the upper signs are taken, for the opposite direction—the lower ones. In this case, for the second Eq. (1.196), the perturbation of the initial velocity field is determined by the value −2u0 plus a small correction compared to u0 . The pair Eq. (1.196) mutually pass into each other when replacing μ ↔ −μ (that is, when changing the direction of the original flow), which eliminates artificial spatial anisotropy. At the same time, the use of (1.194) is more appropriate when describing the type of motion under study, since this equation, in contrast to (1.196) and the equation of the original work [12], does not degenerate in the limit case u0 → 0.

1.6 Alternative Approach to the Description of Surface Waves

59

Envelope soliton and nonlinear Schrödinger equation. The propagation of a modulated Stokes wave of small but finite amplitude on infinitely deep water is considered as another example of the efficiency of the relations (1.171–1.173, 1.179). Following [14], the representation of this surface wave is given ζ = a cos  +

ka 2 3k 2 a 3 k3a4 cos 2 + cos 3 + cos 4 + . . . . 2 8 3

(1.197)

In the classical Stokes wave a = a0 , k = k0 are constants, and the phase function is defined by the expression  = k0 x − ω0 t. In the representation (1.197) the amplitude, wave number, and phase are considered to be slowly changing functions of the coordinate x and time t. The rate of change of these characteristics is set by the value of a small parameter ε = k0 a0 1 of the problem. This allows you to set the wave characteristics as a = a(x1 , x2 , . . . ; t1 , t2 , . . .), k = k0 + κ(x1 , x2 , . . . ; t1 , t2 , . . .) ,  = k0 x − ω0 t + ψ(x1 , x2 , . . . ; t1 , t2 , . . .) where xn = εn x, tn = εn t. Further research is carried out in dimensionless variables x  = k0 x, t  = ω0 t, κ = κ/k0 , a  = a/a0 with the strokes for shortening the entry omitted. A dimensionless surface wave is described by a function η = k0 ζ, and the reduction to a dimensionless form of relations (1.171–1.173,1.179) is achieved by introducing functions ϕn such that n = ω0 k0n−2 ϕn , n ≥ 0. As a result, the surface wave is defined by the expression ε2 (1 + κ) a 2 cos(2(x − t + ψ)) 2 3ε3 ε4 + (1 + κ)2 a 3 cos(3(x − t + ψ)) + (1 + κ)3 a 4 cos(4(x − t + ψ)) + · · · 8 3 (1.198)

η = ε a cos(x − t + ψ) +

and the ratios (1.171–1.173, 1.179) for infinitely deep water (in this case, the value ch(k(ζ + h))/ch(kh) is replaced by exp(kζ)) take the form of ∂ ϕ0 + ∂η ∂ x ∂ t ∂ ∂t

ϕ1

=0 2  1 + ∂∂ ηx +

1 ∂ 2∂x

  2  2 ϕ1 1 + ∂∂ ηx +

   ϕn−1 ηx x +2ϕn−1 x ηx −ϕn−2 x x /(n−1) n (1+ηx2 ) +∞ ∞       f x , t cos k x ϕ0 = π1 exp(k η) −∞ 0 f = ϕ0 − ηϕ1 + η2 ϕ2 − η3 ϕ3 + . . .

∂2η ∂ η ∂ t2 ∂ x

ϕn =

 − x  d x  dk

where σ = gk0 /ω20 in the absence of wave modulations.

+ σ ∂∂ ηx = 0 , (1.199)

60

1 Linear and Nonlinear Surface Waves

To determine the functions a, κ, ψ and ϕn , the expression (1.198) is substituted in the relations (1.199), and, according to the method of many scales [7], the functions ϕn , κ, parameter σ, and differential operators ∂/∂ x, ∂/∂ t are decomposed into series by a small parameter ε ϕn = ε ϕn1 + ε2 ϕn2 + ε3 ϕn3 + . . . , κ = εκ1 + ε2 κ2 + . . . σ = σ0 + ε σ1 + ε2 σ2 + . . . , σ0 = gk0 /ω20 . ∂ ∂ ∂ ∂ ∂ ∂ 2 ∂ 2 ∂ = + ε + ε + . . . , = + ε + ε + . . . ∂x ∂x ∂ x1 ∂ x2 ∂t ∂t ∂ t1 ∂ t2 The integration of the first Eq. (1.199) defines the form of the ϕ0 function, which is not given here because it is somewhat cumbersome. From the second equation follows the form of the function ϕ1 . In addition, the condition for the absence of secular terms in this equation implies σ0 = gk0 /ω20 = 1 (the dispersion equation of small-amplitude waves on the surface of an infinitely deep liquid), σ1 = σ2 = 0, and the amplitude a and phase ψ functions depend on  = 0. the variables a = a(r, t2 ), ψ = ψ(r ), where r = x1 − t1 /2, with ψrr The third Eq. (1.199) allows us to calculate explicitly the necessary functions ϕ2 and ϕ3 in this problem. Thus, they are defined in terms of unknown functions a and ψ and quantities ϕ0 and f , which must be substituted in the integral equation, i.e. into the fourth equation of the system (1.199). This substitution leads to the fact that the integral equation is identically satisfied up to members of the third order of smallness by ε inclusive. In order for a fourth-order member to go to zero, the following conditions must be met κ1 =

∂ψ = const, ∂r

  ∂ 2a ∂ψ 2 3 + 4a − a = 0, ∂ r2 ∂r

∂a ∂a∂ψ −4 = 0. ∂r ∂r ∂ t2 (1.200)

The integration of these equations determines the solution as a soliton of the envelope of the nonlinear Schrodinger equation [14]. At the same time, it should be noted that the use of Eq. (1.199), which initially reflect the specifics of surface waves, allows us to obtain paired Schrodinger equations (1.200) in a significantly simplified form (naturally, this is true only for the problem of waves on water). Description of surface waves using the stream function. The study of twodimensional movements of an incompressible fluid allows for an alternative approach involving the introduction of a stream function ψ such that u = ψz , w = −ψx . The use of decomposition ψ = 0 (x, t) + 1 (x, t)(z − ζ) + 2 (x, t)(z − ζ)2 + 3 (x, t)(z − ζ)3 + . . . for the stream function near the free surface, and the application of the method developed for the potential description of movements leads to a system of co-relations similar to the expressions (1.171–1.173, 1.179).

1.6 Alternative Approach to the Description of Surface Waves

n =

61

  ζ − Ψn−2 /(n − 1) 1 Ψn−1 ζx x + 2Ψn−1 x x xx , n≥2 n 1 + ζx 2

∂Ψ0 + ζt = 0 ∂x  1 ∂  2   ∂   1 1 + ζx2 + Ψ1 1 + ζx 2 + ζtt ζx + gζx = 0, ∂t 2∂x 1 0 = π

∞

sh(k(ζ(x, t) + h)) sh(kh)

0

+∞

(1.201)

F(x  , t) cos(k(x − x  )) d x  dk

−∞

% n where F(x, t) = ∞ n=0 (−ζ) Ψn . The study of the infinitesimal limit of the system (1.201) leads to the same results as in the case of describing movements through the potential. But in the case of long waves of finite amplitude, the system (1.201) is reduced to the equation σ Rxx − Rtt +

  δ2 I V  R ≈ 0, + ε Rx Rtt + 2Rt Rxt 3 tt x x

(1.202)

in this case Z = Rx , all the quantities in (1.202) are dimensionless. Equation (1.202) provides an alternative way to Eq. (1.192) to describe such waves. In the stationary limit, it has the same solutions. Differentiation (1.202) by x leads to the equation σ Z xx − Z tt +

  δ2 I V Z tt x x + ε Z Z tt d x + 2Z t Z t d x ≈ 0, 3 x

(1.203)

alternative to Eq. (1.194). Since Eqs. (1.194) and (1.203) describe the same process, their possible solutions must coincide, and not only in the stationary limit. To prove this fact, a function T is introduced, such that Z = Txt . As a result, the Eqs. (1.194, 1.203) take the form L(Tt ) + L(Tt ) +

δ2 σ V T + 3 x x x xt 2 δ TV +ε 3 x xttt

     ε σ Txx Txt t + σ2 Txx2 x ≈ 0  ,     Txt Ttt t + 21 Ttt 2 x ≈ 0

(1.204)

where L = σ∂ 2 /∂ x 2 − ∂ 2 /∂ t 2 . When substituting the solution T of any of the Eq. (1.204) in the remaining equation, a discrepancy of the form occurs     1 δ2 L(Txxt ) + ε Txt L(T ) t + (L(T )M(T ))x , 3 2

M = σ∂ 2 /∂ x 2 + ∂ 2 /∂ t 2 . (1.205)

62

1 Linear and Nonlinear Surface Waves

According to any Eq. (1.204), the value L(T ) has the order of smallness not lower than δ2 or ε. As a result, the residuum (1.205) has at least the same order of smallness as the discarded terms in the approximate Eq. (1.204). Thus, in the approximation of long waves of finite amplitudes, the solutions of Eqs. (1.194) and (1.203) coincide. Waves on a two-dimensional surface taking into account the surface tension. The description of two-dimensional surface perturbations ζ (x, y, t) (that is, the problem becomes truly three-dimensional) is achievable only on the basis of the velocity potential. Generalization of the previously described approach leads to a system of equations of the form n =

n−1  ζ + 2∇ n−1 · ∇ ζ −  n−2 /(n − 1) , n≥2 nG 2s

ζt + ∇0 · ∇ ζ G 2s  2     ζt + ∇0 · ∇ ζ ∂ Φ0 ∇ζ 1 2 = 0, +gζ+α∇ · + (∇ 0 ) − ∂t 2 G 2s Gs 1 =

(1.206)

0 (x, y, t)   ∞ ∞ ch(k(ζ+h)) +∞  +∞  F(ξ, η, t) cos(k (x − ξ)) cos(k (y − η)) dξ dη dk x dk y = x y 2 π ch(kh) 0 0

−∞ −∞

% 2 n 2 2 2 2 where, as before, F(x, y, t) = ∞ n=0 (−ζ) n , k = k x + k y , G s = 1 + (∇ ζ) , α is the surface tension coefficient normalized to the liquid density ρ. The infinitesimal limit of Eq. (1.206) gives known results for infinitesimal capillary-gravitational waves on water. In the case when the wave amplitude is significantly less than the depth of the liquid, the characteristic spatial scale L x of the waves along the x axis exceeds the depth, but it is less than the spatial scale L y along the y axis, the system (1.206) (ignoring the effects of surface tension) is reduced to an equation of the form   2 σ Z xx +λ2 Z yy − Z tt + δ3σ Z xI xVx x +     σ   2     . ≈0 Z x dt +ε σ Z x Z x dt + Z Z x x dt t + 2

(1.207)

xx

Here the same notation is used here as in (1.194), with λ = L x /L y 1. For λ → 0, Eq. (1.207) becomes into Eq. (1.194). A similar transition in the Kadomtsev-Petviashvili equation [13] leads to the KdV equation, additionally differentiated by x. Just as in the case of the KdV equation, Eq. (1.207) does not violate the isotropy of space, unlike the KP equation. But since the KP equation is derived from the KdV equation [13], its anisotropy is also artificial. The implementation of the approach [12] in relation to the system (1.206) generates paired KP equations

1.6 Alternative Approach to the Description of Surface Waves

63

         σ ∂ σ εσ δ2 σ λ2 σ    2  Z Zx ∓ 1+ − β Z Z , ∓ = ± Zt ± μ − x x x x ∂x μ μ μ 3 μ yy 2μ2

(1.208) which mutually pass into each other when replacing μ ↔ −μ. Due to the degeneration of KP-equations for u0 → 0, Eq. (1.207) is more adequate for describing this type of motion of a two-dimensional surface. Presented integro-differential variants of the description of one-number (1.171– 1.173, 1.179), (1.201) and two-dimensional (1.206) perturbations of the free surface of the liquid represent another approach to the study of wave movements. In particular, the obtained relations allow checking the developed surface wave models by direct substitution.

1.7 The Problem of Exact Solutions Since this section describes the approach to describing surface waves, which is certainly an alternative to the known standard methods, it is necessary to emphasize its difference from the approaches described in the previous paragraphs. The analysis of the relations given earlier shows that the kernels of integro-differential equations that form the results of these alternative approaches include harmonic functions. These functions appear as a result of the chosen representation of solutions of the Laplace equation for the potential (or stream function). Thus, the above methods implicitly include the decomposition of the observed surface wave by sine (or cosine) functions, which is not entirely justified, since these functions cannot be considered the most adequate when describing waves of arbitrary shape. Choosing other functions as eigenfunctions for solving the Laplace equation is no better for the same reason. In this regard, the here proposed approaches to the description of both stationary and non-stationary waves implement the principle of refusing to represent the solution of the Laplace equation based on the decomposition (or integral representation) of any eigenfunctions. This is the first distinctive feature of the method described below. The second distinctive feature is that the approach described in the future is based on the assumption of determinism of laminar fluid flows, the essence of which is the statement that for known physical fields—pressure and velocity field—the shape of the undulated surface is determined unambiguously. Conversely, knowing the shape of a surface leads to an unambiguous definition of the physical fields that generate this shape. The purpose of this section is to obtain an exact equation that includes only the shape of the surface wave, and relations that allow determining all physical fields in the medium based on the shape of the surface. A two-dimensional problem of propagation of potential waves over the surface of an inviscid liquid of finite depth h is considered. In the undisturbed state, the liquid

64

1 Linear and Nonlinear Surface Waves

flows along the horizontal axis x at a constant speed U. A perturbation is a deviation ζ of a free surface from its undisturbed state z = 0. The total velocity of the medium is given by the relation v = (U + u) ex + wez , where ex , ez are the unit orts of a two-dimensional Cartesian coordinate system (the z axis is directed against the gravitational acceleration vector g); u, w are the corresponding components of the velocity field perturbation. It is assumed that the perturbations of all physical fields of the problem and the deviation of the free surface propagate relative to the x axis at a constant speed c, so that the relations p = p(ξ, z), u = u(ξ, z), w = w(ξ, z), ζ = ζ(ξ, z) take place, where ξ = x − ct. To simplify the writing of equations, the constant atmospheric pressure pa and the pressure p in the liquid are normalized to the density of the liquid. The system of equations and boundary conditions of the problem in this statement has the form ut + (U + u)ux + wuz = − px , vt + (U + u)wx + wwz = − pz − g ux + wz = 0, uz − wx = 0,  p|z=ζ = pa , w − (U + u)ζx z=ζ = ζt , w|z=−h = 0. (1.209) For further analysis, the dimensionless variables (denoted by a stroke) x  = x/L, z = z/ h, t  = ct/L, ξ = ξ/L are introduced, where L is the characteristic longitudinal scale of perturbations. In addition, we introduce a dimensionless deviation η = ζ/a of the free surface (a is the characteristic amplitude of surface disturbances), dimensionless pressures p  = p/c2 , pa = pa /c2 and a dimensionless stream function ψ = ψ/ac, with u = ψz , w = −ψx . In the new dimensionless variables, the system (1.209) takes the form (to shorten the record, the strokes of the new variables are omitted) 

 pξ = αλψz ξ − λ2 ψz ψz ξ + μ2 ψξ ψξξ  pz = αλψzz − λ2 ψz ψzz + μ2 ψξ ψz ξ − σ μ2 ψξξ + ψzz = 0,

p|z=λη = pa ,

  ψξ + λψz ηξ 

z=λη

= αηξ ,

  ψξ 

. z=−1

=0 (1.210)

The following symbols are entered here: α = 1 − U/c, λ = a/ h, μ = h/L, σ = gh/c2 . Integration of the first two equations of the system (1.210) gives the result p = αλψz −

 λ2  2  2 μ ψz + ψξ2 − σ z + pa , 2

(1.211)

the use of which finally forms the mathematical model of the problem through the stream function and the free surface perturbation

1.7 The Problem of Exact Solutions

65

  μ2 ψξξ + ψzz = 0, αψz − λ2 μ2 ψz2 + ψξ2  = ση z=λη   .   = αηξ , ψξ  =0 ψξ + λψz ηξ  z=λη

(1.212)

z=−1

Since the Laplace  equationof the system  (6.41) admits a factorization of the form μ2 ψξξ + ψzz = ∂z − i μ ∂ξ ∂z + i μ ∂ξ ψ = 0, where i is an imaginary unit, its solution is represented in the form ψ = F(ξ − i μ(1 + z)) + G(ξ + i μ(1 + z)),

(1.213)

where F(ξ − i μ(1 + z)) and G(ξ + i μ(1 + z)) are arbitrary functions of their arguments. Since the kinematic boundary conditions of the system (1.212) on the free surface and bottom allow the following form   ∂  ψ|z=λη = αηξ , ψξ + λψz ηξ z=λη = ∂ξ

  ∂  ψ|z=−1 = 0, ψξ z=−1 = ∂ξ

they are integrated and converted to a form ψ|z=λη = αη,

ψ|z=−1 = 0.

(1.214)

The integration constant in the first relation (1.214) is set to zero, so that for η = 0 there is ψ = 0 (no perturbations in the medium in the absence of a surface wave). In the second relation (1.214), the choice of the zero constant simply determines the start of the stream function. Substituting (1.213) into the dynamic boundary condition of the system (1.212) and kinematic conditions (1.214) converts them to the form  2λμ2 F  G  + i αμ (F  − G  )z=λη + ση = 0 F(ξ − i μ(1 + λη)) + G(ξ+i μ(1 + λη)) = αη,

F(ξ) + G(ξ) = 0

. (1.215)

The stroke sign in (1.215) indicates the function derivative by its argument. From the first two equations of the system (1.215) follows ⎡ F(ξ − i μ(1 + λη)) =

i ⎣ α(ξ − i μλη) − 2λμ ⎡

ξ

−∞

i ⎣ G(ξ + i μ(1 + λη)) = − α(ξ + i μλη) − 2λμ

ξ

⎤ R(ρ) dρ⎦ + 0 ,

(1.216)

⎤ R(ρ) dρ⎦ − 0 ,

(1.217)

−∞

where R(ξ) =

(α2 − 2σλη)(1 + μ2 λ2 η (ξ)2 ), 0 is some arbitrary constant.

66

1 Linear and Nonlinear Surface Waves

To use the condition F(ξ) + G(ξ) = 0, in Eq. (1.216), a function f (ξ) is added to the variable ξ such that ξ + f (ξ) − i μ(1 + λη(ξ + f (ξ))) = ξ ∼ f (ξ) = i μ(1 + λη(ξ + f (ξ))). (1.218) Similarly, in Eq. (1.217), a function g(ξ) is added to the variable ξ such that ξ + g(ξ) + i μ(1 + λη(ξ + g(ξ))) = ξ ∼ g(ξ) = −i μ(1 + λη(ξ + g(ξ))). (1.219) As a result, the Eqs. (1.216, 1.217) take the form ⎤ ⎡ ξ+ f (ξ) i ⎣ α F(ξ) = R(ρ) dρ⎦ + 0 − αξ − , 2λμ 2λ −∞

⎡ i ⎣ G(ξ) = − αξ − 2λμ

ξ+g(ξ)

(1.220)

⎤ R(ρ) dρ⎦ − 0 −

−∞

α , 2λ

(1.221)

and from the boundary condition F(ξ) + G(ξ) = 0 follows ξ+g(ξ)

2i αμ +

R(ρ) dρ = 0.

(1.222)

ξ+ f (ξ)

Differentiating the Eq. (1.222) with respect to ξ generates the equation (α2 − 2σλη(ξ + f (ξ)))(1 + 2 f  (ξ)) = (α2 − 2σλη(ξ + g(ξ)))(1 + 2g  (ξ)), (1.223) in which there are unknown functions f (ξ) and g(ξ), to exclude which formal solutions of (1.218, 1.219) are used f (ξ) = ξ+ − ξ, g(ξ) = ξ− − ξ,

(1.224)

where ξ± = ξ ± i μ(1 + λη(ξ ± i μ(1 + λη(ξ ± i μ(1 + λη(ξ ± . . .)))))) (three dots mean the infinite repetition). Substituting (1.224) into (1.223) forms the final equation  i μα2 ηξ (ξ+ ) + ηξ (ξ− ) − σ(η(ξ+ ) − η(ξ− ))−  , −2i μλσ η(ξ+ )ηξ (ξ+ ) + η(ξ− )ηξ (ξ− ) = 0

(1.225)

1.7 The Problem of Exact Solutions

67

in which only the shape of the surface wave is present. The solution of the Eq. (1.225) allows us to restore the stream function from the known shape of the surface. Since, according to (1.213), the stream function is defined by the expression ψ = F(ξ − i μ(1 + z)) + G(ξ+i μ(1 + z)), on the basis of (1.220, 1.221) follows i αz − ψ= λ 2λμ

B(ξ,z)

R(ρ) dρ,

(1.226)

A(ξ,z)

where A(ξ, z) = ξ + i μ(1 + z) + g(ξ + i μ(1 + z)) = ξ + i μ z − i λμη(ξ− ), B(ξ, z) = ξ − i μ(1 + z) + f (ξ − i μ(1 + z)) = ξ − i μ z + i λμ η(ξ+ ) = A∗ (ξ, z). The symbol ∗ in the last expression means complex conjugation. And then it follows from (1.226) that i αz − ψ= λ 2λμ

∗ A (ξ,z)

R(ρ) dρ A(ξ,z)

and, since the integrand takes only real values, the stream function is also defined in the real domain. Substitution of z = λη(ξ) in (1.226) (that is, calculating the value of the stream function on a free surface) taking into account the equivalent writing of the Eqs. (1.218, 1.219) f (ξ − i μ(1 + λη(ξ))) = i μ(1 + λη(ξ)), g(ξ + i μ(1 + λη(ξ))) = −i μ(1 + λη(ξ)) results in ψ(ξ, λη(ξ)) = F + G|z=λη = αη, that is in full compliance with the boundary condition (1.214) on the free surface. The expression (1.226) for the stream function defines the components of the velocity field u = ψz =

  1  α i  −  f + g , v = −ψξ =  f − g , λ 2λ 2λμ

(1.227)

where  f = (1 + i μλη (ξ − i μ(1 + z))) R(ξ − i μ(z + λη(ξ − i μ(1 + z)))) . g = (1 − i μλη (ξ+i μ(1 + z))) R(ξ+i μ(z + λη(ξ+i μ(1 + z))))

68

1 Linear and Nonlinear Surface Waves

The pressure in the medium is determined by substituting (1.227) in (1.211). It is important to note that all the relations obtained above are exact, obtained without using any approximations. Thus, we obtain an exact differential-functional equation (1.225) describing stationary waves, the only unknown in which is the shape of the free excited surface [25]. Next, we consider the use of Eq. (1.225) on a number of particular examples. In the case of infinitesimal waves (λ 1), the expansion of (6.54) in a series by this small parameter leads to the equation for infinitesimal waves   i μα2 η (ξ + i μ) + η (ξ − i μ) − σ[η(ξ + i μ) − η(ξ − i μ)] = 0.

(1.228)

If the characteristic longitudinal scale L is significantly greater than the depth of the liquid h (this means that μ 1) from (1.228) follows (α2 − σ)η (ξ) + O(μ2 ) = 0 ⇒ α2 = σ ∼ c2 α2 = gh ∼ c = U +



gh. (1.229)

The obtained ratios determine the known result for waves on shallow water. In this case, as follows from (1.229), the waveform η is an arbitrary function, provided that the smallest longitudinal scale of this function significantly exceeds the depth of the liquid. Thus, a long-wave packet propagates over the surface of a liquid with virtually no dispersion. This is a well-known Stokes result obtained here in general form without reference to Fourier-harmonic decomposition. In the case of arbitrary values of the μvalue, the flatness of the infinitesimal waves allows us to represent (1.228) in the form ∞ 

2n (2n)

(−1) μ η n

n=0



σ α2 − (ξ) (2n)! (2n + 1)!

 = 0,

so a well-known dispersion relation α2 = σ th(kμ)/kμ ∼ ∼ (c − U)2 = g th(kh)/k follows from the characteristic equation (for solutions of the form η = exp(i kξ)). In the case of small gently sloping waves of finite amplitude, a small parameter κ = μλ 1 is introduced, according to which the Eq. (1.225) is decomposed, and the value of σ = σ0 +σ1 κ+σ2 κ2 +σ3 κ3 +. . . is decomposed in a series. The resulting solution in the case of a liquid of finite depth is somewhat cumbersome and is not given here, but its limit for an infinitely deep liquid (μ → ∞) has the form 3κ3 κ4 κ2 cos(2ξ) + cos(3ξ) + cos(4ξ), (1.230) 2 8 3 √ √ where c = U + gL(1 + κ2 /2) ∼ c = U + g/k(1 + k 2 a 2 /2), and k is the wave number (κ = ka). The relations (1.230) describe a well-known weakly nonlinear Stokes wave [6], the speed of which increases with the growth of its amplitude a. η(ξ) = κ cos(ξ) +

1.7 The Problem of Exact Solutions

69

The restoration of the stream function by the formula (1.226) is shown by the example of infinitesimal waves. In this case the approximations are valid A(ξ, z) ≈ξ + i μ z − iμλη(ξ + i μ z), R(ξ) ≈ α 1 − ασ2 λη(ξ)

B(ξ, z) ≈ ξ − i μ z + i μλη(ξ − i μ z)

,

Substituting of which in (1.228) leads to the ratio ⎡ α⎢ iσ ⎣η(ξ − i μ z) + η(ξ+i μ z) + 2 μα2

ψ(ξ, z) =

B(ξ,z)

⎤

⎥ η(ρ) dρ⎦.

A(ξ,z)

For infinitesimal waves η(ξ) = cos(ξ), so that the result has place ψ(ξ, z) = α cos(ξ)ch(μ z) +

sh(μ(1 + z)) σ cos(ξ)sh(μ z) = cos(ξ) , μα sh(μ)

that is, the stream function is restored correctly. Next, we consider the case of localized waves on the surface of shallow water (μ 1). Locality refers to the fulfillment of conditions  η(n) (ξ)ξ=±∞ = 0, n ∈ N .

(1.231)

A condition describing the law of conservation of mass is also imposed +∞ η(ξ) dξ = 0.

(1.232)

−∞

The expansion of the Eq. (1.225) in a row for μ 1 (no restrictions are imposed on the value λ) leads to the ratio μ μ 2  2 (4) (α2 − σ)η + μ3! (σ 5α2 )η(6) + · · ·  − 3α )η2 + 5! (5α  2− σ)η + 7! (σ − 2 2 2 − 3λση + λμ (3σ − 2α )(ηη + η ) + λη(5σ − 2α )(ηη + 2η 2 ) . + 73 λ2 η2 σ(ηη + 3η 2 ) + λ3 μ4 [· · · ] + λ5 μ6 [· · · ] + · · · = 0 (1.233) 2

4

6

Here, in square brackets, dots represent expressions whose structure has integral properties similar to the integral properties of the expression written in the first square brackets. In the infinitesimal limit (λ = 0) of (1.233), the previously obtained relations follow. In the case λ = 0 integration (1.233) over ξ in infinite limits leads, according to the conditions (1.231, 1.232), to the relation

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1 Linear and Nonlinear Surface Waves

+∞ η2 (ξ) dξ = 0.

(1.234)

−∞

This result follows from the integral properties +∞ +∞ +∞  2  2 (ηη + η ) dξ = η(ηη + 2η ) dξ = η2 (ηη + 3η 2 ) d ξ = 0. −∞

−∞

−∞

The integrals of terms denoted by dots are also converted to zero. Since η2 (ξ) ≥ 0 ∀ξ, the ratio (1.234) does not hold for η = 0. Thus, it is impossible for localized stationary waves to exist on the surface of a shallow liquid when the law of conservation of mass is fulfilled. It should be noted that in the case of λ ∼ μ2 , σ = α2 (1 − sμ2 ) (the standard assumption for obtaining soliton solutions [24]) Eq. (1.234) is reduced to the second-order terms of smallness by inclusive to the equation η /3 − sη + 3η2 /2 = 0, the solution of which is the Russell-Rayleigh soliton [11] (also violating the law of conservation of mass). Thus, approximate solutions of the Eq. (1.225) give all known solutions of the theory of stationary waves, including soliton ones. The speed of steady waves. As it follows from (1.224) the relation is valid a

∂ η(ξ± ) ∂ η(ξ± ) ∂ η(ξ± ) +h +L = 0, ∂a ∂h ∂L

(1.235)

which can be proved by the direct substitution of (1.224) into (1.235). The application of (1.235) to the Eq. (1.225) for waveform allows to get some relation for relative speed c˜ = c − U a

∂ c˜ ∂ c˜ ∂ c˜ c˜ +h +L − = 0, ∂a ∂h ∂L 2

(1.236)

or in equivalent formulation c˜ = ga f 1 (λ, μ) = gh f 2 (λ, μ) = gL f 3 (λ, μ).

(1.237)

Here f 1 , f 2 , f 3 are arbitrary functions of λ and μ. Validation of the relations (1.236, 1.237) shows that the speeds of all known wave solutions of the approximate theories satisfy the above equations. The exact solution for complex-valued waves. If in the case of infinitely deep water (μ → ∞) the a priory assumption about exponential decreasing of physical fields with depth is done, i.e. ψ(ξ, z) = (ξ) exp(μ z) ⇒ ψz (ξ, z) = μ ψ(ξ, z),

(1.238)

1.7 The Problem of Exact Solutions

71

then substitution of (1.226) into the second relation of (1.238) leads to equation ψz (ξ, z) =

 i  α + R(A(ξ, z))Az (ξ, z) − R(B(ξ, z)Bz (ξ, z)) . λ 2λμ

(1.239)

On the free surface of water the relations z = η(ξ) and ψ(ξ, η(ξ)) = αη(ξ) have place and the expressions are valid A(ξ, η(ξ)) =ξ  = B(ξ, η(ξ)) η (ξ) , Az (ξ, z)z =η(ξ) = 1+ii μλμ η (ξ)

  (ξ) . Bz (ξ, z)z=η(ξ) = − 1−ii μη λμη (ξ)

(1.240)

Substitution of (1.240) into (1.239) forms equation * α(1 − λμη(ξ)) =

α2 − 2σλμη(ξ) , 1 + λ2 μ2 η 2 (ξ)

from which the equations for waves on infinitely deep water follow ka η (ξ) = ±i

ka η(ξ) . 1 − ka η(ξ)

(1.241)

As a result the exact solutions have the form ζ± (ξ, ζ0 ) = −a± − k1 W(−kζ0 exp(±ikξ)), ψ± (ξ, z) = cζ0 exp(kz) exp(±ikξ) . p± (z, ξ ) = pa − ρ g(z + a± ) + ρ c2 kζ0 exp(ka± ) exp(kz) exp(±ik ξ ) (1.242) Here W(x) is the Lambert function introduced in the previous paragraphs, a± and ζ0 are free parameters. The presented solutions (1.242) are the exact solutions of the problem (1.0). The presented solutions (1.242) under the assumption of an exponential decrease in the flow intensity with depth are the only exact solutions to the problem (1.0) describing the propagation of gravitational waves on infinitely deep water (with the exception of the trivial solution for the still liquid) [26]. Neither real nor imaginary parts of these solutions separately satisfy the equations and boundary conditions. Solutions with the “+” and “ – “ indexes are not subject to the superposition principle. The obtained results allow us to assert that the problem (1.0) of describing steadystate waves decreasing exponentially with depth on infinitely deep water does not have an exact solution in the space of real functions of a real variable. For the purposes of possible comparison of the obtained exact solutions with the experiment, real functions are compiled from linear combinations of complex functions (1.242).

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1 Linear and Nonlinear Surface Waves

The first linear combination is formed by the sum of the obtained independent solutions  ζ0   ζ0  = ζ (ξ, ζ0 ) + a = F(ξ, ζ0 ) = F 2 + ξ, 2 + F− ξ,   . (1.243) k ζ0 1 = − k W − 2 exp(ikξ) + W − k 2ζ0 exp(−ikξ) For a fixed wave number k and a vanishing small value ζ0 , the expression (1.243) (taking into account that W(x) = x − x 2 + 23 x 3 + o(x 3 ) for |x| 1 [8, 9]) is converted to the form F(ξ, ζ0 ) = ζ(ξ, ζ0 ) + a.

(1.244)

Since the fluctuations of the real surface occur without changing the volume and average level of the liquid, the condition is met +∞ ζ(ξ, ζ0 ) d ξ = 0, −∞

and then follows the only choice a = 0 and ζ(ξ, ζ0 ) = ζ0 cos(k ξ). As a result, the standard solution of the system (1.0) for infinitesimal waves is formed ζ = ζ0 cos(kξ), ψ = cζ0 exp(kz) cos(kξ) . p = pa − ρ gz + ρ c2 k ζ0 exp(kz) cos(kξ), c2 k = g

(1.245)

It is the form (1.245) that allows us to associate the value of ζ0 with the amplitude of the wave. In the case of waves of small but finite amplitude, when the ratio 0 < ε = kζ0 1 is true, it follows from (1.243)    2 ε 3 ε2 cos(3k ξ) + o ε . F(ξ, ζ0 ) = ζ(ξ, ζ0 ) + a = ζ0 cos(k ξ) + cos(2kξ) + 2 8 (1.246) The expression (1.246) converts the Eq. (1.0) into an identity only up to and including the first-order terms of smallness in ε. The resulting discrepancy ∼ O(ε2 ) is too large, which calls into question the validity of the decomposition (1.246), starting with second-order terms of smallness by ε. In order to reduce the discrepancy, the solution is searched for in the form F = F0 + B(ξ),

(1.247)

where F0 is the expansion (1.246), and a correction for the wave velocity is introduced

1.7 The Problem of Exact Solutions

73

  c2 k = g 1 + c1 ε + c2 ε2 + · · · .

(1.248)

Substituting expressions (1.247), (1.248) in (1.0) allows you to achieve a discrepancy of ∼ O(ε5 ) if one put B(ξ) =

ε ζ0 k ζ20 ≡ . 2 2

(1.249)

In this case, the speed of propagation of the wave becomes dependent on its amplitude c2 =

 g 1 + ε2 . k

(1.250)

From the condition of immutability of the average liquid level on the wave, it follows that in (1.246) and the elevation of the free surface, the stream function of the and the pressure in the medium are determined by the relations   2 ζ(ξ, ζ0 ) = ζ0 cos(k ξ) + 2ε cos(2k ξ) + 38ε cos(3k ξ) + o ε2     2 2 ψ = ζ0 gk 1 − ε8 exp(kz) cos(k ξ), ξ = x − gk 1 + ε2 t

 2 p = pa − ρ gz + ρ g ζ0 exp(kz) 1 + 3ε8 cos(k ξ) − 2ε exp(2kz) 1 −

ε2 4



,

(1.251) representing the classic Stokes result [6]. Along with the solution (1.243), there is another real linear combination of functions that satisfies the equations (1.0)   ζ0    η(ξ, ζ0 ) = i η+ ξ, ζ20 − η = ζ(ξ, ζ0 ) + A= 2 − ξ, . k ζ0 k ζ0 i = − k W − 2 exp(ik ξ) + W − 2 exp(−ik ξ)

(1.252)

Performing operations with the solution (1.252) by analogy with (1.244)–(1.251) for the solution (1.243) shows that both in the infinitesimal limit and in the case of waves of finite amplitude, expressions of the form (1.245) and (1.251) are obtained, including an additional phase shift by π/2. Of practical interest is the study of the properties of the solution (1.243) when there are no restrictions on the magnitude of the ζ0 amplitude. The calculated free surface forms for different values of the dimensionless parameter ε = k ζ0 = c2 k 2 ζ0 /g, which characterizes the ratio of inertial forces acting on the medium particles to the gravity, are shown in Figs. 1.9 and 1.10. The horizontal coordinate and displacement of the free surface, respectively, normalized by the wave number k, are deposited along the axes in the figures. Depending on the value of ε, the solution (1.243) describes two types of elevations: normal smooth and abnormally steep, pointed, the boundary between which

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1 Linear and Nonlinear Surface Waves

Fig. 1.9 Smooth normal waves at subcritical values ε

Fig. 1.10 Sharp waves at supercritical values ε

is determined by a critical value ε∗ =

2 = 0.735758882 . . . . e

(1.253)

The dependence of the waveform of the first type on the parameter ε < ε∗ is shown in Fig. 1.9 for a fixed wavenumber k. The ridges of all waves of this type are smooth, and the derivatives of displacements along the longitudinal coordinate at the extremum points exist and are equal to zero. As the steepness increases, the ridges sharpen and the depressions widen. Accordingly, there is a growing difference in the values of the maximum deviation of particles on ridges and depressions from the undisturbed level. Waves of the second type (ε > ε∗ ) are characterized by pointed ridges, at the tops of which, as in Rankin waves, there is a break in the water surface. With a decrease in the steepness value up to the critical ε∗ , ridge sharpens, the angle at the top of the wave decreases.

1.7 The Problem of Exact Solutions

75

At a critical value of ε = ε∗ , normal and abnormal waves merge and are characterized by an infinite steepness at the top of the ridge. The estimation of the applicability of the approximate solution (1.251) is based on the restrictions imposed on the value of the small parameter ε by the convergence condition (1.251), and on the wave amplitude by the boundary conditions. Extended by degrees of smallness ε to infinity, the series (1.251) for the ζ waveform converges absolutely for all ξ ∈ (−∞, +∞) values ξ ∈ (−∞, +∞) if ε < ε∗ . From the boundary conditions follows, in particular, the expression for the horizontal component of the velocity of liquid particles on a free surface 

ψz z=ζ

* =c−

c2 − 2g ζ , 1 + ζξ2

which, in turn, imposes a restriction on the deflection of the free surface ζ≤

c2 . 2g

(1.254)

This is a well-known classical result, obtained earlier on the basis of other mathematical methods and arising here naturally without special research. As part of the approximations used to get the solution (1.251), the condition (1.254) is converted to the form 1 + ε2 + 4 W(−ε/2) ≥ 0,

(1.255)

the numerical solution of which has the form εl ≤ 0.443672. From the ratio of critical parameter values, it follows that since the resulting value is less than the critical εl < ε∗ , stationary pointed waves cannot exist. The possibility of their formation in the modes of wave field adjustment, for example, when waves are dispersed by wind, needs experimental verification. As a characteristic angle for the wave crest, we can take the angle between two intersecting tangents to the perturbed surface of the wave passing through its inflection points (points where the sign of the second derivative changes). Then, within the approximation (1.251), the minimum angle at the wave crest is equal to 2π/3. Here it is necessary to emphasize that the angle value obtained by Stokes [6] for the Euler equations and later for the Nekrasov’s equations [16] plays the role of the limit angle only within the framework of the approximation (1.251) for waves on deep water that exponentially decrease with depth. For waves of greater steepness and a different shape than that described by the decomposition (1.251), the restriction loses its force, which makes it possible for more sharpened ridges to exist.

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1 Linear and Nonlinear Surface Waves

References 1. Lamb H (1945) Hydrodynamics, 6th edn. Dover Publications, New York, USA. http://archive. org/details/hydrodynamics00lamb 2. Landau LD, Lifshitz EM (1987) Course of theoretical physics. In: Fluid mechanics vol 6, 2nd edn. Butterworth-Heinemann, Oxford, USA. 3. Lighthill J (1978) Waves in fluids, 2nd edn. Cambridge University Press, Cambridge, UK 4. Thorpe JA (1979) Elementary topics in differential geometry, 1st edn. Springer-Verlag, New York, USA 5. Gerstner FJ (1802) Theorie die wellen//Abh. Kon. Bohm. Gesel. Wiss., 1802, reprinted in Ann. Der Physik, 1809, V. 32, p 412–440 6. Stokes GG (1847) On the theory of oscillatory waves. Philosop Soc 8:441–455 7. Nayfeh AH (1981) Introduction to perturbation techniques. A Wiley-Interscience Publication, Wiley, New York, Chichester, Brisbane, Toronto 8. Valluri SR, Jeffrey DJ, Corless RM (2000) Some applications of the Lambert W function to physics. Canadian J Phys 78:823–831 9. Corless RM, Gonnet GH, Hare DE, Jeffrey DJ, Knuth DE (1996) On the Lambert W function. Adv Computat Maths 5:329–359 10. Boussinesq J (1871) Theorie de l’intumescene liquid appelee onde solitaire ou de translation se propageant dans un canal rectangulaire. Comptes Rendus Acad. Sci., Paris, 1871, V. 72, p 755–759 11. Rayleigh L (1876) On waves. Phil Mag Ser 5(1):257–279 12. Korteweg DJ, de Vries G (1895) On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag. 39(240):422–443 13. Kadomtsev BB, Petviashvili VI (1970) On the stability of solitary waves in weakly dispersing media. Sov Phys Dokl 5:539–542 14. Yuen HC, Lake BM (1975) Nonlinear deep water waves: theory and experiment. Phys Fluids 18(8):956–960 15. Zakharov VE, Shabat AB (1972) Fine theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media. Sov Phys JETP V. 34, p 62–78 16. Nekrasov AI (1967) The exact theory of steady state waves on the surface of a heavy liquid. Technical Summary Report No 813. Mathematical Research center, University of Wisconsin 17. Bayatt-Smith JGB (1971) An integral equation for unsteady surface waves and a comment on the Boussinesq equation. J Fluid Mech 49(pt. 4):625–633 18. Woods LC (1961) The theory of subsonic plane flow. Cambridge Univercity Press 19. Ablowitz MJ, Fokas AS, Musslimani ZH (2006) On a new non-local theory of water waves. J Fluid Mech 421(pt. 3):335–340 20. Kistovich AV, Chashechkin YuD (2008) Integral model of the propagation of stationary potential waves in fluids. Doklady Phys 53(7):395–400 21. Kistovich A.V, Nikishov VI, Chashechkin YD (2013) Integral model of nonlinear surface waves in ideal liquid. Reports of National Academy of Sciences of Ukraine, No. 5, pp 66–71 22. Kistovich AV, Chashechkin YD (2016) Analytical models of stationary nonlinear gravitational waves. Water Res 43(1):86–94 23. Linear Whithem GB (1974), Waves Nonlinear. Wiley, New York, London, Sydney, Toronto 24. Dodd RK, Eilbeck JC, Gibbon JD, Morris HC (1984) Soliton and nonlinear wave equations. Academic Press Inc 25. Kistovich AV (2018) The exact mathematical models of nonlinear surface waves. In: Karev V, Klimov D, Pokazeev K (eds) Physical and mathematical modeling of earth and environment processes. PMMEEP 2017. 2018, pp 305–316. Springer Geology. © Springer International Publishing AG, part of Springer Nature. 2018. https://doi.org/10.1007/978-3-319-77788-7_32 26. Kistovich AV (2013) The exact complex-valued solution for steady surface waves. Procedia IUTAM 8:161–165. https://doi.org/10.1016/j.piutam.2013.04.020

Chapter 2

Applications of the Surface Wave’s Theory to Description of Some Natural Phenomena

This chapter discusses some applications of the theory of surface waves developed in the previous Chap. 1 in relation to observed natural phenomena. Internal waves at the interface of two liquids.

2.1 The Statement of the Problem and its Mathematical Model The actual marine environment is often not nearly as uniform and isotropic as was assumed in the models in the previous chapter. Field observations show the existence of discontinuities in the sea of thermodynamic characteristics of the medium: temperature, salinity, density, etc. Naturally, these discontinuities are not mathematical, but physical, which means that the gradients of thermodynamic parameters are limited and the final thickness of the region where they develop. But at the same time, these gradients are very large, they can exceed the gradients of equilibrium distributions by tens or hundreds of times. The causes of discontinuities in the thermodynamic parameters of the medium are extremely diverse. This is, for example, the flow of layers of water of different origins (fresh water over salt in estuaries of rivers). Another source of discontinuities may be convective mixing of marine layers, accompanied by the formation of a thin structure. Rapid wind mixing of the upper layers of the sea also leads to the formation of breaks at the border with unmixed seawater. Of course, it is impossible to take into account all the manifestations of sharp inhomogeneities in models of ocean and sea waters. Moreover, when studying the properties of certain types of flows, it is even impractical. The main thing is to identify the set of breaks that lead to significant physical effects observed in the nature of the seas and oceans. One of the most important thermodynamic factors affecting the characteristics of natural phenomena is the density of sea water. Density discontinuities in the water © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Kistovich et al., Advanced Studies in Ocean Physics, Springer Oceanography, https://doi.org/10.1007/978-3-030-72269-2_2

77

78

2 Applications of the Surface Wave’s Theory to Description …

column entail the possibility of specific movements that have the characteristics of surface phenomena. An example of such movements are waves propagating along the interface of two horizontal layers of different densities. Naturally, in order to simplify the problem under study, the formulation of the mathematical model forces us to resort to a number of assumptions that cannot be implemented in a real natural environment. Thus, the density jump at the interface of layers is achieved at the region of zero thickness, which means that there is an infinite gradient of this thermodynamic characteristic of water. The presence of diffusion effects leads to the fact that infinite gradients of environmental parameters cannot be formed in nature. Therefore, in order for the mathematical model to preserve the final difference in the density of the two contacting layers, we must abandon the consideration of diffusion processes. As a result, the model problem that allows us to identify the main characteristics of internal waves on the interface is formulated as follows. The initial undisturbed medium, which is in the equilibrium position, is a liquid layer of thickness h and density ρ1 , resting on a liquid semi-infinite layer of density ρ2 . At the same time, ρ2 > ρ1 . We are interested in wave perturbations of the free surface and the interface. To describe the phenomenon under study, we introduce a Cartesian coordinate system (x, z), whose vertical axis z is directed against the gravity vector g. The z = 0 and z = h levels coincide with the positions of the undisturbed interface and free surface, respectively. Deviations of these boundaries from their equilibrium positions are described by the functions η(x, t) and ζ(x, t), as shown in Fig. 2.1.

Fig. 2.1 Schematic representation of one-dimensional internal waves on the interface

2.1 The Statement of the Problem and its Mathematical Model

79

The characteristics of physical fields in the upper layer are indicated by symbols with the lower index “1”, and in the lower-with the index “2”. Atmospheric pressure is considered constant and equal to p0 . Since the densities of liquid layers are constant, the equations of state for them have the form ρ1 = const., ρ2 = const. In full analogy with waves on the sea surface, the continuity and state equations for each layer are replaced by a single incompressibility equation. Thus, the equations of motion of an incompressible fluid in the approximation of infinitesimal perturbations in both layers have the form 1 = −∇ p1 + ρ1 g, ∇ · v1 = 0 ρ1 ∂v ∂t . ∂v2 ρ2 ∂t = −∇ p2 + ρ2 g, ∇ · v2 = 0

(2.1)

One just need to remember that the first two equations describe the movements in the first layer, and the rest—in the second. There is no single scope for equations for variables with different indexes. They have only a common boundary z = η(x, t). It is necessary to complete Eq. (2.1) by boundary conditions. On the free surface z = h + ζ(x, t) the dynamic p1 |z=h+ζ = p0

(2.2)

v1 |z=h+ζ = ζ  t

(2.3)

and kinematic

conditions should be valid, where v1 is the vertical component of velocity in the first medium. On the boundary z = η(x, t) the dynamic p1 |z=η = p2 |z=η

(2.4)

v1 |z=η = ηt = v2 |z=η

(2.5)

and two kinematic

conditions should be valid. At an infinite distance from the interface, the condition of striving for zero perturbations of all physical equilibrium fields must be met p˜ 2 , v2 |z=−∞ = 0,

(2.6)

where the tilde sign indicates the perturbation to the equilibrium pressure value.

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2 Applications of the Surface Wave’s Theory to Description …

So, the task is formulated. Let’s now proceed to its solution. First, let’s pay attention to the fact that the Euler equations of the system (2.1) can be written as ρ1

∂v1 ∂v2 = −∇( p1 + ρ1 gz), ρ2 = −∇( p2 + ρ2 gz). ∂t ∂t

(2.7)

Hence, it is possible to represent the velocity fields in both layers of the liquid by means of potentials ϕ1 and ϕ2 , such that v1 = ∇ϕ1 , v2 = ∇ϕ2 .

(2.8)

Substituting representations (2.8) in (2.7) allows integrating Euler equations by converting them to the form ρ1

∂ϕ1 ∂ϕ2 = − p1 − ρ1 gz + f 1 (t), ρ2 = − p2 − ρ2 gz + f 2 (t), ∂t ∂t

(2.9)

where the constants of integration f 1 (t) and f 2 (t) will be defined further. Substituting representations (2.8) into the incompressibility equations converts them into Laplace equations for potentials ϕ1 = 0, ϕ2 = 0.

(2.10)

Let us now turn to the description of the pressures p1 and p2 . The pressure p1 at the point (x, z) of the upper layer at the time t can be represented as the sum of the atmospheric pressure pa , the hydrostatic contribution ρ1 g(h + ζ − z) of the liquid column from the surface z = h + ζ(x, t) to the observation point z, as well as the addition q1 that occurs due to the movement of the medium p1 = p0 + ρ1 g(h + ζ − z) + q1 .

(2.11)

The pressure p2 at the point (x, z) of the lower layer at a time t can be represented as the sum of atmospheric pressure pa , the hydrostatic contribution ρ1 g(h + ζ − η) of the liquid column from the surface z = h + ζ(x, t) to the top z = η(x, t), the hydrostatic contribution ρ2 g(η − z) of the liquid column from the surface z = η(x, t) to the observation point z, as well as the additive q2 resulting from the movement of the medium p2 = p0 + ρ1 g(h + ζ − η) + ρ2 g(η − z) + q2 .

(2.12)

Substituting (2.11, 2.12) into the equations of the system (2.9) reduces them to the form 1 = − p0 − ρ1 g(h + ζ) − q1 + f 1 (t) ρ1 ∂ϕ ∂t . ∂ϕ2 ρ2 ∂t = − p0 − ρ1 g(h + ζ − η) − ρ2 gη − q2 + f 2 (t)

(2.13)

2.1 The Statement of the Problem and its Mathematical Model

81

In the absence of perturbations of the initial equilibrium state, the media must simultaneously turn to zero ϕ1 , ϕ2 , q1 , q2 , ζ and η. In order for the Eq. (2.13) to remain valid, it is necessary to set the values f 1 (t) and f 2 (t) by expressions f 1 (t) = f 2 (t) = p0 + ρ1 gh.

(2.14)

As a result, expressions (2.13) can be written as q1 = −ρ1

∂ϕ1 ∂ϕ2 − ρ1 gζ, q2 = −ρ2 − ρ1 g(ζ − η) − ρ2 gη, ∂t ∂t

(2.15)

which determines pressure perturbations q1 and q2 using other unknown problems ϕ1 , ϕ2 , ζ and η. In this case, the perturbation p˜ 2 of the equilibrium pressure from the formula (2.6) is determined by the value p˜ 2 = p0 + ρ1 g(h + ζ − η) + ρ2 g(η − z) + q2 − ( p0 + ρ1 gh − ρ2 gz) = ∂ϕ2 . (2.16) = ρ1 g(ζ − η) + ρ2 gη + q2 = −ρ2 ∂t Thus, the original system of equations was reduced to two Laplace equations for potentials ϕ1 , ϕ2 ϕ1 = 0, ϕ2 = 0, and the boundary conditions, after substituting in them representations (2.11, 2.12) for pressures, are transformed, taking into account the definitions (2.8) and expressions (2.15), to the form 



∂ϕ1  ∂z z=h+ζ

∂ϕ1  ∂t z=h+ζ

2 ρ2 ∂ϕ ∂t

= (ρ1 − ρ2 )gη,

= ζ t ,  1 − ρ1 ∂ϕ ∂t 

z=η

= −gζ,



∂ϕ1  ∂z z=η

 ∂ϕ2  ∂x 

z=−∞

=



∂ϕ2  ∂z z=η

 2 = ∂ϕ ∂z 

= η t

z=−∞

=



∂ϕ2  ∂t z=−∞

=0 (2.17)

Part of the boundary conditions (2.17) is set on the time-varying surfaces z = h + ζ(x, t) and z = η(x, t). Since all perturbations are considered infinitesimally small, it is permissible to carry the boundary conditions to constant surfaces z = h and z = 0. As a result of all the transformations and simplifications, the original problem (2.1–2.6) is reduced to the form

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2 Applications of the Surface Wave’s Theory to Description …

ϕ1 = 0, ϕ2 = 0 ∂ϕ1  1 = ζ  t , ∂ϕ = −gζ, ∂z z=h ∂t z=h  ∂ϕ2 ∂ϕ1  ρ2 ∂t − ρ1 ∂t  = (ρ1 − ρ2 )gη, z=0



∂ϕ1  ∂z z=0

 ∂ϕ2  ∂x 

=



∂ϕ2  ∂z z=0

z=−∞

= η t  2 = ∂ϕ = ∂z  z=−∞



∂ϕ2  ∂t z=−∞

. =0 (2.18)

We will look for the solution of the system (2.18) in the form of harmonic traveling waves ζ = ζ0 cos(kx − ω t), η = η0 cos(kx − ω t),

(2.19)

where ζ0 , η0 are their constant amplitudes. A consequence of the boundary conditions on the free surface and the interface is the conclusion about the behavior of potentials ϕ1 and ϕ2 ϕ1 , ϕ2 ∼ sin(kx − ω t). Since these potentials must satisfy the Laplace equation, we will look for them in the form ϕ1 = (A exp(kz) + B exp(−kz)) sin(kx − ω t), ϕ2 = C exp(kz) sin(kx − ω t), (2.20) where A, B and C are constants. By directly substituting representations (2.20) into the Laplace equation, one can directly verify that it turns into an identity. In addition, setting the potential ϕ2 in the presented form leads to automatic satisfaction of the conditions at infinity.

2.2 The Dispersion Equation of Internal Waves at the Interface Substituting (2.19, 2.20) into the remaining boundary conditions transforms the problem (2.18) into a system of relations k(A exp(kh) − B exp(−kh)) = ωζ0 , ω( A exp(kh) + B exp(−kh)) = gζ0 (2.21) k(A − B) = kC = ωη0 , −ωρ2 C + ωρ1 (A + B) = (ρ1 − ρ2 )gη0 We have a system of linear homogeneous algebraic equations with respect to the amplitudes A, B, C, ζ0 and η0 . The condition of nontrivial solvability of such a system (consisting in the equality of its main determinant to zero) generates a dispersion equation that has the form

2.2 The Dispersion Equation of Internal Waves at the Interface

    2 (ρ2 − ρ1 )th(kh) = 0. ω − gk ω2 − gk ρ2 + ρ1 th(kh)

83

(2.22)

For a complete analysis of the properties of wave motion in this problem, one dispersion relation (2.22) is not enough. You also need to determine the relationships between the amplitudes A, B, C, ζ0 and η0 . Since the dispersion equation itself decomposes into two independent equations 1. ω2 = gk, 2. ω2 = gk

(ρ2 − ρ1 )th(kh) , ρ2 + ρ1 th(kh)

(2.23)

then the relations between the amplitudes must be established for each of the Eq. (2.23) separately. Equation (2.23) correspond to different types of medium vibrations. Let us first consider the first dispersion equation ω2 = gk. In its form, it coincides with the dispersion equation of waves on the surface of an infinitely deep liquid. The characteristics of this type of wave are determined by the amplitude of the surface wave. Then, in relation to the case we are considering, we express the amplitudes A, B, C and η0 in terms of the amplitude ζ0 . Simple calculations show that there are relations A=

ω exp(−kh)ζ0 , k

B = 0, C = A, η0 = exp(−kh)ζ0 .

(2.24)

The result obtained makes it possible to combine the potentials ϕ1 and ϕ2 into a single potential ϕ describing the velocity field in the entire space ϕ=

ω ζ0 exp(k(z − h)) sin(kx − ω t), z ∈ (−∞, h]. k

(2.25)

Potential (2.25) describes the vibrations caused by a wave running along the surface of an infinitely deep homogeneous liquid. This will result in the coincidence of the velocity fields in two-layer and homogeneous liquids. But the pressure fields will be different. In the upper layer, the pressure, as follows from (2.11), is determined by the expression p1 = p0 + ρ1 g(h − z) + ρ1

ω2 exp(k(z − h))ζ(x, t), z ∈ [η, h + ζ], k

(2.26)

while in the lower layer, according to (2.12), the pressure is p2 = p1 + (ρ1 − ρ2 )gz − (ρ1 − ρ2 )g exp(kz)η(x, t), z ∈ (−∞, η].

(2.27)

Expressions (2.26) and (2.27) in the limit of infinitesimal perturbations are crosslinked when z = η(x, t) the first dispersion relation is fulfilled, but at the same time expression (2.27) does not coincide with expression (2.26) if the action of the latter is extended to the entire space.

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2 Applications of the Surface Wave’s Theory to Description …

Let’s take a closer look at the expression (2.27). The pressure p1 addition included in it has the form (ρ1 − ρ2 )gz − (ρ1 − ρ2 )g exp(kz)η(x, t) and it consists of two parts: hydrostatic contribution (ρ1 − ρ2 )gz and wave contribution −(ρ1 − ρ2 )g exp(kz)η(x, t). The hydrostatic contribution coincides with the same contribution for the equilibrium state of the medium and has no relation to the wave propagation process. The wave contribution is small due to the fact that η0 = exp(−kh)ζ0 and in addition it decreases exponentially with depth according to the law exp(kz). Thus, waves of the first type can, in the first approximation, be attributed to the class of ordinary surface waves. The second dispersion equation (2.23) relates to a different type of waves–internal waves at the interface. In the case of a small relative difference in the layer densities, that is (ρ2 − ρ1 )/ρ2  1, if the condition is met, the waves at the interface will have a lower frequency of vibrations than surface waves at the same wavelength. This follows from the comparison of dispersion equations of the first and second types. As in the previous case, we define the relations between the amplitudes A, B, C and ζ0 , but we also express them in terms of the amplitude η0 A= C=

g ρ2 − ρ1 ) ρ2th(kh)−1 η , B = ωk 2ω ρ1 (ρ2 +ρ1 th(kh) 0 ρ −ρ ω η , ζ = − 2ρ1 1 exp(−kh)η0 k 0 0



1 ρ2 2 ρ1 (1

 − cth(kh)) − 1 η0

.

(2.28)

The first thing that catches the eye when looking at the relations (2.28) is the smallness of the wave amplitude ζ0 on the surface in comparison with the amplitude η0 of the interface vibrations at a small relative difference in the layer densities. In this case, the wave propagating along the interface does not manifest itself on the surface, that is, it is invisible to an observer standing on the shore or on the deck of a floating ship. It is possible to detect the presence of such a wave only with the help of measuring equipment submerged to a depth of about the thickness of the upper layer. Waves of this type are associated with the phenomenon of “dead water” (a phenomenon experimentally discovered by Nansen [1] and theoretically explained by Ekman [2]). Its essence consists in the fact that a surface vessel, falling into the area of a two-layer medium, cannot effectively increase its speed even with an increase in the speed of the propeller. The costs of the ship’s power plant seem to go nowhere. There is an impression that the engine power has dropped, although in fact this is not the case. Depending on the maximum capabilities of the ship’s power plant, the situation develops in two scenarios. If, all other things being equal (tonnage, draft, length, etc.), a ship with a relatively weak power plant will slowly and painfully move forward, then a ship with high energy capabilities, slowly reaching a certain speed, suddenly, without any visible reasons, as it were, throws off a certain burden, quickly picks up speed and moves further in its usual mode.

2.3 Characteristic Features of the Waves at the Interface

85

2.3 Characteristic Features of the Waves at the Interface The observed phenomenon is caused by the ship generating internal waves at the interface. It is on the generation of these waves that the power plant’s energy costs are spent. Naturally, the ship must have a certain draft in order to disturb the previously stationary interface with its progress. Increasing the speed of the propeller leads mainly to the transfer of energy from the power plant to the internal wave, and a small fraction of the total energy is spent on increasing the speed. The question arises: why does the braking effect disappear when the ship reaches a certain speed? Let’s look at this problem on a qualitative level. Let the characteristic length of the ship be L, and its speed be v. The phase velocity of internal waves, as follows from the second Eq. (2.23), is determined by c ph =

g (ρ2 − ρ1 )th(kh) ≈ k ρ2 + ρ1 th(kh)



ρ2 − ρ1 gL ε, ε = . 2π ρ1

(2.29)

In (2.29), the length of the ship is taken as the wavelength scale. If the speed of the ship v is less than the phase speed c ph of internal waves, then most of the energy of its power plant is spent on generating vibrations of the interface. The ship “drives” an internal wave that is invisible on the surface, while increasing its amplitude. The ship’s speed is growing extremely slowly. When the speed exceeds the value (2.29), the ship moves away from the internal wave, the energy transfer to the vibrations of the interface is almost stopped, and the speed of movement increases rapidly. Let’s compare the kinetic energy reserves in the wave contained in the upper and lower layers. In the case of a wave of the first type from (2.25) for the entire space follows u = ϕx = ωζ0 exp(k(z − h)) cos(kx − ω t) , v = ϕz = ωζ0 exp(k(z − h)) sin(kx − ω t)

(2.30)

ω2 ζ20 v2 = exp(2k(z − h)). 2 2

(2.31)

so that

The kinetic energy density in the vertical section of the upper layer is h T1 = ρ1

v2 ω2 ζ20 dz = ρ1 (1 − exp(−2kh)), 2 4k

0

and in the vertical section of the lower-the value

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2 Applications of the Surface Wave’s Theory to Description …

0 T2 = ρ2 −∞

v2 ω2 ζ20 dz = ρ2 exp(−2kh). 2 4k

Thus, the ratio of the kinetic energy reserves of the wave in the upper and lower layers is determined by the ratio ρ1 T1 = (exp(2kh) − 1). T2 ρ2

(2.32)

If the wavelength is sensitively less than the thickness of the upper layer, and the densities of the upper and lower layers are close, then the main reserve of kinetic energy of the first type of wave is contained in the upper layer. For the second type of wave, based on the relations (2.20), it follows  k2  2 v2 = A exp(2kz) + B 2 exp(−2kz) + k AB cos(2(kx − ω t)), z ∈ [0, h]. 2 2 (2.33) Averaging (2.33) over the oscillation period and integrating along the vertical section of the upper layer determines the kinetic energy reserve when z ∈ [0, h] the expression T1 = ρ1

k2 2 A (exp(2kh) − 1) − B 2 (exp(−2kh) − 1) . 2

(2.34)

In the lower layer, there is k2 v2 = C 2 , z ∈ (−∞, 0], 2 2

(2.35)

so integrating over a vertical slice gives the result T2 = ρ2

k2 2 C . 2

(2.36)

Using the relations (2.28) between the amplitudes A, B and C, of the dispersion Eq. (2.23), it is possible to show using the relations (2.34, 2.36) that for waves of the second type, the relation is fulfilled 2 1 T1 ρ (th(kh) − 1)2 (exp(2kh) − 1) = 2 T2 4ρ1 ρ2 th (kh) 2

− (ρ2 (th(kh) − 1) − 2ρ1 th(kh))2 (exp(−2kh) − 1) .

(2.37)

2.3 Characteristic Features of the Waves at the Interface

87

In the case of a small relative jump in the density, that is, when the condition 1  1 is met, the expression (2.37) is simplified and takes the form ε = ρ2ρ−ρ 1 T1 = cth2 (kh) + O(ε). T2

(2.38)

If the wavelength is significantly less than the thickness of the upper layer, then, as follows from (2.38), the kinetic energies of the first and second layers are almost equal to each other. This is due to the fact that the second type of wave is concentrated near the interface and its intensity decreases exponentially as it moves away from it in both directions. Some basement of the theme presented here may be found in [3]. Transformation of waves on the flow. The process of wave propagation over the surface of a flowing liquid is similar in many ways to the movement of waves through a stationary medium. However, when certain conditions are met, there are significant differences. One example of such differences is the effect of blocking surface waves on a current with a changing flow rate. The essence of this effect is that when moving towards the current, in the direction of increasing its speed, the surface wave reaches a certain point beyond which it is not able to propagate. When approaching this point, called the blocking point, the wave amplitude begins to increase. Downstream, the formation of a reflected wave is observed. Experimental observations show that the blocking effect increases in the presence of wind, the direction of which coincides with the direction of the current. We investigate the issue of blocking surface waves by an oncoming current in more detail. To make the problem statement as close as possible to real field conditions, it is assumed that in addition to the current, there is an additional wind effect.

2.4 The Statement of the Problem of Waves on the Flow Let the equilibrium state of the medium be characterized by a certain wind flow in the atmosphere, a current in the ocean, and a horizontal interface of the z = 0 before the occurrence of surface waves. Such an equilibrium is certainly unstable, but it can be considered acceptable as a reference point for studying a surface wave disturbance. Without specifying for the time being the quantitative characteristics of water and air flows, we denote the stationary wind and current potentials by the symbols a and w , respectively, and the equilibrium pressures, from which hydrostatic components are excluded, by the symbols pa0 and pw0 , so that the system of equations connecting these equilibrium values has the form 1 1 1 1 ∇(∇a )2 = − ∇ pa0 , ∇(∇w )2 = − ∇ pw0 , a = w = 0 2 ρa 2 ρw

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2 Applications of the Surface Wave’s Theory to Description …

   ∂a  ∂w  pa0 − pw0 z=0 = 0, = = 0, ∂z z=0 ∂z z=0

(2.39)

where ρa , ρw are the densities of atmospheric air and water, respectively. Integrating the equations of motion and substituting the results in a dynamic boundary condition finally forms the system of equilibrium state relations used in the future ρa ρw pa0 = − (∇a )2 , pw0 = − (∇w )2 , a = w = 0 2 2        ∂a 2 ∂w 2  ∂a  ∂w  − ρw = 0, = = 0. ρa   ∂x ∂x ∂z z=0 ∂z z=0

(2.40)

z=0

Now let a wave propagate over the water surface and is described by the deflection of the surface ζ(x, t). The linearized system of equations of motion for perturbations of pressure and velocity field has the form   1 ∂a ∂ϕa ∂a ∂ϕa 1 ∂ϕa + + = − ∇ pa , ϕa = ϕw = 0 ∂t 2 ∂x ∂x ∂z ∂z ρa    ∂ϕw 1 ∂w ∂ϕw ∂w ∂ϕw 1 (2.41) ∇ + + = − ∇ pw − gζ(1 − r ). ∂t 2 ∂x ∂x ∂z ∂z ρw 

∇

Here ϕa and ϕw are the perturbation potentials of the velocity fields in the atmosphere and ocean caused by a surface wave, r = ρa /ρw is the ratio of air and water densities. Dynamic and kinematic boundary conditions that are set on the perturbed surface of the z = ζ(x, t) in this case have the form pa − pw |z=ζ = 0

  ∂(ϕa + a ) ∂a ∂ζ  ∂ζ ∂(ϕw + w ) ∂w ∂ζ  ∂ζ − , − = = . (2.42)   ∂z ∂ x ∂ x z=ζ ∂t ∂z ∂ x ∂ x z=ζ ∂t

Integration of the equations of motion (2.41) leads to expressions for pressure perturbations in water and atmosphere 

  ∂ϕa 1 ∂a ∂ϕa ∂a ∂ϕa + + ∂t 2 ∂x ∂x ∂z ∂z     ∂ϕw 1 ∂w ∂ϕw ∂w ∂ϕw + + + gζ(1 − r ) . pw = −ρw ∂t 2 ∂x ∂x ∂z ∂z

pa = −ρa

(2.43)

Since the boundary conditions (2.42) are calculated on the surface z = ζ(x, t), they must first be linearized to the z = 0 level, taking into account that ϕa , ϕw and ζ are the first order quantities of smallness, and a and w are zero-order quantities. In addition, the boundary condition expansion rule has the form

2.4 The Statement of the Problem of Waves on the Flow

f |z=ζ = f |z=0 + ζ

89

  ∂ f  ζ2 ∂ 2 f  + + .... ∂z z=0 2 ∂z 2 z=0

(2.44)

As a result of substituting (2.43) in (2.42) and bringing the boundary conditions to the z = 0 level, a system of relations is formed   1 ∂w ∂ϕw ∂ϕa 1 ∂a ∂ϕa  ∂ϕw + + gζ(1 − r ) − r + =0 ∂t 2 ∂x ∂x ∂t 2 ∂ x ∂ x z=0  ∂ 2 a ∂ζ  ∂a ∂ζ ∂ϕa − −  ζ− =0 ∂z ∂x2 ∂x ∂x ∂t z=0  ∂ 2 w ∂ζ  ∂ϕw ∂w ∂ζ − − ζ − =0 ∂z ∂x2 ∂x ∂x ∂t z=0 ϕa = ϕw = 0.

(2.45)

The solution (2.45) is sought in the class of surface waves, that is, perturbations whose intensity decreases exponentially as they move away from the interface. For this reason, the solution of the Laplace equations for potentials in water and the atmosphere is represented as ϕa±

∞ =

exp(−λ z ± 0

± iλ x)A± (λ,t)dλ, ϕw

∞ =

exp(μ z ± iμ x)W± (μ,t)d μ, 0

(2.46) where the “+”sign refers to waves running to the right, and the “-” sign refers to waves spreading to the left. The using of the properties of the representations (2.46)     ± ± ∂ϕa±  ∂ϕa±  ∂ϕw ∂ϕw   = ± i , = ∓ i ∂z z=0 ∂ x z=0 ∂z z=0 ∂ x z=0

(2.47)

allows us to exclude surface wave potentials from the boundary conditions (2.45) and reduce the problem to the equation for surface elevation ∂ζ± ∂ 2 ζ± ∓ ig(1 − r ) 2 ∂t ∂x          ∂ζ± ∂ζ± ∂ ∂w ∂ ∂w ± ∂a ∂ ∂a ± + ζ +2 +r ζ +2 = 0. ∂x ∂x ∂x ∂x ∂t ∂x ∂x ∂x ∂t (2.48)

(1 + r )

Here the signs for the free surface deflection function denote the same as in (2.46, 2.47). To reduce the calculations, the notations u(x) = ∂w /∂ x, v(x) = ∂a /∂ x are entered.

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2 Applications of the Surface Wave’s Theory to Description …

2.5 The Dispersion Equation of Waves on the Flow Let the source of the surface wave gives rise to harmonic oscillations at the circular frequency ω. If the velocity of air and water flows at the surface does not depend on the x coordinate, it is advisable to look for the solution (2.48) in the form  ±  ζ± = ζ± 0 exp ik x − iω t ,

(2.49)

the substitution of which in (2.48) generates the dispersion equation of linear surface waves. The solutions of this equation, presented in the form of the dependence of the wave number on the frequency and other hydrodynamic parameters, have the form  g 1 ω(u + r v) ± (1 − r ) u2 + r v2 2 

2 g (1 − r )2 ± ω g(1 − r )(u + r v) − r ω2 (u + r v)2 . ∓ 4

k± =

(2.50)

Here it is necessary to emphasize that the sign before the radical is not arbitrary, but is determined by the direction of wave propagation, so for k + the radical is taken with a minus, and for k − with a plus. The choice of signs is due to the fact that in the case of a limit transition of the u → 0, v → 0 expressions for waves at the 2 take place. interface of two resting media k ± = ± ωg 1+r 1−r The blocking condition consists in turning to zero the expression under the sign of the radical in the ratio (2.50), since at negative values of the root expression, the wave number becomes complex, the phase velocity decreases, and the wave amplitude begins to increase exponentially—the wave is blocked. Figure 2.2 shows graphs of the dependence of wave numbers k ± on the frequency of the harmonic surface wave and the speed of water flow in the limit r → 0 (neglect of air density compared to water density) at v = 0 (absence of wind). It can be seen that the blocking points of the wave are different for waves with different frequencies, but in all cases there is a general phenomenon: the wave number increases indefinitely, that is, when approaching the blocking point, the wavelength is sharply shortened. In the case when the effect of blocking waves by current is investigated, the described spectral approach allows us to give only qualitative estimates and only in the case of slow changes in the speeds of air and ocean flows, that is, when the conditions are met                ∂u      ∂ζ ,  ∂v    ∂ζ . (2.51) ∂x  ∂x  ∂x  ∂x  In this case, the wave numbers of surface waves are determined by the approximate relation

2.5 The Dispersion Equation of Waves on the Flow

91

Fig. 2.2 The dependence of the wave numbers k ± on the frequency of the harmonic surface wave and the water flow velocity

  g 3i   1 ω(u + r v) ± (1 − r ) + uu + r vv ∓ κ (x) = 2 2 u + rv 2 2 

2 g (1 − r )2 + ω g(1 − r )(u + r v) − r ω2 (u + r v)2 + f (u,v) , ∓ 4 (2.52) ±

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2 Applications of the Surface Wave’s Theory to Description …

where     2  g f (u,v) = 3i uu + r vv ω(u + r v) ± (1 − r ) − 2i u2 + r v2 u + r v +     2 + u2 + r v2 u2 + uu + r v2 + vv a stroke means differentiation by x. Since the characteristics of air and water flows change slowly, in the relations (2.52), the derivatives of u and v are of the first order, and u and v are of the second order of smallness. The flow rates themselves are not small. Since the velocities of u and v, as well as their derivatives, are slow-changing functions, it follows from (2.52) that the wave numbers κ± (x) are slow-changing functions of the x coordinate. As the surface wave moves against the current, the intensity of which slowly increases, the blocking effect gradually increases (the real part of the subradical expression gradually decreases to zero), the wave length shortens, and the amplitude increases. Simultaneously with this process, a reflected wave occurs, described by the wave number κ− (x), and it is generated not at one point, but in the entire region of the inhomogeneous flow up to the blocking point. The appearance of the reflected wave is confirmed experimentally.

2.6 Reflection of Waves from Flow Inhomogeneities Preliminary results (2.50, 2.52) allow us to outline a model for blocking a surface wave that is close to the real situation. The model flow diagram is shown in Fig. 2.3. The choice of this model is due to its similarity to the shoal, where the well-known Agulhas current is particularly evident. A surface wave propagating from left to right is affected by a current that carries water from right to left. In this case, the flow structure is spatially inhomogeneous.

Fig. 2.3 Model counter-current

2.6 Reflection of Waves from Flow Inhomogeneities

93

The inhomogeneity is defined by the sea depth variability, which changes linearly from its value −H2 in the (−∞, −L] region to the −H1 value in the [0, +∞) region. The flow velocity potential is given by an expression of the form 

    x κ2 z 2 2 2 2 w = −U ϑ (γ x − H1 ) + −μ 1+γ ϑ(H2 + z) H2 H12        γ2 z 2 γ2 z 2  2 2 2 2 2 + ϑ μ 1 + γ − (γ x − H1 ) − − 1+γ ϑ (γ x − H1 ) + H12 H12    2 2 1 + γ2 ln (γ x − H1 )2 + γ z2 H1 × ϑ(z − γ x + H1 ) 2γ      γ2 z 2 x ϑ 1 + γ2 − (γ x − H1 )2 − + z) , (2.53) + ϑ(H 1 H1 H2 1

1 2 where γ = H2 −H ,μ= H and U is the velocity of the incoming flow in the x > 0 L H1 region. In Fig. 2.3, the solid line with arrows shows a separate current line and the direction of flow along it. On the arcs there is a discontinuity of the velocity vector of the flow (but not its absolute value!). The presented distribution (2.53) satisfies the condition of incompressibility of the ∇ · v = 0 and the boundary conditions on the free surface and bottom in the absence of wind action. It is much more difficult to set a wind model, since the question of the interaction of wind with waves has not been resolved, but for simplicity and illustrative purposes, you can put

a = V x,

(2.54)

assuming (in a rough approximation) constancy in space of the air flow velocity. Expressions (2.53) and (2.54) must be substituted into Eq. (2.48) for the surface waveform. Let’s say that a surface wave of the form ζ+ = exp(i(k + x − ω t)) runs over the surface from region x ≤ −L to the inhomogeneous flow region. As a result of interaction with the flow, a reflected wave ζ− = A− exp(−i(k − x + ω t)) will appear in this region, so that the total deviation of the free surface in this region will take the form ζ = exp(i(k + x − ω t)) + A− exp(−i(k − x + ω t)), x ∈ (−∞, −L],

(2.55)

where the values k ± are determined from (2.50) for u = −U H1 /H2 , v = V and A− is a constant amplitude. In the [−L , 0] region, the elevation of the surface is determined by the expression ζ = B+ (x) exp(i(κ+ x − ω t)) + B− (x) exp(−i(κ− x + ω t)), x ∈ [−L , 0], (2.56)

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2 Applications of the Surface Wave’s Theory to Description …

where κ± are defined from (2.52) when substituting in (2.53, 2.54). The amplitudes B± of waves traveling to the right and left are slowly changing functions of the x coordinate. Further downstream, the wave is described by the expression ζ = C+ exp(i(k + x − ω t)), x ∈ [0, +∞),

(2.57)

describing a passing wave of constant amplitude C+ , the wavenumber of which k + is determined from (2.50) at u = − U , v = V . Substitution (2.55–2.57) in Eq. (2.48) with subsequent cross-linking of the results for the values of ζ and ζ  x for x = −L and x = 0 determines the values of the desired amplitudes of waves. If, when solving (2.48), it turns out that to the right of some internal point of the segment [−L , 0], the amplitudes B± turn to zero, then this point will be the point of blocking the flow. The explicit form of expressions (2.55–2.57) is achievable using the method of slowly varying amplitudes, but their value is limited by the approximation of weak interaction of waves with the current, so we can hardly expect an adequate description of the phenomenon of blocking large-amplitude waves in this way. This is just a qualitative approach to the problem. Research in this direction continues.

References 1. Nansen F (1897) Fram over Polhavet. Den norske polarfærd 1893–1896. Kristiania, Aschehoug 2. Jenkins AD, Bye JAT (2006) Some aspects of the work of V.W. Ekman. Polar Record 42(2):15–22 3. Milne-Thomson LM (1960) Theoretical hydrodynamics. Forth edn. London, New York

Chapter 3

Surface Contamination

Currently, almost 20% of the World ocean area is covered with organic films of anthropogenic and biogenic origin. Among the substances of anthropogenic origin, in addition to petroleum products, synthetic surfactants (surfactants) are found in coastal waters. The surface tension in natural films is slightly reduced relative to pure water, by about 0.5–1.5 din/sm. In some cases of large biogenic pollution, lower values of the surface tension coefficient are also observed. Surfactants of anthropogenic origin can both reduce and increase surface tension. It should be emphasized that the effect of surfactants is not limited to changes in the capillary properties of the sea surface. According to the type of interaction with water, surfactants are divided into soluble and insoluble. Soluble surfactants change the structure of the surface substance: in fact, impure (or simply salty) water is located near the surface, but a certain aqueous solution with thermodynamic characteristics that are changed in comparison with water. Insoluble surfactants form visco-elastic films on the sea surface, the thickness of which is not constant along the surface (for example, the film is thinner at wave crests and thicker in depressions). As a result, a two-layer liquid with non-trivial physical properties is formed near the surface. To describe the effects caused by surfactants, it is necessary to formulate the equations of motion and boundary conditions. Since in the presence of surfactants, the influence of tangential stresses on the fluid dynamics increases dramatically, dynamic conditions at the free boundary (in contrast to the situations discussed in the previous sections) should describe the balance of not only normal, but also tangential forces on the sea surface. In the presence of soluble surfactants, the surface tension coefficient of the “sea water—surfactant” solution depends on the concentration of the dissolved substance. This dependence leads to tangential capillary effects that can only be balanced by viscous stresses in the medium. This fact forces us to use the Navier–Stokes equation rather than the Euler equation to describe the phenomena under study.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Kistovich et al., Advanced Studies in Ocean Physics, Springer Oceanography, https://doi.org/10.1007/978-3-030-72269-2_3

95

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3 Surface Contamination

3.1 Basic Equations and Boundary Conditions in the Presence of Surfactants Wave motion in an incompressible fluid is described by the basic equations of hydrodynamics. These include, first, the continuity equations ∇ · v = 0,

(3.1)

and the Navier–Stokes equation ρ

dv = F − ∇ p + η  v, dt

(3.2)

where ρ is the density, v is the velocity, p is the pressure, η is the coefficient of dynamic viscosity, and F is the density of external forces. The problem can be significantly simplified if we consider small-amplitude waves for which the condition ka  1 is met. Another simplification is to study only twodimensional waves, for which v = uex + wez . We assume that the liquid fills the entire lower half-space z < 0, and the air under constant pressure pa fills the upper half-space z > 0. The plane z = 0 corresponds to the undisturbed state of the surface. Within the framework of the model used, the system of linear equations of hydrodynamics for water with the thin surfactant on its surface takes the form ∂u ∂x ∂u ∂t ∂w ∂t

∂w =0 ∂z   2 2 = − ρ1 ∂∂ xp + ν ∂∂ xu2 + ∂∂ zu2 + Fρx   2 2 = − ρ1 ∂∂ zp + ν ∂∂ xw2 + ∂∂ zw2 + Fρz

+

,

(3.3)

where ν = η/ρ is kinematic viscosity of the water. Of all possible external forces, only the force of gravity with Fx = 0, Fz = − ρg is considered. The viscosity of water and the presence of a boundary with air does not allow the representation of the velocity field in the z < 0 region by means of only one potential. Since viscous effects on the free surface of water lead to the generation of vorticity, it is necessary to decompose the velocity field into potential and vortical parts and represent it as a sum of two parts v1 (potential) and v2 (vortical) such that v = v1 + v2 .

(3.4)

Since the density of water is considered a constant value, the system of Eq. (3.3) when using the decomposition (3.4) is divided into two independent systems, one of which describes the potential part of the velocity field ∇p ∂ v1 =− + g, ∂t ρ

∇ · v1 = 0,

(3.5)

3.1 Basic Equations and Boundary Conditions in the Presence of Surfactants

97

and the other describes its vortical part ∂ v2 = ν v2 , ∇ · v2 = 0. ∂t

(3.6)

Because the function v1 is a potential component of the wave motion and function v2 is the vortical part, then let’s present them in the forms v1 = ∇ϕ, v2 = ∇ × ψ,

(3.7)

where ϕ is scalar and ψ is vector potentials correspondingly. For one-dimensional waves the vector-potential ψ has only one component ψ = ψ(x, z)e y , where ψ(x, z) is the stream function. Then the Cartesian velocity components u and w expressed in terms of the introduced potential and stream functions as follows u=

∂ψ ∂ϕ − , ∂x ∂z

w=

∂ϕ ∂ψ + , ∂z ∂x

(3.8)

Substituting (3.7) into (3.5) and (3.6) produces the result ϕ = 0,

(3.9)

and evolution equations p ∂ϕ = − + gz + C1 (t), ∂t ρ

∂ψ = ν ψ + C2 (t), ∂t

(3.10)

The constants C1 and C2 are obtained from the condition that in the absence of movement, the acting force remains only the hydrostatic pressure, so it follows C1 = − pa /ρ, C2 = 0.

3.2 The Dispersion Equation and Features of the Wave Motion in the Presence of Surfactants Solutions of Eqs. (3.9, 3.10) are searched in the forms of harmonic oscillations ϕ = Z 1 (z) exp(i(kx + ω t)),

ψ = Z 2 (z) exp(i(kx + ω t)),

(3.11)

Substitution of (3.11) into (3.10) leads to result d2 Z1 = k2 Z1, d z2

d2 Z2 = m2 Z2, d z2

(3.12)

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3 Surface Contamination

where m 2 = k 2 + iω/ν. The solutions of Eq. (3.12), vanishing for z → −∞, are the following functions Z 1 (z) = A exp(kz),

Z 2 (z) = B exp(mz).

(3.13)

The result received (3.13) shows that in a viscous fluid, in addition to the wave solution Z 1 , there is also a solution Z 2 in the form of a near-surface boundary layer. Thus, for harmonic oscillations with a frequency of ω, there are u = (ik A exp(kz) − m B exp(mz)) exp(i(kx + ω t)) w = (k A exp(kz) + im B exp(mz)) exp(i(kx + ω t)) . p = pa − ρ gz − iωρ A exp(kz) exp(i(kx + ω t))

(3.14)

The relationships between the constants A and B, as well as between the frequency and the wavenumber k, are determined from the boundary conditions, which are formulated as the balance of forces acting on the surface element of the liquid. For an isotropic medium in the linear approximation, the stress tensor has the form  i j = − pδi j + ρν

 ∂ vj ∂ vi , + ∂ xj ∂ xi

(3.15)

where the indexes i, j number the coordinates of x, y, z, so that x x

∂u , x z = zx = ρν = − p + 2ρν ∂x



 ∂u ∂w ∂w + , zz = − p + 2ρν . ∂z ∂x ∂z (3.16)

The normal and tangential components of the stresses on the liquid surface defined by the function ζ(x, t) are expressed in linear approximation in terms of the Cartesian components of the tensor (3.16) n = zz , t = x z .

(3.17)

The normal stress is balanced by the hydrostatic pressure in the air pa and the capillary pressure due to the curvature of the surface, determined by the Laplace formula   1 1 , + α = ρα R1 R2 which in linear approximation is converted to the form α = −ρα

∂ 2ζ , ∂ x2

3.2 The Dispersion Equation and Features of the Wave Motion …

99

so finally the expression for the balance of normal stresses takes the form − p/ρ + 2ν

∂ 2ζ ∂w = − pa /ρ + α 2 . ∂z ∂x

(3.18)

It can be seen from (3.18) that in order to obtain the boundary condition for normal stress in a convenient form, it remains to determine ζ(x, t). Since for small-amplitude waves       ∂ζ dζ ∂ζ ∂ζ + u = ≈ , w|z=ζ = d t z=ζ ∂t ∂ x z=ζ ∂ t z=0 then the w substitution from (3.14) followed by integration leads to the result ζ=

k A + ik B exp(i(kx + ω t)). iω

(3.19)

Substituting (3.14)–(3.16) and (3.19) in (3.18) results in the relation   2 −ω + 2i ν ω k 2 + ω20 A + i ω20 + 2i νω mk B = 0,

(3.20)

where the symbol is entered ω20 = gk + α k 3 .

(3.21)

In the boundary condition for tangential forces in the presence of surfactant, additional forces of the same scale as the capillary stress appear. Indeed, the movement of the liquid and the resulting change in the shape of the surface leads to a change in the surface concentration of surfactant  from point to point. Variations in the surface concentration of surfactant  on the surface result in a restoring tangential force directed from places with a large to places with a lower surface tension. This force, related to the unit of the liquid surface, can be represented by the expression x = −∇(ρα) = − ρ

∂α ∇ . ∂

When the deformation is removed, the relaxation of the surface characteristics of the liquid—surfactant system occurs differently for different surfactants. For insoluble surfactant relaxation process is determined by surface diffusion, for instant relaxation can be carried out by processes of transfer of surfactant molecules from bulk to surface (bulk diffusion), and adsorption–desorption of surfactant molecules on the surface. The rates of these processes are different and the rate of relaxation will be limited to the slowest of them. Taking into account the two-dimensional nature of the problem under consideration, the boundary condition for tangential stresses takes the form

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3 Surface Contamination

 ν

∂u ∂w + ∂z ∂x

 −

∂αd = 0, ∂ dx

(3.22)

that is, you need to know the distribution of surfactants on the surface of the liquid. This distribution is due to several factors. First, surfactants located on the surface of the liquid are attracted by its movement. The associated convective surfactant flow is defined as jc = Vt , where Vt is the tangent to the surface of the velocity component. Due to the fact that the concentration of the substance changes along the surface, surface diffusion of surfactant molecules occurs, described by the expression js = −Ds ∇ , where Ds is the surface diffusion coefficient. Finally, if the surfactant is soluble in a liquid, then the transition of surfactant molecules from the bulk solution to the surface and back can occur. In general, this flow is determined by two processes: the diffusion of surfactant molecules from the depth of the solution on the surface and the adsorption–desorption of molecules on the surface jv = jvd + jva , where jvd is the diffusion flow of the substance from the volume, jva is the flow of the substance due to absorption–desorption. We will consider the specific representation of surfactant flows from the volume below. In the meantime, let’s write out the law of conservation of surfactants on a surface that differs little from the horizontal one   ∂ ∂ ∂ + u − Ds − jv = 0, (3.23) ∂t ∂x ∂x where is the value of the u component on the surface at z = ζ. Further reduction of the boundary condition (3.22) to a user-friendly form based on the solution (3.23) is possible only if the chemical properties of surfactants are known. Insoluble surfactants are characterized by a negligible volume concentration of the substance. Therefore, the process that determines the distribution of surfactants on the surface will be surface diffusion. in Eq. (3.23), the flux jv can be ignored   ∂ ∂ ∂ + u − Ds = 0. ∂t ∂x ∂x For small variations   of the surface concentration near the equilibrium value 0 on an not deformable surface the linear approximation is valid

3.2 The Dispersion Equation and Features of the Wave Motion …

101

∂  ∂u ∂ 2  + 0 − Ds = 0, ∂t ∂x ∂ x2

(3.24)

The solution (3.24) is searched in the form   = ˜  exp(i(kx + ω t)),

(3.25)

where ˜  is the amplitude of the concentration wave. Substituting (3.25) into (3.24) produces the result ˜  = ik0

−ik A + m B . i ω + Ds k 2

(3.26)

Substitution (3.14), (3.25) and (3.26) in (3.22) forms the second equation for determining the parameters A and B     Ek 3 Emk 2 A + ω2 − 2i ν k 2 − B = 0, i 2i ν ω k 2 + ρ ρ

(3.27)

where the notations are entered  E 0 = −ρ0

∂α ∂

 0

,

E=

E0 1+

Ds k 2 iω

.

(3.28)

Due to the smallness of the diffusion coefficient Ds , the value E ≈ E 0 and can be replaced accordingly in (4.42). The value E 0 characterizes the elasticity of the film E 0 = ρS

dα , dS

since for insoluble surfactant films  ∼ 1/S, where S is the element of the surface area. Moving the relative increase in area to the left and replacing d S/S ∼ dl/l leads to the usual formulation of Hooke’s law. Therefore, E 0 is nothing more than the elastic modulus of the surfactant surface film. In the case of soluble surfactants, surface diffusion in comparison with bulk diffusion can be neglected due to the small ratio Ds /Dv . However, there are still two mechanisms for transferring matter to the surface; diffusion from a bulk solution and adsorption–desorption of molecules on the surface. In General, these processes complement each other. The diffusion transfer of a substance works only up to the adsorption layer, whose thickness is about the size of molecules. Then, overcoming the energy barrier, the molecules reach the surface and find themselves in the energy well. It is the decrease in the surface energy of surfactant molecules that makes it difficult to reverse the desorption process and leads to stability of the properties of dissolved surfactants. Further discussion of the features of the adsorption–desorption process is not carried out in this paper due to the need to attract a large amount of

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3 Surface Contamination

information from the field of physical chemistry. The influence of this process on the dynamics of regular waves is important for large volume concentrations of surfactants. Thus, if we limit ourselves to considering only the process of volume diffusion for soluble surfactants, we get a solution that is quite suitable for geophysical applications, since the concentration of pollutants in the ocean is small. The transfer of matter from the volume to the surface is determined by Fick’s law of diffusion jvd = −Dv

∂c , ∂z

where c is the volume concentration of surfactants. We get an analog of the Eq. (3.23) ∂ ∂( u) ∂c + + Dv = 0. ∂t ∂x ∂z

(3.29)

The volume concentration of surfactants is found from the solution of the linearized diffusion equation in the volume  2  ∂ c ∂c ∂ 2c , = Dv + ∂t ∂ x2 ∂ z2

(3.30)

which is searched for in the form c = c0 + c = c0 + c˜ exp(i(kx + ω t)) exp(nz), c  c0 ,

(3.31)

where c˜ is the amplitude of the concentration wave. Substituting this expression in (3.30), we get

n =+ 1+

Ds k 2 . iω

(3.32)

The negative value of the radical is discarded due to the c → c0 boundary condition at z → −∞. Just as before, we rewrite (3.29) for small fluctuations in the concentration near the equilibrium values. Up to terms of the second order of smallness for z = ζ we have ∂u ∂ c ∂  + 0 + Dv = 0. ∂t ∂x ∂z

(3.33)

Due to the fact that the adsorption process proceeds faster than the diffusion process, the local equilibrium condition must be met at any time and for any point (c0 , x, t) = 0 ,

(3.34)

3.2 The Dispersion Equation and Features of the Wave Motion …

(c0 + c , x, t) = 0 +   .

103

(3.35)

Decomposing the left-hand side of (3.35) into a Taylor series, taking into account (3.34) and up to second-order terms of smallness, we obtain the relation between the deviations   and c   ≈ c

d . dc

(3.36)

Substitution (3.14), (3.31), (3.32) and (3.36) in (3.33) determines the amplitude of the volume concentration waves c˜ = 0

ik(−ik A + m B)   , i ω dd c + Dv i ω + Dv k 2

(3.37)

and surface concentration of surfactants ˜  = 0

ik(−ik A + m B)   . i ω + dd c Dv i ω + Dv k 2

(3.38)

Comparing this formula with the similar one for the case of insoluble surfactants (3.26), we can see that they differ only by the second term in the denominator. By analogy with (3.28), neglecting the value of Dv k 2 in comparison with the frequency, we obtain E=

1+

E0 d√ Dv /i dc

ω

.

(3.39)

It is clear that the equation for determining the amplitudes of the surface wave (3.27) remains the same as in the case of insoluble surfactants. The difference between the types of surfactants is the different definition of the elastic modulus. For insoluble substances, this is a real value, and for soluble substances, it is a complex value. Now we obtain the dispersion equation for waves in the presence of surfactants and consider some of its special cases. The amplitudes of waves on the liquid surface A and B are determined from a homogeneous system of Eqs. (3.20) and (3.27). In order for a homogeneous system of linear equations to have a nontrivial solution, its determinant must be zero  −ω2 + 2i ων k 2 + ω2 i ω20 + 4imων 0  (3.40) i 2i ων k 2 + Ek 3 /ρ ω2 − 2i ων k 2 − Ek 2 m/ρ = 0. This is the dispersion equation that relates the frequency and wavenumber of small-amplitude waves in the presence of surfactants.

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3 Surface Contamination

3.3 The Effect of the Surfactants on the Surface Waves Let’s consider two important special cases. For an ideal pure liquid (ν = 0, E = 0) (3.40) is simplified and takes the form  ω2 ω2 − ω20 = 0,

(3.41)

which has 4 roots, two of them ω1,2 = 0 correspond to periodic motion, and the other two ω3,4 = ± ω0 correspond to waves running in the directions +x and −x. The relationship between frequency and wavenumber is determined by the known dispersion relation ω(k) =

gk + α k 3 .

(3.42)

The absence of an imaginary part of the frequency indicates that in an ideal fluid, the wave motion does not decay. Substituting the solution (3.41) in (3.40) leads to the requirement B = 0, i.e. the vortex component of the velocity is zero. Consider another special case—a low-viscosity liquid for which the viscous term in the Navier–Stokes equation is small compared to the derivative ∂ u/∂ t, i.e. the condition is met ν k 2 /ω  1.

(3.43)

The introduction of the β0 = 2ν k 2 notation reduces Eq. (3.40) to the form   ω2 ω2 − 2i ωβ0 − ω20 − β20 (1 − m/k) = 0, where the meaning of the ω1,2 = 0 roots is the same as above (non-periodic motion), and the wave motion corresponds to complex roots  ω3,4 = i β0 ± ω0 1 −



β0 ω0

2

m . k

According to (3.43), the second term under the root can be ignored and ω ≈ ω0 + i β0 for a wave traveling in the direction of −x. Then the frequency of the wave, as for an ideal liquid, is determined by the ratio (3.42), and there is also a damping of the wave amplitude in time ζ = a e−β0 t exp(i(kx + ω0 t)). The vortical component of the velocity is non-zero, but small. Its relation to the potential component is equal to

3.3 The Effect of the Surfactants on the Surface Waves

105

A ≈ β0  1. B ω Let’s now proceed to the solution of Eq. (3.40) for the case E = 0 and introduce dimensionless parameters a1 =

Ek 3 , ρω20

a2 = a1

m . k

(3.44)

When |ω|  ν k 2 then m = k



iω π 1 + 2 ≈ ei 4 νk



ω Ek 3 i π , a ≈ e 4 2 ν k2 ρω20



ω . ν k2

(3.45)

We will solve the dispersion Eq. (3.40) for the case of a low-viscosity liquid β0 = 2ν k 2  |ω| and calculate the determinant (3.40) ω4 − 2i β0 ω3 − (1 + a2 )ω20 ω2 + (a2 − a1 )ω40 = 0

(3.46)

The solution of (3.46) has a form ⎤ ⎡  2 3 /ω4 + 8i β ω ω 4a 1 0 0⎦ . ω2 = 0 ⎣1 + a2 ± (1 − a2 ) 1 + 2 (1 − a2 )2

(3.47)

The right-hand side (3.47) contains the unknown value ω. However, this equation can be solved if we assume that the fraction under the root is a small parameter. The correctness of this assumption is checked a posteriori separately for each of the two families of solutions. Note also that due to the complex nature of the parameter a2 , the denominator of the fraction never turns to zero. In this case the solution (3.47) reforms into  ω2 = ω20

 1 + a2 ± (1 − a2 ) a1 + 2i β0 ω3 /ω40 . ± 2 1 − a2

(3.48)

There is one pair of solutions for the plus sign ωT = ±ω0 ±

a1 + 2i β0 ω3 /ω40 . 1 − a2

(3.49)

Due to this assumption, the fraction represents a small correction to the frequency ωT ≈ ω0 and, making this substitution in the numerator, we finally get ωT = ±ω0 ± i β0

1 − ia1 ω0 /2β0 , 1 − a2

(3.50)

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3 Surface Contamination

from which the damping coefficient follows βT = Im ωT = β0 Re

1 − ia1 ω0 /2β0 . 1 − a2

Thus, using only the assumption of low viscosity of the liquid, we conclude that in the presence of surfactants, there is a solution to the dispersion Eq. (3.40), for which the frequency of free waves is close to ω0 —the frequency of waves at a clean surface. For the second family of solutions, this is generally unfair. Without going into the specifics of surfactants, considering the elastic modulus as a complex quantity E = |E| ei θ , after simple but lengthy algebraic transformations we obtain βT 1 − P(cos θ − sin θ) + P Q + Q sin θ = , β0 1 − 2P(cos θ − sin θ) + 2P 2 |E| k ,Q= where P = √ 3 2

ρ

2νω0

(3.51)

|E| k . 4ρνω0

For the minus sign in (3.48), there is a second pair of roots   a1 + 2iβ0 /ω0 , ω2L = ω20 a2 − 1 − a2

(3.52)

from which the approximation follows  2 4 1/3 |E| k ω ˜L =± exp(i(2θ/3 + π/6)). ρ2 ν

(3.53)

Leaving in (3.53) only the plus sign corresponding to the wave propagating in the direction, we see that the real part (3.53) represents the frequency of the wave, and the imaginary part represents the attenuation ω ˜ L = ωL + i βL , where  ωL =

|E|2 k 4 ρ2 ν

1/3

 cos(2θ/3 + π/6),

βL =

|E|2 k 4 ρ2 ν

1/3 sin(2θ/3 + π/6). (3.54)

As will be shown below, for the diffusion transfer mechanism, the loss angle θ ∈ (0, π/4), so the attenuation coefficient is always positive. Thus, we obtain two families of solutions to the dispersion Eq. (3.40) describing the wave change over time. Meanwhile, the spatial attenuation coefficient is of great interest for the analysis of experimental results. The relationship between the time β and space γ damping coefficients is obtained from the dispersion relation ω = ω(k)

3.3 The Effect of the Surfactants on the Surface Waves

107

for γ  k ω0 + i β = ω(k + i γ) ≈ ω(k0 ) + iγ

dω , dk

from where it follows  γT =

dω dk

−1

=

βT . cg

Here cg is group velocity of wave. For longitudinal waves at a real frequency ω, the wavenumber and the spatial attenuation coefficient γ L are obtained by inverting the formula (3.53)

kL =

ρ ω√ νω cos(θ/2 + π/8), |E|

γL =

ρ ω√ νω sin(θ/2 + π/8). |E|

(3.55)

As shown above, the characteristic Eq. (3.40) has two solutions (ωT , βT ) and (ω L , β), corresponding to two types of wave motion that differ in their physical nature. If for the first type T-waves, the frequency depends on the surface tension coefficient α and does not depend on the elastic modulus E (see Eq. (3.42)), then it follows from (3.53) that L-waves exist only on a surface with non-zero elasticity. In addition, L-waves are characterized by the predominance of the vortex component of motion, while in T-waves the potential component is the determining one. Indeed, from (3.27) for T-waves it follows ω A ≈ m ≈  1, B k ν k2 T

(3.56)

that is, the ratio of the potential component and the vortex component is less than for a pure low-viscosity liquid (there it is ∼ νωk 2 ), but it is quite large. For the L-wave, the picture is reversed. From (3.20) and (3.53) follows

B ω ≈ |a2 | ≈ >> 1. A ν k2 L

(3.57)

The attenuation of the wave energy is proportional to the value of the vortex component, which makes the L-wave more dissipative than the T-wave. From (3.18) and (3.19), taking into account (3.56), (3.57), we obtain the horizontal and vertical displacements of particles in T- and L-waves and, respectively ξT = ωk AB ekz exp(i(kx + ω t)), ζT = −i ωk AB ekz exp(i(kx + ω t)) . ξ L = −i ωm BA emz exp(i(kx + ω t)), ζ L = − ωk BA emz exp(i(kx + ω t))

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3 Surface Contamination

This shows that in the T-wave particles move along orbits close to circles, and in the L-wave along strongly elongated ellipses. The ratio of the longitudinal to the trans verse axis is large ξζLL ≈ mk  1. It is this feature of particle motion that allows us to call L-waves longitudinal, and T-waves in contrast to them—transverse. The transverse waves obey the well-known dispersion Eq. (3.42) and are visible to the naked eye at a sufficient amplitude. The depth of their penetration into the liquid ∼ k −1 . Longitudinal waves have a significantly smaller amplitude of vertical displacements and at the same frequency they fade more quickly with depth (m  k), i.e. they are almost horizontal compression-stretching of the surface. Assuming that all the results obtained are valid under the low-viscosity fluid condition ν k 2 /ω  1, we estimate the limits of applicability of solutions for both types of waves. For transverse waves, the phase velocity (c = ω/k) has a minimum, which for 2 water is ~23 cm/s. Then, to fulfill the condition νωk = νck < cνmink < 1, we obtain that the minimum length of the transverse wave must satisfy the condition λ 

2πν ∼ = 3 · 10−3 sm. cmin

For longitudinal waves, we substitute the corresponding expression for ω L (3.54), so ν k2 = ω



ρν 2 k E

2/3  1, −4

or for water at ρ = 1 g/sm3 , ν = 0.01 sm2 /s,λ  2π·10 . E An interesting feature of the results obtained is the fact that the maximum attenuation of transverse waves is located near the flow of space–time synchronism of two types of waves, when k T ∼ k L , at which quasi-resonant energy transfer occurs, from weakly attenuated transverse waves to strongly dissipative longitudinal waves. When choosing a surfactant with an elasticity of ∼ 1000 din/sm, it is possible to exert a much stronger direct effect on long gravitational waves with frequencies less than 1 Hz, i.e., on the energy-carrying components of the wind wave spectrum. Such surfactants are not a purely mathematical abstraction—some surfactant mixtures may have high elasticity. The targeted use of such substances in Maritime navigation can be of great practical importance.

3.4 Oil Spills in Nature and Methods of Their Elimination Currently, spills of oil and refined products are a great danger, the volume of emergency discharges of which is constantly growing and reaches critical values in manmade disasters (accidents at oil fields, pipeline ruptures). In history, environmental

3.4 Oil Spills in Nature and Methods of Their Elimination

109

disasters have occurred more than once, associated with accidents of large-capacity tankers (“Aegean captain”, “Atlantic Empress”, “prestige”, “Exxon Valdez”, etc.), which were accompanied by large-scale oil leaks and significant consequences for the ecosystems of large areas of the World ocean. The most significant and destructive impact on ecosystems is caused by oil spills in the water area located directly near the coastline—the consequences affect not only the organisms living in the water, but also the flora and fauna of the coastal zone. Eliminating the consequences of accidents in such areas requires much more time (decades) and effort, and consequently, large financial investments. To reduce the possible negative consequences, special attention should be paid to the study of methods for localization and elimination of oil spills and the development of an additional set of measures for the collection and disposal of hydrocarbons that have fallen into the external environment. The problem of eliminating the consequences of accidental oil and oil products spills is not new, but in modern realities it is still relevant. Over the past few years, various States have made great efforts to improve oil spill prevention and response systems. Currently, there are several methods for cleaning up oil pollution in open water and in water areas. The Initial task of all such methods is to detect or localize a contaminated area of water (oil slick). After this stage, they begin to eliminate it—collection and disposal. One of the most common methods is the mechanical collection of petroleum products after the spot is localized by booms that provide a sufficient thickness of the oil layer for collection. This method has a number of positive qualities, namely: a spot surrounded by a similar fence can be towed to a more convenient and, most importantly, safe place to work. Along with the mechanical method, the thermal method of elimination based on oil burning is widely used. However, this method has a serious drawback: the scope of its application is limited to a short period of time immediately after the leak—as long as the layer is sufficiently thick and the oil–water emulsion has not formed. The third method is a physical and chemical method based on the use of dispersants and sorbents. The use of this method assumes that the mechanical recovery of oil is impossible. For example, when there is an immediate threat to ecologically vulnerable areas or the thickness of the oil film is very small. When using dispersants, the natural dispersion of oil is activated, which greatly facilitates its removal from the water surface. Sorbents absorb petroleum products, which leads to the formation of lumps of material, which can then be removed mechanically. At the moment, the most accurate method that allows for fine cleaning of water areas is the biological method. This method is based on the use of special microorganisms that process oil and petroleum products. The biological method is often used when the possibilities of other methods (physical–chemical, mechanical, and other methods) are exhausted. Currently, about two hundred sorbents are produced and used in the world to eliminate oil and oil products spills. These sorbents can be divided into four main types: inorganic, synthetic, organic-mineral and natural organic. In turn, their quality

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Table 3.1 Properties of industrial sorbents for oil collection according to different manufacturers Material

Oil capacity, kg/kg

Water absorption, kg/kg

Degree of spin, %

Sorbent consumption for collecting 1 ton of oil, kg

Polypropylene (fiber)

13–25

3–6

70–80

40

Carbamide styrofoam

40–60

5–10

60–80

25–30

Carbamide-formaldehyde resin

30–50

4.6–10.0

70–80

33

Vermiculite

8–12

2–17

–

100–120

Peat

6–7

1.6

0

110

Moss

2–8

2

10–25

213

Graphite

40–60

0,2

–

25–30

Wool

6–9

1–4

65–80

150–270

Cellulose

7–9

5–7

–

100–200

Bacteria

5–10

–

–

200

and efficiency are determined by the main characteristics, such as: capacity in relation to oil, the degree of hydrophobicity, buoyancy after oil sorption, the degree of hydrophobicity, the presence of mechanisms for regeneration or utilization of the sorbent, the possibility of oil desorption. Technical characteristics of the most popular sorbents that are currently used for the elimination of petroleum products from the water surface are shown in Table 3.1. Natural organic sorbents are widely covered in scientific research. In particular, materials based on rice husks, brown coal, wool, walnut shells, peat, straw, and graphite were considered as such [1, 2]. One of the best natural sorbents is wool. By its sorption capacity, it is comparable to modified peat. One kilogram of wool can absorb up to 8–10 kg of oil, while the natural elasticity of wool allows you to squeeze out most of the light oil fractions. It is also proposed to use pine nut shells as a promising raw material for eliminating oil spills. There is a high-quality sorbent of petroleum products SNC based on cellulose fiber. Special treatment gives cellulose structural stability, buoyancy and hydrophobicity properties, which makes it possible to use it as a sorbent of petroleum products on the surface of land and water. Cellulose fiber has a complex supramolecular structure, the elements of which are microfibrils formed by several tens or hundreds of cellulose macromolecules. Microfibrils are characterized by an amorphouscrystalline structure. The fibers have a disordered structure, which ensures high absorption of organic substances in mixtures with water. Crystalline areas are responsible for strength, amorphous-for the ability to sorption. One weight unit of the SNC sorbent absorbs 8–10 units of oil and oil, the process of oil adsorption from the water surface occurs within 30–60 s. The disadvantage of this sorbent is that the cellulose fiber has a high wetting rate, in addition, the sorbent absorbs both oil and water, so it

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can not stay afloat for a long time. Next, the results of testing the SNC sorbent based on cellulose fiber will be presented. One of the main problems of organic sorbents associated with their use on the water surface is low hydrophobicity and, as a result, low buoyancy of the material. The lack of hydrophobicity also reduces the ability of the sorbent to absorb oil due to high water absorption. The presence of wax coatings, for example, on the surface of some natural sorbents increases the hydrophobicity and sorption capacity. Other related characteristics include the content of functional groups, surface morphology, active surface area, pore size, and interaction surface. It is known that natural sorbents such as zeolites can be successfully used to purify various water media from different types of pollutants. The use of these minerals as sorbents made it possible to purify contaminated water from petroleum products by 70%. A promising direction for cleaning soil ecosystems and man-made soils from petroleum hydrocarbons using a biological method is the use of biosorbents, which are a preparation made on the basis of a natural or artificially synthesized carrier with immobilized microbial cells or enzymes. Currently, various natural and chemical materials are used as a matrix for obtaining a biosorbent. Natural materials include: expanded clay, silica gel, perlite, sapropel, peat, and chemical materials-polyurethane, Teflon, polypropylene, and phenol–formaldehyde foams. We have studied complex bacterial preparations-oil pollution destructors biosorbents, which are produced by the laboratory of Microbial Technologies. “DOP-UNI” is a universal biological product with a high hydrocarbon-oxidizing activity, which is intended for biodegradation of oil and petroleum products in case of contamination of soils, reservoirs and hard surfaces. The biological product is a powder consisting of dry aggregates of viable cells of a specially selected consortium of microorganisms growing on hydrocarbons of various classes and some of their derivatives. Special additives in its composition significantly activate the process of oil destruction. One of the well-known oil collecting devices for removing oil and petroleum products from the water surface is a cable-mop, which includes an infinite belt that adsorbs oil. The belt is made of polyurethane strands that are stretched through the strands of the carrier cable so that they protrude from it in a radial direction around the circumference in the form of a pile, and the adsorbing belt passes between two rotating rollers that squeeze out oil, which is then drained into a tray, from where it is later pumped to the tank. This device has a number of disadvantages: low productivity for the collected oil, the difficulty of moving long (more than 30 m) oil-absorbing elements from one zone of the polluted water area to another. Another well-known device for collecting oil and petroleum products from the water surface includes a partially submerged drum with floats at the ends of the drum and a vacuum receiving chamber adjacent to the drum with a front wall, which is made in the form of a scraper. The disadvantages of this device include the low selectivity of oil collection, with which the predominant amount of water is sucked into the device. A device for removing oil and petroleum products from water or the earth’s surface is also known, which is mats or mats made of a material that adsorbs petroleum

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products [3]. Usually, glass wool is used as such an adsorbent, the fibers of which are coated with a mixture of silicon and starch. One of the disadvantages of the device is its low adsorption capacity. In addition, great care is required when working with glass wool, which significantly reduces performance. A device for removing oil pollution is also used, which consists of two outer fiber layers, each of which is pre-bonded by needle piercing and an intermediate fiber layer is made of synthetic fibers located between the specified outer layers. The intermediate layer is made of polypropylene or polyester fibers. The outer layers are made of polypropylene fibers, and the outer fiber layers are made more dense than the intermediate fiber layer. The disadvantage of the known device can also be attributed to a low adsorption capacity. There is a known method for cleaning the soil and water surface from oil contamination, which provides for its implementation using a device made in the form of mats made of cotton-containing sorbent, the outer side of which is pre-treated by spraying a thin layer of machine or transformer oil. In this case, the layer of cottoncontaining sorbent in the mat is fixed between layers of cotton or synthetic fabric of rare weaving or cotton mesh. The distance between the fabric threads or the size of the mesh cell that fix the sorbent in the Mat does not exceed the size of the particles of the cotton-containing sorbent. One of the disadvantages of the known device is its low adsorption capacity, but the main disadvantage is that cotton absorbs moisture and mats filled with it sink without exhausting their ability to adsorb petroleum products. There is a device for removing oil and petroleum products from the water surface, which includes a rotating drum with a hydrophobic surface and an adjacent system for removing and collecting the product. The disadvantage of this device is the significant influence of ambient temperature on the performance of the device, since during its operation in the cold autumn-spring periods of operation, in winter, at night, due to the increase in the viscosity of the collected petroleum product, the absorption capacity of the hydrophobic shell of oil and petroleum products decreases, which reduces the overall performance of the device. Also, to eliminate oil spills, a product containing an outer shell of wool felt that surrounds a core of finely ground dried pine bark is used. The disadvantage of the known device is a low adsorption capacity.

3.5 Experimental Studies of the Physical Properties of Various Sorbents The following materials were selected as experimentally studied sorbents: sheep wool, cellulose fiber, peat, and the biological product “DOP-UNI". The following types of hydrocarbons were used: oil, diesel fuel, aviation and sunflower oils. Physical characteristics of working environments are given in Table 3.2.

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Table 3.2 Physical characteristics of the liquid media used Marker parameters, when T = 20 ◦ C

Water Sunflower oil Disel fuel Oil

Aviation oil

Density (ρ), kg/m3

998.9 925.0

885.0

Kinematic viscosity (ν), ×

10–6 ,

m2 /s

Coefficient of surface tension on the a ), × 103 , impurity-air boundary (σm N/m

1.05

60.60

73.0

33.06

Coefficient of surface tension on the – w ), × 103 , impurity-water boundary (σm N/m

3.10

840.0 1.50 22.0

7.10

866.0 8.14 30.0

2.80

20.50 32.0

2.45

At the first stage, the buoyancy of sorbing materials of organic origin, which are used to eliminate oil spills in water areas: peat, cellulose and sheep wool, was studied. The experimental method provided for placing various amounts of sorbing material on the surface of a known volume of water and further monitoring the immersion of the sorbent in the liquid thickness. Identical vessels with different sorbents on the water surface were placed side by side to record the dynamics of the process of immersion of sorbing substances, the course of experiments was recorded using a digital camera in automatic mode for 4 h, shooting was conducted at a frequency of 2 frames per minute, the resulting sequences of frames were processed and analyzed. Figure 3.1a and b shows frames from the sequence, the positions of the sorbents were recorded using the photometry method of the processed images (Fig. 3.1c and d), prepared from the original frames using batch processing methods. In the images, three containers contain 100 ml of water each and 1 g of sorbent is added to each.

Fig. 3.1 Fixed positions of the sorbents at the initial a and at the time t = 66 min b after placement on the water surface, processed parts of the image c, d for photometry at the corresponding time points

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Fig. 3.2 Dynamics of sorbent immersion in the water column over time: 1 position of the wool center of mass, 2 position of the pulp center of mass, 3 and 4 interpolations of the corresponding data using the least squares method by polynomials of the 4th degree

From left to right in each image are vessels with wool, cellulose, and peat added, respectively. For the third sorbent (peat), the dynamics of its immersion in the water column was not analyzed, because when comparing successive images, this dynamics could not be detected throughout the experiment (240 min.). based on the obtained data, graphs are constructed that display the dynamics of immersion of sorbents (wool and cellulose), which are shown in Fig. 3.2. The short-term fluctuations in the level of the sorbent center of mass seen in curves 1 and 2 (Fig. 3.2) are associated with local changes in the illumination parameters of the frame field, which lead to errors in registering the position of the center of mass, but these measurement fluctuations did not significantly affect the overall picture of the registered positions of the sorbent centers of mass over time. Both sorbents are submerged in the water column, but the rates of immersion differ significantly. While 60 min after the start of the experiment, the center of mass of the pulp (in the Central vessel) moved down, the volume of wool (in the vessel on the left) only slightly decreased (Fig. 3.1b and d). During the experiments, it was found that untreated sheep wool weighing 1 g is guaranteed to absorb from 6 g of oil and other hydrocarbons (diesel fuel, sunflower oil). Studies of the sorption capacity of wool were carried out in a stationary and rotating liquid consisting of a mixture of water and various hydrocarbons (sunflower oil, diesel fuel, crude oil). Various amounts of wool fibers were placed on the surface of a two-component liquid containing 50 ml of an immiscible impurity. Depending on the weight of the wool used, the values of the typical surface cleaning rate and the

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115

Fig. 3.3 Evolution of the oil distribution pattern over the water surface in a ditch in a current induced by sorption on a fibrous material: a a layer of heavy oil 0.50 mm thick on the water surface 3.5 cm deep, t = 0 s; b partial oil sorption that violates the uniformity of the upper layer t = 10 s after the sorbent outline m = 2 g; c the appearance of pure water spots induced by sorption—induced currents, t = 110 s; d slow evolution of the shape of the oil spot, t = 600 s

residual surface area covered with an immiscible additive differed. Sequential images of the oil absorption process are presented (Fig. 3.3), illustrating its dynamics. After placing the selected amount of sorbing material on the water surface (in this experiment—2 g of natural sheep wool (Fig. 3.3a), the sorption process begins, which violates the uniformity of the thickness of the oil spot on the free surface. The introduction of sorbent water to the surface of a two-layer oil–water system leads to the formation of currents that are visible to the naked eye, which change the distribution of contamination (Fig. 3.3b), and the shape of the spot on the surface changes. In the lower left part of the drawing, in the center at the upper and lower poles, near the wall of the cuvette, the surface areas were noticeably lightened. Each part of the sorbent from the outline is surrounded by a lighter area, which corresponds to the absorption of oil by the wool. The development of the sorption process leads to a more significant change in the shape of the oil slick on the surface (Fig. 3.3c), and its division into separate fragments. The passage of time and the exhaustion of the sorption capacity of the material (or rather its parts in direct contact with

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the contaminant) a stationary pattern of distribution of the remaining portion of the contaminants on the surface (Fig. 3.3d). Additionally, it was found that sheep wool contributes to the coalescence of individual drops of hydrocarbons, and active sorption on the contrary-the appearance of drops of relatively small size, moving under the influence of all factors that determine the process of absorption of pollution by the sorbent. An important advantage of wool can be considered to be the effect of “entrainment” of hydrocarbons by the mass of wool (auto-adhesion) present in the sorption process. These features increase the efficiency of separating hydrocarbons from water and the subsequent collection of the mass of sorbate associated with the sorbent. During the process of sorption of the velocity of emerging currents, the movements of individual oil droplets separated from the main spot were estimated. The calculated displacement rates of individual droplets are shown in Fig. 3.4. Individual drops of various sizes (drop diameter from 0.6 to 3.3 mm). The dynamics of the process of hydrocarbon sorption by sheep wool is shown in Fig. 3.4. The Characteristic times inherent in the absorption of oil by wool are tens of seconds. For given (Fig. 3.3) case the characteristic time of decrease in times square covering the surface of the oil calculated according to the approximation data obtained based on the method of photometry [3], is, with increasing mass of sorbent 1.5 times the characteristic time decreases ten times and is already (see graphs in Fig. 3.5). The graphs show the relative areas of the treated water surface: the ratio of the initial area of the hydrocarbon spot to the residual area of contamination is measured, and then the resulting value is subtracted from one. Depending on the used mass of the sorbent, the residual area of contamination varies within 0.1–0.5 of the original spot area (the relative area of the treated water surface is 0.5–0.9, respectively) and depends not only on the mass of the sorbent, but

Fig. 3.4 Velocities of oil droplets of various sizes in the sorption-induced flow

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117

Fig. 3.5 Time dependence of the area free from surface contamination (relative to the size of the initial spot, 50 ml): a 2 g; b 3 g

also on the features of its placement. If the sorbent is placed in several layers, the upper layers are not involved in the sorption process. It is this feature of the use of sheep wool as a sorbent that formed the basis of the proposed design of the product for collecting liquid hydrocarbons from the water surface. For the cases shown in Fig. 3.5, the threshold values for the relative cleaning area are 0.55 and 0.80, respectively. The largest surface area of water to be cleaned from hydrocarbons, as well as the highest values of the absorption rate, are achieved by increasing the initial surface area of the sorbent, and, of course, by increasing the mass of the sorbent used. It should be emphasized that the method of obtaining results does not provide for external influence on the sorbent during the cleaning process—during the experiment it is not moved, the sorbent follows the flow of the liquid entrapping it or moves only under the influence of currents caused by sorption, which lead to a change in the position of the center of mass of the sorbent outline relative to the vessel (movements in the range of 0.1–2.0 mm). The captured oil binds to the fibers of the sheep’s wool outline quite firmly, the resulting compact volume is easily removed mechanically from the cuvette. All sorbents, regardless of the type, are characterized by sorption capacity in relation to oil (and/or petroleum products), as well as water absorption. The sorption capacity coefficient is measured by placing a known mass of the sorbent on the surface of the liquid, the sorption capacity relative to which is being investigated, placed in a test container, after which, after a selected time interval, the sorbent with the associated sorbate is extracted and the combined mass of both components is measured. Dimensionless coefficient of sorption capacity in relation to oil (water) calculated using the formula K sor b = m sor b /m init ,

(3.58)

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Table 3.3 Coefficient of sorption capacity of sorbents Name Sorbent’s

Sunflower oil

Aviation oil

Disel fuel

Oil

Fleece

8.9

7.8

5.9

8.8

Cellulose fibre

6.6

6.1

3.8

6.7

Peat

6.0

5.0

3.7

5.9

where m sor b is the mass of absorbed substance (oil, water), m init is the initial mass of dry sorbent, which is calculated from a simple ratio of m sor b = M − m init , in which M a total mass of sorbate with sorbent. Based on direct measurement data, the coefficients of sorption capacity relative to oil were calculated for some sorbents that are of interest for the successful implementation of this work. Table 3.3 shows the values of the sorption capacity coefficient in relation to oil for the tested sorbents. To determine the water absorption coefficient of the studied sorbents, samples of materials with a size of 6 × 6 cm were weighed and placed in a test container with a volume of 4 L, half filled with water. The container was installed on a mixing device (shaker) to simulate sea waves. After 15 min, the floating samples were removed, weighed, and the water absorption coefficient was calculated from the data obtained Kw =

Mw , Md

(3.59)

where Md is initial mass of dry sorbent (g), Mw = Msor b − Md is the mass of absorbed water, Mw is the mass of the sorbent with the absorbed water (g). Table 3.4 presents the values of the coefficient of water absorption for tested sorbents. Experiments were also conducted to study the process of sorption of petroleum products with the biological product “DOP-UNI”, as well as by adding it to wool. The culture of bacteria that have the ability to use hydrocarbons for nutrition was transferred in the form of a dry powder, which must be mixed with water and saturate the resulting solution with oxygen under constant stirring to restore the activity of microorganisms. This procedure was conducted within one day immediately before each experiment to study the effects of a biological product (solution cultures of microorganisms in the water) on the sorption process of oil and the value of the coefficient of purification from contaminants (both in terms of the reduction of the Table 3.4 Coefficient of water absorption of sorbents

Name Sorbent’s

Water

Fleece

2.1/1.0 = 2.1

Cellulose fibre

7.7/1.0 = 7.7

Peat

0.7/1.0 = 0.7

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119

area of surface stains, and in part more subtle clearing of a part of light fractions of oil dissolving in water volume). This type of experiment was carried out simultaneously in three identical pools filled with water (10 l), on the surface of which a 30 ml portion of petroleum products was placed (Siberian Light brand, density 845 kg/m3 (36.50 API), sulfur content 0.57%), (Fig. 3.6). 5 g of sheep wool was added to pool 3, 5 g of sheep wool and 500 g of biologics were added to pool 2, and 500 g of biologics were added to pool 1 (Fig. 3.5). The water temperature was 18 °C and the air temperature was 18 °C. Registration of the flow pattern induced by sorption was carried out throughout the experiment (5 h, of which the video was recorded for the first 45 min, then photo recording was carried out with a shooting frequency of 2 frames per second). Due to the duration of the experiment, as well as to improve the quantitative characteristics of the results, the sorbents were added to the containers sequentially, and the interval between the introduction of the sorbent (see Fig. 3.6) in containers numbered 2 and 3 was 6.2 min (for each of the experiments—individually), the entire process of adding sorbent and microorganisms to the oil–water system was recorded using a digital video camera. It should be said that the container number 1 should be excluded from consideration, because adding a solution of microorganisms in the system oil–water (t = 450 °C) leads to an abrupt change in the characteristics of surface interaction between oil spot and water, whereby the spot size is reduced, but the dynamics of decrease of the area of contamination is missing entirely (Fig. 3.7). During the two months during which the situation in container number 1 was monitored, the area of the oil patch on the water surface did not change significantly. The dynamics of the end of the process of active sorption of oil from the water surface with a sorbent (sheep’s wool) is shown in Fig. 3.8. The rate of reduction of the surface area of water covered with oil is per second, which indicates the practical exhaustion of the sorbent capacity in this time interval. The highest rates of absorption

1

2

3

Fig. 3.6 Recording the dynamics of the oil sorption process. Three containers with a portion of oil (30 ml) on the surface: 1 cleaning only with microorganisms, 2 cleaning only with sheep wool, 3 simultaneous use of microorganisms and wool. The image refers to the time t = 4600 s after the start of the sorption process (applying the sorbent to the surface of water with contamination)

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Fig. 3.7 Various stages of the experiment (the time is counted from the moment when the sorbent and microorganisms are placed in the container number 2): a initial contamination with petroleum products, b–d after 70, 380, 750 s

Fig. 3.8 Reduction of the area of contamination under the action of the sorbent in the time interval t = 1200–1250 °C after the introduction of the sorbent in the vessel number 3

of contaminants by the sorbent, and consequently the highest rates of increase in the surface area of water free from contamination, are observed immediately after the introduction of the sorbent (see Fig. 3.7). The main part of the data presented and most of the illustrations reflect the most spectacular part of the experiments: one of the samples of oil used for the experiments has a high optical density (opaque black oil), which makes it easy to identify spots of immiscible impurities on the water surface. All other tested samples of immiscible additives have a significantly lower optical density, so that the visual perception of

3.5 Experimental Studies of the Physical Properties of Various Sorbents

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Fig. 3.9 Sketch of a device for collecting liquid hydrocarbons from the water surface: a diagram, b photo

the flow pattern becomes not too obvious. The processing of such experimental data implies more precise methods for determining the position of different color and intensity areas of the surface corresponding to various impurities and pure water. To conduct experiments on oil collection, models of adsorption media were developed, the schematic image and General view of which is shown in Fig. 3.9. The device for separating from water and collecting liquid hydrocarbons (patent [4]) includes a frame made of hollow elements or mesh 1 filled with natural sheep wool 2 (washed or unwashed). The frame can be made of various geometric shapes— in the form of a Mat in the form of a rectangular parallelepiped (as shown in the drawing) or a Mat. The volume of wool 2 is divided by a waterproof partition 3, located perpendicular to the direction of the smallest frame size. The waterproof partition is made of polyethylene or fluoropolymer, the product frame is removable. In one of the parts of the device separated by a partition, cellulose fiber was added in an amount from 10 to 70% of the wool volume. The use of easily accessible natural sheep wool significantly increases the efficiency of the device. Currently, most of the wool is destroyed due to lack of demand (which is a waste of meat sheep breeding), which significantly reduces the cost of the device. Wool is hydrophobic, almost does not absorb moisture and has a high sorption capacity in relation to hydrocarbons, absorbing them from mixtures with water and even emulsions. At the same time, it was found that wool has a unique property of catalyzing coalescence of liquid hydrocarbons and such a rare variant of coalescence as auto-adhesion. Dividing the wool volume by a waterproof partition located perpendicular to the direction of the smallest frame size will allow the wool fibers to be used most fully for adsorption of hydrocarbons, while on the other side of the device there will be no absorption of both hydrocarbons and water, which will effectively use the second

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side of the device when turning over, as well as increase the overall buoyancy of the device. In particular cases of implementation of the device, to increase the efficiency of collecting liquid hydrocarbons, it is advisable to add cellulose fiber. Adding cellulose fiber to wool increases its sorption capacity, since cellulose fiber absorbs hydrocarbons 9–10 times its weight. Also, the addition of cellulose fiber will increase the temperature range of the device, since water that has penetrated the capillaries of the cellulose fiber freezes at temperatures significantly below 0 °C. It is advisable to add an additional substance in quantities from 10 to 70% of the wool volume. When adding more than 70% of cellulose fibers, the device will become more expensive, and will also be hygroscopic, absorb a large amount of water, which leads to the need for additional elements to maintain buoyancy and increases the cost of the product. When less than 10% of cellulose fibers are added, the sorption capacity of the device increases slightly. Making the frame removable makes it possible to reduce the consumption of material that is not directly involved in the process of cleaning from hydrocarbons, to facilitate and optimize the storage and transportation process. The device for collecting liquid hydrocarbons from the water surface can be used repeatedly: extracted after filling the cells with hydrocarbons and returned to continue collecting after removing the collected material (oils, petroleum products, and other substances), for example, by pressing. Experiments have shown that the ability of natural wool to absorb hydrocarbons is preserved after the excess of the collected material is removed. The use of a frame, tubular or in the form of a grid, provides a more uniform placement of wool and can be used in stationary barriers to separate hydrocarbons from effluents in places of permanent losses (spill stations, gas stations, storm drains, etc.). the. Device is used as follows: Depending on the spill conditions, the device can be used as a barrier or Mat on the water surface when blocking spills of light hydrocarbons, or as a partially submerged barrier or as a screen of a sufficiently high height. When used as a partially submerged barrier or screen, the waterproof partition reduces the through-flow of dirt. During operation, after the lower part of the device (up to the partition) has exhausted its sorption capacity, it turns over and the part that was previously upper becomes working. The partition will prevent the movement of already sorbed hydrocarbons to this part, which will significantly increase the efficiency of the device. The moment when it is necessary to turn to the other side is determined by preliminary experiments based on the study of the sorption capacity of wool. In particular cases of implementation, when cellulose fiber is added to one of the parts of the device, which allows increasing its sorption capacity, it is advisable to operate the device at the first stage when the part that is filled only with wool contacts the water surface, and at the second stage—wool with cellulose. But cellulose also absorbs water in relatively larger quantities than wool. As a result, the buoyancy of the device is reduced, which is an indirect sign that the adsorption process is complete and the device must be removed for washing (in case of reuse) or for disposal.

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Several devices can be connected using any known means, such as one or more ropes, ropes passing through technological reinforced holes in mats or screens, or using a pair of “ring—carabiner”, including formed frame elements. Such devices are preferred when cleaning and removing oil-containing liquids, when heavy equipment is ineffective, or in combination with the latter for posttreatment. The developed device can also be used as a preventive tool at potentially dangerous facilities: oil depots, automobile enterprises, air and sea transport enterprises, and any enterprises whose activities are associated with the use of liquid hydrocarbons, accompanied by potential leaks and spills. The device will be able to operate in two modes: from a surface vessel and from the shore of a reservoir. The cycle of use of the device provides for its cleaning from the collected hydrocarbons and repeated use (at least three times). If the device is stored for a long time (more than 1 month), it must be ensured that there is no contact with the external environment (vacuum packaging). The main advantage of the developed device for collecting liquid hydrocarbons from the water surface is the relatively low cost and environmental safety of its use, as well as the possibility of repeated (2–3 times) use. The device was tested in four pools of various sizes: three made of glass, measuring 0.4 × 0.4 × 0.4 m, 0.6 × 0.6 × 0.6 m, 1.2 × 0.8 × 0.7 m, respectively, and the fourth— inflatable −3 × 4 m, which was filled with tap water. The following hydrocarbons were used: sunflower and aviation oils, diesel fuel (summer and winter) and oil (“Siberian Light” brand, density 845 kg/m3 (36.5 API), sulfur content 0.57%). Samples of the “Liquid hydrocarbon collection device” of three sizes were also made: 0.2 × 0.4 m, 0.4 × 0.8 m, 1 × 2 m several copies of each size, which are mats made of mesh, filled with natural sheep wool. The volume of wool is divided by a waterproof partition made of polyethylene, and cellulose fiber is added to one of the parts in an amount of 20% of the wool volume (Fig. 3.10). To clarify the terminology, the word “mats” will be used later in the description of experiments» Measurements of the weight and size characteristics of product samples (see Table 3.5) were carried out immediately before the tests. Their significant errors are related to various factors: (1) (2)

manual cutting of the frame material leads to inaccuracies in observing its dimensions a small amount of the sorbent placed in the cells of the frame may be lost due to the ratio of the mesh cell size and the main sorbent fibers, which are not calibrated and have a chaotic relationship with each other, 3) the cellulose part of the composite sorbent placed in the cells of the frame is loosely bound to wool fibers, as a result of which it may be partially lost before the start of direct operation/testing of the product.

The method of measuring the sorption capacity coefficient includes placing three test samples on the surface of hydrocarbons and calculating the sorption capacity coefficient K sor b with respect to hydrocarbons using the formula K sor b = m sor b /m init ,

(3.60)

124

3 Surface Contamination

Fig. 3.10 Photos of samples of “devices for collecting liquid hydrocarbons” of three standard sizes: a 0.2 × 0.4 m, b 0.4 × 0.8 m, c 1 × 2 m Table 3.5 Values of weight and size characteristics of device samples of various sizes Standard size

No. 1

No. 2

No. 3

Overall dimensions (error), m

0.2 × 0.4 (0.02)

0.4 × 0.8 (0.05)

1.0 × 2.0 (0.05)

Product weight, g

195 ± 13

515 ± 36

3300 ± 230

320 ± 21

2010 ± 170

33 ± 5

200 ± 35

Mass of the main sorbent (sheep wool), 125 ± 8 g Weight of additional sorbent (cellulose fiber), g

13 ± 2

3.5 Experimental Studies of the Physical Properties of Various Sorbents

125

Table 3.6 The ratio of the sorption capacity of the tested samples No. sample’s

Sunflower oil

Aviation oil

Disel fuel

Oil

Sample No. 1

7.8

7.4

5.2

8.1

Sample No. 2

7.4

7

5

7.7

Sample No. 3

7.2

6.8

4.9

7.5

Table 3.7 The coefficient of water absorption for test specimens

No. Sample’s

K w = Mw /Mc

Sample No. 1

3.5

Sample No. 2

3.33

Sample No. 3

3.21

where m sor b is the mass of the absorbed substance (oil, oils, diesel fuel), m init is the initial mass of the mat, which is calculated from a simple ratio m sor b = M − m init in which M is the total mass of the mat together with the collected substance. Table 3.6 shows the values of the sorption capacity coefficient in relation to hydrocarbons for the tested samples. To determine the water absorption coefficient of the test samples, the mats were weighed and placed in an inflatable pool half filled with water (the volume of water was 4800 L). After 60 min, the floating samples were removed, weighed, and the water absorption coefficient was calculated from the data obtained K w = Mw /Mc ,

(3.61)

where Mc is the initial mass of the sample (mat), Mw = Msor b − Mc is the mass of absorbed water, in which Mc is the total mass of the Mat together with water. Table 3.7 shows the values of the water absorption coefficient for the test samples. To test the “Device for collecting liquid hydrocarbons” in the laboratory, the manufactured samples (mats) were placed on the surface of resting water, to which oil and diesel fuel (winter and summer) were previously added in various concentrations. Before that, liquid samples were taken to determine the degree of purification (Table 3.8). The test samples were left on the water surface for 24 h, after which the mats were taken out, weighed, and wrung out using a wringer (made independently in the form of a wringer attachment from a washing machine on a 200-L barrel. When the material passes through the rollers of the device, the liquid flows by gravity to the bottom of the barrel) and is placed in a vacuum package for further processing or disposal. It should be noted that after 24 h, none of the test samples drowned. After the experiments, water samples were taken again to determine the degree of purification from hydrocarbon contamination. The samples were analyzed by fluorescence spectroscopy (fluorometry) with calibration by Raman scattering of a solvent—water

126

3 Surface Contamination

Table 3.8 The liquid samples The sample’s number

Name

Concentration mg/l, notes

1

Diesel fuel (summer)

–

2

Diesel fuel (winter)

–

3

Oil

Brand “Siberian Light”, density 845 kg/m3 (36.5 API), sulfur content 0.57%)

4

Diesel fuel (summer) + water

25.4

5

Diesel fuel (winter) + water

12.8

6

Oil + water

2.02

7

Diesel fuel (summer) + water

41

8

Diesel fuel (winter) + water

20.6

9

Oil + water

3.23

or hexane. The spectra were taken using a «Horiba Jobin Yvon Fluoromax—4» spectrofluorometer. Processing of the spectra gave the values of the degree of purification given in Table 3.9 for a 1 × 2 m mat. The degree of water purification from hydrocarbon contamination was also determined using the extraction method. The values of oil/diesel fuel concentration in water were determined using the method of oil extraction from water with hexane. The values of the degree of purification obtained by fluorimetry of water samples with oil/diesel fuel are quite satisfactorily consistent with the values obtained by the extraction method. Although the first method contains a preparatory operation (dilution of the sample with distilled water), it is much simpler and more rapid than the extract method, and can also be easily implemented in automatic mode—when the sample flows through the measuring system, in contrast to the extract method. The degree of water purification from oil was twice as high as the degree of purification Table 3.9 Purification’s degrees

Type of pollution

Degree of purification (%)

Summer DF, the concentration 34 25.4 mg/l Winter DF, the concentration of 12.8 mg/l

33

«Siberian Light» oil, concentration 2.02 mg/l

55

Summer DF, the concentration 36 41 mg/l Winter DF, the concentration of 20.6 mg/l

37

«Siberian Light» oil, the concentration of 3.23 mg/l

61

3.5 Experimental Studies of the Physical Properties of Various Sorbents

127

from diesel fuel (DF). The reason for this result, established during measurements by both methods, remains to be found out. The device for collecting liquid hydrocarbons from the water surface can be used repeatedly: the mats are removed, wrung out using a wringer, and re-placed on the water surface. As shown by the experiments, the ability to well Sorb hydrocarbons is preserved at least 3 times. During the experiments, it was found that untreated sheep wool absorbs at least 6 times more than its mass of oil and other hydrocarbons (diesel fuel, aviation and sunflower oil). Additionally, it was found that wool contributes to the coalescence of individual drops of hydrocarbons. An important advantage of wool is the effect of “entrainment” of hydrocarbons by the wool mass (auto-adhesion) present in the sorption process. Due to its hydrophobic properties and low density, sheep wool together with the absorbed substance does not sink into the liquid thickness, therefore, such a sorbent is easy to collect mechanically. It is shown that wool sorbs up to 89% of oil, depending on its initial concentration and the amount of sorbent. Depending on the spill conditions, the device [14] can be used as a barrier or Mat on the water surface when blocking spills of light hydrocarbons, or as a partially submerged barrier or as a screen of a sufficiently high height.

References 1. Bayat A et al (2005) Oil spill cleanup from sea water by sorbent materials. Chem Eng Technol 28(12):1525–1528 2. Rethmeier J, Jonas A (2003) Lignite based oil binder mats: a new absorbent strategy and technology. Spill Sci Technol Bull 8(5–6):565–567 3. Bernhardt B Agent for binding oil present in liquids. Patent DE 4,140,247 4. Chaplina TO, Stepanova EV (2017) Device for collecting liquid hydrocarbons. Utility model Patent No. 169,140, date of state registration 16 March 2017

Chapter 4

Stratification of the Ocean Environment

One of the most important characteristics of an inhomogeneous medium is the stable stratification of its density in the Earth’s gravity field in the absence of introduced disturbances. Knowledge of the initial stable stratification is necessary for correct interpretation of measurement data in field and laboratory conditions. The availability of an adequate stratification model is particularly important when conducting experiments with internal gravitational waves and convective processes in a liquid, as well as when describing the dynamics of the medium in technological processes.

4.1 Model for Stable Stratification of the Marine Environment The actual stratifying density distribution is created by the stratifying distributions of the water temperature and the salt dissolved in it. Therefore, the creation of temperature and salt distribution models is the basis of the density distribution model. The standard approach to creating such models is to study a system of fundamental hydrodynamic equations describing the evolution of the physical fields of the medium. As such physical fields, the velocity field v and the fields of thermodynamic variables p (pressure), T (temperature), S (salinity) and ρ (density) are chosen. As a result, the dynamics of a viscous fluid in the field of gravity, whose gravitational acceleration vector g is directed against the z axis, is described by a set of relations [1], which includes Newton’s second law for a viscous medium (the Navier–Stokes equation), the equation of density evolution, the equations of temperature and salinity transfer, and the equation of state describing the dependence of density on other thermodynamic variables (at the moment it does not have a fundamental explicit form) 

∂ vi + v · ∇vi ρ ∂t

 =−

∂p ∂ xi

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Kistovich et al., Advanced Studies in Ocean Physics, Springer Oceanography, https://doi.org/10.1007/978-3-030-72269-2_4

129

130

4 Stratification of the Ocean Environment

+

      ∂ vj ∂ vi 2 ∂ η + δi j ζ − η ∇ · v + ρ gi + ∂ xj ∂ xj ∂ xi 3

dρ + ρ∇ · v = m dt   dT ρ cp = ∇ · (κ ∇T ) + ρQ T ≡ ∇ · ρ c p χ ∇T + ρQ T dt dS = ∇ · (κs ∇ S) + ρQ S ≡ ∇ · (ρk ∇ S) + ρQ S ρ dt ρ = ρ( p,T, S)

(4.1)

Here ddt = ∂∂t + v · ∇; η, ζ are the first and second dynamic viscosities; κ, χ = κ/ρ c p are the coefficients of thermal conductivity and thermal diffusivity; κs , k = κs /ρ are the coefficients of salt conductivity and salt diffusion; c p is the specific heat capacity of water at constant pressure; Q T (r, t), Q S (r, t), m(r, t) are the sources of heat, salt and mass, respectively. The initial stable stratification is an equilibrium state of the medium that exists in the gravity field only when there are no sources of heat, salt, and mass (Q T = Q S = m = 0) and is characterized by a set of equilibrium values of physical fields v = 0, p = p(z), T = τ(z), S = σ(z), ρ(z) = ρ(0, τ(z), σ(z)). In general, substituting the equation of state into the density evolution equation results in a sequence of relations dρ dt

+ ρ∇ · v =

=

1 dp c2 d t



∂ρ ∂p

− αρ dd Tt +







dp dT + ∂∂ Tρ T,S d t p,S d t βρ dd St + ρ∇ · v = m

+



∂ρ ∂S



dS p,T d t

+ ρ∇ · v

.

(4.2)

 , c is the adiabatic speed of sound; α = − ρ1 ∂∂ Tρ is the T,S p,S  is the coefficient of coefficient of volumetric temperature expansion; β = ρ1 ∂∂ ρS Here

1 c2

=



∂ρ ∂p



p,T

the relative contribution of salinity to the density. Since the stratification model created here assumes that the description of convective processes and the propagation of internal gravitational waves in a liquid medium, it is assumed that the inclusion of heat, salt, and mass sources is not accompanied by an effective generation of acoustic waves, which allows us to use the approximation of an incompressible (with respect to pressure action) liquid, which formally corresponds to the limit transition c → ∞. Then Eq. (4.2) can be divided into two equations dρ dt

− αρ dd Tt + βρ dd St = 0 ∇ · v = α dd Tt − β dd St + mρ =

α ∇ ρ cp

· (κ ∇T ) − βρ ∇ · (κs ∇ S) +

α QT cp

− β QS +

m ρ

.

(4.3)

4.1 Model for Stable Stratification of the Marine Environment

131

The right-hand side of the second equation of the system (4.3) follows when using the temperature and salt transport equations of the system (4.1). The resulting pair of Eqs. (4.3) is equivalent to the pair made up of the density evolution equation and the equation of state of the system (4.1). It is the first equation of the system (4.3) that is necessary when creating a model of stable stratification of the environment. Standard approaches to this problem use the approximation of small temperature superheats (supercooling) of the medium and, as a consequence, the approximation of constancy of all thermodynamic coefficients. In this approximation, the first Eq. (4.3) is easily integrated and sets the density distribution as a function of temperature and salinity explicitly ρ = ρ0 exp(βσ(z) − ατ(z)) · exp(β S˜ − αT˜ ),

(4.4)

where T˜ , S˜ are perturbations of the original stratifying distributions τ(z) and σ(z), ρ0 is a constant. Thus, in the equilibrium state, the system (4.1) (when replacing the already mentioned pair of equations with the equations of the system (4.3)) in the approximation of small temperature superheats takes the form ρ(z) = ρ0 exp(βσ(z) − ατ(z)),

d 2 τ(z) = 0, d z2

d 2 σ(z) dp = −g ρ(z). = 0, 2 dz dz (4.5)

Solutions of the last two equations of the system (4.5) τ(z) = τ0 + G T z, σ(z) = σ0 + G S z; τ0 , σ0 = const

(4.6)

set the stratifying distributions of temperature and salt. In this case, the constant values G T , G S represent the gradients of temperature and salinity in the medium. As a result, the stratifying density distribution is defined by ρ(z) = ρ 0 exp(−z/), ρ 0 = ρ0 exp(βσ0 − ατ0 ) 1 = αG T − βG S = 1T − 1s , T = αG1 T , 1s = 

1 βG S

,

(4.7)

where , T ,  S are the stratification scales of density, temperature, and salt, respectively. The condition for stability of stratification in the field of gravity requires a decrease in density with an increase in the z coordinate, which mathematically means d ρ(z)  0, ∀ z dz

∼

 0

⇒

T  S  S − T

> 0,

(4.8)

132

4 Stratification of the Ocean Environment

at the same time, it is necessary to take into account the fact that both in natural and laboratory conditions, the scales of temperature and salt stratification can take both positive and negative values (phenomena of temperature inversion or salinity). The buoyancy frequency defined by the expression N 2 (z) = −g

d g ln ρ(z) = , dz 

(4.9)

represents, in this approximation, a constant value. As already mentioned above, the ratios (4.5–4.8) are valid for small temperature overheats of the order of degrees. At the same time, in laboratory experiments and technological processes, the initial overheats that create stratification of the medium can reach tens of degrees. In addition, weekly field measurements [2] of the frequency of buoyancy in the World Ocean in the framework of the “Argo” project show its significant dependence on depth. These facts require the construction of a model of the initial stable stratification with the rejection of the approximation of the constancy of the thermodynamic characteristics of the medium. Based on the above, the developed model is based on a system of equations of the equilibrium state of the medium of the form dρ − αρ dd Tt + βρ dd St = 0, dt    d κ dd Tz ≡ ddz ρ c p χ dd Tz dz

dp dz

= −g ρ(z)   = 0, ddz κs dd Sz ≡

d dz

 d S , ρk d z = 0

(4.10)

in general, all the thermodynamic coefficients present in the system (4.10) are considered to be functions of temperature and salinity.

4.2 Thermodynamic Characteristics of the Medium In order to start solving the system (4.10), it is necessary, based on real measurement data, to set the dependencies of the mentioned coefficients on temperature and salinity. Experimental data [3–5] show that the characteristic parameters of the medium (kinetic coefficients, thermal expansion coefficients, etc.) are extremely weakly dependent on salinity, but they are sensitive to temperature. The exception is the salt diffusion coefficient, which depends on both temperature and salinity. The coefficient β of the relative contribution of salinity to density is assumed to be constant, since its relative variations with changes in temperature and salinity are extremely small compared to the variations in measurement data obtained from different sources. For the same reason, the specific heat capacity c p is considered constant (its relative variations with changes in temperature and salinity are fractions of a percent [5]). Figure 4.1 shows tabular data (points) and a graph of the approximate analytical dependence on the temperature of the heat conductivity coefficient of water at normal pressure. The analytical dependence, which is valid in the temperature range from 5

4.2 Thermodynamic Characteristics of the Medium

133

Fig. 4.1 The dependence of the coefficient of thermal conductivity of water κ on temperature

to 90 ◦ C, is described by the relation kg · m ; κ0 ≈ 3.14 · 10−2 , s3 · K κ1 ≈ 4.35 · 10−2 , κ2 ≈ 1.56 · 10−1 .

κ = κ 0 + κ1 y κ 2 ,

(4.11)

Here and further y = T˜ /T0 . The symbol T0 indicates the reference temperature value, which is taken as the value 273.15 K. The symbol T˜ indicates the deviation of the temperature at a given time at a specific point in space from the reference temperature, that is, T˜ = T − T0 , where T is the true value of the thermodynamic temperature. The maximum relative approximation error in the described temperature range does not exceed the value δκ = 10−2 . Figure 4.2 shows tabular data (points) and a graph of the approximate analytical dependence on the temperature of the coefficient of thermal volumetric expansion of water at normal pressure. The analytical dependence is described by the relation α = α0 + α1 y α2 , K −1 ;

α0 ≈ −1.83 · 10−4 , α1 ≈ 1.60 · 10−3 ,

α2 ≈ 0.54. (4.12)

The maximum relative approximation error in the described temperature range does not exceed the value δα = 2.5 · 10−2 . Figure 4.3 shows tabular data (points) and a graph of the approximate analytical

134 Fig. 4.2 The dependence of the coefficient of temperature volumetric expansion of water α on the temperature

Fig. 4.3 The dependence of water density ρ on temperature

4 Stratification of the Ocean Environment

4.2 Thermodynamic Characteristics of the Medium

135

Fig. 4.4 The dependence of the coefficient of thermal conductivity of water χ on the temperature

dependence on the temperature of the water density at normal pressure. The analytical dependence is described by the relation   ρ = 999.87 · exp −T0 α0 y +

α1 y 1+α2 1 + α2

 ,

kg . m3

(4.13)

The maximum relative approximation error in the described temperature range does not exceed the value δρ = 2 · 10−4 . Figure 4.4 shows tabular data (points) and a graph of the approximate analytical dependence on the temperature of the coefficient of thermal conductivity of water at normal pressure. The analytical dependence is described by the relation     κ2 χ ≈ 2.39 · κ0 + κ1 y exp T0 α0 y +

α1 y 1+α2 1 + α2



· 10−7 ,

m2 . s

(4.14)

The maximum relative approximation error calculated for the ejection point at T = 20 ◦ C does not exceed the value δχ = 10−2 in the described temperature range. This dependence (4.14) is not used in the future and is given here as a proof of the quality and consistency of the approximations used, since (4.14) is obtained on the basis of the definition χ = κ/ρ c p , relations (4.11, 4.13) and under the assumption of constancy of the specific heat capacity c p .

136

4 Stratification of the Ocean Environment

Table. 4.1 The values of coefficients a, p in approximation formulas S, 0/00

0

2

4

6

8

10

15

20

25

a

0.374

0.6

0.59

0.63

0.65

0.68

0.74

0.81

0.86

p

3.64

1.52

1.56

1.444

1.40

1.32

1.22

1.12

1.08

Table. 4.2 The values of coefficients b, q in approximation formulas T, ◦ C

18

25

40

60

b

1.15

1.44

1.94

2.57

q

0.0111

0.0046

0.0034

0.0052

Figures illustrating the comparison of tabular data and graphs of approximating analytical dependences on temperature and salinity of the salt diffusion coefficient k at normal pressure are not given here because of their large number. In general, the dependence of the salt diffusion coefficient on temperature and salinity is described by the relation k(S, y) = a · b · (1 + py)(1 + q βS) · 10−7

m2 . s

(4.15)

The values of the a, p and b, q parameters are shown in Tables 4.1 and 4.2, respectively.

4.3 Models of Natural Stratified Media Now, after determining the necessary dependences of the thermodynamic parameters of the medium on temperature and salinity, it is necessary to find a solution to the system (4.10). Since the coefficient of volumetric temperature expansion depends on the temperature, the solution of the first equation of the system (4.10) has the form ⎛ ⎜ ρ(T, S) = ρ0 (T0 , S0 ) exp⎝β S˜ −

T 0 +T˜

⎞ ⎟ α(T ) dT ⎠,

(4.16)

T0

where T0 is the reference temperature, S0 is some reference salinity of the medium (which is usually taken as the average salinity), such that the total salinity S is ˜ and S˜ is the salinity perturbation. determined by the expression S = S0 + S, The equilibrium state of a stratified medium is determined by the following relations v = 0, T˜ = T − T0 + τ(z), S˜ = σ(z), p = p(z).

(4.17)

4.3 Models of Natural Stratified Media

137

Here, T is the average water temperature; τ(z), σ(z) are the stratifying distributions of temperature and salinity, respectively. As a result, the system (4.10) is reduced to the form d dz dp dz

 κ(τ(z)) d dτ(z) = 0, z

d dz

 ρ(τ(z), σ(z)) k(τ(z), σ(z)) d dσ(z) =0 z

= −g ρ(τ(z), σ(z))

.

(4.18)

First, the first equation of the system (4.18) is solved. For the convenience of ˜ 0 further calculations, enter the value θ(z) = TT0 = T −T + τ(z) . Substituting in (4.18) T0 T0 the expression (4.11) for the thermal conductivity coefficient and integrating the first Eq. (4.18) gives rise to the equation for the function θ(z) θ(z) +

κ1 θ(z)1+κ2 = Az + B, κ0 (1 + κ2 )

(4.19)

where A and B are some constants. An exact analytical solution of Eq. (4.19) for arbitrary values of the κi coefficients is unattainable, but for real environments, when the values of the mentioned coefficients given in (4.11) are valid, Eq. (4.19) reduces to the form 4 κ1 θ(z)2+κ2 = (Az + B)2 , κ0 (1 + κ2 )

(4.20)

with a relative error not exceeding the value 10−3 . As a reference point for the stratifying temperature distribution, we select z = 0 and assume that τ(0) = 0. This setting of the reference point determines the value of the constant B by the ratio 2+κ2  T − T0 4 κ1 B = , κ0 (1 + κ2 ) T0 2

as a result, Eq. (4.20) takes the form  θ(z)

2+κ2

=

T − T0 T0

2+κ2 

A z+1 B

2 .

The A/B relation is replaced by an expression of the form λT /T , where T > 0 is the temperature stratification scale, λT = ±1. The choice of a specific sign of the coefficient λT is determined by the stability of the medium density stratification and is carried out later, after determining the type of function σ(z), since the stability of the density stratification is affected by both temperature and salt stratifications. The final expression for the function distribution θ(z) is

138

4 Stratification of the Ocean Environment

θ(z) =

1   1+γ T − T0 z κ2 λT ≈ 0.078. +1 , γ= T0 T 2

(4.21)

The stratifying temperature distribution is also determined from (4.21) 

z λT +1 T

τ(z) = (T − T0 )

1  1+γ

 −1 .

(4.22)

In order to use the second equation of the system (4.10), you must first determine the type of expression for the density based on (4.16)  ρ(τ(z), σ(z)) = ρ0 (T0 , S0 ) exp βσ(z) −  = ρ0 (T0 , S0 ) exp βσ(z) − T0

T +τ(z) 

 α(T ) dT

T0 θ(z) 

.

(4.23)

α2

(α0 + α1 y ) dy

0

Calculating the integral included in (4.23) gives the result 1 δ   1+γ   1+γ

θ(z)   z z α2 α0 + α1 y dy = m λT +1 + n λT +1 , T0 T T

(4.24)

0

 δ α1 T0 T −T0 , δ = 1 + α2 . where m = α0 (T − T0 ), n = 1+α T0 2 Substituting (4.24) into the second equation of the system (4.18) generates an equation for the stratifying salinity distribution d βσ(z) (1 + q βσ(z)) exp( βσ(z)) dz   exp m λT



1 1+γ

 +n λT

δ 1+γ +1



,

(4.25)

(1 + q ( f − 1)) exp( f ) = AI (z) + B,

(4.26)

=A

z T

+1

 1+ p λT

z T

+1



z T

1 1+γ

0 b. where A is some constant, p = T −T T0 Let f (z) = βσ(z). Integration (4.25) leads to the result

where A, B are some constants, and the I (z) function has the form

I (z) =

  1 δ  1+γ  1+γ z z + n λ T T + 1

exp m λT T + 1 1  1+γ 1 + p λT zT + 1

dz.

(4.27)

4.3 Models of Natural Stratified Media

139

The solution of the functional Eq. (4.26) is written in the form  f =W

e

1−q q

q

 (AI (z) + B) −

1−q , q

(4.28)

where W(x) is the Lambert function [6], defined as the solution of the W(x) exp(W(x)) = x equation. This function is well studied and tabulated and is currently included in the set of special functions of mathematical physics. As a reference point for the salt distribution, as well as for the temperature distribution, is selected z = 0. At the same time f (z) = βσ(z) = 0. Then it follows from (4.28) B = 1 − q − AI (0). By definition, the function f defines the structure of the salt stratification, which acts as an independent thermodynamic variable. In the m+n isothermal case (T → ∞) from (4.27) follows I (z) = e1+ p z. Then the value A is chosen as A = λSS (1 + p) e−m−n , where  S is the characteristic scale of the salt stratification, λ S = ±1. The rule for selecting a specific value λ S is determined by the stability of the density stratification and will be presented explicitly later. Finally, the distribution of salt stratification has the form  βσ(z) = W

e

1−q q

q



λS J (z) + 1 − q S

 −

1−q , q

(4.29)

where

J (z) =

  1 δ  1+γ  1+γ

z exp m λT xT + 1 + n λT xT + 1 0

1  1+γ 1 + p λT xT + 1

d x.

(4.30)

Using (4.23, 4.24, 4.29), you can write the stratifying distribution of the medium density in the form 

1 δ    1+γ   1+γ z z ρ(τ(z), σ(z)) = ρ0 (T0 , S0 ) exp βσ(z) − m λT +1 − n λT +1 . T T (4.31)

In the field of gravity, the stratification of the medium will be stable if the condition is met d ρ(τ(z), σ(z)) < 0, ∀ z, dz which means that the density of the medium decreases with height.

(4.32)

140

4 Stratification of the Ocean Environment

Substituting expressions (4.29, 4.31) in (4.32) gives the stability condition an explicit form  W

λS  S

1−q e q q

 1+W



1−q e q q



λS S



J (z)+1−q

λS S



J (z)+1−q

m λT λT zT + 1 − (1+γ) T

  λS

γ − 1+γ

S

Jz (z) J (z)+1−q



δ  1+γ −1 n δ λT z λ − (1+γ) + 1 < 0, ∀ z T  T T

.

(4.33)

In the isothermal case (T → ∞), the form (4.33) is simplified λS S

λS z S

  λS z 1−q + 1 − q W k S 1 e q   < 0, ∀ z, k = . q + 1 − q 1 + W k λS z + 1 − q S

(4.34)

Since the stability condition must be met for all z then for z = 0 from (4.34) it follows λS 1 W(k (1 − q)) λS 1 1 − q 1 λS = = < 0. 1−q  S 1 − q 1 + W(k (1 − q)) S 1 − q q 1 + q S

(4.35)

Since  S > 0 is by definition then from (4.35) follows λ S = −1. The applicability of the stability condition (4.34) depends on the behavior of the Lambert function W(x). Since this function takes valid values for x > − 1e and has the property lim x → − 1e +0 W (x) = −∞, it is necessary to  k

λS z +1−q S

 > −

  q 1 ⇒ z < S 1 + 1 − q . e eq

(4.36)

On the other hand, the physical sense imposes the βσ(z) ≥ −S0 requirement, which means that the total salinity of water cannot take negative values. As follows from (4.29), the fulfillment of the physical requirement ensures the fulfillment of the relation (4.36). This result indicates the operability of condition (4.34) in all physical situations. In the case when the salt distribution is uniform over the space or completely absent ( S → ∞), from (4.33) follows  λT



z +1 m + n δ λT T

  δ+2γ 1+γ

> 0, ∀ z.

(4.37)

Since the values m, n, δ, γ and T included in the expression (4.37) are positive, then λT = 1. The applicability of condition (4.37) is limited by the requirement

4.3 Models of Natural Stratified Media

λT

141

z + 1 ≥ 0 ⇒ z ≥ − T . T

(4.38)

But violation of condition (4.38) means that the water temperature is below the freezing point, therefore, the operability of the ratio (4.37) is erased for the entire range of water existence in the liquid phase. The expression on the left side of the inequality (4.33) is nothing more than the value ddz ln ρ(τ(z), σ(z)). As mentioned earlier, an important characteristic of the stratified media is the buoyancy frequency (Väisälä-Brent frequency), defined by the relation N 2 (z) = −g

d ln ρ(τ(z), σ(z)), dz

(4.39)

where g is the gravitational acceleration. Thus, the expression (4.33) allows us to determine not only the stability of the stratification of the medium, but also the dependence of the buoyancy frequency on the depth. In the case of a marine environment below the level of the near-surface thermocline, when the real scale of temperature and salinity stratification significantly exceeds the depth of the sea, i.e. the relations |z/T |  1, |z/ S |  1 are fulfilled, the condition (4.33) of stability of the density stratification takes the form λS λT 1−q < 0. − (nδ + m) S T 1+γ

(4.40)

Since n, m, δ, γ and 1−q are positive values, the choice λT = 1, λ S = −1 ensures that the condition (4.40) is met. This corresponds to the presence of stable both temperature and salt stratifications. But stability can also be achieved when choosing λT = 1, λ S = 1 (the case of salinity inversion, when salinity increases with height), if T < (nδ + m) 1−q . Stability also holds for λT = −1, λ S = −1 the ratio is correct  S 1+γ

T > (nδ + m) 1−q inequality holds. (the case of temperature inversion), if the  S 1+γ Absolute instability of the density stratification is observed at λT = −1, λ S = 1. As shown by field observations, the first three cases occur in natural systems, and the last one is not observed, in full compliance with the stability condition (4.33).

4.4 Simplified Models of Natural Stratified Media Used in Applications If the temperature overheats (supercooling) near the heat source (drain) are small, then in the approximation of constancy of all thermodynamic coefficients of the medium, Eq. (4.10) can be represented in the form

142

4 Stratification of the Ocean Environment

d (ln ρ + αT − βS) = 0, dt

(4.41)

whence it follows that ln ρ + αT − βS = const,

 . ⇒ ρ = const · exp(βS − αT ) = const · exp(βS0 (z) − αT0 (z)) exp β S˜ − αT˜ (4.42)

In the absence of perturbations, that is for T˜ = S˜ = 0, the expression (4.42) must coincide with the distribution of the stratifying density. Thus, from (4.42) it follows ρ0 (z) = const · exp(βS0 (z) − αT0 (z)).

(4.43)

Taking into account the expressions (4.6) for the stratifying distributions of temperature and salinity, the relation is obtained from (4.43) ρ0 (z) = const · exp(−(ατ − βσ) z) = const · exp(−z/),  =

1 , (4.44) ατ − βσ

which defines an exponential density distribution with a constant buoyancy frequency, which confirms the validity of the model used for an isothermal liquid when describing internal gravitational waves. As a result, it follows from (4.42) and (4.44) that in this model the total density is determined by the ratio  ρ = ρ0 (z) exp β S˜ − αT˜ .

(4.45)

Since for the marine environment β ≈ 7.5·10−4 (0/00)−1 , salinity S does not exceed 400/00, then the relations |βS|  1, |αT |  1 have place and then the approximation is valid  (4.46) ρ ≈ ρ0 (z) 1 + β S˜ − αT˜ , where, in particular, should be   ρ ≈ ρ0 (z) β S˜ − αT˜ .

(4.47)

The relation (4.46) is nothing more than the equation of state of sea water for small variations of thermodynamic fields. So, in the approximation of constancy of all thermodynamic coefficients of the medium, we obtained explicit dependences of the total and stratifying densities, as well as density perturbations on temperature and salt [Formulas (4.45–4.47)]. The

4.4 Simplified Models of Natural Stratified Media Used in Applications

143

most general relations describing the relationships between the thermodynamic parameters of seawater are contained in the International document TEOS-2010, which presents a modern approach to the equation of state of the environment [7]. Useful manuals [8, 9] present methods of approach to the study of materials [7] and computer training programs.

References 1. Landau LD, Lifshitz EM (1987) Course of theoretical physics, vol 6. In: Fluid mechanics, 2nd edn. Butterworth-Heinemann, Oxford. USA 2. https://rus.ferhri.ru/argo/index_r.htm, https://ocean.extech.ru/ioc/programs/argo.php. 3. Tipler PA, Mosca G, Freeman WH, Co (203) Physics for scientists and engineers: with modern physics, 9th edn. 4. Reid RC, Praudnitz JM, Sherwood TK (1976) The properties of gases and liquids, 3rd edn. New York, London. 5. Steele J, Turekian K, Thorpe S (2009) Encyclopedia of ocean sciences, 2nd edn. Academic Press, San Diego 6. Corless RM, Gonnet GH, Hare DE, Jeffrey DJ, Knuth DE (1996) On the Lambert W Function. Adv Computat Maths 5:329–359 7. IOC, SCOR and IAPSO (2010) The international thermodynamic equation of seawater– 2010: calculation and use of thermodynamic properties. Intergovernmental Oceanographic Commission, Manuals and Guides No. 56, UNESCO (English), 196 pp. Printed by UNESCO (IOC/2010/MG/56 Rev.) 8. McDougall TJ, Barker PM Getting started with TEOS-10 and the Gibbs Seawater (GSW) Oceanographic Toolbox. 2010. ISBN: 9780646556215 (pbk.). 9. Pawlowicz R What every oceanographer needs to know about TEOS-10 (The TEOS-10 Primer), 2013. Version 8. 10 pp. www.teos-10.org.

Chapter 5

Internal Gravity Waves in a Stratified Medium

Let us now turn to movements in the sea column, when surface phenomena play a negligible role. As noted in the initial sections, the density of seawater in the local vicinity of a certain observation point is a function of temperature, pressure, and concentration of impurities at that point. In General, the listed thermodynamic characteristics change in space, which entails a change in density from point to point. In a gravitational field, an inhomogeneous density distribution can be in stable equilibrium only if the gradients of the gravitational potential and density are oppositely directed. In other words, equilibrium occurs if the isopycns (levels of constant density) coincide with the levels of the gravitational potential, and the density decreases with increasing potential. In the World Ocean, the Earth’s gravitational field is characterized by extremely small inhomogeneities at distances within a few kilometers. Therefore, to study the basic properties of flows of inhomogeneous liquids in a gravitational field, we will choose a model of constant gravitational acceleration g, which is directed against the selected axis z of the coordinate system used (Cartesian, cylindrical). In order not to complicate the issue we are investigating with conditions at the borders, we will consider the environment unlimited in all directions. In the framework of the chosen model, a stable density distribution can be represented as horizontal layers of constant thickness, located one above the other. The density of the liquid inside the layer is also constant, but it changes from layer to layer, decreasing as it moves along the z axis. This distribution of density across layers is called density stratification, and the fluid is called stratified. If we mentally aim at zero thickness of layers of constant density, while requiring continuity of the ρ(z) distribution, the result is a model of a continuously stratified fluid. It is this model that most adequately describes the density distribution in the seas and oceans outside the mixing region. This spatial distribution of the fluid density in the gravity field allows for the existence of very specific waves called internal gravitational waves. The nature of these waves lies in the occurrence of buoyancy forces acting on an element of the medium that is removed from the equilibrium position, that is, from its density level. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Kistovich et al., Advanced Studies in Ocean Physics, Springer Oceanography, https://doi.org/10.1007/978-3-030-72269-2_5

145

146

5 Internal Gravity Waves in a Stratified Medium

Let’s consider a certain element of the liquid that is at the z level. Its density is equal to ρ(z). Let’s assume that under the influence of some perturbation, this element moved to the z + dz level and was surrounded by liquid elements with a density of ρ(z + dz). According to Archimedes’ law, the buoyancy force Fb = [ρ(z) − ρ(z + dz)]g will begin to act on this separated liquid element. In the case of a stable density distribution, this resulting force will always be directed against the displacement of the liquid element from the equilibrium position. That is, if the value of dz is positive, then Fb is directed against the z axis and vice versa. Therefore, the buoyancy force Fb acts as a restoring force that returns the liquid element to its original position. But this same liquid element enters its equilibrium position at a speed other than zero and skips it by inertia. The resulting returning buoyancy force forces the selected element to return to the starting point again, and again the element overshoots it by inertia. With a weak attenuation of movements, such an oscillatory process can continue for quite a long time. Due to the perturbations of the pressure field introduced by such vibrations, neighboring elements of the medium are drawn into the described process, and the entire set of moving liquid particles forms an internal gravitational wave.

5.1 Mathematical Model of Oscillations of a Stratified Medium To describe the properties of such a wave, you need to create a suitable model. Let the initial equilibrium distribution of the liquid density be given by some known function ρ0 (z). The corresponding distribution of hydrostatic pressure is determined by the expression z p0 (z) = g

ρ0 (ζ)dζ,

(5.1)

0

where the lower limit of integration is taken as the reference level. As a result of some disturbance, the density distribution is disturbed, which also leads to a violation of the pressure field in the medium. Let the deviation of the density ρ, and from its equilibrium distribution ρ0 (z) be described by the perturbation field  the corresponding pressure deviation by the field p, ˜ so that the complete density and pressure in space—time are determined by the relations ρ(x, y, z, t), ρ(x, y, z, t) = ρ0 (z) + 

p(x, y, z, t) = p0 (z) + p(x, ˜ y, z, t). (5.2)

Initially, the liquid was at rest, so that the resulting perturbation of the rest state is described by the velocity field v(x, y, z, t).

5.1 Mathematical Model of Oscillations of a Stratified Medium

147

In general, equations describing the dynamics of a stratified fluid include the Navier-Stokes equation   ∂σ ∂p ∂vi + v · ∇vi = − + ik + ρ gi , ρ ∂t ∂ xi ∂ xk

(5.3)

 where σik is the viscous stress tensor, the continuity equation

∂ρ + ∇ · (ρv) = 0, ∂t

(5.4)

ρ = ρ( p, T, S),

(5.5)

and the state equation

where T and S are the temperature and salinity of the medium. Temperature and salinity are included in the equation of state for the reason that exactly because the distribution of these thermodynamic quantities cause the stratification in the liquid. The total number of equations of the system (5.1–5.5) is two less than the number of unknowns included in it. Therefore, our system needs to be supplemented with temperature and salinity transfer equations. Ignoring the Soret and Dufaur effects, and based on the assumption that the hydrodynamic velocity v of the thermal phenomena is significantly less than the speed of sound in water, which is absolutely true for small perturbations of the medium, the necessary equations have the form   ∂T 1  ∂vi ∇ · (κ∇ S) + σik , (5.6) + v · ∇S = ∂t ρ cp ∂ xk

∂S + v · ∇ S = ∇ · (k∇ S), ∂t

where k is the salt diffusion coefficient, κ is the thermal conductivity coefficient, and c p is the heat capacity of the medium at constant pressure. Now substitute the dependence (5.5) in the Eq. (5.4) ∂ρ ∂t

dρ + ∇ · (ρv) ·v   =dt + ρ∇  dp ∂ρ + ∂∂ρT = ∂p dt T,S

dT p,S dt

+



∂ρ ∂S



dS p,T dt

+ ρ∇ · v = 0

.

(5.7)

Using the obtained result (5.7), we can express ∇ · v by the relation 1 ∇ ·v =− ρ Because



∂ρ ∂p



dp + T,S dt



∂ρ ∂T



dT + p,S dt



∂ρ ∂S



dS . p,T dt

(5.8)

148

5 Internal Gravity Waves in a Stratified Medium



∂ρ ∂p

 = T,S

  1 1 ∂ρ , − = α, c2 ρ ∂ T p,S

  1 ∂ρ = β, ρ ∂ S p,T

(5.9)

where c is the adiabatic speed of sound, α is the coefficient of thermal expansion, and β is the coefficient of the relative contribution of salinity to the liquid density, the expression (5.8) can be given the form ∇ ·v =−

dT dS 1 dp +α −β . 2 ρ c dt dt dt

(5.10)

Using (5.10), we continue the mathematical design of the model of internal gravitational waves. We will consider such fluctuations of the medium, the characteristic time of which is much less than the characteristic times of diffusion of temperature and salt. We will consider all the kinetic coefficients of the medium to be constant values, thus excluding modes with aggravation from consideration. In addition, we will neglect the conversion of mechanical energy into thermal energy due to the forces of viscous friction. These assumptions ensure that the right-hand sides of equations (5.6) vanish, so that the result is dT /dt = 0, d S/dt = 0 in our model . Another assumption is that the fluid is considered incompressible. This means that the all-round compression ratio is λ = ∞, from which c = ∞ follows. As a result, Eq. (5.10) takes the form ∇ · v = 0.

(5.11)

In our model, Eq. (5.11) plays the role of a differential analog of the equation of state. Equation (5.7) also implies that ∂ρ dρ = + v · ∇ρ = 0. dt ∂t

(5.12)

This result is also obtained by direct substitution (5.11) in (5.4). Thus, the sequential formation of the mathematical description of the model we use shows that four Eqs. (5.4–5.6) can be replaced by two Eqs. (5.11, 5.12). In addition, condition (5.11) simplifies the form of the viscous stress tensor, so that Eq. (5.3), taking into account the constancy of the dynamic viscosity coefficient, is transformed into the equation 1 ∂v + (v∇)v = − ∇ p + νv + g, ∂t ρ

(5.13)

where ν is the kinematic viscosity of the liquid. The next simplification of our model is that the perturbations of the initial equilibrium state of the medium are small, that is, the relations are valid

5.1 Mathematical Model of Oscillations of a Stratified Medium



p˜

p





ρ μ . Then and (5.56) follows μ = 2λ

N 2 sin ϕ cos ϕ 4ων N 4 sin2 ϕ cos2 ϕ 3  , λ =  2 λ . ω2 − N 2 cos2 ϕ ω2 − N 2 cos2 ϕ

(5.57)

The last relation indicates that λ >> λ . Thus, the perturbation is exponentially damped in all directions. This solution is not of interest. Let μ = μ . Then there is a solution of the form









λ = λ = 0, μ = μ =

ω2 − N 2 cos2 ϕ . 2ων

(5.58)

The resulting solution describes a periodic Stokes boundary layer, in which the motions are damped as they move along the normal to the surface, and the direction of oscillations periodically changes its sign. In order to maintain this boundary movement, the bounding plane must oscillate along itself at a frequency of ω. Let μ >> μ . Then the system (5.56) is reduced to the form ω2 − N 2 cos2 ϕ μ = 2N 2 sin ϕ cos ϕ λ , ων μ3 = 2N 2 sin ϕ cos ϕ λ



(5.59)

from which it follows 3

λ =

3  2 ω − N 2 cos2 ϕ 4ων N 4 sin2 ϕ cos2 ϕ

λ .

(5.60)

162

5 Internal Gravity Waves in a Stratified Medium

Such a solution can exist if the bounding surface has finite dimensions, that is, it is the surface of the source of vibrations. At the same time, a new interesting phenomenon related to the viscosity of the medium appears. If we use (5.60) as an estimation formula and put λ ∼ 1/L, ω ∼ N , λ ∼ 1/, then a characteristic viscous wave scale appears [4]

L∼

3

ν . N

(5.61)

With respect to this scale, internal gravitational waves exhibit the following properties: if the characteristic size of the emitter is smaller than the viscous scale L, then the entire length of the wave cone is a single-modal structure, similar to that shown in Fig. 5.7. If the source size exceeds the viscous wave scale, then the transverse structure has a two-modal appearance from the source for a certain distance (see Fig. 5.7), then a restructuring occurs, after which the structure becomes single-modal. In addition to viscous effects, a significant role is played by the dependence of the buoyancy frequency on the vertical coordinate z. To understand the essence of the phenomenon, it is necessary to write the dispersion Eq. (5.28) in the form cos2 θ =

ω2 . N 2 (z)

(5.62)

As follows from (5.62), the local angle, which is a monochromatic internal gravitational wave with a vertical, is a function of the z coordinate. As the wave propagates, Fig. 5.7 Two-modal transverse structure of the wave cone

5.4 Some Features of the Generation of Internal Gravity Waves

163

Fig. 5.8 Refraction of the internal gravitational wave

it changes according to the N (z) distribution. If the wave moves in an area with an increase in the buoyancy frequency, the wave beam deviates from the vertical, becoming more horizontal (Fig. 5.8a). If the frequency of buoyancy decreases as you move, the wave approaches the vertical. When the wave reaches the area where ω > N (z), it is reflected from its boundary and, going down (that is, in the direction of lowering the buoyancy frequency), deviates from the vertical (Fig. 5.8b). The phenomenon described here is called refraction of an internal gravitational wave.

5.5 Separation of Internal Gravity and Surface Waves To date, we have studied two types of waves—surface and internal gravitational. This raises some questions, for example: can waves of one type generate waves of another type, is it possible to study waves of different types separately? In order to answer these questions, we consider a two-dimensional problem of propagation of harmonic oscillations in an infinitely deep ideal incompressible stratified fluid with a free surface. A schematic representation of the problem is shown in Fig. 5.9. In the coordinate system (x, z), whose vertical axis is directed against the gravity vector g the fluid density and pressure in it are described by the functions ρ(x, z, t) and p(x, z, t), the velocity field v has two components vx = u, vz = w. The deviation of the free surface from the undisturbed state z = 0 is given by the function ζ(x, t). Fig. 5.9 Flow pattern

164

5 Internal Gravity Waves in a Stratified Medium

The system of equations of motion and boundary conditions have the form  ρ g, ρt + v · ∇ρ = 0, ∇ · v = 0 ρ vt + (v∇)v = −∇ p +

. p|z=ζ = pa , wv − vζx z=ζ = ζt

(5.63)

Here pa is the atmospheric pressure. Setting the pressure in a liquid as the sum of atmospheric, hydrostatic, and wave pressures ζ p = pa + g

ρ(x, ξ, t) dξ + p˜

(5.64)

z

converts the system (5.63) to the form ⎛ ⎞ ζ   ρ ut + uux + wuz = − p˜ x − g ⎝ρ(x, ζ, t)ζx + ρx (x, ξ, t) dξ⎠  ρ wt + uwx + wwz = − p˜ z ρt + v · ∇ρ = 0, ∇ · v = 0

p| ˜ z=ζ = 0, w − uζ = ζ x z=ζ

z

t

(5.65)

The system of waves propagating in the medium under consideration does not allow the introduction of a velocity potential, since even when using the Boussinesq approximation, when the density is represented as ρ = ρ0 (1 + r (x, z, t)), |r (x, z, t) N , the solution (5.72) is searched in the form ψ = exp(k z z)(A exp(ik x x) + B exp(−ik x x)).

(5.73)

166

5 Internal Gravity Waves in a Stratified Medium

Substituting (5.73) into (5.72) gives the dispersion equation  ω2 k z2 = ω2 − N 2 k x2 .

(5.74)

Substituting (5.73) into the boundary conditions (5.71) leads to the relations ik x (A exp(ik x x) − B exp(−ik x x)) exp(−iω t) + ζt = 0 , −iω k z (A exp(ik x x) + B exp(−ik x x)) exp(−iω t) + gζx = 0

(5.75)

the solution of which is in the form of a surface wave ζ=

kx (A exp(ik x x) − B exp(−ik x x)) exp(−iω t) ω

(5.76)

exists when the condition is met ω2 k z = gk x2 .

(5.77)

Simultaneous satisfaction of conditions (5.74) and (5.77) leads to solutions for wave numbers kx =

ω2  ω2  ω2 − N 2 = 1 − N 2 /ω2 . 1 − N 2 /ω2 , k z = g g g

(5.78)

The solution (5.73, 5.76, 5.78) is a surface wave. The applicability of the obtained solutions is limited to the fulfillment of the condition 3/2  2 ω − N2 kx kz F= F 1, ω g2

(5.79)

where F is the amplitudes A, B of surface waves. If ω < N , the solution (5.72) is searched in the form ψ = (α exp(ik z z) + β exp(−ik z z))(A exp(ik x x) + B exp(−ik x x)).

(5.80)

Substituting (5.80) into (5.72) gives the dispersion equation  ω2 k z2 = N 2 − ω2 k x2 ,

(5.81)

and substituting (5.73) in the boundary conditions (5.71) leads to the relations ik x (α + β)(A exp(ik x x) − B exp(−ik x x)) exp(−iω t) + ζt = 0 , ω k z (α − β)(A exp(ik x x) + B exp(−ik x x)) exp(−iω t) + gζx = 0 the solution of which is in the form of surface undulation

(5.82)

5.5 Separation of Internal Gravity and Surface Waves

ζ=

kx (α + β)(A exp(ik x x) − B exp(−ik x x)) exp(−iω t) ω

167

(5.83)

exists when the condition is met ω2 k z (α − β) + igk x2 (α + β) = 0.

(5.84)

From condition (5.84), taking into account the dispersion Eq. (5.81), the relationship between the amplitudes α and β follows β=−

2i sin ϑ/(μ ) 1 − i sin ϑ/(μ ) α, α + β = α, 1 + i sin ϑ/(μ ) 1 + i sin ϑ/(μ )

(5.85)

where the notation is entered cos ϑ =

ω , μ2 = k x2 + k z2 . N

(5.86)

The obtained relations describe internal gravitational waves that experience reflection from a free surface. Since these relations are valid for short internal waves, the 1/(μ) 0, Im(λ) = 0. These conditions are met when the ratio is true c
χ and k < χ cases, respectively, the border c = 0 is shown in blue. Another characteristic boundary b = 0 that separates the regions of zero (b ≥ 0) and positive values on the right side (6.34) is represented in red. In the c > 0 ∪ b > 0 region, the ratio (6.34) is impossible, which corresponds to the suppression of convective movements. This leaves the c > 0 ∪ b < 0 area where the criterion (6.34) can be met. In the case of k > χ, the point where the borders of c = 0 and b = 0 intersect k+ν , q∗ = k 2 χ+ν . If we place the origin of the (x, y) has coordinates p∗ = χ2 k−χ k−χ Fig. 6.3 The regions of convective instability at k>χ

Fig. 6.4 Regions of convective instability at k 1.

(6.36)

If there is no impurity in the medium (k = ∞), the criterion (6.36) becomes the criterion for a temperature-stratified medium −

gατL 2χ L 2ν χν

> 1,

(6.37)

which implies the need for temperature inversion. For the isothermal case (χ = ∞), the ratio has place gβσL 2k L 2ν > 1, νk

(6.38)

which implies the need for salinity inversion. It should also be noted that in the limit case (k = ∞, χ = ∞), the area enclosed between the blue and green borders disappears, which means that conditions (6.37) and (6.38) for single-component convection are complete. The obtained criteria describe the conditions under which the developing convection gradually covers the entire area occupied by the medium. However, these relations cannot describe the occurrence of localized convection, for example, convection in the form of a “Christmas tree” from a point heat source in a liquid with salt stratification. Criterion (6.34) is too strong a condition for such processes. Since absolute convective instability of the medium is forbidden in this case, the relation (6.34) must be reversed, so that the necessary condition for localized convection has the form c>

  a3  3 2x − 3x 2 + 1 ϑ(−b), x = 1 − 3b/a 2 27

(6.39)

6.3 The Problem of Convective Instability of the Marine Environment

181

The second condition sets a lower limit on the intensity of convection. If you determine the height of the convective rise by the limit value limt→∞ |z(t)| for the solution (6.32) and to divide it on the radius of the liquid particle R, then you can select the condition as the second condition included in the criterion of localized convection g αk L 2χ T − βχL 2k S L 2ν limt→∞ |z(t)| = B > 1.  (6.40) =  R R g αk L 2 τ − βχL 2 σ L 2 + kχν χ

k

ν

In the case of only one temperature stratification of the medium, the relations (6.39, 6.40) are simplified to the form −

gατL 2ν L 2χ νχ

= A < 1,

αg|T |L 2ν L 2χ νχR(1 − A)

= B > 1,

(6.41)

and in an isothermal environment there is gβσL 2k L 2ν = A < 1, νk

βg|S|L 2ν L 2k = B > 1. νk R(1 − A)

(6.42)

The B value in all ratios sets the intensity of localized convection. As can be seen from the results obtained, the proposed criteria include not only the gradients of equilibrium stratifications of temperature and salinity, but also the values of initial overheating T and excess salinity S, the average size R of the perturbation region, as well as the characteristic scales of viscous L ν and diffusive L χ , L k processes that affect the dynamics of heat and mass transfer. From the relation (6.36), it can be seen that a stable salinity distribution (σ < 0) suppresses convective processes when a heat source is turned on in the medium. This conclusion is also confirmed experimentally.

6.4 The Physical Approach to the Problem of Convection Onset The problem of the onset of convection depends to a large extent on the type of heat source that occurs in the marine environment. Since it is not possible to describe all types of heat sources in a limited volume manual, the main focus here will be on a flat horizontal source that generates Bénard type convection. It was the Bénard [1] whose regular experimental studies gave the stimulus to the first theoretical descriptions done Lord Rayleigh [2] waked up the interest to investigations of convective problems. On the early stage of the theoretical investigations the basic efforts were focused on the linear stability analysis of the convective problem. Then the works aroused in which the authors attempted to introduce in the mathematical methods the

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6 Convective Phenomena in the Sea

elements of the weak nonlinear analysis [3, 4]. The general results of the methods mentioned above were presented in the well-known books [5, 6]. As the experimental results [7, 8] had shown, the adequate approach to construction of the mathematical models of convection should take into account the dependences of the characteristic parameters of the liquid on the temperature. Especially it is important in the convective regimes when the sufficient great local overheating of the substance has place. The system of equations and boundary conditions of the convective problem for the media, which kinematic coefficients depend on the temperature, has the form  /∂ xk − ρgδik ρ(∂vi /∂t + v · ∇vi ) = −∂ p/∂ xi + ∂σik  ρT (∂s/∂t + v · ∇s) = σik ∂vi /∂ xk + ∇ · (κ∇T ) ∂ρ/∂t + ∇ · (ρv) = 0, ρ = ρ( p, s).

(6.43)

All designations in (6.43) were introduced earlier in the previous sections. On the base of the state equation one can wright  dρ =

∂ρ ∂p



 dp +

s

∂ρ ∂s



1 ds = 2 dp + c p



∂ρ ∂T

  p

∂T ∂s

 ds = p

1 ρT dp − αds, 2 c cp

where α = −(∂ρ/∂ T ) p /ρ depends on the temperature (see Fig. 6.2). In the absence of acoustic effects (this assumption is confirmed by natural and laboratory experiments for ordinary convection flows) the continuity equation has the form ∇ ·v =

 ρα   ραT ds = σik ∂vi /∂ xk + ∇ · (κ∇T ) . c p dt cp

(6.44)

The usage of the relation between entropy and temperature, namely s = c p dT /T , and also Eq. (6.44) permits to give the system (6.43) in the other form  /∂ xk − ρgδik ρ(∂vi /∂t + v · ∇vi ) = −∂ p/∂ xi + ∂σik ραT ds ρα  ∇ · v = c p dt = c p σik ∂vi /∂ xk + ∇ · (κ∇T ) .

(6.45)

In the viscous liquid, the velocity field should be vanished on the rigid walls. This requirement forms the kinematic boundary condition for velocity on motionless and impenetrable boundaries. The boundary condition for the temperature may be presented in two variants (its depend on the real physical situation). The first variant reflects the fact that the temperature is constant on the boundary and described by formula T | = T .

(6.46)

6.4 The Physical Approach to the Problem of Convection Onset

183

The second variant realized for the case of the fixed thermal flux χ∂ T /∂n + γT T | = qT .

(6.47)

Here γT is the coefficient of heat exchange, qT is the value of heat flux along normal to the boundary n. We will consider the first variant, when the temperatures on the bottom and cover are constant T |z=0 = Tb ,

T |z=d = Tc , Tb > Tc ,

(6.48)

where Tb and Tc are the temperatures of bottom and cover respectively. As stated earlier, the cause of convection is the heterogeneity of the temperature distribution over the surface of the heaters or coolers, which may be located on the bottom or/and cover. For any convective situation it is possible to split the heater’s (cooler’s) temperature on the two parties including the mean temperature Tb and (x, y, t) which mean value equals to zero disturbance T (x, y, t), T = Tb + T



 (x, y, t) = lim T

S→∞

 S

(x, y, t)d xd y T , S

(6.49)

Here S is a square of the heater’s surface. Accordingly to the data presented in the Chap. 4 the density of pure water varies with the temperature T by the following manner  α2    T α1 , (6.50) ρ(T ) = ρ0 exp T α0 − 1 + α2 Tr with the relative error not greater than 10−3 for whole diapason of liquid water phase. Here the water temperature is measured in centigrade degree, Tr = 273.15 K is reference temperature (ice melting temperature in absolute thermodynamic scale). The constants used in (6.50) have the following values ρ0 = 999.87 kg/m3 , α0 = 1.83 · 10−4 K−1 , α1 = 1.6 · 10−3 K−1 , α2 = 0.54. The result of (6.49) and (6.50) combination describes water density near the heater by the relation    α2    /Tb 1 − β T /Tb ρ = ρ0 exp α 1 + T ≈ R0 (1 + γx + δx 2 ), x = T Tb  α2 Tb α1 R0 = ρ0 exp(α(1 − β)) = ρ|T=0 , α = α0 Tb , β = α0 (1 + α2 ) Tr 

γ = α(1 − (1 + α2 )β), δ = α(1 − (1 + α2 )(α2 + 2α)β + α(α + α2 )2 )/2, (6.51) which relative error is smaller than 10−2 .

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6 Convective Phenomena in the Sea

The key idea of the approach developing here is that the onset of convection is due to heater’s temperature dispersion along its own surface. For the quantitative estimation of the values under interest it is necessary to calculate in the vicinity of heater’s surface the dispersion and root mean square (rms) deviation of density from its ideal value R0 . As it follows from (6.51) the deviation of density from its ideal value describes by relation ρ = ρ − R0 ≈ R0 (γx + δx 2 ).

(6.52)

On the base of (6.49) and (6.51) the result has place x = 0. Thus one can proclaim that the relations are valid 2

2

ρ = R0 δx 2 ⇒ ρ = R02 δ2 x 2 .

(6.53)

At the same time, in accordance to experimental and natural regimes of convection, the values of x and δ/γ are very small (|x|  1, |δ/γ|  1) and one can wright the approximate relations (ρ)2 ≈ R02 (γx + δx 2 )2 = R02 (γ2 x 2 + 2γδx 3 + δ2 x 4 ) ≈ R02 γ2 x 2 .

(6.54)

The results of calculations give the dispersion of density 2

2

σρ2 = (ρ) − (ρ)2 = R02 γ2 x 2 − R02 δ2 x 2 ≈ R02 γ2 x 2

(6.55)

and rms deviation of the density σρ = R0 |γ|σT .

(6.56)

 Here σT = x 2 is rms deviation of heater’s temperature. The eventuated buoyancy force is a product of the gravity acceleration on the water density deviation from its mean value. Thus the rms of buoyancy force along the heater surface may be estimated by the relation σb = gσρ = g R0 |γ|σT , ain = g|γ|σT .

(6.57)

Here ain is the inertial acceleration of liquid element forcing at initial time moments by the buoyancy effects. The sufficient increase of the buoyancy force dispersion with the growth of the mean surface temperature under fixed temperature dispersion σT is shown on the Fig. 6.5. The presented dispersion is normalized by the constant value g R0 σT . As it is explicit from the Fig. 6.5 the dispersion of the buoyancy force inside the heat source temperature diapason Tb ∈ [10.80]C◦ varies more than in 60 times! This is how many times the force in the first equation of the system (6.43) changes, which leads to the appearance of convective motion!

6.4 The Physical Approach to the Problem of Convection Onset

185

Fig. 6.5 The normalized buoyancy force dispersion

From the relations (6.56, 6.57) the significant conclusion follows: If the root mean square deviation of temperature is fixed then the higher temperature of the heater, the greater is the root mean square of buoyance force and the more intensive are convective flows. This intermediate result is in a good agreement with experimental observations, which show that in the condition of the same temperature stabilization the more intensive convective heat flux has place for the heater with higher temperature. Now, on the base of the results received, it is possible to modify the approach of the § 6.2 to the problem of the heated water element floating-up. In accordance to the probability theory the more probable temperature of the spherical liquid element with radius R and mean temperature Tb is Tb + σT . As a result, under action of buoyancy force (6.57) this water’s sphere will start to float up. After development of the buoyance flow the viscose force should be take into account. For the goals of further description the mean temperature T (t) of liquid element is introduced. In accordanceto (6.50)  the mean density is the function of this mean temperature, i.e. ρ(t) = ρ T (t) . The initial state of the liquid environment is specified by the temperature stratification Tst (z) = Tb − |∇T |z, |∇T | =

Tb − Tc . d

(6.58)

Here d is the distance between bottom and cover. Let us designate the vertical coordinate of the center of floating-up liquid element by the symbol z(t). This element during its up floating will follows past motionless elements, which densities can be defined by the relation ρst (z) = ρ(Tst (z(t))) accordingly to Eq. (6.50). Due to the mechanism of the thermal diffusion with the ambient liquid the mean temperature of the considered element will be changed. As a result the mean density of this element will also be changed. Taking into account the mentioned mechanisms one can wright

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6 Convective Phenomena in the Sea

the system of equations describing the liquid element’s up floating ρ(t)Vz  = (ρ(t) − ρst (t))Vg − πK f ρ(t)Rνz  , z  t=0 = z|t=0 = 0    T (t) = −χ T (t) − Tst (t) , T (t) t=0 = Tb + σT , (6.59) where V = 4πR 3 /3 is the up floating element’s volume and K f is the coefficient of profile’s self drag of liquid drop in the same liquid [see designations to (6.27–6.30)]. The solution of the initial problem of (6.59) for the mean temperature (the second line of the system) is the following ⎛ T (t) = ⎝Tb + σT +  χ ⎛

t

⎞ Tst (τ)eχτ dτ⎠

0

e−χt = Tst + ⎝σT + |∇T |

t

⎞ z  (τ )eχτ dτ⎠e−χt .

(6.60)

0

The process of the solution was done with regard of Eqs. (6.58). The usage of the solution (6.60) in the formula (6.50) for stratified density and substitution of this density into the first line equations of the system (6.59) forms the equation for evolution of vertical coordinate z(t) of up floating liquid element ⎛ z  = α0 g ⎝σT + |∇T |

t

⎞ z  (τ )eχτ dτ⎠e−χt −  νz  ,

0

z

t=0

(6.61)

= z|t=0 = 0;  ν = 3K f ν/4R 2 .

The received solution of the Eq. (6.61) permits to define the required relations for position and velocity of the element on the initial stage of motion    k α0 gσT −kt 1 − e cosh(t) + sinh(t) k 2 − 2  gσ α 0 T e−kt sinh(t), z  (t) =  z(t) =

(6.62)

where  ν + χ , = k= 2



 ν − χ 2

2 + α0 g|∇T |.

(6.63)

The results (6.62) and (6.63) show that the characteristic features of the floatingup process sufficiently depends on the values of parameters k and . So in the case when  > k, the height of up floating is rigorously restricted by the cover’s position

6.4 The Physical Approach to the Problem of Convection Onset

187

so z max (t) = d. It should be mentioned increase of the ratio /k corresponds to increase of the value of standard Rayleigh number Ra. If k >  (the opposite case to the previous one) the maximum value of the upfloating height obeys to the condition z  (t) = 0

⇒

t →∞

from which it follows z max = lim z(t) = t→∞

α0 gσT . k 2 − 2

(6.64)

The increase of the value k/ corresponds to the decrease of the liquid element’s height. This height always is smaller than d. By means of (6.63) one can rewrite the condition  > k in the form = Ra

α0 g|∇T | > 1,  χ ν

(6.65)

a. which defines the modified Rayleigh number R This number is proportional to the old classical number Ra = α0 g|∇T |d 4 /χν.  has the other fundamental physical sense. But at the same time the new number Ra By usage of differentiation of the governing equation of the system (6.61) in consideration of initial conditions of (6.61), it is possible to get new form of equation for the displacement z(t) of the upfloating element ν + χ)z  + ( ν χ − α0 g|∇T |)z = const. z  + (

(6.66)

It is easy to construct the appropriate to (6.66) characteristic equation and to get its solution ν + χ)λ  + ν χ − α0 g|∇T | = 0 λ2 + (  2 χ  ν− χ λ± = − ν+ ± + α0 g|∇T |. 2 2

(6.67)

a > 1 hen from (6.67) follows that λ+ > 0 and the instability of the solution If R of (6.61) has place. Hence, one can define the critical value of the modified Rayleigh number by condition c = 1. Ra

(6.68)

a defines One should to make a special emphasis that modified Rayleigh number R the rate of the increasing of temperature’s convective transport and has no any connection to the convection’s onset. We proclame that convection’s onset is caused only by the rms deviation σT of the heater’s surface temperature from its mean value Tb but no

188

6 Convective Phenomena in the Sea

other reason. When σT = 0 then no convective flux may be observed in the rigorous corresponding to the natural physical sense. But when σT = 0 then convection, even the smallest and very hard observed, has place. Exactly the problem of observability of convection motion converts the criterion of the convection onset into subjective conception. We propose to give the following definition of the convective motion existing: The convective motion in the some space point is proclaimed as existing if the absolute value of convective heat flux is not less than conductive one. The mathematical formulation of this criterion has the form (Tb − Tc )z  (t) ≥ χ|∇T |

⇔

z  (t) ≥

χ , d

(6.69)

where z  (t) is the upfloating velocity averaged through the time of upfloating. In the above relation, the residue Tb − Tc is introduced for the reason that the cool cover also may the source of convection like heated bottom. To show the action of the condition (6.69) one need to investigate two reliable situations. In the first situation (k > ) the liquid element, in accordance to Eq. (6.64), doesn’t reach the cover. The time of rising t∗ is defined by the condition of the sign change of the liquid element acceleration from positive to negative value. For this reason the application of (6.69) is allowed only for the time period t ∈ [0, t∗ ]. The time of the positive acceleration is defined on the base of equation z  (t∗ ) = 0 from which it follows t∗ =

  k exp(−kt∗ ) arctanh(k/) α0 gσT 1 − , , z(t∗ ) = 2 √  k − 2 k 2 − 2

(6.70)

so one can to transform (6.69) into new inequality z  (t) =

z(t∗ ) χ ≥ . t∗ d

(6.71)

In the opposite case (k < ) the maximum displacement is z max = d. Hence the time t∗ of the liquid element’s up floating is the solution of the equation d=

   k α0 gσT −kt ∗ ∗ ∗ 1 − e cosh(t sinh(t ) + ) , k 2 − 2 

(6.72)

and the condition (6.69) transforms into the form tχ /t ∗ ≥ 1,

(6.73)

6.4 The Physical Approach to the Problem of Convection Onset

189

where tχ = d 2 /χ is the characteristic diffusive time of the liquid layer of the d thickness. There exists the well-known results for the dynamic regime of the sphere cooling due to the contuct with ambient environment: «The time of heat losing is proportional to square of sphere radius». Hence it is possible to write χ = Kχ Lχ ∼ R ⇒ 

χ , R2

(6.74)

where K χ is some constant defined by scale proportionality, so the modified Rayleigh a may be presented in the following manner number R a = K Ra R

α0 g|∇T |R 4 , χν

K Ra =

4 3K f K χ

.

(6.75)

From (6.75) one general conclusion may be drown: «The smaller is up floating aand the smaller is the liquid element the smaller is the modified Rayleigh number R convective flux». As it follows from (6.61) the convective motion can exist even in the case of homogeneously distribution of the initial temperature when classical Rayleigh number Ra = 0. The displacement, velocity and acceleration for this case (normalized for the convenience on its own maximum) are presented the Fig. 6.6 [9]. The analysis of the relations (6.71, 6.73) is some sufficiently complicated problem for analytical approaches, but it can be solved by numerical methods for each concrete case. And it necessary to remember that the relations (6.62–6.64, 6.70) describe the Fig. 6.6 The acceleration z  (t) (dot), velocity z  (t) (dash) and height z(t) (solid) of the floating liquid element when Ra = 0

190

6 Convective Phenomena in the Sea

characteristics of limited convection, and relations (6.62–6.64, 6.72) corresponds to fully developed convection’s regime. The standard scientific approach to the investigation of the complex non-linear equations includes the estimations of the terms contribution into mathematical model, their behavior under action of differential operators and the introduction of dimensionless variables. At the first stage it is necessary to neglect the heat generation due to the viscous  ∂vi /∂ xk in the entropy evolution equation of the system (6.45). term σik The second step is the splitting of the velocity field by two parts, namely solenoidal v and potential  v v = v + v, ∇ · v = 0, ∇ ·  v=

α ∇ · (κ∇T ). ρc p

(6.76)

The basic contribution into convective temperature flux and advective transport of the velocity field is due to solenoidal part v. Also this part plays the main role in the viscous tensor  σik ≈ η(∂vi /∂ xk + ∂vk /∂ xi ).

(6.77)

so the system (6.45) acquires new approximate form  /∂ xk − ρgδik ρ(∂vi /∂t + v · vi ) = −∂ p/∂ xi + ∂σik ∂ T /∂t + v · ∇T = ∇ · (χ∇T ), ∇ · v = 0, ∇ ·  v = α∇ · (χ∇T ).

(6.78)

As it was shown earlier the term ρc p practically doesn’t depend on temperature and for this reason doesn’t vary under action of Hamilton operator. This fact allows ) by the term ∇ · (χ∇T ). to change the term ∇·(κ∇T ρc p For the goal of reforming of (6.78) to no dimensional form it is necessary to introduce as spatial and temporal scales so the scales for physical fields temperature, density, pressure etc. The adequate estimation of the mentioned scales values is one of the oldest and most complex hydrodynamics problem. For example, the choice of χ/d [3–5] as a scale for the solenoidal part v leads to incompatible contradiction with experimental data. Analogous situation has place for the scales originated by other kinematic coefficients introduced in many other works devoted to convective problems. This fact obliges to solve the problem of spatial and temporal scales definitions. The temporal and spatial scales for the cases  > k and k >  will by defined separately. The characteristic temporal scale for the case  > k is defined by means of the relation t  = ( − k)t, where t  is the dimensionless time.

(6.79)

6.4 The Physical Approach to the Problem of Convection Onset

191

So in the initial time moments the up floating height is sufficiently less than the layer thickness d then the differential operators acts in accordance to the rules ∂ ( − k) −t  ∂ e , = ∂z α0 gσT ∂t 

∂ ∂ = ( − k)  . ∂t ∂t

(6.80)

On the base of the solution (6.62) one can get the scale for the vertical component of velocity vz =

α0 gσT t   e v, 

(6.81)

where v is the dimensionless velocity. Also let’s introduce the dimensionless temperature perturbation θ into water’s temperature presentation T = Tst (z) + (T + σT )θ.

(6.82)

The both of the above introduced perturbations values have the same order of smallness v , θ ∼ O(1). By means of the relations (6.79–6.82) it is possible to transform the heat transport Eq. (6.78) into dimensionless equation   θt  + λv (1 + λ−1 θt  ) = λ−2 χθt  t  + χT (θt  )2 − χλ−1 θt  ,

(6.83)

where the symbol λ = α0 gσT exp(t  )/( − k) is used. It follows from the just received Eq. (6.83) that the ratio of convective to conductive fluxes may be estimated by the formula Jconv ≈ χλ2 v . J cond

(6.84)

The standard experiments on convection in the water are characterized by the temperature dispersion σT > 10−2 K. For the such situation the ratio (6.84) at the initial moment t = 0 exceeds the value 1 and grows with the time. For the late time moments the height of the up floating element is equal to d. This value is chosen as the vertical spatial scale and the corresponding relations purchase the form

θt  +

z = dz  .

(6.85)

   λv 1 (1+λ−1 θz  ) ≈ 2 χθz  z  + χT T (1 + θz  )2 , d d ( − k)

(6.86)

from which one can get the result

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6 Convective Phenomena in the Sea

Jconv λv d( − k) ≈ , J χ cond

(6.87)

which for water always is greater than 1 for any late time moments and manifests monotonic growth with a time. For the case k >  the calculations done on the base of (6.62) give the result z=

α0 gσT  z. k 2 − 2

(6.88)

The time scale is defined by the relation t = m t t  m t > 0,

(6.89)

where m t is the some unknown temporal scale. The scales for velocity and temperature are the same as in the previous case. The result of transformation of the system (6.78) to dimensionless form is presented by the equation m t θt  +

d(k 2 − 2 )  α0 gσT v (1 + θz  ) = d α0 gσT



k 2 − 2 α0 gσT

2

   χθz  z  + χT T (1 + θz  )2 . (6.90)

As it follows from (6.90) the ratio of convective and conductive fluxes is described by the formula Jconv (α0 gσT )2 v  ≈ dχ( − k) exp(( − k)m t t ). J cond

(6.91)

This value for water always not greater than 1 at the initial moment and tends to monotonic decreasing with a time. It is necessary to point to the fact that if  > k no perturbation methods, which are suitable for the opposite case k > , can be applied to analyze the heat transport equation. The such conclusion is due to strong nonlinearity of the governing equations in the case  > k. The alternative approach to the problem of the beginning of convection presented here is described in more detail in [9]. In addition to the physical conclusions already given, we can also draw a formal mathematical conclusion that the value of the c = 1 separates the applicability of perturbation modified critical Rayleigh number Ra    a < 1 to the problem of convection from the situation R a > 1 when methods R these methods are not applicable.

References

193

References 1. Bénard H (1901) Les tourbillons une nappe liquide transportant de la chaleur par convection en régime permanent. Ann Chim Phys s7(23):62–144 2. Rayleigh L (1916) On convection currents in a horizontal layer of fluid when the higher temperature is on the underside. Phil Mag 32(6):529–546 3. Malkus WVR, Veronis G (1958) Finite amplitude cellular convection. J Fluid Mech 4:225–260 4. Serrin J (1959) On the stability of viscose fluid motions. Arch Rat Mech Anal 3:1–119 5. Chandrasekhar S (1961) Hydrodynamic and hydromagnetic stability. Clarendon, Oxford 6. Joseph DD (1965) On the stability of the Boussinesq equations. Arsh. Rat. Math. Anal. 20:59–87 7. Berg JC, Boudart M, Acrivos A (1966) Natural convection in pools of evaporating liquids. J Fluid Mech 24(Pt. 4):721–735 8. White DB (1988) The planforms and onset of convection with temperature-dependent viscosity. J Fluid Mech 191:247–286 9. Kistovich AV (2018) Convective motions in water: linear and nonlinear models, Criteria of convection onset. In: Karev V, Klimov D, Pokazeev K (eds) Physical and mathematical modelling of earth and environmental processes. Springer Proceedings in Earth and Environmental Sciences. Springer, Cham, pp 174–188. https://doi.org/10.1007/978-3-030-115333_18

Chapter 7

Interaction of Surface Waves With Regions of Near-surface Convection

Despite the fact that the study of the dynamics of short waves, the length of which is substantially less than the thickness of the fluid layer on the surface of which they are the subject of numerous theoretical, experimental, and observational, and synthesis works, some of the key problems of generation, distribution, and interaction of these waves with other types of flows, as well as processes of their disintegration remain. In recent years, interest in the in-depth study of the short-wave part of the sea wave spectrum has been stimulated by the search for physical mechanisms for generating signals and signs that indicate topography, internal waves, and other phenomena in the ocean column in free-surface optical and radar images [1]. When studying wave dynamics, the main attention is usually paid to the analysis of the influence of nonlinear effects, viscosity, and surface tension (assuming the constancy of the corresponding coefficients) on the attenuation and generation of short waves [2, 3]. In real conditions, thermodynamic parameters—the temperature of the medium, the concentration of surfactants—are not constant in space and are variable over time. The processes of near-surface temperature convection create gradients of the coefficients of kinematic viscosity and surface tension, which in turn cause specific types of near-surface convection (Marangoni convection) that affect short-wave waves. Of particular interest is the study of the interaction of waves with regular structures that arise as a result of near-surface convection or falling rain drops. In this connection, it is necessary to construct a mathematical model of the propagation of gravitational-capillary waves in a viscous temperature-inhomogeneous medium based on the complete equations of motion [4], taking into account both regular (wave) and singular (boundary-layer) flow elements.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Kistovich et al., Advanced Studies in Ocean Physics, Springer Oceanography, https://doi.org/10.1007/978-3-030-72269-2_7

195

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7 Interaction of Surface Waves With Regions of Near-surface …

7.1 Statement of the Problem of the Propagation of Waves on a Surface with Varying Properties One-dimensional approximation We consider the transformation of a surface wave with a frequency ω falling from the left on a region x ∈ [0 , D] containing N identical cells of size L, D = N L with a quasi-stationary temperature distribution. In the absence of waves, the water surface is considered flat: z = 0, z is the vertical axis directed against the gravity vector g. The liquid surface temperature, which determines the kinematic viscosity ν(x) = ν(T (x)) and surface tension α(x) = α(T (x)), is given by the spectral decomposition ⎧ ∞ ⎨ T  c sin(2μnx), x ∈ [0, D] n , T (x) = T0 + n=1 ⎩ 0, x ∈ (−∞, 0] ∪ [D, +∞)

(7.1)

where T is the overheating inside the cell, μ = π/L is the wavenumber of the periodic convective structure, and cn is the spectral coefficients. For small deviations of the excited surface ζ(t; x) the system of equations of motion and boundary conditions is linearized and takes the form [3]      ut = − px + 2 ν ux x + ν uz + wx z      wt = − pz + ν uz + wx x + 2 ν wz z − g, ux + wz = 0



. p − p0 + αζx x − 2ν wz z=ζ = 0 p − p0 + αζx x − 2ν wz z=ζ = 0



 ν uz + wx + αx z=ζ = 0, w − ζt − uζx z=ζ = 0

(7.2)

Here v = (u,w) is the velocity vector of the liquid; water pressure p, atmospheric pressure p0 , and surface tension coefficient α are normalized to the average density of the liquid ρ. In real conditions, the temperature variations T are small and the values ν and α inside the convection region are given as decompositions in the vicinity of the base temperature T0 [4] ν(T (x)) = ν(T0 + T α(T (x)) = α(T0 + T

∞  n=1 ∞  n=1

cn sin(2μnx)) ≈ ν(T0 ) + νT (T0 )T cn sin(2μnx)) ≈ α(T0 ) + αT (T0 )T

∞ 

cn sin(2μnx)

n=1 ∞ 

. cn sin(2μnx)

n=1

(7.3) or in a more compact form using small parameters









ν (T0 )T

0. ν

(7.7)

The choice of signs of the real and imaginary parts of the wave numbers k and kb is made from the condition of damping perturbations at x → ±∞ and z → −∞. Substituting (7.7) into the boundary conditions of system (7.6) generates two systems of equations that relate the amplitudes A± and B±  2k 2 A± − kb2 + k 2 B± = 0  k ω2 − gk − α k 3 + 2iων k 2 A± +  + ω2 kb − gk 2 − αk 4 + 2iων kb (3k 2 − kb2 ) B± = 0,

(7.8)

condition for compatibility of which  2k ω2 kb − gk 2 − αk 4 + 2iωνkb (3k 2 − kb2 ) −   − kb2 + k 2 f 7.ω2 − gk − αk 3 + 2iωνk 2 = 0 it is the dispersion equation of capillary-gravitational waves.

(7.9)

198

7 Interaction of Surface Waves With Regions of Near-surface …

It follows from (7.8) that waves and boundary layers appear or disappear simultaneously, and the amplitudes of the wave (A± ) and boundary layer (B± ) parts of surface vibrations are rigidly connected to each other. Taking into account the values of the thermodynamic parameters of water [8], Eq. (7.9) has the only physically acceptable solution     ν ν 4ων k∗2 , −k∗ k = k∗ + + i 1 + k∗ 2 g + 3α k∗ 2ω 2ω

(7.10)

where k∗ is the root of the dispersion equation of capillary—gravitational waves in an ideal liquid ω2 − gk − α k 3 = 0 (only if the conditions Rek, Imk > 0 [3] are met, the dispersion curve is shown in figure 7.1 as a dotted line). From (7.8) also follows A− B− 2ν k 2 = =β= . A+ B+ iω − 2ν k 2 2

(7.11)

In the case of water (ν = 10−2 sm2 /s, α ≈ 70 sm3 /s ), the absolute contribution of the attached boundary layer to the total flow is not so great, as can be seen directly from figure 7.2, where the dependence of | β| on the wavelength λ = 2π/k∗ is presented. As follows from the presented results, a significant contribution of the attached boundary layer is shown at wavelengths less than a micron. The influence

Fig. 7.1 Dispersion curves

7.1 Statement of the Problem of the Propagation of Waves …

199

Fig. 7.2 The dependence |β| on the wavelength

of this part of the flow increases for media with higher viscosity and lower surface tension.  zone is partially The wave in =  ekz + β ekb z eikx falling on the convective  reflected by r = R ekz + β ekb z e−ikx and passes t = T ekz + β ekb z eik(x−D) , here R, T are the coefficients to be determined. Taking into account the smallness of δ and ε, the solution (7.6) is sought as a decomposition  = 0 + ε ν + δ α + . . .

(7.12)

    ˜ ˜ ˜ ˜ ˜ ˜ 0 = w + b = ekw z G + ei kx + G − e−i kx + ekb z H+ ei kx + H− e−i kx (7.13) k˜ = k + εkν + δkα , k˜w = k + εkwν + δkwα , k˜b = kb + εkbν + δkbα in this case, k is determined by the ratio (7.10). Substituting (7.12) and (7.13) in (7.6) gives rise to two systems of equations for the functions ν and α , which are not given here because of their bulkiness. Their solutions depend on the relation between the incident wave wavenumber k˜ and the Bragg wavenumbers μm, m = 1, 2, . . . of the periodic distributions (7.3).

200

7 Interaction of Surface Waves With Regions of Near-surface …

In the non-resonant case k˜ = μm the secular terms in the expansions (7.12, 7.13) do not appear when the relations are fulfilled kwν = kν , kb kbν = kkν , kwα = kα , kb kbα = kkα which, taking into account the second boundary conditions, take the form kν = kα = kwν = kwα = kbν = kbα = 0.

(7.14)

Variations in kinematic viscosity with temperature changes have little effect on the dynamics of non-resonant wave scattering. The amplitudes of the forward (G + , H+ ) and reverse (G − , H− ) waves are weakly related, and in the zero (with respect to δ and ε) approximation, the reflection and transmission coefficients are determined by approximate expressions R ≈ 0, T ≈ exp(ik D) ⇒ |T | ≈ 1

(7.15)

the effect of the convection zone on surface waves is small. Near the resonance, the solution is sought by the method of slowly varying amplitudes  = w + b

 w = eλw z e−sx G + (x)ei(μ m+)x + G − (x)e−i(μ m+)x  b = eλb z e−sx H+ (x)ei(μ m+)x + H− (x)e−i(μ m+)x .

(7.16)

˜  = k˜ − μm is the detuning from the resonance, and λw and λb are where s = Imk, to be determined. In the first approximation in terms of viscosity, the action of the operator in the boundary conditions (7.6) is described by the relations ∂  λw z −sx e e G ± (x)e±i(σ+)x ∂z   2νε cm σ3  G ∓ exp(∓2ix) eλw z e−sx e±i(σ+)x ⇒ (σ ± is)G ± ∓ i G ± ∓ ω   ∂  λb z −sx σ ±i(σ+)x e e H± (x)e H± ⇒ σ2 − iω/νH± ∓ i  ∂z σ2 − iω/ν   2νε cm σ2 σ2 − iω/ν H∓ exp(∓2ix) eλb z e−sx e±i(σ+)x ∓ (7.17) ω  where λw = σ ± is, λb = (σ ± is)2 − iω/ν, σ = μm. Substituting (7.16) in the boundary conditions (7.6) with regard of (7.17) leads to the solution

7.1 Statement of the Problem of the Propagation of Waves …

201

G ± = G 0± exp(±ζ x) exp(∓ix), H± = H±0 exp(±ζ x) exp(∓ix)  cm α|σ|3 ζ = δ2 |km |2 − 2 , |km | = 2(g + 3α|σ|2 )  4ων(μ m)2 |σ| = (μ m)2 + s 2 , s = . (7.18) g + 3α(μ m)2 The multiplier exp(−sx) that characterizes the viscous attenuation is the same for waves traveling in the positive and negative directions. In the resonant case, the surface wave is effectively reflected from the convection region. In the zero approximation with respect to δ, ε the values of the reflection and transmission coefficients |R| ≈ 1, |T | ≈ 0 are typical for Bragg scattering. The dependence of the attenuation coefficient at the wavelength on the wavelength itself (i.e., the value s λ from λ) in the non-resonant case is shown in Fig. 7.3, which clearly shows that for wavelengths longer than 2 mm, viscous attenuation has a rather weak effect on the propagation characteristics. In the inviscid limit there is a set of wavenumbers σ = μ m near which the m ω Bm never overlap properties of surface waves change. The band gaps ωm = δk μm and are highlighted on the axes of Fig. 7.1. The dispersion curve k(ω) of surface waves calculated on the basis of (7.9) is shown as solid lines in Fig. 7.1. It is divided into separate sections, inside which lie the Bragg frequencies ω Bm determined by

Fig. 7.3 The attenuation coefficient at the wavelength

202

7 Interaction of Surface Waves With Regions of Near-surface …

the ratio ω2Bm = μ m(g + α(μ m)2 ). The incident wave is effectively reflected from the area of convection when the condition is met 2 = (k − μ m)2 < δ2 |km |2 . It follows from (7.18) that the viscosity expands the opacity zones. In low-viscosity media (7.18) takes the form  ζ = ± δ2 |σ|2 (1 + ν2 ϕ(σ )) − 2 ϕ(σ) = 24(g + α(μ m)2 )2 (μ m)3 /(g + 3α(μ m)2 )3 ,

(7.19)

where σ = ±cm α(μm)3 /2(g + 3α(μm)2 ). The shorter the incident wave length, the stronger is the influence of viscosity in the resonant case, which is confirmed by the dependence of the value ϕ(λm ) included in the ratio (7.19) on the wavelength λm = μ2πm at resonance, shown in Fig. 7.4. The part of the incident surface wave packet that satisfies the Bragg resonance condition is effectively reflected from the convection zone and generates standing waves in the region x < 0. The remaining part comes out of the convection region with a thinned spectral characteristic. So in relation to surface waves, the Marangoni convection region acts as an effective band-pass filter. The quasi-stationary approximation of the temperature distribution used in the calculations is performed when u, χ k 0

gives the decomposition (7.31) the form +∞ +∞ +∞ =

   A ω, k x , k y exp(kb z) exp i k x x + k y y − ω t dk x dk y dω,

0 −∞ −∞

(7.33) which describes the motion of a liquid as an independent boundary layer, which in its structure coincides with the boundary layer attached to the surface wave. But such a boundary layer, as previously stated, cannot exist separately from the surface wave. This is indeed the case, since substituting (7.33) in the second Eq. (7.30) leads to the relation

206

7 Interaction of Surface Waves With Regions of Near-surface …

⊥ z +∞ +∞ +∞    2 =− kb k⊥ A ω, k x , k y exp(kb z) exp i k x x + k y y − ω t 0 −∞ −∞

dk x dk y dω = 0,

(7.34)

 from which it follows that A ω, k x , k y = 0, which leads to the result  = 0. Thus, there is no need to introduce a toroidal part in the formulation under consideration, and it is sufficient to use only the poloidal potential , which greatly simplifies the further solution. In the light of the obtained result, it makes sense to return to the original record of the equations of motion and consider them from a slightly different point of view. Because now the representation of the velocity field (7.24) is simplified u = x z , v = yz , w = −⊥  the equations of motion of the system (7.21) can be given the form 

p˜ + gζ + L ν z

 x

  = 0, p˜ + gζ + L ν z y = 0, p˜ z − L ν ⊥  = 0.

(7.35)

From (7.35), taking into account the previously obtained equation of motion (3.29) (which is valid in the most General case), it follows that the pressure perturbation p˜ is represented as p˜ = −gζ − L ν z .

(7.36)

Using (7.36) allows to exclude p˜ from the second boundary condition (7.26)

α⊥ ζ − gζ − L ν z + 2ν⊥ z z=0 = 0.

(7.37)

Differentiating (7.37) in time and substituting the ζ t value from the first (kinematic) boundary condition (7.26) leads to the final formulation of the equations of motion and boundary conditions L ν  = 0







∂    ∂    ν zz − ⊥  + α

ν zz − ⊥  + α

= 0, =0 ∂x ∂y z=0 z=0

= 0. (7.38) α2  − g⊥  + L ν  − 2ν⊥ 

⊥

tz

t z z=0

In this case, as follows from (7.38), the second compatibility condition (7.27) is fulfilled automatically and can be excluded from consideration. After solving the problem (7.38), the physical fields and the elevation of the free surface are determined from the relations

7.1 Statement of the Problem of the Propagation of Waves …

u = x z , v = yz , w = −⊥ , p˜ = −gζ − L ν z , ζt = −⊥ |z=0

207

(7.39)

by means of elementary operations.

7.2 Spatial Structure of Gravitational-capillary Waves in the Presence of Convective Flows Statement of the problem of surface wave scattering Let a gravitational capillary wave fall on the left side of the two-dimensional Marangoni convection region, which is schematically viewed from above in Fig. 7.5. The convection region occupies the band x ∈ (−∞, +∞), y ∈ [0, D], where D is the width of the convection region, which can be infinite, and θ is the angle of incidence of the surface wave. The process of scattering and passing of the incident wave depends significantly on the geometry of the temperature distribution inside the convection region. Figure 7.6 shows the most characteristic forms of convective cells [5] that form regular patterns on the liquid surface. The study of various geometries of the convective structures presented here leads to the conclusion that a single description of the interaction process for all possible variants is unattainable. Therefore, each structure must be studied as part of a separate task. The simplest analytical description of the properties of the convective structure, revealing the main characteristic features of interaction with the surface wave, is achieved for the case of rectangular cells shown in Fig. 7.6 b. for this variant, the scheme of Fig. 7.5 is specified and takes the form shown in Fig. 7.7. Fig. 7.5 The scheme of interaction of surface waves with two-dimensional convection area

208

7 Interaction of Surface Waves With Regions of Near-surface …

(a) Triangularcells. © [5]

(b) Rectangular cells. © [5]

(c) Regular hexagonal cells. © [5]

Geometry of the triangular cells borders.

Geometry of the rectangular cells borders.

The geometry of the regular hexagonal cell borders.

Fig. 7.6 a Triangular cells. © [5], Geometry of the triangular cells borders. b Rectangular cells. © [5], Geometry of the rectangular cells borders. c Regular hexagonal cells. © [5], The geometry of the regular hexagonal cell borders. d Regular and irregular hexagonal cells. © [5], Geometry of the borders of regular and irregular hexagonal cells. e Irregular octagonal and square cells. © [5], Geometry of the borders of irregular octagonal and square cells

7.2 Spatial Structure of Gravitational-capillary Waves …

(d) Regular and irregular hexagonal cells. © [5]

Geometry of the borders of regular and irregular hexagonal cells.

(e) Irregular octagonal and square cells. © [5]

209

Geometry of the borders of irregular octagonal and square cells.

Fig. 7.6 (continued)

In the diagram, a, b are the cell sizes along the x and y axes, respectively, and N is the number of cells that fit on the width D of the convective structure. Now that the geometry of the problem has been determined, it is necessary to describe the temperature distribution on the liquid surface. Outside the convection region, the temperature is assumed to be constant T = T0 . Inside the convective region, the temperature distribution is given by a two-periodic function T (x, y) = T (x + a, y) = T (x, y + b) = T (x + a, y + b). The condition of continuity of the temperature distribution T (x, 0) = T (x, D) = T0 is imposed on the boundaries of the convective structure. Representation of the distribution T (x, y) in the form T (x, y) = T0 + T τ(x, y),

(7.40)

210

7 Interaction of Surface Waves With Regions of Near-surface …

Fig. 7.7 The scheme of interaction of surface waves with the convection area, formed of rectangular cells

where T is the value of the characteristic temperature superheat inside the cell, imposes conditions that are satisfied by the temperature superheat distribution function τ(x, 0) = τ(x, D) = 0, τ(x, y) = τ(x + a, y) = τ(x, y + b) = τ(x + a, y + b). (7.41) From (7.41), in particular, it follows that at the boundaries of convective cells parallel to the x axis, that is, at y = mb, m = 0, . . . , N the value of temperature overheating is zero. For a complete description of the convective structure, the relations (7.41) describing two-period properties are not sufficient, since a separate convective cell has an internal structure that generates additional symmetries of the problem. Analysis of the temperature distribution inside rectangular cells [11] shows that in the local coordinate system (ξ, η) of an individual cell, as shown in Fig. 7.5, the distribution τ(ξ, η) is characterized by symmetry relations τ(ξ, η) = τ( − ξ, η) = τ(ξ, − η) = τ( − ξ, − η), ξ ∈ [−a/2], η ∈ [−b/2, b/2],

(7.42)

which highlight local properties of the convective structure. In Fig. 7.8, dark circles mark the points of the same temperature overheating. As can be seen from the figure, in the special case of square cells, the number of local symmetries doubles. An analytical description of this fact is given by an additional relation

7.2 Spatial Structure of Gravitational-capillary Waves …

211

Fig. 7.8 a Internal symmetries of a rectangular cell. b Internal symmetries of a square cell

τ(ξ, η) = τ(η, ξ), ξ ∈ [−a/2, a/2], η ∈ [−a/2, a/2].

(7.43)

It follows from (7.43) that at all boundaries of square cells, the temperature overheating is zero, i.e. τ(na, y) = τ(x, ma) = 0, n ∈ Z , m = 0, . . . , N . We extend this property to rectangular cells, assuming that for x = na, n ∈ Z , there is no temperature overheating, so that as a result τ(na, y) = τ(x, mb) = 0, n ∈ Z , m = 0, . . . , N .

(7.44)

Meeting the conditions (7.22, 7.23, 7.25) allows you to set the temperature superheat distribution within the convective region in the form τ(x, y) =

 y    x   sin (2m + 1)μ b cnm (−1)n+m sin (2n + 1)λ a a b n,m=0 ∞ 

λ = π/a, μ = π/b, cnm

4 = N ab

a N b τ(x, y) cos(2λ nx) cos(2μ my)d xd y, 0

0

(7.45) where the curly brackets { } represent the remainder of the division. The representation (7.45) takes into account both internal symmetries of convective flows in rectangular cells and symmetries in square cells at a = b. The presence of a temperature distribution on the liquid surface leads to the appearance of spatial distributions of its thermodynamic parameters. As shown in the previous section, the main influence on the properties of reflected and transmitted surface vibrations is the distribution of the surface tension coefficient. In this case, the kinematic viscosity coefficient can be considered a constant value. In this case, the problem under consideration is characterized by one small parameter



δ = α T (T0 )T /α(T0 ) , and the spatial distribution of the surface tension coefficient up to the second-order terms of smallness is described by the relation

212

7 Interaction of Surface Waves With Regions of Near-surface …

α(T (x, y)) = α(1 + δτ (x, y)), α = α(T0 ).

(7.46)

Given the fact that the first two boundary conditions of the system (7.38) derived α x and α y describe the deviation of the free surface from the flat undisturbed position due to convective phenomena, and themselves the boundary conditions are written in the approximation of a flat free surface, the system of equations of the problem can be transformed to mind L ν  = 0







∂   ∂  

 − ⊥ 

 − ⊥ 

= 0, ∴ =0 ν ∂ x zz ∂ y zz z=0 z=0

α(1 + δτ(x, y))2⊥  − g⊥  + L ν t z − 2ν⊥ t z z=0 = 0.

(7.47)

The constant factor v (kinematic viscosity) is left in the first two boundary conditions in order to explicitly show their automatic satisfaction in the inviscid fluid approximation. Now, after all the necessary preliminary actions have been performed, the problem under study is formulated in the following form: Let a surface gravitational-capillary wave fall on a convective region x ∈ (−∞, +∞), y ∈ [0, D] consisting of rectangular cells with dimensions [a × b] from the half-space y < 0 at an angle θ. In the region of convection, the distribution of the surface tension coefficient is given by the relations (7.26, 7.27), and the viscosity of the liquid is considered constant. It is necessary to determine the nature of reflected and transmitted vibrations, as well as fluctuations within the convection region. Approximation of an inviscid liquid In order to understand in detail the structure of the scattered and transmitted fields of surface vibrations, we first consider the inviscid fluid approximation. In this case, the system (7.47) is reduced to the form



t = 0, α(1 + δτ(x, y))2⊥  − g⊥  +  tt z z=0 = 0.

(7.48)

Outside the region of convection, the dynamics of the incident and reflected waves obey, in the case of harmonic oscillations, the relations

 = 0, α2⊥  − g⊥  − ω2 z z=0 = 0.

(7.49)

In the y < 0 region, the incident wave is given by the ratio in = ekz exp(ik(x sin θ + y cos θ)),

(7.50)

where θ is the angle of incidence, k is the real solution of the dispersion equation

7.2 Spatial Structure of Gravitational-capillary Waves …

D(ω, k) = α k 3 + gk − ω2 = 0.

213

(7.51)

The reflected wave is set as r e f = ekz exp(i(k x x + k y y)),

(7.52)

moreover, in (7.52) k is a valid solution of (7.51) Imk x = 0, Re, Imk y ≤ 0, which ensures that the reflected wave is limited at x → ±∞, y → −∞. Let’s represent the k y component of the reflected wave as k y = κ + iκ .

(7.53)

Substituting (7.33, 7.34) into the equation of motion of the system (7.49) leads to the relation k 2 − k x2 − κ2 + κ2 − 2iκ κ  = 0, which splits into two expressions k 2 − k x2 − κ2 + κ2 = 0, κ κ  = 0.

(7.54)

Satisfying the second relation (7.54), namely κ κ  = 0, splits the solution (7.54) into two different variants: 1. κ = 0 and 2. κ = 0. 1.

κ = 0.

In this case (7.54) takes the form of an equation k 2 = k x2 + κ 2 , the solution of which is given in the form κ = −k cos ϕ, k x = −k sin ϕ, ϕ ∈ [−π/2, π/2]. The reflected wave r e f (ϕ) = R(ϕ)ekz exp(−ik(x sin ϕ + y cos ϕ)) at a fixed ϕ is the angular spectral component of the scattered surface wave field. Depending on the sign of the angle ϕ, scattered waves can be divided into two types—reflected “forward” and reflected “backward”, as shown in Fig. 7.9. The spectral components of the waves reflected “forward “ are proportional to the values of r+e f (ϕ) ∼ ekz exp(ik(x sin ϕ − y cos ϕ)), ϕ ∈ [0, π/2]

(7.55)

Since the convective structure has the property of periodicity in the direction x with the period a, taking into account the phase incursion ka sin θ of the incident wave imposes an additional condition on the relation (7.55) r e f (x + a) = exp(ika sin θ)r e f (x).

(7.56)

Substituting (7.55) in (7.56) results in a condition imposed on the value of the angle ϕ for waves reflected “forward”

214

7 Interaction of Surface Waves With Regions of Near-surface …

Fig. 7.9 Schematic of the reflection of surface waves from a convective structure

exp(ika(sin ϕ − sin θ)) = 1.

(7.57)

Satisfying the condition (7.57) generates a discrete set of reflection angles “forward”

    ka sin θ ka 2π n , n ∈ Z, − ≤n≤ (1 − sin θ) , ϕn+ = arcsin sin θ + ka 2π 2π (7.58) where the square brackets in the expression (7.58) mean the integer part of the number. The spectral components of the waves reflected “backward”, in turn, are proportional to the values of r−e f (ϕ) ∼ ekz exp(−ik(x sin ϕ + y cos ϕ)), ϕ ∈ [0, π/2].

(7.59)

Substituting (7.59) in (7.56) leads to a condition imposed on the value of the angle ϕ for waves reflected “backward” exp(−ika(sin ϕ + sin θ)) = 1.

(7.60)

Satisfying the condition (7.60) generates a discrete set of reflection angles “backward”

7.2 Spatial Structure of Gravitational-capillary Waves … − ϕm = arcsin



215

    ka sin θ ka 2π m − sin θ , m ∈ Z , ≤m≤ (1 + sin θ) , ka 2π 2π (7.61)

where the square brackets in the expression (7.61) also mean the integer part of the number. The appearance of a whole set of reflection angles, as well as the manifestation of the “backward” reflection effect, are the result of the interaction of the incident wave with the spatially periodic boundary [6]. If the properties of the convective structure were uniform along the x axis, the conditions (7.38, 7.41) would have to be met for any values of the spatial period a. For waves reflected “forward”, this would mean fulfilling the condition sin ϕ − sin θ = 0 ⇒ ϕ = θ,

(7.62)

that is, the wave is reflected at a single angle equal to the angle of incidence. For waves reflected “backward”, from (7.60), the relation follows sin ϕ + sin θ = 0.

(7.63)

Since ϕ, θ ∈ [0, π/2], the condition (7.63) is never fulfilled and when scattering on a homogeneous boundary, waves reflected “backward” are not formed. Combining the results in an expression for the reflected wave field at κ = 0 ⎧ ka ⎨ [ 2π (1−sin  θ)]

kz r(1) e f = e exp(ikx sin θ)

⎩

n=−[

[ +

ka 2π (1+sin θ)



m=[

ka 2π

ka 2π

+ A+ n exp(2iπ nx/a) exp(−iky cos ϕn )

sin θ]

]

sin θ]

⎫ ⎬

− A− m exp(−2iπ mx/a) exp(−iky cos ϕm ) , ⎭

(7.64)

where ϕn± = arcsin



2π n ± sin θ . ka

(7.65)

Let’s now consider the second variant. 2.

κ = 0.

2 2 2 In this case (7.54) takes the form of an equation √ k = k x − κ , the solution of which 2 2 is conveniently presented in the form k x = ± k + κ . As in the previous version, the solution is divided into waves reflected “forward” and “backward”. The wave reflected “forward “ is given by the relation of the form

216

7 Interaction of Surface Waves With Regions of Near-surface …

  r+e f = ekz exp(i k 2 + κ2 x)eκ y , k, κ > 0.

(7.66)

Substituting (7.66) into (7.56) imposes a condition that the value κ must satisfy  2π n k 2 + κ2 − k sin θ = , a whence it follows that  κ = κ+ n =k

2π n + sin θ ka

2

 − 1, n ≥

(7.67)

 ka (1 − sin θ) . 2π

(7.68)

The field of the wave reflected “backward” is described by the expression   r−e f = ekz exp(−i k 2 + κ2 x)eκ y , k, κ > 0.

(7.69)

Satisfying the condition (7.56) is equivalent to fulfilling the condition 

k 2 + κ2 + k sin θ =

2π m , a

(7.70)

as a result, for the wave reflected “backward”, the relation takes place 

κ =

κ− m

 =k

2π m − sin θ ka

2

 ka (1 + sin θ) . − 1, m ≥ 2π 

(7.71)

The relations (7.47, 7.50) describe very specific surface waves propagating along the boundary of the convective structure y = 0. Their amplitudes decrease exponentially both when moving into the liquid thickness at z < 0 (the property of the surface wave itself) and when moving away from the boundary of the convection region at y < 0 (the property of the wave running along the selected direction). The group velocity of these waves is directed along the y = 0 boundary, so that the transfer of energy and momentum in them occurs inside a thin “beam” adjacent to the convective region. Combining the results for the variant κ = 0 results in an expression like ⎧ ⎨

kz r(2) e f = e exp(ikx sin θ)

⎩

∞ 

+ m=[

where

ka 2π (1+sin θ)

]

∞  ka n=[ 2π (1−sin θ)]

Bn+ exp(2iπ nx/a) exp(κ+ n y) ⎫ ⎬

Bm− exp(−2iπ mx/a) exp(κ− m y) , ⎭

(7.72)

7.2 Spatial Structure of Gravitational-capillary Waves …

κ± n

 =k

2π m ± sin θ ka

217

2 − 1.

(7.73)

The full reflected field is a superposition of the fields of both variants, so that (2) r e f = r(1) e f + r e f ,

(7.74)

(2) where r(1) e f and r e f are set by the expressions (7.45, 7.46) and (7.53, 7.54), respectively. An obvious property of trigonometric functions, namely

 cos ϕn±

=



1−

2 2π n ± sin θ = σn± , Re, Imσn± ≥ 0 ka

(7.75)

allows to write (7.74) in general form r e f = A0 ekz exp(ik(x sin θ − y cos θ)) ⎧ ∞ ⎨  + + ekz exp(ikx sin θ) A+ n exp(2iπ nx/a) exp(−ikσn y) ⎩ ka n=−[ 2π sin θ] ⎫ ∞ ⎬  − + (7.76) A− n exp(−2iπ nx/a) exp(−ikσn y) . ⎭ ka n=[ 2π sin θ] In (7.76), summation under the signs of both sums is performed for all n = 0. In the presented relation, the A0 ekz exp(ik(x sin θ − y cos θ)) term is singled out specifically, since it describes the usual reflection of a surface wave, which obeys the Snellius law. In the obtained expressions, the unknown ones are the spectral amplitudes A± n, the values of which are determined when physical fields are cross-linked at the front boundary of the convection region at y = 0. Behind the convective region, at y > D, the field of the transmitted wave is described by similar expressions (2) tr = (1) tr + tr ,

(7.77)

where kz (1) tr = e exp(ikx sin θ)

⎧ ka ⎨ [ 2π (1−sin  θ)] ⎩

ka n=−[ 2π sin θ]

Cn+ exp(2iπ nx/a) exp(ik(y − D) cos ϕn+ )

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7 Interaction of Surface Waves With Regions of Near-surface …

+

ka (1+sin θ)] [ 2π 

ka m=[ 2π sin θ]

⎫ ⎬ − ) , Cm− exp(−2iπ mx/a) exp(ik(y − D) cos ϕm ⎭ ⎧ ⎨

kz (2) tr = e exp(ikx sin θ)

⎩

∞ 

(7.78)

Dn+ exp(2iπ nx/a) exp(κ+ n (D − y))

ka n=[ 2π (1−sin θ)]

∞ 

+

ka m=[ 2π (1+sin θ)]

⎫ ⎬ Dm− exp(−2iπ mx/a) exp(κ− m (D − y)) , ⎭

(7.79)

or combined tr = C0 ekz exp(ik(x sin θ + (y − D) cos θ)) + ekz exp(ikx sin θ) ⎧ ∞ ⎨  Cn+ exp(2iπ nx/a) exp(ikσn+ (y − D))+ ⎩ ka n=−[ 2π sin θ] ⎫ ∞ ⎬  + Cn− exp(−2iπ nx/a) exp(ikσn− (y − D)) , ⎭ ka n=[ 2π sin θ]

(7.80)

where summation is also performed for n = 0. The first term in (7.80) describes the usual field of the passed surface wave. Unknown spectral amplitudes Cn± are determined by crosslinking physical fields at the back boundary of the convection region at y = D. Let us now consider the structure of the wave field inside the convection region. Since for 0 < y < D the system of equations for harmonic oscillations has the form

 = 0, α(1 + δτ(x, y))2⊥  − g⊥  − ω2 z z=0 = 0

(7.81)

we will look for a solution (7.81) in the form  = 0 + δ1 + . . . .

(7.82)

Substituting (7.82) in (7.81) and selecting terms of the same order of smallness with respect to the parameter δ leads to successive systems of equations

0 = 0, α2⊥ 0 − g⊥ 0 − ω2 0z z=0 = 0

1 = 0, α2⊥ 1 − g⊥ 1 − ω2 1z + ατ(x, y)2⊥ 0 z=0 = 0

(7.83)

The solution of the zero approximation is represented in general form   − 0 = ekz ψ+ 0 exp(ik(x sin θ + y cos θ)) + ψ0 exp(ik(x sin θ − y cos θ)) , (7.84)

7.2 Spatial Structure of Gravitational-capillary Waves …

219

so that in the absence of temperature overheating in the convection region, that is, at δ = 0, the surface wave propagates without distortion. On the right side of the boundary condition for the first approximation correction 1 is a term of the form

F(x, y) = −ατ(x, y)2⊥ 0 z=0   − = −α k 4 ψ+ 0 exp(ik(x sin θ + y cos θ)) + ψ0 exp(ik(x sin θ − y cos θ)) ∞  y    x    sin (2m + 1)μ b , cnm (−1)n+m sin (2n + 1)λ a × a b n, m=0 (7.85) which is convenient to present in a slightly different form. Because x  x   x  x  (2n + 1)λ a = (2n + 1)π = (2n + 1)π − a a ay  a y  y  y  (2m + 1)μ b = (2m + 1)π = (2m + 1)π − , b b b b where square brackets mean the integer part of the number, then   x  x    x  = sin (2n + 1)π − = (±1)[x/a] sin((2n + 1)λ x) sin (2n + 1)λ a a a a   y  y    y  = sin (2m + 1)π = (±1)[y/b] sin((2m + 1)μ y), sin (2m + 1)μ b − b b b

(7.86) the symbols (±1)[x/a] and (±1)[y/b] represent +1 or −1. The plus sign is used if [x, a], [y/b] are even numbers, and the minus sign is used if they are odd numbers. Substituting (7.86) into (7.85) converts the expression for the right-hand side to the form  α k4  + ψ0 exp(ik(x sin θ + y cos θ)) + ψ− 0 exp(ik(x sin θ − y cos θ)) 4 ∞  × cnm (−1)n+m (±1)[x/a] (±1)[y/b]

F(x, y) =

n, m=0

× (exp(i(2n + 1)λ x) − exp(−i(2n + 1)λ x)) (exp(i(2m + 1)μ y) − exp(−i(2m + 1)μ y)) ∞

α k4  cnm (−1)n+m (±1)[x/a] (±1)[y/b] 4 n, m=0  × ψ+ 0 (exp(i((2n + 1)λ − k sin θ)x) − exp(−i((2n + 1)λ + k sin θ)x)) × (exp(i((2m + 1)μ − k cos θ)y) − exp(−i((2m + 1)μ + k cos θ)y))

=

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7 Interaction of Surface Waves With Regions of Near-surface …

+ ψ− 0 (exp(i((2n + 1)λ − k sin θ)x) − exp(−i((2n + 1)λ + k sin θ)x))  × (exp(i((2m + 1)μ + k cos θ)y) − exp(−i((2m + 1)μ − k cos θ)y)) . (7.87) Since the solution 0 is an eigenfunction of the operator applied to the function 1 on the left side of the boundary condition (7.84), secular terms can occur in the solution for 1 if, as follows from (7.87), at least one of the conditions is met (2n + 1)λ = 2k sin θ, (2m + 1)μ = 2k cos θ,

(7.88)

which are the conditions of spatial resonance. Expressions (7.88) can be written in general vector form if we introduce the concept of the inverse lattice vector [13] using the relations b p,q = pb1 + qb2 , b1 =

2π 2π ex , b2 = e y , p, q ∈ Z . a b

(7.89)

Then (7.88) takes the form bn+1/2,m+1/2 = 2k,

(7.90)

where k = k(sin θ ex + cos θ e y ) is the wave vector of the surface wave. If the convective cells did not have the internal structure described by the relations (7.23, 7.25), the expressions (7.69, 7.71) would have the form nλ = k sin θ, mμ = k cos θ, bn,m = 2k

(7.91)

and would exactly coincide with the Laue equation [7] for a wave inside a periodic structure. This immediately makes it possible to conduct a qualitative analysis of the process of surface wave scattering on a two-dimensional convective structure. As the object of this analysis, we choose a two-dimensional convective lattice whose internal structure is described by the relations (7.91). For such a lattice, each vector b of the inverse lattice defines a family of planes according to the equationr ·b = 2π s, s ∈ Z . These planes are perpendicular to the direction of the b vector, and the k and k vectors of the incident and reflected waves form the same angles with them. Figure 7.10 shows the individual bn,m vectors and their corresponding pn,m planes. The wave vector k of the incident wave is also shown there. The Laue equation relates the wave vectors of the incident and reflected waves to any vector b of the inverse lattice by the relation k = k + b.

(7.92)

7.2 Spatial Structure of Gravitational-capillary Waves …

221

Fig. 7.10 The scheme of vectors of the inverse lattice of the convective structure

Consider how the relation (7.92) applies to the inverse lattice vectors shown in Fig. 7.10.

Vector b1,0 .

Vector b _1,0 .

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7 Interaction of Surface Waves With Regions of Near-surface …

Vector b1,_ 1 .

Vector b1,_ 6 .

The k vectors corresponding to the cases b1,0 , b−1,0 and b1,−1 are directed inside the convective structure and do not form reflected waves. In the case of b−1,−6 , the vector k is directed away from the convection region and, at first

glance, generates a reflected wave. However, this is not the case, since the |k| = k relation does not hold. The fulfillment of this relation is a prerequisite, since outside the convection

 3 3 2



region,

 the2 wave vectors obey the dispersion relation α|k| + g|k| − ω = α k + g k − ω = 0, which has a unique solution [8]. In order to answer the question whether it is possible to effectively reflect a surface wave from a convective structure, from a formal point of view, it is necessary to iterate over all the vectors of the inverse lattice bn,m for n ∈ Z , m = −N , −N +1, . . . , N − 1, N . The restriction for the m index occurs due to the finiteness of the convection region in the direction of the y axis. Such a search is a time-consuming and irrational process. It’s much easier to do something else. In order to determine the direction of the resonant reflection, it is necessary to draw an inverse lattice based on its elementary translation vectors b1,0 and b0,1 . Then you need to align the end of the k vector with an arbitrary lattice node, then draw a circle of radius |k| from the beginning of the k vector and draw k vectors from the center of the circle to the intersection points of this circle with the lattice nodes, as shown in Fig. 7.11. The k vectors constructed in this way obviously satisfy both the Laue equation (7.92) and the dispersion relation. After that, from the resulting set of vectors, you must discard those that are directed inside the convection region. For example, Fig. 7.11 shows vectors that have a positive component in the direction (they are underlined in the figure). The remaining k vectors form an indicatrix of the resonant scattering of the surface wave. When the frequency of the incident wave changes, but with the direction of its propagation in relation to the convection region, the scattering indicator changes, since the radius of the circle on which the ends of the scattered wave vectors that meet the Laue equation can be located decreases. This fact is illustrated in Fig. 7.12. The decrease in the radius of the circle is associated with a decrease in the frequency of the incident wave, therefore, according to the dispersion relation, with

7.2 Spatial Structure of Gravitational-capillary Waves …

223

Fig. 7.11 The construction of the vectors of the reflected waves

Fig. 7.12 The change of the indicatrix of scattering with frequency change

a decrease in the absolute value of the wave number |k|, which is equivalent to an increase in the wavelength. The upper circle corresponds to two main maxima in the scattering indicatrix. As the frequency decreases further (lower circle), scattering will occur along one selected direction. If, with a further decrease in frequency, a situation occurs when the ratio turns out to be fair



!

2|k| < min b1,0 , b0,1 ,

(7.93)

224

7 Interaction of Surface Waves With Regions of Near-surface …

which is a condition for the absence of solutions to the Laue equation, the incident wave does not experience effective reflection from the convection region and passes through it almost without distortion. Consideration of the geometry of Fig. 7.11 shows that the vectors k, k incident and reflected waves form an angle ϕ with each other, which satisfies the relation 2|k| sin

ϕ = |b| 2

(7.94)

which is called the Bragg-Wulf equation. The set of Laue and Bragg-Wolf equations determine the qualitative form of the scattering indicatrix. The intensity of the reflected waves should be calculated if the boundary conditions at the boundaries of the convection region are satisfied.

7.3 Experimental Characteristics of Scattering of Surface Waves by Convective Structures In the considered situation, the intensity of convective flows was considered to be significantly higher than the perturbations introduced by the surface wave, which allowed us to consider the problem as a process of wave propagation over a surface with thermodynamic characteristics that regularly change in space but are stationary in time. In real conditions, there is often an inverse relationship between the intensities of convective and wave flows. In this case, the spreading wave intensively mixes the surface layer, breaking the established convective pattern. Experimental studies of this phenomenon were carried out in such a way that the characteristic length of excited capillary-gravitational waves (KGW) ranged from values smaller than the characteristic scale of convective cells to values significantly larger. The diagram of the laboratory experiment is shown in Fig. 7.13. The studies were

Fig. 7.13 Diagram of the laboratory installation

7.3 Experimental Characteristics of Scattering of Surface Waves …

225

carried out in a glass tray (1) 7 m long and 50 cm wide. Wave dampers were installed at both ends of the tray (2). the KGW amplitude was Measured by an electrode waveform detector (4), with five sensors (3) located along the axis of the tray. The signal from the sensors was fed to the spectrum analyzer (5) via the channel switch of the waveform detector (4). Regular KGW were excited by a cylindrical semi-submerged plunger (6) oscillating in a vertical plane. During the experiments, the steepness of the waves ka did not exceed the 0.05 value. To measure the vertical temperature profile in the water boundary layer, a falling thermoprobe (7) with a differential copper-constantan thermocouple was used, the time constant of which did not exceed 1 ms, the junction diameter 10 μm, and the vertical resolution of the order 0.1 mm. The temperature profiles in the skin layer were recorded by a memory oscilloscope, while the error in measuring the temperature from the screen did not exceed 0.05, 0.1 ◦ C. To observe horizontal inhomogeneities of water surface temperature, an infrared thermal imager was used (8). The information was displayed on a television screen, from which images of water temperature fields were taken. With small temperature differences Twa between water and air, the image on the screen had small contrasts. For this reason, the structure of convective cells or their absence was determined by applying a dye to the water surface—an alcohol solution of brilliant greens. Within one series, the change in the temperature regime in the tray did not exceed 0.5 ◦ C, which indicates sufficient stability of the thermal regime. Water and air temperatures and humidity in the laboratory were also measured. The cleanliness of the water surface was checked by measuring the surface tension. Its values during the experiments were 72 ± 0.5 din/sm, which corresponds to the values for pure water and indicates the absence of surfactants. With a positive difference Twa , which changed within −3 + 10 ◦ C during the experiment, convection began in the fresh water cooling from the surface. Figure 7.14 shows the structure of cells drawn from images from the thermal imager screen. Dark lines indicate the location of convective convergence zones. The temperature difference at the surface of a single cell is of the order 0.4..0.5◦ C. In the absence of waves, the cells have a complex random shape with randomly oriented borders of different lengths. After excitation on the surface of the KGW, as the spatial dimensions of the KGW and the cells approached the area of coincidence, the structure began to be rearranged—the polygonal cells were transformed into rollers stretched parallel to the front of the traveling surface wave. At the same time, changes in the spectrum of cell sizes were observed. The wide spectrum peculiar to the case without waves (the component in the direction of wave propagation is shown Fig. 7.15) under the action of KGW was contracted to the vicinity of half the wavelength. At the same time, the proportion of cells with the same size increased. Thus, the effect of waves on convective cells consists not only in ordering their orientation, but also in regularizing the cell sizes. With a further decrease of the wavelength, the cell structure back to its original state (see Figs. 7.14 and 7.15).

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7 Interaction of Surface Waves With Regions of Near-surface …

0.0

5.2

0.4

0.5

0.8

10.0

Fig. 7.14 The structure of the convective cells at Twa = 7 ◦ C. The numbers represent the frequency of waves in Hz

Fig. 7.15 Histogram of the size distribution of convective cells in the direction of wave propagation. Solid line—if there are no waves, dotted line-waves with a frequency of 5 Hz

Estimates of the Rayleigh number for thermal conditions in Figs. 7.14 and 7.15, made according to the temperature profiles in the boundary layer, showed that in the initial situation (F = 0.0 Hz) Ra = 5.5 · 109 , in the field of regularization (F = 5.6 Hz) Ra = 109 , and at higher frequencies the Rayleigh number increases again Ra = 3.5 · 109 . Thus, KGW blurs sharp temperature profiles in the boundary layer

7.3 Experimental Characteristics of Scattering of Surface Waves …

227

Fig. 7.16 Dependence of the relative attenuation coefficient of waves on the wavelength and temperature difference  Twa . Dashed lines show theoretical dependences corresponding to experimental temperature differences

and the resulting decrease in heat flow leads to a transition from a three-dimensional polygonal structure to a quasi-two-dimensional roller structure. The results of estimates of the wave attenuation coefficient for several thermal modes are shown in Fig. 7.16, where the graphs of the ratio of the attenuation coefficients βc for convection in the liquid and β0 in its absence (i.e., for Twa = −3 ◦ C) are shown. In Fig. 7.16, dashed lines show theoretical curves corresponding to the same thermal conditions calculated by taking into account temperature changes in the viscosity, density, and surface tension of water. Comparison of Fig. 7.16 with Figs. 7.14 and 7.15 shows that deviations of experimental points from theoretical curves lie in the area of cell structure rearrangement. While the difference  Twa is small, convection is not developed and its influence on the KGW is weakly expressed (curve I, Fig. 7.16). As the difference  Twa increases, the convection increases, and as long as it remains laminar, the effect increases (I → II). an increase in instability leads to turbulization of the boundary layer and saturation is observed (II → III). Apparently, with further growth of  Twa the detected phenomenon should disappear. Figure 7.16 shows another feature—the growth of the  Twa leads to a shift of the minimum attenuation of the KGW to the short-wave region. This is due to the fact that simultaneously with the increase in convection, the cell size decreases. If for mode I the characteristic size of temperature inhomogeneities determined from the images from the thermal imager screen was 5–6 cm, for mode II 3–4 cm, then for mode III 2–3 cm. Figure 7.16 shows that the given values of the characteristic cell sizes are approximately equal to half the length of the KGW with minimal attenuation. At the same time, in the developed convection modes, the dissipation of surface waves decreases by 2–2.5 times. The width of the region of quasi-resonant reduction of the KGW attenuation coefficient corresponds to the size dispersion of the ordered structure, which is confirmed

228

7 Interaction of Surface Waves With Regions of Near-surface …

by the results presented in Figs. 7.15 and 7.16 (curve III). The size spectrum of ordered cells in Fig. 7.15 is concentrated in the area of 0.5–3 cm. Taking into account the halfwave nature of the interaction, this interval corresponds to a band of 1–6 cm KGW lengths. Figure 7.16 shows that the differences between experimental and theoretical results lie in this wavelength range. When the attenuation of waves increases at the edges of the operating frequency range, the cell structure is not adjusted (Fig. 7.14). A passive chaotic structure only scatters the waves. It is expected that if the spatial scales of convective cells and surface waves differ significantly, their interaction should not be observed.

References 1. Summerhayes CP, Thorpe SA (1996) Oceanography. Manson Publishing, Southhampton 2. Dias F, Kharif C (1988) Nonlinear gravity and capillary-gravity waves. Ann Rev Fluid Mech 31:301–346 3. Perlin M, Schultz WW (2000) Capillary effects on surface waves. Ann Rev Fluid Mech 32:241– 274 4. Landau LD, Lifshitz EM (1987) Course of theoretical physics. In: Fluid Mechanics, Vol 2, 2nd edn. Butterworth-Heinemann, Oxford, USA. 5. White DB (1988) The planforms and onset of convection with a temperature-dependent viscosity. J Fluid Mech 191:247–286 6. Landau LD, Lifshitz EM, Pitaevskii LP (1984) Course of theoretical physics. In: Electrodynamics of continuous media, vol 8. 2nd edn. Butterworth-Heinemann, Oxford, USA. 7. Landau LD, Lifshitz EM (1980) Course of theoretical physics. In: Statistical physics, vol 5, 3rd edn. Butterworth-Heinemann, Oxford USA

Chapter 8

Anomalous Waves

There is a great and incalculable variety of waves running on the surface of the seas, oceans, lakes and rivers. Parallel waves running up the shore, swell extending to the horizon generated by a powerful continuous storm, staggered horseshoe-shaped waves or porridge of pointed splashes-this is an incomplete list of pictures of waves that occur on the surface of the water. But, of course, the highest and most dangerous waves attract the eye first. And among these high waves are real giants. People encounter this amazing phenomenon much less often than with powerful storm surges, but the impression of meetings persists until the end of life, sometimes in the truest sense of the word. But what is even more surprising is that giant waves often appear quite suddenly, as if out of nowhere, and at first glance for no reason. This disorderly behavior of huge waves of various shapes and lifespans has given rise to a whole host of “terms” in English-language literature, which they are awarded for lack of real names: “freak, rogue, giant, monster, killer and extreme waves”. None of these names covers the full range of these unusual manifestations of surface agitation. Therefore, in most cases we will call them anomalous waves, which allows us not to determine in advance any specific properties that can be refined from situation to situation. For a long time, abnormal waves were the subject of marine folklore or were considered something rare. When describing these unusual waves, eyewitnesses either talk about a “wall of water”, or a “hole in the sea”, or about several large waves (“three sisters”). In engineering applications, an anomalous wave refers to a large ridge that is accompanied by a shallow but extended depression that is in front of or behind a huge wave. The most important property of anomalous waves is their belonging to the class of wind waves, and the named characteristic periods of killer waves coincide with the characteristic periods of wind waves and swells. Sometimes the term abnormal waves is used to refer to all catastrophic waves in the ocean, for example, to describe tsunami waves that reach a height of 30 m. Currently, tsunami waves are not included in the class of anomalous waves described here. Now some researchers have established the idea of these waves as waves that arise from wind

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Kistovich et al., Advanced Studies in Ocean Physics, Springer Oceanography, https://doi.org/10.1007/978-3-030-72269-2_8

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waves or swells, since their characteristic scales (length, period) fall within the range of wind waves. Modern instrumental measurements made on platforms and buoys in various areas of the World ocean, as well as satellite measurements, have translated anomalous waves from the category of folklore into reality. At the same time, it is noticed that giant waves occur in any weather, both in a storm, and in calm water, when their appearance causes special surprise, in deep water and in shallow water. They also appear on the shore in the form of sudden splashes on a steep Bank or rapid flooding of the beach. The successful use of spacecraft for recording anomalous waves can dramatically increase the amount of data received and provide objective estimates of their geographical distribution. So the existence of anomalous waves after receiving numerous field data can be considered proven. Moreover, the extensive field data obtained show a more frequent prevalence of this phenomenon than previously assumed. The study of anomalous waves is fundamentally important for shipbuilding and marine hydraulic engineering, which is engaged in the design of platforms on the sea shelves, as well as port facilities. The ultimate goal of the research is to develop methods for predicting giant waves. This task involves studying the statistics (frequency of occurrence) of killer waves based on long series of observations that gradually accumulate for different areas of the World’s oceans. Statistics of abnormal waves in each area is different and depends on many geographical factors: the topography of the sea floor, the shape of the coastline, currents, the passage of cyclones or hurricanes. It is necessary to carry out zoning of the World ocean according to the degree of risk, to identify the conditions that precede the appearance of giant waves.

8.1 The Concept of Abnormal Wave In the following descriptions, it is easy to distinguish several types of anomalous waves that got in the way of their unwitting witnesses. On what basis were these waves then assigned to the category of anomalous waves, and are there any scientific definitions that allow us to distinguish an anomalous wave from the entire set of surface waves? There are several engineering criteria for an abnormal wave. The first definition is statistical and is based on the randomness of wind waves. Anomalous surges are always possible in a random field, so anomalous waves can be considered as tails of typical wind wave distribution functions and, consequently, the probability of occurrence of abnormally high waves is easily determined. Since waves with a given statistics are predictable, they are called extreme waves, and anomalous waves are those waves whose statistics do not satisfy the tails of typical wind wave distributions, in particular the Rayleigh distribution. Other definitions suggest that an anomalous wave is a wave that has a large amplitude, is asymmetric and steep, and is accompanied by a depression. This definition

8.1 The Concept of Abnormal Wave

231

was influenced by many descriptions of anomalous waves, but later it turned out that these features are not always the main ones, even for catastrophic consequences. Therefore, only the amplitude criterion of the killer wave was fixed: its height should exceed the “significant” height of wind waves by 2–2.2 times. Significant height refers to the average height of a third of the highest waves. This value is usually indicated by a symbol Hs , while the height of the anomalous wave is assigned a symbol Ha .

8.2 History of Observation of Anomalous Waves The story of the gradual realization of the existence of giant waves in the ocean at its early stage is a chain of more or less true stories of eyewitnesses. Perhaps the first official report about the existence of huge waves was the report of the 1840s outstanding Navigator and Explorer Dumont-d’Jurville in the French Academy of Sciences about his observation in 1826 in the South seas of a single wave height of more than 30 m. The speaker was ridiculed, despite the fact that three other people confirmed the truth of his words. At about the same time, or rather in 1839, captain Robert Fitzroy of the British Navy published a report on his observation of abnormal waves in the Atlantic ocean near the Bay of Biscay, generated by unprecedented gusts of wind. The crests of the waves were higher than the middle of the mainmast, the top of which rose 18 m above the waterline. At the same time, according to the author of the report, getting into the depressions between the ridges, the ship was in a calm zone. Never before or since, even on the traverse of Cape horn and the Cape of Good Hope, had Robert Fitzroy encountered waves of this height [1]. In the same XIX century, there are reports that in 1861 the Bishop lighthouse on the Scilly Islands lost its bell, suspended at a height of 30 m, which was torn off by a wave, according to eyewitnesses, which collapsed on top of it. There is evidence that the waves rolled over the lighthouse on Minot’s ledge, Massachusetts, 23 m high, and also broke the glass in the lighthouse on Tillamook cliff in Oregon, reaching a height of 40 m [2]. Among the suspicious events, provoked, perhaps, by a giant wave, include the disappearance without a trace in 1909 near Durban, in the Agulhas current zone, of the passenger liner “Varata”. In 1921 and 1923, waves of 20 and 25 m high were observed in the North Pacific and North Atlantic, respectively [3]. The following observation dates back to 1933, when the tanker Ramapo was caught in a system of periodic giant waves in the Northern Pacific ocean. According to Lieutenant commander R. P. Whitmarsh, who commanded the ship, on 7 February 1933, the tanker Ramapo was caught in a storm in the Pacific ocean. During this storm, the wind impact covered a huge area (several thousand miles across) and lasted for several days. The wind speed reached 60–66 knots. The tanker was turned exactly astern to the wind, which allowed Whitmarsh and his team to estimate the

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8 Anomalous Waves

length of the resulting waves in the range of 300–450 m (measurements were based on knowledge of the length of the vessel ~143 m). A stopwatch was used to determine the wave period of ~14.8 s. Crew members from the bridge were able to measure the height of the wave by matching the horizon line with the line between its crest and the top of the mast, while the ship’s stern was at the very bottom of the wave trough. Triangulation was used to determine the largest wave span (the distance from the trough to the top) of the waves rolling on the ship ~34 m [4]. In 1945 “The Queen Mary” was struck amidships South of Newfoundland. In 1956, captain Grant, commanding the freighter “Junior”, encountered a 30-m wave 100 miles off Cape Hatteras. French Navy cadet Frederic-Moreau in 1963 during the circumnavigation of the cruiser “Joan of Arc” was caught in a seven-point storm southwest of Tokyo and witnessed the phenomenon, later called “three sisters”, when for 30 s the ship was under the successive blows of three abnormal waves. These waves followed each other at a speed of at least 10 m/s and an interval of about 100 m. They were distinguished by the high steepness of the ridge (almost vertical), which somewhat decreased at the edges of the wave front, the width of which reached 800 m, a large height (from 15 up to 20 m), and, what is especially interesting, the direction of propagation of this triple was an angle of order in relation to the direction of the rest of the curtain wave [5]. According to the report of captain Biles, his cargo ship “Castle Edinburgh” in 1964 fell into a “hole in the ocean”, the slope of the walls to the horizon was at least 30◦ . This report to some extent led to believe in the truth of the words of captain Johnson, who claimed that during the second world war, his cruiser “Birmingham” fell into the analog hole [6]. In 1966, the Italian cruise ship “Michelangelo” was hit by a giant wave, also in the form of a sinkhole in the ocean, which smashed the portholes located 24 m above the waterline, killing one crew member and two passengers. The supertanker “Glory of the world” sank in the Agulhas current in 1968 after being torn in half by a single sudden wave. In the same year, Soviet stereo photographs of the surface of Antarctic waters recorded waves with a span of 25 m. In the Gulf of Mexico in 1969, at a sea depth of 100–350 m, abnormal waves with a height of 20–23 m were recorded. At the same time, the ratio of the height of the anomalous wave to the height of significant wind waves reached 2.3. In the period from 1969 to 1971, waves with a span of 15–22 m were repeatedly recorded in the North Atlantic [7]. In 1973, in the same current of Agulhas, a giant wave almost sank the supertanker “Sapphire of Neptune”, tearing off 60 m of its bow. There, off the coast of South Africa, in 1974, a huge wave severely damaged the Norwegian tanker “Wilstar”, almost completely destroying its bow. In November 1975, not in the ocean, but in lake Superior of the Great lakes system, the ship “Anderson” was hit by two giant waves. And although everything ended well for the “Anderson’s” sailors, the fate of the crew of the “Edmund Fitzgerald” cargo ship, which was located nearby, was tragic—everyone was killed. Although, due to

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233

the death of the crew, there is no reliable evidence of the “guilt” of giant waves in the fate of the cargo ship, they are the main “suspects”. In December 1978, the German cargo ship “Munich” disappeared without a trace in the Atlantic, believed to have been hit by a giant wave. One of the rare photos of giant waves was taken in 1980 by Phillip Lijour (Fig. 8.1) when the “Esso Languedoc” tanker he was on was hit by it near Durban. It is estimated that the wave height exceeded 30 m. In the same year, a huge wave sank the English cargo ship “Derbyshire” near the coast of Japan. In the North sea in 1981 recorded almost symmetrical, single hump height of about 15 m, at this time the height of the surrounding waves never exceeded 8 m. Statistics show that in this region at depths of 20–70 m during 1969–1985, the maximum waves also reached 20 m, and the ratio Ha /Hs was 2.1–2.7. On April 27, 1984, the Soviet ship “Taganrog Bay”, with a wave not exceeding 6 points, hit a “hole in the sea” with a depth of about 16 m. On January 1, 1995, an oil platform in the North sea was hit by a giant wave with a span of 26 m, called the “new year’s wave”. In February 2000, an Oceanographic research vessel withstood the impact and measured the characteristics of a 29-m wave west of the coast of Scotland.

Fig. 8.1 A giant wave rolls over the “Esso Languedoc” tanker (photo by Philippe Lijour ©)

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8 Anomalous Waves

In March 2001, in the South Atlantic, a 30-m wave severely damaged and was on the verge of death cruise ships “Bremen” and “Star of Caledonia”. At the same time, there are no currents in this part of the Atlantic! Not only on the open ocean expanses, but also in the inland seas, giant waves are observed. For example, a phenomenal recording of an anomalous wave was obtained from a buoy near Gelendzhik on the Black sea on November 22, 2001. At a sea depth of 85 m, the wave height left about 10 m, but the Ha /Hs ratio reached a record value of 4–5. In the same area of the Black sea, on February 1, 2003, a wave with a maximum height of 12.3 m was registered, and the probability of such waves appearing according to the Rayleigh distribution is once every 50 years. In September 2004, during the passage of hurricane “Ivan” through the Gulf of Mexico, a group of American scientists led by Wang recorded waves with a span of more than 27 m. On April 16, 2005, the “Norwegian Dawn” cruise ship collided with a wave near the coast of Georgia, which engulfed her flesh up to the tenth deck. The wave appeared suddenly, out of nowhere. And just as suddenly disappeared. In January 2006, the “Norwegian Soul” was hit by a 15-m wave off the coast of Tortola island in the North Atlantic group of the virgin Islands In March 2007, the cruise ship “Prinsendam” was hit by a 21-m wave in the Antarctic segment of the ocean off the tip of South America. The “Riverdance” ferry was stranded on the Lancashire coast after colliding with a giant wave in the Irish sea on February 5, 2008 (Fig. 8.2).

Fig. 8.2 The ferry “Riverdance” on the shore © 2008 Shutterstock

8.2 History of Observation of Anomalous Waves

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This is not all evidence of dangerous encounters of ships with giant waves. According to the data of the company “Pentov”, engaged in insurance of sea transportation, only for the period from 1981 to 1991, 29 ship owners applied for compensation for damage due to the collision of their ships with abnormal waves [8]. Every year, the frequency of such evidence only increases. But with all this, there is a fragmentary nature of observations and the subjectivity of eyewitness reports. This state of affairs led to the proposal of the MaxWave (Maximal Wave) program, implemented in cooperation with the European Space Agency, Whose essence is to measure wave parameters using radars installed on satellites orbiting in near-earth space. Satellite observations presented by W. Rosenthal, gave a stunning result: during a three-week observation period, 10 giant waves were recorded, the height of which was at least 25 m. These facts allow us to reconsider the approach to abnormal waves as something out of the ordinary and recognize them as an integral part of the ocean (and, possibly, sea) waves. Currently, the study of anomalous waves is carried out in three main directions: construction of theoretical approaches, laboratory modeling and field observations.

8.3 Researches and Various Models of Anomalous Waves Since field observations are only a record of accomplished events and, first of all, are valuable as a source of data accumulation, on the basis of which new hypotheses are born and existing theoretical predictions are verified, the main interest is caused by the results of mathematical and laboratory modeling. Being under the impression of the catastrophic consequences of ship encounters with abnormal waves, the vast majority of researchers perceive them as largeamplitude waves moving at high speed on the ocean surface. It is clear that such waves must be the focus of a large amount of energy. Therefore, the main question that researchers of these waves are trying to answer is: how does energy focus in a relatively small space and then transfer it? The vast majority of researchers start from some method that seems most likely to them to generate anomalous waves, which, upon closer examination, is more likely a way to focus energy. Everyone sees the nature of things in their own way, so the complete collection of the mentioned methods of focusing consists of separate parts that seem most correct to their adherents. For example, a group of Norwegian scientists [9] identifies three main types of energy focusing: 1. 2. 3. 4. 5.

Focusing due to dispersion compression of wave packets. Focusing by currents. Nonlinear focusing (or focusing due to the Benjamin-fair instability). Other groups of researchers indicate additional methods: Focusing due to the nonlinear interaction of waves. Focusing due to topography.

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6.

8 Anomalous Waves

In a separate group is sometimes taken out Focusing in storm zone.

In addition to focusing ideas, the idea of generating anomalous waves when a part of the sea floor suddenly sinks is also put forward [10, 11]. The correct choice of focusing mechanism depends on the type of anomalous waves being studied. If we take all the reports of eyewitnesses on faith, we should recognize the existence of three main types of waves: periodic [1, 4], amplitudemodulated and soliton-like [5, 6]. The fourth, non-basic type should include those, sometimes incomplete, and often indistinct, descriptions of waves that do not fit into the generally accepted classification. Regardless of the chosen focusing model and the type of anomalous waves studied, there is a common basis for all hypotheses and approaches, which is simply impossible to abandon. This basis is the physical laws expressed in the form of a fundamental system of hydrodynamic equations, partially simplified in relation to the problems of wave formation, propagation, and interaction. When evaluating the proposed models, one should be guided by the principle that if any theoretical construction violates physical laws, it should be discarded. An equivalent criterion for formal (or even informal) mathematical results of the accepted theory is the condition that they satisfy the system of hydrodynamic equations within the framework of the accepted approximations. Thus, it is necessary first to formulate explicitly the mentioned basis for further research. First, in relation to the problem of anomalous waves, water in the seas, oceans, lakes and other reservoirs is considered a homogeneous isotropic medium with constant thermodynamic parameters over time. Secondly, water is considered incompressible and inviscid, since acoustic and viscous phenomena contain a negligible part of the total energy contained in the anomalous wave. The system of equations describing the fluid motion in the accepted constraints has the form 1 v t + (v∇)v = − ∇ p + g, ∇ · v = 0. ρ

(8.1)

These equations are supplemented by boundary conditions on the free surface and bottom p − pa | S=0 = 0,

v · ∇ S| S=0 = ζt ,

v · ∇ B| B=0 = 0,

(8.2)

where pa is the atmospheric pressure, ζ (x, y, t) is the deviation of the free surface from its equilibrium horizontal state z = 0; S = z − ζ(x, y, t), B = z − η(x, y, t), with S = 0 and B = 0 are the equations of the free surface and bottom, respectively.

8.3 Researches and Various Models of Anomalous Waves

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Since some proposed mechanisms for focusing energy in anomalous waves rely on linear models of surface waves, it makes sense to give a linearized limit of relations (8.1, 8.2) and conditions under which the linear approximation is valid. The nonlinearity of the system (8.1, 8.2) lies in member of the (v∇)v equation of Euler, in specifying boundary conditions for pressure and velocity on a timevariable boundary S, the v·∇⊥ S member in the kinematic boundary condition, where ∇⊥ = ex ∂/∂ x + e y ∂/∂ y is the horizontal gradient, ex , e y are single unit vectors of the Cartesian coordinate system, the beginning of which lies on the undisturbed surface, and the direction of the ort ez is opposite to the direction of gravity. If the model allows bottom movements, then the nonlinearity also consists in setting the boundary conditions on the moving bottom B and in the v · ∇⊥ B term of the boundary condition on the bottom. If the bottom is stationary, i.e. η =η(x, y), then the boundary condition at the bottom is linear. Before starting to linearize the equations, it is useful to exclude the hydrostatic part from the pressure p, that is, to represent the pressure in the liquid as the sum of the hydrostatic part and the perturbation caused by the surface wave p = ρgz + p. ˜

(8.3)

Substituting (8.3) in (8.1) reduces the Euler equation to the form 1 ˜ v t + (v∇)v = − ∇ p, ρ

(8.4)

a dynamic boundary condition is converted to the form p˜ − pa | S=0 = ρ gζ.

(8.5)

The linearization of the Euler equation is valid if both of the two conditions are met    1      |(v∇)v|