A single incompressible, inviscid, irrotational fluid medium bounded by a free surface and varying bottom is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet-Neumann operators. The equations for the surface waves are presented in Hamiltonian form. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types taking into account the effect of the slowly varying bottom. The arising KdV equation with variable coefficients is studied numerically when the initial condition is in the form of the one soliton solution for the initial depth.
Hamiltonian models for the propagation of irrotational surface gravity waves over a variable bottom
Alan Compelli, Rossen I. Ivanov and
Michail D. Todorov
|School of Mathematical Sciences, Dublin Institute of Technology,|
|Kevin Street, Dublin 8, Ireland|
|Erwin Schrödinger Int. Institute for Mathematics and Physics,|
|University of Vienna, Vienna, Austria,|
|Department of Differential Equations,|
|Faculty of Applied Mathematics and Informatics,|
|Technical University of Sofia, 8 Kl. Ohridski Blvd., 1000 Sofia, Bulgaria|
Mathematics Subject Classification (2010): 35Q53, 35Q35, 37K05, 37K40
Keywords: Dirichlet-Neumann Operators, KdV equation, Water waves, Solitons
In 1968 V. E. Zakharov in his work  demonstrated that the equations for the surface waves of a deep inviscid irrotational water have a canonical Hamiltonian formulation. The result has been extended to models with finite depth and flat bottom [9, 13], for internal waves between layers of different density  as well as waves with added shear for constant vorticity [7, 5, 6, 3, 4]. A multilayer model based on the Green-Naghdi approximation has been proposed in .
The Hamiltonian approach to the wave motion has been extended by inclusion of variations of the bottom surface in several papers [11, 12, 2]. The Hamiltonian framework allows for approximations taking into account different scales when considering shallow water and long-wave regimes. As a result the arising model equations are of the type of well known shallow water equations like the Korteweg-de Vries (KdV) and Boussinesq equations.
The problem of waves with a variable bottom has a long history. In the pioneering work of Johnson  a perturbed KdV equation is derived as a model for surface waves from Euler’s governing equations for irrotational inviscid fluid (cf. also with ). The problem has been studied further by Johnson [18, 19] and several other authors, e.g. in [23, 24, 25, 26].
In this review paper we consider the scaling regime adopted in  within the Hamiltonian framework of waves under the variable bottom of Craig et al [11, 12, 2]. We use numerical solutions of the Johnson equation in order to analyse the effects of the propagation of solitary waves over a variable bottom.
An inviscid, incompressible system is presented with a surface wave and variable bottom as shown in Fig. 1.
The average water surface is at and the wave elevation is given by the function Therefore we have
The average bottom depth is at and the bottom elevation from the average depth is given by the function Hence the function that represents the local depth is
The body of the fluid which occupies the domain is defined as
The subscript notation will be used to refer to valuation on the surface and will refer to valuation on the variable bottom .
3 Governing equations
Let us introduce the velocity field The incompressibility and the irrotationality of the flow allow the introduction of a stream function and velocity potential as follows:
in This leads to
The Euler equations written in terms of the velocity potential produce the Bernoulli condition on the surface
where is the acceleration due to gravity.
There is a kinematic boundary condition on the wave surface
and on the bottom
We make the assumption that all considered functions , , are in the Schwartz class for the variable, that is declining fast enough when (for all values of the other variables). In other words we describe the propagation of solitary waves.
4 Hamiltonian formulation
The Hamiltonian of the system will be represented as the total energy of the fluid
It can be written in terms of the variables as
We introduce as usual the variable proportional to the potential evaluated on the surface
and the Dirichlet-Neumann operator given by
where is the outward-pointing unit normal vector (with respect to ) to the wave surface.
With Green’s Theorem (Divergence Theorem) the Hamiltonian can be written as
On the bottom the outward-pointing unit normal vector is
thus and no bottom-related terms are present. Noting that the term is a constant and will not contribute to , we renormalise the Hamiltonian to
The variation of the Hamiltonian can be evaluated as follows. Green’s Theorem transforms the following expression to contributions from the surface and the bottom:
Noting that the variation of the potential on the wave surface is given as
It is noted from (7) that
which represents the bottom condition in variational form. It also explains the fact that does not depend on . Evaluating we remember that and therefore
due to (6).
Next we compute
Noting that, using the kinematic boundary condition (6),
by the virtue of the Bernoulli equation (5).
Thus we have canonical equations of motion
5 Taylor expansion of the Dirichlet-Neumann operator
where The surface waves are assumed small, i.e. and one can expand with respect to as follows:
The operator has an eigenvalue for a given wavelength , when acting on monochromatic plane wave solutions proportional to In the long-wave case has an eigenvalue
thus when is small, one can formally expand in powers of .
For the operator the expansion is
where . It is assumed that can be of the order of provided that the bed stays away from the surface, that is of with This way one can expand in powers of Using the recursive formulae in ,
The Dirichlet-Neumann operator is
Keeping expansions up to and in we need
and hence, the truncated expansion is
In the case when with of order (or smaller) the commutator of and is proportional to which is of order (or smaller) and therefore small with respect to or . Therefore we take the truncated expansion
where is the local depth and indicates that the bottom depth varies slowly with .
6 Boussinesq and KdV approximations
We introduce, as usual, the small scale parameters
where is the wave amplitude and is the wavelength, and consider the long-wave and shallow water scaling regime. We consider the wave propagation regime with of order ; with its eigenvalue of order The quantity has the magnitude of a velocity (multiplied by ) and therefore is of order ; is of order 1 and most importantly, which usually leads to the Boussinesq and KdV propagation regimes.
The magnitudes of the quantities can be made explicit by the change
where now all quantities like and are . We write the Hamiltonian (13) in terms of the scaled variables, using the expansion for the Dirichlet-Neumann operator given in (32), and keeping only terms of order :
As is an overall scale factor we can work with the rescaled Hamiltonian
In the case of constant depth =const ( , , and can be scaled out) the system becomes
This system is integrable, known as the Kaup - Boussinesq system. Indeed, the transformation eliminates the term. A further transformation with brings the system to the form
which has a scalar Lax pair representation (due to D.J. Kaup, ) with a spectral parameter :
Returning back to (36), it is noted that the leading order terms satisfy the system of equations
which shows that satisfies the wave equation for a wavespeed , that is the wavespeed is nearly constant, depending on the slowly varying variable . Thus, in the leading order, the initial disturbance propagates (to the right) with a slowly varying speed :
In the leading order approximation also
The observations from the leading order approximation indicate that the so-called slow variables, like the characteristic might be more adequate for the analysis of the model. Following Johnson [18, 20] we select a slow variable of the form
reminiscent of the far field variable . The function will be determined in what follows so that will be the characteristic for the right running wave. As a next step, the equations will be written in terms of the slow variables . The derivatives are related as follows:
The operator is hence
Note that the leading order approximation in the new variables from (36) is
giving or so we define
Hence we write the equations (36) in the new variables
We observe that all terms are of smaller orders and . Rescaling appropriately the variables to remove the scale parameters and and introducing the variable we write the equation in (36) in the form of a KdV equation with variable coefficients:
This is exactly the equation obtained by Johnson  via appropriate expansions from the governing equations of the fluid motion, (see also the derivation in ) so we can refer to it as Johnson’s equation.
In the propagation regime where clearly the -term in (46) has to be neglected and one obtains the dispersionless Burgers equation
This equation does not have globally smooth solutions, its solutions always form a vertical slope and break.
7 The Johnson equation and soliton propagation
and rescaling of transforms the equation to the form of a perturbed KdV equation
where the perturbation is on the right hand side with
The perturbed KdV equation can be treated within the framework of the inverse scattering approach for the KdV equation. The basics of this approach are outlined for example in [27, 23, 15]. If the initial condition is a pure soliton solution for the KdV equation, due to the perturbation, waves of radiation will appear and will decrease the energy of the initial soliton. The soliton perturbation theory however is unwieldy and in the case when the perturbation gives birth to new solitons is even more problematic. In our study, as we shall see below, new solitons are born when the initial soliton travels over an uneven bottom.
The model equation (47) can be written in terms of the local depth , that is (taking for simplicity ):
We observe that plays the role of the time-like variable in the usual KdV setting and - the space-like variable. In order to keep this analogy in what follows we replace with i.e. while is a space-like variable. This surprising outcome is due to the fact that the original variables are both of order (while combinations like the characteristics are of order 1) and at leading order both and can measure time, see also the explanation in . Thus we consider now the KdV with “time”-dependent coefficients
According to the classification of the KdV equations with variable coefficients, (51) does not appear to be transformable to the KdV equation for any choice of , see for example . We proceed by specifying the -dependent function
where and are appropriate constants. This function represents a ramp at between two constant values (two depths), describes the steepness of the ramp at . Thus, the initial condition will be taken at before the ramp is switched on at . The initial profile then is taken as the KdV 1-soliton when is constant: () and
where is a constant amplitude. For this of course is not the 1- soliton solution for the new depth, and we are interested in investigating the changes in the behaviour triggered at . With rescaling
(and appropriate rescaling of ) the KdV-type equation (50) can be written as a KdV in a canonical form
with initial condition taken as the one soliton solution for depth at
Let us introduce the constant
then the initial condition acquires the form
Recall that the initial condition for the one-soliton is therefore this initial condition can produce several solitons, waves of radiation, or, in any case can change the shape of the incoming soliton.
For the 1-soliton solution is of the form
Let us denote the constant
It is important to know how many solitons () will be born with the initial condition (55). Clearly, will be a function of the threshold . (In the case we have obviously and .)
where is a spectral parameter, which however is time-independent. Therefore it is sufficient to study the spectral problem at
where has been given in (55). Change of variables leads to the associated Legendre equation for
It has solutions on the interval if
where is an integer, the number of the discrete eigenvalues of the spectral problem and thus the number of the solitons. is another integer, labelling the negative discrete eigenvalues, . The solutions to this equation are called the associated Legendre polynomials . Therefore there are special depths (eigendepths) leading to the appearance of solitons for :
We study numerically the evolution of an incoming soliton
which enters a new depth at and and can transform into a multi-soliton solution for the new depth (if respectively ). To this end we consider the fully implicit finite-difference implementation of (51) complemented by an inner iteration with respect to the nonlinear term (for more details see, for example ). In Fig. 2, where , and above the value necessary for the emergence of the two soliton solution (Table 1) the birth of the second soliton (of a much smaller amplitude) is visible. The difference between and increases with and so does the amplitude and the velocity of the second soliton, e.g. when for the propagation from Fig. 3. The increase in the amplitude of the incoming soliton only increases the reflected waves which are waves of radiation. They are highly unstable and decay rapidly with , which can be seen from Fig. 4. Note that if the incoming soliton moves from shallow to deep region ( and ) then new solitons do not appear, due to (56). Then only reflected waves of radiation reduce the energy of the incoming soliton. This is illustrated in Fig. 5. Note that the dispersive radiation waves of small amplitude move to the left. This is because their phase velocity determined from is and becomes significant for the short waves where is not small. This effect is unphysical since the KdV model does not work as a water-wave model for short waves. The soliton velocity is positive as it can be seen e.g. from (57).
As expected from (56) and Table 1 the increase of the threshold (the increase of ) increases the number of the emerging solitons, Fig. 6. Although qualitatively the numerical results are in an agreement with the theory, the exact values of from (56) are not matched. The possible reason is that strictly speaking (56) is valid for a rapid jump from to at while our assumption is for slow (and smooth) bottom variations which we model via the profile (52).
The motion of the wave surface is determined by two functions – the Hamiltonian variables and the potential on the surface . However the fluid motion in the entire domain of the fluid can only be recovered from the entire boundary of which includes the bottom. An expression for the continuation of the potential from the surface into the bulk of the fluid is provided in  in terms of the Dirichlet-Neumann operator. This at least formally determines the dynamic of the fluid in .
There are of course many other possibilities for the nature of the bottom variation, e.g. random topography studied in [2, 12], as well as for the propagation regimes which will be studied in forthcoming publications. The wave dynamics in the presence of shear currents (vorticity) with a variable bottom is another very important and interesting possibility for future research.
AC and RI acknowledge funding from the Erwin Schrödinger International Institute for Mathematics and Physics (ESI), Vienna (Austria) as participants in the Research in Teams Project Hamiltonian approach to modelling geophysical waves and currents with impact on natural hazards, where a big part of this work has been done. AC is funded by a Fiosraigh fellowship at Dublin Institute of Technology (Ireland). MT acknowledges financial support from the Bulgarian Science Fund under grant DFNI I-02/9.
The authors are thankful to Prof. Adrian Constantin, Prof. Robin Johnson, Dr Calin I. Martin and Prof. André Nachbin for many valuable discussions. The authors are also thankful to two anonymous referees for their very constructive remarks and suggestions.
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