Abstract
We establish existence and uniqueness of solutions to the Cauchy problem associated with a new one-dimensional weakly-nonlinear, weakly-dispersive system which arises as an asymptotical approximation of the full potential theory equations for modelling propagation of small amplitude water waves on the surface of a shallow channel with variable depth, taking into account the effect of surface tension. Furthermore, numerical schemes of spectral type are introduced for approximating the evolution in time of solutions of this system and its travelling wave solutions, in both the periodic and nonperiodic case.
1. Introduction
In this paper we study the propagation of water waves on the surface of a shallow channel with variable depth, considering the effect of surface tension. To describe this phenomenon we will derive a new water wave model from Euler’s equations (in dimensionless variables) for an inviscid, incompressible liquid bounded above by a free surface and bounded below by an impermeable bottom topography [1]:with the nonlinear free surface conditionsat . Here denotes the potential velocity and the wave elevation measured with respect to the undisturbed free surface . The dimensionless parameters and are small positive real numbers which measure the strength of nonlinear and dispersive effects, respectively. The parameter measures the ratio inhomogeneities/wavelength and the parameter is associated with the surface tension. The Neumann condition at the impermeable bottom isThe bottom topography is described by , where We will see in Section 3 that system (1)–(3) is equivalent to leading order provided that , to the weakly-nonlinear, weakly-dispersive systemThe variables and are the space and time coordinates, respectively, with denoting the elevation of the free surface, and represents the potential fluid velocity measured at the channel’s bottom. The metric coefficient is related to the channel’s bottom and it is defined aswhere is the coordinate transformation used in the derivation of system (5) to map the original physical channel with variable depth onto a strip in the complex plane. This strategy of change of variable was introduced by Hamilton [2] and later used successfully by Nachbin in [3] to study wave propagation over a channel with a highly variable topography. Observe from (6) that the coefficient is infinitely differentiable although the function describing the channel’s bottom is not smooth. In case of constant depth, the coefficient is identically one and system (5) reduces to a system derived by Quintero and Montes in [4].
Wave-topography interaction has been the subject of considerable mathematical research [5–18]. The physical applications range from coastal surface waves [19] to atmospheric flows over mountain ranges [20, 21]. In particular, the interaction of waves with fine features of the topography is of great interest. As pointed out in the introduction to the Orography proceedings [21] of the European Centre for Medium-Range Weather Forecasts (ECMWF), “the representation … of subgrid-scale orographic processes is recognized as crucial to numerical weather prediction at all time ranges.” In the atmospheric literature, orography implies mountain ranges [20].
In previous works, some weakly-nonlinear, weakly-dispersive models have been developed to describe the interaction of a long pulse with small amplitude that propagates on the surface of a channel with a variable bottom [2, 3, 5, 7–9, 22–24]. However, these models either neglect the effect of surface tension on the free surface where the wave propagates or are not applicable to bottoms described by a discontinuous/nondifferentiable function or the wave elevation is removed in their physical derivation. For instance, Milewski [23] derived a bidirectional scalar Benney-Luke type equation in terms of the potential velocity which includes the surface tension effect and the influence of a variable bottom. However, the asymptotical derivation of this model implies eliminating the wave elevation and consequently neglecting several second-order terms in the parameters and .
Some of the features of new formulation (5) are the following:(i)It can be applied to study wave propagation over a shallow channel with a discontinuous or nondifferentiable bottom, provided that the bottom’s fluctuations satisfy . This is a consequence of introducing the conformal mapping mentioned above. Observe that all coefficients of the reduced equations (5) in the new coordinate system result in being infinitely differentiable since they depend on the smooth function .(ii)One-dimensional system (5) models bidirectional waves and it incorporates the simultaneous effects of surface tension and variable depth upon the shape of a water wave that propagates on the surface of an irregular shallow channel.(iii)Furthermore, in the derivation of system (5), we do not eliminate the wave elevation (which prevents us from neglecting additional terms of order ), as done, for example, in [23]. Therefore, the new formulation (5) is expected to be a more accurate approximation of the full potential theory equations.
In the present paper, we establish existence and uniqueness of a solution to Cauchy problem (5) using classical semigroup theory and Banach’s fixed point principle. In [4] the well-posedness of system (5) was analyzed but only in the case that . In second place, we formulate a Galerkin-spectral numerical scheme of spectral accuracy in space to approximate the solutions of system (5) based on the Fourier basis and using an implicit-explicit (IMEX) second-order strategy for time stepping. This type of temporal discretization is described, for instance, in [25], and it has been used in conjunction with spectral methods [26, 27] for the time integration of spatially discretized PDEs of diffusion-convection type. IMEX schemes have also been successfully applied to the incompressible Navier-Stokes equations [28] and in environmental modelling studies [29]. On the other hand, we develop a numerical solver to compute travelling wave solutions of system (5) by using a Fourier-collocation strategy combined with a Newton-type iterative procedure. We also indicate how to determine appropriate starting points for this iterative process in order to achieve convergence. Existence results on travelling wave solutions (in the periodic and nonperiodic case) of system (5) with have also been established in [4]. Travelling wave solutions exist as a consequence of a balance between nonlinear and dispersive effects present in a system; these waves travel with a constant speed without any temporal evolution in shape or size when the frame of reference moves with the same speed of the wave. In the last decades, the study of travelling waves has grown enormously because they appear in several and varied fields of application, such as fluid mechanics, optics, acoustics, oceanography, and astronomy. Thus, to determine existence and properties of such type of solutions is a fundamental problem in the theory of ordinary and partial differential equations of great interest for both pure and applied mathematicians.
The rest of this paper is organized as follows. In Section 2, we introduce the functional spaces and notation to be employed in the paper. In Section 3, we present in detail the derivation of model (5) starting from the potential theory equations including the surface tension effect. In Section 4, we discuss existence and uniqueness of a solution of the Cauchy problem associated with system (5) by using Banach’s fixed point principle and semigroup theory. In Section 5, we introduce the numerical schemes for approximating the evolution of a solution of system (5) and computing their travelling wave solutions (periodic and nonperiodic cases). Section 6 presents a set of numerical simulations to check the accuracy of the numerical schemes developed in the paper. Furthermore, some numerical experiments are included to illustrate the interaction between the wave-topography effects and surface tension. Finally in Section 7 are the conclusions of the paper.
2. Preliminaries
To analyze existence of solutions of problem (5), we will use the standard notation. For , we will denote by (or simply ) the Banach space of measurable functions in such that if and , if . We define the norm in for byand in by . is a Hilbert space for the scalar productWe set . For function , the Fourier transform is defined asand the inverse Fourier transform is defined byWe will also denote by (or ) and (or ) the extensions of these operators to . The convolution of two functions, , is defined asWe recall that . For , we define the Sobolev space (sometimes written for simplicity as ) as the completion of the Schwartz space (rapidly decaying functions) defined as with respect to the normFor simplicity, we also denote this norm by . The inner product in is defined asThe product norm in the space is defined by for . Sometimes, we will also use the equivalent product normFor , we will denote by the space of continuous functions , that is, the space of continuous functions , , with the supremum norm and the product normfor .
3. Governing Equations
We start by presenting the potential theory formulation for Euler’s equations (in dimensionless variables) for an inviscid, incompressible liquid bounded above by a free surface and bounded below by an impermeable bottom topography and including the effect of surface tension [1]:with the nonlinear free surface conditionsat . Here denotes the potential velocity and the wave elevation measured with respect to the undisturbed free surface . The dimensionless parameters and measure the strength of nonlinear and dispersive effects, respectively. The parameter measures the ratio inhomogeneities/wavelength and is related to the surface tension effects. The Neumann condition at the impermeable bottom isThe bottom topography is described by , whereThe bottom profile is described by the (possibly rapidly varying) function . The topography is rapidly varying when . The scale represents the total length of the irregular section of the coast. The undisturbed depth is given by and the topography can be of large amplitude provided that . The fluctuations are not assumed to be small, nor continuous, nor slowly varying.
Let us consider a symmetric flow domain by reflecting the original one about the undisturbed free surface (cf. Figure 1). This domain is denoted by where and can be considered as the conformal image of the strip where with . Then with and , a pair of harmonic functions on . Following the strategy suggested by Hamilton in [2] and Nachbin [3] within the weakly-nonlinear, weakly-dispersive regime (, ) and using the relationships whereand the variable free surface coefficient defined asthe potential theory equations can be approximated to order in the orthogonal curvilinear coordinates with by the equationwith conditions at the free surface ,and condition at the channel’s bottomObserve that the change of variables lets the origin of the curvilinear coordinate system at the bottom. The Jacobian for the coordinate transformation is represented by , and corresponds to the position of the free surface in the curvilinear coordinate system. By performing an asymptotic simplification as in [1] (page 464) through a power series expansion in terms of the dispersion parameter near the bottom of the channel in the forminto (26)–(28) for being small, we find that free surface conditions (27) can be approximated to order , by the equationsHere denotes the potential fluid velocity at the channel’s bottom . We point out that (30) and (31) with (no surface tension) correspond to those derived in [3] (bottom of page 915 and top of page 916). In the derivation of (30)-(31), we used the relationshipwhich means that, at leading order, the Jacobian of the conformal coordinate transformation is time independent.
Let us introduce the new variable . Thus observe that from (30) This relationship allows us to change the form of dispersive terms in the equations above. In particular, by using the decomposition in (30), we obtain that system (30)-(31) can be approximated to order by the following equations:We point out that the system above is applicable to modelling of propagation of water waves over an arbitrary rapidly varying depth. In case of a slowly varying channel’s topography and with being a stationary random process with standard deviation and correlation length of order one, the metric coefficient can be expanded as (see [9])Thusand system (35) can be approximated to order byNote that the coefficient is smooth even when the function describing the bottom is discontinuous or nondifferentiable. The function is time independent and becomes identically one for a channel with constant depth. In this case, system (38)-(39) reduces to that studied in [4]. Moreover, in applications, this coefficient is bounded and . These properties will be important to obtain existence and uniqueness results for system (38)-(39). We also point out that the function actually depends on the dispersion parameter . For this reason, it will be denoted by whenever we need to emphasize this dependence.
4. Existence and Uniqueness
System (38)-(39) can be written aswhereSystem (40) is supplemented with the initial conditionsTaking Fourier transform in the spatial variable in system (40), we havewhere
4.1. Analysis of the Linear Semigroup
Consider system (40) with ,which has unique solution in the formwhereBy using the inverse Fourier transform in (46), we have thatwherewith
Theorem 1. The family of linear operators is a -semigroup in . Furthermore is its infinitesimal generator in .
Proof. Let . ThenHereafter will denote a generic constant and we recall that . Therefore , , is a family of continuous linear operators in . On the other hand, it is easy to see that , , . Finally Observe that , as and , , as . By using Lebesgue’s dominated convergence theorem, we can conclude thatas . We conclude that , , is a strongly continuous semigroup in .
On the other hand, To see that is the infinitesimal generator of the semigroup , , observe that, for , But by virtue of and using again Lebesgue’s dominated convergence theorem, we get that
4.2. Analysis of the Nonlinear Term
Theorem 2. Let . The application maps on itself andFurthermore,where is a constant and , .
Proof. In the first place, let : Taking into account the inequalities valid for (applying Corollary in [31]), and using the fact that the function is bounded, we get that On the other hand, for , we have that
4.3. Local Existence and Uniqueness
Theorem 3. Let and . Then there exists and unique , which satisfies the integral equation
Proof. Let be fixed constants and consider the nonlinear operatordefined in the complete metric spaceLet us prove that if , then . Indeed if , then We remark that if , thenGiven that is a -semigroup, then as . To analyze the second expression, let us suppose that . We obtain that Observe that since as , , and using Lebesgue’s dominated convergence theorem, we get that as .
On the other hand, due to Theorems 1 and 2, we arrive at Therefore we get that as . This means that .
Let us prove existence of small time such that . Let . For any , for being small enough. Therefore .
Finally, let us show that time exists so that the operator