Models for damped water waves

In this paper we derive some new weakly nonlinear asymptotic models describing viscous waves in deep water with or without surface tension effects. These asymptotic models take into account several different dissipative effects and are obtained from the free boundary problems formulated in the works of Dias, Dyachenko and Zakharov (Physics Letters A, 2008), Jiang, Ting, Perlin and Schultz (Journal of Fluid Mechanics,1996) and Wu, Liu and Yue (Journal of Fluid Mechanics, 2006).


Introduction
The motion of a free boundary irrotational and incompressible flow is a classical research topic [35]. In most applications, the flow is also assumed to be inviscid [7,22]. However, even if in most situations in coastal engineering the assumption of inviscid flow leads to very accurate results, there are other physical scenarios where the viscosity needs to be taken into account. Moreover, there are many situations in which the viscosity is very large and the vorticity is small and its effect negligible. Actually, certain discrepancies between experiments and inviscid theory have been previously reported in the literature. For instance, Wu [39] found that From this comparison the theory appears quite satisfactory in predicting the wave phases during the inward focusing and the subsequent reflection within a radial distance as far as r = 20, while the peak amplitudes observed in the experiments are slightly smaller than those predicted by the theory. This discrepancy can be ascribed to the neglect of the viscous effects in the theory and to the approximation that the initial wave generated in the tank was not cylindrical in shape and departed slightly from a perfect solitary wave profile in the experiment.
In addition to this, Zabusky and Galvin [40] wrote A laboratory-data/numerical-solution comparison of the number of crests and troughs and their phases (or relative locations within a period) shows only negligible difference. As one expects, the crest-to-trough amplitudes differ somewhat more because they are more sensitive to dissipative forces. To quantify some of the details we recommend a study including dissipation. and, furthermore, Longuet-Higgins [24] stated that For certain applications, however, viscous damping of the waves is important, and it would be highly convenient to have equations and boundary conditions of comparable simplicity as for undamped waves.
The purpose of this paper is to derive new weakly nonlinear asymptotic models (in the spirit of [6,13,14,[25][26][27]31]) describing damped water waves and, at the same time, keeping the features of potential flows. We observe that, at fist sight, the idea of viscous damping of potential flows is somehow paradoxical since the hypothesis of irrotational velocity implies that the viscous term in the Navier-Stokes equations vanishes. The equations describing the motion of a irrotational, incompressible, inviscid and homogeneous fluid with a free surface are [41] where are the the region occupied by the fluid and the surface wave, respectively. We write the unit normal to the surface wave, 2Lπ to denote the characteristic wavelength of the surface wave, φ for the scalar potential of the flow, i.e. the velocity field u satisfies u = ∇φ, ρ is the density of the fluid, γ for the surface tension coefficient and is the curvature of the surface wave.
The first attempts to include viscosity effects go back as far as to the works of Boussinesq [5] and Lamb [21]. Later on, Ruvinsky & Freidman [34] formulated a system of equations for weakly damped surfaces waves in deep water and used this system to compute capillary-gravity ripples riding on the forward face of steep gravity waves (see also [33]). Then, these first results were generalized by Ruvinsky, Feldstein & Freidman [32] and the following system is proposed where v and µ denote the vertical component of the vortex part of fluid velocity and the dynamic viscosity. Equation (4) was also studied by Khariff, Skandrani & Poitevin [20]. Using a clever change of variables, Longuet-Higgins [24] simplified the previous system and obtained A similar model was also studied by Jiang, Ting, Perlin & Schultz [16] and Wu, Liu & Yue [38], namely where the dissipative terms are chosen as Another similar model where the dissipation acts only on the velocity is the one by Joseph & Wang [17, Equation (6.7) and (6.8)] (see also Wang & Joseph [37]). We would like to remark that, in the models of damped water waves mentioned so far, there are no dissipative effects acting on the free surface.
In a more recent paper, Dias, Dyachenko & Zakharov [9] proposed a system where the free surface experiments dissipative effects. In particular, based on the linear problem, these authors derived as a model of viscid water waves. This model was also considered by several other authors. Dutykh & Dias [12] obtain a new set of viscous potential free-surface flow equations in the spirit of (8) taking into account the effects of the bottom topography. These authors also derived a long wave approximation. This approximate model takes the form of a nonlocal (in time) Boussinesq system (see also [10,11]). Kakleas & Nicholls [18], using the analytic dependence of the Dirichlet-Neumann operator, derived a system of two equations modelling (8). These equations are the viscid analog of the classical Craig-Sulem WW2 model and were mathematically studied by Ambrose, Bona & Nicholls [3]. The well-posedness of the full (8) was studied very recently by Ngom & Nicholls [30]. In particular these authors proved global existence of solutions starting from a small enough initial data for the case of non-vanishing surface tension γ = 0. Some other related results are those by Kharif [38] in Section 3. After neglecting errors of O(ε 2 ) we find the nonlocal wave equation modelling the case s = 0. After that we consider the case s = 2 and, following a similar approach, find the nonlocal wave equation for the model of Dias, Dyachenko, and Zakharov [9]. Finally, we conclude with a parabolic system of Craig-Sulem flavour in section 5.

Notation. Let
A be a matrix, and b be a column vector. Then, we write A i j for the component of A, located on row i and column j. We will use the Einstein summation convention for expressions with indexes.
We write for the space derivative in the j−th direction and for a time derivative, respectively.
Let f (x 1 ) denote a L 2 function on S 1 (as usually, identified with the interval [−π, π] with periodic boundary conditions). We define the Hilbert transform H and the Dirichlet-to-Neumann operator Λ and its powers, respectively, using Fourier series wheref In particular, for zero-mean functions, we note that These last equalities will be used extensively through the whole text. Finally, we define the commutator as

Damped water waves
2.1. The equations in the Eulerian formulation. We consider system the system where δ i ≥ 0 are constant, the dissipative terms are as in (7), φ is the scalar potential (units of length 2 /time), h denotes the surface wave (units of length) and G (units of length/time 2 ) is the gravity acceleration. The constant δ 1 has units of mass/(length 2 · time) (when D 0 φ = φ) and of mass/time (when D 2 φ = ∂ 2 2 φ) while δ 2 has units of length 2 /time. We observe that, for appropriate choice of δ i and s we recover (exactly) (6) and (8). Indeed, if δ 2 = 0 we obtain the same model by Jiang, Ting, Perlin & Schultz [16] (s = 0) and Wu, Liu & Yue [38] (s = 0 and s = 2), while if δ 2 = δ 1 /ρ and s = 2 we recover the model by Dias, Dyachenko & Zakharov [9]. Following the pioneer work of Zakharov [41], we use the trace of the velocity potential ξ(t, x) = φ(t, x, h(t, x)) (units of length 2 /time). Now we observe that Thus, (10) can be written as The system (11) is supplemented with an initial condition for h and ξ: 2.2. Nondimensional Eulerian formulation. We denote by H and L the typical amplitude and wavelength of the water wave. We change to dimensionless variables (denoted with·) and unknowns with the non-dimensionalized fluid domain We find the following dimensionless parameters: where The first parameter is known as the steepness parameter and measures the ratio between the amplitude and the wavelength of the wave. The α ′ s consider the ratio between gravity and viscosity forces. Finally, the fourth one is the Bond number that compares gravity forces with capillary forces. Dropping the tildes for the sake of clarity, we have the following dimensionless form of the damped water waves problem where we have used the nondimensional parameters (16).

The equations in the Arbitrary Lagrangian-Eulerian formulation.
In the present section we want to express system (17) on the reference domain Ω and reference interface Γ The easiest way to do so is, supposing that h is regular, defining the following family (parametrized in t ∈ [0, T ]) of diffeomorphisms t)) .
We compute With such map we can define the push-back of any application θ defined on Ω (t) simply as Θ = θ • ψ, whence in particular we define We let N = e 2 denote the outward unit normal to Ω at Γ. We also recall that, if Θ = θ • ψ, the following formula holds where Einstein convention is used. Then, we can rewrite (17) as the following system of variable coefficients nonlinear PDEs posed on a fixed reference domain where the operator D s is Next we explicit the values of the A i j 's in the above system (see (19)) obtaining hence where 3. The asymptotic model for damped water waves when s = 0 In this section we consider the case s = 0 (the model by Jiang, Ting, Perlin & Schultz [16] and Wu, Liu & Yue [38]). In this case we have that We introduce the following ansatz: With this ansatz we can re-profile the nonlinear system (21) in an equivalent sequence of linear systems where the evolution of the n-th profile is determined by the evolution of the preceding n − 1 profiles. We are interested in a model approximating (21) with an error O(ε 2 ). Using that we obtain that For the case n = 0, we have that Recalling that we find that (23d) can be equivalently written as We note that (23d) can be equivalently wwritten as thus, Similarly, in the case n = 1, we find that , We now use Lemma A.1 in order to compute We want to provide an explicit expression for the term 0 −∞b (k, y 2 , t) e |k|y 2 dy 2 considering the form of b. We compute that Thus, we find that The evolution equations for h (1) and ξ (1) become hence Using the above equation for h (1) t (28) we can express ξ (1) as a function of h (0) , ξ (0) and h (1) as follows Time differentiating (28) and inserting (29), we deduce Recalling the definition of the Riesz potential Λ −1 and using (23c) and (24) in order to express ξ (0) and ξ (0) t in terms of h (0) , we find that Using Tricomi identity the previous equation can be further simplified and we find that We can express α 0 1 Φ (0) in terms of h (0) as follows and, inserting the previous formula into (30), we find that
4. The asymptotic model for damped water waves when s = 2 In this section we focus on the case s = 2 (the model by Dias,Dyachenko,and Zakharov [9]). In this case we have that D 2 Φ = ∂ 2 2 Φ. We use the ansatz (22) and follow the previous steps. The first term in the series solves Taking a time derivative of the equation (42d), using the fact that and substituting (42c), we find that Similarly, due to the fact that we find that the previous equation for h tt can be written as Analogously as in (26), for n = 1, we find that We use Lemma A.1 and (27) to find that Equation (59) is an asymptotic model of the damped water waves system proposed by Jiang, Ting, Perlin & Schultz [16] and Wu, Liu & Yue [38], while equation (60) is an asymptotic model of the water waves with viscosity system proposed by Dias, Dyachenko, and Zakharov [9].
Appendix A. The explicit solution of an elliptic problem Lemma A.1. Let us consider the Poisson equation where we assume that the forcing b and the boundary data g are smooth and decay sufficiently fast at infinity. Then, the unique solution u of (61) is given by where the operator· denotes the Fourier transform in the variable x 1 . In particular Proof. Let us apply the Fourier transform to the equation (61), this transforms the PDE (61) in the following series of second-order inhomogeneous costant coefficients ODE's The generic solution of (65) can be deduced using the variation of parameters method, whencê The boundary conditions determine the values of the C i 's: (k, y 2 ) e |k|y 2 dy 2 , C 1 (k) = −C 2 (k) +ĝ (k) .