A HAMILTONIAN APPROACH FOR NONLINEAR ROTATIONAL CAPILLARY-GRAVITY WATER WAVES IN STRATIFIED FLOWS

. Under consideration here are two-dimensional rotational stratiﬁed water ﬂows driven by gravity and surface tension, bounded below by a rigid ﬂat bed and above by a free surface. The distribution of vorticity and of density is piecewise constant– with a jump across the interface separating the ﬂuid of bigger density from the lighter ﬂuid adjacent to the free surface. The main result is that the governing equations for the two-layered rotational stratiﬁed ﬂows, as described above, admit a Hamiltonian formulation.

Equatorial flows present several specific features. Apart from a strong stratification (with an interface, called thermocline, separating two layers of constant density), one is also confronted with the presence of strong depth-dependent underlying currents, cf. [23], with flow reversal at a depth of about 100 − 200 m; in the Pacific these currents are realistically modelled by two-dimensional flows with piecewise constant vorticity (see the discussion in [6]). The vorticity is a characteristic of wave-current interactions and captures the swirling motion beneath the surface wave, cf. [9,33,34].
Another key-feature of most water flows is their nonlinear character. While the linear wave theory predicts sinusoidal wave profiles, one needs a nonlinear setting to explain observed wave trains that are almost flat near the trough and exhibit a pronounced elevation near the crest.
The substantial difficulties caused by the presence of (piecewise constant) vorticity, by the fluid stratification and by the nonlinearities occurring in the governing equations can be mitigated if, for instance, a concise formulation that underlines the structural properties of the governing equations becomes available. A first step in this direction was made by Zakharov [38], who showed that the governing equations of two-dimensional irrotational gravity water waves over a flow of infinite depth possess a Hamiltonian structure with the canonical variables given by the surface elevation and the velocity potential evaluated at the free surface. A nearly-Hamiltonian formulation for two-dimensional periodic water waves with constant vorticity over flows with finite depth was achieved in [8]. The extent by which the nearly-Hamiltonian formulation in [8] fails to be Hamiltonian is given by the vorticity, the latter hindrance being dealt with in [35] where it was proved that the formulation in [8] is in fact Hamiltonian with respect to some non-canonical symplectic structure.
While a Hamiltonian formulation for water waves in stratified irrotational flows was achieved in [14], the corresponding formulation for stratified gravity flows with a piecewise constant vorticity distribution, thus allowing for linearly sheared currents, is very recent, cf. [12,13].
These underlying currents are generated by the prevalent regular wind patterns (see the discussion in [6,24,25,31]). Irregular wind bursts generate capillary waves that interact with these currents, and in their modelling it is reasonable to consider the inviscid setting since Reynolds numbers in geophysical fluid dynamics are very large, cf. [28].
While the Hamiltonian formulation for water waves over stratified flows driven only by capillarity and with piecewise constant vorticity was achieved in [26], we aim to show here that the nonlinear governing equations of water waves driven by gravity and surface tension over stratified flows with piecewise constant distribution of vorticity also have a Hamiltonian formulation. Note that wave-current interactions give rise to the possible appearance of critical layers, that is, surfaces where the wave speed equals the mean flow speed, see [10,11,21,27,29,36] for a comprehensive account on critical layers in the context of a homogeneous fluid, and [6] for a discussion about their relevance for equatorial flows.
The Hamiltonian formulation is important in the study of the complicated equatorial ocean dynamics. We intend to pursue this direction in a future work, where we plan to include geophysical effects stemming from the rotation of the Earth, combined with the effects of gravity and surface tension, stratification and the presence of depth-dependent current fields.
2. The governing equations. Under consideration here is a two-dimensional periodic water flow, acted upon by gravity and surface tension. The water domain is bounded below by the rigid flat bed y = −h, (with h > 0), and above by the free surface y = η 1 (x, t) + h 1 , which is a perturbation of the flat free surface y = h 1 . The free surface wave propagates in the positive x-direction, while the y axis points vertically upwards. Here, h 1 > 0 is a constant, t stands for time and x → η 1 (x, t) is a periodic function in the spatial variable x, of principal period L, and which has mean zero, that is, The stratification of the fluid is as follows: we assume that, neighboring the flat bed y = −h, the water domain consists of a layer of constant density ρ, separated by the interface y = η(x, t) from the free-surface adjacent layer x ∈ R, t ∈ R, η(x, t) < y < h 1 + η 1 (x, t)}, of constant density ρ 1 < ρ. The interface x → η(x, t) is, at any fixed time t, an L periodic function with zero mean. Remark 1. The bold text notation used in due course of the paper refers to quantities that are defined in both layers Ω * and Ω * 1 . Denoting with (u(x, y, t), v(x, y, t)) the velocity field and with P = P (x, y, t) the pressure, the equations of motion are Euler's equations supplemented by the equation of mass conservation In order to include underlying currents in our discussion and to treat wave-current interactions (see [9]) we need to allow the presence of vorticity in the flow. The vorticity is defined through and measures local rotation. In our setting the vorticity is constant throughout each layer, but discontinuous across the interface. Therefore, it takes the form To specify the water wave problem we impose appropriate boundary conditions as follows: at the free surface, the dynamic boundary condition incorporates surface tension effects, reads as (with P atm being the constant atmospheric pressure at the surface of the ocean and σ 1 the coefficient of surface tension at the free surface) and decouples the motion of the water from that of the air above. The kinematic boundary conditions, which refer to the flat bed y = −h, the interface y = η(x, t) and the free surface y = h 1 + η 1 (x, t), ensure that a particle once on one of the three boundaries will remain confined to it; they read as with (u, v) and (u 1 , v 1 ) being the velocity fields in Ω * 1 and Ω * , respectively. The balance of forces at the interface y = η(x, t) is expressed by the continuity of the pressure along this internal boundary, that is, 2.1. Mathematical reformulation of the physical problem. We intend to write the velocity field with the help of the stream function and of the (generalized) velocity potential, to be defined in due course. The latter quantities will be then instrumental in bringing the physical problem into a mathematical form that can be handled easier. and 2.1.1. The generalized velocity potential. Due to (3) we are able to introduce in each layer a (generalized) velocity potential, denoted ϕ in Ω and ϕ 1 in Ω 1 , that satisfies u = ϕ x + γy and v = ϕ y , in Ω, u 1 = ϕ 1,x + γ 1 y and v 1 = ϕ 1,y , in Ω 1 .
Up to functions that depend only on time, the generalized velocity potentials are given by while, for (x, y) ∈ Ω 1 , we have It follows from (10) that satisfies κ (t) = 0 for all t, as can be seen from the first equation in (1), (7) and the periodicity of the velocity field and of the pressure P . As a consequence, the function (x, y) → ϕ(x, y, t) − κx is periodic in the x-variable, of period L.
The quantity κ is also related to the current underlying the interface, denoted by U (y, t) and defined at the level y below the trough of the internal wave y = η(x, t) by The above mentioned relation reads as and it is a consequence of the definition of γ and the periodicity of v. An analogous discussion can be performed for ϕ 1 , the potential associated to the upper layer Ω 1 , where the underlying current at level y, above the crest of the internal wave y = η(x, t) and below the trough of the surface wave y = h 1 + η 1 (x, t), is defined by We set As with κ, we see that κ 1 is also independent of t, as it was showed in [13]. We also have that As a consequence, the function (x, Motivated by the discussions in the previous remarks and in view of (9), we may write u =φ x + γy + κ and v =φ y , in Ω, where the functionsφ(x, y) Thus, equation (18) represents a splitting of the velocity field into an underlying steady current component and a periodic harmonic wave velocity field. The kinematic boundary conditions (5) and (6) can now be written as and respectively; the subscript s 1 stands for traces on the free surface y = h 1 + η 1 (x, t). (2) we conclude the existence of two stream functions denoted ψ in Ω, and ψ 1 in Ω 1 , satisfying

The stream function. From the equation of mass conservation
In fact, from above and using also (7) we see that the expressions of the stream functions are for some smooth functions ψ − , ψ + , that depend only on time.
Equalities (6) and (21) show that the stream functions ψ and ψ 1 differ only by a function of time at the interface y = η(x, t). Therefore, we can choose ψ − and ψ + above such that there exists ψ ∈ C(Ω ∪ Ω 1 ) with ψ = ψ in Ω and ψ = ψ 1 in Ω 1 . The latter remark justifies the notation Setting also we are now able to rewrite the kinematic boundary conditions (5) and (6) in a more concise form. Namely, they can be written as formulas that will be useful later on.

CALIN IULIAN MARTIN
Note also that, cf. [13], ψ is periodic of period L in the x-variable. The stream functions are related to the vorticity by the following relations The generalized velocity potentials and the stream functions are beneficial in recasting Euler's equations as and thereforeφ for some functions f and f 1 . Using (4) and (29), we have that on With the help of the function χ 1 introduced in (24) we can express (30) as (32) Moreover, utilizing the function χ from (23) we are able to reformulate (8) as provided Remark 2. The pressure in the fluid can be recovered by means of the stream functions ψ, ψ 1 , of the perturbed velocity potentials ϕ, ϕ 1 and of η and η 1 by setting 3. The Hamiltonian by means of the Dirichlet-Neumann operator. The main object of study in this section is the Hamiltonian functional given by the total energy of the flow by means of the formula where the first term in the double integral represents the kinetic energy (energy of motion), ρgy is the gravitational potential energy (energy of position), while the second integral above stands for the free energy of the surface.
In due course we will be concerned with proving the existence of a density function h which depends solely on the scalar variables ξ, ξ 1 , η, η 1 , on the trace of the (generalized) velocity potentialφ 1 on the free surface, on the combined action of the two (generalized) potentialsφ andφ 1 on the interface, as well as on the spatial derivatives of the mentioned quantities and which has the property that Formula (35) will be instrumental in proving the nearly-Hamiltonian formulation of the governing equations in Theorem 4.1.
Owing to the stratification of the fluid, the functional H equals which, by (18), can be rewritten as We proceed now in proving the claim made in (35). In doing so, we will make use of the Dirichlet-Neumann operators associated to the lower layer and to the upper layer, respectively. For computational aspects pertaining to the Dirichlet-Neumann operators related to water waves we refer the reader to the recent survey paper [37] and to [30]. Given smooth, L-periodic scalar functions Φ and η such that η(x) > −h for all x ∈ [0, L], we denote withφ the unique L-periodic in the x variable smooth solution of the boundary value problem where Ω * (η) = {(x, y) : x ∈ R, −h < y < η(x)}. Denoting with n the outward pointing normal vector along the upper boundary y = η(x) of the domain Ω, the Dirichlet-Neumann operator G = G(η) associated to Ω * = Ω * (η) is defined by mapping the Dirichlet data Φ to the normal derivative of the solution on the upper boundary, To define the Dirichlet-Neumann operator associated to the upper layer, we consider L-periodic functions η, η 1 , Φ 1 , Φ 2 , such that η(x) < h 1 + η 1 (x) for all x ∈ [0, L]. We then setφ 1 to be the unique L-periodic in the x variable solution of the Dirichlet boundary value problem    ∆φ 1 = 0 in Ω * 1 (η, η 1 ), where Ω * 1 (η, where n 1 denotes the outward unit normal vector along the upper boundary y = h 1 + η 1 (x) and n has the same meaning as in the definition of G. The entries of the matrix operator G 1 (η, η 1 ) are denoted by For an account on the properties of the Dirichlet-Neumann operators we refer the reader to [10,14,22]. We will now specify the functions Φ, Φ 1 , and Φ 2 to represent the restriction of the (generalized) velocity potentialsφ andφ 1 to the free surface and to the interface, respectively. More precisely we set To capture the effect of the two (generalized) velocity potentialsφ andφ 1 along the interface y = η(x, t) we set We will also need to introduce the variable From the definition of the Dirichlet-Neumann operators and (20) we have that and Moreover, from (19) we have Adding up the relations (46) and (47), we obtain Denoting B = B(η, η 1 ) := ρ 1 G + ρG 11 and making use of (44), (45) and (49) we can express Φ, Φ 1 , Φ 2 in terms of ξ and ξ 1 as follows With the help of the Dirichlet-Neumann operators, using Green's second identity and (46)-(48) we obtain, Using now the periodicity ofφ,φ 1 , η, η 1 we obtain where, to pass to the second equality, we have used the rule for differentiable functions f 1 , f 2 and F . Similarly, as for the calculation of K 2 , we obtain Summing up the relations for K 1 , K 2 and K 3 , we get (57) With the notation taking into account (50)-(52), and in view of the equalities we obtain that where, in the second equality, we have used the definition of B to derive that and the last equality is obtained by employing again the definition of the operator B, and using that the operators B −1 , G 11 are self-adjoint, while G * 12 = G 21 , cf. [14].
The proof of the claim about the functional dependence of the total energy H made in (35) emerges now by summarizing the above computations in the formula 4. Re-formulation of the governing equations and of their boundary conditions. The purpose of this section is to show that the governing equations for water waves, given in the Section 2, may be reduced to a Hamiltonian system involving functions of one variable. The latter are the free surface η 1 , the interface η and the evaluations of the perturbed velocity potentialsφ andφ 1 .

4.1.
The nearly-Hamiltonian formulation. We first compute the variations of the Hamiltonian functional with respect to η, η 1 , ξ, and ξ 1 , respectively. To this end, we recall that the functions Φ, Φ 1 and Φ 2 , which will appear below, have the same meaning as in (43). Moreover, ξ and ξ 1 are defined in (44) and (45). In order to compute the variations of H we need a couple of formulas concerning harmonic functions and variational calculus, which we state in the next remark.
Throughout this section we rely on formula (37). For computing the variation of the first term in (37), we use rule (63), apply the divergence theorem in the domain D := {(x, y) : 0 < x < L, 0 < y < η(x)} (with n being the outward unit normal to its boundary) and owing to the periodicity ofφ obtain where, for the last equality above, we used Thus, from (62) and since the flat bed y = −h is a fixed boundary, we have Analogously, we have To compute the variations of the other terms in (37) we utilize (7), (62), (55), (65), the periodicity of η, η 1 ,φ andφ 1 , and get as well as Moreover, and Note also that, employing the periodicity of η 1 and the chain rule, we have where, in the second row of the above equality, δ stands for the Dirac point mass.
Having in mind the definitions of ξ and ξ 1 from (44)-(45) we can state the following preliminary result.
Proof. Throughout the proof we will use the variational formulas and as well as the property established in (35). We start by computing the variations of the functional H with respect to the variable η.