Wave-current interaction on a free surface

The classic evolution equations for potential flow on the free surface of a fluid flow are not closed and the wave dynamics do not cause circulation of the fluid velocity on the free surface. The equations for free-surface motion we derive here are closed and they are not restricted to potential flow. Hence, true wave-current interaction occurs. In particular, the Kelvin-Noether theorem demonstrates that wave activity can induce fluid circulation and vorticity dynamics on the free surface. The wave-current interaction equations introduced here open new vistas for both the deterministic and stochastic analysis of nonlinear waves on free surfaces.


Introduction
Waves are disturbances in a medium which propagate due to a restoring force, such as gravity. Currents are flows which transport physical properties, such as mass and heat. When waves propagate in a moving medium, the motion of the medium can affect the waves and, vice versa, the waves can affect the motion of the medium, as they both respond to the same force. The primary example is wavecurrent interaction on the free surface of a moving fluid body under the influence of gravity. This mutual interaction is the province of nonlinear water-wave theory.
Water waves -waves on the surface of a body of water -have fascinated observers over the ages, not only because water waves are so easily observed, and not only because they move; but primarily INTRODUCTION because they form coherent moving deformations of the water surface which can interact with each other in a multitude of ways. Any disturbance -even scooping your hand in a narrow channel of shallow water, for example -will resolve itself into a train of coherent solitary waves with a few extra ripples which are left behind as the coherent solitary waves propagate away from the disturbance. The fascination in observing the creation of coherent water waves from arbitrary disturbances was captured in the famous report by the Victorian engineer John Scott Russell in August 1834, when he saw a solitary wave create itself from an impulse of current and then start propagating along a Scottish canal. As he wrote [44], I followed it on a horseback, and . . . after a chase of one or two miles I lost it in the windings of the channel. Such, . . . was my first chance interview with that singular and beautiful phenomenon.
Although the classic water-wave theory introduced by Stokes [46] has a long history, see, e.g., [16], the excitement in the chase for mathematical understanding of water waves has continued. In particular, John Scott Russell's "singular and beautiful phenomenon" is now called a soliton. The word 'soliton' was coined in a 1965 paper by Zabusky and Kruskal [51] and this word has more than 6 million Google hits, as of this writing. The sequence of approximate shallow water equations exhibiting soliton behaviour includes the Korteweg-de Vries (KdV) equation in 1D [35], as well as the Kadomtsev-Petviashvili (KP) [34] equation, which extends the KdV equation to allow weak transverse spatial dependence. Indeed, the solution behaviour of soliton water wave equations continues to inspire mathematical progress in the theory of integrable Hamiltonian systems of nonlinear partial differential equations and their discretizations in space and time. For a good summary of the early developments of soliton theory, see Ablowitz and Segur [4]. For modern discussions of the classic water-wave theory introduced by Stokes [46], see, e.g., [3,36,37,38].
Paper objectives and methodology. The classic water wave equations (CWWE) for potential flow on a free surface comprise the kinematic constraint at the free boundary and the horizontal gradient of Bernoulli's Law for the case of potential flow restricted to the surface. This paper has two primary objectives. The first objective is to augment the CWWE to include fundamental physical aspects of wave-current interaction on a free surface (WCIFS). These physical aspects include vorticity, wave-current coupling in which the wave activity creates fluid circulation, non-hydrostatic pressure, incompressibility, and horizontal gradients of buoyancy. Now, the multi-scale, fast-slow aspects of wave-current interaction comprise a grand challenge in computational simulation, particularly in ocean physics. In ocean physics, the fast-slow aspects of wave-current interaction tend to introduce irreducible imprecision even beyond computational uncertainty because of unresolvable, or even unknown, processes [43]. This situation leads to the paper's second objective. Namely, the paper also aims to introduce stochastic transport of wave activity by fluid circulation which is intended to be used in combination with data assimilation to model uncertainty due to the effects of fast, computationally unresolvable, or unknown effects of WCI on its slower, computationally resolvable aspects.
To pursue these objectives, we will begin by using the Dirichlet-Neumann operator for 3D potential flow of a homogeneous Euler fluid to impose the kinematic and dynamic boundary conditions of CWWE as constraints on the motion of the free surface in the Euler-Poincaré (EP) variational principle for ideal fluids, [31]. The EP formulation is an extension of earlier variational principles for the CWWE, [7,41]. In using the CWWE as constraints, the EP variational principle introduces additional dynamical equations for the Lagrange multipliers. The Lagrange multipliers are interpreted as the vertical velocity p w and the areal mass density D, arising as Hamiltonian variables canonically conjugate to the elevation ζ and the surface velocity potential p φ, respectively. The resulting Hamiltonian equations are referred to as extended CWWE, abbreviated ECWWE.
After a discussion of alternative formulations which elicit a variety of properties of the ECWWE solutions, we use the EP approach to add further aspects of wave-current interaction, which include vorticity, as well as non-hydrostatic pressure and buoyancy gradients in the free surface flow.
We also introduce a wave-current minimal coupling (WCMC) term into the action integral for the EP variational principle which generates fluid circulation from wave activity and vice-versa. The EP variational equations are referred to here as wave-current interaction on a free surface (WCIFS) equations. Finally, to model the uncertainties associated with the computations of these multi-scale fast-slow WCIFS equations, we introduce stochastic advection by Lie transport (SALT) of the wave activity by the current flow, again following the EP variational approach, as in [25]. In the analytical sections of the paper, both the deterministic and SALT versions of our WCIFS equations are shown to be locally well-posed, in the sense of existence, uniqueness, and continuous dependence on initial conditions.  Dashed arrows represent connections of these exact models to their stochastic versions which are not derived explicitly. However, the missing derivations follow the same patterns as that described in full for the stochastic wave-current interaction model in section 6.3.

Roadmap of the paper
Plan of the paper.
• Section 2 reviews the problem statement, boundary conditions and key relations in the classical framework of three-dimensional fluid flows under gravity with a free surface. The free surface elevation is measured from its rest position, which defines the origin of the vertical coordinate z " 0. The elevation, z " ζpr, tq, is a function of the horizontal position vector r " px, y, 0q and time t. A key relation is stated in equation (2.7). Namely, when evaluated on the free surface, the material time derivative of a function f pr, z, tq and its projection onto the free surface f pr, ζpr, tq, tq are equal. The projection relation in equation (2.7) then leads to Choi's relation (2.9) for the dynamics of the free surface, [14].
• Section 3.1 reviews the formulation of the classic water-wave equations (CWWE) for free surface dynamics in terms of the Dirichlet-Neumann operator (DNO). Section 3.2 then derives the INTRODUCTION extended CWWE (ECWWE) which include equations for the vertical velocity p w and preserved area measure D on the free surface. The derivation of ECWWE proceeds by regarding the CWWE as constraints imposed by Lagrange multipliers p w and D in a new variational principle in equation (3.19), defined in terms of functions on the horizontal mean level of the free surface. The Lie-Poisson Hamiltonian form of the ECWWE is derived in section 3.5. In section 3.3, nonhydrostatic pressure is incorporated into the ECWWE and the comparison to the key relation in (2.9) is shown in Theorem 1. The ECWW theory satisfies the well-known non-acceleration theorem by which the the fluid and wave circulations are preserved separately. Hence, it does not qualify as genuine wave-current interaction.
• Section 4 introduces the augmented CWW system (ACWW) which include wave-current interaction via a minimal coupling (WCMC) construction. The Kelvin-Noether circulation theorem for the ACWW model in Theorem 2 shows that the wave dynamics of the ACWW model can create circulation, as shown locally in Theorem 3 and Corollary 4. In addition, as shown in Theorem 5, the ACWW model with genuine wave-current interaction preserves the same physical energy as the ECWW model does, for which the fluid and wave circulations are preserved separately. That is, the ACWW model with its wave-current minimal coupling (WCMC) term preserves the same energy as the ECWW model.
Section 5 includes additional physical properties into the ACWWE to produce our final model of wave-current interaction on a free surface (WCIFS). The WCIFS model is derived by modifying Hamilton's principle for ACWW with its wave-fluid coupling term, to add an advected scalar buoyancy variable ρ with nonzero horizontal gradients, as well as non-hydrostatic pressure in (5.9). The Kelvin-Noether circulation theorem for the WCIFS model is given in Theorem 6.
• Section 6 introduces a method of incorporating stochastic noise into this theory which preserves its variational structure. This is achieved by applying the method of Stochastic Advection by Lie Transport (SALT) [25]. Within this section, we derive a stochastically perturbed version of the classical water-wave equations, as well as a stochastic variational models of the ECWW, ACWW, and WCIFS models of wave-current interaction of free surfaces.
• Section 7 lays out the analytical framework for dealing with the compressible and incompressible ECWWE equations discussed in sections 3.1 and 3.3, respectively.
• Section 8 discusses potential future research directions and identifies new problems opened up by this work.
• Appendix A discusses transformation theory for ideal fluid dynamics and derives the Kelvin circulation theorem by using the Lie chain rule.
• Appendix B reviews the 3D inhomogeneous Euler equations for incompressible fluid flow under gravity with a free upper surface and a fixed bottom topography. This is done by deriving these equations using a constrained variational principle.
• Appendix C treats the reduced Legendre transformation from Hamilton's principle to the Hamiltonian formulation for ECWWE in the Eulerian fluid representation.
List of abbreviations.

Problem statement, Boundary Conditions and Key Relations
Problem statement. We study the dynamics of fluid parcels which are constrained to remain on the free surface of a three dimensional fluid with coordinates x " pr, zq. Here r " px, yq (resp. z) denotes horizontal (resp. vertical) Eulerian spatial coordinates in an inertial (fixed) domain. The fluid domain is bounded below by a rigid bottom at z "´Bprq and is bounded above by the free surface of the fluid at z " ζpr, tq, which is measured from its rest position at z " 0 as a function of the horizontal position vector r " px, y, 0q and time t.
Three-dimensional fluid equations. The fluid moves in three dimensions with velocity upx, tq " pvpx, tq, wpx, tqq in which vpx, tq and wpx, tq denote, respectively, the horizontal and vertical velocity fields. Incompressible and inviscid fluid motion is governed by the Euler equations of horizontal and vertical momentum dynamics under the constant acceleration of gravity, g. The equations are given by with D :" B t`v¨∇r`w B z and ∇ r¨v`wz " 0 . (2.1) We denote by π the pressure with three dimensional spatial dependence. The volume element is d 3 x " d 2 r^dz, and its measure Dd 3 x is preserved under the incompressible fluid flow. The mass density is given by ρ " ρ 0 p1`bpx, tqq ą 0, in which bpx, tq :" pρ´ρ 0 q{ρ 0 is the fluid buoyancy and ρ 0 ą 0 is the (constant, positive) reference value of mass density. The mass in each fluid volume element is given by ρDd 3 x. The condition ζpr, tq´z " 0, which defines the free surface, is assumed to be preserved under the flow. This condition ensures that a particle initially on the free surface will remain on it. These three preservation relationships may be expressed as the following three advection relations, pB t`Lu qpζpr, tq´zq " pB t`u¨∇ qpζpr, tq´zq " 0 , where the operator pB t`Lu q is the advection operator (see Appendix A). Setting the constant volume measure to D " 1 in the first advection relation implies the continuity equation, ∇¨u " 0. The motion equations in (2.1) and the initial values for the advected quantities ρpr, z, tq and pζpr, tq´zq in the advection relations in (2.2) provide a complete specification of the initial value problem for the fluid motion with appropriate boundary conditions in three dimensions.
Boundary conditions. Whilst the horizontal boundary conditions are yet to be specified and can be chosen to suit specific problems, the vertical boundary conditions must be carefully defined.
The kinematic boundary condition on the free surface is given by PROBLEM STATEMENT, BOUNDARY CONDITIONS AND KEY RELATIONS where the p f notation in p D, p v, and p w is defined for an arbitrary flow variable f to represent evaluation on the free surface, namely, p f pr, tq " f pr, z, tq on z " ζpr, tq . (2.4) Notice that evaluating on the free surface before taking derivatives is not equivalent to taking the derivative before evaluating. In particular, B t p f pr, tq ‰ x B t f pr, tq and ∇ r p f pr, tq ‰ z ∇ r f pr, tq, where x B t f pr, tq " rB t f pr, z, tqs z"ζpr,tq and z ∇ r f pr, tq " r∇ r f pr, z, tqs z"ζpr,tq . (2.5) Instead, from the chain rule we have Consequently, the advection operator on the free surface is given by the following remarkable identity where D :" pB t`v¨∇ q and p D :" pB t`p v¨∇ r q. That is, on the free surface, the 3D material time derivative of f pr, z, tq equals the 2D surface material derivative of the surface evaluation of f .
Choi's relation at the free surface. One may use the relation (2.7) to evaluate the horizontal and vertical coordinates of the motion equation (2.1) onto the free surface. Hence, one finds for constant buoyancy ρ " ρ 0 on the free surface ζpx, y, tq´z " 0 that where, in the last step, we have used the vertical motion equation in (2.1) to evaluate π z for ρ " ρ 0 and the relation (2.7) for the vertical acceleration of the free surface, dw{dt z"ζpr,tq " y Dw " p D p w. In conclusion, upon using the boundary condition p w " p Dζ in (2.3) we find Choi's relation at the free surface [14], This fundamental relation is not restricted to irrotational flows. However, at this stage, the dynamical system comprising equations (2.3) and (2.9) for the motion of the free surface is not yet closed since, (i) the pressure gradient ∇ r p π is still unknown and, (ii) an evolutionary equation for p D p w is missing.
Choi's relation at the bottom boundary. One may also consider boundary conditions at either a lower free surface, z "´Bpr, tq, or at fixed bathymetry, z "´Bprq. Denote by q f the evaluation on the bathymetry, i.e. q f pr, tq " f pr, z, tq on z "´Bpr, tq . (2.10) The bottom boundary condition iś q DBpr, tq "´pB t`q v¨∇ r qBpr, tq " q w , on z "´Bpr, tq . We may now evaluate equations (2.1) onto the lower surface z "´Bpr, tq in the same manner as we have evaluated onto the upper surface z " ζpr, tq to give q Dq v`p q D q w`gq∇ r Bpr, tq "´1 ρ 0 ∇ r q π , on z "´Bpr, tq . (2.13) By the bottom boundary condition (2.11) we then find q Dq v`p´q D 2 q Bpr, tq`gq∇ r Bpr, tq "´1 ρ 0 ∇ r q π , on z "´Bpr, tq . (2.14) When Bpr, tq a time-dependent variable, then equation (2.14) is not closed. However, when the bottom boundary is taken to be time-independent, so that z "´Bprq and q D " q v¨∇ r there, then equation (2.14) would be closed, provide either the bottom pressure q π were prescribed, or the bottom velocity q v were taken to be divergence-free.

Mean continuity relation.
Another exact result about the dynamics of the free surface elevation ζpx, y, tq should be mentioned. This result is the following mean continuity relation for the elevation in terms of the vertically averaged horizontal velocity components, [49]. , see e.g., [8,14,49].
Proof. By direct computation, using the advection condition for ζ and the vertical integral of the divergence free condition divu " 0, and upon noticing that no contribution arises from the flat bottom boundary, one finds that where we have added and subtracted wpr, ζpr, tq, tq and applied a tangential flow condition at the bottom boundary. Thus, the boundary conditions and the divergence-free nature of the three dimensional flow combine to produce the mean continuity relation in (2.15).

FREE SURFACE DYNAMICS
The route to a closure scheme for (2.3) and (2.9) which includes the CWWE. The closure problem for equations (2.3) and (2.9) will be resolved in section 3.2 in the context of the CWWE, which will imply p D p w "´g. In section 3.5, the pair of wave variables p w and ζ will be understood as a canonically conjugate subset of a Hamiltonian system of Eulerian equations for planar fluid motion. This system will also contain the hydrostatic CWWE introduced in section 3.1. In section 3.3, nonhydrostatic pressure p π will be incorporated into the CWW problem to complete the closure of Choi's relation in (2.9). The rest of the paper will then build additional physics into the resulting system of planar fluid equations, e.g., by including horizontal gradients of buoyancy on the free surface. Refer to Figure 1 for more perspective.
3 Free surface dynamics 3

.1 The Classic Water-Wave equations (CWWE)
The Dirichlet-Neumann operator (DNO). In this section, we consider the much studied potential flow governed by the CWWE. The qualitative information obtained here from the CWWE will inspire our derivation of a constrained variational principle below. The CWWE are derived from the free surface three dimensional Euler equations via the Dirichlet-Neumann operator (DNO). The DNO maps the solution of Laplace's equation in an external domain with a Dirichlet boundary conditions to its solution on the boundary with a Neumann flux condition (see e.g. [36]). In particular, the CWWE assume that the flow is incompressible and irrotational and, thus, there exists some φpr, z, tq such that u " ∇φ, where u is the three-dimensional velocity field throughout the domain.
In the hat-notation of (2.4), the variable p φpr, tq " φpr, ζpr, tq, tq evaluates the velocity potential φpr, z, tq on the free surface z " ζpr, tq. The action of the DNO Gpζq on p φpr, tq is defined as the normal component of the three-dimensional velocity field for the potential flow u " ∇φ evaluated at the free surface z " ζpr, tq. Namely, Gpζq p φ :" p´∇ r ζ, 1q¨y ∇φ :"´∇ r ζpr, tq¨z ∇ r φ`p w , (3.1) in which the horizontal gradient of the velocity potential φpr, z, tq is first taken, then evaluated at the surface z " ζpr, tq, cf. equation (2.8).
Thus, the DNO in (3.1) takes Dirichlet data for p φ on z " ζpr, tq, solves Laplace's equation ∆φ " 0 for φpr, z, tq together with the condition that the velocity u " ∇φ have no normal component on the fixed parts of the boundary of the full domain volume, and then returns the corresponding Neumann data, i.e., the three-dimensional fluid normal velocity on the free surface, z " ζpr, tq.
The classical water-wave equations (CWWE). The classical water-wave equations (CWWE), as stated in [36], can be written in terms of the DNO as From the chain rules in (2.6), we have, in the hat notation of equation (2.4), that In terms of the Dirichlet-Neumann operator these are expressed as Gpζq p φ :"´∇ r ζpr, tq¨∇ r p φpr, tq`y B z φ|∇ r ζ| 2`p w . We consider (3.4) together with p w " p Dζ. Observe that in the hat notation p v :" z ∇ r φ, V :" ∇ r p φ and 3.1 THE CLASSIC WATER-WAVE EQUATIONS (CWWE) 9 and hence p w " p Dζ " B t ζ`p v¨∇ r ζ " B t ζ`p∇ r p φ´p w∇ r ζq¨∇ r ζ , (3.8) which after rearranging is equivalent to Thus, applying the chain rule in the Dirichlet-Neumann operator appearing in the kinematic boundary condition has implied the alternative expression for p w in equation (3.9). The alternative equations for p w in (3.9) will be used next in section 3.2 to close the system defined by Choi's relation (2.9) by using a variational principle reminiscent of the approach in [41] to derive an evolutionary equation for p w. The alternative expressions in (3.9) obtained from the Dirichlet-Neumann operator will also inspire a wave-current coupling term in section 4.1.
Evaluating (3.10) on z " ζpr, tq with the constraint π| z"ζ " 0 that the non-hydrostatic pressure vanishes on the free surface yields Upon adding and subtracting y B z φ 2 |∇ r ζ| 2 in the previous equation, one finds Considering this in tandem with equation (3.8) for p w yields which one observes is equivalent to (3.3).
Remark 3.2. Equation (3.9) expresses p w in terms of the time derivative of ζ in the frame of reference moving with horizontal velocity ∇ r p φ rather than with velocity z ∇ r φ. We recall the relation (3.7) and write Physically, p v :" z ∇ r φ may be interpreted as the fluid transport velocity relative to a Galilean frame moving with velocity s :" p w∇ r ζ, while the quantity V :" ∇ r p φ is the total fluid velocity in the inertial frame of the Eulerian fluid description. In fact, the variational formulation taken below will show that the quantity V is the momentum per unit mass given by the variational derivative with respect to transport velocity p v of the Lagrangian in Hamilton's principle for the wave-current dynamics. Likewise, the quantity s " p w∇ r ζ will turn out to be the CWW wave momentum per unit fluid mass derived from Hamilton's principle.
The surface boundary condition (3.9) yields the evolution equation for the elevation ζ written in the two different frames of motion as, Equating p w in these two expressions then yields

IMPOSING CWWE AS CONSTRAINTS IN HAMILTON'S PRINCIPLE
Likewise, equations (3.13) and (3.14) yield the Bernoulli evolution equation for zero non-hydrostatic pressure in terms of the rotational fluid velocities p v and V , This completes the direct derivation of the CWWE.

Imposing CWWE as constraints in Hamilton's principle
Let us introduce the following dimension-free action integral for a variational principle, δS " 0, which comprises the sum of the kinematic boundary condition for the elevation ζ in (3.15) and the Bernoulli equation for zero non-hydrostatic pressure in (3.18). These two CWWE conditions are constrained to hold by the two Lagrange multipliers λ and D, respectively, For spatial integration by parts, we take natural boundary conditions so the boundary terms vanish. The temporal integration by parts introduces a total time derivative, so it also does not contribute to the equations of motion. In equation (3.19), r " px, yq denotes horizontal Eulerian spatial coordinates in an inertial (fixed) domain. We have integrated by parts in time after the second line, and then in space after the third line, dropping boundary terms both times. The constants σ 2 and F r 2 here are squares of the aspect ratio and the Froude number, respectively, which are obtained in making the expression dimension-free. Finally, we make the distinction between the wave variables p w and ζ, and the current variables D and p v. The remaining variables λ and p φ in the final form of the action integral (3.19) are Lagrange multipliers, to be determined from the others. Remark 3.3 (Non-dimensional parameters). Explicitly, the action integral for free surface motion in (3.19) has been cast into dimension-free form by introducing natural units for horizontal length, rLs, horizontal velocity, rV s, time, rT s " rLs{rV s, vertical velocity, rW s and vertical wave elevation, rζs. In terms of these units we have defined the following dimension-free parameters: aspect ratio, rW s{rV s " σ " rζs{rLs and Froude number, F r 2 " rV s 2 {prgsrζsq, for typical wave elevation scale rζs.
Interpreting the three equivalent forms of the action integral in (3.19).
• The second line of the action integral in (3.19) may be regarded as a variant of the action integral in Luke [41]. An action integral for CWWE in [41] was derived in terms of vertically integrated expressions. In contrast, here the action integral in (3.19) has been made two-dimensional by using the Dirichlet-Neumann operator relation to project out the third (vertical) dimension. The Lagrange multiplier λ enforces the kinematic boundary condition for the elevation ζ in (3.15). Likewise, the Lagrange multiplier D enforces the zero-pressure Bernoulli law (3.18) obtained from the Dirichlet-Neumann operator.
• In the third line of (3.19), we have expressed ℓpp v, D, p φ, p w, ζ; λq as a phase-space Lagrangian, by re-writing it as a Legendre transform. The phase-space form of the Lagrangian immediately identifies the canonically conjugate pairs of field variables p p φ, Dq and pσ 2 λ, ζq and determines the Hamiltonian as (3.20)

11
The variation of the Lagrangian in any of its equivalent representations in (3.19) with respect to the vector-field velocity p v yields the momentum density relation This expression provides a (cotangent lift) momentum map from the canonically conjugate pairs of field variables p p φ, Dq and pσ 2 λ, ζq to the momentum density m :" Dp v which is in concert with equation (3.7).
• In the last line of (3.19), we rearrange the constraints in the action integral in the second line into the standard Clebsch advection form for two-dimensional fluid motion, by integrating by parts in time and (horizontal) space. We may then regard the quantity D d 2 r as the area measure on the horizontal domain. That is, the area measure D d 2 r is advected by p v, which is imposed in the last line by regarding the trace of the velocity potential on the free surface p φ as a Lagrange multiplier.
Consequently, the canonical constraint equations appearing in the last line of (3.19) are not potential flows. In contrast, equation (3.14) shows that V " ∇ r p φ is indeed a potential velocity.

Variational formulas. Taking variations of the action integral (3.19) yields
(3.23) Applying the Lagrangian time derivative pB t`Lp v q to the first relation in (3.23) yields the ECWW motion equation, See appendix A for more discussion of the Lie derivative notation (as in L p v ) which is defined by the Lagrangian time derivative. The quantity d̟ is the spatial differential (i.e., the gradient) of Bernoulli's law in the last line of (3.23).
Kelvin circulation theorems for ECWWE in their dimensional form. Moving to the dimensional form and continuing to calculate from the ECWW motion equation in (3.24), we have Remarkably, the p p w, ζq equations in (3.23) imply that the previous equation separates into two transport equations, namely, (3.26)

DERIVATION OF ECWW EQUATIONS WITH NON-HYDROSTATIC PRESSURE
Thus, the wave and current circulations are conserved separately, in a mutual non-acceleration pact, The separation of conservation laws in (3.27) means that the two degrees of freedom do not influence each other's circulation. Actually, this separation is a general feature of wave-current interaction theories which arise from Hamilton's principle with a phase-space Lagrangian, [32].

Reduction of the ECWW motion equation to the pressureless Euler fluid equation.
Because of a cancellation of 1 2 d|p v| 2 in equation (3.26) with the Lie derivative term, the p v-equation simplifies further to produce the following pressureless Euler fluid equation for the transport velocity Thus, while the vector p v transports the density D, it also transports itself as though it were an array of two advected scalars, pp v 1 , p v 2 q. This feature further simplifies the interpretation of the p v-equation, because it can now be seen as an inviscid Burgers equation. However, note that the compressible "Burgers velocity" p v in (3.28) has vorticity p ω :" p z¨curlp v "´Jp p w, ζq which does not vanish, in general. However, the relation p v " ∇ p φ´p w∇ζ and the second equation in (3.26) do imply that Hence, if the vorticity p ω vanishes initially, it will remain so. In this case, the pressureless 2D Euler equation in (3.29) reduces to the well-studied two-dimensional Hamilton-Jacobi equation for p φ. See, e.g., [33,22] for reviews.
Back to the ECWWE in their dimensional forms. We may restore the ECWWE to their dimensional forms as where p v evolves according to (3.28). One observes that the first two equations in (3.30) are equivalent to the CWWE discussed in section 3.1, since p v " z ∇ r φ.

Derivation of ECWW equations with non-hydrostatic pressure
In the standard derivation of the ECWW equations (3.2) and (3.3), the three dimensional pressure π is taken to be zero on the surface and thus the resulting equations of motion have no pressure term. In order for the variational equations we have derived in section 3.2 to match equations (3.2) and (3.3), we have derived compressible equations and thus the system also contains the additional equation for D. Should we want to model an incompressible flow, and avoid having an equation for D, we must introduce pressure as a Lagrange multiplier which enforces that D " 1. Of course, such a non-hydrostatic pressure would be incompatible with assuming the pressure is zero on the surface. We derive the ECWW equations with non-hydrostatic pressure and incompressible transport velocity by varying the action integral defined by Proceeding in the same manner as in section 3.2, and omitting the calculations since they are very much alike, we derive the following system of equations Here, the divergence-free transport velocity p v satisfies a two dimensional Euler equation We may understand the structure of this problem further by comparing it to equation (2.8) with ρ 0 " 1. Noting that p D p w "´g, equations (2.8) and (3.33) together imply This comparison implies the following remarkable observation.
Theorem 1. The pressure, p, in the two dimensional model (3.32) is equivalent to the pressure of the three dimensional fluid evaluated on the free surface, π, up to the addition of a spatial constant.

Conservation laws for the compressible ECWW dynamical system
From here, we return to the compressible ECWW equations without the pressure term.

Eulerian conservation laws
The system of ECWWE in (3.28) and (3.30) possesses the following obvious Eulerian conservation laws in a domain Ω with fixed boundaries.
1. The last equation in (3.30) implies conservation of mass, D :" for p v¨p n on the boundary BΩ with normal vector p n.

Equation (3.28) and the last equation in (3.30) imply conservation of momentum, defined by
3. Combining the curl r of equation (3.28) and the last equation in (3.30) implies conservation of mass-weighted enstrophy, defined by for any differentiable function Φ of vorticity, p ω, which itself is defined by Thus, upon noticing that vorticity p ω is advected as a scalar by the flow of p v, we find advection of any function of Φpp ωq, as well, by the chain rule and linearity of the advection operator for scalars. Namely, pB t`p v¨∇ r qΦpp ωq " 0 .
Thus, we obtain conservation of mass-weighted enstrophy from the continuity equation, the chain rule and integration by parts, as follows, for p v¨p n on the fixed boundary BΩ. Thus, the D-weighted L p norm of the vorticity p ω " curlp v is controlled.

Moment dynamics of a Lagrangian fluid blob under the ECWWE
We re-write the continuity equation in (3.30) and its associated motion equation in (3.28) as Lagrangian conservation laws for mass and momentum, Consider a two dimensional 'blob' of fluid mass occupying a Lagrangian domain of fluid Ωptq which is deforming under the ECWWE flow of the free-surface fluid velocity p v so that no fluid material enters or leaves through its moving boundary BΩptq. In this situation, we have the following Reynolds transport relations for the dynamics of the spatial moments of the mass distribution within the blob.
1. The total mass of a Lagrangian blob is conserved: 2. The rate of change of the centre of mass of the blob is its conserved momentum: Conservation of the blob momentum is shown by a direct computation, 3. The moment of inertia I ij " ş Ωptq r i r j D d 2 r represents the elliptical shape of the blob. Its rate of change may be computed as 4. The acceleration of the elliptical shape of the blob is governed by

5.
Remarkably, the acceleration of the trace of the moment of inertia trpIq is positive-definite

THREE HAMILTONIAN FORMULATIONS OF THE ECWWE USING FREE-SURFACE VARIABLES 15
This is a simple version of the tensor virial theorem [12]. Here, the tensor virial theorem implies that under ECWWE flow equations in (3.35) any initial distribution of mass will expand outward at an acceleration rate proportional to the kinetic energy within its Lagrangian boundary. Because this result holds for every Lagrangian blob of fluid undergoing this motion it follows that the mass density cannot become singular in an infinite flow domain. This means that the measure Dd 2 r cannot become a Dirac measure.
6. Finally, we notice that blob angular momentum L ij :" 3.5 Three Hamiltonian formulations of the ECWWE using free-surface variables

Canonical Hamiltonian formulation of the ECWWE
In the canonical Hamiltonian field variables for currents p p φ, Dq and for waves pλ, ζq, the Bernoulli function ̟ in (3.23) is expressed as (3.37) One observes that the symplectic Poisson operator for two independent degrees of freedom in (3.37) appears in the standard block-diagonal form.

Entangled Hamiltonian formulation of the ECWWE
In terms of the canonical Hamiltonian field variables p p φ, Dq and pλ, ζq, the total momentum density of the fluid x m :" Dp v is defined as The Legendre transform with respect to both pairs of canonical wave variables defining the momentum density m leads to the following Hamiltonian, (3.39) The corresponding conserved energy is given by  where ̟ is defined in equation (3.36). Here, the Poisson operator is the direct sum of the usual semidirect-product Lie-Poisson bracket for ideal fluids [31] and a symplectic Poisson bracket for the canonical wave variables, pλ, ζq. The Poisson operator in (3.41) is said to entangle the dynamics of the wave variables pλ, ζq and fluid variables pm i , Dq. Next, we will untangle the entangled Poisson operator to put it back into block-diagonal form by considering only the momentum density of the potential part of the fluid flow.

Untangled Hamiltonian formulation of the ECWWE
The momentum density of only the purely potential part of the fluid flow is given in terms of the canonical wave variables and the transport velocity The Legendre transform with respect to only the D and p φ variables corresponding to the potential part of the flow leads to the following Hamiltonian, Variations of the Hamiltonian in (3.43). In the Hamiltonian variables, the Bernoulli function ̟ in (3.23) is denoted as   This dual wave-current contribution is already clear from the coordinate-free form of the motion equation in (3.24), since it immediately implies conservation of a two-component Kelvin circulation integral involving both types of fields present in the momentum density, where cpp vq denotes a closed material loop moving with the fluid transport velocity, p v. However, a closer look at the separate equations of motion for λ and ζ in the Hamiltonian form in equation (3.45) shows that evolution of the wave variables is rather passive. After all, the wave dynamics takes place independently in the moving frame the fluid flow, without influencing the flow. Indeed, a closer look at the circulation dynamics in (3.46) shows that the two components of the circulation are conserved separately, as shown in equation (3.26).
Unfortunately, this passive two-fluid, freely-interpenetrating interpretation of the solutions of the ECWWE fails to represent genuine wave-current interaction. For a review and bibliography of previous work in Hamiltonian formulations of the wave-current interaction, see [6]. Apparently, to develop a model in which the wave activity can directly influence the fluid circulation one must augment the interaction.
How to introduce genuine wave-current interaction? How should we introduce a mechanism for genuine wave-current interaction within the variational framework we have developed here for classic water-waves on free surfaces? The approach we propose next is reminiscent of the Craik-Leibovich approach for wind-wave generation of Langmuir circulations [18,28,26,27]. The Craik-Leibovich approach introduces the sum of the transport velocity and the Stokes drift velocity into the integrand of Kelvin's circulation theorem for fluid motion, while retaining the original transport velocity for the Kelvin loop. This approach thus measures the Lagrangian transport velocity of the loop relative to the frame of motion of the Stokes drift. In Hamilton's principle, this is done by adding the product of the fluid momentum density times the Stokes drift velocity, similarly to the "Jaydot-Ay" minimal coupling approach by which electromagnetic interactions are introduced into the dynamics of charged fluids [21]. In the next section 4.1 we will introduce a term into the CWW action integral (3.19) which is intended to model wave-current minimal coupling (WCMC). The WCMC term will modify the transport velocity of the wave momentum density, to allow genuine wave-current interaction, so the free-surface water waves will no longer be passively advected by the fluid velocity.

Variational derivation of the ACWWE
To introduce genuine wave-current interaction for the ACWWE, we propose to augment the Lagrangian in the action integral (3.19) by adding a term proportional to the slip velocity in (3.14) which is related to the gradient of the velocity potential by, cf. equation (3.7), We pair the slip velocity s with the wave momentum density λ∇ r ζ to create a quantity with the dimension of wave energy 1 s¨λ∇ r ζ " p∇ r p φ´z ∇ r φq¨λ∇ r ζ " p wλ|∇ r ζ| 2 .

VARIATIONAL DERIVATION OF THE ACWWE
We propose to augment the action integral in equation (3.19) by adding the wave-current minimal coupling (WCMC) term in (4.2), as follows.
S " where we have retained the same non-dimensional parameters as in remark 3.3 and ǫ is a non-dimensional coupling constant. According to remark 3.3 each wave variable is multiplied by the aspect ratio σ relative the fluid variables. Stationarity of the augmented action integral in (4.3) now yields the variational equations, Notice that the ǫ term in the modified Lagrangian in (4.3) does not affect variations in the fluid variables, pp v, D, p φq. However, it changes the previous relationship between λ and p w by a term proportional to ǫ, which means it yields a different definition of V , as seen in the first and second lines of (4.4). Of course, the previous wave system in (3.23) is recovered when one sets ǫ Ñ 0.
ACWW motion equation. We may write out the ACWW motion equation obtained by substituting the variational results into the application of the advective time derivative pB t`Lp v q on the first line of the system (4.4), to find in the notation λ " D r w that DpB t`Lp v q`p v¨dx`σ 2 r w dζ˘" DdpB t`Lp v q p φ " Dd̟ . Recall from (4.4) that Bernoulli function ̟ and vertical wave momentum density λ are defined as At this point, let us collect the ACWWE in terms of (i) fluid variables, comprising velocity p v and area density D, and (ii) wave variables, comprising surface elevation ζ, and vertical wave momentum density λ. Namely, the ACWWE are given by, (4.7) The equation set (4.7) recovers the ECWWE equations (3.30) when ǫ Ñ 0. The first of these equations implies Kelvin's circulation theorem for the ACWW model, as follows.  This shift in wave velocity introduces wave dynamics into the transport velocity of the wave variables, as follows, B t ζ``p v´ǫσ 2 p w∇ r ζ˘¨∇ r ζ " p w , B t λ`div r´λ`p v´ǫσ 2 p w∇ r ζ˘¯"´1 σ 2 F r 2`ǫ σ 2 div r`λ p w ∇ r ζ˘, (4.11) where one recalls that the canonical momentum density λ conjugate to the elevation ζ is defined in terms of the other wave variables in (4.6).
Remark 4.2. The difference in the wave momentum transport velocity relative to the fluid velocity in equation (4.10) will turn out to produce an important effect by which the waves will generate fluid circulation. To compute this effect on the circulation of the fluid we subtract the fluid transport of the wave momentum from the total momentum transport by the fluid in (4.9). The fluid velocity transport of the wave momentum is found from the wave dynamical equations in (4.11), as Proof. The proof follows by applying the Stokes theorem to the right-hand side of the material loop cpp vq in equation (4.14) in Theorem 3.

Theorem 5 (Total energy conservation is independent of ǫ).
The energy conserved by modified equations (4.12) and (4.13) is independent of ǫ.
Proof. The modified constrained action integral in equation (4.3) may be rewritten equivalently as a phase-space Lagrangian, upon rearranging as follows, Theorem 5 shows that the equations resulting from the modified Lagrangian in (4.3) conserve the same energy as for the unmodified Lagrangian in (3.19). Thus, the modification in (4.3) which was obtained by introducing the ǫ term produces the wave-current interaction in equations (4.12) and (4.13) while also preserving the original physical energy density. What depends on ǫ is the definition of the vertical wave momentum density λ canonically conjugate to the elevation ζ depends on ǫ, as well as the definition of the velocity V in terms of the wave variables, as seen in the first and second lines of (4.4). Remark 4.3 (Tensor virial theorem for a Lagrangian fluid blob under the ACWWE). Although the conserved energy remains the same for the ECWW and ACWW models for any value of the coupling constant ǫ, the tensor virial theorem for a Lagrangian fluid blob under the ACWWE is considerably more intricate than in section 3.4.2 for the ECWW model.  The Hamiltonian in (4.17) is also the conserved energy (3.40) for the system of ECWWE in (3.30), as proven in Theorem 5.

Variations of the Hamiltonian in (4.17). In the Hamiltonian variables, the Bernoulli function ̟ in (4.4) is denoted as
After evaluating the corresponding variational derivatives of the Hamiltonian in (4.17), the system of equations in (4.7) may be written in the untangled block-diagonal form, as Casimir functions. The Casimir functions, conserved by the relation tC Φ , hu " 0 with any Hamiltonian hpM , Dq for the block-diagonal Lie-Poisson bracket in equation (4.19) are given by As one may verify, the C Φ are conserved for any differentiable function, Φ, provided the velocity p v is tangent on the two-dimensional boundary.
The proof of the constancy of the family of functions C Φ is straightforward and well-known. That the C Φ comprise a family of Casimirs so that tC Φ , hu " 0 for any Hamiltonian hpM , Dq is a standard result for semidirect-product Lie-Poisson brackets. See, e.g., [30].
Remark 4.4 (Consequences of introducing the wave-current minimal coupling term (WCMC)). The consequences of introducing the slip velocity s :" p w∇ r ζ in (3.14) into the variational principle in (4.3) as a WCMC term are evident in the Bernoulli function in (4.18) and in the transport velocities of the wave dynamics in (4.19), upon comparing them with (3.44) and (3.45), respectively. In contrast to the complexity of the separate relations for wave and current circulation laws in (4.12) and (4.13), the simplicity of the conservation of the total circulation in equation (4.8) for ACWW dynamics seems to be a more meaningful statement about WCI than in the ECWWE, where the wave and current circulations are conserved separately in equation (3.27), as a mutual non-acceleration pact. In the next section of the paper, we will explore the further ramifications of introducing the WCMC term, by adding non-hydrostatic pressure, buoyancy and other physics to the ACWW system. 5 Hamilton's principle for wave-current interaction on a free surface (WCIFS)

Adding buoyancy and other physics to the ACWW system
This section further augments the ACWWE set (4.7) to add more physical aspects to the wavecurrent interaction on a free surface (WCI FS). These physical aspects include wave-current coupling, non-hydrostatic pressure, incompressibility, and horizontal gradients of buoyancy.
Hamilton's principle for WCIFS. Let us modify the action integral (4.3) for the system of ACWWE in (4.7) to encompass the following aspects of wave-current interaction on a free surface. As in (4.3), we will impose the surface boundary condition (3.17) and the continuity equation for the areal density variable D as constraints. We will also include the wave-current minimal coupling (WCMC) term via the slip velocity, as in section 4.1. In addition, we will introduce an advected scalar 5.1 ADDING BUOYANCY AND OTHER PHYSICS TO THE ACWW SYSTEM buoyancy variable ρ with nonzero horizontal gradients. Finally, we will allow non-hydrostatic pressure, p. To determine the pressure, p, we will constrain the two-dimensional fluid transport velocity p v to be divergence-free. 2 To include these various physical effects, we will apply Hamilton's principle with the following dimension-free action integral, Here, we recall that the quantity s :" p w∇ r ζ is the slip velocity, defined in equation (3.14).
Remark 5.1. Note that by including only certain terms in the above action integral, we may derive equations for the dynamics of subsystems with any combination of these additional properties (wavecurrent coupling, non-hydrostatic pressure, incompressibility, buoyancy).
The passive wave case, ǫ " 0. Taking variations of the dimensional version of action integral in (5.1) with ǫ " 0 yields Applying pB t`Lp v q to the first relation in (5.2) yields pB t`Lp v qpp v¨dr`p wdζq " d̟´1 ρ dp , so we obtain the following Kelvin circulation theorem, As expected, the momentum per unit mass in the motion equation is p v`p w∇ζ ": V . The result has the same right-hand side as for the two-dimensional inhomogeneous Euler equation. Here, the total momentum now is the sum of the fluid momentum and the wave momentum, whose evolution is obtained as a separate degree of freedom appearing in the third and fourth lines of the equation set (5.2).
However, continuing to calculate from (5.3) yields a non-acceleration result as in equation (3.26), in the sense that the wave momentum evolves passively with the flow of the fluid, and the fluid momentum evolves independently of the wave variables, Thus, the momentum equation in the passive wave case is simply a 2D Euler equation to be considered in tandem with the remaining identities from (5.2). Note that if ∇ρ ‰ 0 then the righthand side of the previous equation generates circulation in p v; so, in this case p v cannot produce potential flow.
Next, we will pursue the implications when the wave-current minimal coupling (WCMC) parameter ǫ does not vanish and the wave variables do not interact passively.

Derivation of the WCIFS equations for active waves
To derive a WCIFS model system of equations for the motion of free surface with active waves and spatially varying buoyancy, we will apply the free-surface condition (3.17) and incompressibility of the p v-flow as constraints in the action integral, while also including the minimal coupling term with nondimensional parameter ǫ ‰ 0 in the action integral (5.1). Then, upon restoring dimensionality to the variables in (5.1), we obtain the following action principle for the free-surface motion, The Lagrange multipliers r µ, p, p φ and γ, apply, respectively, the free-surface condition (3.17), incompressibility of the p v-flow, mass preservation, and buoyancy advection. Hamilton's principle, δS " 0, for the restricted free-surface action integral in (5.5) yields the following independent relations, δp v : Dρp v¨dx`r µ dζ " Dd p φ´γdρ , δ p w : Dρ p w´r µ`1`ǫ|∇ζ| 2˘" 0 , δr µ : B t ζ`p v¨∇ζ´p wp1`ǫ|∇ζ| 2 q " 0 , δζ : B t r µ`divpr µp vq "´Dρg`2ǫ div`p w r µ ∇ζ˘, Before enforcing the pressure constraint D " 1, we write out the fluid motion equation obtained by substituting the variational results into the application of the advective time derivative pB t`Lp v q on the first line of the system (5.6), to find, upon writing r µ " Dρ r w DρpB t`Lp v q`p v¨dx`r w dζ˘" DdpB t`Lp v q p φ´D´pB t`Lp v q γ D¯d ρ " Dρ´d̟´1 ρ dp¯.

(5.7)
Recall that the Bernoulli function ̟ and vertical wave momentum density r µ are defined as

COMPARISON OF WCIFS SYSTEM TO OTHER KNOWN SYSTEMS
At this point, let us collect the WCIFS equations in terms of (i) fluid variables, comprising velocity, p v, area density, D, buoyancy, ρ, and (ii) wave variables, comprising surface elevation, ζ and vertical wave momentum density r µ. The WCIFS equations are, pB t`Lp v q`p v¨dx`r w dζ˘" d̟´1 ρ dp , In its role as a Lagrange multiplier in the action integral (5.5), the pressure p enforces the constraint D " 1. In turn, persistence of the condition D " 1 along the flow implies that the fluid motion generated by p v is incompressible. In particular, setting D " 1 in the continuity equation in (5.9) above implies that the free surface fluid velocity p v is divergence free, divp v " 0. The pressure p is then determined by requiring that p v remain divergence free, which implies the following elliptic equation for p,´p Thus, the pressure p depends on the horizontal flow velocity p v of the surface current and fluid buoyancy ρ, as well as the wave elevation ζ and the vertical velocity w. We stress that the flow variables, pp v, ρq, and the wave variables, pζ, r wq, comprise two separate Eulerian degrees of freedom at each point r " px, yq in the two-dimensional domain of flow.
Remark 5.2 (Making the action integral stochastic in section 6). In section 6 the action integral (5.1) will be made stochastic, following [25], and we will to derive a stochastic generalisation of the water-wave equations.

Comparison of WCIFS system to other known systems
Remark 5.3 (Comparison of system (5.9) to the John-Sclavounos (JS) model equations). The JS model comprises a dynamical system of ordinary differential equations for the motion of a single particle which is constrained to remain upon the free surface ζpx, y, tq´z " 0, with prescribed ζpx, y, tq. This dynamical system has recently been derived from a variational principle using the Euler-Lagrange methodology [13]. This variational principle raises the question of whether the particle dynamics of JS model may be associated with Lagrangian fluid trajectory dynamics in the present continuum framework.

25
Choi's relation. One may immediately make the connection between the JS equations (5.13) and Choi's relation (2.9). Naturally, since the JS equations represent a single particle's motion whereas Choi's relation is a statement about continuum flows, (2.9) features a pressure term on the right hand side. However, the two equations are otherwise strikingly similar.
Comparison with the JS equations. To make the comparison between the system of equations derived in this paper with the JS equations in vector form (5.13), we combine the last two equations of the system (5.9) to write, Consequently, the motion equation in system (5.9) may be expressed as Thus, the present form of the WCIFS fluid equations (5.15) does seem to have some kinematic resemblance to the JS equations, although the two types of dynamics also have major physical and mathematical differences in their interpretations.
For example, one may write the WCIFS motion equation (5.15) equivalently in more compact form, as In this compact form, which is also reminiscent of Choi's relation (2.9), the geometric, coordinate-free nature of the WCIFS equation begins to emerge upon writing (5.15) equivalently as the advective Lie derivative of a 1-form, which also arises in the Kelvin circulation theorem below, cf. (5.21), pB t`Lp v q´p v¨dx`r wdζ¯" d̟´1 ρ dp . (5.17) In one dimension with constant buoyancy ρ " ρ 0 and p " p s " ρ 0 gζ, this formula becomes, pB t`Lv q´vdx`r wdζ¯" d̟´g ρ 0 dζ , (5.18) where ̟ is defined in (5.8). In this geometric form, the JS and WCIFS models look rather more distant.
Balance relations required for significant wave-current interaction. For small Froude number, F r 2 ! 1, equation (5.19) approaches hydrostatic balance, and for small aspect ratio σ 2 ! 1, equation (5.19) suppresses wave activity. When Froude number F r 2 and aspect ratio σ are both of order Op1q, then equation (5.19) admit order Op1q significant non-hydrostatic wave activity. Likewise, the r w equation in (5.9) in dimensionless form for the same scaling parameters becomes Dρ div`Dρ p w r w ∇ζ˘. The balance between current and wave properties in the dimension-free r w equation (5.20) also requires both Froude number F r 2 and aspect ratio σ to be of order Op1q for significant wave activity to occur.
Only the motion equation and the r w equation in (5.9) change their coefficients for these scaling parameters. The coefficients of the others remain unchanged.
Remark 5.5. To prove the following Kelvin-Noether circulation theorem for the system of WCIFS equations in (5.9) it is convenient to return to the variational equations in (5.6) and the notation introduced in (5.7) and (5.8).
Theorem 6 (Kelvin-Noether theorem for the WCIFS model). For every closed loop cpp vq moving with the WCIFS velocity p v for the system of WCIFS equations in (5.9) the Kelvin circulation relation holds, Proof. From the variational equations in (5.6), we have Hence, we find pB t`Lp v q´p v¨dx`r µ Dρ dζ¯" d̟´1 ρ dp , (5.22) and the result (5.21) follows from the standard relation for the time derivative of an integral around a closed moving loop, cpp vq, Remark 5.6 (Interpretation of WCIFS as a compound fluid system). The compound circulation of the WCIFS wave-fluid system in (5.9) obeys the same dynamical equations as the planar incompressible flow description of a single-component flow with horizontal buoyancy gradient, except for two features associated with the wave degrees of freedom. First, the presence of the wave field contributes to the solution for the pressure from the condition that the velocity of the fluid component remains incompressible. Second, the presence of the wave field is a source of circulation for the fluid component of this compound system. Both of these features are due to the momentum of the waves, defined using the notation r w defined in (5.9) as r µ{pDρq∇ζ " r w∇ζ, which is proportional to the wave slope, ∇ζ. In particular, the momentum r w∇ζ appears in both the pressure equation in (5.10) and the Kelvin-Noether integrand in (5.21).
Corollary 7 (Total wave-fluid potential vorticity (PV) for WCIFS). The evolution equation for the total wave-fluid potential vorticity (PV) follows by taking the exterior derivative of equation (5.22) in the proof of the Kelvin circulation theorem for WCIFS. Namely, If we introduce a stream function ψ, so that p v " p zˆ∇ψ, then the previous equation can be written formally in terms of the 2D Laplacian ∆ and the Jacobian Jpψ, ρq :" p z¨∇ψˆ∇ρ " p v¨∇ρ between functions ψ and ρ, then we have B t q`Jpψ , qq "´J´ρ´1, ∇p¯, with PV defined as q :" ∆ψ`J`r w, ζ˘.

Integral conservation laws for the WCIFS equations
Spatially varying specific buoyancy. The system of equations for the PV and specific buoyancy pq, ρ´1q is given by B t q`Jpψ , qq "´J´ρ´1, ∇p¯, B t ρ´1`Jpψ , ρ´1q " 0 . (5.25) This system of equations implies that the following integral quantity is conserved under the pq, ρ´1q dynamics, for arbitrary differentiable functions Φ and Ψ.
Spatially homogeneous specific buoyancy. In the case that the specific buoyancy is initially constant, ρ´1 " ρ´1 0 , then it will remain constant, and ∇ρ´1 " 0 will persist throughout the WCIFS domain of flow. In this case, the pq, ρ´1q system (5.25) will reduce to a single equation, B t q`Jpψ , qq " 0, describing simple advection of the PV quantity, q. Hence, the conserved integral quantities are the familiar vorticity functionals from the 2D Euler equations, or potential vorticity functionals from the quasigeostrophic (QG) equation. Namely, for a spatially homogeneous initial specific buoyancy, the WCIFS system in (5.25) will conserve the following class of integral quantities for an arbitrary differentiable function Φ. Thus, the WCIFS integral conservation laws for PV in equations (5.26) and (5.27) depend on whether the specific buoyancy gradient p∇ρ´1q vanishes at the initial time.
Energy. The conserved WCIFS integrated energy is given by While the energy conservation law may be proven directly from the WCIFS equations in (5.9), it may be more enlightening to discover this energy via the Legendre transformation of the Lagrangian in the action integral S in (5.5) and thereby determine the Hamiltonian formulation and its remarkable properties for the WCIFS system. In particular, the Lie-Poisson bracket in the Hamiltonian formulation of the WCIFS system in the next section will explain the source of the WCIFS conservation laws and their relationships among each other from the viewpoint of the Hamiltonian structure for the WCIFS system.
Proposition 2. Sufficient conditions for pq e , ζ e , r w e q to be an equilibrium solution of (5.32) arise by requiring that the functional H Φ " epq, ζ, r wq`C Φ pqq would have a critical point at pq e , ζ e , r w e q.
Proof. Evaluated at pq e , ζ e , r w e q a critical point the functional H Φ satisfies ż p´ψ e`Φ 1 pq e qδq`δ e δζˇˇˇˇp qe,ζe, r weq δζ`δ e δ r wˇˇˇˇp qe,ζe, r This is sufficient for the right-hand side of equation (5.32) to vanish and thereby produce an equilibrium solution.
The Lie-Poisson brackets in the Hamiltonian formulations of the system provided here for the case of constant buoyancy, ρ " ρ 0 , and in the next section for the general case explain the sources of the conservation laws and their relationships among each other from the viewpoint of the Hamiltonian structure for the system.

Hamiltonian formulation of the WCIFS equations
Legendre transformation. By considering the Lagrangian function in the action integral (5.5), ℓpp v, D, ρ, ζ, w; pq, one may define the Legendre transformation as the variation with respect to the velocity p v in (5.6). Namely, Upon defining the fluid momentum as m " Dρp v, the Hamiltonian in these variables is obtained via the following calculation hpm, D, ρ, ζ, r µ; pq :" xm, p vy`r µB t ζ´DB t φ`γB t ρ´ℓpp v, D, ρ, ζ, w; pq in the case of zero surface tension. There are some similarities between this Hamiltonian structure of the water-wave equations and the full system of equations we have derived. However, the Hamiltonian for the water-wave equations and one of the canonical variables are vertically integrated compared to (5.34), which is evaluated as projections on the free surface. Lewis et al. [39] generalized the previous canonical structure of Zakharov [52] for irrotational flow to obtain Hamiltonian structures for 2-or 3-dimensional incompressible flows with a free boundary. The Poisson bracket in [39] was determined using reduction from canonical variables in the Lagrangian (material) description. The corresponding Hamiltonian form for the equations of a liquid drop with a free boundary having surface tension was also demonstrated, as was the structure of the bracket in terms of a reduced cotangent bundle of a principal bundle was explained. In the case of twodimensional flows, a vorticity bracket was determined and the generalized enstrophy was shown to be a Casimir function.
A Hamiltonian description of free boundary fluids has also been studied in Mazer and Ratiu [42]. In [42], the Hamiltonian formulation of adiabatic free boundary inviscid fluid flow using only physical variables was presented in both the material and spatial formulation. By using the symmetry of particle relabeling, the noncanonical Poisson bracket in Eulerian representation was derived as a reduction from the canonical bracket in the Lagrangian representation. When the free boundary of the fluid was specified as the zero level set of an array of advected functions (e.g., Lagrangian labels carried by the fluid flow), the formulation of [42] recovered the Lie Poisson bracket of [1], as well as the corresponding potential vorticity and other conserved quantities found in [2].
Finally, in a tour-de-force, Gay-Balmaz and Marsden [23] carried out Lagrangian reduction for free boundary fluids and deduced both the equations of motion and their associated constrained variational principles in both the convective and spatial representations.
None of these previous Hamiltonian formulations of fluid dynamics with free-boundaries represented the free-boundary dynamics in terms of projection properties of the Dirichlet-Neumann operator (DNO) representation of CWW theory, as done in the present approach.

HAMILTONIAN FORMULATION OF THE WCIFS EQUATIONS
Variations of the Hamiltonian. In the Hamiltonian variables, the Bernoulli function ̟ in (5.8) is denoted as The corresponding variational derivatives of the Hamiltonian in (5.34) for the system of equations in (5.7) are given by The system of equations in (5.7) may now be written in Lie-Poisson form, augmented by a symplectic 2-cocycle in the elevation ζ and its canonical momentum density r µ in its entangled form as  Thus, the Poisson bracket block-diagonalises when it is written in terms of the total fluid plus wave momentum, M " m`r µ ∇ζ.

A one dimensional WCIFS equation
We begin by deriving the one dimensional equation by applying Hamilton's principle to the following action integral, which is the one dimensional version of (5.5), where ζpx, tq and upx, tq are one dimensional functions of one dimension in space and we consider the volume form Dρ to be a single variable. Taking variations with respect to each variable gives, δζ : pB t`Lp v qr µ "´Dρg`2ǫB x pr µ p wB x ζq . (5.41) These relations imply a fluid motion equation where r µ " Dρ r w. Thus we have B t p v`p vB x p v "´1 2 r w 2 B x`p 1`ǫ|B x ζ| 2 q 2˘´2 ǫ Dρ B x`D ρ r w 2 p1`ǫ|B x ζ| 2 qB x ζ˘B x ζ , (5.42) and this equation is to be considered together with These one dimensional WCIFS equations will be investigated, elsewhere.

.1 Stochastic Advection by Lie Transport (SALT)
Stochastic advection by Lie transport [25] provides a methodology of stochastically perturbing a continuum model at the level of the action integral, in a manner whereby the Kelvin-Noether circulation theorem is preserved. Consider first the three dimensional case where we have a three dimensional fluid velocity field, evaluated on the free surface, denoted by p u. For a deterministic (unconstrained) Lagrangian, ℓpp u, qq, depending on the velocity field p u and advected quantities q, we 32 6.2 STOCHASTIC ECWWE constrain the advected quantities to follow a stochastically perturbed path via a Lagrange multiplier. For models where we are considering incompressible flow, the pressure must act as a Lagrange multiplier to enforce the advected quantity D, the volume element, to be constant. More specifically, the advection constraint enforces that the advected quantities obey a stochastic partial differential equation given by pd`L dxt qq :" dq`L p u q dt`ÿ i Lξ i q˝dW i t " 0 , (6.1) where the vector field p u has been perturbed in the following way The action integral is, after the introduction of the stochastic transport constraint, thus in the form of a semi-martingale driven variational principle [47] and thus the pressure constraint must be compatible with the noise introduced in the advection. This is since one cannot enforce a variable in a stochastic system to remain constant without the Lagrange multiplier controlling both the deterministic part of the system as well as the random fluctuations. With these constraints, the action integral takes the form S " ż ℓpp v, qq dt`xdp, D´1y`xλ, dq`L dxt qy . (6. 3) The application of Hamilton's principle implies an Euler-Poincaré equation and, as in [25] we have a Kelvin-Noether circulation theorem for the stochastic system which is analogous to that of the deterministic system. For the purposes of our variational wave models, we need a notation for two dimensional advection as well as three dimensional. We recall the notation for the two dimensional velocity field and introduce a new notation for the first two components of the stochastic perturbation as follows p u " pp v, p wq , andξ i " pξ i , ξ 3i q . (6.4) The perturbation of vector field p v is therefore given by dr t " p vpr t , tq dt`ÿ i ξ i pr t q˝dW i t . (6.5)

Stochastic ECWWE
Firstly, we derive the free surface boundary condition (2.3) in the stochastic case by applying the operator d`L drt to z´ζpr, tq to obtain 0 " pd`L drt qpz´ζpr, tqq " p wpr, tq dt`ÿ i ξ 3i prq˝dW i t´d ζpr, tq´L dr ζpr, tq , and hence dζpr, tq`L drt ζpr, tq " p wpr, tq dt`ÿ i ξ 3i prq˝dW i t . (6.6) Where the notation dr t in (6.5) is the path of a Lagrangian coordinate. When we write dependence on r, we mean that r is an Eulerian point which is the pullback of the path defined by (6.5). In more informal language, r is an Eulerian point along the Lagrangian path r t . We may derive the stochastic ECWW equations by considering the dimensional version of the action integral (3.19) where the transport velocity p v has been perturbed as in (6.5). The stochastic action integral is then S " ż ż Dˆ1 2`| p v| 2`p w 2˘´g ζ˙dt`λ´dζ`L drt ζ´p w dt´ÿ i ξ 3i˝d W i tp φ`dD`L drt D˘d 2 r . (6.7)

Stochastic WCIFS equations
Similarly to the stochastic ECWW equations, we may define stochastic versions of any of the wavecurrent models we have derived, including the MCWW equations. Here we will demonstrate this for our most complete wave-current model corresponding to the action integral (5.5). We may again define the equivalent action integral featuring SALT in order to derive the corresponding stochastic system of equations.
In the stochastic case, in order to couple the waves and currents we consider the insertion of the stochastic vector field dx 3 ∇ r ζ, where dx 3 " p w dt`ř i ξ 3i˝d W i t , into the 1-form r µ∇ r ζ.
(6.11) variables. After introducing a wave-current "minimal-coupling" Ansatz reminiscent of the coupling of a charged fluid to an electromagnetic field [31], we will show that this system of equations can be closed and that the wave variables will be able to generate circulation in the fluid. The resulting coupled equations will model a sort of Craik-Leibovich wave-current interaction on the free surface.
Consider an action integral defined by S " ż ż Dρˆ1 2 |u| 2´g z˙´ppD´1q´µpB t`u¨∇ qpζ´zq ϕpB t D`divpDuqq`γpB t ρ`u¨∇ρq d 3 x dt . (B.1) The Lagrange multipliers µ, ϕ, and γ impose the dynamical constraints in (2.2) as pioneered in Clebsch [15]. From left to right, the terms in (B.1) are: the difference between kinetic and potential energies, the incompressible flow constraint, the Clebsch constraint that the quantity pζ´zq is advected (i.e. particles on the surface remain so), and two more Clebsch constraints which impose advection dynamics on D as a density and ρ as a scalar function, respectively.
Remark B.1. The action integral in (B.1) makes sense physically, as long as the free surface z " ζpr, tq is a graph, so that the magnitude of the elevation slope |∇ r ζ| remains bounded. Hence, we assume that no wave breaking will occur in the underlying fluid model during the temporal interval of the flow.
The variations with respect to each dynamical variable yields the following independent relations, written in the coordinate-free Lie derivative notation discussed in Appendix A, δD : pB t`Lu qϕ " ρˆ1 2 |u| 2´g z˙´p δρ : pB t`Lu q´γ D¯" where L u denotes Lie derivative with respect to the three-dimensional velocity vector field. We have also imposed natural homogeneous boundary conditions. Assembling these variational equations leads