Introduction
Waves are ubiquitous in ocean dynamics and play a central role in air-sea interaction, coastal protection, marine navigation, and climate system dynamics [37 , 68 ] . Wave models are essential for hazard monitoring associated with wave submersion, coastal erosion, and tsunamis [29 ] . While important insights can be derived from linear wave theory [2 ] , most wave systems are intrinsically nonlinear due to the coupling between surface elevation and fluid velocity, the occurrence of breaking, and the transfer of energy across scales [69 , 40 ] . Nonlinear effects manifest in various forms, such as nonlinear dispersion relations, amplitude-dependent wave speeds, and energy transfers among different components of the wave system [31 , 32 ] . However, fully nonlinear wave models, whether based on the Euler equations or on the Zakharov–Craig–Sulem Hamiltonian formulation [69 , 13 ] , remain challenging to solve numerically, especially when broadband spectra and long-time integrations are required [40 ] . Moreover, surface waves constitute a vast energy reservoir [21 ] that is only weakly and indirectly represented in models, owing to the difficulty of coupling waves and currents in numerical models [7 , 60 , 66 ] .
Statistical spectral wave models, which describe the evolution of the wave action spectrum and associated statistical moments – usually referred to as phase-averaged spectral wave models – are often more tractable and computationally efficient [54 ] . However, they do not accurately represent phase evolution and rely heavily on semi-empirical parameterizations of nonlinear source terms [4 , 34 , 35 , 37 ] . The classical Hasselmann kinetic equation [31 ] , derived from weak turbulence theory, has become the cornerstone of operational wave forecasting [62 , 64 ] ; yet it assumes random phases and weak nonlinearity, and its numerical implementation requires careful treatment of the Boltzmann collision integral [51 ] . Moreover, the role of unresolved scales, such as internal waves, submesoscale eddies, and background turbulence, in redistributing wave energy remains poorly understood and is largely absent from operational models [46 , 63 ] .
Here, we aim to derive stochastic dynamical models for wave evolution that bridge the gap between phase-resolved and phase-averaged descriptions. In such models, waves are represented as stochastic processes combining a phase-resolved component with a noise term associated with unresolved wave components. In this sense, the approach reconciles phase-averaged and phase-resolved descriptions, offering a unified framework in which the effects of unresolved scales are represented through a systematic stochastic closure. The formalism is fully stochastic and introduces stochastic partial differential equations to describe wave evolution, thereby providing a principled alternative to empirical source-term parameterizations. This approach builds upon recent advances in stochastic ocean dynamics modeling [6 , 41 , 48 , 52 , 53 , 65 ] and stochastic variational principles for geophysical flows [12 , 16 , 36 , 57 ] . These approaches, incorporating small-scale noise advection – often referred to as transport noise – are supported by extensive recent mathematical analysis; see, for example, [1 , 8 , 9 , 14 , 15 , 19 , 22 , 23 , 24 , 25 , 27 , 28 , 30 , 39 , 18 ] and references therein.
In contrast to stochastic parameterizations involving ad hoc additive or multiplicative forcing, fluctuation-dissipation balance, together with noise-induced advection consistent with Generalized Lagrangian Mean (GLM) theory, are naturally embedded within this formalism [6 ] . The introduction of time-correlated noise models and multiscale variational principles [16 ] has recently helped connect small-scale dynamics driven by large-scale processes to Kraichnan-Tennekes turbulence [10 , 38 , 61 ] , as well as to the seminal multiscale ocean-atmosphere stochastic parameterization proposed by K. Hasselmann [26 , 33 ] and MTV semi-empirical approaches [44 , 45 ] . The introduction of regularized noise processes for wave modeling also provides a methodological framework that is much easier to handle than directly considering delta-correlated noise processes [20 ] , which is particularly advantageous in the wave context.
The present paper establishes the theoretical foundations of such a stochastic approach for waves. Starting from Luke’s variational principle for irrotational, incompressible free-surface flow [42 ] , we introduce a decomposition of the velocity potential into a large-scale deterministic component and a regularized stochastic noise term representing unresolved scales. A pathwise variational principle, inspired by the stochastic variational framework of [16 ] , yields stochastic counterparts of the Laplace, Bernoulli, and kinematic boundary conditions. To close the system, a second variational principle in expectation is formulated, providing evolution equations for the noise correlation functions. The resulting coupled system preserves the Hamiltonian structure of the Zakharov–Craig–Sulem formulation [13 , 59 , 69 ] , ensuring that important invariants such as total energy are respected in a statistical sense. We further analyze explicit solutions via WKB approximations [55 , 67 ] and show that the noise correlation functions satisfy a Hamilton-Jacobi equation with ray-tracing dynamics, linking the stochastic framework to geometric optics and wave packet propagation. A companion paper [49 ] develops, for a linear stochastic coupled system, the asymptotic approximations, kinetic theory, and geophysical implications, including a detailed comparison with Hasselmann-type nonlinearities using JONSWAP observations [34 ] , and suggests that stochastic transport by unresolved scales can dominate classical four-wave interactions across a broad range of realistic ocean conditions. This is reminiscent to the attempt of defining simpler stochastic linear models in wavenumber space to represent wave interactions [3 ] .
The remainder of the paper is organized as follows. Section 2 recalls deterministic preliminaries, including Luke’s principle, the Dirichlet-to-Neumann operator, and the Zakharov–Craig–Sulem Hamiltonian formulation. Section 3 develops the stochastic extension through two coupled variational principles. Section 4 derives asymptotic approximations (Airy, Saint-Venant, Serre-Green-Naghdi, and Boussinesq). Section 5 presents explicit WKB and Ornstein-Uhlenbeck solutions. Appendices provide technical background on the Dirichlet-to-Neumann operator.
Deterministic Preliminaries: Luke's Principle and the Zakharov–Craig–Sulem Hamiltonian
In this section, we introduce two equivalent deterministic variational principles for surface-wave dynamics.
Luke's functional and its critical points
Luke’s variational principle [43 ] for irrotational, incompressible free-surface flow is:
S ( ϕ , η ) = ∫ t 0 t 1 ∫ D h ∫ b η ( x , t ) − ( ∂ t ϕ + 1 2 ∣ ∇ ϕ ∣ 2 + g z ) ρ d z d x d t (1) \mathcal{S}(\phi,\eta)=\int_{t_0}^{t_1}\int_{\mathcal{D}^h}\int_b^{\eta(x,t)}-\left(\partial_t\phi+\frac{1}{2}|\nabla\phi|^2+gz\right)\rho\,\mathrm{d}z\,\mathrm{d}x\,\mathrm{d}t \tag*{(1)} S ( ϕ , η ) = ∫ t 0 t 1 ∫ D h ∫ b η ( x , t ) − ( ∂ t ϕ + 2 1 ∣∇ ϕ ∣ 2 + g z ) ρ d z d x d t (1) where η ( x , t ) \eta(x,t) η ( x , t ) ϕ ( x , z , t ) \phi(x,z,t) ϕ ( x , z , t ) g g g ρ \rho ρ D h \mathcal{D}^h D h b b b [11 ] .
This action constitutes a Lagrangian variational principle and one can show it is possible to transition equivalently to the Zakharov–Craig–Sulem Hamiltonian principle [13 , 69 ] . Both the Lagrangian and Hamiltonian formulations are useful in their own right for deriving the water-wave evolution equations. We first characterise the critical points of Luke’s functional.
Let us compute the variations of the functional (1) with respect to ϕ \phi ϕ η \eta η η \eta η ϕ \phi ϕ φ ( x , t ) = ϕ ( x , η ( x , t ) , t ) \varphi(x,t)=\phi(x,\eta(x,t),t) φ ( x , t ) = ϕ ( x , η ( x , t ) , t ) ( η , φ ) (\eta,\varphi) ( η , φ ) η ( x , t ) \eta(x,t) η ( x , t )
First, keeping η \eta η S \mathcal{S} S δ ϕ \delta\phi δ ϕ Ω ( t ) = { ( x , z ) ∈ D h × R : b ≤ z < η ( x , t ) } \Omega(t)=\{(x,z)\in\mathcal{D}^h\times\mathbb{R}:b\le z<\eta(x,t)\} Ω ( t ) = {( x , z ) ∈ D h × R : b ≤ z < η ( x , t )}
δ ϕ S = − ∫ t 0 t 1 ∫ Ω ( ρ ∂ t δ ϕ + ρ ∇ ϕ ∇ δ ϕ ) d x d z d t . (2) \delta_{\phi}\mathcal{S}=-\int_{t_0}^{t_1}\int_{\Omega}\left(\rho\partial_t\delta\phi+\rho\nabla\phi\nabla\delta\phi\right)\,\mathrm{d}x\,\mathrm{d}z\,\mathrm{d}t. \tag*{(2)} δ ϕ S = − ∫ t 0 t 1 ∫ Ω ( ρ ∂ t δ ϕ + ρ ∇ ϕ ∇ δ ϕ ) d x d z d t . (2) The first term reduces to a total time derivative, which vanishes for test functions satisfying δ ϕ ( t 0 ) = 0 \delta\phi(t_0)=0 δ ϕ ( t 0 ) = 0 δ ϕ ( t 1 ) = 0 \delta\phi(t_1)=0 δ ϕ ( t 1 ) = 0 δ ϕ \delta\phi δ ϕ
Δ ϕ = 0 ∀ ( x , z ) ∈ Ω , ∂ z ϕ = 0 ∀ z = b , (3) \Delta\phi=0\quad\forall(x,z)\in\Omega,\qquad\partial_z\phi=0\quad\forall z=b, \tag*{(3)} Δ ϕ = 0 ∀ ( x , z ) ∈ Ω , ∂ z ϕ = 0 ∀ z = b , (3) together with Neumann, periodic or null Dirichlet boundary condition on the lateral domain. This yields the harmonicity condition satisfied by the velocity potential in the fluid interior.
The upper boundary conditions are obtained through the variations with respect to φ \varphi φ η \eta η
δ η S ∣ z = η ( x , t ) = − ∫ t 0 t 1 ∫ D h ρ ( ∂ t ϕ + g η + 1 2 ∣ ∇ ϕ ∣ 2 ) δ η d x d t , (4) \left.\delta_{\eta}\mathcal{S}\right|_{z=\eta(x,t)}=-\int_{t_0}^{t_1}\int_{\mathcal{D}^h}\rho\left(\partial_t\phi+g\eta+\frac{1}{2}|\nabla\phi|^2\right)\delta\eta\,\mathrm{d}x\,\mathrm{d}t, \tag*{(4)} δ η S ∣ z = η ( x , t ) = − ∫ t 0 t 1 ∫ D h ρ ( ∂ t ϕ + g η + 2 1 ∣∇ ϕ ∣ 2 ) δ η d x d t , (4) Setting this variation to zero for arbitrary test functions yields the Bernoulli equation at the free surface:
∂ t ϕ + g η + 1 2 ∣ ∇ ϕ ∣ 2 = 0 , at z = η ( x , t ) . (5) \partial_t\phi+ g\eta+ \frac{1}{2}\lvert\nabla\phi\rvert^2 = 0,\qquad\text{at } z=\eta(x,t). \tag*{(5)} ∂ t ϕ + g η + 2 1 ∣ ∇ ϕ ∣ 2 = 0 , at z = η ( x , t ) . (5) The trace of the velocity potential and the Dirichlet-to-Neumann operator
To write this equation in terms of the trace of the potential at the surface, we introduce the Dirichlet-to-Neumann operator [13 , 69 ] :
G [ η ] φ = ( ∂ z φ − ∇ x φ ⋅ ∇ η ) ∣ z = η . (6) G[\eta]\varphi= \left.(\partial_z\varphi-\nabla_x\varphi\cdot\nabla\eta)\right|_{z=\eta}. \tag*{(6)} G [ η ] φ = ( ∂ z φ − ∇ x φ ⋅ ∇ η ) ∣ z = η . (6) This operator maps a Dirichlet boundary value φ \varphi φ [40 ] . In particular, for sufficiently smooth functions satisfying appropriate decay or homogeneous boundary conditions at the bottom, this operator is non negative and self adjoint on a dense subspace of L 2 ( R d ) L^2(\mathbb{R}^d) L 2 ( R d ) [40 ] . By application of the chain rule, the expressions of the velocity potential temporal derivative, horizontal gradients and vertical derivative at the free surface can be written, respectively as:
∂ t φ ∣ z = η = ∂ t ϕ − ∂ z φ ∣ z = η ∂ t η , (7) \left.\partial_t\varphi\right|_{z=\eta}=\partial_t\phi-\left.\partial_z\varphi\right|_{z=\eta}\partial_t\eta, \tag*{(7)} ∂ t φ ∣ z = η = ∂ t ϕ − ∂ z φ ∣ z = η ∂ t η , (7) ∇ x φ ∣ z = η = ∇ ϕ − ∂ z φ ∣ z = η ∇ η , (8) \left.\nabla_x\varphi\right|_{z=\eta}=\nabla\phi-\left.\partial_z\varphi\right|_{z=\eta}\nabla\eta, \tag*{(8)} ∇ x φ ∣ z = η = ∇ ϕ − ∂ z φ ∣ z = η ∇ η , (8) ∂ z φ ∣ z = η = G [ η ] ϕ + ∇ η ⋅ ∇ ϕ 1 + ∣ ∇ η ∣ 2 . (9) \left.\partial_z\varphi\right|_{z=\eta}=\frac{G[\eta]\phi+\nabla\eta\cdot\nabla\phi}{1+\lvert\nabla\eta\rvert^2}. \tag*{(9)} ∂ z φ ∣ z = η = 1 + ∣ ∇ η ∣ 2 G [ η ] ϕ + ∇ η ⋅ ∇ ϕ . (9) To proceed further in rewriting the Bernoulli equation, we need the variation with respect to the trace. To obtain this variation, we rewrite the kinetic energy term in terms of φ \varphi φ
∫ Ω h ∫ b η ( x , t ) 1 2 ∣ ∇ φ ∣ 2 d z = 1 2 ∫ S ϕ ( s , t ) ∂ n φ ( s , t ) d s , (10) \int_{\Omega_h}\int_b^{\eta(x,t)}\frac{1}{2}\lvert\nabla\varphi\rvert^2\,dz=\frac{1}{2}\int_S\phi(s,t)\,\partial_n\varphi(s,t)\,ds, \tag*{(10)} ∫ Ω h ∫ b η ( x , t ) 2 1 ∣ ∇ φ ∣ 2 d z = 2 1 ∫ S ϕ ( s , t ) ∂ n φ ( s , t ) d s , (10) using the harmonicity of the velocity potential (i.e. Δ φ = 0 \Delta\varphi=0 Δ φ = 0 ∂ n φ = 0 \partial_n\varphi=0 ∂ n φ = 0 S = { ( x , z ) ∈ R d × R : z − η ( x , t ) = 0 } S=\{(x,z)\in\mathbb{R}^d\times\mathbb{R}:z-\eta(x,t)=0\} S = {( x , z ) ∈ R d × R : z − η ( x , t ) = 0 }
∂ n φ = ∇ φ ⋅ n = ( 1 + ∣ ∇ η ∣ 2 ) − 1 / 2 G [ η ] φ . (11) \partial_n\varphi=\nabla\varphi\cdot n=(1+\lvert\nabla\eta\rvert^2)^{-1/2}G[\eta]\varphi. \tag*{(11)} ∂ n φ = ∇ φ ⋅ n = ( 1 + ∣ ∇ η ∣ 2 ) − 1/2 G [ η ] φ . (11) Using the surface area element d s = ( ( 1 + ∣ ∇ η ∣ 2 ) 1 / 2 ) d x ds=((1+\lvert\nabla\eta\rvert^2)^{1/2})dx d s = (( 1 + ∣ ∇ η ∣ 2 ) 1/2 ) d x
∫ Ω h ∫ b η ( x , t ) 1 2 ∣ ∇ φ ∣ 2 d z = 1 2 ∫ Ω h ϕ G [ η ] ϕ d x . (12) \int_{\Omega_h}\int_b^{\eta(x,t)}\frac{1}{2}\lvert\nabla\varphi\rvert^2\,dz=\frac{1}{2}\int_{\Omega_h}\phi\,G[\eta]\phi\,dx. \tag*{(12)} ∫ Ω h ∫ b η ( x , t ) 2 1 ∣ ∇ φ ∣ 2 d z = 2 1 ∫ Ω h ϕ G [ η ] ϕ d x . (12) The variation with respect to the potential trace, reads now
δ ϕ S = − ρ ∫ t 0 t 1 ( d d t ∫ Ω ( t ) δ ϕ d x d z − ∫ Ω h ∂ t η δ ϕ d x + ∫ Ω h G [ η ] ϕ δ ϕ d x ) d t , (13) \delta_\phi\mathcal{S}=-\rho\int_{t_0}^{t_1}\left(\frac{d}{dt}\int_{\Omega(t)}\delta\phi\,dx\,dz-\int_{\Omega_h}\partial_t\eta\,\delta\phi\,dx+\int_{\Omega_h}G[\eta]\phi\,\delta\phi\,dx\right)dt, \tag*{(13)} δ ϕ S = − ρ ∫ t 0 t 1 ( d t d ∫ Ω ( t ) δ ϕ d x d z − ∫ Ω h ∂ t η δ ϕ d x + ∫ Ω h G [ η ] ϕ δ ϕ d x ) d t , (13) where the self-adjointness of the Dirichlet-to-Neumann operator has been used and δ ϕ \delta\phi δ ϕ δ φ \delta\varphi δ φ
∂ t η = G [ η ] φ . (14) \partial_t\eta= G[\eta]\varphi. \tag*{(14)} ∂ t η = G [ η ] φ . (14) or with the definition of the Dirichlet-to-Neumann operator
∂ t η + ∇ x φ ∣ z = η ⋅ ∇ η = ∂ z φ ∣ z = η , (15) \partial_t\eta+ \left.\nabla_x\varphi\right|_{z=\eta}\cdot\nabla\eta= \left.\partial_z\varphi\right|_{z=\eta}, \tag*{(15)} ∂ t η + ∇ x φ ∣ z = η ⋅ ∇ η = ∂ z φ ∣ z = η , (15) which is the familiar kinematic free-surface condition. This condition enforces that fluid particles initially on the free surface remain on it and provides the evolution of the free surface.
The Bernoulli equation can now be rewritten in terms of the surface trace and the Dirichlet-to-Neumann operator. Using the expression of potential surface partial derivatives (7)–(9) and the kinematic condition (14) in (5), the Bernoulli equation at the surface becomes:
∂ t φ + g η + 1 2 ∣ ∇ φ ∣ 2 − 1 2 ( G [ η ] φ + ∇ η ⋅ ∇ φ ) 2 1 + ∣ ∇ η ∣ 2 = 0. (16) \partial_t\varphi+ g\eta+ \frac{1}{2}\lvert\nabla\varphi\rvert^2 - \frac{1}{2}\frac{\left(G[\eta]\varphi+\nabla\eta\cdot\nabla\varphi\right)^2}{1+\lvert\nabla\eta\rvert^2}=0. \tag*{(16)} ∂ t φ + g η + 2 1 ∣ ∇ φ ∣ 2 − 2 1 1 + ∣ ∇ η ∣ 2 ( G [ η ] φ + ∇ η ⋅ ∇ φ ) 2 = 0. (16) In the next section, we show how to recover the Zakharov–Craig–Sulem Hamiltonian formulation from (1).
Reduction to the Zakharov–Craig–Sulem Hamiltonian
To pass from Luke’s action principle to the Zakharov–Craig–Sulem Hamiltonian formulation, we first note, using Leibniz’s rule, that Luke’s functional can be equivalently rewritten as
S ( φ , η ) = − ∫ t 0 t 1 ρ ( ∂ t ∫ Ω h ∫ b η ( x , t ) φ ( x , z , t ) d x d z − ∫ Ω h ∂ t η ( x , t ) φ ( x , η ( x , t ) , t ) d x + ∫ Ω h ( ∫ b η ( x , t ) 1 2 ∣ ∇ φ ∣ 2 d z + 1 2 g η 2 ( x , t ) − 1 2 g b 2 ) d x ) d t . (17) \mathcal{S}(\varphi,\eta)=-\int_{t_0}^{t_1}\rho\left(\partial_t\int_{\Omega_h}\int_b^{\eta(x,t)}\varphi(x,z,t)\,dx\,dz-\int_{\Omega_h}\partial_t\eta(x,t)\varphi(x,\eta(x,t),t)\,dx+\int_{\Omega_h}\left(\int_b^{\eta(x,t)}\frac{1}{2}\lvert\nabla\varphi\rvert^2\,dz+\frac{1}{2}g\eta^2(x,t)-\frac{1}{2}gb^2\right)\,dx\right)\,dt. \tag*{(17)} S ( φ , η ) = − ∫ t 0 t 1 ρ ( ∂ t ∫ Ω h ∫ b η ( x , t ) φ ( x , z , t ) d x d z − ∫ Ω h ∂ t η ( x , t ) φ ( x , η ( x , t ) , t ) d x + ∫ Ω h ( ∫ b η ( x , t ) 2 1 ∣ ∇ φ ∣ 2 d z + 2 1 g η 2 ( x , t ) − 2 1 g b 2 ) d x ) d t . (17) The variation of the first term according to φ \varphi φ
− ρ ∫ Ω t 1 δ φ ( x , z , t 0 ) d x d z + ρ ∫ Ω t 1 δ φ ( x , z , t 1 ) d x d z (18) -\rho\int_{\Omega_{t_1}}\delta\varphi(x,z,t_0)\,dx\,dz+\rho\int_{\Omega_{t_1}}\delta\varphi(x,z,t_1)\,dx\,dz \tag*{(18)} − ρ ∫ Ω t 1 δ φ ( x , z , t 0 ) d x d z + ρ ∫ Ω t 1 δ φ ( x , z , t 1 ) d x d z (18) and vanishes since the test functions considered vanish at the boundaries of the time domain. This applies also to the variation with respect to the potential trace. As for the variation with respect to the surface elevation, we have
∫ Ω h ( φ ( x , η , t 0 ) δ η ( x , t 0 ) − φ ( x , η , t 1 ) δ η ( x , t 1 ) ) d x , (19) \int_{\Omega_h}\left(\varphi(x,\eta,t_0)\delta\eta(x,t_0)-\varphi(x,\eta,t_1)\delta\eta(x,t_1)\right)\,dx, \tag*{(19)} ∫ Ω h ( φ ( x , η , t 0 ) δ η ( x , t 0 ) − φ ( x , η , t 1 ) δ η ( x , t 1 ) ) d x , (19) which also vanishes for the same reason. Thus the first term in (17) does not contribute to the variational equations; it may therefore be discarded. Luke’s functional can hence be rewritten as:
S ( φ , η ) = ∫ t 0 t 1 ρ ( ∫ Ω h ∂ t η ( x , t ) φ ( x , η ( x , t ) , t ) d x − ∫ Ω h ( ∫ b η ( x , t ) 1 2 ∣ ∇ φ ∣ 2 d z + 1 2 g η 2 ( x , t ) ) d x ) d t . (20) \mathcal{S}(\varphi,\eta)=\int_{t_0}^{t_1}\rho\left(\int_{\Omega_h}\partial_t\eta(x,t)\varphi(x,\eta(x,t),t)\,dx-\int_{\Omega_h}\left(\int_b^{\eta(x,t)}\frac{1}{2}\lvert\nabla\varphi\rvert^2\,dz+\frac{1}{2}g\eta^2(x,t)\right)\,dx\right)\,dt. \tag*{(20)} S ( φ , η ) = ∫ t 0 t 1 ρ ( ∫ Ω h ∂ t η ( x , t ) φ ( x , η ( x , t ) , t ) d x − ∫ Ω h ( ∫ b η ( x , t ) 2 1 ∣ ∇ φ ∣ 2 d z + 2 1 g η 2 ( x , t ) ) d x ) d t . (20) where the constant gravitational potential at the bottom has been removed. The second line of this functional represents the Hamiltonian H \mathcal{H} H ρ \rho ρ ρ \rho ρ
H ( η , φ ) = 1 2 ∫ D h φ G [ η ] φ d x + 1 2 ∫ D h g η 2 d x , (21) \mathcal{H}(\eta,\varphi)=\frac{1}{2}\int_{\mathcal{D}^{h}}\varphi G[\eta]\varphi\,\mathrm{d}x+\frac{1}{2}\int_{\mathcal{D}^{h}}g\eta^{2}\,\mathrm{d}x, \tag*{(21)} H ( η , φ ) = 2 1 ∫ D h φG [ η ] φ d x + 2 1 ∫ D h g η 2 d x , (21) which corresponds to the Hamiltonian introduced by Zakharov and Craig-Sulem for the surface water waves. In compact form this action functional reads
S ( φ , η ) = ∫ t 0 t 1 ∫ D h ( φ ∂ t η − 1 2 φ G [ η ] φ − 1 2 g η 2 ) d x d t . (22) \mathcal{S}(\varphi,\eta)=\int_{t_{0}}^{t_{1}}\int_{\mathcal{D}^{h}}\left(\varphi\partial_{t}\eta-\frac{1}{2}\varphi G[\eta]\varphi-\frac{1}{2}g\eta^{2}\right)\,\mathrm{d}x\,\mathrm{d}t. \tag*{(22)} S ( φ , η ) = ∫ t 0 t 1 ∫ D h ( φ ∂ t η − 2 1 φG [ η ] φ − 2 1 g η 2 ) d x d t . (22) This is precisely the canonical action associated with the Legendre transform
∫ t 0 t 1 ( ⟨ p , q ˙ ⟩ − H ( q , p ) ) d t , \int_{t_{0}}^{t_{1}}\left(\langle p,\dot{q}\rangle-\mathcal{H}(q,p)\right)\,\mathrm{d}t, ∫ t 0 t 1 ( ⟨ p , q ˙ ⟩ − H ( q , p ) ) d t , where η \eta η φ \varphi φ
The Euler–Lagrange equations (14) –(16) may equivalently be written as the canonical Hamilton equations:
∂ t η = δ H δ φ , (23) \partial_{t}\eta=\frac{\delta\mathcal{H}}{\delta\varphi}, \tag*{(23)} ∂ t η = δ φ δ H , (23) ∂ t φ = − δ H δ η . (24) \partial_{t}\varphi=-\frac{\delta\mathcal{H}}{\delta\eta}. \tag*{(24)} ∂ t φ = − δ η δ H . (24) These equations coincide with the critical points obtained from Luke’s variational functional. This equivalence follows from the canonical form of the reduced action.
Non-dimensionalization of the wave system
Before deriving a stochastic formulation of the water-wave equations, we first introduce a dimensionless form of the system using scaling: x = L x ~ x=L\widetilde{x} x = L x z = H z ~ z=H\widetilde{z} z = H z ⋅ ~ \widetilde{\mathord{\cdot}} ⋅ L L L H H H L L L H H H η = A η ~ \eta=A\widetilde{\eta} η = A η A A A
Wave dynamics is typically characterized by the dimensionless parameters
ϵ = A H , μ = H 2 L 2 , ν = tanh ( 2 π μ ) 2 π μ , \epsilon=\frac{A}{H},\qquad\mu=\frac{H^{2}}{L^{2}},\qquad\nu=\frac{\tanh(2\pi\sqrt{\mu})}{2\pi\sqrt{\mu}}, ϵ = H A , μ = L 2 H 2 , ν = 2 π μ tanh ( 2 π μ ) , where ϵ \epsilon ϵ μ \mu μ ν \nu ν μ ≪ 1 \mu\ll1 μ ≪ 1 ν ≈ 1 \nu\approx1 ν ≈ 1 μ ≫ 1 \mu\gg1 μ ≫ 1 ν ≈ ( 2 π μ ) − 1 \nu\approx(2\pi\sqrt{\mu})^{-1} ν ≈ ( 2 π μ ) − 1 [40 ] the parameter ν \nu ν
t = L g H ν t ~ t = \frac{L}{\sqrt{gH\nu}}\tilde{t} t = g H ν L t ~ and the velocity potential is accordingly rescaled as ϕ = Φ 0 ϕ ~ \phi= \Phi_0\tilde{\phi} ϕ = Φ 0 ϕ ~
Φ 0 = ϵ L g H ν . \Phi_0 = \epsilon L\sqrt{\frac{gH}{\nu}}. Φ 0 = ϵ L ν g H . With these scalings, the Dirichlet-to-Neumann operator becomes
G [ η ] ϕ = Φ 0 H G μ , ϵ [ η ~ ] ϕ ~ , with G μ , ϵ [ η ~ ] ϕ ~ = ( ∂ z ϕ ~ − μ ϵ ∇ x ϕ ~ ⋅ ∇ η ~ ) ∣ z = ϵ η ~ , (25) G[\eta]\phi= \frac{\Phi_0}{H}G^{\mu,\epsilon}[\tilde{\eta}]\tilde{\phi}, \qquad\text{with}\qquad G^{\mu,\epsilon}[\tilde{\eta}]\tilde{\phi} = \left(\partial_z\tilde{\phi}-\mu\epsilon\nabla_x\tilde{\phi}\cdot\nabla\tilde{\eta}\right)\bigg|_{z=\epsilon\tilde{\eta}}, \tag*{(25)} G [ η ] ϕ = H Φ 0 G μ , ϵ [ η ~ ] ϕ ~ , with G μ , ϵ [ η ~ ] ϕ ~ = ( ∂ z ϕ ~ − μ ϵ ∇ x ϕ ~ ⋅ ∇ η ~ ) z = ϵ η ~ , (25) the dimensionless Dirichlet-to-Neumann operator associated with the scaled Laplace problem. In what follows, unless otherwise specified, all variables are understood to be dimensionless, and tildes are omitted for simplicity. Under these scalings, the velocity potential satisfies the dimensionless Laplace equation
μ Δ x ϕ + ∂ z z ϕ = 0 , \mu\Delta_x\phi+\partial_{zz}\phi=0, μ Δ x ϕ + ∂ z z ϕ = 0 , together with the corresponding bottom boundary condition. The dimensionless Zakharov–Craig–Sulem system then reads
∂ t η = 1 μ ν G μ , ϵ [ η ] ϕ , (26) \partial_t\eta=\frac{1}{\mu\nu}G^{\mu,\epsilon}[\eta]\phi, \tag*{(26)} ∂ t η = μν 1 G μ , ϵ [ η ] ϕ , (26) ∂ t ϕ + η + ϵ 2 ν ∣ ∇ ϕ ∣ 2 − 1 2 ϵ μ ν ( 1 μ G μ , ϵ [ η ] ϕ + ϵ ∇ η ⋅ ∇ ϕ ) 2 1 + ϵ 2 μ ∣ ∇ η ∣ 2 = 0. (27) \partial_t\phi+\eta+\frac{\epsilon}{2\nu}|\nabla\phi|^2-\frac{1}{2}\frac{\epsilon\mu}{\nu}\frac{\left(\frac{1}{\mu}G^{\mu,\epsilon}[\eta]\phi+\epsilon\nabla\eta\cdot\nabla\phi\right)^2}{1+\epsilon^2\mu|\nabla\eta|^2}=0. \tag*{(27)} ∂ t ϕ + η + 2 ν ϵ ∣∇ ϕ ∣ 2 − 2 1 ν ϵ μ 1 + ϵ 2 μ ∣∇ η ∣ 2 ( μ 1 G μ , ϵ [ η ] ϕ + ϵ ∇ η ⋅ ∇ ϕ ) 2 = 0. (27) As discussed above, the water-wave problem admits an equivalent Hamiltonian formulation in terms of the total energy of the fluid. Luke’s variational principle yields the Euler equations as its Euler–Lagrange equations. In dimensional variables, the incompressible Euler equations for an irrotational flow read
∂ t ∇ ϕ + ( ∇ ϕ ⋅ ∇ ) ∇ ϕ + g e z + ∇ π = 0 , (28) \partial_t\nabla\phi+(\nabla\phi\cdot\nabla)\nabla\phi+g e_z+\nabla\pi=0, \tag*{(28)} ∂ t ∇ ϕ + ( ∇ ϕ ⋅ ∇ ) ∇ ϕ + g e z + ∇ π = 0 , (28) Δ ϕ = 0 , (29) \Delta\phi=0, \tag*{(29)} Δ ϕ = 0 , (29) ∂ t η = G [ η ] ϕ , (30) \partial_t\eta=G[\eta]\phi, \tag*{(30)} ∂ t η = G [ η ] ϕ , (30) ∂ z ϕ = 0 , z = b (31) \partial_z\phi=0,\quad z=b \tag*{(31)} ∂ z ϕ = 0 , z = b (31) where e z = ( 0 , 0 , 1 ) e_z=(0,0,1) e z = ( 0 , 0 , 1 ) π \pi π
∂ t ϕ + 1 2 ∣ ∇ ϕ ∣ 2 + g z + π = 0. (32) \partial_t\phi+\frac{1}{2}|\nabla\phi|^2+gz+\pi=0. \tag*{(32)} ∂ t ϕ + 2 1 ∣∇ ϕ ∣ 2 + g z + π = 0. (32) Here π \pi π [5 , 40 ] and stochastic contexts [17 , 58 ] .
Using the same scaling as above, the dimensionless Euler equations, separated into horizontal and vertical components, read:
∂ t ∇ x ϕ + ϵ ν ( ∇ x ϕ ⋅ ∇ x + 1 μ ∂ z ϕ ∂ z ) ∇ x ϕ = − 1 ϵ ∇ x π , (33) \partial_t\nabla_x\phi+\frac{\epsilon}{\nu}\left(\nabla_x\phi\cdot\nabla_x+\frac{1}{\mu}\partial_z\phi\,\partial_z\right)\nabla_x\phi=-\frac{1}{\epsilon}\nabla_x\pi, \tag*{(33)} ∂ t ∇ x ϕ + ν ϵ ( ∇ x ϕ ⋅ ∇ x + μ 1 ∂ z ϕ ∂ z ) ∇ x ϕ = − ϵ 1 ∇ x π , (33) ∂ t ∂ z ϕ + ϵ ν ( ∇ x ϕ ⋅ ∇ x + 1 μ ∂ z ϕ ∂ z ) ∂ z ϕ = − 1 ϵ ( ∂ z π + 1 ) , (34) \partial_t\partial_z\phi+\frac{\epsilon}{\nu}\left(\nabla_x\phi\cdot\nabla_x+\frac{1}{\mu}\partial_z\phi\,\partial_z\right)\partial_z\phi=-\frac{1}{\epsilon}\left(\partial_z\pi+1\right), \tag*{(34)} ∂ t ∂ z ϕ + ν ϵ ( ∇ x ϕ ⋅ ∇ x + μ 1 ∂ z ϕ ∂ z ) ∂ z ϕ = − ϵ 1 ( ∂ z π + 1 ) , (34) μ Δ x ϕ + ∂ z 2 ϕ = 0 , (35) \mu\Delta_x\phi+\partial_z^2\phi=0, \tag*{(35)} μ Δ x ϕ + ∂ z 2 ϕ = 0 , (35) ∂ t η = 1 μ ν G μ , ϵ [ η ] ϕ , (36) \partial_t\eta=\frac{1}{\mu\nu}G^{\mu,\epsilon}[\eta]\phi, \tag*{(36)} ∂ t η = μν 1 G μ , ϵ [ η ] ϕ , (36) ∂ z ϕ = 0 , z = b , (37) \partial_z\phi=0,\quad z=b, \tag*{(37)} ∂ z ϕ = 0 , z = b , (37) where the pressure has been non-dimensionalized using the hydrostatic scaling P 0 = ρ g H P_0=\rho gH P 0 = ρ g H
Stochastic water-waves
In this section, we develop a stochastic formulation of the water-wave problem within the stochastic variational framework proposed in [16 ] .
The velocity potential is assumed to decompose into a large-scale component and a fluctuating stochastic component:
ϕ ( x , z , t ) = ϕ ‾ ( x , z , t ) + ϕ τ ( x , z , t ) , with ϕ τ ( x , z , t ) = ∫ t − τ t + τ h τ ( t − s ) ϕ i ( x , z , t ) d β s i . (38) \phi(x,z,t)=\overline{\phi}(x,z,t)+\phi^\tau(x,z,t),\qquad\text{with}\qquad\phi^\tau(x,z,t)=\int_{t-\tau}^{t+\tau}h_\tau(t-s)\phi_i(x,z,t)\,\mathrm{d}\beta_s^i. \tag*{(38)} ϕ ( x , z , t ) = ϕ ( x , z , t ) + ϕ τ ( x , z , t ) , with ϕ τ ( x , z , t ) = ∫ t − τ t + τ h τ ( t − s ) ϕ i ( x , z , t ) d β s i . (38) The flow remains irrotational, with both the large-scale and fluctuating velocity components deriving from their respective velocity potentials.
The noise term ϕ τ \phi^\tau ϕ τ h τ h_\tau h τ h τ ( t ) = 1 τ h ( t / τ ) h_\tau(t)=\frac{1}{\tau}h(t/\tau) h τ ( t ) = τ 1 h ( t / τ ) [ − τ , τ ] [-\tau,\tau] [ − τ , τ ] ϕ i \phi_i ϕ i
E ( ∥ ϕ τ ( t ) ∥ L 2 ( Ω ( t ) ) 2 ) < ∞ . \mathbb{E}\left(\|\phi^\tau(t)\|_{L^2(\Omega(t))}^2\right)<\infty. E ( ∥ ϕ τ ( t ) ∥ L 2 ( Ω ( t )) 2 ) < ∞. The β i \beta_i β i h τ h_\tau h τ τ → 0 \tau\to0 τ → 0
Consequently, the regularized noise process
W t τ = ∫ t − τ t + τ h τ ( t − s ) d β s (39) W_t^\tau=\int_{t-\tau}^{t+\tau}h_\tau(t-s)\,\mathrm{d}\beta_s \tag*{(39)} W t τ = ∫ t − τ t + τ h τ ( t − s ) d β s (39) has variance E [ ( W t τ ) 2 ] = τ − 1 \mathbb{E}[(W_t^\tau)^2]=\tau^{-1} E [( W t τ ) 2 ] = τ − 1 W t τ = O ( τ − 1 / 2 ) W_t^\tau=\mathcal{O}(\tau^{-1/2}) W t τ = O ( τ − 1/2 ) ϕ τ \phi^\tau ϕ τ ( L 2 T − 1 ) (L^2T^{-1}) ( L 2 T − 1 ) ϕ i \phi_i ϕ i L 2 T − 1 / 2 L^2T^{-1/2} L 2 T − 1/2
The free surface is transported by the fluid velocity. Consequently, the kinematic boundary condition expresses the fact that particles initially located on the interface remain on the free surface. This corresponds to a transport constraint posed on the moving interface. We assume that the surface remains sufficiently smooth at all times, with no wave breaking. Accordingly, in the decorrelation limit the free-surface elevation is naturally modeled as a semimartingale of the form
d η = η ~ d t + d η τ . (40) \mathrm{d}\eta= \widetilde{\eta}\,\mathrm{d}t + \mathrm{d}\eta^\tau. \tag*{(40)} d η = η d t + d η τ . (40) In contrast to the velocity, whose fluctuating component becomes white in time in the decorrelation limit, the free-surface elevation remains a semimartingale with continuous trajectories. Consequently, evaluations of the velocity potential at the moving interface remain well defined. In what follows, we introduce two coupled variational principles governing respectively the large-scale variables ( φ ‾ , η ) (\overline{\varphi},\eta) ( φ , η ) φ τ \varphi^\tau φ τ
Path-wise variational principle
A pathwise variational principle for stochastic water waves is obtained by considering the same Luke functional as in the deterministic case:
S ( φ , η ) = ∫ t 0 t 1 ∫ R d ∫ b η ( x , t ) ( ∂ t φ + 1 2 ∣ ∇ φ ∣ 2 + g z ) ρ d z d x d t . (41) \mathcal{S}(\varphi,\eta) = \int_{t_0}^{t_1} \int_{\mathbb{R}^d} \int_b^{\eta(x,t)} \left(\partial_t\varphi+ \frac{1}{2} \lvert\nabla\varphi\rvert^2 + gz\right)\rho\,\mathrm{d}z\,\mathrm{d}x\,\mathrm{d}t . \tag*{(41)} S ( φ , η ) = ∫ t 0 t 1 ∫ R d ∫ b η ( x , t ) ( ∂ t φ + 2 1 ∣ ∇ φ ∣ 2 + g z ) ρ d z d x d t . (41) For simplicity, we assume constant density (i.e. incompressible flow without stochastic density fluctuations). Its Euler–Lagrange equations are formally identical to those of the deterministic case, since the Stratonovich differential product satisfies the usual chain and Leibniz rules.
Interior equations and boundary conditions. The total velocity potential satisfies the Laplace equation in the fluid domain:
Δ φ ‾ + Δ φ τ = 0 ∀ ( x , z ) ∈ Ω t , ∂ z ( φ ‾ + φ τ ) = 0 at z = b , φ ‾ ( x , η ( x , t ) , t ) + φ τ ( x , η ( x , t ) , t ) = ϕ ‾ ( x , t ) + ϕ τ ( x , t ) at z = η ( x , t ) . (42) \begin{aligned}
\Delta\overline{\varphi} + \Delta\varphi^\tau&= 0 && \forall(x,z) \in\Omega_t, \\
\partial_z(\overline{\varphi}+\varphi^\tau) &= 0 && \text{at } z=b, \\
\overline{\varphi}(x,\eta(x,t),t) + \varphi^\tau(x,\eta(x,t),t) &= \overline{\phi}(x,t) + \phi^\tau(x,t) && \text{at } z=\eta(x,t). \tag*{(42)}
\end{aligned} Δ φ + Δ φ τ ∂ z ( φ + φ τ ) φ ( x , η ( x , t ) , t ) + φ τ ( x , η ( x , t ) , t ) = 0 = 0 = ϕ ( x , t ) + ϕ τ ( x , t ) ∀ ( x , z ) ∈ Ω t , at z = b , at z = η ( x , t ) . (42) Under a scale-separation assumption, we treat φ ‾ \overline{\varphi} φ φ τ \varphi^\tau φ τ
Δ φ ‾ = 0 , Δ φ τ = 0 ∀ ( x , z ) ∈ Ω t , ∂ z φ ‾ = 0 , ∂ z φ τ = 0 at z = b , φ ‾ ( x , η ( x , t ) , t ) = ϕ ‾ ( x , t ) , φ τ ( x , η ( x , t ) , t ) = ϕ τ ( x , t ) at z = η ( x , t ) . (43) \begin{aligned}
\Delta\overline{\varphi} &= 0, &\quad\Delta\varphi^\tau&= 0 && \forall(x,z) \in\Omega_t, \\
\partial_z\overline{\varphi} &= 0, &\quad\partial_z\varphi^\tau&= 0 && \text{at } z=b, \\
\overline{\varphi}(x,\eta(x,t),t) &= \overline{\phi}(x,t), &\quad\varphi^\tau(x,\eta(x,t),t) &= \phi^\tau(x,t) && \text{at } z=\eta(x,t). \tag*{(43)}
\end{aligned} Δ φ ∂ z φ φ ( x , η ( x , t ) , t ) = 0 , = 0 , = ϕ ( x , t ) , Δ φ τ ∂ z φ τ φ τ ( x , η ( x , t ) , t ) = 0 = 0 = ϕ τ ( x , t ) ∀ ( x , z ) ∈ Ω t , at z = b , at z = η ( x , t ) . (43) By linearity of the Laplace problem, the Dirichlet-to-Neumann operator satisfies
G [ η ] ϕ = G [ η ] ϕ ‾ + G [ η ] ϕ τ , (44) G[\eta]\phi= G[\eta]\overline{\phi} + G[\eta]\phi^\tau, \tag*{(44)} G [ η ] ϕ = G [ η ] ϕ + G [ η ] ϕ τ , (44) and the Dirichlet-to-Neumann operator acts linearly on this decomposition.
Surface boundary conditions. The kinematic boundary condition and the Bernoulli equation at the free surface read, respectively,
∂ t η + ∇ φ ⋅ ∇ η = ∂ z φ ∣ z = η , (45) \partial_t\eta+ \nabla\varphi\cdot\nabla\eta= \left.\partial_z\varphi\right|_{z=\eta}, \tag*{(45)} ∂ t η + ∇ φ ⋅ ∇ η = ∂ z φ ∣ z = η , (45) ∂ t ϕ + g η + 1 2 ∣ ∇ ϕ ∣ 2 − 1 2 ( G [ η ] φ + ∇ η ⋅ ∇ φ ) 2 1 + ∣ ∇ η ∣ 2 = 0. (46) \partial_t\phi+ g\eta+ \frac{1}{2}\lvert\nabla\phi\rvert^2 - \frac{1}{2}\frac{\left(G[\eta]\varphi+\nabla\eta\cdot\nabla\varphi\right)^2}{1+\lvert\nabla\eta\rvert^2}=0. \tag*{(46)} ∂ t ϕ + g η + 2 1 ∣ ∇ ϕ ∣ 2 − 2 1 1 + ∣ ∇ η ∣ 2 ( G [ η ] φ + ∇ η ⋅ ∇ φ ) 2 = 0. (46) These evolution equations now define a stochastic system due to the presence of the fluctuating component. For finite correlation time, the system is formally well-defined and retains the same structure as the deterministic equations. However, in the decorrelation limit, it becomes singular, since quadratic nonlinearities involving gradients of the noise no longer admits a straightforward interpretation within the variational formulation (or in the Bernoulli equation at the free surface).
To approach the zero-correlation limit, it is possible to construct approximate solutions to the above system of critical point equations such that the limit remains well-defined. One possible approximation consists of prescribing the fluctuating potential such that it satisfies a small-scale Bernoulli equation of the form:
∂ t ϕ τ + 1 2 ∣ ∇ ϕ τ ∣ 2 = 0 , (47) \partial_t\phi^\tau+\frac{1}{2}\lvert\nabla\phi^\tau\rvert^2=0, \tag*{(47)} ∂ t ϕ τ + 2 1 ∣ ∇ ϕ τ ∣ 2 = 0 , (47) where the large-scale surface dynamics is treated as frozen, yielding
∂ t ϕ τ ( x , t ) = ∂ t ϕ τ ∣ z = η + ∂ z ϕ τ ∣ z = η G [ η ] ϕ ‾ . (48) \partial_t\phi^\tau(x,t)=\left.\partial_t\phi^\tau\right|_{z=\eta}+\left.\partial_z\phi^\tau\right|_{z=\eta}G[\eta]\overline{\phi}. \tag*{(48)} ∂ t ϕ τ ( x , t ) = ∂ t ϕ τ ∣ z = η + ∂ z ϕ τ ∣ z = η G [ η ] ϕ . (48) Removing the large-scale vertical contribution at the surface, G [ η ] ϕ ‾ G[\eta]\overline{\phi} G [ η ] ϕ [48 ] .
From a geometrical viewpoint, the condition
∂ z ϕ τ G [ η ] ϕ ‾ = 0 , \partial_z\phi^\tau G[\eta]\overline{\phi}=0, ∂ z ϕ τ G [ η ] ϕ = 0 , expresses orthogonality between fluctuating vertical motions and resolved normal transport at the free surface. It removes stochastic work in the normal direction, so unresolved fluctuations act tangentially to the evolving interface and do not modify the large-scale Hamiltonian flux through the boundary.
Using the identities
∂ t ϕ ‾ ( x , t ) = ∂ t ϕ ‾ ∣ z = η + ∂ z ϕ ‾ ∣ z = η ∂ t η = ∂ t ϕ ‾ ∣ z = η + ∂ z ϕ ‾ ∣ z = η ( G [ η ] ϕ ‾ + G [ η ] φ τ ) , (49) \partial_t\overline{\phi}(x,t)=\left.\partial_t\overline{\phi}\right|_{z=\eta}+\left.\partial_z\overline{\phi}\right|_{z=\eta}\partial_t\eta=\left.\partial_t\overline{\phi}\right|_{z=\eta}+\left.\partial_z\overline{\phi}\right|_{z=\eta}\left(G[\eta]\overline{\phi}+G[\eta]\varphi^\tau\right), \tag*{(49)} ∂ t ϕ ( x , t ) = ∂ t ϕ z = η + ∂ z ϕ z = η ∂ t η = ∂ t ϕ z = η + ∂ z ϕ z = η ( G [ η ] ϕ + G [ η ] φ τ ) , (49) ∇ x ϕ ‾ ∣ z = η = ∇ ϕ ‾ − ∂ z ϕ ‾ ∣ z = η ∇ η , ∇ x φ τ ∣ z = η = ∇ φ τ − ∂ z φ τ ∣ z = η ∇ η , (50) \left.\nabla_x\overline{\phi}\right|_{z=\eta}=\nabla\overline{\phi}-\left.\partial_z\overline{\phi}\right|_{z=\eta}\nabla\eta,\qquad\left.\nabla_x\varphi^\tau\right|_{z=\eta}=\nabla\varphi^\tau-\left.\partial_z\varphi^\tau\right|_{z=\eta}\nabla\eta, \tag*{(50)} ∇ x ϕ z = η = ∇ ϕ − ∂ z ϕ z = η ∇ η , ∇ x φ τ ∣ z = η = ∇ φ τ − ∂ z φ τ ∣ z = η ∇ η , (50) together with the Dirichlet-to-Neumann relations
∂ z ϕ ‾ = G [ η ] ϕ ‾ + ∇ η ⋅ ∇ ϕ ‾ 1 + ∣ ∇ η ∣ 2 , ∂ z φ τ = G [ η ] φ τ + ∇ η ⋅ ∇ φ τ 1 + ∣ ∇ η ∣ 2 , (51) \partial_z\overline{\phi}=\frac{G[\eta]\overline{\phi}+\nabla\eta\cdot\nabla\overline{\phi}}{1+\lvert\nabla\eta\rvert^2},\qquad\partial_z\varphi^\tau=\frac{G[\eta]\varphi^\tau+\nabla\eta\cdot\nabla\varphi^\tau}{1+\lvert\nabla\eta\rvert^2}, \tag*{(51)} ∂ z ϕ = 1 + ∣ ∇ η ∣ 2 G [ η ] ϕ + ∇ η ⋅ ∇ ϕ , ∂ z φ τ = 1 + ∣ ∇ η ∣ 2 G [ η ] φ τ + ∇ η ⋅ ∇ φ τ , (51) the noise constraint (47) reads
∂ t φ τ + 1 2 ∣ ∇ φ τ ∣ 2 − G [ η ] φ τ + ∇ η ⋅ ∇ φ τ 1 + ∣ ∇ η ∣ 2 ( G [ η ] φ τ ) + 1 2 ( G [ η ] φ τ + ∇ η ⋅ ∇ φ τ ) 2 1 + ∣ ∇ η ∣ 2 − G [ η ] φ τ + ∇ η ⋅ ∇ φ τ 1 + ∣ ∇ η ∣ 2 ( ∇ η ⋅ ∇ φ τ ) = 0 , (52) \partial_t\varphi^\tau+\frac{1}{2}\lvert\nabla\varphi^\tau\rvert^2-\frac{G[\eta]\varphi^\tau+\nabla\eta\cdot\nabla\varphi^\tau}{1+\lvert\nabla\eta\rvert^2}\left(G[\eta]\varphi^\tau\right)+\frac{1}{2}\frac{\left(G[\eta]\varphi^\tau+\nabla\eta\cdot\nabla\varphi^\tau\right)^2}{1+\lvert\nabla\eta\rvert^2}-\frac{G[\eta]\varphi^\tau+\nabla\eta\cdot\nabla\varphi^\tau}{1+\lvert\nabla\eta\rvert^2}\left(\nabla\eta\cdot\nabla\varphi^\tau\right)=0, \tag*{(52)} ∂ t φ τ + 2 1 ∣ ∇ φ τ ∣ 2 − 1 + ∣ ∇ η ∣ 2 G [ η ] φ τ + ∇ η ⋅ ∇ φ τ ( G [ η ] φ τ ) + 2 1 1 + ∣ ∇ η ∣ 2 ( G [ η ] φ τ + ∇ η ⋅ ∇ φ τ ) 2 − 1 + ∣ ∇ η ∣ 2 G [ η ] φ τ + ∇ η ⋅ ∇ φ τ ( ∇ η ⋅ ∇ φ τ ) = 0 , (52) which simplifies as the symmetric constraint
∂ t φ τ + 1 2 ∣ ∇ φ τ ∣ 2 − 1 2 ( G [ η ] φ τ + ∇ η ⋅ ∇ φ τ ) 2 1 + ∣ ∇ η ∣ 2 = 0. (53) \partial_t\varphi^\tau+ \frac{1}{2}\lvert\nabla\varphi^\tau\rvert^2 - \frac{1}{2}\frac{\left(G[\eta]\varphi^\tau+\nabla\eta\cdot\nabla\varphi^\tau\right)^2}{1+\lvert\nabla\eta\rvert^2}=0. \tag*{(53)} ∂ t φ τ + 2 1 ∣ ∇ φ τ ∣ 2 − 2 1 1 + ∣ ∇ η ∣ 2 ( G [ η ] φ τ + ∇ η ⋅ ∇ φ τ ) 2 = 0. (53) This yields the large-scale Bernoulli equation for the surface potential:
∂ t φ ‾ + g η + 1 2 ∣ ∇ φ ‾ ∣ 2 − 1 2 ( G [ η ] φ ‾ + ∇ η ⋅ ∇ φ ‾ ) 2 1 + ∣ ∇ η ∣ 2 + ∇ φ ‾ ⋅ ∇ φ τ − ( G [ η ] φ ‾ + ∇ η ⋅ ∇ φ ‾ ) G [ η ] φ τ + ∇ η ⋅ ∇ φ τ 1 + ∣ ∇ η ∣ 2 = 0. (54) \partial_t\overline{\varphi}+g\eta+\frac{1}{2}\lvert\nabla\overline{\varphi}\rvert^2-\frac{1}{2}\frac{\left(G[\eta]\overline{\varphi}+\nabla\eta\cdot\nabla\overline{\varphi}\right)^2}{1+\lvert\nabla\eta\rvert^2}+\nabla\overline{\varphi}\cdot\nabla\varphi^\tau-\left(G[\eta]\overline{\varphi}+\nabla\eta\cdot\nabla\overline{\varphi}\right)\frac{G[\eta]\varphi^\tau+\nabla\eta\cdot\nabla\varphi^\tau}{1+\lvert\nabla\eta\rvert^2}=0. \tag*{(54)} ∂ t φ + g η + 2 1 ∣ ∇ φ ∣ 2 − 2 1 1 + ∣ ∇ η ∣ 2 ( G [ η ] φ + ∇ η ⋅ ∇ φ ) 2 + ∇ φ ⋅ ∇ φ τ − ( G [ η ] φ + ∇ η ⋅ ∇ φ ) 1 + ∣ ∇ η ∣ 2 G [ η ] φ τ + ∇ η ⋅ ∇ φ τ = 0. (54) This equation is complemented by the potential flow conditions for both large- and small-scale components (43) , while the kinematic boundary condition reads
∂ t η + ∇ ( φ ‾ + φ τ ) ⋅ ∇ η = ∂ z φ ‾ ∣ z = η + ∂ z φ τ ∣ z = η . (55) \partial_t\eta+\nabla\left(\overline{\varphi}+\varphi^\tau\right)\cdot\nabla\eta=\left.\partial_z\overline{\varphi}\right|_{z=\eta}+\left.\partial_z\varphi^\tau\right|_{z=\eta}. \tag*{(55)} ∂ t η + ∇ ( φ + φ τ ) ⋅ ∇ η = ∂ z φ ∣ z = η + ∂ z φ τ ∣ z = η . (55) In (54) , the first line contains only large-scale contributions, whereas the second line represents interactions between the resolved and fluctuating components. The surface dynamics retains the symmetric coupling structure between φ ‾ \overline{\varphi} φ φ τ \varphi^\tau φ τ
∂ t φ ‾ + 1 2 ∣ ∇ φ ‾ ∣ 2 + ∇ φ ‾ ⋅ ∇ φ τ + g z = 0. \partial_t\overline{\varphi}+\frac{1}{2}\lvert\nabla\overline{\varphi}\rvert^2+\nabla\overline{\varphi}\cdot\nabla\varphi^\tau+gz=0. ∂ t φ + 2 1 ∣ ∇ φ ∣ 2 + ∇ φ ⋅ ∇ φ τ + g z = 0. Keeping the large-scale surface contribution, G [ η ] φ ‾ G[\eta]\overline{\varphi} G [ η ] φ (48) would generally destroy this symmetry.
The approximation developed above corresponds to the prescribed-noise setting underlying Location Uncertainty models. In the next subsection, we remove this assumption and derive an evolution equation for the fluctuating potential from a second variational principle.
In the zero-correlation limit, this approximation corresponds to the LU solution [48 ] , in which the large-scale surface elevation is transported by a random flow with a transport noise defined in the Stratonovich sense. This formulation is consistent with the model proposed in [58 ] for potential flows, both for the large-scale component and for the stochastic contribution. However, an important difference should be noted. In [58 ] , the noise is introduced directly at the level of the surface through the trace of the potential noise correlation functions. In the present formulation, by contrast, the noise is defined in the bulk through velocity potentials satisfying a Laplace equation over the entire fluid domain. As a consequence, the induced noise at the free surface involves the Dirichlet-to-Neumann operator via the vertical derivative of the potential, see (51) . Furthermore, the model in [58 ] is derived as a stochastic approximation of the Euler equations from an underlying variational principle. However, the associated stochastic water wave formulation is not obtained directly from the Luke or Zakharov–Craig–Sulem variational principles. In that framework, the noise is prescribed, and thus implicitly satisfies a balance of the form (47) .
An alternative modeling strategy, still assuming prescribed noise, is to retain a finite correlation time and work directly with the system (43) –(46) . From a physical standpoint, this approach may provide a more consistent representation of the underlying scale interactions.
Large scale Zakharov–Craig–Sulem formulation Under the constraint (47) on the fluctuations, and applying the same reduction as in the deterministic case of the Lagrangian (41) yields the canonical action
S ( φ ‾ , η ) = ∫ t 0 t 1 ∫ D h ( φ ‾ ∂ t η − 1 2 ( φ ‾ + 2 φ τ ) G [ η ] φ ‾ − 1 2 g η 2 ) d x d t , (56) \mathcal{S}(\overline{\varphi},\eta)=\int_{t_0}^{t_1}\int_{\mathcal{D}_h}\left(\overline{\varphi}\partial_t\eta-\frac{1}{2}\left(\overline{\varphi}+2\varphi^\tau\right)G[\eta]\overline{\varphi}-\frac{1}{2}g\eta^2\right)\mathrm{d}x\,\mathrm{d}t, \tag*{(56)} S ( φ , η ) = ∫ t 0 t 1 ∫ D h ( φ ∂ t η − 2 1 ( φ + 2 φ τ ) G [ η ] φ − 2 1 g η 2 ) d x d t , (56) with the modified large-scale Zakharov–Craig–Sulem Hamiltonian
H ‾ ( φ ‾ , η ) = ∫ D h ( 1 2 φ ‾ G [ η ] φ ‾ + φ τ G [ η ] φ ‾ + 1 2 g η 2 ) d x . \overline{\mathcal{H}}(\overline{\varphi},\eta)=\int_{\mathcal{D}_h}\left(\frac{1}{2}\overline{\varphi}G[\eta]\overline{\varphi}+\varphi^\tau G[\eta]\overline{\varphi}+\frac{1}{2}g\eta^2\right)\mathrm{d}x. H ( φ , η ) = ∫ D h ( 2 1 φ G [ η ] φ + φ τ G [ η ] φ + 2 1 g η 2 ) d x . The corresponding Hamilton equations are
∂ t η = δ H ‾ δ φ ‾ = G [ η ] ( φ ‾ + φ τ ) , (57) \partial_t\eta=\frac{\delta\overline{\mathcal{H}}}{\delta\overline{\varphi}}=G[\eta](\overline{\varphi}+\varphi^\tau), \tag*{(57)} ∂ t η = δ φ δ H = G [ η ] ( φ + φ τ ) , (57) ∂ t φ ‾ = − δ H ‾ δ η = − ( g η + 1 2 ∣ ∇ φ ‾ ∣ 2 − 1 2 ( G [ η ] φ ‾ + ∇ η ⋅ ∇ φ ‾ ) 2 1 + ∣ ∇ η ∣ 2 + ∇ φ ‾ ⋅ ∇ φ τ − ( G [ η ] φ ‾ + ∇ η ⋅ ∇ φ ‾ ) ( G [ η ] φ τ + ∇ η ⋅ ∇ φ τ ) 1 + ∣ ∇ η ∣ 2 ) , (58) \partial_t\overline{\varphi}=-\frac{\delta\overline{\mathcal{H}}}{\delta\eta}=-\left( g\eta+\frac{1}{2}\lvert\nabla\overline{\varphi}\rvert^2-\frac{1}{2}\frac{\left(G[\eta]\overline{\varphi}+\nabla\eta\cdot\nabla\overline{\varphi}\right)^2}{1+\lvert\nabla\eta\rvert^2}+\nabla\overline{\varphi}\cdot\nabla\varphi^\tau-\frac{\left(G[\eta]\overline{\varphi}+\nabla\eta\cdot\nabla\overline{\varphi}\right)\left(G[\eta]\varphi^\tau+\nabla\eta\cdot\nabla\varphi^\tau\right)}{1+\lvert\nabla\eta\rvert^2}\right), \tag*{(58)} ∂ t φ = − δ η δ H = − ( g η + 2 1 ∣ ∇ φ ∣ 2 − 2 1 1 + ∣ ∇ η ∣ 2 ( G [ η ] φ + ∇ η ⋅ ∇ φ ) 2 + ∇ φ ⋅ ∇ φ τ − 1 + ∣ ∇ η ∣ 2 ( G [ η ] φ + ∇ η ⋅ ∇ φ ) ( G [ η ] φ τ + ∇ η ⋅ ∇ φ τ ) ) , (58) which coincide exactly with the large-scale dynamics (54)-(55). This shows that the large scale dynamics, together with the constraint (48), preserves a Hamiltonian structure. Hence, the noise constraint plays a dual role. It regularizes the decorrelation limit while preserving the Hamiltonian structure inherited from the original finite-correlation stochastic water-wave system.
Hence, the noise constraint enables not only to obtain an expression that remain valid at the decorrelation limit, but also keeps the Hamiltonian structure of the original (time correlated) stochastic surface wave representation.
Nondimentionalisation
In order to nondimensionalise the stochastic water-wave equations associated with pathwise variational principle, we must specify the scaling of the noise term. We therefore introduce characteristic length scale for the unresolved motion, denoted by L σ L_\sigma L σ H H H μ σ = H 2 / L σ 2 \mu_\sigma=H^2/L_\sigma^2 μ σ = H 2 / L σ 2 ϵ σ = A σ / H \epsilon_\sigma=A_\sigma/H ϵ σ = A σ / H A σ A_\sigma A σ
A word of caution is necessary at this stage, as the interpretation of the noise amplitude depends on the physical situation. One possible interpretation is that the noise represents the mean cumulative energy of all unresolved waves. In this averaged description, the noise amplitude remains comparable to that of the resolved waves. To make this precise, consider an omnidirectional one-dimensional power-law spectrum[50 , 70 ] of the form E ( k ) = C k − m E(k)=Ck^{-m} E ( k ) = C k − m k c = 2 π / L I k_c=2\pi/L_I k c = 2 π / L I
E σ = ∫ k c ∞ C k − m d k = C m − 1 k c − ( m − 1 ) . (59) E_\sigma=\int_{k_c}^{\infty}Ck^{-m}\,\mathrm{d}k=\frac{C}{m-1}k_c^{-(m-1)}. \tag*{(59)} E σ = ∫ k c ∞ C k − m d k = m − 1 C k c − ( m − 1 ) . (59) The energy of a typical resolved wave of wavenumber k p k_p k p E p = 1 2 ρ g A 2 E_p=\frac{1}{2}\rho g A^2 E p = 2 1 ρ g A 2 k p k_p k p Δ k p \Delta k_p Δ k p E p = E ( k p ) Δ ( k p ) E_p=E(k_p)\Delta(k_p) E p = E ( k p ) Δ ( k p ) A = 2 E p / ( ρ g ) A=\sqrt{2E_p/(\rho g)} A = 2 E p / ( ρ g ) k p k_p k p Δ k p = β k p \Delta k_p=\beta k_p Δ k p = β k p
E p = E ( k p ) Δ k p = C k p − m β k p = C β k p 1 − m . E_p=E(k_p)\Delta k_p=Ck_p^{-m}\beta k_p=C\beta k_p^{1-m}. E p = E ( k p ) Δ k p = C k p − m β k p = C β k p 1 − m . Consequently,
C = E p β k p m − 1 , (60) C=\frac{E_p}{\beta}k_p^{m-1}, \tag*{(60)} C = β E p k p m − 1 , (60) and the unresolved tail energy becomes
E σ = E p β ( m − 1 ) ( k c k p ) 1 − m . (61) E_\sigma= \frac{E_p}{\beta(m-1)}\left(\frac{k_c}{k_p}\right)^{1-m}. \tag*{(61)} E σ = β ( m − 1 ) E p ( k p k c ) 1 − m . (61) Using
E p = 1 2 ρ g A 2 , E_p = \frac{1}{2}\rho g A^2, E p = 2 1 ρ g A 2 , we define an equivalent unresolved-wave amplitude by
A σ = 2 E σ ρ g = A 1 β ( m − 1 ) ( k c k p ) 1 − m . (62) A_\sigma= \sqrt{\frac{2E_\sigma}{\rho g}} = A\sqrt{\frac{1}{\beta(m-1)}\left(\frac{k_c}{k_p}\right)^{1-m}}. \tag*{(62)} A σ = ρ g 2 E σ = A β ( m − 1 ) 1 ( k p k c ) 1 − m . (62) As a numerical example, taking β = 0.1 \beta= 0.1 β = 0.1 m = 5 / 2 m = 5/2 m = 5/2 m = 3 m = 3 m = 3 k c / k p = 1 k_c/k_p = 1 k c / k p = 1 A σ ≃ 2.2 A A_\sigma\simeq2.2A A σ ≃ 2.2 A 2.6 A 2.6A 2.6 A k c / k p = 10 k_c/k_p = 10 k c / k p = 10 A σ ≃ 0.2 A A_\sigma\simeq0.2A A σ ≃ 0.2 A 0.5 A 0.5A 0.5 A A σ ∼ A A_\sigma\sim A A σ ∼ A
A second interpretation corresponds to out-of-average conditions, such as noise representing strong swell generated by storms or tropical cyclones. In this case, the typical amplitude of the unresolved motions may significantly exceed that of the resolved large-scale waves, A σ ≫ A A_\sigma\gg A A σ ≫ A
The parameter ϵ σ = A σ / H \epsilon_\sigma= A_\sigma/H ϵ σ = A σ / H ϵ \epsilon ϵ ϵ \epsilon ϵ ϵ σ \epsilon_\sigma ϵ σ
Although the free-surface dynamics is driven by both resolved and unresolved motions, the interface itself is represented by a single elevation field. We therefore retain the deterministic scaling η = A η ~ \eta= A\tilde{\eta} η = A η ~
Following the noise definition (38) , we scale the fluctuating potential as ϕ τ = Φ 0 σ ϕ ~ τ \phi^\tau= \Phi_0^\sigma\tilde{\phi}^\tau ϕ τ = Φ 0 σ ϕ ~ τ Φ 0 σ = ϵ σ L σ g H ν σ − 1 \Phi_0^\sigma= \epsilon_\sigma L_\sigma\sqrt{gH\nu_\sigma^{-1}} Φ 0 σ = ϵ σ L σ g H ν σ − 1 ν σ = tanh ( 2 π μ σ ) / ( 2 π μ σ ) \nu_\sigma= \tanh(2\pi\sqrt{\mu_\sigma})/(2\pi\sqrt{\mu_\sigma}) ν σ = tanh ( 2 π μ σ ) / ( 2 π μ σ ) ν ≈ 1 \nu\approx1 ν ≈ 1 ν σ ≈ ( 2 π μ σ ) − 1 \nu_\sigma\approx(2\pi\sqrt{\mu_\sigma})^{-1} ν σ ≈ ( 2 π μ σ ) − 1
The characteristic small-scale time T σ = L σ / g H ν σ T_\sigma= L_\sigma/\sqrt{gH\nu_\sigma} T σ = L σ / g H ν σ ω 2 = g ∣ k ∣ tanh ( h ∣ k ∣ ) \omega^2 = g|k|\tanh(h|k|) ω 2 = g ∣ k ∣ tanh ( h ∣ k ∣ ) k = 2 π / L k = 2\pi/L k = 2 π / L c = g H ν c = \sqrt{gH\nu} c = g H ν τ = T σ \tau= T_\sigma τ = T σ ν σ = 1 \nu_\sigma= 1 ν σ = 1 L σ = τ g H L_\sigma= \tau\sqrt{gH} L σ = τ g H ν σ = ( 2 π μ σ ) − 1 \nu_\sigma= (2\pi\sqrt{\mu_\sigma})^{-1} ν σ = ( 2 π μ σ ) − 1 L σ = 1 2 π g τ 2 L_\sigma= \frac{1}{2\pi}g\tau^2 L σ = 2 π 1 g τ 2 ν = 1 \nu= 1 ν = 1 ν σ ≈ 1 / ( 2 π μ σ ) \nu_\sigma\approx1/(2\pi\sqrt{\mu_\sigma}) ν σ ≈ 1/ ( 2 π μ σ ) T σ 2 ≈ 2 π L σ L μ T 2 T_\sigma^2 \approx2\pi\frac{L_\sigma}{L}\sqrt{\mu}\,T^2 T σ 2 ≈ 2 π L L σ μ T 2 T σ < T T_\sigma< T T σ < T T σ / T ∼ L σ / L T_\sigma/T \sim L_\sigma/L T σ / T ∼ L σ / L T σ / T ∼ L σ / L T_\sigma/T \sim\sqrt{L_\sigma/L} T σ / T ∼ L σ / L r = L σ / L r = L_\sigma/L r = L σ / L ϵ r σ = ϵ σ r \epsilon_r^\sigma= \epsilon_\sigma r ϵ r σ = ϵ σ r
The stochastic water-wave equations depend not only on the amplitude and spatial scale of the unresolved motions, but also on their temporal persistence relative to the intrinsic resolved-wave time scale. We therefore introduce the dimensionless memory parameter m = T σ / T m = T_\sigma/T m = T σ / T L σ ≤ L L_{\sigma} \le L L σ ≤ L m ≤ 1 m \le1 m ≤ 1
The noise correlation modes do not scale in the same way as the regularised fluctuating potential. Since W t τ = O ( T σ − 1 / 2 ) W_{t}^{\tau} = \mathcal{O}(T_{\sigma}^{-1/2}) W t τ = O ( T σ − 1/2 ) ϕ τ = ϕ j W t τ , j \phi^{\tau} = \phi_j W_{t}^{\tau,j} ϕ τ = ϕ j W t τ , j Φ σ , 0 \Phi_{\sigma,0} Φ σ , 0 ϕ j = Φ j , 0 σ ϕ ~ j \phi_j = \Phi_{j,0}^{\sigma}\widetilde{\phi}_j ϕ j = Φ j , 0 σ ϕ j Φ j , 0 σ = Φ 0 σ T σ \Phi_{j,0}^{\sigma} = \Phi_{0}^{\sigma}\sqrt{T_{\sigma}} Φ j , 0 σ = Φ 0 σ T σ [ ϕ j ] = L 2 T − 1 / 2 [\phi_j] = L^2T^{-1/2} [ ϕ j ] = L 2 T − 1/2 [ ϕ τ ] = L 2 T − 1 [\phi^{\tau}] = L^2T^{-1} [ ϕ τ ] = L 2 T − 1
Motivated by these considerations, we introduce the noise-scaling parameter γ σ = ϵ σ ν σ \gamma_{\sigma} = \frac{\epsilon_{\sigma}}{\sqrt{\nu_{\sigma}}} γ σ = ν σ ϵ σ γ σ r = γ σ r \gamma_{\sigma}^{r} = \gamma_{\sigma}r γ σ r = γ σ r
m = T σ T = L σ L ν ν σ = r ν ν σ , m = \frac{T_{\sigma}}{T} = \frac{L_{\sigma}}{L}\sqrt{\frac{\nu}{\nu_{\sigma}}} = r\sqrt{\frac{\nu}{\nu_{\sigma}}}, m = T T σ = L L σ ν σ ν = r ν σ ν , we obtain
γ σ r = ϵ σ ν σ r = ϵ σ ν m . \gamma_{\sigma}^{r} = \frac{\epsilon_{\sigma}}{\sqrt{\nu_{\sigma}}}r = \frac{\epsilon_{\sigma}}{\sqrt{\nu}}m. γ σ r = ν σ ϵ σ r = ν ϵ σ m . Thus, the effective strength of the unresolved component increases linearly with both its amplitude and its memory and is further enhanced when the resolved waves lie in a deep-water regime, for which ν ≪ 1 \nu\ll1 ν ≪ 1
The scaling of the Dirichlet-to-Neumann operator associated with the fluctuating potential is identical to that of the deterministic problem:
G [ η ] ϕ τ = Φ 0 σ H G μ , ϵ [ η ] ϕ ~ τ , (63) G[\eta]\phi^{\tau} = \frac{\Phi_{0}^{\sigma}}{H}G^{\mu,\epsilon}[\eta]\widetilde{\phi}^{\tau}, \tag*{(63)} G [ η ] ϕ τ = H Φ 0 σ G μ , ϵ [ η ] ϕ τ , (63) where (dropping the tilde accentuation)
G μ , ϵ [ η ] ϕ τ = ( ∂ z ϕ τ − μ ϵ ∇ x ϕ τ ⋅ ∇ η ) z = ϵ η . (64) G^{\mu,\epsilon}[\eta]\phi^{\tau} = \left(\partial_z\phi^{\tau} - \mu\epsilon\nabla_x\phi^{\tau}\cdot\nabla\eta\right)_{z=\epsilon\eta}. \tag*{(64)} G μ , ϵ [ η ] ϕ τ = ( ∂ z ϕ τ − μ ϵ ∇ x ϕ τ ⋅ ∇ η ) z = ϵη . (64) The unresolved horizontal scale enters only through the scaling of Φ 0 σ \Phi_{0}^{\sigma} Φ 0 σ ϵ σ \epsilon_{\sigma} ϵ σ m m m
∂ t ϕ ‾ + η + ϵ 2 ν ∣ ∇ ϕ ‾ ∣ 2 − ϵ μ 2 ν ( 1 μ G μ , ϵ [ η ] ϕ ‾ + ϵ ∇ η ⋅ ∇ ϕ ‾ ) 2 1 + ϵ 2 μ ∣ ∇ η ∣ 2 + ϵ σ m ν ∇ ϕ ‾ ⋅ ∇ ϕ τ − ϵ σ m μ ν ( 1 μ G μ , ϵ [ η ] ϕ ‾ + ϵ ∇ η ⋅ ∇ ϕ ‾ ) ( 1 μ G μ , ϵ [ η ] ϕ τ + ϵ ∇ η ⋅ ∇ ϕ τ ) 1 + ϵ 2 μ ∣ ∇ η ∣ 2 = 0. (65) \partial_t\overline{\phi}+\eta+\frac{\epsilon}{2\nu}\lvert\nabla\overline{\phi}\rvert^2-\frac{\epsilon\mu}{2\nu}\frac{\left(\frac{1}{\mu}G^{\mu,\epsilon}[\eta]\overline{\phi}+\epsilon\nabla\eta\cdot\nabla\overline{\phi}\right)^2}{1+\epsilon^2\mu\lvert\nabla\eta\rvert^2}+\frac{\epsilon_{\sigma}m}{\nu}\nabla\overline{\phi}\cdot\nabla\phi^{\tau}-\frac{\epsilon_{\sigma}m\mu}{\nu}\frac{\left(\frac{1}{\mu}G^{\mu,\epsilon}[\eta]\overline{\phi}+\epsilon\nabla\eta\cdot\nabla\overline{\phi}\right)\left(\frac{1}{\mu}G^{\mu,\epsilon}[\eta]\phi^{\tau}+\epsilon\nabla\eta\cdot\nabla\phi^{\tau}\right)}{1+\epsilon^2\mu\lvert\nabla\eta\rvert^2}=0. \tag*{(65)} ∂ t ϕ + η + 2 ν ϵ ∣ ∇ ϕ ∣ 2 − 2 ν ϵ μ 1 + ϵ 2 μ ∣ ∇ η ∣ 2 ( μ 1 G μ , ϵ [ η ] ϕ + ϵ ∇ η ⋅ ∇ ϕ ) 2 + ν ϵ σ m ∇ ϕ ⋅ ∇ ϕ τ − ν ϵ σ m μ 1 + ϵ 2 μ ∣ ∇ η ∣ 2 ( μ 1 G μ , ϵ [ η ] ϕ + ϵ ∇ η ⋅ ∇ ϕ ) ( μ 1 G μ , ϵ [ η ] ϕ τ + ϵ ∇ η ⋅ ∇ ϕ τ ) = 0. (65) and
∂ t η = 1 μ ν G μ , ϵ [ η ] ϕ ‾ + ϵ σ m ϵ μ ν G μ , ϵ [ η ] ϕ τ . (66) \partial_t\eta=\frac{1}{\mu\nu}G^{\mu,\epsilon}[\eta]\overline{\phi}+\frac{\epsilon_{\sigma}m}{\epsilon\mu\nu}G^{\mu,\epsilon}[\eta]\phi^{\tau}. \tag*{(66)} ∂ t η = μν 1 G μ , ϵ [ η ] ϕ + ϵ μν ϵ σ m G μ , ϵ [ η ] ϕ τ . (66) As noted in [58 ] , the stochastic system (43)-(54)-(55) associated to full potential flow, can be extended to consider rotational noise and a large-scale potential flow only. Writing u τ = ( u h , τ , u z , τ ) u^{\tau}=(u^{h,\tau},u^{z,\tau}) u τ = ( u h , τ , u z , τ )
S ( ϕ , η ) = ∫ t 0 t 1 ∫ D h ∫ b η ( x , t ) ( ∂ t ϕ + 1 2 ∣ ∇ ϕ ‾ ∣ 2 + ∇ ϕ ‾ ⋅ u τ + g z ) ρ d z d x d t (67) \mathcal{S}(\phi,\eta)=\int_{t_0}^{t_1}\int_{\mathcal{D}_h}\int_b^{\eta(x,t)}\left(\partial_t\phi+\frac{1}{2}\lvert\nabla\overline{\phi}\rvert^2+\nabla\overline{\phi}\cdot u^{\tau}+gz\right)\rho\,dz\,dx\,dt \tag*{(67)} S ( ϕ , η ) = ∫ t 0 t 1 ∫ D h ∫ b η ( x , t ) ( ∂ t ϕ + 2 1 ∣ ∇ ϕ ∣ 2 + ∇ ϕ ⋅ u τ + g z ) ρ d z d x d t (67) In that case the stochastic system still involves a Laplace equation for the large-scale flow:
Δ ϕ ˉ = 0 ∀ ( x , z ) ∈ Ω t , with (68) \Delta\bar{\phi}=0 \quad\forall(x,z)\in\Omega_t,\quad\text{with} \tag*{(68)} Δ ϕ ˉ = 0 ∀ ( x , z ) ∈ Ω t , with (68) ∂ z ϕ ˉ = 0 ∀ z = b and (69) \partial_z\bar{\phi}=0 \quad\forall z=b \quad\text{and} \tag*{(69)} ∂ z ϕ ˉ = 0 ∀ z = b and (69) ϕ ˉ ( x , η ( x , t ) , t ) = ϕ ˉ ( x , t ) ∀ z = η ( x , t ) . (70) \bar{\phi}(x,\eta(x,t),t)=\bar{\phi}(x,t) \quad\forall z=\eta(x,t). \tag*{(70)} ϕ ˉ ( x , η ( x , t ) , t ) = ϕ ˉ ( x , t ) ∀ z = η ( x , t ) . (70) This system is complemented by the kinematic boundary condition
∂ t η + ( ∇ x ϕ ˉ ∣ z = η + u ∣ z = η h , τ ) ⋅ ∇ η = ∂ z ϕ ˉ ∣ z = η + u ∣ z = η z , τ . (71) \partial_t\eta+\left(\left.\nabla_x\bar{\phi}\right|_{z=\eta}+u^{h,\tau}_{|z=\eta}\right)\cdot\nabla\eta=\left.\partial_z\bar{\phi}\right|_{z=\eta}+u^{z,\tau}_{|z=\eta}. \tag*{(71)} ∂ t η + ( ∇ x ϕ ˉ z = η + u ∣ z = η h , τ ) ⋅ ∇ η = ∂ z ϕ ˉ z = η + u ∣ z = η z , τ . (71) Equivalently, using the Dirichlet-to-Neumann operator, this condition reads
∂ t η = G [ η ] ϕ ˉ + u ∣ z = η z , τ − u ∣ z = η h , τ ⋅ ∇ η . (72) \partial_t\eta=G[\eta]\bar{\phi}+u^{z,\tau}_{|z=\eta}-u^{h,\tau}_{|z=\eta}\cdot\nabla\eta. \tag*{(72)} ∂ t η = G [ η ] ϕ ˉ + u ∣ z = η z , τ − u ∣ z = η h , τ ⋅ ∇ η . (72) The Bernoulli equation written in terms of the trace of the large-scale velocity potential is
∂ t ϕ ˉ + g η + 1 2 ∣ ∇ ϕ ˉ ∣ 2 − 1 2 ( G [ η ] ϕ ˉ + ∇ η ⋅ ∇ ϕ ˉ ) 2 1 + ∣ ∇ η ∣ 2 + ∇ ϕ ˉ ⋅ u h , τ ∣ z = η = 0. (73) \partial_t\bar{\phi}+g\eta+\frac{1}{2}\lvert\nabla\bar{\phi}\rvert^2-\frac{1}{2}\frac{\left(G[\eta]\bar{\phi}+\nabla\eta\cdot\nabla\bar{\phi}\right)^2}{1+\lvert\nabla\eta\rvert^2}+\left.\nabla\bar{\phi}\cdot u^{h,\tau}\right|_{z=\eta}=0. \tag*{(73)} ∂ t ϕ ˉ + g η + 2 1 ∣ ∇ ϕ ˉ ∣ 2 − 2 1 1 + ∣ ∇ η ∣ 2 ( G [ η ] ϕ ˉ + ∇ η ⋅ ∇ ϕ ˉ ) 2 + ∇ ϕ ˉ ⋅ u h , τ z = η = 0. (73) Indeed, when the bulk Bernoulli equation is expressed in terms of the surface trace and the kinematic condition (70) is used, the terms involving u ∣ z = η z , τ u^{z,\tau}_{|z=\eta} u ∣ z = η z , τ
This system has the important conceptual advantage of allowing the fluctuating velocity field to possess non-zero vorticity. However, it requires the fluctuating velocity at the free surface to be prescribed. For this reason, the quadratic small-scale contribution 1 2 ∣ u τ ∣ 2 \frac{1}{2}\lvert u^\tau\rvert^2 2 1 ∣ u τ ∣ 2
To infer these noise functions, we now consider a variational functional in expectation.
Variational principle in expectation
As mentioned earlier, the gradient of the velocity potential satisfies the incompressible Euler equations. As shown in [16 ] , the stochastic decomposition
∇ ϕ ( x , t ) = ∇ ϕ ˉ ( x , t ) + ∇ ϕ τ ( x , t ) , ∇ ϕ τ ( x , t ) = ∫ t − τ t + τ h τ ( t − s ) ∇ ϕ i ( x , t ) d β s i (74) \nabla\phi(x,t)=\nabla\bar{\phi}(x,t)+\nabla\phi^\tau(x,t),\qquad\nabla\phi^\tau(x,t)=\int_{t-\tau}^{t+\tau}h_\tau(t-s)\nabla\phi_i(x,t)\,\mathrm{d}\beta^i_s \tag*{(74)} ∇ ϕ ( x , t ) = ∇ ϕ ˉ ( x , t ) + ∇ ϕ τ ( x , t ) , ∇ ϕ τ ( x , t ) = ∫ t − τ t + τ h τ ( t − s ) ∇ ϕ i ( x , t ) d β s i (74) satisfies an Euler equation of the form (28)–(31). As already outlined, this Euler equation is not well defined in the decorrelation limit. Assuming a given well-defined approximation for the evolution of ϕ ˉ \bar{\phi} ϕ ˉ ϕ i \phi_i ϕ i [16 ] :
∂ t ϕ i + ∇ ϕ ˉ ⋅ ∇ ϕ i + p i τ = 0 , (75) \partial_t\phi_i+\nabla\bar{\phi}\cdot\nabla\phi_i+p_i^\tau=0, \tag*{(75)} ∂ t ϕ i + ∇ ϕ ˉ ⋅ ∇ ϕ i + p i τ = 0 , (75) p i τ = − ∇ ϕ i ⋅ ∇ ϕ ˉ + 1 2 ∑ ℓ ∇ ϕ ℓ ⋅ ∇ ( ∇ ϕ ℓ ⋅ ∇ ϕ i ) − 1 2 Υ τ − 1 ∑ j ∂ t ( ∇ ϕ ˉ ⋅ ∇ ϕ j ) . (76) p_i^\tau=-\nabla\phi_i\cdot\nabla\bar{\phi}+\frac{1}{2}\sum_\ell\nabla\phi_\ell\cdot\nabla\left(\nabla\phi_\ell\cdot\nabla\phi_i\right)-\frac{1}{2}\Upsilon_\tau^{-1}\sum_j\partial_t\left(\nabla\bar{\phi}\cdot\nabla\phi_j\right). \tag*{(76)} p i τ = − ∇ ϕ i ⋅ ∇ ϕ ˉ + 2 1 ℓ ∑ ∇ ϕ ℓ ⋅ ∇ ( ∇ ϕ ℓ ⋅ ∇ ϕ i ) − 2 1 Υ τ − 1 j ∑ ∂ t ( ∇ ϕ ˉ ⋅ ∇ ϕ j ) . (76) Since Υ τ − 1 ≪ 1 \Upsilon_\tau^{-1}\ll1 Υ τ − 1 ≪ 1
As previously indicated, the surface elevation η \eta η
∂ t η = η ~ + η t τ , η t τ ( x , t ) = ∫ t t + τ h ~ τ ( t − s ) η j ( x , t ) d β s j . (77) \partial_t\eta= \widetilde{\eta} + \eta^\tau_t,\qquad\eta^\tau_t(x,t) = \int_t^{t+\tau} \widetilde{h}_\tau(t-s)\eta_j(x,t)\,\mathrm{d}\beta^j_s. \tag*{(77)} ∂ t η = η + η t τ , η t τ ( x , t ) = ∫ t t + τ h τ ( t − s ) η j ( x , t ) d β s j . (77) where η ~ \widetilde{\eta} η
d η = η ~ d t + η j d β t j . \mathrm{d}\eta= \widetilde{\eta}\,\mathrm{d}t + \eta_j\mathrm{d}\beta^j_t. d η = η d t + η j d β t j . We consider the Zakharov–Craig–Sulem action in expectation
S ( ϕ τ , η τ ) = E ∫ t 0 t 1 ∫ D h ( ϕ ∂ t η − 1 2 ϕ G [ η ] ϕ − 1 2 g η 2 ) d x d t . (78) S(\phi^\tau,\eta^\tau) = \mathbb{E}\int_{t_0}^{t_1}\int_{\mathcal{D}_h}\left(\phi\,\partial_t\eta- \frac{1}{2}\phi G[\eta]\phi- \frac{1}{2}g\eta^2\right)\,\mathrm{d}x\,\mathrm{d}t. \tag*{(78)} S ( ϕ τ , η τ ) = E ∫ t 0 t 1 ∫ D h ( ϕ ∂ t η − 2 1 ϕG [ η ] ϕ − 2 1 g η 2 ) d x d t . (78) Since the gravitational potential energy does not depend on the potential modes φ i \varphi_i φ i φ i \varphi_i φ i
Using the decorrelation properties of the Ito processes, and denoting
E [ W t τ , i W t τ , j ] = Υ τ δ i j , (79) \mathbb{E}\left[W_t^{\tau,i}W_t^{\tau,j}\right] = \Upsilon_\tau\delta_{ij}, \tag*{(79)} E [ W t τ , i W t τ , j ] = Υ τ δ ij , (79) the part of the functional in expectation that depends on the potential modes may be written as
S φ = Υ τ ∫ t 0 t 1 ∫ D h ∑ i ( φ i η i − 1 2 φ i G [ η ] φ i ) d x d t , (80) S_\varphi= \Upsilon_\tau\int_{t_0}^{t_1}\int_{\mathcal{D}_h}\sum_i\left(\varphi_i\eta_i-\frac{1}{2}\varphi_iG[\eta]\varphi_i\right)\,\mathrm{d}x\,\mathrm{d}t, \tag*{(80)} S φ = Υ τ ∫ t 0 t 1 ∫ D h i ∑ ( φ i η i − 2 1 φ i G [ η ] φ i ) d x d t , (80) up to terms independent of the potential modes φ i \varphi_i φ i φ i \varphi_i φ i
δ φ i S φ = Υ τ ∫ t 0 t 1 ⟨ δ φ i , η i − G [ η ] φ i ⟩ d t . (81) \delta_{\varphi_i}S_\varphi= \Upsilon_\tau\int_{t_0}^{t_1}\left\langle\delta\varphi_i,\eta_i-G[\eta]\varphi_i\right\rangle\,\mathrm{d}t. \tag*{(81)} δ φ i S φ = Υ τ ∫ t 0 t 1 ⟨ δ φ i , η i − G [ η ] φ i ⟩ d t . (81) Since Υ τ > 0 \Upsilon_\tau> 0 Υ τ > 0 δ φ i \delta\varphi_i δ φ i
η i = G [ η ] φ i . (82) \eta_i = G[\eta]\varphi_i. \tag*{(82)} η i = G [ η ] φ i . (82) which is fully consistent with the surface transport equation
∂ t η = 1 μ ν G μ , ϵ [ η ] φ ‾ + ϵ σ m ϵ μ ν G μ , ϵ [ η ] φ τ . (83) \partial_t\eta= \frac{1}{\mu\nu}G^{\mu,\epsilon}[\eta]\overline{\varphi}+\frac{\epsilon_\sigma m}{\epsilon\mu\nu}G^{\mu,\epsilon}[\eta]\varphi^\tau. \tag*{(83)} ∂ t η = μν 1 G μ , ϵ [ η ] φ + ϵ μν ϵ σ m G μ , ϵ [ η ] φ τ . (83) already obtained for the large-scale equations (recalled here in nondimensional form).
Expressing the Bernoulli equation for the velocity potential correlation functions at the surface, we obtain:
∂ t φ i + ∇ φ ‾ ⋅ ∇ φ i − ( ∇ φ i ⋅ ∇ η ) ( G [ η ] φ ‾ + ∇ φ ‾ ⋅ ∇ η ) 1 + ∣ ∇ η ∣ 2 − ( G [ η ] φ i + ∇ φ i ⋅ ∇ η ) G [ η ] φ τ 1 + ∣ ∇ η ∣ 2 = 0. (84) \partial_t\varphi_i+\nabla\overline{\varphi}\cdot\nabla\varphi_i-\frac{(\nabla\varphi_i\cdot\nabla\eta)(G[\eta]\overline{\varphi}+\nabla\overline{\varphi}\cdot\nabla\eta)}{1+\lvert\nabla\eta\rvert^2}-\frac{(G[\eta]\varphi_i+\nabla\varphi_i\cdot\nabla\eta)G[\eta]\varphi^\tau}{1+\lvert\nabla\eta\rvert^2}=0. \tag*{(84)} ∂ t φ i + ∇ φ ⋅ ∇ φ i − 1 + ∣ ∇ η ∣ 2 ( ∇ φ i ⋅ ∇ η ) ( G [ η ] φ + ∇ φ ⋅ ∇ η ) − 1 + ∣ ∇ η ∣ 2 ( G [ η ] φ i + ∇ φ i ⋅ ∇ η ) G [ η ] φ τ = 0. (84) It can be observed that while the Bernoulli equation for the potential is a linear PDE, its expression on the surface is a nonlinear SPDE, since the last term depends on the full noise.
In nondimensional form, using the scaling introduced above, this equation reads
∂ t φ i + ϵ ν ∇ φ ‾ ⋅ ∇ φ i − ϵ 2 μ ν ( ∇ φ i ⋅ ∇ η ) ( 1 μ G μ , ϵ [ η ] φ ‾ + ϵ ∇ φ ‾ ⋅ ∇ η ) 1 + ϵ 2 μ ∣ ∇ η ∣ 2 − ϵ σ m μ ν ( 1 μ G μ , ϵ [ η ] φ i + ϵ ∇ φ i ⋅ ∇ η ) 1 μ G μ , ϵ [ η ] φ τ 1 + ϵ 2 μ ∣ ∇ η ∣ 2 = 0. (85) \partial_t\varphi_i+\frac{\epsilon}{\nu}\nabla\overline{\varphi}\cdot\nabla\varphi_i-\frac{\epsilon^2\mu}{\nu}\frac{(\nabla\varphi_i\cdot\nabla\eta)\left(\frac{1}{\mu}G^{\mu,\epsilon}[\eta]\overline{\varphi}+\epsilon\nabla\overline{\varphi}\cdot\nabla\eta\right)}{1+\epsilon^2\mu\lvert\nabla\eta\rvert^2}-\frac{\epsilon\sigma_m\mu}{\nu}\frac{\left(\frac{1}{\mu}G^{\mu,\epsilon}[\eta]\varphi_i+\epsilon\nabla\varphi_i\cdot\nabla\eta\right)\frac{1}{\mu}G^{\mu,\epsilon}[\eta]\varphi^\tau}{1+\epsilon^2\mu\lvert\nabla\eta\rvert^2}=0. \tag*{(85)} ∂ t φ i + ν ϵ ∇ φ ⋅ ∇ φ i − ν ϵ 2 μ 1 + ϵ 2 μ ∣ ∇ η ∣ 2 ( ∇ φ i ⋅ ∇ η ) ( μ 1 G μ , ϵ [ η ] φ + ϵ ∇ φ ⋅ ∇ η ) − ν ϵ σ m μ 1 + ϵ 2 μ ∣ ∇ η ∣ 2 ( μ 1 G μ , ϵ [ η ] φ i + ϵ ∇ φ i ⋅ ∇ η ) μ 1 G μ , ϵ [ η ] φ τ = 0. (85) It can be noted that the two last terms correspond respectively to interactions between the large-scale vertical component and the angle between the horizontal noise basis and the surface normal, and to interactions between the vertical component of the noise and the normal derivative of the noise potential at the surface. In other words, the third term represents a large-scale/small-scale interaction, while the last term corresponds to a small-scale wave–wave interaction.
Final stochastic water wave models
Gathering the elements obtained so far, we obtain the following multiscale water-wave representation. The first system is associated with a finite decorrelation time of the small-scale structures.
System with τ \tau τ For a nonzero correlation time of the small-scale components, we consider the following system:
μ Δ x ϕ + ∂ z 2 ϕ = 0 in Ω t , ∂ z ϕ = 0 at z = − 1 , ϕ ( x , ϵ η ( x , t ) , t ) = φ ( x , t ) at z = ϵ η ( x , t ) . (86) \begin{aligned}
\mu\Delta_x\phi+\partial_z^2\phi&=0 \quad\text{in }\Omega_t,\\
\partial_z\phi&=0 \quad\text{at }z=-1,\\
\phi(x,\epsilon\eta(x,t),t)&=\varphi(x,t) \quad\text{at }z=\epsilon\eta(x,t). \tag*{(86)}
\end{aligned} μ Δ x ϕ + ∂ z 2 ϕ ∂ z ϕ ϕ ( x , ϵη ( x , t ) , t ) = 0 in Ω t , = 0 at z = − 1 , = φ ( x , t ) at z = ϵη ( x , t ) . (86) The kinematic boundary condition
∂ t η = 1 μ ν G μ , ϵ [ η ] φ , (87) \partial_t\eta=\frac{1}{\mu\nu}G^{\mu,\epsilon}[\eta]\varphi, \tag*{(87)} ∂ t η = μν 1 G μ , ϵ [ η ] φ , (87) For the dynamic boundary condition two regimes must be distinguished. For a sufficiently large correlation time, the full nonlinear water-wave Bernoulli equation is retained:
∂ t φ + η + ϵ 2 ν ∣ ∇ φ ∣ 2 − 1 2 ϵ μ ν ( 1 μ G μ , ϵ [ η ] φ + ϵ ∇ η ⋅ ∇ φ ) 2 1 + ϵ 2 μ ∣ ∇ η ∣ 2 = 0. (88) \partial_t\varphi+\eta+\frac{\epsilon}{2\nu}\lvert\nabla\varphi\rvert^2-\frac{1}{2}\frac{\epsilon\mu}{\nu}\frac{\left(\frac{1}{\mu}G^{\mu,\epsilon}[\eta]\varphi+\epsilon\nabla\eta\cdot\nabla\varphi\right)^2}{1+\epsilon^2\mu\lvert\nabla\eta\rvert^2}=0. \tag*{(88)} ∂ t φ + η + 2 ν ϵ ∣ ∇ φ ∣ 2 − 2 1 ν ϵ μ 1 + ϵ 2 μ ∣ ∇ η ∣ 2 ( μ 1 G μ , ϵ [ η ] φ + ϵ ∇ η ⋅ ∇ φ ) 2 = 0. (88) This formulation relies on the decomposition
φ ( x , t ) = φ ‾ ( x , t ) + φ τ ( x , t ) , with φ τ ( x , t ) = ∫ t − τ t + τ h τ ( t − s ) φ i ( x , t ) d β s i . (89) \varphi(x,t)=\overline{\varphi}(x,t)+\varphi^\tau(x,t),\quad\text{with }\varphi^\tau(x,t)=\int_{t-\tau}^{t+\tau}h_\tau(t-s)\varphi_i(x,t)\,\mathrm{d}\beta_s^i. \tag*{(89)} φ ( x , t ) = φ ( x , t ) + φ τ ( x , t ) , with φ τ ( x , t ) = ∫ t − τ t + τ h τ ( t − s ) φ i ( x , t ) d β s i . (89) and
∂ t η ( x , t ) = η ~ ( x , t ) + η τ ( x , t ) , with η τ ( x , t ) = ∫ t − τ t + τ h ~ τ ( t − s ) η i ( x , t ) d β s i . (90) \partial_t\eta(x,t)=\widetilde{\eta}(x,t)+\eta^\tau(x,t),\quad\text{with }\eta^\tau(x,t)=\int_{t-\tau}^{t+\tau}\widetilde{h}_\tau(t-s)\eta_i(x,t)\,\mathrm{d}\beta_s^i. \tag*{(90)} ∂ t η ( x , t ) = η ( x , t ) + η τ ( x , t ) , with η τ ( x , t ) = ∫ t − τ t + τ h τ ( t − s ) η i ( x , t ) d β s i . (90) The regularisation kernels are assumed to satisfy the normalization conditions ∫ t − τ t h ~ τ d s = 1 \int_{t-\tau}^{t}\widetilde{h}_\tau\,\mathrm{d}s=1 ∫ t − τ t h τ d s = 1
When the correlation time becomes small, the nonlinear evolution of the fluctuating components no longer admits a direct interpretation in the decorrelation limit. Assuming the LU-type approximation with the balance relation (47) for the fluctuating potential, the large-scale kinematic condition becomes
∂ t η = 1 μ ν G μ , ϵ [ η ] ϕ ‾ + ϵ σ m ϵ μ ν G μ , ϵ [ η ] φ τ , (91) \partial_t\eta= \frac{1}{\mu\nu}G^{\mu,\epsilon}[\eta]\overline{\phi} + \frac{\epsilon_{\sigma m}}{\epsilon\mu\nu}G^{\mu,\epsilon}[\eta]\varphi^\tau, \tag*{(91)} ∂ t η = μν 1 G μ , ϵ [ η ] ϕ + ϵ μν ϵ σ m G μ , ϵ [ η ] φ τ , (91) while the large-scale dynamic boundary condition reads
∂ t ϕ ‾ + η + ϵ 2 ν ∣ ∇ ϕ ‾ ∣ 2 − ϵ μ 2 ν ( 1 μ G μ , ϵ [ η ] ϕ ‾ + ϵ ∇ η ⋅ ∇ ϕ ‾ ) 2 1 + ϵ 2 μ ∣ ∇ η ∣ 2 + ϵ σ m ν ∇ ϕ ‾ ⋅ ∇ φ τ − ϵ σ m μ ν ( 1 μ G μ , ϵ [ η ] ϕ ‾ + ϵ ∇ η ⋅ ∇ ϕ ‾ ) ( 1 μ G μ , ϵ [ η ] φ τ + ϵ ∇ η ⋅ ∇ φ τ ) 1 + ϵ 2 μ ∣ ∇ η ∣ 2 = 0. (92) \partial_t\overline{\phi}+\eta+\frac{\epsilon}{2\nu}\lvert\nabla\overline{\phi}\rvert^2-\frac{\epsilon\mu}{2\nu}\frac{\left(\frac{1}{\mu}G^{\mu,\epsilon}[\eta]\overline{\phi}+\epsilon\nabla\eta\cdot\nabla\overline{\phi}\right)^2}{1+\epsilon^2\mu\lvert\nabla\eta\rvert^2}
+\frac{\epsilon_{\sigma m}}{\nu}\nabla\overline{\phi}\cdot\nabla\varphi^\tau-\frac{\epsilon_{\sigma m}\mu}{\nu}\frac{\left(\frac{1}{\mu}G^{\mu,\epsilon}[\eta]\overline{\phi}+\epsilon\nabla\eta\cdot\nabla\overline{\phi}\right)\left(\frac{1}{\mu}G^{\mu,\epsilon}[\eta]\varphi^\tau+\epsilon\nabla\eta\cdot\nabla\varphi^\tau\right)}{1+\epsilon^2\mu\lvert\nabla\eta\rvert^2}=0. \tag*{(92)} ∂ t ϕ + η + 2 ν ϵ ∣ ∇ ϕ ∣ 2 − 2 ν ϵ μ 1 + ϵ 2 μ ∣ ∇ η ∣ 2 ( μ 1 G μ , ϵ [ η ] ϕ + ϵ ∇ η ⋅ ∇ ϕ ) 2 + ν ϵ σ m ∇ ϕ ⋅ ∇ φ τ − ν ϵ σ m μ 1 + ϵ 2 μ ∣ ∇ η ∣ 2 ( μ 1 G μ , ϵ [ η ] ϕ + ϵ ∇ η ⋅ ∇ ϕ ) ( μ 1 G μ , ϵ [ η ] φ τ + ϵ ∇ η ⋅ ∇ φ τ ) = 0. (92) μ Δ x ϕ i + ∂ z 2 ϕ i = 0 in Ω t , \mu\Delta_x\phi_i+\partial_z^2\phi_i=0\quad\text{in }\Omega_t, μ Δ x ϕ i + ∂ z 2 ϕ i = 0 in Ω t , ∂ z ϕ i = 0 at z = − 1 , \partial_z\phi_i=0\quad\text{at }z=-1, ∂ z ϕ i = 0 at z = − 1 , ϕ i ( x , ϵ η ( x , t ) , t ) = φ i ( x , t ) at z = ϵ η ( x , t ) . (93) \phi_i(x,\epsilon\eta(x,t),t)=\varphi_i(x,t)\quad\text{at }z=\epsilon\eta(x,t). \tag*{(93)} ϕ i ( x , ϵη ( x , t ) , t ) = φ i ( x , t ) at z = ϵη ( x , t ) . (93) Constraint on the surface elevation correlation functions:
η i = G μ , ϵ [ η ] φ i , (94) \eta_i=G^{\mu,\epsilon}[\eta]\varphi_i, \tag*{(94)} η i = G μ , ϵ [ η ] φ i , (94) where any dimensional prefactors have been absorbed into the definition of η i \eta_i η i
The dynamic (Bernoulli) condition for the noise trace velocity potential (in nondimensional form):
∂ t φ i + ϵ ν ∇ ϕ ‾ ⋅ ∇ φ i − ϵ 2 μ ν ( ∇ φ i ⋅ ∇ η ) ( 1 μ G μ , ϵ [ η ] ϕ ‾ + ϵ ∇ ϕ ‾ ⋅ ∇ η ) 1 + ϵ 2 μ ∣ ∇ η ∣ 2 − ϵ σ m μ ν ( 1 μ G μ , ϵ [ η ] φ i + ϵ ∇ φ i ⋅ ∇ η ) 1 μ G μ , ϵ [ η ] φ τ 1 + ϵ 2 μ ∣ ∇ η ∣ 2 = 0. (95) \partial_t\varphi_i+\frac{\epsilon}{\nu}\nabla\overline{\phi}\cdot\nabla\varphi_i-\frac{\epsilon^2\mu}{\nu}\frac{(\nabla\varphi_i\cdot\nabla\eta)\left(\frac{1}{\mu}G^{\mu,\epsilon}[\eta]\overline{\phi}+\epsilon\nabla\overline{\phi}\cdot\nabla\eta\right)}{1+\epsilon^2\mu\lvert\nabla\eta\rvert^2}
-\frac{\epsilon_{\sigma m}\mu}{\nu}\frac{\left(\frac{1}{\mu}G^{\mu,\epsilon}[\eta]\varphi_i+\epsilon\nabla\varphi_i\cdot\nabla\eta\right)\frac{1}{\mu}G^{\mu,\epsilon}[\eta]\varphi^\tau}{1+\epsilon^2\mu\lvert\nabla\eta\rvert^2}=0. \tag*{(95)} ∂ t φ i + ν ϵ ∇ ϕ ⋅ ∇ φ i − ν ϵ 2 μ 1 + ϵ 2 μ ∣ ∇ η ∣ 2 ( ∇ φ i ⋅ ∇ η ) ( μ 1 G μ , ϵ [ η ] ϕ + ϵ ∇ ϕ ⋅ ∇ η ) − ν ϵ σ m μ 1 + ϵ 2 μ ∣ ∇ η ∣ 2 ( μ 1 G μ , ϵ [ η ] φ i + ϵ ∇ φ i ⋅ ∇ η ) μ 1 G μ , ϵ [ η ] φ τ = 0. (95) An idealised time-decorrelated system is obtained by taking the vanishing-correlation-time limit of the preceding approximation.
System with time-decorrelated velocity noise We now return to dimensional variables. In the zero-correlation-time limit, the finite-correlation potential contribution is replaced by the Stratonovich stochastic potential increment
d ϕ σ = ϕ i ∘ d β t i . (96) \mathrm{d}\phi^\sigma=\phi_i\circ\mathrm{d}\beta_t^i. \tag*{(96)} d ϕ σ = ϕ i ∘ d β t i . (96) The large-scale potential satisfies the Laplace potential flow condition
Δ φ ˉ = 0 in Ω t , ∂ z φ ˉ = 0 at z = b , φ ˉ ( x , η ( x , t ) , t ) = φ ˉ ( x , t ) at z = η ( x , t ) . (97) \begin{aligned}
\Delta\bar{\varphi} &= 0 \text{ in } \Omega_t,\\
\partial_z\bar{\varphi} &= 0 \text{ at } z=b,\\
\bar{\varphi}(x,\eta(x,t),t) &= \bar{\varphi}(x,t) \text{ at } z=\eta(x,t). \tag*{(97)}
\end{aligned} Δ φ ˉ ∂ z φ ˉ φ ˉ ( x , η ( x , t ) , t ) = 0 in Ω t , = 0 at z = b , = φ ˉ ( x , t ) at z = η ( x , t ) . (97) The kinematic boundary condition is
d η = G [ η ] φ ˉ d t + G [ η ] φ i ∘ d β t i . (98) \mathrm{d}\eta=G[\eta]\bar{\varphi}\,\mathrm{d}t+G[\eta]\varphi_i\circ\mathrm{d}\beta_t^i. \tag*{(98)} d η = G [ η ] φ ˉ d t + G [ η ] φ i ∘ d β t i . (98) The large-scale dynamic boundary condition is
d φ ˉ + [ g η + 1 2 ∣ ∇ φ ˉ ∣ 2 − 1 2 ( G [ η ] φ ˉ + ∇ η ⋅ ∇ φ ˉ ) 2 1 + ∣ ∇ η ∣ 2 ] d t + [ ∇ φ ˉ ⋅ ∇ φ i − ( G [ η ] φ ˉ + ∇ η ⋅ ∇ φ ˉ ) ( G [ η ] φ i + ∇ η ⋅ ∇ φ i ) 1 + ∣ ∇ η ∣ 2 ] ∘ d β t i = 0. (99) \begin{aligned}
\mathrm{d}\bar{\varphi}
&+\left[g\eta+\frac{1}{2}\lvert\nabla\bar{\varphi}\rvert^2-\frac{1}{2}\frac{\left(G[\eta]\bar{\varphi}+\nabla\eta\cdot\nabla\bar{\varphi}\right)^2}{1+\lvert\nabla\eta\rvert^2}\right]\mathrm{d}t\\
&+\left[\nabla\bar{\varphi}\cdot\nabla\varphi_i-\frac{\left(G[\eta]\bar{\varphi}+\nabla\eta\cdot\nabla\bar{\varphi}\right)\left(G[\eta]\varphi_i+\nabla\eta\cdot\nabla\varphi_i\right)}{1+\lvert\nabla\eta\rvert^2}\right]\circ\mathrm{d}\beta_t^i=0. \tag*{(99)}
\end{aligned} d φ ˉ + [ g η + 2 1 ∣ ∇ φ ˉ ∣ 2 − 2 1 1 + ∣ ∇ η ∣ 2 ( G [ η ] φ ˉ + ∇ η ⋅ ∇ φ ˉ ) 2 ] d t + [ ∇ φ ˉ ⋅ ∇ φ i − 1 + ∣ ∇ η ∣ 2 ( G [ η ] φ ˉ + ∇ η ⋅ ∇ φ ˉ ) ( G [ η ] φ i + ∇ η ⋅ ∇ φ i ) ] ∘ d β t i = 0. (99) The noise potential modes satisfy
Δ φ i = 0 in Ω t , ∂ z φ i = 0 at z = b , φ i ( x , η ( x , t ) , t ) = φ i ( x , t ) at z = η ( x , t ) . (100) \begin{aligned}
\Delta\varphi_i &= 0 \text{ in } \Omega_t,\\
\partial_z\varphi_i &= 0 \text{ at } z=b,\\
\varphi_i(x,\eta(x,t),t) &= \varphi_i(x,t) \text{ at } z=\eta(x,t). \tag*{(100)}
\end{aligned} Δ φ i ∂ z φ i φ i ( x , η ( x , t ) , t ) = 0 in Ω t , = 0 at z = b , = φ i ( x , t ) at z = η ( x , t ) . (100) The correlation modes of the surface elevation martingale term satisfies
η i = G [ η ] φ i . (101) \eta_i=G[\eta]\varphi_i. \tag*{(101)} η i = G [ η ] φ i . (101) The surface equation for the noise potential modes is
d φ i + [ ∇ φ ˉ ⋅ ∇ φ i − ( ∇ φ i ⋅ ∇ η ) ( G [ η ] φ ˉ + ∇ φ ˉ ⋅ ∇ η ) 1 + ∣ ∇ η ∣ 2 ] d t − G [ η ] φ i + ∇ φ i ⋅ ∇ η 1 + ∣ ∇ η ∣ 2 η j ∘ d β t j = 0. (102) \mathrm{d}\varphi_i+\left[\nabla\bar{\varphi}\cdot\nabla\varphi_i-\frac{(\nabla\varphi_i\cdot\nabla\eta)(G[\eta]\bar{\varphi}+\nabla\bar{\varphi}\cdot\nabla\eta)}{1+\lvert\nabla\eta\rvert^2}\right]\mathrm{d}t-\frac{G[\eta]\varphi_i+\nabla\varphi_i\cdot\nabla\eta}{1+\lvert\nabla\eta\rvert^2}\eta_j\circ\mathrm{d}\beta_t^j=0. \tag*{(102)} d φ i + [ ∇ φ ˉ ⋅ ∇ φ i − 1 + ∣ ∇ η ∣ 2 ( ∇ φ i ⋅ ∇ η ) ( G [ η ] φ ˉ + ∇ φ ˉ ⋅ ∇ η ) ] d t − 1 + ∣ ∇ η ∣ 2 G [ η ] φ i + ∇ φ i ⋅ ∇ η η j ∘ d β t j = 0. (102) Approximated systems
The previous sections provide general coupled surface-wave systems formulated either with finite-time correlated noise or in the time-decorrelated limit. These systems rely on the full nonlinear water-wave description and are therefore numerically demanding. It is both necessary and informative to derive simplified models. In the following, we obtain such approximations through asymptotic assumptions applied to the nondimensional equations. We focus on the system with finite correlation time, (91) –(92) –(95) ; the corresponding Stratonovich formulations are then obtained directly by taking the decorrelation limit.
The approximations considered here are restricted to the shallow-water regime, characterized by the small shallowness parameter μ = H 2 L 2 ≪ 1 \mu=\frac{H^2}{L^2}\ll1 μ = L 2 H 2 ≪ 1 ϵ \epsilon ϵ
For μ ≪ 1 \mu\ll1 μ ≪ 1
V ( x , t ) = 1 h ∫ − 1 ϵ η ( x , t ) ∇ ϕ ( x , z , t ) d z , (103) V(x,t) = \frac{1}{h} \int_{-1}^{\epsilon\eta(x,t)} \nabla\phi(x,z,t)\,\mathrm{d}z, \tag*{(103)} V ( x , t ) = h 1 ∫ − 1 ϵη ( x , t ) ∇ ϕ ( x , z , t ) d z , (103) where h = 1 + ϵ η ( x , t ) h = 1 + \epsilon\eta(x,t) h = 1 + ϵη ( x , t )
G μ , ϵ [ η ] φ = − μ ∇ ⋅ ( h V ) . (104) G^{\mu,\epsilon}[\eta]\varphi= -\mu\nabla\cdot(hV). \tag*{(104)} G μ , ϵ [ η ] φ = − μ ∇ ⋅ ( hV ) . (104) In the shallow-water regime, the depth-averaged velocity admits the following asymptotic expansion in powers of μ \mu μ
V = V 0 + ∑ j = 1 n μ j V j + O ( μ n + 1 ) , (105) V = V_0 + \sum_{j=1}^{n} \mu^j V_j + \mathcal{O}(\mu^{n+1}), \tag*{(105)} V = V 0 + j = 1 ∑ n μ j V j + O ( μ n + 1 ) , (105) with convergence in suitable Sobolev spaces [40, Prop. 3.37]. The leading-order term is V 0 = ∇ φ V_0 = \nabla\varphi V 0 = ∇ φ V 1 = − T [ h ] ∇ φ V_1 = -\mathcal{T}[h]\nabla\varphi V 1 = − T [ h ] ∇ φ
T [ h ] V = − 1 3 h ∇ ( h 3 ∇ ⋅ V ) . (106) \mathcal{T}[h]V = -\frac{1}{3h}\nabla(h^3\nabla\cdot V). \tag*{(106)} T [ h ] V = − 3 h 1 ∇ ( h 3 ∇ ⋅ V ) . (106) In the stochastic setting, the depth-averaged velocity decomposes naturally as V = V ‾ + V τ V = \overline{V} + V^\tau V = V + V τ V τ = ∫ t − τ t + τ h τ ( t − s ) V i d β s i V^\tau= \int_{t-\tau}^{t+\tau} h_\tau(t-s)V_i\,\mathrm{d}\beta_s^i V τ = ∫ t − τ t + τ h τ ( t − s ) V i d β s i
In the following, we derive simplified models under different asymptotic assumptions on the large-scale and small-scale parameters. We begin with the long-wave regime.
Long wave approximation
The cases considered here correspond to weakly nonlinear regimes for small values of the nonlinear parameter ϵ = A / H \epsilon= A/H ϵ = A / H
Airy waves: low amplitude regime
The first approximation of interest is associated with very long waves, μ ≪ 1 \mu\ll1 μ ≪ 1 ν ≈ 1 \nu\approx1 ν ≈ 1 ϵ ≪ 1 \epsilon\ll1 ϵ ≪ 1 ν σ ≈ 1 \nu_\sigma\approx1 ν σ ≈ 1
Since the ratio r < 1 r < 1 r < 1 ϵ σ ∼ ϵ \epsilon_\sigma\sim\epsilon ϵ σ ∼ ϵ A σ > A A_\sigma> A A σ > A A σ ≫ L A_{\sigma} \gg L A σ ≫ L
∂ t ϕ ‾ + η = 0. (107) \partial_t\overline{\phi}+\eta=0. \tag*{(107)} ∂ t ϕ + η = 0. (107) The kinematic boundary condition becomes
∂ t η = 1 μ ∂ z ϕ ‾ ∣ z = 0 + r μ ∂ z ϕ τ ∣ z = 0 , (108) \partial_t\eta=\frac{1}{\mu}\partial_z\overline{\phi}\big|_{z=0}+\frac{r}{\mu}\partial_z\phi^\tau\big|_{z=0}, \tag*{(108)} ∂ t η = μ 1 ∂ z ϕ z = 0 + μ r ∂ z ϕ τ z = 0 , (108) where we used the shallow-water approximation G μ , ϵ ≃ ∂ z G^{\mu,\epsilon}\simeq\partial_z G μ , ϵ ≃ ∂ z ϵ σ ∼ ϵ \epsilon_\sigma\sim\epsilon ϵ σ ∼ ϵ
The kinematic boundary condition is
∂ t η = 1 μ ∂ z ϕ ∣ z = 0 + r μ ∂ z ϕ τ ∣ z = 0 , (109) \partial_t\eta=\frac{1}{\mu}\partial_z\phi\big|_{z=0}+\frac{r}{\mu}\partial_z\phi^\tau\big|_{z=0}, \tag*{(109)} ∂ t η = μ 1 ∂ z ϕ z = 0 + μ r ∂ z ϕ τ z = 0 , (109) The velocity potentials satisfy Laplace equations in the fluid domain,
Δ ϕ ‾ = Δ ϕ i = 0 , (110) \Delta\overline{\phi}=\Delta\phi^i=0, \tag*{(110)} Δ ϕ = Δ ϕ i = 0 , (110) with boundary conditions
∂ z ϕ ‾ ( b ) = ∂ z ϕ i ( b ) = 0 , ϕ ‾ ( x , 0 , t ) = ϕ ‾ ( x , t ) , ϕ i ( x , 0 , t ) = ϕ i ( x , t ) . (111) \partial_z\overline{\phi}(b)=\partial_z\phi^i(b)=0,\quad\overline{\phi}(x,0,t)=\overline{\phi}(x,t),\quad\phi^i(x,0,t)=\phi^i(x,t). \tag*{(111)} ∂ z ϕ ( b ) = ∂ z ϕ i ( b ) = 0 , ϕ ( x , 0 , t ) = ϕ ( x , t ) , ϕ i ( x , 0 , t ) = ϕ i ( x , t ) . (111) With this scaling, the noise correlation functions are time-independent:
∂ t ϕ i = 0. (112) \partial_t\phi_i=0. \tag*{(112)} ∂ t ϕ i = 0. (112) This system can be solved explicitly in Fourier space (assuming lateral periodic condition), or equivalently through a slow-envelope approximation and a WKB expansion. Introducing, U = ( ϕ ‾ , η ) t r U=(\overline{\phi},\eta)^{tr} U = ( ϕ , η ) t r
U ( t ) = e t A U ( 0 ) + r ∫ 0 t e ( t − s ) A ( 0 G [ 0 ] ϕ s τ ) d s , A = ( 0 − 1 G [ 0 ] 0 ) . (113) U(t)=e^{tA}U(0)+r\int_0^t e^{(t-s)A}
\begin{pmatrix}
0\\
G[0]\phi_s^\tau
\end{pmatrix}\,ds,\qquad
A=\begin{pmatrix}
0 & -1\\
G[0] & 0
\end{pmatrix}. \tag*{(113)} U ( t ) = e t A U ( 0 ) + r ∫ 0 t e ( t − s ) A ( 0 G [ 0 ] ϕ s τ ) d s , A = ( 0 G [ 0 ] − 1 0 ) . (113) The linearized Dirichlet-to-Neumann operator at the flat surface is
G [ 0 ] ϕ ( x ) = ∫ R d ∣ k ∣ tanh ( ∣ k ∣ H ) ϕ ^ ( k ) e i k ⋅ x d k . (114) G[0]\phi(x)=\int_{\mathbb{R}^d}|k|\tanh(|k|H)\widehat{\phi}(k)e^{ik\cdot x}\,dk. \tag*{(114)} G [ 0 ] ϕ ( x ) = ∫ R d ∣ k ∣ tanh ( ∣ k ∣ H ) ϕ ( k ) e ik ⋅ x d k . (114) The eigenvalues of A A A λ ± = ± i ω k \lambda_\pm=\pm i\omega_k λ ± = ± i ω k
ω k 2 = g G k [ 0 ] = g ∣ k ∣ tanh ( ∣ k ∣ H ) . (115) \omega_k^2=gG_k[0]=g|k|\tanh(|k|H). \tag*{(115)} ω k 2 = g G k [ 0 ] = g ∣ k ∣ tanh ( ∣ k ∣ H ) . (115) The associated eigenvectors satisfy
v ± ± = ± i ω k G k [ 0 ] v n . (116) v_\pm^\pm=\pm i\frac{\omega_k}{G_k[0]}v^n. \tag*{(116)} v ± ± = ± i G k [ 0 ] ω k v n . (116) The solution may be written in normal-mode form
U = ∑ k a k v − + a − k ∗ v + , with v n = 1 , (117) U=\sum_k a_kv_-+a^*_{-k}v_+,\qquad\text{with }v^n=1, \tag*{(117)} U = k ∑ a k v − + a − k ∗ v + , with v n = 1 , (117) which gives
η k = a k + a − k ∗ and ϕ k = − i ω k G k [ 0 ] ( a k − a − k ∗ ) . (118) \eta_k=a_k+a^*_{-k}\quad\text{and}\quad\phi_k=-\frac{i\omega_k}{G_k[0]}(a_k-a^*_{-k}). \tag*{(118)} η k = a k + a − k ∗ and ϕ k = − G k [ 0 ] i ω k ( a k − a − k ∗ ) . (118) The complex wave amplitude is therefore
a k ( t ) = 1 2 ( η k + i G k [ 0 ] ω k φ k ) . (119) a_k(t) = \frac{1}{2}\left(\eta_k + \frac{iG_k[0]}{\omega_k}\varphi_k\right). \tag*{(119)} a k ( t ) = 2 1 ( η k + ω k i G k [ 0 ] φ k ) . (119) For each wave number, the Duhamel solution is
a k ( t ) = e − i ω k t a k ( 0 ) + r 2 ∫ 0 t e − i ω k ( t − s ) G k [ 0 ] φ s τ d s . (120) a_k(t) = e^{-i\omega_k t}a_k(0) + \frac{r}{2}\int_0^t e^{-i\omega_k(t-s)}G_k[0]\varphi_s^\tau\,\mathrm{d}s. \tag*{(120)} a k ( t ) = e − i ω k t a k ( 0 ) + 2 r ∫ 0 t e − i ω k ( t − s ) G k [ 0 ] φ s τ d s . (120) The surface elevation is therefore
η ( x , t ) = ∑ k a k ( 0 ) e i ( k ⋅ x − ω k t ) + r 2 ∫ 0 t e i ( k ⋅ x − ω k ( t − s ) ) G k [ 0 ] φ s τ d s + c.c. , (121) \eta(x,t) = \sum_k a_k(0)e^{i(k\cdot x-\omega_k t)} + \frac{r}{2}\int_0^t e^{i(k\cdot x-\omega_k(t-s))}G_k[0]\varphi_s^\tau\,\mathrm{d}s + \text{c.c.}, \tag*{(121)} η ( x , t ) = k ∑ a k ( 0 ) e i ( k ⋅ x − ω k t ) + 2 r ∫ 0 t e i ( k ⋅ x − ω k ( t − s )) G k [ 0 ] φ s τ d s + c.c. , (121) where c.c. means complex conjugate, while the surface potential reads
φ ( x , t ) = ∑ k − i ω k G k [ 0 ] a k ( 0 ) e i ( k ⋅ x − ω k t ) − i ω k 2 r ∫ 0 t e i ( k ⋅ x − ω k ( t − s ) ) φ s τ d s + c.c. (122) \varphi(x,t) = \sum_k \frac{-i\omega_k}{G_k[0]}a_k(0)e^{i(k\cdot x-\omega_k t)} - i\frac{\omega_k}{2}r\int_0^t e^{i(k\cdot x-\omega_k(t-s))}\varphi_s^\tau\,\mathrm{d}s + \text{c.c.} \tag*{(122)} φ ( x , t ) = k ∑ G k [ 0 ] − i ω k a k ( 0 ) e i ( k ⋅ x − ω k t ) − i 2 ω k r ∫ 0 t e i ( k ⋅ x − ω k ( t − s )) φ s τ d s + c.c. (122) The velocity potential in the fluid domain is
ϕ ( x , z , t ) = ∑ k − i ω k a k ( t ) G k [ 0 ] cosh k ( z + H ) cosh k H e i ( k ⋅ x ) + c.c. (123) \phi(x,z,t) = \sum_k -i\omega_k\frac{a_k(t)}{G_k[0]}\frac{\cosh k(z+H)}{\cosh kH}e^{i(k\cdot x)} + \text{c.c.} \tag*{(123)} ϕ ( x , z , t ) = k ∑ − i ω k G k [ 0 ] a k ( t ) cosh k H cosh k ( z + H ) e i ( k ⋅ x ) + c.c. (123) The noise correlation functions take the form
ϕ j ( x , z ) = ∑ j b j cosh k j ( z + H ) cosh k j H e i ( k j ⋅ x ) + c.c. (124) \phi_j(x,z) = \sum_j b_j\frac{\cosh k_j(z+H)}{\cosh k_jH}e^{i(k_j\cdot x)} + \text{c.c.} \tag*{(124)} ϕ j ( x , z ) = j ∑ b j cosh k j H cosh k j ( z + H ) e i ( k j ⋅ x ) + c.c. (124) The stochasticity of the model is therefore entirely driven by perturbations of the surface evolution. This provides a minimal stochastic extension of the classical deterministic Airy-wave model for very long waves.
Considering unresolved components in the deep-water regime (L σ < H ≪ L L_\sigma< H \ll L L σ < H ≪ L
A σ ∼ H L σ L , A_\sigma\sim\sqrt{\frac{H}{L_\sigma}}L, A σ ∼ L σ H L , which greatly exceeds the characteristic amplitude of the resolved waves and lies outside the regime of validity of the asymptotic expansion.
Consequently, within the shallow-water long-wave scaling considered here, the Airy model is remarkably robust. Unresolved fluctuations contribute only through an additive linear random forcing of the free surface. Small-scale stochastic perturbations act as a superposition of linear random waves on top of a deterministic carrier, without nonlinear energy transfer through advection of large scales by small scales.
Stochastic Saint-Venant equations: stronger amplitude regime
We now consider waves of larger amplitude, with ϵ ∼ O ( 1 ) \epsilon\sim\mathcal{O}(1) ϵ ∼ O ( 1 ) H ≪ L σ H \ll L_\sigma H ≪ L σ ν σ ≈ 1 \nu_\sigma\approx1 ν σ ≈ 1 γ σ ∼ O ( 1 ) \gamma_\sigma\sim\mathcal{O}(1) γ σ ∼ O ( 1 ) r ∼ O ( 1 ) r \sim\mathcal{O}(1) r ∼ O ( 1 ) A σ ∼ A A_\sigma\sim A A σ ∼ A γ σ r ∼ ϵ \gamma_\sigma^r \sim\epsilon γ σ r ∼ ϵ
At leading order μ \mu μ
∂ t φ ‾ + η + 1 2 ϵ ∇ φ ‾ ⋅ ∇ φ ‾ + ϵ σ m ∇ φ τ ⋅ ∇ φ ‾ = 0. (125) \partial_t\overline{\varphi}+\eta+\frac{1}{2}\epsilon\nabla\overline{\varphi}\cdot\nabla\overline{\varphi}+\epsilon_\sigma m\nabla\varphi^\tau\cdot\nabla\overline{\varphi}=0. \tag*{(125)} ∂ t φ + η + 2 1 ϵ ∇ φ ⋅ ∇ φ + ϵ σ m ∇ φ τ ⋅ ∇ φ = 0. (125) together with
∂ t η = − ∇ ⋅ ( h ∇ φ ‾ ) − ϵ σ m ϵ ∇ ⋅ ( h ∇ φ τ ) . (126) \partial_t\eta=-\nabla\cdot\left(h\nabla\overline{\varphi}\right)-\frac{\epsilon_\sigma m}{\epsilon}\nabla\cdot\left(h\nabla\varphi^\tau\right). \tag*{(126)} ∂ t η = − ∇ ⋅ ( h ∇ φ ) − ϵ ϵ σ m ∇ ⋅ ( h ∇ φ τ ) . (126) Under this scaling, the noise correlation functions satisfy the linear transport equation with random coefficients
∂ t φ i + ϵ ∇ φ ‾ ⋅ ∇ φ i = 0. (127) \partial_t\varphi_i+\epsilon\nabla\overline{\varphi}\cdot\nabla\varphi_i=0. \tag*{(127)} ∂ t φ i + ϵ ∇ φ ⋅ ∇ φ i = 0. (127) The correlation functions are therefore passively transported by the large-scale flow. This is consistent with the Kraichnan–Tennekes hypothesis, according to which small-scale structures are advected by the random energy-containing large-scale flow.
Retaining additional stochastic terms in (127) would require the stronger scaling γ σ = O ( μ − 1 / 2 ) \gamma_\sigma=\mathcal{O}(\mu^{-1/2}) γ σ = O ( μ − 1/2 ) r = O ( μ ) r=\mathcal{O}(\mu) r = O ( μ )
Higher-order shallow-water models are obtained by retaining additional terms in the μ \mu μ
Coastal wave shallow water approximation
Within the long-wave regime considered above, the Saint–Venant approximation remains valid as long as the shallowness parameter μ \mu μ μ \mu μ
Large amplitude Serre-Green-Naghdi approximation
Retaining terms up to first order in μ \mu μ
∇ φ = V − μ 3 h ∇ ( h 3 ∇ ⋅ ∇ φ ) . \nabla\varphi=V-\frac{\mu}{3h}\nabla\left(h^3\nabla\cdot\nabla\varphi\right). ∇ φ = V − 3 h μ ∇ ( h 3 ∇ ⋅ ∇ φ ) . Introducing the operators
T [ h ] V = − 1 3 h ∇ ( h 3 ∇ ⋅ V ) , U = I + μ T , \mathbb{T}[h]V=-\frac{1}{3h}\nabla\left(h^3\nabla\cdot V\right),\qquad\mathbb{U}=I+\mu\mathbb{T}, T [ h ] V = − 3 h 1 ∇ ( h 3 ∇ ⋅ V ) , U = I + μ T , we obtain, up to terms of order O ( μ 2 ) \mathcal{O}(\mu^2) O ( μ 2 )
∇ φ = U V . \nabla\varphi=\mathbb{U}V. ∇ φ = U V . The momentum equation then reads
∂ t ( U V ‾ ) + ∇ η + 1 2 ϵ ∇ ( U V ‾ ⋅ U V ‾ ) + ϵ σ m ∇ ( V ‾ τ ⋅ U V ‾ ) + Q ( V ‾ , V τ ) = 0 , (128) \partial_t(\mathbb{U}\overline{V})+\nabla\eta+\frac{1}{2}\epsilon\nabla(\mathbb{U}\overline{V}\cdot\mathbb{U}\overline{V})+\epsilon_{\sigma m}\nabla(\overline{V}^{\tau}\cdot\mathbb{U}\overline{V})+Q(\overline{V},V^{\tau})=0, \tag*{(128)} ∂ t ( U V ) + ∇ η + 2 1 ϵ ∇ ( U V ⋅ U V ) + ϵ σ m ∇ ( V τ ⋅ U V ) + Q ( V , V τ ) = 0 , (128) where
Q ( V ‾ , V τ ) = − 1 2 μ ϵ ∇ ( − ∇ ⋅ ( h V ‾ ) + ϵ ∇ η ⋅ U V ‾ ) 2 − ϵ σ m μ ∇ ( ( − ∇ ⋅ ( h V ‾ ) + ϵ ∇ η ⋅ U V ‾ ) ( − ∇ ⋅ ( h V τ ) + ϵ ∇ η ⋅ U V τ ) ) , (129) Q(\overline{V},V^{\tau})=-\frac{1}{2}\mu\epsilon\nabla\left(-\nabla\cdot(h\overline{V})+\epsilon\nabla\eta\cdot\mathbb{U}\overline{V}\right)^2
-\epsilon_{\sigma m}\mu\nabla\left(\left(-\nabla\cdot(h\overline{V})+\epsilon\nabla\eta\cdot\mathbb{U}\overline{V}\right)\left(-\nabla\cdot(hV^{\tau})+\epsilon\nabla\eta\cdot\mathbb{U}V^{\tau}\right)\right), \tag*{(129)} Q ( V , V τ ) = − 2 1 μ ϵ ∇ ( − ∇ ⋅ ( h V ) + ϵ ∇ η ⋅ U V ) 2 − ϵ σ m μ ∇ ( ( − ∇ ⋅ ( h V ) + ϵ ∇ η ⋅ U V ) ( − ∇ ⋅ ( h V τ ) + ϵ ∇ η ⋅ U V τ ) ) , (129) where all O ( μ 2 ) \mathcal{O}(\mu^2) O ( μ 2 )
The surface elevation evolution remains
∂ t η = − ∇ ⋅ ( h V ‾ ) − ϵ σ m ϵ ∇ ⋅ ( h V τ ) . (130) \partial_t\eta=-\nabla\cdot(h\overline{V})-\frac{\epsilon_{\sigma m}}{\epsilon}\nabla\cdot(hV^{\tau}). \tag*{(130)} ∂ t η = − ∇ ⋅ ( h V ) − ϵ ϵ σ m ∇ ⋅ ( h V τ ) . (130) The momentum equation governs the evolution of U V ‾ \mathbb{U}\overline{V} U V V ‾ \overline{V} V U \mathbb{U} U η \eta η
The velocity correlation modes satisfy
∂ t ( U V i ) + ϵ ∇ ( U V ‾ ⋅ U V i ) − ϵ 2 μ ( U V i ⋅ ∇ η ) ( − ∇ ⋅ ( h V ‾ ) + ϵ U V ‾ ⋅ ∇ η ) + ϵ σ m μ ∇ ( ( − ∇ ⋅ ( h V i ) + ϵ U V i ⋅ ∇ η ) ∇ ⋅ ( h V τ ) ) = 0. (131) \partial_t(\mathbb{U}V_i)+\epsilon\nabla(\mathbb{U}\overline{V}\cdot\mathbb{U}V_i)-\epsilon^2\mu(\mathbb{U}V_i\cdot\nabla\eta)\left(-\nabla\cdot(h\overline{V})+\epsilon\mathbb{U}\overline{V}\cdot\nabla\eta\right)
+\epsilon_{\sigma m}\mu\nabla\left(\left(-\nabla\cdot(hV_i)+\epsilon\mathbb{U}V_i\cdot\nabla\eta\right)\nabla\cdot(hV^{\tau})\right)=0. \tag*{(131)} ∂ t ( U V i ) + ϵ ∇ ( U V ⋅ U V i ) − ϵ 2 μ ( U V i ⋅ ∇ η ) ( − ∇ ⋅ ( h V ) + ϵ U V ⋅ ∇ η ) + ϵ σ m μ ∇ ( ( − ∇ ⋅ ( h V i ) + ϵ U V i ⋅ ∇ η ) ∇ ⋅ ( h V τ ) ) = 0. (131) Unlike the Saint–Venant approximation, the dynamics of the velocity correlation modes is now governed by a nonlinear stochastic partial differential equation rather than by a linear transport equation with random coefficient.
Weakly nonlinear Boussinesq regime
We now consider the weakly nonlinear scaling ϵ = O ( μ ) \epsilon=\mathcal{O}(\mu) ϵ = O ( μ ) ϵ σ m = O ( ϵ ) \epsilon_{\sigma m}=\mathcal{O}(\epsilon) ϵ σ m = O ( ϵ ) ∂ t U = O ( μ 2 ) \partial_t\mathbb{U}=\mathcal{O}(\mu^2) ∂ t U = O ( μ 2 )
U ∂ t V ‾ μ + ∇ η + 1 2 ϵ ( V ‾ ⋅ ∇ ) V ‾ + ϵ σ m ( V τ ⋅ ∇ ) V ‾ + ϵ σ m ( V ‾ ⋅ ∇ ) V τ = 0 , (132) \mathbb{U}\partial_t\overline{V}^{\mu}+\nabla\eta+\frac{1}{2}\epsilon(\overline{V}\cdot\nabla)\overline{V}+\epsilon_{\sigma m}(V^{\tau}\cdot\nabla)\overline{V}+\epsilon_{\sigma m}(\overline{V}\cdot\nabla)V^{\tau}=0, \tag*{(132)} U ∂ t V μ + ∇ η + 2 1 ϵ ( V ⋅ ∇ ) V + ϵ σ m ( V τ ⋅ ∇ ) V + ϵ σ m ( V ⋅ ∇ ) V τ = 0 , (132) with
∂ t η = − ∇ ⋅ ( h V ‾ μ ) − ϵ σ m ϵ ∇ ⋅ ( h V τ ) . (133) \partial_t\eta=-\nabla\cdot(h\overline{V}^{\mu})-\frac{\epsilon_{\sigma m}}{\epsilon}\nabla\cdot(hV^{\tau}). \tag*{(133)} ∂ t η = − ∇ ⋅ ( h V μ ) − ϵ ϵ σ m ∇ ⋅ ( h V τ ) . (133) The small-scale velocity correlation functions satisfy
U ∂ t ∇ φ i + ϵ ∇ ( V ‾ ⋅ ∇ φ i ) = 0. (134) \mathbb{U}\partial_t\nabla\varphi_i+\epsilon\nabla(\overline{V}\cdot\nabla\varphi_i)=0. \tag*{(134)} U ∂ t ∇ φ i + ϵ ∇ ( V ⋅ ∇ φ i ) = 0. (134) Weakly nonlinear Boussinesq regime with stronger noise
For stronger unresolved fluctuation satisfying ϵ σ m = O ( ϵ / μ ) \epsilon_{\sigma m}=\mathcal{O}(\epsilon/\sqrt{\mu}) ϵ σ m = O ( ϵ / μ )
U ∂ t V ‾ μ + ∇ η + 1 2 ϵ ( V ‾ ⋅ ∇ ) V ‾ + ϵ σ m ( V μ τ ⋅ ∇ ) V ‾ + ϵ σ m ( V ‾ ⋅ ∇ ) V τ + Q σ ( V ‾ , V τ ) = 0 , (135) \mathbb{U}\partial_t\overline{V}^{\mu}+\nabla\eta+\frac{1}{2}\epsilon(\overline{V}\cdot\nabla)\overline{V}+\epsilon_{\sigma m}(V^{\mu\tau}\cdot\nabla)\overline{V}+\epsilon_{\sigma m}(\overline{V}\cdot\nabla)V^{\tau}+Q_{\sigma}(\overline{V},V^{\tau})=0, \tag*{(135)} U ∂ t V μ + ∇ η + 2 1 ϵ ( V ⋅ ∇ ) V + ϵ σ m ( V μτ ⋅ ∇ ) V + ϵ σ m ( V ⋅ ∇ ) V τ + Q σ ( V , V τ ) = 0 , (135) where
Q σ ( V ‾ , V τ ) = − ϵ σ m μ ∇ [ ( ∇ ⋅ ( h V ‾ ) ) ( ∇ ⋅ ( h V τ ) ) ] . (136) Q_{\sigma}(\overline{V},V^{\tau})=-\epsilon_{\sigma m\mu}\nabla\left[(\nabla\cdot(h\overline{V}))(\nabla\cdot(hV^{\tau}))\right]. \tag*{(136)} Q σ ( V , V τ ) = − ϵ σ m μ ∇ [ ( ∇ ⋅ ( h V )) ( ∇ ⋅ ( h V τ )) ] . (136) This model incorporates, through the stochastic forcing, part of the Serre–Green–Naghdi structure into the Boussinesq equations. It is therefore well suited to represent waves in the grey-zone regime, where Boussinesq dynamics remain appropriate but unresolved fluctuations are no longer negligible.
The corresponding dynamics of the velocity correlation modes becomes
U ∂ t ( V ‾ i μ σ ) + ϵ ∇ ( V ‾ μ ⋅ V i μ σ ) − ϵ σ m μ ∇ ( ( ∇ ⋅ ( h V σ ) ) ∇ ⋅ ( h V τ ) ) = 0. (137) \mathcal{U}\partial_t(\overline{V}_i^{\mu\sigma})+\epsilon\nabla(\overline{V}^{\mu}\cdot V_i^{\mu\sigma})-\epsilon_{\sigma m\mu}\nabla\left((\nabla\cdot(hV^{\sigma}))\nabla\cdot(hV^{\tau})\right)=0. \tag*{(137)} U ∂ t ( V i μ σ ) + ϵ ∇ ( V μ ⋅ V i μ σ ) − ϵ σ m μ ∇ ( ( ∇ ⋅ ( h V σ )) ∇ ⋅ ( h V τ ) ) = 0. (137) This yields a simplified nonlinear stochastic model retaining the dominant stochastic dispersive corrections while remaining significantly simpler than the full Serre–Green–Naghdi system.
Geometric-optics solution for the correlation modes
In this section, we derive geometric-optics solutions for simplified models governing the noise correlation modes. We focus on the small-noise-amplitude regime, in which the correlation modes satisfy
Δ ϕ i = 0 in Ω t , ∂ z ϕ i = 0 at z = − h , (138) \begin{aligned}
\Delta\phi_i &= 0 \text{ in } \Omega_t,\\
\partial_z\phi_i &= 0 \text{ at } z=-h, \tag*{(138)}
\end{aligned} Δ ϕ i ∂ z ϕ i = 0 in Ω t , = 0 at z = − h , (138) together with the linear surface condition
( ∂ t ϕ i + ∇ x , z ϕ ‾ ⋅ ∇ x , z ϕ i ) ∣ z = η = 0. (139) \left(\partial_t\phi_i+\nabla_{x,z}\overline{\phi}\cdot\nabla_{x,z}\phi_i\right)\big|_{z=\eta}=0. \tag*{(139)} ( ∂ t ϕ i + ∇ x , z ϕ ⋅ ∇ x , z ϕ i ) z = η = 0. (139) For a flat free surface and periodic lateral boundary conditions, the noise potential may be expanded as
ϕ i ( x , z , t ) = ∫ R d ϕ ^ i ( k , z , t ) e i k ⋅ x d k . (140) \phi_i(x,z,t)=\int_{\mathbb{R}^d}\widehat{\phi}_i(k,z,t)e^{i k\cdot x}\,\mathrm{d}k. \tag*{(140)} ϕ i ( x , z , t ) = ∫ R d ϕ i ( k , z , t ) e ik ⋅ x d k . (140) The Fourier modes satisfy
∂ z 2 ϕ ^ i − ∣ k ∣ 2 ϕ ^ i = 0 , (141) \partial_z^2\widehat{\phi}_i-|k|^2\widehat{\phi}_i=0, \tag*{(141)} ∂ z 2 ϕ i − ∣ k ∣ 2 ϕ i = 0 , (141) with boundary conditions
ϕ ^ i ( k , η , t ) = φ ^ i ( k , t ) , at z = η and ∂ z ϕ ^ i ( k , − h , t ) = 0 at z = − h . (142) \widehat{\phi}_i(k,\eta,t)=\widehat{\varphi}_i(k,t),\quad\text{at }z=\eta\qquad\text{and}\qquad\partial_z\widehat{\phi}_i(k,-h,t)=0\quad\text{at }z=-h. \tag*{(142)} ϕ i ( k , η , t ) = φ i ( k , t ) , at z = η and ∂ z ϕ i ( k , − h , t ) = 0 at z = − h . (142) In the deep-water limit, the decaying solution is
ϕ ^ i ( k , z , t ) = φ ^ i ( k , t ) e ∣ k ∣ ( z − η ) , (143) \widehat{\phi}_i(k,z,t)=\widehat{\varphi}_i(k,t)e^{|k|(z-\eta)}, \tag*{(143)} ϕ i ( k , z , t ) = φ i ( k , t ) e ∣ k ∣ ( z − η ) , (143) and therefore
∂ z ϕ ^ i ∣ z = η = ∣ k ∣ φ ^ i . (144) \left.\partial_z\widehat{\phi}_i\right|_{z=\eta}=|k|\widehat{\varphi}_i. \tag*{(144)} ∂ z ϕ i z = η = ∣ k ∣ φ i . (144) For a slowly varying free surface, the same relation holds at leading geometric-optics order, with ∣ k ∣ |k| ∣ k ∣
We seek WKB solutions of the form
ϕ j ( x , t ) = b j ( ϵ x , ϵ t ) exp ( i P j ( x , t ) ) , 0 < ϵ ≪ 1. (145) \phi_j(x,t)=b_j(\epsilon x,\epsilon t)\exp(iP_j(x,t)),\qquad0<\epsilon\ll1. \tag*{(145)} ϕ j ( x , t ) = b j ( ϵ x , ϵ t ) exp ( i P j ( x , t )) , 0 < ϵ ≪ 1. (145) where for the rapidly varying phase P j P_j P j
k j = ∇ P j , ω j = − ∂ t P j . (146) k_j=\nabla P_j,\qquad\omega_j=-\partial_tP_j. \tag*{(146)} k j = ∇ P j , ω j = − ∂ t P j . (146) The compatibility of the (sufficiently regular) phase implies
∂ t k j = − ∇ ω j , ∂ i ( k ℓ j ) = ∂ j ( k ℓ i ) . (147) \partial_t k_j = -\nabla\omega_j,\qquad\partial_i\left(k_\ell^j\right) = \partial_j\left(k_\ell^i\right). \tag*{(147)} ∂ t k j = − ∇ ω j , ∂ i ( k ℓ j ) = ∂ j ( k ℓ i ) . (147) Substituting the WKB ansatz into the transport equation and collecting leading-order terms gives
− i b j ω j + i b j k j ⋅ u ‾ ∣ z = η + ∂ z ϕ ‾ ∣ z = η b j ∣ k j ∣ = 0 , (148) -i b_j\omega_j + i b_j k_j\cdot\left.\overline{u}\right|_{z=\eta} + \left.\partial_z\overline{\phi}\right|_{z=\eta}b_j\lvert k_j\rvert= 0, \tag*{(148)} − i b j ω j + i b j k j ⋅ u ∣ z = η + ∂ z ϕ z = η b j ∣ k j ∣ = 0 , (148) where u ‾ ∣ z = η = ∇ ϕ ‾ \left.\overline{u}\right|_{z=\eta}=\nabla\overline{\phi} u ∣ z = η = ∇ ϕ
Introducing the intrinsic frequency λ j = ω j − k j ⋅ u ‾ \lambda_j=\omega_j-k_j\cdot\overline{u} λ j = ω j − k j ⋅ u
λ j = − i ∣ k j ∣ ∂ z ϕ ‾ ∣ z = η . (149) \lambda_j=-i\lvert k_j\rvert\left.\partial_z\overline{\phi}\right|_{z=\eta}. \tag*{(149)} λ j = − i ∣ k j ∣ ∂ z ϕ z = η . (149) This yields the following complex dispersion (eikonal) relation:
ω j = k j ⋅ u ‾ ∣ z = η − i ∣ k j ∣ ∂ z ϕ ‾ ∣ z = η . (150) \omega_j=k_j\cdot\left.\overline{u}\right|_{z=\eta}-i\lvert k_j\rvert\left.\partial_z\overline{\phi}\right|_{z=\eta}. \tag*{(150)} ω j = k j ⋅ u ∣ z = η − i ∣ k j ∣ ∂ z ϕ z = η . (150) The real part of ω j \omega_j ω j
∂ t P j + u ‾ ∣ z = η ⋅ ∇ P j − i ∣ ∇ P j ∣ ∂ z ϕ ‾ ∣ z = η = 0. (151) \partial_t\mathcal{P}_j+\left.\overline{u}\right|_{z=\eta}\cdot\nabla\mathcal{P}_j-i\lvert\nabla\mathcal{P}_j\rvert\left.\partial_z\overline{\phi}\right|_{z=\eta}=0. \tag*{(151)} ∂ t P j + u ∣ z = η ⋅ ∇ P j − i ∣ ∇ P j ∣ ∂ z ϕ z = η = 0. (151) This equation can be written in the standard Hamilton–Jacobi form
∂ t P j + ω j ( x , ∇ P j ) = 0 , \partial_t\mathcal{P}_j+\omega_j(x,\nabla\mathcal{P}_j)=0, ∂ t P j + ω j ( x , ∇ P j ) = 0 , where the Hamiltonian is identified with the local complex dispersion relation
ω j ( x , k ) = k ⋅ u ‾ ( x ) − i ∣ k ∣ ∂ z ϕ ‾ ( x ) , \omega_j(x,k)=k\cdot\overline{u}(x)-i\lvert k\rvert\partial_z\overline{\phi}(x), ω j ( x , k ) = k ⋅ u ( x ) − i ∣ k ∣ ∂ z ϕ ( x ) , with k = ∇ P j k=\nabla\mathcal{P}_j k = ∇ P j
d k j d t = − ∇ x j ω j , (152) \frac{\mathrm{d}k_j}{\mathrm{d}t}=-\nabla_{x_j}\omega_j, \tag*{(152)} d t d k j = − ∇ x j ω j , (152) d x j d t = ∇ k j ω j , (153) \frac{\mathrm{d}x_j}{\mathrm{d}t}=\nabla_{k_j}\omega_j, \tag*{(153)} d t d x j = ∇ k j ω j , (153) which describe the evolution of the wavevector and the propagation velocity of the unresolved wave packet, respectively. The complex Hamiltonian structure highlights that unresolved correlation modes are active geometrical objects rather than passively advected fluctuations. The first Hamilton equation
d x j d t = u ‾ ( x j ) − i k j ∣ k j ∣ ∂ z ϕ ‾ ( x j ) , \frac{\mathrm{d}x_j}{\mathrm{d}t}=\overline{u}(x_j)-i\frac{k_j}{\lvert k_j\rvert}\partial_z\overline{\phi}(x_j), d t d x j = u ( x j ) − i ∣ k j ∣ k j ∂ z ϕ ( x j ) , drives both the transport and the complex modulation of the wave packets, while the second
d k j d t = − ( ∇ u ‾ ( x j ) ) t r k j + i ∣ k j ∣ ∇ ∂ z ϕ ‾ ( x j ) , \frac{\mathrm{d}k_j}{\mathrm{d}t}=-\left(\nabla\overline{u}(x_j)\right)^{\mathrm{tr}}k_j+i\lvert k_j\rvert\nabla\partial_z\overline{\phi}(x_j), d t d k j = − ( ∇ u ( x j ) ) tr k j + i ∣ k j ∣ ∇ ∂ z ϕ ( x j ) , governs the evolution of their wavevectors through stretching, refraction and additional modulation induced by the resolved flow. In contrast with the classical Kraichnan–Tennekes random-sweeping hypothesis, the unresolved modes continuously adapt their geometry and amplitude to the large-scale dynamics.
Since the Hamiltonian is complex, it is useful to distinguish the propagation of the wave packet from its modulation. The leading-order ray trajectories are determined by the real part of the Hamiltonian,
d x j d t = ∇ k ℜ ( ω j ) = u ‾ ∣ z = η , (154) \frac{d x_j}{d t} = \nabla_k \Re(\omega_j) = \overline{u}\big|_{z=\eta}, \tag*{(154)} d t d x j = ∇ k ℜ ( ω j ) = u z = η , (154) which coincides with advection by the resolved horizontal flow. The imaginary part,
ℑ ( ω j ) = − ∣ k j ∣ ∂ z φ ‾ ∣ z = η , \Im(\omega_j) = -|k_j|\partial_z\overline{\varphi}\big|_{z=\eta}, ℑ ( ω j ) = − ∣ k j ∣ ∂ z φ z = η , does not alter the propagation direction but instead governs the phase and amplitude modulation of the unresolved wave packet along the rays. The large-scale vertical velocity therefore continuously redistributes the energy of the unresolved modes while they are transported by the resolved circulation.
The slowly varying envelope is transported along the large-scale characteristics:
∂ t b j + u ‾ ⋅ ∇ b j = 0. (155) \partial_t b_j+\overline{u}\cdot\nabla b_j=0. \tag*{(155)} ∂ t b j + u ⋅ ∇ b j = 0. (155) For constant amplitude and wavenumber, this reduces to traveling-wave solutions whose phase is modulated by the large-scale flow.
Ornstein-Uhlenbeck process for the regularized noise
As noted in [16 ] , a simple explicit closure is obtained by modeling the regularized noise through an Ornstein–Uhlenbeck (OU) process, denoted Z t T Z_t^{T} Z t T
φ t T = σ t Z t T = ∑ i φ i ( t ) Z t T , i = ∑ i φ i ( t ) e − t / τ Z t − τ T , i + φ i ( t ) 1 τ ∫ t − τ t e − ( t + τ − s ) / τ d β s i . (156) \varphi_t^{T}=\sigma_t Z_t^{T}=\sum_i\varphi_i(t)Z_t^{T,i}=\sum_i\varphi_i(t)e^{-t/\tau}Z_{t-\tau}^{T,i}+\varphi_i(t)\frac{1}{\tau}\int_{t-\tau}^{t}e^{-(t+\tau-s)/\tau}\,d\beta_s^i. \tag*{(156)} φ t T = σ t Z t T = i ∑ φ i ( t ) Z t T , i = i ∑ φ i ( t ) e − t / τ Z t − τ T , i + φ i ( t ) τ 1 ∫ t − τ t e − ( t + τ − s ) / τ d β s i . (156) Although the exponential kernel does not have compact support, this regularization still belongs to the class of regularized noises considered in this work. For this choice of noise, the small-scale velocity satisfies
d t φ T = ( ∑ i d t φ i ) Z t T − 1 τ ∑ i φ i Z t T , i d t + 1 2 τ ( 1 − 1 e ) ∑ i φ i d β t i . (157) d_t\varphi^{T}=\left(\sum_i d_t\varphi_i\right)Z_t^{T}-\frac{1}{\tau}\sum_i\varphi_i Z_t^{T,i}\,dt+\frac{1}{\sqrt{2\tau}}\left(1-\frac{1}{e}\right)\sum_i\varphi_i\,d\beta_t^i. \tag*{(157)} d t φ T = ( i ∑ d t φ i ) Z t T − τ 1 i ∑ φ i Z t T , i d t + 2 τ 1 ( 1 − e 1 ) i ∑ φ i d β t i . (157) In this closure, the nonlinear dynamics of the unresolved fluctuations is replaced by a linear mean-reverting Ornstein–Uhlenbeck process, while the correlation functions (or potential traces) continue to satisfy the nonlinear SPDE derived previously. If the correlation functions are furthermore assumed stationary, i.e.
d t φ i = 0 , d_t\varphi_i=0, d t φ i = 0 , the unresolved component reduces to
d t φ T = − 1 τ ∑ i φ i Z t T , i d t + 1 2 τ ( 1 − 1 e ) ∑ i φ i d β t i = − 1 τ φ T d t + 1 2 τ ( 1 − 1 e ) ∑ i φ i d β t i . (158) d_t\varphi^{T}=-\frac{1}{\tau}\sum_i\varphi_i Z_t^{T,i}\,dt+\frac{1}{\sqrt{2\tau}}\left(1-\frac{1}{e}\right)\sum_i\varphi_i\,d\beta_t^i=-\frac{1}{\tau}\varphi^{T}\,dt+\frac{1}{\sqrt{2\tau}}\left(1-\frac{1}{e}\right)\sum_i\varphi_i\,d\beta_t^i. \tag*{(158)} d t φ T = − τ 1 i ∑ φ i Z t T , i d t + 2 τ 1 ( 1 − e 1 ) i ∑ φ i d β t i = − τ 1 φ T d t + 2 τ 1 ( 1 − e 1 ) i ∑ φ i d β t i . (158) The unresolved fluctuations therefore evolve as mean-reverting stochastic modes with correlation time τ \tau τ [33 , 47 , 56 ] .
Conclusion
In this paper we have developed a stochastic variational framework for surface gravity waves that bridges deterministic phase-resolved and stochastic phase-averaged descriptions. Starting from Luke’s classical variational principle for irrotational incompressible free-surface flow, we introduced a decomposition of the velocity potential into a large-scale deterministic component and a regularized stochastic term representing unresolved scales. Combined with a pathwise variational principle, this decomposition yielded stochastic counterparts of the Laplace equation, Bernoulli condition, and kinematic boundary condition. The resulting system preserves the Hamiltonian structure of the Zakharov–Craig–Sulem formulation, so that fundamental invariants, including energy, are maintained.
To close the system and provide a dynamics for the unresolved scales, we formulated a second variational principle in expectation governing the evolution of the noise correlation functions. The resulting coupled system, combining large-scale dynamics with correlation-mode evolution, provides a systematic framework for deriving stochastic models of ocean waves. Through a WKB analysis we further showed that these correlation modes satisfy a Hamilton–Jacobi equation with associated ray-tracing dynamics, linking the stochastic formulation to geometric optics and providing an interpretation of the unresolved modes as dynamically evolving wave packets that are transported, stretched and modulated by the large-scale flow.
A systematic asymptotic analysis then yielded simplified models appropriate to several physical regimes. In the long-wave, small-amplitude limit, the theory reduces to a stochastic Airy-wave model in which unresolved fluctuations act as a superposition of random linear waves on a deterministic carrier. In shallow water with stronger nonlinearity, stochastic Saint–Venant equations emerge with transport noise representing unresolved advection. For coastal regimes with weaker scale separation, stochastic Serre–Green–Naghdi and Boussinesq-type systems were obtained, showing how different noise scalings determine whether additional interaction terms must be retained. In appropriate limits, the correlation dynamics reduces further to an Ornstein–Uhlenbeck process, providing a simple colored-noise closure for unresolved intermittency.
Several consequences follow from this formulation. First, stochastic parameterization of unresolved scales do not need to be introduced ad hoc but are rather constrained by variational structure and conservation principles, thereby limiting spurious dissipation or artificial growth. Second, the framework accommodates rotational stochastic forcing while preserving a potential-flow description for the large scales, suggesting a route toward coupling surface-wave dynamics and oceanic turbulence. Third, the geometric-optics interpretation of the noise modes provides a direct connection with ray-tracing and spectral numerical methods, opening a practical path toward efficient implementations.
The framework also opens several directions for future work. The companion paper [49 ] develops the asymptotic approximations, kinetic implications, and geophysical consequences further, including comparison with Hasselmann-type nonlinearities using JONSWAP observations. Those results suggest that stochastic transport by unresolved scales may contribute significantly alongside to classical four-wave interaction mechanisms in realistic ocean conditions, pointing toward a possible revision of standard spectral-wave closures. From a mathematical standpoint, the well-posedness of the coupled system at vanishing correlation time, and the rigorous justification of the geometric-optics approximation remain open questions. From a physical standpoint, extensions to turbulence spectra, wave breaking, surf-zone dynamics, and wind input or dissipation within the same variational structure remain natural developments. Finally, numerical implementation of the stochastic water-wave models derived here, in both phase-resolved and spectral settings, is a necessary step toward operational applications.
Taken together, these results suggest that stochastic variational modelling provides a unified framework in which unresolved dynamics become explicit dynamical variables rather than prescribed stochastic forcings. By combining Hamiltonian structure, multiscale asymptotics and dynamically evolving correlation modes within a single variational formulation, the present work establishes a foundation for a new class of stochastic water-wave models capable of consistently representing transport, uncertainty and wave interactions across scales.
Dirichlet-to-Neumann operator and its shape derivative
Dirichlet-to-Neumann operator
Let
Ω η = { ( x , z ) ∈ R d × R : b ≤ z ≤ η ( x ) } \Omega_\eta= \{(x,z) \in\mathbb{R}^{d} \times\mathbb{R} : b \le z \le\eta(x)\} Ω η = {( x , z ) ∈ R d × R : b ≤ z ≤ η ( x )} be the fluid domain with free surface η ( x ) \eta(x) η ( x ) z = b z=b z = b φ ( x ) \varphi(x) φ ( x ) ϕ \phi ϕ
Δ ϕ = 0 in Ω η , (159) \Delta\phi= 0 \quad\text{in } \Omega_\eta, \tag*{(159)} Δ ϕ = 0 in Ω η , (159) ϕ ∣ z = η ( x ) = φ ( x ) , (160) \left.\phi\right|_{z=\eta(x)} = \varphi(x), \tag*{(160)} ϕ ∣ z = η ( x ) = φ ( x ) , (160) ∂ z ϕ ∣ z = b = 0. (161) \left.\partial_z\phi\right|_{z=b} = 0. \tag*{(161)} ∂ z ϕ ∣ z = b = 0. (161) The Dirichlet-to-Neumann operator is defined as
G [ η ] φ = ( ∂ n ϕ ) ∣ z = η ( x ) , (162) G[\eta]\varphi= \left.(\partial_n\phi)\right|_{z=\eta(x)}, \tag*{(162)} G [ η ] φ = ( ∂ n ϕ ) ∣ z = η ( x ) , (162) where ∂ n \partial_n ∂ n
G [ η ] φ = ∂ z ϕ − ∇ η ⋅ ∇ ϕ at z = η ( x ) . (163) G[\eta]\varphi= \partial_z\phi- \nabla\eta\cdot\nabla\phi\quad\text{at } z=\eta(x). \tag*{(163)} G [ η ] φ = ∂ z ϕ − ∇ η ⋅ ∇ ϕ at z = η ( x ) . (163) The operator G [ η ] G[\eta] G [ η ]
∫ φ G [ η ] ψ d x = ∫ ψ G [ η ] φ d x . (164) \int\varphi G[\eta]\psi\, dx = \int\psi G[\eta]\varphi\, dx. \tag*{(164)} ∫ φG [ η ] ψ d x = ∫ ψ G [ η ] φ d x . (164) Shape derivative
We consider a perturbation of the free surface:
η ↦ η + ϵ ζ , \eta\mapsto\eta+ \epsilon\zeta, η ↦ η + ϵ ζ , and denote by δ G [ η ] φ ⋅ ζ \delta G[\eta]\varphi\cdot\zeta δ G [ η ] φ ⋅ ζ ϕ \phi ϕ φ \varphi φ
B = ∂ z ϕ ∣ z = η , V = ∇ x ϕ ∣ z = η . (165) B = \left.\partial_z\phi\right|_{z=\eta}, \qquad V = \left.\nabla_x\phi\right|_{z=\eta}. \tag*{(165)} B = ∂ z ϕ ∣ z = η , V = ∇ x ϕ ∣ z = η . (165) Then the shape derivative of the Dirichlet-to-Neumann operator is given by
δ G [ η ] φ ⋅ ζ = − G [ η ] ( ζ B ) − ∇ ⋅ ( ζ V ) . (166) \delta G[\eta]\varphi\cdot\zeta= -G[\eta](\zeta B) - \nabla\cdot(\zeta V). \tag*{(166)} δ G [ η ] φ ⋅ ζ = − G [ η ] ( ζ B ) − ∇ ⋅ ( ζ V ) . (166) Equivalently, using the identities
B = G [ η ] φ + ∇ η ⋅ ∇ φ 1 + ∣ ∇ η ∣ 2 , V = ∇ φ − B ∇ η , (167) B = \frac{G[\eta]\varphi+ \nabla\eta\cdot\nabla\varphi}{1 + \lvert\nabla\eta\rvert^{2}}, \qquad V = \nabla\varphi- B\nabla\eta, \tag*{(167)} B = 1 + ∣ ∇ η ∣ 2 G [ η ] φ + ∇ η ⋅ ∇ φ , V = ∇ φ − B ∇ η , (167) the variation can be expressed purely in terms of surface quantities with V V V B B B U = ∇ x , z ϕ U = \nabla_{x,z}\phi U = ∇ x , z ϕ ϕ \phi ϕ
As a consequence, the variation of the quadratic form associated with the Dirichlet-to-Neumann operator satisfies
δ δ η ( 1 2 ϕ ‾ G [ η ] ϕ ) = 1 2 ∣ ∇ ϕ ∣ 2 − 1 2 ( G [ η ] ϕ + ∇ η ⋅ ∇ ϕ ) 2 1 + ∣ ∇ η ∣ 2 . (168) \frac{\delta}{\delta\eta}\left(\frac{1}{2}\overline{\phi}G[\eta]\phi\right)=\frac{1}{2}|\nabla\phi|^2-\frac{1}{2}\frac{\left(G[\eta]\phi+\nabla\eta\cdot\nabla\phi\right)^2}{1+|\nabla\eta|^2}. \tag*{(168)} δ η δ ( 2 1 ϕ G [ η ] ϕ ) = 2 1 ∣∇ ϕ ∣ 2 − 2 1 1 + ∣∇ η ∣ 2 ( G [ η ] ϕ + ∇ η ⋅ ∇ ϕ ) 2 . (168) This identity is a key ingredient in the derivation of the Zakharov–Craig–Sulem formulation of the water wave equations.
Hamiltonian reduction under the stochastic Bernoulli constraint
In the main text, the stochastic closure
∂ t ϕ T + 1 2 ∣ ∇ ϕ T ∣ 2 = 0 (169) \partial_t\phi^T+\frac{1}{2}|\nabla\phi^T|^2=0 \tag*{(169)} ∂ t ϕ T + 2 1 ∣∇ ϕ T ∣ 2 = 0 (169) was introduced to obtain a well-defined decorrelation limit while preserving the Hamiltonian structure of the stochastic water-wave system.
In the prescribed-noise setting considered here, the fluctuating potential ϕ T \phi^T ϕ T
A further compatibility condition is imposed at the free surface:
∂ z ϕ T ∣ z = η G [ η ] ϕ ‾ = 0. (170) \left.\partial_z\phi^T\right|_{z=\eta}G[\eta]\overline{\phi}=0. \tag*{(170)} ∂ z ϕ T z = η G [ η ] ϕ = 0. (170) This condition removes the coupling between fluctuating vertical motions and large-scale normal transport at the interface. It restricts the admissible stochastic coupling without introducing an additional dynamical degree of freedom.
Reduction of Luke’s action
Consider Luke’s action for the decomposed velocity potential
ϕ = ϕ ‾ + ϕ T , (171) \phi=\overline{\phi}+\phi^T, \tag*{(171)} ϕ = ϕ + ϕ T , (171) namely
S ( ϕ ‾ , η ; ϕ T ) = − ρ ∫ t 0 t 1 ∫ Ω h ∫ b η ( x , t ) ( ∂ t ( ϕ ‾ + ϕ T ) + 1 2 ∣ ∇ ( ϕ ‾ + ϕ T ) ∣ 2 + g z ) d z d x d t . (172) S(\overline{\phi},\eta;\phi^T)=-\rho\int_{t_0}^{t_1}\int_{\Omega_h}\int_b^{\eta(x,t)}\left(\partial_t(\overline{\phi}+\phi^T)+\frac{1}{2}|\nabla(\overline{\phi}+\phi^T)|^2+gz\right)\,dz\,dx\,dt. \tag*{(172)} S ( ϕ , η ; ϕ T ) = − ρ ∫ t 0 t 1 ∫ Ω h ∫ b η ( x , t ) ( ∂ t ( ϕ + ϕ T ) + 2 1 ∣∇ ( ϕ + ϕ T ) ∣ 2 + g z ) d z d x d t . (172) The notation S ( ϕ ‾ , η ; ϕ T ) S(\overline{\phi},\eta;\phi^T) S ( ϕ , η ; ϕ T ) ϕ T \phi^T ϕ T ϕ ‾ \overline{\phi} ϕ η \eta η
Applying Leibniz’s rule to the remaining large-scale temporal derivative gives
∫ Ω h ∫ b η ( x , t ) ∂ t ϕ ‾ d z d x = d d t ∫ Ω h ∫ b η ( x , t ) ϕ ‾ d z d x − ∫ Ω h ϕ ‾ ∂ t η d x . (173) \int_{\Omega_h}\int_b^{\eta(x,t)}\partial_t\overline{\phi}\,dz\,dx=\frac{d}{dt}\int_{\Omega_h}\int_b^{\eta(x,t)}\overline{\phi}\,dz\,dx-\int_{\Omega_h}\overline{\phi}\partial_t\eta\,dx. \tag*{(173)} ∫ Ω h ∫ b η ( x , t ) ∂ t ϕ d z d x = d t d ∫ Ω h ∫ b η ( x , t ) ϕ d z d x − ∫ Ω h ϕ ∂ t η d x . (173) After discarding the resulting temporal boundary term and the constant gravitational contribution associated with the flat bottom, the reduced action per unit density becomes
S ′ ρ = ∫ t 0 t 1 ∫ D h φ ‾ , ∂ t η d x d t − ∫ t 0 t 1 ∫ Ω ( t ) ( 1 2 ∣ ∇ φ ‾ ∣ 2 + ∇ φ ‾ ⋅ ∇ φ τ + g z ) d x d z d t , (174) \frac{\mathcal{S}'}{\rho} = \int_{t_0}^{t_1} \int_{\mathcal{D}^h} \overline{\varphi},\partial_t\eta\,\mathrm{d}x\,\mathrm{d}t - \int_{t_0}^{t_1} \int_{\Omega(t)} \left(\frac{1}{2}\lvert\nabla\overline{\varphi}\rvert^2 + \nabla\overline{\varphi}\cdot\nabla\varphi^\tau+ gz\right)\,\mathrm{d}x\,\mathrm{d}z\,\mathrm{d}t, \tag*{(174)} ρ S ′ = ∫ t 0 t 1 ∫ D h φ , ∂ t η d x d t − ∫ t 0 t 1 ∫ Ω ( t ) ( 2 1 ∣ ∇ φ ∣ 2 + ∇ φ ⋅ ∇ φ τ + g z ) d x d z d t , (174) where
Ω ( t ) = { ( x , z ) : x ∈ D h , b ≤ z < η ( x , t ) } . (175) \Omega(t) = \left\{(x,z): x \in\mathcal{D}^h,\ b \le z < \eta(x,t)\right\}. \tag*{(175)} Ω ( t ) = { ( x , z ) : x ∈ D h , b ≤ z < η ( x , t ) } . (175) Because both φ ‾ \overline{\varphi} φ φ τ \varphi^\tau φ τ
∫ Ω ( t ) 1 2 ∣ ∇ φ ‾ ∣ 2 d x d z = 1 2 ∫ D h φ ‾ G [ η ] φ ‾ d x . (176) \int_{\Omega(t)} \frac{1}{2}\lvert\nabla\overline{\varphi}\rvert^2\,\mathrm{d}x\,\mathrm{d}z = \frac{1}{2}\int_{\mathcal{D}^h}\overline{\varphi}G[\eta]\overline{\varphi}\,\mathrm{d}x. \tag*{(176)} ∫ Ω ( t ) 2 1 ∣ ∇ φ ∣ 2 d x d z = 2 1 ∫ D h φ G [ η ] φ d x . (176) and
∫ Ω ( t ) ∇ φ ‾ ⋅ ∇ φ τ d x d z = ∫ D h φ τ G [ η ] φ ‾ d x . (177) \int_{\Omega(t)} \nabla\overline{\varphi}\cdot\nabla\varphi^\tau\,\mathrm{d}x\,\mathrm{d}z = \int_{\mathcal{D}^h}\varphi^\tau G[\eta]\overline{\varphi}\,\mathrm{d}x. \tag*{(177)} ∫ Ω ( t ) ∇ φ ⋅ ∇ φ τ d x d z = ∫ D h φ τ G [ η ] φ d x . (177) The latter expression is symmetric because the Dirichlet-to-Neumann operator is self-adjoint:
∫ D h φ τ G [ η ] φ ‾ d x = ∫ D h φ ‾ G [ η ] φ τ d x . (178) \int_{\mathcal{D}^h}\varphi^\tau G[\eta]\overline{\varphi}\,\mathrm{d}x = \int_{\mathcal{D}^h}\overline{\varphi}G[\eta]\varphi^\tau\,\mathrm{d}x. \tag*{(178)} ∫ D h φ τ G [ η ] φ d x = ∫ D h φ G [ η ] φ τ d x . (178) The gravitational contribution satisfies
∫ D h ∫ b η ( x , t ) g z d z d x = 1 2 ∫ D h g η 2 d x − 1 2 ∫ D h g b 2 d x . (179) \int_{\mathcal{D}^h}\int_b^{\eta(x,t)} gz\,\mathrm{d}z\,\mathrm{d}x = \frac{1}{2}\int_{\mathcal{D}^h}g\eta^2\,\mathrm{d}x - \frac{1}{2}\int_{\mathcal{D}^h}gb^2\,\mathrm{d}x. \tag*{(179)} ∫ D h ∫ b η ( x , t ) g z d z d x = 2 1 ∫ D h g η 2 d x − 2 1 ∫ D h g b 2 d x . (179) The second term is constant and can therefore be discarded. The reduced action consequently takes the canonical form
S ′ ρ = ∫ t 0 t 1 [ ∫ D h φ ‾ , ∂ t η d x − H ‾ ( η , φ ‾ ; φ τ ) ] d t , (180) \frac{\mathcal{S}'}{\rho} = \int_{t_0}^{t_1}\left[\int_{\mathcal{D}^h}\overline{\varphi},\partial_t\eta\,\mathrm{d}x-\overline{\mathcal{H}}(\eta,\overline{\varphi};\varphi^\tau)\right]\mathrm{d}t, \tag*{(180)} ρ S ′ = ∫ t 0 t 1 [ ∫ D h φ , ∂ t η d x − H ( η , φ ; φ τ ) ] d t , (180) where
H ‾ ( η , φ ‾ ; φ τ ) = ∫ D h ( 1 2 φ ‾ G [ η ] φ ‾ + φ τ G [ η ] φ ‾ + 1 2 g η 2 ) d x . (181) \overline{\mathcal{H}}(\eta,\overline{\varphi};\varphi^\tau) = \int_{\mathcal{D}^h}\left(\frac{1}{2}\overline{\varphi}G[\eta]\overline{\varphi}+\varphi^\tau G[\eta]\overline{\varphi}+\frac{1}{2}g\eta^2\right)\,\mathrm{d}x. \tag*{(181)} H ( η , φ ; φ τ ) = ∫ D h ( 2 1 φ G [ η ] φ + φ τ G [ η ] φ + 2 1 g η 2 ) d x . (181) The second term in (180) represents the interaction energy between the prescribed fluctuating potential and the large-scale motion.
Preservation of the Poisson structure
Since φ τ \varphi^\tau φ τ φ τ \varphi^\tau φ τ ( η , φ ‾ ) (\eta,\overline{\varphi}) ( η , φ ) F F F K K K
{ F , K } = ∫ D h ( δ F δ η δ K δ φ ‾ − δ K δ η δ F δ φ ‾ ) d x . (182) \{F,K\} = \int_{\mathcal{D}^h}\left(\frac{\delta F}{\delta\eta}\frac{\delta K}{\delta\overline{\varphi}}-\frac{\delta K}{\delta\eta}\frac{\delta F}{\delta\overline{\varphi}}\right)\,\mathrm{d}x. \tag*{(182)} { F , K } = ∫ D h ( δ η δ F δ φ δ K − δ η δ K δ φ δ F ) d x . (182) The stochastic Bernoulli constraint (168) and the compatibility condition (169) restrict the admissible prescribed fluctuations, but do not generate independent phase-space constraints. Consequently, they do not deform the canonical bracket. Their effect is entirely contained in the modified Hamiltonian (180). Hamilton’s equations retain the canonical form
∂ t η = δ H ‾ δ φ ‾ , (183) \partial_t\eta= \frac{\delta\overline{\mathcal{H}}}{\delta\overline{\varphi}}, \tag*{(183)} ∂ t η = δ φ δ H , (183) ∂ t φ ‾ = − δ H ‾ δ η . (184) \partial_t\overline{\varphi} = -\frac{\delta\overline{\mathcal{H}}}{\delta\eta}. \tag*{(184)} ∂ t φ = − δ η δ H . (184) Using the self-adjointness of G [ η ] G[\eta] G [ η ]
∂ t η = G [ η ] φ ‾ + G [ η ] φ τ = G [ η ] ( φ ‾ + φ τ ) . (185) \partial_t\eta= G[\eta]\overline{\varphi} + G[\eta]\varphi^\tau= G[\eta]\left(\overline{\varphi}+\varphi^\tau\right). \tag*{(185)} ∂ t η = G [ η ] φ + G [ η ] φ τ = G [ η ] ( φ + φ τ ) . (185) The second equation yields
∂ t φ ‾ = − ( g η + 1 2 ∣ ∇ φ ‾ ∣ 2 − 1 2 ( G [ η ] φ ‾ + ∇ η ⋅ ∇ φ ‾ ) 2 1 + ∣ ∇ η ∣ 2 + ∇ φ ‾ ⋅ ∇ φ τ − ( G [ η ] φ ‾ + ∇ η ⋅ ∇ φ ‾ ) ( G [ η ] φ τ + ∇ η ⋅ ∇ φ τ ) 1 + ∣ ∇ η ∣ 2 ) , (186) \partial_t\overline{\varphi} = -\left(g\eta+ \frac{1}{2}\lvert\nabla\overline{\varphi}\rvert^2 - \frac{1}{2}\frac{\left(G[\eta]\overline{\varphi}+\nabla\eta\cdot\nabla\overline{\varphi}\right)^2}{1+\lvert\nabla\eta\rvert^2}+\nabla\overline{\varphi}\cdot\nabla\varphi^\tau-\frac{\left(G[\eta]\overline{\varphi}+\nabla\eta\cdot\nabla\overline{\varphi}\right)\left(G[\eta]\varphi^\tau+\nabla\eta\cdot\nabla\varphi^\tau\right)}{1+\lvert\nabla\eta\rvert^2}\right), \tag*{(186)} ∂ t φ = − ( g η + 2 1 ∣ ∇ φ ∣ 2 − 2 1 1 + ∣ ∇ η ∣ 2 ( G [ η ] φ + ∇ η ⋅ ∇ φ ) 2 + ∇ φ ⋅ ∇ φ τ − 1 + ∣ ∇ η ∣ 2 ( G [ η ] φ + ∇ η ⋅ ∇ φ ) ( G [ η ] φ τ + ∇ η ⋅ ∇ φ τ ) ) , (186) which coincides with the large-scale stochastic Zakharov–Craig–Sulem equation obtained in the main text.
Interpretation
The closure (168) should therefore be understood as a Hamiltonian reduction constraint imposed on the prescribed fluctuating potential. Together with the compatibility condition (169), it restricts the admissible stochastic coupling so that the decorrelation limit remains well defined.
Because the fluctuating potential is prescribed rather than promoted to an independent canonical variable, the reduction modifies the Hamiltonian but leaves the Zakharov symplectic structure unchanged. The resulting stochastic water-wave model therefore remains canonical, with interaction energy
∫ D h φ τ G [ η ] φ ‾ d x , (187) \int_{\mathcal{D}_h}\varphi^\tau G[\eta]\overline{\varphi}\,dx, \tag*{(187)} ∫ D h φ τ G [ η ] φ d x , (187) and the original Poisson bracket on the large-scale variables ( η , φ ‾ ) (\eta,\overline{\varphi}) ( η , φ )
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