Introduction
The companion paper [3 ] developed a stochastic variational framework for surface gravity waves in which unresolved scales enter through a regularized stochastic transport acting on the velocity potential. Starting from Luke’s variational principle and a decomposition into resolved and unresolved components, that work led to a coupled multiscale system governed by two intertwined principles: a pathwise variational principle for the large-scale wave dynamics and a variational principle in expectation governing the evolution of the noise modal functions. The resulting formulation preserves the Hamiltonian Zakharov–Craig–Sulem structure while providing a systematic framework for stochastic reduced models of ocean waves.
In this second paper, we investigate, for the deep-water regime, the physical consequences of that framework from a kinetic and modeling perspective. In particular, we examine how stochastic transport induced by unresolved motions modifies spectral wave evolution, and how this mechanism compares with the classical weak-turbulence picture embodied in the Hasselmann–Zakharov equation [6 , 8 , 20 , 21 , 19 , 22 ] .
This question is motivated by a long-standing gap in wave modeling. Resonant four-wave interactions form a cornerstone of spectral wave theory, yet transport by unresolved currents, turbulence, internal waves, and submesoscale motions is generally represented only indirectly or neglected, despite classical physical arguments—notably the Kraichnan–Tennekes sweeping picture—suggesting that random advection may strongly influence decorrelation and spectral transfer. The stochastic variational framework of the companion paper provides a natural setting in which this mechanism can be studied systematically.
The present work has two main objectives:
Develop a kinetic theory for stochastic transport. Starting from the linearized stochastic water-waves equations, we derive a wave-action kinetic equation exhibiting two complementary regimes: a diffusive scattering regime and a quartic interaction regime structurally reminiscent of the Hasselmann–Zakharov equation. In contrast to classical weak-turbulence theory, these effective quartic interactions arise here from stochastic transport by unresolved motions rather than intrinsic resonant nonlinearity.
Quantify the relative importance of stochastic transport and Hasselmann–Zakharov four-waves resonant interactions. To compare these mechanisms, we introduce the dimensionless transport-to-resonance interaction ratio
Γ = k 2 σ u 2 τ c ϵ μ 4 ω k , (1) \Gamma= \frac{k^2 \sigma_u^2 \tau_c}{\epsilon_\mu^4 \omega_k}, \tag*{(1)} Γ = ϵ μ 4 ω k k 2 σ u 2 τ c , (1) where σ u 2 \sigma_u^2 σ u 2 τ c \tau_c τ c ϵ μ \epsilon_\mu ϵ μ 1 ω k \omega_k ω k Γ > 1 \Gamma> 1 Γ > 1
Using JONSWAP spectral parameterizations, we evaluate this ratio for different sea-states and over realistic oceanic ranges of unresolved transport (eddy viscosity) intensity
S = σ u 2 τ c . S = \sigma_u^2 \tau_c. S = σ u 2 τ c . Our analysis suggests that, for physically realistic values of S S S
These findings motivate a re-examination of spectral wave closures in which unresolved transport acts as an explicit source of spectral redistribution, alongside resonant nonlinear interactions. Beyond the kinetic theory itself, the analysis developed here yields implications for peak dynamics, swell decay, and wave forecasting, while also suggesting new parameterizations for stochastic source terms in operational spectral models.
The remainder of the paper is organized as follows. Section 2 recalls the stochastic water-wave formulation developed in the companion paper. Section 3 derives wave-action kinetic equation, for the linear stochastic water-wave equation with prescribed (stationnary) noise. Its diffusive continuous limit is inferred together with the effects of forcing and dissipation. Section 4 develops the quartic kinetic interaction term associated to the complete linear stochastic water-wave system. Section 5 presents the comparison with JONSWAP spectra and the resulting regime analysis. Section 6 discusses implications for wave modeling and concludes.
In this section we briefly recall the stochastic water-wave model derived in the companion paper [3 ] ; see also [2 ] . The model is based on two coupled variational principles. The first is a pathwise variational principle governing the evolution of the large-scale velocity potential ϕ ˉ \bar{\phi} ϕ ˉ ϕ τ \phi^\tau ϕ τ
ϕ τ ( x , t ) = ∫ t − τ t + τ h τ ( t − s ) ϕ i ( x ) d β s i (2) \phi^\tau(x,t) = \int_{t-\tau}^{t+\tau} h_\tau(t-s)\phi_i(x)\,\mathrm{d}\beta_s^i \tag*{(2)} ϕ τ ( x , t ) = ∫ t − τ t + τ h τ ( t − s ) ϕ i ( x ) d β s i (2) where h τ h_\tau h τ ϕ i \phi_i ϕ i
The full velocity potential is decomposed as
ϕ ( x , t ) = ϕ ˉ ( x , t ) + ϕ τ ( x , t ) . (3) \phi(x,t) = \bar{\phi}(x,t) + \phi^\tau(x,t). \tag*{(3)} ϕ ( x , t ) = ϕ ˉ ( x , t ) + ϕ τ ( x , t ) . (3) Because the temporal regularization is symmetric in past and future times, the decorrelation limit τ → 0 \tau\to0 τ → 0
W t τ = ∫ t − τ t + τ h τ ( t − s ) d β s (4) W_t^\tau= \int_{t-\tau}^{t+\tau} h_\tau(t-s)\,\mathrm{d}\beta_s \tag*{(4)} W t τ = ∫ t − τ t + τ h τ ( t − s ) d β s (4) has, under the kernel normalization considered here, 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 )
The free-surface elevation satisfies the stochastic kinematic boundary condition expressing transport of particles along the moving interface. Throughout, we assume the free surface remains sufficiently smooth and exclude wave breaking. Consistent with the decorrelation limit, the surface elevation is modeled as a semimartingale,
d η = η ~ d t + d η τ . (5) \mathrm{d}\eta= \widetilde{\eta}\,\mathrm{d}t + \mathrm{d}\eta^\tau. \tag*{(5)} d η = η d t + d η τ . (5) Although η \eta η
Stochastic water-wave system
The water waves dynamics involves a classical kinematic condition for the free surface η \eta η
∂ t η = G [ η ] ( ϕ ˉ + ϕ τ ) , (6) \partial_t\eta= G[\eta](\bar{\phi}+\phi^\tau), \tag*{(6)} ∂ t η = G [ η ] ( ϕ ˉ + ϕ τ ) , (6) where we introduced the Dirichlet-to-Neumann operator [1 , 19 ] :
G [ η ] φ = ( ∂ z ϕ − ∇ x ϕ ⋅ ∇ η ) ∣ z = η . (7) G[\eta]\varphi= \left(\partial_z\phi-\nabla_x\phi\cdot\nabla\eta\right)\big|_{z=\eta}. \tag*{(7)} G [ η ] φ = ( ∂ z ϕ − ∇ x ϕ ⋅ ∇ η ) z = η . (7) This operator maps a Dirichlet boundary value φ \varphi φ [13 ] and references therein).
A pathwise variational principle provides the evolution of the trace at the surface of the large-scale velocity 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 ) , (8) \partial_t\overline{\varphi}=-\left(g\eta+\frac{1}{2}|\nabla\varphi|^2-\frac{1}{2}\frac{\left(G[\eta]\overline{\varphi}+\nabla\eta\cdot\nabla\overline{\varphi}\right)^2}{1+|\nabla\eta|^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+|\nabla\eta|^2}\right), \tag*{(8)} ∂ 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 ) ) , (8) These evolution equations can be shown to conserve a Hamiltonian structure derived from a stochastic extension of the Craig–Sulem–Zakharov formulation, so that the resolved dynamics remains consistent with the underlying variational structure.
As for the unresolved component, its dynamics follows from the coupled variational principle in expectation associated with the Euler equations. The evolution of the correlation functions of the trace of the velocity potential at the free surface reads
∂ 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. (9) \partial_t\varphi_i+\nabla\overline{\varphi}\cdot\nabla\varphi_i-\frac{\left(\nabla\varphi_i\cdot\nabla\eta\right)\left(G[\eta]\overline{\varphi}+\nabla\overline{\varphi}\cdot\nabla\eta\right)}{1+|\nabla\eta|^2}-\frac{\left(G[\eta]\varphi_i+\nabla\varphi_i\cdot\nabla\eta\right)G[\eta]\varphi^\tau}{1+|\nabla\eta|^2}=0. \tag*{(9)} ∂ 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. (9) Equations above together with the free-surface transport, define a coupled stochastic water-wave system for the resolved free surface dynamics and the unresolved correlation modes.
In the present work we focus on a simplified regime obtained by linearizing the large-scale dynamics. In the following section we first examine a frozen-correlation approximation for the noise modes, in which the functions φ i \varphi_i φ i
Linearization of the large-scale water-wave system under fixed noise
In the following we provide in Fourier form the solution of the water-wave system linearized around flat surface at rest.
∂ t φ ‾ + η + ∇ φ ‾ τ ⋅ ∇ φ ‾ = 0 , (10) \partial_t\overline{\varphi}+\eta+\nabla\overline{\varphi}^{\tau}\cdot\nabla\overline{\varphi}=0, \tag*{(10)} ∂ t φ + η + ∇ φ τ ⋅ ∇ φ = 0 , (10) ∂ t η = ∂ z ϕ ∣ z = 0 + ∂ z ϕ τ ∣ z = 0 . (11) \partial_t\eta=\left.\partial_z\phi\right|_{z=0}+\left.\partial_z\phi^\tau\right|_{z=0}. \tag*{(11)} ∂ t η = ∂ z ϕ ∣ z = 0 + ∂ z ϕ τ ∣ z = 0 . (11) In nondimensional form, the advection of the large-scale velocity potential by the noise in (10) – referred to as transport-noise – scales in ϵ σ m / ν \epsilon_\sigma m/\nu ϵ σ m / ν ϵ σ = A σ / H \epsilon_\sigma=A_\sigma/H ϵ σ = A σ / H ν = tanh ( 2 π μ ) / ( 2 π μ ) \nu=\tanh(2\pi\sqrt{\mu})/(2\pi\sqrt{\mu}) ν = tanh ( 2 π μ ) / ( 2 π μ ) μ = H 2 / L 2 \mu=H^2/L^2 μ = H 2 / L 2 [3 ] . In deep-water, (for finite depth) ν = ( 2 π μ ) − 1 \nu=(2\pi\sqrt{\mu})^{-1} ν = ( 2 π μ ) − 1 ∇ φ τ ⋅ ∇ φ ‾ \nabla\varphi^\tau\cdot\nabla\overline{\varphi} ∇ φ τ ⋅ ∇ φ ϵ μ σ = A σ / L \epsilon_\mu^\sigma=A_\sigma/L ϵ μ σ = A σ / L ϵ μ σ m \epsilon_\mu^\sigma m ϵ μ σ m
In this first case, the noise is assumed to be given and stationary. We will also assume lateral periodic conditions. As such, the noise can be specified in terms of Fourier basis functions as
φ τ = ∑ j c k j e i k j ⋅ x W j τ ( t ) with W j τ ( t ) = ∫ t − τ t + τ h τ ( t − s ) d β s j . (12) \varphi^\tau= \sum_j c_{k_j} e^{i k_j \cdot x} W_j^\tau(t) \text{ with } W_j^\tau(t) = \int_{t-\tau}^{t+\tau} h_\tau(t-s)\,\mathrm{d}\beta_s^j. \tag*{(12)} φ τ = j ∑ c k j e i k j ⋅ x W j τ ( t ) with W j τ ( t ) = ∫ t − τ t + τ h τ ( t − s ) d β s j . (12) The wavenumbers k j k_j k j k k k c k j c_{k_j} c k j τ \tau τ W j τ = O ( τ − 1 / 2 ) W_j^\tau= \mathcal{O}(\tau^{-1/2}) W j τ = O ( τ − 1/2 ) φ τ ∼ L 2 / T \varphi^\tau\sim L^2/T φ τ ∼ L 2 / T c k j ∼ ∣ k j ∣ − 2 ω j c_{k_j} \sim\lvert k_j\rvert^{-2}\sqrt{\omega_j} c k j ∼ ∣ k j ∣ − 2 ω j ω j = ω ( k j ) \omega_j = \omega(k_j) ω j = ω ( k j ) c k j c_{k_j} c k j
The transport-noise term in the linearized large-scale expression (10) reads:
∇ φ ⋅ ∇ φ τ = ∑ k , j − c k j ( k − k j ) ⋅ k j φ ‾ k − k j W j τ ( t ) e i k ⋅ x , (13) \nabla\varphi\cdot\nabla\varphi^\tau= \sum_{k,j} -c_{k_j}(k-k_j)\cdot k_j\overline{\varphi}_{k-k_j}W_j^\tau(t)e^{ik\cdot x}, \tag*{(13)} ∇ φ ⋅ ∇ φ τ = k , j ∑ − c k j ( k − k j ) ⋅ k j φ k − k j W j τ ( t ) e ik ⋅ x , (13) For a fixed horizontal wavenumber k k k z = 0 z=0 z = 0 U k = ( φ k , η k ) t r U_k=(\varphi_k,\eta_k)^{\mathrm{tr}} U k = ( φ k , η k ) tr
∂ t U k = A k U k + ∑ j W j τ ( t ) B k , k j U k − k j + F k ( t ) , (14) \partial_t U_k=A_kU_k+\sum_j W_j^\tau(t)B_{k,k_j}U_{k-k_j}+F_k(t), \tag*{(14)} ∂ t U k = A k U k + j ∑ W j τ ( t ) B k , k j U k − k j + F k ( t ) , (14) where
A k = ( 0 − g G k [ 0 ] 0 ) , B k , k j = ( c k j k j ⋅ ( k − k j ) 0 0 0 ) , F k = ( 0 ∑ j W j τ ( t ) G k [ 0 ] c k j δ k , k j ) , (15) A_k=\begin{pmatrix}0&-g\\G_k[0]&0\end{pmatrix},\quad B_{k,k_j}=\begin{pmatrix}c_{k_j}k_j\cdot(k-k_j)&0\\0&0\end{pmatrix},\quad F_k=\begin{pmatrix}0\\\sum_j W_j^\tau(t)G_k[0]c_{k_j}\delta_{k,k_j}\end{pmatrix}, \tag*{(15)} A k = ( 0 G k [ 0 ] − g 0 ) , B k , k j = ( c k j k j ⋅ ( k − k j ) 0 0 0 ) , F k = ( 0 ∑ j W j τ ( t ) G k [ 0 ] c k j δ k , k j ) , (15) denotes respectively, the Airy matrix, a convolution multiplicative operator and an additive stochastic forcing applying only at noise modes as seen from the Kronecker δ \delta δ
Denoting the semi-group S k ( t ) = e t A k S_k(t)=e^{tA_k} S k ( t ) = e t A k
U k ( t ) = S k ( t ) U k ( 0 ) + ∑ j ∫ 0 t S k ( t − s ) B k , k j U k − k j ( s ) W j τ ( s ) d s + ∫ 0 t S k ( t − s ) F k ( s ) d s . (16) U_k(t)=S_k(t)U_k(0)+\sum_j\int_0^t S_k(t-s)B_{k,k_j}U_{k-k_j}(s)W_j^\tau(s)\,\mathrm{d}s+\int_0^t S_k(t-s)F_k(s)\,\mathrm{d}s. \tag*{(16)} U k ( t ) = S k ( t ) U k ( 0 ) + j ∑ ∫ 0 t S k ( t − s ) B k , k j U k − k j ( s ) W j τ ( s ) d s + ∫ 0 t S k ( t − s ) F k ( s ) d s . (16) The Dirichlet-to-Neumann at the surface at rest, is given through Fourier multipliers by
G k [ 0 ] = ∣ k ∣ tanh ( ∣ k ∣ H ) . G_k[0]=\lvert k\rvert\tanh(\lvert k\rvert H). G k [ 0 ] = ∣ k ∣ tanh (∣ k ∣ H ) . The eigenvalues of matrix A A A λ ± = ± i ω k \lambda_\pm=\pm i\omega_k λ ± = ± i ω k v ± = ( ± i ω k G k [ 0 ] , 1 ) t r \boldsymbol{v}^\pm=\left(\frac{\pm i\omega_k}{G_k[0]},1\right)^{\mathrm{tr}} v ± = ( G k [ 0 ] ± i ω k , 1 ) tr
ω k 2 = g G k [ 0 ] = g ∣ k ∣ tanh ( ∣ k ∣ H ) . (17) \omega_k^2=gG_k[0]=g\lvert k\rvert\tanh(\lvert k\rvert H). \tag*{(17)} ω k 2 = g G k [ 0 ] = g ∣ k ∣ tanh (∣ k ∣ H ) . (17) Noting that even power of the Airy matrix are A 2 n = ( − ω 2 ) n I 2 A^{2n}=(-\omega^2)^n I_2 A 2 n = ( − ω 2 ) n I 2
S k ( t ) = ( cos ( ω k t ) − g ω k sin ( ω k t ) G k [ 0 ] ω k sin ( ω k t ) cos ( ω k t ) ) . (18) S_k(t)=\begin{pmatrix}
\cos(\omega_k t)&-\frac{g}{\omega_k}\sin(\omega_k t)\\
\frac{G_k[0]}{\omega_k}\sin(\omega_k t)&\cos(\omega_k t)
\end{pmatrix}. \tag*{(18)} S k ( t ) = ( cos ( ω k t ) ω k G k [ 0 ] sin ( ω k t ) − ω k g sin ( ω k t ) cos ( ω k t ) ) . (18) Through normal mode expression of the deterministic solution, we obtain η k = a k + a − k ∗ \eta_k = a_k + a^*_{-k} η k = a k + a − k ∗ φ ‾ k = − i ω k G k [ 0 ] ( a k − a − k ∗ ) \overline{\varphi}_k = -\frac{i\omega_k}{G_k[0]}(a_k-a^*_{-k}) φ k = − G k [ 0 ] i ω k ( a k − a − k ∗ )
a k = 1 2 ( η k + i G k [ 0 ] ω k φ ‾ k ) = 1 2 ( η k + i ω k g φ ‾ k ) . (19) a_k = \frac{1}{2}\left(\eta_k+\frac{iG_k[0]}{\omega_k}\overline{\varphi}_k\right)=\frac{1}{2}\left(\eta_k+\frac{i\omega_k}{g}\overline{\varphi}_k\right). \tag*{(19)} a k = 2 1 ( η k + ω k i G k [ 0 ] φ k ) = 2 1 ( η k + g i ω k φ k ) . (19) Note that usually a rescaling a ~ k = g ω a k \widetilde{a}_k=\frac{\sqrt{g}}{\sqrt{\omega}}a_k a k = ω g a k ω ∣ a ~ k ∣ 2 \omega|\widetilde{a}_k|^2 ω ∣ a k ∣ 2 L 3 / T 2 L^3/T^2 L 3 / T 2 ∣ a ~ k ∣ 2 |\widetilde{a}_k|^2 ∣ a k ∣ 2 N k N_k N k N k = E k / ω N_k=E_k/\omega N k = E k / ω
Taking the time derivative and substituting the expressions of φ \varphi φ η \eta η
∂ t a k = − i ω a k + 1 2 ∑ j C k , k j ( a k − k j − a − ( k − k j ) ∗ ) W j T ( t ) + f k , (20) \partial_t a_k=-i\omega a_k+\frac{1}{2}\sum_j C_{k,k_j}\left(a_{k-k_j}-a^*_{-(k-k_j)}\right)W_j^T(t)+f_k, \tag*{(20)} ∂ t a k = − iω a k + 2 1 j ∑ C k , k j ( a k − k j − a − ( k − k j ) ∗ ) W j T ( t ) + f k , (20) with
C k , k j = c k j k j ⋅ ( k − k j ) G k [ 0 ] ω k ω k − k j G k − k j [ 0 ] and f k = ∑ j G k [ 0 ] c k j W j T ( t ) δ k , k j . (21) C_{k,k_j}=c_{k_jk_j}\cdot(k-k_j)\frac{G_k[0]}{\omega_k}\frac{\omega_{k-k_j}}{G_{k-k_j}[0]}
\quad\text{and}\quad
f_k=\sum_j G_k[0]c_{k_j}W_j^T(t)\delta_{k,k_j}. \tag*{(21)} C k , k j = c k j k j ⋅ ( k − k j ) ω k G k [ 0 ] G k − k j [ 0 ] ω k − k j and f k = j ∑ G k [ 0 ] c k j W j T ( t ) δ k , k j . (21) Introducing the interaction representation b k ( t ) = a k ( t ) e i ω k t b_k(t)=a_k(t)e^{i\omega_k t} b k ( t ) = a k ( t ) e i ω k t a − ( k − k j ) ∗ a^*_{-(k-k_j)} a − ( k − k j ) ∗ ω k + ω k − k j \omega_k+\omega_{k-k_j} ω k + ω k − k j
∂ t a k = − i ω k a k + 1 2 ∑ j C k , k j a k − k j W j T ( t ) + f k . (22) \partial_t a_k=-i\omega_k a_k+\frac{1}{2}\sum_j C_{k,k_j}a_{k-k_j}W_j^T(t)+f_k. \tag*{(22)} ∂ t a k = − i ω k a k + 2 1 j ∑ C k , k j a k − k j W j T ( t ) + f k . (22) A Duhamel solution reads then
a k = e − i ω k t a k ( 0 ) + ∑ j ∫ 0 t e − i ω k ( t − s ) [ 1 2 C k , k j a k − k j W j T ( s ) + f k ( s ) ] d s . (23) a_k=e^{-i\omega_k t}a_k(0)+\sum_j\int_0^t e^{-i\omega_k(t-s)}\left[\frac{1}{2}C_{k,k_j}a_{k-k_j}W_j^T(s)+f_k(s)\right]ds. \tag*{(23)} a k = e − i ω k t a k ( 0 ) + j ∑ ∫ 0 t e − i ω k ( t − s ) [ 2 1 C k , k j a k − k j W j T ( s ) + f k ( s ) ] d s . (23) Equation (23) has the structure of a random linear scattering equation, with multiplicative transport-noise generating mode coupling and additive forcing injecting energy at unresolved modes.
We obtain for the surface elevation
η ( x , t ) = − ∑ k e i k ⋅ x − i ω k t a k ( t ) + e i k ⋅ x + i ω k t a − k ∗ ( t ) (24) \eta(x,t)=-\sum_k e^{i k\cdot x-i\omega_k t}a_k(t)+e^{i k\cdot x+i\omega_k t}a^*_{-k}(t) \tag*{(24)} η ( x , t ) = − k ∑ e ik ⋅ x − i ω k t a k ( t ) + e ik ⋅ x + i ω k t a − k ∗ ( t ) (24) and for the velocity potential
φ ‾ ( x , t ) = ∑ k i ω k G k [ 0 ] [ e i k ⋅ x − i ω k t a k ( t ) + e i k ⋅ x + i ω k t a − k ∗ ( t ) ] . (25) \overline{\varphi}(x,t)=\sum_k\frac{i\omega_k}{G_k[0]}\left[e^{i k\cdot x-i\omega_k t}a_k(t)+e^{i k\cdot x+i\omega_k t}a^*_{-k}(t)\right]. \tag*{(25)} φ ( x , t ) = k ∑ G k [ 0 ] i ω k [ e ik ⋅ x − i ω k t a k ( t ) + e ik ⋅ x + i ω k t a − k ∗ ( t ) ] . (25) This representation describes linear Airy waves coupled through a stochastic mode-scattering operator induced by transport-noise, together with an additive forcing acting at the unresolved noise wavenumbers. It provides the random scattering dynamics underlying the kinetic description developed in the next section.
Kinetic equation for the linear stochastic water-wave dynamics with fixed noise
In the following, we separate the transport-noise term from the additive forcing. In the scattering regime established below, the former redistributes wave action among the resolved modes, while the latter corresponds to wave-action injection at the noise wavenumbers. We therefore first drop the additive-noise contribution and retain both terms of the multiplicative coupling:
∂ t a k = − i ω k a k + 1 2 ∑ j C k , k j ( a k − k j − a − ( k − k j ) ∗ ) W j T ( t ) . (26) \partial_t a_k = -i\omega_k a_k + \frac{1}{2}\sum_j C_{k,k_j}\left(a_{k-k_j}-a^*_{-(k-k_j)}\right)W_j^T(t). \tag*{(26)} ∂ t a k = − i ω k a k + 2 1 j ∑ C k , k j ( a k − k j − a − ( k − k j ) ∗ ) W j T ( t ) . (26) Here and below, we understand the conventional action normalization a k A = g / ω k a k a_k^A = \sqrt{g/\omega_k}\,a_k a k A = g / ω k a k ω k − k j / ω k \sqrt{\omega_{k-k_j}/\omega_k} ω k − k j / ω k C k , k j C_{k,k_j} C k , k j
In this analysis, we replace the smoothed noise W j T W_j^T W j T Z j Z_j Z j h T h^T h T
d Z j = − γ j Z j d t + σ j d B j ( t ) , (27) \mathrm{d}Z_j = -\gamma_j Z_j\,\mathrm{d}t + \sigma_j\,\mathrm{d}B_j(t), \tag*{(27)} d Z j = − γ j Z j d t + σ j d B j ( t ) , (27) where the B j B_j B j
E [ Z j ( t ) Z j ′ ( s ) ] = δ j j ′ σ j 2 2 γ j e − γ j ∣ t − s ∣ . (28) \mathbb{E}[Z_j(t)Z_{j'}(s)] = \delta_{jj'}\frac{\sigma_j^2}{2\gamma_j}e^{-\gamma_j\lvert t-s\rvert}. \tag*{(28)} E [ Z j ( t ) Z j ′ ( s )] = δ j j ′ 2 γ j σ j 2 e − γ j ∣ t − s ∣ . (28) Since W j T W_j^T W j T Z j Z_j Z j T − 1 / 2 T^{-1/2} T − 1/2 γ j \gamma_j γ j σ j \sigma_j σ j τ − 1 \tau^{-1} τ − 1 τ \tau τ γ j = O ( τ − 1 ) \gamma_j = \mathcal{O}(\tau^{-1}) γ j = O ( τ − 1 ) σ j = O ( τ − 1 ) \sigma_j = \mathcal{O}(\tau^{-1}) σ j = O ( τ − 1 )
We now remove the fast oscillations by defining
b k ( t ) = a k ( t ) e i ω k t . (29) b_k(t) = a_k(t)e^{i\omega_k t}. \tag*{(29)} b k ( t ) = a k ( t ) e i ω k t . (29) With this change of variables, the evolution equation becomes
∂ t b k = 1 2 ∑ j C k , k j b k − k j Z j ( t ) e i Ω k , − k j t − 1 2 ∑ j C k , k j b − ( k − k j ) ∗ Z j ( t ) e i Ω k , k j t , (30) \partial_t b_k = \frac{1}{2}\sum_j C_{k,k_j}b_{k-k_j}Z_j(t)e^{i\Omega_{k,-k_j}t} - \frac{1}{2}\sum_j C_{k,k_j}b^*_{-(k-k_j)}Z_j(t)e^{i\Omega_{k,k_j}t}, \tag*{(30)} ∂ t b k = 2 1 j ∑ C k , k j b k − k j Z j ( t ) e i Ω k , − k j t − 2 1 j ∑ C k , k j b − ( k − k j ) ∗ Z j ( t ) e i Ω k , k j t , (30) where
Ω k , − k j = ω k − ω k − k j , Ω k , k j = ω k + ω k − k j . (31) \Omega_{k,-k_j} = \omega_k-\omega_{k-k_j}, \qquad\Omega_{k,k_j} = \omega_k+\omega_{k-k_j}. \tag*{(31)} Ω k , − k j = ω k − ω k − k j , Ω k , k j = ω k + ω k − k j . (31) With the action normalization specified above, this transformation preserves the modal wave action
N k = E ( b k b k ∗ ) . (32) N_k = \mathbb{E}\left(b_k b_k^*\right). \tag*{(32)} N k = E ( b k b k ∗ ) . (32) Its evolution is given by
∂ t N k = E ( ( ∂ t b k ) b k ∗ ) + E ( b k ∂ t b k ∗ ) , = 1 2 ∑ j E ( C k , k j Z j b k − k j b k ∗ e i Ω k , − k j t ) − 1 2 ∑ j E ( C k , k j Z j b − ( k − k j ) ∗ b k ∗ e i Ω k , k j t ) + c . c . (33) \begin{aligned}
\partial_t N_k &= \mathbb{E}\left((\partial_t b_k)b_k^*\right) + \mathbb{E}\left(b_k\partial_t b_k^*\right),\\
&= \frac{1}{2}\sum_j \mathbb{E}\left(C_{k,k_j}Z_jb_{k-k_j}b_k^*e^{i\Omega_{k,-k_j}t}\right) - \frac{1}{2}\sum_j \mathbb{E}\left(C_{k,k_j}Z_jb^*_{-(k-k_j)}b_k^*e^{i\Omega_{k,k_j}t}\right) + \mathrm{c.c.} \tag*{(33)}
\end{aligned} ∂ t N k = E ( ( ∂ t b k ) b k ∗ ) + E ( b k ∂ t b k ∗ ) , = 2 1 j ∑ E ( C k , k j Z j b k − k j b k ∗ e i Ω k , − k j t ) − 2 1 j ∑ E ( C k , k j Z j b − ( k − k j ) ∗ b k ∗ e i Ω k , k j t ) + c.c. (33) The right-hand side does not vanish because the wave amplitudes b b b Z j Z_j Z j
We assume (i) weak coupling, so that C k , k j \mathcal{C}_{k,k_j} C k , k j b b b ∂ t N k \partial_t N_k ∂ t N k
b k ( t ) = b k ( 0 ) + 1 2 ∑ j C k , k j ∫ 0 t Z j ( s ) b k − k j ( s ) e i Ω k , − k j s d s − 1 2 ∑ j C k , k j ∫ 0 t Z j ( s ) b − ( k − k j ) ∗ ( s ) e i Ω k , k j s d s . (34) b_k(t) = b_k(0) + \frac{1}{2} \sum_j \mathcal{C}_{k,k_j} \int_0^t Z_j(s)b_{k-k_j}(s)e^{i\Omega_{k,-k_j}s}\,\mathrm{d}s
- \frac{1}{2} \sum_j \mathcal{C}_{k,k_j} \int_0^t Z_j(s)b^*_{-(k-k_j)}(s)e^{i\Omega_{k,k_j}s}\,\mathrm{d}s . \tag*{(34)} b k ( t ) = b k ( 0 ) + 2 1 j ∑ C k , k j ∫ 0 t Z j ( s ) b k − k j ( s ) e i Ω k , − k j s d s − 2 1 j ∑ C k , k j ∫ 0 t Z j ( s ) b − ( k − k j ) ∗ ( s ) e i Ω k , k j s d s . (34) and similarly
b k ∗ ( t ) = b k ∗ ( 0 ) + 1 2 ∑ j C k , k j ∗ ∫ 0 t Z j ( s ) b k − k j ∗ ( s ) e − i Ω k , − k j s d s − 1 2 ∑ j C k , k j ∗ ∫ 0 t Z j ( s ) b − ( k − k j ) ( s ) e − i Ω k , k j s d s . (35) b_k^*(t) = b_k^*(0) + \frac{1}{2} \sum_j \mathcal{C}^*_{k,k_j} \int_0^t Z_j(s)b^*_{k-k_j}(s)e^{-i\Omega_{k,-k_j}s}\,\mathrm{d}s
- \frac{1}{2} \sum_j \mathcal{C}^*_{k,k_j} \int_0^t Z_j(s)b_{-(k-k_j)}(s)e^{-i\Omega_{k,k_j}s}\,\mathrm{d}s . \tag*{(35)} b k ∗ ( t ) = b k ∗ ( 0 ) + 2 1 j ∑ C k , k j ∗ ∫ 0 t Z j ( s ) b k − k j ∗ ( s ) e − i Ω k , − k j s d s − 2 1 j ∑ C k , k j ∗ ∫ 0 t Z j ( s ) b − ( k − k j ) ( s ) e − i Ω k , k j s d s . (35) We first consider the contribution involving the frequency difference Ω k , − k j \Omega_{k,-k_j} Ω k , − k j
E [ Z j ( t ) b k − k j ( t ) b k ∗ ( t ) ] ≈ 1 2 C k , k j ∗ N k − k j ( t ) × ∫ 0 t E [ Z j ( t ) Z j ( s ) ] e − i Ω k , − k j s d s . (36) \mathbb{E}\left[Z_j(t)b_{k-k_j}(t)b_k^*(t)\right] \approx\frac{1}{2}\mathcal{C}^*_{k,k_j}N_{k-k_j}(t) \times\int_0^t \mathbb{E}\left[Z_j(t)Z_j(s)\right]e^{-i\Omega_{k,-k_j}s}\,\mathrm{d}s . \tag*{(36)} E [ Z j ( t ) b k − k j ( t ) b k ∗ ( t ) ] ≈ 2 1 C k , k j ∗ N k − k j ( t ) × ∫ 0 t E [ Z j ( t ) Z j ( s ) ] e − i Ω k , − k j s d s . (36) After multiplication by the external factors appearing in (33), we obtain
1 2 E ( C k , k j Z j ( t ) b k − k j ( t ) b k ∗ ( t ) e i Ω k , − k j t ) ≈ 1 4 ∣ C k , k j ∣ 2 N k − k j ( t ) × ∫ 0 t E [ Z j ( t ) Z j ( s ) ] e i Ω k , − k j ( t − s ) d s . (37) \frac{1}{2}\mathbb{E}\left(\mathcal{C}_{k,k_j}Z_j(t)b_{k-k_j}(t)b_k^*(t)e^{i\Omega_{k,-k_j}t}\right) \approx\frac{1}{4}\lvert\mathcal{C}_{k,k_j}\rvert^2N_{k-k_j}(t)
\times\int_0^t \mathbb{E}\left[Z_j(t)Z_j(s)\right]e^{i\Omega_{k,-k_j}(t-s)}\,\mathrm{d}s . \tag*{(37)} 2 1 E ( C k , k j Z j ( t ) b k − k j ( t ) b k ∗ ( t ) e i Ω k , − k j t ) ≈ 4 1 ∣ C k , k j ∣ 2 N k − k j ( t ) × ∫ 0 t E [ Z j ( t ) Z j ( s ) ] e i Ω k , − k j ( t − s ) d s . (37) Setting u = t − s u=t-s u = t − s t ≫ γ j − 1 t\gg\gamma_j^{-1} t ≫ γ j − 1
R e j ∫ 0 ∞ R j ( u ) e i Ω k , − k j u d u = σ j 2 2 γ j γ j 2 + Ω k , − k j 2 = 1 2 R ^ j ( Ω k , − k j ) . (38) \mathop{\mathrm{Re}}_j \int_0^\infty R_j(u)e^{i\Omega_{k,-k_j}u}\,\mathrm{d}u = \frac{\sigma_j^2}{2}\frac{\gamma_j}{\gamma_j^2+\Omega_{k,-k_j}^2} = \frac{1}{2}\widehat{R}_j(\Omega_{k,-k_j}) . \tag*{(38)} Re j ∫ 0 ∞ R j ( u ) e i Ω k , − k j u d u = 2 σ j 2 γ j 2 + Ω k , − k j 2 γ j = 2 1 R j ( Ω k , − k j ) . (38) where
R j ( u ) = σ j 2 2 γ j e − γ j u , u ≥ 0. (39) R_j(u)=\frac{\sigma_j^2}{2\gamma_j}e^{-\gamma_j u},\qquad u\ge0 . \tag*{(39)} R j ( u ) = 2 γ j σ j 2 e − γ j u , u ≥ 0. (39) Including the complex-conjugate contribution, we define the incoming scattering rate
G k , k j = 1 4 ∣ C k , k j ∣ 2 R ^ j ( Ω k , − k j ) . (40) \mathcal{G}_{k,k_j}=\frac{1}{4}\lvert\mathcal{C}_{k,k_j}\rvert^2\widehat{R}_j(\Omega_{k,-k_j}) . \tag*{(40)} G k , k j = 4 1 ∣ C k , k j ∣ 2 R j ( Ω k , − k j ) . (40) The coefficient G k , k j \mathcal{G}_{k,k_j} G k , k j T − 1 T^{-1} T − 1
We now return to the second expectation in (33). It involves the anomalous correlation
E [ Z j ( t ) b − ( k − k j ) ∗ ( t ) b k ∗ ( t ) ] (41) \mathbb{E}\left[Z_j(t)b^*_{-(k-k_j)}(t)b_k^*(t)\right] \tag*{(41)} E [ Z j ( t ) b − ( k − k j ) ∗ ( t ) b k ∗ ( t ) ] (41) and is weighted by the sum frequency Ω k , k j = ω k + ω k − k j \Omega_{k,k_j}=\omega_k+\omega_{k-k_j} Ω k , k j = ω k + ω k − k j
R ^ j ( ω k + ω k − k j ) ≪ R ^ j ( ω k − ω k − k j ) . (42) \widehat{R}_j\left(\omega_k+\omega_{k-k_j}\right) \ll\widehat{R}_j\left(\omega_k-\omega_{k-k_j}\right). \tag*{(42)} R j ( ω k + ω k − k j ) ≪ R j ( ω k − ω k − k j ) . (42) It therefore averages out on the slow kinetic time scale and is neglected. If retained, it would describe a parametric exchange of wave action with the prescribed random medium rather than conservative mode-to-mode scattering. In other words, the OU process is fast compared with the kinetic evolution but slow enough that its spectrum is negligible at the sum frequency.
Repeating the same second-order calculation for transitions leaving mode k k k
∂ t N k = ∑ j [ G k , k j N k − k j − G k + k j , k j N k ] . (43) \partial_t N_k=\sum_j\left[\mathcal{G}_{k,k_j}N_{k-k_j}-\mathcal{G}_{k+k_j,k_j}N_k\right]. \tag*{(43)} ∂ t N k = j ∑ [ G k , k j N k − k j − G k + k j , k j N k ] . (43) The first term represents action received by mode k k k k − k j k-k_j k − k j k k k k + k j k+k_j k + k j
d d t ∑ k N k = ∑ k , j G k , k j N k − k j − ∑ k , j G k + k j , k j N k = 0 , (44) \begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t}\sum_k N_k=\sum_{k,j}\mathcal{G}_{k,k_j}N_{k-k_j}-\sum_{k,j}\mathcal{G}_{k+k_j,k_j}N_k \\
&=0, \tag*{(44)}
\end{aligned} d t d k ∑ N k = k , j ∑ G k , k j N k − k j − k , j ∑ G k + k j , k j N k = 0 , (44) after the change of summation variable k ↦ k + k j k\mapsto k+k_j k ↦ k + k j
In the long-wave asymptotic regime ∣ k ∣ / ∣ k j ∣ → 0 \lvert k\rvert/\lvert k_j\rvert\to0 ∣ k ∣ / ∣ k j ∣ → 0
Ω k + k j , − k j ≈ − Ω k , − k j . (45) \Omega_{k+k_j,-k_j}\approx-\Omega_{k,-k_j}. \tag*{(45)} Ω k + k j , − k j ≈ − Ω k , − k j . (45) Since the stationary covariance is even, the frequency-dependent part of the transition rate is consequently the same for the above forward and backward transitions. We additionally assume that the complete coupling coefficient has the same leading symmetry, so that
G k + k j , k j = G k , k j + o ( 1 ) . (46) \mathcal{G}_{k+k_j,k_j}=\mathcal{G}_{k,k_j}+o(1). \tag*{(46)} G k + k j , k j = G k , k j + o ( 1 ) . (46) To leading order, (43) therefore reduces to
∂ t N k = ∑ j G k , k j ( N k − k j − N k ) . (47) \partial_t N_k=\sum_j\mathcal{G}_{k,k_j}\left(N_{k-k_j}-N_k\right). \tag*{(47)} ∂ t N k = j ∑ G k , k j ( N k − k j − N k ) . (47) This equation describes conservative mode-to-mode scattering by a prescribed stationary random medium. The kernel G k , k j \mathcal{G}_{k,k_j} G k , k j
Diffusive continuous limit
We now consider the continuous wavenumber limit of the master equation (47). Assuming that the unresolved scattering wavevectors form a sufficiently dense set and that the discrete scattering kernel admits a regular continuum representation, the discrete sum may be replaced by an integral, yielding
∂ t N ( k ) = ∫ G ( k , q ) [ N ( k − q ) − N ( k ) ] d q . (48) \partial_t N(k) = \int\mathcal{G}(k,q)[N(k-q)-N(k)]\,\mathrm{d}q. \tag*{(48)} ∂ t N ( k ) = ∫ G ( k , q ) [ N ( k − q ) − N ( k )] d q . (48) Here q q q G ( k , q ) ≥ 0 \mathcal{G}(k,q) \ge0 G ( k , q ) ≥ 0
When the dispersion relation varies slowly in space and time, the wave action also depends on the physical position x x x P ( x , t ) \mathcal{P}(x,t) P ( x , t )
∂ t P + ω ( x , ∇ P , t ) = 0 \partial_t\mathcal{P} + \omega(x,\nabla\mathcal{P},t) = 0 ∂ t P + ω ( x , ∇ P , t ) = 0 where the local wavevector is k = ∇ x P k = \nabla_x\mathcal{P} k = ∇ x P
x ˙ = ∇ k ω and k ˙ = − ∇ x ω . \dot{x} = \nabla_k\omega\quad\text{and}\quad\dot{k} = -\nabla_x\omega. x ˙ = ∇ k ω and k ˙ = − ∇ x ω . The corresponding wave action density then reads
∂ t N ( x , k , t ) + ∇ k ω ⋅ ∇ x N ( x , k , t ) − ∇ x ω ⋅ ∇ k N ( x , k , t ) = ∫ G ( k , q ) [ N ( x , k − q , t ) − N ( x , k , t ) ] d q . (49) \partial_tN(x,k,t)+\nabla_k\omega\cdot\nabla_xN(x,k,t)-\nabla_x\omega\cdot\nabla_kN(x,k,t) = \int\mathcal{G}(k,q)[N(x,k-q,t)-N(x,k,t)]\,\mathrm{d}q. \tag*{(49)} ∂ t N ( x , k , t ) + ∇ k ω ⋅ ∇ x N ( x , k , t ) − ∇ x ω ⋅ ∇ k N ( x , k , t ) = ∫ G ( k , q ) [ N ( x , k − q , t ) − N ( x , k , t )] d q . (49) This coincides with the linear radiative transfer equation derived by [16 ] . Under the reciprocity condition
G ( k , q ) = G ( k − q , − q ) , (50) \mathcal{G}(k,q) = \mathcal{G}(k-q,-q), \tag*{(50)} G ( k , q ) = G ( k − q , − q ) , (50) the scattering operator conserves total wave action:
d d t ∫ R d N ( k , t ) d k = 0. \frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R}^d}N(k,t)\,\mathrm{d}k = 0. d t d ∫ R d N ( k , t ) d k = 0. The following argument concerns the scattering dynamics locally in physical space, with the slow spatial and temporal variables frozen. In the long-wave regime, we retain the leading-order translation-invariant part of the scattering kernel,
G ( ϵ κ , q ) = G 0 ( q ) + o ( 1 ) . \mathcal{G}(\epsilon\kappa,q) = \mathcal{G}_0(q) + o(1). G ( ϵ κ , q ) = G 0 ( q ) + o ( 1 ) . We assume that this leading-order kernel is symmetric,
G 0 ( − q ) = G 0 ( q ) , \mathcal{G}_0(-q) = \mathcal{G}_0(q), G 0 ( − q ) = G 0 ( q ) , as follows, in particular, under isotropy assumption.
We now interpret the scattering dynamics as a random jump process in spectral space. Let K t K_t K t
( L f ) ( k ) = ∫ R d G 0 ( q ) [ f ( k − q ) − f ( k ) ] d q . (51) (\mathcal{L}f)(k) = \int_{\mathbb{R}^d}\mathcal{G}_0(q)[f(k-q)-f(k)]\,\mathrm{d}q. \tag*{(51)} ( L f ) ( k ) = ∫ R d G 0 ( q ) [ f ( k − q ) − f ( k )] d q . (51) Thus, the process undergoes spectral jumps
K t ⟼ K t − q K_t \longmapsto K_t-q K t ⟼ K t − q with rate measure G 0 ( q ) d q G_0(q)\,dq G 0 ( q ) d q
To describe the dynamics in the stretched long-wave spectral coordinate, we first set
k = ε κ (52) k = \varepsilon\kappa \tag*{(52)} k = ε κ (52) and define
κ t ε : = ε − 1 K t . (53) \kappa_t^\varepsilon:= \varepsilon^{-1}K_t. \tag*{(53)} κ t ε := ε − 1 K t . (53) At this stage, only the spectral variable is rescaled; the time variable is unchanged. An original jump
K t ⟼ K t − q , K_t \longmapsto K_t-q, K t ⟼ K t − q , therefore becomes
κ t ε ⟼ K t − q ε = κ t ε − ε − 1 q . (54) \kappa_t^\varepsilon\longmapsto\frac{K_t-q}{\varepsilon}=\kappa_t^\varepsilon-\varepsilon^{-1}q. \tag*{(54)} κ t ε ⟼ ε K t − q = κ t ε − ε − 1 q . (54) The generator of κ ε \kappa^\varepsilon κ ε
( L κ ε f ) ( κ ) = ∫ R d G 0 ( q ) [ f ( κ − ε − 1 q ) − f ( κ ) ] d q . (55) (\mathcal{L}_{\kappa^\varepsilon}f)(\kappa)=\int_{\mathbb{R}^d}G_0(q)\left[f\left(\kappa-\varepsilon^{-1}q\right)-f(\kappa)\right]\,dq. \tag*{(55)} ( L κ ε f ) ( κ ) = ∫ R d G 0 ( q ) [ f ( κ − ε − 1 q ) − f ( κ ) ] d q . (55) Hence the jumps are of order ε − 1 \varepsilon^{-1} ε − 1 κ \kappa κ κ ε \kappa^\varepsilon κ ε
A Brownian limit requires the accumulation of many centered scattering events over a longer time scale, together with the corresponding central-limit normalization. We therefore introduce
Y τ ε : = ε 2 K τ / ε 2 ε . (56) Y_\tau^\varepsilon:=\varepsilon^2K_{\tau/\varepsilon^2}^\varepsilon. \tag*{(56)} Y τ ε := ε 2 K τ / ε 2 ε . (56) Equivalently,
Y τ ε = ε K τ / ε 2 . (57) Y_\tau^\varepsilon=\varepsilon K_{\tau/\varepsilon^2}. \tag*{(57)} Y τ ε = ε K τ / ε 2 . (57) This two-step construction separates two distinct operations. The definition κ t ε = ε − 1 K t \kappa_t^\varepsilon=\varepsilon^{-1}K_t κ t ε = ε − 1 K t t = τ / ε 2 t=\tau/\varepsilon^2 t = τ / ε 2 O ( ε − 2 ) \mathcal{O}(\varepsilon^{-2}) O ( ε − 2 ) ε 2 \varepsilon^2 ε 2 ε − 1 \varepsilon^{-1} ε − 1 κ ε \kappa^\varepsilon κ ε
A jump of κ ε \kappa^\varepsilon κ ε − ε − 1 q -\varepsilon^{-1}q − ε − 1 q Y ε Y^\varepsilon Y ε
Δ Y ε = − ε q . (58) \Delta Y^\varepsilon=-\varepsilon q. \tag*{(58)} Δ Y ε = − εq . (58) Thus, although the jumps are large in the intermediate coordinate κ \kappa κ Y ε Y^\varepsilon Y ε ε − 2 \varepsilon^{-2} ε − 2 O ( ε 2 ) \mathcal{O}(\varepsilon^2) O ( ε 2 ) O ( ε ) \mathcal{O}(\varepsilon) O ( ε ) O ( ε − 2 ) \mathcal{O}(\varepsilon^{-2}) O ( ε − 2 ) Y ε Y^\varepsilon Y ε
( L Y ε g ) ( y ) = ε − 2 ∫ R d G 0 ( q ) [ g ( y − ε q ) − g ( y ) ] d q . (59) (\mathcal{L}_{Y^\varepsilon}g)(y)=\varepsilon^{-2}\int_{\mathbb{R}^d}G_0(q)\left[g(y-\varepsilon q)-g(y)\right]\,dq. \tag*{(59)} ( L Y ε g ) ( y ) = ε − 2 ∫ R d G 0 ( q ) [ g ( y − εq ) − g ( y ) ] d q . (59) This describes jumps − ε q -\varepsilon q − εq O ( ε ∣ q ∣ ) \mathcal{O}(\varepsilon|q|) O ( ε ∣ q ∣ ) ε − 2 G 0 ( q ) d q \varepsilon^{-2}G_0(q)\,dq ε − 2 G 0 ( q ) d q
∫ R d q G 0 ( q ) d q = 0 , (60) \int_{\mathbb{R}^d}q\,G_0(q)\,dq=0, \tag*{(60)} ∫ R d q G 0 ( q ) d q = 0 , (60) and has finite second moment,
∫ R d ∣ q ∣ 2 G 0 ( q ) d q < ∞ . (61) \int_{\mathbb{R}^{d}} |q|^{2}\mathcal{G}_{0}(q)\,\mathrm{d}q < \infty. \tag*{(61)} ∫ R d ∣ q ∣ 2 G 0 ( q ) d q < ∞. (61) For every δ > 0 \delta> 0 δ > 0 Y ε Y^{\varepsilon} Y ε
ε − 2 ∫ ∣ ε q ∣ > δ ∣ ε q ∣ 2 G 0 ( q ) d q = ∫ ∣ q ∣ > δ / ε ∣ q ∣ 2 G 0 ( q ) d q ⟶ 0. (62) \varepsilon^{-2}\int_{|\varepsilon q|>\delta} |\varepsilon q|^{2}\mathcal{G}_{0}(q)\,\mathrm{d}q = \int_{|q|>\delta/\varepsilon} |q|^{2}\mathcal{G}_{0}(q)\,\mathrm{d}q \longrightarrow0. \tag*{(62)} ε − 2 ∫ ∣ εq ∣ > δ ∣ εq ∣ 2 G 0 ( q ) d q = ∫ ∣ q ∣ > δ / ε ∣ q ∣ 2 G 0 ( q ) d q ⟶ 0. (62) Hence the jumps of the rescaled process are asymptotically negligible, even though the jumps in the intermediate coordinate κ \kappa κ ε − 1 \varepsilon^{-1} ε − 1
Applying Dynkin’s formula to the coordinate functions g i ( y ) = y i g_i(y)=y_i g i ( y ) = y i
L Y ε g i = 0. \mathcal{L}_{Y}^{\varepsilon}g_i=0. L Y ε g i = 0. Consequently,
Y τ ε , i = Y 0 ε , i + M τ ε , i , (63) Y_{\tau}^{\varepsilon,i}=Y_{0}^{\varepsilon,i}+M_{\tau}^{\varepsilon,i}, \tag*{(63)} Y τ ε , i = Y 0 ε , i + M τ ε , i , (63) where M ε M^{\varepsilon} M ε
⟨ M ε , i , M ε , j ⟩ τ = ε − 2 ∫ 0 τ ∫ R d ( ε q i ) ( ε q j ) G 0 ( q ) d q d s = τ ∫ R d q i q j G 0 ( q ) d q . (64) \left\langle M^{\varepsilon,i},M^{\varepsilon,j}\right\rangle_{\tau}
=\varepsilon^{-2}\int_{0}^{\tau}\int_{\mathbb{R}^{d}}(\varepsilon q_i)(\varepsilon q_j)\mathcal{G}_{0}(q)\,\mathrm{d}q\,\mathrm{d}s
=\tau\int_{\mathbb{R}^{d}}q_iq_j\mathcal{G}_{0}(q)\,\mathrm{d}q. \tag*{(64)} ⟨ M ε , i , M ε , j ⟩ τ = ε − 2 ∫ 0 τ ∫ R d ( ε q i ) ( ε q j ) G 0 ( q ) d q d s = τ ∫ R d q i q j G 0 ( q ) d q . (64) Assuming additionally that Y 0 ε ⟹ Y 0 Y_{0}^{\varepsilon}\Longrightarrow Y_0 Y 0 ε ⟹ Y 0 [4 ] (see appendix for a statement of this theorem),
Y ε ⟹ Y 0 + 2 D W , (65) Y^{\varepsilon}\Longrightarrow Y_0+\sqrt{2D}\,W, \tag*{(65)} Y ε ⟹ Y 0 + 2 D W , (65) in the space D ( [ 0 , T ] ; R d ) \mathbb{D}([0,T];\mathbb{R}^{d}) D ([ 0 , T ] ; R d )
D i j = 1 2 ∫ R d q i q j G 0 ( q ) d q . (66) D^{ij}=\frac{1}{2}\int_{\mathbb{R}^{d}}q_iq_j\mathcal{G}_{0}(q)\,\mathrm{d}q. \tag*{(66)} D ij = 2 1 ∫ R d q i q j G 0 ( q ) d q . (66) To translate the convergence of the jump process into a limit for the wave-action density, we introduce the rescaled density associated with Y ε Y^{\varepsilon} Y ε
N ε ( y , τ ) : = ε − d N ( y ε , τ ε 2 ) . (67) N^{\varepsilon}(y,\tau):=\varepsilon^{-d}N\left(\frac{y}{\varepsilon},\frac{\tau}{\varepsilon^{2}}\right). \tag*{(67)} N ε ( y , τ ) := ε − d N ( ε y , ε 2 τ ) . (67) The factor ε − d \varepsilon^{-d} ε − d
∂ τ N ε = ε − 2 ∫ R d G 0 ( q ) [ N ε ( y − ε q , τ ) − N ε ( y , τ ) ] d q . (68) \partial_{\tau}N^{\varepsilon}=\varepsilon^{-2}\int_{\mathbb{R}^{d}}\mathcal{G}_{0}(q)\left[N^{\varepsilon}(y-\varepsilon q,\tau)-N^{\varepsilon}(y,\tau)\right]\mathrm{d}q. \tag*{(68)} ∂ τ N ε = ε − 2 ∫ R d G 0 ( q ) [ N ε ( y − εq , τ ) − N ε ( y , τ ) ] d q . (68) Equivalently, for every smooth compactly supported test function φ \varphi φ
d d τ ∫ R d φ ( y ) N ε ( y , τ ) d y = ∫ R d N ε ( y , τ ) ( Q ε , ∗ φ ) ( y ) d y , (69) \frac{\mathrm{d}}{\mathrm{d}\tau}\int_{\mathbb{R}^{d}}\varphi(y)N^{\varepsilon}(y,\tau)\,\mathrm{d}y
=\int_{\mathbb{R}^{d}}N^{\varepsilon}(y,\tau)(\mathcal{Q}^{\varepsilon,*}\varphi)(y)\,\mathrm{d}y, \tag*{(69)} d τ d ∫ R d φ ( y ) N ε ( y , τ ) d y = ∫ R d N ε ( y , τ ) ( Q ε , ∗ φ ) ( y ) d y , (69) where the adjoint Q ε , ∗ \mathcal{Q}^{\varepsilon,*} Q ε , ∗ Q ε \mathcal{Q}^{\varepsilon} Q ε
( Q ε , ∗ φ ) ( y ) = ε − 2 ∫ R d G 0 ( q ) [ φ ( y + ε q ) − φ ( y ) ] d q . (70) (\mathcal{Q}^{\varepsilon,*}\varphi)(y)=\varepsilon^{-2}\int_{\mathbb{R}^{d}}\mathcal{G}_{0}(q)\left[\varphi(y+\varepsilon q)-\varphi(y)\right]\mathrm{d}q. \tag*{(70)} ( Q ε , ∗ φ ) ( y ) = ε − 2 ∫ R d G 0 ( q ) [ φ ( y + εq ) − φ ( y ) ] d q . (70) If the scattering kernel is symmetric, G 0 ( − q ) = G 0 ( q ) \mathcal{G}_{0}(-q)=\mathcal{G}_{0}(q) G 0 ( − q ) = G 0 ( q ) Q ε , ∗ = L Y ε \mathcal{Q}^{\varepsilon,*}=\mathcal{L}_{Y}^{\varepsilon} Q ε , ∗ = L Y ε
L Y ε φ ⟶ D i j ∂ i j φ , \mathcal{L}_{Y}^{\varepsilon}\varphi\longrightarrow D^{ij}\partial_{ij}\varphi, L Y ε φ ⟶ D ij ∂ ij φ , where D D D N ε N^\varepsilon N ε N ‾ \overline{N} N
d d τ ∫ φ ‾ N d y = ∫ N ‾ D i j ∂ i j φ d y = ∫ φ ∂ i ∂ j ( D i j N ‾ ) d y . \frac{\mathrm{d}}{\mathrm{d}\tau}\int\overline{\varphi}N\,\mathrm{d}y=\int\overline{N}D^{ij}\partial_{ij}\varphi\,\mathrm{d}y=\int\varphi\,\partial_i\partial_j\left(D^{ij}\overline{N}\right)\,\mathrm{d}y. d τ d ∫ φ N d y = ∫ N D ij ∂ ij φ d y = ∫ φ ∂ i ∂ j ( D ij N ) d y . Relabelling the macroscopic variables ( y , τ , N ‾ ) (y,\tau,\overline{N}) ( y , τ , N ) ( k , t , N ) (k,t,N) ( k , t , N )
∂ t N = ∇ k ⋅ ( D ∇ k N ) . (71) \partial_tN=\nabla_k\cdot(D\nabla_kN). \tag*{(71)} ∂ t N = ∇ k ⋅ ( D ∇ k N ) . (71) It describes a random walk in spectral space generated by repeated scattering events. No corrector is required here: because the jump kernel is translation invariant and centered, the coordinate process is already a martingale. The diffusion therefore arises directly from its quadratic variation.
The present diffusion is obtained in a long-wave, long-time central-limit regime. The microscopic scattering wavevector q q q ε q \varepsilon q εq O ( ε − 2 ) O(\varepsilon^{-2}) O ( ε − 2 )
Scaling analysis
In the deep-water regime, the dispersion relation scales as ω ∼ ∣ k ∣ 1 / 2 \omega\sim|k|^{1/2} ω ∼ ∣ k ∣ 1/2 G k [ 0 ] ∼ ∣ k ∣ G_k[0]\sim|k| G k [ 0 ] ∼ ∣ k ∣ ∣ C k , q ∣ ∼ ∣ c q ∣ ∣ k ∣ 1 / 2 ∣ q ∣ 3 / 2 |C_{k,q}|\sim|c_q||k|^{1/2}|q|^{3/2} ∣ C k , q ∣ ∼ ∣ c q ∣∣ k ∣ 1/2 ∣ q ∣ 3/2 c k , q ∼ ∣ q ∣ − 2 ω 1 / 2 = ∣ q ∣ − 7 / 4 c_{k,q}\sim|q|^{-2}\omega^{1/2}=|q|^{-7/4} c k , q ∼ ∣ q ∣ − 2 ω 1/2 = ∣ q ∣ − 7/4 ∣ C k , q ∣ ∼ ∣ k ∣ 1 / 2 ∣ q ∣ − 1 / 4 |C_{k,q}|\sim|k|^{1/2}|q|^{-1/4} ∣ C k , q ∣ ∼ ∣ k ∣ 1/2 ∣ q ∣ − 1/4
G ( k , q ) = ∣ C k , q ∣ 2 σ q 2 γ q 2 + Ω k , − q 2 ∼ ∣ k ∣ ∣ q ∣ − 1 / 2 σ q 2 γ q 2 + ∣ q ∣ . (72) \mathcal{G}(k,q)=\frac{|C_{k,q}|^2\sigma_q^2}{\gamma_q^2+\Omega_{k,-q}^2}\sim\frac{|k||q|^{-1/2}\sigma_q^2}{\gamma_q^2+|q|}. \tag*{(72)} G ( k , q ) = γ q 2 + Ω k , − q 2 ∣ C k , q ∣ 2 σ q 2 ∼ γ q 2 + ∣ q ∣ ∣ k ∣∣ q ∣ − 1/2 σ q 2 . (72) The diffusion coefficient in wave-number space then scales as
D ( k ) ∼ ∣ k ∣ ∫ ∣ k ∣ ∞ ∣ q ∣ 3 / 2 σ q 2 γ q 2 + ∣ q ∣ d ∣ q ∣ . (73) D(k)\sim|k|\int_{|k|}^{\infty}\frac{|q|^{3/2}\sigma_q^2}{\gamma_q^2+|q|}\,\mathrm{d}|q|. \tag*{(73)} D ( k ) ∼ ∣ k ∣ ∫ ∣ k ∣ ∞ γ q 2 + ∣ q ∣ ∣ q ∣ 3/2 σ q 2 d ∣ q ∣. (73) If the integral is k k k D ( k ) ∼ ∣ k ∣ D(k)\sim|k| D ( k ) ∼ ∣ k ∣ k k k Π ( k ) = − D ( k ) ∂ k N ( k ) \Pi(k)=-D(k)\partial_kN(k) Π ( k ) = − D ( k ) ∂ k N ( k )
∂ k N ( k ) ∼ Π D ( k ) . \partial_kN(k)\sim\frac{\Pi}{D(k)}. ∂ k N ( k ) ∼ D ( k ) Π . To further specify the spectrum, we assume OU process parameters follow power laws γ q 2 ∼ ∣ q ∣ β \gamma_q^2\sim|q|^\beta γ q 2 ∼ ∣ q ∣ β σ q 2 ∼ ∣ q ∣ α \sigma_q^2\sim|q|^\alpha σ q 2 ∼ ∣ q ∣ α ∣ k ∣ − 4 |k|^{-4} ∣ k ∣ − 4 2 2 2
D ( k ) ∼ ∣ k ∣ ∫ ∣ k ∣ ∞ ∣ q ∣ 5 / 2 + α ∣ q ∣ β + ∣ q ∣ d ∣ q ∣ (74) D(k)\sim|k|\int_{|k|}^{\infty}\frac{|q|^{5/2+\alpha}}{|q|^\beta+|q|}\,\mathrm{d}|q| \tag*{(74)} D ( k ) ∼ ∣ k ∣ ∫ ∣ k ∣ ∞ ∣ q ∣ β + ∣ q ∣ ∣ q ∣ 5/2 + α d ∣ q ∣ (74) Two asymptotic regimes arise: (i) memory dominated regime with γ q 2 ≫ Ω k , q 2 \gamma_q^2 \gg\Omega_{k,q}^2 γ q 2 ≫ Ω k , q 2
D ( k ) ∼ ∣ k ∣ ∫ ∣ q ∣ 5 / 2 + α − β d ∣ q ∣ , (75) D(k) \sim|k| \int|q|^{5/2+\alpha-\beta_d}|q|\,, \tag*{(75)} D ( k ) ∼ ∣ k ∣ ∫ ∣ q ∣ 5/2 + α − β d ∣ q ∣ , (75) and (ii) resonnance dominated γ q 2 ≪ Ω k , q 2 \gamma_q^2 \ll\Omega_{k,q}^2 γ q 2 ≪ Ω k , q 2
D ( k ) ∼ ∣ k ∣ ∫ ∣ q ∣ 3 / 2 + α d ∣ q ∣ . (76) D(k) \sim|k| \int|q|^{3/2+\alpha_d}|q|\,. \tag*{(76)} D ( k ) ∼ ∣ k ∣ ∫ ∣ q ∣ 3/2 + α d ∣ q ∣ . (76) For the diffusion approximation to be controlled by the lower cutoff q min ∼ k q_{\min} \sim k q m i n ∼ k q q q α < − 5 2 \alpha< -\frac{5}{2} α < − 2 5 α − β < − 7 / 2 \alpha-\beta< -7/2 α − β < − 7/2 N ( k ) ∼ ∣ k ∣ − ( 7 / 2 + α − β ) N(k) \sim|k|^{-(7/2+\alpha-\beta)} N ( k ) ∼ ∣ k ∣ − ( 7/2 + α − β ) N ( k ) ∼ ∣ k ∣ − ( 5 / 2 + α ) N(k) \sim|k|^{-(5/2+\alpha)} N ( k ) ∼ ∣ k ∣ − ( 5/2 + α )
Matching the Kolmogorov–Zakharov spectrum N ( k ) ∼ ∣ k ∣ − 4 N(k) \sim|k|^{-4} N ( k ) ∼ ∣ k ∣ − 4 α = 3 2 \alpha= \frac{3}{2} α = 2 3 α − β = 1 2 \alpha-\beta= \frac{1}{2} α − β = 2 1 α < − 5 2 \alpha< -\frac{5}{2} α < − 2 5 α − β < − 7 2 \alpha-\beta< -\frac{7}{2} α − β < − 2 7
Surface-induced additive forcing
We now reintroduce the additive forcing produced by the free surface previously neglected. The amplitude evolution reads
∂ t a k = − i ω k a k + 1 2 ∑ j C k , k j ( a k − k j − a − ( k − k j ) ∗ ) W j T ( t ) + f k . (77) \partial_t a_k = -i\omega_k a_k + \frac{1}{2}\sum_j C_{k,k_j}\left(a_{k-k_j} - a^*_{-(k-k_j)}\right)W_j^T(t) + f_k. \tag*{(77)} ∂ t a k = − i ω k a k + 2 1 j ∑ C k , k j ( a k − k j − a − ( k − k j ) ∗ ) W j T ( t ) + f k . (77) with forcing
f k ( t ) = ∑ j G k [ 0 ] c k j W j T ( t ) δ k , k j . (78) f_k(t) = \sum_j G_k[0]c_{k_j}W_j^T(t)\delta_{k,k_j}. \tag*{(78)} f k ( t ) = j ∑ G k [ 0 ] c k j W j T ( t ) δ k , k j . (78) After removing fast oscillations, we obtain
∂ t b k = 1 2 ∑ j C k , k j b k − k j Z j ( t ) e i Ω k , − k j t − 1 2 ∑ j C k , k j b − ( k − k j ) ∗ Z j ( t ) e i Ω k , k j t + f ~ k ( t ) , (79) \partial_t b_k = \frac{1}{2}\sum_j C_{k,k_j}b_{k-k_j}Z_j(t)e^{i\Omega_{k,-k_j}t} - \frac{1}{2}\sum_j C_{k,k_j}b^*_{-(k-k_j)}Z_j(t)e^{i\Omega_{k,k_j}t} + \widetilde{f}_k(t), \tag*{(79)} ∂ t b k = 2 1 j ∑ C k , k j b k − k j Z j ( t ) e i Ω k , − k j t − 2 1 j ∑ C k , k j b − ( k − k j ) ∗ Z j ( t ) e i Ω k , k j t + f k ( t ) , (79) where f ~ k ( t ) = f k ( t ) e i ω k t \widetilde{f}_k(t) = f_k(t)e^{i\omega_k t} f k ( t ) = f k ( t ) e i ω k t
∂ t N k = E ( ∂ t b k b k ∗ ) + E ( b k ∂ t b k ∗ ) , \partial_t N_k = \mathbb{E}(\partial_t b_k b_k^*) + \mathbb{E}(b_k\partial_t b_k^*), ∂ t N k = E ( ∂ t b k b k ∗ ) + E ( b k ∂ t b k ∗ ) , which we decompose as
∂ t N k = T k + I k , \partial_t N_k = \mathcal{T}_k + \mathcal{I}_k, ∂ t N k = T k + I k , where T k \mathcal{T}_k T k
I k = E ( f ~ k b k ∗ ) + E ( b k f ~ k ∗ ) . (80) \mathcal{I}_k = \mathbb{E}(\widetilde{f}_k b_k^*) + \mathbb{E}(b_k\widetilde{f}_k^*). \tag*{(80)} I k = E ( f k b k ∗ ) + E ( b k f k ∗ ) . (80) represents additive energy injection.
Using Duhamel expansion,
b k ( t ) = b k ( 0 ) + ∫ 0 t f ~ k ( s ) d s + O ( C ) , b_k(t) = b_k(0) + \int_0^t \tilde{f}_k(s)\,ds + \mathcal{O}(\mathcal{C}), b k ( t ) = b k ( 0 ) + ∫ 0 t f ~ k ( s ) d s + O ( C ) , and the same weak coupling argument as before, we obtain
E ( f k ( t ) b k ∗ ( t ) ) ≈ ∫ 0 t E ( f k ( t ) f k ∗ ( s ) ) d s . (81) \mathbb{E}\left(f_k(t)b_k^*(t)\right) \approx\int_0^t \mathbb{E}\left(f_k(t)f_k^*(s)\right)\,ds. \tag*{(81)} E ( f k ( t ) b k ∗ ( t ) ) ≈ ∫ 0 t E ( f k ( t ) f k ∗ ( s ) ) d s . (81) Since W τ W^\tau W τ
G k = ∑ j ∣ G k j [ 0 ] ∣ 2 σ k j 2 γ j 2 + ω k 2 E ∣ χ k j ∣ 2 δ k , k j . (82) \mathcal{G}_k = \sum_j \lvert G_{k_j}[0]\rvert^2 \frac{\sigma_{k_j}^2}{\gamma_j^2+\omega_k^2}\mathbb{E}\lvert\chi_{k_j}\rvert^2\delta_{k,k_j}. \tag*{(82)} G k = j ∑ ∣ G k j [ 0 ] ∣ 2 γ j 2 + ω k 2 σ k j 2 E ∣ χ k j ∣ 2 δ k , k j . (82) The full modal energy equation becomes
∑ k ∂ t N k = ∑ k , k j G k , k j ( N k − k j − N k ) + G k , (83) \sum_k \partial_t N_k = \sum_{k,k_j}\mathcal{G}_{k,k_j}\left(N_{k-k_j}-N_k\right)+\mathcal{G}_k, \tag*{(83)} k ∑ ∂ t N k = k , k j ∑ G k , k j ( N k − k j − N k ) + G k , (83) The transport-noise is conservative, while the additive forcing provides localized energy injection in wave-number space. The total wave action density therefore grows according to
d d t ∑ k N k = ∑ k G k . (84) \frac{\mathrm{d}}{\mathrm{d}t}\sum_k N_k = \sum_k \mathcal{G}_k. \tag*{(84)} d t d k ∑ N k = k ∑ G k . (84) When G k [ 0 ] φ τ = 0 G_k[0]\varphi^\tau=0 G k [ 0 ] φ τ = 0 1 / 2 1/2 1/2
G k [ 0 ] φ τ = α χ k τ . (85) G_k[0]\varphi^\tau= \alpha\chi_k^\tau. \tag*{(85)} G k [ 0 ] φ τ = α χ k τ . (85) At the energy level, the additive forcing appears at order α 2 \alpha^2 α 2
G k = α 2 σ k 2 γ k 2 + ω k 2 E ∣ χ k ∣ 2 . (86) \mathcal{G}_k = \alpha^2\frac{\sigma_k^2}{\gamma_k^2+\omega_k^2}\mathbb{E}\lvert\chi_k\rvert^2. \tag*{(86)} G k = α 2 γ k 2 + ω k 2 σ k 2 E ∣ χ k ∣ 2 . (86) This injection is balanced by dissipation due to whitecapping and wave breaking, modeled by a linear damping term, − μ k b k -\mu_kb_k − μ k b k − 2 μ k N k -2\mu_kN_k − 2 μ k N k μ k ∼ ∣ k ∣ p \mu_k\sim\lvert k\rvert^p μ k ∼ ∣ k ∣ p p ≥ 2 p\ge2 p ≥ 2
∂ t N k = ∑ j G k , k j ( N k − k j − N k ) + α 2 σ k 2 γ k E ∣ χ k ∣ 2 − 2 μ k N k . (87) \partial_tN_k = \sum_j\mathcal{G}_{k,k_j}\left(N_{k-k_j}-N_k\right)+\alpha^2\frac{\sigma_k^2}{\gamma_k}\mathbb{E}\lvert\chi_k\rvert^2-2\mu_kN_k. \tag*{(87)} ∂ t N k = j ∑ G k , k j ( N k − k j − N k ) + α 2 γ k σ k 2 E ∣ χ k ∣ 2 − 2 μ k N k . (87) The multiplicative transport noise redistributes wave action conservatively among modes, whereas the additive contribution originates from stochastic forcing of the free-surface kinematic condition and injects wave action at the unresolved forcing wavenumbers. The resulting kinetic equation therefore contains both conservative scattering and surface-induced production. In a statistically steady regime, injection and dissipation balance; the forcing fixes the flux amplitude, while dissipation arrests the cascade at small scales. The injection rate depends not only on the noise amplitude but also on the memory parameter: shorter correlation times suppress energy input, whereas longer correlation times enhance stochastic production.
In the following we now derive a kinetic equation for the complete linear water-waves system.
Linear stochastic water-wave system with nonstationary correlation modes noise
In this second case, we consider the complete linear stochastic water-wave system in which the noise correlation modes evolve dynamically. Linearizing equations (6)–(8) and (9) about a quiescent free surface, and retaining terms up to first order in the wave-slope parameter ϵ μ \epsilon_\mu ϵ μ
∂ t φ ‾ + η + ∇ φ ‾ ⋅ ∇ φ ‾ = 0 , (88) \partial_t\overline{\varphi}+\eta+\nabla\overline{\varphi}\cdot\nabla\overline{\varphi}=0, \tag*{(88)} ∂ t φ + η + ∇ φ ⋅ ∇ φ = 0 , (88) ∂ t η = ∂ z φ ∣ z = 0 + ∂ z φ T ∣ z = 0 , (89) \partial_t\eta=\left.\partial_z\varphi\right|_{z=0}+\left.\partial_z\varphi^{T}\right|_{z=0}, \tag*{(89)} ∂ t η = ∂ z φ ∣ z = 0 + ∂ z φ T z = 0 , (89) ∂ t φ i + ϵ μ ∇ φ ‾ ⋅ ∇ φ i = 0. (90) \partial_t\varphi_i+\epsilon_\mu\nabla\overline{\varphi}\cdot\nabla\varphi_i=0. \tag*{(90)} ∂ t φ i + ϵ μ ∇ φ ⋅ ∇ φ i = 0. (90) The first two equations coincide with the linear stochastic water-wave system considered in the fixed-noise case. The third equation describes the evolution of the spatial noise correlation functions φ i \varphi_i φ i φ ‾ \overline{\varphi} φ
Equation (90) governs the deformation of the spatial correlation modes only. We now additionally assume that each unresolved mode possesses an intrinsic Airy carrier. This carrier is introduced through its temporal stochastic coefficient and not through the transported spatial mode itself.
Wave-amplitude evolution equation
We derive the wave-amplitude equation for the linear stochastic system coupled to the small-scale transport equation (90). Let Y ( s ; t , x ) Y(s;t,x) Y ( s ; t , x ) x x x t t t
d d s Y ( s ; t , x ) = ϵ μ ∇ φ ‾ ( Y ( s ; t , x ) , s ) , Y ( t ; t , x ) = x , 0 ≤ s ≤ t . (91) \frac{\mathrm{d}}{\mathrm{d}s}Y(s;t,x)=\epsilon_\mu\nabla\overline{\varphi}(Y(s;t,x),s),\qquad Y(t;t,x)=x,\qquad0\le s\le t. \tag*{(91)} d s d Y ( s ; t , x ) = ϵ μ ∇ φ ( Y ( s ; t , x ) , s ) , Y ( t ; t , x ) = x , 0 ≤ s ≤ t . (91) We define the inverse flow map by
X t − 1 ( x ) : = Y ( 0 ; t , x ) . X_t^{-1}(x):=Y(0;t,x). X t − 1 ( x ) := Y ( 0 ; t , x ) . The solution of (90) is then
φ j ( x , t ) = φ j ( X t − 1 ( x ) , 0 ) . \varphi_j(x,t)=\varphi_j\left(X_t^{-1}(x),0\right). φ j ( x , t ) = φ j ( X t − 1 ( x ) , 0 ) . For the initial spatial mode
φ j ( x , 0 ) = c k j e i k j ⋅ x , \varphi_j(x,0)=c_{k_j}e^{\mathrm{i}k_j\cdot x}, φ j ( x , 0 ) = c k j e i k j ⋅ x , we obtain
φ j ( x , t ) = c k j e i k j ⋅ X t − 1 ( x ) . (92) \varphi_j(x,t)=c_{k_j}e^{\mathrm{i}k_j\cdot X_t^{-1}(x)}. \tag*{(92)} φ j ( x , t ) = c k j e i k j ⋅ X t − 1 ( x ) . (92) To attach an Airy carrier to this transported spatial pattern, let W j τ ( t ) W_j^\tau(t) W j τ ( t )
Z j τ ( t ) : = e − i ω k j t W j τ ( t ) , ω k j 2 = g G k j [ 0 ] . (93) Z_j^\tau(t):=e^{-\mathrm{i}\omega_{k_j}t}W_j^\tau(t),\qquad\omega_{k_j}^{2}=gG_{k_j}[0]. \tag*{(93)} Z j τ ( t ) := e − i ω k j t W j τ ( t ) , ω k j 2 = g G k j [ 0 ] . (93) The corresponding real-valued noise is obtained by adding the complex-conjugate component:
φ τ ( x , t ) = ∑ j ∈ J + [ c k j e i k j ⋅ X t − 1 ( x ) Z j τ ( t ) + c k j ∗ e − i k j ⋅ X t − 1 ( x ) Z j τ , ∗ ( t ) ] . (94) \varphi^\tau(x,t)=\sum_{j\in\mathcal{J}_+}\left[c_{k_j}e^{\mathrm{i}k_j\cdot X_t^{-1}(x)}Z_j^\tau(t)+c_{k_j}^{*}e^{-\mathrm{i}k_j\cdot X_t^{-1}(x)}Z_j^{\tau,*}(t)\right]. \tag*{(94)} φ τ ( x , t ) = j ∈ J + ∑ [ c k j e i k j ⋅ X t − 1 ( x ) Z j τ ( t ) + c k j ∗ e − i k j ⋅ X t − 1 ( x ) Z j τ , ∗ ( t ) ] . (94) where J + \mathcal{J}_{+} J + { k j , − k j } \{k_j,-k_j\} { k j , − k j }
In the Ornstein–Uhlenbeck approximation used below for the kinetic closure, definition (93) is equivalently realized in law by the complex oscillatory OU process
d Z j = − ( γ j + i ω k j ) Z j d t + σ j d B j C ( t ) , (95) \mathrm{d}Z_j=-(\gamma_j+\mathrm{i}\omega_{k_j})Z_j\,\mathrm{d}t+\sigma_j\,\mathrm{d}B_j^{\mathrm{C}}(t), \tag*{(95)} d Z j = − ( γ j + i ω k j ) Z j d t + σ j d B j C ( t ) , (95) where B j C B_j^{\mathrm{C}} B j C
E [ Z j ( t ) Z ℓ ∗ ( s ) ] = δ j ℓ σ j 2 2 γ j e − γ j ∣ t − s ∣ e − i ω k j ( t − s ) . (96) \mathbb{E}[Z_j(t)Z_\ell^*(s)]=\delta_{j\ell}\frac{\sigma_j^2}{2\gamma_j}e^{-\gamma_j\lvert t-s\rvert}e^{-\mathrm{i}\omega_{k_j}(t-s)}. \tag*{(96)} E [ Z j ( t ) Z ℓ ∗ ( s )] = δ j ℓ 2 γ j σ j 2 e − γ j ∣ t − s ∣ e − i ω k j ( t − s ) . (96) Thus the covariance remains stationary, but its spectrum is a Lorentzian centered at the Airy frequency ω k j \omega_{k_j} ω k j c k j c_{k_j} c k j a k j ( 0 ) a_{k_j}^{(0)} a k j ( 0 )
c k j = − i ω k j G k j [ 0 ] a k j ( 0 ) . (97) c_{k_j}=-\frac{\mathrm{i}\omega_{k_j}}{G_{k_j}[0]}a_{k_j}^{(0)}. \tag*{(97)} c k j = − G k j [ 0 ] i ω k j a k j ( 0 ) . (97) These noise modes coefficients are constant in time, the time dependency of the noise is indicated through the temporal factor e − i ω k j t e^{-\mathrm{i}\omega_{k_j}t} e − i ω k j t Z j Z_j Z j
The spatial pattern of each noise mode is transported and deformed by the large-scale flow through X t − 1 X_t^{-1} X t − 1
∇ ( e i k j ⋅ X t − 1 ( x ) ) = i J t t r ( x ) k j e i k j ⋅ X t − 1 ( x ) , J t ( x ) : = ∇ x X t − 1 ( x ) . \nabla\left(e^{\mathrm{i}k_j\cdot X_t^{-1}(x)}\right)=\mathrm{i}J_t^{\mathrm{tr}}(x)k_j e^{\mathrm{i}k_j\cdot X_t^{-1}(x)},\qquad J_t(x):=\nabla_xX_t^{-1}(x). ∇ ( e i k j ⋅ X t − 1 ( x ) ) = i J t tr ( x ) k j e i k j ⋅ X t − 1 ( x ) , J t ( x ) := ∇ x X t − 1 ( x ) . Consequently, the positive-frequency contribution to the transport-noise term is
∇ φ T ( x , t ) ⋅ ∇ φ ( x , t ) = ∑ j c k j Z j T ( t ) e i k j ⋅ X t − 1 ( x ) i ( J t t r ( x ) k j ⋅ ∇ φ ( x , t ) ) + c . c . (98) \nabla\varphi^{\mathsf{T}}(x,t)\cdot\nabla\varphi(x,t)=\sum_j c_{k_j}Z_j^{T}(t)e^{\mathrm{i}k_j\cdot X_t^{-1}(x)}\mathrm{i}\left(J_t^{\mathrm{tr}}(x)k_j\cdot\nabla\varphi(x,t)\right)+\mathrm{c.c.} \tag*{(98)} ∇ φ T ( x , t ) ⋅ ∇ φ ( x , t ) = j ∑ c k j Z j T ( t ) e i k j ⋅ X t − 1 ( x ) i ( J t tr ( x ) k j ⋅ ∇ φ ( x , t ) ) + c.c. (98) Compared with the fixed-correlation-mode system, the phase is deformed by the flow map and the local noise wavenumber is transformed by the inverse-flow gradient. In Fourier space, the original convolution is therefore replaced, before expansion, by a time-dependent integral transform that mixes modes.
This expression can be made explicit for a smooth deformation close to the identity. Define
B ( s ; t , x ) : = ∇ x Y ( s ; t , x ) , J t ( x ) = B ( 0 ; t , x ) . B(s;t,x):=\nabla_xY(s;t,x),\qquad J_t(x)=B(0;t,x). B ( s ; t , x ) := ∇ x Y ( s ; t , x ) , J t ( x ) = B ( 0 ; t , x ) . Differentiating (91) with respect to x x x
d d s B ( s ; t , x ) = ϵ μ ∇ 2 φ ˉ ( Y ( s ; t , x ) , s ) B ( s ; t , x ) , B ( t ; t , x ) = I . (99) \frac{\mathrm{d}}{\mathrm{d}s}B(s;t,x)=\epsilon_\mu\nabla^2\bar{\varphi}(Y(s;t,x),s)B(s;t,x),\qquad B(t;t,x)=\mathbb{I}. \tag*{(99)} d s d B ( s ; t , x ) = ϵ μ ∇ 2 φ ˉ ( Y ( s ; t , x ) , s ) B ( s ; t , x ) , B ( t ; t , x ) = I . (99) Its formal solution is
B ( s ; t , x ) = T o exp ( ϵ μ ∫ t s ∇ 2 φ ˉ ( Y ( r ; t , x ) , r ) d r ) . (100) B(s;t,x)=\mathcal{T}\mathrm{o}\exp\left(\epsilon_\mu\int_t^s\nabla^2\bar{\varphi}(Y(r;t,x),r)\,\mathrm{d}r\right). \tag*{(100)} B ( s ; t , x ) = T o exp ( ϵ μ ∫ t s ∇ 2 φ ˉ ( Y ( r ; t , x ) , r ) d r ) . (100) where T o \mathcal{T}_o T o ϵ μ \epsilon_\mu ϵ μ
X t − 1 ( x ) = x − ϵ μ ∫ 0 t ∇ φ ‾ ( x , s ) d s + O ( ϵ μ 2 ) , (101) X_t^{-1}(x)=x-\epsilon_\mu\int_{0}^{t}\nabla\overline{\varphi}(x,s)\,\mathrm{d}s+\mathcal{O}(\epsilon_\mu^{2}), \tag*{(101)} X t − 1 ( x ) = x − ϵ μ ∫ 0 t ∇ φ ( x , s ) d s + O ( ϵ μ 2 ) , (101) J t ( x ) = I − ϵ μ ∫ 0 t ∇ 2 φ ‾ ( x , s ) d s + O ( ϵ μ 2 ) . (102) J_t(x)=\mathbb{I}-\epsilon_\mu\int_{0}^{t}\nabla^{2}\overline{\varphi}(x,s)\,\mathrm{d}s+\mathcal{O}(\epsilon_\mu^{2}). \tag*{(102)} J t ( x ) = I − ϵ μ ∫ 0 t ∇ 2 φ ( x , s ) d s + O ( ϵ μ 2 ) . (102) In particular,
e i k j ⋅ X t − 1 ( x ) = e i k j ⋅ x [ 1 − i ϵ μ k j ⋅ ∫ 0 t ∇ φ ‾ ( x , s ) d s ] + O ( ϵ μ 2 ) . (103) e^{i k_j\cdot X_t^{-1}(x)}=e^{i k_j\cdot x}\left[1-i\epsilon_\mu k_j\cdot\int_{0}^{t}\nabla\overline{\varphi}(x,s)\,\mathrm{d}s\right]+\mathcal{O}(\epsilon_\mu^{2}). \tag*{(103)} e i k j ⋅ X t − 1 ( x ) = e i k j ⋅ x [ 1 − i ϵ μ k j ⋅ ∫ 0 t ∇ φ ( x , s ) d s ] + O ( ϵ μ 2 ) . (103) Introduce the Lagrangian displacement and its gradient,
χ ( x , t ) : = ∫ 0 t ∇ φ ‾ ( x , s ) d s , H ( x , t ) : = ∇ χ ( x , t ) = ∫ 0 t ∇ 2 φ ‾ ( x , s ) d s . (104) \chi(x,t):=\int_{0}^{t}\nabla\overline{\varphi}(x,s)\,\mathrm{d}s,\qquad H(x,t):=\nabla\chi(x,t)=\int_{0}^{t}\nabla^{2}\overline{\varphi}(x,s)\,\mathrm{d}s. \tag*{(104)} χ ( x , t ) := ∫ 0 t ∇ φ ( x , s ) d s , H ( x , t ) := ∇ χ ( x , t ) = ∫ 0 t ∇ 2 φ ( x , s ) d s . (104) Inserting (102) and (103) into (98) gives, for the displayed positive-frequency branch,
∇ φ τ ⋅ ∇ φ ‾ = ∇ φ ‾ ⋅ ∑ j c k j Z j τ ( t ) e i k j ⋅ x [ i k j − ϵ μ ( i H ( x , t ) k j − ( k j ⋅ χ ( x , t ) ) k j ) ] + c.c. + O ( ϵ μ 2 ) . (105) \nabla\varphi^\tau\cdot\nabla\overline{\varphi}=\nabla\overline{\varphi}\cdot\sum_j c_{k_j}Z_j^\tau(t)e^{i k_j\cdot x}\left[i k_j-\epsilon_\mu\left(iH(x,t)k_j-\left(k_j\cdot\chi(x,t)\right)k_j\right)\right]+\text{c.c.}+\mathcal{O}(\epsilon_\mu^{2}). \tag*{(105)} ∇ φ τ ⋅ ∇ φ = ∇ φ ⋅ j ∑ c k j Z j τ ( t ) e i k j ⋅ x [ i k j − ϵ μ ( i H ( x , t ) k j − ( k j ⋅ χ ( x , t ) ) k j ) ] + c.c. + O ( ϵ μ 2 ) . (105) The leading term is the fixed-correlation-mode contribution, now carrying the Airy phase through Z j τ Z_j^\tau Z j τ
( ∇ φ τ ⋅ ∇ φ ‾ ) ( 0 ) = ∇ φ ‾ ⋅ ∑ j c k j Z j τ ( t ) e i k j ⋅ x i k j + c.c. (106) \left(\nabla\varphi^\tau\cdot\nabla\overline{\varphi}\right)^{(0)}=\nabla\overline{\varphi}\cdot\sum_j c_{k_j}Z_j^\tau(t)e^{i k_j\cdot x}i k_j+\text{c.c.} \tag*{(106)} ( ∇ φ τ ⋅ ∇ φ ) ( 0 ) = ∇ φ ⋅ j ∑ c k j Z j τ ( t ) e i k j ⋅ x i k j + c.c. (106) For a fixed resolved wavenumber k k k
( ∇ φ τ ⋅ ∇ φ ‾ ) k ( 0 ) = − ∑ j ( k − k j ) ⋅ k j c k j φ ‾ k − k j ( t ) Z j τ ( t ) . (107) \left(\nabla\varphi^\tau\cdot\nabla\overline{\varphi}\right)^{(0)}_k=-\sum_j(k-k_j)\cdot k_j\,c_{k_j}\overline{\varphi}_{k-k_j}(t)Z_j^\tau(t). \tag*{(107)} ( ∇ φ τ ⋅ ∇ φ ) k ( 0 ) = − j ∑ ( k − k j ) ⋅ k j c k j φ k − k j ( t ) Z j τ ( t ) . (107) The first-order correction contains two effects: deformation of the noise phase by the Lagrangian displacement and deformation of the noise wavenumber by the inverse-flow Jacobian. To express them in Fourier space, write
φ ‾ ( x , t ) = ∑ p φ ‾ p ( t ) e i p ⋅ x , \overline{\varphi}(x,t)=\sum_p\overline{\varphi}_p(t)e^{i p\cdot x}, φ ( x , t ) = p ∑ φ p ( t ) e i p ⋅ x , ∇ φ ‾ ( x , t ) = ∑ p i p φ ‾ p ( t ) e i p ⋅ x , \nabla\overline{\varphi}(x,t)=\sum_p i p\,\overline{\varphi}_p(t)e^{i p\cdot x}, ∇ φ ( x , t ) = p ∑ i p φ p ( t ) e i p ⋅ x , ∇ 2 φ ‾ ( x , t ) = ∑ p − p ⊗ p φ ‾ p ( t ) e i p ⋅ x , \nabla^2\overline{\varphi}(x,t)=\sum_p-p\otimes p\,\overline{\varphi}_p(t)e^{i p\cdot x}, ∇ 2 φ ( x , t ) = p ∑ − p ⊗ p φ p ( t ) e i p ⋅ x , χ ( x , t ) = ∑ q i q ( ∫ 0 t φ ‾ q ( s ) d s ) e i q ⋅ x . \chi(x,t)=\sum_q i q\left(\int_{0}^{t}\overline{\varphi}_q(s)\,\mathrm{d}s\right)e^{i q\cdot x}. χ ( x , t ) = q ∑ i q ( ∫ 0 t φ q ( s ) d s ) e i q ⋅ x . The phase-deformation factor is
( k j ⋅ ∇ φ ‾ ( x , t ) ) ( k j ⋅ χ ( x , t ) ) = − ∑ p , q ( k j ⋅ p ) ( k j ⋅ q ) φ ‾ p ( t ) ( ∫ 0 t φ ‾ q ( s ) d s ) e i ( p + q ) ⋅ x . (108) \left(k_j\cdot\nabla\overline{\varphi}(x,t)\right)\left(k_j\cdot\chi(x,t)\right)=-\sum_{p,q}(k_j\cdot p)(k_j\cdot q)\overline{\varphi}_p(t)\left(\int_{0}^{t}\overline{\varphi}_q(s)\,\mathrm{d}s\right)e^{i(p+q)\cdot x}. \tag*{(108)} ( k j ⋅ ∇ φ ( x , t ) ) ( k j ⋅ χ ( x , t ) ) = − p , q ∑ ( k j ⋅ p ) ( k j ⋅ q ) φ p ( t ) ( ∫ 0 t φ q ( s ) d s ) e i ( p + q ) ⋅ x . (108) After multiplication by e i k j ⋅ x e^{i k_j \cdot x} e i k j ⋅ x k k k
− ϵ μ ∑ j c k j Z j T ( t ) ∑ p + q = k − k j ( k j ⋅ p ) ( k j ⋅ q ) φ ‾ p ( t ) ∫ 0 t φ ‾ q ( s ) d s . (109) -\epsilon_\mu\sum_j c_{k_j} Z_j^T(t) \sum_{p+q=k-k_j} (k_j \cdot p)(k_j \cdot q)\overline{\varphi}_p(t) \int_0^t \overline{\varphi}_q(s)\,\mathrm{d}s. \tag*{(109)} − ϵ μ j ∑ c k j Z j T ( t ) p + q = k − k j ∑ ( k j ⋅ p ) ( k j ⋅ q ) φ p ( t ) ∫ 0 t φ q ( s ) d s . (109) Similarly,
∇ φ ‾ ( x , t ) ⋅ H ( x , t ) k j = ∑ p , q ( p ⋅ q ) ( q ⋅ k j ) φ ‾ p ( t ) ( ∫ 0 t φ ‾ q ( s ) d s ) e i ( p + q ) ⋅ x , (110) \nabla\overline{\varphi}(x,t) \cdot H(x,t)k_j = \sum_{p,q} (p \cdot q)(q \cdot k_j)\overline{\varphi}_p(t) \left(\int_0^t \overline{\varphi}_q(s)\,\mathrm{d}s\right)e^{i(p+q)\cdot x}, \tag*{(110)} ∇ φ ( x , t ) ⋅ H ( x , t ) k j = p , q ∑ ( p ⋅ q ) ( q ⋅ k j ) φ p ( t ) ( ∫ 0 t φ q ( s ) d s ) e i ( p + q ) ⋅ x , (110) and its contribution to the transport term is
− ϵ μ ∑ j c k j Z j T ( t ) ∑ p + q = k − k j ( p ⋅ q ) ( q ⋅ k j ) φ ‾ p ( t ) ∫ 0 t φ ‾ q ( s ) d s . (111) -\epsilon_\mu\sum_j c_{k_j} Z_j^T(t) \sum_{p+q=k-k_j} (p \cdot q)(q \cdot k_j)\overline{\varphi}_p(t)\int_0^t \overline{\varphi}_q(s)\,\mathrm{d}s. \tag*{(111)} − ϵ μ j ∑ c k j Z j T ( t ) p + q = k − k j ∑ ( p ⋅ q ) ( q ⋅ k j ) φ p ( t ) ∫ 0 t φ q ( s ) d s . (111) Define
Λ ( k j , p , q ) : = ( k j ⋅ p ) ( k j ⋅ q ) + ( p ⋅ q ) ( q ⋅ k j ) (112) \Lambda(k_j,p,q) := (k_j \cdot p)(k_j \cdot q) + (p \cdot q)(q \cdot k_j) \tag*{(112)} Λ ( k j , p , q ) := ( k j ⋅ p ) ( k j ⋅ q ) + ( p ⋅ q ) ( q ⋅ k j ) (112) and
S k , k j [ φ ‾ ] ( t ) : = c k j ∑ p + q = k − k j Λ ( k j , p , q ) φ ‾ p ( t ) ∫ 0 t φ ‾ q ( s ) d s . (113) \mathcal{S}_{k,k_j}[\overline{\varphi}](t) := c_{k_j}\sum_{p+q=k-k_j}\Lambda(k_j,p,q)\overline{\varphi}_p(t)\int_0^t\overline{\varphi}_q(s)\,\mathrm{d}s. \tag*{(113)} S k , k j [ φ ] ( t ) := c k j p + q = k − k j ∑ Λ ( k j , p , q ) φ p ( t ) ∫ 0 t φ q ( s ) d s . (113) The Fourier coefficient of the complete transport term can then be written as
( ∇ φ τ ⋅ ∇ φ ‾ ) k = − ∑ j ( k − k j ) ⋅ k j c k j φ ‾ k − k j Z j T − ϵ μ ∑ j Z j T S k , k j [ φ ‾ ] + O ( ϵ μ 2 ) , (114) (\nabla\varphi^\tau\cdot\nabla\overline{\varphi})_k = -\sum_j (k-k_j)\cdot k_j c_{k_j}\overline{\varphi}_{k-k_j}Z_j^T-\epsilon_\mu\sum_j Z_j^T\mathcal{S}_{k,k_j}[\overline{\varphi}]+\mathcal{O}(\epsilon_\mu^2), \tag*{(114)} ( ∇ φ τ ⋅ ∇ φ ) k = − j ∑ ( k − k j ) ⋅ k j c k j φ k − k j Z j T − ϵ μ j ∑ Z j T S k , k j [ φ ] + O ( ϵ μ 2 ) , (114) with the conjugate noise branch understood.
Using the Airy variable
a k = 1 2 ( η k + i G k [ 0 ] ω k φ ‾ k ) , φ ‾ p = − i ω p G p [ 0 ] ( a p − a − p ∗ ) , (115) a_k=\frac{1}{2}\left(\eta_k+\frac{iG_k[0]}{\omega_k}\overline{\varphi}_k\right),\qquad\overline{\varphi}_p=-\frac{i\omega_p}{G_p[0]}(a_p-a^*_{-p}), \tag*{(115)} a k = 2 1 ( η k + ω k i G k [ 0 ] φ k ) , φ p = − G p [ 0 ] i ω p ( a p − a − p ∗ ) , (115) and projecting (114) onto a k a_k a k
∂ t a k = − i ω k a k + 1 2 ∑ j C k , k j ( a k − k j − a − ( k − k j ) ∗ ) Z j T ( t ) − ϵ μ R k [ a ] ( t ) + f k ( t ) + O ( ϵ μ 2 ) , (116) \partial_t a_k=-i\omega_k a_k+\frac{1}{2}\sum_j C_{k,k_j}\left(a_{k-k_j}-a^*_{-(k-k_j)}\right)Z_j^T(t)-\epsilon_\mu\mathcal{R}_k[a](t)+f_k(t)+\mathcal{O}(\epsilon_\mu^2), \tag*{(116)} ∂ t a k = − i ω k a k + 2 1 j ∑ C k , k j ( a k − k j − a − ( k − k j ) ∗ ) Z j T ( t ) − ϵ μ R k [ a ] ( t ) + f k ( t ) + O ( ϵ μ 2 ) , (116) where C k , k j C_{k,k_j} C k , k j
R k [ a ] ( t ) : = ∑ j Z j T ( t ) R k , k j [ a ] ( t ) , (117) \mathcal{R}_k[a](t):=\sum_j Z_j^T(t)\mathcal{R}_{k,k_j}[a](t), \tag*{(117)} R k [ a ] ( t ) := j ∑ Z j T ( t ) R k , k j [ a ] ( t ) , (117) R k , k j [ a ] ( t ) : = − i G k [ 0 ] 2 ω k S k , k j [ φ ‾ ( a ) ] ( t ) . (118) \mathcal{R}_{k,k_j}[a](t):=-\frac{iG_k[0]}{2\omega_k}\mathcal{S}_{k,k_j}[\overline{\varphi}(a)](t). \tag*{(118)} R k , k j [ a ] ( t ) := − 2 ω k i G k [ 0 ] S k , k j [ φ ( a )] ( t ) . (118) The factor in (118) results from the projection of the Bernoulli equation onto the Airy variable; it also fixes the sign convention in (116).
Under the same rotating-wave approximation as in the fixed-mode case, the counter-rotating term proportional to a − ( k − k j ) ∗ a^*_{-(k-k_j)} a − ( k − k j ) ∗
∂ t a k = − i ω k a k + 1 2 ∑ j C k , k j a k − k j Z j T ( t ) − ϵ μ R k [ a ] ( t ) + f k ( t ) + O ( ϵ μ 2 ) . (119) \partial_t a_k=-i\omega_k a_k+\frac{1}{2}\sum_j C_{k,k_j}a_{k-k_j}Z_j^T(t)-\epsilon_\mu\mathcal{R}_k[a](t)+f_k(t)+\mathcal{O}(\epsilon_\mu^2). \tag*{(119)} ∂ t a k = − i ω k a k + 2 1 j ∑ C k , k j a k − k j Z j T ( t ) − ϵ μ R k [ a ] ( t ) + f k ( t ) + O ( ϵ μ 2 ) . (119) The additive forcing f k f_k f k
The Airy carrier is particularly transparent in the interaction representation b k ( t ) = a k ( t ) e i ω k t b_k(t)=a_k(t)e^{i\omega_k t} b k ( t ) = a k ( t ) e i ω k t
e i ( ω k − ω k − k j ) t e − i ω k j t = e i ( ω k − ω k − k j − ω k j ) t . (120) e^{i(\omega_k-\omega_{k-k_j})t}e^{-i\omega_{k_j}t}=e^{i(\omega_k-\omega_{k-k_j}-\omega_{k_j})t}. \tag*{(120)} e i ( ω k − ω k − k j ) t e − i ω k j t = e i ( ω k − ω k − k j − ω k j ) t . (120) More generally, (96) replaces each bare frequency mismatch Ω \Omega Ω Ω − ω k j \Omega-\omega_{k_j} Ω − ω k j
Ω k , p , q , k j = ω p + ω q − ω k − ω k j . (121) \Omega_{k,p,q,k_j}=\omega_p+\omega_q-\omega_k-\omega_{k_j}. \tag*{(121)} Ω k , p , q , k j = ω p + ω q − ω k − ω k j . (121) The term R k [ a ] \mathcal{R}_k[a] R k [ a ] ϵ μ \epsilon_\mu ϵ μ
Kinetic theory for transported Airy-carried noise
We now derive the contribution of the transported noise modes to the wave-action equation. As in the fixed-mode case, we introduce the interaction variables
b k ( t ) = a k ( t ) e i ω k t , N k ( t ) = E [ ∣ b k ( t ) ∣ 2 ] . (122) b_k(t)=a_k(t)e^{i\omega_k t},\qquad N_k(t)=\mathbb{E}\left[|b_k(t)|^2\right]. \tag*{(122)} b k ( t ) = a k ( t ) e i ω k t , N k ( t ) = E [ ∣ b k ( t ) ∣ 2 ] . (122) The temporal coefficient of an unresolved positive-frequency Airy mode is written
Z ℓ τ ( t ) = e − i ω ℓ t W ℓ τ ( t ) , ω ℓ 2 = g G ℓ [ 0 ] , (123) Z_\ell^\tau(t)=e^{-i\omega_\ell t}W_\ell^\tau(t),\qquad\omega_\ell^2=gG_\ell[0], \tag*{(123)} Z ℓ τ ( t ) = e − i ω ℓ t W ℓ τ ( t ) , ω ℓ 2 = g G ℓ [ 0 ] , (123) where W ℓ τ W_\ell^\tau W ℓ τ Z ℓ τ Z_\ell^\tau Z ℓ τ
d W ℓ = − γ ℓ W ℓ d t + σ ℓ d B ℓ C ( t ) , (124) \mathrm{d}W_\ell=-\gamma_\ell W_\ell\,\mathrm{d}t+\sigma_\ell\,\mathrm{d}B_\ell^{\mathsf{C}}(t), \tag*{(124)} d W ℓ = − γ ℓ W ℓ d t + σ ℓ d B ℓ C ( t ) , (124) and has stationary covariance
E [ W ℓ ( t ) W m ∗ ( s ) ] = δ ℓ m R ℓ ( t − s ) , R ℓ ( r ) = σ ℓ 2 2 γ ℓ e − γ ℓ ∣ r ∣ , (125) \mathbb{E}[W_\ell(t)W_m^*(s)]=\delta_{\ell m}R_\ell(t-s),\qquad R_\ell(r)=\frac{\sigma_\ell^2}{2\gamma_\ell}e^{-\gamma_\ell|r|}, \tag*{(125)} E [ W ℓ ( t ) W m ∗ ( s )] = δ ℓ m R ℓ ( t − s ) , R ℓ ( r ) = 2 γ ℓ σ ℓ 2 e − γ ℓ ∣ r ∣ , (125) E [ W ℓ ( t ) W m ( s ) ] = 0. (126) \mathbb{E}[W_\ell(t)W_m(s)]=0. \tag*{(126)} E [ W ℓ ( t ) W m ( s )] = 0. (126) Equivalently, Z ℓ Z_\ell Z ℓ
d Z ℓ = − ( γ ℓ + i ω ℓ ) Z ℓ d t + σ ℓ d B ~ ℓ C ( t ) , \mathrm{d}Z_\ell=-(\gamma_\ell+i\omega_\ell)Z_\ell\,\mathrm{d}t+\sigma_\ell\,\mathrm{d}\widetilde{B}_\ell^{\mathsf{C}}(t), d Z ℓ = − ( γ ℓ + i ω ℓ ) Z ℓ d t + σ ℓ d B ℓ C ( t ) , and
E [ Z ℓ ( t ) Z m ∗ ( s ) ] = δ ℓ m R ℓ ( t − s ) e − i ω ℓ ( t − s ) . (127) \mathbb{E}[Z_\ell(t)Z_m^*(s)]=\delta_{\ell m}R_\ell(t-s)e^{-i\omega_\ell(t-s)}. \tag*{(127)} E [ Z ℓ ( t ) Z m ∗ ( s )] = δ ℓ m R ℓ ( t − s ) e − i ω ℓ ( t − s ) . (127) Thus, contractions of the temporal noise must pair Z ℓ Z_\ell Z ℓ Z ℓ ∗ Z_\ell^* Z ℓ ∗ E [ Z ℓ ( t ) Z m ( s ) ] \mathbb{E}[Z_\ell(t)Z_m(s)] E [ Z ℓ ( t ) Z m ( s )]
The linear stochastic-scattering term produces the same master operator as in the previous section, with the OU spectrum now shifted by the Airy frequency. For the positive-frequency branch, its rate is of the form
G k , k j A = 1 4 ∣ C k , k j ∣ 2 R ^ j ( ω k − ω k − k j − ω k j ) , (128) \mathcal{G}^{\mathrm{A}}_{k,k_j} = \frac{1}{4}|C_{k,k_j}|^2\widehat{R}_j(\omega_k-\omega_{k-k_j}-\omega_{k_j}), \tag*{(128)} G k , k j A = 4 1 ∣ C k , k j ∣ 2 R j ( ω k − ω k − k j − ω k j ) , (128) where
R ^ j ( Ω ) : = ∫ R R j ( s ) e i Ω s d s = 2 Re ∫ 0 ∞ R j ( s ) e i Ω s d s . (129) \widehat{R}_j(\Omega) := \int_{\mathbb{R}} R_j(s)e^{i\Omega s}\,\mathrm{d}s = 2\operatorname{Re}\int_0^\infty R_j(s)e^{i\Omega s}\,\mathrm{d}s. \tag*{(129)} R j ( Ω ) := ∫ R R j ( s ) e i Ω s d s = 2 Re ∫ 0 ∞ R j ( s ) e i Ω s d s . (129) The conjugate carrier gives the corresponding negative-frequency branch. With both branches included, the linear scattering operator can be written in the gain-loss transition-rate form
( Q s c N ) k = ∑ j [ G k , k j A N k − k j − G k + k j , k j A N k ] . (130) (\mathcal{Q}_{\mathrm{sc}}N)_k = \sum_j \left[\mathcal{G}^{\mathrm{A}}_{k,k_j}N_{k-k_j}-\mathcal{G}^{\mathrm{A}}_{k+k_j,k_j}N_k\right]. \tag*{(130)} ( Q sc N ) k = j ∑ [ G k , k j A N k − k j − G k + k j , k j A N k ] . (130) If the incoming and outgoing rates coincide at leading order G k , k j A = G k + k j , k j A \mathcal{G}^{\mathrm{A}}_{k,k_j}=\mathcal{G}^{\mathrm{A}}_{k+k_j,k_j} G k , k j A = G k + k j , k j A
∑ j G k , k j A ( N k − k j − N k ) . \sum_j \mathcal{G}^{\mathrm{A}}_{k,k_j}\left(N_{k-k_j}-N_k\right). j ∑ G k , k j A ( N k − k j − N k ) . We next consider the quadratic correction R k [ a ] \mathcal{R}_k[a] R k [ a ]
∫ 0 t φ ‾ q ( s ) d s , φ ‾ q ( s ) = − i ω q G q [ 0 ] a q ( s ) + counter-rotating component . (131) \int_0^t \overline{\varphi}_q(s)\,\mathrm{d}s,\qquad\overline{\varphi}_q(s)=-\frac{i\omega_q}{G_q[0]}a_q(s)+\text{counter-rotating component}. \tag*{(131)} ∫ 0 t φ q ( s ) d s , φ q ( s ) = − G q [ 0 ] i ω q a q ( s ) + counter-rotating component . (131) An additional approximation is required before this nonlocal term can be represented by a time-homogeneous kinetic equation. If b q b_q b q T N T_N T N ω q T N ≫ 1 \omega_qT_N\gg1 ω q T N ≫ 1
∫ 0 t φ ‾ q ( s ) d s = a q ( t ) G q [ 0 ] − b q ( 0 ) G q [ 0 ] + O ( ( ω q T N ) − 1 ) , (132) \int_0^t \overline{\varphi}_q(s)\,\mathrm{d}s=\frac{a_q(t)}{G_q[0]}-\frac{b_q(0)}{G_q[0]}+\mathcal{O}\left((\omega_qT_N)^{-1}\right), \tag*{(132)} ∫ 0 t φ q ( s ) d s = G q [ 0 ] a q ( t ) − G q [ 0 ] b q ( 0 ) + O ( ( ω q T N ) − 1 ) , (132) for the positive-frequency component. We discard the lower-limit term, which retains an explicit dependence on the arbitrary initial time and therefore generates nonstationary initial-layer contributions involving b q ( 0 ) b_q(0) b q ( 0 ) a q ( t ) a_q(t) a q ( t )
The resonance with two incoming resolved waves is carried by the complex-conjugate Airy branch of the real noise. Let ℓ \ell ℓ e − i ℓ ⋅ x + i ω ℓ t e^{-i\ell\cdot x+i\omega_\ell t} e − i ℓ ⋅ x + i ω ℓ t
p + q = k + ℓ . (133) p+q=k+\ell. \tag*{(133)} p + q = k + ℓ . (133) In the notation used in the preceding spatial calculation, this corresponds to the relabelling ℓ = − k j \ell=-k_j ℓ = − k j
Λ ℓ − ( p , q ) : = Λ ( − ℓ , p , q ) = ( ℓ ⋅ p ) ( ℓ ⋅ q ) − ( p ⋅ q ) ( q ⋅ ℓ ) . (134) \Lambda_\ell^{-}(p,q):=\Lambda(-\ell,p,q)=(\ell\cdot p)(\ell\cdot q)-(p\cdot q)(q\cdot\ell). \tag*{(134)} Λ ℓ − ( p , q ) := Λ ( − ℓ , p , q ) = ( ℓ ⋅ p ) ( ℓ ⋅ q ) − ( p ⋅ q ) ( q ⋅ ℓ ) . (134) Substituting the positive-frequency component
φ ‾ p ( t ) = − i ω p G p [ 0 ] a p ( t ) \overline{\varphi}_p(t)=-\frac{i\omega_p}{G_p[0]}a_p(t) φ p ( t ) = − G p [ 0 ] i ω p a p ( t ) and the local-displacement approximation
∫ 0 t φ ‾ q ( s ) d s ≃ a q ( t ) G q [ 0 ] \int_0^t \overline{\varphi}_q(s)\,\mathrm{d}s \simeq\frac{a_q(t)}{G_q[0]} ∫ 0 t φ q ( s ) d s ≃ G q [ 0 ] a q ( t ) into the equation for a k a_k a k R k [ a ] \mathcal{R}_k[a] R k [ a ] i G k [ 0 ] / ( 2 ω k ) iG_k[0]/(2\omega_k) i G k [ 0 ] / ( 2 ω k ) (118) . The retained positive-frequency contribution to the amplitude equation for R k [ a ] \mathcal{R}_k[a] R k [ a ]
∂ t a k ∣ R = ϵ μ ∑ ℓ ∑ p + q = k + ℓ c ℓ ∗ W ℓ T , ∗ ( t ) e i ω ℓ t A k ℓ ( p , q ) a p ( t ) a q ( t ) , (135) \left.\partial_t a_k\right|_{\mathcal{R}}=\epsilon_\mu\sum_{\ell}\sum_{p+q=k+\ell}c_\ell^{*}W_\ell^{\mathsf{T},*}(t)e^{i\omega_\ell t}A_{k\ell}(p,q)a_p(t)a_q(t), \tag*{(135)} ∂ t a k ∣ R = ϵ μ ℓ ∑ p + q = k + ℓ ∑ c ℓ ∗ W ℓ T , ∗ ( t ) e i ω ℓ t A k ℓ ( p , q ) a p ( t ) a q ( t ) , (135) where
A k ℓ ( p , q ) : = G k [ 0 ] ω p 2 ω k G p [ 0 ] G q [ 0 ] Λ ℓ ( p , q ) . (136) A_{k\ell}(p,q):=\frac{G_k[0]\omega_p}{2\omega_kG_p[0]G_q[0]}\Lambda_\ell^{ }(p,q). \tag*{(136)} A k ℓ ( p , q ) := 2 ω k G p [ 0 ] G q [ 0 ] G k [ 0 ] ω p Λ ℓ ( p , q ) . (136) Here the conjugate Airy carrier has spatial–temporal dependence e − i ℓ ⋅ x + i ω ℓ t e^{-i\ell\cdot x+i\omega_\ell t} e − i ℓ ⋅ x + i ω ℓ t k k k
p + q = k + ℓ . p+q=k+\ell. p + q = k + ℓ . The convolution sum in (135) is taken over ordered pairs. Since a p a q = a q a p a_pa_q=a_qa_p a p a q = a q a p p p p q q q
∑ p + q = k + ℓ A k ℓ ( p , q ) a p a q = 1 2 ∑ p + q = k + ℓ [ A k ℓ ( p , q ) + A k ℓ ( q , p ) ] a p a q = ∑ p + q = k + ℓ V k ℓ p q a p a q , (137) \begin{aligned}
\sum_{p+q=k+\ell}A_{k\ell}(p,q)a_pa_q
=\frac{1}{2}\sum_{p+q=k+\ell}\left[A_{k\ell}(p,q)+A_{k\ell}(q,p)\right]a_pa_q \\
&=\sum_{p+q=k+\ell}V_{k\ell pq}a_pa_q, \tag*{(137)}
\end{aligned} p + q = k + ℓ ∑ A k ℓ ( p , q ) a p a q = 2 1 p + q = k + ℓ ∑ [ A k ℓ ( p , q ) + A k ℓ ( q , p ) ] a p a q = p + q = k + ℓ ∑ V k ℓ pq a p a q , (137) where
V k ℓ p q : = G k [ 0 ] 4 ω k G p [ 0 ] G q [ 0 ] [ ω p Λ ℓ ( p , q ) + ω q Λ ℓ ( q , p ) ] . (138) V_{k\ell pq}:=\frac{G_k[0]}{4\omega_kG_p[0]G_q[0]}\left[\omega_p\Lambda_\ell^{ }(p,q)+\omega_q\Lambda_\ell^{ }(q,p)\right]. \tag*{(138)} V k ℓ pq := 4 ω k G p [ 0 ] G q [ 0 ] G k [ 0 ] [ ω p Λ ℓ ( p , q ) + ω q Λ ℓ ( q , p ) ] . (138) It follows directly that
V k ℓ p q = V k ℓ q p . V_{k\ell pq}=V_{k\ell qp}. V k ℓ pq = V k ℓ q p . The additional factor 1 / 2 1/2 1/2 (138) therefore results from symmetrization of the ordered convolution; the factor G k [ 0 ] / ( 2 ω k ) G_k[0]/(2\omega_k) G k [ 0 ] / ( 2 ω k )
The corresponding contribution to the complete amplitude equation can thus be written as
∂ t a k = − i ω k a k + ϵ μ ∑ ℓ ∑ p + q = k + ℓ c ℓ ∗ V k ℓ p q W ℓ T , ∗ ( t ) e i ω ℓ t a p ( t ) a q ( t ) + ⋯ , (139) \partial_t a_k=-i\omega_k a_k+\epsilon_\mu\sum_{\ell}\sum_{p+q=k+\ell}c_\ell^{*}V_{k\ell pq}W_\ell^{\mathsf{T},*}(t)e^{i\omega_\ell t}a_p(t)a_q(t)+\cdots, \tag*{(139)} ∂ t a k = − i ω k a k + ϵ μ ℓ ∑ p + q = k + ℓ ∑ c ℓ ∗ V k ℓ pq W ℓ T , ∗ ( t ) e i ω ℓ t a p ( t ) a q ( t ) + ⋯ , (139) where the ellipsis denotes the linear scattering, additive forcing, counter-rotating terms, and initial-layer contributions. As previously, introducing b k ( t ) = a k ( t ) e i ω k t b_k(t)=a_k(t)e^{i\omega_k t} b k ( t ) = a k ( t ) e i ω k t (139) and gives
∂ t b k ∣ R = ϵ μ ∑ ℓ ∑ p + q = k + ℓ c ℓ ∗ V k ℓ p q W ℓ T , ∗ ( t ) b p ( t ) b q ( t ) e − i Ω k ℓ p q t , (140) \left.\partial_t b_k\right|_{\mathcal{R}}=\epsilon_\mu\sum_{\ell}\sum_{p+q=k+\ell}c_\ell^{*}V_{k\ell pq}W_\ell^{\mathsf{T},*}(t)b_p(t)b_q(t)e^{-i\Omega_{k\ell pq}t}, \tag*{(140)} ∂ t b k ∣ R = ϵ μ ℓ ∑ p + q = k + ℓ ∑ c ℓ ∗ V k ℓ pq W ℓ T , ∗ ( t ) b p ( t ) b q ( t ) e − i Ω k ℓ pq t , (140) where
Ω k ℓ p q : = ω p + ω q − ω k − ω ℓ . (141) \Omega_{k\ell pq}:=\omega_p+\omega_q-\omega_k-\omega_\ell. \tag*{(141)} Ω k ℓ pq := ω p + ω q − ω k − ω ℓ . (141) The wavenumber relation p + q = k + ℓ p+q=k+\ell p + q = k + ℓ Ω k ℓ p q = 0 \Omega_{k\ell pq}=0 Ω k ℓ pq = 0 2 ↔ 2 2\leftrightarrow2 2 ↔ 2 ℓ \ell ℓ
The wave-action balance associated with (140) starts from
∂ t N k ∣ R = 2 ϵ μ Re ∑ ℓ ∑ p + q = k + ℓ c ℓ ∗ V k ℓ p q ( ? ) × E [ W ℓ τ , ∗ ( t ) b p ( t ) b q ( t ) b k ∗ ( t ) ] e − i Ω k ℓ p q t . (142) \partial_t N_k\big|_{\mathcal{R}} = 2\epsilon_\mu\operatorname{Re} \sum_{\ell} \sum_{p+q=k+\ell} c_\ell^* V_{k\ell pq}^{(\,?\,)} \times\mathbb{E}\left[W_\ell^{\tau,*}(t)b_p(t)b_q(t)b_k^*(t)\right]e^{-i\Omega_{k\ell pq}t}. \tag*{(142)} ∂ t N k R = 2 ϵ μ Re ℓ ∑ p + q = k + ℓ ∑ c ℓ ∗ V k ℓ pq ( ? ) × E [ W ℓ τ , ∗ ( t ) b p ( t ) b q ( t ) b k ∗ ( t ) ] e − i Ω k ℓ pq t . (142) We assume weak coupling, separation between the OU correlation time and the kinetic evolution time, and the quasi-Gaussian closure
E [ b p b p ′ ∗ ] = N p δ p p ′ , E [ b p b p ′ ] = 0 , (143) \mathbb{E}[b_p b_{p'}^*] = N_p\delta_{pp'}, \qquad\mathbb{E}[b_p b_{p'}] = 0, \tag*{(143)} E [ b p b p ′ ∗ ] = N p δ p p ′ , E [ b p b p ′ ] = 0 , (143) E [ b p b q b p ′ ∗ b q ′ ∗ ] = N p N q ( δ p p ′ δ q q ′ + δ p q ′ δ q p ′ ) . (144) \mathbb{E}[b_p b_q b_{p'}^* b_{q'}^*] = N_pN_q\left(\delta_{pp'}\delta_{qq'}+\delta_{pq'}\delta_{qp'}\right). \tag*{(144)} E [ b p b q b p ′ ∗ b q ′ ∗ ] = N p N q ( δ p p ′ δ q q ′ + δ p q ′ δ q p ′ ) . (144) Under this closure, all third-order amplitude moments vanish. In particular, the cross term between the linear scattering operator and R k \mathcal{R}_k R k R k \mathcal{R}_k R k ϵ μ 2 \epsilon_\mu^2 ϵ μ 2 b k ∗ b_k^* b k ∗ b p b_p b p b q b_q b q b k ∗ b_k^* b k ∗ b p b_p b p b q b_q b q
The direct contraction contributing to mode k k k
Q R ( + ) [ N ] ∣ k = ϵ μ 2 ∑ ℓ ∑ p + q = k + ℓ K k ℓ p q ( k ) N p N q , (145) \mathcal{Q}_{\mathcal{R}}^{(+)}[N]\big|_k = \epsilon_\mu^2 \sum_{\ell}\sum_{p+q=k+\ell}\mathcal{K}_{k\ell pq}^{(k)}N_pN_q, \tag*{(145)} Q R ( + ) [ N ] k = ϵ μ 2 ℓ ∑ p + q = k + ℓ ∑ K k ℓ pq ( k ) N p N q , (145) where the two Wick pairings have been included in the nonnegative rate
K k ℓ p q ( k ) : = 2 ∣ c ℓ ∣ 2 ∣ V k ℓ p q ∣ 2 R ^ ℓ ( Ω k ℓ p q ) ≥ 0. (146) \mathcal{K}_{k\ell pq}^{(k)} := 2|c_\ell|^2|V_{k\ell pq}|^2\widehat{R}_\ell(\Omega_{k\ell pq}) \ge0. \tag*{(146)} K k ℓ pq ( k ) := 2∣ c ℓ ∣ 2 ∣ V k ℓ pq ∣ 2 R ℓ ( Ω k ℓ pq ) ≥ 0. (146) For the OU covariance (125),
R ^ ℓ ( Ω ) = σ ℓ 2 γ ℓ 2 + Ω 2 . (147) \widehat{R}_\ell(\Omega) = \frac{\sigma_\ell^2}{\gamma_\ell^2+\Omega^2}. \tag*{(147)} R ℓ ( Ω ) = γ ℓ 2 + Ω 2 σ ℓ 2 . (147) The imaginary part of the one-sided integral ∫ 0 ∞ R ℓ ( s ) e i Ω s d s \int_0^\infty R_\ell(s)e^{i\Omega s}\,\mathrm{d}s ∫ 0 ∞ R ℓ ( s ) e i Ω s d s
The remaining response terms cannot be obtained by the change ( p , q ) ↦ ( − p , − q ) (p,q)\mapsto(-p,-q) ( p , q ) ↦ ( − p , − q )
Λ ℓ − ( − p , − q ) = Λ ( − ℓ , − p , − q ) = Λ ( ℓ , p , q ) , (148) \Lambda_\ell^{-}(-p,-q)=\Lambda(-\ell,-p,-q)=\Lambda(\ell,p,q), \tag*{(148)} Λ ℓ − ( − p , − q ) = Λ ( − ℓ , − p , − q ) = Λ ( ℓ , p , q ) , (148) which is not equal, in general, to − Λ ℓ − ( p , q ) -\Lambda_\ell^{-}(p,q) − Λ ℓ − ( p , q ) b p b_p b p b q b_q b q p + q = k + ℓ p+q=k+\ell p + q = k + ℓ
∂ t b p ∣ R = 2 ϵ μ c ℓ V k ℓ p q ( p ) W ℓ τ ( t ) b k ( t ) b q ∗ ( t ) e i Ω k ℓ p q t , (149) \partial_t b_p\big|_{\mathcal{R}} = 2\epsilon_\mu c_\ell V_{k\ell pq}^{(p)}W_\ell^\tau(t)b_k(t)b_q^*(t)e^{i\Omega_{k\ell pq}t}, \tag*{(149)} ∂ t b p R = 2 ϵ μ c ℓ V k ℓ pq ( p ) W ℓ τ ( t ) b k ( t ) b q ∗ ( t ) e i Ω k ℓ pq t , (149) ∂ t b q ∣ R = 2 ϵ μ c ℓ V k ℓ p q ( q ) W ℓ τ ( t ) b k ( t ) b p ∗ ( t ) e i Ω k ℓ p q t , (150) \partial_t b_q\big|_{\mathcal{R}} = 2\epsilon_\mu c_\ell V_{k\ell pq}^{(q)}W_\ell^\tau(t)b_k(t)b_p^*(t)e^{i\Omega_{k\ell pq}t}, \tag*{(150)} ∂ t b q R = 2 ϵ μ c ℓ V k ℓ pq ( q ) W ℓ τ ( t ) b k ( t ) b p ∗ ( t ) e i Ω k ℓ pq t , (150) where
V k ℓ p q ( p ) : = G p [ 0 ] 4 ω p G k [ 0 ] G q [ 0 ] [ ω k Λ ( ℓ , k , − q ) − ω q Λ ( ℓ , − q , k ) ] , (151) V_{k\ell pq}^{(p)} := \frac{G_p[0]}{4\omega_pG_k[0]G_q[0]}\left[\omega_k\Lambda(\ell,k,-q)-\omega_q\Lambda(\ell,-q,k)\right], \tag*{(151)} V k ℓ pq ( p ) := 4 ω p G k [ 0 ] G q [ 0 ] G p [ 0 ] [ ω k Λ ( ℓ , k , − q ) − ω q Λ ( ℓ , − q , k ) ] , (151) V k ℓ p q ( q ) : = G q [ 0 ] 4 ω q G k [ 0 ] G p [ 0 ] [ ω k Λ ( ℓ , k , − p ) − ω p Λ ( ℓ , − p , k ) ] . (152) V_{k\ell pq}^{(q)} := \frac{G_q[0]}{4\omega_qG_k[0]G_p[0]}\left[\omega_k\Lambda(\ell,k,-p)-\omega_p\Lambda(\ell,-p,k)\right]. \tag*{(152)} V k ℓ pq ( q ) := 4 ω q G k [ 0 ] G p [ 0 ] G q [ 0 ] [ ω k Λ ( ℓ , k , − p ) − ω p Λ ( ℓ , − p , k ) ] . (152) For the equation of b p b_p b p b k b q ∗ b_k b_q^* b k b q ∗ ( k , − q ) (k,-q) ( k , − q ) ( − q , k ) (-q,k) ( − q , k )
G p [ 0 ] 2 ω p G k [ 0 ] G q [ 0 ] [ ω k Λ ( ℓ , k , − q ) − ω q Λ ( ℓ , − q , k ) ] = 2 V k ℓ p q ( p ) . \frac{G_p[0]}{2\omega_p G_k[0]G_q[0]}\left[\omega_k\Lambda(\ell,k,-q)-\omega_q\Lambda(\ell,-q,k)\right]=2V_{k\ell pq}^{(p)}. 2 ω p G k [ 0 ] G q [ 0 ] G p [ 0 ] [ ω k Λ ( ℓ , k , − q ) − ω q Λ ( ℓ , − q , k ) ] = 2 V k ℓ pq ( p ) . Here the factor two restores the two ordered placements of the pair ( k , − q ) (k,-q) ( k , − q ) ( k , − p ) (k,-p) ( k , − p ) ϕ ‾ − q \overline{\phi}_{-q} ϕ − q ϕ ‾ − p \overline{\phi}_{-p} ϕ − p
Inserting (149) and (150) into the mixed moment in (142) gives the signed response coefficients
K k ℓ p q ( p ) : = − 2 ∣ c ℓ ∣ 2 Re ( V k ℓ p q V k ℓ p q ( p ) ) R ^ ℓ ( Ω k ℓ p q ) , (153) \mathcal{K}_{k\ell pq}^{(p)}:=-2|c_\ell|^2\operatorname{Re}\left(V_{k\ell pq}V_{k\ell pq}^{(p)}\right)\widehat{R}_\ell(\Omega_{k\ell pq}), \tag*{(153)} K k ℓ pq ( p ) := − 2∣ c ℓ ∣ 2 Re ( V k ℓ pq V k ℓ pq ( p ) ) R ℓ ( Ω k ℓ pq ) , (153) K k ℓ p q ( q ) : = − 2 ∣ c ℓ ∣ 2 Re ( V k ℓ p q V k ℓ p q ( q ) ) R ^ ℓ ( Ω k ℓ p q ) . (154) \mathcal{K}_{k\ell pq}^{(q)}:=-2|c_\ell|^2\operatorname{Re}\left(V_{k\ell pq}V_{k\ell pq}^{(q)}\right)\widehat{R}_\ell(\Omega_{k\ell pq}). \tag*{(154)} K k ℓ pq ( q ) := − 2∣ c ℓ ∣ 2 Re ( V k ℓ pq V k ℓ pq ( q ) ) R ℓ ( Ω k ℓ pq ) . (154) The factor two has the same origin as in (146): the two ordered placements of the resolved pair give identical contractions after symmetrization. In contrast to K ( k ) \mathcal{K}^{(k)} K ( k ) K ( p ) \mathcal{K}^{(p)} K ( p ) K ( q ) \mathcal{K}^{(q)} K ( q )
The kinetic operator obtained directly from the three Duhamel responses is therefore
( Q R N ) k = ϵ μ 2 ∑ ℓ ∑ p + q = k + ℓ [ K k ℓ p q ( k ) N p N q − K k ℓ p q ( p ) N k N q − K k ℓ p q ( q ) N k N p ] . (155) (Q_{\mathcal{R}}N)_k=\epsilon_\mu^2\sum_\ell\sum_{p+q=k+\ell}\left[\mathcal{K}_{k\ell pq}^{(k)}N_pN_q-\mathcal{K}_{k\ell pq}^{(p)}N_kN_q-\mathcal{K}_{k\ell pq}^{(q)}N_kN_p\right]. \tag*{(155)} ( Q R N ) k = ϵ μ 2 ℓ ∑ p + q = k + ℓ ∑ [ K k ℓ pq ( k ) N p N q − K k ℓ pq ( p ) N k N q − K k ℓ pq ( q ) N k N p ] . (155) This is a bath-mediated quadratic response operator. In general, it cannot be written in a single-kernel gain–loss form, since no equality between its three kernels has been imposed. If, for each admissible quartet ( k , ℓ , p , q ) (k,\ell,p,q) ( k , ℓ , p , q )
K k ℓ p q ( p ) = K k ℓ p q ( q ) = K k ℓ p q ( k ) , (156) \mathcal{K}_{k\ell pq}^{(p)}=\mathcal{K}_{k\ell pq}^{(q)}=\mathcal{K}_{k\ell pq}^{(k)}, \tag*{(156)} K k ℓ pq ( p ) = K k ℓ pq ( q ) = K k ℓ pq ( k ) , (156) then (155) reduces to the quadratic gain–loss bracket
( Q R N ) k = ϵ μ 2 ∑ ℓ ∑ p + q = k + ℓ K k ℓ p q ( k ) [ N p N q − N k ( N p + N q ) ] . (157) (Q_{\mathcal{R}}N)_k=\epsilon_\mu^2\sum_\ell\sum_{p+q=k+\ell}\mathcal{K}_{k\ell pq}^{(k)}\left[N_pN_q-N_k(N_p+N_q)\right]. \tag*{(157)} ( Q R N ) k = ϵ μ 2 ℓ ∑ p + q = k + ℓ ∑ K k ℓ pq ( k ) [ N p N q − N k ( N p + N q ) ] . (157) Condition (156) expresses equality of the elementary production and depletion rates associated with each interaction quartet. It does not imply that the corresponding action transfers balance, since they are weighted by the generally different products N p N q N_pN_q N p N q N k N q N_kN_q N k N q N k N p N_kN_p N k N p
A sufficient, but stronger and phase-convention-dependent, condition at the level of the vertices is
V k ℓ p q ( p ) = V k ℓ p q ( q ) = − V k ℓ p q ∗ . (158) V_{k\ell pq}^{(p)}=V_{k\ell pq}^{(q)}=-V_{k\ell pq}^*. \tag*{(158)} V k ℓ pq ( p ) = V k ℓ pq ( q ) = − V k ℓ pq ∗ . (158) Relations of this type arise in a closed, canonically normalized four-wave Hamiltonian system from the reality and permutation symmetries of a single quartic Hamiltonian. They have not been established for the present bath-mediated stochastic interaction and are not assumed below. No factor N ℓ N_\ell N ℓ ∣ c ℓ ∣ 2 R ^ ℓ |c_\ell|^2\widehat{R}_\ell ∣ c ℓ ∣ 2 R ℓ N ℓ N_\ell N ℓ
Combining the linear scattering, the transported-mode correction, the additive injection I k \mathcal{I}_k I k
∂ t N k = ∑ j [ G k , k j A N k − k j − G k + k j , k j A N k ] + ϵ μ 2 ∑ ℓ ∑ p + q = k + ℓ [ K k ℓ p q ( k ) N p N q − K k ℓ p q ( p ) N k N q − K k ℓ p q ( q ) N k N p ] + I k − 2 μ N k . (159) \begin{aligned}
\partial_tN_k={}&\sum_j\left[\mathcal{G}_{k,k_j}^{\mathrm{A}}N_{k-k_j}-\mathcal{G}_{k+k_j,k_j}^{\mathrm{A}}N_k\right]\\
&+\epsilon_\mu^2\sum_\ell\sum_{p+q=k+\ell}\left[\mathcal{K}_{k\ell pq}^{(k)}N_pN_q-\mathcal{K}_{k\ell pq}^{(p)}N_kN_q-\mathcal{K}_{k\ell pq}^{(q)}N_kN_p\right]+\mathcal{I}_k-2\mu N_k. \tag*{(159)}
\end{aligned} ∂ t N k = j ∑ [ G k , k j A N k − k j − G k + k j , k j A N k ] + ϵ μ 2 ℓ ∑ p + q = k + ℓ ∑ [ K k ℓ pq ( k ) N p N q − K k ℓ pq ( p ) N k N q − K k ℓ pq ( q ) N k N p ] + I k − 2 μ N k . (159) With the normalization used in the preceding additive-forcing calculation, one may set
I k = α 2 σ k 2 γ k ∣ χ k ∣ 2 . \mathcal{I}_k = \alpha^2 \frac{\sigma_k^2}{\gamma_k} \lvert\chi_k\rvert^2. I k = α 2 γ k σ k 2 ∣ χ k ∣ 2 . For the continuum description, we replace the sums by integrals. The wavenumber constraint is imposed by a Dirac distribution:
∂ t N k = ( Q s c N ) k + ϵ μ 2 ∫ R 3 d [ R k ℓ p q ( k ) N p N q − R k ℓ p q ( p ) N k N q − R k ℓ p q ( q ) N k N p ] × δ ( d ) ( p + q − k − ℓ ) d p d q d ℓ + I k − 2 μ k N k . (160) \partial_t N_k = (\mathcal{Q}_{\mathrm{sc}}N)_k + \epsilon_\mu^2 \int_{\mathbb{R}^{3d}} \left[\mathcal{R}^{(k)}_{k\ell pq}N_pN_q - \mathcal{R}^{(p)}_{k\ell pq}N_kN_q - \mathcal{R}^{(q)}_{k\ell pq}N_kN_p\right] \times\delta^{(d)}(p+q-k-\ell)\,\mathrm{d}p\,\mathrm{d}q\,\mathrm{d}\ell+ \mathcal{I}_k - 2\mu_kN_k. \tag*{(160)} ∂ t N k = ( Q sc N ) k + ϵ μ 2 ∫ R 3 d [ R k ℓ pq ( k ) N p N q − R k ℓ pq ( p ) N k N q − R k ℓ pq ( q ) N k N p ] × δ ( d ) ( p + q − k − ℓ ) d p d q d ℓ + I k − 2 μ k N k . (160) For an OU envelope with fixed stationary variance D ℓ = σ ℓ 2 / ( 2 γ ℓ ) D_\ell=\sigma_\ell^2/(2\gamma_\ell) D ℓ = σ ℓ 2 / ( 2 γ ℓ )
R ^ ℓ ( Ω ) = 2 D ℓ γ ℓ γ ℓ 2 + Ω 2 → γ ℓ → 0 2 π D ℓ δ ( Ω ) . (161) \widehat{R}_\ell(\Omega) = \frac{2D_\ell\gamma_\ell}{\gamma_\ell^2+\Omega^2} \xrightarrow{\gamma_\ell\to0} 2\pi D_\ell\delta(\Omega). \tag*{(161)} R ℓ ( Ω ) = γ ℓ 2 + Ω 2 2 D ℓ γ ℓ γ ℓ → 0 2 π D ℓ δ ( Ω ) . (161) This is the long-correlation, narrow-band limit of the Airy carrier, not the white-noise decorrelation limit. In this limit,
( Q R N ) k ∣ r e s = 4 π ϵ μ 2 ∫ R 3 d D ℓ ∣ c ℓ ∣ 2 [ ∣ V k ℓ p q ∣ 2 N p N q + Re ( V k ℓ p q V k ℓ p q ( p ) ) N k N q + Re ( V k ℓ p q V k ℓ p q ( q ) ) N k N p ] × δ ( d ) ( p + q − k − ℓ ) δ ( ω p + ω q − ω k − ω ℓ ) d p d q d ℓ . (162) (\mathcal{Q}_{\mathrm{R}}N)_k\big|_{\mathrm{res}} = 4\pi\epsilon_\mu^2 \int_{\mathbb{R}^{3d}} D_\ell\lvert c_\ell\rvert^2\left[\lvert V_{k\ell pq}\rvert^2N_pN_q+\operatorname{Re}(V_{k\ell pq}V^{(p)}_{k\ell pq})N_kN_q+\operatorname{Re}(V_{k\ell pq}V^{(q)}_{k\ell pq})N_kN_p\right] \times\delta^{(d)}(p+q-k-\ell)\delta(\omega_p+\omega_q-\omega_k-\omega_\ell)\,\mathrm{d}p\,\mathrm{d}q\,\mathrm{d}\ell. \tag*{(162)} ( Q R N ) k res = 4 π ϵ μ 2 ∫ R 3 d D ℓ ∣ c ℓ ∣ 2 [ ∣ V k ℓ pq ∣ 2 N p N q + Re ( V k ℓ pq V k ℓ pq ( p ) ) N k N q + Re ( V k ℓ pq V k ℓ pq ( q ) ) N k N p ] × δ ( d ) ( p + q − k − ℓ ) δ ( ω p + ω q − ω k − ω ℓ ) d p d q d ℓ . (162) By contrast, the white-noise limit is obtained by letting γ ℓ → ∞ \gamma_\ell\to\infty γ ℓ → ∞ Q ℓ Q_\ell Q ℓ R ℓ γ ( s ) = Q ℓ γ ℓ 2 e − γ ℓ ∣ s ∣ R^\gamma_\ell(s)=\frac{Q_\ell\gamma_\ell}{2}\mathrm{e}^{-\gamma_\ell\lvert s\rvert} R ℓ γ ( s ) = 2 Q ℓ γ ℓ e − γ ℓ ∣ s ∣
R ℓ γ → γ ℓ → ∞ Q ℓ δ and R ^ ℓ γ ( Ω ) = Q ℓ γ ℓ 2 γ ℓ 2 + Ω 2 ⟶ Q ℓ R^\gamma_\ell\xrightarrow{\gamma_\ell\to\infty} Q_\ell\delta\quad\text{and}\quad\widehat{R}^\gamma_\ell(\Omega)=Q_\ell\frac{\gamma_\ell^2}{\gamma_\ell^2+\Omega^2}\longrightarrow Q_\ell R ℓ γ γ ℓ → ∞ Q ℓ δ and R ℓ γ ( Ω ) = Q ℓ γ ℓ 2 + Ω 2 γ ℓ 2 ⟶ Q ℓ for every fixed Ω \Omega Ω Ω k ℓ p q = 0 \Omega_{k\ell pq}=0 Ω k ℓ pq = 0 γ ℓ → 0 \gamma_\ell\to0 γ ℓ → 0 γ ℓ \gamma_\ell γ ℓ
The operator (155) shares the 2 ↔ 2 2\leftrightarrow2 2 ↔ 2 k + ℓ = p + q k+\ell=p+q k + ℓ = p + q K ( k ) \mathcal{K}^{(k)} K ( k ) K ( p ) \mathcal{K}^{(p)} K ( p ) K ( q ) \mathcal{K}^{(q)} K ( q )
The mechanism may consequently be interpreted as stochastic wave–wave scattering mediated by unresolved velocity fluctuations carrying an Airy phase, whose spatial patterns are swept and deformed by the resolved flow. This connects the construction with the Kraichnan–Tennekes sweeping picture [12 , 17 ] through the advection and temporal decorrelation of unresolved patterns. The scattering mechanism additionally involves their deformation and coupling to the resolved waves, and remains distinct from the intrinsic Hamiltonian four-wave interaction of classical weak-turbulence theory [6 , 8 , 20 , 21 , 19 ] .
Remark: Relation to four-wave kinetics and higher-order closure
The distinction can be made explicit by considering the classical Hasselmann–Zakharov cubic spectral bracket:
B k ℓ p q H Z [ N ] = N p N q ( N k + N ℓ ) − N k N ℓ ( N p + N q ) = N ℓ [ N p N q − N k ( N p + N q ) ] + N k N p N q . (163) \mathcal{B}^{\mathrm{HZ}}_{k\ell pq}[N] = N_pN_q(N_k + N_\ell) - N_kN_\ell(N_p + N_q)
= N_\ell[N_pN_q - N_k(N_p + N_q)] + N_kN_pN_q. \tag*{(163)} B k ℓ pq HZ [ N ] = N p N q ( N k + N ℓ ) − N k N ℓ ( N p + N q ) = N ℓ [ N p N q − N k ( N p + N q )] + N k N p N q . (163) When N ℓ N_\ell N ℓ N ℓ N_\ell N ℓ N k N p N q N_kN_pN_q N k N p N q
Under the assumed common scaling, with normalized spectra and unresolved covariance held O ( 1 ) O(1) O ( 1 ) O ( ϵ 2 ) O(\epsilon^2) O ( ϵ 2 ) O ( ϵ 4 ) O(\epsilon^4) O ( ϵ 4 )
To account for this additional transfer, the collision operator may be extended schematically as
∂ t N ∣ c o l l = ϵ 2 Q ( 2 ) [ N ; C σ ] + ϵ 4 Q ( 4 ) [ N ; C σ ] + ⋯ , (164) \left.\partial_t N\right|_{\mathrm{coll}} = \epsilon^2 Q^{(2)}[N;C^\sigma] + \epsilon^4 Q^{(4)}[N;C^\sigma] + \cdots, \tag*{(164)} ∂ t N ∣ coll = ϵ 2 Q ( 2 ) [ N ; C σ ] + ϵ 4 Q ( 4 ) [ N ; C σ ] + ⋯ , (164) where C σ C^\sigma C σ Q ( 2 ) Q^{(2)} Q ( 2 ) ϵ 2 \epsilon^2 ϵ 2 (163) and a quartet-symmetric interaction kernel. The explicit route includes higher-order corrections to the inverse flow map and its Jacobian, together with all other contributions required at the same order. In the modelling approach, the Hamiltonian four-wave structure provides an additional closure for the higher-order transfer; it is not claimed to follow directly from the leading stochastic operator.
Asymptotic scaling
We now examine the scaling consequences of the kinetic equation (159), or of its continuum counterpart (160). Two contributions must be distinguished: the linear scattering operator and the quadratic bath-mediated interaction generated by deformation of the transported correlation modes.
For the scattering contribution, we assume that the incoming and outgoing transition rates are asymptotically equal:
G k + k j , k j A = G k , k j A + o ( 1 ) . (165) \mathcal{G}^{\mathrm{A}}_{k+k_j,k_j} = \mathcal{G}^{\mathrm{A}}_{k,k_j} + o(1). \tag*{(165)} G k + k j , k j A = G k , k j A + o ( 1 ) . (165) The conservative scattering operator
( Q s c N ) k = ∑ j [ G k , k j A N k − k j − G k + k j , k j A N k ] (Q_{\mathrm{sc}}N)_k = \sum_j \left[\mathcal{G}^{\mathrm{A}}_{k,k_j}N_{k-k_j} - \mathcal{G}^{\mathrm{A}}_{k+k_j,k_j}N_k\right] ( Q sc N ) k = j ∑ [ G k , k j A N k − k j − G k + k j , k j A N k ] then reduces at leading order to
( Q s c N ) k = ∑ j G k , k j A ( N k − k j − N k ) + o ( 1 ) . (166) (Q_{\mathrm{sc}}N)_k = \sum_j \mathcal{G}^{\mathrm{A}}_{k,k_j}\left(N_{k-k_j} - N_k\right) + o(1). \tag*{(166)} ( Q sc N ) k = j ∑ G k , k j A ( N k − k j − N k ) + o ( 1 ) . (166) We refer to (165) as translation invariance of the rates at leading order. It should not be confused with the stronger condition G A ( k , q ) = G 0 ( q ) \mathcal{G}^{\mathrm{A}}(k,q)=\mathcal{G}_0(q) G A ( k , q ) = G 0 ( q ) k k k
Diffusive scattering regime. In the continuum limit, (166) becomes
( Q s c A N ) ( k ) = ∫ R d A ( k , q ) [ N ( k − q ) − N ( k ) ] d q . (167) (\mathcal{Q}_{\mathrm{sc}}^{A}N)(k)=\int_{\mathbb{R}^{d}}\mathcal{A}(k,q)\left[N(k-q)-N(k)\right]\,dq. \tag*{(167)} ( Q sc A N ) ( k ) = ∫ R d A ( k , q ) [ N ( k − q ) − N ( k ) ] d q . (167) We consider a scale-local finite-variance regime. The rescaled scattering variable may satisfy
∣ q ∣ ∼ k , k = ∣ k ∣ , |q|\sim k,\qquad k=|k|, ∣ q ∣ ∼ k , k = ∣ k ∣ , while the actual spectral displacement is δ k = ε q \delta k=\varepsilon q δ k = εq ε ≪ 1 \varepsilon\ll1 ε ≪ 1 ∣ δ k ∣ ≪ k |\delta k|\ll k ∣ δ k ∣ ≪ k ∣ q ∣ ∼ k |q|\sim k ∣ q ∣ ∼ k ε q \varepsilon q εq ε − 2 \varepsilon^{-2} ε − 2
We assume that the kernel has a finite second moment,
∫ R d ∣ q ∣ 2 A ( k , q ) d q < ∞ . (168) \int_{\mathbb{R}^{d}}|q|^{2}\mathcal{A}(k,q)\,dq<\infty. \tag*{(168)} ∫ R d ∣ q ∣ 2 A ( k , q ) d q < ∞. (168) For example, a tail A ( k , q ) = O ( ∣ q ∣ − α ) \mathcal{A}(k,q)=O(|q|^{-\alpha}) A ( k , q ) = O ( ∣ q ∣ − α ) α > d + 2 \alpha>d+2 α > d + 2
Reciprocity and Taylor expansion. When the rate depends on k k k q q q W ε ( k , k ′ ) W_{\varepsilon}(k,k') W ε ( k , k ′ )
W ε ( k , k ′ ) = W ε ( k ′ , k ) . W_{\varepsilon}(k,k')=W_{\varepsilon}(k',k). W ε ( k , k ′ ) = W ε ( k ′ , k ) . Writing k ′ = k − ε q k'=k-\varepsilon q k ′ = k − εq
W ε ( k , k − ε q ) = A ( k − ε q 2 , q ) , A ( k , − q ) = A ( k , q ) . W_{\varepsilon}(k,k-\varepsilon q)=\mathcal{A}\left(k-\frac{\varepsilon q}{2},q\right),\qquad\mathcal{A}(k,-q)=\mathcal{A}(k,q). W ε ( k , k − εq ) = A ( k − 2 εq , q ) , A ( k , − q ) = A ( k , q ) . The midpoint is therefore not a numerical quadrature rule; it simply labels the reciprocal transition by the midpoint of its two end states. The conservative small-jump operator reads
( Q s c ε N ) ( k ) = ε − 2 ∫ R d A ( k − ε q 2 , q ) [ N ( k − ε q ) − N ( k ) ] d q , (169) (\mathcal{Q}_{\mathrm{sc}}^{\varepsilon}N)(k)=\varepsilon^{-2}\int_{\mathbb{R}^{d}}\mathcal{A}\left(k-\frac{\varepsilon q}{2},q\right)\left[N(k-\varepsilon q)-N(k)\right]\,dq, \tag*{(169)} ( Q sc ε N ) ( k ) = ε − 2 ∫ R d A ( k − 2 εq , q ) [ N ( k − εq ) − N ( k ) ] d q , (169) where A ( k , q ) = G q A ( k , q ) + o ( 1 ) \mathcal{A}(k,q)=\mathcal{G}^{A}_{q}(k,q)+o(1) A ( k , q ) = G q A ( k , q ) + o ( 1 )
N ( k − ε q ) − N ( k ) = − ε q i ∂ i N + ε 2 2 q i q j ∂ i j N + o ( ε 2 ∣ q ∣ 2 ) , (170) N(k-\varepsilon q)-N(k)=-\varepsilon q_i\partial_iN+\frac{\varepsilon^2}{2}q_iq_j\partial_{ij}N+o(\varepsilon^2|q|^2), \tag*{(170)} N ( k − εq ) − N ( k ) = − ε q i ∂ i N + 2 ε 2 q i q j ∂ ij N + o ( ε 2 ∣ q ∣ 2 ) , (170) A ( k − ε q 2 , q ) = A ( k , q ) − ε 2 q i ∂ i A ( k , q ) + o ( ε ∣ q ∣ ) (171) \mathcal{A}\left(k-\frac{\varepsilon q}{2},q\right)=\mathcal{A}(k,q)-\frac{\varepsilon}{2}q_i\partial_i\mathcal{A}(k,q)+o(\varepsilon|q|) \tag*{(171)} A ( k − 2 εq , q ) = A ( k , q ) − 2 ε q i ∂ i A ( k , q ) + o ( ε ∣ q ∣ ) (171) make the diffusion limit explicit. Evenness of A \mathcal{A} A ε − 1 \varepsilon^{-1} ε − 1
Q s c ε N ⟶ D i j ( k ) ∂ i j N + ∂ i D i j ( k ) ∂ j N = ∂ i ( D i j ( k ) ∂ j N ) , D i j ( k ) = 1 2 ∫ R d q i q j A ( k , q ) d q . (172) \mathcal{Q}_{\mathrm{sc}}^{\varepsilon}N\longrightarrow D^{ij}(k)\partial_{ij}N+\partial_iD^{ij}(k)\partial_jN=\partial_i\left(D^{ij}(k)\partial_jN\right),\qquad D^{ij}(k)=\frac{1}{2}\int_{\mathbb{R}^{d}}q_iq_j\mathcal{A}(k,q)\,dq. \tag*{(172)} Q sc ε N ⟶ D ij ( k ) ∂ ij N + ∂ i D ij ( k ) ∂ j N = ∂ i ( D ij ( k ) ∂ j N ) , D ij ( k ) = 2 1 ∫ R d q i q j A ( k , q ) d q . (172) Here symmetry gives D i j = D j i D^{ij}=D^{ji} D ij = D j i k k k ∂ i D i j ∂ j N \partial_iD^{ij}\partial_jN ∂ i D ij ∂ j N D i j ∂ i j N D^{ij}\partial_{ij}N D ij ∂ ij N D D D
The leading-order equality of the incoming and outgoing rates does not force D D D d d d
[ G A ] = ( wavenumber ) − d ( time ) − 1 , [ D ] = ( wavenumber ) 2 ( time ) − 1 . [\mathcal{G}^{A}]=(\text{wavenumber})^{-d}(\text{time})^{-1},\qquad[D]=(\text{wavenumber})^{2}(\text{time})^{-1}. [ G A ] = ( wavenumber ) − d ( time ) − 1 , [ D ] = ( wavenumber ) 2 ( time ) − 1 . In two dimensions, [ G A ] = L 2 T − 1 [\mathcal{G}^{A}]=L^{2}T^{-1} [ G A ] = L 2 T − 1 σ u 2 τ e f f \sigma_u^{2}\tau_{\mathrm{eff}} σ u 2 τ eff
Scale-local estimate and transfer rate. We now make the physical scaling of the diffusion tensor explicit. Assume in two horizontal dimensions that the kernel is concentrated near its lower characteristic scale ∣ q ∣ ∼ k |q|\sim k ∣ q ∣ ∼ k
A ( k , q ) ∼ σ u 2 ( k ) τ e f f ( k ) Φ ( ∣ q ∣ k ) , (173) \mathcal{A}(k,q)\sim\sigma_u^{2}(k)\tau_{\mathrm{eff}}(k)\Phi\left(\frac{|q|}{k}\right), \tag*{(173)} A ( k , q ) ∼ σ u 2 ( k ) τ eff ( k ) Φ ( k ∣ q ∣ ) , (173) where Φ ( ξ ) \Phi(\xi) Φ ( ξ ) ξ = O ( 1 ) \xi=O(1) ξ = O ( 1 ) ξ → ∞ \xi\to\infty ξ → ∞ k j k_j k j
τ e f f , j ( Ω k , k j ) : = Re ∫ 0 ∞ R q ( s ) R q ( 0 ) e i Ω k , k j s d s = γ j γ j 2 + Ω k , j 2 = γ j − 1 1 + ( Ω k , j γ j − 1 ) 2 , (174) \tau_{\mathrm{eff},j}(\Omega_{k,k_j}):=\operatorname{Re}\int_{0}^{\infty}\frac{R_q(s)}{R_q(0)}e^{i\Omega_{k,k_j}s}\,\mathrm{d}s=\frac{\gamma_j}{\gamma_j^{2}+\Omega_{k,j}^{2}}=\frac{\gamma_j^{-1}}{1+(\Omega_{k,j}\gamma_j^{-1})^{2}}, \tag*{(174)} τ eff , j ( Ω k , k j ) := Re ∫ 0 ∞ R q ( 0 ) R q ( s ) e i Ω k , k j s d s = γ j 2 + Ω k , j 2 γ j = 1 + ( Ω k , j γ j − 1 ) 2 γ j − 1 , (174) where
Ω k , j = ω k − ω k − q − ω q . \Omega_{k,j}=\omega_k-\omega_{k-q}-\omega_q. Ω k , j = ω k − ω k − q − ω q . In the scaling estimates, τ e f f ( k ) \tau_{\mathrm{eff}}(k) τ eff ( k ) τ e f f , j ( Ω k , j ) \tau_{\mathrm{eff},j}(\Omega_{k,j}) τ eff , j ( Ω k , j ) k j ∼ k k_j\sim k k j ∼ k ∣ Ω k , j ∣ γ j − 1 ≪ 1 |\Omega_{k,j}|\gamma_j^{-1}\ll1 ∣ Ω k , j ∣ γ j − 1 ≪ 1 τ e f f ( k ) ≃ τ c ( k ) \tau_{\mathrm{eff}}(k)\simeq\tau_c(k) τ eff ( k ) ≃ τ c ( k )
If the scattering modes satisfy ∣ q ∣ ≳ k |q|\gtrsim k ∣ q ∣ ≳ k
λ ( k ) : = ∫ R 2 A ( k , q ) d q \lambda(k):=\int_{\mathbb{R}^{2}}\mathcal{A}(k,q)\,\mathrm{d}q λ ( k ) := ∫ R 2 A ( k , q ) d q ∼ 2 π σ u 2 ( k ) τ e f f ( k ) k 2 ∫ 1 ∞ ξ Φ ( ξ ) d ξ ∼ σ u 2 ( k ) τ e f f ( k ) k 2 . (175) \sim2\pi\sigma_u^{2}(k)\tau_{\mathrm{eff}}(k)k^{2}\int_{1}^{\infty}\xi\Phi(\xi)\,\mathrm{d}\xi\sim\sigma_u^{2}(k)\tau_{\mathrm{eff}}(k)k^{2}. \tag*{(175)} ∼ 2 π σ u 2 ( k ) τ eff ( k ) k 2 ∫ 1 ∞ ξ Φ ( ξ ) d ξ ∼ σ u 2 ( k ) τ eff ( k ) k 2 . (175) Keeping the decaying shape Φ \Phi Φ
Define the conditional second moment of the rescaled jump by
⟨ q i q j ⟩ k : = ∫ R 2 q i q j A ( k , q ) d q ∫ R 2 A ( k , q ) d q . (176) \langle q_iq_j\rangle_k:=\frac{\int_{\mathbb{R}^{2}}q_iq_j\mathcal{A}(k,q)\,\mathrm{d}q}{\int_{\mathbb{R}^{2}}\mathcal{A}(k,q)\,\mathrm{d}q}. \tag*{(176)} ⟨ q i q j ⟩ k := ∫ R 2 A ( k , q ) d q ∫ R 2 q i q j A ( k , q ) d q . (176) Isotropy and dominance by ∣ q ∣ ∼ k |q|\sim k ∣ q ∣ ∼ k
⟨ q i q j ⟩ k ∼ k 2 δ i j . (177) \langle q_iq_j\rangle_k\sim k^{2}\delta_{ij}. \tag*{(177)} ⟨ q i q j ⟩ k ∼ k 2 δ ij . (177) Consequently,
D i j ( k ) = 1 2 λ ( k ) ⟨ q i q j ⟩ k ∼ k 4 σ u 2 ( k ) τ e f f ( k ) δ i j . (178) D^{ij}(k)=\frac{1}{2}\lambda(k)\langle q_iq_j\rangle_k\sim k^{4}\sigma_u^{2}(k)\tau_{\mathrm{eff}}(k)\delta_{ij}. \tag*{(178)} D ij ( k ) = 2 1 λ ( k ) ⟨ q i q j ⟩ k ∼ k 4 σ u 2 ( k ) τ eff ( k ) δ ij . (178) The time required to diffuse across a spectral distance of order k k k
τ s c − 1 ( k ) ∼ D ( k ) k 2 ∼ k 2 σ u 2 ( k ) τ e f f ( k ) . (179) \tau_{\mathrm{sc}}^{-1}(k) \sim\frac{D(k)}{k^2} \sim k^2\sigma_u^2(k)\tau_{\mathrm{eff}}(k). \tag*{(179)} τ sc − 1 ( k ) ∼ k 2 D ( k ) ∼ k 2 σ u 2 ( k ) τ eff ( k ) . (179) The transfer-rate estimate in (179) will be compared below with the Hasselmann–Zakharov quartic transfer rate. It is compatible with (165), because equality of the incoming and outgoing rates does not eliminate their common slow dependence on k k k
Radial diffusion and constant action flux. For an isotropic kernel, D i j ( k ) = D ( k ) δ i j D^{ij}(k) = D(k)\delta_{ij} D ij ( k ) = D ( k ) δ ij N ( k , t ) = N ( k , t ) N(k,t) = N(k,t) N ( k , t ) = N ( k , t )
∂ t N = 1 k ∂ k [ k D ( k ) ∂ k N ] , (180) \partial_t N = \frac{1}{k}\partial_k\left[kD(k)\partial_kN\right], \tag*{(180)} ∂ t N = k 1 ∂ k [ k D ( k ) ∂ k N ] , (180) and the shell-integrated action flux is
Π N ( k ) = − 2 π k D ( k ) ∂ k N . (181) \Pi_N(k) = -2\pi kD(k)\partial_kN. \tag*{(181)} Π N ( k ) = − 2 π k D ( k ) ∂ k N . (181) Here 2 π k N ( k ) 2\pi kN(k) 2 π k N ( k )
− 2 π k D ( k ) ∂ k N = Π N 0 . -2\pi kD(k)\partial_kN = \Pi_N^0. − 2 π k D ( k ) ∂ k N = Π N 0 . If D ( k ) ∼ k m D(k) \sim k^m D ( k ) ∼ k m
N ( k ) ∼ k − m , m ≠ 0. (182) N(k) \sim k^{-m}, \qquad m \ne0. \tag*{(182)} N ( k ) ∼ k − m , m = 0. (182) The limiting case m = 0 m = 0 m = 0 G A ( k , q ) = G 0 ( q ) \mathcal{G}^{A}(k,q) = \mathcal{G}_0(q) G A ( k , q ) = G 0 ( q )
If
σ u 2 ( k ) ∼ k − 2 r , γ k ∼ k β , (183) \sigma_u^2(k) \sim k^{-2r}, \qquad\gamma_k \sim k^\beta, \tag*{(183)} σ u 2 ( k ) ∼ k − 2 r , γ k ∼ k β , (183) then, near resonance,
D ( k ) ∼ k 4 − 2 r − β , (184) D(k) \sim k^{4-2r-\beta}, \tag*{(184)} D ( k ) ∼ k 4 − 2 r − β , (184) τ s c − 1 ( k ) ∼ k 2 − 2 r − β , (185) \tau_{\mathrm{sc}}^{-1}(k) \sim k^{2-2r-\beta}, \tag*{(185)} τ sc − 1 ( k ) ∼ k 2 − 2 r − β , (185) N ( k ) ∼ k − ( 4 − 2 r − β ) , (186) N(k) \sim k^{-(4-2r-\beta)}, \tag*{(186)} N ( k ) ∼ k − ( 4 − 2 r − β ) , (186) E ( k ) = ω k N ( k ) ∼ k − 7 / 2 + 2 r + β in deep water . (187) E(k) = \omega_kN(k) \sim k^{-7/2+2r+\beta} \qquad\text{in deep water}. \tag*{(187)} E ( k ) = ω k N ( k ) ∼ k − 7/2 + 2 r + β in deep water . (187) Accordingly, the exponent N ∼ k − 23 / 6 N \sim k^{-23/6} N ∼ k − 23/6
2 r + β = 1 6 . (188) 2r + \beta= \frac{1}{6}. \tag*{(188)} 2 r + β = 6 1 . (188) For r = 1 / 3 r = 1/3 r = 1/3 β = − 1 / 2 \beta= -1/2 β = − 1/2 k 1 / 2 k^{1/2} k 1/2 γ k ∼ ω k ∼ k 1 / 2 \gamma_k \sim\omega_k \sim k^{1/2} γ k ∼ ω k ∼ k 1/2 β = 1 / 2 \beta= 1/2 β = 1/2
τ s c − 1 ∼ k 5 / 6 , N ( k ) ∼ k − 17 / 6 , E ( k ) ∼ k − 7 / 3 . \tau_{\mathrm{sc}}^{-1} \sim k^{5/6}, \qquad N(k) \sim k^{-17/6}, \qquad E(k) \sim k^{-7/3}. τ sc − 1 ∼ k 5/6 , N ( k ) ∼ k − 17/6 , E ( k ) ∼ k − 7/3 . Under the conditional choice β = − 1 / 2 \beta= -1/2 β = − 1/2 r = 1 / 4 r = 1/4 r = 1/4
N ( k ) ∼ k − 4 , E ( k ) ∼ k − 7 / 2 . N(k) \sim k^{-4}, \qquad E(k) \sim k^{-7/2}. N ( k ) ∼ k − 4 , E ( k ) ∼ k − 7/2 . These exponents coincide with familiar Kolmogorov–Zakharov exponents only at the level of their power laws. Here they arise from a forced constant-action-flux solution of the stochastic diffusion operator, rather than from a conservative resonant four-wave collision integral.
For completeness, multiplying the shell action balance by the deep-water frequency gives
∂ t E s h + ∂ k ( ω k Π N ) = ω k ′ Π N , E s h : = 2 π k ω k N . (189) \partial_t \mathcal{E}_{\mathrm{sh}}+\partial_k\left(\omega_k\Pi_N\right)=\omega'_k\Pi_N,\qquad\mathcal{E}_{\mathrm{sh}}:=2\pi k\omega_kN. \tag*{(189)} ∂ t E sh + ∂ k ( ω k Π N ) = ω k ′ Π N , E sh := 2 π k ω k N . (189) Thus ω k Π N \omega_k\Pi_N ω k Π N
Finally, subject to decay at infinity or no-flux boundary conditions,
d d t ( 1 2 ∫ R d N 2 d k ) = − ∫ R d ∇ k N ⋅ D ( k ) ∇ k N d k ≤ 0. (190) \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{2}\int_{\mathbb{R}^d}N^2\,\mathrm{d}k\right)=-\int_{\mathbb{R}^d}\nabla_kN\cdot D(k)\nabla_kN\,\mathrm{d}k\le0. \tag*{(190)} d t d ( 2 1 ∫ R d N 2 d k ) = − ∫ R d ∇ k N ⋅ D ( k ) ∇ k N d k ≤ 0. (190) This decay identity describes homogenization in a closed system and must not be confused with a forced constant-flux stationary state. The diffusion coefficient may depend parametrically, or through a closure, on the transported small-scale variance; positivity of D D D
Scaling of the transported correlation modes. We next consider the scalar transport equation
∂ t φ j + ϵ μ U ⋅ ∇ φ j = 0 , U = ∇ φ ‾ . (191) \partial_t\varphi_j+\epsilon_\mu U\cdot\nabla\varphi_j=0,\qquad U=\nabla\overline{\varphi}. \tag*{(191)} ∂ t φ j + ϵ μ U ⋅ ∇ φ j = 0 , U = ∇ φ . (191) The value of φ j \varphi_j φ j L 2 L^2 L 2
d d t ∫ ∣ φ j ∣ 2 d x = ϵ μ ∫ ( ∇ ⋅ U ) ∣ φ j ∣ 2 d x , (192) \frac{\mathrm{d}}{\mathrm{d}t}\int\lvert\varphi_j\rvert^2\,\mathrm{d}x=\epsilon_\mu\int(\nabla\cdot U)\lvert\varphi_j\rvert^2\,\mathrm{d}x, \tag*{(192)} d t d ∫ ∣ φ j ∣ 2 d x = ϵ μ ∫ ( ∇ ⋅ U ) ∣ φ j ∣ 2 d x , (192) for periodic boundary conditions. Thus, conservation of the spatially integrated variance requires a volume-preserving flow, ∇ ⋅ U = 0 \nabla\cdot U=0 ∇ ⋅ U = 0
Under a geometrical-optics approximation, the phase satisfies the same transport equation as φ j \varphi_j φ j
d k d t = − ϵ μ ( ∇ U ) t r k . (193) \frac{\mathrm{d}k}{\mathrm{d}t}=-\epsilon_\mu(\nabla U)^{\mathrm{tr}}k. \tag*{(193)} d t d k = − ϵ μ ( ∇ U ) tr k . (193) Writing k ^ = k / ∣ k ∣ \widehat{k}=k/\lvert k\rvert k = k / ∣ k ∣ S S S ∇ U \nabla U ∇ U
d k d t = − ϵ μ k k ^ ⋅ S k ^ . (194) \frac{\mathrm{d}k}{\mathrm{d}t}=-\epsilon_\mu k\,\widehat{k}\cdot S\widehat{k}. \tag*{(194)} d t d k = − ϵ μ k k ⋅ S k . (194) The sign of the radial transfer is therefore determined by the strain sampled along each ray; it is not fixed by the transport equation alone.
To obtain an isotropic one-dimensional scaling, we now make two additional phenomenological assumptions: angular averaging can be represented by a positive stretching rate Σ ( k ) \Sigma(k) Σ ( k )
k ˙ = ϵ μ Σ ( k ) k , ∂ t E φ + ∂ k ( k ˙ E φ ) = 0. (195) \dot{k}=\epsilon_\mu\Sigma(k)k,\qquad\partial_tE_\varphi+\partial_k\left(\dot{k}E_\varphi\right)=0. \tag*{(195)} k ˙ = ϵ μ Σ ( k ) k , ∂ t E φ + ∂ k ( k ˙ E φ ) = 0. (195) If Σ ( k ) ∼ k s \Sigma(k) \sim k^{s} Σ ( k ) ∼ k s Π φ \Pi_{\varphi} Π φ
E φ ( k ) = Π φ ϵ μ Σ ( k ) k s ∼ k − ( 1 + s ) . (196) E_{\varphi}(k) = \frac{\Pi_{\varphi}}{\epsilon_{\mu}\Sigma(k)k^{s}} \sim k^{-(1+s)}. \tag*{(196)} E φ ( k ) = ϵ μ Σ ( k ) k s Π φ ∼ k − ( 1 + s ) . (196) (196) is consequently an isotropic strain closure, rather than a direct consequence of (191).
The Airy carrier does not change the equal-time variance of the OU envelope. If
σ κ ∼ κ − α , γ κ ∼ κ β , E ∣ W κ ∣ 2 = σ κ 2 2 γ κ , (197) \sigma_{\kappa} \sim\kappa^{-\alpha}, \qquad\gamma_{\kappa} \sim\kappa^{\beta}, \qquad\mathbb{E}|W_{\kappa}|^{2} = \frac{\sigma_{\kappa}^{2}}{2\gamma_{\kappa}}, \tag*{(197)} σ κ ∼ κ − α , γ κ ∼ κ β , E ∣ W κ ∣ 2 = 2 γ κ σ κ 2 , (197) and correlations between the envelope and the transported spatial mode are neglected at leading order, then
E φ τ ( k ) ∼ k − ( 1 + s + 2 α + β ) . (198) E_{\varphi_{\tau}}(k) \sim k^{-(1+s+2\alpha+\beta)}. \tag*{(198)} E φ τ ( k ) ∼ k − ( 1 + s + 2 α + β ) . (198) The corresponding surface horizontal-velocity spectrum is
E u τ ( k ) = k 2 E φ τ ( k ) ∼ k 1 − s − 2 α − β . (199) E_{u_{\tau}}(k) = k^{2}E_{\varphi_{\tau}}(k) \sim k^{1-s-2\alpha-\beta}. \tag*{(199)} E u τ ( k ) = k 2 E φ τ ( k ) ∼ k 1 − s − 2 α − β . (199) It is important to distinguish this surface-velocity variance from Airy wave action and energy. With the normalization used above,
N φ τ ( k ) = ω k g E φ τ ( k ) , E A ( k ) = ω k N φ τ ( k ) = G k [ 0 ] E φ τ ( k ) . (200) N_{\varphi_{\tau}}(k) = \frac{\omega_{k}}{g}E_{\varphi_{\tau}}(k), \qquad E_{\mathrm{A}}(k) = \omega_{k}N_{\varphi_{\tau}}(k) = G_{k}[0]E_{\varphi_{\tau}}(k). \tag*{(200)} N φ τ ( k ) = g ω k E φ τ ( k ) , E A ( k ) = ω k N φ τ ( k ) = G k [ 0 ] E φ τ ( k ) . (200) For deep-water gravity waves, ω k ∼ k 1 / 2 \omega_{k} \sim k^{1/2} ω k ∼ k 1/2 G k [ 0 ] ∼ k G_{k}[0] \sim k G k [ 0 ] ∼ k
N φ τ ( k ) ∼ k − 1 / 2 − s − 2 α − β , (201) N_{\varphi_{\tau}}(k) \sim k^{-1/2-s-2\alpha-\beta}, \tag*{(201)} N φ τ ( k ) ∼ k − 1/2 − s − 2 α − β , (201) E A ( k ) ∼ k − s − 2 α − β . (202) E_{\mathrm{A}}(k) \sim k^{-s-2\alpha-\beta}. \tag*{(202)} E A ( k ) ∼ k − s − 2 α − β . (202) The carrier phase affects temporal resonance through (174) , but not these equal-time relations. A comparison with a Kolmogorov–Zakharov spectrum is meaningful only after using the same convention for modal versus shell-integrated spectra and after identifying a conserved flux.
Scaling of the bath-mediated quadratic interaction. We finally consider the transported-mode contribution in (160). For the scaling analysis, we impose the quartet-wise common-rate condition (156). Under this condition, the general three-kernel operator (155) reduces to a single-kernel gain–loss operator that is quadratic in the resolved action, and hence quartic in the resolved wave amplitudes. In the long-correlation, narrow-band limit (161), it becomes
( Q R N ) k = 4 π ϵ μ 2 ∫ R 3 d A ℓ ∣ V k ℓ p q ∣ 2 [ N p N q − N k ( N p + N q ) ] δ ( d ) ( p + q − k − ℓ ) δ ( ω p + ω q − ω k − ω ℓ ) d p d q d ℓ . (203) (Q_{\mathrm{RN}})_{k} = 4\pi\epsilon_{\mu}^{2}\int_{\mathbb{R}^{3d}} \mathcal{A}_{\ell}|V_{k\ell pq}|^{2}\left[N_{p}N_{q}-N_{k}(N_{p}+N_{q})\right]\delta^{(d)}(p+q-k-\ell)\delta(\omega_{p}+\omega_{q}-\omega_{k}-\omega_{\ell})\,dp\,dq\,d\ell. \tag*{(203)} ( Q RN ) k = 4 π ϵ μ 2 ∫ R 3 d A ℓ ∣ V k ℓ pq ∣ 2 [ N p N q − N k ( N p + N q ) ] δ ( d ) ( p + q − k − ℓ ) δ ( ω p + ω q − ω k − ω ℓ ) d p d q d ℓ . (203) where
A ℓ : = D ℓ ∣ c ℓ ∣ 2 (204) \mathcal{A}_{\ell} := D_{\ell}|c_{\ell}|^{2} \tag*{(204)} A ℓ := D ℓ ∣ c ℓ ∣ 2 (204) is the prescribed carrier spectral density. The factor A ℓ \mathcal{A}_{\ell} A ℓ − N k ( N p + N q ) -N_{k}(N_{p}+N_{q}) − N k ( N p + N q ) − N k N ℓ -N_{k}N_{\ell} − N k N ℓ
The following dimensional estimate applies only to (203); a finite-width OU spectrum introduces the additional scale γ ℓ \gamma_{\ell} γ ℓ
V λ k , λ ℓ , λ p , λ q = λ 3 V k ℓ p q , (205) V_{\lambda k,\lambda\ell,\lambda p,\lambda q}=\lambda^{3}V_{k\ell pq}, \tag*{(205)} V λk , λ ℓ , λ p , λ q = λ 3 V k ℓ pq , (205) and hence ∣ V ∣ 2 |V|^2 ∣ V ∣ 2 − 2 -2 − 2 − 1 / 2 -1/2 − 1/2
A λ ℓ = λ − a A ℓ , N λ k = λ − x N k . (206) \mathcal{A}_{\lambda\ell}=\lambda^{-a}\mathcal{A}_{\ell},\qquad N_{\lambda k}=\lambda^{-x}N_k. \tag*{(206)} A λ ℓ = λ − a A ℓ , N λk = λ − x N k . (206) Then
( Q R N ) k ∼ ϵ μ 2 k 19 / 2 − a − 2 x , τ R − 1 ( k ) ∼ ϵ μ 2 k 19 / 2 − a − x . (207) (Q_{\mathcal{R}}N)_k\sim\epsilon_{\mu}^{2}k^{19/2-a-2x},\qquad\tau_{\mathcal{R}}^{-1}(k)\sim\epsilon_{\mu}^{2}k^{19/2-a-x}. \tag*{(207)} ( Q R N ) k ∼ ϵ μ 2 k 19/2 − a − 2 x , τ R − 1 ( k ) ∼ ϵ μ 2 k 19/2 − a − x . (207) If A k \mathcal{A}_k A k E φ τ ( k ) = 2 π k A k E_{\varphi\tau}(k)=2\pi k\mathcal{A}_k E φ τ ( k ) = 2 π k A k
a = 2 + s + 2 α + β . (208) a=2+s+2\alpha+\beta. \tag*{(208)} a = 2 + s + 2 α + β . (208) Equation (207) provides the intrinsic quartic transfer-rate estimate and remains valid when a stationary resolved–unresolved equilibrium is imposed. It does not, by itself, determine x x x
∫ R 2 ( Q R N ) k d k ≠ 0 , ∫ R 2 ω k ( Q R N ) k d k ≠ 0 (209) \int_{\mathbb{R}^{2}}(Q_{\mathcal{R}}N)_k\,\mathrm{d}k\ne0,\qquad\int_{\mathbb{R}^{2}}\omega_k(Q_{\mathcal{R}}N)_k\,\mathrm{d}k\ne0 \tag*{(209)} ∫ R 2 ( Q R N ) k d k = 0 , ∫ R 2 ω k ( Q R N ) k d k = 0 (209) in general. The carrier index ℓ \ell ℓ N p N q − N k ( N p + N q ) N_pN_q-N_k(N_p+N_q) N p N q − N k ( N p + N q )
The carrier is nevertheless not necessarily frozen for all time. Its spectrum is prescribed through its initial condition and the OU parameters, while its spatial correlation modes are transported and deformed by the resolved flow. The carrier spectrum A ℓ ( t ) \mathcal{A}_{\ell}(t) A ℓ ( t )
To state this closure without mixing modal and radial spectra, define the shell-integrated resolved action and carrier action by
N s h ( k ) = 2 π k N ( k ) , C N , s h ( k ) = 2 π k ω k g A k . (210) N_{\mathrm{sh}}(k)=2\pi kN(k),\qquad C_{N,\mathrm{sh}}(k)=2\pi k\frac{\omega_k}{g}\mathcal{A}_k. \tag*{(210)} N sh ( k ) = 2 π k N ( k ) , C N , sh ( k ) = 2 π k g ω k A k . (210) The corresponding shell-integrated quartic exchange is
T R , s h ( k ) = 2 π k ( Q R N ) k . (211) \mathcal{T}_{\mathcal{R},\mathrm{sh}}(k)=2\pi k(Q_{\mathcal{R}}N)_k. \tag*{(211)} T R , sh ( k ) = 2 π k ( Q R N ) k . (211) An augmented mean budget can then be written
∂ t N s h + ∂ k Π N = T R , s h , (212) \partial_tN_{\mathrm{sh}}+\partial_k\Pi_N=\mathcal{T}_{\mathcal{R},\mathrm{sh}}, \tag*{(212)} ∂ t N sh + ∂ k Π N = T R , sh , (212) ∂ t C N , s h + ∂ k Π C = − T R , s h + F O U − D O U . (213) \partial_tC_{N,\mathrm{sh}}+\partial_k\Pi_C=-\mathcal{T}_{\mathcal{R},\mathrm{sh}}+F_{\mathrm{OU}}-D_{\mathrm{OU}}. \tag*{(213)} ∂ t C N , sh + ∂ k Π C = − T R , sh + F OU − D OU . (213) The second equation constitutes an additional exchange-budget closure for the transported bath; it does not follow from the prescribed OU process alone. In a stationary range in which OU injection and damping balance, this closure gives
T R , s h = − ∂ k Π C , ∂ k ( Π N + Π C ) = 0. (214) \mathcal{T}_{\mathcal{R},\mathrm{sh}}=-\partial_k\Pi_C,\qquad\partial_k\left(\Pi_N+\Pi_C\right)=0. \tag*{(214)} T R , sh = − ∂ k Π C , ∂ k ( Π N + Π C ) = 0. (214) Thus the total flux of the augmented mean budget is constant, even though Q R \mathcal{Q}_{R} Q R (214) ; it changes the amplitude but not the exponent derived below. Its magnitude must be chosen so that the carrier-flux convergence and the quartic term occur on the same kinetic time scale.
We now estimate the stationary spectrum selected by (214) . From (207) ,
T R , s h ( k ) ∼ ϵ μ 2 k 21 / 2 − a − 2 x . (215) \mathcal{T}_{R,\mathrm{sh}}(k) \sim\epsilon_{\mu}^{2} k^{21/2-a-2x}. \tag*{(215)} T R , sh ( k ) ∼ ϵ μ 2 k 21/2 − a − 2 x . (215) On the other hand, C N , s h ∼ k 3 / 2 − a \mathcal{C}_{N,\mathrm{sh}} \sim k^{3/2-a} C N , sh ∼ k 3/2 − a k ˙ ∼ k 1 + s \dot{k} \sim k^{1+s} k ˙ ∼ k 1 + s
Π C = k ˙ C N , s h ∼ k 5 / 2 + s − a , ∂ k Π C ∼ k 3 / 2 + s − a . (216) \Pi_{C} = \dot{k}\mathcal{C}_{N,\mathrm{sh}} \sim k^{5/2+s-a}, \qquad\partial_{k}\Pi_{C} \sim k^{3/2+s-a}. \tag*{(216)} Π C = k ˙ C N , sh ∼ k 5/2 + s − a , ∂ k Π C ∼ k 3/2 + s − a . (216) Matching the homogeneities in (214) gives
21 2 − a − 2 x = 3 2 + s − a , x = 9 − s 2 . (217) \frac{21}{2}-a-2x = \frac{3}{2}+s-a, \qquad x = \frac{9-s}{2}. \tag*{(217)} 2 21 − a − 2 x = 2 3 + s − a , x = 2 9 − s . (217) The dependence on a a a α \alpha α β \beta β s = 4 / 3 s = 4/3 s = 4/3 x = 23 / 6 x = 23/6 x = 23/6 s = 1 s = 1 s = 1 x = 4 x = 4 x = 4
The same exponent follows from a consistent energy balance. Indeed, the carrier shell energy and its flux scale as
C E , s h = 2 π k G k [ 0 ] A k ∼ k 2 − a , Π C E = k ˙ C E , s h ∼ k 3 + s − a . (218) \mathcal{C}_{E,\mathrm{sh}} = 2\pi k G_{k}[0]\mathcal{A}_{k} \sim k^{2-a}, \qquad\Pi_{C}^{E} = \dot{k}\mathcal{C}_{E,\mathrm{sh}} \sim k^{3+s-a}. \tag*{(218)} C E , sh = 2 π k G k [ 0 ] A k ∼ k 2 − a , Π C E = k ˙ C E , sh ∼ k 3 + s − a . (218) The resolved shell-energy exchange 2 π k ω k Q R 2\pi k\omega_{k}\mathcal{Q}_{R} 2 π k ω k Q R k 11 − a − 2 x k^{11-a-2x} k 11 − a − 2 x ∂ k Π C E ∼ k 2 + s − a \partial_{k}\Pi_{C}^{E} \sim k^{2+s-a} ∂ k Π C E ∼ k 2 + s − a x = ( 9 − s ) / 2 x = (9-s)/2 x = ( 9 − s ) /2
The quartic transfer-rate estimate itself remains (207) . After using (208) and (217) , it becomes
τ R − 1 ( k ) ∼ ϵ μ 2 k 3 − s / 2 − 2 α − β . (219) \tau_{R}^{-1}(k) \sim\epsilon_{\mu}^{2} k^{3-s/2-2\alpha-\beta}. \tag*{(219)} τ R − 1 ( k ) ∼ ϵ μ 2 k 3 − s /2 − 2 α − β . (219) Equivalently, the carrier surface-velocity variance per logarithmic wavenumber interval satisfies, up to numerical constants,
σ u 2 ( k ) ∼ k 4 A k . (220) \sigma_{u}^{2}(k) \sim k^{4}\mathcal{A}_{k}. \tag*{(220)} σ u 2 ( k ) ∼ k 4 A k . (220) The intrinsic resonant quartic rate can therefore also be written, at the level of spectral homogeneity,
τ R − 1 ( k ) ∼ ϵ μ 2 k 11 / 2 σ u 2 ( k ) N ( k ) . (221) \tau_{R}^{-1}(k) \sim\epsilon_{\mu}^{2} k^{11/2}\sigma_{u}^{2}(k)N(k). \tag*{(221)} τ R − 1 ( k ) ∼ ϵ μ 2 k 11/2 σ u 2 ( k ) N ( k ) . (221) Under (217) , this becomes τ R − 1 ∼ ϵ μ 2 k 1 + s / 2 σ u 2 ( k ) \tau_{R}^{-1} \sim\epsilon_{\mu}^{2} k^{1+s/2}\sigma_{u}^{2}(k) τ R − 1 ∼ ϵ μ 2 k 1 + s /2 σ u 2 ( k ) g g g (179) is not automatic; it requires an additional relation among the strain, OU-memory, and scattering scalings.
Production–dissipation decomposition. For clarity, the bath-mediated collision operator may be written
( Q R N ) k = P N ( k ) − Γ N ( k ) N k , (222) (Q_{\mathcal R}N)_k = P_N(k) - \Gamma_N(k)N_k, \tag*{(222)} ( Q R N ) k = P N ( k ) − Γ N ( k ) N k , (222) where, in the colored-noise formulation,
P N ( k ) = ϵ μ 2 ∫ R 3 d R k ℓ p q N p N q δ ( d ) ( p + q − k − ℓ ) d p d q d ℓ , (223) P_N(k) = \epsilon_\mu^2 \int_{\mathbb{R}^{3d}} \mathcal{R}_{k\ell pq}N_pN_q\delta^{(d)}(p+q-k-\ell)\,\mathrm{d}p\,\mathrm{d}q\,\mathrm{d}\ell, \tag*{(223)} P N ( k ) = ϵ μ 2 ∫ R 3 d R k ℓ pq N p N q δ ( d ) ( p + q − k − ℓ ) d p d q d ℓ , (223) Γ N ( k ) = ϵ μ 2 ∫ R 3 d R k ℓ p q ( N p + N q ) δ ( d ) ( p + q − k − ℓ ) d p d q d ℓ . (224) \Gamma_N(k) = \epsilon_\mu^2 \int_{\mathbb{R}^{3d}} \mathcal{R}_{k\ell pq}(N_p+N_q)\delta^{(d)}(p+q-k-\ell)\,\mathrm{d}p\,\mathrm{d}q\,\mathrm{d}\ell. \tag*{(224)} Γ N ( k ) = ϵ μ 2 ∫ R 3 d R k ℓ pq ( N p + N q ) δ ( d ) ( p + q − k − ℓ ) d p d q d ℓ . (224) Both P N P_N P N Γ N \Gamma_N Γ N N k > 0 N_k>0 N k > 0
( Q R N ) k = γ e f f ( k ) N k , γ e f f ( k ) = P N ( k ) N k − Γ N ( k ) , (225) (Q_{\mathcal R}N)_k = \gamma_{\mathrm{eff}}(k)N_k,\qquad\gamma_{\mathrm{eff}}(k) = \frac{P_N(k)}{N_k} - \Gamma_N(k), \tag*{(225)} ( Q R N ) k = γ eff ( k ) N k , γ eff ( k ) = N k P N ( k ) − Γ N ( k ) , (225) is useful diagnostically, but it does not make the interaction local. For scale-local quartets, both terms have the same homogeneity,
P N ( k ) N ( k ) ∼ Γ N ( k ) ∼ ϵ μ 2 k 19 / 2 − a − x . (226) \frac{P_N(k)}{N(k)} \sim\Gamma_N(k) \sim\epsilon_\mu^2 k^{19/2-a-x}. \tag*{(226)} N ( k ) P N ( k ) ∼ Γ N ( k ) ∼ ϵ μ 2 k 19/2 − a − x . (226) Thus production and loss act on the quartic time scale (207) . Their difference, rather than either contribution separately, determines wave growth.
Nonlocal wave-growth mechanism. For an individual quartet, the sign of its contribution is controlled by
R k ℓ p q : = N p N q N k ( N p + N q ) . (227) \mathcal{R}_{k\ell pq} := \frac{N_pN_q}{N_k(N_p+N_q)}. \tag*{(227)} R k ℓ pq := N k ( N p + N q ) N p N q . (227) It contributes to growth of mode k k k R k ℓ p q > 1 \mathcal{R}_{k\ell pq}>1 R k ℓ pq > 1 R k ℓ p q < 1 \mathcal{R}_{k\ell pq}<1 R k ℓ pq < 1 R k ℓ p q \mathcal{R}_{k\ell pq} R k ℓ pq (227) determine the sign of each contribution.
Assume N r ∼ r − x N_r\sim r^{-x} N r ∼ r − x x > 0 x>0 x > 0
R k ℓ p q ∼ 1 2 ( k K ) x ≪ 1. \mathcal{R}_{k\ell pq}\sim\frac{1}{2}\left(\frac{k}{K}\right)^x\ll1. R k ℓ pq ∼ 2 1 ( K k ) x ≪ 1. Interactions with much shorter resolved and unresolved modes therefore tend to remove action from the long resolved mode k k k
R k ℓ p q ∼ ( k K ) x ≪ 1. \mathcal{R}_{k\ell pq}\sim\left(\frac{k}{K}\right)^x\ll1. R k ℓ pq ∼ ( K k ) x ≪ 1. With the corrected loss N k ( N p + N q ) N_k(N_p+N_q) N k ( N p + N q ) N k N p N_kN_p N k N p
R k ℓ p q ∼ 1 2 ( k κ ) x \mathcal{R}_{k\ell pq} \sim\frac{1}{2}\left(\frac{k}{\kappa}\right)^{x} R k ℓ pq ∼ 2 1 ( κ k ) x Consequently, two energetic lower-wavenumber modes feed mode k k k
κ < 2 − 1 / x k , \kappa< 2^{-1/x}k, κ < 2 − 1/ x k , provided that ℓ \ell ℓ
These estimates are conditional on the resonance manifold
p + q = k + ℓ , ω p + ω q = ω k + ω ℓ . p+q=k+\ell,\qquad\omega_p+\omega_q=\omega_k+\omega_\ell. p + q = k + ℓ , ω p + ω q = ω k + ω ℓ . In particular, the asymptotic configuration p , q ≪ k p,q\ll k p , q ≪ k ℓ ∼ k \ell\sim k ℓ ∼ k p p p q q q R ^ ℓ ( Ω ) ∝ ( γ ℓ 2 + Ω 2 ) − 1 \widehat{R}_{\ell}(\Omega)\propto\left(\gamma_{\ell}^{2}+\Omega^{2}\right)^{-1} R ℓ ( Ω ) ∝ ( γ ℓ 2 + Ω 2 ) − 1
The sign of the integrated growth rate γ e f f ( k ) \gamma_{\mathrm{eff}}(k) γ eff ( k ) (227) exhibits a clear turbulence-mediated growth mechanism. Energetic lower-wavenumber resolved modes provide the quadratic production N p N q N_pN_q N p N q A ℓ \mathcal{A}_{\ell} A ℓ k k k p p p q q q
Writing ∂ t N k = γ e f f ( k ) N k \partial_tN_k=\gamma_{\mathrm{eff}}(k)N_k ∂ t N k = γ eff ( k ) N k γ e f f \gamma_{\mathrm{eff}} γ eff
The strain exponent controls the strength of this mechanism through the transported carrier density
A k ∼ k − a , a = 2 + s + 2 α + β . \mathcal{A}_k\sim k^{-a},\qquad a=2+s+2\alpha+\beta. A k ∼ k − a , a = 2 + s + 2 α + β . Increasing s s s s s s (227) .
The equilibrium closure (212) –(213) does not require the carrier to become a fully Hamiltonian wave field. It is compatible with a bath initialized through A ℓ ( 0 ) \mathcal{A}_{\ell}(0) A ℓ ( 0 ) (163) . The OU forcing and damping would still appear as external injection and dissipation in such an enlarged system.
Spectral-peak evolution
While the preceding analysis characterizes stationary power-law solutions, many ocean-wave spectra remain narrow-banded. It is therefore useful to derive an equation for the spectral peak, which provides a reduced description of the competition between stochastic scattering, bath-mediated nonlinear interactions, and external forcing and dissipation. We consider an isotropic modal action density N ( k , t ) N(k,t) N ( k , t ) k p ( t ) k_p(t) k p ( t )
∂ k N ( k p , t ) = 0 , ∂ k k N ( k p , t ) < 0. (228) \partial_k N(k_p,t)=0,\qquad\partial_{kk}N(k_p,t)<0. \tag*{(228)} ∂ k N ( k p , t ) = 0 , ∂ k k N ( k p , t ) < 0. (228) Differentiating the first relation with respect to time gives the exact kinematic identity
k ˙ p = − ∂ k ( ∂ t N ) ∂ k k N ∣ k = k p . (229) \dot{k}_p=-\left.\frac{\partial_k(\partial_tN)}{\partial_{kk}N}\right|_{k=k_p}. \tag*{(229)} k ˙ p = − ∂ k k N ∂ k ( ∂ t N ) k = k p . (229) For the radial kinetic equation
∂ t N = 1 k ∂ k ( k D ( k ) ∂ k N ) + Q R [ N ] + G ( k ) − 2 μ k N , (230) \partial_tN=\frac{1}{k}\partial_k\left(kD(k)\partial_kN\right)+\mathcal{Q}_R[N]+\mathcal{G}(k)-2\mu_kN, \tag*{(230)} ∂ t N = k 1 ∂ k ( k D ( k ) ∂ k N ) + Q R [ N ] + G ( k ) − 2 μ k N , (230) where G \mathcal{G} G
Transport-noise scattering. Denote the radial diffusion operator by
L D N = 1 k ∂ k ( k D ∂ k N ) . \mathcal{L}_D N=\frac{1}{k}\partial_k\left(kD\partial_kN\right). L D N = k 1 ∂ k ( k D ∂ k N ) . It can be expanded as
L D N = D N ′ ′ + ( D ′ + D k ) N ′ . (231) \mathcal{L}_D N=DN^{\prime\prime}+\left(D^\prime+\frac{D}{k}\right)N^\prime. \tag*{(231)} L D N = D N ′′ + ( D ′ + k D ) N ′ . (231) Consequently, its derivative at the peak is
∂ k [ 1 k ∂ k ( k D ∂ k N ) ] ∣ k p = ( 2 D ′ + D k ) N ′ ′ + D N ′ ′ ′ ∣ k p . (232) \left.\partial_k\left[\frac{1}{k}\partial_k\left(kD\partial_kN\right)\right]\right|_{k_p}
=\left.\left(2D^\prime+\frac{D}{k}\right)N^{\prime\prime}+DN^{\prime\prime\prime}\right|_{k_p}. \tag*{(232)} ∂ k [ k 1 ∂ k ( k D ∂ k N ) ] k p = ( 2 D ′ + k D ) N ′′ + D N ′′′ k p . (232) and the corresponding peak velocity is
k ˙ p ( s c ) = − ( 2 D ′ ( k p ) + D ( k p ) k p ) − D ( k p ) N ′ ′ ′ ( k p ) N ′ ′ ( k p ) . (233) \dot{k}_p^{(\mathrm{sc})}=-\left(2D^\prime(k_p)+\frac{D(k_p)}{k_p}\right)-D(k_p)\frac{N^{\prime\prime\prime}(k_p)}{N^{\prime\prime}(k_p)}. \tag*{(233)} k ˙ p ( sc ) = − ( 2 D ′ ( k p ) + k p D ( k p ) ) − D ( k p ) N ′′ ( k p ) N ′′′ ( k p ) . (233) Here and below, primes denote radial derivatives evaluated at the spectral peak when no argument is displayed.
Under the stronger k k k G A ( k , q ) = G 0 ( q ) \mathcal{G}^{A}(k,q)=\mathcal{G}_0(q) G A ( k , q ) = G 0 ( q ) D = D 0 D=D_0 D = D 0
k ˙ p ( s c ) = − D 0 k p − D 0 N ′ ′ ′ ( k p ) N ′ ′ ( k p ) . (234) \dot{k}_p^{(\mathrm{sc})}=-\frac{D_0}{k_p}-D_0\frac{N^{\prime\prime\prime}(k_p)}{N^{\prime\prime}(k_p)}. \tag*{(234)} k ˙ p ( sc ) = − k p D 0 − D 0 N ′′ ( k p ) N ′′′ ( k p ) . (234) For a locally symmetric peak, the second term vanishes. The remaining inverse displacement is a consequence of the radial geometry of two-dimensional diffusion. This radial-geometrical contribution would not appear in a one-dimensional Cartesian diffusion equation.
For a scale-local diffusivity
D ( k ) ≃ D ∗ k m D , D ∗ > 0 , (235) D(k)\simeq D_\ast k^{m_D},\qquad D_\ast>0, \tag*{(235)} D ( k ) ≃ D ∗ k m D , D ∗ > 0 , (235) equation (233) becomes
k ˙ p ( s c ) = − ( 2 m D + 1 ) D ( k p ) k p − D ( k p ) N ′ ′ ′ ( k p ) N ′ ′ ( k p ) . (236) \dot{k}_{p}^{(\mathrm{sc})}=-(2m_D+1)\frac{D(k_p)}{k_p}-D(k_p)\frac{N'''(k_p)}{N''(k_p)}. \tag*{(236)} k ˙ p ( sc ) = − ( 2 m D + 1 ) k p D ( k p ) − D ( k p ) N ′′ ( k p ) N ′′′ ( k p ) . (236) In particular, if the velocity variance and correlation time vary negligibly across the narrow peak, the estimate (179) gives D ( k ) ≃ σ u 2 τ e f f k 4 D(k)\simeq\sigma_u^2\tau_{\mathrm{eff}}k^4 D ( k ) ≃ σ u 2 τ eff k 4
k ˙ p ( s c ) ≃ − 9 σ u 2 τ e f f k p 3 − σ u 2 τ e f f k p 4 N ′ ′ ′ ( k p ) N ′ ′ ( k p ) . (237) \dot{k}_{p}^{(\mathrm{sc})}\simeq-9\sigma_u^2\tau_{\mathrm{eff}}k_p^3-\sigma_u^2\tau_{\mathrm{eff}}k_p^4\frac{N'''(k_p)}{N''(k_p)}. \tag*{(237)} k ˙ p ( sc ) ≃ − 9 σ u 2 τ eff k p 3 − σ u 2 τ eff k p 4 N ′′ ( k p ) N ′′′ ( k p ) . (237) The factor 9, rather than 4, results from the two derivatives of the variable diffusivity together with the radial-geometrical term D / k D/k D / k D D D k 4 k^4 k 4 m D > − 1 / 2 m_D>-1/2 m D > − 1/2
Bath-mediated production–dissipation contribution. From (222) , define the net local exchange
Δ N ( k ) : = P N ( k ) − Γ N ( k ) N ( k ) = ( Q R N ) ( k ) . (238) \Delta_N(k):=P_N(k)-\Gamma_N(k)N(k)=(\mathcal{Q}_R N)(k). \tag*{(238)} Δ N ( k ) := P N ( k ) − Γ N ( k ) N ( k ) = ( Q R N ) ( k ) . (238) Its exact contribution to the peak velocity is
k ˙ p ( R ) = − ∂ k Q R [ N ] N ′ ′ ∣ k p . (239) \dot{k}_{p}^{(R)}=-\left.\frac{\partial_k\mathcal{Q}_R[N]}{N''}\right|_{k_p}. \tag*{(239)} k ˙ p ( R ) = − N ′′ ∂ k Q R [ N ] k p . (239) Because Q R \mathcal{Q}_R Q R Δ N ( k p ) \Delta_N(k_p) Δ N ( k p ) ∂ k Q R [ N ] \partial_k\mathcal{Q}_R[N] ∂ k Q R [ N ]
The earlier production–dissipation interpretation is recovered under an additional scale-local homogeneity approximation. Factoring the homogeneous kernel from the collision operator, write
( Q R N ) ( k ) ≃ ϵ μ 2 k 2 ν Δ ^ N ( k ) , ν = 19 2 − a , (240) (\mathcal{Q}_R N)(k)\simeq\epsilon_\mu^2 k^{2\nu}\widehat{\Delta}_N(k),\qquad\nu=\frac{19}{2}-a, \tag*{(240)} ( Q R N ) ( k ) ≃ ϵ μ 2 k 2 ν Δ N ( k ) , ν = 2 19 − a , (240) where Δ ^ N \widehat{\Delta}_N Δ N N k u ∼ k − y N_k^{\mathrm{u}}\sim k^{-y} N k u ∼ k − y A k = ( g / ω k ) N k u \mathcal{A}_k=(g/\omega_k)N_k^{\mathrm{u}} A k = ( g / ω k ) N k u a = y + 1 / 2 a=y+1/2 a = y + 1/2 ν = 9 − y \nu=9-y ν = 9 − y y = 1 + s y=1+s y = 1 + s ν = 8 − s \nu=8-s ν = 8 − s
Differentiating (240) yields
k ˙ p ( R ) ≃ − ϵ μ 2 k p 2 ν − 1 ν Δ ^ N ( k p ) + k p Δ ^ N ′ ( k p ) N ′ ′ ( k p ) . (241) \dot{k}_{p}^{(R)}\simeq-\epsilon_\mu^2 k_p^{2\nu-1}\frac{\nu\widehat{\Delta}_N(k_p)+k_p\widehat{\Delta}'_N(k_p)}{N''(k_p)}. \tag*{(241)} k ˙ p ( R ) ≃ − ϵ μ 2 k p 2 ν − 1 N ′′ ( k p ) ν Δ N ( k p ) + k p Δ N ′ ( k p ) . (241) Since N ′ ′ ( k p ) < 0 N''(k_p)<0 N ′′ ( k p ) < 0
∣ k p Δ ^ N ′ ( k p ) ∣ ≪ ν ∣ Δ ^ N ( k p ) ∣ , ν > 0 , (242) \left|k_p\widehat{\Delta}'_N(k_p)\right|\ll\nu\left|\widehat{\Delta}_N(k_p)\right|,\qquad\nu>0, \tag*{(242)} k p Δ N ′ ( k p ) ≪ ν Δ N ( k p ) , ν > 0 , (242) then production in excess of loss gives forward migration, whereas loss in excess of production gives inverse migration. For y = 1 + s y=1+s y = 1 + s (241) reads
k ˙ p ( R ) ≃ − ϵ μ 2 k p 7 − s ( 8 − s ) Δ ^ N ( k p ) + k p Δ ^ N ′ ( k p ) N ′ ′ ( k p ) . (243) \dot{k}_{p}^{(R)}\simeq-\epsilon_\mu^2 k_p^{7-s}\frac{(8-s)\widehat{\Delta}_N(k_p)+k_p\widehat{\Delta}'_N(k_p)}{N''(k_p)}. \tag*{(243)} k ˙ p ( R ) ≃ − ϵ μ 2 k p 7 − s N ′′ ( k p ) ( 8 − s ) Δ N ( k p ) + k p Δ N ′ ( k p ) . (243) This is the qualified version of the production–dissipation cascade interpretation: the reduction is not valid when nonlocal resonance geometry makes Δ ^ N \widehat{\Delta}_N Δ N k k k
Carrier-flux closure. Under the equilibrium closure (214), the transported-carrier flux closes the bath-mediated interaction rather than supplying a second independent source. In terms of the shell flux,
Q R [ N ] ( k ) = − 1 2 π k ℏ ∂ k Π C ( k ) . (244) Q_{\mathcal{R}}[N](k)=-\frac{1}{2\pi k^{\hbar}}\partial_k\Pi_{\mathcal{C}}(k). \tag*{(244)} Q R [ N ] ( k ) = − 2 π k ℏ 1 ∂ k Π C ( k ) . (244) and hence
k ˙ p ( R ) = 1 2 π N ′ ′ ( k p ) [ Π C ′ ′ ( k p ) k p − Π C ′ ( k p ) k p 2 ] . (245) \dot{k}_p^{(\mathcal{R})}=\frac{1}{2\pi N''(k_p)}\left[\frac{\Pi_{\mathcal{C}}''(k_p)}{k_p}-\frac{\Pi_{\mathcal{C}}'(k_p)}{k_p^2}\right]. \tag*{(245)} k ˙ p ( R ) = 2 π N ′′ ( k p ) 1 [ k p Π C ′′ ( k p ) − k p 2 Π C ′ ( k p ) ] . (245) Thus the sign of the carrier contribution is determined by the convergence and radial variation of its shell flux. (245) replaces (239) when the equilibrium closure is used; adding the two expressions would count the same exchange twice. Outside this closure, the full nonlocal collision integral must be retained. A genuinely independent unresolved-wave injection may still be included in I \mathcal{I} I
External forcing and dissipation. The remaining contribution is
k ˙ p ( e x t ) = − I ′ − 2 μ k ′ N N ′ ′ ∣ k p . (246) \dot{k}_p^{(\mathrm{ext})}=-\left.\frac{\mathcal{I}'-2\mu_k' N}{N''}\right|_{k_p}. \tag*{(246)} k ˙ p ( ext ) = − N ′′ I ′ − 2 μ k ′ N k p . (246) The term − 2 μ k N ′ -2\mu_k N' − 2 μ k N ′ k p k_p k p
Resulting peak dynamics and regimes. Combining the exact contributions gives
k ˙ p = − ( 2 D ′ ( k p ) + D ( k p ) k p ) − D ( k p ) N ′ ′ ′ ( k p ) N ′ ′ ( k p ) − ∂ k Q R [ N ] ( k p ) + I ′ ( k p ) − 2 μ k p ′ N ( k p ) N ′ ′ ( k p ) . (247) \dot{k}_p=-\left(2D'(k_p)+\frac{D(k_p)}{k_p}\right)-D(k_p)\frac{N'''(k_p)}{N''(k_p)}
-\frac{\partial_k Q_{\mathcal{R}}[N](k_p)+\mathcal{I}'(k_p)-2\mu_{k_p}'N(k_p)}{N''(k_p)}. \tag*{(247)} k ˙ p = − ( 2 D ′ ( k p ) + k p D ( k p ) ) − D ( k p ) N ′′ ( k p ) N ′′′ ( k p ) − N ′′ ( k p ) ∂ k Q R [ N ] ( k p ) + I ′ ( k p ) − 2 μ k p ′ N ( k p ) . (247) When the external contribution is negligible over an inertial range, the relative importance of scattering and bath-mediated interaction can be measured by
R p : = ∣ ∂ k Q R [ N ] ( k p ) ∣ ∣ ( 2 D ′ + D / k ) N ′ ′ + D N ′ ′ ′ ∣ k p . (248) \mathfrak{R}_p:=\frac{\left|\partial_k Q_{\mathcal{R}}[N](k_p)\right|}{\left|(2D'+D/k)N''+DN'''\right|_{k_p}}. \tag*{(248)} R p := ∣ ( 2 D ′ + D / k ) N ′′ + D N ′′′ ∣ k p ∣ ∂ k Q R [ N ] ( k p ) ∣ . (248) This identifies three dynamical regimes:
If R p ≪ 1 \mathfrak{R}_p\ll1 R p ≪ 1 D D D k 4 k^4 k 4
If R p ≫ 1 \mathfrak{R}_p\gg1 R p ≫ 1 ∂ k Q R \partial_k Q_{\mathcal{R}} ∂ k Q R
If R p = O ( 1 ) \mathfrak{R}_p=\mathcal{O}(1) R p = O ( 1 )
Peak migration and cascade should nevertheless be distinguished: a displacement of the maximum diagnoses spectral evolution, whereas a cascade additionally requires a nonzero flux across a wavenumber range. The peak equation therefore supplies a reduced diagnostic of the competition between the two mechanisms, rather than by itself a proof of a constant-flux cascade.
Under the leading-order rate balance (165) , the robust conclusions are the variable-coefficient diffusion tensor (172) , the radial flux relation (181) , and the general peak formula (233) . A power-law spectrum additionally requires the scale-local estimate and explicit velocity and memory scalings in (179) –(183) . For the diffusive branch, the particular exponent 23/6 is obtained only under condition (188) . For the interaction branch supplemented by the resolved–carrier equilibrium closure, the conditional exponent is instead x = ( 9 − s ) / 2 x = (9 - s)/2 x = ( 9 − s ) /2 x = 23 / 6 x = 23/6 x = 23/6 s = 4 / 3 s = 4/3 s = 4/3 (248) .
Numerical comparison between stochastic transport and Hasselmann interactions
The objective of this section is to compare the characteristic transfer rates associated with stochastic transport and with the classical Hasselmann four-wave interaction mechanism under representative deep-water ocean conditions. The comparison is performed using the asymptotic transfer rates derived in the previous sections.
For weakly nonlinear gravity waves, the resonant transfer rate associated with the Hasselmann–Zakharov equation scales as
τ H Z − 1 ∼ ϵ μ 4 ω k , \tau_{HZ}^{-1} \sim\epsilon_{\mu}^{4}\omega_k, τ H Z − 1 ∼ ϵ μ 4 ω k , where ϵ μ \epsilon_{\mu} ϵ μ ω k = g k \omega_k = \sqrt{gk} ω k = g k
The stochastic transport mechanism is characterized by the diffusive transfer rate
τ t r − 1 ∼ k 2 σ u 2 τ c , \tau_{tr}^{-1} \sim k^2\sigma_u^2\tau_c, τ t r − 1 ∼ k 2 σ u 2 τ c , where σ u \sigma_u σ u τ c \tau_c τ c
Throughout this section we compare both mechanisms through the dimensionless ratio
Γ = τ H Z τ t r = k 2 σ u 2 τ c ϵ μ 4 ω k , \Gamma= \frac{\tau_{HZ}}{\tau_{tr}} = \frac{k^2\sigma_u^2\tau_c}{\epsilon_{\mu}^{4}\omega_k}, Γ = τ t r τ H Z = ϵ μ 4 ω k k 2 σ u 2 τ c , Large values (Γ > 1 \Gamma> 1 Γ > 1 Γ < 1 \Gamma< 1 Γ < 1
The Regime Diagram: Transport-Dominated Regimes Under Typical Ocean Parameters
Figure 1 summarizes the competition between the two transfer mechanisms over the parameter space spanned by the wave period T T T S = σ u 2 τ c S = \sigma_u^2\tau_c S = σ u 2 τ c Γ = 1 \Gamma= 1 Γ = 1
Figure 1. Regime diagram and quantitative analysis of wave dynamics. (a) Regime diagram in the parameter space of wave period T T T S = σ u 2 τ c S = \sigma_u^2\tau_c S = σ u 2 τ c Γ = 1 \Gamma= 1 Γ = 1 Γ > 1 \Gamma> 1 Γ > 1 Γ < 1 \Gamma< 1 Γ < 1 σ u \sigma_u σ u τ c = 600 \tau_c = 600 τ c = 600 Γ \Gamma Γ T T T S S S S = 0.1 S = 0.1 S = 0.1 2 ^2 2 Γ \Gamma Γ S S S T = 6 , 8 , 12 , 16 T = 6, 8, 12, 16 T = 6 , 8 , 12 , 16 Γ ∝ S \Gamma\propto S Γ ∝ S Γ \Gamma Γ S = 0.1 S = 0.1 S = 0.1 2 ^2 2
For representative oceanic values S ≃ 0.1 m 2 s − 1 S \simeq0.1\,\mathrm{m}^{2}\,\mathrm{s}^{-1} S ≃ 0.1 m 2 s − 1 σ u ≃ 0.1 m s − 1 \sigma_u \simeq0.1\,\mathrm{m}\,\mathrm{s}^{-1} σ u ≃ 0.1 m s − 1 τ c ≃ 10 s \tau_c \simeq10\,\mathrm{s} τ c ≃ 10 s
Only for substantially weaker unresolved motions (S ≲ 10 − 2 m 2 s − 1 S \lesssim10^{-2}\,\mathrm{m}^{2}\,\mathrm{s}^{-1} S ≲ 1 0 − 2 m 2 s − 1
Figure 1 illustrates the dependence of Γ \Gamma Γ T ≃ 8 s T \simeq8\,\mathrm{s} T ≃ 8 s ϵ μ 4 \epsilon_{\mu}^{4} ϵ μ 4 Γ \Gamma Γ
Sensitivity to transport intensity
The dependence upon unresolved motions is shown in fig:1c –fig:1d .
fig:1c confirms the expected linear dependence Γ ∝ σ u 2 τ c \Gamma\propto\sigma_u^{2}\tau_c Γ ∝ σ u 2 τ c Γ \Gamma Γ
fig:1d summarizes the minimum values of Γ \Gamma Γ 10 3 10^{3} 1 0 3
Spectral dependence
The dependence upon wavenumber is analysed in Figure 2 . For every sea state investigated, Γ ( k ) \Gamma(k) Γ ( k ) τ t r − 1 ∼ k 2 \tau_{\mathrm{tr}}^{-1} \sim k^{2} τ tr − 1 ∼ k 2 τ H Z − 1 ∼ ϵ μ 4 ω k \tau_{\mathrm{HZ}}^{-1} \sim\epsilon_{\mu}^{4}\omega_k τ HZ − 1 ∼ ϵ μ 4 ω k
Figure 2. Multi-panel regime analysis of ocean wave dynamics. (a) Regime indicator Γ = τ H Z / τ t r a n s \Gamma= \tau_{HZ}/\tau_{trans} Γ = τ H Z / τ t r an s k k k Γ = 1 \Gamma= 1 Γ = 1 Γ > 1 \Gamma> 1 Γ > 1 Γ < 1 \Gamma< 1 Γ < 1 Γ > 1 \Gamma> 1 Γ > 1 Γ \Gamma Γ k = 0.2 k = 0.2 k = 0.2 Γ \Gamma Γ H s = 8 H_s = 8 H s = 8 T p = 12 T_p = 12 T p = 12 τ H Z \tau_{HZ} τ H Z τ t r a n s \tau_{trans} τ t r an s ( k , σ u ) (k, \sigma_u) ( k , σ u ) σ u = 0.1 \sigma_u = 0.1 σ u = 0.1 Γ > 1 \Gamma> 1 Γ > 1 σ u \sigma_u σ u Γ < 1 \Gamma< 1 Γ < 1 σ u < 0.03 \sigma_u < 0.03 σ u < 0.03 ϵ μ k = k E k Δ k / ω k \epsilon_{\mu k} = k\sqrt{E_k\Delta k}/\omega_k ϵ μ k = k E k Δ k / ω k τ c = 600 \tau_c = 600 τ c = 600
Near the spectral peak, the increase of steepness enhances nonlinear interactions, leading to the observed minimum. Away from the peak, the rapid decrease of steepness causes Γ \Gamma Γ
The time-scale comparison shown in fig:2c illustrates the resulting separation. For the representative extreme-storm case, transport acts over time scales of a few hours, whereas resonant interactions require several days or longer over most of the spectrum.
Physical interpretation
The numerical results reveal a simple physical picture. Resonant four-wave interactions are intrinsically weak, their characteristic rate varying as the fourth power of the wave steepness. Conversely, stochastic transport depends primarily upon the intensity and persistence of unresolved velocity fluctuations.
Because energetic oceanic flows continuously contain internal waves, submesoscale motions and turbulent currents, the resulting transport intensity σ u 2 τ c \sigma_u^{2}\tau_c σ u 2 τ c
This does not imply that resonant interactions become negligible. Rather, the present analysis suggests that stochastic transport may provide a comparable – or even dominant – mechanism for redistributing wave action under representative ocean conditions. Both processes are expected to coexist, their relative importance being determined by the dimensionless parameter Γ \Gamma Γ
Implications
These results suggest several consequences for ocean-wave modelling. The comparison indicates that unresolved-current scattering may contribute significantly to spectral evolution alongside resonant four-wave interactions. Transport-induced redistribution offers a possible mechanism contributing to the relatively rapid swell evolution observed in the open ocean. In particular, the long-standing problem of underpredicted swell decay may admit a complementary interpretation through scattering by the ocean’s turbulent velocity field, which can redistribute energy across scales and directions. Finally, the present framework predicts enhanced transport effects in regions of strong internal-wave activity, intense submesoscale turbulence, or energetic boundary currents, providing several observational and numerical tests for future work.
Conclusion
We have developed a kinetic description for a coupled multiscale wave system in which large-scale linear surface waves interact with unresolved stochastic fluctuations transported by the resolved flow. Starting from a coupled multiscale stochastic water-wave system, we derived a closed kinetic equation containing two distinct mechanisms: a conservative scattering operator, analogous to wave propagation in random media, and an effective quartic interaction term generated by the transport of unresolved correlation modes. Unlike the classical Hasselmann–Zakharov collision operator, which originates from intrinsic nonlinear wave interactions, the quartic contribution derived here emerges entirely from stochastic multiscale transport.
The resulting kinetic equation exhibits two asymptotic regimes. The scattering operator gives rise to a diffusive redistribution of wave action in wavenumber space, while the quartic transport term produces nonlocal production–dissipation mechanisms and cascade dynamics through its coupling with the transported unresolved spectrum. Depending on the statistical properties of the unresolved fluctuations, the model predicts inverse transfers of wave action together with direct transfers of energy, thereby recovering cascade behaviours analogous to those of weak turbulence, although generated by a fundamentally different physical mechanism.
A numerical comparison with the classical Hasselmann–Zakharov transfer rate was then carried out using representative deep-water sea states. Introducing the dimensionless ratio Γ = τ H Z / τ t r \Gamma= \tau_{HZ}/\tau_{tr} Γ = τ H Z / τ t r Γ \Gamma Γ
This behaviour results from three complementary effects: the persistent scattering generated by unresolved currents, the relatively weak quartic nonlinear transfer associated with realistic wave steepness, and the favourable k 2 k^2 k 2
These results motivate a broader view of wave modelling. Rather than considering unresolved currents solely as perturbations of wave dynamics, they suggest treating stochastic transport as an active mechanism capable of generating spectral redistribution, cascade dynamics, and wave growth. This opens the possibility of extending operational spectral wave models by incorporating transport-induced scattering together with classical resonant interaction source terms. The framework also offers a possible explanation for rapid swell evolution and predicts enhanced transport effects in regions of intense internal-wave activity, submesoscale turbulence, and energetic boundary currents.
More generally, the present work suggests that stochastic transport is not merely a correction to weak turbulence theory, but provides an alternative kinetic pathway through which multiscale ocean dynamics can shape wave spectra. Although the comparison with the Hasselmann equation is based on asymptotic transfer-rate ordering rather than on a direct equivalence between collision operators, it provides a first step toward a unified stochastic kinetic theory integrating resonant wave interactions and transport by unresolved oceanic motions.
The JONSWAP Spectrum
The Joint North Sea Wave Project (JONSWAP) remains one of the most comprehensive field experiments in ocean wave physics. Conducted in 1968-1969 along a 160 km profile extending westward from the island of Sylt into the North Sea, the experiment involved up to thirteen simultaneous wave stations measuring spectra under varying fetch conditions [9 ] . The resulting spectral formulation has become a cornerstone of ocean engineering and wave modeling [11 , 18 ] .
The JONSWAP spectrum is an extension of the Pierson-Moskowitz spectrum for fetch-limited, developing seas [15 ] . It is expressed as:
S ( ω ) = α g 2 ω − 5 exp [ − 5 4 ( ω p ω ) 4 ] γ exp [ − ( ω − ω p ) 2 2 σ 2 ω p 2 ] (249) S(\omega) = \alpha g^2 \omega^{-5} \exp\left[-\frac{5}{4}\left(\frac{\omega_p}{\omega}\right)^4\right]\gamma^{\exp\left[-\frac{(\omega-\omega_p)^2}{2\sigma^2\omega_p^2}\right]} \tag*{(249)} S ( ω ) = α g 2 ω − 5 exp [ − 4 5 ( ω ω p ) 4 ] γ e x p [ − 2 σ 2 ω p 2 ( ω − ω p ) 2 ] (249) where ω \omega ω ω p \omega_p ω p g g g α \alpha α [14 ] .
The key parameters are:
σ = { 0.07 for ω ≤ ω p 0.09 for ω > ω p (250) \sigma=
\begin{cases}
0.07 & \text{for } \omega\le\omega_p \\
0.09 & \text{for } \omega> \omega_p
\end{cases} \tag*{(250)} σ = { 0.07 0.09 for ω ≤ ω p for ω > ω p (250) This formulation reflects the steeper low-frequency face and gentler high-frequency decay observed in measured spectra [18 ] .
α = 0.076 ( U 10 2 g F ) 0.22 (251) \alpha= 0.076\left(\frac{U_{10}^{2}}{gF}\right)^{0.22} \tag*{(251)} α = 0.076 ( g F U 10 2 ) 0.22 (251) where U 10 U_{10} U 10 F F F [9 ] .
Physical Significance
The JONSWAP spectrum captures several fundamental aspects of wind-wave generation [11 ] :
Fetch-limited growth: Unlike the fully-developed Pierson-Moskowitz spectrum, JONSWAP explicitly accounts for the distance over which wind blows, making it applicable to coastal and regional seas.
Nonlinear interactions: The characteristic overshoot of the spectral peak and its downshift with increasing fetch were explained by Hasselman et al [9 ] as resulting from resonant four-wave interactions [7 ] .
Energy balance: The JONSWAP measurements revealed that approximately 80-90% of the momentum transferred from wind to waves is subsequently transferred to currents via nonlinear interactions and dissipation [9 ] .
Parameterization for This Study
For the present analysis, we adopt the widely-used parameterization in terms of significant wave height H s H_s H s T p = 2 π / ω p T_p = 2\pi/\omega_p T p = 2 π / ω p γ = 3.3 \gamma= 3.3 γ = 3.3 [5 , 18 ] . The spectrum is computed as:
S H ( f ) = α H s 2 T p ( f f p ) − 5 exp [ − 1.25 ( f f p ) − 4 ] γ exp [ − ( f − f p ) 2 2 σ f 2 ] (252) S_H(f) = \alpha H_s^2 T_p \left(\frac{f}{f_p}\right)^{-5} \exp\left[-1.25\left(\frac{f}{f_p}\right)^{-4}\right]\gamma^{\exp\left[-\frac{(f-f_p)^2}{2\sigma_f^2}\right]} \tag*{(252)} S H ( f ) = α H s 2 T p ( f p f ) − 5 exp [ − 1.25 ( f p f ) − 4 ] γ e x p [ − 2 σ f 2 ( f − f p ) 2 ] (252) where the normalization factor α \alpha α H s H_s H s [5 ] . Following Goda’s approximation [5 ] , this formulation provides a robust basis for deriving wave steepness across the frequency range.
The steepness distribution ϵ μ ( k ) \epsilon_\mu(k) ϵ μ ( k )
ϵ μ k = k E k Δ k ω k (253) \epsilon_{\mu k} = k\sqrt{\frac{E_k\Delta k}{\omega_k}} \tag*{(253)} ϵ μ k = k ω k E k Δ k (253) where E k = S H ( f ) ⋅ d f d k = S H ( f ) ⋅ c g ω k E_k = S_H(f)\cdot\frac{\mathrm{d}f}{\mathrm{d}k} = S_H(f)\cdot\frac{c_g}{\omega_k} E k = S H ( f ) ⋅ d k d f = S H ( f ) ⋅ ω k c g c g c_g c g Δ k \Delta k Δ k
Functional Martingale central limit theorem (7.1.4) Ethier-Kurtz-86
Let
M n = ( M n , 1 , … , M n , d ) M^n = (M^{n,1},\ldots,M^{n,d}) M n = ( M n , 1 , … , M n , d ) be a sequence of R d \mathbb{R}^d R d M 0 n ⇒ 0 M^n_0 \Rightarrow0 M 0 n ⇒ 0
C ( t ) = ( C i j ( t ) ) 1 ≤ i , j ≤ d , t ≥ 0 , C(t) = \left(C^{ij}(t)\right)_{1\le i,j\le d}, \qquad t \ge0, C ( t ) = ( C ij ( t ) ) 1 ≤ i , j ≤ d , t ≥ 0 , be a deterministic continuous matrix-valued function such that C ( 0 ) = 0 C(0) = 0 C ( 0 ) = 0 C ( t ) − C ( s ) C(t) - C(s) C ( t ) − C ( s ) t ≥ s t \ge s t ≥ s
Assume that one of the following two sets of conditions holds. (a) Optional quadratic-covariation formulation. For every T > 0 T > 0 T > 0
E [ sup 0 ≤ t ≤ T ∣ Δ M t n ∣ ] ⟶ 0 , \mathbb{E}\left[\sup_{0 \le t \le T} \left|\Delta M_t^n\right|\right] \longrightarrow0, E [ 0 ≤ t ≤ T sup ∣ Δ M t n ∣ ] ⟶ 0 , and, for every ( i , j ) (i,j) ( i , j ) t ≥ 0 t \ge0 t ≥ 0
[ M n , i , M n , j ] t ⟶ C i j ( t ) [M^{n,i}, M^{n,j}]_t \longrightarrow C^{ij}(t) [ M n , i , M n , j ] t ⟶ C ij ( t ) in probability.
(b) Predictable quadratic-covariation formulation. For every ( T ≥ 0 ) (T \ge0) ( T ≥ 0 )
E [ sup 0 ≤ t ≤ T ∣ Δ M t n ∣ 2 ] ⟶ 0 , \mathbb{E}\left[\sup_{0 \le t \le T} \left|\Delta M_t^n\right|^2\right] \longrightarrow0, E [ 0 ≤ t ≤ T sup ∣ Δ M t n ∣ 2 ] ⟶ 0 , and, for every ( i , j ) (i,j) ( i , j ) t ≥ 0 t \ge0 t ≥ 0
⟨ M n , i , M n , j ⟩ t ⟶ C i j ( t ) \langle M^{n,i}, M^{n,j}\rangle_t \longrightarrow C^{ij}(t) ⟨ M n , i , M n , j ⟩ t ⟶ C ij ( t ) in probability.
Then
M n ⟹ M in D ( [ 0 , ∞ ) , R d ) , M^n \Longrightarrow M \qquad\text{in} \qquad D([0,\infty),\mathbb{R}^d), M n ⟹ M in D ([ 0 , ∞ ) , R d ) , where D D D M M M
E [ M ( t ) M ( t ) T ] = C ( t ) . \mathbb{E}\left[M(t)M(t)^{\mathsf{T}}\right] = C(t). E [ M ( t ) M ( t ) T ] = C ( t ) . 
