Stochastic Variational Formulations of Fluid Wave–Current Interaction

Abstract

We are modelling multiscale, multi-physics uncertainty in wave–current interaction (WCI). To model uncertainty in WCI, we introduce stochasticity into the wave dynamics of two classic models of WCI, namely the generalised Lagrangian mean (GLM) model and the Craik–Leibovich (CL) model. The key idea for the GLM approach is the separation of the Lagrangian (fluid) and Eulerian (wave) degrees of freedom in Hamilton’s principle. This is done by coupling an Euler–Poincaré reduced Lagrangian for the current flow and a phase-space Lagrangian for the wave field. WCI in the GLM model involves the nonlinear Doppler shift in frequency of the Hamiltonian wave subsystem, which arises because the waves propagate in the frame of motion of the Lagrangian-mean velocity of the current. In contrast, WCI in the CL model arises because the fluid velocity is defined relative to the frame of motion of the Stokes mean drift velocity, which is usually taken to be prescribed, time independent and driven externally. We compare the GLM and CL theories by placing them both into the general framework of a stochastic Hamilton’s principle for a 3D Euler–Boussinesq (EB) fluid in a rotating frame. In other examples, we also apply the GLM and CL methods to add wave physics and stochasticity to the familiar 1D and 2D shallow water flow models. The differences in the types of stochasticity which arise for GLM and CL models can be seen by comparing the Kelvin circulation theorems for the two models. The GLM model acquires stochasticity in its Lagrangian transport velocity for the currents and also in its group velocity for the waves. However, the CL model is based on defining the Eulerian velocity in the integrand of the Kelvin circulation relative to the Stokes drift velocity induced by waves driven externally. Thus, the Kelvin theorem for the stochastic CL model can accept stochasticity in its both its integrand and in the Lagrangian transport velocity of its circulation loop. In an “Appendix”, we also discuss dynamical systems analogues of WCI.

Wave–current interaction on a free surface

“…for v ⋅ n on the boundary 𝜕Ω with normal vector n. 2. Equation ( 44) and the last equation in (46) imply conservation of momentum, defined by…”

“…3. Combining the curl 𝒓 of Equation ( 44) and the last equation in (46) implies conservation of mass-weighted enstrophy, defined by…”

“…We rewrite the continuity equation in (46) and its associated motion equation in (44) as Lagrangian conservation laws for mass and momentum,…”

Semi-martingale driven variational principles

“…As introduced in [44], the action integral for SALT can be coupled to a stochastic wave action which introduces a nonlinear wave propagation. Maintaining notation from the previous section, we introduce conjugate wave variables ( p , q ) 13 and define a coupled action integral in terms of the wave Hamiltonian by right left right left right left right left right left right left3pt0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278emA(u,a,Λ,p,q)=∫t0t1ℓ~(u,a) dt+∫t0t1⟨Λ,da+Ldxta⟩V−∫t0t1⟨p⋄q,dxt⟩frakturX+∫t0t11.623em1.623em(⟨p,dq⟩V−dscriptJ(p,q)1.623em1.623em), where the first integral corresponds to the deterministic fluid action, the second to the stochastic advection constraint, the third to a form of minimal coupling and the fourth to the phase-space wave Lagrangian.…”

Stochastic modelling in fluid dynamics: Itô versus Stratonovich

“…The effects of uncertainty in the statistics of the Stokes mean drift velocity u S in the context of the CL model has been treated in [27], as well. However, no self consistent dynamical theory of the Stokes drift u S has been developed yet, to our knowledge.…”