Stochastic Variational Formulations of Fluid Wave-Current Interaction

Abstract

We are dealing with multi-scale, multi-physics uncertainty in modelling wave-current interaction (WCI) in a stochastic fluid flow. The objective is to introduce stochasticity into WCI for the purpose of quantifying the uncertainty associated with the wave physics in either the generalised Lagrangian mean (GLM) model, or the alternative 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 vector field Lagrangian for the current flow and a phase-space Lagrangian for the wave field. The wave-current coupling is accomplished for GLM by pairing the Lagrangian-mean velocity of the current flow with the momentum map of the Hamiltonian wave system. To demonstrate the applicability of this hybrid approach, we use our wave-current Hamilton's principle approach to close both the deterministic and stochastic GLM equations for a 3D Euler-Boussinesq (EB) fluid. We also apply the method to add wave physics and stochasticity to the familiar 1D shallow water flow model.The appendices discuss finite dimensional analogues of WCI as well as comparing the differences in approaches for the stochastic GLM and CL models. The differences in the types of stochasticity can be seen in the Kelvin circulation theorems for the two theories. The GLM model acquires stochasticity in its Lagrangian transport velocity for the currents and in its group velocity for the waves. However, the CL model is based on modifying the Eulerian velocity in the integrand of the Kelvin circulation. Appendix C shows that 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.