Stochastic Variational Formulations of Fluid Wave-Current Interaction.

Area of Science:

• Fluid Dynamics
• Wave Propagation
• Stochastic Modeling

Background:

• Wave-current interaction (WCI) is a complex multiscale, multi-physics phenomenon.
• Existing models often lack comprehensive uncertainty quantification.

Purpose of the Study:

• To introduce stochasticity into two classic WCI models: generalized Lagrangian mean (GLM) and Craik-Leibovich (CL).
• To compare the distinct ways stochasticity manifests in the GLM and CL frameworks.
• To analyze WCI within a generalized stochastic Hamilton's principle for 3D Euler-Boussinesq fluids.

Main Methods:

• Incorporating stochasticity into the wave dynamics of GLM and CL models.
• Separating Lagrangian (fluid) and Eulerian (wave) degrees of freedom using Hamilton's principle for GLM.
• Coupling Euler-Poincaré reduced Lagrangian for currents with a phase-space Lagrangian for waves.
• Applying GLM and CL methods to 1D and 2D shallow water flow models.
• Comparing Kelvin circulation theorems for stochastic GLM and CL models.

Main Results:

• GLM introduces stochasticity in Lagrangian transport velocity and wave group velocity.
• CL allows stochasticity in both the integrand and circulation loop's Lagrangian transport velocity.
• Differences in stochasticity arise from the definition of Eulerian velocity relative to Stokes drift.
• The study provides a unified framework using stochastic Hamilton's principle for 3D Euler-Boussinesq fluids.

Conclusions:

• The GLM and CL models offer distinct yet complementary approaches to modeling WCI uncertainty.
• Stochasticity in WCI can be systematically incorporated into fluid dynamics models.
• Understanding these stochastic differences is crucial for accurate multiscale WCI simulations.