Variational principles for fully coupled stochastic fluid dynamics across scales

TL;DR

This paper develops a variational framework to couple large-scale stochastic fluid dynamics with evolving small-scale noise, yielding a linearized Euler equation for the small-scale velocity with random coefficients driven by the large-scale transport. It clarifies the connections and distinctions between LU and SALT, and introduces a two-stage variational approach that determines noise directions via expectation and decorrelated regularized noise, including an OU-based closure. The resulting coupled system provides a rigorous basis for representing large-scale flow with embedded small-scale uncertainties, linking to Kraichnan's sweeping hypothesis and established stochastic climate models. The approach offers a path toward more reliable multi-scale geophysical predictions and sets the stage for extensions to primitive equations and wave–current coupling, along with mathematical analysis of weak solutions.

Abstract

This work investigates variational frameworks for modeling stochastic dynamics in incompressible fluids, focusing on large-scale fluid behavior alongside small-scale stochastic processes. The authors aim to develop a coupled system of equations that captures both scales, using a variational principle formulated with Lagrangians defined on the full flow, and incorporating stochastic transport constraints. The approach smooths the noise term along time, leading to stochastic dynamics as a regularization parameter approaches zero. Initially, fixed noise terms are considered, resulting in a generalized stochastic Euler equation, which becomes problematic as the regularization parameter diminishes. The study then examines connections with existing stochastic frameworks and proposes a new variational principle that couples noise dynamics with large-scale fluid motion. This comprehensive framework provides a stochastic representation of large-scale dynamics while accounting for fine-scale components. Our main result is that the evolution of the small-scale velocity component is governed by a linearized Euler equation with random coefficients, influenced by large-scale transport, stretching, and pressure forcing.