Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics

Abstract

In {\em{Holm}, Proc. Roy. Soc. A 471 (2015)} stochastic fluid equations were derived by employing a variational principle with an assumed stochastic Lagrangian particle dynamics. Here we show that the same stochastic Lagrangian dynamics naturally arises in a multi-scale decomposition of the deterministic Lagrangian flow map into a slow large-scale mean and a rapidly fluctuating small scale map. We employ homogenization theory to derive effective slow stochastic particle dynamics for the resolved mean part, thereby justifying stochastic fluid partial equations in the Eulerian formulation. To justify the application of rigorous homogenization theory, we assume mildly chaotic fast small-scale dynamics, as well as a centering condition. The latter requires that the mean of the fluctuating deviations is small, when pulled back to the mean flow.

11 pages, geometric mechanics, stochastic fluid models, stochastic processes, multi-scale fluid dynamics, symmetry reduced variational principles, homogenisation

References in corpus (10)

• Variational Principles for Stochastic Fluid Dynamics
• Almost Sure Invariance Principle For Nonuniformly Hyperbolic Systems
• Fluid flow dynamics under location uncertainty
• Solution properties of a 3D stochastic Euler fluid equation
• Geophysical flows under location uncertainty, Part II: Quasi-geostrophy and efficient ensemble spreading
• Geophysical flows under location uncertainty, Part I Random transport and general models
• A Note on Diffusion Limits of Chaotic Skew Product Flows
• Homogenization for Deterministic Maps and Multiplicative Noise
• Geophysical flows under location uncertainty, Part III: SQG and frontal dynamics under strong turbulence conditions
• A Huygens principle for diffusion and anomalous diffusion in spatially extended systems

Cited by in corpus (29)

• Solution properties of a 3D stochastic Euler fluid equation
• Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model
• Theoretical tools for understanding the climate crisis from Hasselmann's program and beyond
• Noise and dissipation on coadjoint orbits
• Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows
• 2D Euler equations with Stratonovich transport noise as a large scale stochastic model reduction
• Semi-martingale driven variational principles
• Implications of Kunita-Itô-Wentzell formula for $k$-forms in stochastic fluid dynamics
• Global existence and non-uniqueness of 3D Euler equations perturbed by transport noise
• Stochastic effects of waves on currents in the ocean mixed layer
• Stochastic Closures for Wave--Current Interaction Dynamics
• Circulation and Energy Theorem Preserving Stochastic Fluids
• Data-driven stochastic Lie transport modelling of the 2D Euler equations
• Lagrangian averaged stochastic advection by Lie transport for fluids
• Lyapunov Exponents of Two Stochastic Lorenz 63 Systems
• Stochastic Parametrization of the Richardson Triple
• New variational and multisymplectic formulations of the Euler-Poincaré equation on the Virasoro-Bott group using the inverse map
• Stochastic variational principles for the collisional Vlasov-Maxwell and Vlasov-Poisson equations
• Deterministic and Stochastic Euler-Boussinesq Convection
• Stochastic Modelling in Fluid Dynamics: Itô vs Stratonovich
• A Hamiltonian Mean-Field System for the Navier-Stokes Equation
• Variational Stochastic Parameterisations and their Applications to Primitive Equation Models
• Dynamics of non-holonomic systems with stochastic transport
• Effect of Transport Noise on Kelvin-Helmholtz instability
• An eddifying Stommel model: Fast eddy effects in a two-box ocean
• A local-in-time theory for singular SDEs with applications to fluid models with transport noise
• A Hamiltonian interacting particle system for compressible flow
• A model under location uncertainty to predict the mean velocity in wall bounded flows
• Well-posedness by noise for linear advection of $k$-forms