Kinetic theory of jet dynamics in the stochastic barotropic and 2D Navier-Stokes equations

Abstract

We discuss the dynamics of zonal (or unidirectional) jets for barotropic flows forced by Gaussian stochastic fields with white in time correlation functions. This problem contains the stochastic dynamics of 2D Navier-Stokes equation as a special case. We consider the limit of weak forces and dissipation, when there is a time scale separation between the inertial time scale (fast) and the spin-up or spin-down time (large) needed to reach an average energy balance. In this limit, we show that an adiabatic reduction (or stochastic averaging) of the dynamics can be performed. We then obtain a kinetic equation that describes the slow evolution of zonal jets over a very long time scale, where the effect of non-zonal turbulence has been integrated out. The main theoretical difficulty, achieved in this work, is to analyze the stationary distribution of a Lyapunov equation that describes quasi-Gaussian fluctuations around each zonal jet, in the inertial limit. This is necessary to prove that there is no ultraviolet divergence at leading order in such a way that the asymptotic expansion is self-consistent. We obtain at leading order a Fokker--Planck equation, associated to a stochastic kinetic equation, that describes the slow jet dynamics. Its deterministic part is related to well known phenomenological theories (for instance Stochastic Structural Stability Theory) and to quasi-linear approximations, whereas the stochastic part allows to go beyond the computation of the most probable zonal jet. We argue that the effect of the stochastic part may be of huge importance when, as for instance in the proximity of phase transitions, more than one attractor of the dynamics is present.

References in corpus (8)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Spectral gaps in Wasserstein distances and the 2D stochastic Navier--Stokes equations
• Growing condensate in two-dimensional turbulence
• Random changes of flow topology in two dimensional and geophysical turbulence
• Direct Statistical Simulation of Out-of-Equilibrium Jets
• Statistics of the General Circulation from Cumulant Expansions
• Algebraic Correlation Function and Anomalous Diffusion in the HMF model
• Large deviations from a stationary measure for a class of dissipative PDE's with random kicks

Cited by in corpus (40)

• Mathematical and Physical Ideas for Climate Science
• The Generalized Quasilinear Approximation: Application to Zonal Jets
• A rare event algorithm links transitions in turbulent flows with activated nucleations
• Perturbative calculation of quasi-potential in non-equilibrium diffusions: a mean-field example
• Zonal Flow as Pattern Formation
• A statistical state dynamics-based study of the structure and mechanism of large-scale motions in plane Poiseuille flow
• Global climate modeling of Saturn's atmosphere. Part II: multi-annual high-resolution dynamical simulations
• Turbulence statistics in a 2D vortex condensate
• Fluctuation-induced phase separation in metric and topological models of collective motion
• Why Eddy Momentum Fluxes are Concentrated in the Upper Troposphere
• Recent Developments in Theories of Inhomogeneous and Anisotropic Turbulence
• Langevin dynamics, large deviations and instantons for the quasi-geostrophic model and two-dimensional Euler equations
• Vortex-density fluctuations, energy spectra and vortical regions in superfluid turbulence
• Large Deviations in Fast-Slow Systems
• A theory for the emergence of coherent structures in beta-plane turbulence
• Statistical state dynamics of jet/wave coexistence in barotropic beta-plane turbulence
• Cumulant expansions for atmospheric flows
• Multistability and rare spontaneous transitions in barotropic $\beta$-plane turbulence
• Angular momentum transport by the GSF instability: nonlinear simulations at the equator
• S3T stability of the homogeneous state of barotropic beta-plane turbulence
• The culmination of an inverse cascade: mean flow and fluctuations
• The Acceleration of Superrotation in Simulated Hot Jupiter Atmospheres
• Theoretical prediction of Reynolds stresses and velocity profiles for barotropic turbulent jets
• Generalised quasi-linear approximation of the HMRI
• Statistical state dynamics based theory for the formation and equilibration of Saturn's north polar jet
• Dynamical large deviations for homogeneous systems with long range interactions and the Balescu--Guernsey--Lenard equation
• Multiple scales analysis of slow--fast quasilinear systems
• Statistical state dynamics of weak jets in barotropic beta-plane turbulence
• Direct Statistical Simulation of a Jet
• Numerical simulation of the geometrical-optics reduction of CE2 and comparisons to quasilinear dynamics
• Sample-path large deviations for stochastic evolutions driven by the square of a Gaussian process
• Lyapunov exponents of stochastic systems---from micro to macro
• Fluctuations and large deviations of Reynolds stresses in zonal jet dynamics
• Statistics of inhomogeneous turbulence in large scale quasi-geostrophic dynamics
• A large deviation principle for Markovian slow-fast systems
• The quasilinear theory in the approach of long-range systems to quasi-stationary states
• On statistical zonostrophic instability and the effect of magnetic fields
• Fluctuations of large-scale jets in the stochastic 2D Euler equation
• Zonal Flows and Turbulence in Fluids and Plasmas
• Large Deviations of Irreversible Processes