Ergodicity of Truncated Stochastic Navier Stokes with Deterministic Forcing and Dispersion

Andrew J. Majda, Xin T. Tong

Research output: Contribution to journalArticlepeer-review


Turbulence in idealized geophysical flows is a very rich and important topic. The anisotropic effects of explicit deterministic forcing, dispersive effects from rotation due to the β-plane and F-plane, and topography together with random forcing all combine to produce a remarkable number of realistic phenomena. These effects have been studied through careful numerical experiments in the truncated geophysical models. These important results include transitions between coherent jets and vortices, and direct and inverse turbulence cascades as parameters are varied, and it is a contemporary challenge to explain these diverse statistical predictions. Here we contribute to these issues by proving with full mathematical rigor that for any values of the deterministic forcing, the β- and F-plane effects and topography, with minimal stochastic forcing, there is geometric ergodicity for any finite Galerkin truncation. This means that there is a unique smooth invariant measure which attracts all statistical initial data at an exponential rate. In particular, this rigorous statistical theory guarantees that there are no bifurcations to multiple stable and unstable statistical steady states as geophysical parameters are varied in contrast to claims in the applied literature. The proof utilizes a new statistical Lyapunov function to account for enstrophy exchanges between the statistical mean and the variance fluctuations due to the deterministic forcing. It also requires careful proofs of hypoellipticity with geophysical effects and uses geometric control theory to establish reachability. To illustrate the necessity of these conditions, a two-dimensional example is developed which has the square of the Euclidean norm as the Lyapunov function and is hypoelliptic with nonzero noise forcing, yet fails to be reachable or ergodic.

Original languageEnglish (US)
Pages (from-to)1483-1506
Number of pages24
JournalJournal of Nonlinear Science
Issue number5
StatePublished - Oct 1 2016


  • Beta plane
  • Exponential attraction
  • General dispersion
  • Topography
  • Unique stochastic invariant measure

ASJC Scopus subject areas

  • Modeling and Simulation
  • General Engineering
  • Applied Mathematics


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