2014/09/02 by Sergei Kuksin, Kuksin, Sergei, Alberto Maiocchi +1
Mathematics · #FOS: Physical sciences #Mathematical Physics (math-ph) #Navier-Stokes equation solutions #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1409.0652
openalex publication_date 2014/09/02 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
We consider the 2d quasigeostrophic equation on the β-plane for the stream function ψ, with dissipation and a random force: (*) (-Δ+K)ψt - ρJ(ψ, Δψ) -βψx= ⟨ random force⟩ -κΔ2ψ+Δψ, where ψ=ψ(t,x,y), x∈ℝ/2πLℤ, y∈ ℝ/2πℤ. For typical values of the horizontal period L we prove that the law of the action-vector of a solution for (*) (formed by the halves of the squared norms of its complex Fourier coefficients) converges, as β→∞, to the law of an action-vector for solution of an auxiliary effective equation, and the stationary distribution of the action-vector for solutions of (*) converges to that of the effective equation. Moreover, this convergence is uniform in κ∈(0,1]. The effective equation is an infinite system of stochastic equations which splits into invariant subsystems of complex dimension ≤3; each of these subsystems is an integrable hamiltonian system, coupled with a Langevin thermostat. Under the iterated limits limL=ρ→∞ limβ→∞ and limκ→ 0 limβ→∞ we get similar systems. In particular, none of the three limiting systems exhibits the energy cascade to high frequencies.