English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider the 2d quasigeostrophic equation on the $\beta$-plane for thestream function $\psi$, with dissipation and a random force: $$ (*)\qquad(-\Delta +K)\psi_t - \rho J(\psi, \Delta\psi) -\beta\psi_x= \langle\text{random force}\rangle -\kappa\Delta^2\psi +\Delta\psi, $$ where$\psi=\psi(t,x,y), \ x\in\mathbb{R}/2\pi L\mathbb{Z}, \ y\in \mathbb{R}/2\pi\mathbb{Z}$. For typical values of the horizontal period $L$ we prove that thelaw of the action-vector of a solution for $(*)$ (formed by the halves of thesquared norms of its complex Fourier coefficients) converges, as$\beta\to\infty$, to the law of an action-vector for solution of an auxiliaryeffective equation, and the stationary distribution of the action-vector forsolutions of $(*)$ converges to that of the effective equation. Moreover, thisconvergence is uniform in $\kappa\in(0,1]$. The effective equation is aninfinite system of stochastic equations which splits into invariant subsystemsof complex dimension $\le3$; each of these subsystems is an integrablehamiltonian system, coupled with a Langevin thermostat. Under the iteratedlimits $\lim_{L=\rho\to\infty} \lim_{\beta\to\infty}$ and $\lim_{\kappa\to 0}\lim_{\beta\to\infty}$ we get similar systems. In particular, none of the threelimiting systems exhibits the energy cascade to high frequencies.

The limit of small Rossby numbers for randomly forced quasi-geostrophic equation on $\beta$-plane