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arXiv · 1802.08644

Navier--Stokes equations on the $\beta$-plane: determining modes and nodes

Naoko Miyajima·Djoko Wirosoetisno
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Abstract

We revisit the 2d Navier--Stokes equations on the periodic $\beta$-plane, with the Coriolis parameter varying as $\beta y$, and obtain bounds on the number of determining modes and nodes of the flow. The number of modes {and nodes} scale as $cG_0^{1/2} + c'(M/\beta)^{1/2}$ and $cG_0^{2/3} + c'(M/\beta)^{1/2}$ respectively, where the Grashof number $G_0=|f_v|_{L^2}^{}/(\mu^2\kappa_0^2)$ and $M$ involves higher derivatives of the forcing $f_v$. For large $\beta$ (strong rotation), this results in fewer degrees of freedom than the classical (non-rotating) bound that scales as $cG_0$.

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Naoko Miyajima, Djoko Wirosoetisno. 2018-02-23. Navier--Stokes equations on the $\beta$-plane: determining modes and nodes. https://doi.org/10.1016/j.physd.2018.08.005

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