Navier--Stokes equations on the $β$-plane: determining modes and nodes
We revisit the 2d Navier--Stokes equations on the periodic $β$-plane, with the Coriolis parameter varying as $βy$, and obtain bounds on the number of determining modes and nodes of the flow. The number of modes {and nodes} scale as $cG_0^{1/2} + c'(M/β)^{1/2}$ and $cG_0^{2/3} + c'(M/β)^{1/2}$ respectively, where the Grashof number $G_0=|f_v|_{L^2}^{}/(μ^2κ_0^2)$ and $M$ involves higher derivatives of the forcing $f_v$. For large $β$ (strong rotation), this results in fewer degrees of freedom than the classical (non-rotating) bound that scales as $cG_0$.
math.AP↗