Characteristic length scales of strongly rotating Boussinesq flow in variable-aspect-ratio domains

Abstract

We quantify the variability of the characteristic length scales of isotropically forced Boussinesq flows with stratification and frame rotation, as functions of the ratio $N/f$ of the Brunt–Väisälä frequency to the Coriolis frequency. The parameter ranges $0<N<f$, domain aspect ratio $1\leqslant \unicode[STIX]{x1D6FF}_{d}\leqslant 32$ and Burger number $Bu=\unicode[STIX]{x1D6FF}_{d}N/f\leqslant 1$ are explored for two values of $f$, one resulting in linear potential vorticity and the other in nonlinear potential vorticity. Characteristic length scales of the wave and vortical linear eigenmodes are separately quantified using $n$th-order spectral moments in both horizontal and vertical directions, for integer $n\leqslant 3$. In flows with linear potential vorticity, the horizontal vortical length scale $L_{0}$, characterizing a typical width of columnar structures, grows as ${\sim}(N/f)^{1/2}$ at all orders of $n$, regardless of domain aspect ratio. In unit-aspect-ratio domains, when intermediate scales are measured by filtering out the largest scales and using higher-order moments $n>1$, the vortical-mode aspect ratio $\unicode[STIX]{x1D6FF}_{0}$ asymptotes to a scaling of ${\sim}(N/f)^{-1}$, in agreement with quasi-geostrophic estimates. In contrast, the $\unicode[STIX]{x1D6FF}_{0}$ in tall-aspect-ratio domain flows yields a decay rate of at most ${\sim}(N/f)^{-1/2}$ after large-scale filtering. Flows with nonlinear potential vorticity display consistently weaker dependence of the characteristic scales on $N/f$ than the corresponding ones with linear potential vorticity. The wave-mode aspect ratios for all flows are essentially independent of $N/f$. We highlight the differences of these flow structure scalings relative to those expected for quasi-geostrophic flows, and those observed in strongly stratified, non-quasi-geostrophic flows.