Quasi-geostrophic Rayleigh-Bénard convection on the tilted $f$-plane

Abstract

Rapidly rotating Rayleigh-Bénard convection on a $f$-plane at colatitude $\vartheta_f$ is investigated numerically using an asymptotically reduced equation set valid in the limit of very rapid rotation. The equations provide a non-hydrostatic but quasi-geostrophic description in a non-orthogonal coordinate system. The tilt changes the structure of the large-scale barotropic condensate from large-scale vortices to zonal flows as the colatitude of the $f$-plane increases, with bistable states present for certain parameter ranges, extending prior work to a geophysically significant parameter regime. This behaviour is understood through the impact of broken rotation symmetry on the barotropic source terms resulting from baroclinic vortical stresses and baroclinic torque. As the tilt angle $\vartheta_f$ increases, global heat and momentum transport is reduced relative to upright-polar convection, a result that is explained through linear theory and nonlinear power maps both of which demonstrate increased attenuation of the domain of dynamically active spatial scales as the convective modes depart from a North-South alignment in the horizontal plane. A key finding is that the predominance of lateral thermal mixing allows for the maintenance of a persistent unstable mean temperature gradient that saturates at increasing forcing levels and remains insensitive to the colatitude.

Figures (20)

• Figure 1: Local area $f$-plane approximation on the sphere at colatitude $\vartheta_f$ (left plot) and the explicit computational domain (right plot) highlighting the non-orthogonality of the coordinate directions. The rotation vector is given by $\boldsymbol{\Omega}$ and $\eta$ is the coordinate in this direction. Coordinates $(x,y,\eta)$ increase in the eastward, northward and axial directions, respectively.
• Figure 2: (a) Marginal stability curves showing the reduced latitudinal Rayleigh number $\widetilde{Ra}$ vs the horizontal wavenumber $k_\perp$: North-South rolls (black curve, $\chi = 0$), East-West rolls (gray dashed curves, $\chi = \pi/2$) at colatitudes $\vartheta_{f} =\left ( 15^\circ, 30^\circ, 45^\circ, 60^\circ\right )$. (b) Contour plot of the linear growth rate at $\widetilde{Ra}=120, \vartheta_f=60^\circ$ with zero contour: dashed-dotted line, separatrix contour: dashed line.
• Figure 3: Snapshot of the fluctuating temperature field $\theta$ at $\widetilde{Ra} = 20$, $\vartheta_f = 30^\circ$. Cross-sections in (a) the virtual $y$-$\eta$ meridional plane as defined in equation (\ref{['def:sheared_y']}) and (b) the physical $Y$-$Z$ meridional plane. 3D volume renderings: (c) virtual, and (d) physical. All visualizations are for domain dimensions $10L_c\times 10L_c\times 1$ and are not to scale: horizontal spatial scales are in units of $E_f^{1/3}$.
• Figure 4: As in fig. \ref{['fig:theta_slice_30deg']} but for $\widetilde{Ra} = 120$, $\vartheta_f = 60^\circ$.
• Figure 5: Snapshot of the simulation grid in $\widetilde{Ra}$-$\vartheta_f$ parameter space. In all cases the temperature fluctuation $\theta$ is shown in the $(x,y)$ plane at the edge of the upper thermal boundary layer. Columns (rows) represent fixed $\widetilde{Ra}$ ($\vartheta_f$). The color maps are adjusted for each value of $\widetilde{Ra}$, as indicated in the bottom row.