Symmetric instability in a Boussinesq fluid on a rotating planet

Abstract: Symmetric instability has broad applications in geophysical and planetary fluid dynamics. It plays a crucial role in the formation of mesoscale rainbands at mid-latitudes on Earth, instability in the ocean's mixed layer, and slantwise convection on gas giants and icy moon oceans. Here, we apply linear instability analysis to an arbitrary zonally symmetric Boussinesq flow on a rotating spherical planet, with applicability to icy moon oceans. We characterize the instabilities into three types: (1) gravitational instability, occurring when stratification is unstable along angular momentum surfaces, (2) inertial instability, occurring when angular momentum shear is unstable along buoyancy surfaces, and (3) a mixed symmetric instability, occurring when neither of the previous conditions are fulfilled, but the potential vorticity has the opposite sign as planetary rotation. We note that $N^2<0$ where $N$ is the Brunt--V\"ais\"al\"a frequency -- a typical criterion used to trigger convective adjustment in global ocean models -- is neither necessary nor sufficient for instability. Instead, $b_z \sin{\theta_0}<0$, where $b_z$ is the stratification along the planetary rotation axis and $\theta_0$ is the local latitude, is always sufficient for instability and also necessary in the low Rossby number limit. In this limit, relevant for convection in icy moon oceans, the most unstable mode is slantwise convection parallel to the planetary rotation axis. This slantwise convection differs from the parameterized convection in existing general circulation models, whose convective schemes parameterize convection in the direction of gravity. Our results suggest that convection schemes in global ocean models must be revised before being applied to icy moon oceans.