Bistability and hysteresis of axisymmetric thermal convection between differentially rotating spheres

• Published in: Journal of fluid mechanics 2021-01, Vol.911, Article A12
• Main Authors: Mannix, P.M., Mestel, A.J.
• Format: Article
• Language: English
• Subjects: Angular momentum; Asymmetry; Bistability; Cellular convection; Convection; Convection heating; Differential rotation; Diffusion coefficients; Diffusivity; Equatorial regions; Experiments; Fluid dynamics; Fluid flow; Fluid mechanics; Fluid motion; Free convection; Heat transfer; Heated water; Heating; Ice cover; JFM Papers; Jupiter satellites; Maximization; Momentum; Oceans; Optimization; Prandtl number; Reynolds number; Rotating spheres; Rotation; Shells; Simulation; Spheres; Spherical shells; Symmetry; Temperature gradients; Thermal diffusivity; Thermal plumes; Thin films; Torque; Viscous fluids
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2020.1042

Heating a quiescent fluid from below gives rise to cellular convective motion as the temperature gradient becomes sufficiently steep. Typically, this transition increases heat transfer. Differentially rotating spherical shells also generate a state of cellular motion, which in this case transports angular momentum. When both effects are present, it is often assumed that the fluid adopts a configuration which maximises the transfer of angular momentum and heat. Depending on how the equilibrium is reached, however, this maximisation may not always be achieved, with two different stable equilibria often co-existing for the same heating and rotation strengths. We want to understand why the fluid motion in a spherical shell is bistable, and how this scenario might arise. We consider a deep, highly viscous fluid layer, of relevance to the ice shells of Saturn's and Jupiter's moons. We find that bistability depends largely on the relative strength of heating and differential rotation, as characterised by the Rayleigh number $Ra$ and inner sphere Reynolds number $Re_1$, and that the nature of the transition between bistable states depends strongly on the ratio of momentum diffusivity $\nu$ to thermal diffusivity $\kappa$ defined by the Prandtl number ${\textit {Pr}} = \nu /\kappa$. In particular, we find that the transition between solutions at large ${\textit {Pr}}$, depends on the strength of thin thermal layers and can occur either due to the destabilisation of an equatorial jet by buoyancy forces, or alternatively of a polar thermal plume by differential rotation. Our results demonstrate that, although bistability in this system cannot be simply explained by the flow maximising its torque or heat transfer, the polar and equatorial regions are of particular significance.