Spanwise rotating convection in an ideal gas

Abstract

Spanwise rotating convection is a model for the equator region of planets like Jupiter or Earth's outer liquid core. Direct numerical simulations of the Navier-Stokes, and continuity equation as well as an equation for the conservation of internal energy are performed in two and three dimension. This system is investigated regarding the effects of the Coriolis force in combination with the Taylor-Proudman theorem. The effect of the Taylor-Proudman theorem is to suppress fluid motion along the direction of rotation making the flow effectively two-dimensional. A two-dimensional flow can be numerically computed faster and therefore it is beneficial to know under which parameters this restriction occurs. This investigation is done with an incompressible liquid for simplicity. The result is a scaling law for Reynolds and Rossby and can be explained by the elliptical instability theory. Afterwards, the incompressible liquid is exchanged with an ideal gas, which changes the equations of motion, meaning the Navier-Stokes and continuity equation. Two-dimensional simulations without the Coriolis force are compared with three-dimensional simulations from a previous publication and are found to be very similar in the thermal transport but different in the kinetic energy density. Finally, in an analogy to an recently published experiment done in a centrifuge, the Coriolis force is added to the equations again. It is found that the onset of convection changes, which is further investigated with linear equations. Additionally, the experiment found a hysteresis with respect to changes in temperature, meaning the thermal transport depends on a change towards higher or lower temperature differences with consecutive experimental runs. This effect could not be reproduced in the two-dimensional simulations. The result of this work on the one hand is a scaling law that allows researchers to find parameters in spanwise rotating convection that can be perfectly represented with two-dimensional simulations. On the other hand, this work showed effects of compressible convection with and without the effects of the Coriolis force. Further work on the convection in an ideal gas can be used to better compare with present and future experiments