Wall modes for conducting sidewall boundary conditions in rotating Rayleigh-Bénard convection

Rotating Rayleigh-Bénard convection is a model experimental, computational, and theoretical system with which to help understand and characterize a broad range of natural phenomena. These involve rotation and buoyancy including fluid flows in the Earth’s atmosphere, oceans, and deep interior as well as in planetary atmospheres and in stars. The control parameters for rotating convection are the Rayleigh Number Ra ~ ΔT H³, the Ekman Number Ek ~ H^-2/Ω, and the Prandtl Number Pr = ν/κ where ΔT is the temperature difference across a layer of height H,  Ω is the angular rotation rate,  ν is the kinematic viscosity, and κ is the thermal diffusivity.  Numerous investigations have revealed bifurcations to wall localized states (wall modes) occurring at Rayleigh number Ra_w ~ Ek^-1  [1] when the sidewalls are perfectly insulating or weakly thermally conducting.  At higher Ra_c ~ Ek^-4/3, bulk rotating convection begins [2] characterized for rapid rotation by a fascinating interplay of vortex states in near nonhydrostatic geostrophic balance.  When the sidewalls are perfectly conducting, there is a distinct set of wall modes [3] that have very different properties and scalings compared to wall modes with perfectly insulating sidewalls. For example, one has an onset Rayleigh Number Ra_wc ~ Ek^-4/3, the same as bulk mode scaling but with a smaller proportionality constant.  We present direct numerical simulations of rapidly rotating Rayleigh-Bénard convection in small aspect ratio cylindrical convection cells with perfectly conducting boundary conditions for Pr = 0.8 and Ek = 10^-6. We compare and contrast the properties of these conducting-boundary-condition wall modes with those for insulating sidewalls and discuss the importance of these states in experimental studies of rotating convection of compressed gases and cryogenic helium where sidewall conductivity is much greater than fluid conductivity.

1. X. Zhang, P. Reiter, O. Shishkina, and R.E. Ecke, Phys. Rev. Fluids 9, 053501 (2024). 

2.  R.E. Ecke and O. Shishkina, Annu. Rev. Fluid Mech. 55, 603 (2023).

3. J. Herrmann and F.H. Busse, J. Fluid Mech. 255, 183 (1993).