Bounds for rotating convection at infinite Prandtl number from semidefinite programs

TL;DR

This paper develops a numerical method using semidefinite programming to establish bounds on kinetic energies and heat transport in rotating convection at infinite Prandtl number, revealing how these bounds depend on Rayleigh and Taylor numbers.

Contribution

It introduces a semidefinite programming approach to compute bounds for rotating convection at infinite Prandtl number, incorporating boundary conditions and analyzing their behavior with Rayleigh number.

Findings

Bounds match qualitative behavior of physical quantities with Rayleigh number.

Bounds are zero below critical Rayleigh number and increase above it.

Power law exponents from bounds differ from actual Nusselt number dependence.

Abstract

Bounds for the poloidal and toroidal kinetic energies and the heat transport are computed numerically for rotating convection at infinite Prandtl number with both no slip and stress free boundaries. The constraints invoked in this computation are linear or quadratic in the problem variables and lead to the formulation of a semidefinite program. The bounds behave as a function of Rayleigh number at fixed Taylor number qualitatively in the same way as the quantities being bounded. The bounds are zero for Rayleigh numbers smaller than the critical Rayleigh number for the onset of convection, they increase rapidly with Rayleigh number for Rayleigh numbers just above onset, and increase more slowly at large Rayleigh numbers. If the dependencies on Rayleigh number are approximated by power laws, one obtains larger exponents from bounds on the Nusselt number for Rayleigh numbers just above onset than from the actual Nusselt number dependence known for large but finite Prandtl number. The wavelength of the linearly unstable mode at the onset of convection appears as a relevant length scale in the bounds.