Upper-bound problem for a rotating system

Abstract

Recent work (Hunter & Riahi 1975) on nonlinear convection in a rotating fluid is extended to a multi-modal regime. The schematic multi-boundary-layer method of Busse (1969) and the upper-bound technique of Howard (1963) are used to obtain upper bounds on the Nusselt number N. It is shown that there are infinitely many modes in the range $Ta \ll R^{\frac{3}{2}}$, where Ta is the Taylor number and R is the Rayleigh number, and different types of mode optimize N in different regions of the parameter space (R, Ta). While the optimal N is independent of Ta for Ta [Lt ] R, it is found that it increases with Ta in $R \ll Ta \ll (R \log R)^{\frac{4}{3}}$ and decreases as Ta increases in $(R \log R)^{\frac{4}{3}} \ll Ta \ll R^{\frac{3}{2}}$, and that the functional dependence of the optimal N on R and Ta is continuous (within a logarithmic term) throughout the region of R, Ta space.