Convection in a rotating annulus uniformly heated from below. Part 2. Nonlinear results

Abstract

Davies-Jones & Gilman (1971) reported in this Journal on a linear stability analysis and second-order initial-tendency calculation for thermal convection in a thin rotating annulus heated uniformly from below. The present work gives extensive numerical calculations for an annulus model with realistic laboratory boundary conditions which extends the previous work into the finite amplitude range. The model allows the mode with the longitudinal wavenumber predicted to occur at the onset of instability to grow to a finite amplitude and induce secondary axisymmetric differential rotation and meridional circulation through its Reynolds and thermal stresses, these circulations in turn modifying the forcing cells. No higher harmonics are allowed. The Boussinesq equations for the fundamental mode and secondary circulations are solved on a staggered finite-difference grid in the annulus cross-section, mostly following the scheme laid out by Williams (1969).

Stationary convection is studied for Taylor numbers T < 3 × 10³. The profile predicted for the mean circulations in the earlier analysis, that of fast rotation in the outer half, slow rotation in the inner half of the annulus (except at very high Prandtl number), together with a four-celled meridional circulation, is confirmed. For a given Taylor number T, the differential-rotation energy peaks relative to the cell energy at a modest Prandtl number P somewhat less than unity, and its production is driven mostly by the Reynolds stresses in the cells. Meridional circulation is driven by potential-energy conversion at high P and by Reynolds stresses at low P, but retains the same form for all P with increased amplitude relative to the cells at low P. For Rayleigh numbers R a given percentage above the critical value Rc the differential-rotation energy increases like T, while the meridional circulation remains nearly constant. For given P and T differential-rotation relative energy peaks at modest R/R, while the meridional circulation continues to increase. The induced differential rotation tends to distort the cell boundaries from tilted straight lines produced by Coriolis forces into S shapes qualitatively like the profile of differential rotation itself.

One example of nonlinear overstable convection is given. The induced differential rotation has time-independent and oscillatory parts. The redistribution of momentum is carried out by periodically reversing horizontal and vertical Reynolds stresses.