Large-amplitude Bénard convection in a rotating fluid

Abstract

Linear stability theory of Bénard convection in a rotating fluid (Chandrasekhar 1961) has shown that fluids with large ([ges ] 1) Prandtl number, σ, exhibit behaviour markedly different from that of fluids with σ [les ] 1. This difference in behaviour extends also into the finite-amplitude range (Veronis 1959, 1966I). In this paper we report a numerical study of two-dimensional Be´nard convection in a rotating fluid confined between free boundaries, with σ = 6·8 and σ = 0·2 for the range of Taylor number 0 [les ] [Fscr ]2 [Lt ] 105 and for Rayleigh numbers, R, extending an order of magnitude from the critical value of linear stability theory. The behaviour of water (σ = 6·8) is dominated by the rotational constraint even for relatively moderate values (∼ 103) of [Fscr ]2. A study of the resultant velocity and temperature fields shows how rotation controls the system, with the principal behaviour reflected by the thermal wind balance; i.e. the horizontal temperature gradient is largely balanced by the vertical shear of the velocity component normal to the temperature gradient. A fluid with a small Prandtl number (σ = 0·2) becomes unstable to finite-amplitude disturbances at values of the Rayleigh number significantly below the critical value of linear stability theory. The subsequent steady vorticity and temperature fields exhibit a structure which is quite different from that of fluids with large σ. The rotational constraint is balanced primarily by non-linear processes in a limited range of Taylor number ([Fscr ]2 [les ] 103·6). For larger values of [Fscr ]2 the system first becomes unstable to infinitesimal oscillatory disturbances but a steady, finite-amplitude flow is established at supercritical values of R which are none the less smaller than the values that one would expect from linear theory. The ranges of Taylor number in which the above phenomena occur are different from those which were estimated on the basis of an earlier study (Veronis 1966 I) which made use of a minimal representation of the finite-amplitude velocity and temperature fields. No subcritical, finite-amplitude oscillatory motions were found in the present study. Comparison with some of the experimental features observed and reported by Rossby (1966) is also discussed and it is pointed out that some of the differences between theory and experiment may be traced to the restrictive conditions (two-dimensionality and free boundaries) of the present study.