Rossby number regimes for isolated convection in a homogeneous, rotating fluid

Abstract

Isolated convection in a rotating fluid of constant depth H which is initially homogeneous in density is considered. It is shown that three regimes can be expected, depending on the initial parameters: a rotationally controlled regime, a baroclinically unstable regime and a stable regime. The transitions between these regimes can be described by critical values of two Rossby numbers, based on the ratio of two horizontal and two vertical length scales: the Rossby number Ro R = (B 0 /f 3 R 2 ) 1/4 describes the transition between the baroclinically stable and unstable regimes, while the value of the 'natural' Rossby number Ro * = (B 0 /f 3 H 2 ) 1/2 determines whether rotation or buoyancy forces control the small-scale turbulence. Here B 0 is the buoyancy flux applied over a circular area with radius R and f is the Coriolis parameter. The present study is comparable to the one studied by Jacobs and Ivey [Jacobs, P., Ivey, G.N., 1998. The influence of rotation on shelf convection. J. Fluid Mech. 369, 23-48], except for the bottom topography (constant depth vs. shelf and slope). For the regime relevant to oceanic conditions (the baroclinically unstable regime), the steady state density difference g' f and the exchange or eddy velocity υ flux characterising the exchange of heat between the convecting region and the surroundings have been measured in a series of laboratory experiments. Both these quantities depend on the strength of the background rotation, but the product of these which characterises the lateral buoyancy flux out of the convecting region, is independent of f as predicted by the overall buoyancy balance in the steady state. Results show that in experimental models it is crucial to monitor the density increase in the ambient fluid which can occur due to the finite lateral extent of the working fluid. The steady state density difference between convecting and ambient fluids can then be described by g' f = (1.9 ± 0.2)(B 0 f) 1/2 (R/H), the characteristic radial velocity by υ flux = (1.0 ± 0.2)(B 0 /f) 1/2 , while steady state is reached at a time T D = (1.9 ± 0.2)(f/B 0 ) 1/2 R. The typical diameter of the baroclinic vortices is given by D eddy = (2.25 ± 0.50)R D , with R D the Rossby radius of deformation, based on the steady state density difference g' f and the total fluid depth H. These results are consistent with those of Jacobs and Ivey, although the constants of proportionality for the steady state time scale and the vortex size are slightly different.