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

Isolated convection in a rotating fluid of constant depth H which is initia lly homogeneous in density is considered. It is shown that three regimes ca n be expected, depending on the initial parameters: a rotationally controll ed regime, a baroclinically unstable regime and a stable regime. The transi tions between these regimes can be described by critical values of two Ross by numbers, based on the ratio of two horizontal and two vertical length sc ales: the Rossby number Ro(R) = (B-0/f(3)R(2))(1/4) describes the transitio n between the baroclinically stable and unstable regimes, while the value o f 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 J acobs 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 topogra phy (constant depth vs. shelf and slope). For the regime relevant to oceani c conditions (the baroclinically unstable regime), the steady state density difference g(f)' and the exchange or eddy velocity upsilon(flux) character ising the exchange of heat between the convecting region and the surroundin gs have been measured in a series of laboratory experiments. Both these qua ntities depend on the strength of the background rotation, but the product of these which characterises the lateral buoyancy flux out of the convectin g 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 t o 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 b y upsilon(flux) = (1.0 +/- 0.2)(B-0/f)(1/2), while steady state is reached at a time tau(D) = (1.9 +/- 0.2)(f/B-0)R-1/2. The typical diameter of the b aroclinic vortices is given by D-eddy = (2.25 +/- 0.50)R-D, with R-D the Ro ssby 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. (C) 1999 Elsev ier Science B.V. All rights reserved.