Regimes and scaling laws for rotating deep convection in the ocean

Numerical experiments are presented which explore the dependence of the scale and intensity of convective elements in a rotating fluid on variations in external parameters in a regime relevant to open ocean deep convection. Conditions inside a convection region are idealized by removing buoyancy at a uniform rate B from the surface of an initially homogeneous, motionless, incompressible ocean of depth H with a linear equation of state, at a latitude where the Coriolis parameter is tf. The key nondimensional parameters are the natural Rossby number Ro ∗ =(B/f 3H 2) 1 2 and the flux Rayleigh number Ra f = BH 4/( κ 2 ν), where κ and ν are (eddy) diffusivities of heat and momentum. Ro ∗ is set to values appropriate to open ocean deep convection (0.01 < Ro ∗ < 1), and moderately high values of Ra f (10 4 < Ra f < 10 13) were chosen to produce flows in which nonlinear effects are significant. The experiments are in the ‘geostrophic turbulence’ regime. As Ro ∗ and Ra f are reduced the convective elements become increasingly quasi-two-dimensional and can be described as a field of interacting ‘hetons’. The behavior of the flow statistics—plume horizontal length scale L, speed scale U and buoyancy scale G, and the magnitude of the mean adverse density gradient measured by the stratification parameter H—are studied as a function of Ro ∗ and Ra f. Physically motivated scaling laws are introduced, which, when appropriate, employ geostrophic and hydrostatic contraints. They are used to interpret the experiments. In the heton regime, in which the motion is predominantly geostrophic and hydrostatic, the observed scales are sensitive to moderate variations in Ro ∗ and large variations in Ra f. We demonstrate broad agreement between our numerical experiments and previous laboratory studies. The lateral scale of the convective elements and the (adverse) stratification in which they exist adjust to one another so that NH/ fL≈ 1; the horizontal scale of the hetons is thus controlled by a pseudo Rossby radius based on the unstable stratification parameter N, the scale at which the overturning forces associated with N are balanced by the counter-overturning forces associated with rotation.