A simple criterion to determine the transition from a localized convection to a distributed convection regime

A recent numerical study by Noh et al. of open-ocean deep convection in the presence of a single geostrophic eddy showed that two possible regimes exist: 1) the localized convection regime in which baroclinic instability of the eddy dominates, with slantwise fluxes and restratification, and 2) the distributed convection regime in which vertical mixing dominates. Noh et al. found that localized convection dominates for relatively small buoyancy forcing, strong eddies, and strong surface ambient stratification. Here it is shown that this regime transition can be expressed in terms of a ratio of time scales: the localized convection regime appears when the time scale for lateral fluxes from eddy interior to exterior t_L is short in comparison with the time scale for convective erosion of the exterior stratification t_c. Scaling arguments give this ratio of time scales as t_L/t_c ∼ fβ²R ²B/(A²γ) where f is the Coriolis parameter, R is the radius of the eddy, B is the buoyancy forcing, 1 /β is the depth scale of the exponentially decaying surface-intensified stratification, γ is the relative amplitude of the eddy, and Aβ is the value of the surface stratification N ₀². Comparison with the numerical simulations of Noh et al. shows that this parameter does indeed separate the localized and distributed convection regimes, with the transition occurring at t_L/t_c, ≈ 0.05-0.1.