Energy and variance budgets of a diffusive staircase with implications for heat flux scaling

Diffusive convection, the mode of double-diffusive convection that occur when both temperature and salinity increase with increasing depth, is commonplace throughout the high latitude oceans and diffusive staircases constitute an important heat transport process in the Arctic Ocean. Heat and buoyancy fluxes through these staircases are often estimated using flux laws deduced either from laboratory experiments, or from simplified energy or variance budgets. We have done direct numerical simulations of double-diffusive convection at a range of Rayleigh numbers and quantified the energy and variance budgets in detail. This allows us to compare the fluxes in our simulations to those derived using known flux laws and to quantify how well the simplified energy and variance budgets approximate the full budgets. The fluxes are found to agree well with earlier estimates at high Rayleigh numbers, but we find large deviations at low Rayleigh numbers. The close ties between the heat and buoyancy fluxes and the budgets of thermal variance and energy have been utilized to derive heat flux scaling laws in the field of thermal convection. The result is the so called GL-theory, which has been found to give accurate heat flux scaling laws in a very wide parameter range. Diffusive convection has many similarities to thermal convection and an extension of the GL-theory to diffusive convection is also presented and its predictions are compared to the results from our numerical simulations.