The convection induced by double-diffusive instability can be a dominant source for the vertical mixing in, for example, the Arctic Ocean where turbulence is weak. It is for this reason that a proper parametrization for double-diffusion induced diapycnal diffusivities are critical to increasing the performance of the global climate model. However, quantifying such a parametrization scheme is a hard task since the dynamics of double-diffusion involves a complex interplay between a large range of different length scales: the centimetre scale double-diffusive instability can generate a series of vertically well-mixed layers of tens of meters separated by sharp interfaces, which is known as the thermohaline staircases. The mechanism that leads to the formation of staircase is still unsolved, although the gamma instability mechanism proposed by Radko (2003) started to shed light on the problem.

Through a series of direct numerical simulations (DNS), we have parametrized the irreversible fluxes of the homogeneous salt-fingering field in its equilibrium state. The distinction between irreversible and reversible flux has been, for the first time, introduced in a double diffusion system. Based on a combination of theoretical analysis as well as mean-field simulations that utilizes these parametrizations, we further explained a mechanism that leads to different step-sizes that form in different vertical regions of a staircase. Our explanation is then tested by making comparisons with the observed staircase data in the Tyrrhenian Sea.