Applicability and failure of the flux-gradient laws in double-diffusive convection

Abstract

Double-diffusive flux-gradient laws are commonly used to describe the development of large-scale structures driven by salt fingers -thermohaline staircases, collective instability waves and intrusions. The flux-gradient model assumes that the vertical transport is uniquely determined by the local background temperature and salinity gradients. While flux-gradient laws adequately capture mixing characteristics on scales that greatly exceed those of primary double-diffusive instabilities, their accuracy rapidly deteriorates when the scale separation between primary and secondary instabilities is reduced. This study examines conditions for the breakdown of the flux-gradient laws using a combination of analytical arguments and direct numerical simulations. The applicability (failure) of the flux-gradient laws at large (small) scales is illustrated through the example of layering instability, which results in the spontaneous formation of thermohaline staircases from uniform temperature and salinity gradients. Our inquiry is focused on the properties of the 'point-of-failure' scale (H pof ) at which the vertical transport becomes significantly affected by the non-uniformity of the background stratification. It is hypothesized that H pof can control some key characteristics of secondary double-diffusive phenomena, such as the thickness of high-gradient interfaces in thermohaline staircases. A more general parametrization of the vertical transport -the flux-gradient-aberrancy lawis proposed, which includes the selective damping of relatively short wavelengths that are inadequately represented by the flux-gradient models. The new formulation is free from the unphysical behaviour of the flux-gradient laws at small scales (e.g. the ultraviolet catastrophe) and can be readily implemented in theoretical and large-scale numerical models of double-diffusive convection.

Thermo-compositional Diabatic Convection in the Atmospheres of Brown Dwarfs and in Earth’s Atmosphere and Oceans

“…These relations correspond also to the flux-gradient laws used for thermohaline convection (e.g. Radko 2014).…”

“…The diabatic convective transport is similar to the parametrization used for moist convection (Arakawa & Jung 2011). In the thermohaline context, using the standard potential temperature and composition in the flux-gradient laws can lead to inconsistencies such as an ultraviolet catastroph when trying to predict the growth rates of the staircase modes at small scales (Radko 2014). The new parameterization proposed here could provide a way to solve these issues.…”

Multiple states and transport properties of double-diffusive convection turbulence

“…The local density ratio alone does not seem to be sufficient to describe the fluxes. Modification of the commonly used flux gradient law has been proposed and tested for fully periodic domain (34). Another related open question is what controls the height of the finger interface, especially those within staircases.…”

Control of baroclinic instability by submesoscale topography

“…The majority of multiscale models consider steady analytical small-scale patterns, exemplified by the Kolmogorov model (e.g. Meshalkin & Sinai 1961; Manfroi & Young 1999, 2002; Balmforth & Young 2002, 2005; Radko 2014) which represents the background fields by a single Fourier harmonic. Such studies have been generalized to incorporate more complicated two-dimensional small-scale structures (e.g.…”