Equilibration of weakly nonlinear salt fingers

Abstract

An analytical model is developed to explain the equilibration mechanism of the salt finger instability in unbounded temperature and salinity gradients. The theory is based on the weakly nonlinear asymptotic expansion about the point of marginal instability. The proposed solutions attribute equilibration of salt fingers to a combination of two processes: (i) the triad interaction and (ii) spontaneous development of the mean vertical shear. The non-resonant triad interactions control the equilibration of linear growth for moderate and large values of Prandtl number (Pr) and for slightly unstable parameters. For small Pr and/or rigorous instabilities, the mean shear effects become essential. It is shown that, individually, neither the mean field nor the triad interaction models can accurately describe the equilibrium patterns of salt fingers in all regions of the parameter space. Therefore, we propose a new hybrid model, which represents both stabilizing effects in a single framework. The resulting solutions agree with the fully nonlinear numerical simulations over a wide range of governing parameters.

A Reduced Model for Salt-Finger Convection in the Small Diffusivity Ratio Limit

“…Our IFSC model (20) is linked to the weakly nonlinear model derived by Radko [7] (see Equations (8) and (11) of [7]). Even though different parameter regimes are considered in the two models-our reduced model for τ → 0 captures a fully-nonlinear regime where dynamics are not confined to the onset of instability (Ra ranges from 1-∞), while Radko's [7] model relaxes the small τ assumption, but is restricted to dynamics near the onset: the two models match in the overlapping regime τ 1 and Ra = 1 + R with 1 and R = O(1).…”

“…Even though different parameter regimes are considered in the two models-our reduced model for τ → 0 captures a fully-nonlinear regime where dynamics are not confined to the onset of instability (Ra ranges from 1-∞), while Radko's [7] model relaxes the small τ assumption, but is restricted to dynamics near the onset: the two models match in the overlapping regime τ 1 and Ra = 1 + R with 1 and R = O(1). In this regime, the optimal scale from Equation (24) is of O( −1/4 ), suggesting the rescaling:…”

“…Except for weakly nonlinear approaches, e.g., [7,12], most researchers employ numerical studies of the primitive equations consisting of the Navier-Stokes equation coupled to scalar equations for heat and salinity transport. Such studies, particularly in three dimensions, have proven invaluable and identified a number of novel processes, e.g., [13].…”

Mechanics of thermohaline interleaving: beyond the empirical flux laws

“…The growth rate in (2.18) is completely independent of the lateral temperature-salinity gradients, which are an essential ingredient of the classical intrusion theory. This mode is more akin to the lateral shear instability identified in Holyer (1984) and Radko (2010). While this mode is also manifested by the vertically alternating system of currents and corresponding T-S signature -which makes it difficult to discriminate between intrusive and shear modes on the basis of observations -the shear instability is largely driven by salt fingers themselves and not by their interaction with lateral gradients.…”

“…Salt fingers equilibrate through the nonlinear triad interaction with secondary instabilities operating on the scale of salt fingers themselves (Holyer 1984;Radko & Stern 1999;Stern et al 2001;Radko 2010). This mechanism is not described explicitly by our model but can be represented by introducing an appropriate forcing, which balances growth of otherwise linearly unstable modes.…”