Dynamics of fingering convection. Part 2 The formation of thermohaline staircases

Abstract

Regions of the ocean's thermocline unstable to salt fingering are often observed to host thermohaline staircases, stacks of deep well-mixed convective layers separated by thin stably-stratified interfaces. Decades after their discovery, however, their origin remains controversial. In this paper we use 3D direct numerical simulations to shed light on the problem. We study the evolution of an analogous double-diffusive system, starting from an initial statistically homogeneous fingering state and find that it spontaneously transforms into a layered state. By analysing our results in the light of the mean-field theory developed in Paper I, a clear picture of the sequence of events resulting in the staircase formation emerges. A collective instability of homogeneous fingering convection first excites a field of gravity waves, with a well-defined vertical wavelength. However, the waves saturate early through regular but localized breaking events, and are not directly responsible for the formation of the staircase. Meanwhile, slower-growing, horizontally invariant but vertically quasi-periodic γ-modes are also excited and grow according to the γ−instability mechanism. Our results suggest that the nonlinear interaction between these various mean-field modes of instability leads to the selection of one particular γ−mode as the staircase progenitor. Upon reaching a critical amplitude, this progenitor overturns into a fully-formed staircase. We conclude by extending the results of our simulations to real oceanic parameter values, and find that the progenitor γ−mode is expected to grow on a timescale of a few hours, and leads to the formation of a thermohaline staircase in about one day with an initial spacing of the order of one to two metres.

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

“…The parametric flux-gradient model predicts that the growth rate of the unstable gamma-modes monotonically increases without bound with increasing vertical wavenumber, which is clearly unphysical. This ultraviolet catastrophe in the parametric model also contradicts the direct numerical simulations (DNS) showing that the wavelength of the fastest-growing gamma-instability mode exceeds the salt-finger scale by more than an order of magnitude (Radko 2003;Stellmach et al 2011). This apparent discrepancy signals that the flux-gradient laws fail at relatively small scales.…”

“…Particularly significant is the sign of A γ . Theoretical arguments and numerical simulations (Schmitt 1979a,b;Radko 2003;Stellmach et al 2011) indicate that the γ (R ρ ) dependence is non-monotonic. As the density ratio increases from unity, the flux ratio first decreases, as shown in figure 1, reaches a minimum value (at R ρ = R ρ,min ) and then starts to increase.…”

“…Mirouh et al 2012). Several problems have been successfully treated using flux-gradient laws, including theoretical investigations of thermohaline interleaving (Stern 1967;Merryfield 2000;Walsh & Ruddick 2000;Ruddick & Kerr 2003;Mueller, Smyth & Ruddick 2007;Smyth & Ruddick 2010), collective instability waves (Stern et al 2001;Stern & Simeonov 2002;Radko & Stern 2011) and thermohaline staircases (Radko 2003;Stellmach et al 2011). Attempts have also been made to validate the flux-gradient laws numerically (e.g.…”

“…Attempts have also been made to validate the flux-gradient laws numerically (e.g. Stern et al 2001;Stellmach et al 2011) by comparing finger-resolving solutions with their parametric counterparts. These studies document the adequate performance of flux-gradient models in configurations characterized by the significant scale separation between salt fingers, which flux laws parametrize, and fine-scale phenomena, which are modelled using these laws.…”

“…A case in point is the problem of spontaneous layering in an unbounded horizontally homogeneous and doubly stratified fluid (Radko 2003;Stellmach et al 2011). Linear stability analysis of the parametrized mean-field equations reveals the existence of gamma-instability modes, which ultimately transform the uniform background gradient into a series of mixed layers separated by high-gradient interfaces.…”