Stratified Kolmogorov flow

Abstract

In this study we investigate the Kolmogorov flow (a shear flow with a sinusoidal velocity profile) in a weakly stratified, two-dimensional fluid. We derive amplitude equations for this system in the neighbourhood of the initial bifurcation to instability for both low and high Péclet numbers (strong and weak thermal diffusion, respectively). We solve amplitude equations numerically and find that, for low Péclet number, the stratification halts the cascade of energy from small to large scales at an intermediate wavenumber. For high Péclet number, we discover diffusively spreading, thermal boundary layers in which the stratification temporarily impedes, but does not saturate, the growth of the instability; the instability eventually mixes the temperature inside the boundary layers, so releasing itself from the stabilizing stratification there, and thereby grows more quickly. We solve the governing fluid equations numerically to compare with the asymptotic results, and to extend the exploration well beyond onset. We find that the arrest of the inverse cascade by stratification is a robust feature of the system, occurring at higher Reynolds, Richards and Péclet numbers – the flow patterns are invariably smaller than the domain size. At higher Péclet number, though the system creates slender regions in which the temperature gradient is concentrated within a more homogeneous background, there are no signs of the horizontally mixed layers separated by diffusive interfaces familiar from doubly diffusive systems.

The stability of stratified spatially periodic shear flows at low Péclet number

“…In both unstratified and stratified cases studied so far, the first unstable modes at the instability threshold have the same periodicity in z as that of the background shear, so that a = 0. 24,26,31 We verified that this is indeed the case here as well. In what follows, we therefore restrict the presentation of our results to the case a = 0.…”

“…The linear stability of the stratified Kolmogorov flow u L to 2D perturbations was studied in detail by Balmforth and Young. 26 The 2D case can be made more generally relevant by noting that Squire's transformation 25 for the viscous unstratified case can be extended to the stratified case with thermal diffusion to argue that the linear stability of any 3D mode can equivalently be studied by considering that of a 2D mode at lower or equal Reynolds and Péclet numbers, and higher or equal Richardson number. This result, which was summarily discussed by Yih 27 and clarified by Smyth, Klaassen, and Peltier 28 and Smyth and Peltier 29,30 states that the growth rate λ 3 of the 3D normal mode q 3 (x, y, z,t) =q 3 (z) exp(il x + im y + λ 3 t) at parameters (Re, Pe, Ri) is related to that of the 2D normal mode q 2 (x, y, z,t) =q 2 (z) exp(iLx + λ 2 t) at suitably rescaled parameters via…”

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

“…Perhaps an even broader theoretical significance of multiscale analyses lies in their ability to adequately represent a remarkably wide range of layering phenomena. Multiscale solutions have been used to conceptualize dynamics of planetary jets, generated and maintained by mesoscale variability (Manfroi & Young 1999, layering forced by small-scale turbulence in stratified one-component fluids (Balmforth & Young 2002, 2005, thermohaline interleaving driven by lateral property gradients (Radko 2011) and, in this study, the formation of double-diffusive staircases. Their ability to explain the physics of such dissimilar systems is truly remarkable, and this investigation is yet another testament to the power and generality of multiscale modelling.…”

“…Our layering model is analogous to the multiscale models of collective instability (Holyer 1981(Holyer , 1985 and thermohaline interleaving (Radko 2011). It also bears resemblance to the multiscale analyses of mixing in one-component flows (Balmforth & Young 2002, 2005 and to models illustrating spontaneous generation of planetary-scale flows by mesoscale variability (Manfroi & Young 1999. The starting point for such analyses is the choice of the periodic small-scale pattern, which is frequently represented by the Kolmogorov solution -the steady sinusoidal background flow.…”

Mechanics of thermohaline interleaving: beyond the empirical flux laws

“…It is perhaps ironic that multiscale techniques have been used more frequently in configurations where the scale separation is not nearly as well-defined or physically justified as in the salt finger problem. For instance, Balmforth & Young (2002 used a technique analogous to ours to explore the consequences of small-scale mixing in stratified one-component flows. The background field on the molecular dissipation scale in their models was represented by the Kolmogorov flow (Meshalkin & Sinai 1961;Sivashinsky 1985), making it possible to describe the evolution of large-scale fields analytically.…”