# Stably stratified exact coherent structures in shear flow: the effect of Prandtl number

## TL;DR

This paper investigates how stable density stratification affects shear flow equilibria at different Prandtl numbers, revealing distinct physical mechanisms and boundary layer structures in the limits of low and high Prandtl numbers.

## Contribution

It provides a detailed analysis of the effects of Prandtl number on stable stratified shear flows, including asymptotic regimes and boundary layer characteristics, extending understanding of stratified turbulence.

## Key findings

- Stable equilibria exist at high stratification for all Prandtl numbers.
- Different physical mechanisms dominate in the limits Pr→0 and Pr→∞.
- Interior density layers can form away from boundaries at high Reynolds numbers.

## Abstract

We examine how known unstable equilibria of the Navier-Stokes equations in plane Couette flow adapt to the presence of an imposed stable density difference between the two boundaries for varying values of the Prandtl number $Pr$, the ratio of viscosity to density diffusivity, and fixed moderate Reynolds number, $Re=400$. In the two asymptotic limits $Pr \to 0$ and $Pr \to \infty$, it is found that such solutions exist at arbitrarily high bulk stratification but for different physical reasons. In the $Pr \to 0$ limit, density variations away from a constant stable density gradient become vanishingly small as diffusion of density dominates over advection, allowing equilibria to exist for bulk Richardson number $Ri_b \lesssim O(Re^{-2}Pr^{-1})$. Alternatively, at high Prandtl numbers, density becomes homogenised in the interior by the dominant advection which creates strongly stable stratified boundary layers that recede into the wall as $Pr\to\infty$. In this scenario, the density stratification and the flow essentially decouple, thereby mitigating the effect of increasing $Ri_b$. An asymptotic analysis is presented in the passive scalar regime $Ri_b \lesssim O(Re^{-2})$, which reveals $O(Pr^{-1/3})$-thick stratified boundary layers with $O(Pr^{-2/9})$-wide eruptions, giving rise to density fingers of $O(Pr^{-1/9})$ length and $O(Pr^{-4/9})$ width that invade an otherwise homogeneous interior. Finally, increasing $Re$ to $10^5$ in this regime reveals that interior stably stratified density layers can form away from the boundaries, separating well-mixed regions.