Second‐moment closures and length scales for weakly stratified turbulent shear flows

Abstract

For the special hydrodynamic situation of unbounded homogeneous shear layers, turbulence closure models of Mellor‐Yamada type (MY) and k‐ε type are put into a single canonical form. For this situation we show that conventional versions of MY and various k‐ε versions lack a proper steady state, and are unable to simulate the most basic properties of stratified shear flows exemplified in, for example, the Rohr et al. [1988] experiments: exponential growth at sufficiently low gradient Richardson number (Rg), exponential decay at sufficiently large Rg, and a steady state in between. Proper choice of one special model parameter readily solves the problems. In the fairly general case of structural equilibrium (state of exponential evolution) in weakly to moderately stratified turbulence (Rg ≲ 0.25), the ratio between the Thorpe scale (or Ellison scale) and the Ozmidov scale varies like the gradient Richardson number (Rg) to the power 3/4, and the ratio of the Thorpe scale to the buoyancy scale varies like Rg1/2. Length scales predicted by our current model are consistent with laboratory measurements of Rohr et al. [1988], with large‐eddy numerical simulations of Schumann and Gerz [1995], and with microstructure measurements from the 1987 Tropic Heat Experiment in the equatorial Pacific by Peters et al. [1995].