Asymptotic Dynamics of High Dynamic Range Stratified Turbulence

Direct numerical simulations of homogeneous sheared and stably stratified turbulence are considered to probe the asymptotic high dynamic range regime suggested by Gargett et al. J. Fluid Mech. 144, 231 (1984) and Shih et al. J. Fluid Mech. 525, 193 (1999). We consider statistically stationary configurations of the flow that span three decades in dynamic range defined by the separation between the Ozmidov length scale $L_O=\sqrt{\epsilon /N^3}$ and the Kolmogorov length scale $L_K=({\nu }^3/\epsilon {)}^{1/4}$, up to ${\mathrm{Re}}_b\equiv (L_O/L_K{)}^{4/3}=\epsilon /(\nu N^2)\sim O(1000)$, where $\epsilon$ is the mean turbulent kinetic energy dissipation rate, $\nu$ is the kinematic viscosity, and $N$ is the buoyancy frequency. We isolate the effects of ${\mathrm{Re}}_b$, particularly on irreversible mixing, from the effects of other flow parameters of stratified and sheared turbulence. Specifically, we evaluate the influence of dynamic range independent of initial conditions. We present evidence that the flow approaches an asymptotic state for ${\mathrm{Re}}_b⪆300$, characterized both by an asymptotic partitioning between the potential and kinetic energies and by the approach of components of the dissipation rate to their expected values under the assumption of isotropy. As ${\mathrm{Re}}_b$ increases above 100, there is a slight decrease in the turbulent flux coefficient $\mathrm{\Gamma }=\chi /\epsilon$, where $\chi$ is the dissipation rate of buoyancy variance, but, for this flow, there is no evidence of the commonly suggested $\mathrm{\Gamma }\propto {{\mathrm{Re}}_b}^{-1/2}$ dependence when $100\le {\mathrm{Re}}_b\le 1000$.

Figure 1

Energy partitions as a function of ${\mathrm{Re}}_b$. Bars indicate upper and lower quartile measurements of instantaneous quantities.

Figure 2

Instantaneous measurements of small-scale anisotropy. The dashed line represents perfect isotropy at small scales.

Figure 3

(a) Variation of turbulent flux coefficient $\mathrm{\Gamma }$ with ${\mathrm{Re}}_b$. The upper bound proposed by Osborn is indicated by the gray dashed line where $\mathrm{\Gamma }\approx 0.2$, and the ${{\mathrm{Re}}_b}^{-1/2}$-based parametrization suggested in Shih et al. [9] is plotted for their energetic regime of ${\mathrm{Re}}_b>100$. (b) Variation of turbulent Prandtl number ${\text{Pr}}_T$ with ${\mathrm{Re}}_b$.