Diapycnal diffusivity, turbulent Prandtl number and mixing efficiency in Boussinesq stratified turbulence

Abstract

In order that it be correctly characterized, irreversible turbulent mixing in stratified fluids must distinguish between adiabatic ‘stirring’ and diabatic ‘mixing’. Such a distinction has been formalized through the definition of a diapycnal diffusivity, $K_{{\it\rho}}$ (Winters & D’Asaro, J. Fluid Mech., vol. 317, 1996, pp. 179–193) and an appropriate mixing efficiency, $\mathscr{E}$ (Caulfield & Peltier, J. Fluid Mech., vol. 413, 2000, pp. 1–47). Equivalent attention has not been paid to the definitions of a corresponding momentum diffusivity $K_{m}$ and hence an appropriately defined turbulent Prandtl number $\mathit{Pr}_{t}=K_{m}/K_{{\it\rho}}$. In this paper, the diascalar framework of Winters & D’Asaro (1996) is first reformulated to obtain an ‘Osborn-like’ formula in which the correct definition of irreversible mixing efficiency $\mathscr{E}$ is shown to replace the flux Richardson number which Osborn (J. Phys. Oceanogr., vol. 10, 1980, pp. 83–89) assumed to characterize this efficiency. We advocate the use of this revised representation for diapycnal diffusivity since the proposed reformulation effectively removes the simplifying assumptions on which the original Osborn formula was based. We similarly propose correspondingly reasonable definitions for $K_{m}$ and $\mathit{Pr}_{t}$ by eliminating the reversible component of the momentum production term. To explore implications of the reformulations for both diapycnal and momentum diffusivity we employ an extensive series of direct numerical simulations (DNS) to investigate the properties of the shear-induced density-stratified turbulence that is engendered through the breaking of a freely evolving Kelvin–Helmholtz wave. The DNS results based on the proposed reformulation of $K_{{\it\rho}}$ are compared with available estimations due to the mixing length model, as well as both the Osborn–Cox and the Osborn models. Estimates based upon the Osborn–Cox formulation are shown to provide the closest approximation to the diapycnal diffusivity delivered by the exact representation. Through compilation of the complete set of DNS results we explore the characteristic dependence of $K_{{\it\rho}}$ on the buoyancy Reynolds number $\mathit{Re}_{b}$ as originally investigated by Shih et al. (J. Fluid Mech., vol. 525, 2005, pp. 193–214) in their idealized study of homogeneous stratified and sheared turbulence, and show that the validity of their results is only further reinforced through analysis of the turbulence produced in the more geophysically relevant Kelvin–Helmholtz wave life-cycle ansatz. In contrast to the results described by Shih et al. (2005) however, we show that, besides $\mathit{Re}_{b}$, a vertically averaged measure of the gradient Richardson number $\mathit{Ri}_{b}$ may equivalently characterize the turbulent mixing at high $\mathit{Re}_{b}$. Based on the dominant driving processes involved in irreversible mixing, we categorize the intermediate (i.e. $\mathit{Re}_{b}=O(10^{1}{-}10^{2})$) and high (i.e. $\mathit{Re}_{b}>O(10^{2})$) range of $\mathit{Re}_{b}$ as ‘buoyancy-dominated’ and ‘shear-dominated’ mixing regimes, which together define a transition value of $\mathit{Ri}_{b}\sim 0.2$. Mixing efficiency varies non-monotonically with both $\mathit{Re}_{b}$ and $\mathit{Ri}_{b}$, with its maximum (on the order of 0.2–0.3) occurring in the ‘buoyancy-dominated’ regime. Unlike $K_{{\it\rho}}$ which is very sensitive to the correct choice of $\mathscr{E}$ (i.e. $K_{{\it\rho}}\propto \mathscr{E}/(1-\mathscr{E})$), we show that $K_{m}$ is almost insensitive to the choice of $\mathscr{E}$ (i.e. $K_{m}\propto 1/(1-\mathscr{E})$) so long as $\mathscr{E}$ is not close to unity, which implies $K_{m}\approx \mathit{Ri}_{b}\mathit{Re}_{b}$ for the entire range of $\mathit{Re}_{b}$. The turbulent Prandtl number is consequently shown to decrease monotonically with $\mathit{Re}_{b}$ and may be (to first order) simply approximated by $\mathit{Re}_{b}$ itself. Assuming $\mathit{Pr}_{t}=1$, or $\mathit{Pr}_{t}=10$ (as is common in large-scale numerical models of the ocean general circulation), is also suggested to be a questionable assumption.

Robust identification of dynamically distinct regions in stratified turbulence

“…Such sensitivity is also observed in the results reported in both Hebert & de Bruyn Kops (2006a) and Hebert & de Bruyn Kops (2006b). More recently, de Bruyn Kops (2015) found that the dynamics of stratified flows are different at Gn = 48 and Gn = 220, which suggests another threshold at Gn ∼ O(100), consistent with the findings of Shih et al (2005) and Salehipour & Peltier (2015).…”

Mixing efficiency in stratified turbulence

“…A constant mixing efficiency Ri f = 0.17 was assumed by Osborn (1980) leading to a mixing coefficient Γ = Ri f /(1 − Ri f ) = 0.2, a value which has been used in oceanographic applications ever since. More recently, Salehipour and Peltier (2015) have suggested to use the potential energy dissipation rate ǫ p instead of the buoyancy flux when calculating Γ because in steady-state stratified turbulence B = ǫ p and the irreversible conversion of available potential energy into background potential energy due to mixing is given by ǫ p . The potential energy dissipation rate is defined as ǫ p = (D/N 2 ) ∇b · ∇b since the Lindborg and Brethouwer (2008) who derive an analytical expression for the mean square particle displacement 1/2 δz 2 , which increases linearly in time with a constant of proportionality equal to K ρ .…”

A statistical mechanics approach to mixing in stratified fluids

“…Even if it is natural to expect that the dissipation rate of buoyancy and kinetic energy become independent from the value of molecular viscosity and diffusivity when they are sufficiently weak, numerical and laboratory experiments are often performed in intermediate regimes for which those parameters may influence the mixing efficiency, see e.g. Shih et al (2005); Lozovatsky & Fernando (2013) ;Bouffard & Boegman (2013); Salehipour & Peltier (2015).…”

“…Winters et al (1995); Peltier & Caulfield (2003); Davis Wykes et al (2015); Salehipour & Peltier (2015) and references therein. The traditional approach involves direct analyses of the diffusive destruction of smallscale density variance as the experiment proceeds, which in turn requires a separation of the influence of stirring from that of irreversible mixing through application of the Lorenz concept of available potential energy that can be converted into kinetic energy and a base-state potential energy which can not.…”

“…It has been demonstrated that the diffusive destruction of small-scale density variance may be represented by the time derivative M of base-state potential energy plus a small correction due to the action of molecular diffusion on the initial density stratification, a correction that becomes negligible in the limit of high Reynolds number (Winters et al 1995). The time dependent efficiency of turbulent mixing may be then computed from the direct numerical simulations as η t = M/(M + ǫ) where ǫ is the rate of viscous kinetic energy dissipation in the fluid domain (Peltier & Caulfield 2003;Salehipour & Peltier 2015). This definition of mixing efficiency is global in space since the computation of the base-state potential energy requires a rearrangement of the fluid particle at the domain scale.…”