Mixing efficiency in stratified turbulence

Abstract

We consider mixing of the density field in stratified turbulence and argue that, at sufficiently high Reynolds numbers, stationary turbulence will have a mixing efficiency and closely related mixing coefficient described solely by the turbulent Froude number$Fr={\it\epsilon}_{k}/(Nu^{2})$, where${\it\epsilon}_{k}$is the kinetic energy dissipation,$u$is a turbulent horizontal velocity scale and$N$is the Brunt–Väisälä frequency. For$Fr\gg 1$, in the limit of weakly stratified turbulence, we show through a simple scaling analysis that the mixing coefficient scales as${\it\Gamma}\propto Fr^{-2}$, where${\it\Gamma}={\it\epsilon}_{p}/{\it\epsilon}_{k}$and${\it\epsilon}_{p}$is the potential energy dissipation. In the opposite limit of strongly stratified turbulence with$Fr\ll 1$, we argue that${\it\Gamma}$should reach a constant value of order unity. We carry out direct numerical simulations of forced stratified turbulence across a range of$Fr$and confirm that at high$Fr$,${\it\Gamma}\propto Fr^{-2}$, while at low$Fr$it approaches a constant value close to${\it\Gamma}=0.33$. The parametrization of${\it\Gamma}$based on$Re_{b}$due to Shihet al.(J. Fluid Mech., vol. 525, 2005, pp. 193–214) can be reinterpreted in this light because the observed variation of${\it\Gamma}$in their study as well as in datasets from recent oceanic and atmospheric measurements occurs at a Froude number of order unity, close to the transition value$Fr=0.3$found in our simulations.

A statistical mechanics approach to mixing in stratified fluids

“…This linear scaling has been also reported by Maffioli et al (2016) in forced-dissipative numerical experiments, who also provide complementary arguments based on cascade phenomenology.…”

“…This scaling law can † Maffioli et al (2016) report a mixing coefficient Γ = η/(1 − η) = 0.33 in the limit of small Froude numbers, which corresponds to η = 0.25 in the limit of large Richardson numbers.…”

“…Maffioli et al (2016); Venayagamoorthy & Koseff (2016) and references therein †. It is remarkable that the statistical mechanics theory in the large Richardson number limit also yields η = 0.25, provided that the initial buoyancy is strictly monotonic.…”

Helicity Dynamics, Inverse, and Bidirectional Cascades in Fluid and Magnetohydrodynamic Turbulence: A Brief Review

“…The presence of waves affects directly turbulent flows in lessening the rate at which energy is dissipated. It has been observed in several instances (Alexakis, ; Campagne et al, ; Feraco et al, ; Maffioli et al, ; Pouquet et al, ), within the case of rotating stratified flows, a clear linear dependence of the measured dissipation on the control parameter, the Froude number in an intermediate regime of eddy‐wave interactions (Marino et al, ; Pouquet et al, ). On the other hand, the Rossby number controls the amount of energy going to the large scales in a simultaneous inverse cascade, whereas for MHD the control parameter is the magnitude of the large‐scale imposed magnetic field.…”

Mixing Efficiency in the Presence of Stratification: When Is It Constant?

“…Thus, if the only relevant variables are B , ε , ν, and N 2 , then Ri f must be only a function of Re B . If S , TKE, k , and turbulent length scale l t are also thought to be important, then three other parameters upon which Ri f could depend would be Ri , the Froude number Fr k = ε / Nk (Maffioli et al, ), and Fr t = ( L O / L t ) 2/3 (Ivey & Imberger, ). We note that few data sets include S and fewer still include k ; neither is reported in the microstructure database.…”