---

Motivation

Parameterizing mixing in stratified shear flows is essential in many geophysical and engineering problems. A lot of data are now available from laboratory experiments, field observations, and numerical simulations; however, it is still common to use simple regression of a mixing parameter against a nondimensional parameter to develop mixing parameterization. While it may yield statistically satisfactory results, bigger uncertainty comes from the assumption of single independent parameter, and the choice of it, which is debated for stratified shear flows. Also, such parameterizations for multiple variables are more likely to be inconsistent. Contrarily, in (probably) the original paper that proposed common mixing parameterization based on the gradient Richardson number $\mathrm{Ri}_g$, Munk and Anderson (1948) developed good parameterizations from only 8 data points based on physical arguments and intuition. This study was my attempt to make a step towards more physically consistent, comprehensive parameterizations of mixing in stably stratified shear flows.

Approach

I focused on homogeneous stably stratified shear flows at high Reynolds number, because many data sets are available and it is the simplest case in stratified shear flows. To choose nondimensional parameters, note that the effects of stratification in unsheared stratified flows are characterized by the inverse turbulent Froude number $\mathrm{Fr}^{-1}$, whereas those of shear in unstratified shear flows by the nondimensional shear number $\mathrm{Sh}$. So it seemed natural to use both $\mathrm{Fr^{-1}}$ and $\mathrm{Sh}$ to parameterize the effects of stratification and shear in stratified shear flows²⁾. Such parameterizations based on experimental results were not available at the time of this study. (Apparently, this idea was followed by Mater and Venayagamoorthy^6),7).)

To develop physically consistent parameterization, I used the governing equations for the total kinetic energy (TKE) and the total available potential energy (TPE) in the well-developed or exponentially decaying state. The equations in a nondimensional form are

\begin{align} &\frac{1}{2} \gamma = -B_{13} \mathrm{Sh} - \Gamma - 1, \\ &\frac{1}{2} \alpha \gamma = \Gamma - \Gamma_d, \end{align}

where $\Gamma$ is the so-called mixing efficiency (the turbulent buoyancy flux $b$ divided by the TKE dissipation rate $\epsilon$), $B_{13}$ is a kind of mixing efficiency for momentum (twice the turbulent momentum flux divided by TKE), $\gamma$ is the exponential growth rate, $\alpha$ is the ratio of TPE to TKE, and $\Gamma_d$ is the ratio of dissipation rate of TPE to that of TKE. Excluding $\mathrm{Sh}$ that is assumed to be an independent variable, there are 5 variables constrained by the two equations. So to parameterize the 5 variables including mixing efficiencies, we need to fit 3 independent functions, $f_1$, $f_2$, and $f_3$, to experimental data as functions of $\mathrm{Fr^{-1}}$ and $\mathrm{Sh}$. Advantages of this approach are that the resulting parameterizations for the 5 variables are consistent, and that we can choose variables that show clearer dependence on $\mathrm{Fr^{-1}}$ and $\mathrm{Sh}$ with less scatter for fitting.

Data Sets

I collected publicly available results of previous laboratory and numerical experiments on (nearly) homogeneous stably stratified shear flows at the time of this study (2010-2011). The data sets include 'self-similar flows', 'decaying shear flows', and 'unsheared flows', all of which show exponential growth or decay of TKE and TPE. To make the data sets as homogeneous as possible and to minimize viscous effects, I used results with high Reynolds numbers and in the well-developed stage as much as possible.

Proposed Parameterizations

Fig. 1 show proposed parameterizations for mixing efficiencies (note $P/\epsilon=-B_{13} \mathrm{Sh}$). They are parameterizations derived from $f_1$, $f_2$, and $f_3$ fitted to other variables with less scatter. There are relatively large scatter in the data, which is partly due to the error in calculating turbulent fluxes; however, the parameterizations capture overall dependence on $\mathrm{Fr}^{-1}$ and $\mathrm{Sh}$.

Comparison with Turbulent Closure Schemes

Fig. 2 shows comparisons of the proposed parameterizations and the stability functions by Kantha and Clayson (1994) and Canuto et al. (2001). Although these stability functions are commonly used in ocean modelling, they clearly overestimate $P/\epsilon$ and $b/\epsilon$ under strong stratification.

Comparisons with Monin-Obukov Theory

To see the applicability to wall turbulence, I compared the propose parameterizations with the Monin-Obukov theory (Fig. 3). The parameterizations are found to be consistent with the theory. The major difference is that $\mathrm{Sh}$ tends to be ~10 for homogeneous shear flows, but ~6 for wall turbulence.

Comparisons with Field Data

Fig. 4 shows comparison of the proposed parameterizations and field observations in the equatorial Pacific¹⁰⁾. Although the scatter in the data is large, it suggests that the parameterizations based on laboratory and numerical experiments are roughly consistent with field data. I hoped to constrain the range of $\mathrm{Sh}$ in the field, but it was impossible with the data.

Reduction to $\mathrm{Ri}_g$-based Parameterizations

The proposed parameterizations can be reduced to $\mathrm{Ri}_g$-based parameterizations for turbulent viscosity and diffusivity. To do so, we need to estimate turbulence intensity, or the turbulent Reynolds number $\mathrm{Re}$. The relationship suggested by Saggio and Imberger (2001) for energetic flows in a lake leads to $\mathrm{Re} \sim \mathrm{Ri}_g^{-1/2}$, and this appears to hold in estuaries as well^9),18). Fig. 5 shows the resulting parameterization for Lake Kinneret, Israel, assuming $\mathrm{Sh} = 10$. The proposed parameterization lies within the range of measured eddy diffusivity (i.e., directly measured turbulent buoyancy flux divided by buoyancy frequency squared) in the lake²²⁾. This result is encouraging considering the assumptions made in this calculation.

---

Related Publications

Details are available in

Shimizu, K. 2012. Parameterizing individual effects of shear and stratification on mixing in stably stratified shear flows. Journal of Geophysical Research – Oceans, 117: C03030. doi:10.1029/2011JC007514.

---