Parameterizing Mixing in Stably Stratified Shear Flows


Fig. 1. Comparison of measured and parameterized (a) P/ϵ and (b,c) Γ=b/ϵ. For ease of distinguishing Sh dependence, panels b) and c) show Γ for Sh<11 and >9, respectively. Reformatted from Shimizu (2012, J. Geophys. Res.-Oceans).

Fig. 2. Comparison of proposed parameterizations and stability functions by Kantha and Clayson (1994) and Canuto et al. (2001) at . Black lines correpond to those in Fig. 1. Reformatted from Shimizu (2012, J. Geophys. Res.-Oceans).

Fig. 3. Comparison of proposed parameterization (for homogeneous turbulence) and Monin-Obukhov theory (wall turbulence). in Monin-Obukhov theory is converted to corresponding for comparison, where is the Monin-Obukhov length. Symbols correspond to those in Fig. 1.

Fig. 4. Comparisons of proposed parameterizations and field data from equatorial Pacific by Peters et al. (1995). (a) Turbulent Froude number based on the Thorpe scale, and (b) ratio of dissipation rate of TPE to that of TKE. Dotted, dash-dotted, solid, and dashed lines show proposed parameterization at , , , and , respectively. Figure taken from Shimizu (2012, J. Geophys. Res.-Oceans).

Fig. 5. Normalized eddy diffusivity in Lake Kinneret. Vertical bars show measured eddy diffusivity that encompasses 66% of the dissipation observations by Yeates (2008). Solid curve shows proposed parameterization assuming . Figure taken from Shimizu (2012, J. Geophys. Res.-Oceans).


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 , 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 , whereas those of shear in unstratified shear flows by the nondimensional shear number . So it seemed natural to use both and 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 Venayagamoorthy6),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

where is the so-called mixing efficiency (the turbulent buoyancy flux divided by the TKE dissipation rate ), is a kind of mixing efficiency for momentum (twice the turbulent momentum flux divided by TKE), is the exponential growth rate, is the ratio of TPE to TKE, and is the ratio of dissipation rate of TPE to that of TKE. Excluding 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, , , and , to experimental data as functions of and . 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 and 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 ). They are parameterizations derived from , , and 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 and .

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 and 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 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 in the field, but it was impossible with the data.

Reduction to -based Parameterizations

The proposed parameterizations can be reduced to -based parameterizations for turbulent viscosity and diffusivity. To do so, we need to estimate turbulence intensity, or the turbulent Reynolds number . The relationship suggested by Saggio and Imberger (2001) for energetic flows in a lake leads to , and this appears to hold in estuaries as well9),18). Fig. 5 shows the resulting parameterization for Lake Kinneret, Israel, assuming . 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.
Acknowledgements

Special thanks to Lucinda H. Shih for kindly providing the DNS results of Shih et al. (2000, 2005), Parviz Moin for allowing me to use the DNS results of Rogers and Moin (1987), and Peter S. Yeates for helping me to plot a figure using the data presented in his PhD thesis. I thank Jeffrey R. Koseff, Gregory N. Ivey, and Keisuke Nakayama for their help to access some of the data sets. This study was finalized when I was at Max Planck Institute for Meteorology with the financial support of Klaus Hasselmann Postdoctoral Fellowship.

References
  1. Canuto, V. M., A. Howard, Y. Cheng, and M. S. Dubovikov. 2001. Ocean turbulence Part I: One-point closure model—Momentum and heat vertical diffusivities. Journal of Physical Oceanography, 31: 1413–1426.
  2. Kaltenbach, H.-J., T. Gerz, and U. Schumann. 1994. Large-eddy simulation of homogeneous turbulence and diffusion in stably stratified shear flow. Journal of Fluid Mechanics, 280: 1–40.
  3. Kantha, L. H., and C. A. Clayson. 1994. An improved mixed layer model for geophysical applications. Journal of Geophysical Research - Oceans, 99: 25235–25266.
  4. Itsweire, E. C., K. N. Helland, and C. W. van Atta. 1986. The evolution of grid-generated turbulence in a stably stratified fluid. Journal of Fluid Mechanics, 162: 299–338.
  5. Lienhard V, J. H., and C. W. van Atta. 1990. The decay of turbulence in thermally stratified flow. Journal of Fluid Mechanics, 210: 57–112.
  6. Mater, B. D., and S. K. Venayagamoorthy. 2014. A unifying framework for parameterizing stably stratified shear-flow turbulence, Physics of Fluids, 26: 036601.
  7. Mater, B. D., and S. K. Venayagamoorthy. 2014. The quest for an unambiguous parameterization of mixing efficiency in stably stratified geophysical flows. Geophysical Research Letters, 41: 4646–4653.
  8. Munk, W. H., and E. R. Anderson. 1948. Notes on a theory of the thermocline. Journal of Marine Research, 3: 276–295.
  9. Peters, H. 1997. Observations of stratified turbulent mixing in an estuary: Neap-to-spring variations during high river flow. Estuarine Coastal and Shelf Sciences, 45: 69–88.
  10. Peters, H., M. C. Gregg, and T. B. Sanford. 1995. Detail and scaling of turbulent overturns in the Pacific Equatorial Undercurrent. Journal of Geophysical Research - Oceans, 100: 18349–18368.
  11. Rogers, M. M., and P. Moin. 1987. The structure of the vorticity field in homogeneous turbulent flows. Journal of Fluid Mechanics, 176: 33–66.
  12. Rohr, J. J., E. C. Itsweire, K. N. Helland, and C. W. van Atta. 1988a. An investigation of the growth of turbulence in a uniform-mean-shear flow. Journal of Fluid Mechanics, 187: 1–33.
  13. Rohr, J. J., E. C. Itsweire, K. N. Helland, and C. W. van Atta. 1988b. Growth and decay of turbulence in a stably stratified shear flow. Journal of Fluid Mechanics, 195: 77–111.
  14. Saggio, A., and J. Imberger. 2001. Mixing and turbulent fluxes in the metalimnion of a stratified lake. Limnology and Oceanography, 46: 392–409.
  15. Shih, L. H., J. R. Koseff, J. H. Ferziger, and C. R. Rehmann. 2000. Scaling and parameterization of stratified homogeneous turbulent shear flow. Journal of Fluid Mechanics, 412: 1–20.
  16. Shih, L. H., J. R. Koseff, G. N. Ivey, and J. H. Ferziger. 2005. Parameterization of turbulent fluxes and scales using homogeneous sheared stably stratified turbulence simulations. Journal of Fluid Mechanics, 525: 193–214.
  17. 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.
  18. Stacey, M. T., S. G. Monismith, and J. R. Burau. 1999. Observations of turbulence in a partially mixed estuary. Journal of Physical Oceanography, 29: 1950–1970.
  19. Stillinger, D. C. 1981. An experimental study of the transition of grid turbulence to internal waves in a salt-stratified water channel. PhD thesis, 216 pp. University of California, San Diego.
  20. Tavoularis, S., and S. Corrsin. 1981. Experiments in nearly homogeneous turbulent shear flow with a uniform mean temperature gradient. Part 1. Journal of Fluid Mechanics, 104: 311–347.
  21. Tavoularis, S., and U. Karnik. 1989. Further experiments on the evolution of turbulent stresses and scales in uniformly sheared turbulence. Journal of Fluid Mechanics, 204: 457–478.
  22. Yeates, P. S. 2008. Deep mixing in stratified lakes and reservoirs. PhD thesis, 127 pp. University of Western Australia, Crawley, Western Australia, Australia.