References

Ocean Science

Ocean Sci.

1812-0792

Copernicus Publications

Göttingen, Germany

10.5194/os-5-47-2009

Turbulence closure: turbulence, waves and the wave-turbulence transition – Part 1: Vanishing mean shear

Baumert

H. Z.

¹ Peters

Freie Universität, Dept. Mathematics and Computer Science, Berlin, and Institute for Applied Marine and Limnic Studies, Hamburg, Germany

Earth and Space Research, Seattle, USA

06 03 2009

5 1 47 58

2009

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This paper extends a turbulence closure-like model for stably stratified flows into a new dynamic domain in which turbulence is generated by internal gravity waves rather than mean shear. The model turbulent kinetic energy (TKE, K) balance, its first equation, incorporates a term for the energy transfer from internal waves to turbulence. This energy source is in addition to the traditional shear production. The second variable of the new two-equation model is the turbulent enstrophy (Ω). Compared to the traditional shear-only case, the Ω-equation is modified to account for the effect of the waves on the turbulence time and space scales. This modification is based on the assumption of a non-zero constant flux Richardson number in the limit of vanishing mean shear when turbulence is produced exclusively by internal waves. This paper is part 1 of a continuing theoretical development. It accounts for mean shear- and internal wave-driven mixing only in the two limits of mean shear and no waves and waves but no mean shear, respectively. The new model reproduces the wave-turbulence transition analyzed by D'Asaro and Lien (2000b). At small energy density E of the internal wave field, the turbulent dissipation rate (ε) scales like ε~E2. This is what is observed in the deep sea. With increasing E, after the wave-turbulence transition has been passed, the scaling changes to ε~E1. This is observed, for example, in the highly energetic tidal flow near a sill in Knight Inlet. The new model further exhibits a turbulent length scale proportional to the Ozmidov scale, as observed in the ocean, and predicts the ratio between the turbulent Thorpe and Ozmidov length scales well within the range observed in the ocean.

References 1

Abarbanel, H. D I., Holm, D D., Marsden, J E., and Ratio, T.: Richardson number criterion for the nonlinear stability of three-dimensional stratified flow, Phys. Rev. Lett., 26, 2352–2355, 1984.

Baumert, H.: A Novel Two-Equation Turbulence Closure for high Reynolds numbers. Part B: Spatially Non-Uniform Conditions, in: Marine Turbulence, Theories, Observations and Models, edited by Baumert, H., Simpson, J., and Sündermann, J., Cambridge University Press, Cambridge, UK, 4, 31–43, 2005.

Baumert, H. and Peters, H.: Second Moment Closures and Length Scales for Stratified Turbulent Shear Flows, J. Geophys. Res., 105, 6453–6468, 2000.

Baumert, H. and Peters, H.: Turbulence Closure, Steady State, and Collapse into Waves, J. Phys. Oceanogr., 34, 505–512, 2004.

Baumert, H. and Peters, H.: A Novel Two-Equation Turbulence Closure for high Reynolds numbers. Part A: Homogeneous, Non-Rotating Stratified Shear Layers, in: Marine Turbulence, Theories, Observations and Models, edited by Baumert, H., Simpson, J., and Sündermann, J., Cambridge University Press, Cambridge, UK, 14–30, 2005.

Canuto, V M., Howard, A., Cheng, Y., and Dubovikov, M S.: Ocean turbulence I: One-point closure model. Momentum and heat vertical diffusivities, J. Phys. Oceanogr., 31, 1413–1426, 2001.

Crawford, W R.: A Comparison of Length Scales and Decay Times of Turbulence in Stratified Flows, J. Phys. Oceanogr., 16, 1847–1854, 1986.

D'Asaro, E A. and Lien, R C.: Lagrangian measurements of waves and turbulence in stratifed flows, J. Phys. Oceanogr., 30, 641–655, 2000a.

D'Asaro, E A. and Lien, R C.: The Wave-Turbulence Transition for Stratified Flows, J. Phys. Oceanogr., 30, 1669–1678, 2000b.

Dickey, T D. and Mellor, G L.: Decaying Turbulence in Neutral and Stratified Fluids, J. Fluid Mech., 99, 37–48, 1980.

Dillon, T M.: Vertical Overturns: A Comparison of Thorpe and Ozmidov Length Scales, J. Geophys. Res., 87, 9601–9613, 1982.

Foken, T.: 50 years of the Monin-Obukhov similarity theory, Bound.-Lay. Meteorol., 119, 431–447, 2006.

Fritts, D. and Alexander, J.: Middle atmosphere gravity wave dynamics, Rev. Geophys., 41, 1–64, 2003.

Fritts, D C.: A review of gravity wave saturation processes, effects and variability in the middle atmosphere, Pure Appl. Geophys., 130, 242–371, 1989.

Galperin, B., Sukoriansky, S., and Anderson, P. S.: On the critical Richardson number in stably stratified turbulence, Atmos. Sci. Lett., 8, 65–69, 2007.

Garrett, C. J R. and Munk, W H.: Space-Time Scales of Internal Waves, Geophys. Astrophys. Fluid Dyn., 3, 225–264, 1972.

Garrett, C. J R. and Munk, W H.: Space-Time Scales of Internal Waves: A Progress Report, J. Geophys. Res., 80, 291–297, 1975.

Gill, A E.: Atmosphere-Ocean Dynamics, Academic Press, New York, 662~pp., 1982.

Gregg, M C.: Scaling of Turbulent Dissipation in the Thermocline, J. Geophys. Res., 94, 9686–9697, 1989.

Gregg, M C., Winkel, D P., and Sanford, T B.: Varieties of Fully Resolved Spectra of Vertical Shear, J. Phys. Oceanogr., 23, 124–141, 1993.

Henyey, F S., Wright, J., and Flatté, S M.: Energy and Action Flow Through the Internal Wave Field: An Eikonal Approach, J. Geophys. Res., 91, 8487–8495, 1986.

Howard, L N.: Notes on a paper of John W. Miles, J. Fluid. Mech., 10, 509–512, 1961.

Li, M., Zhong, L., and Boicourt, W C.: Simulations of Chesapeake Bay estuary: Sensitivity to turbulence mixing parameterizations and comparison with observations, J. Geophys. Res.-Oceans, 110, C12004, https://doi.org/10.1029/2004JC002585, 2005.

MacKinnon, J A. and Gregg, M C.: Mixing on the Late-Summer New England Shelf – Solibores, Shear, and Stratification, J. Phys. Oceanogr., 33, 1476–1492, 2003.

McComas, C H. and Müller, P.: The Dynamic Balance of Internal Waves, J. Phys. Oceanogr., 11, 970–986, 1981.

Mellor, G L. and Yamada, T.: A Hierarchy of Turbulence Closure models for Planetary Boundary Layers, J. Atmos. Sci., 31, 1791–1806, 1974.

Mellor, G L. and Yamada, T.: Development of a Turbulence Closure Model for Geophysical Fluid Problems, Rev. Geophys., 20(4), 851–875, 1982.

Miles, J W.: On the stability of heterogeneous shear flow, J. Fluid. Mech., 10, 496–508, 1961.

Moum, J N.: The Quest for $K_\rho$ –- Prelimiary Results From Direct Measurements of Turbulent Fluxes in the Ocean, J. Phys. Oceanogr., 20, 1980–1984, 1990.

Moum, J N.: Efficiency of Mixing in the Main Thermocline, J. Geophys. Res., 101, 12 057–12 070, 1996a.

Moum, J N.: Energy-Containing Scales of Stratified Turbulence, J. Geophys. Res., 101, 14 095–14 109, 1996b.

Oakey, N S.: Determination of the Rate of Dissipation of Turbulent Energy From Simultaneous Temperature and Velocity Shear Microstructure Measurements, J. Phys. Oceanogr., 12, 256–271, 1982.

Osborn, T R.: Estimates of the Local Rate of Vertical Diffusion from Dissipation Measurements, J. Phys. Oceanogr., 10, 83–89, 1980.

Peters, H. and Baumert, H Z.: Validating a Turbulence Closure Against Estuarine Microstructure Measurements, Ocean Modelling, 19, 183–203, 2007.

Polzin, K., Toole, J M., and Schmitt, R W.: Finescale Parameterizations of Turbulent Dissipation, J. Phys. Oceanogr., 25, 306–328, 1995.

Richardson, L F.: Weather prediction by numerical processes, Cambridge Univ. P., London, 236~pp., 1922.

Rodi, W.: Examples of Calculation Methods for Flow and Mixing in Stratified Fluids, J. Geophys. Res., 92, 5305–5328, 1987.

Rohr, J J., Itsweire, E C., Helland, K N., and Van Atta, C W.: Growth and Decay of Turbulence in a Stably Stratified Shear Flow, J. Fluid Mech., 195, 77–111, 1988.

Schumann, U. and Gerz, T.: Turbulent Mixing in Stably Stratified Shear Flows, J. Appl. Meteorol., 34, 33–48, 1995.

Shih, L H., Koseff, J R., Ferziger, J H., and Rehmann, C R.: Scaling and Parameterization of Stratified Homogeneous Turbulent Shear Flow, J. Fluid Mech., 412, 1–20, 2000.

Sukoriansky, S.: On the critical Richardson number in stably stratified turbulence, Atmospheric Science Letters, 8(3), 65–69, 2007.

Sukoriansky, S., Galperin, B., and Perov, V.: A quasi-normal scale elimination model of turbulence and its application to stably stratified flows, Nonlinear. Proc. Geoph., 13, 9–22, 2006.

Tjernström, M.: Turbulence length scales in stably stratied flows analyzed from slant aircraft proiles, J. Climate, 32, 948–963, 1993.

Weinstock, J.: Vertical Turbulence Diffusivity for Weak and Strong Stable Stratification, J. Geophys. Res., 86, 9925–9928, 1981.

Wijesekera, H W. and Dillon, T M.: Shannon Entropy as an Indicator of Age for Turbulent Overturns in the Oceanic Thermocline, J. Geophys. Res., 102, 3279–3291, 1997.

Wilcox, D C.: Turbulence modeling for CFD, DCW Industries, Inc., La Canada, CA, 2nd edn., 540~pp., 1998.

Zilitinkevich, S S., Elperin, T., Kleeorin, N., and Rogachevskii, I.: Energy- and flux-budget (EFB) turbulence closure model for the stably stratified flows. Part I: Steady-state, homogeneous regimes, Bound.-Lay. Meteorol., 125, 167–191, 2007.