Closure of the Reynolds equations for stably stratified turbulent flows in the atmosphere and the ocean
Authors: Zilitinkevich S.S., Elperin T., Kleeorin N., Rogachevsky I.
This paper presents a new turbulence closure model based on the budget equations for the key second
moments: turbulent kinetic and potential energies: TKE and TPE (comprising the turbulent total energy:
TTE = TKE + TPE) and vertical turbulent fluxes of momentum and buoyancy (proportional to potential
temperature). Besides the concept of TTE, we essentially use the non-gradient correction to the traditional
buoyancy flux formulation (disregarded in the traditional formulations for the eddy conductivity). In the
proposed model, turbulent motions exist at any values of the gradient Richardson number, Ri. Instead of its
critical value separating – as usually assumed – the turbulent and the laminar regimes, the model, as well as
experimental data, reveals a transition interval, 0.1<Ri<1, which separates two regimes of principally
different nature but both turbulent: strong turbulence at Ri«1; and weak turbulence, capable of transporting
momentum but much less efficient in transporting heat, at Ri>1. Predictions from the model are consistent with
available data from atmospheric and laboratory experiments, direct numerical simulation (DNS) and large-eddy
Tags: anisotropy; closure of turbulent motion equations; critical Richardson number; kinetic turbulent energy; mixing length; potential turbulent energy; stable stratification; total turbulent energy; turbulent transport; turbulent viscosity
- Курбацкий А.Ф. Лекции по турбулентности. – Новосибирск, Изд-во Новосибирского государственного университета, 2000.
- Agrawal, A., Djenidi and R.A. Antobia (2004) URL http://in3.dem.ist.utl.pt/lxlaser2004/pdf/ paper_28_1.pdf
- Banta R.M., R.K. Newsom, J.K. Lundquist, Y.L. Pichugina, R.L. Coulter and L. Mahrt (2002) Nocturnal low-level jet characteristics over Kansas during CASES-99. Boundary-Layer Meteorol., 105, pp. 221–252.
- Beare, R.J., M.K. MacVean, A.A.M. Holtslag, J. Cuxart, I. Esau, J.C. Golaz, M.A. Jimenez, M. Khairoudinov, B. Kosovic, D. Lewellen, T.S. Lund, J.K. Lundquist, A. McCabe, A.F. Moene, Y. Noh, S. Raasch and P. Sullivan (2006) An intercomparison of large eddy simulations of the stable boundary layer. Boundary-Layer Meteorol., 118, pp. 247– 272.
- Bertin, F., J. Barat and R. Wilson (1997) Energy dissipation rates, eddy diffusivity, and the Prandtl number: An in situ experimental approach and its consequences on radar estimate of turbulent parameters. Radio Science, 32, pp. 791–804.
- Canuto, V.M., F. Minotti (1993) Stratified turbulence in the atmosphere and oceans: a new sub-grid model. J. Atmos. Sci. 50, pp. 1925–1935.
- Canuto, V.M., A. Howard, Y. Cheng, M.S. Dubovikov (2001) Ocean turbulence. Part I: One-point closure model – momentum and heat vertical diffusivities, J. Phys. Oceanogr., 31, pp. 1413–1426.
- Cheng, Y., V.M. Canuto, A.M. Howard (2002) An improved model for the turbulent PBL. J. Atmosph. Sci., 59, pp. 1550–1565.
- Churchill, S.W. (2002) A reinterpretation of the turbulent Prandtl number. Ind. Eng. Chem. Res., 41, pp. 6393 –6401.
- Cuxart, J. and 23 co-authors (2006) Single-column model intercomparison for a stably stratified atmospheric boundary layer. Boundary-Layer Meteorol., 118, pp. 273–303.
- Dalaudier, F. and C. Sidi (1987) Evidence and interpretation of a spectral gap in the turbulent atmospheric temperature spectra. J. Atmos. Sci. 44, pp. 3121–3126.
- Elperin, T., N. Kleeorin and I. Rogachevskii (1996) Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow. Phys. Rev. E, 53, pp. 3431–3441.
- Elperin, T., N. Kleeorin, I. Rogachevskii and S.S. Zilitinkevich (2002) Formation of large-scale semi-organized structures in turbulent convection. Phys. Rev. E, 66, 066305 (1–15).
- Elperin, T., N. Kleeorin, I. Rogachevskii and S.S. Zilitinkevich (2006) Turbulence and coherent structures in geophysical convection. Boundary-Layer Meteorol. 119, pp. 449–472.
- Esau, I. (2004) Simulation of Ekman boundary layers by large eddy model with dynamic mixed sub-filter closure, Environmental Fluid Mech., 4, pp. 273–303.
- Esau, I.N. and Zilitinkevich S.S. (2006) Universal dependences between turbulent and mean flow parameters in stably and neutrally stratified planetary boundary layers. Nonlin. Processes Geophys., 13, pp. 135–144.
- Foken, T. (2006) 50 years of the Monin–Obukhov similarity theory. Boundary-Layer Meteorology, 119, pp. 431–447.
- Hanazaki, H. and J.C.R. Hunt (1996) Linear processes in unsteady stably stratified turbulence. J. Fluid Mech., 318, pp. 303–337.
- Hanazaki, H. and J.C.R. Hunt (2004) Structure of unsteady stably stratified turbulence with mean shear. J. Fluid Mech., 507, pp. 1–42.
- Hunt, J.C.R., J.C. Kaimal and J.E. Gaynor (1985) Some observations of turbulence in stable layers. Quart, J. Roy. Meteorol. Soc., 111, pp. 793–815.
- Hunt, J.C.R., D.D. Stretch and R.E. Britter (1988) Length scales in stably stratified turbulent flows and their use in turbulence models. In: Proc. I.M.A. Conference on “Stably Stratified Flow and Dense Gas Dispersion” (J. S. Puttock, Ed.), Clarendon Press, pp. 285–322.
- Holton, J.R. (2004) An Introduction to Dynamic Meteorology. Academic Press, New York, 535 pp.
- Jin, L.H., R.M.C. So and T.B. Gatski (2003) Equilibrium states of turbulent homogeneous buoyant flows. J. Fluid Mech., 482, pp. 207–233.
- Kays, W.M. (1994) Turbulent Prandtl number – where are we? J. Heat Transfer, 116, pp. 284–295.
- Keller, K. and C.W. Van Atta (2000) An experimental investigation of the vertical temperature structure of homogeneous stratified shear turbulence. J. Fluid Mech., 425, pp. 1–29.
- Kraus, E.B. and J.A. Businger (1994) Atmosphere-Ocean Interaction. Oxford University Press, Oxford, 362 pp.
- Kolmogorov, A.N. (1941) Energy dissipation in locally isotropic turbulence. Doklady AN SSSR, 32, No.1, pp. 19–21.
- Kondo, J., O. Kanechika and N. Yasuda (1978) Heat and momentum transfer under strong stability in the atmospheric surface layer. J. Atmos. Sci., 35, pp. 1012–1021.
- Luyten, P.J., S. Carniel and G. Umgiesser (2002) Validation of turbulence closure parameterisations for stably stratified flows using the PROVESS turbulence measurements in the North Sea. J. Sea Research, 47, pp. 239–267.
- L’vov, V.S., A. Pomyalov, I. Procaccia and S.S. Zilitinkevich (2006) Phenomenology of wall bounded Newtonian turbulence. Phys. Rev., E 73, 016303, pp. 1–13.
- Mahrt, L. and D. Vickers (2005) Boundary layer adjustment over small-scale changes of surface heat flux. Boundary-Layer Meteorol., 116, pp. 313–330.
- Mauritsen, T., and G. Svensson (2007) Observations of stably stratified shear-driven atmospheric turbulence at low and high Richardson numbers. J. Atmos. Sci., 64, 645–655.
- Mauritsen, T., G. Svensson, S.S. Zilitinkevich, E. Esau, L. Enger and B. Grisogono. (2007) A total turbulent energy closure model for neutrally and stably stratified atmospheric boundary layers. J. Atmos. Sci., 64, pp. 4117–4130.
- Mellor, G.L. and T. Yamada (1974) A hierarchy of turbulence closure models for planetary boundary layers. J. Atmos. Sci., 31, pp. 1791–1806.
- Moser, R.G., J. Kim and N.N. Mansour (1999) Direct numerical simulation of turbulent channel flow up to Re = 590. Phys. Fluids, 11, pp. 943–945.
- Monin, A.S. and A.M. Obukhov (1954) Main characteristics of the turbulent mixing in the atmospheric surface layer. Trudy Geophys. Inst. AN. SSSR, 24(151), pp. 153–187.
- Monin, A.S. and A.M. Yaglom (1971) Statistical Fluid Mechanics. Vol. 1. MIT Press, Cambridge, Massachusetts, 769 pp.
- Ohya, Y. (2001) Wind-tunnel study of atmospheric stable boundary layers over a rough surface, Boundary-Layer Meteorol., 98, pp. 57–82.
- Ozmidov, R.V. (1990) Diffusion of Contaminants in the Ocean. Kluwer Academic Publishers, Dordrecht, The Netherlands, 283 pp.
- Poulos, G.S., W. Blumen, D.C. Fritts, J.K. Lundquist, J. Sun, S.P. Burns, C. Nappo, R. Banta, R. Newsom, J. Cuxart, E. Terradellas, B. Balsley and M. Jensen (2002) CASES–99: A Comprehensive Investigation of the Stable Nocturnal Boundary Layer, Bull. Amer. Meteorol. Soc., 83, pp. 555–581.
- Rehmann, C.R. and J.H. Hwang (2005) Small-scale structure of strongly stratified turbulence, J. Phys. Oceanogr., 32, pp. 154–164.
- Rehmann, C.R., J.R. Koseff (2004) Mean potential energy change in stratified grid turbulence Dynamics of Atmospheres and Oceans, 37, pp. 271–294.
- Richardson, L.F. (1920) The supply of energy from and to atmospheric eddies. Proc. Roy. Soc. London, A 97, pp. 354–373.
- Rotta, J.C. (1951) Statistische theorie nichthomogener turbulenz. Z. Physik, 129, pp. 547–572.
- Schumann, U. and T. Gerz (1995) Turbulent mixing in stably stratified sheared flows. J. App. Meteorol., 34, pp. 33–48.
- Strang, E.J. and H.J.S. Fernando (2001) Vertical mixing and transports through a stratified shear layer. J. Phys. Oceanogr., 31, pp. 2026–2048.
- Stretch, D.D., J.W. Rottman, K.K. Nomura, and S.K. Venayagamoorthy (2001) Transient mixing events in stably stratified turbulence. In: 14 th Australasian Fluid Mechanics Conference, Adelaide, Australia, 10–14 December 2001.
- Umlauf, L. (2005) Modelling the effects of horizontal and vertical shear in stratified turbulent flows, Deep-Sea Research, 52, pp. 1181–1201.
- Umlauf, L. and H. Burchard (2005) Second-order turbulence closure models for geophysical boundary layers. A review of recent work. Continental Shelf Research, 25, pp. 725–827.
- Uttal, T., J.A. Curry, M.G. McPhee, D.K. Perovich and 24 other co-authors (2002) Surface Heat Budget of the Arctic Ocean, Bull. Amer. Meteorol. Soc., 83, pp. 255–276.
- Weng, W. and P. Taylor (2003) On modelling the one-dimensional Atmospheric Boundary Layer, Boundary-Layer Meteorol., 107, pp. 371–400.
- Zilitinkevich, S.S., T. Elperin, N. Kleeorin, I. Rogachevskii (2007) Energy- and flux budget (EFB) turbulence closure model for stably stratified flows. Part I: Steady-state, homogeneous regimes. Boundary-Layer Meteorol., 125, pp. 167–192.
- Zilitinkevich, S.S., T. Elperin, N. Kleeorin, I. Rogachevskii, I. Esau, T. Mauritsen, M. Miles (2008) Turbulence energetics in stably stratified geophysical flows: strong and weak mixing regimes. Quart. J. Roy. Meteorol. Soc., 134, pp. 793–799.
- Zilitinkevich, S.S. and I.N. Esau (2007) Similarity theory and calculation of turbulent fluxes at the surface for the stably stratified atmospheric boundary layers. Boundary-Layer Meteorol., 125, pp. 193–296.
- Zilitinkevich, S.S., V.M. Gryanik, V.N. Lykossov and D.V. Mironov (1999) Third-order transport and non-local turbulence closures for convective boundary layers. J. Atmosph. Sci., 56, pp. 3463–3477.
- Zilitinkevich, S.S., V.L. Perov and J.C. King (2002) Near-surface turbulent fluxes in stable stratification: calculation techniques for use in general circulation Quart. J. Roy. Meteorol. Soc.,128, pp. 1571–1587.