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Statistical state dynamics of vertically sheared horizontal flows in two-dimensional stratified turbulence

  • Joseph G. Fitzgerald (a1) and Brian F. Farrell (a1)


Simulations of strongly stratified turbulence often exhibit coherent large-scale structures called vertically sheared horizontal flows (VSHFs). VSHFs emerge in both two-dimensional (2D) and three-dimensional (3D) stratified turbulence with similar vertical structure. The mechanism responsible for VSHF formation is not fully understood. In this work, the formation and equilibration of VSHFs in a 2D Boussinesq model of stratified turbulence is studied using statistical state dynamics (SSD). In SSD, equations of motion are expressed directly in the statistical variables of the turbulent state. Restriction to 2D turbulence facilitates application of an analytically and computationally attractive implementation of SSD referred to as S3T, in which the SSD is expressed by coupling the equation for the horizontal mean structure with the equation for the ensemble mean perturbation covariance. This second-order SSD produces accurate statistics, through second order, when compared with fully nonlinear simulations. In particular, S3T captures the spontaneous emergence of the VSHF and associated density layers seen in simulations of turbulence maintained by homogeneous large-scale stochastic excitation. An advantage of the S3T system is that the VSHF formation mechanism, which is wave–mean flow interaction between the emergent VSHF and the stochastically excited large-scale gravity waves, is analytically understood in the S3T system. Comparison with fully nonlinear simulations verifies that S3T solutions accurately predict the scale selection, dependence on stochastic excitation strength, and nonlinear equilibrium structure of the VSHF. These results constitute a theory for VSHF formation applicable to interpreting simulations and observations of geophysical examples of turbulent jets such as the ocean’s equatorial deep jets.


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Ascani, F., Firing, E., McCreary, J. P., Brandt, P. & Greatbatch, R. J. 2015 The deep equatorial ocean circulation in wind-forced numerical solutions. J. Phys. Oceanogr. 45, 17091734.
Bakas, N. A., Constantinou, N. C. & Ioannou, P. J. 2018 Statistical state dynamics of weak jets in barotropic beta-plane turbulence. J. Atmos. Sci. (in review), arXiv:1708.03031.
Bakas, N. A. & Ioannou, P. J. 2011 Structural stability theory of two-dimensional fluid flow under stochastic forcing. J. Fluid Mech. 682, 332361.
Bakas, N. A. & Ioannou, P. J. 2013 On the mechanism underlying the spontaneous emergence of barotropic zonal jets. J. Atmos. Sci. 70, 22512271.
Bakas, N. A., Ioannou, P. J. & Kefaliakos, G. E. 2001 The emergence of coherent structures in stratified shear flow. J. Atmos. Sci. 58 (18), 27902806.
Berloff, P., Kamenkovich, I. & Pedlosky, J. 2009 A mechanism of formation of multiple zonal jets in the oceans. J. Fluid Mech. 628, 395425.
Brandt, P., Funk, A., Hormann, V., Dengler, M., Greatbatch, R. J. & Toole, J. M. 2011 Interannual atmospheric variability forced by the deep equatorial Atlantic Ocean. Nature 473, 497500.
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J. M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343368.
Constantinou, N.2015 Formation of large-scale structures by turbulence in rotating planets. PhD thesis, National and Kapodistrian University of Athens, also at arXiv:1503.07644.
Constantinou, N. C., Farrell, B. F. & Ioannou, P. J. 2014 Emergence and equilibration of jets in beta-plane turbulence: applications of stochastic structural stability theory. J. Atmos. Sci. 71, 18181842.
Constantinou, N. C. & Parker, J. B. 2018 Magnetic suppression of zonal flows on a beta plane. Astrophys. J. 863 (1), 46.
Davis, P. J. 1978 Circulant Matrices. Wiley-Interscience.
Eden, C. & Dengler, M. 2008 Stacked jets in the deep equatorial Atlantic Ocean. J. Geophys. Res. 113, C04003.
Farrell, B. F., Gayme, D. F. & Ioannou, P. J. 2017a A statistical state dynamics approach to wall turbulence. Phil. Trans. R. Soc. Lond. A 375, 20160081.
Farrell, B. F. & Ioannou, P. J. 1993a Stochastic dynamics of baroclinic waves. J. Atmos. Sci. 50, 40444057.
Farrell, B. F. & Ioannou, P. J. 1993b Transient development of perturbations in stratified shear flow. J. Atmos. Sci. 50, 22012214.
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory. Part I. Autonomous operators. J. Atmos. Sci. 53, 20252040.
Farrell, B. F. & Ioannou, P. J. 2002 Perturbation growth and structure in uncertain flows. Part II. J. Atmos. Sci. 59, 26472664.
Farrell, B. F. & Ioannou, P. J. 2003 Structural stability of turbulent jets. J. Atmos. Sci. 60, 21012118.
Farrell, B. F. & Ioannou, P. J. 2007 Structure and spacing of jets in barotropic turbulence. J. Atmos. Sci. 64, 36523665.
Farrell, B. F. & Ioannou, P. J. 2008 Formation of jets by baroclinic turbulence. J. Atmos. Sci. 65, 33533375.
Farrell, B. F. & Ioannou, P. J. 2009a A stochastic structural stability theory model of the drift wave–zonal flow system. Phys. Plasmas 16, 112903.
Farrell, B. F. & Ioannou, P. J. 2009b A theory of baroclinic turbulence. J. Atmos. Sci. 66, 24442454.
Farrell, B. F. & Ioannou, P. J. 2009c Emergence of jets from turbulence in the shallow-water equations on an equatorial beta plane. J. Atmos. Sci. 66, 31973207.
Farrell, B. F. & Ioannou, P. J. 2012 Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow. J. Fluid Mech. 708, 149196.
Farrell, B. F. & Ioannou, P. J. 2017a Statistical state dynamics: a new perspective on turbulence in shear flow. In Zonal Jets: Phenomenology, Genesis, Physics (ed. Galperin, B. & Read, P. L.). Cambridge University Press.
Farrell, B. F. & Ioannou, P. J. 2017b Statistical state dynamics-based analysis of the physical mechanisms sustaining and regulating turbulence in Couette flow. Phys. Rev. Fluids 2 (8), 084608.
Farrell, B. F. & Ioannou, P. J. 2017c Statistical state dynamics based theory for the formation and equilibration of Saturn’s north polar jet. Phys. Rev. Fluids 2 (7), 073801.
Farrell, B. F., Ioannou, P. J. & Nikolaidis, M.-A. 2017b Instability of the roll-streak structure induced by background turbulence in pretransitional Couette flow. Phys. Rev. Fluids 2 (3), 034607.
Fitzgerald, J. G. & Farrell, B. F. 2014 Mechanisms of mean flow formation and suppression in two-dimensional Rayleigh–Bénard convection. Phys. Fluids 26, 054104.
Fitzgerald, J. G. & Farrell, B. F. 2018 Vertically sheared horizontal flow-forming instability in stratified turbulence: linear stability analysis using the analytical approach to statistical state dynamics. J. Atmos. Sci. (in review), arXiv:1803.00847v1.
Fjørtoft, R. 1953 On the changes in the spectral distribution of kinetic energy for two-dimensional, non-divergent flow. Tellus 5A, 225230.
Galmiche, M. & Hunt, J. C. R. 2002 The formation of shear and density layers in stably stratified turbulent flows: linear processes. J. Fluid Mech. 455, 243262.
Galmiche, M., Thual, O. & Bonneton, P. 2002 Direct numerical simulation of turbulence–mean field interactions in a stably stratified fluid. J. Fluid Mech. 455, 213242.
Galperin, B., Sukoriansky, S. & Anderson, P. S. 2007 On the critical Richardson number in stably stratified turbulence. Atmos. Sci. Lett. 8 (3), 6569.
Galperin, B., Sukoriansky, S. & Dikovskaya, N. 2010 Geophysical flows with anisotropic turbulence and dispersive waves: flows with a beta-effect. Ocean Dyn. 60 (2, SI), 427441.
Galperin, B., Young, R. M. B., Sukoriansky, S. & Dikovskaya, N. 2014 Cassini observations reveal a regime of zonostrophic macroturbulence on Jupiter. Icarus 229, 295320.
Herbert, C., Marino, R., Rosenberg, D. & Pouquet, A. 2016 Waves and vortices in the inverse cascade regime of stratified turbulence with or without rotation. J. Fluid Mech. 806, 165204.
Herring, J. R. & Métais, O. 1989 Numerical experiments in forced stably stratified turbulence. J. Fluid Mech. 202, 97115.
Holloway, G. 1986 Eddies, waves, circulation, and mixing: statistical geofluid mechanics. Annu. Rev. Fluid Mech. 18, 91147.
Hua, B. L., D’orgeville, M. & Fruman, M. D. 2008 Destabilization of mixed Rossby gravity waves and the formation of equatorial zonal jets. J. Fluid Mech. 610, 311341.
Huang, H.-P., Galperin, B. & Sukoriansky, S. 2001 Anisotropic spectra in two-dimensional turbulence on the surface of a rotating sphere. Phys. Fluids 13 (1), 225240.
Kaminski, A. K., Caulfield, C. P. & Taylor, J. R. 2014 Transient growth in strongly stratified shear layers. J. Fluid Mech. 758, R4.
Kraichnan, R. H. 1957 Relation of fourth-order to second-order moments in stationary isotropic turbulence. Phys. Rev. 107 (6), 14851490.
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10 (7), 14171423.
Kumar, A., Verma, M. K. & Sukhatme, J. 2017 Phenomenology of two-dimensional stably stratified turbulence under large-scale forcing. J. Turbul. 18 (3), 219239.
Laval, J. P., McWilliams, J. & Dubrulle, B. 2003 Forced stratified turbulence: successive transitions with Reynolds number. Phys. Rev. E 68 (3), 036308.
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207242.
Majda, A. J., Timofeyev, I. & Vanden Eijnden, E. 1999 Models for stochastic climate prediction. Proc. Natl Acad. Sci. USA 96, 1468714691.
Marino, R., Mininni, P. D., Rosenberg, D. L. & Pouquet, A. 2014 Large-scale anisotropy in stably stratified rotating flows. Phys. Rev. E 90 (2), 023018.
Marston, J. B. 2010 Statistics of the general circulation from cumulant expansions. Chaos 20, 041107.
Marston, J. B. 2012 Planetary atmospheres as nonequilibrium condensed matter. Annu. Rev. Condens. Matter Phys. 3, 285310.
Marston, J. B., Conover, E. & Schneider, T. 2008 Statistics of an unstable barotropic jet from a cumulant expansion. J. Atmos. Sci. 65, 19551966.
McCreary, J. P. 1984 Equatorial beams. J. Mar. Res. 42, 395430.
Ménesguen, C., Hua, B. L., Fruman, M. D. & Schopp, R. 2009 Intermittent layering in the Atlantic equatorial deep jets. J. Mar. Res. 67, 347360.
Muench, J. E. & Kunze, E. 1999 Internal wave interactions with equatorial deep jets. Part I. Momentum-flux divergences. J. Phys. Oceanogr. 29, 14531467.
Ogura, Y. 1963 A consequence of the zero-fourth-cumulant approximation in the decay of isotropic turbulence. J. Fluid Mech. 16 (1), 3340.
Parker, J. B. & Krommes, J. A. 2013 Zonal flow as pattern formation. Phys. Plasmas 20, 100703.
Parker, J. B. & Krommes, J. A. 2014 Generation of zonal flows through symmetry breaking of statistical homogeneity. New J. Phys. 16, 035006.
Remmel, M., Sukhatme, J. & Smith, L. M. 2013 Nonlinear gravity–wave interactions in stratified turbulence. Theor. Comput. Fluid Dyn. 28, 131145.
Riley, J. J. & Lelong, M.-P. 2000 Fluid motions in the presence of strong stable stratification. Annu. Rev. Fluid Mech. 32, 613657.
Rorai, C., Mininni, P. D. & Pouquet, A. 2015 Stably stratified turbulence in the presence of large-scale forcing. Phys. Rev. E 92 (1), 013003.
Salmon, R. 1982 Geostrophic turbulence. In Topics in Ocean Physics (ed. Osborne, A. R. & Rizzoli, P. M.). Italian Physical Society.
Smith, L. M. 2001 Numerical study of two-dimensional stratified turbulence. In Advances in Wave Interaction and Turbulence (ed. Milewski, P. A., Smith, L. M., Waleffe, F. & Tabak, E. G.). American Mathematical Society.
Smith, L. M. & Waleffe, F. 2002 Generation of slow large scales in forced rotating stratified turbulence. J. Fluid Mech. 451, 145168.
Squire, J. & Bhattacharjee, A. 2015 Statistical simulation of the magnetorotational dynamo. Phys. Rev. Lett. 114 (8), 085002.
Srinivasan, K. & Young, W. R. 2012 Zonostrophic instability. J. Atmos. Sci. 69, 16331656.
St-Onge, D. A. & Krommes, J. A. 2017 Zonostrophic instability driven by discrete particle noise. Phys. Plasmas 24, 042107.
Sukoriansky, S., Dikovskaya, N. & Galperin, B. 2009 Transport of momentum and scalar in turbulent flows with anisotropic dispersive waves. Geophys. Res. Lett. 36, L14609.
Sukoriansky, S. & Galperin, B. 2016 QNSE theory of turbulence anisotropization and onset of the inverse energy cascade by solid body rotation. J. Fluid Mech. 805, 384421.
Taylor, J. R.2008 Numerical simulations of the stratified oceanic bottom boundary layer. PhD thesis, University of California, San Diego.
Thomas, V. L., Farrell, B. F., Ioannou, P. J. & Gayme, D. F. 2015 A minimal model of self-sustaining turbulence. Phys. Fluids 27, 105104.
Thomas, V. L., Lieu, B. K., Jovanovic, M. R., Farrell, B. F., Ioannou, P. J. & Gayme, D. F. 2014 Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow. Phys. Fluids 26, 105112.
Tobias, S. M., Dagon, K. & Marston, J. B. 2011 Astrophysical fluid dynamics via direct statistical simulation. Astrophys. J. 727, 127.
Tobias, S. M. & Marston, J. B. 2013 Direct statistical simulation of out-of-equilibrium jets. Phys. Rev. Lett. 110, 104502.
Vasavada, A. R. & Showman, A. P. 2005 Jovian atmospheric dynamics: an update after Galileo and Cassini. Rep. Prog. Phys. 68, 19351996.
Waite, M. L. & Bartello, P. 2004 Stratified turbulence dominated by vortical motion. J. Fluid Mech. 517, 281308.
Waite, M. L. & Bartello, P. 2006 Stratified turbulence generated by internal gravity waves. J. Fluid Mech. 546, 313339.
Wunsch, C. 1977 Response of an equatorial ocean to a periodic monsoon. J. Phys. Oceanogr. 7, 497511.
Youngs, M. & Johnson, G. 2015 Basin-wavelength equatorial deep jet signals across three oceans. J. Phys. Oceanogr. 45, 21342148.
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