Diffusive effects in local instabilities of a baroclinic axisymmetric vortex. (English) Zbl 1481.76101

Summary: We present a local stability analysis of an idealized model of the stratified vortices that appear in geophysical settings. The base flow comprises an axisymmetric vortex with background rotation and an out-of-plane stable stratification, and a radial stratification in the thermal wind balance with the out-of-plane momentum gradient. Solving the local stability equations along fluid particle trajectories in the base flow, the dependence of short-wavelength instabilities on the Schmidt number \(Sc\) (ratio between momentum and mass diffusivities) is studied, in the presence of curvature effects. In the diffusion-free limit, the well-known symmetric instability is recovered. In the viscous, double-diffusive regime, instability characteristics are shown to depend on three non-dimensional parameters (including \(Sc\)), and two different instabilities are identified: (i) a monotonic instability (same as symmetric instability at \(Sc = 1\)), and (ii) an oscillatory instability (absent at \(Sc = 1\)). Separating the base flow and perturbation characteristics, two each of base flow and perturbation parameters (apart from \(Sc\)) are identified, and the entire parameter space is explored for the aforementioned instabilities. In comparison with \(Sc = 1\), monotonic and oscillatory instabilities are shown to significantly expand the instability region in the space of base flow parameters as \(Sc\) moves away from unity. Neutral stability boundaries on the plane of \(Sc\) and a modified gradient Richardson number are then identified for both these instabilities. In the absence of curvature effects, our results are shown to be consistent with previous studies based on normal mode analysis, thus establishing that the local stability approach is well suited to capturing symmetric and double-diffusive instabilities. The paper concludes with a discussion of curvature effects, and the likelihood of monotonic and oscillatory instabilities in typical oceanic settings.


76E20 Stability and instability of geophysical and astrophysical flows
76E07 Rotation in hydrodynamic stability
76D17 Viscous vortex flows
76R50 Diffusion
76U60 Geophysical flows
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[1] Bachman, S.D., Fox-Kemper, B., Taylor, J.R. & Thomas, L.N.2017Parameterization of frontal symmetric instabilities. I: theory for resolved fronts. Ocean Model.109, 72-95.
[2] Baker, D.J.1971Density gradients in a rotating stratified fluid: experimental evidence for a new instability. Science172 (3987), 1029-1031.
[3] Billant, P. & Gallaire, F.2005Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities. J. Fluid Mech.542, 365-379. · Zbl 1078.76030
[4] Buckingham, C.E., Gula, J. & Carton, X.2021aThe role of curvature in modifying frontal instabilities. Part I: review of theory and presentation of a nondimensional instability criterion. J. Phys. Oceanogr.51 (2), 299-315.
[5] Buckingham, C.E., Gula, J. & Carton, X.2021bThe role of curvature in modifying frontal instabilities. Part II: application of the criterion to curved density fronts at low Richardson numbers. J. Phys. Oceanogr.51 (2), 317-341.
[6] Calman, J.1977Experiments on high Richardson number instability of a rotating stratified shear flow. Dyn. Atmos. Oceans1 (4), 277-297.
[7] Chandrasekhar, S.1961 Hydrodynamic and Hydromagnetic Stability, Courier Corporation. · Zbl 0142.44103
[8] Danabasoglu, G., Bates, S.C., Briegleb, B.P., Jayne, S.R., Jochum, M., Large, W.G., Peacock, S. & Yeager, S.G.2012The CCSM4 ocean component. J. Clim.25 (5), 1361-1389.
[9] D’Asaro, E., Lee, C., Rainville, L., Harcourt, R. & Thomas, L.2011Enhanced turbulence and energy dissipation at ocean fronts. Science332 (6027), 318-322, https://science.sciencemag.org/content/332/6027/318.full.pdf.
[10] Elliott, Z.A. & Venayagamoorthy, S.K.2011Evaluation of turbulent Prandtl (Schmidt) number parameterizations for stably stratified environmental flows. Dyn. Atmos. Oceans51 (3), 137-150.
[11] Emanuel, K.A.1979Inertial instability and mesoscale convective systems. Part I: linear theory of inertial instability in rotating viscous fluids. J. Atmos. Sci.36 (12), 2425-2449.
[12] Emanuel, K.A., Fantini, M. & Thorpe, A.J.1987Baroclinic instability in an environment of small stability to slantwise moist convection. Part I: two-dimensional models. J. Atmos. Sci.44 (12), 1559-1573.
[13] Fox-Kemper, B., Danabasoglu, G., Ferrari, R., Griffies, S.M., Hallberg, R.W., Holland, M.M., Maltrud, M.E., Peacock, S. & Samuels, B.L.2011Parameterization of mixed layer eddies. III: implementation and impact in global ocean climate simulations. Ocean Model.39 (1-2), 61-78.
[14] Fox-Kemper, B., Ferrari, R. & Hallberg, R.2008Parameterization of mixed layer eddies. Part I: theory and diagnosis. J. Phys. Oceanogr.38 (6), 1145-1165.
[15] Gerkema, T., Zimmerman, J.T.F., Maas, L.R.M. & Van Haren, H.2008Geophysical and astrophysical fluid dynamics beyond the traditional approximation. Rev. Geophys.46, RG2004.
[16] Godeferd, F.S., Cambon, C. & Leblanc, S.2001Zonal approach to centrifugal, elliptic and hyperbolic instabilities in stuart vortices with external rotation. J. Fluid Mech.449, 1-37. · Zbl 1053.76022
[17] Haine, T.W.N. & Marshall, J.1998Gravitational, symmetric, and baroclinic instability of the ocean mixed layer. J. Phys. Oceanogr.28 (4), 634-658.
[18] Holmes, R.M., Thomas, L.N., Thompson, L. & Darr, D.2014Potential vorticity dynamics of tropical instability vortices. J. Phys. Oceanogr.44 (3), 995-1011.
[19] Hoskins, B.J.1974The role of potential vorticity in symmetric stability and instability. Q. J. R. Meteorol. Soc.100 (425), 480-482.
[20] Hoskins, B.J. & Bretherton, F.P.1972Atmospheric frontogenesis models: mathematical formulation and solution. J. Atmos. Sci.29 (1), 11-37.
[21] Hua, B.L., Ménesguen, C., Le Gentil, S., Schopp, R., Marsset, B. & Aiki, H.2013Layering and turbulence surrounding an anticyclonic oceanic vortex: in situ observations and quasi-geostrophic numerical simulations. J. Fluid Mech.731, 418-442. · Zbl 1294.76267
[22] Itano, T. & Maruyama, K.2009Symmetric stability of zonal flow under full-component coriolis force—effect of the horizontal component of the planetary vorticity—. J. Meteorol. Soc. Jpn.87 (4), 747-753.
[23] Kays, W.M. & Crawford, M.E.1993 Convective heat and mass transfer. In Heat Transfer: The Laminar External Boundary Layer, chap. 10, pp. 159-191. McGraw-Hill.
[24] Kays, W.M.1994Turbulent Prandtl number. Where are we?Trans. ASME J. Heat Transfer116 (2), 284-295.
[25] Kirillov, O.N. & Mutabazi, I.2017Short-wavelength local instabilities of a circular Couette flow with radial temperature gradient. J. Fluid Mech.818, 319-343. · Zbl 1383.76096
[26] Kloosterziel, R.C. & Carnevale, G.F.2007Generalized energetics for inertially stable parallel shear flows. J. Fluid Mech.585, 117-126. · Zbl 1119.76028
[27] Kloosterziel, R.C., Carnevale, G.F. & Orlandi, P.2017Equatorial inertial instability with full coriolis force. J. Fluid Mech.825, 69-108. · Zbl 1374.86012
[28] Kloosterziel, R.C. & Van Heijst, G.J.F.1991An experimental study of unstable barotropic vortices in a rotating fluid. J. Fluid Mech.223, 1-24.
[29] Kuzmina, N. & Zhurbas, V.2000Effects of double diffusion and turbulence on interleaving at baroclinic oceanic fronts. J. Phys. Oceanogr.30 (12), 3025-3038.
[30] Landman, M.J. & Saffman, P.G.1987The three-dimensional instability of strained vortices in a viscous fluid. Phys. Fluids30 (8), 2339-2342.
[31] Le Duc, A. & Leblanc, S.1999A note on Rayleigh stability criterion for compressible flows. Phys. Fluids11 (11), 3563-3566. · Zbl 1149.76448
[32] Leblanc, S.1997Stability of stagnation points in rotating flows. Phys. Fluids9 (11), 3566-3569. · Zbl 1185.76635
[33] Lifschitz, A. & Hameiri, E.1991Local stability conditions in fluid dynamics. Phys. Fluids A: Fluid Dyn.3 (11), 2644-2651. · Zbl 0746.76050
[34] Mahadevan, A.2006Modeling vertical motion at ocean fronts: are nonhydrostatic effects relevant at submesoscales?Ocean Model.14 (3-4), 222-240.
[35] Mahadevan, A. & Tandon, A.2006An analysis of mechanisms for submesoscale vertical motion at ocean fronts. Ocean Model.14 (3-4), 241-256.
[36] Mathur, M., Ortiz, S., Dubos, T. & Chomaz, J.-M.2014Effects of an axial flow on the centrifugal, elliptic and hyperbolic instabilities in Stuart vortices. J. Fluid Mech.758, 565-585.
[37] Mcintyre, M.E.1970Diffusive destabilisation of the baroclinic circular vortex. Geophys. Astrophys. Fluid Dyn.1 (1-2), 19-57.
[38] Mcwilliams, J.C.1985Submesoscale, coherent vortices in the ocean. Rev. Geophys.23 (2), 165-182.
[39] Meunier, P., Miquel, B., Le Dizès, S., Chowdhury, H. & Alam, F.2014 Instabilities around a rotating ellipsoid in a stratified rotating flow. In 19th Australasian Fluid Mechanics Conference, Melbourne, Australia (ed. H. Chowdhury & F. Alam). RMIT University.
[40] Miyazaki, T.1993Elliptical instability in a stably stratified rotating fluid. Phys. Fluids A: Fluid Dyn.5 (11), 2702-2709. · Zbl 0791.76030
[41] Miyazaki, T. & Fukumoto, Y.1992Three-dimensional instability of strained vortices in a stably stratified fluid. Phys. Fluids A: Fluid Dyn.4 (11), 2515-2522. · Zbl 0762.76027
[42] Nagarathinam, D., Sameen, A. & Mathur, M.2015Centrifugal instability in non-axisymmetric vortices. J. Fluid Mech.769, 26-45. · Zbl 1431.76038
[43] Negretti, M.E. & Billant, P.2013Stability of a gaussian pancake vortex in a stratified fluid. J. Fluid Mech.718, 457-480. · Zbl 1284.76179
[44] Nguyen, H.Y., Hua, B.L., Schopp, R. & Carton, X.2012Slow quasigeostrophic unstable modes of a lens vortex in a continuously stratified flow. Geophys. Astrophys. Fluid Dyn.106 (3), 305-319.
[45] Ooyama, K.1966On the stability of the baroclinic circular vortex: a sufficient criterion for instability. J. Atmos. Sci.23 (1), 43-53.
[46] Pedlosky, J.1987Geophysical Fluid Dynamics, vol. 710. Springer. · Zbl 0713.76005
[47] Peltier, W.R. & Caulfield, C.P.2003Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech.35 (1), 135-167. · Zbl 1041.76024
[48] Pierrehumbert, R.T. & Swanson, K.L.1995Baroclinic instability. Annu. Rev. Fluid Mech.27 (1), 419-467.
[49] Ruddick, B.1992Intrusive mixing in a mediterranean salt lens—intrusion slopes and dynamical mechanisms. J. Phys. Oceanogr.22 (11), 1274-1285.
[50] Salehipour, H. & Peltier, W.R.2015Diapycnal diffusivity, turbulent Prandtl number and mixing efficiency in Boussinesq stratified turbulence. J. Fluid Mech.775, 464-500. · Zbl 1403.76028
[51] Salehipour, H., Peltier, W.R., Whalen, C.B. & Mackinnon, J.A.2016A new characterization of the turbulent diapycnal diffusivities of mass and momentum in the ocean. Geophys. Res. Lett.43 (7), 3370-3379.
[52] Sarkar, S., Pham, H.T., Ramachandran, S., Nash, J.D., Tandon, A., Buckley, J., Lotliker, A.A. & Omand, M.M.2016The interplay between submesoscale instabilities and turbulence in the surface layer of the bay of bengal. Oceanography29 (2), 146-157.
[53] Shakespeare, C.J.2016Curved density fronts: cyclogeostrophic adjustment and frontogenesis. J. Phys. Oceanogr.46 (10), 3193-3207.
[54] Shcherbina, A.Y., et al.2015The latmix summer campaign: submesoscale stirring in the upper ocean. Bull. Am. Meteorol. Soc.96 (8), 1257-1279.
[55] Singh, S. & Mathur, M.2019Effects of Schmidt number on the short-wavelength instabilities in stratified vortices. J. Fluid Mech.867, 765-803. · Zbl 1430.76147
[56] Sipp, D. & Jacquin, L.2000Three-dimensional centrifugal-type instabilities of two-dimensional flows in rotating systems. Phys. Fluids12 (7), 1740-1748. · Zbl 1184.76511
[57] Solberg, M.1936 Le mouvement d’inertie de l’atmosphere stable et son role dans le theorie des cyclones. In Meteor. Assoc. U.G.G.I., pp. 66-82. Dupont.
[58] Storer, B.A., Poulin, F.J. & Ménesguen, C.2018The dynamics of quasigeostrophic lens-shaped vortices. J. Phys. Oceanogr.48 (4), 937-957.
[59] Thomas, L.N., Tandon, A. & Mahadevan, A.2008Submesoscale processes and dynamics. Ocean Model. Eddying Regime177, 17-38.
[60] Thomas, L.N., Taylor, J.R., Ferrari, R. & Joyce, T.M.2013Symmetric instability in the gulf stream. Deep Sea Res. Part II: Top. Stud. Oceanogr.91, 96-110.
[61] Thompson, A.F., Lazar, A., Buckingham, C., Garabato, A.C.N., Damerell, G.M. & Heywood, K.J.2016Open-ocean submesoscale motions: a full seasonal cycle of mixed layer instabilities from gliders. J. Phys. Oceanogr.46 (4), 1285-1307.
[62] Turner, J.S.1973Double-Diffusive Convection, pp. 251-287. Cambridge University Press.
[63] Vallis, G.K.2017Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press. · Zbl 1374.86002
[64] Venayagamoorthy, S.K. & Stretch, D.D.2010On the turbulent Prandtl number in homogeneous stably stratified turbulence. J. Fluid Mech.644, 359. · Zbl 1189.76253
[65] Weber, J.E.1980Symmetric instability of stratified geostrophic flow. Tellus32 (2), 176-185.
[66] Xui, Q. & Clark, J.H.E.1985The nature of symmetric instability and its similarity to convective and inertial instability. J. Atmos. Sci.42 (24), 2880-2883.
[67] Yim, E. & Billant, P.2016Analogies and differences between the stability of an isolated pancake vortex and a columnar vortex in stratified fluid. J. Fluid Mech.796, 732-766. · Zbl 1462.76060
[68] Yim, E., Billant, P. & Ménesguen, C.2016Stability of an isolated pancake vortex in continuously stratified-rotating fluids. J. Fluid Mech.801, 508-553. · Zbl 1462.76061
[69] Zeitlin, V.2018Symmetric instability drastically changes upon inclusion of the full coriolis force. Phys. Fluids30 (6), 061701.
[70] Zhang, Z., Wang, W. & Qiu, B.2014Oceanic mass transport by mesoscale eddies. Science345 (6194), 322-324.
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