Influence of Bottom Topography on Vortex Stability

Bowen Zhao Department of Geology and Geophysics, Yale University, New Haven, Connecticut

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Emma Chieusse-Gérard ENSTA ParisTech, Paris, France

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Glenn Flierl Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts

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Abstract

The effects of topography on the linear stability of both barotropic vortices and two-layer, baroclinic vortices are examined by considering cylindrical topography and vortices with stepwise relative vorticity profiles in the quasigeostrophic approximation. Four vortex configurations are considered, classified by the number of relative vorticity steps in the horizontal and the number of layers in the vertical: barotropic one-step vortex (Rankine vortex), barotropic two-step vortex, and their two-layer, baroclinic counterparts with the vorticity steps in the upper layer. In the barotropic calculation, the vortex is destabilized by topography having an oppositely signed potential vorticity jump while stabilized by topography of same-signed jump, that is, anticyclones are destabilized by seamounts while stabilized by depressions. Further, topography of appropriate sign and magnitude can excite a mode-1 instability for a two-step vortex, especially relevant for topographic encounters of an otherwise stable vortex. The baroclinic calculation is in general consistent with the barotropic calculation except that the growth rate weakens and, for a two-step vortex, becomes less sensitive to topography (sign and magnitude) as baroclinicity increases. The smaller growth rate for a baroclinic vortex is consistent with previous findings that vortices with sufficient baroclinic structure could cross the topography relatively easily. Nonlinear contour dynamics simulations are conducted to confirm the linear stability analysis and to describe the subsequent evolution.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Bowen Zhao, bowen.zhao@yale.edu

Abstract

The effects of topography on the linear stability of both barotropic vortices and two-layer, baroclinic vortices are examined by considering cylindrical topography and vortices with stepwise relative vorticity profiles in the quasigeostrophic approximation. Four vortex configurations are considered, classified by the number of relative vorticity steps in the horizontal and the number of layers in the vertical: barotropic one-step vortex (Rankine vortex), barotropic two-step vortex, and their two-layer, baroclinic counterparts with the vorticity steps in the upper layer. In the barotropic calculation, the vortex is destabilized by topography having an oppositely signed potential vorticity jump while stabilized by topography of same-signed jump, that is, anticyclones are destabilized by seamounts while stabilized by depressions. Further, topography of appropriate sign and magnitude can excite a mode-1 instability for a two-step vortex, especially relevant for topographic encounters of an otherwise stable vortex. The baroclinic calculation is in general consistent with the barotropic calculation except that the growth rate weakens and, for a two-step vortex, becomes less sensitive to topography (sign and magnitude) as baroclinicity increases. The smaller growth rate for a baroclinic vortex is consistent with previous findings that vortices with sufficient baroclinic structure could cross the topography relatively easily. Nonlinear contour dynamics simulations are conducted to confirm the linear stability analysis and to describe the subsequent evolution.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Bowen Zhao, bowen.zhao@yale.edu
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  • Adams, D. K., and G. R. Flierl, 2010: Modeled interactions of mesoscale eddies with the east Pacific rise: Implications for larval dispersal. Deep-Sea Res. I, 57, 11631176, https://doi.org/10.1016/j.dsr.2010.06.009.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Beismann, J.-O., R. H. Käse, and J. R. Lutjeharms, 1999: On the influence of submarine ridges on translation and stability of agulhas rings. J. Geophys. Res., 104, 78977906, https://doi.org/10.1029/1998JC900127.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Biebuyck, G. L., 1986: Self propagation of a barotropic circular eddy. Woods Hole Oceanographic Institution Tech. Rep. WHOI-86-45, 193–197.

  • Bretherton, F. P., and D. B. Haidvogel, 1976: Two-dimensional turbulence above topography. J. Fluid Mech., 78, 129154, https://doi.org/10.1017/S002211207600236X.

    • Search Google Scholar
    • Export Citation
  • Carnevale, G., R. Kloosterziel, and G. van Heijst, 1991: Propagation of barotropic vortices over topography in a rotating tank. J. Fluid Mech., 233, 119139, https://doi.org/10.1017/S0022112091000411.

    • Search Google Scholar
    • Export Citation
  • Flierl, G. R., 1988: On the instability of geostrophic vortices. J. Fluid Mech., 197, 349388, https://doi.org/10.1017/S0022112088003283.

  • Ford, R., M. E. McIntyre, and W. A. Norton, 2000: Balance and the slow quasimanifold: Some explicit results. J. Atmos. Sci., 57, 12361254, https://doi.org/10.1175/1520-0469(2000)057<1236:BATSQS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Garzoli, S. L., P. L. Richardson, C. M. Duncombe Rae, D. M. Fratantoni, G. J. Goñi, and A. J. Roubicek, 1999: Three Agulhas rings observed during the Benguela Current Experiment. J. Geophys. Res., 104, 20 97120 985, https://doi.org/10.1029/1999JC900060.

    • Search Google Scholar
    • Export Citation
  • Goni, G. J., S. L. Garzoli, A. J. Roubicek, D. B. Olson, and O. B. Brown, 1997: Agulhas ring dynamics from TOPEX/POSEIDON satellite altimeter data. J. Mar. Res., 55, 861883, https://doi.org/10.1357/0022240973224175.

    • Search Google Scholar
    • Export Citation
  • Gould, W. J., R. Hendry, and H. E. Huppert, 1981: An abyssal topographic experiment. Deep-Sea Res., 28A, 409440, https://doi.org/10.1016/0198-0149(81)90135-7.

    • Search Google Scholar
    • Export Citation
  • Grimshaw, R., X. He, P. Sun, and D. Broutman, 1994a: Analytical and numerical study of a barotropic eddy on a topographic slope. J. Phys. Oceanogr., 24, 15871607, https://doi.org/10.1175/1520-0485(1994)024<1587:AANSOA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Grimshaw, R., Y. Tang, and D. Broutman, 1994b: The effect of vortex stretching on the evolution of barotropic eddies over a topographic slope. Geophys. Astrophys. Fluid Dyn., 76, 4371, https://doi.org/10.1080/03091929408203659.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hart, J., 1975a: Baroclinic instability over a slope. Part I: Linear theory. J. Phys. Oceanogr., 5, 625633, https://doi.org/10.1175/1520-0485(1975)005<0625:BIOASP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hart, J., 1975b: Baroclinic instability over a slope. Part II: Finite-amplitude theory. J. Phys. Oceanogr., 5, 634641, https://doi.org/10.1175/1520-0485(1975)005<0634:BIOASP>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Huppert, H. E., and K. Bryan, 1976: Topographically generated eddies. Deep-Sea Res. Oceanogr. Abstr., 23, 655679, https://doi.org/10.1016/S0011-7471(76)80013-7.

    • Search Google Scholar
    • Export Citation
  • Johnson, E., 1978: Trapped vortices in rotating flow. J. Fluid Mech., 86, 209224, https://doi.org/10.1017/S0022112078001093.

  • Kamenkovich, V. M., Y. P. Leonov, D. A. Nechaev, D. A. Byrne, and A. L. Gordon, 1996: On the influence of bottom topography on the Agulhas eddy. J. Phys. Oceanogr., 26, 892912, https://doi.org/10.1175/1520-0485(1996)026<0892:OTIOBT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kizner, Z., V. Makarov, L. Kamp, and G. van Heijst, 2013: Instabilities of the flow around a cylinder and emission of vortex dipoles. J. Fluid Mech., 730, 419441, https://doi.org/10.1017/jfm.2013.345.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • LaCasce, J. H., 1998: A geostrophic vortex over a slope. J. Phys. Oceanogr., 28, 23622381, https://doi.org/10.1175/1520-0485(1998)028<2362:AGVOAS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lahaye, N., and V. Zeitlin, 2015: Centrifugal, barotropic and baroclinic instabilities of isolated ageostrophic anticyclones in the two-layer rotating shallow water model and their nonlinear saturation. J. Fluid Mech., 762, 534, https://doi.org/10.1017/jfm.2014.631.

    • Search Google Scholar
    • Export Citation
  • Lahaye, N., and V. Zeitlin, 2016: Understanding instabilities of tropical cyclones and their evolution with a moist convective rotating shallow-water model. J. Atmos. Sci., 73, 505523, https://doi.org/10.1175/JAS-D-15-0115.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McDonagh, E. L., K. J. Heywood, and M. P. Meredith, 1999: On the structure, paths, and fluxes associated with agulhas rings. J. Geophys. Res., 104, 21 00721 020, https://doi.org/10.1029/1998JC900131.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Meacham, S., 1991: Meander evolution on piecewise-uniform, quasi-geostrophic jets. J. Phys. Oceanogr., 21, 11391170, https://doi.org/10.1175/1520-0485(1991)021<1139:MEOPUQ>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Nycander, J., and J. Lacasce, 2004: Stable and unstable vortices attached to seamounts. J. Fluid Mech., 507, 7194, https://doi.org/10.1017/S0022112004008730.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Olson, D. B., 1980: The physical oceanography of two rings observed by the cyclonic ring experiment. Part II: Dynamics. J. Phys. Oceanogr., 10, 514528, https://doi.org/10.1175/1520-0485(1980)010<0514:TPOOTR>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Olson, D. B., 1991: Rings in the ocean. Annu. Rev. Earth Planet. Sci., 19, 283311, https://doi.org/10.1146/annurev.ea.19.050191.001435.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Olson, D. B., and R. H. Evans, 1986: Rings of the Agulhas current. Deep-Sea Res., 33A, 2742, https://doi.org/10.1016/0198-0149(86)90106-8.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Polvani, L., N. Zabusky, and G. Flierl, 1988: Applications of contour dynamics to two layer quasigeostrophic flows. Fluid Dyn. Res., 3, 422424, https://doi.org/10.1016/0169-5983(88)90103-7.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pullin, D., 1992: Contour dynamics methods. Annu. Rev. Fluid Mech., 24, 89115, https://doi.org/10.1146/annurev.fl.24.010192.000513.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rabinovich, M., Z. Kizner, and G. Flierl, 2018: Bottom-topography effect on the instability of flows around a circular island. J. Fluid Mech., 856, 202227, https://doi.org/10.1017/jfm.2018.705.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ribstein, B., and V. Zeitlin, 2013: Instabilities of coupled density fronts and their nonlinear evolution in the two-layer rotating shallow-water model: Influence of the lower layer and of the topography. J. Fluid Mech., 716, 528565, https://doi.org/10.1017/jfm.2012.556.

    • Search Google Scholar
    • Export Citation
  • Richardson, P. L., 1981: Anticyclonic eddies generated near the corner rise seamounts. Tech. Rep. WHOI-81-20, Woods Hole Oceanographic Institution, 20 pp., https://doi.org/10.1575/1912/10240.

    • Crossref
    • Export Citation
  • Schmidt, G., and E. Johnson, 1997: Instability in stratified rotating shear flow along ridges. J. Mar. Res., 55, 915933, https://doi.org/10.1357/0022240973224201.

    • Search Google Scholar
    • Export Citation
  • Schouten, M. W., W. P. Ruijter, P. J. Leeuwen, and J. R. Lutjeharms, 2000: Translation, decay and splitting of agulhas rings in the southeastern atlantic ocean. J. Geophys. Res., 105, 21 91321 925, https://doi.org/10.1029/1999JC000046.

    • Search Google Scholar
    • Export Citation
  • Smith, D. C., and J. O’Brien, 1983: The interaction of a two-layer isolated mesoscale eddy with bottom topography. J. Phys. Oceanogr., 13, 16811697, https://doi.org/10.1175/1520-0485(1983)013<1681:TIOATL>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Stern, M. E., 1987: Horizontal entrainment and detrainment in large-scale eddies. J. Phys. Oceanogr., 17, 16881695, https://doi.org/10.1175/1520-0485(1987)017<1688:HEADIL>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stern, M. E., and T. Radko, 1998: The self-propagating quasi-monopolar vortex. J. Phys. Oceanogr., 28, 2239, https://doi.org/10.1175/1520-0485(1998)028<0022:TSPQMV>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • van Aken, H., A. van Veldhoven, C. Veth, W. De Ruijter, P. van Leeuwen, S. Drijfhout, C. Whittle, and M. Rouault, 2003: Observations of a young Agulhas ring, Astrid, during MARE in March 2000. Deep-Sea Res. II, 50, 167195, https://doi.org/10.1016/S0967-0645(02)00383-1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • van Geffen, J., and P. Davies, 2000: A monopolar vortex encounters a north–south ridge or trough. Fluid Dyn. Res., 26, 157179, https://doi.org/10.1016/S0169-5983(99)00022-2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Verron, J., and C. Le Provost, 1985: A numerical study of quasi-geostrophic flow over isolated topography. J. Fluid Mech., 154, 231252, https://doi.org/10.1017/S0022112085001501.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, X., 1992: Interaction of an eddy with a continental slope. Ph.D. thesis, Massachusetts Institute of Technology, 216 pp., https://doi.org/10.1575/1912/5490.

    • Crossref
    • Export Citation
  • Wei, J., D.-P. Wang, and C. N. Flagg, 2008: Mapping gulf stream warm core rings from shipboard ADCP transects of the Oleander Project. J. Geophys. Res., 113, C10021, https://doi.org/10.1029/2007JC004694.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zabusky, N. J., M. Hughes, and K. Roberts, 1979: Contour dynamics for the euler equations in two dimensions. J. Comput. Phys., 30, 96106, https://doi.org/10.1016/0021-9991(79)90089-5.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zavala Sansón, L., 2002: Vortex–ridge interaction in a rotating fluid. Dyn. Atmos. Oceans, 35, 299325, https://doi.org/10.1016/S0377-0265(02)00014-3.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zavala Sansón, L., A. B. Aguiar, and G. van Heijst, 2012: Horizontal and vertical motions of barotropic vortices over a submarine mountain. J. Fluid Mech., 695, 173198, https://doi.org/10.1017/jfm.2012.9.

    • Crossref
    • Search Google Scholar
    • Export Citation
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