Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-20T01:32:51.747Z Has data issue: false hasContentIssue false
  • English
  • Français

Shallow-water modons on the f-plane

Published online by Cambridge University Press:  30 April 2008

Z. KIZNER ,
G. REZNIK ,
B. FRIDMAN ,
R. KHVOLES  and
J. McWILLIAMS
Z. KIZNER*
Affiliation:
Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
G. REZNIK
Affiliation:
P.P. Shirshov Institute of Oceanology, 36 Nakhimovsky Prosp., Moscow 117997, Russia
B. FRIDMAN
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
R. KHVOLES
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
J. McWILLIAMS
Affiliation:
Department of Atmospheric and Oceanic Sciences and Institute of Geophysics and Planetary Physics, University of California, Los Angeles 405 Hilgard Avenue Los Angeles, CA 90095-1565, USA
*
†Author to whom correspondence should be addressed: zinovyk@mail.biu.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

Solutions for steadily translating localized vortical structures, or modons, are sought in the framework of a 1½-layer rotating shallow-water (RSW) model on the f-plane. In this model, the fluid is assumed to rotate at a constant rate and to be composed of an active finite-depth layer and a passive infinitely deep layer. The focus is on the smooth intense modons, in which the potential vorticity field is continuous, and the pressure (hence, the active-layer thickness) and velocity are smooth, while inertial effects and deviations of the active-layer thickness from the static level are considerable. The problem is solved numerically employing a Newton--Kantorovich iterative procedure, Fourier–Chebyshev spectral expansion and collocations. The numerics are preceded by a theoretical modon design discussion that includes: derivation of fundamental modon invariants; distinction between the flow in the trapped-fluid region and the flow outside it; and the boundary conditions at the separatrix, the streamline demarcating the two regions. Also, some basic distinctions from the quasi-geostrophic modons are discussed, and an asymptotic analysis of the RSW modon far-field characteristics is carried out. This analysis reveals that an RSW modon must propagate more slowly than inertia–gravity waves. In smooth modons, the requirement that the active-layer thickness should be positive imposes an even stronger restriction on the allowed translational speed. To enable the use of Fourier–Chebyshev series, only the modons with circular separatrices are considered. The numerical iterative procedure is initialized by an analytical quasi-geostrophic dipolar modon solution; accordingly, the obtained RSW modons appear as cyclone–anticyclone pairs. Computations show that the allowed maximal translational speed monotonically decreases as a function of the modon size and, for reasonable sizes, is appreciably smaller than the gravity-wave limit. As distinct from quasi-geostrophic modons, the RSW modons with circular separatrices display nonlinearity of the potential vorticity (PV) vs. streamfunction relation, and the cyclone–anticyclone asymmetry: while the integral mass anomaly in the modon is zero, the cyclone is more intense and compact than the anticyclone.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 603 , 25 May 2008 , pp. 305 - 329
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Berestov, A. L. 1979 Solitary Rossby waves. Izv. Acad. Sci. USSR. Atmos. Ocean Phys. 15, 648–654.Google Scholar
Berestov, A. L. 1981 Some new solutions for the Rossby solitons. Izv. Acad. Sci. USSR. Atmos. Ocean. Phys. 17, 82–87.Google Scholar
Boyd, J. P. & Ma, H. 1990 Numerical study of elliptical modons using spectral methods. J. Fluid Mech. 221, 597–611.CrossRefGoogle Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods, 2nd edn, Dover.Google Scholar
Chaplygin, S. A. 1903 One case of vortex motion in fluid. Trans. Phys. Sect. Imperial Moscow Soc. Friends of Natural Sciences 11 (N 2), 11–14.Google Scholar
Flierl, G. R., Larichev, V. D., McWilliams, J. C. & Reznik, G. M. 1980 The dynamics of baroclinic and barotropic solitary eddies. Dyn. Atmos. Oceans 5, 1–41.CrossRefGoogle Scholar
Flierl, G. R., Stern, M. E. & Whitehead, J. A. Jr 1983 The physical significance of modons: laboratory experiments and general integral constraints. Dyn. Atmos. Oceans 7, 233–263.CrossRefGoogle Scholar
Flór, J. B. & van Heijst, G. J. F. 1994 An experimental study of dipolar vortex structures in a stratified fluid. J. Fluid Mech. 279, 101–133.CrossRefGoogle Scholar
Haupt, S. E., McWilliams, J. C. & Tribbia, J. J. 1993 Modons in shear flow. J. Atmos. Sci. 50, 1181–1198.2.0.CO;2>CrossRefGoogle Scholar
Hesthaven, J. S., Lynov, J. P., Nielsen, A. H., Rassmussen, J. J., Schmidt, M. R., Shapiro, E. G. & Turitsyn, S. K. 1995 Dynamics of a nonlinear dipole vortex. Phys. Fluids 7, 2220–2229.CrossRefGoogle Scholar
Khvoles, R., Berson, D. & Kizner, Z. 2005 The structure and evolution of barotropic elliptical modons. J. Fluid Mech. 530, 1–30.CrossRefGoogle Scholar
Khvoles, R., McWilliams, J. C. & Kizner, Z. 2007 Non-coincidence of separatrices in two-layer modons. Phys. Fluids 19, 056602/1–14.CrossRefGoogle Scholar
Kizner, Z. 1984 Rossby solitons with axisymmetric baroclinic modes. Dokl. USSR Acad. Sci. 275, 1495–1498.Google Scholar
Kizner, Z. 1986 a Strongly nonlinear solitary Rossby waves. Oceanology 26, 382–388.Google Scholar
Kizner, Z. 1986 b Intensive synoptic eddies and the quasi-geostrophic approximation. Oceanology 26, 28–35.Google Scholar
Kizner, Z. 1988 On the theory of intrathermocline eddies. Dokl. USSR. Acad. Sci. 300, 453–457.Google Scholar
Kizner, Z. 1997 Solitary Rossby waves with baroclinic modes. J. Mar. Res. 55, 671–685.CrossRefGoogle Scholar
Kizner, Z. & Berson, D. 2000 Emergence of modons from collapsing vortex structures on the β-plane. J. Mar. Res. 58, 375–403.CrossRefGoogle Scholar
Kizner, Z., Berson, D. & Khvoles, R. 2002 Baroclinic modon equilibria on the beta-plane: stability and transitions. J. Fluid Mech. 468, 239–270.CrossRefGoogle Scholar
Kizner, Z., Berson, D. & Khvoles, R. 2003 a Non-circular baroclinic beta-plane modons: constructing stationary solutions. J. Fluid Mech. 489, 199–228.CrossRefGoogle Scholar
Kizner, Z., Berson, D., Reznik, G. & Sutyrin, G. 2003 b The theory of beta-plane baroclinic topographic modons. Geophys. Astrophys. Fluid Dyn. 97, 175–211.CrossRefGoogle Scholar
Konshin, V. N. & Shapiro, G. I. 1988 The evolution of two-dimensional Rossby solitons under the influence of finite amplitude effects. Dokl. USSR Acad. Sci. 300, 1461–1465.Google Scholar
Kuo, A. C. & Polvani, L. M. 2000 Nonlinear geostrophic adjustment, cyclone–anticyclone asymmetry, and potential vorticity rearrangement. Phys. Fluids 12, 1087–1100.CrossRefGoogle Scholar
Lamb, H. 1895 Hydrodynamics, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Larichev, V. D. & Reznik, G. M. 1976 a Two-dimensional solitary Rossby waves. Dokl. USSR. Acad. Sci. 231, 1077–1080.Google Scholar
Larichev, V. D. & Reznik, G. M. 1976 b Strongly nonlinear two-dimensional solitary Rossby waves. Oceanology 16, 961–967.Google Scholar
Li, C. 2004 A geosolitary wave solution on the f plane. J. Phys. Oceanogr 34, 856–864.2.0.CO;2>CrossRefGoogle Scholar
Malanotte-Rizzoli, P. 1982 Planetary solitary waves in geophysical flows. Adv. Geophys. 24, 147–224.CrossRefGoogle Scholar
Meleshko, V. V. & van Heijst, G. J. F. 1994 On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid. J. Fluid Mech. 272, 157–182.CrossRefGoogle Scholar
Neven, E. C. 1994 Baroclinic modons on a sphere. J. Atmos. Sci. 51, 1447–1464.2.0.CO;2>CrossRefGoogle Scholar
Nycander, J. 1988 New stationary vortex solutions of the Hasegawa–Mima equation. J. Plasma Phys. 39, 418–428.CrossRefGoogle Scholar
Pallas-Sanz, E. & Viudez, A. 2007 Three-dimensional ageostrophic motion in mesoscale vortex dipoles. J. Phys. Oceanogr. 37, 84–105.CrossRefGoogle Scholar
Pakyari, A. & Nycander, J. 1996 Steady two-layer vortices on the beta-plane. Dyn. Atmos. Oceans 25, 67–86.CrossRefGoogle Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer.CrossRefGoogle Scholar
Perret, G., Stegner, A., Farge, M. & Pichon, T. 2006 Cyclone–anticyclone asymmetry of large-scale wakes in the laboratory. Phys. Fluids 18, 036603/1–11.CrossRefGoogle Scholar
Reznik, G. M. 1992 Dynamics of singular vortices on a beta-plane. J. Fluid Mech. 40, 405–432.CrossRefGoogle Scholar
Reznik, G. M. & Sutyrin, G. G. 2001 Baroclinic topographic modons. J. Fluid. Mech. 437, 121–142.CrossRefGoogle Scholar
Reznik, G. M., Zeitlin, V. & Ben, J elloul, M. 2001 Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model. J. Fluid Mech. 445, 93–120.CrossRefGoogle Scholar
Sous, D., Bonneton, N. & Sommeria, J. 2004 Turbulent vortex dipoles in a shallow water layer. Phys. Fluids 16 (8), 2886–2898.CrossRefGoogle Scholar
Stern, M. E. 1975 Minimal properties of planetary eddies. J. Mar. Res. 33, 1–13.Google Scholar
Swenson, M. 1987. Stability of equivalent-barotropic riders. J. Phys. Oceanogr. 17, 492–506.2.0.CO;2>CrossRefGoogle Scholar
Tribbia, J. J. 1984. Modons in spherical geometry. Geophys. Astrophys. Fluid Dyn. 30, 131–168.CrossRefGoogle Scholar
Verkley, W. T. M. 1984. The construction of Barotropic modons on a sphere. J. Atmos. Sci. 41, 2492–2504.2.0.CO;2>CrossRefGoogle Scholar
Verkley, W. T. M. 1987 Stationary barotropic modons in westerly background flows. J. Atmos. Sci. 44, 2383–2398.2.0.CO;2>CrossRefGoogle Scholar
Verkley, W. T. M. 1993 A numerical method to find form-preserving free solutions of the barotropic vorticity equation on a sphere. J. Atmos. Sci. 50, 1488–1503.2.0.CO;2>CrossRefGoogle Scholar