Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-28T01:22:56.392Z Has data issue: false hasContentIssue false
  • English
  • Français

On the stability of large-amplitude geostrophic flows in a two-layer fluid: the case of ‘strong’ beta-effect

Published online by Cambridge University Press:  26 April 2006

E. S. Benilov
E. S. Benilov
Affiliation:
School of Mathematics, University of New South Wales, PO Box 1, Kensington, NSW 2033, Australia Present address: Department of Applied Computing and Mathematics, University of Tasmania, PO Box 1214, Launceston, Tasmania 7250, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper examines the stability of two-layer geostrophic flows with large displacement of the interface and strong β-effect. Attention is focused on flows with non-monotonic interface profiles which are not covered by the Rayleigh-style stability theorems proved by Benilov (1992a, b) and Benilov & Cushman-Roisin (1994). For such flows the coefficient of the highest derivative in the corresponding boundary-value problem vanishes at the point where the depth profile has an extremum. Although this singularity is similar to a critical level, it cannot be regularized by the simplistic introduction of infinitesimal viscosity through the assumption that the phase speed of the disturbance is complex. In order to regularize the singularity properly, one should consider the problem within the framework of the original ageostrophic viscous equations and, having obtained the boundary-value problem for harmonic disturbance, take the limit Rossby number → 0, viscosity → 0.

The results obtained analytically and (for special cases) numerically indicate that the stability of flows with non-monotonic profiles strongly depends on the depth of the upper layer. If the upper layer is ‘thick’ (i.e. if the average depth H1 of the upper layer is of the order of the total depth of the fluid H0), the stability boundary-value problem does not have any solutions at all, which means stability (however, this stability is structurally unstable, and the flow, generally speaking, can be made weakly unstable by any small effect such as external forcing, viscosity, or ageostrophic corrections). In the case of ‘thin’ upper layer (H1/H0 [lsim ] Ro), the order of the singularity changes and all non-monotonic flows are unstable regardless of their profiles. It is also demonstrated that thin-upper-layer flows do not have to be non-monotonic to be unstable: if u–βR20 (where u is the zonal velocity, β is the β-parameter, and R0 is the deformation radius) changes sign somewhere in the flow, the stability boundary-value problem has another singular point which leads to instability.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 284 , 10 February 1995 , pp. 137 - 158
Copyright
© 1995 Cambridge University Press

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

Benilov, E. S. 1992a Large-amplitude geostrophic dynamics: the two-layer model. Geophys. Astrophys. Fluid Dyn. 66, 67–79.Google Scholar
Benilov, E. S. 1992b A note on the stability of one-layer geostrophic fronts. Geophys. Astrophys. Fluid Dyn. 66, 81–86.Google Scholar
Benilov, E. S. 1994 Dynamics of large-amplitude geostrophic flows: the case of ‘strong’ beta-effect. J. Fluid Mech. 262, 157–169.Google Scholar
Benilov, E. S. & Cushman-Roisin, B. 1994 On the stability of two-layered large-amplitude geostrophic flows with thin upper layer. Geophys. Astrophys. Fluid Dyn. To appear.Google Scholar
Benilov, E. S. & Reznik, G. M. 1994 The complete classification of large-amplitude geostrophic flows. Geophys. Astrophys. Fluid Dyn. (submitted).Google Scholar
Cushman-Roisin, B. 1986 Frontal geostrophic dynamics. J. Phys. Oceanogr. 16, 132–143.Google Scholar
Cushman-Roisin, B., Sutyrin, G. G. & Tang, B. 1992 Two-layer geostrophic dynamics. Part 1: Governing equations. J. Phys. Oceanogr. 22, 117–127.Google Scholar
Dikiy, L. A. 1976 Hydrodynamic Stability and Dynamics of the Atmosphere. Gidrometeoizdat, Leningrad. (In Russian).
Griffiths, R. W., Killworth, P. D. & Stern, M. E. 1982 Ageostrophic instability of ocean currents. J. Fluid Mech. 117, 343–377.Google Scholar
Nowlin, W. D. & Klinck, J. M. 1986 The physics of the Antarctic Circumpolar Current. Rev. Geophys. 24, 469–491.Google Scholar
Pavia, E. 1992 The breakup of frontal filaments. J. Phys. Oceanogr. 22, 399–403.Google Scholar
Roden, G. I. 1975 On North Pacific temperature, salinity, sound velocity and density fronts and their relation to the wind and energy flux fields. J. Phys. Oceanogr. 5, 557–571.Google Scholar
Swaters, G. E. 1993 On the baroclinic dynamics, Hamiltonian formulation and general stability characteristics of density-driven surface currents and fronts over a sloping continental shelf. Phil. Trans. R. Soc. Lond. A 345, 295–325.Google Scholar
Wasow, W. 1953 Asymptotic solution of the differential equation of hydrodynamic stability in a domain containing a transitional point. Ann. Maths 58, 222–252.Google Scholar
Williams, G. P. & Yamagata, T. 1984 Geostrophic regimes, intermediate solitary vortices and Jovian eddies. J. Atmos. Sci. 41, 453–478.Google Scholar