Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-20T15:57:25.789Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of a potential vorticity front: from quasi-geostrophy to shallow water

Published online by Cambridge University Press:  26 April 2006

E. Boss ,
N. Paldor  and
L. Thompson
E. Boss
Affiliation:
School of Oceanography, University of Washington, Seattle, WA 98195-7940, USA
N. Paldor
Affiliation:
Department of Atmospheric Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel
L. Thompson
Affiliation:
School of Oceanography, University of Washington, Seattle, WA 98195-7940, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear stability of a simple two-layer shear flow with an upper-layer potential vorticity front overlying a quiescent lower layer is investigated as a function of Rossby number and layer depths. This flow configuration is a generalization of previously studied flows whose results we reinterpret by considering the possible resonant interaction between waves. We find that instabilities previously referred to as ‘ageostrophic’ are a direct extension of quasi-geostrophic instabilities.

Two types of instability are discussed: the classic long-wave quasi-geostrophic baroclinic instability arising from an interaction of two vortical waves, and an ageostrophic short-wave baroclinic instability arising from the interaction of a gravity wave and a vortical wave (vortical waves are defined as those that exist due to the presence of a gradient in potential vorticity, e.g. Rossby waves). Both instabilities are observed in oceanic fronts. The long-wave instability has length scale and growth rate similar to those found in the quasi-geostrophic limit, even when the Rossby number of the flow is O(1).

We also demonstrate that in layered shallow-water models, as in continuously stratified quasi-geostrophic models, when a layer intersects the top or bottom boundaries, that layer can sustain vortical waves even though there is no apparent potential vorticity gradient. The potential vorticity gradient needed is provided at the top (or bottom) intersection point, which we interpret as a point that connects a finite layer with a layer of infinitesimal thickness, analogous to a temperature gradient on the boundary in a continuously stratified quasi-geostrophic model.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 315 , 25 May 1996 , pp. 65 - 84
Copyright
© 1996 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

Barth, J. A. 1989 Stability of a coastal upwelling front 2. Model results and comparison with observations. J. Geophys. Res. 94, 10857–10883.Google Scholar
Barth, J. A. 1994 Short-wavelength instabilities on coastal jets and fronts. J. Geophys. Res. 99, 16095–16115.Google Scholar
Bretherton, F. P. 1966a Critical layer instability in baroclinic flows. Q. J. R. Met. Soc. 22, 325–334.Google Scholar
Bretherton, F. P. 1966b Baroclinic instability and the short wavelength cut-off in terms of potential vorticity. Q. J. R. Met. Soc. 22, 335–345.Google Scholar
Ford, R. 1993 Gravity wave generation by vortical flows in a rotating frame. PhD thesis, University of Cambridge.
Griffiths, R. W., Killworth, P. D. & Stern, M. E. 1982 Ageostrophic instability of ocean currents. J. Fluid Mech. 117 343–377 (referred to herein as GKS).Google Scholar
Griffiths, R. W. & Linden, P. F. 1982 Laboratory experiments on fronts. Geophys. Astrophys. Fluid Dyn. 19, 159–187.Google Scholar
Hall, M. M. & Fofonoff, N. P. 1993 Downstream development of the Gulf Stream from 68 degrees to 55 degrees W. J. Phys. Oceanogr. 23, 225–249.Google Scholar
Hayashi, Y. Y. & Young, W. R. 1987 Stable and unstable shear modes of rotating parallel flows in shallow water. J. Fluid Mech. 184, 477–504.Google Scholar
Kamke, E. 1951 Differentialgleichungen Lösungsmethoden und Lösungen. Akademische Verlagsgesellschaft.
Killworth, P. D. 1983 Long-wave instability of an isolated front. Geophys. Astrophys. Fluid Dyn. 25, 235–258.Google Scholar
Killworth, P. D., Paldor, N. & Stern, M. E. 1984 Wave propagation and growth on a surface front in a two-layer geostrophic current. J. Mar. Res. 42 761–785 (referred to herein as KPS).Google Scholar
Meacham, S. P. 1991 Meander evolution on piece wise-uniform, quasi-geostrophic jets. J. Phys. Oceanogr. 21, 1139–1170.Google Scholar
Nakamura, N. 1988 The scale selection of baroclinic instability - effect of stratification and non-geostrophy. J. Atmos. Sci. 45, 3253–3267.Google Scholar
Nakamura, N. 1993 An illustrative model of instabilities in meridionally and vertically sheared flows. J. Atmos. Sci. 50, 357–375.Google Scholar
Paldor, N. 1983 Linear stability and stable modes of geostrophic fronts. Geophys. Astrophys. Fluid Dyn. 24, 299–326.Google Scholar
Paldor, N. & Ghil, M. 1991 Shortwave instabilities of coastal currents. Geophys. Astrophys. Fluid Dyn. 58, 225–241.Google Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer.
Pratt, L. J. & Stern, M. E. 1986 Dynamics of potential vorticity fronts and eddy detachment. J. Phys. Oceanogr. 16, 1101–1120.Google Scholar
Ripa, P. 1983 General stability conditions for zonal flows in a one-layer model on the β-plane or the sphere. J. Fluid Mech. 126, 463–489.Google Scholar
Sakai, S. 1988 Rossby-Kelvin instability: a new type of ageostrophic instability caused by a resonance between Rossby waves and gravity waves. J. Fluid Mech. 202, 149–176.Google Scholar
Stone, P. 1970 On non-geostrophic baroclinic stability: part II. J. Atmos. Sci. 27, 721–726.Google Scholar
Swaters, G. E. 1991 On the baroclinic instability of cold-core coupled density fronts on a sloping continental shelf. J. Fluid Mech. 224, 361–382.Google Scholar
Tracey, K. L. & Watts, D. R. 1986 On Gulf stream meander characteristics near Cape Hatteras. J. Geophys. Res. 91, 7587–7602.Google Scholar
Watts, D. R. & Johns, W. E. 1982 Gulf stream meanders: Observations on propagation and growth. J. Geophys. Res. 87, 9467–9476.Google Scholar
Williams, R. 1991 Primitive equation models of Gulf stream meanders. Woods Hole Ocean. Inst. Tech. Rep. WHOI-92–16, pp. 272–285.