Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-19T15:00:26.005Z Has data issue: false hasContentIssue false
  • English
  • Français

General stability conditions for zonal flows in a one-layer model on the β-plane or the sphere

Published online by Cambridge University Press:  20 April 2006

P. Ripa
P. Ripa
Affiliation:
C.I.C.E.S.E., Ensenada, B.C.N., México
Get access Rights & Permissions [Opens in a new window]

Abstract

Sufficient stability conditions are derived for a zonal flow on the β-plane or the sphere. Two conditions guarantee both shear stability (to perturbations with vanishing zonal average) and inertial stability (to longitude-independent perturbations). These conditions are not restricted to normal-mode disturbances, and are derived without making use of the quasi-geostrophic approximation. The main limitation of the model is to have only one layer.

On the β-plane, the conditions are: (i) that the product of the meridional gradient of potential vorticity and the difference between an arbitrary constant and the zonal velocity be everywhere non-negative; and (ii) that the absolute value of this difference be nowhere larger than the local phase speed of long gravity waves. Inertial stability is independently assured if the Cariolis parameter and the potential vorticity are everywhere of the same sign (this well-known condition can be easily violated near the equator, but the flow may nonetheless be stable).

If the meridional gradient of potential vorticity has everywhere the same sign, then conditions (i) and (ii) can be shown to be consequences of the conservation of a total pseudo-energy E0 and pseudomomentum P0, defined so that their lowest-order contribution is quadratic in the deviation from the fundamental state (even in the case that the perturbation is longitude-independent). Thus, if there exists a value of α such that the integral of E0 − αP0 is positive-definite, then the flow is stable. In this case, the stability conditions are valid for small, rather than infinitesimal, perturbations.

The parameters of stable flows, as guaranteed by these conditions, are investigated for the family of Gaussian jets centred at the equator; both the cases of an unbounded ocean and a semi-infinite ocean, poleward from a zonal wall, are considered. Easterlies with the width of a Kelvin wave and westerlies with that width or wider may be unstable, even though the gradient of potential vorticity is positive for any strength of the jet.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 126 , January 1983 , pp. 463 - 489
Copyright
© 1983 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

Arnold, V. I. 1965 Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid Dokl. Akad. Nauk SSSR 162, 975–978. (English transl. in Sov. Math. 6, 331–334.)Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Blumen, W. 1968 On the stability of quasi-geostrophic flow J. Atmos. Sci. 25, 929–931.Google Scholar
Blumen, W. 1970 Shear layer instability of an inviscid compressible fluid J. Fluid Mech. 40, 769–781.Google Scholar
Blumen, W. 1978 A note on horizontal boundary conditions and stability of quasi-geostrophic flow J. Atmos. Sci. 35, 1314–1318.Google Scholar
Bretherton, F. P. 1966 Critical layer instability in baroclinic flows Q. J. R. Met. Soc. 92, 325–334.Google Scholar
Busse, F. H. & Chen, W. L. 1981a On the (nearly) symmetric instability. J. Atmos. Sci. 38, 877–880.Google Scholar
Busse, F. H. & Chen, W. L. 1981b Shear flow instabilities in a rotating system. Geophys. Astrophys. Fluid Dyn. 17, 199–214.Google Scholar
Cairns, R. A. 1979 The role of negative energy waves in some instabilities of parallel flows J. Fluid Mech. 92, 1–14.Google Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluid Adv. Appl. Mech. 9, 1–89.Google Scholar
Dunkerton, T. J. 1981 On the inertial stability oif the equatorial middle atmosphere J. Atmos. Sci. 38, 2354–2364.Google Scholar
Emanuel, K. A. 1979 Inertial instability and mesoscale convective systems. Part I: Linear theory of inertial instability in rotating viscous fluids. J. Atmos. Sci. 36, 2425–2449.Google Scholar
FJØRTOFT, R. 1950 Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex Geophys. Publ. 17, 1–52.Google Scholar
Gill, A. E. 1974 The stability of planetary waves on an infinite beta-plane Geophys. Fluid Dyn. 6, 29–47.Google Scholar
Griffiths, R. W., Killworth, P. D. & Stern, M. E. 1982 Ageostrophic instability of ocean currents J. Fluid Mech. 117, 343–377.Google Scholar
Hughes, R. L. 1979 On the dynamics of the equatorial undercurrent Tellus 31, 447–455.Google Scholar
Hughes, R. L. 1981 On inertial instability of the equatorial undercurrent Tellus 33, 291–300.Google Scholar
Kuo, H. L. 1949 Dynamic instability of two-dimensional non-divergent flow in a barotropic atmosphere J. Met. 6, 105–122.Google Scholar
Lipps, F. B. 1963 Stability of jets in a divergent barotropic fluid J. Atmos. Sci. 20, 120–129.Google Scholar
Marinone, S. G. & Ripa, P. 1983 Energetics of the instability of a depth independent equatorial jet. In preparation.
Miyata, M. 1981 A criterion for barotropic instability at the equator. Tropical Ocean–-Atmosphere Newsletter no. 5 (January), University of Washington (unpublished manuscript).Google Scholar
Paldor, N. 1982 Stable and unstable modes of surface fronts. Ph.D. thesis, University of Rhode Island.
Philander, S. G. H. 1976 Instabilities of zonal equatorial currents J. Geophys. Res. 81, 3725–3735.Google Scholar
Philander, S. G. H. 1978 Instabilities of zonal equatorial currents. 2 J. Geophys. Res. 83, 3679–3682.Google Scholar
Rayleigh, J. W. S. 1880 On the stability, or instability of certain fluid motions Proc. Lond. Math. Soc. 9, 57–70.Google Scholar
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond A 93, 148–154.Google Scholar
Ripa, P. 1981a On the theory of nonlinear interactions among geophysical waves. J. Fluid Mech. 103, 87–115.Google Scholar
Ripa, P. 1981b Symmetries and conservation laws for internal gravity waves. In Nonlinear Properties of Internal Waves (ed. B. West), pp. 281–306. A.I.P. Proceedings.
Ripa, P. 1982 Nonlinear wave–-wave interactions in one-layer reduced gravity model on the equatorial beta-plane J. Phys. Oceanogr. 12, 97–111.Google Scholar
Ripa, P. & Marinone, S. G. 1983 On the stability of an equatorial jet. In Hydrodynamics of the Equatorial Ocean (ed. J. C. J. Nihoul). Elsevier Oceanographic Series (in press).
Semtner, A. J. & Holland, W. R. 1980 Numerical simulation of equatorial ocean circulation. Part I: A basic case in turbulent equilibrium. J. Phys. Oceanogr. 10, 667–693.Google Scholar