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Three-dimensional baroclinic instability of a Hadley cell for small Richardson number

  • Basil N. Antar (a1) and William W. Fowlis (a2)

Abstract

A three-dimensional linear stability analysis of a baroclinic flow for Richardson number Ri of order unity is presented. The model considered is a thin, horizontal, rotating fluid layer which is subjected to horizontal and vertical temperature gradients. The basic state is a Hadley cell which is a solution of the Navier–Stokes and energy equations and contains both Ekman and thermal boundary layers adjacent to the rigid boundaries; it is given in closed form. The stability analysis is also based on the Navier–Stokes and energy equations; and perturbations possessing zonal, meridional and vertical structures were considered. Numerical methods were developed for the solution of the stability problem, which results in an ordinary differential eigenvalue problem. The objectives of this work were to extend the previous theoretical work on three-dimensional baroclinic instability for small Ri to a more realistic model involving the Prandtl number σ and the Ekman number E, and to finite growth rates and a wider range of the zonal wavenumber. The study covers ranges of 0.135 [les ] Ri [les ] 1.1, 0.2 [les ] σ [les ] 5.0, and 2 × 10−4 [les ] E [les ] 2 σ 10−3. For the cases computed for E = 10−3 and σ ≠ 1, we found that conventional baroclinic instability dominates for Ri > 0.825 and symmetric baroclinic instability dominates for Ri < 0.675. However, for E [ges ] 5 × 10−4 and σ = 1 in the range 0.3 [les ] Ri [les ] 0.8, conventional baroclinic instability always dominates. Further, we found in general that the symmetric modes of maximum growth are not purely symmetric but have weak zonal structure. This means that the wavefronts are inclined at a small angle to the zonal direction. The results also show that as E decreases the zonal structure of the symmetric modes of maximum growth rate also decreases. We found that when zonal structure is permitted the critical Richardson number for marginal stability is increased, but by only a small amount above the value for pure symmetric instability. Because these modes do not substantially alter the results for pure symmetric baroclinic instability and because their zonal structure is weak, it is unlikely that they represent a new type of instability.

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Antar, B. N. & Fowlis, W. W. 1981 Baroclinic instability of a rotating Hadley cell J. Atmos. Sci. 38, 21302141.
Antar, B. N. & Fowlis, W. W. 1982 Symmetric baroclinic instability of a Hadley cell J. Atmos. Sci. 39, 12801289.
Bennets, D. A. & Hoskins, B. J. 1979 Conditional symmetric instability a possible explanation for frontal rainbands Q. J. R. Met. Soc. 105, 945962.
Busse, F. H. & Chen, W. L. 1981 On the (nearly) symmetric instability J. Atmos. Sci. 38, 877880.
Calman, J. 1977 Experiments on high Richardson number instability of a rotating stratified shear flow Dyn. Atmos. Oceans 1, 277297.
Charney, J. G. 1947 The dynamics of long waves in a baroclinic westerly current J. Met. 4, 135162.
Conte, S. D. 1966 The numerical solution of linear boundary value problems S.I.A.M. Rev. 8, 309320.
Eady, E. T. 1949 Long waves and cyclone waves Tellus 1, 3552.
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, 2425810.
Hadlock, R. K., Na, J. Y. & Stone, P. H. 1972 Direct thermal verification of symmetric baroclinic instability J. Atmos. Sci. 29, 13911393.
Hide, R. & Mason, P. J. 1975 Sloping convection in a rotating fluid Adv. Phys. 24, 47100.
Kuo, H. L. 1956 Forced and free axially symmetric convection produced by differential heating in a rotating fluid J. Met. 13, 521527.
Lorenz, E. N. 1967 The Nature and Theory of the General Circulation of the Atmosphere. World Meteorological Organization.
Mcintyre, M. E. 1970 Diffusive destabilization of the baroclinic circular vortex Geophys. Fluid Dyn. 1, 1957.
Pedlosky, J. 1979 Geophysical Fluid Dynamics. Springer.
Solberg, H. 1936 Le Mouvement d'inertie de l'atmosphère stable et son rôle dans le théorie des cyclones. In Proc. Union Géodesique et Géophysique Internationale VIième Assemblée, Edinburgh, vol. II, pp. 6682.
Stone, P. H. 1966 On non-geostrophic baroclinic stability J. Atmos. Sci. 23, 390400.
Stone, P. H. 1967 An application of baroclinic stability theory to the dynamics of the Jovian atmosphere J. Atmos. Sci. 24, 642652.
Stone, P. H. 1970 On non-geostrophic baroclinic stability: Part II. J. Atmos. Sci. 27, 721810.
Stone, P. H. 1971 Baroclinic stability under non-hydrostatic conditions J. Fluid Mech. 45, 659671.
Stone, P. H., Hess, S., Hadlock, R. & Ray, P. 1969 Preliminary results of experiments with symmetric baroclinic instability J. Atmos. Sci. 26, 991996.
Tokioka, T. 1970 Non-geostrophic and non-hydrostatic stability of a baroclinic fluid J. Met. Soc. Japan 48, 503520.
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