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  • Author or Editor: Basil N. Antar x
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Timothy L. Miller and Basil N. Antar

Abstract

Calculations have been performed of the (linear) stability of a baroclinic flow to three-dimensional perturbations. Both the simple Eady basic state and the rotating Hadley cell of Antar and Fowlis are considered. The independent influences of the Richardson (Ri), thermal Rossby (baroclinicity), Ekman, and Prandtl numbers are examined, as well as the influences of the angle of orientation of the horizontal wave vector and the wavelength.

It is shown that if the wavelength is allowed to vary freely, disturbances of the Eady type are preferred (i.e., have greatest growth rate) unless Ri and Ekman numbers are small enough and the thermal Rossby number is large enough. In the latter case, disturbances whose angles of orientation are almost symmetric and whose wavelengths are mesoscale are preferred. If, on the other hand, the wavelength is fixed at a mesoscale size, only the symmetric and almost symmetric modes have growth. By allowing the wave vector orientation to deviate from purely symmetric, we note that the region of instability (i.e., critical Ri) is increased, the extent of which is greater for longer wavelength. For Prandtl number = 1, permitting the angle to be nonsymmetric demonstrates the existence of two maxima in growth rate at opposite angles of orientation and with very different energetics. For Prandtl number far enough from one and for large enough dissipation, only one of these two modes has positive growth rates. Growing oscillatory modes were found for some cases.

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Basil N. Antar and William W. Fowlis

Abstract

The stability of a thin fluid layer between two rotating plates which are subjected to a horizontal temperature gradient is studied. First, the solution for the stationary basic state is obtained in a closed form. This solution identifies Ekman and thermal layers adjacent to the plates and interior temperature and velocity fields which are almost linear functions of height. Then the stability of that basic state with respect to infinitesimal zonal waves is analyzed via the solution of the complete viscous linear equations for the perturbations. The character of the growth rates is found to be similar to the growth rates of the classical baroclinic waves. The neutral stability curves for these waves possessed a knee in the Rossby-Taylor number plane to the left of which all perturbations are stable. The region of instability is found to depend on the Prandtl number, the vertical stratification parameter, and both the meridional and zonal wavenumbers. It is found in general that the flow is unstable for small enough Ekman numbers and for Rossby numbers less than 10. It is also found that increased vertical stable stratification and increased Prandtl number stabilize the flow.

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Basil N. Antar and William W. Fowlis

Abstract

A Stability analysis of a thin horizontal rotating fluid layer which is subjected to arbitrary horizontal and vertical temperature gradients is presented. The basic state is a nonlinear Hadley cell which contains both Ekman and thermal boundary layers; it is given in closed form. The stability analysis is based on the linearized Navier-Stokes equations, and zonally symmetric perturbations in the form of waves propagating in the meridional direction are considered. Numerical methods were used for the stability problem. The objective of this investigation was to extend previous work on symmetric baroclinic instability with a more realistic model. Hence, the study deals with flows for which the Richardson number (based on temperature and flow gradients at mid-depth) is of order unity and less. The computations cover ranges of Prandtl number 0.2 ≤ σ ≤ 5, Rossby number 10−2 ≤ ε ≤ 102 and Ekman number 10−4 ≤ E ≤ 10−1. It was found, in agreement with previous work, that the instability sets in when the Richardson number is close to unity and that the critical Richardson number is a non-monotonic function of the Prandtl number. Further, it was found that the critical Richardson number decreases with increasing Ekman number until a critical value of the Ekman number is reached beyond which the fluid is stable. The principal of exchange of stability was not assumed and growth rates wore calculated. A wavelength of maximum growth rate was found. For our model overstability was not found. Some computations were performed for Richardson numbers less than zero. No discontinuities in growth rates are noticeable when the Richardson number changes sign. This result indicates a smooth transition from symmetric baroclinic instability to a convective instability.

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