Symmetric Baroclinic Instability of a Hadley Cell

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  • 1 The University of Tennessee Space Institute, Tullahoma 37388
  • | 2 Space Sciences Laboratory, NASA, Marshall Space Flight Center, AL 35812
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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.

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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