Instabilities in a Stratified Fluid Having One Critical Level. Part II: Explanation of Gravity Wave Instabilities Using the Concept of Overreflection

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  • 1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139
  • | 2 Center for Earth and Planetary Physics. Harvard University, Cambridge, MA 02138
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Abstract

We show that there is a one-to-one correspondence between the values of the parameters Ri (Richardson number at the critical level, where mean flow equals phase speed), k (horizontal wavenumber) and cr (phase speed) for which gravity wave instabilities in the presence of a rigid lower boundary were found in Part I and the parameters Ri, k and cr for which overreflection of a neutral wave incident on the shear zone from below would occur. A simple formula involving the reflection coefficient, vertical group velocity of the mode, and the ground to shear layer distance can give quantitative estimates of the growth rate that are especially accurate for small growth rates. The maximum growth rates and corresponding reflection coefficients can be parameterized in a more physically meaningful manner in terms of the Richardson number at the critical level than in terms of the minimum Richardson number in the background wind profile.

Abstract

We show that there is a one-to-one correspondence between the values of the parameters Ri (Richardson number at the critical level, where mean flow equals phase speed), k (horizontal wavenumber) and cr (phase speed) for which gravity wave instabilities in the presence of a rigid lower boundary were found in Part I and the parameters Ri, k and cr for which overreflection of a neutral wave incident on the shear zone from below would occur. A simple formula involving the reflection coefficient, vertical group velocity of the mode, and the ground to shear layer distance can give quantitative estimates of the growth rate that are especially accurate for small growth rates. The maximum growth rates and corresponding reflection coefficients can be parameterized in a more physically meaningful manner in terms of the Richardson number at the critical level than in terms of the minimum Richardson number in the background wind profile.

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