Instabilities in a Stratified Fluid Having One Critical Level. Part I: Results

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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 consider instabilities in a stratified Boussinesq fluid with basic velocity Uo(z). We initiate this study by considering the detailed stability properties of a profile where Uo(z) is constant in a top and bottom layer and varies linearly with z in an intermediate layer. For an infinite fluid, we find, in addition to Kelvin-Helmholtz instabilities similar to those found by Miles and Howard (1964), an unstable gravity wave mode propagating energy away from exactly one side of the shear zone if and only if the Richardson number in the shear zone is less than 0.1164±0.0001. With a lower boundary present, we find gravity wave instabilities propagating above as well as below the shear zone can exist only when the Richardson number is less than 0. 116. The ability of a shear zone to sustain instabilities propagating energy to infinity on one side is hence only present when the Richardson number is less than 0.116, whether or, not the ground is present.

With a lower boundary present, we also find gravity wave instabilities evanescent above the shear zone for Richardson numbers up to 0.2499. After an exhaustive search involving varying the height of the shear zone above the ground and the wavenumber k, we conclude that, for any Richardson number less than ¼, the growth rates of gravity wave instabilities can be as much as, but not greater than, 26% of the maximal growth rates for Kelvin-Helmholtz instabilities having a critical level with the same Richardson number as the gravity wave instabilities have. Even though the relative importance of gravity wave instabilities can be significantly greater than found by other investigators (e.g., Davis and Peltier, 1976), the largest growth rates are still associated with Kelvin-Helmholtz instabilities.

In order to understand the basis for this behavior we will analyze, in subsequent parts, both types of instability in terms of wave overreflection. This will enable us to show that the relative importance of Kelvin-Helmholtz and gravity wave instabilities is by no means universal.

Abstract

We consider instabilities in a stratified Boussinesq fluid with basic velocity Uo(z). We initiate this study by considering the detailed stability properties of a profile where Uo(z) is constant in a top and bottom layer and varies linearly with z in an intermediate layer. For an infinite fluid, we find, in addition to Kelvin-Helmholtz instabilities similar to those found by Miles and Howard (1964), an unstable gravity wave mode propagating energy away from exactly one side of the shear zone if and only if the Richardson number in the shear zone is less than 0.1164±0.0001. With a lower boundary present, we find gravity wave instabilities propagating above as well as below the shear zone can exist only when the Richardson number is less than 0. 116. The ability of a shear zone to sustain instabilities propagating energy to infinity on one side is hence only present when the Richardson number is less than 0.116, whether or, not the ground is present.

With a lower boundary present, we also find gravity wave instabilities evanescent above the shear zone for Richardson numbers up to 0.2499. After an exhaustive search involving varying the height of the shear zone above the ground and the wavenumber k, we conclude that, for any Richardson number less than ¼, the growth rates of gravity wave instabilities can be as much as, but not greater than, 26% of the maximal growth rates for Kelvin-Helmholtz instabilities having a critical level with the same Richardson number as the gravity wave instabilities have. Even though the relative importance of gravity wave instabilities can be significantly greater than found by other investigators (e.g., Davis and Peltier, 1976), the largest growth rates are still associated with Kelvin-Helmholtz instabilities.

In order to understand the basis for this behavior we will analyze, in subsequent parts, both types of instability in terms of wave overreflection. This will enable us to show that the relative importance of Kelvin-Helmholtz and gravity wave instabilities is by no means universal.

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