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K. K. Tung and A. J. Rosenthal

Abstract

Previous results claiming the existence of multiple equilibria based on simple barotropic or two-layer models of the atmosphere are reexamined. While not ruling out the existence of multiple equilibria we find that the application of these results to the atmosphere, especially with regard to high/low zonal index and zonal/blocked situations, is probably problematic. Results based on truncated low-order models are found to change drastically when full nonlinearity is retained. Although multiple equilibria may still exist in some nonlinear models in some restricted parameter regimes, it is argued that the parameter values adopted in previous studies are probably not physically based.

Mathematically interesting properties of the nonlinear system, such as resonance bending, hysteresis, bifurcation and multiplicity of solutions are also discussed.

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K. K. Tung and A. J. Rosenthal

Abstract

Using a nonlinear model of stationary long waves which incorporates wave-wave and wave-mean flow interactions, we assess the variability of the stationary long waves in the stratosphere and troposphere. This note supplements the paper by Tung and Rosenthal, where the formulation was given in more detail, but the climatology and variability of stationary waves (No. 1 and No. 2) were not discussed.

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A. J. Rosenthal and R. S. Lindzen

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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A. J. Rosenthal and R. S. Lindzen

Abstract

We consider instabilities in a stratified Boussinesq fluid with basic velocity U o(z). We initiate this study by considering the detailed stability properties of a profile where U o(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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R. S. Lindzen, A. J. Rosenthal, and B. Farrell

Abstract

It is suggested that barotropically unstable easterly jets may be approximated by broken line profiles. It is then possible to use solutions to the Charney problem for baroclinic instability in order to solve the barotropic problem. Results are presented for easterly jets with an arbitrary degree of asymmetry.

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R. S. Lindzen, B. Farrell, and A. J. Rosenthal

Abstract

We have approached the barotropic instability of mean zonal flows over the Bay of Bengal for the months June, July and August from the perspective of pulse asymptotics rather than most rapidly growing normal modes. The results are in good agreement with observations of monsoon depressions.

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