Effect of Zonal Flows on the Free Oscillations of a Barotropic Atmosphere

Akira Kasahara National Center for Atmospheric Research, Boulder, CO 80307

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Abstract

Solutions of the linearized global shallow-water equations (Laplace tidal equations) including the effect of a mean zonal flow are obtained by the Galerkin-transform method. Free oscillations of the first kind (gravity-inertia modes) are little affected by the zonal flow. Solutions of the second kind (rotational modes of the Rossby-Haurwitz type) are significantly affected by a zonal flow different from solid rotation. Only a few lowest rotational modes, whose angular phase velocities are less than the minimum velocity in the zonal flow, appear as discrete. The remaining angular phase velocities fall into a continuous spectrum which covers the interval between the minimum and maximum zonal velocities. An approximate, but accurate, frequency formula is obtained for the discrete modes of free oscillations under the effect of a mean zonal flow.

The frequencies and latitudinal structures of a few lowest rotational modes under the effect of a mean zonal flow are examined in detail and compared to observational evidence of westward propagating wavenumber 1 long-period oscillations in the atmosphere. The 5-day wavenumber 1 oscillation (the lowest symmetric rotational mode IR+1) is found insensitive to the presence of zonal flows. Other discrete modes are relatively sensitive to and their periods increased by the zonal flow effect. In particular, the period of the second symmetric rotational mode IR+3 (zonal wavenumber 1) increases to about 16–19 days in favor of the observations summarized by Madden (1978).

Abstract

Solutions of the linearized global shallow-water equations (Laplace tidal equations) including the effect of a mean zonal flow are obtained by the Galerkin-transform method. Free oscillations of the first kind (gravity-inertia modes) are little affected by the zonal flow. Solutions of the second kind (rotational modes of the Rossby-Haurwitz type) are significantly affected by a zonal flow different from solid rotation. Only a few lowest rotational modes, whose angular phase velocities are less than the minimum velocity in the zonal flow, appear as discrete. The remaining angular phase velocities fall into a continuous spectrum which covers the interval between the minimum and maximum zonal velocities. An approximate, but accurate, frequency formula is obtained for the discrete modes of free oscillations under the effect of a mean zonal flow.

The frequencies and latitudinal structures of a few lowest rotational modes under the effect of a mean zonal flow are examined in detail and compared to observational evidence of westward propagating wavenumber 1 long-period oscillations in the atmosphere. The 5-day wavenumber 1 oscillation (the lowest symmetric rotational mode IR+1) is found insensitive to the presence of zonal flows. Other discrete modes are relatively sensitive to and their periods increased by the zonal flow effect. In particular, the period of the second symmetric rotational mode IR+3 (zonal wavenumber 1) increases to about 16–19 days in favor of the observations summarized by Madden (1978).

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