Wavenumber Selection for Single-Wave Steady States in a Nonlinear Baroclinic System

H-Y. Weng Florida State University, Tallahassee, Florida

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A. Barcilon Florida State University, Tallahassee, Florida

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Abstract

The mechanism of nonlinear wavenumber selection for single-wave steady states is examined analytically and numerically in an Eady-type model with uneven Ekman dissipation. The spectral solution is truncated to three zonal waves, no −1, n0, n0 + 1, with the lowest meridional mode in both wave and mean flow correction fields. For a given parameter setting, we determine the preferred wavenumber by testing the stability of finite-amplitude, single-wave steady states corresponding to the waves found in such a triad. Comparison between analytical and numerical results show that the preferred wavenumber of a steady wave is unique for a given parameter setting in such an f-plane model, and that the wavenumber selected by strongly nonlinear theory is lower than the most unstable wavenumber predicted by linear theory, bat may be higher than the one selected by weakly nonlinear theory.

The physical processes of nonlinear wavenumber selection are described in terms of linear and nonlinear Eady angles as well as the relative positions of the wave heat flux vectors within these angles. Two competing, scale-dependent processes, similar to the finding of Gall et al., are at work in nonlinear wavenumber selection: the wave-mean flow interaction reduces the basic state shear to an elective shear, while the Ekman pumping increases the minimum critical shear to an effective critical shear. These time-dependent shears control the nonlinear Eady angles. In a given triad for a given parameter setting, the wave with the nonlinear Eady angle that is last to vanish is the preferred wave.

Abstract

The mechanism of nonlinear wavenumber selection for single-wave steady states is examined analytically and numerically in an Eady-type model with uneven Ekman dissipation. The spectral solution is truncated to three zonal waves, no −1, n0, n0 + 1, with the lowest meridional mode in both wave and mean flow correction fields. For a given parameter setting, we determine the preferred wavenumber by testing the stability of finite-amplitude, single-wave steady states corresponding to the waves found in such a triad. Comparison between analytical and numerical results show that the preferred wavenumber of a steady wave is unique for a given parameter setting in such an f-plane model, and that the wavenumber selected by strongly nonlinear theory is lower than the most unstable wavenumber predicted by linear theory, bat may be higher than the one selected by weakly nonlinear theory.

The physical processes of nonlinear wavenumber selection are described in terms of linear and nonlinear Eady angles as well as the relative positions of the wave heat flux vectors within these angles. Two competing, scale-dependent processes, similar to the finding of Gall et al., are at work in nonlinear wavenumber selection: the wave-mean flow interaction reduces the basic state shear to an elective shear, while the Ekman pumping increases the minimum critical shear to an effective critical shear. These time-dependent shears control the nonlinear Eady angles. In a given triad for a given parameter setting, the wave with the nonlinear Eady angle that is last to vanish is the preferred wave.

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