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Large-Scale Dynamical Response to Differential Heating: Statistical Equilibrium States and Amplitude Vacillation

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  • 1 National Center for Atmospheric Research, Boulder CO 80307
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Abstract

This is a study of the statistical behavior of a very low order “general circulation model,” consisting of a single finite-amplitude baroclinic wave interacting with a mean zonal shear flow, maintained against dissipation by differential heating. Starting with the equations for two-level quasi-geostrophic flow in a β-plane channel, we derive a closed system of evolution equations for five zonally-averaged quantities at 45° latitude—including net poleward heat transport, meridional kinetic energy and mean vertical shear (or mean horizontal temperature gradient).

This system of equations enables us to relate the equilibrium values of mean vertical shear between 250 and 750 mb and rms northward component of velocity at 500 mb to the rate of differential heating. The former is estimated to be 10.7 m s−1 at 45°S: Oort's observed statistics show that it is actually about 12 m s−1. The theoretical estimate of the rms northward component of velocity at 45°S is 11.0 m s−1. Oort's and Trenberth's statistics also give a value of about 11 m s−1.

For conditions at 45°S during the Southern Hemisphere summer season, numerical integrations of our model equations show that amplitude vacillations around the equilibrium state have a period of about 22.7 days, which compares favorably with the observed periods reported by Webster and Keller in addition to Randel and Stanford. This is added confirmation that a very simple model may provide a physically valid basis for understanding the dominant large-scale dynamical response to differential heating.

Abstract

This is a study of the statistical behavior of a very low order “general circulation model,” consisting of a single finite-amplitude baroclinic wave interacting with a mean zonal shear flow, maintained against dissipation by differential heating. Starting with the equations for two-level quasi-geostrophic flow in a β-plane channel, we derive a closed system of evolution equations for five zonally-averaged quantities at 45° latitude—including net poleward heat transport, meridional kinetic energy and mean vertical shear (or mean horizontal temperature gradient).

This system of equations enables us to relate the equilibrium values of mean vertical shear between 250 and 750 mb and rms northward component of velocity at 500 mb to the rate of differential heating. The former is estimated to be 10.7 m s−1 at 45°S: Oort's observed statistics show that it is actually about 12 m s−1. The theoretical estimate of the rms northward component of velocity at 45°S is 11.0 m s−1. Oort's and Trenberth's statistics also give a value of about 11 m s−1.

For conditions at 45°S during the Southern Hemisphere summer season, numerical integrations of our model equations show that amplitude vacillations around the equilibrium state have a period of about 22.7 days, which compares favorably with the observed periods reported by Webster and Keller in addition to Randel and Stanford. This is added confirmation that a very simple model may provide a physically valid basis for understanding the dominant large-scale dynamical response to differential heating.

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