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The Gill Model and the Weak Temperature Gradient Approximation

Christopher S. BrethertonDepartment of Atmospheric Sciences, University of Washington, Seattle, Washington

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Adam H. SobelDepartment of Applied Physics and Applied Mathematics, and Department of Earth and Environmental Sciences, Columbia University, New York, New York

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Abstract

The authors investigate the accuracy of the weak temperature gradient (WTG) approximation, in which the divergent flow is computed by an assumed balance between adiabatic cooling and diabatic heating, for a prototype linear problem, the Gill model of a localized tropical heat source in a zonally periodic domain. As in earlier work by Neelin using a realistic, spatially distributed forcing, WTG is found to be a reasonable approximation to the full Gill model even when fairly large values of the thermal damping are used. Use of the local forcing and consideration of the different dispersion relations of free modes in the WTG and full Gill systems helps to clarify differences between the two solutions. WTG does not support an equatorial Kelvin wave. Instead, the Kelvin wave speed can be regarded as infinite in WTG. Hence, WTG becomes highly accurate when the thermal damping is sufficiently weak (0.1 day−1 or less) that an equatorial Kelvin wave can propagate around the globe without substantial loss of amplitude. Free Rossby waves are not equatorially trapped under WTG, and this increases the magnitude of the response far from the equator.

Corresponding author address: Dr. Adam H. Sobel, Department of Applied Physics and Applied Mathematics, Columbia University, 500 West 120th Street, Room 217, New York, NY 10027. Email: sobel@appmath.columbia.edu

Abstract

The authors investigate the accuracy of the weak temperature gradient (WTG) approximation, in which the divergent flow is computed by an assumed balance between adiabatic cooling and diabatic heating, for a prototype linear problem, the Gill model of a localized tropical heat source in a zonally periodic domain. As in earlier work by Neelin using a realistic, spatially distributed forcing, WTG is found to be a reasonable approximation to the full Gill model even when fairly large values of the thermal damping are used. Use of the local forcing and consideration of the different dispersion relations of free modes in the WTG and full Gill systems helps to clarify differences between the two solutions. WTG does not support an equatorial Kelvin wave. Instead, the Kelvin wave speed can be regarded as infinite in WTG. Hence, WTG becomes highly accurate when the thermal damping is sufficiently weak (0.1 day−1 or less) that an equatorial Kelvin wave can propagate around the globe without substantial loss of amplitude. Free Rossby waves are not equatorially trapped under WTG, and this increases the magnitude of the response far from the equator.

Corresponding author address: Dr. Adam H. Sobel, Department of Applied Physics and Applied Mathematics, Columbia University, 500 West 120th Street, Room 217, New York, NY 10027. Email: sobel@appmath.columbia.edu

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