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Long Atmospheric Waves and the Polar-Plane Approximation to the Earth’s Spherical Geometry

Alison F. C. BridgerDepartment of Atmospheric Science, Colorado State University, Fort Collins 80523

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Duane E. StevensDepartment of Atmospheric Science, Colorado State University, Fort Collins 80523

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Abstract

The spherical geometry of the earth is replaced by polar cylindrical geometry, with a plane tangential to the earth at the pole. The resulting frequency and structure of free motions in an isothermal, adiabatic atmosphere with a resting basic state is studied in both geometries. The solutions for ν (meridional wind) may be written as a single Bessel function if certain approximations are made. For positive equivalent depths, the geometrical approximation is best when the Lamb parameter ε≳ 10, so that Rossby waves are well modeled, while fast moving gravity waves are not well approximated. The impact of setting f to a constant value when undifferentiated, as in the usual midlatitude beta-plane approximation, is examined. It is found that the value of f is as important in determining how well the model behaves as are the geometrical and other approximations.

Abstract

The spherical geometry of the earth is replaced by polar cylindrical geometry, with a plane tangential to the earth at the pole. The resulting frequency and structure of free motions in an isothermal, adiabatic atmosphere with a resting basic state is studied in both geometries. The solutions for ν (meridional wind) may be written as a single Bessel function if certain approximations are made. For positive equivalent depths, the geometrical approximation is best when the Lamb parameter ε≳ 10, so that Rossby waves are well modeled, while fast moving gravity waves are not well approximated. The impact of setting f to a constant value when undifferentiated, as in the usual midlatitude beta-plane approximation, is examined. It is found that the value of f is as important in determining how well the model behaves as are the geometrical and other approximations.

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