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Normal Modes of a Global Nonhydrostatic Atmospheric Model

Akira KasaharaNational Center for Atmospheric Research, Boulder, Colorado

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Jian-Hua QianNational Center for Atmospheric Research, Boulder, Colorado

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Abstract

Anticipating use of a very high resolution global atmospheric model for numerical weather prediction in the future without a traditional hydrostatic assumption, this article describes a unified method to obtain the normal modes of a nonhydrostatic, compressible, and baroclinic global atmospheric model.

A system of linearized equations is set up with respect to an atmosphere at rest. An eigenvalue–eigenfunction problem is formulated, consisting of horizontal and vertical structure equations with suitable boundary conditions. The wave frequency and the separation parameter, referred to as “equivalent height,” appear in both the horizontal and vertical equations. Hence, these two equations must be solved as a coupled problem.

Numerical results are presented for an isothermal atmosphere. Since the solutions of the horizontal structure equation can only be obtained numerically, the coupled problem is solved by an iteration method. In the primitive-equation (hydrostatic) models, there are two kinds of normal modes: The first kind consists of eastward and westward propagating gravity–inertia oscillations, and the second kind consists of westward propagating rotational (Rossby–Haurwitz type) oscillations. In the nonhydrostatic model, there is an additional kind of eastward and westward propagating acoustic–inertia oscillations. The horizontal structures of the third kind are distinguished from those of the first kind by large differences in the equivalent height. The second kind is hardly affected by nonhydrostatic effects. In addition, there are so-called external inertia–gravity mode oscillations (Lamb waves), which propagate horizontally with almost constant speed of sound. Also, there are external rotational mode oscillations that correspond to equivalent barotropic planetary waves. Those two classes of oscillations are identical to those in the hydrostatic version of model. Nonhydrostatic effects on the first kind of oscillations become significant for smaller horizontal and deeper vertical scales of motion.

* Current affiliation: IRI/LDEO, Columbia University, Palisades, New York.

Corresponding author address: Dr. Akira Kasahara, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000.

Email: kasahara@ncar.ucar.edu

Abstract

Anticipating use of a very high resolution global atmospheric model for numerical weather prediction in the future without a traditional hydrostatic assumption, this article describes a unified method to obtain the normal modes of a nonhydrostatic, compressible, and baroclinic global atmospheric model.

A system of linearized equations is set up with respect to an atmosphere at rest. An eigenvalue–eigenfunction problem is formulated, consisting of horizontal and vertical structure equations with suitable boundary conditions. The wave frequency and the separation parameter, referred to as “equivalent height,” appear in both the horizontal and vertical equations. Hence, these two equations must be solved as a coupled problem.

Numerical results are presented for an isothermal atmosphere. Since the solutions of the horizontal structure equation can only be obtained numerically, the coupled problem is solved by an iteration method. In the primitive-equation (hydrostatic) models, there are two kinds of normal modes: The first kind consists of eastward and westward propagating gravity–inertia oscillations, and the second kind consists of westward propagating rotational (Rossby–Haurwitz type) oscillations. In the nonhydrostatic model, there is an additional kind of eastward and westward propagating acoustic–inertia oscillations. The horizontal structures of the third kind are distinguished from those of the first kind by large differences in the equivalent height. The second kind is hardly affected by nonhydrostatic effects. In addition, there are so-called external inertia–gravity mode oscillations (Lamb waves), which propagate horizontally with almost constant speed of sound. Also, there are external rotational mode oscillations that correspond to equivalent barotropic planetary waves. Those two classes of oscillations are identical to those in the hydrostatic version of model. Nonhydrostatic effects on the first kind of oscillations become significant for smaller horizontal and deeper vertical scales of motion.

* Current affiliation: IRI/LDEO, Columbia University, Palisades, New York.

Corresponding author address: Dr. Akira Kasahara, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000.

Email: kasahara@ncar.ucar.edu

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