A Semigeostrophic Eady-Wave Frontal Model Incorporating Momentum Diffusion. Part I: Model and Solutions

William Blumen Department of Astrophysical, Planetary, and Atmospheric Sciences, University of Colorado, Boulder, Colorado

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Abstract

The Hoskins and Bretherton two-dimensional semigeostrophic and uniform potential vorticity model is modified by the incorporation of momentum diffusion in a thin layer—the frontal transition zone. The derived solutions are valid for an extended period following the critical time that the inviscid solution for the cross-frontal geostrophic velocity v becomes discontinuous. This discontinuous behavior is removed by momentum diffusion. The evolution of frontal development is described until equilibration is attained, a quasi-steady state exists, or until decay occurs.

Principal features of the solutions are the lifting of the warm sector above the ground, and the interplay between unstable growth of the baroclinic Eady wave and momentum diffusion that acts as a dissipative mechanism. The semigeostrophic ageostrophic circulation is characterized by a broad clockwise cell. The narrow counterclockwise direct circulation, that encompasses the frontal zone before v becomes discontinuous, is not described by semigeostrophic model dynamics when the front has equilibrated. Similarities and differences between results obtained in primitive equation numerical model experiments, presented by both Williams and by Nakamura and Held, are discussed and analyzed. Nakamura and Held find a change in the vertical structure of the baroclinic wave, that becomes prominent as equilibration is reached. This feature does not emerge as a characteristic of the present model solutions. It is concluded that ageostrophic effects that have been omitted in the semigeostrophic formulation are responsible for this discrepancy between the model results. However, the lifting of the warm air sector above the ground, the widening of the frontal transition zone with time and the magnitudes of the velocities predicted by the primative equation model are all replicated by the semigeostrophic model solutions. Means to control the excessive velocity amplitudes, that are common to all the two-dimensional models, are discussed.

Abstract

The Hoskins and Bretherton two-dimensional semigeostrophic and uniform potential vorticity model is modified by the incorporation of momentum diffusion in a thin layer—the frontal transition zone. The derived solutions are valid for an extended period following the critical time that the inviscid solution for the cross-frontal geostrophic velocity v becomes discontinuous. This discontinuous behavior is removed by momentum diffusion. The evolution of frontal development is described until equilibration is attained, a quasi-steady state exists, or until decay occurs.

Principal features of the solutions are the lifting of the warm sector above the ground, and the interplay between unstable growth of the baroclinic Eady wave and momentum diffusion that acts as a dissipative mechanism. The semigeostrophic ageostrophic circulation is characterized by a broad clockwise cell. The narrow counterclockwise direct circulation, that encompasses the frontal zone before v becomes discontinuous, is not described by semigeostrophic model dynamics when the front has equilibrated. Similarities and differences between results obtained in primitive equation numerical model experiments, presented by both Williams and by Nakamura and Held, are discussed and analyzed. Nakamura and Held find a change in the vertical structure of the baroclinic wave, that becomes prominent as equilibration is reached. This feature does not emerge as a characteristic of the present model solutions. It is concluded that ageostrophic effects that have been omitted in the semigeostrophic formulation are responsible for this discrepancy between the model results. However, the lifting of the warm air sector above the ground, the widening of the frontal transition zone with time and the magnitudes of the velocities predicted by the primative equation model are all replicated by the semigeostrophic model solutions. Means to control the excessive velocity amplitudes, that are common to all the two-dimensional models, are discussed.

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