# Transformed Eliassen Balanced Vortex Model

Wayne H. Schubert Department of Atmospheric Sciences, Colorado State University, Fort Collins, CO 80523

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James J. Hack IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598

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## Abstract

We consider the axisymmetric balanced flow occurring in a thermally forced vortex in which the frictional inflow is confined to a thin boundary layer. Above the boundary layer the absolute angular momentum ½fR2=rvfr2 is conserved. We refer to R as the potential radius, i.e., the radius to which a particle must be moved (conserving absolute angular momentum) in order to change its tangential component v to zero. Using R as one of the dependent variables we review the equations of the Eliassen balanced vortex model.

We next reverse the roles of the actual radius r and the potential radius R, i.e., we treat R as an independent variable and r as a dependent variable. Introducing transformed components (u*, w*) of the transverse circulation we obtain the transformed Eliassen balanced vortex equations, which differ from the original equations in the following respects: 1) the radial coordinate is R which results in a stretching of positive relative vorticity regions and a shrinking of negative relative vorticity regions; 2) the thermodynamic equation contains only the transverse circulation component w*, the coefficient of which is the potential vorticity q; 3) the equation for r contains only the transverse circulation component u*; 4) the transverse circulation equation contains only two vortex structure functions, the potential vorticity q and the inertial stability s, where pq=(ζ/f)(g0)(∂θ/∂Z) and ρs=f2R4/r4.

The form of the transverse circulation equation leads naturally to a generalized Rossby radius proportional to (q/s)½. A typical distribution Of (q/s)½ is calculated using the composite tropical cyclone data of Gray. The fundamental dynamical role of (q/s)½ is then illustrated with a simple analytical example.

## Abstract

We consider the axisymmetric balanced flow occurring in a thermally forced vortex in which the frictional inflow is confined to a thin boundary layer. Above the boundary layer the absolute angular momentum ½fR2=rvfr2 is conserved. We refer to R as the potential radius, i.e., the radius to which a particle must be moved (conserving absolute angular momentum) in order to change its tangential component v to zero. Using R as one of the dependent variables we review the equations of the Eliassen balanced vortex model.

We next reverse the roles of the actual radius r and the potential radius R, i.e., we treat R as an independent variable and r as a dependent variable. Introducing transformed components (u*, w*) of the transverse circulation we obtain the transformed Eliassen balanced vortex equations, which differ from the original equations in the following respects: 1) the radial coordinate is R which results in a stretching of positive relative vorticity regions and a shrinking of negative relative vorticity regions; 2) the thermodynamic equation contains only the transverse circulation component w*, the coefficient of which is the potential vorticity q; 3) the equation for r contains only the transverse circulation component u*; 4) the transverse circulation equation contains only two vortex structure functions, the potential vorticity q and the inertial stability s, where pq=(ζ/f)(g0)(∂θ/∂Z) and ρs=f2R4/r4.

The form of the transverse circulation equation leads naturally to a generalized Rossby radius proportional to (q/s)½. A typical distribution Of (q/s)½ is calculated using the composite tropical cyclone data of Gray. The fundamental dynamical role of (q/s)½ is then illustrated with a simple analytical example.

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