The time-dependent transformed Eliassen balanced vortex model of a tropical cyclone.

Abstract

The time-dependent, transformed Eliassen balanced vortex model of a tropical cyclone is analyzed mathematically and integrated numerically. The simulated vortices have characteristics resembling aspects of real tropical cyclones. Based on the principle of potential vorticity invertibility, the baroclinic model vortex evolves on an f-plane, assumes Boussinesq and hydrostatic approximations, and is posed in absolute angular momentum coordinates with circular symmetry. Theoretical analysis of the model equations is used to derive efficient numerical methods. Linear operator theory is applied to elliptic, diagnostic equations for the tangential velocity potential function and the transverse circulation streamfunction, which are then solved by successive line overrelaxation methods. Equations for the potential vorticity and potential temperature are solved using a fourth-order Runge Kutta method. A maximum growth rate estimated for the potential vorticity equation limits the size of the timestep, yet reveals the presence of exponentially growing modes for given profiles of the thermal forcing. Initial potential vorticity and potential temperature distributions are based on a mesocyclone study. Vortex evolution is computed for a specified forcing that emulates condensational heating and a heating function parameterized in terms of the model variables. The specified heating produces a realistic circulation but does not allow the vortex to decay. When a parameterized heating function is used, the transverse circulation requires external forcing through frictionally induced vertical motion at the surface boundary. Mathematical and physical descriptions of the boundary conditions are discussed and compared; the surface conditions are insufficient for maintenance or amplification of the vortex under the parameterized heating. In addition to analyzing physical and dynamical relationships among model variables, the simulations can be used as a 2-D basic state for a perturbation study using 3-D primitive equations. Such an investigation would illuminate nonaxisymmetric features of hurricanes such as rain bands and outflow jets.