Nonhydrostatic, three-dimensional perturbations to balanced, hurricane-like vortices. Part I: Linearized formulation, stability, and evolution

In this paper, the first of two parts, the dynamics of linearized perturbations to hurricane-like vortices are studied. Unlike previous studies, which are essentially two-dimensional or assume that the perturbations are quasi-balanced, the perturbations are fully three-dimensional and nonhydrostatic. The vortices used as basic states are also three-dimensional (though axisymmetric), with wind fields modeled closely after observations of hurricanes and tropical storms, and are initially in hydrostatic and gradient wind balance. The equations of motion, computational methods for solving them, and methods for generating the basic-state hurricane-like vortices are presented. In particular, three basic states are studied: a vortex modeled after an intense (category 3) hurricane, a moderate (category 1) hurricane, and a weak tropical storm. The stability of each vortex is considered. The category 3 vortex is found to be rather unstable, with its fastest growing mode occurring for azimuthal wavenumber three with an e-folding time of approximately 1 h. The category 1 vortex is less unstable, as its most unstable mode occurs for wavenumber two with an e-folding time of 5 h. In both cases, these unstable modes are found to be close analogs of their strictly two-dimensional counterparts, and essentially barotropic in nature. The tropical storm-like vortex is found to be stable for all azimuthal wavenumbers. For this vortex, the evolution of purely thermal, unbalanced perturbations in the vortex environment are studied; such disturbances might be the result of asymmetric bursts of convection in the vicinity of the vortex, which are typical for developing storms. The evolution of these perturbations goes through two phases. First, there is substantial gravity wave radiation and rapid adjustment to quasi-gradient wind balance. In the second phase, the quasi-balanced perturbations are axisymmetrized by the shear of the basic-state vortex, and cause localized accelerations of the symmetric vortex via eddy momentum and heat fluxes. The full response of the symmetric vortex and comparisons to fully nonlinear simulations are the topics of the second part.