The structure and dynamics of axisymmetric tornado-like vortices are explored with a numerical model of 
axisymmetric incompressible flow based on recently developed numerical methods. The model is first shown 
to compare favorably with previous results and is then used to study the effects of varying the major parameters 
controlling the vortex: the strength of the convective forcing, the strength of the rotational forcing, and the 
magnitude of the model eddy viscosity. Dimensional analysis of the model problem indicates that the results 
must depend on only two dimensionless parameters. The natural choices for these two parameters are a convective 
Reynolds number (based on the velocity scale associated with the convective forcing) and a parameter analogous 
to the swirl ratio in laboratory models. However, by examining sets of simulations with different model parameters 
it is found that a dimensionless parameter known as the vortex Reynolds number, which is the ratio of the far-
field circulation to the eddy viscosity, is more effective than the conventional swirl ratio for predicting the 
structure of the vortex.

As the value of the vortex Reynolds number is increased, it is observed that the tornado-like vortex transitions from a smooth, steady flow to one with quasiperiodic oscillations. These oscillations, when present, are caused by axisymmetric disturbances propagating down toward the surface from the upper part of the domain. Attempts to identify these oscillations with linear waves associated with the shears of the mean azimuthal and vertical winds give mixed results.

The parameter space defined by the choices for model parameters is further explored with large sets of numerical simulations. For much of this parameter space it is confirmed that the vortex structure and time-dependent behavior depend strongly on the vortex Reynolds number and only weakly on the convective Reynolds number. The authors also find that for higher convective Reynolds numbers, the maximum possible wind speed increases, and the rotational forcing necessary to achieve that wind speed decreases. Physical reasoning is used to explain this behavior, and implications for tornado dynamics are discussed.

The dynamics of both transient and exponentially growing disturbances in two-dimensional vortices that are 
maintained by the radial inflow of a fixed cylindrical deformation field are investigated. Such deformation fields 
are chosen so that both one-celled and two-celled vortices may be studied. The linearized evolution of asymmetric 
perturbations is expressed in the form of a linear dynamical system dx/dt 5 Ax. The shear of the mean flow 
results in a nonnormal dynamical operator A, allowing for the transient growth of perturbations even when all 
the modes of the operator are decaying. It is found that one-celled vortices are stable to asymmetric perturbations 
of all azimuthal wavenumbers, whereas two-celled vortices can have low-wavenumber instabilities. In all cases, 
generalized stability analysis of the dynamical operator identifies the perturbations that grow the fastest, both 
instantaneously and over a finite period of time. While the unstable modal perturbations necessarily convert 
mean-flow vorticity to perturbation vorticity, the perturbations with the fastest instantaneous growth rate use 
the deformation of the mean flow to rearrange their vorticity fields into configurations with higher kinetic energy. 
Also found are perturbations that use a hybrid of these two mechanisms to achieve substantial energy growth 
over finite time periods.

Inclusion of the dynamical effects of radial inflow—vorticity advection and vorticity stretching—is found to be extremely important in assessing the potential for transient growth and instability in these vortices. In the two-celled vortex, neglecting these terms destabilizes the vortex for azimuthal wavenumbers one and two. In the one-celled vortex, neglect of the radial inflow terms results in an overestimation of transient growth for all wavenumbers, and it is also found that for high wavenumbers the maximum transient growth decreases as the strength of the radial inflow increases.

The effects of these perturbations through eddy flux divergences on the mean flow are also examined. In the one-celled vortex it is found that for all wavenumbers greater than one the net effect of most perturbations, regardless of their initial configuration, is to increase the kinetic energy of the mean flow. As these perturbations are sheared over they cause upgradient eddy momentum fluxes, thereby transferring their kinetic energy to the mean flow and intensifying the vortex. However, for wavenumber one in the one-celled vortex, and all wavenumbers in the two-celled vortex, it was found that nearly all perturbations have the net effect of decreasing the kinetic energy of the mean flow. In these cases, the kinetic energy of the perturbations accumulates in nearly neutral or unstable modal structures, so that energy acquired from the mean flow is not returned to the mean flow but instead is lost through dissipation.

The effects of stochastically excited asymmetric disturbances on two-dimensional vortices are investigated. 
These vortices are maintained by the radial inflow of fixed cylindrical deformation fields, which are chosen so 
that both one-celled and two-celled vortices may be studied. The linearized perturbation equations are reduced 
to the form of a linear dynamical system with stochastic forcing, that is, dx/dt 5 Ax 1 Fj, where the columns 
of F are forcing functions and the elements of j are Gaussian white-noise processes. Through this formulation 
the stochastically maintained variance of the perturbations, the structures that dominate the response (the empirical 
orthogonal functions), and the forcing functions that contribute most to this response (the stochastic optimals) 
can be directly calculated.

For all cases the structures that most effectively induce the transfer of energy from the mean flow to the perturbation field are close approximations to the global optimals (i.e., the initial perturbations with the maximum growth in energy in finite time), and that the structures that account for most of the variance are close approximations to the global optimals evolved forward in time to when they reach their maximum energy. For azimuthal wavenumbers in each vortex where nearly neutral modes are present (k 5 1 for the one-celled vortex and 1 # k # 4 for the two-celled vortex), the variance sustained by the stochastic forcing is large, and in these cases the variance may be greatly overestimated if the radial inflow that sustains the mean vortex is neglected in the dynamics of the perturbations.

Through a modification of this technique the ensemble average eddy momentum flux divergences associated with the stochastically maintained perturbation fields can be computed, and this information is used to determine the perturbation-induced mean flow tendency in the linear limit. Examination of these results shows that the net effect of the low wavenumber perturbations is to cause downgradient eddy fluxes in both vortex types, while high wavenumber perturbations cause upgradient eddy fluxes. However, to determine how these eddy fluxes actually change the mean flow, the local accelerations caused by the eddy flux divergences must be incorporated into the equation for the steady-state azimuthal velocity. From calculations of this type, it is found that the effect of the radial inflow can be crucial in determining whether or not the vortex is intensified or weakened by the perturbations: though the net eddy fluxes are most often downgradient, the radial inflow carries the transported angular momentum back into the vortex core, resulting in an increase in the maximum wind speed. In most cases for the vortex flows studied, the net effect of stochastically forced asymmetric perturbations is to intensify the mean vortex. Applications of the same analysis techniques to vortices with azimuthal velocity profiles more like those used in previous studies give similar results.