Generalized stability analyses of asymmetric disturbances in one- and two-celled vortices maintained by radial inflow

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.