Editorial Type:
Article
Article Type:
Research Article

The Intensification of Two-Dimensional Swirling Flows by Stochastic Asymmetric Forcing

David S. Nolan Department of Mathematics, Lawrence Berkeley National Laboratory, Berkeley, California

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Brian F. Farrell Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts

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Print Publication:
01 Dec 1999
DOI:
https://doi.org/10.1175/1520-0469(1999)056<3937:TIOTDS>2.0.CO;2
Page(s):
3937–3962
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Abstract

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 = Ax + Fξ, where the columns of F are forcing functions and the elements of ξ 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 = 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.

* Current affiliation: Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado.

Corresponding author address: Dr. David S. Nolan, Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523.

Abstract

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 = Ax + Fξ, where the columns of F are forcing functions and the elements of ξ 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 = 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.

* Current affiliation: Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado.

Corresponding author address: Dr. David S. Nolan, Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523.

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