### Abstract

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 = 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 wave-numbers 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1282-1307 |

Number of pages | 26 |

Journal | Journal of the Atmospheric Sciences |

Volume | 56 |

Issue number | 10 |

State | Published - May 15 1999 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Atmospheric Science

### Cite this

*Journal of the Atmospheric Sciences*,

*56*(10), 1282-1307.

**Generalized stability analyses of asymmetric disturbances in one- and two-celled vortices maintained by radial inflow.** / Nolan, David S; Farrell, Brian F.

Research output: Contribution to journal › Article

*Journal of the Atmospheric Sciences*, vol. 56, no. 10, pp. 1282-1307.

}

TY - JOUR

T1 - Generalized stability analyses of asymmetric disturbances in one- and two-celled vortices maintained by radial inflow

AU - Nolan, David S

AU - Farrell, Brian F.

PY - 1999/5/15

Y1 - 1999/5/15

N2 - 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 = 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 wave-numbers 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.

AB - 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 = 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 wave-numbers 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.

UR - http://www.scopus.com/inward/record.url?scp=0033562288&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033562288&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033562288

VL - 56

SP - 1282

EP - 1307

JO - Journals of the Atmospheric Sciences

JF - Journals of the Atmospheric Sciences

SN - 0022-4928

IS - 10

ER -