### Abstract

An exact solution for the evolution of linearized perturbations of azimuthal wavenumber one on inviscid vortices was previously discovered for nondivergent vorticity dynamics on an f plane. The longtime asymptotics for this exact solution have been shown to allow an algebraic instability with unbounded growth even in the absence of exponentially growing modes. The necessary requirement for this instability is that there exist a local maximum in the basic-state angular velocity other than at the center of circulation. Hurricanes are naturally occurring examples of such vortices, due to the relatively calm eye and intense vorticity in the eyewall region. In this paper, the dynamics of this algebraic instability are studied in the context of the near-core dynamics of hurricanes. The longtime asymptotic solution can be written as a sum of three parts: a discrete mode whose amplitude grows in time as t^{1/2}, an excitation of the neutral pseudomode (vortex displacement) that is constant in time, and residual terms that decay in time as t^{-1/2}. A remarkable feature of the solution is that the discrete mode requires the decaying residuals to support its growth; without them it remains constant in amplitude. These residuals are shown to be a collection of sheared vortex-Rossby waves that are trapped in the core of the vortex. The explicit mechanism by which these waves sustain the longtime growth of the instability is investigated. The decaying vortex-Rossby waves are found to continuously amplify the growing discrete mode through the production of perturbation vorticity via interaction with the basic-state vorticity gradient. This is fundamentally different from the classic instability mechanism in barotropic shear flows, often interpreted in terms of discrete, counterpropagating vortex-Rossby waves. The instability manifests itself as a growing wobble of the low-vorticity core of the vortex, resulting in a net transport of high vorticity to the vortex center. A fully nonlinear simulation is performed to study the behavior of the instability as its amplitude becomes large. The steady growth of the instability leads to secondary instabilities and vorticity mixing in the vortex core. The implications for hurricanes of the algebraic instability and its large-amplitude nonlinear dynamics are discussed.

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

Pages (from-to) | 3514-3538 |

Number of pages | 25 |

Journal | Journal of the Atmospheric Sciences |

Volume | 57 |

Issue number | 21 |

State | Published - Nov 2000 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Atmospheric Science

### Cite this

*Journal of the Atmospheric Sciences*,

*57*(21), 3514-3538.

**The algebraic growth of wavenumber one disturbances in Hurricane-like vortices.** / Nolan, David S; Montgomery, M. T.

Research output: Contribution to journal › Article

*Journal of the Atmospheric Sciences*, vol. 57, no. 21, pp. 3514-3538.

}

TY - JOUR

T1 - The algebraic growth of wavenumber one disturbances in Hurricane-like vortices

AU - Nolan, David S

AU - Montgomery, M. T.

PY - 2000/11

Y1 - 2000/11

N2 - An exact solution for the evolution of linearized perturbations of azimuthal wavenumber one on inviscid vortices was previously discovered for nondivergent vorticity dynamics on an f plane. The longtime asymptotics for this exact solution have been shown to allow an algebraic instability with unbounded growth even in the absence of exponentially growing modes. The necessary requirement for this instability is that there exist a local maximum in the basic-state angular velocity other than at the center of circulation. Hurricanes are naturally occurring examples of such vortices, due to the relatively calm eye and intense vorticity in the eyewall region. In this paper, the dynamics of this algebraic instability are studied in the context of the near-core dynamics of hurricanes. The longtime asymptotic solution can be written as a sum of three parts: a discrete mode whose amplitude grows in time as t1/2, an excitation of the neutral pseudomode (vortex displacement) that is constant in time, and residual terms that decay in time as t-1/2. A remarkable feature of the solution is that the discrete mode requires the decaying residuals to support its growth; without them it remains constant in amplitude. These residuals are shown to be a collection of sheared vortex-Rossby waves that are trapped in the core of the vortex. The explicit mechanism by which these waves sustain the longtime growth of the instability is investigated. The decaying vortex-Rossby waves are found to continuously amplify the growing discrete mode through the production of perturbation vorticity via interaction with the basic-state vorticity gradient. This is fundamentally different from the classic instability mechanism in barotropic shear flows, often interpreted in terms of discrete, counterpropagating vortex-Rossby waves. The instability manifests itself as a growing wobble of the low-vorticity core of the vortex, resulting in a net transport of high vorticity to the vortex center. A fully nonlinear simulation is performed to study the behavior of the instability as its amplitude becomes large. The steady growth of the instability leads to secondary instabilities and vorticity mixing in the vortex core. The implications for hurricanes of the algebraic instability and its large-amplitude nonlinear dynamics are discussed.

AB - An exact solution for the evolution of linearized perturbations of azimuthal wavenumber one on inviscid vortices was previously discovered for nondivergent vorticity dynamics on an f plane. The longtime asymptotics for this exact solution have been shown to allow an algebraic instability with unbounded growth even in the absence of exponentially growing modes. The necessary requirement for this instability is that there exist a local maximum in the basic-state angular velocity other than at the center of circulation. Hurricanes are naturally occurring examples of such vortices, due to the relatively calm eye and intense vorticity in the eyewall region. In this paper, the dynamics of this algebraic instability are studied in the context of the near-core dynamics of hurricanes. The longtime asymptotic solution can be written as a sum of three parts: a discrete mode whose amplitude grows in time as t1/2, an excitation of the neutral pseudomode (vortex displacement) that is constant in time, and residual terms that decay in time as t-1/2. A remarkable feature of the solution is that the discrete mode requires the decaying residuals to support its growth; without them it remains constant in amplitude. These residuals are shown to be a collection of sheared vortex-Rossby waves that are trapped in the core of the vortex. The explicit mechanism by which these waves sustain the longtime growth of the instability is investigated. The decaying vortex-Rossby waves are found to continuously amplify the growing discrete mode through the production of perturbation vorticity via interaction with the basic-state vorticity gradient. This is fundamentally different from the classic instability mechanism in barotropic shear flows, often interpreted in terms of discrete, counterpropagating vortex-Rossby waves. The instability manifests itself as a growing wobble of the low-vorticity core of the vortex, resulting in a net transport of high vorticity to the vortex center. A fully nonlinear simulation is performed to study the behavior of the instability as its amplitude becomes large. The steady growth of the instability leads to secondary instabilities and vorticity mixing in the vortex core. The implications for hurricanes of the algebraic instability and its large-amplitude nonlinear dynamics are discussed.

UR - http://www.scopus.com/inward/record.url?scp=0034319003&partnerID=8YFLogxK

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M3 - Article

AN - SCOPUS:0034319003

VL - 57

SP - 3514

EP - 3538

JO - Journals of the Atmospheric Sciences

JF - Journals of the Atmospheric Sciences

SN - 0022-4928

IS - 21

ER -