### Abstract

To study a limit of validity of balanced models, stability of a zonal jet is investigated both linearly and nonlinearly in an f-plane shallow water system for a wide range of parameter. It is shown that quasi-geostrophic approximation gives not only a good estimation of maximum growth rate for high Rossby number, Ro, but also is valid even in the nonlinear phase of instability for high Ro as long as Froude number, Fr, is low. While the maximum growth rate of unstable modes is well estimated by the quasi-geostrophic approximation, dominant balance is different between high and low Ro. In the low Ro regime (Ro < 5), geostrophic balance is dominant in the perturbation field, the ratio ∥ φ ∥ / ∥ ψ ∥ of the amplitudes of divergent flow to that of rotational flow is proportional to Fr^{2} / Ro, where φ and ψ are velocity potential and streamfunction, respectively. On the other hand, in the high Ro regime (Ro > 5), cyclostrophic balance with basic shear is dominant, ∥ φ ∥ / ∥ ψ ∥ ∝ Fr^{2}. Considering that the barotropic instability is caused by the resonance of neutral Rossby wave modes, we can explain the difference of the ratio in each regime. Using the ratio ∥ φ ∥ / ∥ ψ ∥ being small, different approximation of the linear shallow water equations for each regime is deduced. Properties of the linear unstable modes are explained with these approximations.

Original language | English |
---|---|

Pages (from-to) | 353-377 |

Number of pages | 25 |

Journal | Fluid Dynamics Research |

Volume | 39 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2007 May |

Externally published | Yes |

### Fingerprint

### Keywords

- Balanced models
- Jet
- Linear stability analysis
- Nonlinear simulation
- Quasi-geostrophic approximation
- Shallow water system
- Shear instability

### ASJC Scopus subject areas

- Mechanical Engineering
- Statistical and Nonlinear Physics

### Cite this

*Fluid Dynamics Research*,

*39*(5), 353-377. https://doi.org/10.1016/j.fluiddyn.2006.07.004

**Balance regimes for the stability of a jet in an f-plane shallow water system.** / Sugimoto, Norihiko; Ishioka, Keiichi; Yoden, Shigeo.

Research output: Contribution to journal › Article

*Fluid Dynamics Research*, vol. 39, no. 5, pp. 353-377. https://doi.org/10.1016/j.fluiddyn.2006.07.004

}

TY - JOUR

T1 - Balance regimes for the stability of a jet in an f-plane shallow water system

AU - Sugimoto, Norihiko

AU - Ishioka, Keiichi

AU - Yoden, Shigeo

PY - 2007/5

Y1 - 2007/5

N2 - To study a limit of validity of balanced models, stability of a zonal jet is investigated both linearly and nonlinearly in an f-plane shallow water system for a wide range of parameter. It is shown that quasi-geostrophic approximation gives not only a good estimation of maximum growth rate for high Rossby number, Ro, but also is valid even in the nonlinear phase of instability for high Ro as long as Froude number, Fr, is low. While the maximum growth rate of unstable modes is well estimated by the quasi-geostrophic approximation, dominant balance is different between high and low Ro. In the low Ro regime (Ro < 5), geostrophic balance is dominant in the perturbation field, the ratio ∥ φ ∥ / ∥ ψ ∥ of the amplitudes of divergent flow to that of rotational flow is proportional to Fr2 / Ro, where φ and ψ are velocity potential and streamfunction, respectively. On the other hand, in the high Ro regime (Ro > 5), cyclostrophic balance with basic shear is dominant, ∥ φ ∥ / ∥ ψ ∥ ∝ Fr2. Considering that the barotropic instability is caused by the resonance of neutral Rossby wave modes, we can explain the difference of the ratio in each regime. Using the ratio ∥ φ ∥ / ∥ ψ ∥ being small, different approximation of the linear shallow water equations for each regime is deduced. Properties of the linear unstable modes are explained with these approximations.

AB - To study a limit of validity of balanced models, stability of a zonal jet is investigated both linearly and nonlinearly in an f-plane shallow water system for a wide range of parameter. It is shown that quasi-geostrophic approximation gives not only a good estimation of maximum growth rate for high Rossby number, Ro, but also is valid even in the nonlinear phase of instability for high Ro as long as Froude number, Fr, is low. While the maximum growth rate of unstable modes is well estimated by the quasi-geostrophic approximation, dominant balance is different between high and low Ro. In the low Ro regime (Ro < 5), geostrophic balance is dominant in the perturbation field, the ratio ∥ φ ∥ / ∥ ψ ∥ of the amplitudes of divergent flow to that of rotational flow is proportional to Fr2 / Ro, where φ and ψ are velocity potential and streamfunction, respectively. On the other hand, in the high Ro regime (Ro > 5), cyclostrophic balance with basic shear is dominant, ∥ φ ∥ / ∥ ψ ∥ ∝ Fr2. Considering that the barotropic instability is caused by the resonance of neutral Rossby wave modes, we can explain the difference of the ratio in each regime. Using the ratio ∥ φ ∥ / ∥ ψ ∥ being small, different approximation of the linear shallow water equations for each regime is deduced. Properties of the linear unstable modes are explained with these approximations.

KW - Balanced models

KW - Jet

KW - Linear stability analysis

KW - Nonlinear simulation

KW - Quasi-geostrophic approximation

KW - Shallow water system

KW - Shear instability

UR - http://www.scopus.com/inward/record.url?scp=34247232522&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34247232522&partnerID=8YFLogxK

U2 - 10.1016/j.fluiddyn.2006.07.004

DO - 10.1016/j.fluiddyn.2006.07.004

M3 - Article

AN - SCOPUS:34247232522

VL - 39

SP - 353

EP - 377

JO - Fluid Dynamics Research

JF - Fluid Dynamics Research

SN - 0169-5983

IS - 5

ER -