### Abstract

The propagation properties and stability of wave motions in a spherical fluid shell, rotating uniformly in a co-rotating zonal magnetic field, are studied. Various profils of magnetic field and temperature gradient are examined. The fluid is viscous and electrically and thermally conducting. The analysis is applicable when the rotation rate is very high and the Elsasser number A- 1. (A measures the ratio of Lorentz to Coriolis forces). The wave motions occur in the form of annular cylindrical cells whose thickness is determined by the rotation rate and magnetic field amplitude (Eltayeb and Kumar, 1977). It is shown that an infinity of modes exists and both westward and eastward phase propagation is possible. The sloping boundary of the spherical shell tends to oppose westward propagation by imparting an eastward phase speed, of the same magnitude for all modes of the same wavenumber, whose magnitude is proportional to the colatitude of the annular cell. The infinity of modes can be divided into two distinct categories depending on their parity with respect to the equatorial plane. The even (odd) solution is associated with a temperature perturbation and axial vorticity which are symmetric (anti-symmetric) with respect to the equatorial plane. Both categories of solutions are present outside the annular cylindrical surface, C touching the inner core at its equator but only the analytic continuation of the odd mode is present inside Cc. This result is used to clarify the relationship between wave motions in thin and thick spherical shells. The stability of the waves is investigated and the effects of the profiles of the ambient magnetic field and basic temperature gradient as well as the presence of the inner core on the preferred mode of convection, are studied. The influence of the magnetic number q(= k/n, where k and n are the thermal and magnetic diffusivities, respectively) on the location and direction of phase propagation of the unstable wave is also examined. In particular the changeover of the critical mode from an eastward to a westward propagating magnetic mode at a value qo of q is examined in detail. It is found that qc, which signifies a change from buoyancy Rossby waves (q

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

Pages (from-to) | 211-239 |

Number of pages | 29 |

Journal | Geophysical and Astrophysical Fluid Dynamics |

Volume | 67 |

Issue number | 1-4 |

DOIs | |

Publication status | Published - Dec 1 1992 |

### Fingerprint

### Keywords

- Hydromagnetic-rotating fluids
- planetary dynamos
- Rossby buoyancy waves
- slow MHD waves

### ASJC Scopus subject areas

- Geochemistry and Petrology
- Geophysics
- Computational Mechanics
- Astronomy and Astrophysics
- Mechanics of Materials

### Cite this

**The Propagation and Stability of Linear Wave Motions in Rapidly Rotating Spherical Shells : Weak Magnetic Fields.** / Eltayeb, I. A.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The Propagation and Stability of Linear Wave Motions in Rapidly Rotating Spherical Shells

T2 - Weak Magnetic Fields

AU - Eltayeb, I. A.

PY - 1992/12/1

Y1 - 1992/12/1

N2 - The propagation properties and stability of wave motions in a spherical fluid shell, rotating uniformly in a co-rotating zonal magnetic field, are studied. Various profils of magnetic field and temperature gradient are examined. The fluid is viscous and electrically and thermally conducting. The analysis is applicable when the rotation rate is very high and the Elsasser number A- 1. (A measures the ratio of Lorentz to Coriolis forces). The wave motions occur in the form of annular cylindrical cells whose thickness is determined by the rotation rate and magnetic field amplitude (Eltayeb and Kumar, 1977). It is shown that an infinity of modes exists and both westward and eastward phase propagation is possible. The sloping boundary of the spherical shell tends to oppose westward propagation by imparting an eastward phase speed, of the same magnitude for all modes of the same wavenumber, whose magnitude is proportional to the colatitude of the annular cell. The infinity of modes can be divided into two distinct categories depending on their parity with respect to the equatorial plane. The even (odd) solution is associated with a temperature perturbation and axial vorticity which are symmetric (anti-symmetric) with respect to the equatorial plane. Both categories of solutions are present outside the annular cylindrical surface, C touching the inner core at its equator but only the analytic continuation of the odd mode is present inside Cc. This result is used to clarify the relationship between wave motions in thin and thick spherical shells. The stability of the waves is investigated and the effects of the profiles of the ambient magnetic field and basic temperature gradient as well as the presence of the inner core on the preferred mode of convection, are studied. The influence of the magnetic number q(= k/n, where k and n are the thermal and magnetic diffusivities, respectively) on the location and direction of phase propagation of the unstable wave is also examined. In particular the changeover of the critical mode from an eastward to a westward propagating magnetic mode at a value qo of q is examined in detail. It is found that qc, which signifies a change from buoyancy Rossby waves (q

AB - The propagation properties and stability of wave motions in a spherical fluid shell, rotating uniformly in a co-rotating zonal magnetic field, are studied. Various profils of magnetic field and temperature gradient are examined. The fluid is viscous and electrically and thermally conducting. The analysis is applicable when the rotation rate is very high and the Elsasser number A- 1. (A measures the ratio of Lorentz to Coriolis forces). The wave motions occur in the form of annular cylindrical cells whose thickness is determined by the rotation rate and magnetic field amplitude (Eltayeb and Kumar, 1977). It is shown that an infinity of modes exists and both westward and eastward phase propagation is possible. The sloping boundary of the spherical shell tends to oppose westward propagation by imparting an eastward phase speed, of the same magnitude for all modes of the same wavenumber, whose magnitude is proportional to the colatitude of the annular cell. The infinity of modes can be divided into two distinct categories depending on their parity with respect to the equatorial plane. The even (odd) solution is associated with a temperature perturbation and axial vorticity which are symmetric (anti-symmetric) with respect to the equatorial plane. Both categories of solutions are present outside the annular cylindrical surface, C touching the inner core at its equator but only the analytic continuation of the odd mode is present inside Cc. This result is used to clarify the relationship between wave motions in thin and thick spherical shells. The stability of the waves is investigated and the effects of the profiles of the ambient magnetic field and basic temperature gradient as well as the presence of the inner core on the preferred mode of convection, are studied. The influence of the magnetic number q(= k/n, where k and n are the thermal and magnetic diffusivities, respectively) on the location and direction of phase propagation of the unstable wave is also examined. In particular the changeover of the critical mode from an eastward to a westward propagating magnetic mode at a value qo of q is examined in detail. It is found that qc, which signifies a change from buoyancy Rossby waves (q

KW - Hydromagnetic-rotating fluids

KW - planetary dynamos

KW - Rossby buoyancy waves

KW - slow MHD waves

UR - http://www.scopus.com/inward/record.url?scp=84972866698&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84972866698&partnerID=8YFLogxK

U2 - 10.1080/03091929208201845

DO - 10.1080/03091929208201845

M3 - Article

AN - SCOPUS:84972866698

VL - 67

SP - 211

EP - 239

JO - Geophysical and Astrophysical Fluid Dynamics

JF - Geophysical and Astrophysical Fluid Dynamics

SN - 0309-1929

IS - 1-4

ER -