Hydromagnetic waves in rapidly rotating spherical shells generated by magnetic toroidal decay modes

Zhang, K.; Fearn, David R.

doi:10.1080/03091929408203679

Cited by 33 publications

(20 citation statements)

References 20 publications

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“…Moffatt, 1978, pp. 38,39), which is consistent with the observed dipolar field at leading order, and BT is assumed to be the toroidal decay mode of the largest scale satisfying boundary condition (2.9) (Zhang and Fearn, 1994), and ~~~~~~~ = 1 and ~~~~~~~ = 1. The most unstable modes are always associated with the lower azimuthal wavenumbers (Feam, 1993).…”

Section: Modelswithcomparabletoroidalandpoloidalfieldmentioning

confidence: 78%

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Estimates for the effective electrical conductivity of the core in the interior of Jupiter and Saturn

1996

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In the deep interior of the giant planets Jupiter and Saturn, ordinary hydrogen and helium are transformed into a conducting metallic liquid at extremely high pressure. It is likely that the giant planets' observed magnetic field is constantly generated in the metallic fluid core by magnetohydrodynamic processes, converting mechanic energy in the form of convection into magnetic energy. The maximum strength of their magnetic fields is likely to be limited by magnetic field instabilities which convert the magnetic energy back into convection. The parameter which governs the occurrence of magnetic instabilities is the Elsasser number, A = B2a/2Q, where B is the field strength, 0 is the electrical conductivity, Q is the rotation rate and p is the density. Since magnetic instability will be very active when A exceeds a critical value A, N 10 (the precise value depending on the magnetic field distribution), this imposes an upper bound on the effective electrical conductivity of the metallic fluid which comprises the bulk of Jupiter's interior and much of Saturn's Stability calculations including both toroidal (model) and poloidal (observed) components of the magnetic field in a rapidly rotating spherical shell, have been performed. The most stable configuration of the field is when the poloidal component of field is strong and the toroidal field is weak, in this case we obtain an upper bound for electrical conductivity of 0-3x 106S/m; while the most unstable configuration of the field is when the toroidal and poloidal fields are comparable, giving rise to UM -3 x lo5 S/m. The implications of the results for general dynamo theory are also discussed.

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Section: Modelswithcomparabletoroidalandpoloidalfieldmentioning

confidence: 78%

“…This is important because of the likely formation of boundary layers in the limit E + 0 (Zhang and Fearn, 1994). It is found that the effect of those boundary layers is always passive and does not prevent us from achieving a good accurate solution for an asymptotically small Ekman number that is relevant to the metallic core.…”

Section: Accuracyofthesolutionsmentioning

confidence: 99%

Estimates for the effective electrical conductivity of the core in the interior of Jupiter and Saturn

1996

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“…One of the primary objectives is to test the approach of magnetic field stability analysis. It has been argued that stability analysis of a toroidal field can provide an important upper bound and other information on the invisible toroidal field (Fearn, 1993;Zhang and Fearn, 1994). Unlike the stability results for a toroidal field, the stability properties of a poloidal field can be checked with observation.…”

Section: Introductionmentioning

confidence: 96%

“…A particularly interesting feature is that, for the most unstable mode, contours of the non-axisymmetric azimuthal flow are closely aligned with the basic axisymmetric poloidal magnetic field lines. As a result, the amplitude of the azimuthal component of the instability is smaller than or comparable with that of the poloidal component, in contrast with the instabilities generated by toroidal decay modes (Zhang and Fearn, 1994). It is shown, by examining the same system with and without fluid inertia, that fluid inertia plays a secondary role when the magnetic Taylor number T , 2 lo5.…”

mentioning

confidence: 93%

“…An efficient and useful way to study some important aspects of the dynamics of the Earth's fluid core without solving the extremely difficult problem of geodynamo, is to examine the stability of magnetic fields likely to be generated by the geodynamo (Fearn, 1993). Though it is the poloidal component that can be observed, past investigations of magnetic stabilities have concentrated almost exclusively on the toroidal component (see for example Acheson, 1972Acheson, 1978Fearn and Weiglhofer, 199 1 a, b, 1992;Schubert, 1968;Zhang and Fearn, 1994). For this, the different types of instability together with the conditions required for their existence have been well studied.…”

Section: Introductionmentioning

confidence: 98%

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Hydromagnetic waves in rapidly rotating spherical shells generated by poloidal decay modes

Zhang

Fearn

1995

Geophysical & Astrophysical Fluid Dynamics

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This paper presents the first attempt to examine the stability of a poloidal magnetic field in a rapidly rotating spherical shell of electrically conducting fluid. We find that a steady axisymmetric poloidal magnetic field loses its stability to a non-axisymmetric perturbation when the Elsasser number A based on the maximum strength of the field exceeds a value about 20. Comparing this with observed fields, we find that, for any reasonable estimates of the appropriate parameters in planetary interiors, our theory predicts that all planetary poloidal fields are stable, with the possible exception of Jupiter. The present study therefore provides strong support for the physical relevance of magnetic stability analysis to planetary dynamos. We find that the fluid motions driven by magnetic instabilities are characterized by a nearly two-dimensional columnar structure attempting to satisfy the Proudman-Taylor theorm. This suggests that the most rapidly growing perturbation arranges itself in such a way that the geostrophic condition is satisfied to leading order. A particularly interesting feature is that, for the most unstable mode, contours of the non-axisymmetric azimuthal flow are closely aligned with the basic axisymmetric poloidal magnetic field lines. As a result, the amplitude of the azimuthal component of the instability is smaller than or comparable with that of the poloidal component, in contrast with the instabilities generated by toroidal decay modes (Zhang and Fearn, 1994). It is shown, by examining the same system with and without fluid inertia, that fluid inertia plays a secondary role when the magnetic Taylor number T , 2 lo5. We find that the direction of propagation of hydromagnetic waves driven by the instability is influenced strongly by the size of the inner core.

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1996

Surv Geophys

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Hydromagnetic waves in rapidly rotating spherical shells generated by magnetic toroidal decay modes

Cited by 33 publications

References 20 publications

Estimates for the effective electrical conductivity of the core in the interior of Jupiter and Saturn

Estimates for the effective electrical conductivity of the core in the interior of Jupiter and Saturn

Hydromagnetic waves in rapidly rotating spherical shells generated by poloidal decay modes

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