Stability Of Toroidal Magnetic Field Research Articles

The kink-type instability of toroidal stellar magnetic fields with thermal diffusion

The stability of toroidal magnetic fields in rotating radiative stellar zones is studied for realistic values of both the Prandtl numbers. The two considered models for the magnetic geometry represent fields with odd and even symmetry with respect to the equator. In the linear theory in Boussinesq approximation the resulting complex eigenfrequency (including growth rate and drift rate) are calculated for a given radial wave number of a nonaxisymmetric perturbation with m = 1. The ratio of the Alfvén frequency, ΩA, to the rate of the basic rotation, Ω, controls the eigenfrequency of the solution. For strong fields with ΩA > Ω the solutions do not feel the thermal diffusion. The growth rate runs with ΩA and the drift rate is close to −Ω so that the magnetic pattern will rest in the laboratory system. For weaker fields with ΩA < Ω the growth rate strongly depends on the thermal conductivity. For fields with dipolar parity and for typical values of the heat conductivity, the resulting very small growth rates are almost identical with those for vanishing gravity. For fields with dipolar symmetry, the differential rotation of any stellar radiative zone (like the solar tachocline) is shown as basically stabilizing the instability independent of the sign of the shear. Finally, the current-driven kink-type instability of a toroidal background field is proposed as a model for the magnetism of Ap stars. The recent observation of a lower magnetic field threshold of about 300 Gauss for Ap stars is understood as corresponding to the minimum magnetic field producing the instability. This article is dedicated to Andrew Soward on the occasion of his 65th birthday.

Stability of toroidal magnetic fields in the radiation zone of a star

The stability of a toroidal magnetic field in the rotating radiation zone of a star is analyzed to estimate the maximum possible magnitude of relic fields. Equations for small perturbations are obtained taking into account the finite diffusivity and the stabilizing effect of the subadiabatic stratification. The numerical solution of the eigenvalue problem indicates that the threshold field strength for the onset of instability in the radiation zone of the Sun is about 600 G. This figure sets an upper bound for the strength of the relic field. The assumption that magnetic instabilities are present in the solar radiation zone disagrees with the observed abundance of lithium. Our analysis of joint stability of toroidal field and nonuniform rotation shows that two-dimensional MHD solutions for the solar tachocline are stable against three-dimensional perturbations.

Stability of toroidal magnetic fields in the solar tachocline and beneath

AbstractStability of toroidal magnetic field in a stellar radiation zone is considered for the cases of uniform and differential rotation. In the rigidly rotating radiative core shortly below the tachocline, the critical magnetic field for instability is about 600 G. The unstable disturbances for slightly supercritical fields have short radial scales ∼1 Mm. Radial mixing produced by the instability is estimated to conclude that the internal field of the sun can exceed the critical value of 600 G only marginally. Otherwise, the mixing is too strong and not compatible with the observed lithium abundance. Analysis of joint instability of differential rotation and toroidal field leads to the conclusion that axisymmetric models of the laminar solar tachocline are stable to nonaxisymmetric disturbances. The question of whether sun‐like stars can posses tachoclines is addressed with positive answer for stars with rotation periods shorter than about two months. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Stability of toroidal magnetic fields in rotating stellar radiation zones

Aims. Two questions are addressed as to how strong magnetic fields can be stored in rotating stellar radiation zones without being subjected to pinch-type instabilities and how much radial mixing is produced if the fields are unstable. Methods. Linear equations are derived for weak disturbances of magnetic and velocity fields, which are global in horizontal dimensions but short–scaled in radius. The linear formulation includes the 2D theory of stability to strictly horizontal disturbances as a special limit. The eigenvalue problem for the derived equations is solved numerically to evaluate the stability of toroidal field patterns with one or two latitudinal belts under the influence of rigid rotation. Results. Radial displacements are essential for magnetic instability. It does not exist in the 2D case of strictly horizontal disturbances. Only stable (magnetically modified) r-modes are found in this case. The instability recovers in 3D. The minimum field strength Bmin for onset of the instability and radial scales of the most rapidly growing modes are strongly influenced by finite diffusion, the scales are indefinitely short if diffusion is neglected. The most rapidly growing modes for the Sun have radial scales of about 1 Mm. In the upper part of the solar radiation zone, Bmin � 600 G. The toroidal field can exceed this value only marginally, for otherwise the radial mixing produced by the instability would be too strong to be compatible with the observed lithium abundance.

Stability of toroidal magnetic fields in the solar tachocline

The stability of toroidal magnetic fields in the radiative interior of the Sun is studied in a three-dimensional environment. The underlying differential rotation in the tachocline depends on both latitude and radius. We found stability limits for toroidal fields of only 100 G even in the simplified, incompressible, but stably stratified case. Nonaxisymmetric modes are linearly unstable with a weak influence on the amplitude of the underlying differential rotation. We conclude that the tachocline is not a suitable location for the storage of magnetic fields as strong as 100 kG. Since the results are based on an extrapolation of the numerical computations to lower solar diffusivities, high-resolution calculations at higher magnetic Reynolds numbers are required.

The stability of toroidal magnetic fields with equatorial symmetry: evolution of instabilities

We examine the nonlinear evolution of instabilities of toroidal magnetic fields that are either antisymmetric or symmetric with respect to the equatorial plane. Strong antisymmetric toroidal fields will be present in planetary interiors because of strong differential rotation, and the stability of such fields will be an important consideration for the overall dynamo mechanism. An earlier detailed linear stability analysis has provided the foundation for the calculations, and previous nonlinear stability calculations of z-independent fields form the basis for our numerical approach. The magnetic field strength is parameterised by the Elsasser number, Λ, such that Λ c is the critical field strength, and we use an Ekman number, ϵ = 10 −4. We find that finite amplitude solutions exist for Λ > Λ c, and that no such solutions exist for Λ < Λ c (a weakly nonlinear analysis in the magnetostrophic regime had previously shown that subcritical finite amplitude solutions do exist). By investigating the dominant nonlinear effects, we conclude that our solutions are influenced by the viscosity of the model fluid, and that at ϵ = 10 −4 the solutions are not yet dominated by the geostrophic flow, as we might expect in the small ϵ limit.

The stability of toroidal magnetic fields with equatorial symmetry: implications for the Earth's magnetic field

Geophysical observation suggests that the symmetry of the Earth's magnetic field has been predominantly dipolar over the last 2.5 m.y. Given that such an antisymmetric field will vanish at the equatorial plane, we might expect antisymmetry to be a source of magnetic field instability, as the presence of so-called critical surfaces are robust destabilising features, independent of field morphology. To examine the influence of antisymmetry on the stability of magnetic fields a detailed investigation of the stability of model toroidal fields is presented. Using a cylindrical geometry ( s ∗,ϕ,z ∗ ), a basic model field B(s ∗,z ∗) ϕ is subject to an infinitesimal perturbation which is allowed to grow (or decay) in the presence of B ϕ . The addition of an antisymmetric z-dependence to a previously stable s-dependent field, B′(s ∗) ϕ , causes the new field, B(s ∗,z ∗) ϕ , to destabilise; the addition of an antisymmetric z-dependence to an already unstable field significantly lowers Λ c (where Λ c is a measure of the critical field strength). The addition of antisymmetric z-dependence gives an extra critical surface at the equator ( z ∗ = 0 ), which, in turn, gives an extra degree of freedom for field line reconnection, thus leading to a lower Λ c. The use of a cylindrical geometry permits examination of the spatial structure and complex growth rate of the instability in the large field strength, small ohmic diffusion regime (defined by large Λ). In this regime, the most unstable perturbations are found to belong to the resistive class: their mechanism is reconnection and breaking of field lines near a critical surface. Comparisons between results in cylindrical and spherical geometry in the low field strength regime are favourable. Results are also given for the ideal instability which can exist in the absence of diffusion. Our model shows strong localisation of the ideal mode far from the critical surfaces.

Adiabatic stability of toroidal magnetic fields in differentially rotating stars

Adiabatic stability of toroidal magnetic fields in differentially rotating stars

The Stability of Toroidal Magnetic Fields in Stellar Interiors

view Abstract Citations (27) References (8) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Stability of Toroidal Magnetic Fields in Stellar Interiors Schubert, Gerald Abstract The stability of a toroidal magnetic field, B~, in the radiative zones of stars is investigated In cylindri- cal coordinates ~ a necessary condition for stability, for sufficiently large x/n, is given by -g~(a/az)1n(B~/p) > 0 and [-gm - (2c2/~)](ö/0t~)1n(B~/pz~) > 0, where x is the thermal diffusivity, n the magnetic diffusivity, g the gravitational acceleration, p the density, and c2 the isothermal sound speed. Since g2 is negative or positive according as z is positive or negative, B~/p must be an increasing function of z for positive z and an increasing function of - z for negative z, in order to insure stability. In those regions of the star where g~ > 2c2/w, B~/pzz~ will have to be an increasing function of ~ for the configuration to be stable Where g~ < 2c2/w, stability will require B~j,/pw to decrease with increasing t~. In the absence of a gravitational field, stability requires that (9/ôz)1n(B~/p) = 0 and - (3,/&~) in (B~/p~) > 0 (with no restriction on the magnitude of x/n) Stability criteria are also presented for several interesting limiting situations, e g, the adiabatic one, x = n = 0. Even though a magnetic field configuration may be stable to adiabatic perturbations, the non-zero thermal conductivity of the medium may promote an instability Such a diffusive instability is studied in detail Gravitational potential energy, which under adiabatic conditions normally provides a stabilizing influence (Schwarzschild's stability criterion), can reverse its role and be the source of the magnetic in- stability Publication: The Astrophysical Journal Pub Date: March 1968 DOI: 10.1086/149508 Bibcode: 1968ApJ...151.1099S full text sources ADS |