Zur Dynamotheorie kosmischer Magnetfelder. I. Gleichungen für sphärische Dynamomodelle

This paper deals with the mathematical treatment of special models of hydromagnetic dynamos. For the models considered here the conducting medium occupies a spherical region surrounded by vacuum. Both laminar and turbulent dynamos are included. The partial vector differential equations governing the models are transformed into an infinite set of differential equations for scalar functions by applying a method previously used by BULLARD and GELLMAN and on the basis of a representation of vector fields as a sum of a poloidal and a toroidal part. The scalar functions depend only on a radial coordinate and possibly on the time. In both the stationary and the periodic case, an infinite set of ordinary differential equations results which may be treated numerically. A series of relations for computing various dynamo models is prepared.     

The dynamics of a toroidal magnetic ring

Solving the nonlinear partial differential equations of magnetohydrodynamics numerically, we examine (1) the time development of a purely toroidal magnetic field (a magnetic ring) and (2) the interaction of a magnetic ring with a poloidal magnetic field. Axisymmetry and incompressibility are assumed. Parameters are chosen to correspond to photospheric conditions. In case (1), the magnetic ring contracts to the axis and then splits in two with one ring travelling up along the axis and the other down. In case (2), a large toroidal velocity field is generated which has opposite direction of flow above and below the magnetic ring. The magnetic and flow patterns of case (2) may persist with little change for a relatively long time. We conjecture that toroidal magnetic fields may be involved in the bright rings of sunspots or in the dynamics of spicules.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Gravitational collapse of polytropic, magnetized, filamentary clouds

For the case in which the gas of a magnetized filamentary cloud obeys a polytropic equation of state, gravitational collapse of the cloud is studied using a simplified model. We concentrate on the radial distribution and restrict ourselves to a purely toroidal magnetic field. If the axial motions and poloidal magnetic fields are sufficiently weak, we could reasonably expect our solutions to be a good approximation. We show that while the filament experiences gravitational condensation and the density at the centre increases, the toroidal flux-to-mass ratio remains constant. A series of spatial profiles of density, velocity and magnetic field for several values of the toroidal flux-to-mass ratio and the polytropic index, is obtained numerically and discussed.     

On axisymmetric hydromagnetic equilibria

The physical characteristics of possible axisymmetric equilibria are examined on the basis of the integrals of hydromagnetic equations. It is shown for nearly spherical configurations that a surface differential rotation is possible only in the absence of a meridional circulation with either purely toroidal or purely poloidal magnetic field. In the presence of a meridional circulation, it is shown that no surface rotation or constant rotation is possible if the magnetic field is purely toroidal, and that no rotation is possible if the magnetic field is purely poloidal. A brief discussion is given on the possible solutions including the case of stellar winds with force-free magnetic fields.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Effect of Induced Toroidal Rotation on Poloidal Rotation and Ion Heat Conductivity of Tokamak Edge Plasmas

The transport processes in edge (collisional) plasmas of tokamaks with smooth profiles of macroscopic plasma parameters and induced poloidal and toroidal plasma flows, are considered. The toroidal and poloidal velocities of particles, the radial electric field and the ion heat flux are derived. It is shown that forces, induced by radio frequency waves, plasma turbulence or neutral beam injection, can be used to control the poloidal and toroidal plasma velocities, as well as ion heat conductivity, in a wide range of these values.     

The periods of stellar models with linked poloidal-toroidal magnetic fields

First-order perturbation theory results for the changes in pulsation frequencies of a Cowling model star containing a magnetic field with both poloidal and toroidal components are presented. A toroidal field large enough to stabilize the poloidal field may reverse the sign of the frequency change caused by a purely poloidal field for some modes, including the fundamental radial mode.     

微椭弹性地球运动的一个边界条件的推导

在研究地球章动或潮汐理论时，常常需要对均匀自转、微椭、弹性、自引力的地球的运动方程组积分，并通过选取一组恰当的边界条件来定解。在阶扁率近似下，先将椭球形参考边界上一个有关形变的连续量转化到等效球面上，然后作广义面球谐函数展开进行标量化，并分解为球形与环形部分，截断后可导出3个标量常微分形式的边界条件。     

Cooling of neutron stars with strong toroidal magnetic fields

We present models of temperature distribution in the crust of a neutron star in the presence of a strong toroidal component superposed to the poloidal component of the magnetic field. The presence of such a toroidal field hinders heat flow toward the surface in a large part of the crust. As a result, the neutron star surface presents two warm regions surrounded by extended cold regions and has a thermal luminosity much lower than in the case the magnetic field is purely poloidal. We apply these models to calculate the thermal evolution of such neutron stars and show that the lowered photon luminosity naturally extends their life-time as detectable thermal X-ray sources. Work partially supported by UNAM-DGAPA grant #IN119306.     

Axisymmetric stars with both poloidal and toroidal magnetic fields

Chandrasekhar and Prendergast have established a result which has been assumed to imply that axisymmetric stars with an internal toroidal magnetic field should have zero external poloidal field. By considering mildly singular functions, the range of solutions is increased, and models can then be constructed which have toroidal and poloidal fields in the interior and a non-zero, external, poloidal field. Both the magnetic field and its associated current are continuous everywhere.     

A new numerical scheme for structures of rotating magnetic stars

We have developed a new numerical scheme for obtaining structures of rapidly rotating stars with strong magnetic fields. In our scheme, both poloidal and toroidal magnetic fields can be treated for stars with compressibility and infinite conductivity. By introducing the vector potential and its integral representation, we can treat the boundary condition for the magnetic fields across the surface properly. We show structures and distributions of magnetic fields as well as the distributions of the currents of rotating magnetic polytropic stars with polytropic index   N = 1.5  . The shapes of magnetic stars are oblate as long as the magnetic vector potential decreases as 1/ r when   r →∞  . For extremely strong magnetic fields, equilibrium configurations can be of toroidal shapes.     

Simulating the generation of the solar toroidal magnetic field by differential rotation

Within the kinematic dynamo theory, we construct a mathematical model for the evolution of the solar toroidal magnetic field, excited by the differential rotation of the convective zone in the presence of a poloidal field of a relic origin. We use a velocity profile obtained by decoding the data of helioseismological experiments. For the model of ideal magnetic hydrodynamics, we calculate the latitudinal profiles of the increasing-with-time toroidal field at different depths in the solar convection zone. It is found that, in the region of differential rotation, the excited toroidal field shows substantial fluctuations in magnitude with depth. Based on the simulations results, we propose an explanation for the “incorrect polarity” of magnetic bipolar sunspot groups in solar cycles.     

Axisymmetric magnetic fields in stars: relative strengths of poloidal and toroidal components

In this third paper in a series on stable magnetic equilibria in stars, I look at the stability of axisymmetric field configurations and, in particular, the relative strengths of the toroidal and poloidal components. Both toroidal and poloidal fields are unstable on their own, and stability is achieved by adding the two together in some ratio. I use Tayler's stability conditions for toroidal fields and other analytic tools to predict the range of stable ratios and then check these predictions by running numerical simulations. If the energy in the poloidal component as a fraction of the total magnetic energy is written as E_p / E , it is found that the stability condition is a ( E / U ) < E_p / E ≲ 0.8 where E /U is the ratio of magnetic to gravitational energy in the star and a is some dimensionless factor whose value is of order 10 in a main-sequence star and of order 10³ in a neutron star. In other words, whilst the poloidal component cannot be significantly stronger than the toroidal, the toroidal field can be very much stronger than the poloidal–given that in realistic stars we expect E / U < 10⁻⁶. The implications of this result are discussed in various contexts such as the emission of gravitational waves by neutron stars, free precession and a 'hidden' energy source for magnetars.     

Convective stability in magnetic stars

Dynamical stability of a static axisymmetrical magnetic star with respect to high-order modes of oscillation is investigated by means of the energy method, neglecting the Eulerian perturbation of gravity. The magnetic field is assumed to be continuous across the surface of the star and its first-order spatial derivatives, but it may have both toroidal and poloidal components.The second variation of the potential energy is written in a way which, in the case of apurely toroidal field, and for axisymmetrical and non-axisymmetrical modes, yields Tayler's local stability criteria which are necessary and sufficient conditions for convective stability, and in the case of ageneral field yields a single local stability criterion, which is a sufficient condition for convective stability.     

The Scalar Boundary Conditions For The Motion Of The Elastic Earth To Second Order In Ellipticity

The scalar equations of infinitesimal elastic gravitational motion for a rotating, slightly elliptical Earth are always used to study the Earth's nutation and tides theoretically, while the determination of the integration of the equations depends, to a certain extent, on the choice of a set of appropriate boundary conditions. In this paper, a continuity quantity related to the displacement is first transformed from the elliptical reference boundary to the corresponding effective spherical domain, and then converted from a vector (or tensor) form to a scalar form by generalized surface spherical harmonics expansion. All the related components, including the displacement vector field (or the stress tensor field), are then decomposed into the poloidal and toroidal field using the symmetry restrictions on the normal mode eigenfunctions. After truncation, the boundary conditions are finally derived, in a scalar ordinary differential form. The process of the derivation is second order in ellipticity and in full detail. Moreover, the other boundary conditions are also presented as second order in ellipticity at the end of this paper. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Rossby-wave dynamo for the Sun,II

To make the analysis more tractable, we simplify the equations of Part I to apply to two superposed layers of fluid, with horizontal variations in the motion and magnetic field represented by a small number of Fourier harmonics. The resulting set of eighteen ordinary nonlinear differential equations in time for the Fourier amplitudes is integrated numerically. We analyze in detail the dynamo action from a typical Rossby wave motion and compare it with the solar cycle.The field reversal process is similar in some respects to that put forth by Babcock. Toroidal fields are dragged up by vertical motions in the Rossby waves to form large-scale vertical fields, whose polarities alternate with longitude roughly like bipolar magnetic regions. Vertical fields of preferentially one polarity are carried toward the pole by the meridional motion in the wave to form an axisymmetric poloidal field. This poloidal field is then stretched out by the differential rotation into a new toroidal field of the opposite sign from the original. The poloidal field changes sign when the toroidal and bipolar region like fields are maximum, and vice versa.For the case studied, the reversal period is too short ( 2 years) and the poloidal fields too large ( 40 G) for the sun. Improvements for the model are discussed.Part I has been published in Solar Phys. 8, 316.     

Thermal generation of magnetic fields with radiation pressure in rotating stars

It has been suggested by Biermann that in rotating stars the electron partial pressure could generate a toroidal magnetic field of a considerable strength. However, Mestel and Roxburgh have shown recently that the generation of such a toroidal magnetic field could almost completely be suppressed when a weak primodial poloidal magnetic field exists in the star. In this paper it is shown that a toroidal magnetic field of a moderate strength could be generated even in the presence of a primodial poloidal magnetic field, if the effect of radiation pressure is taken into consideration. This considered mechanism is effective for moderately massive stars, and numerical estimate indicates that in A type stars a toroidal magnetic field of the order of a thousand gauss can be generated near the surface within the time scale of the evolution of the star.Visiting Scientist to the High Altitude Observatory on leave of absence from the Department of Astronomy, University of Tokyo, Japan.     

Magnetization of celestial bodies with special application to the primeval Earth and Moon

The magnetic fields of celestial bodies are usually supposed to be due to a ‘hydromagnetic dynamo’. This term refers to a number of rather speculative processes which are supposed to take place in the liquid core of a celestial body. In this paper we shall follow another approach which is more closely connected with hydromagnetic processes well-known from the laboratory, and hence basically less speculative. The paper should be regarded as part of a general program to connect cosmical phenomena with phenomena studied in the laboratory. As has been demonstrated by laboratory experiments, a poloidal magnetic field may be increased by the transfer of energy from a toroidal magnetic field through kink instability of the current system. This mechanism can be applied to the fluid core of a celestial body. Any differential rotation will produce a toroidal field from an existing poloidal field, and the kink instability will feed toroidal energy back to the poloidal field, and hence amplify it. In the Earth-Moon system the tidal braking of the Earth's mantle acts to produce a differential angular velocity between core and mantle. The braking will be transferred to the core by hydromagnetic forces which at the same time give rise to a strong magnetic field. The strength of the field will be determined by the rate of tidal braking. It is suggested that the magnetization of lunar rocks from the period ?4 to ?3 Gyears derives from the Earth's magnetic field. As the interior of the Moon immediately after accretion probably was too cool to be melted, the Moon could not produce a magnetic field by hydromagnetic effects in its core. The observed lunar magnetization could be produced by such an amplified Earth field even if the Moon never came closer than 10 or 20 Earth's radii. This hypothesis might be checked by magnetic measurements on the Earth during the same period.     

A dipole field model for axisymmetric alfvén waves with finite ionosphere conductivities

An axisymmetric model for approximate solution of the magnetospheric Alfvén wave problem at latitudes above the plasmapause is proposed, in which a realistic dipole geometry is combined with finite anisotropic ionosphere conductivities, thus bringing together various ideas of previous authors. It is confirmed that the axisymmetric toroidal and poloidal modes interact via the ionospheric Hall effect, and an approximate method of solution is suggested using previously derived closed solutions of the uncoupled wave equations.A solution for zero Hall conductivity is obtained, which consists of sets of independent shell oscillations, regardless of the magnitude of the Pedersen conductivity. One set reduces to the classical solutions for infinite Pedersen conductivity, while another predicts a new set of harmonics of a quarter-wave fundamental, with longer eigenperiods than the classical solutions for a given L-shell.     

Hydrodynamic model of spatial and temporal variations of poloidal and Toroidal components of three-dimensional solar flows

Numerical simulation results of the global solar flows are presented. The conclusion on the common hydrodynamic nature of the torsional oscillations and spatial-temporal variations of the poloidal flow was made. Both processes were shown to be toroidal and poloidal components of a single hydrodynamic oscillatory flow that is asymmetric about the solar equator. The basis for these processes is the physical mechanism of the loss of stability of the solar differential rotation.     

The effect of rotation on the luminosity of magnetic stars

The effect of rotation and a general magnetic field on the luminosity, radius, and effective temperature of the upper Main-Sequence stars has been investigated using a perturbation analysis. The magnetic field profile prevailing inside the star is assumed to have both poloidal and toroidal components. The case of constant as well as differential rotation is admitted. Model calculations indicate that these stellar parameters modify considerably as a result of coupling between rotation and the magnetic field.