The effect of gravity on the stability of a line-tied coronal magnetohydrostatic equilibrium

Abstract

The magnetohydrodynamic stability of a class of magnetohydrostatic equilibria is investigated. The effect of gravity is included as well as the stabilising influence of the dense photospheric line-tying.

Although the two-dimensional equilibria exhibit a catastrophe point, when the ratio of plasma pressure to magnetic pressure exceeds a critical value, arcade structures, with both footpoints connected to the photosphere, become unstable to three-dimensional disturbances before the catastrophe point is reached.

Numerical results for field lines that are open into the solar corona suggest that they are completely stable. Although there is no definite proof of stability, this would allow the point of non-equilibrium to be reached.     

Equilibrium of magnetic fields with arbitrary interweaving of the lines of force i. discontinuities in the torsion

Abstract

This paper considers the static force-free equilibrium V×B=αB of a magnetic field in which all of the lines of force connect without knotting between parallel planes. The field is formed by continuous deformation from an initial uniform field, and is conveniently described in terms of the scalar function ψ, which is effectively the stream function for the incompressible wrapping and interweaving of the lines of force, and the scalar function θ, which describes the local compression and expansion. Equilibrium requires satisfaction of two independent equations (the third equation defines α), which cannot be accomplished without the full freedom of both functions ψ and θ. It is shown by integration along the characteristics of the equilibrium equations that, when ψ is predetermined by an arbitrary winding pattern, there appear discontinuities in α. Discontinuities in α have discontinuities in the field (i.e. current sheets) associated with them.

We expect such discontinuities to be produced in the magnetic fields extending outward from the convecting surfaces of the cooler stars.     

The rapid dissipation of magnetic fields in highly conducting fluids

Abstract

This paper treats the dynamical conditions that obtain when long straight parallel twisted flux tubes in a highly conducting fluid are packed together in a broad array. It is shown that there is generally no hydrostatic equilibrium. In place of equilibrium there is a dynamical nonequilibrium, leading to neutral point reconnection and progressive coalescence of neighboring tubes (with the same sense of twisting), forming tubes of larger diameter and reduced twist. The magnetic energy in the twisting of each tube declines toward zero, dissipated into small-scale motions of the fluid and thence into heat.

The physical implications are numerous. For instance, it has been suggested that the subsurface magnetic field of the sun is composed of close-packed twisted flux tubes. Any such structures are short lived, at best.

The footpoints of the filamentary magnetic fields above bipolar magnetic regions on the sun are continually shuffled and rotated by the convection, so that the fields are composed of twisted rubes. The twisting and mutual wrapping is converted directly into fluid motion and heat by the dynamical nonequilibrium, so that the work done by the convection of the footpoints goes directly into heating the corona above. This theoretical result is the final step, then, in understanding the assertion by Rosner, Tucker, and Valana, and others, that the observed structure of the visible corona implies that it is heated principally by direct dissipation of the supporting magnetic field. It is the dynamical nonequilibrium that causes the dissipation, in spite of the high electrical conductivity. It would appear that any bipolar magnetic field extending upward from a dense convective layer into a tenuous atmosphere automatically produces heating, and a corona of some sort, in the sun or any other convective star.     

A class of analytic solutions of the magnetic force-free field equations

Abstract

Formation of electric current sheets in the corona is thought to play an important role in solar flares, prominences and coronal heating. It is therefore of great interest to identify magnetic field geometries whose evolution leads to variations in B over small length-scales. This paper considers a uniform field B ₀[zcirc], line-tied to rigid plates z = ±l, which are then subject to in-plane displacements modeling the effect of photospheric motion. The force-free field equations are formulated in terms of field-line displacements, and when the imposed plate motion is a linear function of position, these reduce to a 4 × 4 system of nonlinear, second-order ordinary differential equations. Simple analytic solutions are derived for the cases of plate rotation and shear, which both tend to form singularities in certain parameter limits. In the case of plate shear there are two solution branches—a simple example of non-uniqueness.     

Tangential discontinuities and the optical analogy for stationary fields. v. formal integration of the force-free field equations

Abstract

This paper demonstrates the appearance of tangential discontinuities in deformed force-free fields by direct integration of the field equation ? x B = αB. To keep the mathematics tractable the initial field is chosen to be a layer of linear force-free field B_x = + B ₀cosqz, B_y = — B ₀sinqz, B_z = 0, anchored at the distant cylindrical surface ? = (x ² + y ²)^1/2 = R and deformed by application of a local pressure maximum of scale l centered on the origin x = y = 0. In the limit of large R/l the deformed field remains linear, with α = q[1 + O(l ²/R ²)]. The field equations can be integrated over ? = R showing a discontinuity extending along the lines of force crossing the pessure maximum. On the other hand, examination of the continuous solutions to the field equations shows that specification of the normal component on the enclosing boundary ? = R completely determines the connectivity throughout the region, in a form unlike the straight across connections of the initial field. The field can escape this restriction only by developing internal discontinuities.

Casting the field equation in a form that the connectivity can be specified explicitly, reduces the field equation to the eikonal equation, describing the optical analogy, treated in papers II and III of this series. This demonstrates the ubiquitous nature of the tangential discontinuity in a force-free field subject to any local deformation.     

The propagation of torsion along flux tubes subject to dynamical nonequilibrium

Abstract

The dynamical nonequilibrium of close-packed flux tubes is driven by the torsion in the individual tubes so that, wherever tubes with the same sense of twisting come into contact, there is reconnection of their azimuthal field components. The reconnection consumes the local torsion, causing the propagation of torsional Alfven waves into the region from elsewhere along the tubes.

The formal problem of the propagation of the torsion along twisted flux tubes is presented and some of the basic physical properties worked out in the limit of small torsion.

It is pointed out that in tubes with finite twisting the propagation of torsional Alfven waves can be a more complicated phenomenon.

Application to the sun suggests that the propagation of torsion from below the visible surface up into the corona is an important energy supply to the corona for a period of perhaps 10–20 hours after the emergence of the flux tubes through the surface of the sun, bringing up torsion from depths of 10⁴km or more. Torsion is continually supplied by the manipulation and shuffling of the field by the convection, of course.     

Spontaneous tangential discontinuities and the optical analogy for static magnetic fields. VI. Topology of current sheets

Abstract

The electric surface current in a tangential discontinuity in a force-free magnetic field is conserved. The direction of the current is halfway between the direction of the continuous fields on either side of the surface of discontinuity. Hence the current sheets, i.e. the surface of tangential discontinuity, have a topology that is distinct from the lines of force of the field. The precise nature of the topology of the current sheet depends upon the form of the winding patterns in the field. Hence, invariant winding patterns and random winding patterns are treated separately. Current sheets may have edges, at the junction of two or more topological separatrices. The current lines may, in special cases, be closed on themselves. The lines of force that lie on either side of a current sheet somewhere pass off the sheet across a junction onto another sheet. In most cases the current sheets extending along a field make an irregular honeycomb.

The honeycomb pattern varies along the field if the winding pattern of the field varies. The surface current density in a tangential discontinuity declines inversely, or faster, with distance from its region of origin. The edges of weaker tangential discontinuities (originating in more distant regions) are bounded by the stronger tangential discontinuities (of nearby origin).

An examination of the force-free field equations in a small neighborhood of the line of intersection of two tangential discontinuities shows that the lines of force twist around to cross the line of intersection at right angles. If the angle between the tangential discontinuities exceeds π/2, there is also the possibilitity that the lines twist around so as to come tangent to the line of intersection as they cross it.     

Relativistic magnetohydrodynamic winds of finite temperature

Abstract

The singular differential equations for finite temperature relativistic magnetohydrodynamic (MHD) winds integrate to two algebraic equations when the source magnetic field is a monopole. This simplification enables an extensive characterization of the asymptotic wind solutions in terms of source parameters. We will consider only the critical solutions-those that pass smoothly through both an intermediate (Alfvenic) and a fast MHD critical point and expand to zero pressure at infinite radial distance from the source. Because the constants of motion must be specified to extremely high accuracy, the critical solutions cannot be found analytically. Synopsis of many numerical solutions reveals a uniform parametric characterization of the asymptotic wind in terms of one combination of source parameters, Z, the mean source particle energy divided by mc²[sgrave]^½, where [sgrave] is a generalization of Michel's (1969) cold relativistic wind strength parameter. Cool winds, with Z<1, behave asymptotically much as Michel's cold wind minimum torque solution; Z1 hot winds have quite different, but simply characterized, asymptotic solutions. Thus, the strength of magnetized relativistic outflows can depend critically upon the temperature of the source.     

Effect of pressure gradients and line-tying on the magnetic stability of coronal loops

Abstract

In this paper we study the stability of an idealised magnetostatic coronal loop, incorporating both the effect of line-tying, due to the dense photosphere, and of pressure gradients. The stability equations may be solved analytically for our particular equilibrium. From the marginally stable case, the critical conditions separating instability from stability are derived. It is found that stretching or twisting a loop eventually makes it kink unstable.     

Nonlinear dynamics of a stellar dynamo in a spherical shell

Abstract

A simple mean-field model of a nonlinear stellar dynamo is considered, in which dynamo action is supposed to occur in a spherical shell, and where the only nonlinearity retained is the influence of the Lorentz forces on the zonal flow field. The equations are simplified by truncating in the radial direction, while full latitudinal dependence is retained. The resulting nonlinear p.d.e.'s in latitude and time are solved numerically, and it is found that while regular dynamo wave type solutions are stable when the dynamo number D is sufficiently close to its critical value, there is a wide variety of stable solutions at larger values of D. Furthermore, two different types of dynamo can coexist at the same parameter values. Implications for fields in late-type stars are discussed.     

Periodic and aperiodic dynamo waves

Abstract

In order to show that aperiodic magnetic cycles, with Maunder minima, can occur naturally in nonlinear hydromagnetic dynamos, we have investigated a simple nonlinear model of an oscillatory stellar dynamo. The parametrized mean field equations in plane geometry have a Hopf bifurcation when the dynamo number D=1, leading to Parker's dynamo waves. Including the nonlinear interaction between the magnetic field and the velocity shear results in a system of seven coupled nonlinear differential equations. For D>1 there is an exact nonlinear solution, corresponding to periodic dynamo waves. In the regime described by a fifth order system of equations this solution remains stable for all D and the velocity shear is progressively reduced by the Lorentz force. In a regime described by a sixth order system, the solution becomes unstable and successive transitions lead to chaotic behaviour. Oscillations are aperiodic and modulated to give episodes of reduced activity.     

On the coalescence of twisted flux tubes

Abstract

We study the problem of the coalescence of twisted flux tubes by assuming that the azimuthal field lines reconnect at a current sheet during the coalescence process and everywhere else the magnetic field is frozen in the fluid. We derive relations connecting the topology of the coalesced flux tube with the topologies of the initial flux tubes, and then obtain a structure equation for calculating the field configuration of the coalesced flux tube from the given topology. Some solutions for the two extreme cases of low-β plasma and high-β plasma are discussed. The coalesced flux tube has less twist than the initial flux tube. Magnetic helicity is found to be exactly conserved during the coalescence, but the assumptions in the model put a constraint on the energy dissipation so that we do not get a relaxation to the minimum-energy Taylor state in the low-β case. It is pointed out that the structure equation connecting the topology and the equilibrium configuration is quite general and can be of use in many two-dimensional flux tube problems.     

Coronal magnetic arcades with a helical structure

A family of force-free magnetic fields, describing multipolar magnetic arcades is described. The family allows for twisted magnetic ropes, where magnetic field lines have a helical shape. These helical structures are specific to filament channels of active regions. In their structure, arcades belong to normal-polarity fields and correspond to prominences of active regions according to available observational data. A potential field is found that is external with respect to an arcade. The constructed force-free fields can be used to further develop the MHD model of filaments that takes into account the gas pressure gradient and gravity.     

Convective dynamos with intermediate and strong fields

Abstract

The weak-field Benard-type dynamo treated by Soward is considered here at higher levels of the induced magnetic field. Two sources of instability are found to occur in the intermediate field regime M ～ T ^1/12, where M and T are the Hartmann and Taylor numbers. On the time scale of magnetic diffusion, solutions may blow up in finite time owing to destabilization of the convection by the magnetic field. On a faster time scale a dynamic instability related to MAC-wave instability can also occur. It is therefore concluded that the asymptotic structure of this dynamo is unstable to virtual increases in the magnetic field energy.

In an attempt to model stabilization of the dynamo in a strong-field regime we consider two approximations. In the first, a truncated expansion in three-dimensional plane waves is studied numerically. A second approach utilizes an ad hoc set of ordinary differential equations which contains many of the features of convection dynamos at all field energies. Both of these models exhibit temporal intermittency of the dynamo effect.     

Internal waves in a rotating stratified fluid in an arbitrary gravitational field

Abstract

Small amplitude oscillations of a uniformly rotating, density stratified, Boussinesq, non-dissipative fluid are examined. A mathematical model is constructed to describe timedependent motions which are small deviations from an initial state that is motionless with respect to the rotating frame of reference. The basic stable density distribution is allowed to be an arbitrary prescribed function of the gravitational potential. The problem is considered for a wide class of gravitational fields. General properties of the eigenvalues and eigenfunctions of square integrable oscillations are demonstrated, and a bound is obtained for the magnitude of the frequencies. The modal solutions are classified as to type. The eigenfunctions for the pressure field are shown to satisfy a second-order partial differential equation of mixed type, and the equation is obtained for the critical surfaces which delineate the elliptic and hyperbolic regions. The nature of the problem is examined in detail for certain specific gravitational fields, e.g., a radially symmetric field. Where appropriate, results are compared with those of other investigations of waves in a rotating fluid of spherical configuration and the novel aspects of the present treatment are emphasized. Explicit modal solutions are obtained in the specific example of a fluid contained in a rigid cylinder, stratified in the presence of vertical gravity, with the buoyancy frequency N being an arbitrary prescribed function of depth.     

Dynamo models with magnetic diffusivity-quenching

Abstract

The magnetic influence on a turbulent plasma also produces a complicated structure of the eddy diffusivity tensor rather than a simple and traditional quenching of the eddy diffusivity. Dynamo models in plane (galaxy) and spherical (star) geometries with differential relation are developed here to answer the question whether the dynamo mechanism is still yielding stable configurations. Magnetic saturation of the dynamos is always found—at magnetic energies exceeding the energy-equipartition value.

We find that the effect of magnetic back-reaction on the turbulent diffusivity depends highly on whether the dynamo is oscillatory or not. The steady modes are extremely influenced. They saturate at field strengths strongly exceeding its turbulence-equipartition value. Subcritical excitation is even found for strong seed fields. The oscillating dynamos, however, only provide a small effect. Hence, the strong over-equipartition of the internal solar magnetic fields revealed by studies of flux-tube dynamics cannot be explained with the theory presented. Also the run of the cycle frequency with the dynamo number is too smooth in order to explain observations of stellar chromospheric activity.     

Magnetic equilibria resulting from a helically symmetric kink instability

Abstract

This paper explores magnetic equilibria which could result from the kink instability in a cylindrical magnetic flux tube. We examine a variety of cylindrical magnetic equilibria which are susceptible to the kink, and simulate its evolution in a frictional fluid. We assume that the evolution takes place under conditions of helical symmetry, so the problem becomes effectively two-dimensional. The initial cylindrical equilibrium field is specified in terms of its twist function k(r) = B _θ/(rB_z ) and for a variety of k(r) functions we calculate linear growth rates for the kink instability, assuming that it develops under helical symmetry with pitch τ. We find that the growth rate is sensitive to the value of τ.

We simulate nonlinear evolution of the kink using a Lagrangian frictional code which constrains the field to have helical symmetry of a given pitch τ. Ideal MHD is assumed and the plasma pressure is taken to be small in order to mimic conditions in the solar corona. In some cases the flux tube evolves to a new smooth helically symmetric equilibrium which involves a relatively small change in the maximum electric current. In other cases there is evidence of current-sheet formation.     

Dielectrophoretic,thermal instability in a spherical shell of fluid

Abstract

This paper is concerned with the dielectrophoretic instability of a spherical shell of fluid. A dielectric fluid, contained in a spherical shell, with rigid boundaries is subjected to a simultaneous radial temperature gradient and radial a.c. electric field. Through the dependence of the dielectric constant on temperature, the fluid experiences a body force somewhat analogous to that of gravity acting on a fluid with density variations. Linear perturbation theory and the assumption of exchange of stabilities lead to an eighth order differential equation in radial dependence of the perturbation temperature. The solution to this equation, satisfying appropriate boundary conditions, yields a critical value of the electrical Rayleigh number and corresponding critical wave number at which convective motion begins. The dependence of each critical number is presented as a function of the gap size and temperature gradient. In the limit of zero shell thickness both the critical Rayleigh number and critical wave number agree with results for the case in the infinite plane problem.