The question what geometrical restrictions are imposed on static magnetic fields by the magnetohydrostatic (MHS) equation is addressed. The general mathematical problem is therefore to determine the solutions of the MHS equations in the corona subject to an arbitrary normal component of the magnetic field at the boundary and arbitrary connectivity. What constraints the MHS equations impose on the geometry of the solutions, expressed in metric tensors, will be determined

Martens, P. C. H.

English

NASA Technical Reports Server

N87-20937
A COORDINATE FREE DESCR_PTTON OF
HAGNETOHYDROSTATIC EQUILIBRIA
P.C.H. Mmrtens
NAS/NRC Research Associate
Laboratory for Astronomy and Solar Physics
NASA/Goddard Space Flight Center
Greenbelt, Maryland 20771
INTRODUCTION
This paper addresses the question what geometrical restrictions are
imposed on static magnetic fields by the MagnetoHydro-Static (MHS) equations.
This question is of obvious importance for the problem of coronal heating,
since it has been argued by Parker (1972, 1979 and 1986; see references
therein) that the MHS-equations in general overdetermine the magnetic field
structure and that consequently the field needs to have some sort of symmetry
to satisfy all the constraints imposed on it by the equations. The field in
the solar corona is determined by the MHS equations and the boundary
conditions at the corona/photosphere interface. The latter are the normal
component of the magnetic field at the boundary (to ensure the continuity of
the magnetic field) and the connectivity of the field lines, defined as the
positions of all field llne footpolnts at the boundary (because the field
lines are frozen in, Sturrock and Woodbury, 1967). These boundary conditions
are completely arbitrary, because they are determined by the magnetic fields
and the fluid motions in the photosphere and convection zone, that cannot be
altered by the relatively weak forces from the coronal magnetic field. The
general mathematical problem is therefore to determine the solutions of the
MHS-equations in the corona subject to an arbitrary normal component of the
magnetic field at the boundary and arbitrary connectivity.
It is very unlikely that these boundary conditions would conspire to
satisfy any symmetry requirement that the MHS equations might impose. And even
if they would at a given moment, only minor footpoint displacements - as a
result of the photospheric velocity field - would destroy the symmetry. Hence
the coronal magnetic field cannot be in static equilibrium at any time, and,
according to Parker (1983), the force free condition will break down at some
locations in the corona, where current sheets will form. In these sheets the
dissipation of magnetic field is much larger than that calculated with
classical resistivity and the resulting heating rate may be large enough to
explain the observed non-thermal heating of the corona. This process is called
topological heating.
Recently, however, Parker's hypothesis has been challenged by Van
Ballegooijen (1985) and Antiochos (1986); see also their contributions in this
chapter. Van Ballegooijen points out an error in Parker's (1972) original
demonstration of the need for an ignorable coordinate and furthermore, by
improving upon Parker's analysis, gives an algorithm for calculating solutions
to the MHS equations, subject to arbitrary boundary conditions. Antiochos
argues that the problem is generally well posed by showing that when the
magnetic field is expressed in Euler potentials, the topology of the field in
the corona is completely determined by the values of the potential at the
boundary. Consequently there is no need for the formation of current sheets in
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Lthese analyses.
Clearly the question whether there is the requirement of some sort of
symmetry in the solutions of the MHS equations deserves further attention.
This symmetry has to be of a subtler form than that of an ignorable coordinate
as once proposed (Parker, 1972), since recently explicit analytical examples
have been given of fully 3-D magnetostatic equilibria (Low, 1985). The latter
solutions still show some form of symmetry, however.
The problem is the more intriguing because it has recently been shown
quite convincingly by Tsinganos et al. (1984) and Moffat (1985) - in very
different ways - that magnetostatic equilibria in Tokamak type structures do
lead to topological dissipation, when they do not exhibit symmetry. However,
in Tokamaks the boundary conditions are quite different: here the requirement
is that the normal component of the field vanishes everywhere at the surface
of the containment vessel and hence the field is self contained (see Martens,
1985, for a comparison). The field lines in a Tokamak are either infinite in
length, or close in themselves, quite the contrary of the structure of closed
coronal fields, where the field lines are anchored at both ends. If there is a
difference between closed coronal magnetic fields and Tokamak-type fields with
regard to their intrinsic symmetry, this difference must be caused by the
nature of the boundary conditions. The difference then is probably related to
the fact that the corona/photosphere interface takes up the stresses from the
coronal field, while the containment vessel of a Tokamak obviously doesn't.
In this paper I will take up the issue of the geometrical constraints on
magnetostatic equilibria from a somewhat different point of view. I will write
the MHS equations in a general coordinate system - not necessarily orthogonal
- and then choose the coordinates in such a way that the pressure gradients
and current density vector are along coordinate lines, which makes their
expressions very simple. I will then try to determine what constraints the MHS
equations impose on the geometry of the solutions, that is expressed in the
metric tensor. The first results do indicate some restrictions to the possible
geometries of the solutions, but these do not seem to represent some sort of
symmetry. The analysis of this paper cannot be regarded as completed, and more
definite results will be published in the literature.
THE NHS EQUATIONS IN ARBITRARY COORDINATE SYSTEMS
The basic equations governing magnetostatic equilibria are well known
(i),
_._ = 0 (2).
There are no general solutions known to this deceivingly simple looking set of
equations. The system is nonlinear because of the Lorentz-force term in Eq.
(I): the sum of two solutions in general does not represent a third solution.
All particular analytical solutions that are known to date have some sort of
symmetry (Low, 1985).
An alternative notation of Eqs. (I) and (2) in an arbitrary orthogonal
coordinate system is found after introduction of the metric tensor
432
h12 0 0 ]gij = I0 h22 00 h3 2
(3),
where the h i represent the length of the unit vectors. The components of the
magnetic field are identi_%ed with the three independent components of an
antisymmetric 3-D tensor B 3,
B3 _ BI , B2
B 12 - B 23 =-- B 31 =-- (4).
hlh 2 h_3 hlh3
The identification of the components of the magnetic field with those of an
antisymmetric tensor, instead of with the components of a contravariant
vector, will lead to a particulary simple formulation of the equations.
Moreover, it accounts properly for the fact that the magnetic field is a
pseudo-vector, instead of a real vector: by inspection of the expression for
the Lorentz-force one finds that the magnetic field must remain the same under
a mirror transformation of the the coordinates (x ÷ -x, etc.), since the
Lorentz force and the current both will change sign.
The contravariant components of the current density vector are given by
+ A
ji _ (J'xi)
hi (5),
while the covariant components of the pressure gradient are given in the same
way.
The basic equations in this notation are
P¶k = jiBkl (6),
ji BiJ¶¶J _g(/_BiJ= = )lJ (7),
{Bij¶k} = 0 (8).
Here ¶¶i denotes covariant differentiation and .. ordinary differentiation
with respect to the variable denoted by the index.l%..} means a summation over
all permutations of the indices of the tensor within the brackets. For the
antisymmetric magnetic tensor it reduces to
B12¶3 + B23¶I + B31¶2 = 0 (9)•
FLUX SURFACE COORDINATES
Now I generalize the MHS equations (6) to (8) and suppose their validity
in non-orthogonal coordinate systems. The expressions remain the same of
course, only the metric tensor has off-dlagonal elements. This allows one to
433
choose a coordinate system that makes the equations particulary simple. First
I choose the pressure gradient parallel to the first unit vector and hence
P * P(xI) (10).
The isobaric surfaces are now by definition surfaces of constant xI. Further
I choose x2 and x3 perpendicular to Xl, but not necessarily perpendicular to
each other. This means that the metric tensor has the form
!ii 0 0 ]gi_ = g22 g23J
g23 g33
(II),
and has therefore 4 independent components. From the force balance equations
(7) and Eq. (Ii) it can now easily be shown that
jl:O
B23 = -B 32 = 0
B23 = -B32 = 0 (12).
The equation expressing the divergencelessness of the magnetic field, Eq. (8),
reduces to
B12¶3 + B3112 : 0 (13),
with the solution,
BI2 : A¶2
BI3 = A¶3 (14),
and A(Xl,X2,X 3) an arbitrary function. Now that the three components of the
magnetic field have been reduced to dne unknown function only dne component of
the force-balance equations remains to be satisfied,
P¶I = j2A¶2 + j3A¶3 (15),
while the demand jl = 0 leads to a second constraint on the solutions
(reminiscent of Low's (1980) compatibility relation).
So far the non-orthogonality of the coordinates has not been used. I make
a small digression from my main argument now to iexplore somewhat further the
compatability constraint in orthogonal systems, j = 0.
(_B12)12 + (_B13)13 : 0 (16).
In an orthogonal coordinate system I can now reduce the force balance equation
and the compatability constraint to:
(17),
434
and,
hlh2h3P¶l = A¶2[hlh 2 _I +
(18).
By trial and error one may find that it is extraordinarily difficult to obtain
functions A that satisfy both constraints Eqs. (17) and (18). To give a short
and very simple example I shall investigate the case where the isobaric
surfaces are cylinders. We have (Xl,X2,X 3) = (r,_,z), hl=h3=l, h2=r, and
therefore Eqs. (17) and (18) reduce to
AI@1@ + rZAlz¶z = 0 (19).
rPlr = A, afA1__l
• v[ r J¶r + A1zIA¶zr)¶r
The only solution of Eq.(19) that is consistent with P = P(r) is
(2o).
A(r,@,z) = f(r)z + g(r)@ (21),
and Eq. (20) takes the well known form
P¶r =I/_B# 2 + BzZ)¶r + B_ 2/r
after the identification
(22),
g(r) = B r , f(r) = -B (23).
z
The solutions of Eq. (22) are well known (L_st and Schl_ter, 1954). I conclude
that the requirement of cylindrical isobares introduces the necessity of
cylindrical symmetry of the magnetic field. However, the general question one
would llke to answer remains: what are the restrictions imposed on the
function A by Eqs. (17) and (18) in any coordinate system?
NON-ORTHOGONAL C00RDINAI"E$
I will proceed with the main line of my argument now and specify further
the choice of the coordinate system. I choose the direction of the second unit
vector along the current density vector, i.e.
i
j = (0,j 2,0) (24).
It can be shown that with this choice of the second unit vector the coordinate
system cannot be orthogonal anymore. In Fig. I a cylindrical surface is drawn
that contains a set of field lines and a set of current density vectors. The
first unit vector is by definition perpendicular to this surface, while the
other two must lie within the surface. In the figure a current line is drawn
which has the second unit vector everywhere parallel to it. In an orthogonal
coordinate system the third unit vector must be everywhere perpendicular to
the second one, but if one follows the third coordinate from point (Xl,X2,X 3)
435
one finds that it intersects again the current line at the point
(xl,x2,x3+Ax3). However, one may also follow the current llne from the point
(x ,x ,x ) and then the second intersection is reached at (x ,x +Ax ,x ). This
2 3 + I 2 2 3po}nt is physically the same as (xl,Xg,X _ Axe) and hence it must have the same
coordinates; if not, the eoordinat_ s_stem b_eomes double (and multi-) valued.
This requirement can be met by having x 2 and x 3 as cyclic coordinates: x 9 +
Ax 2 E x2, and x + Ax 3 E x 3 (just as in spherical coordinates). However, now
one runs into t_e contradfction that two physically different points (the
first and the second intersection) have identical coordinates; a situation
which is also undesirable. I conclude that it is impossible to choose a
consistent isobaric orthogonal coordinate system if _//x 2.
VALID CHOICE FOR X 3 -'-"
f
J
STARTING POINT _- X_
-,--.- SECOND INTERSECTION
"'"_CURRENT DENSITY VECTOR
Figure i. A demonstration of the inconsistency of an orthogonal coordinate
system with one unit vector parallel to the pressure gradient and
another with the current density vector.
A valid choice for the third coordinate is indicated in Fig. I. It is
clear that as one follows the third unit vector, the line will close in itself
and there is no inconsistency. With this choice the third vector is not
perpendicular anymore to the second and hence the term g23=g32 in the metric
tensor must be nonzero.
Eqs. (14), (15) and (16) were derived for the metric Eq. (II). B_ the
spec[al choice for the second coordinate one finds in addition, because j =0,
(#g B31) II = 0 (25).
Eqs. (16) and (25) are satisfied when
(_gB 21) = f(xl,x3) + m¶3(x2,x 3) (26),
436
and
(_gB 31) = _mlz(x2,x3) (27),
with f and m arbitrary functions. The force balance equation (15) now reduces
to
JgP¶l = A¶2f¶l (28).
The contravariant components of the field tensor may be eliminated with
A¶2 = gllg22 BI2 + gllg23B 13 (29),
A¶3 = gllg32 BI2 + gllg33B 13 (30),
and one is left with three equations, (26), (27) and (28), for the function A.
For a given geometry g..,
is clearly overdetermin_ and arbitrary functions f, m and P, the function Aand consequently some restrictions must apply to the
possible choices of the geometry. In this stage of the research it is not
clear yet what these restrictions are, although it seems that the restrictions
do not necessarily impose an ignorable coordinate.
More work along the lines of this paper is needed to shed light on the
geometries that are consistent with MHS-equilibrla. In particular the
restrictions that the boundary conditions impose on the possible solutions
will be investigated.
EEFERENCES
Antiochos, S.K. 1986: "The Topology of Force-Free Magnetic Field and
its Implications for Coronal Activity", Astrophys. J., in press.
Low, B.C. 1980: Solar Phys. 67, 57.
Low, B.C. 1985: Astrophys. J. 293 ,31.
L_st, R., Schl_ter, A. 1954: Z. Astrophys. 34, 263.
Martens, P.C.H. 1985: "Magnetic Reconuectlon & Turbulence', M.A. Dubois,
D. Gresillon, M.N. Bussac (eds.), p.79, Editions de Physique, Orsay.
Moffat, H.K. 1985: J. Fluid Mech. 159, 359.
Parker, E.N. 1972: Astrophys. J. 174, 499.
Parker, E.N. 1979: "Cosmical Magnetic Fields', pp. 359-391, Oxford,
Clarendon Press.
Parker, E.N. 1983: Astrophys. J. 264, 642.
Parker, E.N. 1986: Geophys. Astrophys. Fluid Dyn. 35, in press.
Solov'ev, L.S. 1972: "Revlews of Plasma Physics', Vol. 6, p.239, M.A.
Leontovich (ed.), Consultants Bureau, New York
Sturrock, P.A., Woodbury, E.T. 1967: "Plasma Astrophysics', p.155, P.A.
Sturrock (ed.), Academic Press, New York.
Tsinganos, K.C., Distler, J., Rosner, R. 1984: Astrophys. J. 278, 409.
Van BallegooiJen, A.A. 1985: Astrophys. J. 298, 421.
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A coordinate free description of magnetohydrostatic equilibria

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