A&A 417, 333-340 (2004)
DOI: 10.1051/0004-6361:20031776
L. J. November
The Light Physics, PO Box 217, La Luz, NM 88337-0217, USA
Received 29 January 2003 / Accepted 18 December 2003
Abstract
We show that the atmospheric and magnetic height variations are
coupled in general MHS equilibria with gravity when isolated thin
non-force-free flux tubes are present. In gas-dominated
environments, as in stellar photospheres, flux tubes must expand
rapidly with height to maintain pressure balance with the cool
surroundings. But in magnetically dominated environments, as in
stellar coronae, the large-scale background magnetic field determines
the average spreading of embedded flux tubes, and rigidly held flux
tubes require a specific surrounding atmosphere with a unique
temperature profile for equilibrium. The solar static equilibrium
atmosphere exhibits correct transition-region properties and the
accepted base coronal temperature for the sun's main magnetic
spherical harmonic. Steady flows contribute to the overall pressure,
so equilibria with accelerated wind outflows are possible as well.
Flux tubes reflect a mathematical degeneracy in the form of
non-force-free fields, which leads to coupling in general equilibrium
conditions. The equilibrium state characterizes the system average
in usual circumstances and dynamics tend to maintain the MHS
atmosphere. Outflows are produced everywhere external to rigidly held
flux tubes that refill a depleted or cool atmosphere to the
equilibrium gas profile, heating the gas compressively.
Key words: magnetohydrodynamics (MHD) - Sun: atmosphere - Sun: corona - Sun: transition region - stars: coronae - stars: winds, outflows
In our observation of solar "threads'' (November & Koutchmy 1996, hereafter NK96) we puzzled over whether coronal flux tubes should expand with height to maintain pressure balance with the ambient gas-pressure decrease or follow the lines of the external magnetic field. After all, if the corona is independently heated, the gas and magnetic height variations should be generally different. The many observations of EUV and X-ray loops as well as the purely density-sensitive observations of white-light >15 Mm "voids'' (MacQueen et al. 1974), >2 Mm threads, and scintillation >1 km filamentary microstructure (Coles & Harmon 1978; Woo et al. 1995) all suggest that the large-scale solar coronal magnetic field is interspersed with flux tubes with a great range of sizes. Figure 1 shows dark and bright threads in a square region about a solar radius on a side over the west limb taken with the Canada-France-Hawaii Telescope (CFHT) at the unique total-eclipse opportunity on Mauna Kea on July 11, 1991. The dark and bright threads, which were essentially unchanged over the 4-min eclipse duration, appear to be aligned and organized in arched and radial surfaces, probably current sheets, overlapping in their projection onto the plane of the sky in the line-of-sight view.
We know that isolated thin flux tubes must expand with height in
balance with the ambient gas-pressure decrease. The flux-tube gas
pressure difference, external minus internal, is related to the
magnetic pressure difference by the static equilibrium relation
We can argue too that the radial expansion of isolated thin flux
tubes must conform to the external magnetic field.
Parker (1972) shows that
Therefore both ways of looking at the problem are essentially
correct. The coronal flux-tube field must comply with the external
field
,
and the radial
variation of the external gas pressure outside isolated thin
evacuated coronal flux tubes is proportional to the magnetic pressure
difference
.
Hence,
the external gas and magnetic pressures must be coupled with
outside and along isolated thin evacuated
flux tubes. The background magnetic field is dominant in lower
stellar coronae and must determine the expansion of embedded flux
tubes. The expansion of evacuated flux tubes then defines the radial
variation of the ambient gas pressure for equilibrium, the gas
pressure being horizontally stratified in the external force-free or
potential field. More generally, flux tubes of arbitrary base
pressure but in thermal balance with the local ambient atmosphere
satisfy
,
giving the same external atmosphere
.
Thus an atmosphere containing
isolated thin non-force-free flux tubes in different forms of local
balance must exhibit a temperature profile defined by the dominant
background magnetic field for a perfect gas in hydrostatic
equilibrium.
The properties of flux tubes reflect a natural degeneracy in the form
of non-force-free magnetic fields, and imply the existence of an
underlying mathematical relationship between an equilibrium
atmosphere and large-scale magnetic field. In Sect. 2, we
show that the height variations of the gas pressure and magnetic
field amplitude are directly coupled in the most-significant-order
terms of the gravitational perturbation in general non-force-free MHS
equilibria, with smaller field divergence and curl effects. In
low-
systems, with strong magnetic fields, the external gas
pressure corresponds to a small difference
,
which allows large pressure deviations for a
given relative flexibility of the large-scale magnetic field around
individual flux tubes. Magnetic fields can exhibit and persist in
significant deviations from equilibrium, however the coupling
describes the general equilibrium condition
for arbitrary
.
Historical MHS equilibrium solutions with gravity have also suggested that a general relationship exists between the large magnetic and gas-pressure scales. Dungey (1953) derived plane-parallel MHS equilibria for a magnetic field with its degenerate axis directed perpendicular to gravity (see also: Cheng & Choe 1998; Low 1975). His solutions are represented in a modified Grad-Shafranov (GS) equation that retains the gas-pressure radial scale-height variation as a separable factor in the pressure term. Dungey showed that when the magnetic field is also zero in its degenerate axis, the separable factor can be absorbed into the coordinates, giving the classical GS equation in a remapped coordinate system. The new GS coordinates are defined by the gas-pressure scale height, meaning that magnetic fields in a prespecified atmosphere must be spatially distorted. Horizontal flux tubes exhibit substantial radial deformation in a solar corona of given constant temperature (Low 1992; Zweibel & Hundhausen 1982). Depending upon their size compared to the gas-pressure scale height, flux tubes are either compressed or expanded in the gravitational direction, and approximately undistorted only in a specific atmosphere.
In Sect. 3, we examine equilibrium radial profiles of temperature and flux-tube diameter using an approximate solution to the equilibrium equations. The profiles reproduce the salient features of the solar atmosphere: the rapid expansion of flux tubes in the photosphere where the atmospheric gas pressure is dominant, a transition region of reasonable thickness located just above the height where the gas and magnetic energy densities are equal, and a corona of correct base temperature for the sun's main magnetic spherical harmonic. Steady flows do not alter the coupling, but may produce a Bernoulli pressure, which modifies the gas pressure, and leads to possible accelerated steady wind solutions.
Although the dynamics of the system can be quite complicated, the equilibrium state characterizes the average in many types of MHD systems, e.g. "quasi-steady'' (Low 1980). Hydrodynamics in such systems tend to maintain the MHS atmosphere, as we discuss in Sect. 4. The possible importance of MHS atmospheres for other problems in physics and astrophysics beyond the formation of solar and stellar coronae and in more marginal non-quasi-steady physical conditions needs to be considered, but discussion lies beyond the scope of this paper.
The equations for static equilibrium in a gravitational field are the
MHS equation and Gauss' Law
The set of four vector-element Eqs. (3) and (4)
contain five unknowns, p, h, and the three vector elements of
.
However, the solution space for
is more
restrictive than would be obtained with linear relations, as we
discuss in this section and elaborate in Appendix A.
The gravitational term in Eq. (3) compared to the pressure gradient is of order d/h, where d is a characteristic flux-tube thickness; for the main power in threads in the solar corona d/h < 10-3. In the vicinity of non-force-free fields gas-pressure changes must be mainly magnetically determined, and only far away can the gravitational gradient be significant. Where the Lorentz term is negligible, the gas pressure is hydrostatic and defined by a single radial temperature profile. Since the gravitational term is small in non-force-free conditions, the classical equilibrium solutions without gravity described in Appendices A and B are applicable, and Eqs. (1) and (2) are valid in every small volume, but with the allowance that quantities may vary slowly spatially. Thus we obtain coupled atmospheric and magnetic height variations, at least if certain types of flux tubes are present as described in the Introduction.
Conditions along field lines are represented by a parallel-field
equation
Substituting Eq. (6) into the MHS Eq. (3) and
using Eq. (8) gives the cross-field equation
Since the base pressure
is constant along field lines, the
variation in the scale-height function
on the left side of
Eq. (11) must be reproduced on the right side in the
Lorentz term, and so in
too. Thus
must vary
like
along field lines in non-force-free fields
within d/h, which is the order of the derivatives of the
scale-height function. Formally we can absorb
into the
magnetic field on the right side by decomposing
into the
general product
Additional perturbative equations for the base functions and
originate with the rescaling of the magnetic
field vector in r as discussed in Appendix C. The
rescaling leads to the small term
in both
Eqs. (13) and (14). The term appears on
the right side of Eq. (14) and is needed since a slow
radial decrease in
cannot occur without a divergence of
field lines; it represents a compensating creation of flux in
with r. The term is subtracted from the curl in
the Lorentz force in Eq. (13) and corresponds to a
small residual pseudo-force directed between
and the
outward radial
;
it represents a progressive
left-handed twist of field lines at the rate of 1 rad in every
magnetic scale height 2h(r). In MHS equilibria with gravity, the
amplitude, the divergence, and the curl of
all vary on the
large scale 2h(r).
In principle anyway, a single thin flux tube somewhere in the atmosphere, evacuated or in thermal balance with its surroundings, fixes the equilibrium gas pressure for the entire atmosphere, but all non-force-free fields must be consistent to satisfy mutually the horizontal pressure boundary condition. In reality of course, magnetic fields may exhibit local deviations, and the equilibrium state might only be reflected in the average in an atmosphere containing many flux tubes.
A flux tube in thermal balance satisfies the condition
whether of increased or decreased field strength
compared to the local background magnetic field. A flux tube with
increased field strength
,
must
exhibit a decreased internal gas pressure in Eq. (1),
consistent with the extreme dark-thread evacuated
case of
,
which requires the minimum surrounding atmosphere
for a given magnetic pressure difference. Flux tubes with less field
strength than the external magnetic field
,
must have an increased gas pressure
,
consistent with bright threads.
Threads appear to be isolated thin flux tubes aligned with the
background magnetic field and embedded in current sheets in a bimodal
amplitude distribution, consistent with the natural equilibrium form
discussed in Appendix B. The temperature of bright
threads was indeed found to be the same as the external surroundings
within observational uncertainties (NK96). The solar EUV coronal
temperature for unresolved quiet and active regions is relatively
uniform, 1.5-
K (Withbroe 1975), and a
narrow temperature range around
K is found for a broad
range of sizes and densities of X-ray loops for all but prominence
flare loops (Vaiana et al. 1976; Davis et al. 1975; Rosner et al. 1978). Large-area coronal temperatures derived
from the solar white-light scale height and forbidden line ratios
give temperatures somewhat lower than the EUV and X-ray temperatures,
in the range of 1-
K at the solar radial height
(Guhathakurta et al. 1992).
A slightly heated (or cooled) flux tube exhibits an increased (or
decreased) internal pressure
with height and thereby must
affect a decreased (or increased) internal field strength
or an increased (or decreased) external pressure
to satisfy
the equilibrium Eq. (1). Small changes in internal field
strength might be accommodated by changes in the flux-tube size as a
state of local stress in the large-scale magnetic field, and changes
in the surrounding ambient pressure
might be modified by
compensating external dynamics like we describe in Sect.
4. Hot flux tubes should tend to cool to the temperature
of their immediate surroundings, bringing them back to the normal MHS
equilibrium condition. A relatively cool nonevacuated flux tube with
excess field strength tends to evacuation with height as a different
equilibrium condition, and so might just persist out of equilibrium
within about one internal scale height of its base.
Continuous 3D non-force-free equilibrium solutions are claimed
(Low 1985,1991), but these allow an arbitrary
hydrostatic 1D gas pressure independent of the magnetic field (see
the use of p0(r) and discussion around Eq. (24) of
Low 1991, in Sect. III in Bogdan & Low 1986, or
Sect. 4 in Neukirch 1997). In the general force balance,
that is the cross-field Eq. (11) with
allowed to
vary from field line to field line, the pressure and magnetic
variations cannot be separated except in force-free regions
unconstrained by a horizontal pressure boundary condition, and so
necessarily lacking isolated thin non-force-free flux tubes evacuated
or in thermal balance with their surroundings.
We have confined our study here to the static solutions, but it is
straightforward to broaden consideration to include certain types of
steady flows. Steady uniform flows along field lines add a flow
pressure to the parallel-field Eq. (5), which leads to a
modified hydro-steady relation in Eqs. (9) and (10), but the same cross-field Eq. (11)
results. Thus we obtain a gas pressure containing static and wind
components coupled to the large-scale magnetic field outside flux
tubes. For a given gas pressure variation or scale-height function
as is defined by the magnetic field in a magnetically
dominated environment, the static contribution to the pressure scale
height h(r) and corresponding temperature
are always larger with a steady flow than without
(Parker 1960). Steady flux-tube flows can modify the
coupling too.
The derivation of an explicit static equilibrium solution is a
formidable task because such an analysis must consider the specific
form that distorted magnetic fields take in the presence of a gas.
However an approximate coupled scale-height function can be written
At a base height
taken as the stellar photospheric surface, we
assume that magnetic fields are relatively weak,
and
.
There the gas pressure is dominant
and must follow its nominal atmospheric form not being much affected
by the magnetic field, whereas the magnetic field lines must be
highly distorted. The nominal atmospheric-pressure scale-height
function
is defined by the hydrostatic relation Eq. (10)
with a temperature found using a transfer equation
that includes all of the usual energy input and loss mechanisms.
The atmospheric contribution
characteristically falls off
much more rapidly than the magnetic pressure
,
so
around some transition height
where
,
changes from
to
over about one gas-pressure scale height.
Above
the magnetic field is dominant
,
and the field must follow its nominal form
irrespective of the gas pressure. There the gas pressure profile may
be very distorted from its nominal form, representing an added
atmospheric heating produced by MHS restoring flows as we discuss in
Sect. 4.
Both the magnetic and gas pressure variations must be distorted from
their nominal separate forms around the intermediate transition
height .
Preserving the total pressure
gives an approximate flux-conserving extrapolation below the
magnetically dominated corona and an ostensible gas-pressure
extrapolation above the photosphere and chromosphere. The total
relative pressure around the intermediate transition height goes from
for a nominal
superposition of independent gas and magnetic pressures
and
with
the total surface
area, to
for the coupled pressure
with
the
evacuated area and AB the magnetically filled area. The two
pressure totals are equal in general only with
from
Eq. (17) and when the field-filling regions are evacuated
with
.
Anyway the choice of coupled scale-height
function is not too critical for our demonstration, as smooth
switching functions with correct asymptotic behavior in the
photosphere and corona exhibit similar temperature profiles even
around the transition height.
For demonstration purposes, we take
for a polytrope
atmosphere as derived in Appendix D
A nominal scale-height function
is defined for a
single-spherical-harmonic potential magnetic field. The potential
magnetic form seems to be consistent with what is observed in the
lower solar corona for
(Schatten et al. 1969; Altschuler & Newkirk 1969). The
potential magnetic field vector is the gradient of a sum of scalar
spherical-harmonic component functions, each of which is the
separable product of a 2D surface function and radial multiplier
.
The resulting magnetic field in each spherical harmonic
goes like
in all its vector elements; a monopole
exhibits a 1/r2 radial falloff, a dipole field
,
a 1/r3 falloff, etc. For a single spherical harmonic
,
the
magnetic energy density falls off like
The temperature is written from the hydrostatic Eq. (9)
Figure 2 shows some example scale-height functions
from Eqs. (17)-(19) in the
lowest part of the solar atmosphere around the transition height
.
Scale-height functions with the polytrope adiabat
and spherical harmonic
are plotted for different
,
using the solar parameters and surface photospheric
temperature
K from the VAL model atmospheres
(Vernazza et al. 1981), which defines the surface scale height
for
in Eq. (18). As all
of the curves are based on the same
,
they coincide until the
transition height
for the model is reached, and then switch
rapidly to the nominal magnetic
,
which appears to be
relatively flat on the log scale.
![]() |
Figure 2:
Log of scale-height function
![]() ![]() ![]() ![]() |
The radial expansion of a flux tube is another way to visualize the
field strength decrease in the atmosphere and the properties of the
coupled scale-height function. For conservation of the total flux
through the cross-sectional area of a flux tube
,
the flux-tube diameter d(r) must increase with radial
distance r,
.
Figure 3 portrays the relative
flux-tube diameter as a function of height
for
different solar atmospheric models denoted by
and
.
For a given polytrope adiabat ,
all flux tubes exhibit
approximately the same relative shape up to a height that depends
upon
,
where the field lines straighten up. Stronger fields
exist with a lower transition height and exhibit less overall
relative expansion before straightening up. The canopy depends upon
the surface distribution of the fields, but strong fields might give
the appearance of a lower canopy height too consistent with
observations of sunspots (Giovanelli & Jones 1982). The
straightening height is somewhat below the location of the fastest
temperature change due to the differing dependencies: the field
strength, which goes like
,
corrects from its
atmospherically determined form at
,
but the most rapid
temperature change occurs significantly higher where the slope of the
scale-height function flattens out, as
.
Figure 4 illustrates the temperature variation in the
lower solar atmosphere using Eq. (20) with
K from
the VAL atmospheres. The upper panel shows the model atmospheres
with
and
for different polytrope adiabats
,
and the lower panel the model atmospheres with
and
for different
.
The VAL A and F model
temperatures for relatively cool inner network and hot network bright
points are also shown.
In the solar photosphere, the nominal atmospheric polytrope follows
the VAL temperature roughly, giving the best tradeoff between
photospheric and chromospheric temperatures with
.
Of course the features of a real solar atmosphere can never be well
approximated by a polytrope. The shape of the temperature function
through the chromosphere and transition height, where the
scale-height function
goes from
to
,
depends upon
and
but not
on
,
which separately determines the base coronal temperature.
Temperature profiles with
or 1000 give best
agreement around the transition height with the VAL A or F
atmospheres, respectively, averaging approximately through the
Ly
plateau.
For flux-tube evacuation in the photosphere
,
and we obtain a maximum field strength of
1715 G using
dyn cm-2 from the VAL
solar models;
might be taken as the fractional area
covered by evacuated magnetic fields. Strictly the coefficient
represents the coronal base gas or non-force-free
magnetic pressure extrapolated back to
,
which can be no larger
than the average surface magnetic pressure. Thus the fractional area
covered by evacuated magnetic fields at
must be at least
,
or 1/71 for
and 1/32 for
,
which corresponds to a minimum base coronal field
strength of
24 G for
and
54 G for
,
within the range of
solar observations (Lin et al. 2000). A lower
transition region occurs where the photospheric flux-tube areal
coverage and coronal magnetic field strength are larger near active
regions, consistent with the known tendency.
The solar atmospheric model of Fontenla et al. (1990),
which includes particle diffusion and conduction effects, lacks the
Ly
plateau and exhibits a much more abrupt transition region
than what we obtain. While radiative, diffusive, and conductive
losses must be largely balanced by MHS restoring flows in the corona,
as we discuss in Sect. 4, loss mechanisms can influence the
detailed shape of transition-region profiles. Real radiative models
coupled to more general magneto-hydro-steady equilibrium solutions
need to be developed, but such work may be complicated by the
intrinsic limitations and uncertainties in atmospheric modeling,
especially associated with inhomogeneous magnetic structure
(Ayres 1981; Carlsson & Stein 1995). It is possible that
general methods based upon transition-region emissivity profiles,
which have been used to determine the nonradiative atmospheric
heating contribution (Craig & Brown 1976; Anderson et al. 1996), might be able to
distinguish coronal heating by MHS restoring flows from other heating
mechanisms.
Figure 5 illustrates the solar atmospheric temperature
function T(r) for various values of
with the coronal
temperatures inferred from the white-light-intensity radial gradient
taken from eclipse photographs (Newkirk et al. 1970)
and inferred from FeXIV 5303Å line-width measurements
(Jarrett & von Klüber 1958). The limited resolution in r in
the figure hides the transition region near
.
The coronal model
for the spherical harmonic
seems to give the best overall
agreement with the measured radial profiles, but
better matches the base coronal temperature. However all of the model
curves show a more rapid falloff than the measured temperature
profiles. Models containing an outward wind should give
systematically higher temperatures with height in the corona.
![]() |
Figure 5:
Static solar corona T(r) for ![]() |
Under usual conditions, e.g. quasi-steady, the large-scale and long-term system average is represented by the equilibrium state. In the quasi-steady approximation, the system evolution can be described by a sequence of nearby equilibria. Low (1977) shows that the quasi-steady approximation is applicable when magnetic adjustments are fast compared to evolutionary processes and the magnetic field is dominant over the gas, as in the lower solar corona.
A qualitative feature of MHD processes in a slowly evolving dominant magnetic field where reconnections are fast is an atmospheric restoring flow. Hydro-steady equilibria are always self-restoring. In the absence of sufficient external gas or flow pressure, a time-dependent outward gas acceleration arises outside and along rigidly held non-force-free flux tubes in the overall MHD balance, as described in Appendix E. Flows in a fixed magnetic field act generally to reestablish the equilibrium atmosphere, heating the gas compressively. As long as the timescale for magnetic evolution is relatively long, such MHD processes can have a large-scale effect since the pressure perturbation that drives the flow propagates horizontally away from flux tubes at the sound speed.
The available energy for outflows must be limited and the atmosphere has to eventually cool, distorting and geometrically stressing the coupled magnetic field, which contributes to magnetic instability. The magnetic field can be perturbed in other ways too, for example by new flux coming up from the convection zone below or by photospheric twisting and flux-tube motions. Magnetic evolution is complicated since unstable equilibria are generally possible where perturbations reduce the overall energy of the system. Also local deviations may persist within field allowances before equilibria can be resolved. However if the new magnetic equilibrium after fast reconnection is mostly force-free containing rigidly held non-force-free flux tubes, outflows again must reestablish the equilibrium gas-pressure profile of a depleted cool atmosphere, applying the argument from Appendix E.
MHS restoring flows may provide an important coronal heating mechanism, which can exist consistently along with other heating processes. By offseting the total coronal energy requirement, other heating sources should add stability to the large-scale magnetic field and thus longevity to the magnetic evolution. The total coronal energy requirement is estimated conventionally using heat-flux scaling relations for loops and homogeneous regions (Hammer 1982; Hearn & Kuin 1981; Rosner et al. 1978). If sufficient magnetic energy is not available from evolutionary processes to offset losses, then pure force-free cool equilibria may be the only possibility.
Localized heating in flux tubes, which is thought to be an important coronal heating process (Vaiana & Rosner 1978; Airapetian & Smartt 1995), may affect ambient MHS restoring flows. Heated flux tubes must exhibit field-line perturbations that stress the large-scale magnetic field and tend to produce outflows in the surroundings at least on the average, which heat the ambient atmosphere above the normal equilibrium temperature. The ambient hydrostatic imbalance can have a relatively large scale of influence since it propagates horizontally away from its source much faster than thermal diffusion effects. Such post-reconnection hydrodynamic heating acting throughout the coronal volume may help overcome the often-cited difficulty that reconnection events are so spatially localized and short-lived that only a larger event rate than what is directly inferred from observations could account for coronal heating (Hudson 1991; Heyvaerts & Priest 1984).
Thus MHD processes may lead to refilling outflows, a "magnetic suction'', which might be in evidence as the large-scale flows seen in the vicinity of coronal voids (Wagner et al. 1983), or be responsible for the supergranular-scale chromospheric upward velocities interpreted from post-flare solar spectral-line blue shifts (Cauzzi et al. 1996; Schmieder et al. 1987), or appear as spicule eruptions in the chromosphere. The upward mass flux produced by solar spicule events is about 100 times the total solar-wind mass flux, but represents only a small fraction of its total energy flux (Pneuman & Kopp 1978).
The remarkable feature of an MHS corona is that the temperature is
essentially a geometric parameter defined by the stellar surface
gravity ,
radius
,
mean particle mass
,
and
magnetic spherical harmonic
Formulae for the average base coronal temperature like Eq. (23) have been suggested already in the hypothesis of a "geometric boundary condition'', relevant for different kinds of heating models where the base pressure scale height is proportional to the stellar radius (Scudder 1992; Menzel 1968). The relation appears to give reasonable coronal temperatures for a range of stellar types (Williams & Mullan 1996).
Acknowledgements
The author is grateful to many for useful discussion during the course of this work, especially to Eric Priest and Ray Smartt for their careful reviews of the manuscript, and to the referee for many insightful remarks and helpful suggestions.
There are well-known solutions to the classical static problem
describing a gas existing in conjunction with elongated magnetic
fields without a perturbing gravitational field (Parker 1979, Chap. 6). It is helpful to revisit the classical problem
here in a general way. The classical MHS equilibrium equations
without gravity are the static equilibrium equation
Plane-parallel solutions can be developed based upon the general
Cartesian vector function
Any solutions to Eq. (A.1) must satisfy the two
conditions
and
,
which are written with
from Eq. (A.2)
The two Eqs. (A.3) and (A.4) can be used to
write two of the functions in terms of the third, for example
and
in terms of
.
The third
relation, Eq. (A.1) in the vector direction perpendicular to
both
and
,
gives a
differential relation for the third quantity. The separate vanishing
of every commutator term is a reduction that is satisfied with
certain general functional dependencies between the variables and
leads to a well-known self-consistent form for the third relation,
the Grad-Shafranov Equation. Other possibilities are not considered
here.
Taking every commutator term to vanish in Eqs. (A.3) and (A.4), but without other reduction, gives
,
,
and spatial derivatives of
and
as functions of
alone, which is only
possible with specific spatial forms for real functions, so other
reductions are required in general. Taking
gives a consistent solution defined by the functional dependencies
a=a(p) in Eq. (A.3) and
in Eq. (A.4). A similar
solution is obtained by taking
.
These two
solutions or linear combinations are equivalent with rotation of the
x - z axes around y. Taking z as the degenerate direction
gives the usual 2D solution, ordinarily written with a as the
implicit function
It is widely believed that this 2D solution is the only plane-parallel one. Using perturbative expansions, Parker (1979, Sect. 14.2) shows that no nearby 3D solutions exist for bounded quantities in an infinite spatial domain. However 3D variations do arise as large-scale deviations from plane parallel (Arendt & Schindler 1988). In the common shorthand, "2.5D'' solutions refer to perturbed solutions in the problem with gravity, which admit large-scale variations in the degenerate direction of the magnetic field.
Substituting
from Eq. (A.5) into Eq. (A.1)
gives the governing equation for the implicit magnetic potential
function a(x,y) for the static equilibrium problem without gravity
The total pressure P(a) and its component functions p(a) and
Bz(a) are all one dimensional and nonlinear, but must all be
well-defined everywhere in the solution domain consistent with their
forms at the boundaries. Strictly, disagreeing z boundary
conditions are inconsistent with the z independence of the
solutions; differences might produce small deviations in the
solutions like twist or divergence or be a source of dynamical
instability. It is popular to restrict consideration to entirely
force-free solutions with p'(a)=0; the restriction does not change
the nature of the basic GS Eq. (B.2) for a(x,y) but requires
the specific current density
for an arbitrary Bz(a), as is evident by expanding
using
from
Eq. (A.5) with
.
The magnetic potential function a can be seen to be constant
everywhere in the local plane of
and
.
From Eq.
(A.1),
.
Then
.
Thus
at least when
,
and when p'(a)=0,
so
anyway.
We think of a as constant on ribbon-like current sheets, the
solution surfaces being more curved along
as it is defined
by the derivatives of
.
The GS Eq. (B.2) has the unusual feature that it contains the
1D filter function P'(a). The Fourier transform of the GS equation
in the linear case, with
P'(a)= k02 a for k0 a constant
wavenumber, requires a transform function
that is
zero everywhere in its 2D wavenumber domain
except on a thin annulus at
,
where arbitrary complex
values Hermitian in
are allowed. Taking the Fourier
transform back gives the potential function a(x,y) as a common
Bessel-function radial kernel times azimuthal factor convolved with a
spatial distribution of delta-function source points. Boundary
conditions on a(x,y) constrain the azimuthal factor, leading to
possible planar solutions defined by all power at one azimuth, or
axisymmetric solutions with power uniformly distributed in azimuth,
consistent with our visualization of current sheets and cylindrical
flux tubes. Physical arguments show that nonlinear kernels are
similarly constrained by boundary conditions
(Vainshtein & Parker 1986).
Random spatial distributions of a common GS kernel are nonlinear GS
solutions too. We take the potential function a(x,y) to be the
convolution of a distribution of sources D(x,y) with a common 2D
kernel function A(x,y),
a(x,y)= D(x,y)* A(x,y). The distribution
is written
counting sources
j of varying strength cj, where
denotes the 2D
delta function. For a spatially incoherent or random distribution
D of equal strength sources with all cj=1, D raised to a power
is D alone; for equal amplitude sources with
,
Draised to an odd power is D alone. For an analytic nonlinear driver
function P(a) the convolution factors out
P'(a)= P'(D*A)=
D*P'(A), when the distribution in D is incoherent, disjoint, and
contains suitably restricted amplitudes. Then the GS Eq. (B.2) reduces to the same GS equation, but for the common
kernel function A in place of a. Dark and bright threads appear
to be isolated thin flux tubes aligned with the background magnetic
field organized in current sheets suggestive of a bimodal amplitude
distribution of a common flux-tube kernel.
Parker (1972) considers MHS equilibria that contain a
relatively strong constant background field. Expanding the Lorentz
term in Eq. (A.1) gives an alternate form for the MHS
equation
The Parker theorem Eq. (B.5) reflects the special features
of the classical equilibria in Eq. (A.5) in the degenerate
direction of the magnetic field z. MHS equilibria allow only
certain alignments for the background field. A constant magnetic
field can enter into the z vector element Bz(a) without
producing other ramifications, whereas an added offset in the x or
y elements of
requires adding a uniformly inclined plane
to a(x,y). Such a plane limits the solutions and thus cannot be
considered general, and even appears to be precluded by the
boundedness of a(x,y): low-wavenumber components in the Fourier
domain needed to represent the added plane are at odds with the
Fourier ring solutions in the linear case of the GS Eq. (B.2), and also contrary to the form of a superposition of
spatially compact axisymmetric nonlinear kernels.
The less-significant order n>0 equations in the perturbative series
developed from Eqs. (13) and (14) are
written for
;
from the MHS Eq. (13)
A wide latitude is allowed for choosing perturbative quantities, and
as long as they are relatively small almost everywhere the standard
procedure remains valid. Van Ballegooijen (1985, Appendix
A) questions the Parker (1972) choice of
perturbative quantity and equation separation procedure, but van Ballegooijen's assigning of approximate magnitudes to individual
terms to separate his equations is not an algebraic procedure, and
his resulting equations have inherent contradictions as described by
Parker (1987). Parker (1972) does not ascribe a
particular functional meaning to his kernel ,
quoting from
the beginning of his Sect. II: "Expand the field
and
pressure p in ascending powers of some parameter
,
which is of the order of
", where
is the small spatially varying magnetic field on the
uniform background field
.
Our choice for perturbative
kernel d/h(r) with d constant leads to a convenient separation of
orders.
The perturbative procedure is a functional separation method rooted
in the natural independence of the basis functions
irrespective of their real amplitudes. If the parameter
is small almost everywhere, the series solutions will be convergent.
There could be locations where the amplitudes of components seem to
be of differing orders as van Ballegooijen contends. Where a
normally small expansion parameter
actually becomes
relatively large series expansions may be divergent, but such
behavior characteristically defines the isolated singularities of
differential equations as discussed in theorems on the Frobinius
method for series solutions.
A polytrope atmosphere is defined by the pressure relation
for the polytrope adiabat
.
Thus
the 1D polytrope scale height
can be
written
Substituting h(r) into the hydrostatic Eq. (9) defines
the scale-height function for a polytrope
If the magnetic field is dominant over the gas, and either the field remains stationary, or the magnetic adjustment time is much shorter than the hydrodynamic adjustment time, then the Lorentz term in the MHD equation can be treated as constant in time and the hydrodynamics separate from the magnetodynamics.
Fast magnetic adjustment is a remarkable feature of solar
reconnection phenomena. Sporadic crossings of coronal loops in
low-energy flare events (Lin et al. 1992) suggest "X''-type
configurations, and the instability of the "X'' configuration is well
known to form a new current sheet at near the Alfvén velocity if
a nonzero resistivity can be affected in the region
(Dungey 1958; Syrovatskii 1981); such a resistivity is
predicted in some models (Petschek & Thorne 1967; Priest 1972). Taking a field strength of
G as
representative of chromospheric magnetic fields near neutral lines in
solar active regions where flares commonly occur, we obtain an
Alfvén speed
cm s-1 (with
g cm-3, using
N=1012 cm-3 and
g). The Alfvén speed
stays fairly constant into the corona giving a collapse time of
s for the propagation of the instability
through a magnetic scale height
cm, using a primary magnetic spherical
harmonic of
in Eq. (21).
The hydrodynamic response to already present or newly formed isolated
thin non-force-free flux tubes can be obtained by perturbative
analysis of the dynamical equations projected along the magnetic
field. Taking the velocity to be aligned with the magnetic field and
of lesser order than the mean quantities adds a single
time-derivative term to the projected MHS Eq. (3), which
is written with the continuity equation
Perturbing the quantities backward in time, p=p0-p1 and
,
from a final equilibrium state
leaves the hydrostatic relation Eq. (9) with p0 coupled
to the magnetic field through the cross-field Eq. (11),
and three coupled perturbative equations for
The unperturbed density
varies on a scale similar
to the gas pressure p0, since the gravity g and temperature
contained in h vary on much larger scales; thus the derivative is
approximated
.
The perturbed quantities must vary on the scale of
the magnetic field
,
giving
.
Substituting for the time derivative
lets us solve for the hydrodynamic timescale