Solar Phys (2007) 246: 31–39
DOI 10.1007/s11207-007-9072-9
How to Reach Superequipartition Field Strengths
in Solar Magnetic Flux Tubes
A. Ferriz-Mas · O. Steiner
Received: 4 February 2007 / Accepted: 5 October 2007 / Published online: 23 November 2007
© Springer Science+Business Media B.V. 2007
Abstract A number of independent arguments indicate that the toroidal flux system responsible for the sunspot cycle is stored at the base of the convection zone in the form of
flux tubes with field strength close to 105 G. Although the evidence for such strong fields
is quite compelling, how such field strength can be reached is still a topic of debate. Flux
expulsion by convection should lead to about the equipartition field strength, but the magnetic energy density of a 105 -G field is two orders of magnitude larger than the mean kinetic
energy density of convective motions. Line stretching by differential rotation (i.e., the “
effect” in the classical mean-field dynamo approach) probably plays an important role, but
arguments based on energy considerations show that it does not seem feasible that a 105 G field can be produced in this way. An alternative scenario for the intensification of the
toroidal flux system in the overshoot layer is related to the explosion of rising, buoyantly
unstable magnetic flux tubes, which opens a complementary mechanism for magnetic-field
intensification. A parallelism is pointed out with the mechanism of “convective collapse”
for the intensification of photospheric magnetic flux tubes up to field strengths well above
equipartition; both mechanisms, which are fundamentally thermal processes, are reviewed.
Keywords Sun: magnetic field · Sun: flux tubes · Sun: convection zone
A. Ferriz-Mas
Departamento de Física Aplicada, Universidad de Vigo, 32004 Orense, Spain
A. Ferriz-Mas ()
Astronomy Division, Faculty of Physical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulu,
Finland
e-mail: antonio.ferriz@oulu.fi
O. Steiner
Kiepenheuer-Institut für Sonnenphysik, Schöneckstr. 6, 79104 Freiburg, Germany
e-mail: steiner@kis.uni-freiburg.de
32
A. Ferriz-Mas, O. Steiner
1. Introduction
Observational and theoretical arguments support the idea (e.g., Spiegel and Weiss, 1980;
Galloway and Weiss, 1981) that the magnetic flux responsible for the cycle of sunspots
is stored in a subadiabatic overshoot layer below the convection zone (for a review of these
arguments see, e.g., Schüssler and Ferriz-Mas, 2003). It seems plausible that, at least at some
stage prior to eruption, this storage takes place in the form of isolated bundles of magnetic
flux.
Numerous studies indicate that the strength of the stored toroidal flux tubes is close to
105 G. As far as we know, this value was first suggested by van Ballegooijen (1982), who
considered flux tubes in hydrostatic equilibrium extending across the convection zone. Flux
tubes with weaker fields would erupt at too high latitudes owing to the action of the Coriolis
force (Choudhuri and Gilman, 1987). Large fields are required to account for the observed
tilt angles of active regions (e.g., Caligari, Moreno-Insertis, and Schüssler, 1995; Fisher et
al., 2000), to avoid excessive weakening of rising flux tubes – the so-called explosion of flux
tubes (Moreno-Insertis, Caligari, and Schüssler, 1995) – and to give account of the coherence
of sunspots. Completely independent support comes from a linear stability analysis of flux
tube equilibria, which shows that a threshold value for buoyancy instabilities (Spruit and
van Ballegooijen, 1982) of thin flux tubes in the overshoot region is close to 105 G (e.g.,
Moreno-Insertis, Schüssler, and Ferriz-Mas, 1992; Ferriz-Mas and Schüssler, 1993, 1995).
Additional and independent support for superequipartition field strengths at the base of
the convection zone may come from helioseismology. Observations show that the frequencies of the solar p modes vary with the solar cycle (e.g., Woodard and Noyes, 1985). Conventional wisdom nowadays (e.g., Dziembowski and Goode, 2005) is that the measured
frequency shift is due to changes in the physical conditions near the solar surface. Nevertheless, because the acoustic frequencies depend on the internal structure of the Sun, it has
also been conjectured that the frequencies of p modes might also be modified by a magnetic
field in the solar interior evolving over the solar cycle (e.g., Roberts and Campbell, 1986).
Chou and Serebryanskiy (2005) infer from the observed frequency shifts a field in the region
near the bottom of the solar convection zone with a strength of a few times 105 G (see also
Serebryanskiy and Chou, 2005); these results are still controversial because the magnetic
field at the bottom of the solar convection zone is rather weak, in the sense that the ratio β
of the gas to the magnetic pressure is very large (β ≈ 105 for B ≈ 105 G).
How can field strengths of 100 kG be reached in flux tubes at the bottom of the convection zone? Moreno-Insertis, Caligari, and Schüssler (1995) and later Rempel and Schüssler
(2001) have investigated a possible intensification mechanism that does not rely on mechanical line stretching through shear motions but is of thermal origin: the explosion of rising
flux tubes. This explosion brings about a matter and an entropy upflow from the overshoot
layer below the convection zone into higher levels of the convection zone and thus produces
an intensification of the magnetic field in the nonexploded, anchored part of the flux tube.
In this paper (Section 2) we discuss some energy estimates for the operation of the dynamo in the subadiabatic overshoot layer – which more or less coincides spatially with the
tachocline – and conclude that an alternative (or, at least, a complementary) mechanism to
magnetic line stretching by shear flows must be considered. In Section 3 the fundamentals of
the explosion of flux tubes rising quasi-statically in a superadiabatically stratified medium
are reviewed. The situation is reminiscent of the intense magnetic elements in the photosphere (Section 4): by mechanical processes alone it is not possible to produce field strengths
of 1.5 kG in photospheric flux tubes. As a possible explanation of how this superequipartition (at photospheric levels) field strength can be reached, the mechanism of convective
Superequipartition Flux Tubes in the Sun
33
collapse was considered by Parker (1978) and others. In Section 5 we propose a parallelism
between the “convective collapse” of photospheric flux tubes and the “explosion” of large
flux ropes that rise from the subadiabatic layer of overshooting convection just below the
solar convection zone. In both cases – convective collapse and explosion mechanism – the
ambient superadiabatic stratification is an essential ingredient and both are fundamentally
thermal processes leading to superequipartition field strengths. We present a brief but selfconsistent summary of previous work on these two mechanisms and put forward the suggestion that the explosion mechanism can be understood as an inverse convective collapse.
2. Energy Budget of Differential Rotation in the Tachocline
Although the existence of strong magnetic flux tubes (of the order of 105 G ) at the bottom
of the convection zone is widely accepted (but see, e.g., Brandenburg, 2005; Arlt, Sule,
and Rüdiger, 2007), it is by no means clear how superequipartition field strengths can be
reached. Flux expulsion by convection should lead to about equipartition field strength, that
is, a magnetic energy density equal to the mean kinetic energy density of convective motions
(equivalent to about 104 G at the bottom of the convection zone), but the magnetic energy
density of a 105 -G field is a factor 100 larger than that of equipartition.
The available kinetic energy for the intensification of the toroidal field is that from differential rotation (i.e., the difference in kinetic energy between the actual rotation and rigid
rotation with equal rotational momentum). Since the layer of overshooting convection below
the convection zone coincides spatially, more or less, with the tachocline, a widespread idea
is that differential rotation in the tachocline can obviously generate a toroidal field (the socalled effect in the classical mean-field approach). But an inspection of the energy budget
shows (e.g., Rempel, 2001) that it is not at all obvious how through pure mechanical stresses
(i.e., line stretching) a field strength of 105 G can be reached.
Consider a thin spherical shell of thickness d at the bottom of a convection zone, located
between the radii R − d and R (d R). If we call Σ a semisection of the spherical shell by
a half-plane containing the rotation axis and n is a normal unit vector, then the (unsigned)
flux of the toroidal field through Σ is
π
|B · n| = f B R 2 − (R − d)2 ≈ πf dRB,
(1)
Φ =f
2
Σ
where B = B is the field strength inside a flux tube and f is the volume filling factor of
the toroidal flux tubes. The magnetic energy contained in form of toroidal flux tubes in the
thin shell is
Emag =
B2
R
4π 3
B2
R − (R − d)3
f ≈ 4πf dR 2
=
ΦB.
3
8π
8π
2π
(2)
Galloway and Weiss (1981) estimated the magnetic flux emerging at the solar surface over
one cycle to be Φ ≈ 1024 Mx and, thus, if one assumes that this flux is stored in tubes with
strength B ≈ 105 G, the estimated magnetic energy over one cycle long is Emag ≈ 1032 J, a
result that is independent of the thickness d (as far as d > dmin ≈ 106 m since it must be
f ≤ 1).
Next we estimate the energy contained in the differential rotation of a layer of thickness
d assuming uniform density ρ0 and the velocity profile of a Couette flow, that is,
v=
v0
x
d
for −
d
d
≤x≤ .
2
2
(3)
34
A. Ferriz-Mas, O. Steiner
The available kinetic energy in the shear layer is then
Ediff ≈ 2πR 2 ρ0
d/2
−d/2
v 2 dx =
π 2
R dρ0 v02 .
6
(4)
By choosing v0 = 100 m s−1 (e.g., Charbonneau et al., 1999), Equation (4) yields a value of
Ediff ≈ 1031 J. But this is an upper bound for the kinetic energy available for conversion into
magnetic energy because not all the tachocline (i.e., the shear layer) is available for the storage of magnetic flux tubes (since the tubes cannot penetrate the part of the tachocline inside
the upper part of the radiative zone). Therefore, because the kinetic energy contained in the
differential rotation of the shear layer is one order of magnitude smaller than the magnetic
energy of the toroidal flux generated within one 11-year cycle, a very efficient mechanism of
angular momentum transport from the bulk of the convection zone into the tachocline would
be required for the continuous replenishment of the shear layer (i.e., a mechanism that would
continuously feed energy into the differential rotation). Although it is not impossible that
this mechanism may exist, an alternative explanation seems to be more feasible.
3. Flux Tube Explosions
An alternative (or complementary) scenario for the intensification of the toroidal flux system
stored in the overshoot layer is related to the explosion of rising magnetic flux tubes. The
phenomenon, which was first described by Moreno-Insertis, Caligari, and Schüssler (1995),
is caused by the nearly adiabatic evolution of a flux tube rising in the convection zone, which
is superadiabatically stratified.
A brief reminder of the concept of superadiabaticity may be in order here. The superadiabaticity is simply a dimensionless measure of the entropy gradient in the vertical (or radial)
direction when pressure is used as vertical (or radial) coordinate, that is,
δ≡
p ds
,
cp dp
(5)
where s is the specific entropy (i.e., per unit mass) and cp is the specific heat at constant
pressure. A better known derivative of this kind is the logarithmic temperature gradient (∇)
defined as
p dT
,
(6)
∇≡
T dp
whereby temperature is considered as a function of pressure (i.e., the vertical coordinate
z has been replaced by p). Observe the parallelism between definitions (5) and (6), but
notice that in definition (5) the specific heat at constant pressure (which has the same units
as the specific entropy) has been chosen to nondimensionalize the derivative. In the case
of a homoentropically stratified medium (inaccurately called “an adiabatic stratification”
in most of the astrophysical and geophysical literature), the corresponding dimensionless
temperature gradient is denoted ∇ad . It is straightforward to show that the superadiabaticity
and the logarithmic temperature gradients ∇ and ∇ad are related by
δ = ∇ − ∇ad ,
which is taken in most textbooks as the definition of the superadiabaticity.
(7)
Superequipartition Flux Tubes in the Sun
35
The explosion mechanism can be qualitatively understood by using the thin-flux-tube
approximation (e.g., Roberts and Webb, 1978), which permits the reduction of the full set of
magnetohydrodynamic equations to a mathematically more tractable form without making
any restriction on the compressibility of the plasma or on the magnetic Lorentz force. One of
the basic assumptions of the approximation is the condition of instantaneous lateral pressure
balance between the interior of the flux tube and surroundings, that is,
B2
+ pi = p e ,
8π
(8)
where pi is the internal gas pressure and pe is the gas pressure of the field-free environment.
In what follows we consider the simplified analytical treatment of Schüssler and FerrizMas (2003) for a flux tube rising quasi-statically in a superadiabatic medium. Consider the
convection zone represented by a plane-parallel medium and call z the vertical coordinate;
if the logarithmic temperature gradient ∇ is uniform throughout the medium, then
z ∇ 1/∇
,
pe (z) = pe,0 1 −
He,0
(9)
where He,0 is the pressure scale height in the convection zone at the level z = 0. Assuming
that the plasma inside the flux tube was initially also homoentropic and that the evolution
of the flux tube is adiabatic (owing to the very long thermal exchange time with its environment), the internal gas pressure as a function of height is given by:
z∇ad 1/∇ad
,
pi (z) = pi,0 1 −
Hi,0
(10)
where Hi,0 is the pressure scale height inside the flux tube. If we further assume that the
plasma may be described thermodynamically by the constitutive relation of the ideal gas,
then it is straightforward to compute the temperature difference between the interior and the
exterior of the flux tube:
T (z) ≡ Ti (z) − Te (z) = Ti0 − Te0 +
μg
R
(∇ − ∇ad )z,
(11)
where R is the universal gas constant, g is the gravitational acceleration, and μ is the mean
mass per mole of free particles. From Equation (11) it follows that, since the ambient stratification is superadiabatic (δ > 0 ⇐⇒ ∇ > ∇ad ), the internal pressure scale height eventually
becomes larger than the external pressure scale height.
If a flux tube rises adiabatically in a superadiabatically stratified medium, the internal
gas pressure decreases less rapidly with height z than the ambient pressure; therefore there
is a critical height zc at which pi (zc ) = pe (zc ). The pressure balance condition (8) implies
B 2 /8π → 0 when z → zc . Conservation of magnetic flux (i.e., πR 2 B = const.) demands
that the radius (R) of the flux tube tends to ∞ for z → zc , which implies that the top of
the rising loop explodes; this is schematically shown in Figure 1 (left panel). Numerical
simulations of rising flux tubes confirm the simple ideas presented here; Figure 1 (right
panel) shows the result of a numerical simulation by Rempel (2001).
The explosion implies an intensification of the magnetic field strength of the part of the
flux tube that remains anchored in the subadiabatic overshoot layer. The mechanism works
as follows: The abrupt weakening of the magnetic field in combination with the dynamic
rise of the loop results in matter being “sucked up” from below into the exploding part and
36
A. Ferriz-Mas, O. Steiner
Figure 1 Explosion of a rising flux tube. If a flux tube rises adiabatically in a superadiabatically stratified
medium, the internal gas pressure decreases less rapidly with height than the ambient pressure and thus for a
critical height (zc ) at which pi (zc ) = pe (zc ) the tube explodes. This is shown schematically in the left panel;
the result from a numerical simulation by Rempel (2001) is presented in the right panel.
therefore a decrease in the gas pressure of the submerged part of the tube ensues. As a consequence, an intensification of the magnetic field strength takes place in the submerged part
according to Equation (8). The explosion implies that the flux tube loses its coherence and
becomes passive with respect to convective motions, so that its further evolution is no longer
describable with the thin-flux-tube approximation. Full two-dimensional numerical simulations of unstable magnetic sheets in Cartesian geometry (Rempel and Schüssler, 2001) find
a significant intensification of the remaining flux tube after the explosion, which closely
follows this picture.
Do all rising flux tubes explode? The value of the critical height (zc ) depends on the initial field strength at z = 0. The explosion mechanism is interesting only if it occurs before a
rising flux tube approaches the photosphere. Stronger fields lead to a higher initial pressure
deficit inside the tube and thus to a larger explosion height. A field strength of B ≈ 104 G
results in the explosion already in the lower half of the convection zone. Flux tubes with
initial B ≈ 105 G can reach the surface without exploding. Since a linear stability analysis
of flux tube equilibria in the overshoot layer shows that a threshold value for instability is
close to ≈ 105 G we have to conclude that some nonlinear mechanisms must be invoked that
destabilize equipartition-strength flux so that the tubes enter the convection zone and buoyantly rise toward the surface so that they can “explode” and thus intensify the submerged
part.
4. Convective Collapse
The magnetic field in the photospheric layers of the Sun is distributed in a very inhomogeneous way, with sunspots being the most relevant manifestation of magnetic flux at the solar
surface, but not the only one. The advent of the magnetograph in the 1950s (Babcock, 1953;
Kiepenheuer, 1953) allowed the discovery of smaller magnetic structures at photospheric
levels. Among photospheric weak-field regions there exist intense small-scale concentrations of magnetic flux, magnetic elements, with a characteristic size of say 100 km and field
strength close to 1.5 kG, generally located in intergranular lanes.
Superequipartition Flux Tubes in the Sun
37
Figure 2 An adiabatic
downflow within an initially
weak magnetic flux tube (dashed
contours) embedded in a
superadiabatically stratified
medium leads to an increasing
pressure deficit with depth, which
must be balanced by a
corresponding field
intensification. As a result, a
strong magnetic flux tube (solid
contours) ensues.
How these field strengths can be achieved in photospheric flux tubes is not a trivial issue:
Flux expulsion by convection should lead to about equipartition field strength (about 700 G),
but the magnetic energy density of a 1.5-kG field is more than a factor four larger than the
mean kinetic energy density of convective motions of granules and therefore the question
arises as to how these high field strengths are attained. The interaction of convective motions
with a magnetic field leads to an intermittent distribution of magnetic flux (see, e.g., Proctor
and Weiss, 1982). Spruit and Zweibel (1979) suggested that the resulting intermittent structures are prone to a convective instability with the consequence that they eventually collapse
to kilogauss intensity.
The instability sets in as a downward flow, the tube partially evacuates and cools with
respect to its surroundings, and its field strength increases; kilogauss field strengths are necessary before the collapse is halted. An exact determination of the field strength required to
quench the instability depends on the precise details of the modeling: the thermodynamic
equation of state, appropriate choice of the boundary conditions, and consideration of nonlinearities. The effects of radiative transfer may be very important (see, e.g., Venkatakrishnan, 1986; Rajaguru and Hasan, 2000).
At the basis of the “convective collapse instability” is, as in the case of the “flux tube
explosion” (Section 3), the superadiabaticity of the ambient stratification. The gas pressure
in the adiabatic downflow increases less rapidly with depth than the gas pressure in the
ambient medium, leading to a field intensification according to Equation (8).
The convective collapse of “weak” photospheric flux tubes has been studied by Webb and
Roberts (1978), Spruit and Zweibel (1979), and a long list of authors. For a general, updated
account of the problem see, e.g., Steiner (1999), Roberts (2001), and Steiner (2003). BellotRubio et al. (2001) find spectropolarimetric evidence of convective collapse in photospheric
flux tubes.
5. Discussion and Conclusions
Although there are strong, independent arguments in favor of intense (superequipartition)
toroidal flux tubes with fields of the order 105 G at the bottom of the convection zone, it
38
A. Ferriz-Mas, O. Steiner
is yet unclear how these high field strengths can be achieved. Arguments based on energy
considerations (Section 2) show that the kinetic energy contained in the differential rotation
of the part of the tachocline available for the storage of these flux tubes could hardly account
for all the magnetic energy generated over one solar cycle. The situation is in some respects
similar to the one posed to solar physicists more than two decades ago when the origin
of superequipartition fields in photospheric magnetic elements had to be accounted for. A
possible explanation for these intense fields in photospheric flux tubes is the mechanism of
the “convective collapse,” which relies on an instability that results in a partial evacuation
of the flux tube through a downflow and a concomitant intensification of the field. Similarly,
the “explosion” mechanism is a process that takes place a in buoyantly unstable flux tube
and involves an upflow and also a partial evacuation of the tube.
In this paper we call attention for the first time to an interesting parallelism between the
“convective collapse” of photospheric flux tubes and the “explosion” of large flux ropes that
rise from the subadiabatic layer of overshooting convection just below the solar convection
zone. In both cases, the ambient superadiabatic stratification is an essential ingredient and
both are fundamentally thermal processes leading to superequipartition field strengths. The
explosion mechanism can be understood as an “inverse convective collapse.” The convective
collapse plays a role in the context of the local dynamo similar to the explosion of large
toroidal flux tubes in the context of the global dynamo.
The existence of superequipartition fields at the bottom of the convection zone requires
a modification of the classical mean-field dynamo theory, whose main ingredients are the
α effect, the effect, and turbulent diffusivity, and which does not involve thermal effects
(and in most studies the plasma is even assumed to be incompressible). The “explosion”
of rising flux tubes (Moreno-Insertis, Caligari, and Schüssler, 1995; Rempel and Schüssler,
2001) is at variance with the classical “ effect,” which is basically a kinematic approach;
the effect probably plays a role in shaping the toroidal structures, but it ceases being
an efficient mechanism of toroidal field intensification when the field strength approaches
equipartition.
Note Added in Proof Clear signature of the “convective collapse” has been recently discovered with Hinode (S. Tsuneta, private communication, to appear as Nagata et al., 2007, in preparation).
Acknowledgements We are grateful for the kind hospitality of the Isaac Newton Institute for Mathematical Sciences, where various aspects of this work were discussed. We thank Michael Stix and John Thomas
for insightful discussions and an anonymous referee for useful comments. Antonio Ferriz-Mas has received
financial support from the Spanish Ministery of Science and Education (project AYA2006-26999-E).
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