Download Coronal activity from dynamos in astrophysical

Mon. Not. R. Astron. Soc. 318, 724±732 (2000)
Coronal activity from dynamos in astrophysical rotators
Eric G. Blackman1,2w and George B. Field3
1
Theoretical Astrophysics, Caltech 130-33 Pasadena, CA 91125, USA
Department of Physics & Astronomy, University of Rochester, Rochester, NY 14627, USA
3
Harvard-Smithsonian Center for Astrophysics (CFA), 60 Garden St., Cambridge, MA 02139, USA
2
Accepted 2000 June 5; Received 2000 June 5; in original form 1999 December 29
A B S T R AC T
We show that a steady mean-field dynamo in astrophysical rotators leads to an outflow of
relative magnetic helicity and thus magnetic energy available for particle and wind
acceleration in a corona. The connection between energy and magnetic helicity arises
because mean-field generation is linked to an inverse cascade of magnetic helicity. To
maintain a steady state in large magnetic Reynolds number rotators, there must then be an
escape of relative magnetic helicity associated with the mean field, accompanied by an equal
and opposite contribution from the fluctuating field. From the helicity flow, a lower limit on
the magnetic energy deposited in the corona can be estimated. Steady coronal activity
including the dissipation of magnetic energy, and formation of multi-scale helical structures
therefore necessarily accompanies an internal dynamo. This highlights the importance of
boundary conditions which allow this to occur for non-linear astrophysical dynamo
simulations. Our theoretical estimate of the power delivered by a mean-field dynamo is
consistent with that inferred from observations to be delivered to the solar corona, the
Galactic corona, and Seyfert 1 AGN coronae.
Key words: accretion, accretion discs ± magnetic fields ± Sun: corona ± stars: coronae ±
stars: magnetic fields ± galaxies: Seyfert.
1
INTRODUCTION
Understanding both the origin and destruction of magnetic fields is of fundamental importance to astrophysics. Not only are magnetic fields
astrophysical entities in and of themselves, but they play an important intermediary role between gravitational energy and radiation in
rotating systems such as the Sun, galaxies, and accretion discs. In this paper we will explore a link between the origin of large-scale
magnetic fields from dynamo action in rotating systems and the export of magnetic energy into a corona which can dissipate and accelerate
particles.
Large-scale magnetic fields are observed in the Sun and in spiral galaxies. The solar field changes on time-scales much shorter than
would be allowed if the time-scale of dissipation were governed by resistivity alone. The presence of a turbulent solar convection zone leads
naturally to the conclusion that an effective turbulent diffusivity must be at work. However, for the field to maintain its strength in the
presence of turbulent diffusion, exponential amplification of the large-scale field must occur. In the Galaxy the argument is similar ± if the
turbulent ISM effectively diffuses magnetic field, then the large-scale micro-gauss fields must be somehow sustained. As the role of
magnetic turbulence in accretion discs is thought to be fundamental for angular momentum transport, the rotating turbulent media of
accretion discs are also plausible sites for a similar mechanism.
The leading, but somewhat controversial, candidate to explain the origin of mean magnetic flux growth in stars and galaxies is the
mean-field turbulent magnetic dynamo theory (cf. Moffatt 1978; Parker 1979; Krause & RaÈdler 1980; Zeldovich et al. 1983; Ruzmaikin,
Shukurov & Sokoloff 1988; Beck et al. 1996). The theory appeals to a combination of helical turbulence (leading to the a effect),
differential rotation (the V effect), and turbulent diffusion to exponentiate an initial seed-mean magnetic field. Steenbeck, Krause & RaÈdler
(1966) developed a formalism for describing Parker's (1955) concept that helical turbulence can twist toroidal fields into the poloidal
direction, where they can be acted upon by differential rotation to regenerate a powerful large-scale magnetic field. Their formalism
Å and a fluctuating scale component b, and similarly for the velocity field
involved breaking the total magnetic field into a mean component B
V. The mean can be a spatial mean or an ensemble average. For comparison with observations of a single astrophysical system, the
ensemble average is approximately equal to the spatial average when there is a scale separation between the mean scale and the fluctuating
w
E-mail: [email protected]
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Coronal activity from dynamos in astrophysical rotators
725
scale. Otherwise, the ensemble average need not correspond to an average that characterizes the specific realization of the ensemble
corresponding to the object of interest. The scale separation is often not as good as the dynamo theorist would like. Nevertheless, we will
proceed to consider the averages as spatial averages to get at the main points.
Å satisfies the induction equation
Steenbeck et al. (1966) showed that B
­B
ˆ 2c7 E;
…1†
­t
where
E ˆ 2…V B†=c 2 kv bl=c ‡ l7 B;
…2†
where the first term describes the effect of differential rotation (`V effect'),
kv bl ˆ aB 2 b7 B
…3†
is the `turbulent emf,' and l ˆ c2 =4ps is the magnetic diffusivity defined with the conductivity s . Here a represents Parker's twisting (`a
Å to first order in BÅ and hence the dynamo coefficients a and b
effect') and b (@l ) is the turbulent diffusivity. Steenbeck et al. calculated E
Å from the statistics of the turbulence. They ignore the Navier±Stokes equation even to linear order in BÅ which is therefore
to zero order in B
Å and V
Å was calculated by Blackman &
a completely kinematic approach. (The back-reaction on the dynamo coefficients to first order in B
Chou (1997), and Field et al. (1999) have calculated a to all orders in BÅ for the case when mean-field gradients are small.)
When (2) is substituted into (1), we have the mean-field dynamo equation:
­B
‡ 7 …aB†
2 7 …b ‡ l†7 B:
ˆ 7 (V B†
…4†
­t
Å which can be solved as an eigenvalue problem for
In the approximation that VÅ, a and b are independent of BÅ, (4) is a linear equation for B
the growing modes in the Sun and other bodies. Actaully, a rapid growth of the fluctuating field necessarily accompanies the mean-field
dynamo. Its impact upon the growth of the mean field, and the impact of the mean field itself on its own growth are controversial.
The controversy results because Lorentz forces from the growing magnetic field react back on and complicate the turbulent motions
driving the field growth (e.g. Cowling 1957, Piddington 1981, Kulsrud & Anderson 1992; Kitchatinov et al. 1994; Cattaneo & Hughes
1996; Vainshtein 1998). It is tricky to disentangle the back-reaction of the mean field from that of the fluctuating field. Analytic studies and
numerical simulations seem to disagree as to the extent to which the dynamo coefficients are suppressed by the back-reaction of the mean
field.
Pouquet, Frisch & Leorat (1976) showed, from the numerical solution of approximate equations describing the spectra of energy and
„
helicity in MHD turbulence, that the a effect conserves magnetic helicity …ˆ …A ´ B† d3 x†; by pumping a positive (negative) amount to
scales .L (the outer scale of the turbulence) and pumping a negative (positive) amount to scales !L, where it is subject to Ohmic
Å of Steenbeck et al. (1966). Thus, dynamo action leading to an ever
dissipation. They identified magnetic energy at the large scale with the B
larger BÅ, hence the creation of ever more large scale helicity, can proceed as long as small-scale helicity of opposite sign can be dissipated
by Ohmic diffusion.
The fate of small-scale helicity is debated. According to the non-linear solutions of Pouquet et al. (1976), it cascades to large
wavenumbers where it is destroyed by Ohmic dissipation. According to several authors (Cattaneo & Hughes 1996; Gruzinov & Diamond
1994; Seehafer 1994) the necessity for this process limits the buildup of large-scale helicity, and hence, large-scale magnetic fields. This
would effectively eliminate the a ±V dynamo as a practical process for creating large-scale magnetic fields in systems having a large
magnetic Reynolds number RM.
As part of an effort to investigate aspects of the back-reaction problem and the apparent differences between different simulation
results, Blackman & Field (2000) have shown that when the scale of the averaging is the scale of the simulation box, the coefficient a
attains substantial values only if field gradients and non-periodic boundary conditions are present. This was shown to be related to a flow of
magnetic helicity through the boundary of the system. In the steady state, an equal and opposite flow of large- and small-scale magnetic
helicity should escape through the boundary since the total magnetic helicity is conserved in ideal MHD.
Blackman & Field (2000) showed that differences in some apparently conflicting simulations may correspondingly result from
whether or not the boundary conditions are periodic (Cattaneo & Hughes 1996) or diffusive (Brandenburg & Donner 1996) and whether
there is significant scale separation. To test astrophysically relevant dynamo issues, periodic boundary conditions can be used if the
averaging scale is significantly smaller than the size of the box (e.g. Pouquet et al. 1976; Meneguzzi, Frisch & Pouquet 1981; Balsara &
Pouquet 1999). This is not the case in Cattaneo & Hughes (1996) who find a reduced a , but for the above reasons this reduction may result
from the boundary conditions rather than from a dynamical suppression.
The importance of boundary conditions makes it natural to estimate the magnitude of quantities deposited through the boundary in a
working dynamo. In this regard, note that the Sun, the Galaxy, and accretion discs in AGN seem to harbour steady active corona, requiring
an energy source for heating, winds, and particle acceleration. Here we point out that steady active coronae, in which magnetic helicity and
energy are deposited, naturally occur when the field of the underlying rotator is sustained by a dynamo.
In this paper we will estimate the helicity and the magnetic energy flow into a corona which accompanies a mean-field dynamo. The
energy deposition rate is bounded below from the helicity deposition rate (Frisch et al. 1975; Field & Blackman 1999). The resulting
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E. G. Blackman and G. B. Field
estimates are roughly consistent with that required of coronae in the Sun, the Galaxy, and AGN accretion discs, all systems in which the
operation of a mean-field dynamo is natural. The existence of and properties of a steadily active corona, therefore provide some selfconsistency checks and signatures of a working dynamo.
2
R E L AT I V E M AG N E T I C H E L I C I T Y F L O W A N D A S S O C I AT E D E N E R G Y F L U X
To explore the role of mean-field gradients and boundary conditions in determining the value of the a dynamo parameter, Blackman &
Field (2000) took Ohm's law
E ˆ 2c21 V B ‡ hJ
…5†
and averaged the dot product with B to find
kE ´ Bl ˆ E ´ B ‡ ke ´ bl ˆ 2c21 kv bl ´ B ‡ hJ ´ B ‡ ke ´ bl
…6†
where J is the current density and h the resistivity.
A second expression for kE´Bl also follows from Ohm's law without first splitting into mean and fluctuating components, that is
kE ´ Bl ˆ hkJ ´ Bl ˆ hJ ´ B ‡ hk j ´ bl ˆ hJ ´ B ‡ c21 lkb ´ 7 bl:
…7†
Using (7) and (6), Blackman & Field (2000) obtain
E ´ B ˆ 2c21 kv bl ´ B ˆ c21 lkb ´ 7 bl 2 ke ´ bl;
…8†
which they used to constrain kv bl:
Now consider E in terms of the vector and scalar potentials A and F:
E ˆ 27F 2 …1=c†­t A:
…9†
Dotting with B ˆ 7 A we have
E ´ B ˆ 27F ´ B 2 …1=c†B ´ ­t A:
…10†
After straightforward algebraic manipulation, application of Maxwell's equations and 7 ´ B ˆ 0; this equation implies
E ´ B ˆ 2…1=2†7 ´ FB ‡ …1=2†7 ´ …A E† 2 …1=2c†­t …A ´ B† ˆ …21=2c†­m H m . 0;
…11†
where H m ˆ …H 0 ; H i † ˆ ‰A ´ B; cFB 2 cA EŠ is the magnetic helicity density 4-vector (Field 1986) and the last similarity follows for
nearly ideal MHD according to (5).
Taking the average of (11) gives
­m H m ˆ 22ckE ´ Bl ˆ 22cE ´ B 2 2cke ´ bl . 0:
…12†
Integrating (11) over all of space, U, gives
…
…
…
…
E ´ B d3 x ˆ 2…1=2† 7 ´ FB d3 x ‡ …1=2† 7 ´ …A E† d3 x 2 …1=2c†­t A ´ B d3 x ˆ 2…1=2c†­t H …B† . 0;
U
U
U
U
…13†
where the divergence integrals vanish when converted to surface integrals. The . follows for large RM, and we have defined the global
magnetic helicity
…
A ´ B d3 x;
…14†
H …B† ;
U
where U allows for scales much larger than the mean-field scales. It is straightforward to show that a parallel argument for the mean and
fluctuating fields, respectively leads to
…
…
ˆ ­t A ´ B d3 x ˆ 22c E ´ B d3 x
­t H …B†
…15†
U
and
ˆ ­t
­t H…b†
…
U
U
ka ´ bl d3 x ˆ 22c
…
U
ke ´ bl d3 x ˆ 22c
…
U
e ´ b d3 x ˆ ­t H …b†;
…16†
where the last two equalities in (16) follow from the redundancy of averages; the volume integral amounts to averaging over a larger scale
than the inside brackets.
To estimate the energy flow implied by the deposition of magnetic helicity, we split (15) and (16) into contributions from inside and
outside the rotator. One must exercise caution in doing so because H is gauge invariant, and hence physically meaningful, only if the
volume U over which H is integrated is bounded by a magnetic surface (i.e. normal component of B vanishes at the surface), whereas the
surface separating the outside from the inside of the rotator is not magnetic in general.
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Berger & Field (1984) (see also Finn & Antonsen 1985) showed how to construct a revised gauge invariant quantity that they called the
relative magnetic helicity. This can be written as
H R;i …Bi † ˆ H …Bi ; Po † 2 H …Pi ; Po †;
…17†
where the two arguments represent inside and outside the body, respectively, and P indicates a potential field. The relative helicity of the
inside region is thus the difference between the actual helicity and the helicity associated with a potential field inside that boundary, so the
use of Pi is not arbitrary in (17). However, (17) is insensitive to the choice of external field (see discussion in Berger & Field 1984), but it is
most convenient to take it to be a potential field as is done in (17) symbolized by Po.
The relative helicity of the outer region, HR,o, is of the form (17) but with the o's and i's reversed. This quantity is invariant even if the
boundary is not a magnetic surface. Berger & Field (1984) also showed that the total global helicity, in a magnetically bounded volume
divided into the sum of internal and external regions, U ˆ U i ‡ U e satisfies
H …B† ˆ H R;o …B† ‡ H R;i …B†;
…18†
when the boundary surfaces are planar or spherical. This latter statement on the boundaries simply leads to the vanishing of an additional
Å and b, so (15) and (16) can be written as
term associated with potential fields that would appear in (18). Similar equations apply for B
ˆ ­t H R;o …B†
‡ ­t H R;i …B†
­t H …B†
…19†
and
­t H …b† ˆ ­t H R;o …b† ‡ ­t H R;i …b†;
…20†
respectively. According to equation (62) of Berger & Field (1984),
…
…
­t H R;i …B† ˆ 22c
E ´ B d3 x ‡ 2c
…Ap E† ´ dS;
Ui
DU i
…21†
where Ap is the vector potential corresponding to a potential field P in Ue, and DUi indicates integration on the boundary surface of the
rotator. Similarly, we have
…
…
ˆ 22c
´ dS
­t H R;i …B†
…A p E†
…22†
E ´ B d3 x ‡ 2c
Ui
and
­t H R;i …b† ˆ 22c
…
Ui
DU i
ke ´ bl d3 x ‡ 2c
…
DU i
…ap e† ´ dS
…23†
Note again that the above internal relative helicity time derivatives are both gauge invariant and independent of the field assumed in the
external region. If we were considering the relative helicity of the external region, it would be independent of the actual field in the internal
region. In this sense, we will explore only the deposition of relative helicity to the exterior and the associated total magnetic energy, but we
will not rigorously study the various ways in which the associated magnetic energy converts to particles or flows there.
We assume that the rotator is in a steady state over the time-scale of interest, then the left-hand sides of (22) and (23) vanish. Note that
in a system like the Sun where the mean-field flips sign every ,11 years, the steady state is relevant for time-scales less than this period,
but greater than the eddy turnover time (,5 104 s†: Over the ,11 year times scales, the mean large- and small-scale relative helicity
contributions need not separately be steady and the left-hand sides need not vanish.
The helicity supply rate, represented by the volume integrals [second terms of (22) and (23)], are then equal to the integrated flux of
relative magnetic helicity through the surface of the rotator. Moreover, from (12), we see that the integrated flux of the large-scale relative
helicity, ;FR,i(BÅ), and the integrated flux of small-scale relative helicity, ;FR,i(b), are equal and opposite. We thus have
…
…
ˆ 22c
e ´ b d3 x ˆ 2F R;i …b†:
…24†
E ´ B d3 x ˆ 2c
F R;i …B†
Ui
Ui
To evaluate this, we use (1) and (2) to find
E ˆ 2c21 …aB 2 b7 B†
throughout Ui. Thus
ˆ 2F R;i …b† ˆ 2
F R;i …B†
…25†
…
Ui
d3 x:
…aB 2 2 bB ´ 7 B†
…26†
This shows the relation between the equal and opposite large- and small-scale relative helicity deposition rates and the dynamo
coefficients.
Now according to Frisch et al. (1975), realizability of a helical magnetic field requires its turbulent energy spectrum, EM
k ; to
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E. G. Blackman and G. B. Field
satisfy
EM
k …b† $
1
kjH k …b†j;
8p
…27†
where the equality applies to force-free fields with 7 B ˆ ^kB: The same argument also applies to the mean-field energy spectrum,
so that
EM
k …B† $
1
kjH k …B†j:
8p
…28†
If we assume that the time and spatial dependences are separable in both EM and H, then a minimum power delivered to the corona
can be derived. For the contribution from the small-scale field, we have
…
…
…
1
kmin
kmin
kmin
jF k;R;i …b†j dk $
kjF
E_ M …b† ˆ E_ M
…b†
dk
$
…b†j
dk
$
…29†
jF R;i …b†j ˆ
jF R;i …B†j;
k;R;i
k
8p
8p
8p
8p
where the last equality follows from the first equation in (26). The last quantity is exactly the lower limit on EÇM(BÅ). Thus the sum of
‡ …kmin =8p†jF R;i …b†j ˆ
the lower limits on the total power delivered from large and small scales is …kmin =8p†jF R;i …B†j
Now for a mode to fit in the rotator, k . kmin ˆ 2p=h; where h is a characteristic scale height of the turbulent
2…kmin =8p†jF …B†j:
layer. Using (26), the total estimated energy delivered to the corona (ˆ the sum of the equal small- and large-scale contributions) is
then
kmin
kmin
ˆ V jaB 2 2 bB ´ 7 Bj;
E_ M $ 2
jF R;i …b†j ˆ 2
jF R;i …B†j
8p
8p
h
…30†
where V is the volume of the turbulent rotator. We will address implications of (30) in the subsequent sections.
3
R E L AT I O N T O T H E P O Y N T I N G F L U X A N D T H E F O R C E F R E E C A S E
Blackman & Field (2000) suggest that the combination of periodic boundary conditions and absence of mean-field gradients does not allow
a significant a when the averaging scale is of the order of the scale of the periodic region. Under these conditions, the last term on the righthand side of (6) was shown to vanish, and the main contribution to the turbulent EMF comes from the second last term on the right. This
term is suppressed by the magnetic Reynolds number. Thus, tests of a suppression when ignoring field gradients and using periodic
boundary conditions may be misleading as the apparent suppression is not from the back-reaction, but is built-in from the boundary
conditions. More generally, periodic boundary conditions are not appropriate for characterizing mean magnetic flux growth unless there are
many scale lengths of the mean field within the simulation box of interest.
When field gradients are allowed (and when the mean-field scale is allowed to be smaller than the overall scale of the system) the
following conclusion applies as a result of (30): the flow of helicity through the boundary is required for substantial a (0) unless the
combination of a…0† B 2 2 b…0† 7 B ˆ 0: Equation (30) thus measures the extent to which a steady-state field structure inside the object
requires the deposition of magnetic helicity into the corona, and gives a lower limit on the flow of magnetic energy to the exterior. Since
there is no physical principle dictating that this quantity should be zero in general, we will later estimate the flow of helicity and magnetic
energy through the boundary in the generic case for which the difference is represented by the order of magnitude of the first term.
There is however, one exceptional case for which the difference in (30) does vanish exactly, and for which the energy deposition also
vanishes exactly. This is the case for which the field is force free. To see this, note that from Maxwell's equations we have
­t B ˆ 2c7 E:
…31†
Dotting with BÅ and using vector identities gives Poynting's theorem (for jBj @ jEj†
2 cE ´ 7 B:
…1=2†­t B 2 ˆ 2c7 ´ …E B†
…32†
If we integrate over all of space, and assume a steady state inside the object as we did for the helicity above, we obtain
…
…
…
…1=2†
­t B 2 d3 x ˆ 2c
E ´ 7 B d3 x 2 c
E ´ 7 B d3 x;
Ue
Ui
Ue
…33†
where the surface term vanished. The internal contribution on the right-hand side represents a source term. Using (3) and the triple product
rule, we obtain for this term, represented as a energy depostion rate,
…
…
2 d3 x 2 c
d3 x:
E_ M jsource ˆ
aB ´ 7 B 2 b…7 B†
V ´ …J B†
…34†
Ui
Ui
B 2 :
Now suppose we demand that the right-hand side of (30) vanish. Then setting that right-hand side equal to zero gives a ˆ bB ´ 7 B=
Plugging this back into (34) shows that the right-hand side of (34) then vanishes completely in the force (density) free case, J B ˆ 0: Note
that last term on the right-hand side of (34) depends on the mean velocity, which is not related to other terms on the right-hand side of (34)
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729
in any obvious way. It thus appears that the only natural case for which both helicity and energy flux vanish exactly has to be force free. But
such a case is unphysical because there are no differentiable field configurations for which J B ˆ 0 and B ± 0 when JÅ is confined to a
finite volume (Moffatt 1978). Because of this, even if F U;i ˆ 0; the energy deposition rate would not vanish in general, which is consistent
with FU,i, representing a lower limit as described in the previous section.
The flow of relative magnetic helicity appears to be generically important for dynamo flux generation, but the exact value of the
Å and the actual values of a and b , and the
difference between the terms in (30) should depend on the solution for the dynamo equations for B
physics of the magnetic diffusion and buoyancy. The application of our result to specific dynamo solutions, and a study of the boundary
physics to see how buoyancy competes with turbulent diffusion in various environments are both necessary components of future work.
In the next section, we discuss evidence that a significant residual value of (30) may be escaping into the coronae of the Sun, the
Galaxy and AGN accretion discs.
4
A P P L I C AT I O N S
Keeping in mind the issues addressed in the previous section, here we assume that the two terms on the right-hand side of (30) do not
cancel, and use the first term of (30) as representative.
4.1
Solar corona
As alluded to above, the steady state is applicable to the Sun over time-scales short compared to the 11-year oscillation, but long compared
to the typical eddy turnover times of 5 104 s: The latter determines the minimum time-scale over which mean fields make sense. Energetic
solar flares occur about once per day, so the intermediate time-scale is relevant.
Assuming that we are working in this allowed time range, if we apply (30) to each hemisphere of the Sun we have, using the first term
as an order of magnitude estimate
E_ M *
2 B 2
2pR2(
R
a
aB 2 ˆ 0:9 1028
erg s21 ;
10
21
3
40 cm s
150 G
7 10 cm
…35†
where we have taken a , 40 cm s21 from Parker (1979), and we have presumed a field of 150 G at a depth of 10 km beneath the solar
surface in the convection zone, which is in energy equipartition with turbulent kinetic motions.
As this energy deposition rate is available for reconnection that can generate AlfveÂn waves and drive winds, and energize particles, we
must compare this limit with the total of downward heat conduction loss, radiative loss, and solar wind energy flux in coronal holes, which
cover ,1/2 the area of the Sun. According to Withbroe & Noyes (1977), this amounts to an approximately steady activity of 2:5 1028 erg s21 ; about three times the predicted value of (35).
Other supporting evidence for deposition of magnetic energy and magnetic helicity in the Sun includes: (1) The related pseudoscalar,
current helicity kB ´ 7 Bl; has been directly measured for flux tube filaments and their overlying loop arcades penetrating the surface into
the solar corona. The filaments, of order 103 ±104 km, have opposite sign of the current helicity associated with the larger scale 105 km
overlying loops (Rust 1994; Rust & Kumar 1996; Martin 1998; Ruzmaikin 1999). These small- and large-scale fields having the opposite
sign are predicted above and is similar to that of Pouquet et al. (1976). (2) The larger scale loops are associated with Coronal Mass Ejections
(CMEs).Reverse `S' shaped CME loops dominate forward `S' shaped CMEs with a 6:1 ratio in the northern hemisphere, and a similar
opposite ratio in the south (Rust & Kumar 1996). (3) There is a correlation between the sign of kB ´ 7 Bl and the sign of the observed twist
of the field-aligned features in the photosphere, implying that the parallel current responsible for the twist originates below the photosphere
and continues into the corona. (4) Laboratory experiments indicate that twisted flux tubes are subject to kink instability, leading to reconnection
and magnetic energy release. (5) The Yohkoh satellite provides some of the most direct evidence for magnetic reconnection in flares of various
sizes (Masuda et al. 1994; Tsuenta 1996). Shibata (1999) has a model in which clouds of plasma (`plasmoids') are confined in twisted flux
tubes which reconnect with nearby flux, ejecting the plasmoid together with its twisted flux, hence magnetic helicity. Ejection of helicity is an
important part of the model. (6) Measurements of magnetic helicity in the Solar Wind seem to show that there is indeed a small-scale magnetic
helicity (Carbone & Bruno 1997) as well as a large-scale magnetic helicity (Bruno & Dobrowolny 1986; Goldstein, Roberts & Fitch 1994) and
also an asymmetry in the handedness between northern and southern helioshpheres at 1 AU (Smith & Bieber 1996).
In summary, currents along B twist emerging flux tubes, endowing them with current helicity and therefore magnetic helicity as well.
Instability leads to reconnection, allowing magnetic flux to escape in CMEs carrying magnetic helicity. The net result is that helical fields
below the photosphere escape the Sun, carrying magnetic helicity with them. Qualitatively, this is what is expected from Blackman & Field
(2000) and shown explicitly in (30).
4.2
Galactic corona
If we apply (30) to the Galaxy, we have for each hemisphere a lower limit on the luminosity delivered to the corona, of
E_ M * …pR2 †aB 2 , 1040
R
12 kpc
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2
a
B
erg s21 :
105 cm s21 5 1026 G
…36†
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E. G. Blackman and G. B. Field
The value of a that we have scaled to is from Ruzmaikin et al. (1988). FerrieÁre (1993) suggests that a , 2 104 cm s21 at maximum would
lower the above estimate of the limit by a factor of 4. The study of Savage (1995) suggests that the required steady energy input to the warm
ionized medium in the Galactic corona (scale height h , 1 kpc† is ,1041 erg s21, whereas that input into the highly ionized coronal gas
…h * few kpc† is ,1040 erg s21. This compares favorably with (36). This is also implied by the study of Reynolds, Haffner & Tufte (1999),
who argue that spatial variations of [S ii]/H-Alpha and [N ii]/H-Alpha line intensity ratios in the halos of our and other galaxies are
inconsistent with pure photoionization models. Instead, a secondary heating mechanism is required that increases the electron temperature
at low densities ne with a dependence on ne to a lower power than the n2e of photoionization. Reynolds et al. (1999) estimate the required
input heating rate of 4:1 1040 erg s21 over a 12 kpc radius. This again compares well with (36).
The observed tangled microGauss field in the halo also suggests that magnetic reconnection and turbulent dissipation may be occurring
(e.g. Beck et al. 1996)
4.3
Active galactic nuclei
For a thin accretion disc in the turbulent viscosity framework, we have roughly the turbulent viscosity as
b , vT l , v2T =V ˆ ass cs h;
…37†
where vT is the turbulent velocity, l the typical correlation scale, V the rotational velocity, h the height (1/2 thickness) , cs is the sound
speed, and a ss the turbulent viscosity parameter (Shakura & Sunyaev 1973). If we assume that the angular momentum transport results from
MHD turbulence driven by a shearing instability (cf. Balbus & Hawley 1998), then vT . vA ; in steady state, where vA is the AlveÂn speed.
We thus have ass , v2A =c2s where we have used the relation Vh , cs ; applicable to pressure supported discs. Then using (37), we also have
ass . …l=h†2 :
Note next that the mass continuity equation for accretion discs gives for the density
_
_
M
M
rˆ
ˆ
4pRhvr 4pass h2 cs
…38†
…39†
Ç is the accretion rate. Using the above equations in (30) we have for
where we have used vR ˆ ass hcs =R for the radial accretion speed, and M
each hemisphere
E_ M * …pR2 †aB 2 =2 , 2pamass c2s pR2 r;
…40†
where m ; B 2 =kB2 l:We write
a ˆ qass cs ;
…41†
where q , 1 is a dimensionless parameter to accommodate the actual value of a in a disk which will depend on the dynamical effects of the
back-reaction as well as on the physical source of a (e.g. Brandenburg & Donner 1996; Moss, Shukurov & Sokoloff 1999; Brandenburg
2000). It should be noted that Brandenburg & Donner (1996) and Brandenburg (2000) find the opposite sign of a as compared to the
standard kinematic interpretation when a is presumed proportional to the negative of the kinetic helicity. This may have to do with the fact
that the a effect is more accurately represented as proportional to the difference between small-scale current and kinetic helicity, the latter
of which can change sign from shear. We restrict further discussion of this point here.
Introducing (41) in (40) and using the continuity equation,
_
_
Rg
MM
M
p
ass m q E_ M * mqass G
erg s21 ;
…42†
, 2 1042
21
R
0:03 0:1 0:1
R
0:1M ( yr
2
where Rg is a gravitational radius. Note that the parameters m, q, a ss should ensure that the MHD luminosity is below that which would be
associated with catastrophic angular momentum transport by the escaping fields. If the product qma ss is not ! 1024 ; then the lower limit
of (42) compares reasonably well with the X-ray luminosities observed in Seyfert 1 galaxies (cf. Mushotzky, Done & Pounds 1993), which
can be as large as 30 per cent of the total luminosity, and variabilites indicate emission from R , 10Rg : The best working paradigm for the
X-ray coronal luminosity is the dissipation of magnetic energy in a corona located above the disc (Galeev, Rosner & Vaiana 1979; Haardt &
Maraschi 1993; Field & Rogers 1993) which is consistent with the above result. Note that the effect of back-reaction on q must be studied
with boundary conditions that allow corona formation in the first place, otherwise the suppression may not be dynamical Blackman & Field
(2000).
It is important to note that like the solar and Galactic coronae, the dissipation is required to be more or less steady. Since dissipation is
an exponentially decaying process, steady dissipation requires a process that feeds this energy exponentially. This is also consistent with the
a ±V dynamo picture.
5
CONCLUSIONS
The a ±V mean-field dynamo in a turbulent rotator generically predicts some escape of large-scale relative magnetic helicity and an equal
q 2000 RAS, MNRAS 318, 724±732
Coronal activity from dynamos in astrophysical rotators
731
and opposite escape of small-scale relative magnetic helicity in the steady state. The helicity escape rate leads to a lower limit on the total
magnetic energy deposition into the corona. Exponential field growth from a dynamo would thus, in the steady state, lead to a steady supply
of magnetic energy into the corona which can feed a steady MHD wind or non-thermal luminosity resulting from dissipation of magnetic
fields. Because of the inverse cascade of magnetic helicity (cf. Frisch et al. 1975, Pouqet et al. 1976, Balsara & Pouquet 1999), which is in
fact at the root of the mean field dynamo, the energy estimates here are relevant for the large scale field deposition to the corona as well as
the small scale field. Further, when the corona itself is turbulent, there should be an inverse cascade of magnetic helicity in any wind driven
outward, thus the dominant magnetic helicity scale would appear to increase on increasingly large distances from the source.
The estimated energy deposition rates are consistent with the coronal + wind power from the Sun, the Galaxy and Seyfert 1s. The
helical properties also seem to agree well in the Solar case where they can be observed. The steady flow of magnetic energy into coronae
thus provides an interesting connection between mean field dynamos and coronal dissipation paradigms in a range of sources. In short, a
reasonably steady (over time-scales long compared to turbulent turnover time-scales), active corona with multi-scale helical structures,
provides a self-consistency check for a dynamo production of magnetic field in which there is exponential field growth inside the body on
small and large scales. If the growth were only linear, the corona and wind output would be predicted to be more episodic. This deserves
further scrutiny amidst the back-reaction debate as it highlights the role and necessity of boundary conditions that allow magnetic helicity to
escape (Blackman & Field 2000).
Future work should consider specific dynamo solutions in different settings to determine more precisely the predicted energy and
helicity deposition rates from (30) for a range of dipole and quadrupole growth modes. A dynamical treatment of buoyancy (e.g. Moss et al.
1999) and field escape should be included, as well as what fraction of energy can be predicted to go into bulk wind versus non-thermal
particles based on the spectrum of the deposited field. Note that in the case of accretion discs, while corona formation in discs has long been
presumed, there has not been enough analytic work demonstrating how the corona forms when the discs are fully turbulent. Recent global
simulations are starting to show such corona formation. (Miller & Stone 2000; Hawley, personal communication). Finally, the connection
between observed oppositely signed large- and small-scale current helicities in e.g. the solar corona, needs to be correlated more precisely
with our predicted oppositely signed large- and small-scale magnetic helicities.
AC K N O W L E D G M E N T S
We thank A. Shukurov for comments. E.B. acknowledges support from NASA grant NAG5-7034 while at Caltech.
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This paper has been typeset from a TEX/LATEX file prepared by the author.
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