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arXiv:astroph/0205557v1 31 May 2002
Problems and Progress in Astrophysical
Dynamos
Ethan T. Vishniac1, A. Lazarian2, and Jungyeon Cho2
1Johns Hopkins Univ., Baltimore MD21218, USA
2Univ. of Wisconsin, Madison WI53706, USA
Abstract. Astrophysical objects with negligible resistivity are often threaded by large
scale magnetic fields. The generation of these fields is somewhat mysterious, since a
magnetic field in a perfectly conducting fluid cannot change the flux threading a fluid
element, or the field topology. Classical dynamo theory evades this limit by assuming
that magnetic reconnection is fast, even for vanishing resistivity, and that the large scale
field can be generated by the action of kinetic helicity. Both these claims have been
severely criticized, and the latter appears to conflict with strong theoretical arguments
based on magnetic helicity conservation and a series of numerical simulations. Here
we discuss recent efforts to explain fast magnetic reconnection through the topological
effects of a weak stochastic magnetic field component. We also show how meanfield
dynamo theory can be recast in a form which respects magnetic helicity conservation,
and how this changes our understanding of astrophysical dynamos. Finally, we com
ment briefly on why an asymmetry between small scale magnetic and velocity fields is
necessary for dynamo action, and how it can arise naturally.
1 Introduction
Magnetic fields have played a curious role in astrophysics, being both common
place and poorly understood. They are ubiquitous in ionized systems, from the
interiors of stars to the hot interstellar medium. The magnetic energy density
is typically roughly comparable to the turbulent kinetic energy density. In stel
lar interiors, this means that magnetic fields tend to play a small role. In the
interstellar medium, and in stellar coronae, their role is large, and consequently
a matter of intense debate. In accretion disks the typical magnetic field energy
density is probably an order of magnitude below the ambient gas pressure (e.g.
[39,40,73,14,15]) but they play a critical role in the outward transfer of angular
momentum and the dissipation of orbital energy. Moreover, in optically thin en
vironments the presence of a strong magnetic field can have a dramatic effect on
the luminosity and spectrum of an object. A clear understanding of the genera
tion and dynamics of magnetic fields is important to astrophysics in many ways.
Unfortunately, their dynamics has not been well understood, at least judging by
the diversity of opinions found in the literature [20,66,50,80,16]. Consequently,
arguments which cite magnetic fields as a dynamically important element in any
particular object have tended to rely on phenomenology, rather than any sort of
fundamental explanation.
Fortunately, over the last ten years, and especially quite recently, there has
been significant progress in this area. First, although direct observations of high
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2 Vishniac, Lazarian & Cho
conductivity magnetic field dynamics are still restricted to the solar wind and
the Sun, improvements in resolution have made it possible to watch magnetic
fields evolve in real time [43], and to measure the power spectrum of magnetohy
drodynamic (MHD) turbulence in the solar wind directly [57]. Second, numerical
simulations have reached the point where it is possible to simulate simple MHD
systems with ∼ 108cells over many dynamical times. Third, a better under
standing has been reached in terms of MHD turbulence theory (for a review see
the chapter by Cho, Lazarian & Vishniac in this volume).
These results encourage us to believe that the many remaining problems
are ripe for further progress. These problems range from the nature of dynamo
processes in stars, accretion disks, and galaxies, to the question of how mag
netic fields reconnect on dynamical time scales, with apparent disregard for the
constraint due to fluxfreezing. To be more precise, in the limit of negligible re
sistivity, the magnetic field in a fluid medium is frozen, the sense that neither
the magnetic flux threading a fluid element, nor the field topology, can change.
Magnetic reconnection, the exchange of partners between adjacent field lines vi
olates the second condition, while the generation of a large scale field through
dynamo action apparently violates the first.
Conventional mean field dynamo theory (see [62,65,49] for reviews) allows a
largescale magnetic field to grow exponentially from a seed field (see [69,51]) at
the expense of smallscale turbulent energy through a process of spiral twisting
and reconnection, illustrated in Fig. 1. This process starts with a set of large
scale parallel field lines pointing in some arbitrary direction. If the underlying
turbulence has a tendency to twist the field lines into spirals with a preferred
handedness (i.e. the velocity field has some net helicity), then reconnection on
two dimensional surfaces between adjacent spirals will produce a new field, at
right angles to the old one, provided that there is a systematic gradient in the
strength of the spirals. The new field component is at right angles to both the
gradient and the old field component. In a differentially rotating system, we can
get a dynamo if the original field direction is in theˆφ direction, and the dynamo
process produces a radial field component. Differential shearing of Brwill then
drive the azimuthal field component, closing the cycle. This is the ‘α − Ω dy
namo’. In the absence of global shear, we need a second round of dynamo action,
which gives an α2dynamo. This process can be given a systematic mathematical
treatment by a suitable choice of averaging procedures.
There are two ways in which this picture ignores, rather than solves, the dif
ficulties imposed by fluxfreezing. The more obvious point is that adjacent spiral
field lines are assumed to reconnect quickly. Without this assumption the field
will accumulate small scale tangled knots which will quickly suppress dynamo
action, and the large scale magnetic field will saturate far below equipartition
with the surrounding turbulence. Unfortunately, if reconnection happens at the
rate allowed by the generally accepted SweetParker model [64,78], it is far too
slow. However, there are observations which suggest that this represents more
of a challenge for theorists than a real constraint on the evolution of magnetic
fields. If reconnection is slow, turbulence would cause many magnetic reversals
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Astrophysical Dynamos3
B
+ twisting motions
+ reconnection
?
+
+
+
+


Fig.1. The meanfield dynamo in action. Anisotropic turbulence twists the field lines
into spirals. Reconnection restores the original field lines, but a vertical gradient in the
strength of the spirals generates a net flux out of the page.
per parsec within the interstellar medium. Observations, on the contrary, show
that magnetic field is coherent over the scales of hundreds of parsecs. This fact, as
well as direct observations of large and small scale Solar flares [27], suggest that
the rate of reconnection is many orders of magnitude more rapid than allowed by
the SweetParker model. As this example shows, the importance of reconnection
in astrophysics is not limited to understanding the dynamo process. The process
of reconnection is an integral part of the transfer of magnetic energy to fluid and
particle motion in stellar coronae and in the interstellar medium. More generally,
it is impossible to claim that we understand MHD unless we can predict whether
crossing magnetic flux tubes will reconnect or bounce from one another.
A more subtle difficulty arises from the process by which straight field lines
are twisted into spirals. This is intuitively appealing if we consider field lines
as isolated strings of infinitesimal radius. More realistically, the field occupies a
nonzero volume. Twisting a tube into a spiral shape requires that we either allow
the ends to slip, or allow parts of the tube to twist in the opposite sense. There is
a geometrical constraint which is ignored in the standard picture. This objection
can be given a rigorous mathematical form, the conservation of magnetic helicity,
which we will describe in §3.
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4 Vishniac, Lazarian & Cho
How do numerical simulations of dynamo activity compare to meanfield
dynamo theory? Computer simulations of dynamos can be divided into two
classes. There are simulations in which some local instability (convection, the
BalbusHawley instability etc.) is allowed to operate, and there are simulations
in which the turbulence is driven externally, usually in such a way as to guarantee
the presence of a net fluid helicity. The former simulations are often successful
at generating large scale magnetic fields whose energy density is at least as great
as the turbulent energy density (e.g.[14,37,32]). The latter are less successful,
in the sense that the energy density of the large scale magnetic field is often
quite modest (e.g. [60,4]). In particular there are simulations ([19,13]) which
produce dynamos in a computational box, with forced heliacal turbulence. These
dynamos show a steep inverse correlation between the dynamo growth rate and
the conductivity. Naively extrapolating to astrophysical regimes suggests that
magnetic dynamos driven by fluid helicity would take enormous amounts of
time to grow. This conclusion is sharply at odds with evidence for rapid and
efficient stellar dynamos.
Here we discuss recent work on the problems of fast reconnection and mag
netic helicity conservation in astrophysical dynamos. For reconnection we con
centrate on a generic reconnection scheme that appeals to magnetic field stochas
ticity as the critical property that accelerates reconnection ([53], see [54] for a
review). Collisionless plasma effects which may also accelerate magnetic recon
nection are addressed in the chapter by Bhattacharjee in this volume. We will
typically assume that the evolution of the magnetic field is described by the
simplest form of the induction equation
∂tB = ∇ × v × B + η∇2B, (1)
although we will make reference to work which includes more realistic treatments
of collisionless plasma effects.
2 Rates of Magnetic Reconnection
A simple dimensionless measure of the importance of resistivity, η, in a con
ducting fluid is the Lundquist number ≡ VAL/η, where VA≡ B/(4πρ)1/2is the
Alfv´ en velocity and L is a typical scale of the system. When fluid velocities are
of order the Alfv´ en speed, as is usual in astrophysics, this is a crude estimate
of the ratio of the first and second terms in equation (1). Typically this number
is very large under most astrophysical circumstances, and flux freezing should
be a good approximation. More precisely, the coefficient of magnetic field dif
fusivity in a fully ionized plasma is η = c2/(4πσ) = 1013T−3/2cm2s−1, where
σ = 107T3/2s−1is the plasma conductivity and T is electron temperature. The
characteristic time for field diffusion through a plasma slab of size L is L2/η,
which is large for any “astrophysical” L.
What happens when magnetic field lines intersect? Do they deform each
other and bounce back or they do change their topology? This is the central
question of the theory of magnetic reconnection. In fact, the whole dynamics of
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Astrophysical Dynamos5
magnetized fluids and the backreaction of the magnetic field depends on the
answer.
2.1 The SweetParker Scheme and its Modifications
The literature on magnetic reconnection is rich and vast (see, for example, [68]
and references therein). We start by discussing a robust scheme proposed by
Sweet and Parker [64,78]. In this scheme oppositely directed magnetic fields are
brought into contact over a region of length Lx(see Fig. 2). In general there will
be a shared component, of the same order as the reversed component. However,
this has only a minor effect on our discussion. The gradient in the magnetic field
is confined to the current sheet, a region of vertical size ∆, within which the
magnetic field evolves resistively. The velocity of reconnection, Vr, is the speed
with which magnetic field lines enter the current sheet, and is roughly η ≈ Vr∆.
Arbitrarily high values of Vr can be achieved (transiently) by decreasing ∆.
However, for sustained reconnection there is an additional constraint imposed
by mass conservation. The plasma initially entrained on the magnetic field lines
must escape from the reconnection zone. In the SweetParker scheme this means
a bulk outflow, parallel to the field lines, within the current sheet. Since the mass
enters along a zone of width Lx, and is ejected within a zone of width ∆, this
implies
ρVrecLx= ρ′VA∆, (2)
where we have assumed that the outflow occurs at the Alfv´ en velocity. This is
actually an upper limit set by energy conservation. If we ignore the effects of
compressibility ρ = ρ′and the resulting reconnection velocity allowed by Ohmic
diffusivity and the mass constraint is
Vrec,sweet−parker≈ VAR−1/2
L
, (3)
where RL is the Lundquist number using the current sheet length. Depending
on the specific astrophysical context, this gives a reconnection speed which lies
somewhere between 10−3(stars) and 10−10(the galaxy) times VA.
It is well known that using the SweetParker reconnection rate it is impos
sible to explain solar flares. For the reasons given in the introduction, it is also
well known that it is impossible to reconcile dynamo theory with observations
without some substantially faster reconnection scheme. Consequently, for forty
years discussions of reconnection speeds have tended to focus on mechanisms that
might give reconnection speeds close to VA, i.e. ‘fast’ reconnection. In general, we
can divide schemes for fast reconnection into those which alter the microscopic
resistivity, broadening the current sheet, and those which change the global ge
ometry, thereby reducing Lx. Ultimately, a successful scheme should satisfy basic
physical constraints without requiring contrived geometries or boundary condi
tions. In the near term, we can gain some insight into the likely nature of the
solution by considering that reconnection is not always fast. Magnetic field lines
in the solar corona and chromosphere which could reach a lower energy config
uration through reconnection do not always immediately do so. Furthermore, a
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6 Vishniac, Lazarian & Cho
x
L
SweetParker model
Turbulent model
blow up
λ
∆
∆
λ
Fig.2. Upper plot: SweetParker scheme of reconnection. Middle plot: illustration of
stochastic reconnection that accounts for field line noise. Lower plot: a closeup of the
contact region. Thick arrows depict outflows of plasma. From [54].
solution which relies entirely on collisionless effects, for example, would imply
that field lines do not reconnect in dense environments, which would leave a
major problem in understanding the nature of stellar dynamos.
Attempts to accelerate SweetParker reconnection are numerous. We start
by considering schemes to broaden the current sheet. Anomalous resistivity is
known to broaden current sheets in laboratory plasmas. It is present in the
reconnection layer when the field gradient is so sharp that the electron drift
velocity is of the order of thermal velocity of ions u = (kT/m)1/2[65]. In other
words, when j > jcr= Neu. If the current sheet has a width δ with a change
in the magnetic field ∆B then 4πj = c∆B/δ. The effective resistivity increases
nonlinearly as j becomes greater than jcr, thereby broadening the current sheet.
We can find an upper limit to this effect by assuming that j never gets very much
larger than jcr, that is δ ≈
radius rc= (muc)/(eBtot), where Btotis the total magnetic field (including any
shared component) we find
c∆B
4πNeu. Expressing δ in terms of the ion cyclotron
δ ≈ rc
?VA
u
?2∆B
Btot
, (4)
which agrees with [65] up to the factor ∆B/Btot, which equals 1 in that treat
ment. Combining (2) and (4) one gets [53]
Vrec,anomalous≈ VArc
Lx
?VA
u
?2∆B
Btot
. (5)
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Astrophysical Dynamos7
Equation (5) shows that the enhanced reconnection velocity is still much less
than the Alfven velocity if Lxis much greater than the ion Larmor (cyclotron)
radius. In general, “anomalous reconnection” is important when the thickness of
the reconnection layer in the SweetParker reconnection scheme is less than δ.
However, for typical interstellar magnetic fields the Larmor radius rcis ∼ 107cm
and anomalous effects are negligible.
Tearing modes are a robust instability connected to the appearance of nar
row current sheets [31]. The resulting turbulence will broaden the reconnection
layer and enhance the reconnection speed. Here we give an estimate of this ef
fect and show that while it represents a significant enhancement of SweetParker
reconnection of laminar fields, it leaves reconnection slow. One difficulty with
many earlier studies of reconnection in the presence the tearing modes stemmed
from the idealized two dimensional geometry assumed for reconnection. In two
dimensions tearing modes evolve via a stagnating nonlinear stage related to the
formation of magnetic islands. This leads to a turbulent reconnection zone [59],
but the current sheet remains narrow and its effects on the overall reconnec
tion speed are unclear. This nonlinear stagnation stage does not emerge when
realistic three dimensional configurations are considered [53]. In any realistic cir
cumstances field lines are not exactly antiparallel. Consequently, we expect that
instead of islands one finds nonlinear Alfv´ en waves in three dimensional recon
nection layers. The tearing instability proceeds with growth rates determined by
the linear growth phase while the resulting magnetic structures propagate out
of the reconnection region at the Alfv´ en speed.
The dominant mode will be the longest wavelength mode, whose growth rate
will be
η
∆2
γ ≈
?VAλ?
η
?2/5
. (6)
The transverse spreading of the plasma in the reconnection layer will start to
stabilize this mode when its growth rate is comparable to the transverse shear
VA/λ?[18]. At this point we have Vrec,local≈ γ∆ and [53]
Vrec,tearing= VA
?
η
VALx
?3/10
, (7)
which is substantially faster than the SweetParker rate, but still very slow in
any astrophysical context. Note that unlike anomalous effects, tearing modes
do not require any special conditions and therefore should constitute a generic
scheme of reconnection.
Finally, we note that there is a longstanding, but controversial suggestion,
that ions tend to scatter about once per cyclotron period, ‘Bohm diffusion’ [12].
Even if this is correct, the effective diffusivity of magnetic field lines would still be
only ηBohm∼ VArc. While this would be a large increase over Ohmic resistivity,
it produces fast reconnection, of order VA, only if rc∼ Lx. It therefore fails as an
explanation for fast reconnection for the same reason that anomalous resistivity
does.
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8 Vishniac, Lazarian & Cho
2.2Xpoint Reconnection
The failure to find fast reconnection speeds through current sheet broadening
has stimulated interest in fast reconnection through radically different global
geometries. Petschek [67] conjectured that reconnecting magnetic fields would
tend to form structures whose typical size in all directions is determined by the
resistivity (‘Xpoint’ reconnection). This results in a reconnection speed of order
VA/lnRL. However, attempts to produce such structures in numerical simu
lations of reconnection have been disappointing. Typically the Xpoint region
collapses towards the SweetParker geometry as the Lundquist number becomes
large [7,8,9,85,58].1One way to understand this collapse is to consider perturba
tions of the original Xpoint geometry. In order to maintain this geometry shocks
are required in the original (Petschek) version of this model. These shocks are,
in turn, supported by the flows driven by fast reconnection, and fade if Lxin
creases. Naturally, the dynamical range for which the existence of such shocks
is possible depends on the Lundquist number and shrinks when fluid conductiv
ity increases. The apparent conclusion is that, at least in the collisional regime,
reconnection occurs through narrow current sheets.
One may invoke collisionless plasma effects to stabilize the Xpoint recon
nection (for collisionless plasma). For instance, a number of authors [71,70,72]
have reported that in a two fluid treatment of magnetic reconnection, a stand
ing whistler mode can stabilize an Xpoint with a scale comparable to the ion
plasma skin depth, c/ωpi ∼ (VA/cs)rL. The resulting reconnection speed is a
large fraction of VA, and apparently independent of Lx, which would suggest
that something like Petschek reconnection emerges in the collisionless regime.
This possibility is discussed at length in the chapter by Bhattacharjee (this vol
ume). However, these studies have not yet demonstrated the possibility of fast
reconnection for generic field geometries, since they assume that there are no
bulk forces acting to produce a large scale current sheet. Similarly, those studies
do not account for fluid turbulence. Magnetic fields embedded in a turbulent fluid
will give fluctuating boundary conditions for the current sheets. On the other
hand, boundary conditions need to be fine tuned for a Petschek reconnection
scheme [68].
Finally, we note that a number of researchers have claimed that turbulence
may accelerate reconnection (for example, [75], where tearing modes are used
as the source of the turbulence). The general idea is that turbulent motions can
provide an effect transport coefficient ∼ ?v2?τ [65]. However, a closer examination
of this process has convincingly demonstrated that an unrealistic amount of
energy is required to mix field lines unless they are almost exactly antiparallel
[66]. In the next section we will discuss a mechanism that, when it works, should
produce reconnection under a broad range of field geometries, without regard to
the particle collision rate.
1Recent plasma reconnection experiments [86] do not support Petschek scheme either.
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Astrophysical Dynamos9
2.3 Stochastic Reconnection
Two idealizations were used in the preceding discussion. First, we considered
reconnection in only two dimensions. Second, we assumed that the magnetized
plasma has laminar field lines. The SweetParker scheme can easily be extended
into three dimensions, in the sense that one can take a crosssection of the
reconnection region such that the shared component of the two magnetic fields is
perpendicular to the crosssection. In terms of the mathematics nothing changes,
but the outflow velocity becomes a fraction of the total VA and the shared
component of the magnetic field will have to be ejected together with the plasma.
This result has motivated researchers to do most of their calculations in 2D,
which has obvious advantages for both analytical and numerical investigations.
However, physics in two and three dimensions are very different. This is true,
for example, in hydrodynamic turbulence, partly because lines of vorticity have
different dynamics when they are free to move around one another. Similarly,
the ability of magnetic field lines to move past one another in three dimensions
dramtically alters the topolical constraints on their dynamics. In [53] we consid
ered three dimensional reconnection in a turbulent magnetized fluid and showed
that reconnection is fast. This result cannot be obtained by considering two di
mensional turbulent reconnection (cf. [59]). This point has been the source of
significant confusion. Turbulent reconnection has usually been used to refer to
reconnection driven by the turbulent transport of magnetic flux, as discussed in
the previous subsection. In other words, one looks for a net flux transport term,
operating on microscales, that is proportional to magnetic field gradients and
has a coefficient which is independent of the resistivity. This process was recently
examined, and severely criticized, in [45], under the mistaken impression that
it the critical physical process in stochastic reconnection. Instead, stochastic re
connection is a geometric effect arising from the appearance of stochastic field
line wandering in three dimensions, which gives rise to a broad outflow from
the current sheet, but has little effect on the current sheet structure. Below we
briefly discuss the idea of stochastic reconnection, while the full treatment of the
problem is given in [53].
MHD turbulence guarantees the presence of a stochastic field component,
although its amplitude and structure clearly depends on the amplitude and the
turbulence driving mechanism. Our model of the field line stochasticity also de
pends on our ability to model generic MHD turbulence. We consider the case
in which there exists a large scale, wellordered magnetic field, of the kind that
is normally used as a starting point for discussions of reconnection. This field
may, or may not, be ordered on the largest conceivable scales. However, we will
consider scales smaller than the typical radius of curvature of the magnetic field
lines, or alternatively, scales below the peak in the power spectrum of the mag
netic field, so that the direction of the unperturbed magnetic field is a reasonably
well defined concept. In addition, we expect that the field has some small scale
‘wandering’ of the field lines. On any given scale the typical angle by which field
lines differ from their neighbors is φ ≪ 1, and this angle persists for a distance
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10 Vishniac, Lazarian & Cho
along the field lines λ?with a correlation distance λ⊥ across field lines (see
Fig. 2).
The modification of the mass conservation constraint in the presence of a
stochastic magnetic field component is selfevident. Instead of being squeezed
from a layer whose width is determined by Ohmic diffusion, the plasma may
diffuse through a much broader layer, Ly∼ ?y2?1/2determined by the diffusion
of magnetic field lines. (Here ‘y’ is the axis perpendicular to the mean field
direction. See Fig. 2.) This suggests an upper limit on the reconnection speed of
∼ VA(?y2?1/2/Lx). This will be the actual speed of reconnection if the progress of
reconnection in the current sheet itself does not impose a smaller limit. The value
of ?y2?1/2can be determined once a particular model of turbulence is adopted,
but it is obvious from the very beginning that this value is determined by field
wandering rather than Ohmic diffusion, as in the SweetParker model.
What about limits on the speed of reconnection that arise from considering
the structure of the current sheet? In the presence of a stochastic field compo
nent, magnetic reconnection dissipates field lines not over their entire length
∼ Lx but only over a scale λ? ≪ Lx (see Fig. 2), which is the scale over
which magnetic field line deviates from its original direction by the thickness
of the Ohmic diffusion layer λ−1
⊥
≈ η/Vrec,local. If the angle φ of field devi
ation did not depend on the scale, the local reconnection velocity would be
∼ VAφ, independent of resistivity. However, for any realistic model of MHD
turbulence, φ (= λ⊥/λ?, does depend on scale. Consequently, the local recon
nection speed Vrec,local is given by the usual SweetParker formula but with
λ?instead of Lx, i.e. Vrec,local ≈ VA(VAλ?/η)−1/2. Also, it is apparent from
Fig. 2 that ∼ Lx/λ?magnetic field lines will undergo reconnection simulta
neously (compared to a one by one line reconnection process for the Sweet
Parker scheme). Therefore the overall reconnection rate may be as large as
Vrec,global ≈ VA(Lx/λ?)(VAλ?/η)−1/2. Whether or not this limit is important
depends on the value of λ?.
The relevant values of λ?and ?y2?1/2depend on the magnetic field statistics.
This calculation was performed in [53] using the GoldreichSridhar model [33] of
MHD turbulence, the Kraichnan model ([41,48]) and for MHD turbulence with
an arbitrary spectrum (limited only some basic physical constraints and which is
in rough agreement with observations [1,52,74]). In all the cases the upper limit
on Vrec,globalwas greater than VA, so that the diffusive wandering of field lines
imposed the relevant limit on reconnection speeds. Among these, the Goldreich
Sridhar model provides the best fit to observations (e.g. [1,74]) and simulations
[22,23]. In this case the reconnection speed was
Vrec,up= VAmin
??Lx
l
?1
2
,
?
l
Lx
?1
2??vl
VA
?2
, (8)
where l and vlare the energy injection scale and turbulent velocity at this scale
respectively. We stress that the use of MHD turbulence models here is solely
for the purpose of providing a welldefined model of field line stochasticity. The
dynamics of the turbulent cascade are largely irrelevant and any process which
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Astrophysical Dynamos11
leads to small scale field line stochasticity (e.g. footpoint motions for solar field
lines) is a possible cause of fast reconnection.
In [53] we also considered other processes that can impede reconnection and
find that they are less restrictive. For instance, the tangle of reconnection field
lines crossing the current sheet will need to reconnect repeatedly before indi
vidual flux elements can leave the current sheet behind. The rate at which this
occurs can be estimated by assuming that it constitutes the real bottleneck in
reconnection events, and then analyzing each flux element reconnection as part
of a selfsimilar system of such events. This turns out to limit reconnection to
speeds less than VA, which is obviously true regardless. As the result equation
(8) is not only an upper limit on the reconnection speed, but is the best estimate
of its value.
Naturally, when turbulence is negligible, i.e. vl→ 0, the field line wandering
is limited to the SweetParker current sheet and the SweetParker reconnection
scheme takes over. However, in practice this requires an artificially low level of
turbulence that should not be expected in realistic astrophysical environments.
Moreover, the release of energy due to reconnection, at any speed, will con
tribute to the turbulent cascade of energy and help drive the reconnection speed
upward. This may be relevant to the slow onset, and rapid acceleration, of the
reconnection process in solar flares.
We stress that the enhanced reconnection efficiency in turbulent fluids is only
present if 3D reconnection is considered. In this case ohmic diffusivity fails to
constrain the reconnection process as many field lines simultaneously enter the
reconnection region. The number of lines that can do this increases with the
decrease of resistivity and this increase overcomes the slow rates of reconnection
of individual field lines. It is impossible to achieve a similar enhancement in 2D
(see [87]) since field lines can not cross each other.
There is a limited analogy one can draw between the enhancement of re
connection speeds in Xpoint models and increased rate of reconnection due to
field line stochasticity. In both cases one gets a boost from a reduced parallel
length scale. In the case of Xpoint models this effect is, usually by design, enor
mous since Lx → ∆. Stochastic reconnection depends on a relatively modest
enhancement, since Lx→ λ?(∆) ≫ ∆. The bulk of the effect comes from the
simultaneous reconnection of many independent flux elements, and the steady
diffusion of the ejected plasma away from the current sheet. The main problem
with Xpoint reconnection models, their tendency to collapse to narrow current
sheets, is absent in stochastic reconnection, since in the latter case the current
sheets stay narrow, and the diverging field lines are separated by other field lines,
rather than by unmagnetized plasma.
A more subtle difficulty arises from our prescription for the structure of the
stochastic field near the current sheet. We have assumed that we can apply the
statistically homogeneous prescription for field line perturbations in a turbulent
medium near planes where there is a dramatic change in the structure of the
large scale magnetic field. This is not obvious. It may be that the presence
of a strong shear in the field acts as a kind of internal surface, producing an
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altered, and perhaps greatly reduced, level of stochasticity. This kind of internal
‘shadowing’ does not appear in current simulations, but there has been little
attempt to look for it, and the issue can only be resolved when detailed numerical
simulations of stochastic reconnection are performed. Similarly, one may wonder
if the systematic ejection of plasma along the field lines might modify their
topological connections. In this case it seems more plausible to suppose that this
would lead to an increase in the diffusion rate, rather than a decrease, but again
no simulations of this process are available.
2.4 Reconnection in Partially Ionized Gas
A substantial fraction of the ISM in our galaxy is partially ionized, as well
as photospheres of most stars. This motivates studies of the effect of neutrals
on reconnection and MHD turbulence. The role of ionneutral collisions is not
trivial. On one hand, neutral particles tend to have a substantially longer mean
free path, so that drag between the neutrals and ions may truncate the turbulent
cascade at a relatively large scale. On the other hand, the ability of neutrals to
diffuse perpendicular to magnetic field lines enhances reconnection rates, at least
in the SweetParker model.
Reconnection in partially ionized gases has been studied by various authors
([63,88,84]) in the context of the SweetParker reconnection model. Our com
ments here are based on [84] where we studied the diffusion of neutrals away
from the reconnection zone. In general, in a partially ionized gas the reconnection
zone consists of two distinct regions. A broad region, which width is determined
by the ambipolar diffusivity, ηambi≈ V2
rate, and a narrow region whose width is determined by the Ohmic diffusivity.
Magnetic reconnection takes place in the narrow region, while the broader region
allows a more efficient ejection of matter.
If the recombination time is short, then ions and neutrals are largely inter
changeable and the reconnection speed is [84]
A/tniwhere tniis the neutralion collision
Vrec≈ VA
?VAtin
Lx
?1/2
. (9)
This is faster than the SweetParker rate, but not fast in the sense of allowing
reconnection speeds close to VA. In practice, even this rate is typically unachiev
able. Under typical interstellar conditions the reconnection speed is limited by
the recombination rate. That is, the rate at which ions recombine and leave the
resistive region determines the speed of the whole process. Consequently, the am
bipolar reconnection rates obtained in [84] are insufficient either for fast dynamo
models or for the ejection of magnetic flux prior to star formation. In fact, the
increase in the reconnection speed stems entirely from the compression of ions in
the current sheet, with the consequent enhancement of both recombination2and
2In the model [84] it is assumed that the ionization is due to cosmic rays. In the case
of photoionization of the heavy species, e.g. carbon, the recombination and therefore
the reconnection rates are lower.
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Astrophysical Dynamos13
ohmic dissipation. This effect is small unless the reconnecting magnetic field lines
are almost exactly antiparallel. As above, we expect that including the effects
of anomalous resistivity and tearing modes may enhance reconnection speeds
appreciably, but not to the extent of producing fast reconnection.
None of this work included the effects of field line stochasticity, which is
critical for producing fast reconnection in ionized plasmas. We expect that in this
case also the presence of turbulence will lead to substantially higher reconnection
speeds. However, whether or not this produces fast reconnection must depend
on the nature of the turbulent cascade in a partially ionized gas. Recent work,
which is discussed in detail in the chapter by Cho, Lazarian & Vishniac in this
volume, show that the magnetic field in a partially ionized gas has a much
more complex structure than it is usually assumed. In fact, in [25] we reported
a new regime of MHD turbulence which is characterized by the existence of
intermittent magnetic structures below the viscous cutoff scale. The root mean
square perturbed magnetic field strength in these structures does not drop at
smaller scales. However, the curvature scale (and therefore the divergence rate)
for these structures does not decrease significantly as their perpendicular scale
decreases. At sufficiently small scales the ions and neutrals will decouple, and
a turbulent cascade, extending down close to resistive scales but involving only
ions will appear.
The existence of strong magnetic field structures on small scales, and the
reappearance of a strong turbulent cascade at very small scales, should lead to
fast reconnection speeds through stochastic reconnection. However, it remains to
be seen whether or not the intermediate scales, characterized by weak divergence
of field lines, will impose a significant bottleneck on the reconnection plasma
outflow. If it does, then the implication is that interstellar clouds with small
ionized fractions may not allow fast reconnection. This conclusion would not
pose any problems with galactic dynamo, but may be extremely important for
other essential processes, e.g. star formation. This issue is examined further in
[56].
3 The Dynamo Process
3.1 Conventional Theory and its Problems
We start this section by briefly reviewing the standard approach to dynamo
theory, and discussing various objections to it. Some of these objections center
around the speed of reconnection, and can be safely ignored if reconnection is
fast in a turbulent environment. In fact, since stochastic reconnection depends on
small scale structure in the magnetic field, the claim that small scale structure
tends to accumulate energy faster than the large scale field [50] can be seen
as selflimiting. A disproportionate growth in power on small scales will only
continue until the reconnection speed is boosted to large fraction of VA. However,
as we have already mentioned, some objections to dynamo theory are more subtle
and require substantial modification to meanfield dynamo theory.
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14 Vishniac, Lazarian & Cho
The usual approach to the dynamo problem is to take equation (1), set η = 0,
and divide the velocity field into small scale turbulence and some large scale
rotational motion. In order to follow the evolution of the large scale magnetic
field, we write
B ≡ ?B? + b. (10)
The brackets here denote averaging over scales somewhat larger than the turbu
lent eddy size. In other words, they indicate a smoothing process which averages
out all small scale features. The field ?B? is the ‘mean field’. The dynamo pro
cess can be written in mathematical terms by approximating the evolution of
the small scale field component, b, as
∂tb ≈ ∇ × v×?B?, (11)
and substituting the result into the evolution equation for the large scale field,
∂t?B? = ∇×?v × b?. (12)
In a turbulent, incompressible and homogeneous plasma this implies
∂t?B? = ∇×(α · B) + ∇·(DT· ∇)?B?. (13)
Here α, the kinetic helicity, and DT, the turbulent diffusion tensor, are dyads
given by
αil≡ ǫijk?vj∂lvk?τc, (14)
and
DT,ij≡ ?vivj?τc, (15)
where τcis the eddy correlation time. The component of the electromotive force
along the large scale field direction, ?ˆB? · ?v × b?, is the piece that can drive an
increase in the large scale magnetic field. (The component perpendicular to ?B?
gives an effective large scale field velocity, that is, it affects the transport of the
field rather than its generation.) The trace of α divided by τcis what is usually
referred to as the kinetic helicity, and it is often assumed for convenience that
α is a scalar times the identity matrix. In symmetric turbulence α vanishes, but
DTdoes not. In fact, since a successful dynamo requires nonvanishing diagonal
components for α, we can see from this expression that a successful dynamo
should require symmetry breaking along all three principal axes.
The appearance of DT in equation (13) would seem to vindicate the use of
turbulent diffusion in astrophysical MHD. There are two reasons why this is not
quite right. First, fast reconnection is implicit in this kind of averaging argu
ment. Rather than appealing to turbulent diffusion as an explanation for fast
reconnection, we are actually using our understanding of fast reconnection to
explain diffusion. The second point is less formal and more important. Equation
(13) is not a realistic description of the evolution of ?B?. As noted in §1, twist
ing magnetic field lines into spirals is not easily accomplished, and numerical
simulations do not support the use of equation (13).
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Astrophysical Dynamos15
3.2Magnetic Helicity Conservation Constraint
The fundamental problem is that there is an important mathematical constraint
that follows from equation (1), which is not respected by equation (13). The
magnetic helicity, defined as H ≡ A · B evolves according to
∂tH = −∇·[A×(v × B + ∇Φ)] − ηB · ∇ × B, (16)
where Φ is an arbitrary function of space and time. For the Coulomb gauge,
which turns out to be a convenient choice, we require
∇2Φ = ∇·(v × B). (17)
In the limit of vanishing resistivity, this not only implies that the volume inte
grated magnetic helicity vanishes, it also implies that the magnetic helicity of
any individual flux tube is separately conserved [79].
For a nonzero, but very small, η, we can transfer magnetic helicity from one
flux tube to another. However, since H is of order LB2, where L is a charac
teristic scale of the field, it takes less energy to hold magnetic helicity on large
scales than on eddy scales, and a divergent amount on infinitesimal scales. Con
sequently, in the limit of vanishing resistivity the resistive term in equation (16)
does not affect the global conservation of helicity, even in the presence of fast
reconnection, as long as reconnection only occurs in an infinitesimal fraction
of the plasma volume. On the other hand, the conservation of magnetic helic
ity for individual flux tubes is completely lost. The implication is that global
magnetic helicity conservation is a good approximation for laboratory plasmas,
a point that was originally stressed by Taylor [79], and an even better one for
astrophysical systems.
How does this affect dynamo theory? The large scale distribution of magnetic
helicity can be divided into a piece carried by large scale magnetic structures
and a piece carried by small scale structures, or
?H? = ?A? · ?B? + ?a · b?. (18)
Henceforth we will use h ≡ ?a·b?. The evolution of the first piece, in a perfectly
conducting fluid, is
∂t(?A? · ?B?) = 2?B? · ?v × b? − ∇·[?A?×(?v × b? + ∇?Φ?)]. (19)
The second term on the right hand side is the magnetic helicity transport driven
by meanfield terms. The first represents the exchange of magnetic helicity be
tween large and small scales. This term is proportional to the component of the
electromotive force which drives the dynamo process. In other words, the gen
eration of a large scale magnetic field is a direct consequence of the transfer of
magnetic helicity between large and small scales.
The point that MHD turbulence transfers magnetic helicity to the largest
available scales, even if that scale is much larger than any eddy scale, is well
known [30,76,77]. We can estimate the rate at which h is transferred to large