Download Dynamos in Accretion Disks:

The Disk Dynamo: How to drive a dynamo
in an accretion disk through shear and a local
instability
Collaborators:
Jungyeon Cho --- Chungnam U.
Dmitry Shapovalov --- Johns Hopkins U.
Princeton, NJ 2005
Outline:
• The - Dynamo
• Magnetic Helicity
• The Nonlinear Dynamo
Making Large Scale Fields in Astrophysical Plasmas
• In the limit of perfect conductivity, we find that the
magnetic field is “flux-frozen”. The magnetic flux through
a fluid element is fixed at all times.


 tB    v  B      B
B 
vL



• The same result guarantees that the topology of a magnetic
field is unchanged, and unchangeable.
• Simple models of magnetic reconnection (topology
changes) when resistivity is merely very small give very
slow reconnection speeds. Need fast reconnection
(collisionless effects, stochastic reconnection).
The - Dynamo
• In a strongly shearing environment radial components of the
magnetic field will be stretched to produce a toroidal field.
(For a disk we invoke cylindrical geometry.)
• The radial field is generated from the toroidal field, through
the `` effect’’ (more later). This requires the surrounding
turbulence to have an asymmetrical effect on the field lines,
twisting them into spirals with a preferred handedness, and
vertical gradients in the field strength.
• The growth rate is the geometric mean of the local shear, ,
and
 
LB
Schematically…
i.e. a 3D process in which new field is generated orthogonal to the old field, and its
gradient. In an accretion disk the radial field component is generated from the
toroidal component, and differential rotation regenerates the toroidal component.
More Mathematically . . .
We divide the field into large and small scale pieces
ur
ur r
Btotal  B  b
which evolve following averaged versions of the
induction equation
r
t B   
r
t b   
r r
vb
r r r r r r
v B vb  vb


r
We can estimate the electromotive force v  b
by setting it equal to zero at some initial time, taking
the time derivative and multiplying it by the eddy
correlation time.
In a nonshearing environment this gives . . .
r
r
r
r
r
r
Dt v  Bg b, and Dt b  Bg v
 
 
so
r
r r
r r
r r r
v  b  v  Bg v-b  Bg b  eddy
 
 
plus advective terms which give rise to turbulent
diffusion effects.
The first term arises from the kinetic helicity tensor.
This can be nonzero, in an interesting way, if the
environment breaks symmetry in all three directions
(which brings
in large length scales). Note that the
r
trace vg is not a conserved quantity in ideal MHD
(and is not a robust conserved quantity in
hydrodynamic turbulence with an infinitesimal
viscosity).
The second term arises from the current helicity tensor.
This can be rnonzero, in an interesting way, if its trace
is nonzero.
in turn will be nonzero if the magnetic
bgThis
j
helicity (in the Coulomb gauge) is nonzero, i.e.
r
r
AgB  0, gA  0
This is interesting because the magnetic helicity is a robustly
conserved quantity. This term gives rise to the early saturation
of kinematic dynamos (where the environment, or the
programmer, enforces some kinetic helicity).
Where does the disk turbulence come from?
--- The magnetorotational instability (MRI)
Radial wiggles in a vertical or azimuthal field, embedded in a
shearing flow, will transfer angular momentum outward
through magnetic field line tension (like the tethered satellite
experiment). This increases the amplitude of the ripples.
3
 max  
4
 : VA 
Numerical simulations indicate a dynamo effect,
in which the amplitude of the large scale field,
and the size of the eddies, increases together with
the small scale magnetic field and kinetic energy.
B2 : b 2
v2
Conserved Quantities from the Induction Equation
ur
r ur
t B    v  B


There are two conserved quantities associated which
follow from this: magnetic flux and magnetic helicity

ur ur
and H  AgB
The former is a gauge-dependent measure of topology.
In the Coulomb gauge we can write:
ur ur
 BgJ
Some useful points about magnetic helicity:
• Magnetic helicity is conserved for all choices of gauge, but
in the coulomb gauge the current helicity and magnetic
helicity have a close connection. Gauge-independent
manifestations of magnetic helicity actually depend on the
current helicity (unfortunately, the latter is not conserved).
• Magnetic helicity has dimensions of
(energy density)x(length scale)
The energy required to contain a given amount of
magnetic helicity increases as we move it to smaller scales.
(Reversed field pinch, flux conversion dynamo, Taylor
states)
Magnetic helicity is a good (approximate) conservation law even
for finite resistivity!
The Inverse Cascade of Magnetic Helicity
We can expect from the energy argument that magnetic
helicity will be stored on the largest scales. This can be
shown analytically in a variety of models for turbulence
(see for example Pouquet, Frisch and Leorat 1976). We
can gain additional insight by looking at a two scale
model, i.e.
so that

r r r
r
r r r
t H  2 Bg v  b  g B  Ag v  b
r r r
r
t h  2 Bg v  b  gjh

If we compare this to the averaged induction equation:
r
r r
t B    v  b
we see that the large scale field is driven by the transfer of
magnetic helicity between scales.
The kinematic dynamo vs. magnetic helicity
The kinematic dynamo drives a large scale magnetic
field by generating magnetic helicities of equal and
opposite signs for the large and small scale fields, that is
H  h
However, the small scale helicity has a much larger
current helicity, and the back-reaction through the
second term in the electromotive force will quickly
overwhelm the kinetic forcing (Gruzinov and
Diamond).
The obvious loophole is that a small scale magnetic
helicity current can prevent a buildup of current
helicity. This implies that jh controls the growth of
the large scale magnetic field.
This is the RIGHT way to twist a flux tube
We need a magnetic helicity current perpendicular to the old and new field components!
The Eddy-Scale Magnetic Helicity Current
If we make the approximation that the inverse cascade
is faster than anything else, we have Boozer 1986; Bhattacharjee 1986;
Kleeorin, Moss, Rogachevskii and
Sokoloff 2000; Vishniac and Cho
2001
The eddy scale magnetic helicity current can be
calculated explicitly. It is
r
J h  hV 
r
r r
r r r r r r
r r r
d 3r 
r r r r r r
2
 4 r   r  sb r  sl jl r  r    b r Bg r  r    vr Bgj r  r  
Here sigma is the symmetrized large scale shear tensor. This
current will be zero in perfectly symmetric turbulence.
However, if we have symmetry breaking in the radial and
azimuthal directions (due to differential rotation) then it will be
non-zero, despite the vertical symmetry.
When is this nonzero?
We can rewrite the last two terms as
r r r
r
2
2 Bg( Bg)v  corr
For a successful dynamo the most important part of the
magnetic helicity current is perpendicular to the mean field
lines. In a cylindrically symmetric system this is the
vertical magnetic helicity current. Then this term can be
rewritten as

2kc 2 A 2 corr vr v  r vr 2

…plus some terms which depend on the vertical
velocity dispersion.
The shear term in the magnetic helicity current is
rkc 4

r br
2
  b
2

whose sign is ambiguous in general. Note that this term
does not depend on the correlation time.
For the MRI this term has the sign of   
N.B. These expressions ignore vertical fields. In otherwise
isotropic turbulence, increasing these will tend to drive an
anti-dynamo.
What Does This Tells Us About the Large Scale
Dynamo?
• In an “-” dynamo


t B  q(r)Br  g DT B ,
(r)  r q
• The radial field must be produced from eddy-scale
motions acting on the azimuthal field. The eddy
scale magnetic helicity is quickly transferred to the
large scale magnetic field. Assuming this process
is fast we can assume that h is stationary and that:
r
r
r
 B

B
B
v  b   2 gjh  t Br   
gjh    z  2  z jhz 
2 
 2B

B
2B
j
This suggests that hc B 2 has a preferred sign in a

successful dynamo.
More…….
• The vertical magnetic helicity current is zero in
perfectly homogeneous turbulence, but nonzero in
the presence of differential rotation.
• It is quadratic in the magnetic field strength. r(r)
• In a successful dynamo it has the same sign as
• When length scales are defined by the magnetic
field, e.g. when the turbulence is driven by a
magnetic instability, the growth rate is a large
fraction of the shear rate and magnetic field
structure grows until the vertical structure is like
the disk thickness.
When does the nonlinear - work?
This discussion has assumed that the transfer of
magnetic helicity to large scales is arbitrarily fast,
or at least faster than the turbulent mixing rate.
However, in practice
the transfer rate is
2
B
v2  corr
which beats turbulent dissipation over the large
scale length L only if
B
DT
eddy
L L v
which is always true for the MRI dynamo.
When Does a Kinematic Dynamo Work?
Suppose we have some imposed kinetic helicity and there is
no significant magnetic helicity current. The generation of
radial field doesn’t have to get anywhere near equipartition to
generate a large azimuthal field. Nevertheless the
backreaction does become important before we reach
equipartition between the magnetic and kinetic energies. The
saturation level for the exponential process is

EB : Ekinetic  eddy

1/2
eddy
Lsymmetry
Where the kinetic energy includes only correlated
pieces. That is, an additional random velocity field
(due for example to the MRI) wouldn’t contribute to
the RHS.
Magnetic Helicity Ejection From Disks
• The magnetic helicity current ejected vertically from a disk
dynamo is of order
& 2 2
J H : B2 T vT ( R2 )  ss MHr
• For a stationary accretion disk this is insensitive to radius,
unless (H/r) varies strongly with radius. Advective regions
will have a disproportionate contribution to the magnetic
helicity flux.
• A typical disk galaxy ejects enough magnetic helicity to fill
its corona with a coherent field of a few tenths of a
microgauss (AGN contribution is small). If this field fills
larger volumes its strength will drop as the inverse length
scale squared. Still large.
Summary:
• Magnetic helicity conservation gives us a
powerful tool to understand the production
of large scale ordered fields.
• The growth rate for the MRI driven dynamo
is some large fraction of the local shear.
Typical domain sizes will increase with the
field strength..
• Magnetic helicity currents are a necessary
part of disk dynamos and these will be
directed along the disk axes.