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.

• 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).

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∂ t

r B =∇ × v ×

r B −∇ × η∇ ×

r B ( )

€

ℜB ≡v Lη

→ ∞

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 φφ

LB

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.

Schematically…

More Mathematically . . .

Bur

total =Bur+b

rWe divide the field into large and small scale pieces

which evolve following averaged versions of the induction equation

∂t

rB = ∇ ×

rv ×

rb

∂t

rb = ∇ ×

rv ×

rB +

rv ×

rb −

rv ×

rb( )

We can estimate the electromotive force by setting it equal to zero at some initial time, taking the time derivative and multiplying it by the eddy correlation time.

rv×

rb

In a nonshearing environment this gives . . .

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 trace is not a conserved quantity in ideal MHD (and is not a robust conserved quantity in hydrodynamic turbulence with an infinitesimal viscosity).

Dt

rv ≈

rBg∇( )

rb, and Dt

rb≈

rBg∇( )

rv

sorv×

rb ≈

rv×

rBg∇( )

rv-

rb×

rBg∇( )

rb τeddy

rvg

rω

The second term arises from the current helicity tensor. This can be nonzero, in an interesting way, if its trace is nonzero. This in turn will be nonzero if the magnetic helicity (in the Coulomb gauge) is nonzero, i.e.

rbg

rj

rAg

rB ≠0, ∇g

rA=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.

Γmax ≈3

4Ω

λ : VAΩ

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 : b2

: v2

Conserved Quantities from the Induction Equation

∂t B

ur= ∇ × v

r× B

ur( )

There are two conserved quantities associated which follow from this: magnetic flux and magnetic helicity

and H ≡Aur

gBur

Φ

The former is a gauge-dependent measure of topology. In the Coulomb gauge we can write:

−ηBur

gJur

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

If we compare this to the averaged induction equation:

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.

∂t H = 2

rBg

rv ×

rb −∇g

rBΦ +

rAg

rv ×

rb( )

so that

∂t

rB = ∇ ×

rv ×

rb

we see that the large scale field is driven by the transfer of magnetic helicity between scales.

∂th = −2

rBg

rv ×

rb −∇g

rjh

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 controls the growth of the large scale magnetic field.

rjh

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

The eddy scale magnetic helicity current can be calculated explicitly. It is

If we make the approximation that the inverse cascade is faster than anything else, we have

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.

Boozer 1986; Bhattacharjee 1986; Kleeorin, Moss, Rogachevskii and Sokoloff 2000; Vishniac and Cho 2001

rJh =h

rV +

d3r′r4π ′r

′r 2∂s

rb

rr( )σ sl

rjl

rr +

r′r( )−

rb

rr( )

rBg

rω r

r +r′r( ) +

rv

rr( )

rBg

rj

rr +

r′r( )⎡⎣ ⎤⎦∫

When is this nonzero?

We can rewrite the last two terms as

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

−2∇−2

rBg

rω(

rBg∇)

rv τ corr

2kc−2ωA

2τ corr vrvφ −r ′ vr2( )

…plus some terms which depend on the vertical velocity dispersion.

The shear term in the magnetic helicity current is

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

r ′ kc−4 ∂rbr

2 − ∂φbφ2

( )

− ′

N.B. These expressions ignore vertical fields. In otherwiseisotropic turbulence, increasing these will tend to drive an anti-dynamo.

What Does This Tells Us About the Large Scale Dynamo?

• In an “-” dynamo

• 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:

∂t Bφ = −qΩ(r)Br +∇g DT∇Bφ( ), (r) ∝ r−q

rv×

rb =−

rBB2 ∇gjh ⇒ ∂tBr =−∇×

rB

2B2 ∇gjh( ) ≈∂z

Bφ

2B2 ∂z jhz

⎛⎝⎜

⎞⎠⎟

This suggests that has a preferred sign in a successful dynamo.

jhc

Bφ2

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.• 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.

−∂rΩ(r)

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

which beats turbulent dissipation over the large scale length L only if

:

B2

v2 τ corr

B ≥DTL ≈

λeddy

Lv

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 τeddy( )

1/2 λeddy

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

• 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.

JH : B2λTvT(πR2 ) : ss&MHr2Ω2

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.