# Making Magnetic Fields: Dynamos in the Nonlinear Regime

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### Transcript of Making Magnetic Fields: Dynamos in the Nonlinear Regime

Making Magnetic Fields: Dynamos in the Nonlinear Regime

Collaborators:

Alex Lazarian --- U. Wisconsin

Jungyeon Cho --- Chungnam U.

Dmitry Shapovalov --- JHU

Baltimore -September 2005

Outline:

Defining the problem

Magnetic Helicity

The Nonlinear Dynamo

Disks and the Magnetization of the Universe

What is a dynamo?

1) The growth of magnetic energy density from some negligible beginning. In turbulence, this comes from the stretching of weak magnetic field lines entrained in a fluid. Since the magnetic field is dragged by the turbulent eddies, it has a typical length scale no larger than the scale of the eddies, and often much smaller. This is the “small scale dynamo”.

2) The growth of magnetic fields with scales much larger than individual eddies, usually comparable to the size of the object generating the field, a planet, a star, an accretion disk, or even a galaxy. This is the “large scale dynamo”.

This is the growth of a magnetic field from some much weaker seed field. It can mean several different things:

What’s so surprising about a dynamo?

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. (However, see papers on collisionless effects and `stochastic’ reconnection.)

€

∂ t

r B =∇ × v ×

r B −∇ × η∇ ×

r B ( )

€

ℜB ≡v Lη

→ ∞

What do we know about the small scale dynamo?

Many papers describing computer simulations of MHD turbulence evolving from a weak initial field, beginning with simulations (!). The current generation of simulations have up to .

323

The conclusion is that MHD turbulence stores a large amount of energy in a disordered magnetic field (roughly in equipartition with the turbulent energy density). The time scale for reaching equipartition is of order an eddy turn over time (perhaps logarithmically longer).

10243

Cosmological Magnetic Fields?

Large scale magnetic fields are generated by both kinds of dynamos:

• Magnetic fields in galaxies (and AGN, stars, and all other rotating objects) are examples of large scale dynamos. The large scale is organized by the system rotation. Such systems typically show small scale magnetic field structures as well. Growth rates depend on the details of the dynamo process.

• Magnetic fields in the intracluster medium are generated by the small scale dynamo, driven turbulence in the intracluster medium. Growth rates at a given scale are given (more or less) by the eddy turn over rate at that scale.

The Standard - Dynamo

• In a strongly shearing environment radial components of the magnetic field will be stretched to produce a toroidal field.

• The radial field is generated from the toroidal field, through the effect. This requires a kinetic helicity in the surrounding turbulence 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 robustly 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).

The Disk Dynamo: Generating B for the MRI

The usual treatment involves keeping only vertical gradients.

An accretion disk dynamo is a standard example of an “-” dynamo, in which the azimuthal field component is generated by shearing the radial field, and the radial field is generated by eddy-scale motions from the azimuthal field.

∂t

rB = ∇ ×

rV ×

rB +

rv ×

rb( )

⇒ ∂t Bθ ≈ −

3

2ΩBr + ∇gDT g∇( ) Bθ

∂t Br = ∇ ×

tα g

rB( ) + ∇gDT g∇( ) Br

rBg

rv×

rb is critical for the dynamo effect.

Where does the disk turbulence come from? --- In an accretion disk it comes 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

In the absence of any magnetic helicity current, the dynamo can only work by creating equal and opposite amounts of magnetic helicity on large and small scales, limiting the large scale magnetic energy to the ratio of the eddy scale to the large scale, times the small scale magnetic field energy.

Even this is unrealistic. The magnetic helicity h will interact coherently with the large scale field, inducing motions (through the current helicity ) which will transform h into H. The rate for this is

A successful dynamo requires a systematic magnetic helicity current, driving local accumulations of “h”, which then drives the dynamo through the nonlinear cascade.

τ −1cascade ≈ k2B2τ corr

The Eddy-Scale Magnetic Helicity Current

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

Unless the large scale field is very weak, the inverse cascade is faster than anything else, so that

This will be zero in perfectly symmetric turbulence. What do we need to get a non-zero effect?

rv×

rb =−

rB

2B2 ∇grJ h

rJh =h

rV +

d3rk

(2π )3k2∫rv(

rBg

rj )−

rb

rBg

rω( )( )−2r ′

d3rk

(2π )3k4∫ jr∂θrb+ jθ∂r

rb

Symmetry Properties of the Magnetic Helicity Flux

This flux is invariant under a reversal of the magnetic field. It will depend on even powers of the large scale magnetic field.

This flux is invariant under a parity transformation. It can be nonzero in mirror-symmetric turbulence.

A non-zero flux requires symmetry breaking in 2 directions. Differential rotation will provide this.

A constant flux is uninteresting, but if the large scale magnetic field varies with position, then div J will not vanish.

A Simple Model of the Accretion Disk Dynamo

In order to see what this does to the accretion disk dynamo, we need to plug in the correlations expected from the MRI. We can gain some insight by simply writing the magnetic helicity current in the vertical direction as Plugging this into the dynamo, and neglecting the dissipative terms, we get

This will give an exponentially growing magnetic field if the magnetic helicity current is positive. Otherwise we get an anti-dynamo.

ABθ2

∂t Br ≈ A∂z2Bθ ∂t Bθ = −

3

2ΩBr

The Magnetic Helicity Flux for the MRI

We can evaluate the magnetic helicity flux to quasilinear order by taking the time derivative of correlations between the fluctuating magnetic and velocity fields and multiplying by the correlation time. In the presence of differential rotation we need to invert the effects of shear and the Coriolis and centrifugal forces. The final expression is quite complicated. However, the MRI produces eddies with

kθ < kr < kz, and τ corr : 3

We can use this to get

Jhz : 2Bθ

2τ corr

kθ2

k2 vrvθ +32

vr2 −6 brbθ

⎡⎣⎢

⎤⎦⎥

The MRI Dynamo

We conclude that the MRI should drive a positive vertical magnetic helicity current, and a dynamo.

By the same line of reasoning, whatever drives turbulence in the Sun drives a dynamo, and the predicted magnetic helicity flux should be negative.

If we treat the accretion disk as a periodic shearing box, then this gives us a dynamo growth rate of

The existence of a dynamo depends on the sign of the magnetic helicity current. If it ran the other way the dynamo would be suppressed (as it is in magnetic Kelvin-Helmholtz simulations).

The implication is that the MRI dynamo field always grows on a time scale of a few eddy turn over times, and is always dominated by scales which are a few eddy scales in size (vertically). This is what is seen in the MRI simulations.

Seeding the Universe If we compare the energy of formation of a galaxy

and its central black hole we that they are (very) roughly comparable. The ability of AGN to project a large fraction of this energy in a high velocity jet implies that they can play a large role in heating the IGM.

The flux carried by an AGN jet is limited by the very small cross section of the jet. On the other hand, the flux ejected in a galactic wind is largely toroidal and may contain many field reversals. It’s not clear who wins here. Neither effect is large.

However, since both the AGN accretion disk and the galactic disk contain dynamos, and will eject magnetic helicity coherently over the lifetime of the disks.

−

Magnetic Helicity Injection From Disk Galaxies

The magnetic helicity current ejected vertically from a disk dynamo is of order

This favors the galactic wind over AGN jets in L* galaxies by a factor of at least

The net effect is that within a single rotational period a typical disk galaxy ejects enough magnetic helicity to fill its corona with a coherent field of a few tenths of a microgauss. If this field fills larger volumes its strength will drop as the inverse length scale squared. Still large.

JH : B2λTvT(πR2 )

105

Magnetic Helicity and Turbulent Diffusion

The electromotive force in a turbulent medium, has a piece of the form:

This is zero in Alfvenic, isotropic turbulence. When the turbulence is anisotropic due to shearing, the diffusion coefficients differ and attach to the appropriate gradient. When the back reaction is large, the diffusion is suppressed by the factor . The radial diffusion of entrained flux is suppressed by the ratio of the perpendicular components to the total field.

−DT

rJ , DT ≈ v2 − b2

( )τ corr 3

I −b̂b̂

: B⊥

2

B2

Outward diffusion = inward drag when….

Balancing the two, as before, we get a critical poloidal field strength

This is just big enough that the radial and vertical field components will be similar (Lubow, Papaloizou and Pringle 1994) and the stress (and angular momentum transport) through the magnetosphere will be comparable to the internal transport. When the poloidal field is weaker than this, it will be dragged inward. Due to this, and the scaling of disk properties, the poloidal field will typically affect the innermost part of the accretion disk, or none of it.

Bcrit : Bθ

hr( )

1/2: cs h

r( )1/2

When does the environment supply enough magnetic flux?

Since the accreted poloidal field has nowhere to go, whether or not one gets an inner annulus dominated by a large scale poloidal field will depend on the environment. In general, it’s not obvious what this means for systems that accrete from the ISM. We can look at binary accreting systems, and assume that the accretion disk can accrete some fraction of the donor star’s poloidal field, determined by the size of the disk. This will be important if

Bprdisk

rinner( )

2

≥Bcrit : &M

r( )

Plugging in Numbers…. Obviously, for otherwise identical accreting systems, this is more

likely to give us a strong poloidal field when the accreting object is a compact object, the smaller the better.

A white dwarf near the Chandrasekhar limit, accreting at 1018 gm/sec, with an externally imposed field of about a gauss, and an inner disk radius of about 109 cm, will fail to acquire a strong poloidal field by a bit more than an order of magnitude. In the same kind of system a black hole will easily acquire a strong poloidal field.

Is this the reason CV’s do not (usually) have jets?

Summary: Magnetic helicity conservation gives us a

powerful tool to understand the production of large scale ordered fields.

Cosmic magnetic fields can be grown from small scale fields. Disk dynamos are an important part of this process, and can produce locally intense fields as soon as plasma disks form.

Accretion disks will accrete poloidal flux from the environment very efficiently, until (and if) the accreted flux at small radii is sufficient to produce strong dynamical effects (including jets?).

The IGM is seeded with significant magnetic fields ( ) on scales of several hundred kpc

10−9 to 10−10