Plans for Dynamo Research

Presented byF. Cattaneo, S. Prager

Outline

• Evidence for dynamo effectsin astrophysics

in the lab

• Nonlinear Features of the dynamostatus and plans

• Dynamo Effects Beyond MHDstatus and plans

Evidence for dynamo effects in astrophysics

IGM

• Typical size: 30 kpc wide, 300 kpc long• Magnetic fields: 0.5 – 5 Gauss• Dynamo action in disk around central

SMBH

Galaxy

• Typical size: 1020 m. Rotation period 108

years• Magnetic fields: 3 Gauss (horizontal

cmpnt)• Turbulence driven by supernovae

explosions• Classical - dynamo

Evidence for dynamo effects in astrophysics

Accretion disks

• Turbulence driven by MRI• Magnetic field necessary to drive MRI, self

consistently generated by dynamo action

Late-type stars (Sun)

• Magnetic activity extremely well documented

• Turbulence driven by convection. • Activity cycles• “Mounder minima”• Classical - dynamo for large-scale field• Evidence for small scale dynamo action

Evidence for dynamo effects in “astrophysics”

Geodynamo • Example of system where dynamo must

operate• Turbulence driven by (compositional?)

convection. Strong rotation• Moderate Rm

• Dipolar field exhibits reversals

Laboratory experiments• Plasma devices (more about it

presently)• Liquid metal experiments

• Experiments with highly constrained geometries have achieved dynamo action

• Experiments with “open” geometries hopefully will achieve dynamo action soon

The Madison Dynamo Experiment

Dynamo Effects in Laboratory Plasmas

The lab plasma dynamo does• Generate current locally • Convert poloidal magnetic flux to toroidal flux (and the inverse)• Increase toroidal magnetic flux• Conserve magnetic helicity• Act through alpha and other effects• Arise from fluctuations superposed on the mean field• Achieve a nonlinearly saturated state with

(with self-generated )

The lab plasma dynamo does NOT• Generate magnetic field from a small seed field• Increase magnetic energy (it redistributes magnetic field) †

B ˜ B

Evidence of field generation

• Cowling’s Theorem

• Toroidal flux generation

• Ohm’s law

Cowling’s theorem applied to the RFP

A time-independent, cylindrically symmetric plasma cannot contain a reversed magnetic field

Proof: assume Bz is reversed.

at the radius where Bz = 0

E dl E ||rd

J r2 dBz

drr2 0

Thus, magnetic flux decays within reversal surface, in constrast to experiment

Bz

r

Toroidal magnetic flux increases(in discrete dynamo events)

ToroidalMagneticFlux(Wb)

MST

time (ms)

in experiment

-0.5

0.5

1.0

1.5

2.0

V/m

0.0

0.0 0.2 0.4 0.6 0.8 1.0/a

E||

neo J||(Zeff = 2)

E j

E||

j||

radius

additional current drive mechanism (dynamo)

Linear and nonlinear dynamos

1 2

1 2

( )

( Re ) ,

0, 0, .

t

t

Rm

p

B u B B u

u u u

u B J

J F

B

B

Kinematic regime • Weak initial field• Lorentz force negligible • Seek “exponentially” growing solutions of the induction equation • Linear eigenvalue problem

Nonlinear regime• Lorentz force dynamically important• Dynamo saturation and stationary MHD state• Self consistent solution of velocity and magnetic field• Nonlinear initial value problem

Large and small scale dynamos

Assume that velocity is characterized by typical scale ℓ

Small scale dynamo • Generation on scales ℓo• Competition between line stretching and enhanced diffusion

• Dynamo generates B2 but not B2

Large scale dynamo• Generation on scales ℓo• Lack of reflectional symmetry important (helicity)• Inverse cascades (magnetic helicity, energy, etc.)• Mean field theory and transport

• Average induction α-effect• Average diffusion β-effect• Average advection γ-effect

From kinematic to nonlinear dynamosMost astrophysical situations:

• Dynamos operate in nonlinear regime• Magnetic fields are in equipartition with velocity on integral

scales• Rotation is present and important

What are the dynamo saturation mechanisms that leads to observed field stregths?

ℓ/ℓo

1B2

kin

em

ati

c m

od

els

non

linear

mod

elsLarge-scale dynamos

Small-scale dynamos

How do dynamos saturate?Small-scale dynamos

• What happens to lagrangian properties of flow?

• What is the structure of resulting MHD turbulence?

• Dependence of dynamo fields on Pm

Large-scale dynamos

• Saturation of turbulent transport• α-effect (strong-weak)• β-effect (strong-weak)

• Role of shear• Role of boundary conditions

Proposed researchBasic phenomena: SSD

• Study development and properties of stationary MHD turbulence state generated and sustained by dynamo action (Turbulence effort)• Eulerian properties• Lagrangian properties

• Study dependence of final state with magnetic Prandtl number.

Requirements:

• Existing codes • Manpower

Proposed researchBasic phenomena: LSD

• Establish existence of inverse cascades in high Rm systems• Establish conditions for strong satruration of α-effect

• Boundary terms (helicity flux)• Time dependence• Relation between DN simulations results and RFP

experiments• Conditions for strong satruration of β-effect• Role of shear • Role of magnetic helicity

Requirements:

• Some modifications of existing codes• Formulation of sensible “model problems”• Manpower

Proposed researchSpecific models:

• The solar dynamo: Develop a self consistent model capable of reproducing basic observed properties• Cyclic activity• Realistic distribution of angular velocity in the CZ• Thin tachocline• Correct migration pattern of magnetic activity

Requirements:

• New code must be developed• Spherical geometry• Incompressible/anelastic• Spatial adaptivity

• Major effort in Sub-Grid-Scale modeling• Better understanding of angular momentum transport (angular

momentum effort)• Manpower

Dynamo Effects Beyond MHD

• In the labstrong indications of importance,a rich, relatively unexplored topic

• In astrophysicsgeneral importance not established,possibly only some “special cases,”depends on scope of “dynamo physics”

Dynamo Effects Beyond MHD

• Hall dynamo

• Diamagnetic dynamo

• Kinetic dynamo (current transport)

time (ms)

r/a = 0.9

˜ v ˜ B

j E

MHD dynamo dominant at some radii, not everywhere

r/a = 0.8

Measurement of MHD dynamo

0

-10

-20

0

-20

-10

Volts m

Volts m

-0.5 0 0.5time (ms)

r/a = 0.9

r/a = 0.8

Hall dynamo: a two-fluid effect

j ˜ v ˜ B ˜ j ˜ B

neMHD

dynamoHall

dynamo

Two fluid effects also alter the <v x B> dynamo

From quasilinear theory for tearing mode dynamo

-1

0

1

2

3

4

5

6

0.001 electron skin depth 0.05 ion Larmor radius 1 3

DISTANCE FROM RESONANCE SURFACE X/L

˜ j ˜ B ||

ne

˜ v ˜ B || 100

distance from resonant surface

de s

Mirnov

100

80

60

40

20

0-2 -1 0 1 2

Time [ms]

Time Evolution of Current Density Fluctuation

Ding et al

-25

-20

-15

-10

-5

0

5

-1 -0.5 0 0.5 1 1.5 2

Hal

l dyn

amo

(V/m

)

Time from crash (ms)

Hall term is significant at r/a = 0.8

time (ms)

V/m

˜ j ˜ B ne

Fiksel, Almagri

The diamagnetic dynamo

j|| E

|| ˜ v ˜ B

||

˜ j ˜ B ||

ne

j E v B j B

ne

pe

ne

j E ˜ E ˜ B

B

˜ p e ˜ B

B

parallel component of mean-fields,

or, writing yields

˜ v ˜ j

ne ˜ E ˜ p e B

B2

MHD dynamo

diamagnetic dynamo

Measurement of diamagnetic dynamo

Ji et al

TPE-1RM20 RFP

Different dynamo mechanisms dominate in different parameter regimes

Kinetic Dynamo

Radial transport of parallel current (electron momentum) by particle motion along stochastic magnetic field

not yet measured

Ready for a comprehensive study via

• Experiment (MST, some SSPX, possibly MRX)

• Analytic theory (quasilinear, early nonlinear stage)

• Computation (NIMROD)

Measure dynamo mechanisms directly

• MHD

• Hall

• Diamagnetic

• Kinetic

• Also measure <E> and <j>

˜ v ˜ B

˜ j ˜ B

˜ B ˜ p e

˜ B || ˜ p e

Measurement Techniques

In the hot core

˜ v passive spectroscopy,

active spectroscopy (charge exchange recombination spectroscopy)

Laser Faraday rotation

Motional Stark effect

˜ B

In the cool edge

Insertable probes: magnetic, Langmuir (E), spectroscopic

Active Spectroscopy

30 keV H Beam

Beam CurrentMonitor

Perpendicular Viewing Chords

22.5° ViewingChordMST Vessel

3-Wave Polarimeter-Interferometer System

MST R0 = 1.50 ma = 0.52 mIp = 400 kAne ~ 1019 m-3

B0 ~ 4 kG

Faraday rotation/interferometer system

Spectroscopic probe

Plannned measurements

MHD DynamoEdge: upgrade spectroscopy probe (6 months)Core: CHERS - operation in 1 year for V fluctuations

physics in 2 years

Hall DynamoEdge: probe measurements in 1 yearCore: improve FIR - 1 year

MSE - design spec for mag fluctuations - 6 months

first operation ~ 1.5 years

Study spectral properties (nonlinear coupling)

Diamagnetic dynamoedge: probes to reversal surface ~ 1.5 yearsCore: need new ideas for pe fluctuations (fast

Thomson)

Kinetic dynamoNeed ideas for pe|| fluctuations

Two-Fluid Dynamo Theory

• Quasilinear theory with p(one year)

• Early nonlinear stage

(1 - 2 years)

Two-Fluid Computation

Nimrod: well-suited to experiment,

two-fluid operation in ~ 1

year,

physics results in 2 - 3 years

In ~ 3 years,expect major advance in understanding two-fluid dynamos in the lab

Flow-driven dynamo

• Drive flow with neutral beam injection orbiased probes (in MST, MRX)

• Establish NBI feasibility for MST (expt) and MRX (calculation) - 6 months

Effects beyond MHD in astrophysics

Important physical parameters:

ion skin depth

ion sound gyroradius

MHD reconnection layer width

di c

pi

s cs

ci

dR L

S 2 / 5

Hall dynamo important if

di >> dR

or s >> dR

satisfied in lab

Venues in astrophysics with Hall effects

•Extra-galactic radio lobes

flux conversion dynamo in relaxing plasmas

•Black hole accretion disks

MRI dynamo, flux conversion

•Protostellar disks

Weakly ionized, charged dust

•Neutron star, white dwarf crusts

ions immobilized

Plans: Computation of disk flux conversion

r

z

0 5 10 15 200

2

4

6

8

10

12

14

16

18

20 (a)

r

z

0 0.25 0.5 0.750

0.3

0.6

0.9

1.2

1.5(b)

disk arcade spheromak

Proceed with Nimrod

Plans:

more completely assess prospects of non-MHD effects in astrophysical dynamo physics

then, construct work plan or de-emphasize (~ 4 months)