Comparing classical and lab plasma dynamos

S. PragerUniversity of Wisconsin

useful discussions with

D. Craig, H. Ji, J. Sarff, E. Zweibel

• The classical dynamowell-posed problem(s)

• The lab plasma dynamowell-posed

• The astrophysical field generation problemmaybe less clear

The classical dynamo problem

V

€

˜ v × ˜ B

€

B

velocity-driven

energy source fluctuations mean, large-scale

(with seed B)

Poynting flux

Poynting flux is outward from plasma volume

€

d

dt

B2

2μ0

∫ dV = − E × B • dS∫ − j • E dV∫

< 0 > 0, source term

€

− η j 2∫ dV − j∫ • ˜ v × ˜ B dV

V

P

> 0

Magnetic helicity flux

Magnetic helicity flux direction is unclear

€

dHm

dt= − ΦB • dS∫ − E • B dV∫

€

dHm

dt= − ΦB • dS∫ − ηJ • B dV∫

using Ohm’s law

< 0 in sodium expts

< 0 in Taylor state

unclear in astrophysics

> 0 in all lab cases

V

P

Hm

or

V

P

Hm

The classical dynamo

The lab plasma dynamo

B

Magnetically-driven

energy source fluctuations mean, large-scale

€

˜ v × ˜ B

€

B

two cases:

•Free relaxation (no energy or helicity injected)

•Driven relaxation (energy and helicity injected)

Free relaxation

Poynting flux = 0 = helicity injection

large-scale field, <B>, transported by fluctuations

( in MHD)

€

˜ v × ˜ B

0

50

100

150

200

-3 -2 -1 0 1 2 3Time (ms, relative to crash)

0.00

0.02

0.04

0.06

0.08

magnetic energy (kJ)

Helicity(Wb)

Time (ms)

Driven relaxation

Poynting flux 0 helicity injection

PHm

Magnetic field grows and redistributes

Experimental examples

in a torus (e.g. reversed field pinch)

€

dHm

dt= ˜ Ψ tor

˜ V tor − 2 ˜ E • ˜ B dV∫

helicity injection through surface

€

˜ Ψ tor= toroidal flux ~

€

˜ ˙ E pol

€

˜ V tor = toroidal loop voltage

€

~ ˜ E tor

€

˜ E pol (ω0)

€

˜ E tor(ω0)dc injection of helicity

fluctuations

€

v(ω,k) × B(ω,k)

€

B

experimental result

MST

McCollam, Blair, Sarff

another experimental example

€

dHm

dt= Φ(B •∫ dS) − ˜ E • ˜ B dV∫

spheromak

One physics link between the classical and lab dynamos

In both cases, can be driven by instability or nonlinear coupling

€

˜ v × ˜ B

lab dynamo shows alpha effect can be large,

Indicates that dynamo quenching predictions are not universal

The astrophysical field generation problem

B fields are observed or deduced to

•Grow from a seed field (Earth, ISM…)

•Oscillate in time (Earth, Sun….)

•Be transported in spatial scale or wavenumber (ISM….)

•Be transported through space (Extragalactic jets…)

What are the most important problems in the generation of magnetic fields in astrophysics?

Lab relaxation processes can contribute to the latter three

Coupling of two dynamo processese.g., discussed by Blackman

velocity-driven

dynamo

magnetic-driven dynamo

(relaxation)P

Hm

velocity-driven dynamo on LHS drives relaxation or field growth on RHS

Coupling of two dynamo processese.g., discussed by Blackman

velocity-driven

dynamo

magnetic-driven dynamo

(relaxation)P

Hm

velocity-driven dynamo on LHS drives relaxation or field growth on RHS

Hm

Solar fields

V dynamo

P

Disk/Jet/lobe system

velocity-driven

dynamo

magnetic-driven dynamo

(relaxation)

P

disk engine Jet/lobe

Disk/Jet/lobe system

magnetic-driven dynamo

PJet/lobe

relaxation, transport of B over over space, transport of B from high to low k

Magnetic energy in the universe

1

2

jets/lobes

other

Magnetic energy in the universe

1

2

jets/lobes

other

is this correct?

so, magnetic transport and consequent creation of large-scale field may be important (the lab plasma dynamo or magnetic dynamo)

SummaryTwo B generation mechanisms can work together

velocity-driven engine (dynamo)internal energy source in flowcontains little magnetic energy (?)occupies small space (?)

magnetically-driven relaxationdriven by boundary conditionproduces large-scale field via transportcontains large magnetic energy (?)occupies large space (?)

SummaryTwo B generation mechanisms can work together

velocity-driven engine (dynamo)internal energy source in flowcontains little magnetic energy (?)occupies small space (?)

magnetically-driven relaxationdriven by boundary conditionproduces large-scale field via transportcontains large magnetic energy (?)occupies large space (?)

Should the astrophysical “dynamo problem” be broadened to include both effects about equally?