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Comparing classical and lab plasma dynamos
S. PragerUniversity of Wisconsin
useful discussions with
D. Craig, H. Ji, J. Sarff, E. Zweibel
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• The classical dynamowell-posed problem(s)
• The lab plasma dynamowell-posed
• The astrophysical field generation problemmaybe less clear
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The classical dynamo problem
V
€
˜ v × ˜ B
€
B
velocity-driven
energy source fluctuations mean, large-scale
(with seed B)
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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
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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
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V
P
Hm
or
V
P
Hm
The classical dynamo
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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)
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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)
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Driven relaxation
Poynting flux 0 helicity injection
PHm
Magnetic field grows and redistributes
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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
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experimental result
MST
McCollam, Blair, Sarff
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another experimental example
€
dHm
dt= Φ(B •∫ dS) − ˜ E • ˜ B dV∫
spheromak
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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
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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
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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
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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
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Solar fields
V dynamo
P
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Disk/Jet/lobe system
velocity-driven
dynamo
magnetic-driven dynamo
(relaxation)
P
disk engine Jet/lobe
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Disk/Jet/lobe system
magnetic-driven dynamo
PJet/lobe
relaxation, transport of B over over space, transport of B from high to low k
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Magnetic energy in the universe
1
2
jets/lobes
other
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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)
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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 (?)
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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?