Why current-carrying magnetic flux tubes gobble up plasma and become thin as a result - model and...
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Transcript of Why current-carrying magnetic flux tubes gobble up plasma and become thin as a result - model and...
Why current-carrying magnetic flux tubes gobble up plasma and become thin as a
result
- model and supporting lab experiments
Paul Bellan
Caltech
Students/Postdocswho worked on experiments
• Freddy Hansen
• Shreekrishna Tripathi
• Scott Hsu
• Sett You
• Eve Stenson
Model:
– ideal MHD (field frozen to plasma)– dynamics, history, non-equilibrium– compressibility– finite pressure gradient, finite – non-conservative property of J x B force– J x B driven flows, flow stagnation
Ideal MHD:
dUdt
J B P
B t
U B
B 0 J
t
U 0
P
Eq. of motion
Induction equation
Ampere’s law
Mass conservation equation
Adiabatic relation
Statement of the problem• Potential flux tubes are not axially uniform
potential flux tube bulges at top because B field weaker at top,and cross section area A~1/B
solar surface
Classic pinch force cannot explain uniform cross-section
• classic pinch fails because J x B ~ 1/r3, so pinch force smaller at axial midpoint (sausaging)
2r
B ~ 1/r Jaxial ~ 1/r2
Simplify analysis by considering straight-axis flux tube
(axis curvature considered later)
footpoint
2r
ToroidalDirection
( )Poloidal direction
(r,z)
footpoint
Electric current is made to flow along flux tube
from one footpoint to the other
Current I
axial flow
1. Twisting (rising current)
2. Axial thrust (steady current)
3. Stagnation (steady current)
Twisting
Thrust Stagnation
I(t)
t
Physics consists of three distinct stages:
i. Incompressible torsional motion (like Alfven wave)
ii. Torque provided by polarization current
iii. No poloidal motion
iv. Profile of flux tube unchanged
v. Toroidal velocity given by
I(t)
tTwisting
risingcurrent
t
I
rB
sU
pol
20
Initially untwisted potential flux loop
sdistance along field line from midplaneSurface of constant poloidal flux, ψ
Axial current twists flux loop, creates B
Finite toroidal fluid velocity (twisting), no poloidal (axial) fluid velocity
B
t rBpol
U r rUpol
B r B Upol
Toroidal component of induction equation
(frozen-in flux condition)
zero during first stage,no poloidal flow
t
I
rB
sU
pol
20
in first stage
Integrate w.r.t. distance s
Second Stage
Axial thrust stage (steady-state current)i. Bidirectional flows accelerated by torque
ii. Non-equilibrium
I
tthrust
To understand 2nd stage physics, first consider simpler situation, namely axially non-uniform current without embedded axial field
current
canted JxB force gives axial thrust
•thrust direction independent of current polarity•flow goes from small to large radius
axial flow
current
canted J x B force gives axial thrust
axial flow
Like squirting toothpaste from a toothpaste tube
Now consider arc between two equal electrodes
current
J x B force gives axial thrust
axial flow axial flow
Current flow along initially potential flux tube
(i.e., now include embedded axial field)
• Current produces B so net field is twisted
(first stage physics)• Current is steady-state so 0U
Axial acceleration
• Any plasma can be decomposed into arbitrarily shaped fluid elements
• Decompose into toroidal fluid elements
• J x B force accelerates toroidal fluid elements axially from footpoints towards midpoint
• Fluid element does not rotate as it moves axially, since current is constant
Third Stage- Stagnationi. Flow stagnation heats plasma
ii. Density accumulation at midplane
iii. Toroidal flux accumulation at midplane
iv. Enhancement of pinch force at midplane
v. Hot, dense, axial uniform equilibrium results
I
tstagnation
Flux conservation
• Induction equation shows:
Magnetic flux linked by any closed material line is conserved
• Material line is a line
that convects with the fluid
Thus, a toroidal fluid element has both its toroidal and poloidal flux individually conserved
material line enclosingtoroidal flux
S
B
material line enclosing poloidal flux
poloidal flux linked by toroidal fluid element is invariant
Poloidal current I linked by toroid must also be invariantsince I I
sB dpol
closed material line
Typical toroidal fluid elements
What happens to toroidal fluid elements accelerated from endsto midplane by J x B force
Small side-effect: Fermi acceleration of small number ofselect particles bouncing between approaching toroidal fluid elements
Collision between toroids
Effect of collision1. Axial translational kinetic energy is converted into heat (stagnation)2. Axial compression of toroidal fluid elements increases B (frozen-in)
B
t rBpol
U r rUpol
B r B Upol
Toroidal component of induction equation in vicinity of stagnation layer in third stage
0U since I is constant
zero atstagnation layer
Induction equation reduces to
B
t B Upol
negative, since flows are converging
Thus, toroidal magnetic field increases at stagnation layer
B
t B
t
B
t B Upol
Upol 1
t
implies
toroidal field grows in proportion to mass accumulation at stagnation layer
induction
mass conservation
Ampere’s law: 2 rB 0I
I is constant
B is increasing at stagnation layer
Therefore, r must decrease at stagnation layer
Flux tube becomes axially uniform
COLLIMATION !!!
Analog model
• Bulged tube wrapped by elastic bands
• Elastic bands represent B field lines
– B field lines are due to axial current I
– B field lines provide pinch force
– Magnetic tension along field line (pinch)– Magnetic pressure perp to field line
low density of bands at middlecorresponding to low B~I/r
higher density of bands at tube endscorresponding to larger B~I/r
accumulation of bands in middle,increases B in middle, pinches middle,
stops when no axial gradient in B2
flow of elastic bands
flow of elastic bands
Current-carrying flux tube gobbles plasma from footpoints,
gets filled up with plasma
and becomes thin (collimated)
Trajectory of toroidal fluid elements(frozen to poloidal flux surface, accelerated axially inwards by MHD force)
force
force
force
force
Grad-Shafranov equation predicts of collimated flux tube
(give quick overview here, details in Bellan Phys. Plasmas 2003)
• Toroidal symmetry causes vector equation
to reduce to the scalar equation
involving poloidal flux
J B P
r2 1r2 4 2r2 0
P 0I
0I 0
Grad-Shafranov analysis shows that I I , P P Simplest non-trivial dependence is linear
Define 0 as the flux surface on which P vanishes
P 1 0
P0 implying P P0
0
Let 0I so 0I 0I
2
Grad-Shafranov equation becomes
r r
1r
r
2z2
2 2 r2
a 02
a 0
2
where
2 0P0
0
a02
2 2 0 peak pressure
average B z2
and the normalized flux is r, z r, z 0
r r
1r
r
2z2
2 2 r2
a 02
a 0
2
If 2a02/2
then the only solution to Grad-Shafranov equation satisfying b.c. that pressure vanishes when r, z 1
is the solution r, z r2
a 02
which is axially uniform
but
2a02/2
is precisely the beta provided by flow stagnation
Thus, flow stagnation should always give axial uniformity,
0I , helicity parameter
Kinking
• Occurs when field line has one complete twist along its length, i.e., when
Bazimuthal/2a=Baxial/L
- Because current system can increase inductance in flux-conserving manner while satisfying periodicity boundary conditions
Lab experiment nominal parameters
• Experiment duration 10 microseconds• Current 30 - 60 kA• Voltage: 3-6 kV at breakdown, < 1 kV after• Input power ~50 megawatts• Gas: hydrogen, argon, neon, or nitrogen• Plasma density ~1014 -1015 cm-3
• Plasma temperature ~2-10 eV• Camera shutter speed: 10 nanoseconds
Supply different gases at two footpoints If jet model is correct, then jets from footpoints
should be distinguishable (different gases)
If not, then plasma should be a mix of two gases (i.e., no jets, gases not distinguishable)
Demonstration that Bidirectional Flows Indeed Come from Footpoints
puff nitrogen
puff hydrogen
Inject different gases at each footpoint
CapacitorBank, 5kV,~40 kA
ignitron
Sequence:1. Establish magnetic field2. Puff in gas3. Fire ignitron
Nitrogen
Hydrogen
If gobble theory is not correct, should get this:
Nitrogen-hydrogenmixture becomesionized
Vacuum field lines unchanged as plasma forms from prefill
Nitrogen
Hydrogen
If gobble theory is correct, should get this:
Nitrogen MHD-driven jet (slow because heavy gas)
Hydrogen MHD-driven jet(fast because light gas)
nitrogen
hydrogen
arched magnetic field
vacuum sideatmosphere side
magnetic field coils
1) Coil-generated potential magnetic field, up to 0.3 T
2) Fast gas valves inject H2, N2 at footpoints
3) 3-6 kV, applied to the electrodes, ionizes the gas and drives a 40-80 kA current
3 s1 s 4.5 s
.
Experimental Result•Hydrogen jet (red) coming from top collides with nitrogen jet (green) coming from bottom
•Jets follow arched expanding magnetic field
•Jets are collimated
Conclusion: MHD-driven jet model is verified
•Distinct nitrogen and hydrogen jets observed
•Heavy MHD jet (nitrogen) moves slower
•Flux tube collimated, interferometer & Stark density measurements show density is strongly peaked in flux tube •Collimated flux tube major radius increases due to hoop force
•Collimated flux tube eventually kinks
•Plasma in bright flux tube not from ionization of neutral prefill, rather is convected in by MHD jet that fills and collimates flux tube
Larger-scale force-free structures
• To an outside observer the collimated flux tube appears as a tube with an axial current, a field “line” with axial current
• This is the building block for larger-scale force-free structures formed from distinct plasma-filled flux tubes