1. Why current-carrying magnetic flux tubes gobble up plasma and become thin as a result - model and supporting lab experiments
Paul Bellan
Caltech

2. Students/Postdocs who worked on experiments
Freddy Hansen
Shreekrishna Tripathi
Scott Hsu
Sett You
Eve Stenson

3. Question:
Why are bright flux tubes collimated?
(observed in lab, solar plasmas)

4. Model: ideal MHD (field frozen to plasma)
dynamics, history, non-equilibrium
compressibility
finite pressure gradient, finite b
non-conservative property of J x B force
J x B driven flows, flow stagnation

5. Ideal MHD: Eq. of motion Induction equation Ampere’s law
Mass conservation equation
Adiabatic relation

6. 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

7. 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
Bf ~ 1/r
Jaxial ~ 1/r2

8. Simplify analysis by considering straight-axis flux tube (axis curvature considered later)
footpoint
footpoint
Toroidal
Direction
( )
Poloidal direction
(r,z)

9. Electric current is made to flow along flux tube from one footpoint to the other
axial flow
Current I

10. Physics consists of three distinct stages:
Stagnation
I(t) t
Twisting
Thrust
Physics consists of three distinct stages:
Twisting (rising current)
Axial thrust (steady current)
Stagnation (steady current)

11. (rising current gives twisting)
First Stage
(rising current gives twisting)

12. Incompressible torsional motion (like Alfven wave)
I(t)
rising
current t
Twisting
Incompressible torsional motion (like Alfven wave)
Torque provided by polarization current
No poloidal motion
Profile of flux tube unchanged
Toroidal velocity given by

13. Surface of constant poloidal flux, ψ
Initially untwisted potential flux loop s
distance along
field line
from midplane
Surface of constant poloidal flux, ψ

14. Finite toroidal fluid velocity (twisting),
Axial current twists flux loop, creates Bf
Finite toroidal fluid velocity (twisting),
no poloidal (axial) fluid velocity

15. Toroidal component of induction equation (frozen-in flux condition)
Integrate
w.r.t. distance s
zero during first stage,
no poloidal flow
in first stage

16. vanishes when I is constant

17. Same ψ(r,z) profile

18. Second Stage Axial thrust stage (steady-state current)
Bidirectional flows accelerated by torque
Non-equilibrium I t
thrust

19. To understand 2nd stage physics, first consider simpler situation, namely axially non-uniform current without embedded axial field
canted JxB force gives axial thrust
axial flow
current
thrust direction independent of current polarity
flow goes from small to large radius

20. canted J x B force gives axial thrust
current
canted J x B force gives axial thrust
axial flow
Like squirting toothpaste from a toothpaste tube

21. Now consider arc between two equal electrodes
J x B force gives axial thrust
axial flow
axial flow
current

22. J x B force gives axial thrust
axial flow
axial flow
current

23. Current flow along initially potential flux tube (i. e
Current flow along initially potential flux tube (i.e., now include embedded axial field)
Current produces Bf so net field is twisted
(first stage physics)
Current is steady-state so

24. 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

25. J x B forces on typical toroidal fluid elements

30. Third Stage- Stagnation
Flow stagnation heats plasma
Density accumulation at midplane
Toroidal flux accumulation at midplane
Enhancement of pinch force at midplane
Hot, dense, axial uniform equilibrium results I t
stagnation

31. 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

32. Thus, a toroidal fluid element has both its toroidal and poloidal flux individually conservedBf
material line enclosing
poloidal flux
material line enclosing
toroidal flux

33. Toroidal flux in fluid element remains invariant during all motions of fluid element

34. closed material line

35. Typical toroidal fluid elements
What happens to toroidal fluid elements accelerated from ends
to midplane by J x B force
Typical toroidal fluid elements

40. Small side-effect: Fermi acceleration of small number of
select particles bouncing between approaching toroidal fluid elements

43. Collision between toroids
Effect of collision
Axial translational kinetic energy is converted into heat
(stagnation)
2. Axial compression of toroidal fluid elements
increases Bf (frozen-in)

44. axial compression in a collision

45. zero at stagnation layer Toroidal component of induction equation
in vicinity of stagnation layer in third stage
zero at
stagnation layer
since I is constant

46. Induction equation reduces to
negative,
since flows are converging
Thus, toroidal magnetic field increases
at stagnation layer

47. implies induction mass conservation
toroidal field grows in proportion to mass accumulation at stagnation layer

48. COLLIMATION !!! I is constant
is increasing at stagnation layer
Therefore, r must decrease at stagnation layer
Flux tube becomes axially uniform
COLLIMATION !!!

49. Analog model Bulged tube wrapped by elastic bands
Elastic bands represent Bf field lines
Bf field lines are due to axial current I
Bf field lines provide pinch force
Magnetic tension along field line (pinch)
Magnetic pressure perp to field line

50. elastic bands representing
Bf magnetic field lines
bulged tube

51. higher density of bands at tube ends corresponding to larger Bf~I/r
low density of bands at middle
corresponding to low Bf~I/r

52. flow from ends to middle driven
by higher magnetic pressure Bf2 at ends

53. accumulation of bands in middle,
flow of
elastic bands
flow of
elastic bands
accumulation of bands in middle,
increases Bf in middle, pinches middle,
stops when no axial gradient in Bf2

54. Current-carrying flux tube gobbles plasma from footpoints,
gets filled up with plasma
and becomes thin (collimated)

55. Trajectory of toroidal fluid elements
(frozen to poloidal flux surface,
accelerated axially inwards by MHD force)
force
force
force
force

59. Grad-Shafranov equation predicts b 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

61. Grad-Shafranov equation becomes
where

62. which is axially uniform
then the only solution to Grad-Shafranov equation
satisfying b.c. that pressure vanishes when
is the solution
which is axially uniform

63. is precisely the beta provided by flow stagnation
but
is precisely the beta provided by flow stagnation
Thus, flow stagnation should always give axial uniformity,

64. Hoop Force Consequence of curvature of flux tube axis
(before had assumed axis was straight)

65. collimated, dense flux tube
field due to current
stronger on inside of curve
than on outside
collimated, dense flux tube
hoop force
electric current

66. hoop force increases major radius of flux tube axis

68. Kinking
Occurs when field line has one complete twist along its length, i.e., when
Bazimuthal/2pa=Baxial/L
- Because current system can increase inductance in flux-conserving manner while satisfying periodicity boundary conditions

69. Kinking

70. Lab experiment nominal parameters
Experiment duration 10 microseconds
Current kA
Voltage: 3-6 kV at breakdown, < 1 kV after
Input power ~50 megawatts
Gas: hydrogen, argon, neon, or nitrogen
Plasma density ~ cm-3
Plasma temperature ~2-10 eV
Camera shutter speed: 10 nanoseconds

71. Lab version of footpoint
METAL ELECTRODE
Lab version of
footpoint

72. Setup
Initial potential
magnetic field

74. 2 meters

76. 20 cm

79. Collimated and kinked

80. Twisted ribbon (gift wrapping)
Experiment

81. that Bidirectional Flows Indeed Come from Footpoints
Demonstration
that Bidirectional Flows Indeed Come from Footpoints
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)

82. Inject different gases at each footpoint
puff
hydrogen
ignitron
Capacitor
Bank, 5kV,
~40 kA
puff
nitrogen
Sequence:
Establish magnetic field
Puff in gas
Fire ignitron

83. Hydrogen Nitrogen If gobble theory is not correct, should get this:
Nitrogen-hydrogen
mixture becomes
ionized
Hydrogen
Nitrogen
Vacuum field lines unchanged as
plasma forms from prefill

84. Hydrogen Nitrogen If gobble theory is correct, should get this:
Hydrogen MHD-driven jet
(fast because light gas)
Hydrogen
Nitrogen
Nitrogen MHD-driven jet (slow because heavy gas)

85. Now do the experiment to see which is correct
Classic prefill model or
2. MHD-driven jet model

86. - (cathode) (anode) + arched magnetic field
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 kA current
(cathode)
magnetic field coils
hydrogen
nitrogen
arched magnetic field
(anode) +
atmosphere side
vacuum side

87. 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
1 ms
3 ms
4.5 ms .

88. 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

89. Breakdown, “spider leg formation”
Gun design features
Cylindrical coordinate system
Experimental operation sequence (1) bias field, (2) gas puff, (3) gun discharge
Describe I-V traces

90. Spider leg

91. MHD physics
Anti-parallel currents repel

94. Hsu/Bellan,
MNRAS 2002
Astrophysical jet experiment

95. kink threshold in good agreement with q=1
Kruskal-Shafranov kink stability theory

96. three million frames per second

97. 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

98. Summary Sequence with α increasing from zero: Potential field
Twisting (Alfven physics)
Upflows, stagnation, heating, filling, pinching (arcjet physics)
Collimation (Grad-Shafranov radial pressure balance)
Kink instability, sigmoids, eruption (Kruskal-Shafranov physics)