Paul Bellan Caltech

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/Postdocswho 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 Jx B force • Jx 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) 2r 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. I(t) t Stagnation • Twisting (rising current) • Axial thrust (steady current) • Stagnation (steady current) Twisting Thrust Physics consists of three distinct stages:

11. First Stage (rising current gives twisting)

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

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

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

19. To understand 2nd stage physics, first consider simpler situation, namely axially non-uniform current withoutembedded 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 axial flow current Like squirting toothpaste from a toothpaste tube

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

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. Bf S Thus, a toroidal fluid element has both its toroidal and poloidal flux individually conserved material line enclosing poloidal flux material line enclosing toroidal flux

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

34. closed material line

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

48. Iis 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