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

Question:

Why are bright flux tubes collimated?

(observed in lab, solar plasmas)

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:

First Stage

(rising current gives twisting)

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

t

I

rB

sU

pol

20 vanishes when I is constant

Same ψ(r,z) profile

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

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

J x B forces on typical toroidal fluid elements

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

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

Toroidalflux

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)

axial compression in a collision

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

elastic bands representingB magnetic field lines

bulged tube

low density of bands at middlecorresponding to low B~I/r

higher density of bands at tube endscorresponding to larger B~I/r

flow from ends to middle drivenby higher magnetic pressure B

2 at ends

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

Hoop Force

Consequence of curvature of flux tube axis

(before had assumed axis was straight)

hoop force

electric current

field due to currentstronger on inside of curvethan on outside

hoop force increases major radius of flux tube axis

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

Kinking

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

METAL ELECTRODE

Lab version offootpoint

Initial potentialmagnetic field

Setup

2 meters

20 cm

Collimated and kinked

Twisted ribbon (gift wrapping)

Experiment

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)

Now do the experiment to see which is correct

1.Classic prefill model or

2. MHD-driven jet model

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

Breakdown, “spider leg formation”

Spider leg

MHD physics

• Anti-parallel currents repel

Hsu/Bellan, MNRAS 2002Astrophysical jet experiment

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

three million frames per second

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

Summary

Sequence with α increasing from zero:1. Potential field

2. Twisting (Alfven physics)

3. Upflows, stagnation, heating, filling, pinching (arcjet physics)

4. Collimation (Grad-Shafranov radial pressure balance)

5. Kink instability, sigmoids, eruption (Kruskal-Shafranov physics)