Why current-carrying magnetic flux tubes gobble up plasma and become thin as a result

- model and supporting lab experimentsPaul BellanCaltech

Students/Postdocswho worked on experimentsFreddy HansenShreekrishna TripathiScott HsuSett YouEve Stenson

Question:

Why are bright flux tubes collimated?

(observed in lab, solar plasmas)

Model:

ideal MHD (field frozen to plasma)dynamics, history, non-equilibriumcompressibilityfinite pressure gradient, finite b non-conservative property of J x B forceJ x B driven flows, flow stagnation

Ideal MHD:Eq. of motionInduction equationAmperes law Mass conservation equationAdiabatic relation

Statement of the problemPotential flux tubes are not axially uniform

potential flux tube bulges at top because B field weaker at top,and cross section area A~1/Bsolar 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 Bf ~ 1/r Jaxial ~ 1/r2

Simplify analysis by considering straight-axis flux tube(axis curvature considered later) footpoint2r

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

Twisting (rising current)Axial thrust (steady current)Stagnation (steady current)

Twisting

Thrust

Physics consists of three distinct stages:

First Stage(rising current gives twisting)

Incompressible torsional motion (like Alfven wave)Torque provided by polarization currentNo poloidal motionProfile of flux tube unchangedToroidal velocity given by

I(t)tTwistingrisingcurrent

Initially untwisted potential flux loopsdistance along field line from midplaneSurface of constant poloidal flux,

Axial current twists flux loop, creates BfFinite toroidal fluid velocity (twisting), no poloidal (axial) fluid velocity

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

vanishes when I is constant

Same (r,z) profile

Second StageAxial thrust stage (steady-state current)Bidirectional flows accelerated by torqueNon-equilibrium

To understand 2nd stage physics, first consider simpler situation, namely axially non-uniform current without embedded axial field

current canted JxB force gives axial thrustthrust direction independent of current polarityflow goes from small to large radiusaxial flow

Like squirting toothpaste from a toothpaste tube

Now consider arc between two equal electrodes

current J x B force gives axial thrustaxial flow

current J x B force gives axial thrustaxial flowaxial flow

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

Axial acceleration

Any plasma can be decomposed into arbitrarily shaped fluid elementsDecompose into toroidal fluid elements J x B force accelerates toroidal fluid elements axially from footpoints towards midpointFluid element does not rotate as it moves axially, since current is constant

J x B forces on typical toroidal fluid elements

Third Stage- Stagnation

Flow stagnation heats plasmaDensity accumulation at midplaneToroidal flux accumulation at midplaneEnhancement of pinch force at midplaneHot, dense, axial uniform equilibrium results

It

Flux conservationInduction 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 conservedmaterial line enclosingtoroidal fluxmaterial line enclosing poloidal flux

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

closed material line

Typical toroidal fluid elementsWhat 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 toroidsEffect of collisionAxial translational kinetic energy is converted into heat (stagnation)2. Axial compression of toroidal fluid elements increases Bf (frozen-in)

axial compression in a collision

Toroidal component of induction equation in vicinity of stagnation layer in third stage since I is constantzero atstagnation layer

Induction equation reduces tonegative, since flows are convergingThus, toroidal magnetic field increases at stagnation layer

impliestoroidal field grows in proportion to mass accumulation at stagnation layerinductionmass conservation

I is constant

is increasing at stagnation layerTherefore, r must decrease at stagnation layerFlux tube becomes axially uniformCOLLIMATION !!!

Analog modelBulged tube wrapped by elastic bands

Elastic bands represent Bf field lines Bf field lines are due to axial current IBf field lines provide pinch forceMagnetic tension along field line (pinch)Magnetic pressure perp to field line

elastic bands representingBf magnetic field linesbulged tube

low density of bands at middlecorresponding to low Bf~I/r higher density of bands at tube endscorresponding to larger Bf~I/r

flow from ends to middle drivenby higher magnetic pressure Bf2 at ends

accumulation of bands in middle,increases Bf in middle, pinches middle, stops when no axial gradient in Bf2 flow of elastic bandsflow 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)forceforceforceforce

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

Grad-Shafranov equation becomeswhere

Ifthen the only solution to Grad-Shafranov equation satisfying b.c. that pressure vanishes when is the solutionwhich is axially uniform

butis precisely the beta provided by flow stagnationThus, flow stagnation should always give axial uniformity,

Hoop ForceConsequence of curvature of flux tube axis (before had assumed axis was straight)

hoop forceelectric currentfield due to currentstronger on inside of curvethan on outside

hoop force increases major radius of flux tube axis

KinkingOccurs 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

Kinking

Lab experiment nominal parametersExperiment duration 10 microsecondsCurrent 30 - 60 kAVoltage: 3-6 kV at breakdown, < 1 kV afterInput power ~50 megawattsGas: hydrogen, argon, neon, or nitrogenPlasma density ~1014 -1015 cm-3Plasma temperature ~2-10 eVCamera shutter speed: 10 nanoseconds

METAL ELECTRODELab version offootpoint

Initial potentialmagnetic fieldSetup

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 nitrogenpuff hydrogenInject different gases at each footpointCapacitorBank, 5kV,~40 kAignitronSequence:Establish magnetic fieldPuff in gasFire ignitron

NitrogenHydrogenIf gobble theory is not correct, should get this:Nitrogen-hydrogenmixture becomesionizedVacuum field lines unchanged as plasma forms from prefill

NitrogenHydrogenIf 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

Classic prefill model or 2. MHD-driven jet model

nitrogenhydrogenarched magnetic fieldvacuum sideatmosphere sidemagnetic field coils1) 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 ms1 ms4.5 ms.

Experimental ResultHydrogen 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 increasing from zero:Potential fieldTwisting (Alfven physics)Upflows, stagnation, heating, filling, pinching (arcjet physics)Collimation (Grad-Shafranov radial pressure balance)Kink instability, sigmoids, eruption (Kruskal-Shafranov physics)

Gun design featuresCylindrical coordinate systemExperimental operation sequence (1) bias field, (2) gas puff, (3) gun dischargeDescribe I-V traces