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Why current-carrying magnetic flux tubes gobble up plasma and become thin as a result - model and supporting lab experiments. Paul Bellan Caltech. Students/Postdocs who worked on experiments. Freddy Hansen Shreekrishna Tripathi Scott Hsu Sett You Eve Stenson. Question : - PowerPoint PPT Presentation
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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