MS T , 2001sprott.physics.wisc.edu/biewer/ttf2001poster.pdf · Pressure Profile Evolution: F~-0.24 time from sawtooth (ms) ρ (m) ρ (m) time from sawtooth (ms) p e Electron pressure

May 17th, 20012001 Joint US-European TTF Workshop

University of Wisconsin Department of Physics

adison ymmetric orusM S T

Sawtooth Sawtooth Cycle Time Evolution ofCycle Time Evolution ofMeasured Energy and Particle Transport in theMeasured Energy and Particle Transport in the

MST Reversed-Field PinchMST Reversed-Field Pinch

T.M. T.M. BiewerBiewer, J.K. Anderson, D.J. Craig, D. Demers, J.K. Anderson, D.J. Craig, D. Demers11, W. Ding, W. Ding22, G. , G. FikselFiksel, B. Hudson, J. Lei, B. Hudson, J. Lei11,,J.C. Reardon, U. ShahJ.C. Reardon, U. Shah11, S.D. Terry, S.D. Terry22, S.C. , S.C. PragerPrager, and C.B. Forest, and C.B. Forest

University of Wisconsin-MadisonUniversity of Wisconsin-Madison

11Rensselaer Polytechnic InstituteRensselaer Polytechnic Institute

22University of California-Los AngelesUniversity of California-Los Angeles

The “sawtooth cycle” is a slowly changing, but very significant aspect of the plasma equilibrium.Observations show that the character of the sawtooth event in the MST changes dramatically

when the q=0 surface is removed from the plasma. Time evolved measurements ofthermodynamic profiles have been obtained in both reversed (F<0) and non-reversed (F=0)

plasmas, leading to the first measurement of radially resolved, time evolving heat transport inthe MST. In both cases, the heat flux is predominantly conductive over the majority of theplasma volume, though convective heat transport becomes significant in the edge. Theobserved heat and particle fluxes cannot be described by a diagonal transport matrix.

However, including cross-terms, such as those consistent with Rechester-Rosenbluth andHarvey, can account for the observed fluxes. The value of magnetic field stochasticity

expected by this analysis agrees very well with predictions from DEBS modeling.

This work was supported by the U.S. D.O.E.This work was supported by the U.S. D.O.E.


OutlineOutline

• Motivation: To measure the coefficients for electron particle andenergy transport with sufficient accuracy to make valid comparisonsto theoretical models.

– Measurements of particle and energy source terms and gradients.

• Experimental Conditions: time dependant transport study of

• “Standard” plasma: ~385 kA, 1.1x1019 m-3, F ~ -0.24, in deuterium.

• “Non-Reversed” plasma: ~385 kA, 1.1x1019 m-3, F = 0, in deuterium.

• Determination of Particle Flux (in flux surface geometry)– ne(ρ,t), Sp(ρ,t), dne/dt Γ(ρ,t)

• Determination of Heat Flux– Te(ρ,t), ne(ρ,t), ηj2(ρ,t) Q(ρ,t)

• Diffusive transport coefficients: D and χ

• The Harvey model of stochastic transport: Dst


MotivationMotivation

• The plasma “sawtooth cycle” is a slowly changing, but very significantaspect of the plasma equilibrium. Examining transport quantities wrt/the sawtooth cycle phase can shed light on important plasmadynamics.

• To accurately study the movement of energy and particles in a MSTplasma it is necessary to have detailed, accurate measurements ofbasic plasma parameters: Te, ne, nD, Ti, j, etc.

• Some of these quantities can be directly measured:

– ne(r,t) from Far Infrared Interferometry

– nD(r,t) from ne and a collinear Dα array

– Te(r,t) from Thomson Scattering

– Ti(~0,t) from Rutherford Scattering

• Others (in particular j) can be found from modeling (MSTFIT)


Particle and Heat Transport EquationsParticle and Heat Transport Equations

•• Including possible sources and sinks of energy and particles, the equations ofIncluding possible sources and sinks of energy and particles, the equations ofcontinuity and energy balance are:continuity and energy balance are:

•• Solving for the particle and heat fluxes:Solving for the particle and heat fluxes:

•• For electronsFor electrons in a 3 component plasma (e in a 3 component plasma (e--, , npnp++, impurity Z), we can, impurity Z), we canwrite:write: S P P P PE e e ei ez rad, = − − −S n n vp e e H i, = σ

P n n e T T m Teimm i e e i e e

e

i= −4

43 3

02ln( ) ( ) /Λ

ε π

P je = η 2

P n n Z e T T m Tezmm z e e z e e

e

z= −2 4

43 3

02ln( ) ( ) /Λ

ε π

P C r a Prad rad e= ( / )8

Q r S nT rr E t

= −( )∂∫ ∂∂

1 32

( )

∂∂

∂∂= − +n

t r r pr S1 ( )Γ

Γe r p ent

r S re= −∫ ∂∂∂

1 ( ),

∂∂

∂∂= − +

t r r EnT rQ S( ) ( )32

1


The Madison Symmetric The Madison Symmetric Torus Torus Reversed Field PinchReversed Field Pinch


PoloidalPoloidal Projection of Diagnostic Locations Projection of Diagnostic LocationsInboard Outboard

HIBP

Toroidal Array of32 Magnetic Coils

Poloidal Array of16 Magnetic Coils

Insertable Langmuir Probe

Toroidal IDS View

MSE

RS

Vie

w

Thomson Scattering

Thomson Scattering

Pol

oida

l ID

S C

hord

s

FIR

Inte

rom

eter

, Far

aday

Rot

atio

n, H

_alp

ha

+

CO

2 In

terf

erom

eter

MachineAxis


Introduction to the Experiment: F~-0.2Introduction to the Experiment: F~-0.2

Motivation: To study theMotivation: To study thedynamics of a fullydynamics of a fully

diagnosed MST plasmadiagnosed MST plasmaover a over a sawtooth sawtooth cycle.cycle.

•• Typical plasma parameters:Typical plasma parameters:

–– ““StandardStandard”” w/ full PFN w/ full PFN

–– IIpp ~ 385 kA ~ 385 kA

–– F ~ -0.24F ~ -0.24

–– nnee ~ 1.1x10 ~ 1.1x101313 cm cm-3-3

–– DeuteriumDeuterium

•• Sawtooth Sawtooth period ~ 5.5 msperiod ~ 5.5 ms

•• Time of interest: ~ 15 ms,Time of interest: ~ 15 ms,i.e. early in the dischargei.e. early in the dischargebut during plasma but during plasma ““flattopflattop””

13-nov-2000 Shot 69


Introduction to the Experiment: F=0Introduction to the Experiment: F=0

Motivation: To study theMotivation: To study theeffect of removing theeffect of removing the

reversal surface from thereversal surface from theplasma.plasma.

•• Typical plasma parameters:Typical plasma parameters:

–– ““Non-reversedNon-reversed”” w/ full PFN w/ full PFN

–– IIpp ~ 385 kA ~ 385 kA

–– F ~ 0F ~ 0

–– nnee ~ 1.1x10 ~ 1.1x101313 cm cm-3-3

–– DeuteriumDeuterium

•• Sawtooth Sawtooth period ~ 5.5 msperiod ~ 5.5 ms

•• Time of interest: ~ 15 ms,Time of interest: ~ 15 ms,i.e. early in the dischargei.e. early in the dischargebut during plasma but during plasma ““flattopflattop””

30-mar-2001 Shot 80


Shots are Shots are Ensembled wrtEnsembled wrt. The . The Sawtooth Sawtooth CrashCrash

•• Plasma shots arePlasma shots areensembledensembled for 2 reasons: for 2 reasons:

–– Want an Want an ““averageaverage””measure of quantities, i.e.measure of quantities, i.e.smooth out the fluctuationssmooth out the fluctuations

–– Need ~400 shots to getNeed ~400 shots to getTTee(r,t) from Thomson (r,t) from Thomson sctsct..

•• This 6 ms time windowThis 6 ms time windowabout the crash is sub-about the crash is sub-divided into 12 time slices,divided into 12 time slices,every 0.5 ms.every 0.5 ms.

–– High time resolutionHigh time resolutionsignals (e.g. FIR) aresignals (e.g. FIR) arecomputed as 0.1 mscomputed as 0.1 msensembles.ensembles.

–– Raw TS data is Raw TS data is ensembledensembledin 0.5 ms bins.in 0.5 ms bins.

nov-2000 ensemble of ~400 shots

-2.75 -2.25 -1.75 -1.25 -0.75 -0.25 +0.25 +0.75 +1.25 +1.75 +2.25 +2.75


The q=0 surface is excluded in F=0 plasmasThe q=0 surface is excluded in F=0 plasmas

•• In MST plasmas BIn MST plasmas BTT~B~BPP and andR/a~3, resulting in q<1R/a~3, resulting in q<1everywhere.everywhere.

•• Core modes are dominantly m=1Core modes are dominantly m=1and n=5,6,7and n=5,6,7

•• Edge modes are dominantly m=0Edge modes are dominantly m=0(at the reversal surface) and(at the reversal surface) andn=1,2,3,n=1,2,3,……

•• In F=0 plasmas, the q=0 surface isIn F=0 plasmas, the q=0 surface isremoved to the extreme edge,removed to the extreme edge,eliminating the m=0 rationaleliminating the m=0 rationalsurface, and presumably reducingsurface, and presumably reducingthe the overal overal level of magneticlevel of magneticfluctuations.fluctuations.

The “Safety Factor” profile -1.75 ms away fromthe nearest sawtooth crash.

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 0.1 0.2 0.3 0.4 0.5

F=0F=-0.2

q=R

BP/r

BT

rho (m)

m/n=1/5

m/n=1/6

m/n=1/8

m/n=1/7

m/n=0/1,2,3,...

etc.


BBPP array signals constrain q array signals constrain q00(t)>1/6 for F~-0.24(t)>1/6 for F~-0.24(G

auss

)

time from sawtooth (ms)

Magnetic mode number


BBPP array signals constrain q array signals constrain q00(t)>1/5 for F=0(t)>1/5 for F=0(G

auss

)


Magnetic mode number


TTii(0,t) from Rutherford Scattering shows ion heating(0,t) from Rutherford Scattering shows ion heating

•• Rutherford Scattering measures the temperature of the majority ionRutherford Scattering measures the temperature of the majority ionspecies, showing evidence of ion heating at the species, showing evidence of ion heating at the sawtooth sawtooth crash tocrash totemperatures hotter than the central electron temperature for F~-0.24.temperatures hotter than the central electron temperature for F~-0.24.

•• Impurity ion (CImpurity ion (CVV) temperatures, measured via an Ion Dynamics) temperatures, measured via an Ion DynamicsSpectrometer, have confirmed this heating in the past. Spectrometer, have confirmed this heating in the past. Data Pending.Data Pending.

•• There is no evidence of this ion heating in F=0 plasmas.There is no evidence of this ion heating in F=0 plasmas.

RS Ti (solid), MSTFIT Te(0) (stars)

~350 Shot Ensemble of Ti ~350 Shot Ensemble of Ti

Standard, F~-0.24 Non-Reversed, F=0


Temperature Profile EvolutionTemperature Profile Evolution

ρ (m)


Te

ρ (m

)


ρ (m

)


MSTFIT splines to 16 chord Thomson Scattering

ρ (m)


Te



Eliminating the Reversal Surface Results in Higher TEliminating the Reversal Surface Results in Higher Tee


0

5 0

100

150

200

250

300

350

0 0.1 0.2 0.3 0.4 0.5

-1.75 ms away-0.25 ms before+0.75 ms after

Te (

eV

)

rho (m)

0

50

100

150

200

250

300

350

0 0.1 0.2 0.3 0.4 0.5


Te (

eV)

rho (m)


0

5 1018

1 1019

1.5 1019

0 0.1 0.2 0.3 0.4 0.5


n e (

m-

3)

rho (m)

0

2 1018

4 1018

6 1018

8 1018

1 1019

1.2 1019

1.4 1019

0 0.1 0.2 0.3 0.4 0.5


n e (m-3)

rho (m)

Rev

ersa

l Sur

face

Rev

ersa

l Sur

face


Density Profile EvolutionDensity Profile Evolution

ρ (m)


ne

ρ (m

)


ρ (m

)


ρ (m)


ne


MSTFIT Abel inverted 11 chord FIR Interferometry


Pressure Profile Evolution: F~-0.24Pressure Profile Evolution: F~-0.24


ρ (m

)

ρ (m)


pe

Electron pressure evolution from ne*Te

•• Before the Before the sawtoothsawtooth crash crashthe edge pressure gradientthe edge pressure gradientcrosses the crosses the SuydamSuydaminterchange mode stabilityinterchange mode stabilitylimit.limit.

•• After the crash, the edgeAfter the crash, the edgepressure gradient haspressure gradient hasrelaxed away from therelaxed away from theSuydamSuydam limit, which itself limit, which itselfhas softened due to anhas softened due to anincrease in the shear of qincrease in the shear of q((ieie.. q q’’/q)./q).


Pressure Profile Evolution: F=0Pressure Profile Evolution: F=0


ρ (m

)

ρ (m)


pe

Electron pressure evolution from ne*Te

•• Similarly, before the Similarly, before the sawtoothsawtoothcrash the edge pressurecrash the edge pressuregradient crosses the gradient crosses the SuydamSuydaminterchange mode stabilityinterchange mode stabilitylimit.limit.

•• In both In both ““StandardStandard”” and and ““Non-Non-ReversedReversed”” plasmas, it appears plasmas, it appearsthat the pressure profilethat the pressure profilebegins to flatten prior to thebegins to flatten prior to theactual actual sawtoothsawtooth crash. crash.

•• Even away from the crash, theEven away from the crash, thepressure gradient is very nearpressure gradient is very nearto the stability limit.to the stability limit.


Particle and Heat FluxParticle and Heat Flux

Q r S nT rr E t

= −( )∂∫ ∂∂

1 32

( )

1018

1019

1020

1021

0 0.1 0.2 0.3 0.4 0.5


gam

ma_

e (1

/m2)

rho (m)

1018

1019

1020

1021

0 0.1 0.2 0.3 0.4 0.5


gam

ma_

e (1

/m2)

rho (m)

Γe r p ent

r S re= −∫ ∂∂∂

1 ( ),Standard, F~-0.24 Non-Reversed, F=0

0

5 104

1 105

1.5 105

2 105

2.5 105

0 0.1 0.2 0.3 0.4 0.5

Total

conductive

convective

Qe (

W/m

2)

rho (m)

0

5 104

1 105

1.5 105

2 105

2.5 105

0 0.1 0.2 0.3 0.4 0.5

totalconductiveconvective

Qe (

W/m

2)

rho (m)


Particle and Heat transport coefficients: D, Particle and Heat transport coefficients: D, χχStandard, F~-0.24 Non-Reversed, F=0

1

10

100

1000

104

0 0.1 0.2 0.3 0.4 0.5


D_d

iff (

m2/s

)

rho (m)

1

10

100

1000

104

0 0.1 0.2 0.3 0.4 0.5


D_d

iff (

m2/s

)

rho (m)

1

10

100

1000

104

0 0.1 0.2 0.3 0.4 0.5


chi_

e (m

2/s

)

rho (m)

1

10

100

1000

104

0 0.1 0.2 0.3 0.4 0.5


chi_

e (m

2/s

)

rho (m)

Γ = − ∇D n

Q T n T= − ∇52 Γ χ


Transport Equations (Harvey)Transport Equations (Harvey)

•• The kinetically derived The kinetically derived FokkerFokker-Plank Equation for a plasma, including an-Plank Equation for a plasma, including anambipolarambipolar, radial electric field is:, radial electric field is:

•• Taking the first two moments of this equation and comparing to the sourceTaking the first two moments of this equation and comparing to the sourcefree continuity ( ) and energy balance ( ) equationsfree continuity ( ) and energy balance ( ) equationsleads again to expressions for the particle and heat flux:leads again to expressions for the particle and heat flux:

•• By making the identification ( ), these expressions can beBy making the identification ( ), these expressions can becombined to yield a relation for the convective and conductive parts of thecombined to yield a relation for the convective and conductive parts of theheat flux:heat flux:

•• Assume for the moment that the Assume for the moment that the ambipolarambipolar terms are small by comparison in terms are small by comparison ineach expression. In this assumption a stochastic diffusion coefficient caneach expression. In this assumption a stochastic diffusion coefficient caneasily be calculated.easily be calculated.

∂∂

∂∂= −n

t r rr1 ( )Γ ∂

∂∂∂= −

t a r rnT qE rQ( ) ( )3

21Γ

χπ

= =22v D Dt m

∂∂

∂∂

∂∂

∂∂

∂∂( ) = +

+

ft m

v

rrD

rqE

mvrDv

fr

qEmv

fv

par m a

par

m

par

a

par par

( ) ( )

Γ = − + −( )∂∂

∂∂Dn n

nr T

Tr

qET

a1 12 Q DnT n

nr T

Tr

qET

a= − + −( )∂∂

∂∂2 1 3

21

Q T nT

rQ Qconv cond= − ∂

∂= +2Γ χ


In F~-0.24 plasmas stochastic diffusion can account for hollow TIn F~-0.24 plasmas stochastic diffusion can account for hollow Tee, , nnee profiles profiles

•• In the approximation that the EIn the approximation that the Eaa terms are small, the derived terms are small, the derivedstochastic diffusion coefficients are well behaved across the plasmastochastic diffusion coefficients are well behaved across the plasmavolume.volume.

•• Note that in this formalism there is the identity that Note that in this formalism there is the identity that χχ = 2D= 2Dstst..

•• Including the EIncluding the Eaa terms effectively leads to terms effectively leads to DDstst==χχMSTFITMSTFIT/2, since /2, since ΓΓ is isrelatively small compared to Q (i.e. conductive heat flux is very muchrelatively small compared to Q (i.e. conductive heat flux is very muchlarger that convective flux) over the majority of the plasma volume.larger that convective flux) over the majority of the plasma volume.

1

1 0

100

1000

1 04

0 0.1 0.2 0.3 0.4 0.5


D_

st

(m2

/s)

rho (m)

1

1 0

100

1000

1 04

0 0.1 0.2 0.3 0.4 0.5


De (

m2/s

)

rho (m)

Γ = − ∇D n Γ = − ∇ ∇D n +n

2TTst ( )


ConclusionsConclusions

•• This represents the first measurement of time resolved TThis represents the first measurement of time resolved Teeprofiles in the MST.profiles in the MST.

•• In both Standard and Non-Reversed plasmas, theIn both Standard and Non-Reversed plasmas, thepressure gradient appears to cross the critical pressure gradient appears to cross the critical SuydamSuydamlimit to interchange instability prior to the limit to interchange instability prior to the sawtoothsawtooth crash. crash.

•• Energy loss is primarily conductive, not convective.Energy loss is primarily conductive, not convective.

•• Confinement is somewhat improved in Non-ReversedConfinement is somewhat improved in Non-Reversedplasmas over Standard plasmas by excluding the q=0plasmas over Standard plasmas by excluding the q=0surface.surface.

•• A reduced Harvey model for transport in a stochasticA reduced Harvey model for transport in a stochasticmagnetic field could account for observed Tmagnetic field could account for observed Tee, , nnee profile profilecharacteristics in Standard plasmas, especially where acharacteristics in Standard plasmas, especially where asimple diffusive model fails.simple diffusive model fails.


Future WorkFuture Work

•• The inclusion of HIBP data is underway.The inclusion of HIBP data is underway.–– An additional constraint on the current profile.An additional constraint on the current profile.

–– An avenue of direct comparison between An avenue of direct comparison between ““independentindependent””calculations of Ecalculations of Eaa and phi. and phi.

•• IDS data yet to be examined.IDS data yet to be examined.–– minority temperature profile, minority temperature profile, TTzz(r,t)(r,t)

–– flow velocity profiles for an ion momentum balance flow velocity profiles for an ion momentum balance estest. of E. of Eaa

–– Possible confirmation of ion heating at the Possible confirmation of ion heating at the sawtoothsawtooth

•• Repeat the analysis for a F>0 plasma experiment.Repeat the analysis for a F>0 plasma experiment.

•• Repeat the analysis for a high confinement, Repeat the analysis for a high confinement, ““PPCDPPCD””experiment.experiment.


ReprintsReprintsFull color version of this poster is available online at:

http://sprott.physics.wisc.edu/biewer/ttf2001poster.pdf

Documents

MS T , 2001sprott.physics.wisc.edu/biewer/ttf2001poster.pdf · Pressure Profile Evolution: F~-0.24 time from sawtooth (ms) ρ (m) ρ (m) time from sawtooth (ms) p e Electron pressure