Plasma Rotation in Tokamaks - University of Texas at...

4th ITER International Summer SchoolAustin, TXMay 31, 2010

Plasma Rotation in Tokamaks

C. C. HegnaDepartments of Engineering Physics and Physics

University of WisconsinMadison, WI

Co-authors: J. D. Callen, A. J. Cole


Toroidal rotation impacts a variety of importantplasmas physics topics

• Determining the magnitude, profile and evolution of toroidal flow isimportant in a number of topical areas– E X B flow shear of anomalous transport– Prevention of locked modes --- penetration of resonant field errors– Control of edge localized modes via resonant magnetic

perturbations– Resistive wall mode

physics

Garofalo et al, PRL ‘99


Mechanisms have been developed tocontrol/affect plasma rotation

• Toroidal rotation is influenced by:– External sources --- neutral beams– Intrinsic rotation --- topic of considerable research

• A number of mechanisms have been proposed for intrinsicrotation --- turbulence, etc.

– 3-D Magnetic fields• Field errors• Due to MHD instabilities• Applied 3-D fields-- both resonant and non-resonant

W. Zhu et al, PRL, 2006


Describing toroidal rotation in tokamaksis a transport problem

• To date, most treatments describing toroidal rotation evolution rely on:– Braginskii formulation (collisional plasma ν > vth/qR) --- in

practice, never rigorously applicable to modern tokamaks– Additional physics added in ad hoc manner

• Sources• Collisional transport• Radial plasma transport due to neoclassical effects• Turbulent transport --- often modeled as anomalous

diffusion/pinch coefficients• 3-D fields• Magnetic field transients• etc.


Thesis

• A new approach to construct transport equations for tokamak plasmasis derived. [Callen et al, NF 49, 085021 (2009), Callen et al PoP 16,082504 (2009)].– Starting point is kinetic equation, not Braginskii– Multiple timescale equilibration processes– Toroidal momentum balance (or Er) derived– Ambipolar particle transport

• Momentum balance equation is derived that accounts for classical,neoclassical collisional transport, anomalous transport, sources andsinks, magnetic field transients, and the effects of 3-D fields(neoclassical toroidal viscosity --- NTV).


Outline

• Fluid moment equations• Expansion procedure• Multiple timescale approach

– Radial momentum balance– Parallel momentum balance– Toroidal momentum balance

• Physics of neoclassical toroidal viscosity (NTV)– Theory– Experimental validation

• Toroidal rotation equation• Radial electric field and ambipolarity


Starting point for the calculation isthe plasma kinetic equation

• Plasma kinetic equation for fs(x,v,t).

– C(fs) = Fokker-Planck collision operator– S(fs) = Kinetic source --- applied RF fields, neutral beams, etc.

• Fluid moment equations are obtained from velocity-space moments ofthe plasma kinetic equation

– Evolution equations for low order velocity space moments(ns,Vs,ps)

!

d3r v (1,msr v ,msv

2

2)" [#fs

#t+

r v $ %fs +qs

ms

(r E + r v &

r B )$ #fs

#r v

= C( f s) + S( f s)]

!

"fs

"t+

r v # $fs +qs

ms

(r E + r v %

r B )# "fs

"r v

= C( fs) + S( f s)


Fluid equations require kineticallydetermined closure moments

• Exact fluid equations– Density

– Momentum

– Energy

• Closure moments are required for a closed set of equations– Heat flux qs, viscous stress tensor πs determined kinetically– Collision operator physics determines Qs, Fs

!

!

""t(msns

r V s) +#$ (msns

r V s

r V s) = nsqs(

r E +

r V s %

r B ) &#ps &#$

t ' s +

r R s +

r S m

!

32"ps

"t+#$ (5

2ps

r V s +

r q s) = Qs +r

V s$#ps %t & s :#

r V s + SE

!

< Q >=

d"d#r B $ %#

Q(&,#,")''d"d#r B $ %#''

!

"ns

"t+#$ (ns

r V s) = Sn


A number of assumptions are madeto make analytic progress

• Small gyroradius expansion– Consequences for how we describe flows

• Lowest order --- MHD force balance• First order flows are within flux surfaces• Second order “transport” fluxes across surfaces

• Lowest order axisymmetric magnetic fields --- nested toroidal flux surfaces• Small 3-D non-axisymmetric magnetic fields --- no magnetic islands

– In practice, many 3-D fields of interest

• Small plasma fluctuations

• Slow magnetic field transients

!

"L

<<1

!

˜ B Bo

~ 10"3 "10"4

!

˜ n sno

~˜ T s

To

<<1


Multi-stage strategy is used todetermine transport equations

• Average the density, momentum and energy equations over fluctuations(average over toroidal angle) and then average over the flux surfaces

• Sequentially consider specific components of the equilibrium force balanceequation and their consequences

– Radial --- zeroth order radial force balance enforced by compressionalAlfven waves to obtain relation between flows, electric field and pressuregradients

– Parallel --- first order poloidal flows, heat fluxes within magneitc surface– Toroidal --- require net radial current from all particle fluxes to vanish -->

establishes flux surface averaged toroidal momentum balance equation• Use results of the net second order ambipolar fluxes back into flux

surface averaged transport equations to obtain comprehensive “radial”transport equations for ns, Ts and toroidal rotation

!

< Q >=

d"d#r B $ %#

Q(&,#,")''d"d#r B $ %#''


Small gyroradius expansion is used

• Gyroradius expansion: order terms and physical processes such asequilibrium, Pfirsch-Schluter flows, non-axisymmetries andfluctuations

– Magnetic field is the sum of an axisymmetric magnetic field andsmall 3-D fluctuations

– Fluctuation derivatives are large perpendicular to B --- ballooning-like ordering

!

p = p(") +#[p 1(",$) + ˜ p (",$,%)]+ O(# 2) # ~ &i /a <<1

EquilibriumPfirsch-Schluter variations

3-D fluctuations

!

r B =

r B o +"

r ˜ B = q(#)$# % $& +$' % $# +"r ˜ B

| B |= Bo(#,&) +"B|| (#,&,')

!

"#˜ p ~ 1$$ ~ 1, "|| ˜ p ~ $ 0$ ~ $


Transport equations for density and pressure areobtained by flux surface averaging

• Flux surface averaging density and energy equations with V’ = dV/dψ

– Cross-field particle/heat fluxes due to collisional and fluctuationprocesses

!

"n0s

"t+

1V '

""#

(V '$s) =< Sn > $s =< (n0s

r V 2 + ˜ n s1

r ˜ V s1)% &# >

32"ps0

"t+

1V '

""#

(V '< [r q 2s +52

( ps0

r V 2s + ˜ p 1s

r ˜ V 1s)]% &# >

=< Qs > ' <r R 1s%

r V 1s +

r ˜ R 1s%r ˜ V 1s > + <

r V 2s% &p0s +

r ˜ V 1s% &˜ p 1s > ' <t ( s :&

r V 1s > + < SE >

Density

EnergyParticle flux


Flux surface average of the momentum balanceequation has three components

• Convenient to consider the “radial” <eψ.(momentum balance)>, parallel<BO

.(momentum balance)>, and toroidal <eζ.(momentum balance)>projections

Radial O(δ0)

Parallel O(δ)

Toroidal O(δ2)

!

msn0s"

r V s"t

= nsoqs(r E +

r V s #

r B ) $%ps

msn0s" <

r B 0 &

r V s >

"t= n0sqs <

r B 0 &

r E > $ <

r B 0 & %&

t ' s > + <

r B 0 &

r R s > $msns <

r B 0 &

r ˜ V s1& %r ˜ V s1 > +...

""t

<r e ( & msns0

r V s >= qs)s$ <

r e ( &%&

t ' s > $ <%& (msns0(r e ( &

r ˜ V 1s)r ˜ V 1s > +...

Radial particle flux enters toroidal momentum balance equation


The different orders of the momentum balanceequation refer to different timescales

• To leading order in δ, MHD force balance– Summing radial momentum balance– Radial force balance produces relationship between toroidal,

poloidal flows, Er and pressure gradient

• First order flows are on magnetic surfaces V1 ~ δ

• Radial flows perpendicular to flux surfaces are second order

!

r J 0 "

r B 0 =#p0

!

0 =r e " # [ni0qi(

r E +

r V i $

r B ) %&pi]

't (r

V i # &) *r

V # &) = %(d+d"

+1

niqi

dpi

d"% q

r V # &,)

Vt *Er

Bp

%1

niqi

dpi

dr+

Bt

Bp

VP

!

r V 1 =

r e "r V # $" +

r e %r V # $% = V||

r B 0B0

+r

V &r V & =

r B 0 ' $(

B02 (d)

d(+1

nsqs

dps0

d()

!

r V 1s" #$ = 0,

r V 2s" #$ % 0


Poloidal flow is obtained from parallel force balance

• Summing the parallel force balance over species

– The poloidal flow is determined mainly by the parallel viscousforce

– Parallel viscosity is calculated from kinetic theory --- collisionalprocess, accounts for the speed dependence of the Fokker-Planckcollision operator

– On times longer than the ion collision time

!

mini0" <

r B 0 #

r V i >

"t= $ <

r B 0 #

s% &#

t ' s > $mini0 <

r B 0 #

r ˜ V i # &r ˜ V i > + <

r B 0 #

r ˜ J (r ˜ B > + <

r B 0 #

r S s >

s%

!

<r B 0 " #"

t $ i|| >%mini0[µi00Ui& + µi01

'25pi

Qi& + ...] < B2 > µi00,µi01 ~ () i

!

Ui"0 #

r V $ %"r B $ %"

= &µi01

µi00

&25pi

Qi" 'cpI

qi < B2 >dTi

d()Vp '

1.17qiB

dTi

drU i"= Ui"

0 + S" Sθ= Sum of sources and turbulent Maxwell/Reynolds stress


Viscous damping occursin the direction of asymmetry

• Viscsous stress tensor can be written in CGL form

• Pressure anisotropies are driven by flows/heat fluxes in the directionof Grad|B|

– To leading order, axisymmetric magnetic fields– Lowest order flows are within magnetic flux surfaces

---> Damping in the poloidal direction

!

t " s|| =

13

(p|| # p$)( ˆ b ̂ b #t I )

!

p|| " p# ~ µr

V $ %B, µr Q $ %B

!

B = B(",# )r

V s1$ %" = 0&r

V s1$ %B ~r

V s1$ %#'B'#


After determining the poloidal flow, there is a uniquerelationship between Er and the toroidal rotation

• Recalling the radial force balance relationship

– Using parallel momentum balance result– Relationship between radial electric field and toroidal flow

– Poloidal flow damping produces parallel plasma flow

!

"t #r V $ %& = '(d(

d)+1

niqi

dpi

d)' q

r V $ %*)

!

<r B 0 " #"

t $ i >% 0

!

Vt "Er

Bp

#1

niqiBP

dpidr

+1.17qiBp

dTidr

Pressure/temperature profiles determined by transport processes

!

r V 1 =

r e "r V # $" +

r e %r V # $% = V||

r B 0B0

+r V &

r V & =

r B 0 ' $(

B02 (d)

d(+1

nsqs

dps0

d()

V|| * +R(d)d(

+1

niqi

dpi0

d() +1.17R0

2

qiRdTi

d(


Electron parallel momentum balance producesparallel Ohm’s law

• Following the same logic for the parallel electron momentum balanceequation

– Using collision friction and neoclassical closure from kinetic theory

– Parallel Ohm’s law

!

0 = "nee <r E #

r B > " <

r B 0 # $#

t % e > + <

r B #

r R e > + <

r B 0 #

r S em >

"mene0 <r B 0 #

r ˜ V e # $r ˜ V e > "ne0e <

r B 0 #

r ˜ V e &r ˜ B >

!

r B 0 "

r R e #$||nee

r J "

r B 0

<r B 0 " %"

t & e|| >= mene0 < B2 > (µe00Ue' + µe01Qe' )

!

<r E "

r B 0 >=#||

nc <r J "

r B 0 > $#||[<

r J "

r B BS > + <

r J "

r B CD > + <

r J "

r B dyn >]

!

"||nc ="|| (1+

"#µe00

"||$ e

) <r J %

r B BS >= &

"#µe00

"||$ e

(I dpd'

& neeUi( < B2 >)

<r J %

r B CD >= &

<r B %

r S em >

nee"||

<r J %

r B dyn >=

1"||

(<r B 0 %

r ˜ V e )r ˜ B > +

me

e<

r B 0 %

r ˜ V e % *r ˜ V e >)

Boostrap current, curent drive from external sources, dynamo due to fluctuations


Toroidal torque from force balance gives radial flows

• Toroidal force balance produces an equation for radial particle flux

– Particle flux is induced by toroidal torques on the plasma

– Flux surface averaged toroidal momentum balance equationproduces equation for particle flux

!

r e " #r

V s $r B = %

r V s#

r e " $r B =

r V s# &'

!

r e " # (nsqs

r V s $

r B +

r F j ) = 0 qs

j% &s = '

r e j #r F j = ' Tj"

j%

j%

!

qs"s = # <r e $ %

r R > + <

r e $ % &%

t ' s > #no <

r e $ %

r E > # <

r e $ %

r S m >

+((t

(msns <r e $ %

r V s >)# <

r e $ % no

r ˜ V )r ˜ B > + <&% msns(

r e $ %

r ˜ V )r ˜ V >

Collisions, sources,Inertia, fluctuations


Parallel Ohm’s law is used to describe collisionalambipolar particle flux

• Consider the particle fluxes from collisional friction

– Vector identity used to facilitate analysis

– Collisional-friction can be decomposed into parallel andperpendicular contributions

!

"sa = #

1qs

<r e $ %

r R s > #n0 <

r e $ %r E >

r R e & nee('||

r J || +'(

r J () = #

r R i

!

r e " =I

B02

r B 0 #

r B 0 $ %&

B02

!

1qs

<r e " #

r R s >= $neI%|| <

r J #

r B 0

B02 > +ne%& <

|'( |2

B02 >

dp0d(

<

r J #

r B 0

B02 >=

<r J #

r B 0 >

< B02 >

+ <r J #

r B 0(

1B02 $

1< B0

2 >) >

Classical transport

Pfirsch-Schluter

Calculated from parallel Ohm’s law


Particle flux has many contributions --- six ambipolarcomponents

• The intrinsically ambipolar contributions to particle fluxes can beidentified

!

"a = "cl + "PS + "bp + "CD + "dyn + "E

"cl = #ne0$% <|&' |2

B02 >

dp0d'

Dcl () e*e2

"PS = #ne0I2$|| <

1B02 (1#

B02

< B02 >)2 >

dp0d'

DPS ~ q2Dcl

"bp =I

e < B02 >

<r B 0 + & +

t , e > Dbp ( µe*ep

2 ~ q2

-3 / 2Dcl

"CD + "dyn =ne0I$||< B0

2 ><

r J +

r B CD +

r J +

r B dyn >

"E = #ne0[<r e . +

r E > (1# I2 < R#2 >

< B02 >

) #I <

r B p +

r E >

< B02 >

)]

Collisional transport from classical, Pfirsch-Schluter,banana-plateau.current drivedynamo andelectric field pinchcontributions


Plasma fluctuations influence particle flux/toroidalmomentum balance

• At O(δ2), plasma fluctuation effects enter into the toroidal momentumbalance– Microturbulence effects --- turbulent Reynolds/Maxwell stresses– 3-D magnetic fields --- error fields, applied 3-D coils

• Resonant magnetic perturbations ---> localized electromagnetictorques

• Non-resonant magnetic perturbations ---> Neoclassical toroidalviscosity

• In general, these effects are not intrinsically ambipolar and hence willaffect toroidal momentum balance.


Resonant magnetic fields produce localizedelectromagnetic torques at rational surfaces

• Inherent magnetic field errors or applied 3-D magnetic coils may havecomponents that are resonant in the plasma

– Two asymptotic limits• Fully penetrated - radial magnetic field produces a magnetic island at

the rational surface• Fully shielded - eddy currents flow in a resistive layer at q = m/n,

magnetic perturbation does not penetrate, current sheet– Sufficient plasma rotation relative to the 3-D field source provides effective

shielding --- rotation sustains current sheet• Produces localized electromagnetic perturbation (Fitzpatrick and

Hender ‘91)

!

r B ~ eim" # in$ %

mn

= q(&)

!

<r ˜ J "

r ˜ B >~ #(r $ rmn ) Brmn2

B02

%&2 + (%' L )2

ˆ b " ˆ n


3-D magnetic fields produce neoclassical toroidalviscous forces (NTV) throughout the plasma

• In an axisymmetric magnetic field, the toroidal component of theparallel viscous stress tensor is zero (µdB/dζ = 0)

– However, in the presence of 3-D magnetic fields, toroidal torqueson toroidally flowing plasmas are generated.

• Physics --- transit-time magnetic pumping, banana-drift, ripple-trapping effects

• Generally, the ion component dominates (the ion root ofstellarator physics)

• Ion viscous damping coefficient µit depends on collisionality, Er• Beff

2 is a weighted average sum over all m and n.

!

<r e " # $ # % ||s

A >= 0

!

<r e " # $ # % ||s >= miniµti

˜ B eff2

B02 [< R2&t > ' < R2&* >]


The NTV force is felt throughout the plasma

• Unlike torques due to resonant 3-D magnetic fields, the NTV force isglobal– Applied 3-D fields on NSTX demonstrated the damping effect of

toroidal flow (Zhu et al, PRL ‘06)

• Favorable comparisonto analytic predictions

• NTV physics has beenseen on NSTX, DIII-D, MAST, JET


Another unusual aspect of NTV is the appearance ofan off-set rotation frequency

• NTV force ~ µi Beff2(Ω − Ω*)

– Offset velocity Ω* is a diamagnetic-type toroidal rotation frequencyproportional to ion temperature gradient

• Physics of the offset is due to ions of different energy havingdifferent radial drift speeds --- produces ct. Poloidal flowdamping coefficient cp due to parallel viscosity.!

"* =cp + ctqi

dTid#


Experiments on DIII-D have demonstrated thepresence of the NTV offset velocity

• Off-set rotation velocity observed on DIII-D (Garofalo et al ‘08)

Initially, slowly rotatingPlasmas sped up to theOffset NTV velocity when3-D fields are applied


Recent experiments on DIII-D have demonstrated apeak in the NTV force at zero radial electric field

• The toroidal damping rate (µti) is sensitive to the value of the radialelectric field– Damping rate corresponds to different collisionality regimes of

stellarator neoclassical transport– Smoothed formula constructed to model different collisionality

regimes (Cole et al, ‘10)

Peaks at ωE ~ 0

Recent experiment on DIII-DDemonstrates peak NTV at ωE ~ 0

!

µti ~ nvti2 " ˆ #

< R2 > [0.3 |$ |%B |ˆ # + 0.04 ˆ v 3 / 2+ |$E |3 / 2]


Particle flux has 8 non-ambipolar contributions

• Not intrinsically ambipolar

!

"na = "# ||NA + "#$ + "pol + "Re y + "Max + "J%B + " ˙ & p

+ "S

"# ||NA =

1qs

<r e ' ( ) (

r # s||

NA >*miniµit

qi

(˜ B eff

2

B02 )(< R2+t > , < R2+* >) +* *

cp + ct

qi

dTi

d&

"#$ =1qs

<r e ' ( ) ( # s$ >~ ,- t)

2+t - t ~ (1+ q2). i/i2

"pol =00t

(msnso

qs

< e' (r V s >)

"Rey =1

qsV '00&

(V '1s&' ) 1s&' = msns <r ˜ V s ( )&

r ˜ V (r e ' > + <)& (

t # s^ (

r e ' >

"Max = , <r e ' ( ns

r ˜ V s1 %r ˜ B >

"J%B =1e

<r e ' (

r ˜ J || %r ˜ B >~ 2(r , rmn ) cA3 minioR

e4

52 + (46 L )2

˜ B rmn

B02

" ˙ & p=

˙ & pqs

00&

(msns0 <r e ' (

r V s >) "S = ,

1qs

<r e ' (

r S sm >

Additional particlefluxes from neoclassicaltoroidal viscosity (NTV),cross field viscosity,polarization flow,microturbulence inducedReynolds and Maxwellstresses, field-error inducedlocalized EM torques,poloidal field transientsand momentum sources.


Zero radial current produces torque balance relation

• Summing radial species currents to obtain net radial plasma current

– Charge continuity requires no net radial current

– Setting radial current equal to zero produces a comprehensivetoroidal torque balance relation

!

<r J " #$ >= qs%s =

s& qs%s

NA

s&

!

""t

< #q >= $1V '

dd%

V '<r J & '% >= 0

!

Lt " mini0 < R2#t >

1V '

$$t

(V 'Lt ) = % <r e & ' ( ' ) ||

NA > % <r e & ' ( ' ) i* > %

1V '

$$+

(V ', i+& )+ <r e & '

r ˜ J -r ˜ B > % ˙ + p

$L$+

+ <r e & '

r S sm >

s.


Toroidal rotation equation includes many differenteffects

• Equation for toroidal angular momentum density Lt = mini0<R2Ωt>

– NTV damping by 3-D magnetic fields

– Collision damping

– Microturbulence-induced ion Reynolds stresses causes radialtransport of Lt (diffusion, pinch, residual stress)

!

1V '

""t

(V 'Lt ) = # <r e $ % & % ' ||

NA > # <r e $ % & % ' i( > #

1V '

"")

(V '* i)$ )+ <r e $ %

r ˜ J +r ˜ B > # ˙ ) p

"L")

+ <r e $ %

r S sm >

s,

!

" <r e # $ % $ & ||s >= "miniµti

˜ B eff2

B02 [< R2't > " < R2'* >] '* =

cp + ct

qi

dTi

d(

!

<r e " # $ # % s& >~ '( t$

2)t ( t ~ (1+ q2)* i+i2

!

"s#$ = msns <r ˜ V s % &#

r ˜ V % r e $ > + <&# %t ' s^ %

r e $ >

~ () t&Lt + LtVpinch +"RS


Toroidal rotation determines radial electric fieldrequired for net ambipolar particle flux

• From toroidal rotation equation, radial electric field is determined

• The resultant electric field causes the electron and ion non-ambipolarradial particle fluxes to be equal (ambipolar)– Hence, the net ambipolar particle flux is the sum of Γa + ΓΝΑ(Εr),

which is easiest to evaluate in ion root [ΓιΝΑ(Εr) ~ 0]

– Procedure is familiar to stellarator researchers• Nonlinear dependences of Γs on Er can lead to different ‘roots’,

transition barriers, etc.

!

Er = "#$d%d$

= "#$(<&t > +1

ni0qidpid$

"cpqidTid$)

!

"enet = "e

A + "eNA (Er) = "i

net


This is approach is different than Braginskii-likeapproach and has some consequences

• Key differences in this new approach for plasma transport equations– First solve for flows of electrons, ions in flux surfaces --- Ohm’s

law, poloidal ion flow– Derivation of particle flux and toroidal flow are naturally joined– Simultaneously solve transport equations for n, T, Ωt

– Effects of micro-turbulence are all included self-consistently– Fluctuation induced particle flux is determined from

Reynolds/Maxwell stresses– Source effects, poloidal field transients are included– Net transport equations follow naturally from extended two-fluid

moment equations --- consistent with formulations of extendedMHD code frameworks (NIMROD, M3D)


Summary

• Comprehensive transport equations for n, T, Ωt have been derived• Radial, parallel and toroidal components of force balance are

considered– Radial force balance --- relationship between Vt, Vp, Er and dpi/dψ– Parallel viscous damping determines neoclassical Ohm’s law and

poloidal ion flow– Radial particle fluxes arise from average toroidal torques on the

plasma• Radial particle flux has many contributions --- ambipolar and non-

ambipolar• Requiring ambipolar particle flux yields evolution equation for toroidal

angular momentum density


Summary

• 3-D magnetic fields have an important effect of flow evolution– Localized EM torques from resonant magnetic fields– Neoclassical toroidal viscosity (NTV) from variations in |B|

• Many aspects of NTV theory are being tested against experiments– Global damping of toroidal flow profile– Appearance of an offset rotation ~ dTi/dψ– Peak of NTV torque near Er ~ 0

Plasma Rotation in Tokamaks - University of Texas at...

Documents

Transcript of Plasma Rotation in Tokamaks - University of Texas at...