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Lecture 4: Tokamak Plasma Transport Modeling

J.D. Callen, University of Wisconsin, Madison, WI 53706-1609 USA

Lectures on “Fluid and transport modeling of plasmas” atCEMRACS 2014 Summer School on Numerical modeling of plasmas, CIRM, Marseille, July 21–25, 2014

Issues To Be Addressed:

1) Many effects influence radial plasma transport in tokamaks.

2) Small 3-D magnetic perturbations have interesting and usefuleffects on toroidal plasmas — directly on Ωt, indirectly ne, pe, pi.

3) What is the general strategy for developing a “predictivecapability” for the behavior and control of ITER plasmas?

Outline:

• Effects of 3-D fields on Ωt rotation — NTV, ripple, resonant FEs, RMPs

• Role of macroscopic plasma responses in determining 3-D ~B fields

• Flutter models of effects of 3-D resonant RMPs on pedestal transport

• There could be many effects of small 3-D fields on plasma transport in ITER

• Extended MHD, gyrokinetic (µturb) & transport approaches should be merged

JDC Lecture 4/CEMRACS 2014, CIRM, Marseille, France — July 21–25, 2014, p 1

Tokamak Plasma Transport Equations Include Many Effects

•With sources of ne, Lt ≡ ρm〈R2〉Ωt and ps, transport equations are

density1

V ′∂

∂t

∣∣∣∣ψp

neV′ + ρψp

∂ne

∂ρ+

1

V ′∂

∂ρ(V ′Γ) = 〈Sn〉,

tor. mom.1

V ′∂

∂t

∣∣∣∣ψp

LtV′ + ρψp

∂Lt

∂ρ+

1

V ′∂

∂ρ(V ′Πρζ) = 〈~eζ ·

(~J× ~B − ~∇·

↔Π +

∑s~Sps

)〉,

energy3

2ps∂

∂t

∣∣∣∣ψp

ln psV′5/3 +

3

2ρψp

∂ps

∂ρ+

1

V ′∂

∂ρ(V ′Υs) + 〈~∇· ~qpc

s∗〉 = Qsnet.

• There are many classes of effects in plasma transport equations:

transients in the poloidal flux ψp via ∂/∂t|ψt and advection of ψp surfacesrelative to the toroidal-flux-based radial coordinate ρ via ρψp ≡ ψp/ψ

′p,

transport fluxes of ambipolar density Γ, toroidal mom. Πρζ and heat Υs, ~qpcs∗

“radially” across ψp poloidal flux surfaces that have many contributionsinduced by various collision and microturbulent processes in each species s,

ambipolar density 〈Sn〉, toroidal momentum 〈~eζ · ~Sp〉 and energy 〈Sε〉 sourceswith energy sources contributing to net energy heating rate Qsnet, whichalso includes external heating sources (NBI, ECH etc.) & radiation losses,

and toroidal torques on the plasma caused by ~J× ~B & viscous stresses ~∇·↔Π.

• This lecture focuses on recent developments: small 3-D field effects.


Tokamak Plasma Transport Equations Include 3-D Field Effects

•With sources of ne, Lt ≡ ρm〈R2〉Ωt and ps, transport equations are

density1

V ′∂

∂t

∣∣∣∣ψp

neV′ + ρψp

∂ne

∂ρ+

1

V ′∂

∂ρ(V ′Γ) = 〈Sn〉,

tor. mom.1

V ′∂

∂t

∣∣∣∣ψp

LtV′ + ρψp

∂Lt

∂ρ+

1

V ′∂

∂ρ(V ′Πρζ) = 〈~eζ ·

(~J× ~B − ~∇·

↔Π +

∑s~Sps

)〉,

energy3

2ps∂

∂t

∣∣∣∣ψp

ln psV′5/3 +

3

2ρψp

∂ps

∂ρ+

1

V ′∂

∂ρ(V ′Υs) + 〈~∇· ~qpc

s∗〉 = Qsnet.

• Small 3-D field (|δ ~B|/B0 ∼ %∗) effects come about in many ways:

externally applied resonantm/n ' q and non-resonant fields cause field error

(FE, 〈~eζ · ~J× ~B〉) and neo. toroidal viscous (NTV, 〈~eζ ·~∇·↔Π〉) damping of Ωt,

toroidal magnetic field “ripple” caused by the finite number of coilsthat produce the toroidal magnetic field which damps Ωt via NTV,

externally applied edge resonant magnetic perturbations (RMPs) used tomodify the pressure profile there and stabilize edge MHD instabilities, and

spontaneous magnetic perturbations in the plasma which are caused byextended MHD macroscopic plasma instabilities that are under control,e.g., neoclassical tearing modes (NTMs) or resistive wall modes (RWMs).


There Are Many Categories Of 3-D Magnetic Field Effects1

• Tokamak magnetic field is axisymmetric (2-D) to lowest order

plus small 3-D magnetic perturbations δ ~B (<∼ 10−2B0) that willbe expanded in a Fourier series — see next viewgraph.

• 3-D field effects can be classified by toroidal mode number n of δ ~B:

low n (∼ 1–3) non-res. =⇒ neoclassical toroidal viscous (NTV) damping of Ωt,

low n resonant (with field lines) =⇒ resonant braking, locking, disruption,

medium n (toroidal field ripple) =⇒ edge ion losses plus NTV braking of Ωt,

high n =⇒ µ-turbulence-induced Reynolds stress, anomalous Ωt transport.

• Radial force balance + poloidal flow damping + toroidal plasmatorques + ambipolarity constraint =⇒ transport equation fortoroidal flow Ωt (→ Eρ) with 3-D effects — see next viewgraph.

1J.D. Callen, topical review paper on “Effects of 3D magnetic perturbations on toroidal plasmas,” Nucl. Fusion 51, 094026 (2011), which will providethe basis for most of the issues discussed in the remainder of this lecture.


Plasma Toroidal Rotation Equation Provides Context

• Magnetic field magnitude will be represented in ψp, θ, ζ coordinates by

| ~B| = | ~B0(ψp, θ)|︸︷︷︸2D, axisymm.

+∑n,m

δBn(ψp,m) cos (mθ − nζ − ϕm,n)︸︷︷︸low m,n resonant, non-resonant

+ δBN(ψp, θ) cos(Nζ)︸︷︷︸medium n, ripple

+ · · ·︸︷︷︸µturb.

.

• On µs time scale compressional Alfven waves enforce radial force balance:

Ωt ≡ ~Vi ·~∇ζ = −(∂Φ0

∂ψp

+1

niqi

∂pi

∂ψp

)+ q ~V i ·~∇θ =⇒ Vt '

Eρ

Bp

−1

niqiBp

dpi

dρ+Bt

Bp

Vp.

• On the ms time scale poloidal flow is damped to Vp ' (cp/qi)(dTi/dψp) + · · · .

• Toroidal plasma torques cause radial particle fluxes: ~eζ·~Force = − qs~Γs·~∇ψp.

• Setting the total radial plasma current induced by sum of the non-ambipolar

particle fluxes to zero yields transport equation2 for plasma toroidal angular

momentum density Lt ≡∑

ionsmini〈R2~V i ·~∇ζ〉, Ωt(ρ, t) ≡ Lt/mini〈R2〉 :

∂Lt

∂t︸︷︷︸inertia

' −〈~eζ ·~∇·↔π

3D

i‖ 〉︸︷︷︸NTV from δB

+ 〈~eζ· δ ~J×δ ~B〉︸︷︷︸resonant FEs

− 〈~eζ·~∇·↔πi⊥〉︸︷︷︸

cl, neo, paleo

−1

V ′∂

∂ρ(V ′Πiρζ)︸︷︷︸

Reynolds stress2

+ 〈~eζ ·∑

s~Sps〉︸︷︷︸

mom. sources

.

• Radial electric field for net ambipolar transport is determined by Ωt:

Eρ ≡ − |~∇ρ| ∂Φ0/∂ρ ' |~∇ρ| [ Ωtψ′p+(1/ni0qi) dpi/dρ− (cp/qi) dTi/dρ], ωE ' −Eρ/RBp.

2a) J.D. Callen A.J. Cole and C.C. Hegna, Nucl. Fusion 49, 085021 (2009); b) Phys. Plasmas 16, 082504 (2009); c) Phys. Pl. 17, 056113 (2010).


Nonaxisymmetric Magnetic Fields Induce NTV

• An axisymmetric (A) ~B0 produces no toroidal torque on a tokamak plasma be-

cause ∂| ~B0|/∂ζ = 0 (i.e., no toroidal bumps in | ~B0| to impede toroidal flow):

〈~eζ ·~∇·↔π

A

‖ 〉 = 0, in which ~eζ ≡ R2~∇ζ = R~eζ.

• However, when | ~B| ' B0 +δB‖ varies toroidally (i.e., ∂ δB‖/∂ζ 6= 0), it causes

a 3-D nonaxisymmetric (NA) neoclassical toroidal viscous (NTV) torque on

a tokamak plasma that can be written in the generic form:2,3

〈~eζ ·~∇·↔π

NA

‖ 〉 ' mini〈R2〉µi t

(δB‖eff

B0

)2

(Ωt − Ω∗) , in which

µi t is parallel-stress-induced toroidal viscous damping frequency (see following viewgraphs),

|δB‖eff/B0| '∑

n |δBn/B0| is the effective variation of magnetic field strength along ~B,

Ωt = −(dΦ0

dψp

+1

niqi

dpi

dψp

)+

cpI2

qi〈R2〉〈B20〉dTi

dψp

, with cp ≡ ki of neoclassical theory,

which is the FSA of R2× the general toroidal rotation frequency Ωt(ρ, θ), and

Ω∗ 'cp + ct

qi

dTi

dψp

is the normalized FSA of R2× the offset rotation frequency.

3For derivations and summary of radial particle fluxes see K.C. Shaing et al., Nuc. Fus. 50, 025022 (2010).


NTV Torque Is Calculated From NA Neoclassical Particle Fluxes

• For 3-D NA toroidal plasmas the FSA toroidal torque is calculated from NA

neoclassical transport theory via the flux-friction (viscous force) relation4

〈~eζ · ~∇ ·↔π

NA

s‖ 〉 = qsψ′p ΓNA

s π‖, obtained from FSA of toroidal torques (~eζ · force balance).

• The fundamental 3-D nonaxisymmetric (NA) particle fluxes have been calcu-

lated for a wide variety of asymptotic regimes by Shaing et al.3,5

• The relevant asymptotic results3,5 have been extended and adapted6 to the

nearly axisymmetric tokamak magnetic geometry by doing the following:

1) recalculating the relevant NA neoclassical particle fluxes in terms of a “smoothed” multi-collisionality energy and (WKB-type) pitch-angle kernal of the transport integrand,

2) determining toroidal torque from the particle flux via the flux-friction relation, and

3) using the definition of the toroidal rotation frequency Ωt on the preceding viewgraph towrite the NTV torque in terms of the damping of Ωt to an offset Ω∗.

4K.C. Shaing and J.D. Callen, Phys. Fluids 26, 3315 (1983); K.C. Shaing, S.P. Hirshman, and J.D. Callen, Phys. Fluids 29, 521 (1986).5K.C. Shaing, Phys. Plasmas 10, 1443 (2003); K.C. Shaing et al., Phys. Plasmas 15, 082506 (2008); Pl. Phys. and Cont. Fus. 51, 035009 (2009).6A.J. Cole, J.D. Callen et al., “Observation of Peak Neoclassical Toroidal Viscous Force in the DIII-D Tokamak,” Phys. Rev. Lett. 106, 225002 (2011);

Phys. Plasmas 18, 055711 (2011).


NTV Is Caused By Radial Drifts Of Ions Induced By 3-D Fields

• In axisymmetric (2-D) neoclassical theory, centers of banana drift orbits do

not move radially =⇒ radial flux is ambipolar =⇒ no 2-D NTV torque.

• Small 3-D δBn causes radial “banana drifts” at speed v3Dd ∼ n (δBn/B0) v

2Dd0 .

• Radial excursions of trapped ions are limited by various physical processes:1

collisions (1/ν regime) =⇒ ∆ρ ∼ v3Dd /(νi/ε),

collisional boundary layer (√ν regime) =⇒∆ρ ∼ v3D

d /(|n|ωE), ωE ≡ dΦ/dψp ∼ −Eρ/RBp,

superbanana plateau (sbp, ωE→0, v2Dd0 limited) =⇒ ∆ρ ∼ v3D

d /[(|n|v2Dd0 /R0)

2/3(νi/ε)1/3],

...

— see next viewgraph.

• Radial ion drifts =⇒ non-ambipolar radial ion flux =⇒ NTV plasma torque.

• Experimental tests have confirmed key NTV predictions at δBn/B0 ∼ 10−3

— see p 11, 12.


NTV Theory: Non-resonant δBn Induces NTV Torque

• Neoclassical toroidal viscous (NTV) torque3,5,6 induced by a single δBn is

−〈~eζ·~∇·↔π

3D

i‖ 〉 ' −mini〈R2〉µi t

(δBn

B0

)2(Ωt − Ω∗) , Ω∗'

cp+ct

qi

dTi

dψp

∼1

qiRBp

dTi

dρ< 0.

• NTV damps Ωt to Ω∗ < 0 at rate µi t(δBn/B0)2 ∼ (Di/%

2i ) (Bp/B0)

2.

D i plateau

banana

P-S

ν

νν

sbp

1/

i

i i

ωωωωωsb sbp n E | | ε

ω E = 0

ti ti ε3/2

,_

transit-resonance (TTMP)

Figure 1: Ion collisionality regimes for particle diffusivity Di ∝ NTV damping frequency µi t. Transitionsoccur at key frequencies: ion transit ωti ≡ vT i/R0q, ~E× ~B-induced ε |nωE|, superbanana-plateau (sbp)radial drift ωsbp ≡ ε |n|ωd0 and superbanana ωsb ≡ ε−1/2(δBn/B0)

3/2(|n|ωd0). The Di and µi t becomelarge when the radial electric field vanishes, i.e., ωE → 0 (short dashes curve). Fig. 1 in Ref. [1] on p 4.


NTV Damping Frequency Includes Many Effects

• A Pade approximation for the NTV damping frequency that encompasses the

relevant collisionality regimes for tokamaks has been constructed:6

µi t P(Ωt) =0.21

√|n|νi v2

T i/〈R2〉|Ωt − Ω0|3/2 + 0.3

√νi/|nε| |ω∇B|+ 0.04 (νi/|nε|)3/2

,

νi is ion collision frequency and vT i is ion thermal speed,

n is toroidal mode number of NA δB‖ perturbation,

Ω0 ≡ Ωt+ωE, in which ωE ≡ dΦ/dψp ' −Eρ/RBp is ~E× ~B toroidal precession frequency,

ω∇B ' (Ti/qi)(dε/dψp) is the superbabana grad-B toroidal drift frequency (ε ≡ ρ/R0).

• This damping frequency encompasses three asymptotic regimes:

νi |nεωE|, 1/νi regime3 where ion 3-D NA radial drifts are limited by collisions,

νi <∼ |nεωE|,√νi regime3 where pitch-angle boundary layer effects dominate, and

Ωt ' Ω0 where ωE ' 0, “superbanana” regime3,5,6 where ω∇B limits radial drifts.

• Note that the peak NTV torque occurs where the ~E× ~B toroidal precession

frequency ωE ≡ dΦ0/dψp ' −Eρ/RBp vanishes, which causes the particle flux

to be determined by the radial width of “superbanana” drift orbits.


NTV Exp. I: NSTX Results Agreed With Early NTV Theory

• There have been many indications of NTV torque effects (δBn/B0 ∼ 10−3):

for 1/3 fields (DIII-D, 2002), in quasi-symmetric stellarator (HSX, 2005), together withresonant n (DIII-D & NSTX, 2006–2010), from rotating MHD modes (MAST, 2010).

• Figures 2, 3 show first detailed comparisons of NTV theory (in 1/ν regime,

neglecting Lagrangian effects) with toroidal torque data from NSTX.7

7W. Zhu, S.A. Sabbagh et al., Phys. Rev. Lett. 96, 225002 (2006); S.A. Sabbagh et al., Nucl. Fusion 46, 435 (2006).

Fig. 3

TN

TV

(N m

)

3

4

2

1

00.9 1.1 1.3 1.5

R (m)

measured

theoryt = 0.360s

116931

axis

Figure 2: NSTX experimental test7 of the spatialprofile and magnitude of NTV-induced torqueTNTV for n=3 non-resonant 3-D field.

Fig. 4

TN

TV

(N m

)

3

4

2

1

0

0.9 1.1 1.3 1.5

R (m)

With RWM

(DCON)6

8

4

2

0

10T

NT

V(N

m)

applied

field

only

axis

t = 0.370s

116939

t = 0.400s

axis

0.30 0.35 0.40 0.45t(s)

0

10

20

30

40

0

1

2

3

4

5

2

1

0IRWM

(kA

)

RWM

RFA

(a)

(b)

With RFA (c)

(d)

!N

|"B

p| (

n=

1)(G

)

!N > !Nno-wall

applied

field

only

With RFA

(DCON)

Figure 3: NSTX experimental test7 of the NTVtorque shows resonant field amplification (RFA)effects are needed for n=1 3-D field.


NTV Exp. II: Offset Ω∗ And ωE → 0 Peak Validated In DIII-D

•When I-coils are turned on in n=3 configuration8 in DIII-D, Ωt is damped to

the diamagnetic-level rotation frequency Ω∗ < 0 — see Fig. 4 below.

• As Ωt is varied6 (via balancing co/ctr NBI beams in DIII-D), peak torque is

where ωE → 0, in agreement with NTV predictions — see Fig. 5 below.

8A.M. Garofalo et al., Phys. Rev. Lett. 101, 195005 (2008); Phys. Plasmas 16, 056119 (2009).

-100

-50

0

5004 n=3 I-coil (kA)

20

0

-20

-40

(a)(b) (c)

Time (s) dL/dt (Nm/m3)0.05-0.05 01.8 1.9 2.0 2.1 2.2

V (km/s)

131331

131320

131861

131400131405131408

131866

131868

OffsetRotation

Figure 4: DIII-D experiments validated8 the NTV-induced damping/braking to offset rotation frequencyΩ∗ < 0, as predicted by the NTV torque formula.

0.0

0.5

1.0

1.5

2.0

−30 −25 −20 −15 −10 −5 0 5 10 15

ExptFitTheoryModel

Ω [krad/s], + is co-Ip

−TNTV

[Nm]

Figure 5: DIII-D experiments validated6

NTV peak caused by µi t P(νi, ωE) occurswhere ωE ' 0 at which Ωt ' − 2 krad/s.


III. Toroidal Field Ripple Causes Many Toroidal Torques

• Magnetic field ripple (δ ≡ δBN/B0<∼ 1%) is caused by the finite number N

(typically 18–32) of coils producing the toroidal magnetic field ~Bt.

• Non-ambipolar particle fluxes and toroidal torques are induced by:

ripple-induced direct ion losses at edge =⇒ radial current =⇒ return current in the plasma=⇒ ~J× ~Bp toroidal torque in counter-current direction =⇒ reduction in Ωt,

NTV damping effects (TTMP,√νi regime) =⇒ braking of Ωt toward Ω∗ < 0,

radially drifting ripple-trapped particles =⇒ NTV torque that scales as (δBN/B0)3/2.

• Experimental tests confirm reduction in Ωt as ripple is increased, with smaller

effects on ne, Te and Ti transport (but slight density “pump-out”)

— see next 2 viewgraphs.

• More modeling needed for NTV effects in core, with self-consistent Eρ (Ωt).


Ripple Theory: Toroidal Field Ripple Causes NTV & Other Effects

• Magnetic field ripple (δBN/B0<∼ 1%) caused by finite number N of toroidal

field coils induces various types of 3-D NTV effects, which are additive:

1) transit-resonance magnetic pumping (TTMP) NTV effects are often dominant;

2) banana-drift NTV effects are likely in√νi regime because usually νi < ε |NωE|; and

3) ions with νi < (δBN/B0)1/2Nωti can be trapped in ripples (if ε | sin θ| < Nqδ), drift

radially and induce an ion particle flux and NTV torque that scales as (δBN/B0)3/2.

• At the edge superthermal ions and NBI fast ions can be ripple trapped or

have asymmetric banana drift orbits and drift out of the plasma, which:

1) causes a radial ion “direct loss” (dl) current 〈 ~Jdl ·~∇ψp〉 that induces a radially inward“return current” in the plasma to preserve quasineutrality,

2) induces a toroidal torque on the edge plasma in the counter-current direction when thisradially inward (negative) plasma return current is crossed with ~Bp, and

3) is represented in the Lt equation by (p 5) a momentum sink 〈~eζ · ~Spi〉 = −〈 ~Jdl ·~∇ψp〉.

• Thus, ripple-induced direct loss and NTV effects should both decrease the

plasma toroidal rotation frequency Ωt; NTV effects damp it toward Ω∗ < 0.

• Ripple-induced reductions in Ωt have been observed in many tokamaks:

ISX-B (1985), JT-60U (2006-08), JET (2008-2010), Tore-Supra (2010).


Ripple Exp.: Increased Ripple Induces Large Reductions In Ωt

• Addition of ferritic steel tiles (FSTs) in JT-60U reduced ripple and increased

Vt;9 modeled edge direct loss effects agree9b — see Fig. 6 below.

• Increasing ripple in JET reduced edge Ωt without changing other plasma

parameters10 (gas puffing prevents density “pump-out”) — see Fig. 7 below.9a) M. Yoshida et al., Plasma Phys. Control. Fusion 48, 1673 (2006); b) M. Honda et al., Nucl. Fusion 48, 085003 (2008).

10a) G. Saibene et al., Paper EX/2-1 at 2008 Geneva IAEA FEC (to be published); b) H. Urano et al., EXC/P8-17, 2010 Daejeon IAEA FEC.

Figure 6: Toroidal plasma flow decreases asfield ripple in JT-60U is increased from 1%(with FSTs) to 2% without (w/o) FSTs.9a

IT/EX2-1

6

become irregular, with mixed Type I, Type III and

long ELM-free phases.

In general, type I ELMs appear to be smaller, at

least in terms of the D! emission from the divertor.

To quantify this observation, the normalized (to the

pedestal energy Wped) type I ELM energy loss is

shown, in figure 8

Figure 7 divertor H-! traces for the 4

plasma discharges in figure 1

Figure 8: energy loss per ELM

(normalized to the pedestal energy

Wped) vs "#ped . 2.6MA/2.2T, fine

ripple scan.

Figure 9: ripple scan at 1MA/1T – From top to bottom:

total NB input power, plasma stored energy, line average

density (core and edge), and plasma toroidal rotation

frequency, at ~3.77m

, as function of "*ped. The plot

R

e

be

Thi lts of a fine ripple scan

(0.3%, 0.5%, 0.7% and 1%), carried out on reference H-

indicates that, for similar collisionality, the size of

the ELM energy loss is reduced as the ripple

increases. In particular, the reduction of the ELM

energy loss is due to a reduction of the relative

prompt Te drop for higher $BT, suggesting that ELM

losses become more convective as the ripple is

increased, even at low "*.

esults at reduced Ip/BT.

effect of ripple on H-mode characteristics seems to

reduced as the plasma current and field are reduced.

s is illustrated by the resu

4.

Th

mode plasma at 1MA/1T (q95 = 3.6), with plasma shape

identical to that of the 2.6MA/2.4T H-modes just

discussed. The method used to test ripple effects was to

select a reference plasma and then, using real time

control of the NB input power to keep constant %N, to

attempt to produce identical plasmas for increasing

ripple. The reference plasma was a 1MA H-mode

obtained in a pedestal identity experiment between JET,

DIII-D and ASDEX-Upgrade. This target plasma was

selected to test the effects of ripple on the identity (since

DIII-D and ASDEX-Upgrade have higher

ripple at the separatrix than JET) as well as

to obtain data on ripple effects at reduced

current and field, or higher &* (the reference

discharge has &*ped ~1.7 10-1

and "*ped ~

0.23 , at the pedestal top). Figure 9 shows

that, in contrast to the results obtained at

higher field and current, the global and

pedestal parameters of the 1MA/1T series of

H-modes do not change significantly for

increasing ripple, and in particular the

plasma stored energy is only slightly

reduced at high ripple at constant input

power. Quite interestingly, the effect of

ripple on VTOR is still observed, and the

plasma rotation is reduced as the ripple is

increased, with VTOR approaching 0 at the

plasma periphery for $BT = 1.0%.

Figure 7: Toroidal plasma rotation decreasesmonotonically with increasing field ripple(% #s at right of 4th panel) in edge of JET.10a


IV. Low n Resonant 3-D Fields Cause Locking, Disruptions

• Field errors (FEs) can introduce resonant 3-D fields that produce localized

torques at rational surfaces [ q(ρm/n) = m/n ] in the plasma.

•Well established (Fitzpatrick) cylindrical model of FE effects predicts:

shielding/screening of FE on and inside of rational surface if plasma rotates fast,

for slow rotation FE “penetrates” =⇒ toroidal torque ∼ δB2ρm/n at ρm/n,

larger FE torque than ⊥ viscosity χt i =⇒ locked mode =⇒ growing island =⇒ disruption

— see next viewgraph.

• Recent developments in FE-induced n=1 locked mode studies:

experiment — FE locking threshold characterized =⇒ δBρ 2/1/B0 ∼ 10−4 ∝ neR0,

theory — two-fluid layer physics plus NTV effects =⇒ closest to ∝ ne scaling but χt i?,

compensation — “correction” of FEs by using additional 3-D fields shows that theplasma response to them is critical and the plasma response increases with β

— see viewgraph on p 18.


FE Theory: Field Errors (FEs) Can Cause Mode Locking

• Field errors introduce low n resonant 3-D fields which

in the dissipationless ideal MHD model cause no toroidal torque on the plasma, but

induces dissipative local torques in thin layers around rational surfaces at q(ρm/n) = m/n.

• The local Maxwell-stress-induced FSA plasma toroidal torque density for a

cylindrical model (with a δ-function resistive singular layer at ρm/n) is11

〈~eζ · δ ~J‖m/n×δ ~Bρm/n〉 ' −mini (4nc2A)

(δBvac

ρm/n

B0

)2 [(−ωτs)

(−∆′)2 + (−ωτs)2

]V δ[ρ− ρm/n]

V ′,

in which δBvacρ m/n ≡ [δ ~B ·~∇ρ]ρm/n

, ω ≡ ~k ·~V i =⇒ n [ωE + (1/niqi)(dpi/dψp)], τs is thesingular-layer diffusion time, and ∆′ ∼ − 2m is the tearing mode instability index.

• This radially localized FE-induced electromagnetic toroidal torque:

competes with the NBI momentum source and radial transport of Ωt induced by χt i,

for ωτs 1 “penetration” of δBρm/n is limited by Ωt and δBvacρm/n vanishes for ρ ' ρm/n,

for large δBvacρm/n the “penetration threshold” is exceeded (e.g., for δBvac

ρ 2/1/B0>∼ 10−4),

Ωt(ρm/n) is no longer restrained by χt i and Ωt no longer “shields” out resonant torque;

then, Ωt solution bifurcates (in a few ms) to no flow at ρm/n, magnetic reconnection occursand a growing m/n “locked mode” is induced, which often leads to a disruption.

11R. Fitzptrick, Nucl. Fusion 33, 1049 (1993). See also http://farside.ph.utexas.edu/papers/lecture.html.


FE Exp. Status: Locking Thresholds, Resonant Field Amplification

• Recent FE theory developments:

two-fluid singular layer effects developed,

NTV showna to increase Ωt shielding effects.

• Recent FE experimental developments:

error field locking thresholds characterized,b

scaling closest to NTV-enhanced scaling,a

validation tests limited by uncertainty in χt i.

• Recent compensations of resonant field

errors show that plasma resonant field

amplification (RFA) is critical.c

• Measured RFA agrees with ideal MHD

MARS-F calculations,d up to near no-

wall β limit where non-ideal MHD ef-

fects become important — see Fig. 8.

aA.J. Cole et al., Phys. Rev. Lett. 99, 065001 (2007).bS.M. Wolfe et al., Phys. Plasmas 12, 056110 (2005).cJ.-K. Park et al., Phys. Rev. Lett. 99, 195003 (2007).dM.J. Lanctot et al., Phys. Plasmas 17, 030701 (2010).

plasma perturbation is modeled using only ideal MHD,which does not provide any free parameters that directly af-fect the amplitude of the plasma response. Since the influ-ence of the background plasma rotation on the RWM stabil-ity is neglected, we focus here only on equilibria that arebelow the no-wall limit. This is reasonable since above thestability threshold, ideal MHD predicts an unstable RWM,which is not experimentally observed in these discharges.

Since the MARS-F code solves for the plasma responsebased on fixed boundary equilibria, multiple equilibria werecomputed for each experimental reconstruction by retainingbetween 99.0 and 99.7% of the total poloidal flux. Theseequilibria were used as input to MARS-F to calculate theplasma perturbation, and to estimate the error introduced bythe flux truncation. At !N=1.7, both the magnitude and tor-oidal phase of the computed n=1 perturbed field at varioussensor locations are in good agreement !within 20%" withthe measured plasma response, Fig. 3, which is obtained bysubtracting the measured coil-sensor coupling from the totalperturbed field. The measured toroidal phase is quoted withrespect to the applied radial field at the midplane, and showsthat "Br

plas is in phase with the applied field, while "Bpplas is

shifted by +90° in the direction of the plasma current. Thephases of the upper and lower radial magnetic probes indi-cate the helical structure of the perturbation. A systematicphase shift of the measurements with respect to the predic-tions in the direction of the plasma rotation is likely causedby the interaction of the mode with the plasma rotation.These magnetic measurements show that ideal MHD is ad-equate to describe the external plasma response for values of!N sufficiently far from !N

NW.At higher pressures, MARS-F tends to overestimate the

perturbed field. In Fig. 4, the measured and modeled magni-tude and phase of "Bp

plas are compared as a function of!N /!N

NW, where !NNW is calculated for each equilibrium. In

the range of 75%–100% of !NNW, the computed "Bp

plas exceedsthe observed magnitude by a factor of 1.5–3, and the phaseshifts in the negative Ip direction. The poor agreement hereindicates that nonideal effects are important even in this re-gime. Above !N

NW, the observed phase of "Bpplas shifts in the

direction of Ip by 50° at !N=2.3, which is inconsistent withideal MHD theory. Although there is an apparent jump in theobserved phase near the no-wall limit for this data set, alarger database of plasma response measurements shows asmooth transition through the stability threshold.

The magnitude of the computed "Bplas depends on thestability of the plasma, which is determined in part by thedetails of the current profile. This is demonstrated usingequilibria obtained by solving the Grad–Shafranov equationindependent of the experimental constraints using a scalarmultiplier to vary the pressure profile determined with EFIT

while keeping the current profile fixed for the lowest andhighest beta equilibria shown in Fig. 1!b". The solid anddashed lines in Fig. 4 mark the computed "Bp

plas based on thelow and high beta discharge, which have a plasma internalinductance of 0.85 and 0.79, respectively. For all values of!N, the computed "Bp

plas is larger for the plasma with a lowerinternal inductance. However, the "Bp

plas for both sets of dis-charges are equal when considered as a function of !N /!N

NW.This indicates that the current profile influences the plasmaresponse, but only insofar as it affects the stability limit,which is proportional to the internal inductance.22 However,the discrepancy between the measurements and the MHDcalculations near the no-wall limit cannot be explained by avariation of the experimentally determined current profilewithin the known uncertainties of the bootstrap current cal-culation.

The plasma response also depends on the safety factorprofile and the structure of "Bext through a resonance withthe unstable kink eigenmode, which has no pitch resonant

FIG. 3. !a" Magnitude and !b" toroidal phase of "Bplas at !N=1.7 fromvarious magnetic diagnostics. Subscripts on "B refer to poloidal !p", orradial !r" field probes, the probe location either internal !no label" or external!e" to the vacuum vessel, and the probe elevation either at !no label", above!up" or below !low" the midplane. Solid lines mark perfect agreement be-tween measured and modeled data. The error bars for the calculated quan-tities are based on a variation of the total poloidal flux between 99.0% and99.7% of the experimentally determined flux.

FIG. 4. !Color online" Comparison of the measured !square" and computed!diamond" !a" magnitude and !b" toroidal phase of "Bp

plas at the midplane asa function of !N /!N

NW. Solid and dashed lines mark the computed "Bpplas for

scaled equilibria based on discharges with !N=1.14 and !N=1.95.

030701-3 Validation of the linear ideal MHD model… Phys. Plasmas 17, 030701 !2010"

Downloaded 23 Sep 2010 to 128.104.1.220. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions

Figure 8: The magnitude of the n= 1 res-onant 3-D field on the DIII-D outer mid-plane increases about linearly with β.d


V. Plasma, Combined Effects Are Complicated, Being Developed

• Plasma response to externally applied 3-D fields

amplifies δ ~B in plasma by coupling to “least stable” n=1 kinks via edge m > q95 which

can be estimated using an ideal MHD model (IPEC); but self-consistent evaluation needsresistive and two-fluid layer physics as in M3D-C1, NIMROD, JOREK, MARS-F etc.

• Plasma instabilities (NTMs, RWMs) cause additional 3-D fields and more

field error (FE) sensitivity for low Ωt, increasing β.

• Resonant magnetic perturbations (RMPs) for ELM control:

was initially based on stochasticity caused by island overlap (Chirikov criterion);

but Ωt “screening” of RMP fields reduces stochastic region width and has other effects;

while RMP effects are not yet fully understood, they are important tools for pedestals.

• 3-D fields directly affect Ωt [global NTV, local FE locking of Ωt(ρm/n) → 0];

ne, Te and Ti transport effects are small, except for ripple- and RMP-induced

slight density “pump-out.”

• Following viewgraphs discuss some of these issues in greater detail.


FE Plasma Effects I: RFA Due To Coupling To n=1 Global Mode

• External 3-D fields can amplify resonant components of δ ~B in plasma by

coupling to “least stable” MHD-type eigenmodesd — n=1 kink-type.

• The largest δBρm/n responses at m/n = 2/1, 3/1 resonant surfaces in the

plasma result fromc (Fig. 10) external δ ~B which is pitch-aligned with edge

n=1 global kink eigenmode components that have m >∼ q >∼ q95 (Fig. 9).

Nucl. Fusion 49 (2009) 115001 H. Reimerdes et al

0 90 180 270 360

-45

0

45

90Perturbed normal field at vessel wall

Polo

idal

ang

le !

(º)

"B (au)

Toroidal angle # (º)

Upper I-coils

Lower I-coils-90 -1.0

-0.5

0.0

0.5

1.0

0.0 0.5 1.0Poloidal flux coordinate s=($/$a)0.5

1.50.00

m=1m=2m=3m=4m=5m=6m=7

Ideal MHD n=1 RWM (MARS-F) 134234 t = 1750 ms

Mode structure

Plas

ma

boun

dary

Vess

el w

all

(a)

|("B

J A)m

1| (a

u)

(b)

0.02

0.04

0.06

Figure 7. (a) The poloidal spectrum (using a straight-field-line coordinate) of the normal component of the perturbed field (multiplied withthe surface Jacobian) (!B · JA)mn of an unstable n = 1 RWM calculated for 134234 at t = 1750 ms with the MARS-F code as a function ofa normalized poloidal flux coordinate s shows global kink mode structure. (b) The corresponding perturbed normal field !B at the outboardwall best matches an I-coil phase difference "#I between the upper and lower I-coil arrays of approximately 300!.

-2 0 2 4 6 8 TNBI (Nm)

0

5

10

15Mode born lockedLocked mode precededby rotating 2/1 mode

n=1 braking experiments

L0 (

Nms)

0.0

0.2

0.4

plas

Fit offset:% (TNBI + 1.4 Nm)

Angular momentum before n=1 braking

Fit exponent:% (TNBI + Tan)0.45

T an

= 1.

4 Nm

Fit offset:% (TNBI + 8.8 Nm)

-2

(a)

(b)

Figure 8. (a) The tolerable n = 1 plasma response !Bplasp,crit evaluated

at the rotation collapse in slow magnetic braking experimentsincreases with increasing NBI torque TNBI. A linear fit (solid) yieldsa torque offset of 8.8 N m. Fitting the measurement to a powerlaw (dashed) while constraining the torque offset using an estimatefor the anomalous torque of Tan = 1.4 N m, which is (b) obtainedfrom the dependence of the angular momentum L0 before anyexternal n = 1 magnetic field is applied on TNBI, yields an exponentof 0.45.

0 1 2 3 4Density ne (1019m-3)

Pertu

rbed

fiel

d (G

)

0

4

8

12

Low & locked modeexperiments

Measured"Bp,crit

plas

IPEC

Calculated (IPEC) totalresonant field "B21,crit

NBI heatedH-mode

Error field tolerance in low and high &N plasmas

Figure 9. Comparison of IPEC calculations of the total resonantn = 1 field at the q = 2 surface !B21,crit (filled circles) immediatelyprior to the rotation collapse in an NBI heated H-mode discharge($N = 1.5, TNBI = 1.8 N m) with the linear density dependence ofthe tolerable total field found in Ohmically heated L-modedischarges (shaded) [25]. The calculated plasma response at thesensors (filled diamond) in the NBI heated H-mode exceeds thecorresponding measurement (open diamond) by 40%.

TNBI = 1.8 N m, shown in figure 3, results in a total resonantfield at the q = 2 surface of !B21 " 12 G, figure 9. With anaverage electron density of #ne$ = 3.8%1019 m&3 this value for!B21 exceeds the extrapolation of the linear density dependenceof the error field tolerance in Ohmic plasmas by more than afactor of two, figure 9. This increase in the error field tolerancecould be attributed to the beneficial effect of NBI torque. Theempirical fit obtained in figure 8 suggests an increase in theerror field tolerance in a plasma with TNBI = 1.8 N m ofapproximately 45% over an Ohmically heated plasma withTNBI = 0, in qualitative agreement with the result in figure 9.

The IPEC calculation also yields a plasma response at thelocation of the poloidal field probes at the outboard mid-planeof !B

plasp " 10 G, which exceeds the measured value of 7 G

by approximately 40%, figure 9. This comparison represents afirst quantitative test of the IPEC model. The agreement can be

7

Figure 9: Poloidal mode spectrum ofδBρm/n=1(ψ) for an unstable n= 1 RWMfor q95 ' 5 in DIII-Dd from MARS-F.

j!21j. Although the C coil greatly enhanced the low har-monics of the external field when the m ! 7, 8, 9 harmon-ics were reduced, this did not prevent a successfulmitigation of error field effects. For NSTX the dominantcoupling is from the m ! 12, 13, 14 harmonics [Fig. 6(a)].Indeed, two out of three of these harmonics are reducedwhen error field effects are mitigated (M" EFC), but twoout of three are increased when error field effects areenhanced (M# EFC) by changing the toroidal phase ofthe EFC coil currents [Fig. 6(b)]. The large shift of thedominant poloidal harmonics to higher modes in bothmachines is the typical characteristic of the toroidal plasmaresponse. Note that the test plasmas in both machines werestable and far from a stability limit.

The description of the external error field in terms of itsFourier harmonics on the plasma surface offers little in-sight since such high harmonics in flux coordinates are noteasily controlled in real space. Much better insight isobtained from the distribution on the plasma surface ofthe normal component of the external field that maximizesthe total resonant fields. This distribution can be written as~bx $ nb ! A%!& cos%"& " B%!& sin%"&, where " is the polartoroidal angle. Figure 7 represents these two functions as adeviation from a surface that represents the plasma edgefor DIII-D [Fig. 7(a)] and NSTX [Fig. 7(b)]. Figure 7implies that the asymmetric variation in external field onthe outboard side is most important and should be con-trolled. The dominant patterns are weakly dependent ontarget plasmas and explains the successful cancellation oferror fields by the C coils in DIII-D and EFC coils inNSTX, despite their limitations in the control of poloidaldistribution.

In summary, a comparison of theory and experiment hasshown that magnetic perturbations that drive islands on loworder rational magnetic surfaces in tokamaks are greatlymodified by the perturbation to the plasma equilibrium.

The plasma is far more sensitive to particular components(Figs. 5 and 6) or distributions (Fig. 7) of the externalnormal field on the plasma boundary, ~bx $ nb. The effectsof field errors can be mitigated by controlling the currentsin external coils to null the drive of these distributions, asverified in locking experiments.

Special thanks to Alan H. Glasser and Morrell S. Chancefor assistance. This work was supported by DOE ContractsNo. DE-AC02-76CH03073 (PPPL), No. DE-FC02-04ER54698 (GA), and No. DE-FG02-03ERS496 (CU).

[1] K. Ikeda, Nucl. Fusion 47, S1 (2007).[2] T. C. Hender et al., Nucl. Fusion 32, 2091 (1992).[3] R. J. La Haye, R. Fitzpatrick, T. C. Hender, A. W. Morris,

J. T. Scoville, and T. N. Todd, Phys. Fluids B 4, 2098(1992).

[4] R. J. Buttery et al., Nucl. Fusion 39, 1827 (1999).[5] J. T. Scoville and R. J. LaHaye, Nucl. Fusion 43, 250

(2003).[6] J. E. Menard et al., Nucl. Fusion 43, 330 (2003).[7] M. G. Bell et al., Nucl. Fusion 46, S565 (2006).[8] J.-K. Park, A. H. Boozer, and A. H. Glasser, Phys. Plasmas

14, 052110 (2007).[9] A. H. Boozer, Phys. Rev. Lett. 86, 5059 (2001).

[10] R. Fitzpatrick, Nucl. Fusion 33, 1049 (1993).[11] R. Fitzpatrick, Phys. Plasmas 5, 3325 (1998).[12] J. L. Luxon and L. G. Davis, Fusion Technol. 8, 441

(1985).[13] J. Spitzer, M. Ono, M. Peng, D. Bashore, T. Bigelow,

A. Brooks, J. Chrzanowski, H. M. Fan, P. Heitzenroeder,and T. Jarboe, Fusion Technol. 30, 1337 (1996).

[14] A. H. Glasser and M. S. Chance, Bull. Am. Phys. Soc. 42,1848 (1997).

[15] A. H. Boozer and C. Nuhrenberg, Phys. Plasmas 13,102501 (2006).

[16] S. Hamada, Nucl. Fusion 2, 23 (1962).

FIG. 6 (color). (a) The poloidal harmonic coupling spectrumfor j!21j and (b) the Fourier components of the external errorfield on the plasma boundary in NSTX for machine intrinsic (M),the best (M" EFC), and the worst (M# EFC) fields. The dottedcircle in (a) shows the most important harmonics of the externalfield.

FIG. 7 (color). The distributions of the external field on theplasma boundary maximizing the total resonant fields on rationalsurfaces, for (a) DIII-D and (b) NSTX. The three-dimensionaldistribution can be constructed by ~bx $ nb!A%!&cos%"&"B%!&'sin%"& relative to the plasma boundary (black line).

PRL 99, 195003 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending9 NOVEMBER 2007

195003-4

Figure 10: Distributions of external 3-D field compo-nents from IPEC that maximizec the total resonantfields on rational surfaces in: a) DIII-D and b) NSTX.


FE Plasma Effects II: RFA Modeling Developments Are Needed

• RFA and its effects can be estimated using an ideal MHD model:

radial component δBρ ≡ δ ~B ·~∇ρ = [~∇×(~ξ× ~B0)]·~∇ρ = ( ~B0 ·~∇)(~ξ ·~∇ρ) must vanish at

rational (resonant) surfaces for finite ~ξ in the plasma because ~B0·~∇ ∼ i(m− nq)/R0q;

in ideal MHD when an external 3-D m/n resonant perturbation is applied to a rapidlyrotating (i.e., ωτs 1) plasma, a delta-function “shielding current” δJ‖m/n is induced;

an ideal perturbed equilibrium code (IPEC12) has been developed to implement thisprocedure for all rational surfaces using the linear ideal MHD stability DCON code;

the shielding current δJ‖m/n and the dissipatively relaxed δBplasmaρ,m/n it could induce are usedc

to estimate the resonant toroidal torque using NTV-enhanced two-fluid layer physics;

this IPEC procedure explainedc field error compensation trends in DIII-D and NSTX;

however, subsequent more precise IPEC evaluations have been less conclusive.12b

• Dissipative, non-ideal singular layer effects are critical for determining rotat-

ing plasma response and achieving stable plasmas above the no-wall limit:

δJ‖m/n, δBplasmaρ,m/n and resonant torque Tζ can be evaluated self-consistently in MARS-F;13

nonlinear 3-D initial value codes (M3D-C1, NIMROD or reduced MHD codes BOUT++,JOREK) can also calculate them and explore dynamics of FE-induced mode locking;

resistive MHD or two-fluid layer models are now used, which should add neoMHD inertia.

12a) J.-K. Park et al., Phys. Plasmas 14, 052110 (2007), b) Phys. Plasmas 16, 056115 (2009), c) paper EXS/P5-12, 2010 Daejeon IAEA FEC.13a) Y.Q. Liu et al., Phys. Plasmas 7, 3681 (2000), Y.Q. Liu et al., THS/P5-10, 2010 Daejeon IAEA FEC; b) M.S. Chu et al., THS/P5-04.


MHD Mode Effects: NTMs And RWMs Interact With 3-D Fields

• There are direct effects of the 3-D fields produced by NTMs and RWMs:

1) RWM-induced δBn(ψ,m) perturbations cause non-resonant low n NTV effects; and

2) resonant classical and neoclassical tearing modes bifurcate the magnetic topology andform magnetic islands within the plasma that complicate and modify NTV effects.3,5

• Ideal MHD-type RWMs are stabilized if Ωt is large enough for the resistive

wall to represent a conducting wall to the rotating plasma:

if Ωt→0 magnetic field perturbations penetrate the resistive wall and RWMs can becomeunstable above the no-wall limit when the toroidal rotation |Ωt| is too small;

recently, low critical Ωt has been explained by14 stabilizing kinetic effects15 of fast and thermalions whose toroidal precessional drifts are resonant with the mode rotation frequency ω;

and even stable RWMs increase RFA of n=1 δ ~B in the plasma as βN increases (see Fig. 8),

which increases NTV damping, sensitivity to low n field errors and excitation of NTMs.

• Tearing modes can be nonlinearly excited by low m/n (typically 3/2 and 2/1)

3-D magnetic perturbations or they can appear “spontaneously.”

Critical issue is: what is threshold βN for given combinations of δBplasmaρm/n and Ωt?

Recent experiments indicate low Ωt via NTV affects ∆′, βN limits and 3-D sensitivity.16

14a) J.W. Berkery et al., Phys. Pl. 17, 082504 (2010); b) H. Reimerdes et al., EXS/5-4, 2010 Daejeon IAEA FEC; c) S.A. Sabbagh et al., EXS/5-5.15B. Hu, R. Betti, Phys. Rev. Lett. 93, 1005002 (2004).16R.J. Buttery et al., EXS/P5-03, 2010 Daejeon IAEA FEC.


RMPs: Resonant Magnetic Perturbations Have Many Effects

• Use of resonant magnetic perturbations (RMPs)17 to control ELMs initially

was based on edge magnetic stochasticity18 to reduce pedestal region gradients:

magnetic field stochasticity is caused by island overlap — Chirikov criterion;

some key RMP effects are explained by resonances — q95 sensitivity, divertor flux patterns;

but some are not — electron heat transport increased at pedestal top, density “pump-out.”

• Many possible RMP effects are currently being explored:

most importantly, “screening” of RMP fields by Ωt reduces stochastic region width;19

ne pump-out via ~E× ~B cells,19a large ~ξ ·~∇ρ19c,d at X-point, q95 resonances19i or µturb?19j

homoclinic tangles (lobes) of magnetic flux in the pedestal, SOL X point regions;20

magnetic-stochasticity-induced radial electron current that causes Eρ to increase;21a and

kinetic simulation of RMP effects on pedestal21b — screening of RMPs, reduced pedestalEρ key for density pump-out, only untrapped particles contribute to RR transport.

recently a “magnetic flutter” transport model has been developed — next few viewgraphs.

• BOTTOM LINE: RMPs have many effects and are interesting tools for mod-

ifying edge plasma transport (ne, Te, Ti, Ωt) and associated edge stability.

17a) T.E. Evans et al., Nat. Phys. 2, 419 (2006); b) Nuc. Fus. 48, 024002; c) M.E. Fenstermacher et al., Phys. Pl. 15, 056122 (2008),18T.E. Evans et al., J. Nucl. Mat. 145-147, 812 (1987); A. Grosman, PPCF 41, A185 (1999); Ph. Ghendrih et al., Nucl. Fus. 42, 1221 (2002).19a) V.A. Izzo and I. Joseph, Nucl. Fus. 48, 115004 (2008); b) M.S. Chu et al., THS/P5-04, 2010 Daejeon IEA FEC; c) Y.Q. Liu et al., THS/P5-10; d)

A. Kirk et al., EXD/8-2; e) H.R. Strauss et al., NF 49, 055025 (2009); f) M. Becoulet et al., 2010 Dublin EPS mtg.; g) L. Sugiyama et al., THS/P3-04;h) Q. Yu and S. Gunter, THS/P3-06; i) Y. Liang et al., EXS/P3-04; j) Z. Yan et al., EXC/P3-05.

20a) O.Schmitz et al., EXD/P3-30, 2010 Deajeon IAEA FEC; b) Phys. Rev. Lett. 103, 165005 (2009); c) Nucl. Fusion 48, 024009 (2008).21a) V. Rozhansky et al., THC/P3-06, 2010 Deajeon IAEA FEC; b) C.S. Chang et al., THC/P4-04, 2010 Daejeon IAEA FEC.


RMP Effects Are Large In DIII-D Discharge 12644022

• The profile changes are large because the I-coil current of 5.2 kAin 12644022 is unusually high — way above the usual threshold.

22P.T. Raum, S.P. Smith, J.D. Callen, N.M. Ferraro and O. Meneghini, “Modeling of Magnetic Flutter-Induced Transport In DIII-D,” GA-A27559,July 2013. S.P. Smith, invited talk NI2.05 on “Magnetic Flutter Plasma Transport Induced by 3D Fields in DIII-D,” APS-DPP mtg., Nov. 11-15, 2013.

0.0

0.5

1.0

1.5

2.0T e [ keV ]

0

2

4

6

8

10 [ m2 s]

0.80 0.85 0.90 0.95 1.001.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0ne [ 1019 m3 ]

0.80 0.85 0.90 0.95 1.000

5

10

15

20

25

30

35p [kPa]

with RMPno RMP

/

/χe

ρN ρN

coretop

edgepedestal

Figure 11: Profiles for 126440 (5.2 kA), 126443 (no RMP). From Fig. 2 in Ref. 22.


Flutter Plasma Transport Model Has Been Developed

• Collisional electron heat conduction along ~B = ~B0+δ ~B producesBraginskii parallel electron heat flux ~qe‖≡− (neχe‖/B

2) ~B ~B ·~∇Te.

• In collisionless limit, χe‖ → ∞ causes ~B ·~∇Te = 0 for |x| δ‖,which forces usual collisional ~qe‖ to vanish off rational surfaces.

• However, kinetic-based irreversible e collisions plus RMP fluttercause23,24 radial electron thermal diffusivity χδBe ∝ χeff

e‖ (δBρ/B0)2.

• Cylindrical23 and toroidal24 models of χeffe‖(ρ) have been developed.

• The most relevant kinetic-based low collisionality toroidal model:24

uses a Lorentz pitch-angle scattering collision operator,

accounts for parallel flows only being carried by untrapped (ut) particles,

resolves untrapped electron collisional boundary layer in velocity space, and

includes near-separatrix toroidal geometry and finite aspect ratio effects.

23J.D. Callen, A.J. Cole, C.C. Hegna, S. Mordijck and R.A. Moyer, “Resonant magnetic perturbation effects on pedestal structure and ELMs,” Nucl.Fusion 52, 114005 (2012).

24J.D. Callen, A.J. Cole, and C.C. Hegna, “Resonant-magnetic-perturbation-induced plasma transport in H-mode pedestals,” Phys. Plasmas 19, 112505(2012).


RMP-Flutter Induces Electron Transport Fluxes

• RMP-flutter-induced non-ambipolar transport fluxes of electron

density Γfluttet ≡〈~Γ

flutt

et · ~∇ρ〉 and heat Υfluttet ≡〈~q

fluttet · ~∇ρ〉 are24

[Γfluttet

Υfluttet /Te

]= −ne

[Det DT

χn χet

]·[d ln pe/dρ

d ln Te/dρ

],

d ln pe

dρ=d ln pe

dρ+eEρ

Te,

in which the total flutter-induced diffusivities are summed over all the m,n components:[Det DT

χn χet

]=∑mn

[Dm/net D

m/nT

χm/nn χm/net

]≡v2Te

νe

1

2

∑mn

(〈δBpl

ρm/n〉Bt0

)2 [K00 K01

K10 K11

].

The kinetically-derived Pade-approximate Kij matrix of coefficients are defined by24[K00 K01

K10 K11

]≡ cK

[G00 G01

G10 G11

], with coefficient cK ≡

Bt0/Bmax

〈v‖|λ=1/v〉13

24π,

in which the matrix Gij(x) of dimensionless, spatially-dependent geometric coefficients are[G00 G01

G10 G11

]≡

4

13 |X|3/2

(|X|3/2

c‖t

∫ 1/|X|1/2

0

dy y3e−y+

∫ ∞ymin

dy e−y

)[1 y − 5

2

y − 52

(y − 52)2

], c‖t =

(3/16)(B2t0/B

2max)

fc 〈v‖|λ=1/v〉,

ymin ≡ max1/|X|1/2, 1/X1/2crit and the normalized radial distance from m/n rational surface is

X ≡x

δ‖t=q(ρ)−m/n

q′ δ‖t'ρ− ρm/nδ‖t

in which δ‖t ≡ ct

LS

kθ λe, with ct ≡ 3

√π |〈v‖|λ=1/v〉|

Bmax

Bt0

.


Role of δBplρmn Is Critical Determinant In Flutter Model

•Main scaling of flutter transport between rational surfaces is

minχflutte eff ' νeλ

1/2e (R0q)

3/2

(δBpl

ρmn

B0

)2

∝(neZeff

Te

)1/2

q3/2

(δBpl

ρmn

B0

)2

.

• Kink-type responses off rational surfaces are dominant in δBplρmn.

•M3D-C1 currently computes linear δBplρmn(ρ) spectra, and then

they are extrapolated to desired I-coil current which

produces Rutherford regime island widths larger than layer widths (∼ mm),

but maxima are limited so kink-type ξρ don’t cause Te contours to overlap.

• Really need nonlinear M3D-C1 (or NIMROD) simulations,

but those are computationally very costly ( >∼ few months).

• Alternatively, explore RMP effects on lower performance H-modes:

via reduced PNBI which yields lower βpedp , further below P-B boundary?


Plasma Response Reduces RMP-Fields At Rational Surfaces

• RMP-induced m/nfields from M3D-C1

are reduced from vac-

uum values on m/n

rational surfaces,

by flow-screening fac-

tor fscr ≡ δBpl/δBvac,

but grow ∼ linearly

away from them.

• Parameters of thehighlighted 11/3RMP field are

fscr ∼ 0.1,

LδB ∼ 0.02 a

∼ 1.6 cm.

ρ

126006@3600 ms

vacuum

Figure 12: Flow-screened RMP-induced 〈Bplρm/n〉. Cour-

tesy of N.M. Ferraro, O. Meneghini & S.P. Smith 2012.


Plasma Response To RMPs Produces Flutter Between

Radially Isolated Island Chains, Edge Direct Losses

magnetic utter

Figure 13: Poincare plots of field lines with vacuum RMP fields (left) and with linear

M3D-C1 flow-screened linear RMP plasma response fields 〈δBplρm/n〉 (right).

Figures courtesy of D. Orlov, R.A. Moyer and N.M. Ferraro 2012.


Flutter Model χe and Te Profiles Using M3D-C1 RMP

Fields Matches 5.2 kA 126440 Data Reasonably Well22

• “Diffusivity hill” at pedestal top reduces |~∇Te|, limits pedestalexpansion =⇒ ?stabilizes P-B instabilities, suppresses ELMs?

• Island model predicts too much transport and too flat Te profile.

0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.000

2

4

6

8

10

12

13/312/311/310/39/3

Experimental N o R M P

E f f e c t i ve

w / I s l a n d s

χ

ρ

e

χe

χeRMP

χeRMP

N

diffusivity hill

Figure 14: Electron thermal diffusivity

profiles for flutter and island models

in DIII-D RMP discharge 126440.

0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.000

500

1000

1500

2000

13/312/311/310/39/3

T e B C

ρ

T (eV)e

No RMP

With RMP

N

Figure 15: Corresponding Te profiles for

flutter and island models in the edge

of DIII-D RMP discharge 126440.


RMPs Increase Pedestal Top Te,ne Gradient Lengths23

• Te, ne gradient length ratios with RMPs to wo (sym) ∝ χe, D:

[LTe]RMP

[LTe]sym

'χRMPe + χsym

e

χsyme

∝ I2coil ; plus similar formula for Ln, DRMP.

• ApparentlyRMPs increase

χe by up to ×6,

D by up to ×2.

• Changes peak atpedestal top:

0.93 <∼ ΨN<∼ 0.97.

• In edge RMPsaffect ne most.

0.9 0.92 0.94 0.96 0.98 1 0 1

2

3

4

5

6

7

8

ΨN

L L

9/3 10/3 11/3 12/3 13/3

/ 0 kA 126443 5.2 kA 126440

nTe

LRMP

symL___

smalleffects

densitypumpout

pedestal top edge~ 1 +

χe__χe

_RMP

sym_

Figure 16: Ratio of Te and ne gradient scale lengths (∼diffusivities) with RMPs to without (sym) vs radius23

for case with strong RMP suppression of ELMs.


RMPs Cause Flutter Transport At Pedestal Top

• χflutte eff is dominant RMP-induced flutter transport effect:

it matches power balance (ONETWO) χRMPe reasonably well for 2 cases;22

islands cause too large χRMPe and Te profile flattening at the pedestal top;

need more cases, with low βpede where linear M3D-C1 might be applicable?

• Next level RMP-induced flutter predictions25 need to be tested:

increases in electric field Eρ in edge region and at the pedestal top,

increase in density diffusivity (∼∆Eρ?) at pedestal top, which may be criticalfor ELM suppression because density growth there precipitates ELM;26

are q95 windows for ELM suppression where Eρ, D increase at pedestal top?

• Two new effects occur “at” separatrix (cause density pump-out?):

X-point flutter transport caused by 3-D homoclinic tangles/lobes, plus

electron heat loss along small fraction of “open” field lines to divertor plates.

25J.D. Callen, C.C. Hegna and A.J. Cole, “Magnetic-flutter-induced pedestal plasma transport,” Nucl. Fusion 53, 113015 (2013).26R.J. Groebner, T.H. Osborne, A.W. Leonard and M.E. Fenstermacher, “Temporal evolution of H-mode pedestal in DIII-D,” NF 49 045013 (2009).


Limiting Transport In Edge Is Due To KBMs Or Paleo

• The temporal evolution27 of the edge (III) plasma gradients uptoward the peeling-ballooning instability boundary

at low Ip (high βpedp ) follow the KBM prediction, while

at high Ip (low βpedp ) they follow paleoclassical prediction better.

• Analysis of the DIII-D pedestal database28 (158 conditions) foundthat the paleoclassical-based pedestal structure transport model29

predicts edge Te gradient well at low βpede but is too large at high βped

p , and

systematically predicts too large edge ne gradient, but only by a factor of 2.

• Do KBMs limit edge pressure gradient at high βpedp but paleo-

classical transport determines edge Te, ne gradients at low βpedp ?

• Both models predict edge Te gradient increases when ne decreases.

27See Figs. 6 and 14 (c), (d) in R.J. Groebner et al., “Improved understanding of physics processes in pedestal structure, leading to improved predictivecapability for ITER,” Nucl. Fusion 53, 2013 (2013).

28S.P. Smith, J.D. Callen, R.J. Groebner, T.H. Osborne, A.W. Leonard, D. Eldon, B.D. Bray and the DIII-D Team, “Comparisons of paleoclassicalbased pedestal model predictions of electron quantities to measured DIII-D H-mode profiles,” Nucl. Fusion 52, 114016 (2012).

29J.D. Callen, J.M. Canik and S.P. Smith, “Pedestal Structure Model,” Phys. Rev. Lett. 108, 245003 (2012).


Edge (III) Transport Diffusivities Are Small, Deff<∼ χe/10

• In edge (III) of 98889 pedestal30 χe ∼ 1 m2/s, but Deff < 0.1 m2/s.

• Thus, DRMP >∼ 0.1 m2/s in edge could change ne gradient “at”separatrix but comparable χe would have no effect on Te gradient.29

30J.D. Callen, R.J. Groebner, T.H. Osborne, J.M. Canik, L.W. Owen, A.Y. Pankin, T. Rafiq, T.D. Rognlien and W.M. Stacey, “Analysis of pedestalplasma transport,” Nucl. Fusion 50, 064004 (2010).

GTEDGEGTEDGE

GTEDGEGTEDGE

ASTRAASTRA

UEDGEUEDGE UEDGEUEDGE

SOLPSSOLPS

SOLPSSOLPSONETWOONETWO

ONETWOONETWO

χ χe i

II IIII IIIIII II IIII IIIIII

a) b) ionelectron

ρ ρN N

Figure 17: Electron thermal diffusivity

profiles in DIII-D pedestal 9888930 ob-

tained from various interpretive codes.

0.85 0.90 0.95 1.000.01

0.10

1.00

ONETWO

SOLPS

UEDGE

GTEDGE

D

ρN

II IIII IIIIII

Figure 18: Corresponding effective density

diffusivity in DIII-D pedestal 9888930

estimated by various codes.


RMPs Increase Density Transport “At” Separatrix

• Preceding graphs show that in the edge region near separatrix

while electron density ne on separatrix is unchanged by RMPs (Fig. 11), in

edge (III) RMPs decrease |~∇ne| (Fig. 11) via increase in Deff there (Fig. 16).

• Analysis will focus on RMP effects on electron transport because

edge radial electric field Eρ increases when RMPs are applied, and

open ~B field line and lobe-induced flutter transport is dominated by electrons.

• “At” separatrix RMPs cause additional transport near X point:

radial lobes produce a new type of magnetic field flutter transport, and

parallel electron heat flux to divertor plates occurs on “open” ~B field lines.

• To develop theory for this additional RMP-induced transport,suitable magnetic flux coordinates are needed in X point region.


RMPs Cause Homoclinic Tangles, Lobes “At” X Point31

Figure 19: In vicinity of the X point of a 2-D axisymmetric divertor magnetic field,

3-D RMPs cause splitting of the magnetic topology into lobe-type structures.31

Red colors indicate essentially closed field lines, blue colors indicate typical

connection lengths Lc to divertor target plates of the order of 60 m or more.

31See Fig. 2 (a) in O. Schmitz, T.E. Evans, M. Fenstermacher, H. Frerichs et al., “Aspects of three dimensional transport for ELM control experimentsin ITER-similar shape plasmas at low collisionality in DIII-D,” Plasma Phys. Control. Fusion 50, 124029 (2008).


Special Coordinates Are Needed In The X Point Region

• A magnetic field, flux coordinate system is needed because chargedparticles travel along ~B much faster than they drift across it.

• There are two types of magnetic-field-based coordinate systems:

ideal MHD in which ~B · ~∇ψ = 0 and ψ flux surfaces include 3-D perturba-

tions — this is not viable in regions with stochastic or rapidly reconnecting

(diffusing) ~B fields, or at an X point unless dP/dψ = 0 in these regions,

* lowest order 2-D (axisymmetric) ~B0 field plus “small” 3D perturbations δ ~B

— viable for tokamaks and meshes well with 1–1/2-D tokamak transport

codes, which explore plasma transport across 2-D axisymmetric flux surfaces.

• The 2-D ~B0 plus small 3-D perturbation δ ~B model (*) will be used:

first step is to develop 2-D coordinate system in the X point region, and

then add effects of a resonant 3-D perturbation δ ~B ≡ δ ~Bmn cos(mθ − nζ).


Cartesian Coordinates Are Useful Near X Point

• Cartesian coordinates will be used near X point (subscript X)tilted a small angle ϑ 1 relative to the vertical (Z) direction,

“radial” (' major radius R), x ≡ (R−RX)/ cosϑ ' R−RX,

“vertical” (' vertical distance), y ≡ (Z − ZX)/ cosϑ ' Z − ZX,

“toroidal” (axisymmetry direction), z ≡ RXζ

Figure 20: Illustration of x, y coordinates in the vicinity of an X point. Red Y =

constant line is a flux surface. Wiggly line shows effect of a resonant 3-D δ ~B.


Poloidal Flux Surfaces Near X Point Are Simple

• In axisymmetric tokamak the toroidal (subscript t) magnetic field~Bt near an X point (subscript X) where B0 ' BtX is (~∇z = ~ez)

~Bt ≡ I ~∇ζ ' BtX~∇z, I ≡ RBt ' constant ≡ RXBtX, ζ ≡ z/RX.

• An X point is produced by saddle point in poloidal magnetic flux:

ψpX(x, y) ≡x2 − y2

2L2X

aRXBpX, ~BpX ≡ ~∇ζ×~∇ψpX =a~∇(xy)

L2X

BpX.

• Some definitions and constants are useful here:

LX ≡[

aRXBpX

∂2ψpX/∂x2|x=0,y=0

]1/2

=[− aRXBpX

∂2ψpX/∂y2|x=0,y=0

]1/2

, X point scale length <∼ a,

a ≡ (ψt sep/πBtX)1/2, average minor radius of separatrix toroidal flux ψt sep,

BpX ≡ (a/q95RX)BtX, a characteristic strength of poloidal magnetic field.

• Defining poloidal flux surfaces with ψpX = −Y 2aRXBpX/2L2X,

the Y poloidal flux surfaces are given by the hyperbolic equation

Y 2 = y2 − x2 = constant =⇒ y(x)2 = Y 2 +x2, or x2(y) = y2−Y 2.


ψpX ∝ Y 2 Flux Surfaces Have Interesting Properties

• They are orthogonal to xy = const hyperbolas: ~∇Y 2·~∇(xy) = 0;

thus, Y and xy could be used as field line coordinates in the x− y plane.

• To asymptote to the flux surfaces defined by ρN at x ∼ LX <∼ awhere the Taylor series expansion breaks down, for small Y

near the 45 line x = y ' LX + Y 2/2LX the small Y surface corresponds to

δρN ≡ (a−ρ)/a ' δy/√

2a ' Y 2/(2√

2 aLX) >∼ Y 2/(23/2a2) near mid-plane,

which implies a flux expansion factor ofY

a δρN=

Y

a− ρ∼ 2√

2a

Y 1.

• Assuming a ∼ 0.8 m for a DIII-D LSN diverted plasma

a ρN ' 0.99 surface is predicted to have Y ' [2√

2 aLX δρN ]1/2 <∼ 0.13 m,

which corresponds to a δρN flux expansion factor of Y/(0.01× 0.8) ∼ 16,

and a distance expansion from mid-plane of about 0.5 cm to Y ' 13 cm.


Field Line Equations Are Simple, Yield Key Length

• Using ~B0 = ~BtX + ~BpX, the ~ex ≡ ~∇x, ~ey ≡ ~∇y, ~ez ≡ ~∇zprojections of the field line equation d~x/d` = ~B0/B0 yield

dx

d`=

y

L‖,

dy

d`=

x

L‖,

dz

d`=BtX

B0

' 1, in which L‖ ≡L2XBtX

aBpX

<∼ q95RX.

• Field line length from y ' LX Y to the X point is obtained byintegrating (dy/d`)−1 along Y flux surface where x(y) ≡

√y2 − Y 2:

d`

dy=L‖

x=⇒ `(y)−`(Y ) = L‖

∫ yY

dy′√y′2−Y 2

= L‖ ln

(y+√y2−Y 2

Y

), which yields

∆` ' κL‖ ln (2LX/Y ) ∼ κ q95RX ln (21/4/√δρN) q95RX, mid-plane to X.

• For DIII-D parameters of κ ' 1.7, RX ∼ 1.6 m and q95 ' 3.5,

length of δρN ' 0.01 field line from mid-plane to X point is ∆` ∼ 24 m,

which is comparable to the collision length λe at this radius.

But for ρN > 0.99, λe will be less than ∆` and ‖ heat flow will be collisional.


Straight Field Line Coordinates Are Most Useful

• The 2-D ~B field can be written in terms of orthogonal coordinates:

~B0 = ~Bt+ ~Bp with ~Bt ' BtX~∇z, ~BpX = (1/RX)~∇z×~∇ψpX = (aBpX/L

2X)~∇(xy).

• However, poloidal and toroidal angle coordinates θ and ζ forwhich the ~B0 field has straight lines on the flux surfaces are betterfor theoretical analyses — especially for identifying resonances.

• The straight field line poloidal angle θ is determined from BtX ≡I ~∇ζ ' BtX

~∇z = ~∇ψpX×q95~∇θ, which for θ ≡ θ(y) yields

dθdy

= BtXq95 ∂ψpX/∂x

=L‖

x q95RX=⇒ θ(y) =

L‖q95RX

∫ yY

dy′√y′2−Y 2

=L‖

q95RXln

(y+√y2−Y 2

Y

).

• Thus, the X point magnetic field has the following properties:

~B0 ≡ ~∇ψpX×~∇(q95θ − ζ) = BtX [ ~∇z + (y/L‖)~∇x+ (x/L‖)~∇y ],

dY 2/d` = 0 =⇒ on Y surfaces ~B0 field lines have angle dζ/dθ = q95.


Resonant 3-D Perturbations Are Most Important

• Next, add 3-D perturbation δ ~B = ~∇ζ×~∇[δψmn(Y ) sin(mθ−nζ)]:

~B = ~B0+δ ~B = BtX

[~∇z +

y

L‖~∇x+

x

L‖~∇y]+~∇ζ×~∇[δψmn(Y ) cos(mθ−nζ)].

• “Radial” motion of field lines from constant Y 2 flux surfaces is

dY 2

d`=~B ·~∇Y 2

BtX

= 0 +a δBx(Y )

BtX

cos(mθ − nζ), δBx =2m

aRX

δψmn.

• Since typical δBx/BtX<∼ 10−3, large radial excursions only occur

for resonant responses where |mθ − nζ| is “small” and

δY 2 =a δBx(Y )

BtX

∫ `

dz cos(mθ − nζ) 'a δBx(Y )

BtX

sin[mθ − nz/RX]

(m− nq95)/q95RX

.

• These oscillatory “radial” excursions can apparently produce lobessuch as those seen in Fig. 19 on p 36 with δY 2 ∼ Y 2 <∼ (7 cm)2 when

δBx

BtX

>∼Y 2

a q95RX

|m− nq95|, e.g., forY 2

a q95RX

< 10−3 and |m− nq95| < 1?


The Lobes Could Induce Flutter-Type Transport

• The key ingredient in flutter transport is that the collision lengthλe be longer than periodicity length L of the δ ~B resonant field.

• From the analysis on the preceding page the periodicity length isL ≡ q95RX/|m−nq95| >∼ 5 m for |m−nq95| <∼ 1, which is shorter

than the separatrix collision length λe >∼ 10 m (longer for ρ < a).

• The flutter transport in this region should be analogous to thatobtained midway between rational surfaces and might be roughly

DRMPX , χRMP

eX ∼ fXνeL2(δBx/BtX)2, in which fX is fraction of flux surface.

• Since in the X point region νe ∼ 106/s, δBx/BtX>∼ 3×10−4 and

perhaps fX ∼ 1/25, it seems DRMPX

>∼ 0.1 m2/s may be possible.


In Summary, RMP Effects Depend On Pedestal Region

• At pedestal top, flow screening can produce separated islands:

RMP-induced flutter transport produces diffusivity hill and dominates?

• In steep gradient edge region close to separatrix:

KBMs limit ~∇P at high βpedp but paleo may limit gradients at low βped

p ;

in absence of RMPs Deff<∼ 0.1χe in edge region close to separatrix; and

small DRMP >∼ 0.1 m2/s decreases |~∇ne|, but similar χRMPe has little effect.

• Quantification of plasma transport “at” separatrix requires useof special flux surface coordinates in the X point region:

Y 2 = y2 − x2 = constant flux surfaces, flux expansion factor ∼ 2√

2 a/Y ,

mid-plane to X field line length ∆` ∼ κL‖ ln(2LX/Y ) ∼ 24 m for ρN ' 0.99.

• “At” separatrix RMPs can cause additional plasma transport:

parallel electron heat flux on “open” field lines, limited by χe⊥, ∆` > λe;

lobes may cause flutter transport DRMPX ∼ fXνeL2(δBx/BtX)2 >∼ 0.1 m2/s.


Transport Effects: 3-D Fields Directly Affect Ωt, Indirectly ne, T

• 3-D fields affect Ωt via

resonant δBρm/n/B0>∼ 10−4,

NTV for δBn/B0>∼ 10−3, and

TF ripple for δBN/B0>∼ 10−2.

• Ωt responds to NTV and

resonant modes differently:

Fig. 21a: NTV damping is global,

Fig. 21b: resonant locally, spreads.

• Theoretically, 3-D induced

Te, Ti, net ne transport are

%2∗ (Bt/Bp)2 smaller.

• Experimentally, 3-D effects

are usually a factor >∼ 3

smaller for Te, Ti transport,

when density held constant.

• But RMPs, ripple cause

density “pump-out”—how?

is constant on a flux surface, each of which is considereda thin, inertial toroidal shell with moment of inertia, I.The plasma mass density, !, is measured at the plasmamidplane. The total integrated torque over the flux sur-face due to NTV, TNTV!TNTV"p#TNTV"$1="%, whereTNTV"p;$1="% !R0Vahet& ~r&!$ip;$1="%, and Va is the vol-ume of the inertial shell. Experimentally, TNTV is isolatedby allowing the plasma rotation profile to reach a quasi-steady state, so that TNBIj0 ! "$TJxB # T#? # TNTV%j0 .After this time, the nonaxisymmetric field is applied. Thedifferences between TJxBj0, T#?j0, TNBIj0, and their valuesat later times of interest are significantly smaller thanTNTV$t%. The experiment is conducted during periodswhen significant tearing instabilities are absent, so TJxBis itself small. The plasma rotation profile evolution due totearing modes is distinct from NTV-induced rotationdamping so their lack of influence on rotation drag canbe verified [15]. This is illustrated in Fig. 2, where therotation damping observed in the present experiments[Fig. 2(a)] is contrasted with rotation damping due toTJxB [Fig. 2(b)]. The damping due to NTV is relativelyrapid, global, and the rotation profile decays in a self-similar fashion. In contrast, the damping due to TJxB isinitially localized near the island, and diffusive from thisradius, leading to a local flattening of "$ and a distinctivemomentum diffusion across the rational surface fromsmaller to larger R as expected by theory [7,9]. Also,T#? ' TNTV can be shown by first measuring the magni-tude of #? in similar plasmas that exhibit radially local-ized tearing modes. Applying the measured value of #?using the plasma illustrated in Fig. 2(b) at t ! 0:395 s tothe NTV momentum dissipation experiment in Fig. 2(a),we find that T#? < 0:15 TNTV at the peak value of TNTV

between t ! 0:355–0:375 s. Finally, (TNBI " TNBIj0% 'TNTV is independently verified using the TRANSP code.The plasma shown in Fig. 3 has $TNBI " TNBIj0%=TNTV !0:02 at the peak value of TNTV. With these assumptions, the

equation of motion during times of interest becomesd$I"$%=dt ! TNTV, which is evaluated on reconstructedequilibrium flux surfaces.

The plasma toroidal-momentum dissipation is first eval-uated at values of %N below the n ! 1 ideal no-wall betalimit, %no-wall

N , the time evolution of which is evaluated bythe DCON [23] ideal MHD stability code. When %N <%no-wall

N , beta effects such as RFA are insignificant, so thenonaxisymmetric field that penetrates the plasma can bemodeled simply as the vacuum field modified by shieldingcaused by plasma rotation. Comparison of the measureddissipation of plasma angular momentum caused by theexternally applied nonaxisymmetric fields to the theoreti-cal NTV torque profile is shown in Fig. 3 for an n ! 3applied field configuration. The measured value ofd$I"$%=dt includes error bars that take into account theuncertainty in "$ and !.

As %N approaches and exceeds %no-wallN , RFA is mea-

sured as the applied nonaxisymmetric field is amplified bythe weakly stabilized RWM and needs to be included in thecalculation of TNTV. The RFA magnitude is defined as the

(deg)0 60 120 180 240 300 360

10

14

12

8

6

4

2

0

|B| (

G)

(a)

1.1 1.51.2 1.3 1.4

R(m)

(b)

(c)

116939t = 0.37s

R = 1.5m1.4m1.3m

R = 1.2,1.1,1.0m

Bns

(G)

3

-2

2

10

-1

0

20

15

10

5

n = 5

n = 1

n = 11

n = 5

n = 1

n = 11n = 3

n2(B

nc2 +

Bns

2)(

G2 )

FIG. 1 (color online). (a) jBj vs toroidal angle and majorradius for nonaxisymmetric field coil currents yielding a radialmagnetic field with dominant n ! 1 component, (b) major radialprofile of toroidal mode number spectrum (sine component) ofthis field, and (c) flux surface-averaged field component sum-mation n2$Bnc

2 # Bns2% relevant to the NTV calculation.

0.365s0.375s0.385s0.395s

0.405s

0.425s

0.375s0.385s0.395s

0.405s

0.425s 116939

(a)

(kH

z)

40

30

20

10

0

0

-40

-30

-20

-10

(kH

z)

0.335s0.325s0.345s0.355s0.365s0.375s0.385s0.395s

0.335s0.345s

0.355s0.365s0.375s0.385s0.395s

115600

(b)

0.9 1.0 1.1 1.2 1.3 1.4 1.50.9 1.0 1.1 1.2 1.3 1.4 1.50.8

R (m)R (m)

FIG. 2 (color online). Toroidal plasma rotation profile vs majorradius, and the difference between initial profile and subsequentprofiles for rotation damping (a) during application of nonaxis-ymmetric field, and (b) during excitation of rotating tearinginstability.

TN

TV

(N m

)

3

4

2

1

00.9 1.1 1.3 1.5

R (m)

measured

theoryt = 0.360s116931

axis

FIG. 3 (color online). Comparison of measured d$I"$%=dtprofile to theoretical integrated NTV torque for an n ! 3 appliedfield configuration.

PRL 96, 225002 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending9 JUNE 2006

225002-3

Figure 21: NSTX toroidal plasma rotation profile versus ma-jor radius, and difference between initial and subsequentprofiles for rotation damping: a) during application of 3-Dfield, and (b) during excitation of rotating tearing instabil-ity. Fig. 2 from Ref. [7] on p 11.


ITER I: Extrapolation Of 3-D Effects To ITER Is Still Embryonic

• Test blanket module (TBM) magnetic field mock-up in DIII-D:

experimental results ∼ consistent with theory — reduced Ωt, slight density pump-out,

but need NTV, FE & RFA theory for δ-function 3-D field (a field mapping calculation).

• Major 3-D field effects in ITER seem to be:

ripple-induced NTV torque likely causes low Ωt ∼ Ω∗ < 0 =⇒ greater 3-D, β sensitivity;

density pump-out, RFA effects on n=1 FEs; locked modes more likely at low Ωt & high β.

• Some needs for improving predictive capability for ITER are

theory: analytic theories of combined 3-D effects (RFA, NTMs, RWMs), mapping calc.,

modeling: Ωt screening of RMPs, RFA with best layer physics, NTV ripple effects,

experiments: screening of RMPs with reduced Ωt in edge, cause of 3-D induceddensity pump-out, low Ωt and high β effects on NTMs and RWMs.


ITER II: Some Common 3-D Physics Is Involved In TBM Tests

• Recent experiments were performed32 on DIII-D to explore possible effects of

field errors introduced by ITER test blanket modules (TBMs, δ ∼ 1.2%):

TBM mock-up was toroidally localized (∆ζ ∼ 2π/24) with δ ≡ δBN/B0 ∼ 1–3%;

main effect was braking of Ωt (∝ Ωt) with increasing δ, causing ∆Ωt/Ωt up to − 50%;

changes in density (slight pump-out), confinement and β were factor >∼ 3 smaller; and

more, but easily compensated, locking sensitivity to n=1 fields at higher β and lower Ωt.

• Major issue for previous theory is that TBM is toroidally localized, repre-

sented by a very large δBn Fourier spectrum (±n up to 2×24 coils):

NTV, FE & RFA theory need to be developed for delta-function-type field ripple;

nonetheless, 3-D effects of TBM test was estimated by summing over all Fourier δBn.

• TBM test results were consistent with32 3-D effects theory, modeling:

global NTV Ωt braking semi-quantitatively predicted32b by IPEC12 calculations— very small n=1 edge FE from TBM is amplified in core by edge coupling to n=1 kink;

I-coil compensation of TBM-induced n=1 FE semi-quantitatively matched32b by IPEC;12

TBM mainly affects Ωt, with lesser effects on n, T transport (slight density pump-out)and has small effects on fast ion confinement.22c

32a) M.J. Schaffer et al., ITR/1-3, 2010 Dajeon IAEA FEC; b) J.K. Park et al., EXS/P5-12; c) G.J. Kramer et al., EXW/P7-10.


ITER III: What Are The Major 3-D Fields Issues For ITER?

• Plasma torques from edge direct ion losses and ripple-induced NTV (from

N = 18 coils + FSTs =⇒ δ <∼ 0.4 % at edge) is likely to cause braking of Ωt

toward Ω∗ < 0 — NBI and other torques are likely to be smaller. Lower,

diamagnetic-level plasma toroidal rotation Ωt could have some effects:

greater sensitivity to n=1 external 3-D field errors that can induce locked modes?

smaller radial electric field shear with less stabilization effects on microturbulence?

reduced βN thresholds for NTMs?

more reliance on kinetic ion effects to stabilize RWMs above no-wall limit?

non-resonant fields in ITER may be able to use NTV effects to control Ωt(ρ, t).

• Some additional important 3-D field effects issues are:

precise 3-D RMP field characteristics required for stabilization or mitigation of ELMs,

density pump-out caused by FEs, RMPs and ripple/TBMs that is not yet understood,

RFA effects on n=1 fields in plasmas including singular layer and kinetic effects,

determination of how small field errors must be to avoid locked modes as β is increased— and assessment of degree to which dynamic compensation coils might be needed.


ITER IV: What’s Needed For Predictive Capability For ITER?

• Theory of 3-D magnetic perturbation effects:

mapping calculation of NTV induced by TBM “delta-function” δB (i.e., not Fourier expanded),

analytic model of n=1 “global” resonant plasma response including layer physics,

theoretical models of 3-D field effects plus toroidal flow and flow shear on tearing modes,

and more development of combined effects of resonant 3-D fields, NTV and kinetic RWMs.

• Modeling of 3-D magnetic perturbation effects:

NTV ripple effects throughout plasma including self-consistent Ωt (radial electric field),

nonlinear modeling of Ωt screening effects on penetration of RMPs into edge plasmas,

more extensive modeling of plasma responses to n=1 global modes including layer physics— with linear eigenmode, reduced MHD and nonlinear initial value codes, and

complete modeling of Ωt evolution using comprehensive Lt transport equation.

• Experimental explorations of effects of 3-D magnetic perturbations:

more validation of NTV effects in core plasma, particularly with FEs, 3-D fields and ripple,

clarification of how 3-D fields from ripple, field errors and RMPs cause “density pump-out,”

elucidation of effects of 3-D fields and rotation on NTMs and RWMs at low Ωt (∼ Ω∗),

validation of methods for minimizing coupling of external 3-D fields to n=1 kink modes,

and studies of screening effects of various toroidal rotation magnitudes on RMPs in edge.


MODELING SUMMARY: Status, Issues And Research Topics

• Modeling of Tokamaks: There are many effects in plasma transport equations:

transients, collision- & microturbulence-induced transport, sources & sinks, small 3-D fields.

• 3-D Field Effects: Fundamental physics of the effects of 3-D magnetic field

perturbations on toroidal plasmas has “come of age” over the past decade:

transport-time-scale equation for Ωt (Eρ) evolution including 3-D effects is now available;

NTV theory is partially validated — torque magnitude, offset frequency, peak at ωE → 0;

toroidal field ripple reduction of Ωt is caused by edge direct ion losses plus NTV effects;

resonant n=1 field error effects can be corrected, mode locking criteria include RFA effects;

resonant field amplification (RFA) occurs via “least stable” n=1 kink MHD plasma responses;

interpreting RMP effects on H-mode pedestals is progressing — flutter at top, ?near separatrix.

• Status: Combination of 3-D field effects with extended MHD being developed:

NTMs and RWMs interact with low n external δ ~B — sensitivity increases at low Ωt, high β,

resonant magnetic perturbations (RMPs) — stochasticity limited by Ωt “flow screening,”

3-D plasma transport effects — directly on Ωt, but mostly indirectly on ne, T transport.

• Possible research topics in fluid and transport modeling:

develop and begin using a self-consistent theoretical framework for combining extendedMHD, CEKE, closures and plasma transport equations for tokamaks — see next viewgraph.


Major Themes Of These CEMRACS 2014 Lectures

• The fundamental 6-D (~x, ~v) plasma kinetic equation (PKE) is mainly a first

order hyperbolic (Vlasov) equation with a small parabolic (diffusive in ~v)

operator due to Coulomb collisions that is critical for temporal irreversibility

and entropy production in kinetic, extended MHD and transport equations.

• Plasma kinetics is fundamental, but for time scales longer than species collision

times (i.e., t > 1/νs) 3-D (~x) fluid equations (extended MHD and transport)

are feasible, appropriate, useful and needed. They are exact to extent that

the relevant Chapman-Enskog kinetic equation (CEKE) can be solved for the

kinetic distortion Fs which yields the needed collisional and closure moments.

• At present a GRAND UNIFIED TOKAMAK SIMULATION (GUTS)?

— for developing “predictive capability” for ITER — seems to require us to:

use small gyroradius expansion to order various tokamak physics effects, especially thosein the radial, parallel, and toroidal components of the species force balance equation,

use extended MHD to check macrostability, obtain ~B field structure including plasmaresponses to 3-D fields, reconnecting regions and stochastic fields near separatrix X points,

determine microturbulence by solving the CEKE in this “distorted” ~B field geometry andproduce the collision- and microturbulence-induced closures and radial transport fluxes,

solve resultant tokamak plasma transport equations simultaneously for ne, Ωt (Eρ), ps, and

then iterate these extended MHD, CEKE, closures and transport steps for self-consistency.


tokamak plasma transport modelinghomepages.cae.wisc.edu › ~callen ›...

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