edge intrinsic rotation: theory and experiment · overview a simple kinetic transport theory...

Theory

Experiment

Edge Intrinsic Rotation:Theory and Experiment

T. Stoltzfus-DueckPrinceton University/PPPL

PPPL R&R SeminarMay 29, 2015

Stoltzfus-Dueck Edge Intrinsic Rotation (1)

Theory

Experiment

OverviewA simple kinetic transport theory predicts strong linear dependenceof edge intrinsic toroidal rotation on R

X

.

= (RX

�Rmid)/a, with RX

the major-radial position of the X-point.I “Edge” means the radial range of a cm or so both inside and

outside the LCFS, where spatial variation is rapid.I An analytic calculation yields a simple formula vpred for the

toroidal rotation at the core-edge boundary.

A series of Ohmic L-mode shots on TCV, scanning RX

, showed:I Entire rotation profile shifts rather rigidly as R

X

changes.I Linear dependence of edge rotation on R

X

(X)I Rotation sign change for adequately outboard X-point (X)I Fits of rotation vs R

X

give reasonable values for theory’s twoinput parameters (X)


Theory

Experiment

Outline

I Theoretical modelI AssumptionsI Ingredients of analytical calculationI Cartoon of modelI Resulting predictions

I ExperimentI TCV featuresI Rotation profiles for different R

X

I Qualitative and quantitative comparison with modelI Consideration of other models


Theory

Experiment

Experimentally, tokamak plasmas rotate spontaneously,without external torque.

0

20

40

60

80

100

V Tor(k

m/s

)

0 50 100 150 200Change in Edge Grad T (keV/m)

H- modeI- mode

Ti (keV)

0.0 0.2 0.4 0.6 0.8 1.0

0

10

20

30

V!(km/s)

co-Ip

Rice et al PRL 2011, Fig. 5b deGrassie et al NF 2009, Fig. 7

I Co-current in the edge. I Spin-up at L–H transition.I v

f

/vti

⇠ O(0.1) at the core-edge bound. I Roughly proportional to W /Ip

.

I Edge rotation proportional to T or —T?Stoltzfus-Dueck Edge Intrinsic Rotation (4)

Theory

Experiment

Model

Rotation

Edge orderings relevant for intrinsic rotation

Influence of SOL=)nonlocal, steep gradients, strong turbulence, very anisotropic

Lengths: L?a

,

a

qR

⌧1, kk⇠ 1

qR

,

kkk?

. kkL?n 1

Rates: D

tur

L

2

?⇠ v

ti

|sep

qR

n w ⇠ v

ti

L?

Dtur

⇠ v2

Er

t

ac

⇠ v2

Er

/k?vEr ⇠ c f/Bdecreases in r near LCFS, on scale L

f

⇠ L?

�vk|turb

:⇣�vk|turb

v

ti

⌘2

⇠ kkk?

�T

e

T

i

ef

T

e

1

k?r

i

|sep

�⌧1

Wide passing-ion orbits: d

.

= qr

i

L

f

⇠O(1)

Absolute Fluctuation Levels

10

vo

lts

0 5 10 15

3

! (mm)

ELM-free

EDA

L-mode

ELM-free

EDA

L-mode

Floating Potential, RMS[Vf ](c)

LaBombard et al NF 2005, Fig. 8.


Theory

Experiment

Model

Rotation

A simple kinetic transport theory models edge intrinsicrotation.

∂

t

fi

+ vk∂

q

fi

�dv2

k (sinq)∂

x

fi

�∂

x

[D (x ,q)∂

x

fi

] = 0

Extremely simple kinetic transport model contains only:

0

2⇡

x

✓

vk>0:fi

=0

periodic

(=f

i

!f

i0

=)f

i

!0

x

✓

I Free flow along the magnetic fieldI Radially-directed curvature driftI Radial diffusion due to turbulence

I Diffusivity stronger outboard, decays in x

I Two-region geometryI Confined edge: periodic in q

I SOL: pure outflow to divertor legs

After some variable transforms, obtain steady-state equation∂

q

fi

= Deff

�vk�

∂

x

�e�x

∂

x

fi

�,

in which Deff

depends on the sign of vk.Stoltzfus-Dueck Edge Intrinsic Rotation (6)

Theory

Experiment

Model

Rotation

Asymmetric diffusivity caused by drift orbits’ interaction withballooning transport and X-point angle.

(a)

co

ctr

HFS L

FS

(b)

co

ctr

HFS L

FS


Theory

Experiment

Model

Rotation

Asymmetric diffusivity caused by drift orbits’ interaction withballooning transport and X-point angle.

(a)

co

ctr

HFS L

FS

(b)

co

ctr

HFS L

FS

Edge rotation may become counter-current for outboard X-point!


Theory

Experiment

Model

Rotation

Suprathermal ions drive a robust momentum flux.

∆"0.28, Θ0"$5Π ! 8

D" 0.033 "1&0.8 cos # Θ$ %

v&

!

!v&2!2

!v&1 2 3 4

0.02

0.04

0.06

0.08

Assume a Maxwellian at the boundary with the core,

fi0

�vk�= (2p)�1/2 exp(�v2

k /2),

then moments of the transport are just

�p.

=Z •

�•��vk�dvk, ⇧

.

=Z •

�•vk�

�vk�dvk, Qk

.

=12

Z •

�•v2

k ��vk�dvk.


Theory

Experiment

Model

Rotation

Vanishing momentum transport: pedestal-top intrinsic rotation.

∆

v int

Θ 0"#

3Π! 4 Θ0

"#5Π!

8

Θ0"#Π! 2

Θ0"#3Π! 8

Θ0 "#Π! 4

0.2 0.4 0.6 0.8 1.0

"0.5

0.5

1.0

0 =R •�•

�vint

+ vk��vk�dvk = v

int

�p +⇧

vpred =� ⇧�p vti ⇡ 0.104

�1

2

dc

� RX

�qT

i

(eV)L

f

(cm)B0

(T)km/s

I D = D0

(1+dc

cosq) I Co-current for typical parametersI 1/B

q

) 1/Ip

I Rotation magnitude O(vti

/10)I X-point angle dependence I L-H spin-up due to *T

i

, + Lf


Theory

Experiment

Description

Results

TCV is well-suited to investigateRX -dependent edge rotation.Extreme geometric flexibility:

I Vary RX

from inner to outer wallI Both LSN and USN

Diagnostic NBI for CXRS on C6+:I applies negligible torque (⇠1%tint)I LFS & HFS viewing chords

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

x(m)

y(m) HFS

LFS

Ip,BT<0

DNBI

0.6 0.8 1 1.2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

R [m]

Z [m

]

Parameter ranges for this experiment:X-point major radius (R

X

) 0.675–1.085m

Major radius (R0

) 0.88–0.89m

Minor radius (a) 0.22–0.23m

Edge safety factor (qeng) 3.6–4

Plasma current (Ip

) 150–155kA

Electron density (ne,avg) 1.4–2.2⇥1019m�3

Elongation (k) 1.35–1.45

Triangularity -0.3 –+0.4 Figures from A. BortolonStoltzfus-Dueck Edge Intrinsic Rotation (11)

Theory

Experiment

Description

Results

Discharges with RX from inner to outer wall, LSN and USN.

RX=−0.75 RX=−0.23 RX=+0.30 RX=+0.87

RX=−0.80 RX=−0.30 RX=+0.35 RX=+0.86


Theory

Experiment

Description

Results

Changing RX indeed shifts the boundary rotation, shiftingthe whole rotation profile with it.

0.2 0.4 0.6 0.8 1 1.2−60

−40

−20

0

20

40

RX=71cm

RX=108cm

ρ

v LFS(km/s)

0.2 0.4 0.6 0.8 1 1.2−60

−40

−20

0

20

40

RX=71cm

RX=108cm

ρ

v HFS(km/s)

Measured carbon rotation profiles for an inboard (RX

= 71cm) and outboard(R

X

= 108cm) X-point.


Theory

Experiment

Description

Results

Theory-Experiment agreement is surprisingly good.

Roughly linear dep of vexp on RX

.I Counter-current for large R

X

.Reasonable fitting parameters:

I dc

⇡ 1.1: outboard ballooningI L

f

⇡ 4.1cm ⇡ 1.5LTe

I Lf

⇡3.8cm from LP measI L

f

⇡ 1–2LTe

on other expts−1 −0.5 0 0.5 1

−15

−10

−5

0

5

10

15

20

25

RXv e

xp(km/s)

LSN

USN

USN⇠ 6km/s more counter-current than LSN.Some mid-range R

X

inaccessible due to machine constraints


Theory

Experiment

Description

Results

The basic trend holds for alternate radial positions.

−10

0

10

20km/s

v0.80 v0.85

−1 −0.5 0 0.5 1 |−1

−10

0

10

20

RX

km/s

v0.90

−0.5 0 0.5 1

RX

v0.95

LSN

USN


Theory

Experiment

Description

Results

Core rotation reversal seems to have little effect on edge rotation.

Spontaneous core rotation reversalwell-known on TCV (Bortolon et alPRL 2006)

Accidentally triggered reversal in shots48152–48153, due to larger I

p

−30

−20

−10

0

10

20

30

40

Norm

al

Reversal

vLFS(k

m/s)

0.2 0.4 0.6 0.8 1−30

−20

−10

0

10

20

30

Norm

al

Reversal

ρ

vHFS(k

m/s)

Theory: in the absence ofactual core torque,rotation peaking does notaffect edge momentumflux, thus intrinsicrotation is maintained. −1 −0.5 0 0.5 1 |−1

−10

0

10

20

RX

km/s

v0.85

−0.5 0 0.5 1

RX

v0.95


Theory

Experiment

Description

Results

Can transport-driven SOL flows drive rotation in theconfined plasma?Intrinsic rotation velocity is determined by vanishing momentum flux.Although transport-driven toroidally-asymmetric flows exist in the theoreticalcalculation, they do not drive rotation at the boundary with the core plasma.

ctr

co

rB (B ·I ·

ctr

co

rB (B ·I ·

v

r

HFS

LFS

!3 !2 !1 0 1 2

0.1

0.2

0.3

0.4

0.5

0.6

v

r

HFS

LFS

!3 !2 !1 1 2

!0.4

!0.3

!0.2

!0.1

0.1

0.2

0.3

vk∂

q

fi

�⇠⇠⇠⇠⇠⇠⇠XXXXXXXdv2

k (sinq)∂

x

fi

�∂

x

[D (x ,q)∂

x

fi

] = 0


Theory

Experiment

Description

Results

Favorable/unfavorable —B comparison can clarify physics.Reverses transport-driven flows but not orbit shifts and their flows.

ctr

co

rB (B ·I ·

ctr

co

rB (B ·I ·

co

ctr

co

ctr

LSN:

v

r

HFS

LFS

!3 !2 !1 0 1 2

0.1

0.2

0.3

0.4

0.5

0.6

v

r

HFS

LFS

!3 !2 !1 1 2

!0.4

!0.3

!0.2

!0.1

0.1

0.2

0.3

:USNStoltzfus-Dueck Edge Intrinsic Rotation (20)

Theory

Experiment

Description

Results

Rotation data consistent with dominant drive by orbit shifts.

−1 −0.5 0 0.5 1−15

−10

−5

0

5

10

15

20

25

RX

v exp(km/s)

LSN

USN

Dominant variation of rotation unaffected by LSN!USN.However, reason for LSN-USN rotation difference remains unexplained, maybegeometry interacts with: collisional effects, trapped particles, particle sources/sinksOr maybe LSN vs USN just affects turbulence properties like d

c

or Lf


Theory

Experiment

Description

Results

Do orbit losses explain rotation at the core-edge boundary?Nucl. Fusion 49 (2009) 085020 J.S. deGrassie et al

midplane to the vicinity of the X-point. The barely trappedorbit (circle) (red online) spends the most time ‘falling’ andthus determines the lower energy boundary of this part of theloss cone. The more deeply trapped orbit (cusp) (blue online)has greater energy and thus a greater vertical drift velocity toescape after just the outboard leg of the orbit. The inboardco-Ip velocity ion (green online) is a mirror situation to thesecond orbit, but the magnetic well force accelerates this ionalong the flux surface from inboard to outboard whereas theoutboard ion (blue online) is decelerated moving inwards inR. If a guiding centre orbit is computed for each of these threeorbits with the initial parallel velocity reflected, the resultingcollisionless orbits are confined because then the drift velocitymoves each ion away from the LCFS as it circulates in theopposite direction poloidally.

We derive in appendix A the approximation used here tocompute the guiding centre loss-orbit boundaries in velocityspace. These boundaries are computed for deuterium ionsstarting at midplane locations, both outboard and inboard. Thepitch angle in velocity space is p, defined by V∥ = V ·B/|B| =|V | cos(p) and the pitch angle at the starting midplane locationis p1. Given the B and Ip directions in figure 8, a counter-Ip

initial V∥ lies in the region 0 < p1 < π/2. In relation tothe toroidal velocity, Vφ , we have Vφ

∼= −V∥. The computedkinetic energy, pitch angle loss boundaries for this dischargeshape and conditions are plotted in figure 9. The horizontal,kinetic energy axis is shown in terms of the scaled variableαE , where αE ≡ W0/Wloss and Wloss ≡ Mω2

θ&2/2. Here,

W0 is the initial kinetic energy of the ion, ωθ is the cyclotronfrequency in the poloidal magnetic field at the starting locationand & is the midplane radial distance between the LCFS, atRL, and the starting location, R1, & = RL − R1.

Orbits which are started in the shaded regions of the phasespace shown in figure 9 are lost near the X-point. The lost co-Ip going ions, p1 > π/2, start from the inboard side, as thetrajectory with diamond symbols (green online) in figure 8.The initial counter-Ip going ions start outboard and are shownby the shaded regions with p1 < π/2, as for the circle (redonline) and cusp (blue online) symbol orbits in figure 8. Thestarting (αE , p1) values for these three orbits from figure 8are shown in figure 9 by the equivalent symbols. The guidingcentre computations were selected to show some orbits nearthe boundaries in this phase space. Two limiting values of p1

are indicated in figure 9. The trapped/passing boundary is atp1 = pt and at p1 = px an outboard starting orbit is trappedwith turning point at the major radius of the X-point, RX. Moredeeply trapped ions, π/2 > p1 > px , do not reach the X-pointand in our approximation are not lost since we do not considerloss at any other boundaries.

The ‘finger’ region for outboard, counter-IpV∥ ionstypically dominates the low collisionality loss cone. This isthe region with pt < p1 < px in figure 9. This region isqualitatively seen in an analytic calculation by Hinton andChu [54] using a very idealized model for trapped and passingion drift orbits. The numerical results of CM [55] show theloss cone to be essentially that of figure 9 for p1 < π/2.The energy loss parameter used by CM is W0/&

2. Likewise,figure 2 of Miyamoto [56] with parameter φ = 0 (i.e. E = 0)

qualitatively shows our finger-like loss region reflected aboutthe αE = 0 and then rotated by 90◦. The outboard loss region

G.C. orbitsD+

Outboard, counter V// canbe lost, co V// confined

Inboard, co V// canbe lost, counter V// confined

B X ∇BBT

Ip

Z (m

)

1.0

0.5

0.0

–0.5

–1.0

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4R (m)

118183.01800

X

Figure 8. Equilibrium flux surfaces for discharge discussed infigures 1–5. Counter-Ip V∥ outboard loss orbits (circle, cusp).Co-Ip V∥ inboard loss orbit (diamond). These are guiding centreorbits computed with the full equilibrium fields determined withEFIT. (Colour online.)

that extends to the lowest, threshold αE for loss, αE = αE min,in figure 9 (red online) is for those trapped orbits that havemade a turn inside Rx and are lost on the way back out. Theother outboard loss region (blue online) is for orbits lost onthe first encounter with R = Rx . This threshold αE min isroughly a factor of 2 smaller than the lowest αE for the inboardloss, at p1 = π/2. However, in absolute terms for the kineticenergy, W0, the inboard loss threshold requires an additionalfactor of ∼4 greater than for the outboard threshold. This canbe seen by noting that Wloss ∼ M[(Ze/M)&ψ/R]2, where&ψ is the starting distance from the edge measured in ψ andZ is the ion charge number. For two orbits starting on thesame ψ surface the inboard Wloss exceeds the outboard valueby (RL-out/RL-in)

2 ∼ 4. So, for a Maxwellian distributionof ion kinetic energies the inboard orbit loss is negligiblecompared with the outboard orbit loss. And the outboard lossis dominated by the finger region for low collisionality orbits.

H-mode discharges exhibit a negative well in E inthe pedestal region [44, 58]. CM determined the effect ofthis E-well on the finger region of the loss cone usingorbit computations with spatially parabolic and linear electricpotential models [55], and Miyamoto [56] incorporated a linearpotential model in an analytic formulation. The loss boundaryminimum for αE min, the finger tip in figure 9, moves downtowards 0 pitch angle as the well depth is increased becausean ion with fixed energy requires more initial parallel velocityto climb out of the potential well. However, we neglect this

8

Nucl. Fusion 49 (2009) 085020 J.S. deGrassie et al

αE

00.0

0.4

0.6p 1/π

0.8

1.0

2 4 6 8 10 12 = { V/[ ωθ (RL-R1 ) ] } 2

pt

px

BV

p

Lostco-V//

inboard start

Counter -V//outboard start

Figure 9. Velocity space loss cone computed with analyticapproximation given in appendix A. Starting pitch angle, p1, versusαE parameter. Orbits with p1 < π/2 have counter-Ip V∥. The filledin symbols are the starting conditions for the three orbits with thesame open symbols shown in figure 8. The trapped/passingboundary is at p1 = pt and at p1 = px an outboard starting ion willhave a trapped turning point at R = Rx , the major radius of theX-point. (Colour online.)

effect for the intrinsic rotation discharge conditions describedhere for DIII-D, given the pedestal region measured Ti and thedepth of the computed potential well using the measured E

profile. For example, the potential well depth is ∼ 100 eVwhile Ti ∼ 300 eV. With these conditions the electric wellmodification to the loss cone is a first order correction. Withthis approximation the kinetic energy of an ion is constantalong the trajectory. Neglecting this effect of the H-mode E

well on the thermal loss cone may not be valid for higher powerNBI-driven H-modes in DIII-D.

4.3. Computation of ⟨V∥⟩ for an empty loss cone

We assume an empty loss cone consisting of the outboard lossfinger region, approximated in pitch angle by pt < p < px .For the lower boundary in kinetic energy for the loss region wetake a constant value, the value of αE min at the average pitch,p1 = (pt + px)/2, which we define as αav

E = αE min(p1). Thenthe velocity integration is taken to be from Vmin to ∞, whereVmin =

!2αav

E Wloss/M . The resulting formula for ⟨V∥⟩ isgiven by equation (A-9) in appendix A, and is

⟨V∥⟩ =−

"2πV (b + 1)e−b(x − r1)/2

D (b, x, r1), (1)

in the co-Ip direction, consistent with the leading minus sign.Here, x = Rx/R1, r1 = R1in/R1, b = αav

E (Wloss/Ti), R1 is theoutboard surface midplane starting major radius and R1in is the

0.00

0.04

0.08

0.12

0.0 1.0 2.0

–<V /

/>(W

loss

/M)1/

2

Ti /Wloss

Figure 10. Scaled distribution-averaged co-Ip velocity, −⟨V∥⟩,computed with equation (1), versus Ti/Wloss, using the equilibriumconditions shown in figure 8. (Colour online.)

inboard midplane major radius on the same flux surface. Thedenominator, D, is given explicitly in appendix A, but differsfrom unity by at most 10% for these DIII-D conditions.

In figure 10 we plot equation (1) as −⟨V∥⟩/√

Wloss/Mversus Ti/Wloss using the values of R1in, Rx and R1 from thedischarge shape shown in figure 8. The approximate linearincrease in |⟨V∥⟩| with Ti is evident, with one factor of

√Ti

from V and roughly one factor of√

Ti from the variation withb in the lower energy limit of the integration in the region ofthreshold kinetic energy for loss. In appendix A we show howthis universally scaled value of ⟨V∥⟩ changes with the majorradius of the X-point, Rx . However, the approximation that theloss cone is empty may not necessarily quantitatively predict⟨V∥⟩ with collisions included. Yet, it may be that the scalingof the intrinsic velocity boundary condition, Vφ ∼ Ti, will holdin the region of low to moderate collisionality. As a measure ofthe relative collisionality we compute that ωbτi ≈ 1.5, whereωb is the bounce frequency for a D+ ion near the top of thepedestal in figure 3, with turning point just outside of R = Rx

and kinetic energy = Ti. The ion–ion collision time is τi.A general scaling of this computed pedestal region intrinsicvelocity taken from approximating as linear the relation shownin figure 9 is −⟨V∥⟩ ∼ εpTi/(Z'Bθ ), where εp ≪ 1 is a smallnumerical value that accounts for the small fraction of velocityspace occupied by the loss cone.

In order to apply equation (1) to obtain an absolutevalue for ⟨V∥⟩ we must know ', the distance between thestarting surface for the orbit and the LCFS. The sensitivity ofequation (1) to ' is significant within the assumed error in theEFIT computation of RL, ±5 mm [45]. In figure 11 we plot Vφ

versus Ti using three values of RL with the starting location, R1,that of the Vφ measurement channel at R = 2.265 m shownin figure 2, which is at the top of the ion pressure pedestal.The middle curve (red online) uses the EFIT-determined valueof RL0 = REFIT = 2.283 m, ' = 18 mm and the othertwo curves are for RL = RL0 + 5 mm (orange online) andRL = RL0 − 5 mm (green online), respectively. The measured

9

deGrassie et al NF 2009, Figs. 8–9

Ion orbit losses are oneway of looking at theorigin of the edge flowlayer. However, the effectof the edge flow layer isdetermined by momentumtransport physics.

v

r

HFS

LFS

q0=-3pê4

-3 -2 -1 0 1 2

0.1

0.2

0.3

0.4

0.5

0.6

vk∂

q

fi

�dv2

k (sinq)∂

x

fi

�((((((((hhhhhhhh∂

x

[D (x ,q)∂

x

fi

] = 0


Theory

Experiment

Description

Results

SummaryI Simple theory for intrinsic rotation due to interaction of:

I spatial variation of turbulenceI different radial orbit excursions for co- and counter-current passing ions

I Predicted rotation depends strongly on RX

I Performed series of Ohmic L-mode shots on TCV, scanning RX

I Change of RX

shifts entire rotation profile, fairly rigidly

I Experiment and theory appear fairly consistentI vexp depends about linearly on R

X

, goes counter-current for large RX

.I Linear fit results in reasonable adjustable parameters d

c

, Lf

.I Basic results hold for various alternate radial positions.I vexp appears fairly insensitive to core rotation reversal.

I Possible further topics:I E⇥B drift, collisions, real magnetic geometry and orbits/trappingI Self-consistent calculation of turbulence properties: d

c

, Lf

, . . .I Why is USN rotation more counter-current than LSN?


edge intrinsic rotation: theory and experiment · overview a simple kinetic transport theory...

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