Intrinsic Rotation and
Toroidal Momentum Transport:
Status and Prospects
(A Largely Theoretical Perspective)
P.H. Diamond
[1]
1
WCI Center for Fusion Theory, NFRI, Korea
[2] CMTFO and CASS, UCSD, USA
Thought for the Day:
• “As an economist in good standing, I am
quite capable of writing things nobody can
read”
– Paul Krugman
2
Driver: Milestone for Theory and Computation, F2010
Wet Ware:
- UCSD: P.H. D, G. Dif-Pradalier, C.J. McDevitt, Y. Kosuga, C. Lee
- PPPL: W.X. Wang, T.S.Hahm, S. Ethier
- NYU/CPES: S. Ku, C.S. Chang
- NFRI: J.M. Kwon, S.S. Kim, H. Jhang, S.M. Yi, T. Rhee
- CEA: Y. Sarazin et. al. GYSELA team
- Ecole Polytechnique: O.D. Gurcan
Hard Ware:
- G. Tynan, J. Rice, W. Solomon, K. Ida, M. Yoshida, S. Kaye, M. Xu, Z. Yan
Soft Ware:
- GTS : Wang, PPPL
- XGC1 : Ku, Chang, NYU
- gKPSP : Kwon, NFRI
- TRB : S.S. Kim, NFRI
- GYSELA : Sarazin, CEA
Presentation: J.M. Kwon
3
Approach: A Critical Appraisal, extended.
• Outline:
I) Some Background
II) What we understand
III) What we think we understand, but would benefit from
more work on
IV) What we don’t understand – and should be studying
4
I) Some Background
5
6
7
8
9
Comparison/Contrast
The Question: Rotation Profile?
Heating → flux driven turbulence →
→ Rotation Profile?
The Theoretical Issue: Mean Field Theory for
represent
The Difference
Negligible feedback
transport coeffs + mean quantities
STRONG Flow feedback on turbulence
II) What We Understand
11
General Structure of Flux
• Toroidal momentum transport is driven by parallel and
perpendicular Reynolds stresses, as well as convection
• Both of the above may be decomposed into a turbulent
viscous piece, a (toroidal) ‘pinch’ piece, and a non-diffusive
residual stress piece, not proportional to velocity or velocity
gradient.
P r ,f = - cf
¶ Vf
¶r
+ V Vf + P resid
r ,f
non-diffusive stress
• Easy Part : χφ ~ χi, with an intrinsic Prandtl number of roughly
0.5 < Pr < 0.7, depending on deviation from threshold. The
detailed physics underpinning of the dependence is the
resonant particle velocity.
12
General Structure of Flux
(cont’d)
V Vf ¹ 0 on boundary
13
The Pinch
• The parallel flow pinch V is toroidal in origin, and consists of
turbulent equipartition (TEP) and thermoelectric pieces. The
thermoelectric pinch is driven by both ∇Ti and ∇n. These tend
to oppose one another for a wide range of cases. For the TEP
pinch, produced by the compressibility of VExB in toroidal
geometry, V/Xφ ~ O(1/R). TEP momentum, particle and heat
pinches are closely related.
14
15
16
The Pinch (cont’)
More Physics of the Flow Pinch
17
18
Residual Stress: Dynamics and Structure
• The residual stress is driven by ∇T, ∇p, ∇n, and produces
a local intrinsic torque by its divergence. Residual stress
can spin up the plasma from rest, acting in concert with
boundary conditions.
N.B. Boundary conditions are unclear … “No Slip” is
intellectual crutch.
19
Residual Stress
Piece of Reynolds Stress without Explicit Dependence on,
(i.e.,ÑP , Ñn , etc)
–
–
�
–
Vf ,
d
Vf
dr
Beyond Diffusion and Pinch
d n
G
=
-D
+V n
�
particle number conserved →
n
�
dr
pinch is only “off-diagonal” for particles
but: wave-particle momentum exchange possible
�
Can accelerate resting plasma:
ÑP,i Ñn→
→
a
0
¶t ò dr Vf = ¶t Vf » -P
resid a
r,f
0
|
P resid
residual stress acts with boundary condition to generate intrinsic rotation
[Gurcan, Diamond, Hahm, Singh, PoP 2007]
� �
How ? Broken Symmetry in Turbulence
akin to -effect
a in dynamo theory
�
�
20
Evidence for intrinsic torque: Exp
0
P rf = - cf
¶uf
¶r
+ Vuf + P resid
rf
Diffusion
0 = -Ñ × P
0
resid
rf
Pinch
Residual
Stress
+ t NBI
[Solomon , 2007]
• Intrinsic rotation à Mechanism?
• DIII-D experiments (W. Solomon, A. Garofalo) have demonstrated that a
finite external NBI/NRMP torque can null out the intrinsic rotation.
• How reconcile zero rotation with NBI? à need introduce intrinsic torque
density ( t = -Ñ × P resid
) (also, [Ida, 2010])
rf
23rd IAEA Fusion Energy Conference, Daejeon, Korea 15 Oct. 2010
21
Evidence for intrinsic torque: Simulation
XGC1p
Decreasing residual stress
XGC1p
Disposed of
by B.C.
•
Residual stress is inward and decreasing towards to edge. (r/a<0.8)
à Co-current intrinsic torque (= - Ñ×P)
•
Counter-current torque r/a>0.8 is disposed of by no-slip B.C.
•
PDF for Heat flux and magnitude of momentum flux are strikingly similar.
à Outward heat avalanches drive inward momentum avalanches.
à Further evidence for non-diffusive, flux – driven nature of momentum flux
23rd IAEA Fusion Energy Conference, Daejeon, Korea 15 Oct. 2010
22
Wave Momentum Flux and Residual Stress
•
Wave momentum flux from radiation hydrodynamics for short mean free path:
P
•
�
�
•
wave
r,||
=
ò
ì
¶ Nk ü
2 ¶ Nk
dkk|| í-t c,k v gr
+ t c,k v gr kq v E '
ý
¶r
¶k r þ
î
First term ↔ radiative diffusion of quanta
¶
– ¶r N k > 0
, universally increasing
– inward scattering from edge
Symmetry Breaking
• Growth asymmetry [Coppi, NF ‘02]
–Bias in g (k|| ) from V|| '
–unlikely in realistic regime
Second term ↔ refraction induced
wave quanta population imbalance:
important for regimes of strong shear,
sharp
relevant to ETB, ITB
– mode dependence, via v* sign
– can flip direction [TCV, C-Mod] in
LOC→SOC
�
• Wind-up by shearing packet
�
→ akin spiral arm formation: requires
and magnetic shear
VE '
• Refractive
Force due to GAMs
–relevant near edge
–polarization effects
�
[McDevitt et al., PoP ‘08]
23
What we understand (cont’)
• Residual parallel stress requires symmetry breaking, so as to
convert radial inhomogeneity into parallel spectral asymmetry.
Symmetry breaking mechanisms include electric field shear
〈VE〉’ and intensity gradient ∂rI – both of which are selfreinforcing and linked to the driving heat flux – as well as updown asymmetry of current density. Additional symmetry
breaking sets the polarization stress (〈krk||〉 ≠ 0) ->
essentially a quadrupole spatial moment of the spectrum is
required) and the poloidal Reynolds stress, which drives
flow through 〈Jr〉Bθ/c (again 〈VE〉’ and ∂rI are critical)
24
Basic Physics of Symmetry Breaking
•
•
Non-diffusive residual part is crucial
→ Broken symmetry is essential
(Diamond, et al., NF’09)
¶ V||
P r || = - cf
+ VPinch V|| + P rRS||
¶r
cf ~ c i → flow damping
Symmetry breaking induced by ExB shear flow
(Dominguez, et al., Phys. Fluids B 1993;
Gurcan, et al., PoP2007)
Symmetry breaking :
Conversion of radial inhomogeneity to
→ parallel asymmetry
→ rotation generic to confinement
P rRS|| µ vth cf / Lsym
•
Well known symmetry breaking by V’ExB
→ Is this universal or fundamental?
1 / Lsym ~ kq d VE¢ / Ls ;
25
dV'E ~ VE '
Broken symmetry in fluctuating potential
for ITG turbulence and Resulting parallel flow
Basic Physics of Symmetry Breaking (cont’d)
• Interesting alternative:
fluctuation intensity gradient
P
RS
r ||
D2 ¶
= kq
Fk
Ls ¶r
2
k||
2
• Fluctuation intensity gradient tied to
mean profile curvature
1 ¶I
1 ¶ 2T
1 ¶c neo
~,
I ¶r
¶T / ¶r ¶r 2 c turb ¶r
•
Symmetry breaking induced by radial
gradient of turbulence intensity
(Gurcan, et al., PoP’10)
Note VExB shear is also related to profile curvature
¶
1 ¶ 2 Pi
1
VE =
Pi¢ni¢ + Vf¢Bq - Vq¢Bf
¶r
n | e | ¶r 2 n 2 | e |
N.B. Profile curvature
⇔ Πresid link
• Fluctuation intensity gradients ubiquitous to confined plasmas
– Robust and Generic mechanism
– Closely related to V’ExB → i.e. transport barrier!
26
Basic Physics of Symmetry Breaking (cont’d)
• Interesting alternative:
fluctuation intensity gradient
P
RS
r ||
D2 ¶
= kq
Fk
Ls ¶r
2
k||
2
• Fluctuation intensity gradient tied to
mean profile curvature
1 ¶I
1 ¶ 2T
1 ¶c neo
~,
I ¶r
¶T / ¶r ¶r 2 c turb ¶r
•
Symmetry breaking induced by radial
gradient of turbulence intensity
(Gurcan, et al., PoP’10)
Note VExB shear is also related to profile curvature
¶
1 ¶ 2 Pi
1
VE =
Pi¢ni¢ + Vf¢Bq - Vq¢Bf
¶r
n | e | ¶r 2 n 2 | e |
N.B. Profile curvature
⇔ Πresid link
• Fluctuation intensity gradients ubiquitous to confined plasmas
– Robust and Generic mechanism
– Closely related to V’ExB → i.e. transport barrier!
27
Testing Symmetry breaking by Intensity Gradient
Intensity front
• Global δf gyrokinetic simulation (gKPSP)
– ITG turbulence with adiabatic electrons
– ∇Φ=0 is imposed on boundaries
– Ti - profile relaxes
r
C[P rRS|| , I ]
• Ti-profile relaxation
→ Propagating fluctuation intensity front
(favorable to test the role of intensity gradient)
C[P rRS|| , I ¢ ]
¢ , I¢
• Time correlations for P rRS|| , VExB
RS
– Strong correlation of C[P r|| , I ¢ ] over most radii
– Lower zonal flow shearing due collisions
C[P rRS|| , VE¢ ]
→ Role of intensity gradient more prominent
r / R0
R/LTi = 10.0, ν* = 0.08
28
More: Intensity gradient - a major symmetry breaker
With residual stress
With intrinsic torque
•
•
•
Investigation of correlations between {Residual stress, Intrinsic torque} and symmetry
breakers
Strong correlation(~0.6) with intrinsic torque, smaller correlation(~0.4) with residual stress
Intensity gradient shows the same level of correlation as usually invoked ExB shear.
23rd IAEA Fusion Energy Conference, Daejeon, Korea 15 Oct. 2010
29
Stress lags symmetry breakers
vr v||
dv||
dt
= -Ñ|| P
• ExB shear and intensity gradient show about p/2 and -p/2 phase lag relative to
residual stress.
• Phase between ExB shear and intensity gradient is about p/2.
è Suggests that drift acoustic response induces phase lag between symmetry
breakers and residual stress.
23rd IAEA Fusion Energy Conference, Daejeon, Korea 15 Oct. 2010
30
Local Physics of Πresid
• For ITG turbulence, Πresid
– increases with R/LT – R/LTc
– is insensitive to collisional zonal flow damping
– couples to the zonal flow-turbulence self regulatory
predator-prey interaction
31
Microscopic Origins of Macro-Scaling?
• Experimentally observed macro scaling
→ Micro Foundations?
|V|| / Vth|
R/LTi - R/LTic = 6.00
0.01
– ΔVφ(0) ~ ΔW/Ip for H-mode plasmas
R/LTi - R/LTic = 3.14
(Rice NF’07)
– Vφ(0) ∝ ∇Ti at transport barrier
1E-3
R/LTi - R/LTic = 1.56
(Rice ’10, this conference; Ida NF’10)
0.00
0.02
0.04
• Stronger net rotation as R/LTi , q-value↑
– Increasing R/LTi : more free energy to drive
stronger turbulence
– Increasing q-value (normal shear)
ü More effective symmetry breaking at higher q-values
ü Weaker turbulence regulation by ZF for increasing
GAM fraction → increase in turbulence and intrinsic
flow level
8.4x10-3
n*
0.08
0.10
|V|| / Vth|
-3
7.2x10
6.0x10-3
4.8x10-3
3.6x10-3
2.4x10-3
1.0
1.5
2.0
2.5
q(0.5a)
32
0.06
3.0
3.5
Flux Driven Rotation: Qi ⇒ Vintrinsic
• Net intrinsic rotation with a peak 〈V||〉/Vth = 0.05 ~
0.15 can be produced in flux driven ITG
simulations with no slip boundary conditions
33
Intrinsic rotation and its cancelation (XGC1p)
External
Torque
Initial flow
• Starting from zero initial flow
à Net intrinsic rotation at 7 ms: co-current, MT @ 0.05 (5% of vth)
• Peak flow: still increasing & moving from edge to core
• With applied external counter-current torque:
à Total rotation reduced by more than factor of 5
(Similar result with counter momentum source in GYSELA)
23rd IAEA Fusion Energy Conference, Daejeon, Korea 15 Oct. 2010
34
Intrinsic Rotation and ITB’s
• Strong intrinsic rotation can be generated by flux
driven ITG turbulence in reverse shear ITBs with offaxis minimum q(r), with no slip boundary conditions
35
Intrinsic rotation occurs in RS ITB plasmas
• Strong (Mth ~ 0.1-0.25, |VITB | >> |VL|) co-current rotation is generated in
heat flux driven ITB plasmas with reversed magnetic shear
• The flow is intrinsic rotation generated via the residual stress
Ion temperature vs. r/a
36
V|| vs. r/a
Formation of intrinsic rotation
•
•
•
Intrinsic rotation is generated near ITB head and, initially, propagates into the core.
The position of maximum intrinsic rotation coincides with the position of maximal
g E´B | f% |2 → symmetry breaking
Reynolds stress v%r v%|| < 0, because of large inward residual stress
V|| evolution during initial phase
Diffusive part
Reynolds stress
Residual stress
r/a
37
III) What We Think We Understand
but Need More Work On
38
Alternative Mechanisms
• The mechanism for generation of toroidal rotation
by fluctuation driven radial currents – i.e. via the
toroidal projection of the perpendicular Reynolds
stress.
⇒ Is the emphasis on k|| symmetry breaking
warranted?!
39
Alternative Mechanisms
-
VEE〉' and intensity gradient still critical
- 〈V
40
Scalings
• The basic structure of the Rice scaling (ΔvT ~
ΔWp/Ip) originates from:
– strong localized temperature gradients, as in the
pedestal and ITBs (i.e. the local origin of ΔWp)
– q(r) scaling
• the origin of Ip dependence
• sensitivity to q(r) structure
41
Simple Illustrative Model – Reduced Model
Conservation Laws:
Fluxes:
TEP Pinch for
Momentum and
Density
Angular Momentum
Residual Stress
due to ExB Shear
Radial Force Balance:
Er @
1 ¶P
- uq ,Neo Bf + uf Bq
ne ¶r
e0
e=
¶uEy 2
1+ b (
)
¶r
L-mode Turbulence Intensity
only adjustable parameter
B.C.'s:
Lf (a),Vf (a),n(a)
e0
: given
Boundary conditions critical!
�
ExB Shear Reduction
[Hinton, PF-B ‘91]
�
f
42
Scaling Trends manifested by Model
Dimensional analysis for pedestal flow velocity
suggests scaling with width:
Scaling with r from
turbulence characteristics
a
vf /vTi µ (DrTurb /a)(D ped /a)µ (r *) (D ped /a)
With the simple model linking width to height,
D ped µ Pped
�
where
a
Dvf /vTi µ (r *) DW p
DW p
: Incremental Stored Energy
Ip scaling not recovered for GB model→
pedestal/edge turbulence issues?
�
�
Model is not quantitatively accurate,
�
predicts a scaling of the pedestal toroidal
velocity with the pedestal width.
Vf , ped ~ D ped ~ Pped
but recovers qualitative behavior
Hahm
43
q(r) Shape Can Enhance Intrinsic Flow
2.1x10-3
|V|| / Vth|
1.8x10-3
1.5x10-3
1.2x10-3
9.0x10-4
0.90
1.05
1.20
rq'/q (0.65a)
44
1.35
⇒ Weak but positive
shear is favored
Observation re: q(r) structure
• Key correlator for residual stress:
~ 2
~ ~
resid
P r ,f ~ Vr Ñ||f ~ kq k|| | f |
2
¢
¢
¢
k|| = nq x + nq x / 2 + K
shear
P
resid
r ,f
2
curvature – flat q(r)
~
~ n q¢ x | f |
2
2
~
2
¢
¢
+ n q x |f | / 2 +K
spectral centroid ~ 〈VE〉’ etc ⇒ sensitive
2
spectral variance ⇒ robust – set by spectral width
• Expect strong Πresid in flat-q regimes (i.e. q’=0, q’’≠0) ⇒
Intrinsic rotation in ITB’s, low torque “de-stiffened”
regimes ?!
45
Aside: Why Care?
• Core stiffness, no man’s land place SEVERE
demands on ITER pedestal
• Need reconsider either:
– ITB
– de-stiffened, hybrid mode approach
• Current understanding suggests:
– intrinsic flow shear will dominate ITB 〈VE〉’, absent
external torque
– intrinsic rotation very sensitive to q(r) profile structure
46
Saturation of Intrinsic Rotation – Especially ITB
• Saturation of Rice scaling in high power ITBs and
the ultimate limit on intrinsic rotation. Of particular
note here is dependence on collisional Prandtl
number or its equivalent.
47
New regime of V||(0) vs ∇Ti scaling found
• ~ Linear V||(0) vs. -∇Ti enters roll-over for χi,turb < χi,neo (strong turbulence
~
suppression in ITB) → Ultimate limitation on intrinsic rotation?
• Why? There are intermediate
states between “active” and
“fully suppressed” turbulent
states → determined by residual
heat and momentum transport
in barrier
Prneo = cf,neo /ci,neo
c i ,turb > c i ,neo
48
c i ,turb <~ c i ,neo
a
I g E æ Qi 1
~
ç
ÑTi
Qi çè c i ,t 1 + cˆ i
ÑVf
b
ö c i ,t 1 + cˆ i
,
÷÷
ø cf ,t 1 + cˆf
ITB – Relative Hysteresis
• Relative hysteresis of ∇Vφ and ∇T in ITG
intrinsic rotation (K. Ida 2010)
49
Hysteresis happens!
Q
DQc
DQc = A / D(ÑTi ),
where A is area spanned
by hysteresis curve
•
50
Strength of hysteresis
increases with Nusselt number
(Nu= χi,turb /χi,neo ) increases
→ agrees with prediction based
on bifurcation theory
•
Open loop hysteresis is
seen in Qi,tot vs. -∇Ti plot
→ Contrast to closed loop
S-curve model
Relative hysteresis between ∇Ti & ∇V||
observed and explained
tokamak
•
•
51
Relatively stronger hysteresis of intrinsic rotation over temperature gradient is
observed → Recovers features of recent experimental observation in LHD
[K. Ida et. al., NF 50 (2010) 064007]
Residual transport (Prneo) governs strength of relative hysteresis
→ Δ(∇V||) decreases as Prneo increases.
• C-Mod: ITBs in q ' > 0 plasmas (C. Fiore, et. al.)
– ITB develops in H-mode with H-mode pedestalgenerated intrinsic rotation
– Intrinsic Vf¢ is largest contributor to VE '
– Core rotation can reverse in ITB
52
53
54
55
What we think we understand but would benefit
from more work on (cont’)
• The theory of poloidal momentum transport by
turbulence
⇒ See C. McDevitt, et. al. PoP 2010 for
discussion
56
Reversals – A New Type of Transport Bifurcation
• The precise relation between turbulence propagation
direction (i.e. v*e vs. v*i) and toroidal rotation direction
⇒ Reversals?
– Observed in TCV, C-Mod
– Appears linked to CTEM-ITG, LOC ⇒ SOC cross over
– Exhibits many features of transport bifurcation without enhanced
energy confinement
– Likely relation to p-ITB of W.W. Xiao et al
57
58
59
Residual ⇔ Mode Propagation
v gr ~ v*
Key parameter : v gr VE '
See P.D. et al PoP 2008
60
Intrinsic Rotation as Heat Engine
• The theory of fluctuation entropy
balance and how it relates to the
notion of intrinsic rotation as the
output of a plasma heat engine
⇒ See: Y. Kosuga, et. al. PoP 2010
J. Rice, et. al. PRL 2011
• Punchlines:
– ∇Ti correlates well with VT(0) in C-Mod H-mode, I-mode
∇P fails for I-mode
– Theory based on heat engine (low efficiency!) picture
calculated C-Mod VT(0) to good accuracy
61
IV) What We Do Not Understand
– and should be studying
62
Boundary Condition
VERY IMPORTANT
• The interplay of turbulence and wave scattering with
neoclassical effects and orbit loss in determining the
boundary condition for intrinsic rotation ⇒ need quantify the
amount of ‘slip’.
• The detailed interplay between core intrinsic torque and the
edge boundary condition, and its role in determining net
rotation direction. The connection between SOL flows and
core rotation.
• The impact of filaments and ELMs on intrinsic rotation
63
�
Physics of Boundary Condition Effects
•
SOL Flows: [LaBombard et al., NF `04]
“ballooning” particle flux produced by outboard source
and SOL symmetry breaking (LSN vs USN)
Influence core
in L-mode
LSN →
USN →
But, in H-mode,
toward X-point →
from X-point →
co
counter
is always CO
•
Key question: How can SOL flow influence core
plasma ?
•
For S|| (r) = speed profile of SOL flow
dS|| (r)
> 0 in SOL ---> Inward viscous stress of SOL flow
dr
�
on core
Key: SOL symmetry breaking sets
direction
Strong for parallel shear flow instability for ÑV|| > Ñn
Alternative: Recoil from Blob Ejection (Myra)
�
�
64
ρ* Scaling
• The apparent absence of ρ* scaling of intrinsic rotation
~ is this real?
• More generally, the obvious problem of:
– the general, qualitative success of drift wave theories of
momentum transport, which universally ρ* predict dependence
– the apparent absence of ρ* dependence in intrinsic rotation
scalings
• Isotope dependence
65
What we do not understand
• The effect of energetic particles on rotation
At least two Issues:
- E.P. momentum scattering – i.e. deposition
- Momentum transport by A.E.’s etc.
66
RF and q(r) Structure
• The effect of q-profile structure on intrinsic rotation
• The detailed dynamics of how RF and current drive (i.e.
LHCD, ECH) affect intrinsic rotation. Is this via q(r)? Is it ν*
change?
• Recent EAST results (Shih, et.al.; PRL 2011) are an
interesting challenge
• Relation to inversion phenomena
67
Dynamical Evolution
• The detailed spatio-temporal dynamics
of intrinsic rotation profile build-up
68
Space-time evolution of intrinsic torque
Residual stress
Intrinsic torque
ExB shear
Intensity gradient
•
The turbulence arises near the outside boundary and propagates inward. [Ku, 2009]
•
ExB shear and intensity gradient show the same inward propagation pattern.
•
Residual stress and intrinsic torque evolution history correlated with that of intensity
gradient.
23rd IAEA Fusion Energy Conference, Daejeon, Korea 15 Oct. 2010
69
Low Torque vs No Torque
• The critical torque-to-power ratio that likely delimits
pinch vs. residual stress dominated momentum
transport regimes
⇒ How Low is “Low” ?
70
Some Long Term Issues
v~r v~^ ⇔ intrinsic rotation connection
~~
• EM fluctuation, Br Bq role in rotation saturation – unexplored!
•
• Elucidate ‘Boundary Condition’ Issue: Degree of ‘slippage’
– SOL flows (USN vs LSN)
– RMP
– nn deposition – SMBI
Insightful experiments
are badly needed!
• Study intrinsic rotation in flat q discharges
• Elucidate physics of residual stress, pinch in electron channel dominated
discharges
• Explore the poloidal ⇔ toroidal rotation synergy, especially in ITB
71