Physics of the Power Threshold Minimum for L-H
Transition
M.A. Malkov1 , P.H. Diamond1 , K. Miki2 , J.E. Rice3 and
G.R. Tynan1
University of California, San Diego, 2 JAEA, 3 MIT
Acknowledgment: This material is based upon work supported by the U.S. Department
of Energy, Oce of Science, Oce of Fusion Energy Sciences, under Award Number
DE-FG02-04ER54738
1
1 / 19
Outline
1 Motivation
Pthr - the LH transition Power Threshold
-Micro - Macro connection → How does physics set Pthr ?
2 Key Questions
3 Reduced Model
Basic Structure
Te and Ti equations
Anomalous Coupling
4 Model Studies
Recovering the Minimum
Towards the Anomalous Regime (Preliminary, if time allows)
5 Conclusions
2 / 19
Motivation and brief history of LH studies
L→H transition is a 33 (!) year-old story (Wagner,
revolutionized connement physics
central to ITER ignition
Underlying ideas
dimensional analysis (e.g.
scalings
et al 1982)
Connor and Taylor, 1977) and simple
in general Pthr ∝ nBS
early phenomenology (t) Pthr ∝ n0.7 - inconsistent with the
minimum in Pthr (n)
connection of the power threshold to the edge parameters (Fukuda
et al 1988): evolving story
Mechanism: shear suppression paradigm (Biglari, Diamond and
Terry, 1990 ++)
3 / 19
Emerging Scenario
LH-triggering sequence of events
Q ↑ =⇒ ñ, ṽ ↑=⇒ < ṽr ṽϑ >;
etc.
< ṽr ṽϑ > d < v >/dr ↑ =⇒ |ñ|2 ↓,
=⇒ ∇Pi | ↑ =⇒ lock in transition (Tynan
et al. 2013)
∇T etc. drives turbulence that generates low frequency shear ow
via Reynolds stress
Reynolds work coupling collapses the turbulence thus reducing
particle and heat transport
Transport weakens → ∇ hPi i builds up at the edge, accompanied
by electric eld shear ∇ hPi i → hVE i0
locks in L → H transition: (see Hinton ,Staebler 1991, 93)
Complex sequence of Transition Evolution and Alternative End
States (I-mode) possible (D. Whyte et al. 2011)
4 / 19
Some Questions:
How does the scenario relate to
the Power Threshold?
Is Pthr (n) minimum
recoverable?
Ryter et al 2013
Micro-Macro connection in
threshold, if any?
How does micro-physics
determine threshold scalings?
What is the physics/origin of
Pthr (n)? Energy coupling?
Will Pmin persist in
collisionless, electron-heated
regimes (ITER)?
5 / 19
Further Questions and important Clue:
J. Hughes, Y. Ma, J. Rice, 2011,12
Rice et al., 2009
Is Pthr set only by local
properties at the edge?
(Common wisdom)
Is Pthr minimum related to
collisional energy transfer? i.e.
νn (Te − Ti ). Low n branch
couples to ions, enables ∇Pi ?
Pthr (n) minimum correlates
with n 'LOC-SOC' transition
⇒ i.e. min power related to
collisional inter-species transfer
Threshold is controlled by
global transport processes!?
6 / 19
Scenario (inspired partly by F. Ryter, 2013-14)
∇Pi |edge essential to 'lock in' transition
to form ∇Pi at low n, etc. need (collisional)
energy transfer from electrons to ions
∂Te
1 ∂
2m
+
r Γe = −
(Te − Ti ) + Qe
∂t
r ∂r
Mτ
1 ∂
2m
∂Ti
+
r Γi = +
(Te − Ti ) + Qi
∂t
r ∂r
Mτ
suggests that the minimum is due to:
Pthr decreases due to increasing heat transfer from electrons to ions
Pthr increases (stronger edge∇Pi driver needed) due to increase in
shear ow damping
Power and edge heat ux are not the only crit. variables: also need
the ratio of electron energy conf. time to exceed that of e − i temp.
equilibration Tr = τEe /τei - most important in pure e-heating regimes
Tr 1
Tr 1
somewhat equivalent to direct ion heating
ions remain cold
anomalous!)
→
no LH transition (or else, it's
7 / 19
Predator-Prey Model Equations
Based on 1-D numerical 5-eld model (Miki & Diamond++
2012,13+)
Currently operates on 6 elds (+Pe ) with self-consistenly evolved
transport coecients, anomalous heat exchange and NL ow
dissipation (MM, PD, K. Miki, J. Rice and G. Tynan, PoP 2015)
Heat transport, + Two species, with coupling, i,e (anomalous heat
exchange in color):
∂Pe
1 ∂
2m
+
r Γe = −
(Pe − Pi ) + Qe − γCTEM · I
∂t
r ∂r
Mτ
∂Pi
1 ∂
2m
+
r Γi =
(Pe − Pi ) + Qi + γCTEM · I + γZFdiss · I
∂t
r ∂r
Mτ
√ 2
2 √ 2
∂P
∂ E0
∂ E0
Γ = − (χneo + χt )
, γZFdiss = γvisc
+ γHvisc
∂r
∂r
∂r 2
I and E0 - DW and ZF energy (next VG), plasma density and the
mean ow, as before
8 / 19
Equations cont'd; Anomalous Heat Exchange
in high Te low n regimes (pure e-heating)
the thermal coupling is anomalous
(through turbulence)
ZF dissip. (KH?) supplies energy to ions,
and returns energy to turbulence
DW turbulence:
∂I
∂ ∂I
= γ − ∆ωI − α0 E0 − αV hVE i02 I + χN I , χN ∼ ω∗ Cs2
∂t
∂r ∂r
Driver : γ = γITG +γCTEM +NL ZF Dissip less Pi Heat (currently balanced)
ZF energy:
∂E0
=
∂t
γZFdiss
α0 I
− γdamp E0 , γdamp = γcol + γZFdiss · I /E0
1 + ζ0 hVE i02
√ 2
2 √ 2
= γvisc ∂ ∂rE0 + γHvisc ∂ ∂r 2E0 - toy model form (work in
progress)
9 / 19
Model studies: Transition (Collisional Coupling)
ion heat dominated transition
Hi/(i+e) = 0.7
0.005
strong pre-transition uctuations of all
quantities
well organized post-transition ow
strong Pe edge barrier
0.004
0.003
0.002
0.001
000
10
10 000
5 000
0 0
0.2
0.4
0.6
0.8
0.0025
6
0.002
0.0015
4
0.001
0.0005
0
1
5 000
10 000
0.5
5 000
2
00
15 000
0.2
10 000
0.4
0.6
0.8
0 0
5 000
1 0
10 / 19
Model Studies: Control Parameters
Heating mix
Hi/(i+e) ≡
Qi
Qi + Qe
(aka Hmix )
Density (center-line averaged) is NOT a control parameter. It is
measured at each transition point
Related control parameter is the reference density given through
BC and fueling rate
There is a complicated relation between density and ref. density
Other control parameters:
fueling depth
heat deposition depth and width, etc.
→they appear less critical than Hi/(i+e)
11 / 19
Pth n, Hi/(i+e)
scans: Recovering the Minimum
Relate Hi/(i+e) and n by a
monotonic Hi/(i+e) (n)
Pthr Hi/(i+e) , n -
electron heating at lower
densities
ion heating at higher densities
Pthr (n) min recovered!
12 / 19
Summary of collisional coupling results
Pthr (n) grows monotonically in both pure ion Hi/(i+e) = 1 and
pure electron Hi/(i+e) = 0 heating regimes with collisional coupling
The descending (low-density) branch, followed by a distinct
minimum, results from a combination of:
1
2
increase in electron-to-ion collisional heat transfer and
growing fraction of heat Hi/(i+e) ↑ deposited to ions (relative to
total heat)
The later upturn of Pthr (n) is due to increase of the shear ow
damping
The heating mix ratio Hi/(i+e) 6= 0 is essential for the heat
transport from the core to build up the ion pressure gradient at the
edge, ∇Pi , which is the primary driver of the LH transition
There are many possibilities to render Hi/(i+e) 6= 0
13 / 19
Anomalous Regime (Preliminary)
Anomalous Regime: νei n (Te − Ti ) < γanom−eicoupl · I
'78; Zhao, PD, 2012; Garbet, 2013)
(Manheimer,
Anomalous regime, strong electron heating (ITER)
n scaling coupling =⇒ Anomalous coupling
∂Te
1 ∂
2m
+
r Γe = −
(Te − Ti ) + Qe
∂t
r ∂r
Mτ
∂Ti
1 ∂
2m
+
r Γi = +
(Te − Ti ) + Qi
∂t
r ∂r
Mτ
Anomalous coupling dominates
scaling + intensity dependence
=⇒coupling
∂Te
1 ∂
+
r Γe = Qe + hE · Je i → (< 0)
∂t
r ∂r
∂Ti
1 ∂
+
r Γi = Qe + hE·Ji i → (> 0)
∂t
r ∂r
14 / 19
LH transition: Anomalous Transfer Dominates
Extreme limit to illustrate temperature relaxation: Pure electron heating,
νei → 0
% turbulence
& ions
Is Pthr set only by local properties
at the edge?
CTEM →Heat Exch
0.8
0.6
2
1.5
0.4
1
0.5
e − i -temperature
0.2
0
0.2
0
0.4
0.6
0.8
equilibration front
Pi ↑ globally→strong ∇Pi at the edge
→ LH transition
0
1
0.25
0.1
0.2
2
0.05
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
0
0.15
2
0.1
1
0.05
0
0
0.2
0.4
0.6
0.8
0
1
15 / 19
Anomalous Regime: Issues
An Issue:
Predator-Prey ⇒ Shear Flow Damping
⇒Anomalous regime: collisional drag problematic
Low collisionality → what controls heat exchange?
NL damping ⇔ mediated by ZF instability (i.e. KH, tertiary;
Rogers et al 2000; Kim, PD, 2003)
⇒ hyperviscosity, intensity dependent
Returns ZF energy to turbulence → Pi
Results so far
transition with anomalous heat exchange happens!
requirements for LH transition in high Te regimes when the
collisional heat exchange is weak:
ecient ion heating by CTEM turbulence
energy return to turbulence by ZF damping (caused by KH
instability?!)
may be related to Ryter 2014. Subcritical ∇Te ↑ states at ultra-low
density
16 / 19
Conclusions
1
density minimum in Pthr (n) is recovered in the extended model
Pthr decrease: due to e → i heat transfer and ion heating increase
Pthr increase: due to increase in ow damping
2
3
ion heat channel (direct or indirect ⇔ through electrons) is
ultimately responsible for LH transitions
The role of Tr = τEe /τequil (global quantity!) in LH is crucial:
a2 /DGB τequil 1 - no electron-heated LH transition
a2 /DGB τequil 1 - LH trans. originated by electron heat. is possible
4
5
Threshold physics requires, but is not limited, to edge physics
anomalous heat exchange important in low collisionality,
anomalous coupling regimes (collisional e − i heat coupling
negligible)
Anomalous exchange ⇔ Fluctuation intensity dependent
CTEM driven turbulence dissipation → ion heating
ITG driven turbulence dissipation → ion heating
ZF dissipation → ion heating
6
Density minimum is TBD
17 / 19
Future (ongoing) work
complete exploration of anomalous regime
explore eects of ZF spreading
back transitions: quantify hysteresis
fate of minimum in anomalous regime
what are relevant global parameters?
toroidal rotation
geometry/conguration (builds on Fedorczak,
PD, et al. 2012)
∇B -drift asymmetry
Collisionless saturation/damping of CTEM-driven ZF is
fundamental issue
18 / 19
Short Conclusions
Pth (n) minimum recovered with collisional coupling
Threshold physics not limited to edge
a2 /χτeq > 1 required for electron heated transitions
=⇒some global dependence
Predictions:
Anomalous heat exchange and shear ow damping initiated in
collisionless, electron heated regimes (ITER).
Transition manifested as propagating thermal equilibration front;
triggers ∇Pi increase at edge.
19 / 19