Physics of the Power Threshold Minimum for L-H
Transition
M.A. Malkov
University of California, San Diego
1
Collaborators: P.H. Diamond , K. Miki
2,
J.E. Rice
3
and G.R. Tynan
1
Supported by US DoE
1 / 18
Outline
1
1
Motivation
P - the LH transition Power Threshold
-Micro - Macro connection → How does physics set Pthr ?
thr
2
3
4
5
Key Questions
Reduced Model
Basic Structure
Te and Ti equations
Anomalous Coupling
Model Studies
Recovering the Minimum
Towards the Anomalous Regime (Preliminary, if time allows)
Conclusions
1
N.B. No discussion of hysteresis, back transition due to time limitation
2 / 18
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
in general Pthr ∝ nBS
et al 1982)
Connor and Taylor, 1977) and simple
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 / 18
Emerging Scenario
LH-triggering sequence of events
Q
↑ =⇒
etc.
ñ, ṽ
↑=⇒ < ṽr ṽϑ >; < ṽr ṽϑ > d < v >/dr ↑ =⇒
=⇒ ∇Pi | ↑ =⇒ lock in transition (Tynan
|ñ|2 ↓,
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 / 18
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 / 18
Further Questions and important Clue:
J. Hughes, Y. Ma, J. Rice, 2011,12
density at LOC/SOC transition (1020/m3)
1.4
H-mode threshold density minimum
1.2
Is P set only by local
properties at the edge?
(Common wisdom)
Is P minimum related to
collisional energy transfer? i.e.
ν n (Te − Ti ). Low n branch
couples to ions, enables ∇Pi ?
P (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!?
thr
FT
1.0
2.8 < q < 3.8
0.8
C-Mod
thr
TCV
0.6
1/R
0.4
TEXT
ASDEX, AUG
JFT-2M
T-10
0.2
JT-60
Tore Supra
DIII-D
JET
0.0
0.0
0.5
1.0
1.5
2.0
R (m)
Rice et al., 2009
2.5
3.0
3.5
thr
6 / 18
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
+
rΓ = −
(T − Ti ) + Qe
∂t
r ∂r e
Mτ e
1 ∂
2m
∂ Ti
+
rΓ = +
(T − Ti ) + Qi
∂t
r ∂r i
Mτ e
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 somewhat equivalent to direct ion heating
Tr 1 ions remain cold → no LH transition (or else, it's
anomalous!)
7 / 18
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 Γ = − (Pe − Pi ) + Qe − γCTEM · I
∂t
r ∂r e
Mτ
1 ∂
2m
∂ Pi
+
r Γi =
(P − Pi ) + Qi + γCTEM · I + γZFdiss · I
∂t
r ∂r
Mτ e
√ 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 / 18
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 − ∆ω I − 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
α0 I
− γdamp E0 , γdamp = γcol + γZFdiss · I /E0
1 + ζ0 hVE i02
√ 2
2 √ 2
γZFdiss = γvisc ∂ ∂ rE0 + γHvisc ∂ ∂ r 2E0 - toy model form (work in
progress)
9 / 18
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 / 18
Model Studies: Control Parameters
Heating mix
Hi /(i +e ) ≡
Qi
Qi + Qe
(aka
H
mix
)
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 / 18
Pth n, Hi /(i +e )
scans: Recovering the Minimum
Relate Hi /(i +e ) and n by a
monotonic Hi /(i +e ) (n)
0.008
0.006
0.004
0.000
2.5
0.0
0.2
2.0
0.4
Hmix
1.5
0.6
1.0
0.8
0.5
1.0
P
thr
n
0.010
1.0
0.008
0.8
Hi/(i+e)
0.006
0.6
0.0
Hi /(i +e ) , n -
Pthr(n)
min
0.004
Hmix
0.002
Pthr
Pthr
0.4
electron heating at lower
densities
ion heating at higher densities
0.002
0.000
0.2
0.5
P
thr
1.0
20 -3
1.5
2.0
0.0
n/10 m
(n) min recovered!
12 / 18
Summary of collisional coupling results
P
(n) grows monotonically in both pure ion Hi /(i +e ) = 1 and
pure electron Hi /(i +e ) = 0 heating regimes with collisional coupling
thr
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 P (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
thr
13 / 18
Anomalous Regime (Preliminary)
Anomalous Regime: νei n (Te − Ti ) < γ
'78; Zhao, PD, 2012; Garbet, 2013)
−eicoupl
anom
·I
(Manheimer,
Anomalous regime, strong electron heating (ITER)
n scaling coupling =⇒ Anomalous coupling
∂ Te
1 ∂
2m
+
r Γ = − (Te − Ti ) + Qe
∂t
r ∂r e
Mτ
∂ Ti
1 ∂
2m
+
r Γ = + (Te − Ti ) + Qi
∂t
r ∂r i
Mτ
Anomalous coupling dominates
scaling + intensity dependence
=⇒coupling
∂ Te
1 ∂
+
r Γ = Qe + hE · Je i → (< 0)
∂t
r ∂r e
∂ Ti
1 ∂
+
r Γ = Qe + hE·Ji i → (> 0)
∂t
r ∂r i
14 / 18
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 equilibration front
Pi ↑ globally→strong ∇Pi at the edge
0.2
0
0
0.2
0
0.4
0.6
0.8
1
→ LH transition
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 / 18
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 / 18
Conclusions
1
2
3
4
5
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
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
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 / 18
Future (ongoing) work
continue exploration of anomalous regime
explore eects of ZF anomalous spreading
back transitions, hysteresis
fate of minimum in anomalous regime
what are relevant global parameters?
toroidal rotation
geometry/conguration (builds on Fedorczak, PD, et al. 2012)
→ Collisionless saturation/damping of CTEM-driven ZF is
fundamental issue
18 / 18