Towards a Model of the L-H Power Threshold
Scaling
Mikhail Malkov
UCSD
Collaborators: P. Diamond and K. Miki
1 / 18
Outline
1
Basic physics of LH transition
2
Recent Incentive Experiments and Shortcomings of Available
Models
3
Available Physical Models and their Retrotting to Studies of
Pth Minimum
4
Model Equations
5
Results and analysis
6
Conclusions
2 / 18
Mechanism and occurrence of LH transition
originates via coupling of turbulence to low frequency shear
ows by Reynolds work
causes collapse of turbulence and turbulent transport
growth of diamagnetic electric eld associated with ∇P
→LH transition
occur via a protracted I-phase or in a single burst of shear
ow
3 / 18
Objectives of this work (ongoing)
establish link between microscopics and macroscopics in
power threshold scaling
reproduce and understand observed threshold Pth (n)
minimum
explore Pth in terms of other parameters, such as e-i
thermal coupling eciency, noise...
investigate the role of heating prole in LH transition
e-i heating split ratio
mean shear in locking-in of the transition
4 / 18
Observations of power threshold minimum
Ryter et al 2013
ion heat ux plays a dominant
role in LH transition
electron channel is ignorable
But: in low-density regimes
with dominant EC heating
electrons must transfer energy
to ions
electron description must be
separated from that of ions
temperature and density
dependence of collision rate
need to be included (average
values do not suce)
5 / 18
Preceding models: Advantages of 0-D
2
N
E
U
E,U,N
1.5
1
0.5
0
0
100
200
300
400
time
(M & Diamond 2009)
similar Predator-Prey
dynamics in 0D+1D-k-space
(M, Diamond & Rosenbluth
2001)
500
0-D KD2003 model captures
pre-transition limit-cycle
oscillations (Kim & Diamond
2003)
reproduces NL period growth
(L. Schmitz, this meeting)
allows simple dynamical
system interpretation of L-I-H
transition as Hopf bifurcation
from unstable xed point to
LC
6 / 18
0-D model prospective
to another unstable
(hyperbolic) FP (H-mode)
nally to stable xed point QH
0.5
E
L
T
H
QH
0
0
1
0.5
N
M & Diamond 2009
7 / 18
New 1-D Numerical Model
Based on 1-D numerical 5-eld model (Miki & Diamond
2012,13+) signicantly extends KD 2003 0D model
MD 2012 captures transition layer evolution but is
incapable of separating species
modify MD 2012 by adding separate electron heat
transport equation
include e-i thermal coupling depending on locally evolving
temperatures and density
include these parameters in ZF damping description
include trapped electron growth
8 / 18
Predator-Prey Model Equations
Heat transport i,e:
∂ Pi ,e
1 ∂ (p)
2m
(r − a)2
+
r Γ = ± (Pe − Pi ) + Q exp
∂t
r ∂ r i ,e
Mτ
2 ∆r 2
Γ = − (χneo + χt )
Density
∂P
∂r
"
#
a−r
∂ n 1 ∂ (n)
(a − r )2
+
r Γ = Γa 2 exp
∂t
r ∂r
Ldep
2L2dep
Γ(n) = − (Dneo + Dt )
χ, Dt =
τc Cs2 I
1 + αt hVE i
02
,
∂n
∂r
1
hVE i0 = ρi Cs L−
p
L−p 1 − L−n 1
−hVϑ i0
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Equations cont'd
DW turbulence
∂I
∂ ∂I
= γL − ∆ω I − α0 E0 − αV hVE i02 I +χN I , χN ∼ ω∗ Cs2
∂t
∂r ∂r
s
Cs R R
L
1
−1
−
−
+ γ0e Cs L−
γL = γ0i
Te + Ln
R
ZF energy
Lp
Ln
R
crit
∂ E0
α0 E0 I
=
− γdamp E0
∂t
1 + ζ 0 h Vϑ i 2
mean ow shear
∂ hVϑ i
a
= −α5 γ0i Cs2
∂t
R
−µneo νii q 2 R 2
s
R
Lp
R
−
Ln
L
R
∂I
crit ∂ r
−
Vϑ − 1.17Cs ρi LT−1
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Transition dynamics, transition criterion
take half-way to clean pedestal
cross-check with DW,ZF,MF
channels
→need transition criterion
to scan
Pth n, He ,i , Ldep , ...
11 / 18
Spatio-temporal dynamics of transition
I-phase persists before
transition
clearly spatio-temporal
behavior beyond 0-D model
ZF signicantly advances into the
core before transition
12 / 18
Spatio-temporal dynamics of transition
slight temperature attening
in the core due to enhanced
turbulent transport
I-phase in density
13 / 18
Accurate transition identication
0.50
<n>
0.45
many transitions are poorly
resolved
select only well resolved
transitions for density and
power scans
n_scale=1.00_Q=0.00050--0.0030.dat
x-aver density
0.40
0.35
0.30
<n>
0.7
Qth
0.20
n_scale=1.50_Q=0.00500--0.0180.dat
0.8
0.25
0.0010
0.0015
0.0020
0.0025
0.0030
0.6
0.5
Shallow Pth (n, . . . .) minimum requires accurate determination of
transition point
0.4
0.3
0.2
Qth
0.006
0.008
0.010
0.012
0.014
0.016
0.018
14 / 18
Identifying transitions, Pth density scans
n_scale=1.00_Q[=0.0248--0.0310].dat
n_scale=1.00_Q[=0.0195--0.0260].dat
1.1
0.9
1.0
0.9
Ldep=0
0.7
<n>
<n>
0.8
0.6
Ldep=0.4
0.8
0.7
0.6
0.5
0.5
0.4
0.4
0.3
0.025
0.026 0.027
0.3
0.028 0.029 0.030
0.020
0.021
0.022
Q(t)
0.9
0.28
0.8
0.26
Ldep=0.3
0.6
0.024
0.025
scan.pdw
0.30
Qth
<n>
n_scale=1.00_Q[=0.0190--0.0210].dat
1.0
0.7
0.023
Q(t)
0.24
0.22
0.5
0.20
0.4
0.18
0.16
0.3
0.0195
0.0200
Q(t)
0.0205
0.80
0.85
0.90
n_h
0.95
1.00
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Pth scans
very bottom of the curve is not
robust in density scans (work
ongoing)
considerably more robust in
other representations (such as
e-i heating ratio)
sharp rise to 0.02
0.0020
PL-H
0.0015
0.0010
0.0005
0.020
0.0000
0.0
0.2
0.4
0.6
0.8
1.0
Hspl=Hi/(Hi+He)
0.016
0.012
Power threshold minimum recovered
PL-H
0.008
0.004
<N>pre-trans
0.000
0.30
0.35
0.40
0.45
0.50
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Analysis
threshold power increase for o-axis electron heat
deposition (reduced electron-ion coupling)
minimum power is predicted for heating mix scan as well as
for density scan →
→possibility of a global minimum in multi-parameter space
no clear threshold minimum for pure ion heat deposition
Ongoing work is concerned with quantifying the strength of hysteresis in terms of multiple macroscopic parameters and with relating this to observed back-transition shear ow and turbulence
dynamics
17 / 18
Conclusions
an extended 6-eld 1-D PDE model is developed (Pe , Pi , n,
DW, ZF, Mean Flow)
link between microscopics (e-i collisional heat exchange,
turbulence) and macroscopics (transport barrier, P-n
proles) in power threshold scaling is established
threshold Pth (n) minimum is reproduced and understood
using a simple model of e-i heat transfer
Pth n, Ldep , . . . . is explored in terms of its dependence on
other parameters, such as e-i thermal coupling eciency
the role of heating prole in LH transition is investigated
the role of e-i heating split ratio is studied, minimum of Pth
predicted
role of mean shear in locking-in of transition is signicant
18 / 18