L-H Transition Dynamics and Power Threshold
Minimum
M.A. Malkov1 , P.H. Diamond1,2 , K. Miki2,3 and G.R. Tynan1
1 University
of California, San Diego, USA;
3
e-mail:
mmalkov@ucsd.edu
2 WCI,
NFRI Daejeon, Korea, JAEA, Kashiwa, Japan
APS 2014 CP8.00048
1 / 23
Abstract
A physical model to explain the minimum in the power threshold
of the L→H transition is suggested. Its crucial new elements,
compared to the prototype due to (Miki & Diamond 2012), are
the separately evolved electron and ion heat transfers and the
independent power supply in each chanel.
The power threshold minimum may be explained as follows. The
low-density branch, where the heat initially goes to the electrons,
is associated with growing eciency of e − i heat transfer and
the ow generation. This suppresses the transport and lowers
the threshold Pth. . The high-density branch is associated with
increased collisional damping of the ow. Model studies also
reveal: (a) an increase in threshold power for o-axis electron
heat deposition and (b) the absence of a clear P (n) minimum
for pure ion heating.
thr
2 / 23
Outline
1
Basic physics of LH transition
2
Recent Experiments and Shortcomings of Available Models
3
Predecessor Model and its Extension to Studies of Pth
Minimum
4
Model Equations
5
Results and analysis
6
Conclusions
3 / 23
Mechanism and occurrence of LH transition
originates via coupling of turbulence to low frequency shear
ows by Reynolds work
Reynolds work causes collapse of turbulence and then
turbulent transport
diamagnetic electric eld grows, associated with ∇P
→LH transition
can occur via a protracted I-phase or in a single burst of
shear ow
4 / 23
Ob jectives 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 branching ratio
role of mean shear in locking-in of the transition
5 / 23
Observations of power threshold minimum
Ryter et al 2013 [1]
ion heat ux plays a dominant
role in LH transition → ∇P
electron channel thought to be
ignorable
But: in low-density regimes
with dominant EC heating
electrons must transfer energy
to ions to steepen ∇P
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)
6 / 23
1-D Numerical Model
Based on 1-D numerical 5-eld model (Miki & Diamond
2012,13+) [2] signicantly extends Kim & Diamond 2003
[3] 0D model
MD 2012 captures transition layer evolution but does not
separate species
modify MD 2012 by adding separate electron heat
transport equation
include e-i thermal coupling depending on locally
evolved temperatures and density
include these parameters in ZF damping description
include trapped electron growth
7 / 23
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
8 / 23
Equations cont'd
DW turbulence with ITG and TEM instabilities
∂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
L
−
−
Ln
R
∂I
crit ∂ r
Vϑ − 1.17Cs ρi LT−1
9 / 23
Control parameters for transition
plasma density
- nref -reference (xed) density
- n -center line averaged (current) density
Heat mix parameter
Qi
Qi
≡
Qi + Qe Q
i.e., Qi = H Q , Qe = (1 − H ) Q , where Q denotes the
Hmix =
mix
mix
total power deposited into the plasma.
widths of the heat sources
∆re = ∆ri = 0.15a,
heat deposition radii ae ,i = 0.3a
10 / 23
Transition morphology
L→H transition event shown in four characteristic variables
center line averaged:
density,
ZF energy,
MF E × B velocity,
DW energy.
shown as functions of heating rate Q (t ). Data points are taken
at equal time intervals so their density indicates both the rate at
which Q is changing and how quickly the changes in the
variables occur.
11 / 23
Transition morphology
0.037
<Vθ > , Mean Flow
<n>, density
2.4
2.2
2.0
1.8
1.6
0.0065
-10
0.0075
0.0085
0.033
0.031
0.029
0.0065
Q(t)
0.0075
0.0085
Q(t)
-11
ln<I>, DW Energy
ln<(E0)>, ZF Energy
-10.3
-12
-13
-10.5
-14
-15
-10.7
-16
-17
0.0065
0.035
0.0075
Q(t)
0.0085
-10.9
0.0065
0.0075
Q(t)
0.0085
12 / 23
Transition morphology
Top row:
- an example of LH transition with an extended
pre-transition I-phase shown for the electron pressure Pe ,
ZF energy and the MF velocity
- strong, edge-localized MF is a marker of the H-mode.
Bottom row:
- an example of a failed transition with inward propagation
- the edge MF jet starts to form but then merge with the
large scale MF
13 / 23
Transition morphology
14 / 23
Identifying transitions
take half-way to clean pedestal
cross-check with DW,ZF,MF
channels
→need transition criterion
to scan
Pth n, He ,i , Ldep , ...
- very weak transitions occur and
are hard to detect
15 / 23
Spatio-temporal dynamics of transition
I-phase persists before
transition
clearly spatio-temporal
behavior beyond 0-D model
(similar to MD 2012)
ZF signicantly advances into the
core before transition
16 / 23
Spatio-temporal dynamics of transition
slight temperature attening
in the core due to enhanced
turbulent transport
I-phase in density
17 / 23
Pth (n)
scans
Squares indicate strong transitions
with the density jumps & 0.1
Circles indicate weaker transitions.
as a function of reference
density
0.008
0.007
0.005
0.004
0.003
0.002
0.008
0.001
0.000
0.8
1.2
1.6
2.0
0.006
n, density
Pth
Pth
0.006
0.004
P , shown against the
center-line averaged n
thr
0.002
0.000
1.0
1.5
2.0
2.5
3.0
Nref , reference density
18 / 23
Pth (n, Hmix )
scans
0.008
0.006
0.004 Pth
0.002
0.000
2.5
0.0
0.2
1.5
0.8
0.5
0.010
1.0
n
0.0
in heating mix -density variables, Hmix and n.
n, Hmix choice: electron/ion biased
heating at lower/higher densities
extended sub-sample at Hmix = 1 is
also included
0.008
0.8
Hmix(n)
0.006
0.6
thr
0.004
0.4
Pth(n)
0.002
0.000
Hmix
1.0
Pth
0.6
1.0
P
thr
2.0
0.4
Hmix
Monotonic dependence of Hmix (n) ,
arbitrarily chosen from the sample
shown in the previous Figure and
the resulting P (n) constrained by
the above relation.
0.2
0.5
1.0
-3
20
n/10 m
1.5
2.0
0.0
19 / 23
Model Findings
no clear threshold minimum for pure ion heat deposition
minimum consistend with observations [1, 4] found for
mixed e − i heating
on low-density branch of P (n) electrons absorb most of
the heat initially and as they transfer it to ions more
eciently with growing density, P (n) decreases
on high-density branch ions are heated and as ZF damping
is growing with n, P must also grow
thr
thr
thr
20 / 23
Model Findings
the overall picture is consistent with the following two
premises:
- L-H transition is locked in by VE0 ∼ (∇Pi /n)0
- DW turbulence coupling to ow is a key trigger
Pth increases for o-axis electron heat deposition (reduced
electron-ion coupling)
shallow minimum power is predicted for heating mix scan
as well as for density scan
from above ndings a global minimum in multi-parameter
space is predicted
quantifying strength of hysteresis in terms of
multiple macroscopic parameters;
relating this to observed back-transition shear ow and turbulence
dynamics
Ongoing work:
21 / 23
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
22 / 23
References
[1] F. Ryter et al., Nucl. Fusion 53, 113003 (2013).
[2] K. Miki et al., Phys. Plasmas 19, 092306 (2012).
[3] E.-J. Kim and P. H. Diamond, Physical Review Letters 90,
185006 (2003).
[4] C. F. Maggi et al., Nuclear Fusion 54, 023007 (2014).
23 / 23