Gyrokinetic Simulations of the Nonlinear Upshift of the
Critical Density Gradient for TEM Turbulence in
Tokamak Fusion Plasmas
Kyle M.Zeller
SUBMITTED TO THE DEPARTMENT OF NUCLEAR SCIENCE AND
ENGINEERING
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE
OF
BACHELOR OF SCIENCE IN NUCLEAR SCIENCE AND ENGINEERING AT
THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
JUNE 2006
@Kyle Zeller. All Rights Reserved
The author herby grants to MIT permission to reproduce and to distribute publicly paper
and electronic copies of this thesis document in whole or in part.
Signature of Author:
-
•y-
Z
Kyle Zeller
Depjý
of Nuclear Science and Engineering
5/12/06
5/12-O6
Certified by:.
Darin Ernst
MIT Plasma Science and Fusion Center
Thesis Supervisor
Accepted by:
David G. Cory
-0sSACHUSERms
Professor of Nuclear Science and Engineering
Chairman, NSE Committee for Undergraduate Students
IN~ fITUIY
I., rIi'67r:
OF TECHNOLOC.'Y
OCT 1i2 2007
LIBRARIES
ARCHIVE~
-1-
Gyrokinetic Simulations of the Nonlinear Upshift of the Critical
Density Gradient for TEM Turbulence in Tokamak Fusion Plasmas
By
Kyle Zeller
Submitted to the Department of Nuclear Science and Engineering on 5/12/06
In Partial Fulfillment of the Requirements for the Degree of
Bachelor of Science in Nuclear Science and Engineering
ABSTRACT
The effect of collisionality on a new nonlinear upshift of the critical density
gradient for onset of Trapped Electron Mode (TEM) turbulence is investigated in detail.
Both linear and nonlinear, high resolution simulations were performed on massively
parallel computers using the gyrokinetic code, GS2. The TEM nonlinear upshift is
analogous to the Dimits Shift for ion temperature gradient driven (ITG) turbulence, but
exists in the density gradient as opposed to the temperature gradient. In the ITG case,
increasing ion-ion collisions damp the zonal flows but have little effect on the linear
growth rate. In contrast, electron-ion collisions strongly damp the TEM growth rate,
while ion-ion collisions weakly damp zonal flows, causing an increase in the TEM
upshift. Numerous simulations were run, scanning different density gradients to
determine the critical density gradients for each collisionality and to examine the upshift
caused by increasing collisionality. The linear critical density gradient was not
significantly affected by collisionality, while both critical density gradients were
determined to be larger for the nonlinear runs.
-2-
Acknowledgements
I would like to thank Dr. Darin Ernst for continual guidance and motivation
throughout the years we have worked together. It has been my pleasure to work with
someone with such a passion for plasma theory. Without his assistance, explanations,
and tireless effort along the way, this project would not have been possible.
Parallel computations of linear growth rates were performed on the 50 processor
MIT PSFC Marshall Theory Cluster maintained by Darin Ernst, John Wright, and Ted
Baker. Massively parallel nonlinear turbulence simulations were performed on the IBM
SP/2 and IBM p575 POWER 5 system at the National Energy Research Supercomputer
Center (NERSC) in Berkeley.
-3-
1
Introduction
Fusion power offers the potential of a nearly limitless source of energy for future
generations. Unlike fossil fuels and nuclear fission, this energy source is inexpensive and
would produce almost no waste products. Fusion is a process where two light nuclei join
together to form a heavier nucleus. Nuclear fusion is accompanied by the release of
neutrons and energy, which can be used to sustain a reaction and create power. Physics
experiments have successfully demonstrated production of fusion power, with the amount
of power produced reaching two-thirds of the input power. Nevertheless, there are many
technical challenges which have yet to be overcome.
It takes large amounts of energy to force two nuclei to fuse, due to the strong
repellent forces of protons. In order to overcome this mutual repulsion, it is necessary to
increase the energy of the nuclei, as the cross-section for fusion increases with energy
and reaches a maximum at 100keV. Though this is a large amount of energy to supply
efficiently, the fusion reaction releases up to 18 MeV; an exothermic reaction with a
substantial energy increase. Currently, the most promising method of supplying the
energy is to heat the fuel (usually deuterium-tritium) to a sufficiently high temperature so
as to increase the thermal velocities of the fuel particles. Good confinement is required to
attain high enough densities and temperatures, as the fusion power produced is
proportional to the pressure squared. The necessary temperature to create this
"thermonuclear" fusion is around 10keV, and at such temperatures, the fuel is a fully
ionized plasma. 1
In the attempt to achieve a useful power source from thermonuclear fusion, the
tokamak has become the predominant research tool. In a tokamak, the plasma particles
-4-
are confined to gyrating orbits around a torus by helical magnetic fields. This technology
has progressed over the past twenty years, and has been successful in reaching both the
required confinement times and densities to sustain significant fusion reactions. The first
high power fusion experiments, in 1994, on the Princeton Tokamak Fusion Test Reactor,
produced 10.7 MW for approximately one second, with 40 MW of heating power. 2 Later,
the Joint European Torus (JET) produced 16 MW of fusion power for 26 MW of heating
power. 3 Today, the international project, ITER, is being developed, and will be
completed by 2017 in Cadarache, France. Together, the countries of China, the European
Union, India, Japan, South Korea, Russia, the USA, and Switzerland will work together
with the IAEA to develop this experimental step between the studies of plasma physics
and an electricity-producing fusion power reactor. This 12 billion dollar project will
produce around 500 MW of fusion power for at least five seconds.4
Since the confinement in all tokamaks is never perfect, some plasma may leak
through the confining fields. This leakage process is turbulent and chaotic, and predicting
the amount of heat and numbers of particles lost is a major theoretical and computational
challenge. This project will focus on the behavior of one of these causes of particle
energy loss, specifically, Trapped Electron Mode (TEM) turbulence in the tokamak
fusion plasma near the threshold of instability. TEM turbulence plays an essential role in
determining the turbulent particle and electron thermal energy transport. This is a major
limiting factor in obtaining the required densities needed for the confinement of plasmas
and the production of fusion power.
-5-
2
Background Information and Previous Works
2.1
Trapped Electron Modes (TEM)
The trapped electron mode (TEM) is one important microinstability which drives
turbulence in plasmas. It is an instability of the electron drift wave and depends on the
presence of magnetically trapped, banana-orbiting electrons in the equilibrium. Drift
waves are waves propagating mainly perpendicular to the magnetic field, whose phase
velocity parallel to the magnetic field is somewhere between that of an ion's and
electron's thermal speeds. A drift wave with the electron diamagnetic frequency, o)•, is
called an "electron drift wave" since it propagates near the electron drift velocity.' In a
collisionless plasma, this electron drift wave is driven unstable by processes that result in
a non-adiabatic electron response to perturbations, such as inverse Landau damping,
resonances with trapped electrons, and bad curvature. In addition to these non-adiabatic
electron responses, plasmas with large collisionalities can be driven unstable by
collisional dissipation of trapped electrons.
The trapping of electrons prevents them from responding adiabatically to a
perturbation in the electrostatic potential,
4.
This prevents the electrons from taking up a
Boltzmann distribution, &ne = enl/Te, and as a result, the perturbed electron density takes
the form:'
S~•, e_ e
Fue
T
n
n
Te
)
2T
iv
me
d- +
En
-6-
2
dv
where:
FMe = Maxwellian distribution with density n and temperature Te
dlnT
dlnn
7li =0
--,
,*e
2
kog
k= is the magnetic curvature, where g = 'h is the effective gravity.
L2O
R
kD
is the electron drift frequency
cT
D = Z ' is the Bohm diffusion coefficent
B
ZeB,
e
E= r/R o is the inverse aspect ratio with major radius, R, and minor radius, r.
dlnn
dr
(•) =perturbation in potential averaged over the banana orbit
Two types of trapped electron modes were initially discovered. 5'6 The first type
of TEM is a collisionless mode, driven unstable by resonance with the trapped electron
toroidal precession drift. The banana orbits of trapped electrons drift toroidally as a result
of magnetic curvature and VB drifts. This collisionless type of TEM is more relevant to
the study conducted, as there is not a very high collision frequency. In this mode, the
non-trapped, circulating electrons are adiabatic; they can instantaneously respond to a
perturbation in the potential. Their perturbed electron density may be described by
lec = (enl/Te), where ý is the perturbed electrostatic potential.7 On the other hand, the
trapped electrons are not entirely free to move along field lines in response to potential
-7-
perturbations, so they are not adiabatic. In the fluid limit, 5 for
De
<<
0, their response
to a perturbation is determined by a combination of equilibrium precession drift, and
perturbed ExB drift: iier = [*e,/()o - •,)] x (ener / Te).
In the more general case,
including the effect of precession drift resonance for trapped electrons, but treating ions
in the slab limit and taking zero ion temperature gradient, the resulting dispersion relation
can be written as 8 .
1-
IOe
n
(o
++
n
*e
-xVa)_
J
2
e
-
[1 + e(x
-3/2)]
x
o_)-
(1)
=ETdx=0
ex+(ive/e)
(1
This yields an unstable mode, even when the electron temperature gradient, rl , is
zero. This mode is driven by the precession drift of trapped electrons and of all the ions
are "caught" by the mode in a region of unfavorable magnetic curvature, in a
combination with their density gradient.'( If the trapped electron response in Eq. (1) is
ignored, the stable electron drift wave with frequency o = oe is obtained. Treating the
trapped electron response as a perturbation, i.e., n, In =
<< 1,and allowing for the
possibility for the denominator of Eq. (1) to go to zero, an approximate expression for the
growth rate can be obtained. Taking (o = oe + iy, where potential and other
perturbations are described by e(a,k 1;t) = oe - i
t+ik±y and
k i = k 9 , yields the simplest
resonant Trapped Electron Mode. 9 This mode has a nonzero growth rate, 7, which can be
written as:
Y=
'
e nT2
n
7qe
(
--
2
32
( 2a)
1,
where E, is defined as:
-8-
en
L
"
R
-
-
ioee
oe
, and
1
L,
=
1 dn
n dr
The real frequency of this mode is approximately, 9
10 = (O,e
(2b)
2- F0 '
where 1o7is defined as FT = o(b)e-b, with b= kp,22 and Iois the Bessel function of the
first kind.
In order for the electron to become "trapped", it must not have enough energy to
complete a total gyro-orbit. Since the electron's energy is given by:
E= mvy + - mv 2
or
1
E=-mv l +pB
where p is defined as
my 12
2B
This means that v11 can be written as:
VI = (E - uB
2
Therefore, if vii never reaches zero, the electron is circulating, completing a full orbit
along filed lines. However, if vRldoes equal zero at some poloidal angle, that angle is a
turning point, and the electron is trapped in a banana orbit, transferring its energy from v,,
to vI.
Because the magnetic field, B = BoR o IR
/ = Bo / (1+ ecos 0) and E = r/R << 1, it
follows that,
-9-
=Bo
=+'E-ol+ecos(O)
v+l
2
m
and with some algebra, the above becomes,
VII =
(- A(1-ecosO)Y), for e<<1,
where X is defined as,
E
Setting the parallel velocity to zero at the outboard midplane yields the maximum
value, Xma. = 1+e, for deeply trapped particles. The critical value of A above which all
particles are trapped is found by taking the parallel velocity to be zero at the inboard
midplane 0 = Ic, where Ac = 1-E . On the outboard midline, the trapped electrons occupy
the region of velocity space lying within a cone as shown in Fig. (1). The angle of this
cone can be calculated as follows:
l_ ,c(-ECOSO)
vii2
- ,I
,
= 2e + O(
_
1-c(
2)
2it
If the equilibrium distribution function is isotropic, this means that the fraction of
trapped electrons, n-, is nearly equal to -X, which is the fractional surface area
n
occupied by trapped particles on a constant energy sphere, as shown in Fig. (1).
-10-
Figure 1: Trapped Region of Velocity Space
UL
The other type of TEM is the "dissipative trapped electron mode," or DTEM.6 In
DTEM, an electron drift wave is driven unstable by the nonadiabatic electron response
produced by detrapping the electrons with effective collision frequency veff = ve /e.This
effective frequency represents the fact that the collisions do not conserve the number,
energy, or momentum of the trapped electrons, and is enhanced from the 90 degree
collision frequency ve to the collision frequency ve le for scattering a trapped electron
through the small angle -v necessary to cause detrapping. Generally the collision
frequency is not large enough in the hot plasma core for this dissipative mode to be
relevant. For more on general TEM theory see Refs. [5-11].
2.2
Zonal Flows: Previous Works
There has been a lot of progress made in the past decade towards the
understanding of particle transport and microturbulence in plasmas. Specifically, the
study of ExB poloidal flows, or zonal flows, has become an integral part of tokamak
- 11 -
plasma research. Zonal flows have been found to be a important part of self-regulation
for drift wave transport and turbulence, as they shear apart eddies associated with
turbulence, reducing radial transport.
In tokamak plasmas, the zonal flows have zero or very low frequency, and are
poloidally symmetric. They are constant on a magnetic surface, but rapidly vary in the
radial direction. According to Ref. [12], zonal flows are not themselves unstable, but
driven only by a nonlinear transfer of energy from finite-n drift waves. This nonlinear
drift wave interaction is a three-wave triad coupling between two high-k drift waves and
one low q=qrr (where q, is the radial wavenumber) zonal flow excitation. Secondary
modes, which interact with the primary mode, drive the zonal flows, and they then
transfer energy from the primaries. Because of this, zonal flow generation both reduces
the intensity and level of transport caused by the primary drift wave turbulence. Most of
the interest in zonal flows comes from this ability to regulate turbulence via shearing.
An example demonstrating the importance of zonal flows is the Dimits Shift. 13
Dimits analysis of the critical ion temperature gradient for toroidal ITG turbulence
showed a nonlinear upshift of the critical temperature gradient of the onset of ITG
turbulence. As shown in Ref. [14], if the temperature gradient is measured by a
dimensionless parameter e, and if the threshold for linear instability is e,, there is a
regime E,< e < & where only zonal flows, but not drift wave activity is observed. The
Dimits shift is then Ae==P - E, where &.is the onset of the driftwave turbulence. 14 In
the interval As, the primaries are completely self-quenched, and the turbulence is
nonlinearly stable, yet linearly unstable. The original demonstration of the Dimits shift is
shown in Fig. (2).
-12-
.4
10,
IC-
+
F-d
*
,.
~
8
+/
S +,
4
-
I
/
+
1 /
/
,/
/.
fit/tottoLLNLGK
U.Col. OK
94 IFSIPPPL -97PPPL.GFL +
~~R
2
_.,
LLNL GK
8SPPPL GR.
SydoraGK
GK
Weilard QL-.TG
A
C'
.0
"
T
R/L
RLTexpt
10
15
20
R/Lyi
Fig. 2: Original Dimits shift, Ref [13]. A range of temperature gradients exists for which
ITG modes are linearly unstable, yet nonlinearly stable.
A proposed theory of this shift is given in Ref. [15] by Rogers, Dorland, and
Kotschenreuther. They focused on the behavior of the zonal flows in marginally stable
conditions just before turbulent saturation, when the secondary modes have large growth
rates compared to primary modes, but relatively small amplitudes. In this regime, they
found that the secondaries grow as the exponential of an exponential, and it is here that
most of the zonal flows are generated. These quickly growing secondary modes are
Kelvin-Helmholz-like, and drive a zonal flow component. These zonal flows can go
undamped for a certain time, and nearly suppress further turbulence in the system.
Rogers et. al. proposed that this shear-flow-dominated state is damped by "tertiary mode"
instabilities, which grow when the zonal flows exceed a certain threshold. These tertiary
modes grow to a nonlinear amplitude large enough to completely suppress the zonal
flows, and allow the primary modes to continue to grow. It is believed that the threshold
when these tertiary modes begin is associated with the boundary of the nonlinear Dimits
shift.
-13-
The secondary modes grow above a threshold described by modulation instability
theory, as reviewed in Diamond et al.12:
22
Yx = qyqx
-q
primary
qX
(3)
y =L
Yx
YZF
Cs
where
q, = kyp, ; ky is the wavenumber of the primary
qx= kz p,; krF is the wavenumber of the zonal flow
'primary
P, T,
2_ T
2ZeB
and p
•
mi (Oci
and c, =
, with o,
mic
mi
From this expression, it is clear that the modulational instability (secondary) has a
growth rate nearly proportional to the amplitude of the primary mode. If the primary
mode is growing exponentially in time with growth rate y, the zonal flows will grow
extremely rapidly, almost explosively, with a growth rate
'z
oc exp(exp(yt)). If the
zonal flows are only weakly damped, it is not hard to see that they can take most of the
energy away from the primary modes, resulting in a rotationally dominated state. The
Dimits shift region is characterized by these rotationally dominated states.
A new nonlinear upshift of the TEM critical density gradient was discovered by
Ernst, during the course of performing nonlinear gyrokinetic simulations of TEM
turbulence in Alcator C-mod internal transport barriers.16 While focusing on particle and
electron thermal transport, this study 6',1 7,18 found that the onset of TEM turbulence
-14-
produces an outflow that strongly increases with the density gradient at a point when the
linear critical density gradient is exceeded by a significant margin. Dr. Ernst suspected
that this was a new nonlinear upshift similar to the Dimits shift, however, this upshift
occurs in TEM rather than toroidal ion temperature gradient driven turbulence. It occurs
for TEM driven solely by the density gradient, resulting in an increase of the critical
density gradient for the onset of turbulent particle and electron heat transport. This
upshift is shown in Fig. (3).
.
.
4
I'T
E
2
S1
0
1.0
1.2
1.4
1.6
1.8
2.0
NL Shift
Figure 3: Nonlinear upshift for the TEM in Alcator C-Mod internal transport barriers.
Ref [16].
Trapped electron modes are known to be strongly damped by collisions. Because
zonal flows are weakly damped by realistic ion-ion collisions, Dr. Ernst hypothesized that
increasing the collisionality would increase the TEM upshift. The work presented here
confirms this hypothesis with a series of detailed nonlinear simulations at two
- 15-
collisionalities. In addition, the role of zonal flows is clearly illustrated by showing the
existence of rotationally dominated states in the upshift region.
2.3
GS2: Gyrokinetic Code
In order to simulate TEM, GS2 was utilized. The GS2 code solves the
gyrokinetic Vlasov-Maxwell equations as an initial value problem, using the ballooning
representation for linear cases, and a radially local, flux tube simulation domain for
nonlinear cases. 16 19 The term "gyrokinetic" refers to a model for charged particles in a
strong magnetic field that evolves gyro-averaged functions of position-velocity phasespace variables and time.
GS2 is based on the electromagnetic nonlinear gyrokinetic equation, which
describes the evolution of fluctuations which satisfy:
Fo
T
Bp
B
Q
L
h E<<1 kpL~ kp1
where:
h = nonadiabatic part of the perturbed distribution function
Fo = equilibrium distribution function
O= perturbed part of the electrostatic vector potential
All= perturbed part of the parallel vector potential
B1= perturbed parallel magnetic field
B= equilibrium magnetic field
L= equilibrium scale length of density, temperature, or magnetic field
eB
mc
V
m
e= charge
-16-
GS2 20 ,22 features both linear and nonlinear simulations, with a solution method
that is fully implicit for linear terms and explicit for nonlinear terms. For linear runs, the
linear microstability growth rates are calculated on a wavenumber-by-wavenumber basis,
and for nonlinear simulations of fully developed plasma, species-by-species anomalous
transport coefficients for particles, fluctuation spectra, heating rates, momentum, and
energy are all provided. 20
Because the simulations are performed in field-line-following coordinates using
toroidal flux tubes, the nonlinear gyrokinetic equation can be shown as:23. 24,25
-' +VI +iod +C
h+-c{,h}=io.x -e-
dt
B
Fm
dE
dt
where the fields are represented by:
S = Jo( b
- A A,(b)
c
b
mvI
e
B
where Jo is the Bessel function
and:
h = h(y, a,0,E,u;t)x eino(a+q°O), where 0 is the poloidal projection of the distance along
field lines, V is constant on the flux surface, and a is constant on the field lines.
b
k.kvi
B= Vax Vy, and Vyfoc k, and Vao ky
B'V V 0
oV xVft.
a=J
a
The self-consistent electromagnetic field fluctuations are computed from the gyrokinetic
Poisson-Ampere equations,
-17-
VI =
dEdlt
v
vii
4e 27trB
J dEd uJ
species
V2 Al VA 11 =
,B
B
C
X
c
,
edF
B
4re
m
2
47e 27rB
B 2m
•
dE
(5)
4
vi1
(4)
b
(6)
Since 3 = 8np/B2 << 1, this implies that electromagnetic effects are unimportant, and
only Eq. (4) is solved for.
-18-
3
Methods
In order to collect the necessary data, nonlinear simulations of micro-turbulence
were performed using a comprehensive kinetic computer code on massively parallel
computers, including the 6,080 processor IBM SP RS/6000 and 888 processor IBM p575
POWER 5 systems at the DOE National Energy Research Supercomputing Center
(NERSC) in Berkeley, California. Several hundred thousand hours of processor time
were utilized, and GS2 20 ,22, a flux-tube nonlinear gyrokinetic simulation program, was
run to scan different collisionalities and density gradients.
Using a simplified test case similar to the Cyclone Base Case 13 (a simplified
model of tokamak with circular cross-section, two species, etc.), but with zero
temperature gradient and finite collisionality, we have carried out more detailed studies
of the role of zonal flows and collisions in the TEM upshift16-18. A concentric, circular
cross-section model equilibrium was used for all the runs, with Te=Ti and ne=ni, where Te
and Ti are the electron and ion temperatures and ne and ni are the electron and ion
densities. All jobs were run with an
= r/R = 0.18,
= (rlq)dq/dr=0.8, and a magnetic
"safety factor" of q = rB, /RBP = 1.4, where Bt and Bp are the toroidal and poloidal
magnetic field components. Each of the simulations were run with the highest practical
resolution, 11 poloidal modes and 85 radial modes, as determined by convergence studies
using: 16 energy grid points, 32 pitch angles, 32 parallel grid points, and 10 circulating
pitch angles.'
6
To determine the linear threshold for the TEM, GS2 was run on the MIT Plasma
Science and Fusion Center cluster, Marshall, on "linear" mode, with R/L~ set to 1.0, 1.2,
-19-
1.4, 1.5, 1.8, and 2.0. GS2_PLOT, a plotting tool written by Darin Ernst and further
developed by the author in the Summer of 2003, was used to plot the growth rate vs.
density gradient. Once it was determined that the linear instability threshold lay between
1.4 and 1.5, more jobs were submitted (R/4L=1.41, 1.44, 1.46, 1.48) and R/Ln=1.44 was
determined to the be the threshold of linear stability when veiRo /c,=.01. This process
was repeated for a collisionality of ve,iR o /c,=0.1, and the linear instability threshold was
determined to be 1.56. These thresholds are closely reproduced by Eq. (3), despite the
many approximations inherent to it, but the fact that t1e40 in this study suggests that this
is probably a coincidence.
Next, GS2 was set to run on "non-linear" mode, and jobs were submitted to
NERSC IBM p575 system. In order to study the existence and effect of collisionality on
the nonlinear upshift, over 20 jobs with combinations of density gradients and
collisionalities were simulated. First, the ve,Rl /c , = 0.01 case was tested, and jobs with
R/L, =1.5 through R/Ln = 20.0 were submitted. For all cases, consistent ion-ion
collisions were included, i.e. vii=vei/ 6 0. Each of these runs were analyzed using
GS2_PLOT, and plots of particle flux vs. time, convergence, and average particle flux
can be seen in the results section. A sample GS2 input file can be found in Appendix A.
The collisionality was then changed to velRo c,--0.1 and again jobs were run with
R/Ln between 2 and 20. These results were plotted and analyzed as described above, and
all results are shown below.
The final graph of average particle flux vs. density gradient was
constructed and the results analyzed to investigate intermittency and the role of zonal
flows.
-20-
4
Results
Table 1: Average particle flux for each density gradient and collisionality
Average Part. Flux
F/[neCs(Pi/Ro) 2]
Density
Gradient
Average Part. Flux
1.8
2
(vei Ro/ck=0.01)
0.002
0.15
F/[neCs(pi/Ro) 2]
(vei Ro/cs=0.1)
0
0
3
5
7
10
15
3.178
14.733
32.55
58.98
141.41
0.04
5.33
16.17
46.16
102.27
Nonlinear Upshift of TEM Critical Density Gradient
t
.
.14
12
10
6
Density Gradient
~'~
Figure R1: The nonlinear upshift of TEM critical density gradient is shown. The linear
threshold is shifted from 1.44 to around 2 for a collisionality of 0.01, and from 1.56 to
around 3 for a collisionality of 0.1.
.....
•
-21 -
Non Linear Upshift of TEI Critical Density
Gradiet Increases with Colaiironality
1SO
135
120
105
45
30
15
0
2
4
6
14
10 12
8
Density Gradient
16
18
20
Figure R2: The average particle flux vs. density gradient is shown. Increasing ion-ion
collisions increases the TEM nonlinear upshift of the critical density gradient.
- 22-
vet R/c,=O.O
1
Ts
O
0.8
0.6
0.4
0.2
0.0
0
0
0
1
2
3
4
5
Figure R3: (Top) The maximum growth rate and real frequency/10 for a collisionality of
0.01 are shown. The critical value of Ro/L, is shown to be 1.44. This is not far from the
1.5 value predicted in Eq. 2. (Bottom) The growth rate of the maximum kePi is shown to
have a value of 1.0 at the critical density gradient. It has a typical value of 0.60.
- 23-
A
v.et R0/c=0.1
s
A
U.8
0.6
0.4
0.2
0.0
1.5
-
1w0 0M I" l
I
I
.•.I
.
.
T I"
I
I 1I I
i
I
.!I
I
I
l
"
II
I"I
I
I I
I
I1
1
I
1"
kePiof Max Growth Rate
1.0
L
-
0.5
n .on'
,
rl_~.....l...,,...,
II~-
I
..............
r............l
.......
;.,
Ro / L,
Figure R4: (Top)The maximum growth rate and real frequency/10 for a collisionality of
0.1 are shown. The critical value of Ro/L, is shown to be 1.56. (Bottom) The growth rate
of the maximum kepi is shown to have a value of 1.5 at the critical density gradient.
- 24-
0.30
Ro/Ln
3.10
On
0.25
0.20
c~c
-J
0.10
0.05
0.00
0.0
0.5
1.0
1.5
2.0
Figure R5: Both the linear growth rate and frequency/10 are shown for a linear GS2
simulation with veiRo/cs =0.01 and R0/L•= 3.0. The growth rate is shown to have a peak
at k0Pi= 0.6. The data approximately agrees with the flux predicted in Eq. (2).
- 25-
25
20
o)
r-"
Ci:
5
0.00
200
400
800
600
cst / R
1000
1200
1400
o
Figure R6: Normalized particle flux vs. normalized time for Ro L ,, = 1.8 simulation with
collisionality of 0.01. Only a single burst of particle flux is observed, though there will
probably be another if run for longer.
10 2
10
0
LI:
-
10
(1)
,,
Il
V
10
-8
0
500
1000
1500
2000
ct /R o
Figure R7: Normalized potential for the range of poloidal modes, kop,, evolved in the
GS2 nonlinear simulation for Ro/L, =1.8 with a collisionality of 0.01. The zonal flow
(kop i -0) dominates.
- 26-
Ro/Ln =2.0
Ev
i
= "0fIi
R
-ei "0"s
.
40
•
.
•
..
0,
c4
..
30
.
.
• .
UL
LI
20
10
0.0
I-
t"
200
400
600
800
ct/R o
Figure R8: Normalized particle flux vs. normalized time for Ro /L,, =2.0 simulation with a
collisionality of 0.01. Only a single burst of particle flux is observed.
Ro/L n = 2.0
v., Rc.= .01
'Jr 10
-2
0a
p
kopi
0.000
0.200
0.400
10
-4
0)1 .000
-
1.200
1.400
10
10
1
1.800
2.000
.10"8.
I,
0
., .
I
200
,
,1 •
,
400
I
,
600
,
,
I
800
,
,
,
I . ..
1000
.
I•
1200
ct R o
Figure R9: Normalized potential for the range of poloidal modes, kp,, evolved in the
GS2 nonlinear simulation for RO/ L , =2.0 with a collisionality of 0.01. The zonal flow
(kop, --0) dominates.
- 27 -
R /Ln = 3.0
6.0
Ve Ro/c,= .01
twA
0u
AhV
4.0
0
2.0
0.0
AA
JV"I
~
c
I
.
.
.
I
.
I
.
.
100
50
c,tt/R
I
.
150
I
,
I
,
I
.
200
o
Figure R10: Normalized particle flux vs. normalized time for RolL,, =3.0 simulation with
collisionality of 0.01.
1021
R/Ln - 3.0
ve. ROIcs= .01
100
CL
10 --2
(D
10 - 4
10-6
,
I
•
· · · · 1 ~~_· _ · · · · _ _ _ · · _ · _ _ ·
50
100
150
200
c~t/ Ro
Figure R11: Normalized potential for the range of poloidal modes, kop,, evolved in the
GS2 nonlinear simulation for Ro L,, =3.0 with a collisionality of 0.01.
- 28 -
Ro/L n = 5.0
40
vei R,/c=-.o1
30
•0~
C.)
C4P
e-'
20
4Z
10
V
0.0
··- · · ·
0.0
L
I:
I
1
i
I
i
60
40
20
80
ct/ Ro
Figure R12: Normalized particle flux vs. normalized time for Ro /L, =5.0 simulation with
collisionality of 0.01.
Ro/L -= 5.0
1o2
Vei R/C=.Ol
100
10-2
kepi
0.000
0.200
0.400
1-
1 000
010-4
1.200
1.400
1.800
2.000
10-6
.
.
.
.
I
I
I
I
20
I -
i
40:
ct// Ro
"1
I
I
60
I
I
I
I
80
Figure R13: Normalized potential for the range of poloidal modes, kop i, evolved in the
GS2 nonlinear simulation for R /L,, =5.0 with a collisionality of 0.01. An unidentified
numerical instability appears for short wavelength for ct/Ro > 65.
- 29-
Ro/L = 7.0
.
100
Sv i Ro/c,=
0
n
.01
80
S60
Q.
Em 40
20
0.0
I.
0
10
20
30
- I
40
c,t / Ro
Figure R14: Normalized particle flux vs. normalized time for Ro / L , =7.0 simulation with
collisionality of 0.01.
RolLn = 7.0
10
:
.
.
.
.
v; Rolc,= .01
o10
0
(rQ
0".7 10
0ePi
000
200
400
-s-o,
000
200
400
I
800
.000
71 . . ..... ,--.".-------.
I....
...
. .I....
20
. .
I .
.-.
30
ct / R
....
.
.
..-
I.....
40
,,-..
I.
I
50
Figure R15: Normalized potential for the range of poloidal modes, kop,, evolved in the
GS2 nonlinear simulation for 7.0 with a collisionality of 0.01. An unidentified numerical
instability appears for short wavelength for ct/Ro > 35.
- 30-
80
60
40
4
6
20
0.0
10
15
20
25
30
ct Ro
Figure R16: Normalized particle flux vs. normalized time for R o /L, = 10.0 simulation
with a collisionality of 0.01.
Ro/L = 10.0
10 2
v1,
~IR,/C = .01
100
0
1o-2
ci
I)
10
-6
10.
.I. .
10
ct / R
20
.
.
30
Figure R17: Normalized potential for the range of poloidal modes, kep ,, evolved in the
GS2 nonlinear simulation for R /L,, = 10.0 with a collisionality of 0.01. An unidentified
numerical instability appears for short wavelength for ct/Ro > 25.
31-
Ro/Ln = 15.0
300
01
200
CL
U,
100
0.0
20
15
10
Cst /R o
Figure R18: Normalized particle flux vs. normalized time for R/IL, =15.0 simulation
with a collisionality of 0.01.
10 2
100
1iO2
r"
I-.
S10-2
"
10
4
10
-6
0
10
5
15
20
ct / Ro
Figure R19: Normalized potential for the range of poloidal modes, kop,, evolved in the
GS2 nonlinear simulation for Ro/L, = 15.0 with a collisionality of 0.01. An unidentified
numerical instability appears for short wavelength for ct/Ro > 16.
- 32-
0.6
0.4
C0,
0.2
0.0
200
400
ct/
600
Ro
800
Figure R20: Normalized particle flux vs. normalized time for Ro/L, =3.0 simulation with
a collisionality of 0.1. Only a single burst of particle flux is observed.
RlLn = 3.0
102
vi R/I= 0.1
d
•
1U
o
re
'
S1U
kep,
0.
r-
).000
.
).200
S1U
.400
8
10 -6
.200
-
000
.400
1uu2.000
10"8
I.
.
0
200
•
I
.
400
ct /
.
.
600
800
1000
Ro
Figure R21: Normalized potential for the range of poloidal modes, koep,, evolved in the
GS2 nonlinear simulation for Ro /L , =3.0 with a collisionality of 0.1.
- 33-
20
15
. 10
5
I...I...I:..I
0.0
20
40
60
100
80
120
140
ct / F
Figure R22: Normalized particle flux vs. normalized time for R o IL , =5.0 simulation with
a collisionality of 0.1. The bursts of particle flux are intermittent.
Rv/L = 5.0
10 2
-41
CE,
01
10
v., Rolc,= .1
)
-2
-eCD 1O-4
10
-6
!
.
.
.
..
50
ct / Ro
.
100
.
.
.
1.50
Figure R23: Normalized potential for the range of poloidal modes, koPi, evolved in the
GS2 nonlinear simulation for R o I L , =5.0 with collisionality of 0.1.
- 34 -
RoLn = 7.0
40
S'..
I.
V• Rt/c=. 1
30
\
d,
20
O10
0.0
,
I
-1
,
40
20
c,t/R
I
60
o
Figure R24: Normalized particle flux vs. normalized time for R0 / L,, =7.0 simulation with
a collisionality of 0.1.
Ro/La = 7.0
102 vo Rolc= .1
0
o
10
kep i
).000
).200
).400
a)
.000
.200
.400
.800
2.000
I
.
,
,
I
20
.
.
.
Cst/ Ro
i
40
,
,
.
I
60
Figure R25: Normalized potential for the range of poloidal modes, kop,, evolved in the
GS2 nonlinear simulation for R IL,, =7.0 with a collisionality of 0.1. An unidentified
numerical instability appears for short wavelength for cst/Ro > 55.
- 35-
80
60
Q.
40
. .
20
0.0
0
20
10
30
cst / Ro
Figure R26: Normalized particle flux vs. normalized time for Ro/L, = 10.0 simulation
with a collisionality of 0.1.
Ro/L = 10.0
10 2
100
a:
R. 10
ci
10 - 4
10-6
0
10
ct / Ro
20
30
Figure R27: Normalized potential for the range of poloidal modes, kop,, evolved in the
GS2 nonlinear simulation for Ro I L , = 10.0 with a collisionality of 0.1. An unidentified
numerical instability appears for short wavelength for ct/Ro > 26.
- 36-
RoLn = 15.0
200
ve Ro/c,= .1
150
03
100
o
I
i.
50
0.0
LL
..
~
I
.
I
I
15
20
,
10
ct tR
I-
.
25
o
Figure R28: Normalized particle flux vs. normalized time for R0 /L, = 15.0 simulation
with a collisionality of 0.1.
R /Ln = 15.0
,I
210
· · · · I · · · · ~····I
v Re
· · · · 1
II
I-
10
)
)
-.
)
)
)
)
10-6
•
I
I
I
10
15
20
25
ct/R o
Figure R29: Normalized potential for the range of poloidal modes, kep, evolved in the
GS2 nonlinear simulation for Ro / L , = 15.0 with a collisionality of 0.1. An unidentified
numerical instability appears for short wavelength for ct/Ro > 17.
- 37 -
In order to investigate the numerical instability at the end of the runs, the energy
grid resolution was doubled, and the Ro / L , = 7.0 case was re-run. The results are shown
below, and, as is obvious from the figure, the numerical instability was unaffected by
energy grid resolution.
100
80
60
S40
20
0.0
25
20
15
30
35
ct/ Ro
Figure R30: Normalized particle flux vs. normalized time for R0 /L , =7.0 simulation with
a collisionality of 0.01 and negrid doubled.
Ro/L - 7.0
10
2
vt Rolc,= .01
u
er-o
'0
'0
0
,,
-
)10
'0
)i
)O
>o
- 10
'(D
o1
0o
0o
10-6
-.I
..- l
.
I
.,l
...
,...,,,,,
I
,,,
20
c,t / Ro
Figure R31: Normalized potential for the range of poloidal modes, kop,, evolved in the
GS2 nonlinear simulation for Ro /L, =7.0 with a collisionality of 0.1 and negrid doubled.
The numerical instability is not corrected, and appears for ct/Ro > 65..
- 38-
Table 2: Onset times for numerical instability
R 0/Ln
1.8
Vei RO/Cs
0.01
cstinst/R o
> t_max
Cstit/L,
n/a
2
0.01
> tmax
n/a
3
0.01
> t_max
n/a
5
0.01
65
325
7
0.01
35
245
10
15
3
5
0.01
0.01
0.1
0.1
25
16
> tmax
> tmax
250
240
n/a
n/a
7
10
0.1
0.1
55
26
385
260
15
0.1
17
255
The ctinst/Ln values are similar for most instances of the numerical instability.
5
Discussion
Our simulations have confirmed the existence of a new nonlinear upshift in the
critical density gradient. 16 This new upshift appears in the density gradient drive for the
TEM, rather than the temperature gradient as in the Dimits Shift for the ITG. Because
zonal flows are only weakly damped by ion-ion collisions, while the TEM is strongly
damped by the electron-ion collisions, the collisions have a net stabilizing effect. This
causes stronger instability drive, or larger density gradients, to be required to attain the
same turbulent response. The result is that at higher collision frequencies, the plot of
particle flux vs. density gradient is shifted to larger density gradients. This is shown in
Figures R1 and R2.
In Fig. R2, error bars have not been added, because the probability distribution of
particle flux is non-Gaussian, as the simulated turbulent fluxes are often intermittent and
- 39-
sometimes quasi-periodic. Therefore, error analysis such as standard deviation can not be
directly applied. Methods for determining the level of intermittency have been proposed
by Mikkelsen 2 1 and Nevins. Mikkelsen's method is based on the central limit theorem,
which states that the distribution of sums of many samples will be a Gaussian. It remains
to apply Mikkelsen's method to determine whether the simulations need to be run longer.
Nevertheless, a consistent trend is shown in the results.
5.1
Linear Simulations
The frequency and growth rates are shown for collisionalities of 0.01 and 0.1 in
the top plots of Figures R3 and R4, respectively. The linear critical density gradient can
be seen to be 1.44 in Fig. R3 (Top) and 1.56 in Fig R4 (Top). These limits were
determined using a range of linear simulations with R0/L from 1.0 to 4. In both figures,
the growth rate increases with the density gradient drive.
The most unstable wavenumber for both the 0.01 and 0.1 cases is kop,=.6. This
value was determined using a range of 20 kopi values from 0.1 to 2.0. Figures R3 and R4
(Bottom) show the wavelength of maximum growth rate for the runs, and the peaks are
1.0 and 1.5 respectively. It seems that in this collisionality range there is not much effect
on the threshold due to collisions.
The growth rate for a linear run with a density gradient of RodLn = 3.0 and
collisionality of Ro/c, = 0.01 is shown in Fig. R5, and a similar figure is obtained for Ro/cs
=0.1 (not shown). The maximum growth rate is determined to be .27 with a k.p, value
of 0.6. A similar figure is obtained for a collisionality of 0.1 (again, not shown). Fig. R5
shows that the spectrum is farther extended than ITG modes, and other linear runs (not
-40-
shown) have been extended to show that the spectrum does not go to zero until kop,= 4.
As mentioned, the linear growth rate peaked at kop,= 0.6, which is in the same range of
wavelengths that the toroidal ITG spectrum peaks.
In addition, Eq. (2) predicts the Ro/Ln instability threshold for finite Tie is et' > 1.5.
In this study, the critical density gradients are 1.44 and 1.56 for the .01 and .1 cases
respectively. Obviously, both thresholds are very close to the predicted values.
However, Eq. (2) applies to the case tie>0, while Ile- 0 in our study.
For the case with critical density gradient, R/L=1.44 shown in Fig. R3, 0o/)oe
can be calculated using:
o*e = (kepi) -h- where, kopi=1.0, Vthi=Cs , and Te=Ti, RILn=1.44.
Substituting in the appropriate values, we found
o = ko p,
c
(1.1)=
R
L
- (1.1)
R
0
1.1
= -= 0.76
o
1.44
Eq. (2b) implies:
_ F 0 _ 0.72
2=
F
-=
o,1e
2-0.72
2-
0.56
This implies that - is not far from the prediction in Eq. (2b), which should be
oe
applicable.
5.2
Nonlinear Simulations
Figures R6, R8, R10,R12, etc. show plots of the particle flux vs. time for
different density gradients and collisionalities. These figures show that for the pure TEM
-41-
case, electron-ion collisions are linearly damping. Figs. R6, R8, and R20 show cases
where only a single burst is observed. In these cases, the zonal flows grow to large
amplitudes and stay there (see Figs. R7, R9, R21). This causes a quenching of the
primaries, which are not able to suppress the very slowly damped zonal flows due to the
small density gradients. This is a demonstration of a TEM which is linearly unstable, but
nonlinearly self stabilizing. Though these figures only show a single burst, more bursts
would be observed farther out in time.
The other plots of particle flux vs. time show bursts that are cyclical and
overlapping. This is due to the continuous back and forth exchange of energy between
the zonal flows and the primaries. Because the density gradients in the simulations are
high, the primaries are able to recover from the zonal flow growth, and another burst of
flux occurs. This causes the zonal flows to be suppressed, starting the cycle over again.
Figures R7, R9, R11, etc. show plots of convergence vs. time. In these plots, the
kopi=O.O (red lines) are the zonal flows. Initially, there is a linear growth phase until the
amplitude of the primaries exceeds the threshold 3 given in Eq. (3) and drives the
secondaries, which cause the zonal flows. Again, these plots show that the zonal flows
begin to blow up after the threshold is reached, causing the primaries to stop growing. At
a certain density gradient (see Fig. R16, R18), the primaries grow fast enough to
compete with the zonal flows, and this is where the particle fluxes become intermittent.
In many of the plots, there is a numerical instability late in the run, for short
wavelengths Times where the instability was present have been excluded form the
average flux values shown. The instability does not affect the final results, apart from
limiting the duration of the simulations. In order to determine if the size of the energy
-42-
grid was causing this problem, the Ro/IL n = 7.0 case was re-run with negrid doubled, and
the results are shown in Fig. R31. The cause of this instability has yet to be determined,
but it is believe that classical diffusion could potentially suppress this instability and a
correction has been implemented into the GS2 code by D. R. Ernst. Results from these
runs will be presented at a future time.
5.3
Future Plans
We surmise that any effect which stables TEM but doesn't damp zonal flows will
result in an increase in the nonlinear upshift. In addition, it remains to be seen if the
upshift has a parametric dependence on quantities such as q, as suggested by Rogers et al.
or magnetic shear. Also, the role of the tertiaries1 5 has yet to be determined in this
upshift.
-43 -
References
[1]
[2]
[3]
[4]
[5]
[6]
J. Wesson, Tokamaks, Clarendon Press, Oxford, England. (1987).
R.J. Hawryluk et. al., Phys. Plasmas 5(5), 1577 (1998).
Joint European Torus: http://www.answers.com/topic/ioint-european-torus
ITER: http://www.iter.org/index.htm (2005)
B. Coppi and G. Rewoldt, Phys Rev. Lett. 33, 1329 (1974).
B. B. Kadomstev and 0. P. Pogutse, Sov. Phys. JETP 24, 1172 (1967).
[7] F. F. Chen, Introduction to PlasmaPhysics and controlledFusion, Plenum Press,
[8]
[9]
[10]
[11]
[12]
New York, 1984.
W. M. Tang, G. Rewoldt, Liu Chen, Phys. Fluids 29, 3715 (1986).
J. C. Adam, W. M. Tang, and P. H. Rutherford, Phys. Fluids 19, 561 (1976).
B. Coppi, S. Migliuolo, and Y.-K. Pu, Phys. Fluids B 2, 2322(1990).
B. Coppi and F. Peoraro, Nucl. Fusion 17, (1977).
P. H. Diamond, S-I Itoh, K Itoh, and T. S. Hahm Plasma Phys. and Cont. Fusion 47,
R35-R161 (2005).
[13] A.M. Dimits et. al., Phys. Plasmas 7, 969 (2000).
[14] R. A. Kolesnikov and J. A. Krommes, The transitionof collisionless ITG plasma
turbulence: A dynamical systems approach
[15] B. N. Rogers, W. Dorland, and M. Kotschenreuther, Phys. Rev. Lett. 85, 5336
(2000).
[16] D. R. Ernst et. al. Phys. Plasmas 11, 2637 (2004).
[17] D. R. Ernst, et. al., including K. Zeller, Proc. 20 th Int'l. Atomic Energy Agency
Fusion Energy Conference, Vilamoura, Portugal, 1-6 November 2004, oral paper
IAEA-CN-1 16/TH/4-1.
(http://www-naweb.iaea.org/napc/physics/fec/fec2004/datasets/TH_4-1.html)
[18] D. R. Ernst et. al., including K. Zeller, "Gyrokinetic Simulations of Trapped
Electron Mode Turbulence in Alcator C-Mod Internal Transport Barriers," oral
presentation 201, 2004 Sherwood International Fusion Theory Conference,
Missoula, Montana, April 25-28. (http://www.sherwoodtheory.org)
[19] M. Beer et al., Phys. Plasmas 2, 2687 (1995).
[20] W. D. Dorland et. al., Phys. Rev. Lett. 85, 5579 (2000).
[21] D.R. Mikkelsen, Private Communication (2005).
[22] M. Kotschenreuther, G Renoldt, and W.M. Tang, Comp. Phys. Com. 88, 128
(1995).
[23] T.M. Antonsen and B. Lane, Phys. Fluids 23, 1205 (1980).
[24] E. A. Frieman and L. Chen, Phys. Fluids 25, 502 (1982).
[25] P.J Catto, W.M. Tang, and D. E. Baldwin Plasma Phys. Cont. Fusion 23, 639
(1981).
-44 -
Appendices
Appendix A: GS2 Input Namelist
&collisions_knobs
conserve_momentum=.false.
collision_model='default'
&source_knobs
source_option="full"
&fields_knobs
field_option= "implicit"
&gs2_diagnostics_knobs
nwrite=
10
navg=
30
print_line=.false.
write_line=.true.
write_phi=.true.
write_apar=.false.
write_aperp=.false.
write_omega=.true.
write_omavg=.true.
write_qheat=.true.
write_pflux=.true.
write_vflux=.false.
write_qmheat=.false.
write_pmflux=.false.
write_vmflux=.false.
write_dmix=.true.
write_kperpnorm=.true.
write_phitot=.true.
write_eigenfunc=.false.
write_final_fields=.true.
write_avg_moments=.true.
writefinal_epar=.true.
write_final_moments=.true.
write_final_antot=.false.
write_lamavg=.true.
-45 -
write_eavg=.true.
write nl flux=.true.
dump_neoclassical_flux=.false.
writeneoclassicalflux=.false.
dump_check 1=.false.
write_fieldcheck=.false.
writefcheck=.false.
writeflux line=.true.
print_flux_line=.true.
print_old_units=.false.
write_ascii=.false.
omegatol= 1.0e-3
500.000
omegatinst=
save_for_restart=.true.
exit_when_converged=.false.
&parameters
beta = 0.00
zeff = 1.0
! 403. * ne_19 * T_i(kev) / 1.e5 / B_T**2
! only used for collisionality
&le_grids_knobs
advanced_egrid=.true.
ngauss = 10
negrid = 16
ecut = 6.0
&dist fn knobs
1.00000
gridfac=
boundary_option="linked"
&kt_grids_knobs
grid_option='box'
norm_option='t_over_m'
&kt_grids_box_parameters
32
ny=
128
nx=
5.00000
yO=
jtwist= 4
/
-46 -
&knobs
1.00000
fphi=
fapar=
0.00000
faperp=
0.00000
delt= 0.2500000
nstep= 60000
wstarunits=.false.
!delt_option='check_restart'
margin= 0.0500000
&nonlinear_terms_knobs
nonlinearmode='on'
cfl= 0.500000
&reinit_knobs
1.50000
delt_adj=
delt_minimum= 1.00000e-04
&kt_grids_specified_parameters
naky=
9
ntheta0=
1
&kt_grids_range_parameters
naky = 20
ntheta0 = 1
aky_min = 0.1
aky_max = 2.0
theta0_min = 0.
theta0_max = 0.
&species_knobs
nspec= 2
&species_parameters_1
z=
-1.0
mass = 0.000278566
dens = 1.0
temp = 1.0
tprim = 0. ! R/Lt=6.9
fprim= 7.00 ! R/Ln=2.2
- 47 -