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Ion temperature gradient turbulence simulations and plasma flux surface
shape
R. E. Waltz and R. L. Miller
Citation: Phys. Plasmas 6, 4265 (1999); doi: 10.1063/1.873694
View online: http://dx.doi.org/10.1063/1.873694
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Ion temperature gradient turbulence simulations and plasma
flux surface shape
R. E. Waltza) and R. L. Millerb)
General Atomics, P.O. Box 85608, San Diego, California 92186-5608
共Received 24 February 1999; accepted 29 July 1999兲
A generalization of the circular sˆ-
␣
local magnetohydrodynamic 共MHD兲equilibrium model to finite
aspect ratio 共A兲, elongation 共
兲, and triangularity 共
␦
兲has been added to a gyrokinetic stability code
and our gyrofluid nonlinear ballooning mode code for ion temperature gradient 共ITG兲turbulence.
This allows systematic studies of stability and transport for shaped flux surfaces with the same
minor midplane radius label 共r兲, plasma gradients, q,sˆ, and
␣
while varying A,
, and
␦
. It is shown
that the 共linear, nonlinear, and sheared兲E⫻Bterms in the equation of motion are unchanged from
a circle at radius rwith an effective field Bunit⫽B0
d
/rdr, where
⫽B0
2/2 is the toroidal flux,
ris the flux surface label, and B0is the magnetic axis field. This leads to a ‘‘natural gyroBohm
diffusivity’’
natural, which at moderate q⫽2 to 3 is weakly dependent on shape 共
兲at fixed Bunit .
Since Bunit /B0⬀
and
具
兩
ⵜr
兩
2
典
⬇(1⫹
2)/(2
2), the label independent
ITER⫽
natural /
具
兩
ⵜr
兩
2
典
at
fixed B0scales as 2/(1⫹
2) with much weaker scaling at high-qand stronger at low-qwhere
increased
is stabilizing. The generalized critical E⫻Bshear rate to be compared to the maximum
linear growth rate is a flux surface quantity (r/q)d/dr(cq/rBunitd
0/dr)⫽(r/q)d(Ex0/
BpR)/dr.©1999 American Institute of Physics. 关S1070-664X共99兲01411-1兴
I. INTRODUCTION AND SUMMARY
The dependence of magnetohydrodynamic 共MHD兲sta-
bility limits on tokamak plasma shape is well known and
long studied both theoretically and experimentally. The de-
pendence of confinement on shape has been studied experi-
mentally, but hereafter theoretical micro-instability growth
rates, and particularly, diffusion loss rates have not been
given the full attention they deserve. A previous study by
Hua, Xu, and Fowler,1using a high-nlinear ballooning mode
code for ion temperature gradient 共ITG兲modes, showed
growth rates in the plasma core decrease with elongation, but
are insensitive to triangularity. A key difficulty with previous
studies has been that high-nballooning instabilities 共micro-
instabilities兲are local and should only depend on local
plasma and flux surface shape parameters, whereas the stud-
ies varied the global equilibrium, making it unclear what
local parameters 共if any兲remained fixed as the shape varied.
Furthermore, no comparison with the standard analytic infi-
nite aspect ratio circular s-
␣
local equilibrium model2was
made.
The present paper uses a generalization of the s-
␣
model
to a finite aspect ratio 共A兲, elongation 共
兲, and triangularity
共
␦
兲local equilibrium model by Miller et al.3This allows sys-
tematic studies of growth rates versus local flux shapes with
the same minor midplane radius label 共r兲, plasma density and
temperature gradients, q,sˆ⫽dlnq/dlnr and
␣
共the MHD
pressure gradient parameter兲while varying A,
, and
␦
.
Growth rates 共
␥
兲can be conveniently normalized to cs/aunit ,
where csis the local ion sound speed and aunit is the unit of
length for measuring rand all gradient lengths. We will take
aunit as the flux surface label rat the last closed flux surface.
Since growth rates can be displayed as a spectrum in k
s,
where k
⬅nq/rspecifies the toroidal mode number nand
s⬅cs/(eBunit /cMi) specifies the norm of the ion gyrora-
dius, growth rates refer only to n/Bunit and there is no sepa-
rate specification for n, and more importantly, the unit of
magnetic field Bunit ; thus Bunit is a free parameter in linear
theory.
While it may be sufficient in some cases 共such as the
core plasma兲to specify growth rates, or more accurately,
critical temperature gradients for marginal stability (
␥
⫽0),
more generally, the local heat diffusivity is needed. Apart
from the possible effects of diamagnetically dominated E
⫻Bshear stabilization effects, the E⫻Bdiffusivity will
have a unit gyroBohm scaling DgB⫽(cs/aunit)
s
2. The latter
is the natural norm for diffusion from a high-nnonlinear
ballooning mode representation.4Thus, diffusivity will scale
as 1/Bunit
2. This paper shows that if we choose Bunit as
B0
d
/rdr, where
is defined through the toroidal flux
⫽B0
2/2 where B0is the magnetic field on the magnetic
axis, then all linear, nonlinear, and sheared E⫻Bphysics
appear to be identical to that in a circular plasma of minor
radius rwith magnetic field Bunit . Thus using ras a flux
surface label, we obtain a heat diffusion loss rate
⫺1/V⬘共r兲
/
rV⬘共r兲n0共
ˆnaturalDgB兲dT0/dr,共1兲
where V⬘⫽dV/dr with Vthe flux surface volume. Also in
Eq. 共1兲n0and T0are the local density and temperature,
respectively, and
ˆnatural is the heat diffusivity normalized to
the unit gyroBohm diffusivity DgB . Using a nonlinear bal-
looning mode gyrofluid code for ITG turbulence4reformu-
lated to shaped geometry with the Miller local equilibrium,
a兲Electronic mail: Waltz@gav.gat.com
b兲Present address: Archimedes Technology Group, Inc., 5405 Oberlin Drive,
San Diego, California 92121.
PHYSICS OF PLASMAS VOLUME 6, NUMBER 11 NOVEMBER 1999
42651070-664X/99/6(11)/4265/7/$15.00 © 1999 American Institute of Physics
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we find for moderate q共2–3兲and well above marginal sta-
bility and in the absence of E⫻Bshear, that
ˆnatural is weakly
dependent on shape despite the fact that the maximum ITG
growth rates can decrease significantly with increasing shap-
ing, or specifically, with increasing elongation.
It was found previously4–6 in circular geometry that
when the E⫻Bshear rate
␥
Ebecomes comparable to the
maximum growth rate
␥
max , the ITG turbulence vanishes
共i.e.,
␥
Ecrit⫽
␣
E
␥
max ;
␣
E⬇1兲. In shaped geometry we find
that the appropriate E⫻Bshear rate is a flux surface variable
reminiscent of the circular formulation
␥
E⬅共r/q兲d/dr共cq/rBunitd
0/dr兲⫽共r/q兲d共Ex0/BpR兲/dr.
共2兲
The factor (r/q) can be significantly smaller in elongated
plasmas than the corresponding factor (BpR/B) in the com-
monly used real geometry Hahm–Burrell formula7共as mea-
sured on the outboard side兲.Bpis the poloidal field and
0(r) is the electrostatic potential and Ex0the radial electric
field. Since (BpR/B) is not a flux function, neither is the
Hahm–Burrell rate. 共We hasten to add that the Hahm–
Burrell formula was derived in the context of a general two-
point renormalization theory and not in the context of the
simulations with a ballooning mode formalism; thus there is
actually no quantitative basis for comparing it with the maxi-
mum growth rate.兲In the limited study of real geometry in
this paper, we find
␣
E⬇0.5 is a better fit than the circular fit
of 1.0.
This simple mnemonic formulation has an immediate
consequence: To convert a circular sˆ-
␣
transport model
共e.g., Ref. 8兲to real geometry, use the circular model and the
circular Eqs. 共1兲and 共2兲, but replace the toroidal magnetic
field by an effective field Bunit(r)⫽B0Sh(r) where Sh(r)
⫽
d
/rdr is the shape factor, and ris the flux surface label.
Well the threshold for instability and ignoring E⫻Bshear
effects, and to the extent that the
ˆnatural is approximately
constant, gyroBohm diffusive confinement time would scale
as (ShB0aunit)2at fixed midplane minor radius. Shcan ac-
quire rather large values 共e.g., 2–10 from the center to the
edge even for
⫽1.6兲. For concentric elliptical shapes
Sh(r)⫽
. It is possible to take a profile of Miller’s local
equilibria from experimental discharges and rescale only the
profile in
. Rescaling a typical discharge to a nearly circular
plasma (
⫽1), we still find the profile of Shcan still vary
from 1.3 to 3 and we also find that Sh⬀
.
To understand how this simple recipe compares with
previous modeling recipes, we need to define diffusivity
more carefully. In terms of Eq. 共1兲, the International Toka-
mak Experimental Reactor 共ITER兲Modeling Group,9defines
a standard diffusivity
具
兩
ⵜr
兩
2
典
ITER⫽
ˆnatural DgB . Substitut-
ing this in Eq. 共1兲, we see that
ITER has the very useful
feature of being independent of flux surface label; that is, we
could replace the rlabel by
,
,or⌿共the poloidal flux兲.We
will continue to use ras a flux surface label, and for illus-
tration, consider concentric ellipses, in which case
具
兩
ⵜr
兩
2
典
⫽(1⫹
2)/(2
2). If
ˆnatural is in fact exactly constant at
fixed Bunit and gyroBohm-like, then
ITER 共at fixed B0,r,
T0,q兲would scale as 2/(1⫹
2). Under the same conditions
and on the basis of empirical low 共L兲-mode confinement time
scaling 共
1/2 at fixed current and power兲, Kinsey et al.10 and
Bateman et al.,11 posited that
ITER would exhibit a much
stronger 1/
4scaling. The models of Refs. 8 and 12 give
ITER⬀
具
兩
ⵜr
兩
典
/
兩
ⵜr
兩
2
典
, which is a weakly decreasing function
of
. As we show in some detail,
ˆnatural is constant only at
moderate q⫽2–3, well away from the critical gradient.
From a survey of our limited ITG simulations, ranging over
⫽1to2,
ITER scales as 2/(1⫹
2)or1/
1.3 at q⫽2–3.At
low q⫽1.5 where increasing
can stabilize the ITG modes,
it scales more strongly, 1/
2.1. At high q⫽4, it scales more
weakly as
0.03. Clearly the dependence on elongation is not
separable from the other dependencies.
In Sec. II we show in detail how the E⫻Bterms in any
equation of motion retain their circular form with an effec-
tive field Bunit , and formulate the components of the balloon-
ing mode representation for Miller’s local equilibrium. Some
details are given in Appendix A. In Sec. III, some illustra-
tions of stability versus the shape variables 共
,
␦
, and A兲
from a full dynamics linear gyrokinetic code are given. We
also show how the code recovers the MHD beta limit. Sec-
tion IV shows the result of our ITG gyrofluid simulations
with varying elongation and triangularity at numerous qval-
ues and temperature gradients, and in comparison to the sˆ-
␣
local equilibrium. Finally we reexamine the rotational shear
stabilization criterion.
II. FORMULATION
To summarize the local equilibrium of Miller et al.3for
finite elongation and triangularity, we specify the shape of a
flux surface by the major radius Rand height Zas a function
of the position poloidal angle
.
R⫽R0⫹rcos关
⫹arcsin共
␦
兲sin共
兲兴,共3兲
Z⫽
rsin共
兲.共4兲
The magnetic-field B⫽ⵜ
xⵜ⌿⫹fⵜ
has poloidal and to-
roidal components Bp⫽
r⌿/R
兩
ⵜr
兩
and Bt⫽f/R, respec-
tively, where ⌿is the poloidal flux and
⫽B0
2/2 is the
toroidal flux per 2
.q⫽d
/d⌿, is the average safety factor,
and
兩
ⵜr
兩
⫽
⫺1
兵
sin2关
⫹arcsin共
␦
兲sin共
兲兴关1⫹arcsin共
␦
兲cos共
兲兴2
⫹
2cos2共
兲
其
1/2/
兵
cos关arcsin共
␦
兲sin共
兲兴⫹
rR0cos共
兲
⫹关s
⫺s
␦
cos共
兲⫹共1⫹s
兲arcsin共
␦
兲cos共
兲兴
⫻sin共
兲sin关
⫹arcsin共
␦
兲sin共
兲兴
其
.共5兲
Note that
兩
ⵜr
兩
→1 in an infinite aspect ratio circle. In addi-
tion to q,sˆ⫽(r/q)dq/dr, and the MHD pressure gradient
variable
␣
⫽q2(⫺
rV/2
2)(V/2
2R0)1/24
rP/(rBunit)2
关
␣
⫽q2R0d

/dr for a circle兴, we need to specify the aspect
ratio A⫽R0/r, the local Shafranov shift rate
rR0, the elon-
gation
, the triangularity
␦
, and the gradients, s
⫽(r/
)
r
,s
␦
⫽关r/(1⫺
␦
2)1/2兴
r
␦
. The latter are essen-
tially degenerate variables which may be approximated as
s
⫽(
⫺
0)/
, where
0is the on axis value, and s
␦
⫽
␦
/(1⫺
␦
2)1/2 when not provided by the profile of elonga-
tion and triangularity. It follows that:
4266 Phys. Plasmas, Vol. 6, No. 11, November 1999 R. E. Waltz and R. L. Miller
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Bp⫽Bunit共r/Rq兲
兩
ⵜr
兩
共6兲
and
Bt⫽Bunit共r/R兲/
冋
冖
d
/2
共dl/d
兲/共R
兩
ⵜr
兩
兲
册
,共7兲
where lis the poloidal arc length and B2⫽Bp
2⫹Bt
2.
Following Bishop et al.,13 the ballooning mode represen-
tation follows entirely from the shape 关see Eqs. 共3兲and 共4兲兴
and the magnetic-field solutions to MHD equilibrium Eqs.
共5兲and 共6兲. This requires that the field perturbations be writ-
ten in eikonal form F⫽Fn,kx0(
)exp(⫺inS⫹ikx0
兩
ⵜr
兩
x). The
eikonal, S, and poloidal flux, ⌿, are expanded in a normal
distance xfrom the flux surfaces: S⫽
⫹S0(
)⫹xS1(
) and
⌿⫽⌿0⫹x⌿1⫹x2⌿2.
is the toroidal angle, nis the tor-
oidal mode number, which we will specify through k
⫽nq/r. The radial Fourier transform wave number kx0is
often written in terms of the ballooning angle label
0:kx0
⫽⫺k
sˆ
0. We can establish a map between the poloidal
arc length laround the flux surface and
, which can be
considered the extended poloidal angle starting at the outer
midplane. At each location
, the magnetic-field direction b
ˆ
and the cross field directions yˆ and xˆ in and out of the sur-
face, are oriented with respect to a locally orthogonal grid
system (
ˆ,lˆ,xˆ). Then ⵜ
⫽
ˆ/R,ⵜl⫽lˆ, and ⵜx⫽xˆ.S
and xdetermine the rapid variation of the perturbations
across the magnetic field, such that b
ˆ•ⵜS⫽0 and b
ˆ•ⵜx
⫽0. The slow variation along the field are determined by the
variations in Fn,kx0(
). The parallel field derivative on a
perturbation is given by
ⵜ
储
⫽共b
ˆ•ⵜ
兲
⫽共Bp/B兲共d
/dl兲
⇒共1/R0q兲
,共8兲
where here and below ⇒refers to the infinite aspect ratio
circular limit. The rapid cross field derivatives are given by
ⵜy⫽⫺in yˆ•ⵜS⫽iky⫽in/R共B/Bp兲⫽ik
共rB/RBpq兲⇒ik
,
共9兲
which follows from S0(
), and
ⵜx⫽⫺inxˆ•ⵜS⫹ikx0
兩
ⵜr
兩
⫽ikx⫽iky关关共kx/ky兲兴兴⫹ikx0
兩
ⵜr
兩
⇒ik
共sˆ
⫺
␣
sin
⫺sˆ
0兲,共10兲
which follows from S1(
) with xˆ•ⵜS⫽RBpS1(q)/RBp.We
define (rB/RBpq)⬅关关qˆ 兴兴(
)⇒1. The explicit forms for the
eikonal components S0and S1taken from Ref. 13 and the
important function 关关kx/ky兴兴(
) are given in Appendix A.
Considering the cross-field derivatives we can illustrate
how the E⫻Bmotion has a circular form with an effective
field Bunit . Apart from this, all formulas here are identical to
the circular analogues of Refs. 4–6. In physical units the
relevant terms in the continuity equation for density pertur-
bations are given by
n
˜
/
t/n0⫹
˜
E•ⵜn0/n0⫹
˜
E•ⵜn
˜
/n0⫹
E0•ⵜn
˜
/n0⫽¯.
共11兲
From
˜
E⫽c(⫺ⵜ
˜
⫻b
ˆ)/Band
0E⫽c(⫺ⵜ
0⫻b
ˆ)/Bit is
straightforward to show that
˜
E•ⵜn0/n0⫽i
¯
*e
/Te
⫽i关⫺cTe/共eBunit兲k
dlnn0/dr兴e
/Te,
共12兲
which clearly identifies the diamagnetic drift frequency
¯
*
as a flux function. The nonlinear term is
˜
E•ⵜn
˜
/n0⫽⌺k共c/Bunit兲共k
共1兲kx0
共2兲⫺k
共2兲kx0
共1兲兲
共1兲n共2兲/n0,
共13兲
where we have made use of the result
兩
ⵜr
兩
关关qˆ 兴兴/B
⫽1/Bunit , and the cancellation of the 关关kx/ky兴兴(
) part of kx
in the cross product of k(1)⫻b
ˆ–k(2). The sum over wave
number, k, satisfies the usual conditions; namely k
(2)⫽k
⫺k
(1) and kx0
(2)⫽kx0⫺kx0
(1) . The Doppler shift and E⫻B
shear terms are given by
E0•ⵜn
˜
/n0⫽⫺in关cq/rBunitdln
0/dr兴
⫹d/dr关共cq/rBunit兲d
0/dr兴
兩
ⵜr
兩
xn
˜
⫽i
En
˜
⫹
␥
Ek
n
˜
/
kx0,共14兲
where we made a Taylor expansion of the Doppler shift as a
flux function. We used dr⫽
兩
ⵜr
兩
•xand the Fourier identity
兩
ⵜr
兩
•x⫽⫺i
/
kx0to define the E⫻Bshear rate
␥
Eas in
Eq. 共2兲.
¯
Eis the Doppler shift frequency, which clearly
must be a flux function as it refers to a single mode. The
shear rate is also a flux function. As we have defined it, it is
smaller by the factor (r/q)/(BprR/B)⫽关关qˆ 兴兴(
) than the
commonly used Hahm–Burrell formula.7On the outboard
side 关关qˆ 兴兴(
) could be O(1/3) for a
⫽2 plasma. The dif-
fusivity is defined from the transport continuity equation
n0/
t⫹ⵜx共
˜
Exn
˜
兲⫽source, 共15兲
where
ⵜx共
˜
Exn
˜
兲⫽⫺1/V⬘共r兲
/
rV⬘共r兲
具
兩
ⵜr
兩
2
典
⫻关Dnatural /
具
兩
ⵜr
兩
2
典
兴dn0/dr,共16兲
with
Dnatural⫽
具
c/Bunitik
˜
n
˜
/n0
典
/共⫺dlnn0/dr兲.共17兲
As we noted in the Introduction, the combination
Dnatural /
具
兩
ⵜr
兩
2
典
⬅DITER is independent of the flux surface
label. V⬘(r)⫽
兰
d
(dS/d
)/
兩
ⵜr
兩
, where the differential sur-
face area is (dS/d
)⫽(dl/d
)2
R(
). In deriving Eq.
共17兲we took the normal flux at the flux surface to be
˜
Exn
˜
⫽⫺ik
(c/Bunit)
˜
n
˜
/
兩
ⵜr
兩
, again making use of the identity
兩
ⵜr
兩
关关qˆ 兴兴/B⫽1/Bunit . Also the surface average of any func-
tion Fis defined to be
具
F
典
⫽
兰
d
F(dS/d
)/
兩
ⵜr
兩
/
兰
d
(dS/d
)/
兩
ⵜr
兩
. The later surface averaging is a very
weak function of
; for perfectly nested ellipses (s
⫽0,s
␦
⫽0,
rR0⫽0), the surface average becomes
具
F
典
⫽
兰
d
FR(
)/
兰
d
R(
), which can be ignored 共i.e., replaced
with a simple
average兲at high aspect ratio. Thus apart from
the special surface average Dnatural , is virtually identical to
the circular formula. The conversion to dimensionless units
proceeds as in Ref. 5 with the unit of macro-length, aunit 共rat
the last closed flux surface兲,
s⬅cs/(eBunit /cMi) the unit of
micro-length 共e.g., k
s→k
兲,DgB⫽(cs/aunit)
s
2the unit of
4267Phys. Plasmas, Vol. 6, No. 11, November 1999 Ion temperature gradient turbulence simulations and...
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diffusion, cs/aunit the unit for rates, and perturbations nor-
malized to
s/aunit 关e.g., (e
/Te)/(
s/aunit)→
兴.
To obtain gyrofluid closures or gyrokinetic equations
reminiscent of the circular limit, we need only specify fur-
ther the arguments of the Bessel functions (z⫽k⬜
i) for the
gyroradius averages and the curvature frequency
D
k⬜
i⫽共2Ti/Te兲1/2共k
s兲共
⑀
关关B
ˆ兴兴兲1/2关关qˆ 兴兴/关关B
ˆ兴兴
⫻
兵
1⫹共关关kx/ky兴兴⫹kx0
兩
ⵜr
兩
/关关qˆ 兴兴兲2
其
1/2,共18兲
D⫽共Ti/Te兲共k
s兲共cs/aunit兲共2aunit /R0兲关关qˆ兴兴/关关B
ˆ兴兴
⫻
兵
关
⑀
共1⫺关关B
ˆ兴兴/2兲共关关cos兴兴⫹共关关kx/ky兴兴
⫹kx0
兩
ⵜr
兩
/关关qˆ 兴兴兲关关sin兴兴兲⫹关
⑀
共1⫺关关B
ˆ兴兴兲
⫻关关cos⫺p兴兴
其
,共19兲
where
⑀
⫽(v/vth)2,
⑀
关关B
ˆ兴兴⫽(v⬜/vth)2with
⑀
and gyro-
average constants of motion and the Maxwellian aver-
age of
⑀
(1⫺关关B
ˆ兴兴/2)⫽1 and
⑀
(1⫺关关B
ˆ兴兴)⫽1
2. Further
关关B
ˆ兴兴(
)⫽B/Bunit⇒1, 关关cos兴兴(
)⇒cos(
) and 关关sin兴兴
⫻(
)⇒sin(
) refer to the normal and geodesic curvature
components, and 关关cos⫺p兴兴(
)⇒0 refers to the finite beta ef-
fect of grad-Bminus curvature drift. These functions of
are
discussed in Appendix A.
III. SOME EXAMPLES OF GYROKINETIC LINEAR
STABILITY IN REAL GEOMETRY
The initial value electromagnetic gyrokinetic code, first
developed by Kotschenreuthers’,14 was modified using the
formulation of Sec. II. In addition, following the gyrokinetic
formulation of Antonsen and Lane,15 parallel magnetic per-
turbations (B
˜
储
) were added to the electrostatic and magnetic
vector potential perturbations (
˜
,A
˜
储
). This code was inde-
pendently modified for real geometry and B
˜
储
by Dorland and
Kotschenreuter16 and several code checks were made. Table
I gives a sample deuterium benchmark for the equilibrium
given in Ref. 3 共A⫽3.17, q⫽3.03,
⫽1.66,
␦
⫽0.416, s
⫽0.7, s
␦
⫽1.37, sˆ⫽2.85 共Ref. 3 sˆMHD⫽2Vq⬘/qV⬘⫽2.47兲,
rR0⫽⫺0.345,
␣
⫽1.22 and r/a⫽0.83, a/LT⫽3, a/Ln
⫽1, Ti/Te⫽1, nei⫽0, k
s⫽0.3兲. In the one fluid MHD
limit (k
s→0) there is a cancellation of terms such that B
˜
储
can be dropped as long as ⵜBdrift is replaced with curvature
drift, i.e. zero p⬘terms in 关关cos兴兴(
) and 关关cosគp兴兴(
) terms
of Eqs. 共A9兲and 共A11兲. Remarkably, this MHD recipe holds
rather well even at finite k
s, as shown by the fourth line of
Table I.16 Figure 1 shows recovery of the ideal critical

at
␣
⫽2.4.
An important feature of the Miller’s local equilibrium is
the ability to scan in parameter space away from the sˆ-
␣
model. Figure 2 shows a scan away from the standard circu-
lar point 关r/a⫽1
2,R/a⫽3共i.e., A⫽6兲,q⫽2, sˆ⫽1, a/LT
⫽3, a/Ln⫽1, Ti/Te⫽1,

⫽0(
␣
⫽0), Vei⫽0兴. In Fig. 2共a兲
we can see the effects of finite aspect ratio, and in Fig. 2共b兲
we see the general improvement with elongation. Figure 2共c兲
shows that reverse triangularity gives an improvement, likely
because of the drift reversal on the trapped electron drive.
This is reminiscent of the ‘‘comet’’ shape effect.17 Figure
2共d兲shows that there can be lower growth rates in going to
low aspect ratio, but only in the absence of trapped electrons.
Aspect ratio scans at fixed

were made in Ref. 18. However,
the main benefit of low aspect ratio Ais seen by scans at
fixed
␣
and fixed shift (
R0) with

⬀1/A. As shown in Fig.
3, growth rates do not increase, yet MHD stability can be
realized at much higher

共i.e.,
␣
crit independent of Aand q兲.
Another important feature of Miller local equilibria is that
they can be reconstructed from a stored plasma profile data
FIG. 1. Growth rate
␣
scan through the MHD critical point
␣
⫽2.4 of Ref.
3 Miller’s sample local equilibrium 共A⫽3.17, q⫽3.03,
⫽1.66,
␦
⫽0.416, s
⫽0.7, s
␦
⫽1.37, sˆ ⫽2.85,
rR0⫽0.345,
␣
⫽1.22, and r/a
⫽0.83, a/LT⫽3, a/Ln⫽1, Ti/Te⫽1,
ei⫽0, k
s⫽0.3兲.
FIG. 2. Growth rate scan in wave number 共a兲elongation 共b兲, triangularity
共c兲, and aspect ratio 共d兲about a standard point 关r/a⫽1/2, R/a⫽3共i.e., A
⫽6兲,q⫽2, sˆ ⫽1, a/LT⫽3, a/Ln⫽1, Ti/Te⫽1,

⫽0(
␣
⫽0), vei⫽0兴.
TABLE I. Bench point test at a reference local equilibrium.
A
储
B
储
共
␥
,
兲Ref. 16 共
␥
,
兲
Off Off electrostatic 共0.29, ⫺0.09兲共0.30, ⫺0.0927兲
On Off 共0.31, ⫺0.16兲共0.327, ⫺0.166兲
On On 共0.40, ⫺0.20兲共0.426, ⫺0.21兲
On Off ⵜB⫽
共0.39, ⫺0.214兲
4268 Phys. Plasmas, Vol. 6, No. 11, November 1999 R. E. Waltz and R. L. Miller
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base having R(
), r(
),
(
),
␦
(
), as well as Ti,e(
) and
ni,e(
)共e.g., the ITER transport profile data base in Ref. 9兲.
Figure 4 shows the stability profile reconstruction of the
DIII-D19 negative central shear Shot No. 84736,20 compar-
ing real geometry to the electrostatic and the finite-

sˆ-
␣
model. Here we can see that there can be substantial finite-

stabilization in the core plasma, yet the sˆ-
␣
model is known
to have a critical-

substantially below that possible with
shaped geometry.
From these examples of linear growth rates versus shape
parameters, we hesitate to make any general or quantitative
inferences about the benefits of shape. As we show in Sec. II
with nonlinear studies of turbulent transport for the ITG
modes with adiabatic electrons, the actual diffusivity well
above the critical temperature gradient does not follow linear
growth rates in a simple way. Documentation of the ITG
critical gradient with shape may be more useful than a docu-
mentation of growth rates.
IV. EXAMPLES OF NONLINEAR ITG MODE
SIMULATIONS IN REAL GEOMETRY
The code with a four-moment gyrofluid closure for adia-
batic electron ITG mode turbulence used in Refs. 4–6 was
reprogrammed with the formulation of Sec. II. Starting from
the standard case, well away from the critical gradient 关r/a
⫽1
2,R/a⫽3(A⫽6), q⫽2, sˆ⫽1, a/LT⫽3, a/Ln⫽1, Ti/Te
⫽1,

⫽0(
␣
⫽0), Vei⫽0兴, we move to finite elongation and
triangularity with s
⫽(r/
)
r
⬇(
⫺1)/
,s
␦
⫽
␦
/(1
⫺
␦
2)1/2. The key results are shown in Table II in dimen-
sionless units.
natural is in gyroBohm units with Bunit fixed.
Surprisingly, the
natural does not reflect the decrease in
␥
max with shape and is apparently offset by the increase in
the mixing length 1/⌬kx
max . For example, if we take ⌬kx
max to
be the spectral width in kx0for growth, and
mix⬀
␥
0/⌬kx
2
⫻
␥
net
␥
max /(
␥
max
2⫹
max
2) with
␥
0the damping rate of the n
⫽0 modes at kx0⫽k
max and
␥
net⫽
␥
max⫺
␥
E共
␥
E⫽0 here兲,
共a formula used in Ref. 8兲, then
mix is 0.24, 0.19, and 0.17
for
⫽1, 1.6, and 2. 共Unfortunately, we have not found an
expression for the mixing length that captures the depen-
dence on shape in a universal way.兲We also find that the
diffusion 共flux兲balloons much more to the outside as elon-
gation is increased. From these limited cases, it appears that
norm
natural is not a strong function of shape. Thus the benefit of
shape in this case comes from the increase in the square of
the effective field (Bunit /B0)2⬇
2. Thus confinement time at
fixed minor radius and B0would exhibit
2scaling well
away from a critical gradient. The last column of Table II
shows
ITER⬇2/(1⫹
2)⬇1/
1.3 at fixed B0. This is not as
strong as 1/
4suggested by Kinsey et al.10 and Bateman
et al.,11 but stronger than the 1/
0posited by the transport
models of Refs. 8 and 12.
We hasten to add, however, that the quantitative depen-
dence on shape 共or more specifically elongation兲is not likely
separable from other parameters. For example, in Table III
we show that the dependence on
is much stronger at low q
(
ITER⬇1/
2.1), where the increasing
drives closer to sta-
bility 共e.g., q⫽1at
⫽2 is stable兲, and much weaker
(
ITER⬇
0.03) at high-q.
Previous work using the sˆ-
␣
model4–6 showed that tur-
bulence vanishes when the shear rate satisfies
␥
E⫽
␣
E
␥
max
with
␣
Enear 1.0 to within 20%–30%. In Sec. II, we showed
FIG. 3. Growth rate scans in aspect ratio at fixed
␣
and
R0⫽⫺0.1,
⫽2, s
⫽0.5,
␦
⫽0 at various

from the standard point of Fig. 2.
FIG. 4. Growth rate 关normalized to cs(0)/a兴versus a normalized
using
local equilibria from a profile data base for DIII-D Shot No. 84 736 共see
Ref. 19兲.
TABLE II. ITG diffusivity at increasing elongation and triangularity.
␦
␥
max ⌬kx
max
natural 1/
具
兩
ⵜr
兩
2
典
(B0/Bunit)2
ITER
sˆ ⫺
␣
0.082 0.417 2.1 1 1 2.1
1 0 0.123 0.286 2.72 1 1 2.72
1.6 0 0.085 0.210 2.47 1.82 0.39 1.72
2.0 0 0.059 0.144 2.78 2.01 0.25 1.46
2.0 0.4 0.057 0.148 3.18 2.00 0.25 1.58
TABLE III. ITG diffusivity at increasing elongation for various q.
q
␥
max ⌬kx
max
natural
Bunit fixed Scaling of
natural
ITER
B0fixed
1.5 1 0.11 0.241 2.26 2.26
1.5 2 0.034 0.117 0.98
⫺1.2 0.516
2.0 1 0.123 0.286 2.72 2.72
2.0 2 0.059 0.144 2.78
01.46
3.0 1 0.15 0.346 4.80 4.80
3.0 2 0.10 0.206 5.04
0.07 2.72
4.0 1 0.16 0.434 5.72 5.72
4.0 2 0.10 0.210 11.18
0.96 5.87
4269Phys. Plasmas, Vol. 6, No. 11, November 1999 Ion temperature gradient turbulence simulations and...
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how to properly generalize the shear rate to a flux function
关Eq. 共2兲兴. In Table IV, we show some examples about the
standard case for real geometry. From Table IV, we conclude
that
␣
Earound 0.5 is more appropriate in real geometry
共even at
⫽1兲. It seems to take less rotational shear to sta-
bilize in real geometry; this may be due to the fact that the
extent of the unstable spectrum region ⌬
0, or the mode
width ⌬
is a smaller fraction of 2
for the finite aspect ratio
as compared to the sˆ-
␣
model. Thus the mode spends less
time in its rotation through the outboard side 共see Ref. 6 for
discussion of the Floquet mode rotation picture兲. Recall that
␥
Eas we have defined it is 关关q兴兴(0) times the commonly
used Hahm–Burrell shear rate at
⫽0. 关关q兴兴(0) can easily
cancel the
␣
E⫽0.5 if one compares the Hahm–Burrell rate
to
␥
max .
ACKNOWLEDGMENTS
This is work is supported by the U.S. Department of
Energy under Grant No. DE-FG03-95ER54309 and addition-
ally by the Grand Challenge Numerical Tokamak Turbulence
Project. We wish to acknowledge many useful discussions
with Dr. W. Dorland at the University of Maryland.
APPENDIX A: BALLOONING MODE LOCAL
EQUILIBRIUM FORMULATION
Field perturbations can be represented F
⫽Fn,kx0(
)exp(⫺inS⫹ikx0
兩
ⵜr
兩
x) with the eikonal Sex-
panded in a normal distance xfrom the flux surfaces as S
⫽
⫹S0(
)⫹xS1(
). Similarly, the poloidal flux is given
by ⌿⫽⌿0⫹x⌿1⫹x2⌿2. Following Ref. 13:
S0共
兲⫽⫺f
冕
0
d
共dl/d
兲/R0
2Bp
2,共A1兲
S1共
兲⫽RBpf
冕
0
d
共dl/d
兲/R3Bp
2
⫻
兵
2 sinu/R⫹2/Rc⫺f⬘/fRBp
⫺4
Rp⬘/Bp⫺ff⬘/RBp
其
共A2兲
1共
兲⫽RBp,共A3兲
2共
兲⫽共1/2兲
兵
sinu⫹R/Rc兲Bp⫺4
R2p⬘⫺ff⬘
其
,共A4兲
where dl2⫽dR2⫹dZ2gives the poloidal arc length with
dR/dl⫽cos(u), dZ/dl⫽⫺sin(u), and du/dl⫽⫺f⬘/Rc
⬅df/dx, and p⬘⬅dp/d
. Explicitly,
ky⫽共⫺n/R兲共Bp/B兲共⫺Bp/R⫹S0f/RBp兲
⫽共n/R兲共B/Bp兲,共A5兲
or ky⫽k
关关qˆ 兴兴(
), defining 关关qˆ 兴兴(
)⫽(rB/RBpq). In ad-
dition, we defined kx⫽ky关关kx/ky兴兴⫹kx0
兩
ⵜr
兩
where
关关kx/ky兴兴共
兲⫽⫺共RBp兲2/B⫻关S1共
兲/RBp兴.共A6兲
Decomposing
关S1共
兲/RBp兴⫽D0共
兲⫹Dp⬘共
兲p⬘⫹Dff⬘共
兲ff⬘
and using q⫽(f/2
)
兰
d
(dl/d
)/R2Bp, it can be shown,
after much tedious algebra, that 2
q⬘⫽⫺D0(2
)
⫺Dp⬘(2
)p⬘⫺Dff⬘(2
)ff⬘, so that
关S1共
兲/RBp兴⫽Dff⬘共2
兲⫺1
兵
关D0共
兲Dff⬘共2
兲
⫺D0共2
兲Dff⬘共
兲兴⫹关Dp⬘共
兲Dff⬘共2
兲
⫺Dp⬘共2
兲Dff⬘共
兲兴p⬘⫺2
Dff⬘q⬘
其
.共A7兲
Using q⬘⫽sˆ(RBp/
兩
ⵜr
兩
)(q/r) and p⬘⫽dp/dr(RBp/
兩
ⵜr
兩
)
Eq. 共A7兲gives 关关kx/ky兴兴 explicitly.
The curvature (
⫽b
ˆ•ⵜb
ˆ) and grad-B(ⵜ
ជ
B/B) drift fre-
quency is given by
¯
Di⫽共kyyˆ⫹kxxˆ兲⫻2共cTi/eB兲•b
ˆ
⫻
兵
ជ
关共1⫺关关B兴兴兲⫹ⵜ
ជ
B/B共关关B兴兴/2兴
其
,共A8兲
This can be decomposed as in Eq. 共19兲with
关关cos兴兴共
兲⫽R0
xB2/2B2
⫽R0Bp/B2关Bp/Rc⫺4
Rp⬘
⫺f2sin共u兲/B2R3兴,共A9兲
关关sin兴兴共
兲⫽共R0f/RB兲共dl/d
兲
B2/2B2,共A10兲
关关cos–p兴兴共
兲⫽R0共4
p⬘RBp/B2兲,共A11兲
where we used
ជ
⫽关
xB2/2B2⫹4
p⬘RBp/B2兴xˆ
⫹共f2/R2B2兲
lB2/2B2lˆ⫺共fBp/RB2兲
lB2/2B2
ˆ
共A12兲
and
ⵜ
ជ
B/B⫽
xB2/2B2xˆ⫹
lB2/2B2lˆ.共A13兲
To force ⵜBdrift to equal curvature drift set the p⬘terms in
Eq. 共A9兲and 共A11兲to zero.
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4 1 0.22 0.09
4 2 0.13 0.08
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