PHYSICS OF PLASMAS
VOLUME 9, NUMBER 4
APRIL 2002
Numerical studies of edge localized instabilities in tokamaks
H. R. Wilson
EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, Oxon OX14 3DB,
United Kingdom
P. B. Snyder
General Atomics, San Diego, California 92186-5608
G. T. A. Huysmans
Association EURATOM-CEA sur la Fusion, CEA Cadarache, F-13108 St Paul lez Durance, France
R. L. Miller
Archimedes Technology Group, 5405 Oberlin Dr., San Diego, California 92121
共Received 16 November 2001; accepted 15 January 2002兲
A new computational tool, edge localized instabilities in tokamaks equilibria 共ELITE兲, has been
developed to help our understanding of short wavelength instabilities close to the edge of tokamak
plasmas. Such instabilities may be responsible for the edge localized modes observed in high
confinement H-mode regimes, which are a serious concern for next step tokamaks because of the
high transient power loads which they can impose on divertor target plates. ELITE uses physical
insight gained from analytic studies of peeling and ballooning modes to provide an efficient way of
calculating the edge ideal magnetohydrodynamic stability properties of tokamaks. This paper
describes the theoretical formalism which forms the basis for the code. 关DOI: 10.1063/1.1459058兴
I. INTRODUCTION
the typical equilibrium length scales兲 and two-dimensional
nature. In this paper we describe a formalism which permits
efficient analysis of this class of instabilities within a linearized ideal MHD model.
The paper is set out as follows. In the next section we
describe the formalism which we have developed for analyzing high n, coupled peeling-ballooning modes in tokamaks.
The technique is applicable to arbitrary tokamak equilibria,
except that at present we have not considered the separatrix.
Then, in Sec. III, we illustrate some results from the code,
which we have called ELITE 共Edge Localized Instabilities in
Tokamak Equilibria兲. We first describe a benchmark case,
which we take to be a circular cross section, aspect ratio A
⫽3 plasma which has an edge pressure pedestal unstable to
n⫽⬁ ballooning modes. We compare the results of ELITE
with those of MISHKA-1,9 and find good agreement for n
⬎4 共ELITE makes use of an expansion for large n兲. Next we
illustrate the results for a shaped plasma equilibrium, in this
case a DIII-D equilibrium re-constructed using EFIT.10 In
Sec. IV we draw some conclusions, and make some suggestions for how the model could be further improved.
In the high confinement H-mode of tokamak operation, a
transport barrier forms at the plasma edge, leading to a steep
pressure gradient, and therefore, a high bootstrap current.
High pressure gradients can lead to high toroidal mode number, n, ballooning instabilities 共e.g., see Ref. 1兲 but, on the
other hand, high edge current reduces the magnetic shear
which helps to stabilize these. However, high edge currents
can drive high n kink or ‘‘peeling’’ modes,2– 4 and these are
stabilized by edge pressure gradient 共due to the effect of
favorable average curvature in a tokamak兲. This interplay
between pressure and current makes edge stability particularly interesting. As a final ‘‘twist’’ to the story, the ballooning and peeling modes can couple,5 leading to particularly
dangerous current-driven instabilities which can extend right
across the transport barrier region, into the plasma core.4 The
result is that a range of instabilities can exist from highly
localized peeling modes, more extended ballooning modes,
extended, coupled peeling-ballooning modes, or, indeed, access to a second stability region6 with correspondingly
higher pressure gradient limits. This range of possible ideal
magnetohydrodynamic 共MHD兲 plasma edge instabilities may
help to explain the wide range of edge localized mode
共ELM兲 phenomena observed in tokamaks, such as relatively
small type III ELMs at low pressure gradient, up to large
type I ELMs, smaller ‘‘grassy’’ ELMs or even ELM-free
regimes at larger pressure gradient 共for a review of ELM
phenomenology, see Ref. 7兲. Understanding these instabilities may then help us to identify how to access regimes
which have tolerable ELMs: A key issue for the International
Thermonuclear Experimental Reactor 共ITER兲.8 It is, therefore, important to be able to analyze these instabilities, which
is complicated by their short wavelength 共much shorter than
1070-664X/2002/9(4)/1277/10/$19.00
II. ELITE FORMALISM
To help improve the efficiency of ELITE we restrict consideration to intermediate to high n modes, and evaluate the
change in energy associated with a radial perturbation of a
plasma fluid element, denoted by X. Following Ref. 1, we
perform an expansion in n ⫺1 , which permits the other two
components of plasma perturbation to be eliminated in favor
of X. After some algebra, neglecting the contribution due to
1277
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1278
Wilson et al.
Phys. Plasmas, Vol. 9, No. 4, April 2002
the inertia for the present, it can be shown that the change in
energy, correct to the first two orders in n ⫺1 can be written in
the form
␦ W⫽ ␲
冕
0
⫺⬁
d␺
冋
冖
d␹
再
JB 2
兩k 储X兩
R 2 B 2p
冉
冊
2
⫹
冏 冏
R 2 B 2p 1 ⳵ Y
JB 2 n ⳵ ␺
2
⫺
⳵
2Jp ⬘
B2
i f ⳵B2 X* ⳵X
兩X兩2
p⫹
⫺
2
B
⳵␺
2
2 JB 2 ⳵␹ n ⳵ ␺
⫺
X*
1
JBk 储 共 ␴ ⬘ X 兲 ⫹ 关 PJBk *
储 Q * ⫹ P * JBk 储 Q 兴
n
n
⫹
⳵ ␴
X *Y
⳵␺ n
冋
册冎
.
册
共1兲
Here we have defined the poloidal magnetic flux, ␺, and
toroidal field function, f ( ␺ ), such that the magnetic field can
be expressed in the form
B⫽ f “ ␾ ⫹“ ␾ ⫻“ ␺ ,
共2兲
so that ␺ is an increasing function of the minor radius of flux
surfaces. We have chosen the gauge for ␺ such that ␺ ⫽0 at
the plasma edge, and takes a negative value everywhere in
the core. In fact, because we are interested in edge localized
modes, whose amplitude X→0 in the core, we can replace
the lower limit of integration in the ␺ variable by ⫺⬁, as
indicated in Eq. 共1兲. The orthogonal 共␺,␹,␾兲 coordinate system we use is similar to that used in Ref. 1, where ␹ is a
poloidal angle and ␾ the toroidal angle; J is the Jacobian in
this system, defined such that Jd ␹ ⫽dl/B p , where dl is the
poloidal arc length element along a flux surface, and B p is
the poloidal component of the magnetic field. The major radius is denoted by R and B is the total magnetic field. The
pressure 共with a factor ␮ 0 absorbed兲 is denoted by p, a prime
denotes a differential with respect to ␺ and a star denotes a
complex conjugate. The variable Y is simply related to X
Y ⫽JBk 储 X,
共3兲
where the parallel gradient operator is defined by
JBk 储 ⫽⫺i
⳵
⫹n ␯ ,
⳵␹
共4兲
with
␯⫽
fJ
.
R2
共5兲
The parallel current density, denoted by ␴, can be written in
the form
␴ ⫽⫺
f p⬘
⫺ f ⬘.
B2
Q⫽
f 1 ⳵Y
p⬘
.
2 X⫹
B
JB 2 n ⳵ ␺
␦ W I⫽ ␲ ␥ 2
冕
0
⫺⬁
冉
d␺
冖
d␹
再
冏 冏
冉 冊
␳ JR 2 B 2p ⳵ X
␳J
2
兩
X
兩
⫹
2
n 2B 2 ⳵ ␺
R 2B p
冊 冉
⫹
⳵X*
⳵X
⳵X
G
H ⳵X*
⫹X *
⫹ 3
JBk 储
2 X
n
⳵␺
⳵␺
n ⳵␺
⳵␺
⫹
⳵X
⳵X*
JBk *
储
⳵␺
⳵␺
冉 冊冊 冎
.
2
共9兲
The equilibrium functions H and G are defined in Eqs. 共A4兲
and 共A5兲 of Appendix A, ␳共␺兲 is the mass density and ␥ 2 is
the eigenvalue of the system, equal to the square of the
growth rate. Again, within our ‘‘incompressible MHD’’
model, the inertia term is correct to the first two orders in
n ⫺1 .
Note that we have performed an ordering in n ⫺1 so that
the Hermitian property of ideal MHD is exactly preserved;
we find this to be important for the numerical results.
The method is now straightforward, but involves a large
amount of tedious algebra. We first Fourier decompose the
poloidal variation of the displacement X, and it is convenient
to do this in terms of a ‘‘straight field line’’ angle, ␻
␻⫽
1
q
冕␯␹
␹
d ,
共10兲
where the safety factor q is simply q⫽(1/2␲ )养 ␯ d ␹ . Thus
we express
X⫽
兺m u m共 x 兲 e ⫺im ␻ ,
共11兲
where m are the poloidal mode numbers, and then we have
共6兲
Finally, P and Q are defined as
f B 2p ⳵ Y
,
P⫽ ␴ X⫹
n␯B2 ⳵␺
Note that Eq. 共1兲 involves contributions to ␦ W due to
surface terms arising from integrations by parts. These were
absent in the ballooning analysis of core modes developed in
Ref. 1, because there the mode amplitude was assumed to be
negligible at the plasma–vacuum interface. It is important to
retain these in our situation as the mode amplitude is not
negligible at the plasma boundary, and indeed is an essential
feature of the peeling modes which we wish to analyze. Also
note that by retaining terms up to, and including, O(n ⫺1 ) we
retain the kink drive, proportional to the radial derivative of
the parallel current density.
The above expression 共1兲 neglects the inertial energy.
Although we are primarily interested in marginal stability, it
is useful to have an indication of the growth rate, so we
employ a simplified model which neglects that inertia associated with displacements parallel to the magnetic field
共equivalent to the assumption of incompressible MHD, valid
at marginal stability兲. Thus, for the inertial energy, we derive
共7兲
共8兲
Y⫽
兺m
冉 冊
⫺
␯
共 m⫺nq 兲 u m 共 x 兲 e ⫺im ␻ .
q
共12兲
For high m modes field line bending will have a strong influence, and will tend to restrict the u m to be localized about
their mode rational surfaces, where m⫽nq. Thus we have
introduced the ‘‘fast’’ radial variable
x⫽m 0 ⫺nq,
共13兲
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Phys. Plasmas, Vol. 9, No. 4, April 2002
Numerical studies of edge localized instabilities . . .
which is the length scale on which we expect the u m to vary.
Here m 0 is the poloidal mode number of some reference
rational surface, and we arbitrarily choose it to be that associated with the first rational surface in the vacuum, i.e.,
m 0 ⫽Int关 nq a 兴 ⫹1,
共14兲
where q a is the edge safety factor. Note that x increases by 1
each time a rational surface is crossed as we go from the
plasma edge towards the core.
We now substitute the forms for X and Y into Eqs. 共1兲
and 共9兲, and perform integrations by parts 共in the radial coordinate兲 to eventually derive a form for ␦ W, which can be
decomposed into a contribution from the core plasma, ␦ W p ,
and a piece arising from the surface terms due to the integrations by parts, ␦ W s
兺m
再
␦ W⫽ ␦ W p ⫹ ␦ W s .
1279
共15兲
For our vacuum model we assume that the vessel wall is far
from the plasma, so that it is effectively surounded by an
infinite vacuum on all sides; this is a good approximation for
the radially localized modes which we consider. In such a
situation the vacuum contribution to ␦ W can be expressed
analytically solely in terms of the plasma displacement at the
plasma–vacuum interface,11 and therefore, can be absorbed
into ␦ W s . The resulting expressions for ␦ W p and ␦ W s are
given in Appendix A.
We stress here that ELITE is not based on a ␦ W numerical approach, but instead solves the full set of Euler equations for the radial variations of the Fourier mode amplitudes. These can be derived trivially from Eq. 共A1兲, and for
each Fourier mode k they take the form
⬙ k,m 共 m⫺nq 兲 2 共 k⫺nq 兲 ⫹A m2k
⬙ k,m 共 m⫺nq 兲共 k⫺nq 兲 2 ⫹A mk
⬙ k,m 共 m⫺nq 兲共 k⫺nq 兲 ⫺ ␥ 2 共 I m⬙ k,m 共 m⫺nq 兲 ⫹I ⬙k k,m 共 k⫺nq 兲
关 A 2mk
⫹I ⬙ k,m 兲兴
d 2u m
⬘ k,m 共 m⫺nq 兲 2 共 k⫺nq 兲 ⫹A m2k
⬘ k,m 共 m⫺nq 兲共 k⫺nq 兲 2 ⫹A mk
⬘ k,m 共 m⫺nq 兲共 k⫺nq 兲 ⫹A 2m
⬘ k,m 共 m⫺nq 兲 2
⫹ 关 A 2mk
d␺2
⬘ k,m 共 k⫺nq 兲 2 ⫹ 共 A m⬘ k,m ⫺ ␥ 2 I m⬘ k,m 兲共 m⫺nq 兲 ⫹ 共 A ⬘k k,m ⫺ ␥ 2 I ⬘k k,m 兲共 k⫺nq 兲 ⫹A ⬘ k,m ⫺ ␥ 2 I ⬘ k,m 兴
⫹A 2k
du m
d␺
k,m
k,m
k,m
k,m
k,m
⫹ 关 A 2mk
共 m⫺nq 兲 2 共 k⫺nq 兲 ⫹A m2k
共 m⫺nq 兲共 k⫺nq 兲 2 ⫹A mk
共 m⫺nq 兲共 k⫺nq 兲 ⫹A 2m
共 m⫺nq 兲 2 ⫹A 2k
共 k⫺nq 兲 2
冎
k,m
k,m
2 k,m
k,m
⫹共 Am
⫺ ␥ 2I m
⫺ ␥ 2 I k,m 兴 u m ⫽0,
兲共 m⫺nq 兲 ⫹ 共 A k,m
k ⫺ ␥ I k 兲共 k⫺nq 兲 ⫹A
where the matrix elements A k,m , A ⬘ k,m , and A ⬙ k,m are given
in Appendix A, together with those associated with the inertia, labeled by I. Note that the number of primes on the
matrix elements is used as a notation to indicate the number
of ‘‘radial’’ derivatives on their associated Fourier amplitudes u m , and does not represent a ␺-derivative of the matrix
elements themselves here 共superscripts label the matrix elements兲.
If we assume that some number, M , of Fourier harmonics is required to describe the modes, then Eq. 共16兲 represents
a system of M coupled, linear differential equations for the
mode amplitudes, u m . To complete the specification of the
system we need to provide a set of boundary conditions. The
first of these is that the u m (x) tend to zero deep inside the
兺m
再
共16兲
plasma core 共i.e., we are interested only in edge-localized
modes兲; thus we have
lim u m 共 x 兲 ⫽0.
共17兲
x→⬁
Another boundary condition is that the surface contribution
to the total energy must be zero, which represents the fact
that the jump in p⫹B 2 /2 across the plasma–vacuum interface is zero.11 The final matching condition, that the radial
component of the magnetic field is continuous across the
interface, is used to express the vacuum magnetic-field perturbation in terms of the u m at the plasma surface. Thus we
arrive at a second set of conditions 共for each k兲 to be applied
at the plasma boundary, x⫽⌬⫽m 0 ⫺nq a , where 0⭐⌬⭐1
⬘ k,m 共 m⫺nq 兲 2 共 k⫺nq 兲 ⫹S m2k
⬘ k,m 共 m⫺nq 兲共 k⫺nq 兲 2 ⫹S mk
⬘ k,m 共 m⫺nq 兲共 k⫺nq 兲 ⫺ ␥ 2 共 J m⬘ k,m 共 m⫺nq 兲
关 S 2mk
⫹J k⬘ k,m 共 k⫺nq 兲 ⫹J ⬘ 兲兴
du m
k,m
k,m
k,m
k,m
⫹ 关 S 2mk
共 m⫺nq 兲 2 共 k⫺nq 兲 ⫹S m2k
共 m⫺nq 兲共 k⫺nq 兲 2 ⫹S mk
共 m⫺nq 兲共 k⫺nq 兲 ⫹S 2k
共 k⫺nq 兲 2
d␺
k,m
2 k,m
k,m
⫹S k,m
兲兴 u m ⫹ ␦ W k,m
k 共 k⫺nq 兲 ⫺ ␥ 共 J m 共 m⫺nq 兲 ⫹J k 共 k⫺nq 兲 ⫹J
v um
冎
⫽0.
共18兲
x⫽⌬
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1280
Phys. Plasmas, Vol. 9, No. 4, April 2002
Again, the matrix elements S and J are defined in Appendix
represents the contribution from the vacuum.
A, and ␦ W k,m
v
The set of Eqs. 共16兲–共18兲 constitutes the complete eigenmode system that ELITE has been designed to solve, either
to determine whether or not a given configuration is stable,
or to find the growth rate of the instability through calculating the eigenvalue ␥ 2 . There are, however, a number of features that greatly increase the efficiency of the code and are
worthwhile noting. First there is a separation of equilibrium
and mode-structure length scales. Because the equilibriumvaries slowly on a length scale of the order of the distance
between rational surfaces 共for intermediate to high n兲 the
matrix elements can be evaluated and tabulated on a relatively coarse radial mesh. The Fourier modes, u m (x), on the
other hand, vary on the length scale comparable to the distance between rational surfaces, and these need a much finer
radial grid. A second feature of ELITE is that it makes use of
the fact that we expect each u m (x) to be highly localized
about its rational surface 关i.e., where m⫽nq( ␺ )兴 due to field
line bending. This means that while many harmonics may be
important to reconstruct the full mode structure, at any one
radial position only a limited subset of these will be significant, and the rest can be set to zero amplitude. This provides
a significant saving of both computing memory and time.
A final feature of ELITE is how it is designed to evaluate
radial derivatives of equilibrium quantities. In principle these
can be calculated numerically if the equilibrium is known at
all flux surfaces. However, by making use of an expansion of
the Grad–Shafranov equation about a flux surface, these derivatives can be calculated analytically in terms of quantities
only on that flux surface12 关specifically the flux surface
shape, the poloidal field variation on the flux surface and the
profiles p( ␺ ) and f ( ␺ ) are all that are required兴. We require
second derivatives 共e.g., of the safety factor q兲 with respect
to ␺ here, and therefore, we need to expand to higher
order than is usually done: The results are presented in
Appendix B.
III. NUMERICAL RESULTS
In this section we illustrate some of the results from
ELITE. Here we restrict consideration to general results, and
will describe more specific tokamak edge stability studies
elsewhere.
We begin by considering the stability of a simple aspect
ratio, A⫽3, circular cross section tokamak. We have constructed a ␤ N ⫽2.1 equilibrium with a steep pressure pedestal
at the plasma boundary 共see Fig. 1兲 so that it is unstable to
n⫽⬁ ballooning modes in the region ␺ / ␺ a ⬎0.82, where ␺ a
is the poloidal flux at the plasma boundary ( ␤ N
⫽ ␤ (%)a(m)B(T)/I p (MA) where ␤ is the ratio of thermal
to magnetic pressure, a is the minor radius, B is the magnetic
field and I p is the plasma current兲. The current profile is
calculated from neoclassical theory, assuming a constant
loop voltage across the plasma, and allowing for the bootstrap current. Figure 1共b兲 shows the profile of the plasma
current density; note that it is strongly peaked on axis, which
is a consequence of neoclassical conductivity, but that there
Wilson et al.
FIG. 1. 共a兲 Pressure profile and 共b兲 current profile for the circular crosssection benchmark study.
is significant current driven in the pedestal region, due
mainly to the bootstrap current.
Analyzing the stability of the equilibrium, we find that
ELITE predicts instability over a range of n. Figure 2 shows
a typical mode structure, in this case for a n⫽10 ballooning
mode. In Fig. 2共a兲 we show the set of curves for the u m ( ␺ ),
which illustrates the feature that each harmonic is more radially localized 共about its rational surface兲 than the whole
mode structure. Note that in this case the mode is essentially
localized within the steep pressure gradient region at the
plasma edge 关compare with Fig. 1共a兲兴. In Fig. 2共b兲 we show
the mode structure in poloidal cross section, where we see a
clear ballooning mode structure.
Going to higher toroidal mode number, n⫽50, the mode
becomes much more radially localized, as expected from
conventional ballooning mode theory 共e.g., see Ref. 1兲. This
can be seen by comparing Fig. 3, which shows the radial
profiles of the Fourier mode amplitudes for this case, with
Fig. 2共a兲.
Using the circular cross-section equilibrium as an example, we have performed a careful benchmark with the
MISHKA-1 code,9 which is also capable of analyzing high n
ideal MHD stability. In Fig. 4 we compare the predictions for
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Phys. Plasmas, Vol. 9, No. 4, April 2002
Numerical studies of edge localized instabilities . . .
1281
FIG. 4. Comparison between ELITE 共diamonds兲 and MISHKA-1 共triangles兲
of the growth rate for ideal MHD instabilities in the circular cross-section
tokamak 共␻ A is the Alfvén frequency兲.
reconstruction of the discharge, limited at the 99% flux surface 共recall that ELITE cannot yet handle the separatrix itself, but can approach it very closely兲. Note the extremely
large number of poloidal harmonics required to analyze these
shaped equilibria 共compare with Figs. 2 and 3 for the circular
cross-section case兲; ELITE is designed to efficiently handle
such cases 共⬃ minute time scales on ⬃1 GHz computers兲.
FIG. 2. 共a兲 The radial profiles of the Fourier harmonics u m ( ␺ ) and a contour
plot of the eigenfunction, X( ␺ , ␹ ), for an n⫽10 instability in the circular
cross section, A⫽3 equilibrium described in the text. In the contour plot,
light and dark shades represent large positive and negative perturbations,
respectively.
the growth rates as a function of n from the two codes, and
obtain good agreement over the full range of nⲏ4. Recall
that ELITE is based on an expansion in n ⫺1 , while
MISHKA-1 is valid over the full range of n. This comparison indicates that ELITE remains valid over the range of
intermediate to high n, which is the range of interest for
studies of ELMs.
As a final example, we show in Fig. 5 the mode structure
in the poloidal cross section of a DIII-D VH-mode discharge
共#97887兲, just before an ELM occurred. In this case the equilibrium is taken from a high resolution, well-converged EFIT
FIG. 3. The radial profiles of the Fourier harmonics for an n⫽50 ballooning
mode for the circular cross-section equilibrium.
FIG. 5. 共a兲 A contour plot of the eigenfunction, X( ␺ , ␹ ), and 共b兲 the radial
profiles of the dominant Fourier amplitudes for an n⫽20 ballooning-type
instability in a DIII-D equilibrium 共shot #97887兲.
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1282
Wilson et al.
Phys. Plasmas, Vol. 9, No. 4, April 2002
IV. CONCLUSIONS
Using our knowledge of the physics underlying ballooning and peeling modes, we have developed a new, efficient
tool for calculating the ideal MHD stability of the tokamak
plasma edge region, which is often two-dimensional in nature. It is anticipated that this will be useful for helping our
understanding of ELMs; in particular, how to control them
and how to identify regimes with benign ELMs. In addition,
ELITE is useful for calculating the pressure limits that can
exist in the edge pedestal region, which is an important ingredient for determining the temperature pedestal 共which influences the confinement time兲.13 The use of the code to interpret ELMs and the temperature pedestal will be described
elsewhere.14
This work represents a first step to develop a tool for
understanding the stability of tokamak edge plasmas, and as
such represents the simplest, useful model of edge plasma
stability, ie ideal MHD. There are a number of other features
which one could consider incorporating into the model, and
these are being considered. One issue is to improve the
plasma geometry, allowing for the presence of a separatrix,
and even a scrape-off layer plasma, where the field lines
connect to material surfaces. The scrape-off layer might be
expected to be important when its width is comparable to the
␦ W p⫽ ␲
冕
␺a
⫺⬁
d␺
u*
兺
k
m,k
再
linear radial mode width, and could be incorporated into
ELITE through modified boundary conditions. Improved
plasma physics models would also help us to understand the
different ELM regimes observed in more detail. For example, it is likely that diamagnetic effects will be important
in the steep gradient regions of the edge transport
barrier,15–17 and also strong sheared plasma flows, as measured in H-mode, might be expected to have an impact on
the radially extended, ballooning-type modes. ELITE has
been developed in such a way that these improvements could
be incorporated as the models are developed.
ACKNOWLEDGMENTS
This work was supported by the U.K. Department of
Trade and Industry and EURATOM, and U.S. DOE contract
No. DE-FG03-95ER54309.
APPENDIX A: ELITE MATRIX ELEMENTS
The Euler equations and their boundary conditions are
derived from the plasma and surface contributions to the perturbed energy, respectively. In terms of the Fourier mode
amplitudes, defined through Eq. 共11兲, these can be written in
the form
⬙ k,m 共 m⫺nq 兲 2 共 k⫺nq 兲 ⫹A m2k
⬙ k,m 共 m⫺nq 兲共 k⫺nq 兲 2 ⫹A mk
⬙ k,m 共 m⫺nq 兲共 k⫺nq 兲 ⫺ ␥ 2 共 I m⬙ k,m 共 m⫺nq 兲
关 A 2mk
⫹I k⬙ k,m 共 k⫺nq 兲 ⫹I ⬙ k,m 兲兴
d 2u m
⬘ k,m 共 m⫺nq 兲 2 共 k⫺nq 兲 ⫹A m2k
⬘ k,m 共 m⫺nq 兲共 k⫺nq 兲 2 ⫹A mk
⬘ k,m 共 m⫺nq 兲共 k⫺nq 兲
⫹ 关 A 2mk
d␺2
⬘ k,m 共 m⫺nq 兲 2 ⫹A 2k
⬘ k,m 共 k⫺nq 兲 2 ⫹ 共 A m⬘ k,m ⫺ ␥ 2 I m⬘ k,m 兲共 m⫺nq 兲 ⫹ 共 A ⬘k k,m ⫺ ␥ 2 I k⬘ k,m 兲共 k⫺nq 兲 ⫹A ⬘ k,m
⫹A 2m
⫺ ␥ 2 I ⬘ k,m 兴
du m
k,m
k,m
k,m
k,m
⫹ 关 A 2mk
共 m⫺nq 兲 2 共 k⫺nq 兲 ⫹A m2k
共 m⫺nq 兲共 k⫺nq 兲 2 ⫹A mk
共 m⫺nq 兲共 k⫺nq 兲 ⫹A 2m
共 m⫺nq 兲 2
d␺
冎
k,m
k,m
k,m
2 k,m
k,m
⫹A 2k
⫺ ␥ 2I m
⫺ ␥ 2 I k,m 兴 u m ,
兲共 m⫺nq 兲 ⫹ 共 A k,m
共 k⫺nq 兲 2 ⫹ 共 A m
k ⫺ ␥ I k 兲共 k⫺nq 兲 ⫹A
共A1兲
for the plasma contribution, and
␦ W s⫽
␲
n
u*
兺
k
m,k
再
⬘ k,m 共 m⫺nq 兲 2 共 k⫺nq 兲 ⫹S m2k
⬘ k,m 共 m⫺nq 兲共 k⫺nq 兲 2 ⫹S mk
⬘ k,m 共 m⫺nq 兲共 k⫺nq 兲 ⫺ ␥ 2 共 J m⬘ k,m 共 m⫺nq 兲
关 S 2mk
⫹J k⬘ k,m 共 k⫺nq 兲 ⫹J ⬘ 兲兴
du m
k,m
k,m
k,m
⫹ 关 S 2mk
共 m⫺nq 兲 2 共 k⫺nq 兲 ⫹S m2k
共 m⫺nq 兲共 k⫺nq 兲 2 ⫹S mk
共 m⫺nq 兲共 k⫺nq 兲
d␺
冎
k,m
k,m
2 k,m
k,m
⫹S 2k
兲兴 u m ⫹ ␦ W k,m
共 k⫺nq 兲 2 ⫹S k,m
k 共 k⫺nq 兲 ⫺ ␥ 共 J m 共 m⫺nq 兲 ⫹J k 共 k⫺nq 兲 ⫹J
v um ,
for the surface contribution 共where quantities are to be evaluated at the plasma surface兲. The vacuum contribution is represented by the terms ␦ W k,m
v
The matrix elements which appear in the above two contributions to the plasma energy and also the set of Euler
equations for the radial variation of the Fourier mode ampli-
共A2兲
tudes solved by ELITE, Eq. 共16兲, are given by
A k,m ⫽⫺2
⫺
m k,m imq ⬘ k,m
p ⬘ q k,m
T ⫹q p ⬘ T 11
⫺ 2 T 24
f 9
n
n q
imq ⬘ k,m iq k,m
T ⫹ T 30 ,
n 2 q 26
n
Downloaded 14 Feb 2006 to 198.129.105.159. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp
Phys. Plasmas, Vol. 9, No. 4, April 2002
A k,m
k ⫽
冉
k,m
mq ⬘ k,m
2im 2 q ⬘ k,m mq ⬘ k,m T 17
⫹ 2 2 T 20
T
⫹
T
⫺
2 2
2 2 16
6
n q
n q
n
n q
⫹2
⫹
k,m
⫽⫺
Am
⫹
m 2 q ⬘ 2 k,m m k,m m k,m 2imq ⬘ k,m
T ⫹ 2 T 25 ⫹ 2 T 27 ⫹ 3 3 T 50
n 3 q 4 23
n
n
n q
k,m
T 17
n
⫺
mq ⬘ k,m m k,m
T ⫹ 2 T 25
n 2 q 2 20
n
k,m
m k,m T 31
i k,m i k,m
T
⫹
⫺ 2 T 58 ,
⫹ 2 T 56
2 27
n
n
n
n
im k,m 2im 2 q ⬘ k,m
m
T ⫹ 3 3 T 21 ⫺ 3 3
n 2 q 19
n q
n q
冉
k,m
A 2m
⫽⫺
⬘ k,m ⫽⫺
A 2k
⬘ k,m ⫽
A 2m
冊
q ⬘ 2 k,m 4mq ⬘ k,m mq ⬘ k,m
T 23 ⫺ 3 3 T 28 ⫺ 3 3 T 53 ,
q
n q
n q
im k,m
T ,
n 2 q 19
T k,m
1
q
⫺
2im k,m m 2 k,m
1
k,m
T
⫹
T ⫹ 2 T7
n 2q 2 2
n 2q 5
n q
1283
1 k,m 2mq ⬘ k,m
T ⫺ 3 3 T 23 ,
n 2 q 20
n q
1 k,m
T ,
n 2 q 20
⬘ k,m ⫽⫺
A mk
i k,m i k,m
T ⫺ 2 T 60 ,
n 2 57
n
⫻ q ⬙ ⫺2
k,m
⫽
A mk
冊
2mq ⬘ k,m
m
q ⬘ 2 k,m
T
⫹
q
⫺2
T4
⬙
n 2q 2 3
n 2q 2
q
⫺
k,m
A 2k
⫽
Numerical studies of edge localized instabilities . . .
⫺
2 k,m 2im k,m
1 k,m
2 T 3 ⫹ 2 T 6 ⫺ 2 T 16
n q
n q
n q
4mq ⬘ k,m
2i k,m
T ⫺ 3 2 T 50
,
n 3 q 3 23
n q
⬘ k,m ⫽⫺
A 2mk
2im k,m
2
1
k,m
k,m
3 2 T 21 ⫹ 3 2 T 28 ⫹ 3 2 T 53 ,
n q
n q
n q
⬘ k,m ⫽⫺
A m2k
2im k,m
4
1
k,m
k,m
3 2 T 21 ⫹ 3 2 T 28 ⫹ 3 2 T 53 ,
n q
n q
n q
⬙ k,m ⫽⫺
A mk
1 k,m
T ,
n 2q 4
⬙ k,m ⫽
A m2k
1
n q
⬙ k,m ⫽
A 2mk
1
T k,m ,
n 3 q 2 23
k,m
3 2 T 23 ,
q k,m imq k,m imq k,m m 2 q k,m
⫺ 2 T 36 ⫺ 2 T 37 ⫺ 2 T 38
I k,m ⫽⫺ T 32
f
n
n f
n f
冉
冊
⫹
im k,m
1 k,m im k,m
m 2 q ⬘ k,m
2 T 8 ⫺ 2 T 14 ⫹ 2 T 15 ⫹4i 3 3 T 21
n q
n q
n q
n q
⫺
2im 2 q ⬘ k,m
m
2q ⬘ 2 k,m m k,m
T 40 ⫹ 3 q ⬙ ⫺
T 41 ⫺ 3 T 42
3
n
q
n q
q
n
⫺
2m
2q ⬘ 2 k,m 4mq ⬘ k,m
2 k,m
q
⫺
T 23 ⫺ 3 3 T 28 ⫺ 2 T 45
⬙
3 3
n q
q
n q
n q
⫹
2mq ⬘ k,m q k,m m k,m mq ⬘ k,m
T ⫹ 2 T 63 ⫺ 3 T 65 ⫹ 3 T 66 ,
n 3 q 44
n
n
n q
⫹
m
n q
⫺
k,m
⫽⫺
A m2k
⫹
k,m
⫽⫺
A 2mk
⫺
冉
冊
I k,m
k ⫽
m 2 k,m im k,m im k,m
T ⫹ 3 T 61 ⫹ 3 T 64 ,
n 3 39
n
n
2mq ⬘ k,m
i k,m
1 k,m
1 k,m
T ⫹ 3 T 54
⫺ 2 T 55
⫺ 2 T 59
,
n 3 q 3 53
n q
n q
n q
k,m
⫽
Im
m 2 k,m im k,m 2im k,m im k,m
T ⫹ 3 T 61 ⫹ 3 T 62 ⫹ 3 T 64 ,
n 3 39
n
n
n
m 2 k,m 4im k,m
im k,m
3 2 T 22 ⫺ 3 2 T 29 ⫺ 3 2 T 46
n q
n q
n q
I ⬘ k,m ⫽⫺
2
im k,m
T k,m ⫺ 3 2 T 51
,
n 3 q 2 47
n q
I ⬘k k,m ⫽
2im k,m 1 k,m
T ⫺ 3 T 66 ,
n 3 40
n
m 2 k,m 2im k,m
im k,m
3 2 T 22 ⫺ 3 2 T 29 ⫺ 3 2 T 46
n q
n q
n q
⬘ k,m ⫽
Im
2im k,m 2 k,m 1 k,m
T ⫺ 3 T 44 ⫺ 3 T 66 ,
n 3 40
n
n
im k,m
2
k,m
T ⫹ 3 2 T 52
,
n 3 q 2 51
n q
I ⬙ k,m ⫽
q k,m
T ,
n 2 f 33
k,m
3 2 T 47 ⫺
2m k,m
i
k
k,m
k,m
3 2 T 48 ⫺ 3 2 T 49 ⫹ 3 2 T 52
n q
n q
n q
2imq k,m q k,m 2mq ⬘ k,m
T ⫹ 2 T 35 ⫹ 3 T 41 ,
n 2 f 34
n
n q
A ⬘ k,m ⫽
iqp ⬘ k,m
T 10 ,
n
I ⬙k k,m ⫽⫺
1 k,m
T ,
n 3 41
A k⬘ k,m ⫽
2mq ⬘ k,m i k,m i k,m
T ⫹ 2 T 24 ⫹ 2 T 26 ,
n 2q 2 4
n
n
⬙ k,m ⫽⫺
Im
1 k,m
T .
n 3 41
⬘ k,m ⫽
Am
i k,m i k,m
T ⫹ 2 T 26 ,
n 2 24
n
Turning now to the matrix elements involved in the boundary
conditions, we have
Downloaded 14 Feb 2006 to 198.129.105.159. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp
1284
Wilson et al.
Phys. Plasmas, Vol. 9, No. 4, April 2002
S k,m
k ⫽⫺
mq ⬘ k,m
k,m
T ⫹ f p ⬘ T 13
nq 2 4
K 7⫽
f B 2p
B2
共␻⬘兲 ,
K 8⫽
冉
冊
2
K 9⫽
k,m
S 2k
⫽
mq ⬘ k,m 1 k,m
T ,
T ⫹
n 2 q 3 23
nq 20
K 10⫽
R2 ⳵B2
,
JB 4 ⳵␹
k,m
⫽
S mk
1 k,m im k,m 1 k,m
T ⫺
T ⫹
T
nq 3
nq 6
nq 18
K 13⫽
1
,
B2
2mq ⬘ k,m
k
i
k,m
k,m
T ⫺ 2 2 T 28
⫹ 2 2 T 50
,
n 2 q 3 23
n q
n q
K 15⫽ ␯␻ ⬘
k,m
⫽
S m2k
im k,m
1
k,m
2 2 T 21 ⫺ 2 2 T 28 ,
n q
n q
K 16⫽ ␯
k,m
⫽
S 2mk
im k,m
1
k,m
2 2 T 21 ⫺ 2 2 T 28 ,
n q
n q
K 19⫽
1 k,m
T ,
nq 4
K 22⫽
⬘ k,m ⫽
S mk
K 11⫽
K 14⫽ ␯ ⬘
冉 冊
冉 冊
⳵ R 2 B 2p
,
⳵ ␺ JB 2
f 2 ␴␻ ⬘
,
R 2B 2
K 20⫽
f 3 ␻ ⬘ 2 B 2p
2
R B
4
K 17⫽
K 23⫽
,
1
T k,m ,
n 2 q 2 23
␻⬘
f 2␴ ⳵
,
K 27⫽
␯ ⳵␹ R 2 B 2
K 28⫽
J k,m
k ⫽
⫺im k,m
T ,
n 2 40
K 31⫽ ␴ ⬘ ,
k,m
⫽
Jm
⫺im k,m
T ,
n 2 40
1 k,m
T .
n 2 41
冖
K 4⫽
f
2,
R 4B p
f B 2p
B2
,
R 2B 4
K 24⫽
4,
K 26⫽
B4
,
,
p ⬘ R 2 B 2p ⳵ B 2
,
JB 6 ⳵␹
冉 冊
1
f 2␴ ⳵
,
␯ ⳵␹ R 2 B 2
K 29⫽
f 3 B 2p ␻ ⬘ ␯ ⬘
,
R 2B 4 ␯
␳
,
B 2p
K 33⫽
K 30⫽
1 ⳵␴ ⬘
,
␯ ⳵␹
␳ R 4 B 2p
,
B2
冉
冊
1 ⳵ ␳ JR 2 B 2p
K 35⫽
,
␯ ⳵␺
B2
冊
␳ R 4 B 2p ␻ ⬙
,
K 37⫽
B2
K 40⫽H ␻ ⬘ ,
K i 共 ␺ , ␻ 兲 e i(k⫺m) ␻ d ␻ ,
K 43⫽
共A3兲
K 46⫽
K 2 ⫽q
K 5⫽
K 32⫽
冉
where
K 1⫽
R B
f 3 B 2p ␻ ⬘
B 2p
␻ ⬘ ⳵ ␳ JR 2 B 2p
,
K 36⫽
␯ ⳵␺
B2
The matrix elements T k,m
are defined in terms of flux surface
i
averages of kernels K i
T k,m
i 共 ␺ 兲⫽
f 3 B 2p ␯ ⬘
,
R 2B 4 ␯
␳ R 4 B 2p ␻ ⬘
K 34⫽
,
B2
q k,m
T ,
n f 33
⬘ k,m ⫽
Jm
f 3 B 2p
2
K 18⫽ f p ⬘
K 21⫽
冉 冊
imq k,m mq ⬘ k,m q k,m m k,m
T ⫺ 2 T 41 ⫺ T 43 ⫹ 2 T 44 ,
n f 34
n q
n
n
1 k,m
T ,
n 2 41
␴p⬘
,
B2
f 2␴
,
R 2B 2
⬘ k,m ⫽⫺
S 2mk
J k,m ⫽
K 12⫽1,
⳵ R 2 B 2p
,
⳵ ␺ JB 2
R 2 B 2p p ⬘ ␻ ⬘ ⳵ B 2
K 25⫽
,
JB 6
⳵␹
J k⬘ k,m ⫽
R 2␻ ⬘ ⳵ B 2
,
JB 4 ⳵␹
冉 冊
1
n q
J ⬘ k,m ⫽⫺
␻ ⬙,
⳵ R 2 B 2p
,
⳵ ␺ JB 2
⬘ k,m ⫽⫺
S m2k
k,m
2 2 T 23 ,
B2
R2 ⳵
B2
p⫹
,
B2 ⳵␺
2
i k,m i k,m
k,m
⫹ f ⬘ T 12
⫺ T 24
⫺ T 26 ,
n
n
⫹
f B 2p
R 2 B 2p
␯ ⬙,
K 3⫽
␯ ⬘␻ ⬘,
K 6⫽
JB 2
R 2 B 2p
JB 2
R 2 B 2p
JB 2
f B 2p
B2
␯ ⬘,
␻ ⬘,
G
,
␯
R 2B 4
H␯⬙
,
␯
K 41⫽H,
K 42⫽
H␯⬘
,
␯
K 45⫽
K 47⫽
f 3 B 2p ␯ ⬙
,
R 2B 4 ␯
K 44⫽
f 3 B 2p ␻ ⬙
␳ R 4 B 2p ␻ ⬘ 2
K 38⫽
,
B2
,
冉 冊
冉 冊
K 48⫽
B 2p
f 3␻ ⬘␯ ⬘ ⳵
,
␯
⳵␻ R 2 B 4
K 50⫽
B 2p
f 3␯ ⬘ ⳵
,
␯ ⳵␻ R 2 B 4
K 39⫽H ␻ ⬘ 2 ,
冉 冊
f 3 B 2p ␯ ⬘
R 2B 4 ␯
K 49⫽
2
,
冉 冊
冉 冊
B 2p
f 3␯ ⬙ ⳵
,
␯ ⳵␻ R 2 B 4
K 51⫽ ␯ 2 ␻ ⬘
f 3 B 2p
⳵
,
⳵ ␺ ␯ 2R 2B 4
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Phys. Plasmas, Vol. 9, No. 4, April 2002
冉
冊
f 3 B 2p
⳵
K 52⫽ ␯␯ ⬘
,
⳵ ␺ ␯ 2R 2B 4
K 54⫽ ␯
冉 冊 冉 冊
冉 冊
冉
冊
冊
f p⬘ ␯⬘ ⳵B
,
JB 6 ␯ ⳵␹
2
2
冉
K 59⫽ ␯
冊
⳵ f 2p⬘ ⳵B2
,
K 60⫽
⳵ ␺ JB 6 ⳵␹
␺ 3共 l 兲 ⫽
1
sin u
1
⫺2B ps sin u
⫹
6
Rc
Rs
冉 冊
⳵H
.
⳵␺
␳ f R 2 B 2p
B4
共A4兲
冉
⫹R s2 p ⬘
␻ ⬘␯ ⬘
K 62⫽
H,
␯
␯⬘ ⳵H
K 65⫽
,
␯ ⳵␺
G⫽
冉
冊
␳ R 2 JB 2p p ⬘ ␯ ⬘ f 2
⫹
.
B2
B 2 ␯ R 2B 2
共A5兲
Note that a prime denotes a derivative with respect to ␺, and
␳ is the plasma mass density.
冉
冊
冊 冉
冊 冉
册
冉
␺ ⫽ ␺ 0 ⫹ ␺ 1 共 l 兲 r⫹ ␺ 2 共 l 兲 r 2 ⫹ ␺ 3 共 l 兲 r 3 ⫹¯ ,
共B1兲
where l is the distance around the flux surface labeled ␺ 0 in
the poloidal cross-section and r is the radial distance from
that flux surface 共see Ref. 12兲. Although we need to expand
to higher order than was done in Ref. 12 共as we need to
evaluate second derivatives with respect to ␺兲, the technique
is basically the same. We, therefore, do not describe it in
共B3兲
冊
sin u
sin u
1
1
⫺
⫹f f⬘
⫹
Rc
Rs
Rc
Rs
冊
⫺R s B ps 共 R s2 p ⬙ ⫹ 共 f f ⬘ 兲 ⬘ 兲 .
共B4兲
All quantities are to be evaluated on the reference flux surface, ␺ ⫽ ␺ 0 共the subscript s on some of the variables denotes
this兲. The angle u is defined by dR s /dl⫽cos u, dZ s /dl⫽
⫺sin u and R c is the radius of curvature, which can also be
defined in terms of u: du/dl⫽⫺R ⫺1
c .
Now that we know ␺ to sufficiently high order we can
evaluate the equilibrium quantities required by ELITE. In
particular, we have
冋
冋
冉 冊 册册
␺2 r2 6␺3 1 ⳵␺1
⫹
⫹ 2
␺1 2 ␺1
␺1 ⳵l
2
, 共B5兲
for the poloidal magnetic field, where
R⫽R s 共 l 兲 ⫹r sin u.
共B6兲
Another important quantity is ␯, as the derivatives of this are
required in order to derive q ⬘ and q ⬙ . Thus we need to evaluate ␯ to O(r 2 ), and Eq. 共B5兲 is required to provide this. Thus
we express
APPENDIX B: EQUILIBRIUM RADIAL VARIATION
The first task of ELITE is to analyze the equilibrium and
evaluate the matrix elements described in Appendix A. It is
convenient to calculate these matrix elements on each flux
surface in turn, without reference to other equilibrium flux
surfaces in their neighborhood. It is therefore necessary to be
able to evaluate radial derivatives on a flux surface in terms
of only the equilibrium parameters on that flux surface. This
is made possible through the technique of expanding the
equilibrium locally about that flux surface, using the information provided by the Grad–Shafranov equation, as described in Ref. 12, for example.
Thus, following Ref. 12, we express:
册
1
⳵ 1 ⳵␺1
sin u
⫹
⫺R s
Rc
Rs
⳵l Rs ⳵l
RB p ⫽R s B ps 1⫹2r
and
冊
共B2兲
Rs
⫹sin u B ps ⫺R s2 p ⬘ ⫺ f f ⬘ ,
Rc
1
2
⫹4 ␺ 2
Here we have defined two additional variables, dependent
upon the equilibrium
H⫽
冋冉
冋
␺ 2共 l 兲 ⫽
⳵ f␴
,
⳵ ␺ JB 2
⳵H
K 64⫽ ␻ ⬘
,
⳵␺
1 ⳵G
,
K 63⫽
␯ ⳵␺
␺ 1 共 l 兲 ⫽R s 共 l 兲 B ps 共 l 兲 ,
K 61⫽H ␻ ⬙ ,
1285
detail, but simply quote some useful results. Thus, by substituting the above form for ␺ into the Grad–Shafranov equation and equating powers in r, we find
p ⬘ R 2 B 2p ␯ ⬘ ⳵ B 2
K 56⫽
,
JB 6 ␯ ⳵␹
⳵ p ⬘ R 2 B 2p ⳵ B 2
K 57⫽
,
⳵␺
JB 6 ⳵␹
K 66⫽
冉
f 3 B 2p
⳵
K 53⫽ ␯
,
⳵ ␺ ␯ 2R 2B 4
2
f 3 B 2p
⳵ ␯⬘
⳵
,
2 2 4
⳵ ␺ ␯ R B ⳵␹ ␯
⳵ f p ⬘ B 2p
,
K 55⫽ ␯
⳵␺ ␯B4
K 58⫽
Numerical studies of edge localized instabilities . . .
␯ ⫽ ␯ s 共 1⫹ ␯ 1 r⫹ ␯ 2 r 2 ⫹¯ 兲
f 共␺兲
,
f 共␺0兲
共B7兲
where
␯ 1 ⫽⫺
␯ 2⫽
冉
1
sin u 2 ␺ 2
⫺
⫺
,
Rc
Rs
␺1
sin u 2 ␺ 2
⫹
Rs
␺1
冊冉
共B8兲
冊
1
sin u
3 ␺ 3 4 ␺ 22
⫹
⫺
⫹ 2 .
Rc
Rs
␺1
␺1
共B9兲
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