iol - t6562 - core
TRANSCRIPT
This IB a preprint of a paper intended tor publication in a journal or proceeding!. Since changes may be made before publication, this preprint is made avuiable with the understanding that it will r.ot be cited or reproduced MKhout the permission of the author.
I O L - T6562 PREPRINT„
LAWRENCE UVERMORE LABORATORY University ol CaUccnia/Livermore. Caktomia
COMPUTER APPLICATIONS
IN CONTROLLED FUSION RESEARCH
John Killeen
February 1975
-NOTICE Tins report was prepared as an account of wort sponsored by the United Stales Govetnmc.-n. Neither the United States nor the 'United States Energy Research and Development Administration, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or ir.ipUed, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe privately uwned rights.
This paper has been prepared for the American Nuclear Society National Topical Meeting on Nuclear Engineering Education, the proceedings of which will appear in a special issue of Nuclear Technology.
%;73 ? "'"BiBurioaoFTwsr.ocuMtKj
UNLIMITED
COMPUTES APPLICATIONS 111 COIJTFIOI.LED Ft!. IN RESEARCH
John Killeen
Department of Applied Science University of California, Davis-Liveroore
and
National CTR Computer Center Lawrence Livermore Laboratory University of California
P.O. lox 8C3, Livermore, California '-hyy
February 197:
Communications should te addressed to
Professor John Killeen L-'.vrence Livenaore Laboratory P. 0. Box 8C8, Livermore 9*5:0
Telephone: {'•*!') 1)47-1100, ext. 5278
Number of pages: 22 Tables: none Figures; none
This paper has been prepared for the American Mucloar Society National Topical Meeting on Nuclear Engineering Education, the proceedings of which will appear in a special issue of Nuclear Teclinology.
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AESTRACT
The role of Nuclear Engineering Education in the application of
computers to controlled fusion research can be a very important one.
In the ne£.r future the use of computers in the numerical modelling of
fusion systems should increase substantially. A recent study group
has identified five categories of computational models to study the
physics of magnetically confined plasmas. A comparable number of types
of models for engineering studies are called for. The development and
application of computer codes to implement these Todels is a vital step
in reaching the goal of fusion power. In order to meet the needs of
the .'usion program the National CTR Computer Center has been estab
lished at the Lawrence Livennore Laboratory. A large central computing
facility is linked to smaller computing centers at each of the roa.ior
CTR laboratories by a communications network. The crucial elemenx that
is needed for success is trained personnel. The number of peopje with
knowledgs of plasma science and engineering that are trained in numeri
cal methods and computer science is quite small, and must be increased
substantially in the next few years. Nuclear Engineering departments
should encourage students to enter this field and provide the necessary
courses and research programs in fusion computing.
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IUTRODUCTIOK
Large-scale digital computers will play an increasingly important
role in future controlled fusion research and in the development of
practical fusion reactors. The plasma state exhibits such a diverse
variety of phycical phenomena that only through extensive use of high
speed computers can the interplay of all factors that affect the per
formance of a fusion system be modelled.
The behavior of a plasma confined by a magnetic field is simu
lated by a variety of numerical models. Some models used on a short
time scale give detailed knowledge of the plasma on a microscopic scale,
while other models used on much longer time scales compute macroscopic
properties of the plasma dynamics. All of these models are under con
tinual development. but in the next few years there should be a sub
stantial increase in the development and use of numerical models in
order to meet the needs of the fusion power program.
A 1971* study gro-.'.p on the ''Applications of Computers to CTB"
rponsored by the AEC/DCTR identified five categories of computer codes
used to model the physics of fusion devices.
1. Time-dependent macroscopic (fluid) codes.
2- Tirce-independent macroscopic codes.
3- Vlasov and particle codes.
•i. Fokker-Planck codes.
p. Hybrid codes.
In later sections of this report we shall examine these problem areas
in more detail and consider their importance to the CTR program and
discuss future goals in each area.
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There are also a wide variety of engineering applications which
require extensive computing. These applications will become more and
and more important as we move into the reactor development phase. We
shall also discuss some of these problems in more detail later in the
report.
Computers are expected tc play an Important role in the acquisi
tion and analysis of experimental data as well as the control of
experiments.
In order to meet the need? of the controlled fusion program the
National CTR Computer Center has teen established at the Lawrence
Livermore Laboratory. This Center will be connected to all major CTR
laboratories, as well as selected universities, by means of a wide
band communications network. The increased computing power made avail
able by the Center will accelerate the development of the theoretical
models and associated computer codes needed to predict the behavior of
plasma confinement systems and the operating characteristics of fusion
power reactors. A more detailed description of the Center will be
given later in this report.
COMPUTER MODELS OF A MAGMETICALLY CONFINED PLASMA
1. Time-dependent macroscopic codes
The complex nature of the MHD fluid equations of motion is such
that our understanding of the macroscopic behavior of realistic
toroidal plasma devices has been most efficiently advanced by numerical
studies of simplified fluid models. When one considers the additional
complicating features Involved in realistic boundary conditions, the
presence of divertors, and extensions to non-axisymmetric systems,
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it becomes clear that many new physical phenomena await investigation.
The range of time and space scales of the various physical phe
nomena leads to a fairly natural division from the physics point of
view—(a) fast time scale; and (b) diffusion time scale. Such a divi-
nion is also natural from the numerical point of view as the techniques
involved in the solution for each category are different. Not only do
the classification of the model equations change (i.e., from hyperbolic
to parabolic), but it is clear that even with Class IV and V computer
hardware, one wouli not, for example, attempt to reach confinement
time scales in a Tokamak simulation by time-stepping a code designed
to follow fast congressional Alfven waves,
a. Simulation on the Fast Time Scale
Detailed comparison of experimental data from Scyllacs and pinches
vith theory, taking iue account of experimental complications (plasma
heating, compression, progression of equilibria, stability, atomic
processes, etc.), vill depend on the development of 2-E and 3-D (two
and three dimensional) versions of codes analogous to the 1-D Hain-
Roberts code. Most of the physical phenomena important here lie in
the fast MKD timescale (nanoseconds to microseconds).
For Tokaraak configurations, the corresponding effects occur
primarily on rather longer time scales, milliseconds, and are discussed
in subsection (b). However, the questions of stability of Tokamak dis
charges toward MHD modes are very important for achieving efficiency
in a reactor system and for minimizing the cost of feasibility devices.
One example of great interest is the area of the early stages of a
Tokamak discharge and the formation and destruction of magnetic surfaces.
Here also the relevant tines are on the fast MHD time scale.
The potentially high inpflct of simulation techniques at th» com-
pressional Alfven tine scale lies In the possibility of determining
the ranges oi plasma parameters In vhich (8© nodes (e.g., local,
hallooning, kink, tearing nodes) »re stable, or sufficiently so to be
tolerable. Factors which affect the existence and severity of ouch
modes are discharge shape, distribution of plasma pressure, current
and magnetic field, location of conducting valla, dlssipative effects
in the plasma and vails, self limitation of nodes from nonlinear ef-
"_vts, possibilities for stabilisation i'je to feedback, finite particle
orbit si£e effects, etc. In addition to studies of specific effectr,
composite models will be needed in which all the effects are treated
simultaneously and accurately to yield results vith all the relevant
physics taken into account.
Some descriptions of codes in this category are given in Refs.
2-16.
b. Simulation on the Diffusion Tine Scale
In order to simulate the time evolution of a plasma in a magnetic
confinement device over most of its lifetime—from tens to hundreds of
milliseconds—a set of partial differential equations of the diffusion
type must be solved, fypical dependent variables are the number densi
ties and temperatures of each particle species, current densities, and
magnetic fields. The transport coefficients such as thermal conduc
tivity, electrical resistivity, and diffusion coefficients are obtained
from the best available theories, e.g., neo-classical, but the codes
should also have the capability of easily changing the form of the
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?or-fficier:ts i:. or ior . ro develop phenomer.ological models or tahe into
iecour:t r e ru t : : of part icle codes.
In the- part feu years, a considerable effort has been devoted, at
. everal laYjoratorie:-, to the n-mierical solution of one-dimensional
(radial) transport eqjitior.s for toroidal plasmas. This effort pro
vides an oxer-He:/, isenr.c of comparing theory with experiment. Recent
jevelopmentr of these coie.- have concentrateu or. the inclusion of
.-"jCJti'als ar.: impurities ir. the mode1..-, and the use of empirical t rans
port coefficient.:.
In the r.c.t five ;.\.arr th"re - i l l be a great increase in the use
•if the. e coi'->." a:.: cor.siueralls refinement of the models will be
required. O'-b-r physical effects will include ar. increase in the
nJsier of icr. -pc-lcv-—en-.. ic-;teror.s, t r i tor .s , and a-partlcles—in
. . :cr t.o slrr. late T.T turning experiments. The effects of radiation
an: atomic ar. i molecular processes will t e included.
The iifvre.t .'har^e. however, ar.d the greatest iemani on the ta lents
of the computational physicist ar..: profirarcer will be the increase in
:imer..-ior.s rco i.-e: ir. there moid.-. Ir. .~rdcr to rea l i s t i ca l ly simu
late actual :evice-" with r.cr.-cire :l^r cross sections, neutral beam
•ines, and :ivcrtors, tvo- ani three-:imer.siona± codes will be required.
ooae iescriptior.r of codes in this category are given in Refs.
17-25.
2. Time-Inieper. :ent Macroscopic Codes
I t i s normally necessary to develop time-independent codes to
support the design and operation of each aa.jor CTR experiment. These
include codes used to compute and study prospective equilibrium plasma
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cor.figuratlons and determine their s tabi l i ty from 5W-type calculations
(especially important ir. high beta applications).
Experimental devices incorporating the idea of axial symmetry in
a torus appear to be capable of pla.-ria confinement for times which are
of great in teres t . One reason for this result i r the assurance of equi-
l i i r i a ir. s.;?h devices as predict? '. ly MHD ani p : i iing-center theories.
T-o-dimer.sior.al .TiO-lels :'or rlasma eq. i l it ria have "teen use i to compute 2 the equililri.;m f ie l :s for varices containment schemes ir.clu :i:.£ Astron,
Levitron, ToxaEah." and the _•'. iffe : cusp."'
Several containment scben/*s. is.ci u iir.g the stabilized mirror
devices, do not possess ar. icr.cratlc coordinate, allowing a reduction
of the equilibrium equations to two dimensions. Codec have leer, de
veloped to handle the general case for open containment—a three-
dimensional code that solves for plasma equil ibria in open-fioli 28 2r* geometries, which allows analysis of most i-inimura-E mirror systems. '
The c r i t i ca l problems in this area are the following:
(a) Develop "wo and three dimensional plasma equilibrium codes.
(b) Levelcp 5W codes to determine the s tab i l i ty of various plasma
cr .-.figurations.
(c) Apply these codes to Jeteraine what plasma equil i tr ie. exist
which look promising from thei r s tab i l i ty properties and from the
plasma that they can confine.
(d) Investigate the effects of divertors and other boundaries.
Some descriptions of equil ibr ia calculations are given in Hcfs.
2^-31 and time-independent s tab i l i ty codes are described in Refs. J52-31*.
VTauov a:: i Particle Code:;
Particle code:; are the most fundamental in that they emulate
mature, following In detail the motion of par t ic les under the influ-
•ricc of their self-consistent e lec t r ic and magnetic fields, as vei l
•i." an;.' externally impose- f i c l i c . These codes give the most detailed
results; the.-,- pive phase-.pace i i r t r lbut ion functions, fluctuation and
wave spectra, anj Ivor, -.he ori it:; sjf individual par t ic les . They are
i leal for providing ie ta i l e : info m a t ! or. or. the growth and saturation
-:' .-trace i:-.'sat il i t i e r ar,: Vr.<: effects of turbulence. Such turbulence
:ar. (rive ri--- to rear.;.- irrportnr.t -ai-roscopls effects—for example, the
'jr.omalo-:- aisorption of VE'/CJ d:i Lhe transport cf plasraa, energy,
an: aonentuc. Ir.for-saticr. on at...-c rptlcn properties helps determine
•he relative- merit.- of iifferer.t '.ype." of turbulent heating, e .g . ,
-.hat type of j i s t r i i ution function.: are produced, what type of tur
bulent spectra are produce.;, ar,: what turbulent transport i s caused
iy various wive heatinc nethods. Information or transport produced
iy turbulence is essential for fluid S..-.-J hy l r i j eoaes because these
coder require a knowledge of transport coefficients which result from
microscopic processes, par t ic le codes are also valuable in checking
theories and the approximat ions that go into t h e e They provide clues
to the important nonlinear nechanisms which saturate ins tab i l i t i e s and
help establish an intui t ive understanding of plasma behavior. Because
these codes must follow the behavior of plasma on the plasma-frequency
time scale and on the Defcye-length space scale they are not suitable
for modeling large systems for long time periods.
Despite the fact that some applications of par t ic le codes to
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r ea l i s t i e protlems will require eonsiJe:*alle development over a low;
period of t i n t , rany urgent prollems can te carried out vith rxason-
at ly straightforward extensions of existing techniques.
The following prollems which are c r i t i c a l to the progress of the
CTR program can te usefully attacked with part icle codes:
(a) Determining the s tab i l i ty of plasmas heatej ly neutral injec
tion or containing energetic reaction products.
( t ) Leterffiinlr.g how the loss rate from mirrors is affected ly
the loss-cone and high-frequency i n s t a t i l i t i e s .
(c) Understanding the operation of i iver tcrs and ietercining
the effects of tour.daries on plasma tehavir r.
(a) Determining the effecti ;eness of :.eat5r.g ly means of para
metric i n s t a t i l i t i e s using various types of waves (lower hybrid, ion
cyclotron, e t ; . ) .
(e) Determining the effects of trapped par t ic le i n s t a l i l i t i e s
in Tokamaks and other closed geocetries.
(f) Simulation of current penetration in fusion disci.irges.
(g) Determining the nature and magnitude of pa r t i c l . diffusion
and other transport processes due to nonlir^ar collective effects.
Three-dimensional coder and calculations will le limited by com
puting time and the size of systems which can be handled. I t is. d i f f i -
"ult to predict the exact extern. :f progress in an effort which is in
i t s infancy and just beginning to make contact with laboratory experi
ments. The size of the effort required to make rapid progress on a l l
of the man;' c r i t i c a l problems i s very large. Availability of people
trained in both plasma physics and computing techniques will be one of
. j:v-- : . -rir.t'.'.:.. "f r*irt-'-Jr- -o :-?: an: rer :Itc are giver: in
i v - : V . : . - • •.
I.". '.:•-• I'.i. ' . i t ! ' : . ' :' •:.'\,-:x-tl:aily confine: rla.tca:: vhere the ion;
••.:••: r.o-l Ka>rv.---'. 1I'»!. '«:'.: -here a /.».r«lejC" of the- : i ; tr lL„tior. function:
:. I.T.V'••:-.»:.• , .••.l:.e\Io •-•o .u". I::.. r. ..:i. to .-Dive;. The proposition that
': .•.a:>- :•..':•:••.;• r!a: 'r . • . . ; ] ! ;•:••:: r.ct thereon .clear power jeper.ii or.
".!.-. :TI'.. •;' •..'..';•?. r'::-:f •:•_•. are lo:•-. c .*. Ihe "r.:.- of the device. At
:.^:.i^:' •.•-:..•!-. ;•• .• •:,.: '.•-••«•!•.•. •yr l :a i :f .-irror cachinez, the c-::i
"?. .e.- 'ire : .•• p.-;~-;: t:. '? the- ; '"atterint cf cha.-je; part icles into
•;:••• : : . . •?:.•.. ;:. \vl ?:•!-;• . r t y ':;. .-iarricai Cy:lcz\ col l i ricr.2. The
.•::;t-M- •? :•-!•. l:r. :•:*. -rl": -.V -:.!.- pr:~e.-r !:• the rcltznonr. equation
v.-1th Fohker-ria:..-;-. ^cLii . ' i ' - -.•:-:-..
The ;•'•• cf ".hi. oc.atio:. '..' no-, restr icted tc Eirror systems. The
hoatinC of r".a,:-.a.- ':;. <..r.vrrerir r.e ; tral ieaxc. the thermalization of
i-part icle: ' in ?T r l a .~a . . :':.: .••:::.•• ef runavay electron; in Tolcaaaks,
'-.:.: the pe."forH2::co of tvo—"::?•".-.'•' "OBroner.t furi^r. reactors are other
•-xanple.: where "-hi -J slut lor. cf the Fok*er-?lar.c>. equation is required.
The proilen: i.- to ;cive a nonlinear part ial differential equation
for the distribution, function of each charge! species in the plasma,
and there are- .-ever. ii:depe::ient variable.: (three spat ial coordinates,
three velocity coordinates, ar.i t ine) . Such an equation, even for a
single species, exceeds the capability of any present computer so
several simplifying assumptions are therefore required to t rea t the
protlem. Typical approximations that are made in present day codes
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are to neglect spatial dependence and to assume that the distribution
functions are azimuthally invariant in velocity space (about the direc
tion of the i-ignetic field) and can be represented by the lowest
angviar eigenfunction.
The goal in this area is to develop multi-species codes which are
fully two-dimensional in velocity space and include the spatial de
pendence of the magnetic field. The development of two-dimensional
(v, u) multi-species codes is already completed. The next step is to
introduce one space dimension (r) to such codes. Such codes should
be in production use in one year. The goal for the five-year pi-riod
(1975-1979) should be to include realistic spatial dependence in
Fokker-Planck codes. This will certainly require Class V and VI
computers.
Descriptions of Fokker-Planck codes and results are given in
Pefs. 40-44.
-j. Hybrid Codes
There will be a critical need for codes which can best be de
scribed as Hybrid Codes; these are codes which combine the good features
of fluid codes with the good features of particle codes, but which can
avoid some of the weaknesses of each. The advantage of a particle code
is that it contains the most complete treatment of the physics. Its
disadvantage also stems from this feature because it is forced to fol
low the development of the plasma on the fastest time scale and short
est space scale at which significant plasma phenomena occur. These
scales are typically much shorter than the time and size scale in
thermonuclear plasma devices. It is clear that even the fastest
•13-
computer would not te able to follow the plasma development in a micro
scopic way over the length of time or over the spatial volume of prac
tical interest. The feature of fluid codes which is attractive is that
they treat the plasma on a coarser scale and hence need many fewer time
steps and spatial points. On the other hand, the usual conditions
required for a fluid treatment (short mean free paths and frequent
collisions) are not realized for a thermonuclear plasma. The motions
of certain classes of particles are often crucial. One example is the
motion of trapped particles, which cause the trapped ^article insta
bility. Further, the generation and damping of waves and the effective
ness of waves in causing diffusion and energy transport will depend
critically on resonance phenomena "between particles and waves. It is
clear that proper treatment of --h phenomena will require some reason
ably accurate description of the important class of particles. On the
other hand, it should he possible to treat the rest of the plasma by
means of fluid equations. Thus some mixed description should be effec
tive and economical.
The relevance of hybrid codes to thermonuclear plasma devices is
really the same as the relevance of fluid codes and particle codes.
It appears that the impact of developments in this area will be large
•snd, perhaps, even crucial. Codes of this type will probably give the
most accurate overall description of the plasma which is practical and
hence may ultimately he called upon to predict the performance of vari
ous plasma devices and reactors. It I& difficult to assess Just what
the developments will be or how soon they will impact the program be
cause this is an area which is still very much in its infancy; not
-Ik.
many things have been tried, and it is not yet known how successful
this approach will be.
Another class of hybrid code which promises to be very useful is
the coupling of a Fokker-Planck code to a plasma transport (diffusion)
code. In neutral-beam-heated Tokaroaks and the two-energy-component
toroidal fusion test reactor There is a warm Maxwellian background
plasma which can be described by macroscopic transport equations and
an energetic species which should be described by the Fokker-Planck
equation. The coupling of these systems is by means of sources of
particles and energy in the multi-species transport code and a time-
dependent Maxwellian target plasma in the multi-species Fokker-Planck
code. Codes of this type are solved en the same time scale (milli
seconds) separately so there does not appear to be any difficulty in
this regard.
The first step is to add r dependence to a (v, u) multi-species
Fokker-Planck code and then couple it with an already existing 1-D
transport code of the type described in lb.
The long-range goal would be the coupling of a 2-D transport
code to a Fokker-Planck code which included two space dimensions.
Some existing hybrid codes are described in Refs. kk-k6.
ENGINEERING AND TECHNOLOGY STUDIES
Engineering and technology studies will become more and more
important as we move towards a DT burner and into the reactor develop
ment phase. Among the critical engineering problems are the following:
(a) Development of 1- and 2-D transport codes, including burning
reaction products and radiation.
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(b) Meutron reaction and transport studies in the blanket and
reactor structure using 2- and 3-D Monte Carlo calculations.
(c) Studies of vail and divertor effects, sputtering, plasma and
heat loss to the divertor, effectiveness of divertors in preventing
the penetration of impurities, etc.
(d) Specific machine design calculations, especially parametric
studies with optimization in view.
(e) Investigation:: of control systems.
(f) Structural .vtudies, stress, deflections, heat loads, induced
activity, etc.
(g) Safety studies, possible faults and failures and their effects,
(h) Analysis of the operation of direct converters.
The near-terra goals are to provide the required calculations for
the design, construction, and operation of upcoming and existing expert-
mental devices. The long-term goals will be similar but, in addition,
we must be sure that the required codes are available for the very
large experiments which will be built and that the required manpower
and computing capability exist.
With proper support there appears to be no reason why the criti
cal engineering calculations cannot be carried out. Class IV computers
are capable of performing nearly all of these studies. Class V com
puters will probably be required for the parameter studies and reactor
system studies needed. Large engineering calculations requiring s.c
least Class IV and possibly Class V computers will continue as long
as machines are designed and built.
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HATIONAL CTR COMPUTER CENTER
The initial configuration of the CTR Computer Center, which will
be operational in late 1975, consists of a Control Data Corporation
7fcO0 computer located at Livent This type of computer is referred
to as a Class IV computer and will have 500K words of large core
memory and (MK words of small core memory as well as disc storage,
'ihts machine will be fully dedicated to fusion computing and will be
used by all of the CTR installations.
The CTR pr^iects at Los Alamos, Oak Ridge, Princeton, General
Atomic Corp., and Liverraore will each have a User Service Center (USC)
which consists of a Digital Equipment Co. FDP10KI computer, disc packs,
printers, and other peripheral equipment. The PDPIO, which is referred,
to as a Class II computer will have 6**K words of fast memory.
Each of the UCS's will be connected to t'.e central computer at LLL
by means of leased 50 kilobit/sec lines. The USC's will serve as local
computers for small .jobs and experimental data processing and as remote
job entry terminals to the 7600 for large numerical calculations of the
type described in previous sections of this report.
Future expansion of the CTR computer network will include smaller
User Service Centers at universities engaged in CTR research. These
minl-USC's will be linked to the network at the nearest USC by means of
communication lines of somewhat smaller capacity. In addition the com
puting capability of the Center at LLL will be increased by memory
expansion of the CDC 7600 and later by another Class IV computer or
possibly a Class VT computer if such a machine is available.
-17-
ACKNOWLEDGMEHTS
I wish to thank Drs. J. H. Dawson, R. A. Dory, R. c. Grur
, H. R. Lewis, Bennett Miller, and A. W. Trivelpiece for
of their views on critical problems in fusion computing.
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REFERENCES
K. Hain, G. Hain, K. V. Rober t s , S. J - Rober t s , and W. KSppendBrfer,
Z. Na tur forsch . l ^a , 1039 ( i 9 6 0 ) .
J . K i l l e e n , in Phys ics of Hot Plasmas, S c o t t i s h U n i v e r s i t i e s '
Summer School , 1968, B. J . Bye and J . C. Tay lor , Eds. ( O l i v e r and
Boyd, Edinburgh, 1970), pp 202-255-
K. V. Rober ts and V. E. P o t t e r , i n Methods In Computational Phys ics
(Academic P r e s s , Hew York, 1°70) , Vol. 9, pp. ik0~kvj.
J . K i l l e e n , i n Informat ion Process ing 71 (North-Holland Pub l i sh ing
Company, Arostevdam, 1972) , p p . 1191-1205.
I . Lindemuth and J . K i l l e e n , J . Comput. Phi 's. 1 3 , 181 (1973)
J . E. Crov, J . K i l l e e n , and D. C. Robinson, P roc . 6 th European
Conference on Con t ro l l ed Fusion and Plasma P h y s i c s , Moscow, X,
269 (1973)-
J . A. D i b i a s e , Numerical S t u d i e s of R e s i s t i v e I n s t a b i l i t i e s i n
Diffuse P inches , Ph. D. T h e s i s , U n i v e r s i t y of C a l i f o r n i a , Davis ,
Lawrence Livermore Labora tory Report UCRL-51591, 1971*.
W. Schneider and 0 . Eateman, MHD I n s t a b i l i t i e s a s an I n i t i a l - V a l u e
Problem for Elongated Cross s e c t i o n s , paper IAEA-CN-33/A12-1 of IAEA
F i f t h Conference on Plasma Phys ics and Cont ro l led Nuclear Fusion
Research, Tokyo, November 11-15 , 1971*.
A. Sykes and J . A. Wesson, Two-Dimensional MHD Computation of
Tokamak S t a b i l i t y , paper IAEA-CN-33/A12-3 of above Conference.
R. White, D. Mont i ce l lo , M. II. Rosenbluth, H. S t r a u s s , and E. B.
Kadomtsev, Numerical S t u d i e s of Non-Linear Evolut ion of Kink and
Tear ing Modes i n Tokamaks, paper IAEA-CN-33/A13-3 of above Conference.
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1 1 . D. E. P o t t e r and G. H. T u t t l e , Two-Dimenslonal S imula t ion of a
F l u i d Model for Tokamak, paper IAEA-CN-33/A17-3 of above Con
ference .
12. E. P. P u t t , e t a l , Confinement and S t a b i l i t y of High-Beta Plasma
i n a Reversed F i e ld Pinch, paper IAEA-CN-53/E9-2 of above Con
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N O T i r t
"Tot* report i n prepared »» JM account of work tpmnorcd hy Ihc Urulo] S l i m Government yctthrr the tinned Slate* nor Ik- Uuilcd State* Atonic tncrgy rontrhnuon. nor any ol thai employee*, not my of Iherr contractor*. *twcorHiacton. 01 thru employee*, make* any n m m y . ctpre** or winked, ot av-aimc* anyk»Wh»hi«ly«ic»pon»iliiliiyforlhe»t5™f«y.conifneIcitc« or inefutnci* of any infornution. apparatus product or pwc** dt*do*cd. or rcprctenl* that nv late would not wfrmfle pmalcly-owncd riffil*."