Advanced methods for space simulations

281 Pages·36.649 MB·English

by  edited by H. Usui and Y. Omura.

Advanced Methods for Space Simulations
Cover: Demonstration that the KEMPO1 code can simultaneously accommodate both an
electrostatic two-stream instability and an electromagnetic instability.
An electron beam with a gyrotropic ring distribution is injected into a thermal plasma
with a drift velocity along the static magnetic field. Both beam electrons and thermal
electrons undergo strong diffusion in the pitch-angle and parallel velocity.
Advanced Methods for
Space Simulations
Edited by
H. Usui and Y. Omura
TERRAPUB, Tokyo
Advanced Methods for Space Simulations
Edited by H. Usui and Y. Omura
ISBN 978-4-88704-138-7
Published by TERRAPUB, 2003 Sansei Jiyuugaoka Haimu, 27-19 Okusawa 5-chome,
Setagaya-ku, Tokyo 158-0083, Japan.
Tel: [phone] Fax: [phone]
URL http://www.terrapub.co.jp/
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© 2007 by TERRAPUB, Tokyo
No part of the material protected by this copyright notice may be reproduced or utilized in
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by any information storage and retrieval system, without written permission from the
copyright owner.
(This book is partly supported by Grant-in-Aid for Publication of Scientific Research Results
of the Japan Society for the Promotion of Science.)
Printed in Japan
Preface
The International School for Space Simulations (ISSS) was established in the
early 1980’s by M. Ashour-Abdalla, R. Gendrin, H. Matsumoto and T. Sato to pro-
mote science and technology related to space plasma physics via computer simula-
tions. The series of ISSSs were held at Kyoto in Japan (1982), Hawaii in U.S.A.
(1985), La Londe-les-Maures/Beaulieu-sur-Mer in France (1987), Kyoto / Nara in
Japan (1990), Uji, Kyoto in Japan (1996), and Garching in Germany (2001). They
have greatly contributed not only to promotion and advancement of space simula-
tions,butalsotoeducationofyoungscientistsandstudents.
InMarch2005theseventhISSS(ISSS-7)washeldatKyoto,Japan. Thepresent
book was inspired by ISSS-7, and it is a collection of extended lecture notes of the
tutorialsgivenatISSS-7bytheinvitedlecturerswhohavebeenactivelyinvolvedin
computersimulationtechniquesinspaceplasmaphysics.
Theaimofthisbookistoprovidetheinformationonthelatestadvancedmeth-
odsforspaceplasmasimulationstothosewhohavebeeninvestigatingspaceplasma
science by numerical simulations. The present book treats a wide variety of issues
on advanced techniques of space simulations. The book contains lecture notes on
advancedschemesofPIC(Particle-In-Cell)simulationnewlyincorporatedinthefull
particlemodelandthehybridmodel,advancedsimulationssuchastheDelta-fmodel,
adaptivePICmodel,Vlasovsimulation,anddiscrete-eventsimulation. Inassociation
with space simulations, techniques on unstructured mesh generation and advanced
visualizationaredescribed. Somesimulationcodesareincludedinthetextattheend
ofthebook.
Before the reader studies the advanced methods in the present book, we recom-
mendthatthereadersfirststudybasicspacesimulationtechniquespublishedinearlier
textbooks or WEB sites (e.g. “Computer Space Plasma Physics: Simulation Tech-
niques and Software” edited by H. Matsumoto and Y. Omura. The contents of the
bookarenowavailableathttp://www.terrapub.co.jp/e-library/cspp/index.html).
Theeditorsaregratefulnotonlytotheimmenseeffortsmadebythelecturerswho
preparedtheextendedmanuscriptsofthelecturenotes, butalsototherefereeswho
contributedtotheimprovementofthemanuscripts.
Finally,ontheretirementofProfessorHiroshiMatsumotofromtheprofessorship
atKyotoUniversityin2005,wewouldliketoexpressourheartygratitudetohimfor
his dedicated contribution to the progress of computer simulations in space plasma
physicsandspaceengineeringthroughISSSformorethan20years.
January,2007
HideyukiUsui
YoshiharuOmura
v
Contents
Preface ................................................................ v
SimulationTechniques
One-dimensional Electromagnetic Particle Code: KEMPO1 A Tutorial on Micro-
physicsinSpacePlasmas
Y.Omura .......................................................... 1
Vlasov-codesimulation
J.Bu¨chner ........................................................ 23
δf Particle-in-CellPlasmaSimulationModel: PropertiesandApplications
R.D.Sydora ...................................................... 47
AutomaticAdaptiveMulti-DimensionalParticleInCell
G.Lapenta ........................................................ 61
GeneralizedCurvilinearCoordinatesinHybridandElectromagneticCodes
D.W.Swift ....................................................... 77
ANewMethodologyforMulti-ScaleSimulationofPlasmas
H.Karimabadi,Y.Omelchenko,J.Driscoll,R.Fujimoto,andK.Perumalla..91
NumericalmethodsusedintheLyon-Fedder-MobarryGlobalcodetomodelthemag-
netosphere
J.G.Lyon ....................................................... 101
UnstructuredMeshesandFiniteElementsinSpacePlasmaModelling:Principlesand
Applications
R.Marchand,J.Y.Lu,K.Kabin,andR.Rankin ..................... 111
VisualizationofTangledVectorFieldTopologyandGlobalBifurcationofMagneto-
sphericDynamics
D.Cai,K.Nishikawa,andB.Lembege ............................. 145
IntroductiontoVirtualRealityVisualizationbytheCAVEsystem
N.OhnoandA.Kageyama ........................................ 167
SimulationSoftware
KEMPO1KyotouniversityElectroMagneticParticlecOde: 1dversion
Y.Omura ........................................................ 209
TheElementsforSettingupaHybridorElectromagneticCodeinCurvilinearCoor-
dinates
D.W.Swift ...................................................... 237
vii
AdvancedMethodsforSpaceSimulations,editedbyH.UsuiandY.Omura,pp.1–21.
(cid:1)c TERRAPUB,Tokyo,2007.
One-dimensionalElectromagneticParticleCode: KEMPO1
ATutorialonMicrophysicsinSpacePlasmas
YoshiharuOmura
ResearchInstituteforSustainableHumanosphere,KyotoUniversity,Japan
[email protected]
1 Introduction
Thebasictechniquesandimportantconceptsoftheone-dimensionalelectromag-
netic particle code: KEMPO1 [Omura and Matsumoto, 1993] are reviewed briefly
inthisarticle. Inthecode,Maxwell’sequationsandequationsofmotionsforalarge
numberofsuperparticlesaresolved.Becauseofthelimitationinthenumberofsuper-
particles, electrostatic thermal fluctuations are enhanced in the particle code, which
ofteninterferewithphysicalprocessestobereproducedinsimulations. Fromasim-
plifiedanalysisofthefluctuations,acriterionfortheratioofthegridspacingtothe
Debye length is given. A modification of the KEMPO1 for the solution of the rel-
ativistic equations of motion is also described. Since essential parts of the code are
very simple and short, it is easy to modify the code regarding the initialization and
theboundaryconditions. Tofacilitatemodificationofthecodeanditsverificationby
graphicoutputs,wehaverewrittentheKEMPO1codeusingtheMATLABprogram-
ming software. MATLAB provides us with a convenient graphic user interface and
flexible graphic diagnostics. Explanation of the input parameters for the relativistic
KEMPO1/MATLABcodeandseveralexamplesofapplicationsaregivenfortutorial
purposes.
2 BasicEquationsandMethodsofComputation
Electromagnetic processes in space plasmas are governed by Maxwell’s equa-
tions:
1 ∂E
∇×B=µ J+ (1)
0 c2 ∂t
∂B
∇×E=− (2)
∂t
ρ
∇·E= (3)
ε
o
∇·B=0 (4)
whereJ,ρ,c,ε ,andµ arethecurrentdensity,chargedensity,lightspeed,electric
o 0
permittivity,andmagneticpermeability,respectively. Insimulationsthevaluesofthe
1
2 Y.Omura
permittivityε andpermeabilityµ canbedefinedarbitrarily,aslongastheysatisfy
0 0
therelation
1
ε µ = . (5)
0 0 c2
IntheKEMPO1,forsimplicity,weadoptthefollowingdefinition
1
ε =1, µ = .
0 0 c2
WesolveMaxwell’sequationsfortheelectricfieldE ≡ (E ,E ,E )andmag-
x y z
neticfieldB≡(B ,B )inaone-dimensionalsystem. Itisnotedthat B isaconstant
y z x
intheone-dimensionalsystembecauseof(4). Inavacuumwithoutanychargedpar-
ticles, we have J = 0. The set of Maxwell’s equations (1) and (2) are solved by
the standard FDTD (Finite Difference Time Domain) method, which has been used
widelyinthevariousfieldofradioscience.
We introduce two sets of spatial grid systems along the x-axis. One is a full-
integergridsystemdefinedati(cid:10)x (i =1,2,3,...,Nx)andtheotherisahalf-integer
grid system at (i +1/2)(cid:10)x. We define E , B , J , and ρ on the full-integer grids,
y y y
and E , E , B , J onthehalf-integergrids. Wereplacespatialandtimederivatives
x z z x
inMaxwell’sequationswiththefollowingcentereddifferencesby(cid:10)x andthetime
step(cid:10)t.
By,i+(cid:10)1−x By,i =µoJz,i+1/2 (6)
Bz,i+1/2(cid:10)−xBz,i−1/2 =−µoJy,i (7)
E(X ,t)= E exp(kX −ωt) (8)
i o i
∂E(X ,t) E(X +(cid:10)x/2,t)−E(X −(cid:10)x/2,t)
i = i i
∂x (cid:10)x
1
= [exp(k(cid:10)x/2)−exp(−k(cid:10)x/2)] E(X ,t)
(cid:10)x i
sin(k(cid:10)x/2)
=i E(X ,t)=iKE(X ,t) (9)
(cid:10)x/2 i i
where
sin(k(cid:10)x/2)
K = (10)
(cid:10)x/2
Becauseofthecentereddifferencescheme,thedispersionrelationofelectromag-
neticwavesinavacuumω2 =c2k2isreplacedbyamodifieddispersionrelation
(cid:13)2 =c2K2 (11)
One-dimensionalElectromagneticParticleCode:KEMPO1 3
where(cid:13)isgivenby
sin(ω(cid:10)t/2)
(cid:13)= . (12)
(cid:10)t/2
ThisgivestheCourantconditionforthetimestepandthegridspacing,
c(cid:10)t <(cid:10)x. (13)
If the Courant condition is violated, the electromagnetic field grows exponentially
becauseoftheimaginarypartofωasasolutionof(12).
Inthepresenceofchargedparticles,weneedtocomputethechargedensityand
the current density to incorporate their effects on the electromagnetic field. The
chargedensityρ onagridpointatx = X iscalculatedby
i i
1 (cid:2)Np
ρ = q W(x −X ) (14)
i (cid:10)x j j i
j
whereW isaparticleshapefunctiongivenby
x
W(x)=1− |x|,|x|≤(cid:10)x
(cid:10)x
= 0, |x|>(cid:10)x (15)
Thesummationin(14)istakenforall N particlesinthesimulationsystem.
p
Theinitialelectricfield E iscalculatedfrom(3)inthedifferenceform
x
Ex,i+1/2−Ex,i−1/2 = ρi (16)
(cid:10)x ε
0
whereε istheelectricpermittivity.
0
ThecurrentdensityJ iscalculatedbasedonthechargeconservationmethod[Vil-
x
lasenorandBuneman,1992;Umedaetal.,2003]satisfyingthecontinuityequations
ofthecharge,
Jt+(cid:10)t/2− Jt+(cid:10)t/2 =−(cid:10)x(ρt+(cid:10)t −ρt). (17)
x,i+1/2 x,i−1/2 (cid:10)t i i
Thecurrentdensities J and J arecalculatedby
y z
Jit++(cid:10)1/t2/2 = (cid:10)1x (cid:2)Np qjvW(xj −Xi+1/2). (18)
j
The values of J calculated at the half-integer grids are relocated to the full-
y
integergridsbythefollowingprocedure:
Jy,i = Jy,i−1/2+ Jy,i+1/2. (19)
2
4 Y.Omura
Fig.1. Dispersionrelationofthethermalnoiseinanunmagnetizedplasma.
With these components of the current density J, we can trace time evolution
of electromagnetic fields E and B by solving Maxwell’s equations with the FDTD
method.
In Fig. 1, we plotted frequency ω and wavenumber k spectra of an electromag-
netic component E obtained by a run of the KEMPO1 code. The electromagnetic
z
fluctuationisinducedbythecurrentdensitiesduetothethermalmotionoftheplasma.
UsingthetechniqueofthefastFouriertransform(FFT),wehaveappliedthediscrete
Fouriertransformtotheelectricfieldcomponent E inspaceandtime. Theforward
z
andbackwardtravelingwavesareseparatedbypositiveandnegativewavenumbers,
respectively. TheseparationtechniqueisdescribedinMatsumotoandOmura[1985].
Aswefindinthemodifieddispersionrelationduetothecentereddifferencescheme
describedabove,thehighfrequencypartofthedispersionrelationofelectromagnetic
waves deviates from the oblique dashed lines representing the speed of light. The
holizontaldashedlineindicatestheelectronplasmafrequency.
SincethecurrentdensityJ exactlysatisfiesthecontinuityequationsofthecharge
x
density, E updatedbythecurrentdensity J automaticallysatisfy(16),ifthediffer-
x x
enceequationissatisfiedinitially.
The equations of motion for a particle with a charge q and a mass m are the
following:
dv q
= (E+v×B), (20)
dt m