Post on 16-Oct-2019
transcript
The
Hyb
ridM
HD
-Gyrokin
eticC
ode
HM
GC
G.V
lad*
*Associazion
eE
uratom
-EN
EA
sulla
Fusion
e,C
.R.Frascati,
Rom
e,Italy
incollab
orationw
ithS.B
riguglio
*,G
.Fogaccia
*,F.Zon
ca*
and
B.D
iM
artino †
†Secon
dU
niversity
ofN
aples,
Nap
les,Italy
IIIC
onvegno
Nazion
alesu
“La
Fisica
del
Plasm
ain
Italia”L’A
quila,
20-22M
aggio2002
Electron
icversion
:http
://fusfi
s.frascati.enea.it/˜vlad
/Miscellan
ea/slides
L’A
quila
2002.pdf
1
1O
utlin
e
•Introd
uction
•T
he
mod
el
•N
um
erics
–Flu
idsection
–G
yrokinetic
section:
particle
simulation
s,Particle-in
-cell(P
IC)
vs.Fin
ite-size-particle
(FSP
)
–Parallelization
:D
omain
vs.Particle
decom
position
–Parallel
architectu
res:D
istributed
Mem
ory,Shared
Mem
ory,H
ierarchical
Distrib
uted
-
Shared
Mem
ory
•E
xamples
2
2In
troductio
n
•T
he
Hyb
ridM
HD
-Gyrokin
eticC
ode
(HM
GC
)has
been
develop
ped
atth
eC
.R.
Frascati,
EN
EA
laboratory
inth
efram
eof
therm
onuclear
fusion
research
•R
ecentexp
erimentald
evicesare
approach
ing
the
socalled
ignition
condition
:fu
sionα
-particles
arecon
fined
inth
etoroid
al(T
okamak)
plasm
aan
dsu
stainth
eburn
ing
plasm
a
•C
onfinem
entprop
ertiesofth
een
ergetic(α
)particles
arecru
cialinob
tainin
ggood
perform
ances
inreactor
relevantregim
es
•Fusion
α-p
articlesare
born
main
lyin
the
plasm
acentre,
and
the
correspon
din
grad
ialprofi
le
ispeaked
:th
eirpressu
regrad
ientis
afree-en
ergysou
rceth
atcan
destab
ilizew
avesw
hich
resonantly
interactw
ithth
eperiod
icm
otionof
the
energetic
particles
•E
nergetic
(hot)
ions
(α-p
articles)in
plasm
asclose
toign
itioncon
dition
shave
vH≈vA
=B/ √
4πni m
i .
•Interaction
betw
eenen
ergeticparticles
and
shear
Alfven
waves
islikely
tooccu
r
•Shear
Alfven
waves
=⇒M
agnetohyd
rodyn
amic
(MH
D)
mod
el
•E
nergetic
particles
(wave-p
articleinteraction
)=⇒
Kin
eticm
odel
•C
onfinem
entprop
ertiesof
energetic
particles
=⇒N
onlin
earm
odel
3
3T
he
model
•B
ulk
plasm
a:describ
edby
Magn
etohydrod
ynam
ic(M
HD
)equ
ations
0R
a
ϑ
r
R
Z
ϕ
0
0.2
0.4
0.6
0.8 1
00
.20
.40
.60
.81
ω
r
m=
2
m=
1
m=
1
m=
2
•Shear
Alfven
waves
innon
-uniform
equilib
riaexh
ibit
acontinu
-
ous
spectru
m:
“local”plasm
aoscillation
sw
ithfrequ
ency
contin-
uou
slych
angin
gth
rough
out
the
plasm
a
•In
toroidal
geometry,
the
poloid
al-symm
etrybreakin
gdue
toth
e
toroidal
field
variationon
agiven
magn
eticflux
surface
cause
diff
erentpoloid
alharm
onics
tobe
coupled
:frequ
ency
“gap”
ap-
pears
inth
eA
lfvencontinu
um
•D
iscrete,glob
alM
HD
mod
es(T
oroidal
Alfven
Eigen
mod
es,or
TA
E’s)
canexist
inth
egap
sof
the
shear-A
lfvenfrequ
ency
spec-
trum
.TA
E’s
arem
arginally
stable
MH
Dm
odes
and
canbe
easily
driven
unstab
leby
the
resonan
cew
ithen
ergeticparticles
•U
sered
uced
MH
Dequ
ations
expan
ded
up
toO
(ε3),
with
ε≡
a/R
0 ,a
andR
0th
em
inor
radiu
san
dth
em
ajor
radiu
sof
the
torus,
respectively,
tokeep
toroidal
effects
inth
em
odel
4
•H
ybrid
MH
D-kin
eticm
odels
–E
nergetic
particle
den
sityis
typically
much
smaller
than
the
bulk
plasm
aden
sity
–O
rderin
g:nH
ni≈O
(ε3),
TH
Ti≈O
(ε −2)
–T
hus,
the
followin
gord
ering
forth
eratio
ofth
een
ergeticto
bulk
ionβ
(β≡
8πP
0 /B20
is
the
ratiobetw
eenth
eplasm
akin
etican
dth
em
agnetic
pressu
res)follow
s:
βH
βi≈O
(ε)
–It
canbe
show
nth
at,m
aking
use
ofth
eab
oveord
ering,
the
MH
Dm
omentu
mequ
ationis
mod
ified
bya
termw
hich
represent
the
perp
endicu
larcom
pon
entof
the
divergen
ceof
the
energetic-p
articlestress
tensor
ΠH
–E
nergetic-p
articlestress
tensor
obtain
edby
solving
Vlasov
equation
5
•Particle
simulation
:gyrokin
eticm
odel
Direct
solution
ofth
eequ
ationdescrib
ing
the
time
evolution
ofth
eparticle
distrib
ution
function
F(t,Z
)for
collisionless
plasm
as:
Vlasov
equation
:
∂F∂t
+dZ
i
dt
∂F
∂Zi
=0,
Equ
ations
ofm
otion:
dZi
dt
=...
.
Discretized
formofF
(t,Z):
F(t,Z
)≡
∫
dZ′F
(t,Z′)δ(Z
−Z′)≈
N∑l=1∆l F
(t,Zl )δ(Z
−Zl ).
Phase-sp
acegrid
points
Zl (t)
evolveaccord
ing
toeqs.
ofm
otion:
num
ericalparticles.
Gyrocenter
coordin
atesZ≡
(R,µ,v‖ ,θ):
Ris
the
gyrocenterposition
,v‖
parallel
(toB
)velocity,
µm
agnetic
mom
ent(exactly
conserved
inth
iscoord
inate
system),θ
the
gyrophase
(does
not
appear
explicitly).
Volu
me
elements
∆l (t)
evolveaccord
ing
to:d∆l
dt
=∆l (t)
∂
∂Zi dZ
i
dt
t,Zl (t)
.
Often
itcou
ldbe
convenient
toevolve
only
the
pertu
rbed
part
δFof
the
distrib
ution
function
:
=⇒F
(t,Z)
=F
0 (t,Zl )
+δF
(t,Zl )
6
4Partic
le-in
-cell
versu
sFin
ite-siz
e-p
artic
le
Plasm
acon
dition
n0 λ
3D
1(collective
interactiondom
inate
overcollision
s)im
plies
ahu
genu
mber
ofsim
ulation
particles.
Even
assum
ingns λ
3D≈
10,typ
ically(L
eqequ
ilibriu
mlen
gth):
Npart ≈
ns L
3eq=ns λ
3D(L
eq /λD
)3≈
1013.
Violation
ofplasm
acon
dition
n0 λ
3D
1:system
toocollision
al,sh
ortran
geinteraction
sbetw
een
particles
dom
inate
overth
elon
gran
geon
es.
Particle-in
-cell(P
IC)
1.E
lectromagn
eticfield
scom
puted
atth
epoints
ofadiscrete
spatial
grid
2.Interp
olationof
the
e.m.
field
sat
the
(continuou
s)particle
posi-
tions
tocom
pute
the
forcesan
dperform
particle
push
ing
3.P
ressure
contribution
ofen
ergeticparticle
calculated
atth
egrid
points
toclose
the
equation
s
=⇒Short-ran
geinteraction
sare
then
cut
offfor
mutu
aldistan
cessh
orterth
anth
etyp
icalsp
acing
–Lc–
betw
eengrid
points,
whilst
the
relevantlon
g-range
interactionsare
not
signifi
cantlyaff
ected.
=⇒P
ICparticle
ensem
ble
beh
avesas
aplasm
aunder
the
much
more
relaxedcon
dition
n0 L
3c 1
(with
Lc
λD
)
7
•Fin
ite-size-particle
(FSP
)
Fin
ite-size-particles
(charge
clouds):
ns (x
)=
∑l∆l δ(x
−xl )
−→∑l
∆l S
(x−xl )
δ(x)
x
Ls S(x)
x
The
spatial
characteristic
wid
thof
the
cloudλD
Ls
Leq
restrictsth
em
aximum
spatial
resolution
attainab
lein
the
simulation
(assum
ingLeq /L
s ≈100,
ns λ
3D≈
10):
ns λ
3D
1−→ns L
3s 1,
Npart ≈
ns L
3s(L
eq /Ls )
3≈10
7.
Ls
plays
the
roleofLc
8
5C
om
puta
tionallo
ads
and
Paralle
lizatio
n
Assu
me
that
field
solveruses
Fou
riertran
sformto
solveth
eM
HD
equation
s.
Serial
code,
num
ber
ofop
erations
(O)
per
time
stepan
dm
emory
(M)
required
:
PIC
:
OPIC≈f
(Nharm
)+nFT ×
Nharm×Ncell +
nint ×
Npart ,
MPIC≈mharm×Nharm
+mcell ×
Ncell +
mpart ×
Npart ,
Nharm:
num
ber
ofFou
rierharm
onics
retained
inth
esim
ulation
;f
(Nharm):
operation
sfor
the
solution
ofth
e
field
solver;n
FT:
num
ber
ofop
erations
need
edto
compute
eachad
den
dum
inth
eFou
riertran
sform;N
cell :
num
ber
ofcells
ofth
esp
atialgrid
;n
int :
operation
sfor
the
field
interp
olation;N
part :
num
ber
ofsim
ulation
particles;
mharm
,m
cell
andm
part :
mem
oryneed
edto
store,resp
ectively,a
single
harm
onic
ofth
ecom
plete
setof
Fou
rier-space
field
s,th
ereal-sp
acefield
sat
eachgrid
poin
tan
dth
ephase-sp
aceco
ordin
atesfor
each
particle.
FSP
:
OFSP≈f
(Nharm
)+nFT ×
Nharm×Npart ,
MFSP≈mharm×Nharm
+mpart ×
Npart .
Typ
ically,f
(Nharm
)negligib
lein
comparison
with
terms∝
Npart ;
forP
ICcod
es,Nppc
≡Npart /N
cell≈
n0 L
3c
1:as
farasnFT×Nharm
nint
the
gridless
FSP
meth
odis
more
expen
siveth
anth
eP
ICon
e,w
ithou
tpresentin
gany
signifi
cantad
vantagein
terms
ofm
emory
requests.
9
Tw
odistin
ctreason
scou
ldhow
everju
stifya
diff
erenttren
d:
1.Interest
insim
plifi
edsim
ulation
sin
which
only
veryfew
mod
esare
evolved:
linear
simulation
s,
orw
eaknon
linear
couplin
g(n
onlin
earm
ode
spectru
mrestricted
toa
limited
num
ber
ofsign
if-
icantharm
onics):
insu
cha
few-h
armon
icfram
ework,
the
condition
nFT ×
Nharm
nintcan
be
violatedor,
atleast,
signifi
cantlyw
eakened
;
2.sch
emes
ofparallelization
(distrib
uted
-mem
oryarch
itectures):
dom
aindecom
position
(d.d.)
versus
particle
decom
position
(p.d.)
iprocs =1
iprocs =3
iprocs =2
iprocs =4
iprocs=1
iprocs=3
iprocs=2
iprocs=4
10
Dom
aindecom
position
,P
IC
iprocs =1
iprocs =3
iprocs =2
iprocs =4
Diff
erentportion
sof
the
physical
dom
ainare
assigned
to
diff
erentprocessors,
together
with
the
particles
that
reside
onth
em.
OPIC
d.d.≈f
(Nharm
)+1
nproc
(nFT ×
Nharm×Ncell +
nint ×
Npart )
,
MPIC
d.d.≈mharm ×
Nharm
+1
nproc
(mcell ×
Ncell +
mpart ×
Npart )
.
Advantages:
almost
linear
scaling
ofth
eattain
able
physical-sp
aceresolu
tion(m
oreprecisely,
the
maxim
um
num
ber
ofsp
atialcells)
with
the
num
ber
ofprocessors.
Disad
vantages:particle
migration
fromon
eportion
ofth
e
gridto
anoth
er,possib
lesevere
load-b
alancin
gprob
lems
=⇒dyn
amical
redistrib
ution
ofgrid
and
particle
quanti-
tiesis
required
,w
hich
makes
the
parallel
implem
entation
ofa
PIC
code
verycom
plicate.
11
Particle
decom
position
,P
IC
iprocs=1
iprocs=3
iprocs=2
iprocs=4
Statically
distrib
utin
gth
eparticle
pop
ulation
amon
gpro-
cessors,w
hile
replicatin
gth
edata
relativeto
gridqu
anti-
ties.B
eforeupdatin
gth
eelectrom
agnetic
field
s,at
each
time
step,partial
contribution
sto
particle
pressu
recom
ing
fromdiff
erentportion
sof
the
pop
ulation
must
be
sum
med
together
(reduction
).
OPIC
p.d.≈f
(Nharm
)+nFT ×
Nharm×Ncell +
nint ×
Npart
nproc ,
MPIC
p.d.≈mharm×Nharm
+mcell ×
Ncell +
mpart ×
Npart
nproc .
Advantages:
loadbalan
cing
isau
tomatically
enforced
;par-
allelizationis,
inprin
ciple,
almost
straightforward
.It
is
veryeffi
cientif
computation
alload
relatedto
particles
dom
inates,
foreach
processor,
the
one
relatedto
the
grid
(nproc <∼
Nppc ).
Disad
vantages:grid
calculation
sdo
not
takead
vantage,
with
regardboth
toth
enu
mber
ofoperation
san
dth
em
em-
oryrequ
ests,fromsu
cha
parallelization
:each
processor
has
tohan
dle
the
whole
spatial
dom
ain.
Even
neglectin
geffi
-
ciency
prob
lems,
high
spatial-resolu
tionlevels
arelim
ited
byth
esin
glenod
eR
AM
.
12
Particle
decom
position
,FP
S
iprocs=1
iprocs=3
iprocs=2
iprocs=4
bottle-n
ecksin
efficien
cyan
dperform
ance
associatedto
grid
quantities
induces
one
toby-p
assth
eintrod
uction
ofa
spa-
tialgrid
,so
resorting
toth
egrid
lessFSP
simulation
:
OFSP
p.d.≈f
(Nharm
)+nFT ×
Nharm×Npart
nproc
,
MFSP
p.d.≈mharm×Nharm
+mpart ×
Npart
nproc
.
Advantages:
Fou
riertran
sforms
aredistrib
uted
amon
gth
e
processors.
High
spatial
resolution
canbe
obtain
ed.
Mas-
sivelyparallel
simulation
scan
yieldsign
ificant
ben
efits
as
faras
the
num
ber
ofm
odes,
Nharm
,retain
edin
the
simula-
tionis
relativelysm
all,in
spite
ofth
ehigh
mod
enu
mbers
consid
ered(h
ighsp
atialresolu
tion).
Disad
vantages:few
-harm
onic
limitation
.
13
6H
MG
CParalle
larchite
ctu
res
•T
he
HM
GC
code
existsin
aP
ICversion
and
ina
gridless
FSP
version.
•Parallel
implem
entations
inclu
de:
–D
istributed
Mem
ory(IB
MSP,clu
sterof
workstation
s),
–Shared
Mem
ory(S
ymm
etricM
ultip
rocessorSystem
s,SM
Ps),
–H
ierarchical
distrib
uted
-shared
mem
orym
ultip
rocessorarch
itectures
(IBM
SP,clu
sterof
SM
Ps).
Level
INT
ER
-NO
DE
Langu
age
HP
FStra
tegyParticle
Decom
position
Level
INT
RA
-NO
DE
Langu
age
Open
MP
Phase
Particle
push
ing
Pressu
reupdatin
gVaria
nt
Versio
n
Stra
tegyParticle
decom
position
Particle
decom
position
Dom
aindecom
position
Critical
Auxiliary
arraypaux
Sortin
gSelective
sorting
–v1v2a
v2b
14
7R
esu
lts
Flu
idnon
linearities:
saturation
ofa
TA
E(ω
=ω
0 ≈0.33
ωA)
10-1
4
10-1
2
10-1
0
10-8
10-6
10-4
0100
200300
400500
WT
OT
m,n
ωA t
(1,0)
(1,1)
(2,1)(3,2)
0.30
0.32
0.34
0.36
0.38
0.40
0.660.68
0.700.72
0.74
ω/ω
A
s
linear phase
non-linear phase
Volu
me
integrateden
ergy(m
agnetic
plu
skin
etic)for
diff
erentFou
riercom
pon
ents(m,n
)vs.
time
fora
non
-linear
simulation
ofan
unstab
ledriven
TA
E.T
heq-p
rofile
has
aparab
olicrad
ialdep
en-
den
cew
ithq(0)
=1.1
andq(a
)=
1.9.T
he
inverseasp
ectratio
isε
=0.075,th
eden
sityis
constant
%=%
0an
dth
eresistivity
correspon
ds
toS−
1=
10 −5.
Blow
-up
ofth
eA
lfvencontinu
um
.T
he
continuou
ssp
ectraob
tained
inth
elin
earlim
itan
dat
the
begin
nin
gof
the
non
-linear
phase
arecom
pared
.15
Kin
eticnon
linearities:
gap-m
ode
saturation
(ω=ω
0 ≈0.33
ωA)
6420200
400600
800
ωA t
ln A (t)
0.8
0.7
0.6
0.5
02π
4π
r/a
0.8
0.7
0.6
0.5
r/a
Ψ2,1
02π
4πΨ
2,1
a)b)
Tim
eevolu
tionof
the
mod
eam
plitu
deA
(t)for
apertu
rbative
non
-linear
simulation
with
γD
=
0.01ωA,βH
(0)=
0.08.
Orb
itin
the
plan
e(Ψ
2,1 ,r)(Ψ
m,n≡ωr t−
mϑ
+nϕ
),for
atest
particle,
inth
etim
eintervals
0<ωAt<
284(a)
(linear
growth
)an
d264
<ωAt<
560(b
)(n
on-lin
earsatu
ration).
The
Ψ2,1
axisis
map
ped
ontoth
einterval
0≤
Ψ2,1
<4π
.T
he
particle
isin
itiallypassin
g,but
becom
es
trapped
asth
em
ode
reaches
acertain
amplitu
de.16
Tran
sitionfrom
Kin
eticTA
Eto
EP
M(E
nergetic
Particle
Mod
e):
0.00
0.04
0.08
0.12
0.0
0.2
0.4
0.6
0.8
0.0050.015
0.0250.035
βH
γ/ω
Aω
r /ωA
ωr K
TA
E
ωr E
PM
γK
TA
E
γE
PM
a)
0.00.2
0.40.6
0.81.0
r/a
b)
β-th
reshold
vs.toroid
alm
ode
num
bern:
0
0,1
0,2
0,3
00
,01
0,0
20
,03
0,0
40
,05
βH
γτA
n=
8n
=4
n=
1
0.0
0
0.0
1
0.0
2
0.0
3
05
10
15
20
n
βH
th
PICF
SP
17
Non
linear
EP
Msatu
rationgen
erating
shear
flow
s(n
=8,
mon
otonicq(r)
profi
le,q(0)
=1.1,
q(a)
=1.9,
Npart ≈
16.7×10
6):see
movie:
http://fu
sfis.frascati.en
ea.it/˜vlad/M
iscellanea/E
PM
MO
VIE
S/n
89
imirr1
13zon
al3x4.m
ov
18
Deep
lyhollow
qprofi
le
•D
eeply
hollow
qprofi
le(q(0)≈
5,qmin
=2.1,
q(a)≈
5,profi
le(a)),
βH
(0)=
2.5%.
•ωgap /ω
A,r=
0=
1/(2q(r)√
ρ/ρ
r=0 ):
assum
efirst
arad
iallycon
stantth
ermal-p
lasma
den
sityρ
⇒rad
iallycon
stantA
lfvenvelocity
(such
anassu
mption
will
be
removed
later).
seem
ovie:
http
://fusfi
s.frascati.enea.it/˜v
lad/M
iscellanea/IA
EA
-Goteb
org/n4
JE
T7.m
ov
•A
ftera
transien
tin
itialphase,
am
ode
localized
around
the
max
imumβ′H
emerges
atr≈
0.35a,
with
frequen
cyw
ellin
side
the
contin
uum
.W
ecan
iden
tifyth
ism
ode
asan
Energetic
Particle
contin
-uum
Mode
(EP
M).
•Its
saturation
takesplaces
becau
seof
astron
g(con
vective)ou
tward
radial
disp
lacemen
tof
the
energetic
ions.
•A
ssu
cha
disp
lacemen
ttakes
place,
the
lo-cal
drive
isred
uced
due
toth
eflatten
ing
ofth
e
energetic-ion
den
sityprofi
le.T
he
drive
isno
longer
able
toovercom
eth
estron
gcon
tinuum
dam
pin
gat
the
original
frequen
cy.•
The
max
imum
ofth
epow
ersp
ectrum
migrates
toward
sth
egap
(inord
erto
min
imize
the
contin
-uum
dam
pin
g),but
italso
moves
outw
ards,
fol-
lowin
gth
edisp
lacedsou
rce,in
order
tom
axim
izeth
edrive.
•T
he
mode
reaches
the
gapan
dit
localizes
around
the
zero-shear,
qm
insu
rface(r≈
0.5a).
19
0R
a
ϑ
r
R
Z
ϕ
Low
-βtokam
akord
ering
(β≡
8πP
0 /B20
isth
e
ratiobetw
eenth
eplasm
akin
etican
dth
em
ag-
netic
pressu
res):
v⊥vA≈B⊥
Bϕ≈
B/B
·∇∇⊥
≈O
(ε),
vϕ
vA≈∇·v⊥
vA/a
≈∇
(RBϕ )
Bϕ
≈O
(ε2),
∂∂t ≈
vAR.
Here,
acylin
drical-coord
inate
system(R,Z,ϕ
)
has
been
used
,an
dth
esu
bscrip
t⊥den
otescom
-
pon
entsperp
endicu
larto∇
ϕ.
The
magn
eticfield
canbe
written
as
B=
(F0+F
)∇ϕ
+R
0 ∇ψ×∇ϕ
+O
(ε3B
ϕ )
where
ψis
the
poloid
alm
agnetic
flux
function
,
F0
=R
0 B0 ,B
0is
the
vacuum
(toroidal)
magn
etic
field
atR
=R
0 ,an
dF≈O
(ε2F
0 )is
given,
at
the
leadin
gord
er,by
equilib
rium
corrections.
20
•R
educed
MH
Dequ
ations:
∂ψ∂t
=−cR
2
R0 B
0 ∇ψ×∇ϕ·∇
φ−
cR0
∂φ
∂ϕ
+ηc2
4π∆∗ψ
+O
(ε4vABϕ ),
%
DDt −
2c
R0 B
0
∂φ
∂Z
∇2⊥φ
+∇%·
DDt −
c
R0 B
0
∂φ
∂Z
∇φ
=
−B
0
4πcB·∇
∆∗ψ−
B0
cR0 ∇
·[R
2(∇P
+∇·ΠH
)×∇ϕ
]
+O
(ε4%v
2ABϕ
a2c
),
DP
Dt
=O
(ε4 v
AB
2ϕ
a),
with
ηth
eplasm
aresistivity,
v⊥
=−cR
2
R0 B
0 ∇φ×∇ϕ
+O
(ε3vA),
%=R
2
R20 %,
DDt
=∂∂t
+v⊥·∇
,
∇2⊥≡
1R
∂∂RR∂∂R
+∂
2
∂Z
2,
∆∗ψ
=R
2∇·
∇ψ
R2
=R∂∂R
1R
∂ψ
∂R
+∂
2ψ
∂Z
2.
21
•Particle
simulation
target:ob
tainin
gfrom
the
num
ericalplasm
ath
esam
ebeh
aviour
ofth
e
physical
one
Itis
impossib
leto
simulate,w
ithtod
aynu
mericalresou
rces,th
enu
mber
ofparticles
ofrealp
lasmas
Con
sider
the
Debye
length
λD
and
the
Larm
orrad
iusρL :
λD
=
T
4πe
2n
2
,ρL
=c(m
T)2
eB
Con
sider
mutu
allyinteractin
gm
acroparticles:
ns
=nf
M,
M
1,
es
=Mef,
ms
=Mmf,
vs
=vf
=⇒Ts ∝
ms v
2s=Mmf v
2f ∝MTf
=⇒λD,s
=λD,f,
ρL,s
=ρL,f
Plasm
acon
dition
n0 λ
3D
1(collective
interactiondom
inate
overcollision
s)im
plies
ahu
genu
mber
ofsim
ulation
particles.
Even
assum
ingns λ
3D≈
10,typ
ically(L
eqequ
ilibriu
mlen
gth):
Npart ≈
ns L
3eq=ns λ
3D(L
eq /λD
)3≈
1013.
Violation
ofplasm
acon
dition
n0 λ
3D
1:system
toocollision
al,sh
ortran
geinteraction
sbetw
een
particles
dom
inate
overth
elon
gran
geon
es.
22
The
pressu
reten
soris:
ΠH
(t,x)
=1
m2H
∫
d6ZDzc →
ZFH
(t,R,M
,U)×
ΩHM
mH
I+
bb
U
2−ΩHM
mH
δ
(x−
R),
Equ
ationof
motion
foren
ergeticparticles:
dRdt
=U
b+
eH
mH
ΩHb×∇φ−
U
mH
ΩHb×∇a‖
+
MmH
+UΩH
U+
a‖mH
b×∇
lnB,
dMdt
=0,
dUdt
=1
mHb·
eH
ΩH
U+
a‖mH
∇φ
+MmH ∇
a‖
×∇
lnB
+eH
mH
ΩH ∇
a‖ ×∇φ
−ΩHM
mH
b·∇
lnB.
23