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