university of colorado at boulder santa fe campus center for integrated plasma studies implicit...

University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies

http://cips.colorado.edu/

Implicit Particle ClosureIMP

D. C. BarnesNIMROD Meeting

April 21, 2007



Outline• The IMP algorithm

– Implicit fluid equations– Closure moments from particles– f with evolving background– Constraint moments

• Symmetry – Energy conservation theorem– Conservation for discrete system– absolutely bounded, no growing weight problem!

• Present restricted implementation• G-mode tests• Future directions

f 2



IMP Algorithm

• Fluid equations– Quasineutral, no displacement current– Electrons are massless fluid (extensions possible)

Pen e BJBuE

0

0

uu

u

eeeeee PPt

P

nDtDn

eneJuu



IMP Algorithm• Fluid equations

– Ions are massive, collisionless, kinetic species– Use ion (actually total) fluid equations w. kinetic

closure

P�

iePMnDtDMn BJgu

BJEB

0

1;t

0

vBvF

xv f

Mef

tf

uvwwww ;P fdMi�



IMP Algorithm• f particle closure algorithm

– Background is fixed T with n, u evolving– Particle advance uses particular velocity w (very

important)

ev

nffff vw

T

T22 2/

32

;~

Axx

uwBww

uwx

~12

nn

vMe

dtddtd

T

wwwAA fdnMn

nTkMn

Bi ~1~;~P�



IMP Algorithm

~

Aw

xu:wwu ~1~

~11

2vdtd

T

ff /~~

• Note: (perturbation) E does not enter closure directly

• There is some kind of symmetry between advance and A~



Constraint Moments

• With infinite precision and particles, should have …

vef

SWfd

SWfd

T

vw

T

pppppT

ppppT

T

2 2/3

2/

2

~0~

~0~

22

xxwww

xxw



Constraint Moments

• Satisfy constraints by shaping particle in both x and w

corrections

sfdsfd

ssW

w

wTwT

ppwppp

www

wwxxwx

0

,~,~



Constraint Moments

• Using Hermite polynomials, find

vfs

T

ppTpw 2

1, wwwww

• Projection of weight equation is then

xu:wwu

Awxu:wwuwww

v

vsd

dtd

T p

Tpw

p

p

2

2

1

~1,~

~11



Constraint Moments

• …and, closure moment has symmetric form

ppppTn

ppppppn

ppwTppn

sWvCsWC

sfdsCn

xxxxww

wwwwwxxA

~~

,~ ~

2



Symmetry Leads to Energy Integral

Aux

x

~

log122

2

0

22

ndM

nnvMPBMnuMnddtd

Te

e

Usual fluid w. isoT ions Interchange w. closure




ppp

Tn

ppppTn WvCM

dtd

dtd

WvCM

ndM

~2

~~1log

~

...

22

2

Aux

Usual fluid w. isoT ions Closure energy




.~~1log

log122

2

2

0

22

constWvCM

nnvMPBMnuMnd

ppppTn

Te

e

E

x

EE

nnvMPBMnuMnd-

WvCMWvCM

Te

e

ppp

Tn

ppppTn

log122

~2

~~1log

2

0

22

22

2

x




• r.m.s. of particle weights absolutely bounded

• Stability comparison theorem– Kinetic system more stable than isoT ion fluid

system– But only for marginal mode at zero frequency

• This is absolutely the most important point!



IMP2 Implementation

• 2D, Cartesian• TE polarization

– B normal to simulation plane– E in plane

• Linearized, 1D equilibrium• Uniform T

0|| k



Time Centering

• Moments use Sovinec’s time-centered implicit leap-frog– Direct solve

• Particles use simple predictor-corrector– Present, use full Lorentz orbits w. orbit

averaging (Anticipating Harris Sheet or FRC calculations)

– Iteration required (3 – 5 typical count)



Time Centering

t u u

n,T,B n,T,B

S.I. MHD step Implicit induction

equation step

Leap-frog particle substeps give orbit averaged A ~

Particles use average of u(depends on A, so need iterate)



Space Differencing

• Use Yee mesh, w. velocity with B

* * *

* * *

* * * ux,Ey

uy, Ex

Bz ~ ~

n,Pe ~ ~ ~

~ ~



G-Mode Tests1

yx

00 0.1

Contours of ux for Roberts-Taylor G-mode

g

• Following Roberts, Taylor, Schnack, Ferraro, Jardin, …



G-Mode Tests

• Two series– Low Hall stabilized – with and w/o closure– High gyro-viscous stabilized (Hall turned off)

• Numerical parameters– Nx x Ny = 30 x 16– 9 – 25 particles/cell– Typically 100 particle steps/fluid step



0 0

.5 1

1.5

2 2

.5 3

3.5

4 0 0

.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

y

pertu

rbed

den

sity

min

= -5

.694

E+16

m

ax=

5.6

94E+

16

0 0

.5 1

1.5

2 2

.5 3

3.5

4 0 0

.5

1 1.5

2 2.5

3 3.5

4

x

y

pertu

rbed

den

sity

min

= -2

.864

E+18

m

ax=

2.8

64E

+18

Perturbed density

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 */ g

M

g/g M

G-Mode Tests

• Low – 0.02– B = 6.0 T– n = 2. x 1020 m-3

– g = 1. x 1012 m/s2

– Ln = 120 m– T = 8.94 keV

– = 2.28 mm

• Arrow marks k = 0.15

Fluid only

Closure



G-Mode Tests• High

– 1.0– B = 0.482 T– n = 5.78 x 1019 m-3

– g = 2.7 x 108 m/s2

– Ln = 10 m– T = 10 keV– g/i = 2.25 x 10-4

– = 2.99 cm• Arrow marks

k = 0.15 0

0.5

1 1

.5 2

2.5

3 3

.5 4

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

y

pertu

rbed

den

sity

min

= -5

.694

E+16

m

ax=

5.6

94E+

16

0 0

.5 1

1.5

2 2

.5 3

3.5

4 0 0

.5

1 1.5

2 2.5

3 3.5

4

x

y

pertu

rbed

den

sity

min

= -2

.864

E+18

m

ax=

2.8

64E+

18

Case10.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

kv*/ G_M

/gM

g/g M

Perturbed density

0 0

.5 1

1.5

2 2

.5 3

3.5

4 0 0

.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

y

pertu

rbed

den

sity

min

= -5

.694

E+16

m

ax=

5.6

94E+

16

0 0

.5 1

1.5

2 2

.5 3

3.5

4 0 0

.5

1 1.5

2 2.5

3 3.5

4

x

y

pertu

rbed

den

sity

min

= -2

.864

E+18

m

ax=

2.8

64E+

18

Case10.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

kv*/ G_M

/gM

g/g M

Perturbed density



G-Mode Tests

• Hall and gyro-viscous stabilization of fundamental G-mode observed– Stability seems consistent with Roberts-

Taylor, modified by Schnack & Ferraro, Jardin

• New, higher kx mode observed at k > 0.2 or so– Drift wave?– Present in fluid (+ Braginskii) also?



Temperature Variation?

• Vlasov equation is linear, so superposition allowed– Superimpose number of uniform T solutions– Slightly different equation

• Mild restriction on equilibrium T– Density must depend only on potential

nnefdf wM 2/2ˆ



Future Directions

• Short-term– Understand high kx mode– Finish manuscript

• Medium-term– Add T variation– Add full polarization, check Landau damping

• Long-term– Gyrokinetic version– Parallel implementation



Preprints

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