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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
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
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
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
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
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
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�
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
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�
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
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~
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
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
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
Constraint Moments
• Satisfy constraints by shaping particle in both x and w
corrections
sfdsfd
ssW
w
wTwT
ppwppp
www
wwxxwx
0
,~,~
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
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
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
Constraint Moments
• …and, closure moment has symmetric form
ppppTn
ppppppn
ppwTppn
sWvCsWC
sfdsCn
xxxxww
wwwwwxxA
~~
,~ ~
2
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
Symmetry Leads to Energy Integral
Aux
x
~
log122
2
0
22
ndM
nnvMPBMnuMnddtd
Te
e
Usual fluid w. isoT ions Interchange w. closure
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
Symmetry Leads to Energy Integral
ppp
Tn
ppppTn WvCM
dtd
dtd
WvCM
ndM
~2
~~1log
~
...
22
2
Aux
Usual fluid w. isoT ions Closure energy
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
Symmetry Leads to Energy Integral
.~~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
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
Symmetry Leads to Energy Integral
• 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!
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
IMP2 Implementation
• 2D, Cartesian• TE polarization
– B normal to simulation plane– E in plane
• Linearized, 1D equilibrium• Uniform T
0|| k
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
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)
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
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)
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
Space Differencing
• Use Yee mesh, w. velocity with B
* * *
* * *
* * * ux,Ey
uy, Ex
Bz ~ ~
n,Pe ~ ~ ~
~ ~
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
G-Mode Tests1
yx
00 0.1
Contours of ux for Roberts-Taylor G-mode
g
• Following Roberts, Taylor, Schnack, Ferraro, Jardin, …
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
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
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
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
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
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
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
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?
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
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
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
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
University of Colorado at BoulderSanta Fe CampusCenter for Integrated Plasma Studies
http://cips.colorado.edu/
Preprints