particle simulation, gyrokinetics, turbulence and beyond w ... · particle simulation,...

Particle Simulation, Gyrokinetics, Turbulence and Beyond

W. W. Lee

Theory Department SeminarPPPL

April 2015

AcknowledgementS. Ethier and R. Ganesh (IPR, India)

1000

Other pioneers: Oscar Buneman

Ned Birdsall

103

104

105

106

1536 3072 6144 12288 24576 49152 98304

com

pute

pow

er (m

illion

s nu

mbe

r of p

artic

les

per s

econ

d pe

r ste

p)

number of nodes

A200$

B200$

C200$

D200$

Sequoia*

BG-Q Performance: Weak Scaling Results

Mira$

$$$$$Mira$

Mira$

*NNSA’s Sequoia (16.3 PF)

x 16 → number of cores

2014

80,000 particles/core

P. J. Catto

Nonlinear

Vlasov equation

Gyrokinetic

Density response due to

+

+

B

W. W. Lee

F = F0 + f

δf simulation for

F (x,v, t) =

NX

j=1

[x xj(t)][v vj(t)]

f(x,v, t) =

NX

j=1

wj(t)[x xj(t)][v vj(t)]

w = f/F

Computational Science & Discovery 1 (2008) 015010 W W Lee et al

Figure 5. Time evolution of (a) the ion thermal flux, (b) the particle weights, (c) the field energy,and (d) the radial modes, as well as (e) the zonal flow structure for a/ρ = 500 including both thenonlinearly generated zonal flows and the velocity space nonlinearity (VNL).

than the quasilinear value

collisions

5vti < vk < 5vti µ < 12.5v2ti/⌦i

⌫ii = 0.001vti/Ln

dRdt

= vkb + vd − @φ

@R⇥ b,

Applications to tokamak transport physics δ

[Lee, Jenkins, and Ethier, CPC 2011; Ganesh, Ethier and Lee, ICPP, 2014]

vd ⇡✓

v2k +

v2?2

◆b ⇥ @

@RlnB

dvk

dt= b ·

✓v2?2

@

@RlnB

◆ b⇤ · @φ

@R

b⇤⇡ b + vkb ⇥ @

@RlnB

↵1 + ↵2 = 1

⇢ = eX

j

[↵1pj + ↵2wj ]δ(x xj)]

dp

dt= (p w)(vd + vE⇥B) ·

dw

dt= (p w)

(vd + vE⇥B) · +

Te

Ti(vkb + vd) · E

�vd

d(p w)

dt= (p w)

Te

Ti(vkb+ vd) ·E

µB ⌘ v2?2B

⇡ const.,

↵1 =|wj |2

|pj |2 + |wj |2↵2 =

|pj |2

|pj |2 + |wj |2

and Krommes (PoP ’94)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 200 400 600 800 1000 1200 1400

sum

w^2

/N (

red)

, sum

p^2

/N-1

(gr

een)

, sum

(p-

w)^

2/N

-1 (

blue

)

time [a/Cs]

Entropies : comparision, p-w system. p-weight not included in dynamics,dt=0.1, micell=100, arho=125

a

⇢s= 125

particle interaction

NL stage.

losing energy in the saturated state

scattering are similar physical processes !

@F↵

@t+ v · @F↵

@x+

q

m

E+

1

cv ⇥ (B0 + δB)

�· @F↵

@v= 0

E = −rφ− (1/c)@A/@t B = r⇥A

L =1

2mv2 q+

q

cv ·A

@F↵

@t+ v · @F↵

@x+

q

m

−r(φ− 1

cv ·A) +

1

cv ⇥B0

�· @F↵

@(v + q↵A/m↵c)= 0

β

v ! v +q↵m↵c

A?

@F↵

@t+ v · @F↵

@x+

q

m

−r(φ− 1

cv? ·A?)−

1

c

@Ak

@t+

1

cv ⇥ (B0 + δB?)

�· @F↵

@(v + q↵A?/m↵c)= 0

to obtain

F ⌘ F (x,v, t)

F ⌘ F (x, vk, µ/B, t)

r2A− 1

v2A

@A?@t2

= −4⇡

c

X

↵

q↵

ZvF↵dvkdµ

v ⇡ vkb+c

B0E⇥ b

E = −r(φ− v? ·A?/c)− (1/c)@Ak/@t

b = b0 + B?/B0 b0 = B0/B0 B? = r⇥Ak

@F↵

@t+

vkb− c

B0r(φ− 1

cv? ·A?)⇥ b0

�· @F↵

@x− q

m

"r(φ− 1

cv? ·A?) · b+

1

c

@Ak

@t

#@F↵

@vk= 0

µ = v2?/2 v

Tp = (mc/eB2)(@2

A?/@2t)

d

dt

⌧Z(1

2v2k + µ)(meFe +miFi)dvkdµ+

!2ci

⌦2i

|r?Φ|2

8⇡+

|rAk|2

8⇡

�

x

= 0

⌘ − v? ·A?/c

v

Lp = −(mc2/eB2)(@r?φ/@t)

k2?⇢2i ⌧ 1r2φ+

!2pi

⌦2i

r2?φ = −4⇡

X

↵

q↵

ZF↵dvkdµ

!2 ⌧ k2?v2A

b

⇤ ⌘ b+vk

⌦↵0b0 ⇥ (b0 ·r)b0 b = b0 +

r⇥ A

B0

F↵ =

N↵X

j=1

(RR↵j)(µ µ↵j)(vk vk↵j)

@F↵

@t+

dR

dt· @F↵

@R+

dvk

dt

@F↵

@vk= 0

⌦↵0 ⌘ q↵B0/m↵c

⌘ − v? ·A?/c

dR

dt= vkb

⇤ +v2?

2⌦↵0b0 ⇥rlnB0 −

c

B0rΦ⇥ b0

dvk

dt= −v2?

2b

⇤ ·rlnB0 −q↵m↵

✓b

⇤ ·r+1

c

@Ak

@t

◆

v? ·A? = 1

2⇡

eB0

mc

Z 2⇡

0

Z ⇢

0

δBkrdrd✓

(m,n) tearing modes [APS 2004, Sherwood 2005] using GTS [Wang et al., PoP 2003].

Gyrokinetic Current Densities

Jgc(x) = Jkgc(x) + J

M?gc(x) + J

d?gc(x)

=X

↵

q↵hZ

F↵gc(R)(vk + v? + vd)δ(R− x+ ⇢)dRdvkdµi'

[Qin, Tang, Rewoldt and Lee, PoP 7, 991 (2000); Lee and Qin, PoP 10, 3196 (2003).]

p↵? = m↵

Z(v2?/2)F↵gc(x)dvkdµ

p↵k = m↵

Zv2kF↵gc(x)dvkdµ

J?gc = J

M?gc + J

d?gc

J?gc =c

B

X

↵

b⇥rp↵

=c

B

X

↵

hb⇥rp↵? + (p↵k − p↵?)(r⇥ b)?

i

J

d?gc =

c

B

X

↵

hp↵k(r⇥ b)? + p↵?b⇥ (rlnB)

i

J

M?gc(x) = −

X

↵

r? ⇥ cb

Bp↵?

vd =v2k

⌦↵b⇥ (b · @

@R)b+

v2?2⌦↵

b⇥ @

@RlnB

p↵ = p↵k = p↵?

ρv - ion

Rx

b - out of the board

k2?⇢2i ⌧ 1 F ! F ! Ak ! Ak v? ·A? ! 0

r2?Ak = −4⇡

cJk

J? =c

B

X

↵

b⇥rp↵

b ⌘ B

BB = r⇥A

B = B0 + B

d

dtr2

?φ− 4⇡v2Ac2

r · (Jk + J?) = 0d

dt⌘ @

@t− c

Brφ⇥ b ·r

dp↵dt

= 0

Ek ⌘ −1

c

@Ak

@t− b ·rφ = ⌘Jk ! 0

! = ±kkvA

r2?A? − 1

v2A

@2A?@t2

= −4⇡

cJ? !2 ⌧ k2?v

2A

J? =c

B

X

↵

b⇥rp↵

r · (Jk + J?) = 0

d

dtr2

?φ+v2Ac(b ·r)r2

?Ak − 4⇡v2Ac2

r · J? = 0

Ek ⌘ −1

c

@Ak

@t− b ·rφ = 0

d

dt⌘ @

@t− c

Brφ⇥ b ·r

@Ak

@t! 0 ! 0

n(x) = n +12⇢2

t

1Tr2

?nT

0 500 1000 1500 2000 2500 3000 3500 4000 45000

0.5

1

1.5

2

2.5

3

t(a/cs)

i (cs

s2 /a)

0 500 1000 1500 2000 2500 3000 3500 4000 4500!18

!16

!14

!12

!10

!8

!6

t(a/cs)

ln(r

adia

l m

ode)

(vE× B

/cs)

ITG simulation using GTC (a/rho =125)

J? =c

Bb ⇥rp? + qnvE⇥B + qn

⇢2t

2

hr2

?vE⇥B +vE⇥B

nTr2

?nTi

p? = m

ZµFgc(x)dvkdµ

J =c

Bb ⇥r(p?e + p?i) + eni

⇢2i

2

r2

?vE⇥B +vE⇥B

niTir2

?niTi

�

VE⇥B

cs=

1

2pi⇢s

Ti

Teb⇥ x

pi⇢s ⌘ −⇢srpipi

J =c

Bb⇥r(p?e + p?i −

1

2⇢2ir2

?p?i)

@⇢m@t

+r · ⇢mV = 0

⇢m

✓@V

@t+V ·rV

◆=

1

cJ⇥B−rp

E+1

cV ⇥B = ⌘J

r⇥E = −1

c

@B

@t

r⇥B =4⇡

cJ

! = ±kvA ! = ±kkvA

⇢/L ⇠ !/⌦ ⇠ o(✏)

Discussions

parameter exchanges.

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