Extension of Drift Kinetic Hot Particles to Full

Orbits in NIMROD

Charlson C. KimRichard D. Milroy

and the NIMROD Team

PSI Center - University of Washington, Seattle

PSI Center Kinetic Group

• Objective: develop the capability to model full kinetic minority species in

NIMROD

• The Plan

– implement full kinetic orbits for particles

– test different coupling schemes (J vs. P)

– improve timestepping → orbit averaging

– implement collisions with backgrounda

– implement multiple ‘species’ e.g. some full kinetic particles some drift

– explore issues concerning open boundary conditionsb

– explore running fluid electrons and kinetic ions

aY. Chen and R. B. White,‘Collisional δf method’, Physics of Plasmas 4 3591 (1997)bibid

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Outline

• Motivation

• the full kinetic equations of motion

• hybrid kinetic MHD model

• δf PIC

• brief description of NIMROD and Extended MHD equations

• finite elements and PIC in the finite element method

• benchmark of hybrid kinetic MHD model

• summary and future plans

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Full Orbit Particles are Necessary for ICC devices

• capture kinetic effects lost in MHD approximation

• lost kinetic effects can significantly affect MHD instabilities

– fraction of plasma executes non-trivial orbits

• experiments show that FRCs are stable to MHD tilt modes, attributed to

FLR effects

• recent simulations have shown this is a possible mechanisma

• kinetic effects are believed ubiquitous in ICC devices

aE. V. Belova, et.al, ‘Kinetic Effects on the Stability of Properties of FRC’, Physics of

Plasmas 10 2361 (2003)

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Full Orbit Equations Well Studied

• equations of motion are Lorentz force equations

v =q

m(E + v× B)

• Buneman (1967)

– subtracts off vE×B

– separates ‖,⊥ motion

– rotation of v⊥

• Boris (1970) - this is what we have initially implemented

– limited in timestep by ωc∆t < 0.35

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Description of the Boris Algorithma

• Boris push utilizes leap frog time centering

• half step push of electric field v− = vt−∆t/2 +qE

m

∆t

2

• increment v− to v′ = v− + v− × ~ωc∆t

2where ~ωc =

qB

m

��� ��� v’

v+

v−

B

• rotate v− to v+ = v− + v′ × s~ωc where s = ∆t

1+ω2c

∆t2

4

s.t. |v+| = |v−|

• remainder of half step push of electric field vt+∆t/2 = v+ +qE

m

∆t

2

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Full Ion Orbits in Tokamak

full orbits has marginal consequences for overall trajectory in tokamaks

nontrivial orbits in ICCs

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The δf PIC method a

• PIC is a Lagrangian simulation of phase space f (x,v)

• PIC evolves f (x(t),v(t))

• spatial grid is not inherantly necessary, but very convenient!

• in principle, f (x(t),v(t)) contains everything

• typically PIC is noisy, can’t beat 1/√

N

• δf PIC reduces the discrete particle noise associated with conventional PIC

• Vlasov Equation∂f (z)

∂t+ z · ∂f (z)

∂z= 0

z is the phase coordinate

• split phase space distribution into steady state and evolving perturbation:

f = feq(z) + δf (z, t)aS. E. Parker and W. W. Lee, ‘A fully nonlinear characteristic method for gyro-kinetic

simulation’, Physics of Fluids B 5, 1993

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• δf evolves along the characteristics z (control variates MCb)

˙δf = −z · ∂feq

∂z

using z = zeq + z and zeq · ∂feq

∂z= 0

• the particle characteristics are taken form the equations of motion

• as particle algorithm advances, may be useful to compare δf PIC

implementation with full PIC

bA. Y. Aydemir, ‘A unified MC interpretation of particle simulations...’, Physics of Plasmas

1, 1994

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The Hybrid Kinetic-MHD Equationsa

• in the limit nh � n0, βh ∼ β0, quasi neutrality, only modification of MHD

equations is addition of the hot particle pressure tensor in the momentum

equation:

ρ

(

∂U

∂t+ U · ∇U

)

= J× B−∇ · pb−∇ · p

h

the subscripts b, h denote the bulk plasma and hot particles

• the steady state equation

J ×B = ∇ · p = ∇ · pb+ ∇ · p

h

• evolved momentum equation is

δρUs · ∇Us + ρs

(

∂δU

∂t+ Us · ∇δU + δU · ∇Us

)

=

Js × δB + δJ × Bs −∇ · δpb−∇ · δp

h

aC.Z.Cheng,‘A Kinetic MHD Model for Low Frequency Phenomena’,J. Geophys. Rev 96,

1991

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NIMRODa (NonIdeal MHD with Rotation - Open Discussion)

• massively parallel 3-D MHD simulation

• finite elements in poloidal plane and Fourier modes in toroidal direction →axisymmetric geometry

• utilizes Lagrange type finite element

• can handle extreme anisotropies,χ‖

χ⊥� 1

• flexibility to model general geometry → real experiments

• model experiment relevant parameters, S > 107

• semi-implicit advance, not restricted by magnetosonic CFL condition

• assumes a steady state background and evolves perturbed quantities

→ A(x, t) = As(x) + δA(x, t)

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NIMROD equations

• NIMROD evolves the extended MHD equations

∂B

∂t= −∇ × E

∇× B = µ0J

E = −U× B+ηJ +1

neJ × B

+me

ne2

[

∂J

∂t+∇ · (JU + UJ)

+∑

α

qα

mα(∇pα + ∇ · Πα)

]

∂n

∂t+ ∇ · (nU) = ∇ · D∇n

mn(∂U

∂t+ U · ∇U) = J× B−∇p −∇ · Π−∇ · ph

nα

γ − 1(∂Tα

∂t+ Uα · ∇Tα) = −∇ · qα + Qα

−pα∇ · Uα − Πα : ∇Uα

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Finite Element representation of NIMROD

• the perturbed NIMROD fields are in FE-Fourier representation

δA(x, t) =∑

j

Aj,0(t)αj,0 +∑

j

∑

n

(Aj,n(t)αj,n + c.c.)

where

αj,n = Nj (η, ξ) exp(inφ)

(η, ξ) are logical coordinates

��

��

��

��

R

Z

� �� ��

χ

η

• PIC becomes nontrivial in FE

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PIC in FEM

• gather/scatter require logical coordinates

• FE representation for (R, Z) need to be inverted

R =∑

j

RjNj (η, ξ), Z =∑

j

ZjNj(η, ξ),

• Use Newton method to solve for (η, ξ) given (R, Z)

ηk+1

ξk+1

=

ηk

ξk

+1

∆k

∂Z∂ξ −∂R

∂ξ

−∂Z∂η

∂R∂η

k

R − Rk

Z − Zk

where ∆k is the determinant and k denotes iteration

• sorting required for parallelization

• load balancing tricky

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Deposition of δph

onto Finite Element grid

• assume CGL-like form δph

=

p⊥ 0 0

0 p⊥ 0

0 0 p‖

• evaluate pressure moment at a position x is

δp(x) =

∫

m(v − Vh)2δf(x,v)d3v

=N

∑

i=1

m(vi − Vh)2g0wiδ3(x − xi)

where sum is over the particles, m mass of the particle, g0wi is the

perturbed phase density, and Vh is the hot flow velocity

• discretize to the poloidal plane by performing a Fourier transform,a

δpn(R, Z) =N

∑

i=1

m(vi − Vh)2g0wiδ(R − Ri)δ(Z − Zi)e−inφi

afor large n runs, a conventional φ deposition with a FFT is performed

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• express lhs δpn(R, Z) in the finite element basis,

δpn(R, Z) =∑

k

δpknNk(η, ξ)

where sum k is over the basis functions and (η, ξ) are functions of (R, Z)

• project onto the finite element space by casting in weak form:

∫

N l∑

k

δpknNkd3x =

∫

N lN

∑

i=1

m(vi − Vh)2g0wiδ(R − Ri)δ(Z − Zi)e−inφid3x

Mδpkn =

∑

l∈k

N∑

i=1

m(vi − Vh)2g0wie−inφiN l(ηi, ξi)

M is the finite element mass matrix

• gather is done using the same shape functions

A(xi) =∑

l

AlN l(ηi, ξi)

for some field quantity A

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Drift Kinetic Benchmark with M3D

• transition from internal kink mode to fishbonea

• monotonic q, q0 = .6, qa = 2.5, β0 = .08, circular tokamak R/a=2.76

• dt=1e-7, τA = 1.e6

aF. Porcelli, ‘Fast Particle Stabilisation’, Plasma Physics and Controlled Fusion 33, 1991

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Summary and Future Plans

• Lorentz equation of motion has been implemented as Boris push

• can push particles multiple times per MHD time step

• orbits calculated for FRC equilibrium

• implement orbit averaged pressure deposition and couple to MHD

• explore alternative coupling schemes

• investigate implementation of collisions and sources/sinks for open b.c.

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