Continuum solution to ChapmanEnskoglike
drift kinetic equation (CELDKE) in NIMROD
Eric Held, JeongYoung Ji, Andy Spencer and Mukta Sharma
Utah State University, Logan, UT 84322
and NIMROD Team
CEMM Meeting
11/16/08
Dallas, TX
Motivation for continuum solution to closure
problem.
Treat time-dependent problems such as Landau damping or coupling
of closures to rapidly evolving instabilities.
Easily incorporate nonlinearities and particle trapping effects.
Increase the efficiency of the closure calculation.
Incorporate accelerations effects.
●
Solve lowest-order Chapman-Enskog-like drift kinetic equation:
∂F v
F q ∥⋅E ∂ − ∂ B ∂ F −C F f
v∥⋅∇
M
=
∂t m v ∂v B ∂t ∂
2 1/2 d ln T 3/ 2
− L 1
V
∇⋅ 1 f M
v
L 1 ∥⋅∇ ln T f M
∥ ∇⋅ − f
v ⋅ R M
3 dt
2
v∥ I
2 s P2 b b− :∇ V1 f
M
v 3
Simplify.
● Consider solving CEL-DKE by expanding F = Fi (x,t) i( v||, v) :
∂F v
F q ∥⋅ E ∂ − ∂ B ∂ F=
v⋅
L L F ∥ ∇
∂t m v ∂v B ∂ t ∂
C −L F f M
CEL drives
● Acceleration and collision terms, C(-L) (F + fM), couple velocity expansion
coefficients in speed variable, v.
●
Preliminary implementation ignores acceleration term and uses
a moment approach for C(-L) (J-Y Ji's work and J. James' thesis).
Existing implementation solves for coefficients
of F expansion on grid in s=v/vT.
Expanding F = Fl (x,v,t) Pl (v|| / v) yields:
∂
I − L L A v b⋅∇ − B v b⋅∇ ln B F = drives ,
F F
∂t
ab 1 E ' s b
where L =∑b 2 − E s b
s 3
a
s 2
b
sb
− Matrices A, B, and L represent free-streaming, |B| and pitch-angle
collisional couplings, respectively.
− With Lorentz pitch-angle scattering operator, speed enters as a
parameter only.
− Solve equation on grid in s.
Coupling to the fluid equations: time
discretization and parallelization issues (I).
Nimrod uses staggered advance.
V
B
n , ,T
●
For parallel heat flow closure, can couple fully implicit solves for T and F.
V
B
n , ,T , F
● Must solve simultaneously for coefficients, Fl , on speed grid, si , i=1,...,n.
● Closure moment q|| couples to T equation.
●
Leads to large system of equations with “ parallelization” performed
inside solver.
Coupling to the fluid equations: time
discretization and parallelization issues (I).
3
b=... ,
n T t ∇⋅q∥
T equation
2
a l ,l / si t L si / s i L l , l F l
t b⋅∇ al , l1 F l 1 a l ,l −1 F l −1
b
t ⋅∇ ln B [ bl , l1 F l 1 b l ,l −1 F l− 1 ]
l1 al , l L 3 /2 f M b⋅ ln T = ... ,
1
coupled equations for F l
3 /2
v
where q∥=−T ∫ d v ∥ L 1
P 1 v∥ / v F 1
Applied to heat transport problem in cylindrical
geometry (Hölzl et al., POP 2008).
2/1 island added into zero pressure, cylindrical eq.
Heat source finite for r < 0.2 and zero outside.
Can solve for T coupled to F equations.
T profiles as function of flux
show flattening across 2/1
pd=2, continuum solution using
island. F1 and F2.
All cases used 3 Fourier
modes and 10 x 10 grid.
Predicted T from continuum
solution spatially accurate for
bicubic (pd=2) finite elements.
T flattening
pd=5, T-only. across island.
Here chosen to yield
heat flow consistent with pd=2, T-only.
Braginskii closure with large
parallel conductivity.
Solution not resolved in
velocity variables!
Generalize velocity space basis to include 2-D
finite elements.
Expand F = Fi (x,t) i( , ) , where and are appropriate velocity
variables and insert into CEL-DKE:
∂F v
F q ∥⋅E ∂ − ∂ B ∂ F −C F f
v
∥⋅∇
M
=drives
∂t m v ∂ v B ∂t ∂
−i n '
x v
Integrate equation using ∫ d ∫ d i ' , j ' R , Z e
Use NIMROD finite element and Gaussian quadrature machinery to
compute velocity integrals.
Consider C0 or possibly discontinuous basis functions.
Requires new data types and solver development.
Future work.
●
At present, can couple solution for coefficients of Legendre
expansion, Fl(x,t), on grid in s to T in fully implicit advance.
●
Future work includes:
- testing convergence of fully implicit advance as Legendre
polynomials and grid points in speed are added.
- implementing staggered advance and comparing with fully
implicit solutions.
- generalizing continuum solution to allow for 2-D finite-element
basis functions for velocity dependence.