Accurate representation of velocity space using
truncated Hermite expansions.
Joseph Parker
Oxford Centre for Collaborative Applied Mathematics
Mathematical Institute, University of Oxford
Wolfgang Pauli Institute, Vienna
23rd March 2012
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
1 / 34
Model for ITG instabilities (from Pueschel et al., 2010)
Test problem for G ENE, simplified gyrokinetic equations:
1D slab, (z, vk )
adiabatic electrons
B = Bz homogeneous and static
drift-kinetic ions
electrostatic: E = −∇Φ
quasineutral
uniform density and temperature gradients in background
perturbation about Maxwellian F0 (vk ) = π −1/2 exp(−vk2 )
nonlinearity dropped
May also derive directly from Vlasov equation and quasineutrality.
Electrostatic limit of Belli & Hammett (2005) model
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
2 / 34
Model for ITG instabilities (from Pueschel et al., 2010)
Linearized kinetic equation
∂F1
∂Φ
∂F1
1
2
+ αi vk
+ ωn + ωTi vk −
iky ΦF0 + τe αi vk
F0 = 0
∂t
∂z
2
∂z
Quasineutrality
Z
∞
Φ=
F1 dvk .
−∞
Parameters
ωn = 1,
ωTi = 10,
ky = 0.3,
τe = 1,
αi = 0.34
normalized density gradient
normalized ion temperature gradient
perpendicular wavenumber
species temperature ratio (Ti = Te )
nondimensional constant: velocity and length scales
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
3 / 34
Wave-like solutions
F1 = f (vk ) exp i kk z − ωt
Leads to eigenvalue problem,
L0 f = ωf
1
2
ky ΦF0 + τe αi kvk ΦF0
where L0 f = αi kvk f + ωn + ωTi vk −
2
L real =⇒ eigenvalues in complex conjugate pairs =⇒ no damping
So actually want to solve,
lim L f = ωf,
→0
for
L = L0 + iC
where C is collisions, e.g. Lénard–Bernstein or BGK.
Breaks symmetry, so decay rate may tend to nonzero limit as → 0.
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
4 / 34
Dispersion relation
0.4
growth rate, γ
0.2
0.0
−0.2
−0.4
−0.6
−0.8
−1.0
0
1
Joseph Parker (University of Oxford)
2
3
4
5
6
parallel wavenumber, kk
Velocity space using Hermite expansions
7
8
23rd March 2012
5 / 34
Discretization
Keep exp(ikk z) behaviour, represent vk on a uniform grid,
Φ=
N
X
∆vf (vi )
i=1
Obtain linear ODE system
∂f
= iM f,
∂t
M real N × N matrix
Matrix eigenvalue problem (quicker than running IVPs):
−iωf =
∂f
= iM f
∂t
Growth rate is γ = max =(ω), for ω ∈ spectrum(M ).
Matrix M depends on:
All parameters, particularly parallel wavenumber.
Discretization and resolution of grid in vk .
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
6 / 34
Growth rate for simple discretized system
0.3
N = 10
N = 100
dispersion relation
growth rate, γ
0.2
0.1
0.0
−0.1
−0.2
0
1
2
3
4
5
6
parallel wavenumber, kk
7
8
Capture growth, but convergence slow with N .
No decay.
I
I
→ 0 at fixed N resolution.
Need to restore collisions.
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
7 / 34
0.14
Distribution function in velocity space
k = 2.75
k = 3.25
k = 3.75
0.12
0.10
f (v)
0.08
0.06
0.04
0.02
0.00
−0.02
−0.04
−3
−2
−1
0
v
1
2
3
Fine scales develop as =(ω) → 0.
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
8 / 34
Hermite expansion in velocity space
Use expansion in asymmetric Hermite functions,
f (v) =
∞
X
Hm (v)
φm (v) = √
,
2m m!
am φm (v),
m=0
φm (v) = F0 (v)φm (v)
Bi-orthogonal polynomials with Maxwellian as weight function
Z ∞
φm φn dv = δmn
−∞
Recurrence relation
r
vφm =
m+1
φm+1 +
2
r
m
φm−1
2
=⇒ particle streaming becomes mode coupling
Represent velocity space scales,
√
mπ Hm (v) ∝ cos v 2m −
, as m → ∞
2
large m =⇒ fine scales
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
9 / 34
Relative Entropy and Free Energy
Relative entropy is
R[F |F0 ] ≡
Z
∞
F log
Z−∞
∞
=
F
F0
− F + F0 dv
F log F dv + 2U + function(density)
−∞
Expansion F = F0 + F1 gives (at leading order)
2
R[F |F0 ] =
2
Z
∞
−∞
∞
F12
2 X 2
dv =
am + O(3 )
F0
2
m=0
Define free energy of each Hermite mode, Em ,
E=
∞
X
m=0
Em ,
1
with Em = |am |2
2
cf. energy spectra in Fourier space for Navier–Stokes turbulence.
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
10 / 34
Moment system
Replace F0 ≡ φ0 and v with Hermite functions
∂F1
ωTi
kτe αi Φ
+ iαi kvF1 + iky Φ ωn φ0 + √ φ2 + i √ φ1 = 0
∂t
2
2
Put F1 = am φm (implicit sum over repeated m)
∂am
ωTi
kτe αi Φ
φm + iαi kvam φm + iky Φ ωn φ0 + √ φ2 + i √ φ1 = 0
∂t
2
2
Use recurrence relation on particle streaming
!
r
r
m+1
m
∂am
φm + iαi kam
φm+1 +
φm−1 + . . . = 0
∂t
2
2
Gives infinite set of coupled algebraic equations for {an }.
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
11 / 34
System of equations for expansion coefficients
r
ωam =
m+1
am+1 +
2
r
m
am−1
2
!
#
"
ky
ωTi a0
+
ωn a0 δm0 + √ δm2 +
αi kk
2
|
{z
}
driving
τe a0
√ δm1
2
| {z }
Boltzmann response
We have also used φ0 ≡ 1, so that,
Z
∞
Φ=
0
F1 φ dv =
−∞
Joseph Parker (University of Oxford)
∞ Z
X
∞
am φm φ0 dv = a0
m=0 −∞
Velocity space using Hermite expansions
23rd March 2012
12 / 34
Matrix equation




















ωn ky
αi k k
τe
√
+ √12
2
ky ω
√ Ti
2αi kk

√1
2
1
q
1
Joseph Parker (University of Oxford)
q
3
2
√
3
2
√
2
..
2
..
.
q
.
q
N
2
N
2
..
Velocity space using Hermite expansions









 a = ωa






.. 
. 

.
23rd March 2012
13 / 34
Simple Truncation









ωn ky
αi kk
τe
√
+ √12
2
ky ω
√ Ti
2αi kk
√1
2

1
q
1
q
3
2
3
2
√
2




 a = ωa

√ 
2 
Assume aN +1 = 0 at point of truncation. Last row of matrix:
r
r
N +1
N
aN +1 +
aN −1 = ωaN
2
2
Equivalent to discretization on Gauss–Hermite points {vj }
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
14 / 34
100
Hermite spectra of eigenfunctions
free energy, |am|2
10−10
10−20
10−30
k = 2, γ > 0
k = 6, γ = 0
10−40 0
10
101
Hermite mode, m + 1
102
Growing mode: am decay as m increases =⇒ aN +1 ≈ 0 is okay
(Not) decaying mode not resolved
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
15 / 34
Collisions
Free energy should cascade to large m (fine scales in v).
Free energy has nowhere to go when aN +1 = 0.
Restore collisions:
∂F1
1
2
+ ωn + ωTi vk −
iky ΦF0 + iαi kvk F1 + iτe αi kvk ΦF0 = C[F1 ]
∂t
2
Desirable properties for C[F1 ]
conserves
mass, Zmomentum and
energy
Z
Z
I
C[F1 ] dv =
vC[F1 ] dv =
v 2 C[F1 ] dv = 0
satisfies a linearized H theorem:
Z ∞
dR
F1 C[F1 ]
2
=
dv 6 0
dt
F0
−∞
represents small-angle collisions (contains v-derivatives)
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
16 / 34
Lénard–Bernstein (1958) collisions
Simple member of linearized Landau/Fokker–Planck class:
∂
1 ∂F1
C [F1 ] = ν
vF1 +
∂v
2 ∂v
collision frequency ν
Hermite functions are eigenfunctions:
C [am φm ] = −νmam φm
=⇒ easy to implement in Hermite space
Conserves mass, but not momentum or energy
Satisfies a linearized R theorem:
X
dR
= −2 ν
m|am |2 6 0
dt
m
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
17 / 34
Lénard–Bernstein in Hermite space














ωn ky
αi kk
τe
√
+ √12
2
ky ω
√ Ti
2αi kk

√1
2
−iν
1
1
−2iν
q
3
2
q
3
2
−3iν
..
.
..
.
q
q
N
2
N
2
−iN ν






 a = ωa






matrix now complex
roots not in complex-conjugate pairs =⇒ can find negative
growth rates
(we can manually conserve momentum and energy)
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
18 / 34
Lénard–Bernstein in Hermite space














ωn ky
αi kk
τe
√
+ √12
2
ky ω
√ Ti
2αi kk

√1
2
H
−iν
H
1
1
X
X
−2iν
X
q
3
2
q
3
2
−3iν
..
.
..
.
q
q
N
2
N
2
−iN ν






 a = ωa






matrix now complex
roots not in complex-conjugate pairs =⇒ can find negative
growth rates
(we can manually conserve momentum and energy)
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
18 / 34
Growth rate, Lenard–Berstein collisions, N = 100
0.4
growth rate, γ
0.2
0.0
−0.2
−0.4
ν = 10−1
−0.6
ν = 10−2
ν = 10−3
dispersion relation
−0.8
−1.0
0
1
2
3
4
5
6
parallel wavenumber, kk
7
8
If collision frequency ν large enough to get kk > 4 correct,
then shape for kk < 4 distorted.
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
19 / 34
Growth rate, Lenard–Berstein collisions, ν = 10−2
0.4
growth rate, γ
0.2
0.0
−0.2
−0.4
N = 100
N = 200
N = 300
dispersion relation
−0.6
−0.8
−1.0
0
1
2
3
4
5
6
parallel wavenumber, kk
7
8
resolves dissipative scales
expensive: 300 modes
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
20 / 34
100
Lenard–Bernstein collisions, ν = 10−2
k=2
k=6
10−1
free energy, |am|2
10−2
10−3
10−4
10−5
10−6
10−7
10−8 0
10
101
Hermite mode, m + 1
Appreciable damping along whole spectrum.
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
102
23rd March 2012
21 / 34
Hypercollisions
Would prefer:
low modes undamped
high modes strongly damped
. . . whatever the truncation point N
Iterate Lénard–Bernstein collisions:
1 ∂F1
∂
vF1 +
,
C[F1 ] = −ν(−N )−n Ln [F1 ]
L[F1 ] =
∂v
2 ∂v
(like hyperdiffusion: −(−∇2 )n in physical space)
Hermite functions are still eigenfunctions:
m n
C [am φm ] = −ν
am φm
N
ν sets the decay rate of the highest mode
∞ X
dR
m n
2
linearized R-theorem:
= − ν
|am |2 6 0
dt
N
m=0
n = 1 corresponds to Lénard–Bernstein collisions.
two parameters: n ≈ 6, ν ≈ 10 (robust to variation)
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
22 / 34
Growth rate with hypercollisions, ν = 10, n = 6
0.2
growth rate, γ
0.0
−0.2
−0.4
−0.6
simple truncation, N = 100
hypercollisions, N = 10
exact dispersion relation
−0.8
−1.0
0
1
2
3
4
5
6
parallel wavenumber, k
7
8
Captures decaying parts of spectrum.
Excellent fit, fast convergence.
Appropriate for nonlinear problems.
Two parameters, n, ν, robust to variation.
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
23 / 34
100
Hermite spectra with hypercollisions, k = 2
10−1
free energy, |am|2/2
10−2
10−3
10−4
10−5
10−6
10−7
10−8 0
10
N = 10
N = 100
101
Hermite mode, m + 1
102
Low moments largely undamped
Damping at high m, for any N .
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
24 / 34
100
Hermite spectra with hypercollisions, k = 6
10−1
free energy, |am|2/2
10−2
10−3
10−4
10−5
10−6
10−7
10−8
10−9
10−10 0
10
N = 10
N = 100
101
Hermite mode, m + 1
102
Low moments largely undamped
Damping at high m, for any N .
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
25 / 34
Energy equations and theoretical spectra
Equation for coefficients, valid for m > 3,
!
r
r
m+1
m
am+1 +
am−1 + driving + Boltzmann response
ωam =
2
2
Treat as a finite difference approximation in continuous m.
Energy equation for Em = |am |2 /2
(Zocco & Schekochihin, 2011)
m n
√
∂Em
∂
Em .
+
2mEm = −2ν
∂t
∂m
N
For a mode with growth rate γ,
Em
C
γ
=√
exp −
|γ|
2m
m
mγ
with the cutoffs,
1
mγ = 2 ,
8γ
Joseph Parker (University of Oxford)
m(n+1/2)
c
1/2
−
m
mc
n+1/2 !
N n (n + 1/2)
√
=
ν 2
Velocity space using Hermite expansions
,
23rd March 2012
26 / 34
Energy equations and theoretical spectra
Equation for coefficients, valid for m > 3,
!
r
r
m+1
m
am+1 +
am−1 + hypercollisions
ωam =
2
2
Treat as a finite difference approximation in continuous m.
Energy equation for Em = |am |2 /2
(Zocco & Schekochihin, 2011)
m n
√
∂Em
∂
Em .
+
2mEm = −2ν
∂t
∂m
N
For a mode with growth rate γ,
Em
C
γ
=√
exp −
|γ|
2m
m
mγ
with the cutoffs,
1
mγ = 2 ,
8γ
Joseph Parker (University of Oxford)
m(n+1/2)
c
1/2
−
m
mc
n+1/2 !
N n (n + 1/2)
√
=
ν 2
Velocity space using Hermite expansions
,
23rd March 2012
26 / 34
Growing mode, k = 2
100
collision
dominated
10−1
free energy, Em
10−2
10−3
10−4
m−1/2
behaviour
10−5
growth rate dominated
10−6
10−7
10−8
10−9
10−10 0
10
theoretical spectrum
numerical spectrum
growth rate cutoff
collisional cutoff
101
102
Hermite mode, m + 1
n+1/2 !
C
γ
m 1/2
m
Em = √
exp −
−
|γ| mγ
mc
2m
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
27 / 34
Decaying mode, k = 6
10−1
free energy, Em
10−2
growth rate dominated
10−3
m−1/2
10
−4
10−5 0
10
theoretical spectrum
numerical spectrum
growth rate cutoff
collisional cutoff
collision
dominated
101
102
Hermite mode, m + 1
n+1/2 !
C
γ
m 1/2
m
Em = √
exp −
−
|γ| mγ
mc
2m
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
28 / 34
How strong should collisions be?
We can find the collision strength required for a given resolution.
Write hypercollisions as,
C[F1 ] = −ν(−L)n [F1 ]
(i.e. remove N −n , damping strength expressed just by ν.)
Need to resolve collisional cutoff,
N > mc =
(n + 1/2)
√
ν 2
1/(n+1/2)
→ ∞ as ν → 0
Need infinite resolution to resolve collisionless case.
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
29 / 34
Hermite
spectra with different ν, k = 6, n =4, N =100
0
10
10−2
free energy, |am|2/2
10−4
10−6
10−8
10−10
10−12
10−14
ν = 0.8 × 10−7
10−16
ν = 1.0 × 10−7
10−18
10
−20
ν = 1.2 × 10−7
0
20
40
60
Hermite mode, m + 1
80
100
Weaker collisions =⇒ finer scales in spectra
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
30 / 34
Growth rate against collision strength, k = 6, n =4
0.0
N = 20
N = 50
N = 100
correct γ
growth rate, γ
−0.1
−0.2
−0.3
−0.4
−0.5 −10
10
10−9
10−8
10−7
10−6
10−5
hypercollision strength, ν
a range of ν give the correct growth rate
range extends to smaller ν as N increases
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
10−4
10−3
23rd March 2012
31 / 34
Hypercollisions on other grids
Easiest to implement in
Hermite space:
Growth rate calculated on uniform grid, N = 16
0.4
0.2
C[am ] = −ν(m/N ) am
C[a] = Da
But may be used on any grid.
Map function values f to
Hermite space by a = M f ,
collide and map back,
C[f ] = M −1 DM f
growth rate, γ
n
0.0
−0.2
−0.4
−0.6
simple truncation
hypercollisions
exact dispersion relation
−0.8
−1.0
0
1
2
3
4
5
6
parallel wavenumber, k
7
8
Example: hypercollisions
implemented on a uniform grid.
Only need to calculate
M −1 DM once.
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
32 / 34
Summary
1D model for ITG instability
I
I
velocity space discretization =⇒ eigenvalue problem
finite composition of normal modes =⇒ no Landau damping
Hermite representation
I
in decaying modes, have energy pile-up at small scales
Damping with collisions
I
Lénard–Bernstein collisions
F
I
finds damping, but requires O(100) terms
hypercollisions
F
F
F
F
F
F
excellent agreement with dispersion relation
theoretical expression for eigenfunctions
only ∼ 10 terms
robust parameters
easy to implement in Hermite space
can use on any grid
“Vanishing collisions” is different from “collisionless”.
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
33 / 34
References
B ELLI , E. A. & H AMMETT, G. W. (2005) A numerical instability in an ADI algorithm for
gyrokinetics. Computer Physics Communications 172 (2), 119–132.
L ANDAU, L. D. (1946) On the Vibrations of the Electronic Plasma. Journal of
Physics-U.S.S.R. 10 (25).
L ÉNARD, A. & B ERNSTEIN , I. B. (1958) Plasma oscillations with diffusion in velocity
space. Physical Review 112 (5), 1456–1459.
P UESCHEL , M. J., DANNERT, T. & J ENKO, F. (2010) On the role of numerical
dissipation in gyrokinetic Vlasov simulations of plasma microturbulence. Computer
Physics Communications 181, 1428–1437.
VAN K AMPEN , N. G. (1955) On the Theory of Stationary Waves in Plasmas. Physica
21, 949–963.
Z OCCO, A. & S CHEKOCHIHIN , A. A. (2011) Reduced fluid-kinetic equations for
low-frequency dynamics, magnetic reconnection, and electron heating in low-beta
plasmas. Physics of Plasmas 18 (10), 102309.
Supported by Award No KUK-C1-013-04 from King Abdullah University of Science
and Technology (KAUST).
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
34 / 34
Drift approximation in the Vlasov equation
v×B
c
= 0,
v = vk ẑ − 2 ∇Φ × B,
B = Bẑ,
E = −∇Φ.
c
B
Vlasov equation becomes
∂f c
q ∂Φ ∂f
+ vk ẑ − 2 ∇Φ × B · ∇f −
= 0.
∂t
B
m ∂z ∂vk
E⊥ +
Perturb about stationary equilibrium, f (x, v, t) = F0 (x, v) + F1 (x, v, t)
Impose density and temperature gradients in x,
"
!#
2
vk2 + v⊥
∂F0
1
3
=−
ωn + ωTi
−
F0
2
∂x
L
2
vth
Assume a Fourier mode in y:
∂y 7→ iky .
Integrate out perpendicular directions in v.
∂F1
1
∂F1
∂Φ
2
+ ωn + ωTi vk −
iky ΦF0 + αi vk
+ τe αi vk
F0 = 0
∂t
2
∂z
∂z
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
1
Quasineutrality
Poisson’s equation
∇2 Φ = 4πq(ni − ne ),
Boltzmann electrons
ne = n̄e exp
qΦ
Te
Nondimensionalize (with = ρs /L)
Z ∞
Te
n̄e
2
∇ Φ= 1+
exp (Φ) .
F1 dv −
2
2
4πn̄i q ρs L
n̄i
−∞
Z ∞
n̄e
n̄e
2 2
∇⊥ Φ = 1 −
+
F1 dv − Φ + O(2 ).
n̄i
n̄i
−∞
Φ, potential for electrostatic perturbation
Z ∞
Φ=
F1 dvk .
−∞
Joseph Parker (University of Oxford)
Velocity space using Hermite expansions
23rd March 2012
2