FullfandfGK particle simulations Fullfandfgyrokinetic particle simulations of Alfv en waves and energetic particle physics

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Full fand δf GK particle simulations
Full fand δf gyrokinetic particle simulations of Alfv´en waves and energetic particle
physics
Zhixin Lu, Guo Meng, Roman Hatzky, Matthias Hoelzl, Philipp Lauber
In this work, we focus on the development of the particle-in-cell scheme and the ap-
plication to the studies of Alfv´en waves and energetic particle physics in tokamak
plasmas. The δf and full fschemes are formulated on the same footing adopt-
ing mixed variables and the pullback scheme for electromagnetic problems. The
TRIMEG-GKX code [Lu et al. J. Comput. Phys. 440 (2021) 110384] has been up-
graded using cubic spline finite elements and full fand δf schemes. The EP-driven
TAE has been simulated for the ITPA-TAE case featured by a small electron skin
depth ∼1.18 ×10−3m, which is a challenging parameter regime of electromagnetic
simulations, especially for the full fmodel. The simulation results using the δf
scheme are in good agreement with previous work. Excellent performance of the
mixed variable/pullback scheme has been observed for both full fand δf schemes.
Simulations with mixed full fEPs and δf electrons and thermal ions demonstrate
the good features of this novel scheme in mitigating the noise level. The full fscheme
is a natural choice for EP physics studies which allows a large variation of EP profiles
and distributions in velocity space, providing a powerful tool for kinetic studies us-
ing realistic experimental distributions related to intermittent and transient plasma
activities.
1
arXiv:2210.04354v1 [physics.plasm-ph] 9 Oct 2022
Full fand δf GK particle simulations
I. INTRODUCTION
The gyrokinetic particle-in-cell (PIC) simulation provides a powerful tool for the studies
of tokamak plasmas.1For improving the simulation quality, especially for electromagnetic
simulations, various schemes have been implemented such as the pkformulae and the iterative
scheme for solving Amp´ere’s law,2,3the noisy matrix4and the implicit scheme.5,6The noise
reduction scheme has been summarized comprehensively recently,3where various numerical
applications have been studied using models with single ion species for linear physics in
GYGLES, which demonstrates the excellent performance of the control variate method in
noise reduction and the enhancement of simulation quality. The pullback scheme using
mixed variables is implemented in ORB5 using the δf scheme for the studies of EP driven
TAEs,7demonstrating its capability in the MHD limit. In Table I, we briefly summarized
the discretization schemes (full f, control variate or δf) and the physics models (“symplectic
(vk)” formula, “Hamiltionian (pk)” formula or mixed variables with pullback scheme) of some
previous works.
While most previous work has adopted the δf scheme,8more effort has recently been
spent on the full fapproach.5,9,10 The full fmethod does not rely on the separation of the
equilibrium and the perturbation, and thus provides a natural way to handle substantial
changes of the profiles in the course of a simulation.9However, the full fsimulations are
more expensive and require more strict noise reduction to make simulation studies of toka-
mak plasma feasible. Especially, the full fmodel in the MHD limit is still a challenge. In this
work, we focus on applications of the noise reduction schemes to the full fand δf simulations
of Alfv´en waves and energetic particle physics. By following the formulation from previous
work,3we implement the pullback scheme using mixed variables in the TRIMEG-(GKX)
code.5,11 TRIMEG (TRIangular MEsh based Gyrokinetic) code was originally developed
using the unstructured triangular meshes for the whole plasma volume simulations of the
electrostatic ion temperature gradient mode,11 and later was extended to study Alfv´en waves
and energetic particle physics using the the full felectromagnetic model.5In the following,
the full fand δf models are formulated and implemented on the same footing. Although the
full fscheme can be applied to all species in the TRIMEG-(GKX) code, a mixed scheme
of full fEPs and δf thermal ions and electrons is proposed and applied in the present
work. A common phenomenon in experiments is that the EP distribution in velocity space
2
Full fand δf GK particle simulations
changes substantially, while the background is well described by the Maxwell distribution.
With this novel mixed full fand δf scheme, the large EP profile variation and the arbitrary
distribution in velocity space can be treated in a natural way and the computational per-
formance is improved compared to using a full fscheme for all species. The mixed scheme
for different species has been implemented in the TRIMEG-(GKX) code benefiting from its
object-oriented programming and modular design of different species.
The paper is organized as follows. In Sec. II, the equations for the discretization of
full fare derived with mixed variables and the pullback scheme adopted. In Sec. III, the
normalized equations are given and the numerical methods are introduced with a rigorous
filter derived. In Sec. IV, the δf and full fsimulations of energetic particle driven toroidicity
induced Alfv´en eigenmode are performed, demonstrating the features of the schemes and
keys issues for the accurate description of the energetic particles in tokamak plasmas.
II. PHYSICS MODELS
A. Discretization of distribution function
Following the formulation in the previous δf work,3,12 Nmarkers are used with a given
distribution,
g(z, t)≈
N
X
p=1
δ[zp−zp(t)]
Jz
,(1)
where zis the phase space coordinate, δis the Dirac delta function, Jzis the correspond-
ing Jacobian and z= (R, vk, µ ≡v2
⊥/(2B)) is adopted in this work, Ris the real space
coordinate. For the full fmodel, the total distribution of particles is represented by the
markers,
f(z, t) = Cg2f Ptot(z, t)g(z, t)≈Cg2f
N
X
p=1
pp,tot(t)δ[zp−zp(t)]
Jz
,(2)
where the constant Cg2f ≡Nf/Ng,Nf/g is the number of particles/markers, and gand f
indicate the markers and physical particles respectively. For each marker,
pp,tot(t) = 1
Cg2f
f(zp, t)
g(zp, t)= const ,(3)
3
Full fand δf GK particle simulations
for collisionless plasmas since
dg(z, t)
dt= 0 ,df(z, t)
dt= 0 .(4)
The expression of Ptot(z, t) (and consequently, pp,tot) can be readily obtained
Ptot(z, t) = 1
Cg2f
f(z, t)
g(z, t)=nf
hnfiV
hngiV
ng
fv
gv
,(5)
where nfis the density profile and fvis the distribution in velocity space, namely, the particle
distribution function f=nf(R)fv(vk, µ), h. . .iVindicates the volume average. There are
different choices of the marker distribution functions as discussed previously.3,12 In this work,
the markers are randomly distributed in toroidal direction and in the (R, Z) plane but the
distribution in velocity space is identical to that of the physical particles, which leads to
Ptot(z, t) = nf
hnfiV
R
R0
.(6)
The density and parallel current are readily obtained from the markers,
{n, jk}ir,iθ,iφ≡ZdV{n, jk}(R)Nir(r)Niθ(θ)Niφ(φ) (7)
=Cg2f Xpp,tot{1, vk}Nir(rp)Niθ(θp)Niφ(φp),(8)
where dV=rRdrdθdφfor an ad-hoc equilibrium, Nir,Niθand Niφare basis functions and
ir,iθand iφindicate the indices in r,θand φdirections.
For the δf model, the total distribution function is decomposed to the background and
perturbed parts, f(z, t) = f0(z, t)+δf(z, t). The background part can be chosen as the time-
independent one, i.e., f0(z, t) = f0(z), and one typical choice is the Maxwellian distribution.
The background and perturbed distribution functions are represented by the markers as
follows,
f0(z, t) = P(z, t)g(z, t)≈
N
X
p=1
pp(t)δ[zp−zp(t)]
Jz
,(9)
δf(z, t) = W(z, t)g(z, t)≈
N
X
p=1
wp(t)δ[zp−zp(t)]
Jz
,(10)
where pp(t) = f0(zp, t)/g(zp, t) and wp(t) = δf(zp, t)/g(zp, t) are time-varying variables. The
4
Full fand δf GK particle simulations
evolution equations are readily obtained,12
d
dtwp(t) = −pi(t)d
dtln f0(zp(t)) ,(11)
d
dtpp(t) = pi(t)d
dtln f0(zp(t)) ,(12)
d
dt=∂
∂t +˙
R· ∇ + ˙vk
∂
∂vk
,(13)
where ˙µ= 0 is used in the last equation. Generally, the guiding center’s equation of motion
can be decomposed to the equilibrium part corresponding to that in the equilibrium magnetic
field, and the perturbed part due to the perturbed field
˙
R=˙
R0+δ˙
R,(14)
˙vk= ˙vk,0+δ˙vk.(15)
For the equilibrium distribution function,
d
dt0
f0=∂
∂t +˙
R0· ∇ + ˙vk,0
∂
∂vkf0= 0 ,(16)
and thus
d
dtf0=δ˙
R· ∇ +δ˙vk
∂
∂vkf0,(17)
where f0is chosen as a steady state solution (∂f0/∂t = 0). In this work, the Maxwell
distribution is chosen (f0=fM),
fM=n0
(2T/m)3π3/2exp (−mv2
k
2T−mµB
T),(18)
and thus
d
dtln fM=δ˙
R·"~κn+ mv2
k
2T+mµB
T−3
2!~κT−mµB
T~κB#−δ˙vk
mvk
T,
(19)
where ~κn,T,B ≡ ∇ln{n, T, B}. Note for the Maxwell distribution, without considering
the neoclassical physics, the following approximation has been made in the traditional δf
scheme,
d
dt0
fM≈0.(20)
5
摘要：

FullfandfGKparticlesimulationsFullfandfgyrokineticparticlesimulationsofAlfvenwavesandenergeticparticlephysicsZhixinLu,GuoMeng,RomanHatzky,MatthiasHoelzl,PhilippLauberInthiswork,wefocusonthedevelopmentoftheparticle-in-cellschemeandtheap-plicationtothestudiesofAlfvenwavesandenergeticparticlephysic...

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