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SUMMARY:Coupling plasma and neutral kinetic models: Considerations and sol
utions
DTSTART;VALUE=DATE-TIME:20210512T162500Z
DTEND;VALUE=DATE-TIME:20210512T164500Z
DTSTAMP;VALUE=DATE-TIME:20211028T112050Z
UID:indico-contribution-17381@conferences.iaea.org
DESCRIPTION:Speakers: George Wilkie (Princeton Plasma Physics Laboratory)\
nDivertor detachment is a scenario characterized by the dominance of neutr
al interactions to mitigate the extreme plasma heat flux that would otherw
ise be incident upon solid walls of fusion reactors. Despite the critical
role theory will play in predicting divertor performance\, rigorous modell
ing of neutrals is plagued by the difficulty of directly solving the nonli
near Boltzmann equation. While several assumptions are appropriate in some
contexts to alleviate this difficulty\, these break down in the detached
regime. Here we present new capabilities in the DEGAS2 neutral transport s
olver [1] that include a rigorous treatment of nonlinear collision operato
rs. This is a unique capability among comprehensive kinetic simulation mod
els and will be an important component in predicting the performance of ad
vanced divertor scenarios in ITER and future magnetically-confined fusion
reactors.\n\nNeutral atoms and molecules\, formed by recombination either
at the solid wall or within the scrape-off layer plasma\, play an importan
t role in advanced divertor concepts. Their interaction with the plasma is
purely through reactions\, charge exchange\, elastic and inelastic scatte
ring. Because of their relatively long mean free path\, neutrals are highl
y non-Maxwellian\, as is the plasma they interact with in the scrape-off l
ayer. Therefore\, to rigorously simulate the dynamics of plasma and neutra
ls in the divertor region\, kinetic models are needed. XGC is a suite of e
dge gyrokinetic solvers [2]\, with early versions fully coupled to DEGAS2.
This identified important issues that need to be addressed for high-fidel
ity simulations of neutral-dominant regimes.\n\nCooling the plasma with a
radiative divertor results in more neutrals generated from plasma recombin
ation. In addition\, the gas can become optically dense\, where the finite
mean free path of photons needs to be accounted for. A cooler plasma furt
her leads to longer lifetimes for hydrogenic molecules. Elastic scattering
between neutrals\, as well as excitation of internal energy states also b
ecome important in these regimes. Beyond this additional complexity\, an i
ncreased neutral density presents several fundamental computational challe
nges compared to the low-recycling regime. Firstly\, since collisions betw
een neutrals become more important\, this necessitates the fully nonlinear
Boltzmann collision operator. In DEGAS2\, neutral-neutral interactions ar
e currently modelled with an iterative BGK operator\, but this is only an
approximation and can be improved. \n\nAnother complication is the lack of
conservation of energy and momentum in charge-exchange collisions when th
e non-Maxwellian nature of ions and neutrals is not respected. Although bo
th XGC and DEGAS2 are fully kinetic solvers\, the field particle species i
n collisions have historically been modelled as equivalent Maxwellians. Pr
evious work identified this approximation as the cause of the lack of ener
gy conservation when the physical charge exchange cross section is used [3
]. Conservation can be recovered if a constant collision kernel is used in
stead [4]. If this approximation is to be relaxed for increased accuracy\,
more information about the neutral and ion distributions need to be accou
nted for in calculating the respective collision operators charge exchange
\, among other interactions.\n\nOne approach is to use DEGAS2 to directly
calculate Monte Carlo estimates of the ion collision operator against neut
rals. The charge-exchange collision operator is an integral operator\, so
for each spatial grid point\, this requires $N_v^2$ Monte Carlo estimates\
, where $N_v$ is the number of velocity space grid points used for collisi
ons in the XGC-1 (typically $N_v \\sim 500$). This is prohibitively expens
ive and is indicative of the challenge in numerically solving even the lin
ear Boltzmann collision operator.\n\nIn recent years\, applied mathematici
ans have risen to this challenge and several methods have been developed t
o efficiently solve the Boltzmann equation. This work builds on one such m
ethod: the conservative spectral scheme of Gamba & Rjasanow [5]. The distr
ibution function is expanded in an orthonormal Burnett basis:\n\n$$\n f\
\left(v\, \\theta\, \\phi\\right) \\approx \\sum\\limits_{i=1}^{N} f_{i}
A_{k_i l_i} \\left(\\frac{v}{v_\\mathrm{ref}} \\right)^{l_i} e^{-v^2/2 v_\
\mathrm{ref}^2} L_{k_i}^{l_i+1/2}\\left(\\frac{v^2}{v_\\mathrm{ref}^2} \\r
ight) Y_{l_im_i} \\left(\\theta\, \\phi \\right)\,\n$$\n\nwhere $i$ is a c
ompound index encoding the triplet $(k_i\, l_i\, m_i)$\, $L_k^{l+1/2}$ are
generalized Laguerre polynomials\, $Y_{lm}$ are spherical harmonics\, $A_
{kl}$ are normalization factors\, and $v_\\mathrm{ref}$ is a characteristi
c speed scale of the basis. The weak form of the Boltzmann equation is sol
ved with test functions chosen to *manifestly* conserve collisional invari
ants. The discrete collision operator is a triply-indexed object of size $
N^3$ where each element is an 8-dimensional integral over two velocities a
nd two scattering angles. These integrals are precomputed with a combinati
on of generalized Gaussian quadrature [6] in energy and Lebedev quadrature
in solid angle\, and these are stored in an online database for efficient
application. A prototype framework for solving the Boltzmann equation wit
h this method has been developed [7].\n\nIn the context of neutrals in div
ertors\, the spectral scheme will be added to supplement the Monte Carlo s
cattering operators in DEGAS2. This is most usefully viewed as a *moment m
ethod*\, wherein the collisional response of several moments (the coeffici
ents $f_{i}$) are robustly calculated and evolved. For neutrals\, these mo
ments are calculated with Monte Carlo estimators in DEGAS2\, the spectral
collision operator finds their respective time derivatives\, and a project
ion onto the velocity space grid provides a conservative collision operato
r for use in the total-f framework of XGC-1 [8]. It is found that Monte Ca
rlo estimates of higher-order moments becomes difficult beyond $N \\sim 27
$ due to sampling noise. With such a few number of moments\, the spectral
scheme is remarkably efficient: requiring only about 0.1 ms per timestep o
f the nonlinear Boltzmann operator. As such\, rigorous nonlinear neutral-n
eutral scattering can be included in the XGC1-DEGAS2 coupling with negligi
ble computational overhead. \n\n[1] D. P. Stotler and C. Karney. "Neutral
Gas Transport Modeling with DEGAS 2". *Contrib. Plasma. Phys.*\, **34**:39
2 (1994)\n[2] C. S. Chang and S. Ku. "Spontaneous rotation sources in a qu
iescent tokamak edge plasma". *Physics of Plasmas*\, **15**:062510 (2008)\
n[3] D. P. Stotler et al. "Energy conservation tests of a coupled kinetic
plasmaâ€“kinetic neutral transport code". *Comput. Sci. Disc.*\, **6**:015
006 (2013)\n[4] R. M. Churchill et al. "Kinetic simulations of scrape-off
layer physics in the DIII-D tokamak". *Nuc. Mat. and Energy*\, **12**:978
(2017)\n[5] I. M. Gamba and S. Rjasanow. "Galerkinâ€“Petrov approach for t
he Boltzmann equation". *Journal of Computational Physics*\, **366**:341 (
2018)\n[6] G. J. Wilkie. *Microturbulent transport of non-Maxwellian alpha
particles*. PhD thesis\, University of Maryland (2015)\n[7] G. J. Wilkie.
"Lightningboltz: A distributed framework for efficient solution of the bo
ltzmann equation". In preparation.\n[8] S. Ku et al. "A new hybrid-Lagrang
ian numerical scheme for gyrokinetic simulation of tokamak edge plasma". *
Journal of Computational Physics*\, **315**:467 (2016)\n\nhttps://conferen
ces.iaea.org/event/214/contributions/17381/
LOCATION:Virtual Event
URL:https://conferences.iaea.org/event/214/contributions/17381/
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