To determine the electron heat flux density on macroscopic scales, the most widely used approach is to solve a diffusion equation through a multi-group technique. This method is, however, restricted to transport induced by temperature gradients without accounting for other sources of fast electrons because the electric field induced by the charge separation is indirectly treated. In addition, significant discrepancies are reported in the underlying distribution function when compared to complete kinetic calculations. These limitations motivate the research for alternative reduced kinetic models. The physical content of one of them is here deepened, its precision is improved, and the benefit of its usage compared to other models is discussed.
References
1.
V.
Tikhonchuk
, “Progress and opportunities for inertial fusion energy in Europe
,” Philos. Trans. R. Soc., A
378
, 20200013
(2020
).2.
W.
Manheimer
and D.
Colombant
, “Analytic insights into nonlocal energy transport. III. Steady state Fokker Planck theory in spherical and planar geometry
,” Phys. Plasmas
28
, 012701
(2021
).3.
4.
A.
Brantov
and V.
Bychenkov
, “Nonlocal transport in hot plasma. Part I
,” Plasma Phys. Rep.
39
, 698
(2013
).5.
A.
Brantov
and V.
Bychenkov
, “Nonlocal transport in hot plasma. Part II
,” Plasma Phys. Rep.
40
, 505
(2014
).6.
R.
Balescu
and A.
Kuszell
, “Kinetic equation for an inhomogeneous plasma far from equilibrium
,” J. Math. Phys.
5
, 1140
(1964
).7.
V.
Senecha
, A.
Brantov
, V.
Bychenkov
, and V.
Tikhonchuk
, “Temperature relaxation in hot spots in a laser-produced plasma
,” Phys. Rev. E
57
, 978
(1998
).8.
J.
Albritton
, E.
Williams
, I.
Bernstein
, and K.
Swartz
, “Nonlocal electron heat transport by not quite Maxwell-Boltzmann distributions
,” Phys. Rev. Lett.
57
, 1887
(1986
).9.
D.
Del Sorbo
, J.-L.
Feugeas
, P.
Nicolaï
, M.
Olazabal-Loumé
, B.
Dubroca
, S.
Guisset
, M.
Touati
, and V.
Tikhonchuk
, “Reduced entropic model for studies of multidimensional nonlocal transport in high-energy-density plasmas
,” Phys. Plasmas
22
, 082706
(2015
).10.
M.
Holec
, P.
Loiseau
, A.
Debayle
, J.
Brodrick
, D.
Del Sorbo
, C.
Ridgers
, V.
Tikhonchuk
, J.-L.
Feugeas
, P.
Nicolai
, B.
Dubroca
, and R. J.
Kingham
, “AWBS kinetic modeling of electrons with nonlocal Ohms law in plasmas relevant to inertial confinement fusion
,” arXiv:1901.11378 (2018
).11.
A.
Chrisment
, A.
Debayle
, J.-L.
Feugeas
, P.
Loiseau
, P.-E.
Masson-Laborde
, J.
Mathiaud
, P.
Nicola
, and V.
Tikhonchuk
, “A deterministic numerical scheme for an electron heat transport model
,” (to be published
).12.
J.-F.
Luciani
, P.
Mora
, and R.
Pellat
, “Quasistatic heat front and delocalized heat flux
,” Phys. Fluids
28
, 835
(1985
).13.
J.-F.
Luciani
and P.
Mora
, “Resummation methods of the Chapman-Enskog expansion for a strongly inhomogeneous plasma
,” J. Stat. Phys.
43
, 281
(1986
).14.
L.
Landau
, “Die kinetische gleiehung fur den fall coulombscher wecgselwirkung
,” Phys. Z. Sowjetunion
10
, 154
(1936);See also
Collected papers of L. D. Landau
, edited by D.
Ter Haar
(Pergamon Press
, Oxford
, 1936
), p. 163
.15.
J.-F.
Luciani
, P.
Mora
, and J.
Virmont
, “Nonlocal heat transport due to steep temperature gradients
,” Phys. Rev. Lett.
51
, 1664
(1983
).16.
P.
Mora
and J.-F.
Luciani
, “Nonlocal electron transport in laser created plasmas
,” Laser Part. Beams
12
, 387
(1994
).17.
L.
Spitzer
, Jr. and R.
Härm
, “Transport phenomena in a completely ionized gas
,” Phys. Rev.
89
, 977
(1953
).18.
E.
Epperlein
and R.
Short
, “A practical nonlocal model for electron heat transport in laser plasmas
,” Phys. Fluids B
3
, 3092
(1991
).19.
A.
Bendib
, J.-F.
Luciani
, and J.-P.
Matte
, “An improvement of the nonlocal heat flux formula
,” Phys. Fluids
31
, 711
(1988
).20.
C.
Lu
, V.
Tikhonchuk
, and S.
Weber
, “Analytic solutions for delocalized heat transport
,” Plasma Phys. Controlled Fusion
63
, 075005
(2021
).21.
G.
Schurtz
, P.
Nicolaï
, and M.
Busquet
, “A nonlocal electron conduction model for multidimensional radiation hydrodynamics codes
,” Phys. Plasmas
7
, 4238
(2000
).22.
M.
Sherlock
, J.
Brodrick
, and C.
Ridgers
, “A comparison of non-local electron transport models for laser-plasmas relevant to inertial confinement fusion
,” Phys. Plasmas
24
, 082706
(2017
).23.
J.
Brodrick
, R. J.
Kingham
, M.
Marinak
, M. V.
Patel
, A.
Chankin
, J.
Omotani
, M.
Umansky
, D.
Del Sorbo
, B.
Dudson
, J.
Parker
, G. D.
Kerbel
, M.
Sherlock
, and C. P.
Ridgers
, “Testing nonlocal models of electron thermal conduction for magnetic and inertial confinement fusion applications
,” Phys. Plasmas
24
, 092309
(2017
).24.
G. N.
Minerbo
, “Maximum entropy Eddington factors
,” J. Quant. Spectrosc. Radiat. Transfer
20
, 541
(1978
).25.
B.
Dubroca
, J.-L.
Feugeas
, and M.
Frank
, “Angular moment model for the Fokker-Planck equation
,” Eur. Phys. J. D
60
, 301
(2010
).26.
W.
Rozmus
, T.
Chapman
, A.
Brantov
, B.
Winjum
, R.
Berger
, S.
Brunner
, V.
Bychenkov
, A.
Tableman
, M.
Tzoufras
, and S.
Glenzer
, “Resonance between heat-carrying electrons and Langmuir waves in inertial confinement fusion plasmas
,” Phys. Plasmas
23
, 012707
(2016
).27.
P.
Bhatnagar
, E.
Gross
, and M.
Krook
, “A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems
,” Phys. Rev.
94
, 511
(1954
).28.
W.
Manheimer
, D.
Colombant
, and V.
Goncharov
, “The development of a Krook model for nonlocal transport in laser produced plasmas. I. Basic theory
,” Phys. Plasmas
15
, 083103
(2008
).29.
L.
Landau
and E.
Lifshitz
, Statistical Physics—Part 1
(Butterworth-Heinemann/Elsevier Ltd
, 1981
).30.
V.
Arnold
, Lectures on Partial Differential Equations
(Springer
, 2004
).31.
E.
Epperlein
, “Effect of electron collisions on ion-acoustic waves and heat flow
,” Phys. Plasmas
1
, 109
(1994
).32.
33.
A.
Gurevich
and Y.
Istomin
, “Thermal runaway and convective heat transport by fast electrons in a plasma
,” Sov. Phys.-JETP
50
, 470
(1979
).34.
S.
Krasheninnikov
, “On nonlocal electron heat conduction
,” Phys. Fluids B
5
, 74
(1993
).35.
A.
Decoster
, P. A.
Markowich
, and B.
Perthame
, Modeling of Collisions
(Elsevier Masson
, 1998
), Vol. 2
.36.
J.
Nikl
, I.
Göthel
, M.
Kuchařík
, S.
Weber
, and M.
Bussmann
, “Implicit reduced Vlasov–Fokker–Planck–Maxwell model based on high-order mixed elements
,” J. Comput. Phys.
434
, 110214
(2021
).37.
Aladin is a Vlasov–Fokker–Planck code developed at cea by D. Deck and K.-C. Le Thanh based on Ref. 40, private communication (November 16, 2021).
38.
M.
Tzoufras
, A.
Tableman
, F.
Tsung
, W.
Mori
, and A.
Bell
, “A multi-dimensional Vlasov-Fokker-Planck code for arbitrarily anisotropic high-energy-density plasmas
,” Phys. Plasmas
20
, 056303
(2013
).39.
A.
Joglekar
, B.
Winjum
, A.
Tableman
, H.
Wen
, M.
Tzoufras
, and W.
Mori
, “Validation of OSHUN against collisionless and collisional plasma physics
,” Plasma Phys. Controlled Fusion
60
, 064010
(2018
).40.
R.
Kingham
and A.
Bell
, “An implicit Vlasov-Fokker-Planck code to model non-local electron transport in 2D with magnetic fields
,” J. Comput. Phys.
194
(1
), 1–34
(2004
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
Author(s)
You do not currently have access to this content.
