General Neoclassical Closure Theory: Diagonalizing the Drift Kinetic Operator

General Neoclassical Closure Theory: Diagonalizing the Drift Kinetic Operator E. D. Held eheld@cc.usu.edu Utah State University General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.1/9

Introduction Goal of a general neoclassical closure theory: Capture collisional, free-streaming, trapping and drift-orbit physics in closure relations for long time scale, electromagnetic fluid simulations of magnetized plasmas. Allow for 1. complicated geometry, 2. varying collisionality, and 3. numerical tractability. General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.2/9

$ "# % /.- - # Solve drift kinetic equation (DKE) in Invert DKE operator Closures of interest for 5-moment fluid model are! () '& Difficulties:, +. * 1. 6-dimensional configuration space,. ( 7 6 5 5 34 21 2. disparate time scales, General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.3/9

] S Q M : G @ G G L < \ D K R F P G S Q b d L @ \ K R S ; g S @ A d DKE operator is complicated Separating out parallel speed dependence, NO HJI CED @Y> N T NZ2[ G X UWV NT S ; B*@ 9*@ A =?> ; 9: 8 X R Ǹ ;: T \ H_I 9 @ ; B*@ 9*@ A =?^ HaI @Y> :NT N Zc[ G X UWV NT S ; B*@ 9*@A = S X R ;f 9: e. b 9 ; X > B X 9 d b < S B*@ where General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.4/9

j u o u u y l ˆ s x t u ~ Ž y o ˆ x s Ž s s o o k o p Ž DKE operator is complicated Separating out parallel speed dependence, { } z vjw res o n { { 2 u Wƒ { ~ k q*o i*o p m?n k ij h k kj { 2Œ u {`2 i ˆ v_w i o k q*o i*o p m?š vaw o n j{ { c u Wƒ { k q*o i*op m j s n op op. i k n q i Ž l q*o where General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.4/9

³ ª ² ž ª ² ª º ¹ º º º ª º ª š ÆÅ ª º º É ¹ É º DKE operator is complicated Separating out parallel speed dependence, J œe «2± W «ª * * š? «2 `«2µ ² _ * * š? «c± W «ª a * * š ª š š» º. ª * where Low-beta, bounce-averaged form is Å * Ä ÀÂÇ * š ª ÀÂÄ ¾ ½ ¼ ÀÂÁ Ã Ë Êš È General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.4/9

Basis function expansion aides diagonalization Ì ÐÏ Ï Ò Î ÐÏ ÑÎ Ó Ø Ý Ó Ú Î ØÙ Ð Ý Þ Ü ëê Ü é Use expansion in spherical harmonics as guide ìîí ç Ü è Ø æ â ãåä Û Ü Þàß á ÓÔÖÕ Í Î General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.5/9

Basis function expansion aides diagonalization ï óò ò õ ñ óò ôñ ö ü ñ úû ù ó ð þ þ Use expansion in spherical harmonics as guide þ ú ö ú ÿ ÿ ý þ ö Öø ð ñ Expand in solutions of separated bounce-averaged eigenvalue equation: ñ ñ ñ ÿ ÿ General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.5/9

Basis function expansion aides diagonalization! #" " %! #" $! L ' B 1 ' L & + & -! +, * #! +E R F 1 + 1 & -! # > > K / <; / : Use expansion in spherical harmonics as guide => 8 / 9 + 7 4 5 6. / 1 2 3 &')( Expand in solutions of separated bounce-averaged eigenvalue equation:! JE HI G!?E!?' D C @A??' Laguerre polynomials serve as energy eigenfunctions:. / +, &' &OQP ( #! M 1S2 3 #!NM General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.5/9

X Y y w g g w Y u r Z ts Œ Perturbative expansion necessary for diagonalization : and integrate over W TVU Multiply through by bv xzy bv loq p imlon h ikj g h fg èd {} ~ c b `a _ W T U ^ \] X [ ˆ ƒ i š i w w Ž Ž œ Š Œ ož electrons frequency i š i w Ž Ž œ w Š ož ions General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.6/9

Æ Ä Ä Â ÁÀ Ò Æ Ä Ä Ê æ á ß í éè ì éè ë éè ê éè æ á ß Perturbative expansion necessary for diagonalization : and integrate over V Multiply through by ³ÃË Ê ÅzÆ ³Ã»o¾ ½ ¹m»o¼ ¹kº ±eµ Ç}ÈÉ ³ ±² ª ª «ÎÑ ÐÏ ÌÍ Î Diagonalizable systems include ³Ã ŠƻӾ ½ ¹m» ¼ ¹ º ÇÔÈÉ ± µ ³ ±² and ¹ä ã ¹ Ù Ü Ù ÄâÛ Ù ÜÞÝ Ù ÄÚÛ Ø Ø å ß à ß ÕÖ æoç electrons frequency ¹ä ã ¹ Ù Ü Ù ÄâÛ Ø Ø å ß à î ß Ù Ü Ý Ù ÄÚÛ ÕÖî æoç ions General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.6/9

ò ó ÿ ó ô! ÿ ÿ > :9 = :9 < :9 ; :9 Perturbative expansion necessary for diagonalization : and integrate over ñ ïvð Multiply through by ü ü úeþ ý ü úû ù ñ ï ð ø ö ò õ Diagonalizable systems include ü # #$ " ú þ ý ü úû ù and ü " ú þ ý ü úû ù 7 7 8 6 5 4 *.- * 3, 2 1 ' *.-#/ * +, ) ) %&(' electrons frequency 7 7 8 6 1? * - / * - * +, 2 5 4 * 3, ) ) %&? ions General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.6/9

Balance electron free-streaming with e-e and e-i collisions C AB @ Electrons have X X SUT NW V SUT O.R O PQ C(N E M L C B GIHKJ DFE General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.7/9

Balance electron free-streaming with e-e and e-i collisions \ Z[ Y \ o n b \ [ q np Electrons have m m iuj dl k iuj e.h e fg \(d ^ c b \ [ _I`Ka ]F^ Expand in eigenvectors of, _I`Ka, and multiply through by : \ w rx b \ i r tvu s r ] ^ \ i r m d ^Y General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.7/9

Balance electron free-streaming with e-e and e-i collisions š y œ ~ Š ~ y { ~ Ž Œ ~ ž Œ Œ ž Electrons have ŠU. ( ŠU Œ ~ ˆ ƒ { I K K} z{ Expand in eigenvectors of, I K, and multiply through by : Š v } Solution to all orders has form ž Ž œ Œ General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.7/9

² Å Ã Balance ion drifting with time dependence. Ions have Å ÁU ÁU ¼Ä ½.À ÁUÂà ²(¼½ ³¾» ²ƒº $ ¹ µi F³ ± ª ª «General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.8/9

Ó æ à ä Î ë Ó é è Ò ÑÐ Ê Ô í Î Balance ion drifting with time dependence. Ions have æ âuã È É âuã Ýå Þ.á âuãä Ó(ÝÞ Ôß Ü ÓƒÛ Ò Ø$ÙÚ ÕÖI ÊFÔ ÐÑ±Ò Ì Ï ËÈ Ì Í Ê Æ Ç È É :, and multiply through by ÓçÛ Ö èê Ø ÙÚ, Expand in eigenvectors of Ó ñ íò Ó Û â í ð ÏÙ È Ì Õ(ï î Ê Ó â í Ì Ï ËÈ Ì Í ÇìÈ É General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.8/9

ÿ û ÿ þý # û * " û Balance ion drifting with time dependence. Ions have õ ö ýþ±ÿ ù ü øõ ù ú ó ô õ ö :, and multiply through by, Expand in eigenvectors of "! õ ù ü ù ü øõ ù ú ôìõ ö ' $ & %$ Expansion in Laguerre polynomials and substitution using diagonalizes system: ýþ,+ * # ( * ( õ ù ü )( ù ü øõ ù ú ô õ ö General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.8/9

? U N R 9 Z? X W = <; 3 @ \ 9 :F f ] 3 3 9 B @ K U U W u u f f BR qf R \ R @ :F f ] 3 q 9 Balance ion drifting with time dependence. Ions have U PQ / 1 PQ KT LO PQSR?KL @M J?IH = E FG BC D 3A@ ;<>= 58: 4/6587 3 -./21 :, and multiply through by?vh C D WY E FG, Expand in eigenvectors of? ` \a? H P \ _ :F / 5 B^ ] 3? P \ 58: 4/65 7.[/21? d b c? H X%b Expansion in Laguerre polynomials and substitution using diagonalizes system: f? ` \ a e Hf? \ e ;<,g / 5 \)e 58: 4/65 7./21 Solution to all orders has form { jz Uy KxT j s \ e ik d q U Kwv c U Kt R s \ e nporq kl m i h H \)e } 1 \ j e. y ~ Z H ;UY / 5 \ e 58: 4/ 587 K ~ Hf \)e where General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.8/9

Conclusions ƒ ƒ ƒ Ž From calculate closures of interest Œ ˆŠ ƒ ˆ Diagonalizing via expansion in basis functions permits 1. integration along essential characteristics of orbits) in complicated magnetic fields, (drift or free-streaming 2. truncation of integration helped by collisional effects, 3. reduction of configuration space variables hence numerical tractability, and, 4. simple prescription for higher-order corrections. Closures include collisional, free-streaming, trapping and drift-orbit physics for ions and electrons. General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator p.9/9