Energy stability analysis for a hybrid fluid-kinetic plasma model

Texte intégral

(1)Energy stability analysis for a hybrid fluid-kinetic plasma model Philip Morrison, Emanuele Tassi, Cesare Tronci. To cite this version: Philip Morrison, Emanuele Tassi, Cesare Tronci. Energy stability analysis for a hybrid fluid-kinetic plasma model. Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations, Nov 2012, Banff, Canada. �hal-01053857�. HAL Id: hal-01053857 https://hal.archives-ouvertes.fr/hal-01053857 Submitted on 2 Aug 2014. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(38) = 86$B606- H.$ 403H8606 9,=G! M$ B$4B2 H.H 0H 6HKB33O $N:B$DD$D H.$ H8H3 $6$B-O 8( H.$ DODH$4! 86D0DH06- 8( H.$ DK4 8( H.$ K32 206$H0 $6$B-O! H.$ 4-6$H0 $6$B-O 6" H.$ 206$H0 $6$B-O 8( H.$ .8H :BH03$ :8:K3H086! 8BB$D:86"06- H8 H.$ H.B$$ H$B4D ::$B06- 06 9,=G! B$D:$H0L$3O= 0H. B$-B" H8 H.$ 80DD86 B2$H! 86 H.$ 8H.$B .6"! M$ 8D$BL$ H.H 0H :8DD$DD$D :KB$ :BH! 86D0DH06- 8( H.$ )BDH HM8 H$B4D 8( 9,=F! M.0. 8BB$D:86" H8 H.$ 80DD86 B2$H 8( B$"K$" &,! &,= H 3D8 :8DD$DD$D :KB$3O 206$H0 :BH! -0L$6 O 0HD 3DH HM8 H$B4D! M.0. 063K"$ H.$ 3D8L B2$H &Q! &J! &J= .$ B$40606- H$B4D! 86 H.$ 8H.$B .6"! B$ H.8D$ B$D:86D03$ (8B H.$ 8K:306- $HM$$6 H.$ 6" H.$ 206$H0 84:86$6HD= !#)#)# D "0DKDD$" 06 D$H086 9,=J! H.$ $6$B-O/D040B 4$H.8" B$>K0B$D H.$ 0"$6H0)H086 8( H.$ D040B 06LB06HD ! 0=$= (K6H0863D DH0D(O06- Q (8B 6O B0HBBO (K6H0863 06 H.$ 3-$B 8( 8D$BL3$D= 8M$L$B! )6"06- H.$ D040BD 0D 68H 3MOD 6 $DO HD2 6" H.$0B 3040H$" L03030HO D84$H04$D DH6"D D H.$ 418B 8DH3$ H8 H.$ ::30H086 8( H.$ 4$H.8"= 8M$L$B! 06 H.$ D$ K6"$B 86D0"$BH086! H.$ $N0DH$6$ 8( B8DD/.$300HO 06LB06H (8B H.$ 403H8606 MD D.8M6 06

(39) 9J 6" H.0D D040B )6"D 0HD MO 06H8 H.$ :B$D$6H HM8/"04$6D0863 H.$8BO! M.$B$ 0H 0D -$6$B30P$"= .$ $N0DH$6$ 8( DK. D040B (8B H.$ H.B$$/"04$6D0863 80DD86 B2$H O0$3"D M.83$ (403O 8( D040B 06LB06HD M.$6 :B81$H$" 86 H.$ :36$= :86 DK4406- 86HB0KH086D B0D06- (B84 H.$ 4-6$H0 .$300HO 6" H.$ 3D8L "O640D! H.$ H8H3 D040B 06LB06H (8B H.$ :36B .OB0" 48"$3 B$"D . !. & ! "J 4 "I 2 . 9,=&. M.$B$ ! 6" B$ B0HBBO (K6H086D! 6" M$ .L$ 06HB8"K$" H.$ D.8BH.6". . 6 0 4 . M0H.. . !. "I 2 ,5 . .H 9,=& 0D 06 (H D040B 06LB06H 0D D.8M6 06 ::$6"0N ;D$H086 9,=&<= .$ $N0DH$6$ 8( DK. D040B 06LB06HD 0D 4$63$ H8 :.OD03 06H$B:B$HH086! M.0. $84$D 3$B$B M.$6 86D0"$B06- D$:BH$3O! H.$ 86HB0KH086D 8406- (B84.

(40) I9&. 86306$B .OD03 ODH$4D. H.$ (K6H086D ! 6" = ( " " Q! H.$ B$40606- D040B $N:B$DD$D H.$ 86D$BLH086 8( 4-6$H0 +KN H.B8K-. DKB($ 48L06- M0H. H.$ K32 +K0" L$380HO= .0D :B8:$BHO 0D H.$ (B8P$6/06 86"0H086 (8B H.$ 4-6$H0 +KN! 06.$B0H$" (B84 0"$3 = ( " " Q! M$ B$HB0$L$ H.$ 86D$BLH086 8( H.$ 06H$-B3 8( 6O (K6H086 8( ! M.0. 0D .BH$B0DH0 8( 3D8L DODH$4D 6" M.8D$ :.OD03 4$606- 06 H$B4D 8( :BH03$ B$BB6-$4$6HD MD -0L$6 06 &F= 8M$L$B! 6$M (403O 8( 06LB06HD DD80H$" M0H. H.0D HM8/"04$6D0863 .OB0" 48"$3 ::$BD M.$6 D$HH06- " " Q= 8B H.0D D$! H.$ D040B (403O B$"K$D H8 . !. & ! "J 4 . 6 0 4 "I 2 ,5 . !. . 4 "J 4 7 . !. & ! "J 4 7 5 4 5 "I 2 ,5 4 6 0 9,=7. ". M.$B$ M$ B$33 H.H "I 2 ,5 ! M.0. 8BB$D:86"D H8 H.$ 484$6HK4 8( H.$ .8H :BH03$ D:$0$D 06 H.$ 45/:36$= >KH086 9,=7 06HB8"K$D .OB0" B8DD/.$300HO . 4 ! $N:B$DD06- H.$ 8BB$3H086 $HM$$6 H.$ 4-6$H0 )$3" 6" "$6D0HO L$380HO )$3" 8H06$" O DKHBH06- (B84 H.$ K32 L$380HO 86HB0KH086 8406(B84 H.$ 206$H0 D:$0$D= :86 D$HH06- " Q! H.0D D040B B$"K$D H8 H.$ B8DD .$300HO (403O 8( 06LB06HD (8B HM8/"04$6D0863 M.0. H8 8KB 268M3$"-$ MD )BDH (8K6" 06 &,= 8B -0L$6 86DH6H Q ! .88D06- Q ! M0H. 06"0H06- H.$ $L0D0"$ (K6H086 M0H. DH$: 38H$" H Q ! $N:B$DD$D H.$ :B8:$BHO H.H 68H 863O 0D H.$ H8H3 -$6$B30P$" B8DD/.$300HO 0D 86D$BL$"! KH 3D8 0HD 06H$-B3 8L$B "8406D Q ! M.0. B$ 8K6"$" O 4-6$H0 +KN DKB($D= 0L$6 H.H H.$D$ (4030$D 8( D040BD .L$ $$6 0"$6H0)$" $N:300H3O! M$ B$ B$"O H8 ::3O H.$ $6$B-O/D040B 4$H.8" 6" )6" DK()0$6H 86"0H086D (8B $6$B-O DH030HO 8( H.$ DODH$4= .0D 0D BB0$" 8KH 06 H.$ 6$NH D$H086= !## $/

(41) %$ %(

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(43)

(44) /%% 68M3$"-$ 8( H.$ D040B 06LB06HD :B8L0"$D LB0H0863 :B060:3$ (8B $>K030B0! Q! M.0. M$ H$6" H8 06 D$H086 9,=,=9= .$6! H.$ 6$NH DH$: 8( H.$ $6$B-O/D040B 4$H.8" 0D H8 86D0"$B H.$ D$86" LB0H086! J Q! M.0. 0D "86$ 06 D$H086 9,=,=J= 8H$ H.H 33 :.OD03 86DH6HD .L$ $$6 D$H H8 K60HO= 48B$ :$BD:0K8KD DHK"O 8( H.$ "O6403 $.L08B 6 $ 8H06$" K:86 B$DH8B06- H.$D$ 86DH6HD=.

(45) 6$B-O H030HO 63OD0D. I97. !##!# . 8 86DHBKH LB0H0863 :B060:3$ (8B H.$ $>K030B0 K6"$B 86D0"$BH086! M$ )BDH "$)6$ H.$ (B$$ $6$B-O (K6H0863 = .$6! M$ H2$ 0HD )BDH LB0H086! M.0. B$"D. . . !!. "J 4"I 2 , 4 6 0 . % , " M.$B$ M$ D:30H/8(( H.$ 3D8L :BH "J 4"I 2 7 ,J

(46) J 6" H.$ :BH. . !. ! , % % , "J 4 . .9 "J 4 7 . 7 . :86 D$HH06- Q! H.$ $>K030B0K4 $>KH086D HKB6 8KH H8 $ Q. . Q. Q. 9,=9Q. . 7 7 7 , 4 6 0 . 9,=99. J. , 7 J. 9,=9J. M.8D$ )BDH B$3H086 B$6"$BD H.$ H.0B" 06 H.$ (8B4 9 9 J . J 7 Q , J J. 9,=9I. 6 H.$ 8L$ $N:B$DD086D! M$ 06HB8"K$" H.$ DKDB0:H H8 06"0H$ $>K030B0K4 >K6H0H0$D= :86 DDK406- 6 06L$BH03$ ! (B84 9,=9I! M$ 8H06 H.$ $>K030B0K4 "0DHB0KH086 (K6H086 06 H.$ (8B4 & & 9 9 9 9 J . J , , 6 0 J . 5 J 9,=9, J J J J D 6 $N4:3$! 86D0"$B H.$ B$3H0L$ KDD06 "0DHB0KH086 & 9 9 J J $N: . , J J.

(47) IJQ. 86306$B .OD03 ODH$4D. H 0D $DO H8 D$$ H.H H.0D O0$3"D !. J. 9. "I 2 , .. J . . 5 4. * * ## J 9 9 . J 5 5 . 6" H.$B$(8B$! 06 H.$ D$6$ " 8( $>K030B0K4 +8M! M$ .L$ Q! M.0. 4$6D H.H H.$ L8BH00HO 6 0 4 I 2 ,5 DD80H$" M0H. H.$ .8H :BH03$ +8M 0D 3D8 P$B8= 6 H.$ -$6$B3 D$ 8( 6 B0HBBO $>K030B0K4 8( H.$ HO:$ 9,=9,! H.0D >K6H0HO 0D 84:KH$" D ! ! 9 9 9 J 2J6 4 6 0 ,5 J . 6 0 4 "I 2 ,5 I 2 7 ,5 J J J ". . 5 4 & 5 . M.$B$ & I 2 0D H.$ .8H :BH03$ $>K030B0K4 "$6D0HO= 6 863KD086! H.$ )63 (8B4 8( H.$ .OB0" $>K030B0K4 B$3H086 B$"D . 7 5 4 9 & 5 7 Q 6 H.$ D$6$ 8( .8H D:$0$D ;& " Q<! H.0D B$"K$D H8 H.$ $3$BH$" B"'.(B68L $>KH086 (8B B$"K$" &,= 8H$! M.$6 & 2" Q! M$ 33 H.$ 8L$ $>K030B0K4 B$3H086 H.$ 5,! , ,&(2 +0.!(&= !##*# KB606- 68M H8 DH030HO B0H$B0! M$ 84:KH$ H.$ D$86" LB0H086. J J J . J .. !!. !!. "J 4"I 2 ,5 4 6 0 . "J 4"I 2 ,5 4 6 0 7 J . M.$B$ M$ .L$ 06HB8"K$". J . !. , % "J 4 . J 9 . 7 J J . . J . !. !. % , "J 4 77 77 7 7 J . "J 4"I 2 77 J . M.0. 8BB$D:86" H8 H.$ D$86" LB0H086 $N:B$DD086D (8B B$"K$" &, 6" H.$ 3D8L $>KH086 GI! GI! &F!

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(49) 6$B-O H030HO 63OD0D. IJ9. (H$B D84$ B$BB6-$4$6H! H.$ $N:B$DD086 (8B H.$ D$86" LB0H086! $L3KH$" H 6 $>K030B0K4 D83KH086 8( 9,=9Q'9,=9J! 6 $ MB0HH$6 D (8338MD. J J J . . !!. !!. !!. .J . . "J 4"I 2 , 4 6 0 77 J. <J < < < "J 4"I 2 .9 < . 5 7 4 6 0 ,5 <. <J < < < "J 4"I 2 .9 < . 7 5 4 6 0 ,5 <. !!. !! !. 9,=9*. "J 4"I 2 .9 J .. !. "J 4 B 5 5 7 J J. % ,J "J 4"I 2 ,5 4 5 7 J. % , "J 4 7J B 5 5 J. !!. "J 4"I 2 7J ,5 4 5 J ". M.$B$ M$ .L$ "$)6$" 5 I 2 ,5 ,5 ! M.03$ B "$68H$D H.$ 8B"06BO 4HB0N HB$= $KD$ $6$B-O DH030HO 0D HH06$" ;O "$)60H086< 0( H.$ D$86" LB0H086! $L3KH$" H H.$ $>K030B0K4! .D "$)60H$ D0-6! (8B 6O :$BHKBH086D ! 6" ! M$ 06($B (B84 9,=9*! H.H DK()0$6H 86"0H086D (8B DH030HO B$ :B8L0"$" O 7 J . 9 9 B 5. 77 77 7 7 . 5 7 J B 5 Q 77 J

(50) . 9,=9G 9,=9F 9,=9&. 8H$ H.H . 04:30$D .7 ! D8 H.H H.$ DH030HO 86"0H086 8( 9,=9G B$"D 9 B 5 = $KD$ 8( H.$ :B$D$6$ 8( H.$ 206$H0 84:86$6H! M.$B$ 6" 8KB DH030HO 86"0H086 B$>K0B$D D38M$B $>K030B0K4 +8MD! 06 84:B0D86 H8 H.$ 8BB$D:86"06- 86"0H086 (8B B$"K$" = .$ 3HH$B 86"0H086 &,! 06"$$"!.

(51) IJJ. 86306$B .OD03 ODH$4D. B$>K0B$D +8MD H8 $ 1KDH DK/3(L%60! M.$B$D H.0D 0D 68 386-$B DK()0$6H H8 DH0D(O 9,=9G 06 H.$ :B$D$6$ 8( .8H :BH03$ :8:K3H086= 4

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(53) ! H.$ 86"0H086 9,=9F 6 $ :86 4206- KD$ 8( 7 . B$(8B4K3H$" 06 H.$ (8338M06- MO & J9 0 6 0 6 4. 4 5 J 5 J J J J < & -<J < 0 6 << < B 5 Q 4. 6 0. 4 . < 5 5 J <. 0 6 4 5 J. &. 9.. J J. -. .. 9,=97 9,=JQ. M.0. :B8L0"$D 6 06H$B:B$HH086 8( H.0D DH030HO 86"0H086 06 H$B4D 8( :.OD03 :B8:$BH0$D 8( H.$ $>K030B0K4 DHH$= .$ H$B4D 8( 306$ 9,=97 8BB$D:86" H8 H.$ D4$ H$B4D ::$B06- 06 H.$ $6$B-O DH030HO 86"0H086 (8B B$"K$" = 6 :BH0K3B! M$ 6 B$8-60P$ 06 H.$ )BDH H$B4 H.$ 8L$/4$6H086$" DK/3(L%60 86"0H086! 06 ""0H086 H8 86"0H086D "$:$6"06- 86 H.$ B$3H0L$ "0B$H086 8( H.$ $>K030B0K4 4-6$H0 )$3"D 6" H.$ -B"0$6H 8( H.$ KBB$6H "$6D0HO= .$D$ .L$ $$6 D.8M6 H8 $ D8KB$ (8B H.$ 2062 6" 06H$B.6-$ 06DH030H0$D 8D$BL$" 06 H8242D 06 H.$ :B$D$6$ 8( 4-6$H0 KBLHKB$! D$$ .:H$B 9J= .$ H$B4D 8( 306$ 9,=JQ 8K6H (8B H.$ 6$M 86HB0KH086D "K$ H8 H.$ 206$H0 D:$0$D= .$D$ B$ "K$ H8 H.$ 84:B$DD0030HO 8( H.$ .8H :BH03$ $>K030B0K4 +8M 6" H8 H.$ .8H :BH03$ $6$B-O= $ 8D$BL$ H.H! 06 H.$ D$ 8( DHH0 $>K030B0K4 ;0=$= Q<! H.$ :B$D$6$ 8( H.$ 206$H0 D:$0$D .D 68 06+K$6$ 86 H.$ 86"0H086 9,=97'9,=JQ= 6 :BH0K3B! 06 H.H 3040H! M$ B$8L$B H.$ :KB$ B$"K$" 86"0H086 &,! 0 6 4 5 Q! M.0. 8BB$D:86"D H8 KBB$6H "$6D0HO :B8)3$ 4868H86033O "$B$D06- M0H. H.$ $>K030B0K4 +KN (K6H086 = 0633O! K:86 "0(($B$6H0H06- H.$ $>K030B0K4 B$3H086 (8B 7 M0H. B$D:$H H8 ,! M$ 8H06 77 .. 9 .J 7 7. 8B $N4:3$! H.$ :BH0K3B KDD06 "0DHB0KH086D DK. H.H 7 . ;K60H LB06$< B$ DH3$ $>K030B0= .0D 0D 48"0)H086 8( B"6$BAD M$33/268M6 4868H8600HO H.$8B$4 GI! GI= H D.8K3" $ $4:.D0P$" H.H 8KB DK()0$6H 86"0H086D B$ 68H 8:H043= 3$B3O! D84$ :8H$6H033O DH030P06- :8D0H0L$ "$)60H$ H$B4D .L$ 68H $$6 KD$"! 6" H$B4D 06L83L06- -B"0$6HD! $=-=! 5 J ! 8K3" $ $DH04H$" 06 861K6H086 M0H. H.8D$ 06L83L06- J O H.$ 806B% 06$>K30HO 06 8B"$B H8 8H06 $HH$B B$DK3HD= 3D8! H.$ 86"0H086D M$ .L$ 8H06$" 86HB83 6" ! KH 6 $N406H086 8( J .

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