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Jα(x, t) = qα nα(x, t)V α(x, t) .$ '( I &

*)2!E * g .()2 - JE )+!%+ 1 s#)+*.()+-UN )+V α

.()2 *- !)2 1 2s#)+*.()+-UN )+. , )+%2 * )+* )+*JE ? %+)2.%2JE *WE )+!%21

v′ = v − V α ,$ '( 5 &

b&* /$2 *3 -/ 97:5)P8 7<&$6+ P$' P

1 - Js#)+**E *mg .1WZ s )+ 6 E * .()2 1*- !)2 * 1 2s#)+*.()+-UN )+ .*V 1\@.(!FN g0.* !TN*1

Pα = pαI + Πα .$ '( : &

)2 1 1\Z 1)2 1.JE 1\N% * !FN $ 3.() - * %2%2JE * & *. .JE _)21\N 1!)+TC1\@*1 $A)2 .() - * %2%2JE * & Z 1EhE *4 1\N% !FN )21!#- - * .(

pα =1

3nαmα

∫v′2 fα(x,v, t) dv = nα(x, t)Tα(x, t) .

$ '(qc &M E )+ )21 .()2 , JE 3 !

Tα(x, t) *( %+%21 g .1yZ s )+ 6 E * . *- !)2 *

1 s#)+*.()+-UN )+ - - *4)2 , JE 3 ! )21! )2#)+* l)+* )+*Qf * %2 *HZ - *4)2S E 3 ** *7. 1b_)21FN 1!)+TC1!*110! PC%+WE * .(

(Πα)kl = mα

∫nα

[(vk − (Vα)l) (vl − (Vα)k) −

1

3(vk − (Vα)l)

2 δkl

]fα(x,v, t) dv .

$ '(Le &M)201!)+*. *-%2)2 4A %+% 1 !* .()2 _!)+%+*- 1 *HN !)2* g.()2#*- !)2 1 2s#)+*.()+-UN )+*)2 )21!3 )21! 1 S * TN R@ ME % )21!0. *)2! %21

δkl =

1

gk = l

0g

k 6= l.

$ '(LO &"$)+! )+* % CG 1!)2 .()27" C\E HG .()2

qαg )+* .(!)+! 1 1 2s#)+*.()+-UN )+1\E E *

. *)2*qα =

1

2nαmα

∫v′2 v fα(x,v, t) dv

$ '(+9 &X * )+*JE p JZ - 11\(1!JE 8.1 1!)2 E )+ .JE ? %+)2.%2JE * yE )+! Js - )+* .JE %21

V = 0-)+%+(3TC1* !)2!qα

.*Ql *!-)2!*1!3 !Hs %2 .( .()2*- !)2 p Js -*- ., )+%2 *

α*1!

V 1!- *- 13*HN.p ! !*- *$1!)+*. .**7.()2%E N1\N 3)21 * , *1!3 !!-%2)2 *- *g l %2JN !** *.X **as %2 %21H !#"$ ! !%'&( )* ,+-/.01/* JN** 1)+*.

dn

dt+ n∇ · ve = 0

$ '(+ &dn

dt+ n∇ · vi = 0

$ '(+' &

me ndve

dt= −∇ · Pe − e n(E + ve × B) + Re

$ '(+7K &

mi ndvi

dt= −∇ · Pi + e n(E + vi × B) + Ri

$ '(+JI &

3

2nd

dtTe = −∇ · qe − Pe : ∇ve +Qe

$ '(+5 &3

2nd

dtTi = −∇ · qi − Pi : ∇vi +Qi ,

$ '(+: &

a # -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

s )8 )20.()2 *)+!)2* .bN *f JN!)^f *V%2)+!*- *\Z s#.d

dt=

∂

∂t+ vα · ∇ $ '(+=c &

17s#)2 .()2S*2f *!)2*vα ≡ V α

.%f * * * * *#* Q? %+)2.(- 1 s#)+*.()+-UN )+* ! a@Ws .*4 )2 )2.*71!*$%2)2 *- * $ '(+ & *. $ '(+' & 3TC1*!)2*$.()2 , )+%2 *a@Z %2! %+!*- .l %2JN !** *. X **0s,C *. .()2 %2)2 *- * $ '(+7K & *. $ '(+JI & .()2XBE 3%21! %+!*- *./.()2%2)2 *- * $ '(+5 & *. $ '(+: & .()2l*!-)2 ! %+!*- 1 )+ *4%2)2 HZ)+!)+-4EYg11* .()2 6 8ias %+%+-%2)2 *- * g%+%+ s .*4

∇ · E =1

ε0

∑

α

nα qα +1

ε0%

$ '(+7e &

∇ × B = µ0 ε0∂E

∂t+ µ0

∑

α

nα qα V α + µ0 J $ '(+7O &

∇ × E = −∂B

∂t

$ '( '9 &∇ · B = 0 ,

$ '( '( &s ) 1!)2 )2 .()2k?%2.

E*.

B1 .%MHEhE /. i !** *. - JE )+!%+* 3% 1R@

E )+* !**?%2. !- *4AC l )+*HZ%+ )+* g .* !- *-f * .yN)+*!)21 * p 1 )+*-$. 1 % 1\E 1Z ,? %+)2.( 1 )+*- 1=s#)2g .()2 !%2)+!*- . ? %+)2.(-%2)2 *- *E )+!%21 6 E * *a@)+%2.(*-g .()2hN )+*!)21 kp Js -*- 1!-%2)2 *-*.E *$)+*]4 !Hg !*s#)2kZ p4)+*Le /O 9 , \E

R)+* .* XBE 3%21)+% *HZ* $ '(+7K & *. $ '(+JI & 3TC1* !)2! .* XBE 3%21Fg !!-

.( QMFG 0)+* , )+%2 *1! $l %2JN !** & E )+ )+* *.* , )+%2 *1! $X ** & V 1.(!TNQ)+*/.* " C\E )+% *HZ -%2)2 *- * $ '(+5 & *. $ '(+: & 1! g .()2 1!- 1 " C\E )+* , )+%2 *1!YE )+ .*.* , )+%2 *1! .()+*- .( %2E 1FG

)2p -)+*1\N )+)j@ %2)2 *- * 1!%+%2* ** )+* - 1 %211* 1 %2)2 *- 11\(1!JE . . .()2FG *

ΠklqR

*.Q)+* p JZ )2 *- Z .* r1!*.1-FG *

nV

*.T- 1\Z ;s .*4

!)+*HZ )+3)2%+% -)+/ 1wf 1 )2 .*m_ * Z M %+)2JG*- )+* 1w? %+)2.(-%2)2 *- 11\(1!JE 1)2 1$ ! * (kN )+*!)21 * , !)2* *. f \s *.*zf 1 )2 .* MGUE .%+%2 Z p 1 )+*- . %2E 1!FG . % 1\E )+%2 *4V %21%p )21!3)2% 1) )2 .()2 -'8C%aG#"S%NXU!-1D/ +8 . .()2 %NXYP*8*CP!-1D0bf * . +' - * ** ?g .()2 f !%+)2- *.kV! )+s .* .()2 )+- * , *1!3 !!-%2)2 *- * $ '(+ & )21 $ '(+: & Z a@1EhE *QE )+#.*mM*..( 1!%+* g0.()2 !"2$!3546 7 * * mM \p -)+*1\N)+) +7K f \s *. )2 , *13 !\N kZ )2* * 1!)+*. %21 3 !!)2* %+)+TC1\N *1* * )+* .* F@G *

Πkl

q

R*.

Q* ! %+*4#)2 1 FG * 1!)+*. * . 6 ! .f * p -)+*1FN)+)

33 !!)2* %,Z .*l JN *s %2 )+*kS !JN ! Z HE ! \E )21 * %2)2 - Js#)2 )j@* 1# % 1\E 1#.1!%+%2*4HXE ! \E )21 * %2)2 - Js#)2 s#)+. .()2N)+*!)21 p Js -*- 1!-%2)j@ *- $p %+\Z E **a@ %2)2 *- & Ps %2 )+* . 6 E * *HE ! . * \Z ,s#)+. .( )+*6 8ias %+%^f !)+%+*- 1 *HN !)2*

f 0

α

- % 1!V%21yN%2)+*S!JN!kZ 678ias %+%^f !)+%+*-s#)+.f 1

α

! s %2 %2N% .( N %2)+* TCHE %+)2 .()2* *Z p 4.()2 . )2 .

b&* /$2 *3 -/ 97:5)P8 7<&$6+ P$' 't

, JE 3 ! . %2JN !!)21 4?%2.88 \f !- !( * s#)+. l 1 -)+%+ |f 1

α| f 0

α )2

1 s#)+*.()+-UN )+*. , JE 3 !s .* .** - * 1hs#)2 gf 0

α%2N%Z p 1!!)ÊhE *-

f *f 1

α *- JZ- *4

)2$* ( %+- *.* , *1!3 !\N kZ )2** 1!)+*. * 7.mV** HE )+* 1mZ %+)+*.(!)21 * % 1\E 1,E )+ )+*JE 1!\N * E - ** 6 -* %2. 1!!)ÊhE s .*4( 1 g !5Z * R@1 )2 .(%+)2 JE , *1!3 ! )+* 3 %+%2%20*. 1*HN )2 !*-wZ HE 67-* %2. V %+%2DMG3Z 11s .* %21 % 1!!)21 %2E a@ MFG ! 1 .G )2 .()2 N2@!)21!!)21 * MGUZ)+*QE G- %+)2 71!)+*.

τe = 3 (2π)3/2ε20m

1/2e T

3/2e

n e4 lnΛ, τi =

(2mi

me

)1/2 (TiTe

)3/2

τe .$ '( '' &

)2! ) )21!τe

.()2 N !)21!!)21 MGUZ)+#gWMFG f *Ql %2JN !**QE )+ X ** *. l %2JN!**E )+ l %2JN !**(*.

τi)21! .()2MGUZ)+ gbMFG .bX ** * )+* *.

%2E a@B]4- !)+! HE 1 )21!0. *)2!%21

lnΛ = 32.2 − 1

2ln(n) + ln(Te) ,

$ '( 'UK &s ) )2 .()2 , )+%2 *.()2

n)+*E −3 *. .()2yl %2JN!**JE 3 ! )+* l %2JN !**2f %+

*HZ - * )21!'?g .()2Dl *!-)2 )+* )+ l %2JN !**2f %+ -)+%+1B_

= 1.602 · 10−19 , 3)21 "$! g , NE N 3% 1\E * 1!)+*.

lnΛ ≈ 10 − 20

?g0.()2 !)2*1! *HZ* .0 )2.* , )+%2 *1!* -)+%+

Ωe =eB

me, Ωi =

eB

mi.

$ '( 'I &)2 * ( %+- *.* , *1!3 !\N kZ )2**ms .* )+*QV*2s 1* )+ )+* 1067-* %2.1bE G@- %+)2 .( .1 .(HN )+ 4MGUZ)+*E )+ )+ !)2*1 *HZ 1!!)ÊhE 5p )1!\N JE 67-* %2. -)+%+

Ωe τe 1*.

Ωi τi 1

.()2 % !)^f - 1 s#)+*.()+-UN )+ .Wl %2JN !** *.mX **s#)+. .()2 1E 1!!E .()2 . *)2!

j = e n (vi − ve) .$ '( '5 &

hXE 3%21\g !!-R

)+* .* XE 3%21!)+% *HZ* $ '(+7K & *. $ '(+JI & 1\Z 1)2 1 . )j@*- 1\N *. . , \E N Z 1EhE *4

R = RF + RT+?gl %2JN !**$*. X **

-)+%+

Re = e nTη · j − e n

TeTζ · ∇Te ,

$ '( ': &Ri = −Re ,

$ '( 'dc &E )+ .JE 1!)21!!)^f )+TC1!*1

Tη*. .JE , \E N !*1

Tζ$ 1!)2 %2)2 *- * $ '(LK' &*. $ '(LKUK & & ?g .()2 1!- 1 " C\E

Q)+* .JE )24 ! * r<s )^N E 3 ***a@ % 1\E ! PC%+

E *Qi =

3me

mi

n

τe(Te − Ti) , Qe = −Qi +

1

e nRe · j .

$ '( 'Ue &

# -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

E )^N 1\N 3)21 $\Z- " C\E HGq.( .()2 *- !)2 p Js -*- . , )+%2 *

)21!0- - * .(

qi = −Tκ,i · ∇Ti , qe = −Tκ,e · ∇Te − Tζ · j ,$ '( 'UO &

E )+ .JE " C\E %2)+!*1Tκ

_)21\N 1)+TC1!*1

Πα = −Tµ,α · Wα$ '(LK9 &

)21! )+*?*HN !)2* . 1WM3 ***- 1!*11

(Wα)kl = ∇k(vα)l + ∇l(vα)k −2

3δkl∇ · vα .

$ '(LK( &)2YS E 3 ** *7. , *1*

TηTζ

*.Tκ

1)+*. s#)2 %+-#- - *4

Tη · j = η‖ j‖ + η⊥ j⊥ ,

Tζ · ∇Te = ζ‖∇‖Te + ζ∧B

B× ∇Te ,

−Tκ,α · ∇Tα = −κα‖∇‖Tα − κα⊥∇⊥Tα − κα∧B

B× ∇Tα ,

$ '(LK' &

E )+ .* , *13 !\N kZ )2* *

η‖ = 0.51me

n e2 τe,

ζ‖ = 0.71Tee,

κi‖ = 3.9nTi τimi

,

κi⊥ = 2.0nmi Tie2 B2 τi

∼κi‖

Ω2i τ

2i

,

κi∧ = −5

2

nTieB

∼κi‖

Ωi τi,

η⊥ =me

n e2 τe,

ζ∧ =3

2

TeΩe τe e

,

κe‖ = 3.2nTe τeme

,

κe⊥ = 4.7nme Tee2 B2 τe

∼κe‖

Ω2e τ

2e

,

κe∧ =5

2

nTeeB

∼κi‖

Ωe τe.

$ '(LKUK &

?g .()2kS E 3 ** * . 1_)21\N 1!)+TC1!*11 $ '(LK9 & E )+ Tµ,α1) *$.()2 1yM%+%2 (0.()2

*HZ +7Kf \s#)2 1*4

"# " +# , " & " !" #%$!&')(

)2 . 1 % 1FE )+*1 %+11 1 )21! g )+* N *E )21 4V 1* \Z *- . 1%?1!)2*1!3@Z 11 1 * 1 )2.*. )2Wl)+*1 %+HGUZ)+ . 1 % 1FE 1 s#)+./)+* )+*#6 -* %2.aN *a-!)2*E G- %+)2 $.( .* , )+%2 *a@0*. " C\E !*1!3 ! 1*HN DZ .*767-* %2.(%+)+*)2* 1!!)ÊhE

|Y&-H9=(-H$ 9=6& *-/ )<(-/&$2w3 -/ 97:5)P8 a

6 *7 ! Z *PC 1! )+*Z %+)+*.(!)21 1 % 1\E 4E )+ E - *JE 6 -* %2. +X *$)+*JE6 -* %2. -U !)2* X ** *.l %2JN!** HE .()2 6 -* %2.(%+)+*)2*7 !HE <E )+ .*$@!)2*1 *HZ* $ '( 'I & *. )+ * !)2*1!.()2*

%i =v iΩi

=

√mi TieB

, %e =v eΩe

=

√me TeeB

,$ '(LKI &

s ) .()2 ! \E )21 1 s#)+*.()+-UN )+ . , )+%2 * E )+v α =

√Tα/mα

. *)2! )21" C *. .()2p Js -*- ./ !)2*1\Z* !* )0*!% *- . 67-* %2.(%+)+*)2* %+-s .* .()2 , )+%2 *7.( mZ ( C%+%+)+- %2E 1!FG %f * )+ * !)2*1! **7- %2*HN * MGUs#)+*HN % !- * 1!)2 * 1 )2 .(%+)2 _1 )2*- * . !)2*1FZ* !*1*HN Z HE 67-* %2. s 1 %2\Z !%+)2 Z )+*JE JN !)^f * , )+%2 *a@*. " C\E !*1R@3 !#)+* .() %20)2 !*- g !

! !#"%$'&'()*+-,."'/0+$%132( "0 $'4))5/0"0)6 "0&'()

180 798 +$'+-*+6:;#</0"0+6 6=+>@? AB C)D; "E 7 GFH$?)IJJ

)2 1 "$ 2#* 6+*- , *13 ! 1!%+%+ .18*2f \E )2.( #6 )+*)Ê HE ._!%+1! )+* 1 )+*1 %+)2 @G *.*~M 1!JE 1 .+l )21! )+*$.() 1)^f ZJG g .* E *$.*7 %+- *.*V*1\Z g.()2(ZJs )+%+)+- , )+%2 *1!%E

Γ(r) = −D(r)dn(r)

dr,

$ '(LK5 &

q(r) = −n(r)χ(r)d T (r)

dr,

$ '(LK: &s ) . .() %2 , )+%2 * !*1!3 ! 33 !!)2* %Z HE )2 -.()2* )21! 3 !!)2@* %+)+TC1 N

D(r))21! . , )+%2 *.() 1!)2*1\N kZ )2* )2 15s#)+. .( .()2MGUZ)+ .

MG3 !!* $ 1!)2 $ '( '' & & *. .* !)2*1!.()2* 1!!)ÊhE

D ∼ %2

τ.

V ( -!*./. XE 3%21! %+!*- !- */*MFG WZ s#)21 * * 1 )2 .(%+)2 * , )+%2 *%21l %2JN!** *.X **'Z HE , )+%2 * !*1!3 ! ) ( .() 1!)^f " C\E %2)+!*- 1\N kZ )2*

# -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

. *)2!1!)2 QE )+χ⊥ =

2

3

κ⊥n(r)

∼ %2

τ *( %+%21bg )+**N% 11!)21 * , *1!3 !\N kZ )2** $ '(LKUK & Pp )4. " C\E %2)+!*- !8@- * * *QZ 1#C\Z %+)2 .()2YMFG . , )+%2 *1!* *)+* *.WZ HE , *1!3 ! ) %E *Z )+*JE )+*$)2.%2* 67-* %2. g 1 * 1 * ( -!*.1f *S \gHEy@E *- *.wX * E - *)+TC* . 167-* %2.1 , )+%2 *.(!)& *'s %2 )+**/ JN !)^f */ % 1\@E )+*1 %+HG *HEk-%+)2 4E *+JI 2X */)+*JE , NE Nks#)+. . .( X *.(HN !)2* )+* 1)2.%2*7% 1\E 1!!E 1)+* 3 %2)2.%2 167-* %2. \Z-.11!)2 .JE i(!** )j@.%2*/67-* %2. g !% - ! )2.( /! PC%+5E */)+*0 %+)^N%2 ?%2.(%+)+*)2*2f .(!)+%+%+*-/s#)21!)2 )+*QV)+%2.(*-/'( ' *- . )21p ! E * .()2 1 E !)Ê 3%21!)+% *HZ )+*.JEE * .()2 XBE 3%21-%2)2 *- * gl%2JN!**$ '(+7K & *.yX ** $ '(+JI & ..()2!1 ! PC%+E * )Ê 1!!)21 */%2)2 - Js#)2 1! %+% )+* .JE ., TC- )+1!\E $S *2f JN !)2*1!\E & .1 %2JN!!)21 k?%2.$*. .()2y_)21\N 1)+TCf !* % C11!)+-s#. .()2 % !)2*

j × B = ∇p .$ '(LKdc & )2! ) )21! . 1E .(!FN . *)2! .(

p = pe+pi2V* %2)2 *-$ '(LKdc & \N ** E *1 ! .GhE )+

B · ∇p = 0 , j · ∇p = 0.()2 67-* %2.(%+)+*)2* *. M!E %+)+*)2* (D? % C * N *1!* * !TN 1 %+)2- *4 )2 1 ? % C8@ * JZ)2 * E */%215E -*!)21 D? %+HG <C *4'X * )+* , NE N@BS *a-!)2*()+* ..1 % 1FE 1\'EhE !!)21 Z -G * , !1 1 )21! $ 8i() %21\'EhE !!)21 & !-)+1)2 . *PC !*- 1\s )21 )+* 67-* %2.(3 %2-)2/s#)2 1!)2 )+*QV )+%2.(*- '( ' .!- 1!%+%+ )21

Magnetfeldlinien

Flußflächenmagnetische

/'/' * "%$0/0# )" D &'()5/0+ ++ 132 " ) $0.-- /'"% 7$0& ()$ ; $'6- / <"%)$'"%) ++ ) )"0$0& ( /%/G! 1 ) "0 "0)$0/0) /') 7 )/0$'&'() D & () )) HD ) 7 5/0"0+- # )& ( $'& ( ) +"0$ )> 5/0)"0$0& ()& ( )"%$'& ()"%H 1> #-D & ()-132( "%/ 4) ))

q ;"0+ ? ABC #"0&'( * )$= D +"%$'&'( $'4))5/0"0 2&'(, H "!#)"#& ( #"% ) /0J

)2 E -*!)21 *? %+HG <C * )+%2.* )+*)+* *.kf 1 %+ , !) ( .** 1!)2 .()26 -* %2.(%+)+*)2*~E )+ * 1 )2 .(%+)2 %+)^Z )+TC !HE 1 *48 E

q. H@

*)2! 1!)2 g .1;_! PC%+!*)21 .5V*HZ % )2.%2 E % C( )+*,?%2.(%+)+*)2 Z .5V *HZ %

|Y&-H9=(-H$ 9=6& *-/ )<(-/&$2w3 -/ 97:5)P8 /

)+ 3 %2)2.%2* WE % C( .()201!)2 */!)+- HE 1!)2 )+*/1)2 /1%+1!;Z /1 %+)2JG * HZ s )+*/)+*3 %2)2.%2*M *)+! *%s#)2 .E 1%+ * *HN *HZ - % *- *4

q =B · ∇ϕ

B · ∇θ=Bϕ r

Bθ R,

$ '(LKUe &E )+ . 3 %2)2.%2* *. )2.%2* 3 1!)^N%+)21 *ySE 3 **0. 1 6 -* %2. 1 $ 1!)2 ,V *a@ *-wV ' & 1 .( 7.* , )2.%21!!E )+*.(HZ )2! %2)2.%& %2.mf!)+)2!E )+.JE N%2)+** % 1\E .()+1

r1 .G .()2yMTC\N .D_1 *- .?%2.(%+)+*)2*$( Z! .? %+HG <C

*.1 *. 1E )+q)+*,?*HN!)2* . 1N %2)+** % 1\E .()+1 )21! E

qs#)+.

M)2 ! )+1 N - * ** . 1)+* "$!E G- %+)2 g .()2bS *!%+%2bf *hE N1FN 3)21 *X *1!)+%+)+TC* )21!Hs %2 .* \sbg*1 *l )+*1 %+HG Z1!F*mNd***4 %+-)2 Dl )+*1 %+HG . 1 % 1\E 1 1!)2! )+* .()2 1 !1!\E )+- * V *.(**- (8.MTC\N .W_1 *- .6 -* %2.(%+)+*)2*4 )2 1C*'+*-*'.10

s% CG 1!)2 7 g

.()2%M !% C*- Ls

1.(\gFN *4

s =r

q

d q

dr, Ls =

q R

s.

$ '(LKUO & - *2g .JE N% 11)21 * , *1!3 ! )+*$- .(%+)+*)+- * 6 -* %2.!* 1)+*. , )+%2 *a@*." C\E !*1!3 ! )+* .()2 1 )2.%2*l )+*1 %+HGUN *a-!)2* 1\N ! / )2 1 2"c ^2*%' B+*-C WNW* )21 ..( .()+*- .G 1!)2 .()2 , )+%2 * ** .( !)& * )+* .*)+* E - **4?%2.!* HE -FG DM! FN * %21)+ D!)2*1!.()2*wf */.*kE -*!)21 *? % C * * !**Nd*** *. .E )+ ) %2E 1!FG * )+*%s 1*!%+)2 -FG hM !)+!R@s )+ *+5 1l!- *)21 ! !)21 6 .%+%2YZ HE * N % 11!)21 * , *1!3 ! .- * 1ks#)2 .N% 11)21 , *1!3 !* %2E s 1%^s#)+\N*- * \gTN 1!)2 !)+- * bp \gFN(1!)2 !)+-*- )+* E - * ?%2..(\gTN#1!)2 *0*.JE )+* .* , *13 !R@N kZ )2** 1Hs %2 * * HE s )+* FG *.(* *-/-FG 1!)+*.

D

⊥ ∼ χ

⊥ ∼ (1 + q2)%2

τ,

$ '( I 9 &. )+* g%+)2 *S *a-!)2**7E % 1\E *.

q2 ∼ 10 − 20)21!

q)21!#.0 * !% C!

M)2 ! )+1 N. 11* .() %2 1 % .1067G . _1 !*- 1!!)ÊhE X * . , 5N *** ) i 3 !)Ê * %+%2* 6 11!*- * . ) 1!)2*1\N kZ )2* */.()20* N% 1R@1!)21 *4_! 1- * )+%^s )21 1!TC!)+-,s .*+ X */-)2** )+* .** .1 % 1FE 1 )^G/)21! %21/)Ê % 1\E N !* N ** *7" C\E .() 1!)2*1Fs ! gX ** - *.*s .*s %2 )+* Bs )+* . FG *.(* *- .#* N% 11)21 * "$! %+)2- *4

χ⊥ i ∼ (1 − 10)χ

⊥ i .

p ) .* l %2JN!** %+)2- * )Ê _!-%2)2 Z .* i 3 !)Ê * %+%2* l !- *)211* .()2 f ! R@- 1-* "$! . 1 , )+%2 *@ *. " C\E !*1!3 !1.!%+)2 HE Z s ) FG *.(**- **)2 .(!)+-

χ ⊥ e ∼ 102χ

⊥ e .l * !* E * 1!)2 Qs )+f E % 1\E N !* )+* 0)2 !*-/. 1 % 1\E *.1 1 % CG 1)2 E )+

.* * N % 11!)21 * , !)2* N )+* 1*HN , *13 ! )+* , )+%2 *1! 1!)2 *.

# -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

\N% C*4 s#)+. .0)21! #*)2 \N % C!V* )+%8. 1 , )+%2 *a@ *. " C\E !*13 !18 )*gC CN - * ** l)+* f)2%^f 1!3 *.%l \N % C!*- 1*1\Z )2 .()2 *)2 !%+)j@* , !)2 . 1 !!%2* * , *1!3 !1" )2 1 * )+* +7K *- ./Nd***X *1)j@%+)+TC* ( 1 bN%2)+** TCHE %+)2 * *.Z)+!%+)2 *MN%2*1- * ** X\ B"$ W BN* ! i ,N=W* s %2 Z p .( .()2** )+*$. )2 . . , JE 3 ! \f !- !( *1s .*%2N% .* , *1!3 ! ! / *4'? %+HN!)2*.%2JN!!)21 ? %2. )Ê r1 EhE * *-%E )+!R@%2* * 6 )^N)+*1!)+%+)+TC*hs .* * * %21 1 g .* *E %2* , *1!3 ! - 1 *)+*.JE .()2 , !%2*HZ)+*bS *f JN!)2*wf * % 1\E 1*HN 5Z .* )+*)+* *.!- 1 %j@*E -*!)21 *? %+HG <C * \f !!( !)& s %+%2*7.l %2JN !** *.QX **QE )+0)+ 1*HN *7!)& !- 1 2s#)+*.()+-UN )+

v⊥ ∼ E⊥

Bg * )+*_!)+*.(*-kE )+ E )^N 1\N 3)21 * )2 1F!*- *

nZ )+*JE`N *2f JN !)^f * , )+%j@

*HGΓ = 〈v⊥n〉 ,s ) 〈〉 .()2 6 )+!!%+*- g /)+*? %+HG <C . l* 1!3 *. 1!!)ÊhE /1!)2 .

!!%2* " C\E HG .( !)& * *. %2N%2 , JE 3 !1 s *HN *- *

q ∼ 3

2n 〈v⊥T 〉 .

)2 .)+%+%+)2!* !*.(% - * . !)& Bs %+%2*HE .%+%2s %2 .1hM!.()+HE . 1 *E %2*, *1!3 !1 !% */s .* )+* .* %+- *.*V1 *)+!* !% C!"$)+! )+*Nd*** 1MF!*- *7. 167-* %2. 1 /s %2 E )^N1\N 3)21 )Ê % 1\@E Yf ! *.* 1)+*./ . s %2 Df * HG * .( i(!*MF %2.#\Z-bs .* $ 1)2 S 3)+%I & #)+** ! / * , *1!3 ! ! )&g * )+*.JE 1)2 .()2 67-* %2.(3 %2-)2f TC*.!*4 * MTC\N . MF!*- f \E * 1!)2 .()2 6 -* %2.(%+)+*)2* )21 )+* Z )+*JE fd%+%+)+- *V ( *7.E -*!)21 *? %+HG <C *4 167-* %2. ! PC%+)+* 8@.() %2YS E 3 ** 1 .G , )+%2 *E )+ )+*3 %+%2%2*7 1 2s#)+*.()+-UN )+DZ HE 67-* %2.0)2 !*-/ % 1FE *. 1!!FE *QNd***

Γ ∼ n

B〈v‖Br〉 .

X *1!- 1 E % 11* 1!)2 .** .()2 !!%2** , *1!3 !!3Z 11 )+*QMHEhE g .*/- 1Ey@* N%2)+**$ % 1FE .()+1f !)+%+ %21 .() 1!)^f , *1!3 !!3ZJG 411*4 D_!-%2)2 E )+ .* i(3 !)Ê *%+%0 1!!)ÊhE * ) 1!)2*1\N kZ )2* * %+)2 ! )+* - )+*1!!)Êy@E *- )+* . FG *.(* *-4f *

D⊥ e ∼ χ

⊥ e ∼ 1

E 2

1 .$ '( I &

ws 1*!%+)2 *1 )2 ..Z HE N % 11!)21 * *. * N% 11)21 * , *1!3 ! %+)2-/.!)+*.G ) !!%2** Z 11* .()2wN %+%2JN!)^f /"$ 1%^s#)+\N*- %+%2 -U !)2*.* , )+%2 *E )+ )+ *$%2JN!E -*!)21 * ?%2.!* 1=s#)2 .* i(!*$\Z-*?%2.!*~p \gTN 1!)2 !)j@-*-D*.)2,N %+%2JN !)^f "$ 1%^s#)+\N *-E 1!)2 .( .()2 l *1! *-Yf 1 )2 .*

t +$ +8$60 7$B 7$ '>

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+ +, ( ( +

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Z.1!%+%2* %21

mndv

dt= −∇ · P + Z en(E + v × B) + R .

$ '( I ' &)21*HN SE 3 ** .# 1 s#)+*.()+-UN )+ ! PC%+ E * * * .( 4_JN \E %+!)+3%+)^N8@!)2*QE )+

B/B

v⊥ =E × B

B2+

B × ∇p

Z enB2+

mB

Z eB2× dv⊥

dt+

B × ∇ · ΠZ enB2

+R × B

Z enB2.

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.() E -*!)21 0!)& v∗

% !)21!)2*1.(!)& v

a_)21\N 1!)+TC1.(!)& vΠ

*. )+*- 1.(!)& v

X * . % !)21!)2*1.(!)& )21!0.()2 %2 r )+%2)+!*- d

dt

as#)2 1)2 )+* $ '(+=c & . *)2!)21!* R@ %+*4 )2 1Ds#)2 .!HE *! PC%+.*V1.(!TN . - 1E * 1*HN *$ 1 2s#)+*.()+-UN )+v⊥

E )+s#)+. .()2 *)2 !%+)+*bA ! . %2)2 *- $ '( I2K & *. 1E )+ . 1 - 1 E * , *1R@3 !!-%2)2 *- 11\(1!JE .!%+)2 . Z p * *- . 1*HN */ 1 2s#)+*.()+-UN )+v⊥

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% # -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

* ( * ( +,* +

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k⊥ k‖ .$ '( IUI &

p ! E * "$%+%2*3Z 11 )+* . AC f *k‖ ≈ 0

1 1!3!)2 wE * f * .f Q.*D/*8=-**.00 X * . !)& !*PC !*-71!)+*.$.()2 3)21 * ? *HZ* . "$%+%2*3Z 11 )ÊhE 1 f )2%N%2)+*%21 .()2X **-U !)2*1 *HZ

∂

∂t∼ ω Ωi .

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ω = k⊥ |v | $ 1!)2 V 1 *)+!'( 5( ' & p ! E * * *mV %&fdo*a@ "$%+%2*/s %2 1)2 QE )+ . V%&fdo*a@ 1 s#)+*.()+-UN )+

v =B√

µ0 nmi

$ '( I : &)Ê % 1\E 1!)+*1 *1 )2. E * Z s ) f 1 )2 .* V !*4WN E 3 11!)2* %2V %&fdo*a@ "$%+%2* *.M R@BV%&fdo*a@ "$%+%2*4 X *V4'(LK 1!)+*. .()2 1 )2.* "$%+%2*B 3 *71 @E !)21 1\N)^Z Z )2!

v

B

v

B

@<5$G 8 + "%#$0$'+ 1 *) 7 ))6- / $0"%# /06$0)<"%)& (/')" & ( 7/0H4# /%13)> )& (/'$# & ()" 7 1 *) 7 ))6- / $0"%# /0- & (5/0 *+

B

S E 3 11!)2* %2mV %&fdo*a@ "$%+%2* * 1 * .( E -*!)21 * !TN s %2 1!)2 1*HN@ yZ HE 67-* %2. .( .()2 % 1\E /g11!)+-UN )+ 1!)+ )2 1 % CG 1!)2 *1 %+)2 %21 r1EhE *.(\gTN * *.wV1)+* *.\Z )2 * Z )+* *. 3 %+%2%2#6 -* %2.(%+)+*)2*wf R@1! *4 1!3!)2 E * 4f */)+* %2*-)+!.()+* %2* "$%+%2* 1!)+!*- 1*HN Z 6 -* %2.(!)2 !*-M R@BV%&fdo*a@ "$%+%2*$*1! * .( mE -*!)21 kM3 ** *- HZ s 0S \gHEhE *- .W?%2.(%+)j@*)2*as %2 Z p(.( %2N%2 " )+! % .0*.Df 1 !p Js -*- * )+* )+*#)+*HN Ey@3 11!)+%2* E -*!)21!)2!*h? %^g11!)+-UN )+ \f !- !( *ys#)+. )2 . ) * 1! *. iHp @BS

1B = /$2 1 * b&*T! J&$6+ P$

f 1! .()2 1S \gHEhE *- s#)2 . 1\Z -%2)2 * *. .E )+ .()2067-* %2.(%+)+*)2* - . Z Z )2 *4/l10*1! )+* !*1\f 1%2 "$%+%2* Js -*- )+* 3 %+%2%20)2 !*-wE )+M s *a@N*- * . 1 6 -* %2.10)+* 1*HN )2 !*-6 )+ .p .()+*-*- $ '( IUI & \N ** QE * .G .()2 ?*HZ* gS E 3 11!)2*1R@BV%&fdo*a@"$%+%2*

ω ∼ k⊥ v f )2%0-FG 1)+*. %21?Z!* g M R@BV %&fdo*a@ "$%+%2*

ω ∼ k‖ v X * .

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ω ∼ k‖ c & @1!!)ÊhE E )+%f !*.* )21 .()2k?.!*- )+* 1YN%2)+** % 1\E 8@Bp 1 gYl%2JN!**E )+ .0 *)+!)2* .M %+%+- 1 s#)+*.()+-UN )+

c $ '(Le5 &

βe =µ0 peB2

=c2

v2 1 .

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A 1$-)+%+ )+* . !)& !*PC !*- .G .()2Z)+!%+)2

V %2)+!*- .1*HN *S E 3 ** . 1b_JN!3 * !) %21Wf 1 2s#)+*.∂

∂tA⊥ ≈ 0 .

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E⊥ = −∇⊥φ*.

E‖ = −∂ A‖

∂t−∇‖φ .

$ '( I2O & )2! ) )21!

φ.101\N% %2JN!!)21 * !) %

V ( TCHE %+)2 * MN%2* *./.()2 1*HN * E )^N )21 )+*.Z ]C*- * 1!!s %2 .( .()2:9?* <, ^=W0/# N!)21)2!1!)+*.

%s =

√mi TeeB

.$ '( 59 &

)2/!)& !% C*- wN EhE )+*JE X **-U !)2*1!.()+1 ),l %2JN!**JE 3 ! -%2)2 *..()25Z ! *.* Z 11 . !)& Bs %+%2*.a * E )^N 1!3)2%2* 1!)2 )+* 1*HN 0)2 !*-Z HE 67-* %2. )+* . FG *.(* *- .()2 1!)& !% C*- Ps %2 3)21 \s )21 )+*)+- 10−4 E !TC- ),s#)+. .()2 !)& .a * E )^N1E )+ )+ * "$%+%2*% C*- * )+* )+*JE p )2 . 10 % 1\E 11!!.()2!. .(

%s ≤ λ⊥ ≤ L⊥$ '( 5( &

-*HZ )21!L⊥

s#)+. %21)+* !-!*.(-.()2**% C*- # JZ)2 * *. !TC-8B 3)21 \s )j@1 )+*)+- JE 6 * 1!%+%+ 1)2 .()2 % 1FE -FG *as#)2 )2 , JE 3 ! 67-* %2.

P # -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

Z 1EhE *- 1\Z 1 )+*JE %2)2 - Js#)2 1! )+*!-!*.* )+% *. )+*hN%2)+** MF!*-f HZ p

n = n0 + nE )+ n

n 1 .

$ '( 5' &6 )+ .JE .()2** . )+* !-!*..()2

∇n0 ∼ n0/L⊥s#)+. .()2 7*HZ . 1

TCHE %+)2 *p )2 1 $ '( 5( & . 1Z * 1! *.* % 1\E 1 1!!- %2-6 )+

δ ∼ %sL⊥

1$ '( 5UK &

%+)2- .** )+*MN%2* !***- Z s#)21 * .* )+* !-!*.(-FG * *. .*N %2)+*1\N%+)+- *MF!*- *f )2 !*.()2. )4.!%2)+!*- .? %+)2.'@ !)& !-%2)2 *- *7)21!.G . *E %2 , *1R@3 ! )Ê s 1*!%+)2 * .( 6 )^N )+*1!)+%+)+TC*f !1 bs#)+. 's %2 .( .()2* *. )+* !-!*.(-FG * - !!)2 * s .*4 )2 * E )^N . N %2)+*1\N%+)+- *MF!*- * (.MN% f *

%s1) .** - . 1 1 * .G 1)2 .()2 .()2* *$. 1 )+* !-!*.1

$Z p . )+*!-!*..()2 & 1FZ -%2)2 *f \E -

∇n ∼ ∇n0 =⇒ n

n0∼ %sL⊥

,$ '( 5I &

s ) .()2 .()2* *% C*- . MF!*- * )+* . FG *.(* *- f *%s

%+)2- )2 % !)^f *VE 3%+)+!.*7%+%2MF!*- * 1\N%+)2*QZ 1EhE * )+* .[.(* *-wf *

δ

n

n0∼ e φ

Te∼ B⊥

B∼u‖

c1 ∼ δ =

%sL⊥

1 .$ '( 55 &

?g.()2 !%2)+!*- . ? %+)2.'@ !)& R@ %2)2 *- *s .* .()2p -)+*1\N )+)j@ %2)2 *- * $ '(+ &)21$ '(+: & Z -!*.#- %2- , TC- )+1\E 2s %2 #33 !!)2* %/Z ;l %2JN !**HE 1101)+*. s .*f !* % C11!)+-.me mi

-)+%+)2 , \E .# ! * *)2 !%+)+** * 8@E )^N 1!)+*.7 )2 11 %+)2JG%+)2 .(!)21 $)+*$.*~MF!*- *44)2yN %2)+**$ E

δ*.

βes .* %21W[.(**- 1!3 E )8. !%2)+!*- f \s *.

X * . !)& !*PC !*-7 1! .()2 1*HN M!E .()2 .** * 1 . .() E -*!)j@1 * M!E .()2 *. .0 % !)21!)2*11!!E .()2

j⊥ = j∗ + j ,$ '( 5: &

. g.()2 1*HN *7!)& !- 1 2s#)+*.()+-UN )+*$.()2Yl × p5@ 1 s#)+*.()+-UN )+ .()2 .() E -*!)j@1 1 2s#)+*.()+-UN )+*. .()2 % !)21!)2*1.(!)& . X ** ! Ws#)+. . 4.()2 1*.() , \E 1/ %2)2 *- $ '( I2K & E )+ Z = 1

"$- * .kS %2)+* )+ .hl %2JN!**HE 11s#)+. .()2 % !)21!)2*11!!E .()2 * 1 .hX **- 1 s#)+*.()+-UN )+ - )+%2. )2wl × p5@ 1 s#)+*.()+-UN )+ )21! g l%2JN!** *.X ** -%2)2

v =B × ∇φ

B2.

$ '( 5dc &

1B = /$2 1 * b&*T! J&$6+ P$ /t

?g0.()2 .() E -*!)21 !)& 0.0 )2.* , )+%2 *1!* -)+%+

v∗ = −B × ∇pee n0B2

*.u∗ =

B × ∇pie n0B2

.$ '( 5Ue &

X * *)2 .(!)+- 1!D[.(* *-δ-)+%+0.** g0.()2 1*HN 1 2s#)+*.()+-UN )+ .l %2JN !**

v⊥ =B × ∇φ

B2− B × ∇pe

e n0B2

$ '( 5UO &*. g0.()2X **

u⊥ =B × ∇φ

B2+

B × ∇pie n0B2

,$ '( :9 &

1 .G g0.()2 .() E -*!)21 hM!E .()2

j∗ = e n0 (u⊥ − v⊥) =1

B2B × ∇(pi + pe)

$ '( :( &*. g0.()2 % !)21!)2*11!E .()2

j =mi n0

B2B × d

dtu⊥

$ '( :' &-)+%+ +=c N E 3 11!)+%2 V* )+% . .() E -*!)21 * M!E .()2 ,s#)+.Yg S \gHEhE *- 1R@\E 1!- .(\gFN ) -)+%+ Z HE p )21!3)2% g0.()2l %2JN !**

∇ · v∗ = −∇ · B × ∇pee n0B2

=1

n0 eK(pe) ,

$ '( :UK &

∇ · v = ∇ · B × ∇φ

B2= −K(φ) .

$ '( :I & )2#)21!0.WS \gHEhE *- 13 K g 0.()2%[3 !)2*

K(f) := −∇ ·(

B

B2× ∇f

)= −

(∇ × B

B2

)· ∇f

$ '( :5 &E )+ .01\N% *?*HN !)2*

f. *)2! s .*4

?g5s )+l)+*HZ%+ )+*wZ !%2)+!*- .* * %+- *.* ? %+)2.'@ !)& R@ %2)2 *- * 1)(.()2 )+%+)+!)2*11 !)& +7e HS 3)+%K f \s#)2 1*4 !

"$- * .n 1!)+*!%+)+TC )21! 1 -%2)2 -Ug%+!)+- .()2 S * !)+*)+TC1!-%2)2 *- .l%2JN!**$ '(+ & . .()2 . X ** $ '(+' & . p -)+*1FN)+)j@ %2)2 *- * *- JZ- * s#)+. ?g .()2S * !)+*)+TC1!-%2)2 *- . , )+%2 *HZ %2! %+!*- )+* .0!)& .(* *- ! PC%+E *

d n

dt= B∇‖

(j‖

eB−n0 u‖

B

)+ n0 K(φ) − 1

eK(pe) .

$ '( :: &

# -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

)2! ) )21!4.()2 %2 r )+%2)+!*-)+* .4.()2;V#.af JN !)2* *8.( .()2;l × p @ 1 2s#)+*.()+-UN )+1!!R*. . *)2!.( d

dt=

∂

∂t+ v · ∇ .

$ '( :dc &?g.()2V %2)+!*-/* !% *-/. 1 6 -* %2. 10-)+%+

∇‖ =B

B· ∇ =

B0

B· ∇ − 1

B2B0 × ∇A‖ · ∇ ,

$ '( :Ue &s )WE * .1 67-* %2.

B = B0 + B⊥)+* )+** *- 1F!* *. - 1!F!* V* )+%

( - 1!3 %+ "$)+! )+* -)+%+#g0.()2 3 %+%2%2%S E 3 ** )+* 1_JN1z

z‖ =B

B· z 17s#)2

∇ ·(

B

Bf

)= B∇‖

f

B

$ '( :UO &g )+* %+)2)+- 1FN% FG

f

?!*;s#. )+* .S * !)+*)+TC1!-%2)2 *- $ '( :: & * .()2# *)+!)2* . 3 %+%2%2*4M!Ey@.()2 1!- *\Z j‖ = e n0 (u‖ − v‖) .

$ '(qc9 &

)2yX **3 % !)21 !)2*1.(!)& *.l )+*- *- )+*7.()2yn 1)+*!%+)+TC1! .()+*-*-s %2 .()2)^f !- *HZ )+ )+ . 1E 1!E .()2 * 1!)2 Z )2

∇ · j = 0 .$ '(qc &

)2\E )+s#)+.7- Js C !%2)21!4.G 1!)2 $.1 % 1\E *)2 %21 *HZ 1 <C.( *. %2JN !!)21 *!%%2)+,X * .4]4)+! )21 .()2 1p .()+*-*- %214VE )+3 % !)+TC1! .()+*-*- JN** +7O '9 )2bE )+)+ ;f !*.** %2JN !!)21 *4?%2. JZ)2 * E */ *( %+%21 %21E )+3 % p ).%M!E .()2 $ '( I & * 1 )2.%E * s#)2 .YZ s#)21 *$3 %+%2%2 *. 1*HN 0)2 !*- Z HE 67-* %2.

∇ · j‖B

B+ ∇ · j∗ + ∇ · j = 0 .

$ '(qc' &6 )+ .JE V1.(!FN g.()2 % !)21!)2*11!!E .()2 $ '( :' & *. . *)+!)2* . 1WS \gHEy@E *- 13 1 $ '( :5 & !-)+ 1!)2 * kp \gFN(1!)2 !)+-*- . !)& .(* *- .()2 _!!)^Z )j@TC1!-%2)2 *-

nmi

B2

d

dt

(∇2

⊥φ+∇2

⊥pie n

)= B∇‖

j‖

B−K(pe + pi) .

$ '(qcK & )2! )4-)+%+#g0.* 1*HN *$.()2** *. .* 1*HN *] 3% @ [3

∇⊥ = ∇ − ∇‖*. ∇2

⊥ = ∇ · ∇⊥ .$ '(qc8I &

1B = /$2 1 * b&*T! J&$6+ P$ P

?!* !!)+! )2 )+* . %2* r )+%2)+!*- $ '(+=c & .()2QV#.af JN !)2* . 1 2s#)+*.()+-UN )+.( .()2 .() E -*!)21 1 2s#)+*.()+-UN )+ *)2 YE ( 4. 1!)2 .( .()2 .() E -*!)j@1 *1SE 3 *** . 1D_)21\N 1!)+TC1!*11DN E 3 *1!)2!s#)+. +ADC 1DZ $.()2 1? WI *%"226`H*6W, 7y*. E * )+* '( ('' /M E )+,*.as#)2 )+* .#)2 -%2)2 *-$ '( :: & .()2bV#.af JN!)2* )+* . %2*4r )+%2)+!*- *;E )+ .;l × p @ 1 2s#)+*.()+-UN )+ 1!!1!)2 *)+!)2* $ '( :dc &

*Y_\s *.(*- . V 1.(\gTN $ '( :dc & $ '( :5 & 1=s#)2 $ '( :Ue & *. $ '(qc9 & !- * 1)2 )+*.!)& !*PC !*- 1.*7" C\E )+% *HZ* $ '(+5 & *. $ '(+: & .()2 , JE 3 !!-%2)2 *- *g0.()2l%2JN!** *.X **

3

2nd

dtTe =

TeB

e∇‖

j‖

B− peB∇‖

u‖

B

$ '(qc5 &

−5

2

peeK(Te) −

TeeK(pe) − peK(φ) −B∇‖

qe‖

B.

3

2nd

dtTi =

TiB

e∇‖

j‖

B− piB∇‖

u‖

B

$ '(qc: &

+5

2

pieK(Ti) −

TieK(pe) + piK(φ) −B∇‖

qi‖

B.

p )4. !%2)+!*- % 11* 1!)2 .()2 " C\E )+% *HZ* $ '(+5 & *. $ '(+: & HE 1 )+ *QZ

3

2n

(∂

∂t+ vα · ∇

)pα +

5

2pα∇ · vα + ∇ · qα = −Πα : ∇vα +Qα .

$ '(qcUc &E )+ . , )+%2 *1!

α4)2 , \E (#. * M )+ .()2 1 %2)2 *-Nd*** .**

s - *ω Ωi

*.me mi

- - *2g . )^f !- *HZ . 1 " C\E 11 1yf !* % C11!)+-s .* $ 1!)2 / 'UK M )+c9 & 6 )+ . *)+!)2* . " C\E %2)+\N kZ )2* * $ '(LKUK &\N ** QE *#.G g )+* 1!\NE -*!)21!)2! 1 % 1\E E )+

Ωα τα 1. 1*HN

V *)+%Z HE " C\E HG )Ê_!-%2)2 Z HE .() E -*!)21 * V* )+%Df !* % C11!)+- )21?!* s#)+. )4.!%2)+!*- * .()2D 76, 6+*-`H*6W, 7 \gFN(1!)2 !)+-Pl1% CG/1!)2 .Z)+- * .G 1!)2 .()2 .() E -*!)21 * V* )+%2 )+* . !)& !- 1 2s#)+*.()+-UN )+ *.)Ê " C\E HG )Ê ? %+%2 )+* 1 E - ** 67-* %2. 1 - - *1)+!)+- ( * 'I l1 -)+%+ )21!3)2%21Fs )21 g0.()2l %2JN !**

3

2nv∗ ·∇Te+pe∇ ·v∗ +∇ ·qe∧ = −5

2

pee

∇ ·(

B × ∇TeB2

)− Tee

∇ ·(

B × ∇peB2

).$ '(qce &

M E )+4!TC- .()2.() E -*!)21 !)& !- 1 2s#)+*.()+-UN )+ )+* )+*JE E - ** 6 -* %2. *)2 Z HE , *1!3 !,f * " C\E ) HZ s 1 %+)2- N )+* .()+JNDV#.af JN !)2* . , JE 3 !bf

# -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

!

)21 )+* %+%2*/%2)2 *- * .()2 3 %+%2%2X **- 1 2s#)+*.()+-UN )+0( */!)+-E * )2g * )+*p 1!!)ÊhE *- 1!-%2)2 *- )2\Z E %+!)+3%+)^Z )2!E * .()2~XBE 3%21!)+% *HZ*$ '(+7K & *.$ '(+JI & gX ** *.yl %2JN !**yE )+ .JEzl)+* )+1\f JN . 1 67-* %2. 1

B/B*. ..()2!*1 %+)2JG *. .()2 )2.* 3 %+%2%2*XE 3%21!)+% *HZ*4 )0f !* % C11!)+- E *s#)2 . .()2 , TC- )+1!\E .5l %2JN !**wE )+

me1=s#)2 .()2#3 %+%2%2l %2JN !**2f )21\N 1!)+TC

E )+! PC%+E * .()2 3 %+%2%2 1E !)Ê 3%21)+% *HZ

n0mid

dtu‖ = −∇‖(pe + pi) − µi‖∇2

‖u‖$ '(qcO &

E )+ .03 %+%2%2*mSE 3 ** . 1_)21\N 1)+TC1!*11 g X ** - JE CG +7K µi‖ = 0.96ni τi .

$ '(Le9 &)2 %2JN !!)21 * V* )+%2 * 1)2 ) .hV#..()+!)2* . 3 %+%2%2* p )+% *HZ* s - * . 1* 1 )2 .(%+)2 * _\Z)2 *1)+* .]8.(*-/*. .()2 )+*- 1\E - JE CG $ '( 'dc & (!

p )+%2.YE * 1 .* 3 %+%2%2* XBE 3%21!)+% *HZ* .Yl %2JN !**$*.1X ** )+*/%2)2 *-g .()2 % !)^f !)& !- 1 2s#)+*.()+-UN )+

u‖ − v‖1 - Js#)+** wE * .1/%+%+- JE )+* [ HE 1

1\Z Hs %2 1.()2 3 %+%2%2%M!E .()2 YE )+.JE 3 %+%2%2* %2JN !!)21 *?%2.f \N *2g3(

e n0

∂A‖

∂t+me

e

d

dtj‖ = ∇‖pe − e n0∇‖φ−Re‖ .

$ '(Le( &)2 3 %+%2%2 M!E .()2 )21!g .1DVE 3 1 1\Z E )+ .JE 3 %+%2%2*_JN !3 @* !) %f !*.*

j‖ = − 1

µ0∇2

⊥A‖ .$ '(Le' &

?!*ks#. )Ê [ HE 1 * 1\Zs#)2 )+* . 3 %+%2%2*XE 3%21!)+% *HZ %2)2 *-$ '(qcO & .()2 -U 7f )21\N 1SE 3 *1!)2* *\Z r1EhE *%E )+ .* 3 %+%2%2* , *1!3 !\N kZ )2**$ '( ': & *. $ '(Le9 & )+%2.* .()2 %2)2 *a@- * $ '( :: & $ '(qcK & $ '(qc5 & $ '(qc: & $ '(qcO & $ '(Le( & *. $ '(Le' & )+* - 1 %211* 1 %2)2 *- 1R@1\ 1JE s %2 1 .()2 * E )^N$. 1\N% *~? %2. (

n, φ,A‖, Te, Ti, u‖) 1 )+ )2 1 1

+*- !<6 ^DD0* B ,s#.$1 .* %+%+- JE )+** , *1!3 !!-%2)2 *- * .( .()27!)& !*PC @!*- - Js ***4 01!)+*. 1)+* %2)2 *- *7 %21 .(HZ )2!p -)+*1\N )+)j@ %2)2 *- * JN** '5

!+"' *& +# #) + & +

X * .()2 1JE V1 *)+!Ws#)+. .1 7B<* DD/* ! Pf !- 1!%+%+<s %2 1 1!)2 1.JE M 1 %2.'@E .%+%4 !%2)+Ps **QE * .()2 , JE 3 !1!F!*- *f !* % C11!)+-V * *. . 10%+)+*!)21!)2!*_)2 %2.aE .%+%21 s .* .()2Z -!*.%+)2- *.* %+)+** Z 1R@1 *. .()2 .E )+hf !*.**.X *1!)+%+)+TC* .()21\N !)2! )bs#)+. .()2 * E )^N )+*)+*HZ%+**s %+%2*( \E )+- *1MF!*- !

5 $2-H$m|Y8) d$2979=$m( $8$&*(1)+*$&& 0

?g .()2Qs )+ * 1! *- . 1 , *1!3 !1 )+* ./% 1\E *.1 )2 1%+%2* )2/.()2MF!*- * )+* . , JE 3 !

Tα*)2 \gFN(1!)2 !)+-;s .*4 ** -)+%+Z p g5MF!*- *

. 1 l %2JN !**.(!FN(1pe = Te n+ n0 Te =⇒ pe = Te n .

$ '(LeUK & 1 ..G/.()2 )2.*mp 1!!)ÊhE *- 1!-%2)2 *- * g .()2 , JE 3 !*. )+ MF@!*- * $ '(qc5 & *. $ '(qc: & *)2 WE 0 */!)+- s .*4?!* s#)+. .()2ADC !*-kf *QN%+*X ** *\Z

Ti TeHZ s

Ti ≈ 0 .$ '(LeI &

E )+ E%g11* )+* .* f !- 1!%+%+* %2)2 *- *N )+*Dl JN HZ -% ()+* 1#*.(%+)2 *X **a@-U!)2*1!.()+1 \gFN(1!)2 !)+-s .*4 )2!)& !% C*- $ '( 59 & N** .** %E )+ ρs=c /Ωi1!- .(\gFN s .*/s )8.()2YM %+%+- 1 s#)+*.()+-UN )+ - - * )210.(

c =

√Temi

.$ '(Le5 &

)2N *1!* )+* !-!*.(JE 3 !Te

. 1 1!)2 )Ê %2)2 - Js#)2 *.*.* % 1R@E 1bs#)+.s#)2 *1 .()2 )+* !-!*..()2

n0 . .1067-* %2.

B%210 E 1!

f !- - *4 , \E .()2 .* X **.(!FNpi*. 1)+*MF!*-/* ! %+*af 1 2s#)+*.* .8@

E )+ *1hs#)2 .V1.(!TN g .()2 3 %+%2%2YX **2f )21\N 1!)+TC )+* . 1E !)Ê 3%21!)+% *HZ$ '(qcO & .()2 1Wf )+*( * %2)2 *- * $ '( :: & $ '(qcK & $ '(qcO & *.$ '(Le( & - % *-E *Z HE 1- * *** N*B<* DD/* ! $Id? 6 & g.()2 1FN% *?%2. (n, φ, u‖, A‖

) )2 %+- *a@.* %2)2 *- * 1 )+ * .()2 *)2 !%+)+* * E )^N . N %2)+*1\N%+)+- *MF!*- *4

d n

dt= −v · ∇n0 +B

(∇‖0

+ ∇‖

)( j‖

eB−n0 u‖

B

)+ n0 K(φ) − Te

eK(n)

$ '(Le: &

mi n0

B2

d

dt∇2

⊥φ = B(∇‖0

+ ∇‖

) j‖B

− TeK(n)$ '(Ledc &

∂ A‖

∂t+

me

e2 n0

d j‖

dt=

Tee n0

∇‖n0 +Tee

(∇‖0

+ ∇‖

)( n

n0− e φ

Te

)− η‖ j‖

$ '(LeUe &

mi n0

d u‖

dt= −Te∇‖n0 − Te

(∇‖0

+ ∇‖

)n .

$ '(LeUO &

)2! ) s#.)+* .* %2)2 *- * $ '( :: & $ '(qcK & $ '(qcO & *. $ '(Le( & .()2 , ** *- . F@G * )+* )+*!-!*. *. MF!*-Z p n = n0+n

i 3%+)^Z )+ \gFN(1!)2 !)+- *. 1s#.* *, \E )+* 1 *.wZ s )+[.(* *- )+* .*QMF!-FG *4E )+!- *EhE *4H?g .10%2)2 - @s#)2 -)+%+HG .JE

φ0 = j‖0 = u‖0 = 0

# -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

r 11* * 1 )2.( \N )+4f * %+)+** *. *)2 !%+)+** , \E * s#. )2 .()2 %+%2% %2)+!*- )+* )+ * *- 1F!* *. - 1!F!*V *)+% ( - 13 %+

∇‖ = ∇‖0+ ∇‖

E )+ ∇‖0=

B0

B· ∇ , ∇‖ = − 1

B2B0 × ∇A‖ · ∇ .

$ '(LO9 & .()2kl × p @ !)& - .E )+ * .()2wMF!*- . 1 %2JN !!)21 * * !) %21 )+* .*1N *a@f JN!)^f *V *)+%8.#%2*r )+%2)+!*-wE )+ )+*

d

dt=

∂

∂t+ v · ∇ , v =

B × ∇φ

B2.

$ '(LO( &WX *. iQl s#)+.Qf * * *7* )+* .r )+%2)+!*-4s -- % 11*4)2A )2 !%+)+*!)+TC*7 @*.* 1!)2 %21 )+*/.* , \E * . V#.af JN !)2*4f * MF!*- *

v ·∇f17s#)2)+*/.* , \E *

E )+ . %+%2% %2)+!*- . 10- 1!F!* 67-* %2.1 ∇‖f‖

?g )+** N \Z* !%+)2FN ! QE * Z *PC 1! )+* % 1\E E )+ )+*JE E - **6 -* %2. /s %2 1)+*N! 1!)21 *S.()+* *E )+

Bx = By = 0*.

Bz = B = *1!

- - * )216 *QE )+** "$%+%2* *1\ZYE )+Ef .1' N**%NXY$D/*

f =∑

kx,ky,kz

fk ei (kx x+ky y+kz z−ω t) ,

$ '(LO' &s ),Z s#)21 * . 1*HN * )2 !*- Z HE 67-* %2.

k⊥ ≡ kx ex + ky ey*. .

3 %+%2%2* 0)2 !*-k‖ ≡ kz ez

* 1 )2 .*ms#)+. 6-/&&"%$'&&$' p ! bE * .1 )+*/ - JN **HZ)2 * * \E .%+% . 1 %+)+*!)21)2!*4_)2 %2.aE .%+%21E )+ .* %2)2 *- * $ '(Le: & *. $ '(LeUO &

∂ n

∂t= −v · ∇n0 +B∇‖0

(j‖

eB−n0 u‖

B

)+n0 K(φ) − Te

eK(n)

$ '(LOUK &

mi n0

∂ u‖

∂t= −Te∇‖n0 − Te∇‖0

n .$ '(LOI &

*. 1\Z .* "$%+%2* *1\Z $ '(LO' & g0.()2YMF!*- * *1 ! PC%+ E *

−i ω n = −i k‖n0 u‖

−mi n0 i ω u‖ = −Te i k‖ n .

5 $2-H$m|Y8) d$2979=$m( $8$&*(1)+*$&& />

)2hMF!*- . 3 %+%2%2*X **- 1 2s#)+*.()+-UN )+u‖

N**7* * .( .()2 )2 1!F!*-n1!- .(\gFN s .* *. HE - JN ! 67* ! PC%+ 1 %+)2JG%+)2 .*V1.(!FN

ω2 =Temi

k2‖ ⇐⇒ ω = ±c k‖ ,

$ '(LO5 &s ) .()2/FG * . MF!*- 1!- JNg\Z %s#.* *.E )+ $ '(Le5 & %21%]1!*- .()2 )j@1!3 1)2*1!% !)2* g M %+%^s %+%2* 1 )+* 1 - JN **HZ)2 * 6 .%+%0 1 )+%21 .()2V1!)+!*-kf *M %+%^s %+%2* )+* 3 %+%2%2 0)2 !*-wZ HE 6 -* %2.

6$7 " & $'&&$' l )+*ys )+ 6 .* 1)+!*- )+*73 %+%2%2 0)2 !*- ! PC%+E *E )+ %+- *.JE * R@E .%+%8. %2)2 *- * $ '(Ledc & *. $ '(LeUe &

mi n0

B2

∂

∂t∇2

⊥φ = B∇‖0

j‖

B−TeK(n)

∂ A‖

∂t+

me

e2 n0

∂ j‖

∂t=

Tee n0

∇‖n0 +Tee∇‖0

(n

n0− e φ

Te

)−η‖ j‖ .

6 )+0.* E \N)2!* , \E * *. .JE "$%+%2* *1\Z ! PC%+E *g

ωµ0mi n0

Bk2⊥ φ = B kz k

2⊥ A‖

−i ω A‖ = −i kz φ.()2 )21!3 1)2*1!% !)2*

ω2 =B2

µ0mi n0k2z ⇐⇒ ω = ±v k‖ .

$ '(LO: & )2! )Hs#. .1;VWE 3 1 1\Z $ '(Le' & g .()2WMF!*- * . 3 %+%2%2* M!E .()2 *\Z 17s#)2 .()2 *)+!)2* .hV %&fdo*a@ 1 2s#)+*.()+-UN )+ $ '( I : & )2 )21!3 1!)2*1!% !)2* 1 )+ )2 .()2 V1!)+!*-f * !*1\f 1%2* V%&fdo*a@ "$%+%2* )+* 3 %+%2%2 0)2 !*-)2 l )+- *1 .3 %+%2%2*1V 1)+!*-/)21! .( .()2 1*HN *MF!*- *$. 16 8@-* %2.1 - - *s %2 **4g

A‖1!- .(\gFN bs .*4()2 1"$%+%2* JZ)2 *bE *

%21WM R@BV%&fdo*a@ "$%+%2* *.s#.* )+10)+*QV)+%2.(*- '(LK 1FN)^Z Z )2!

&$ 7)P97 7-a =9=6$m! #"%$&&$ )2 )+*( 1!?\E )+* Z HE 67-* %2. 1*HN *mV1!)+!*- )+*MF!*- )21!0.()2!)& Bs %+%2 )2\Z ! 4E * .1 - JN **HZ)2 * *\E .%+%0. %2)2 *- *

t) # -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

$ '(Le: & *. $ '(Ledc &

∂ n

∂t= −v · ∇n0 +B∇‖0

(j‖

eB−n0 u‖

B

)+n0 K(φ) − Te

eK(n)

mi n0

B2

∂

∂t∇2

⊥φ = B∇‖0

j‖

B−TeK(n) .

l )+*1\Z* . Z s )+* %2)2 *- )+* .()2 1!N** .()2 3 %+%2%2M!E .()2 j‖%+)Ê )+*)2!Qs .*4r]1!*- . 1 %2)2 *- 11\ 1JE 1 */!)+- E * * )+*s )+

%2)2 *-s %2 .()20)2 @ *. * !) %21F!*-hZ )+* *. )+* % !)2* !)+*- )2gs#)+. .()2V#.() !)^Z )+TC1** HE g l %2JN!** - JE

n

n0=e φ

Te.

$ '(LOdc &XBE %+%+- JE )+**kf 1 2s#)+*. . 3 %+%2%2 .()2* ∇‖0

)Ê [ HE 1 */ 1\Z $ '(LeUe & *)2 1 .G )+*/. \Z %2)+*.* )21!3 1!)2*1!% !)2* .()2V PC*-)+-UN )+ f *kz

( *wsbgR@. . - E * )!)& Bs %+%2* .7f * 1.G 1!)2 .()2 .() E -*!)21 * !)& !3Z 11s 1*!%+)2 % *- 1E 1!3)2%2* %21 .()2 3 %+%2%2 W * E )^N .l %2JN !**42[. *.11\@- .(\gFN )2,V#.() !)^Z )+TC1 .()+*-*- $ '(LOdc & s %2 #*)2 1 *. 1 )21 %21 .()2 %+)+*!)21!)2!?\E . p %+\Z E **-%2)2 *-$gYl%2JN!** )Ê ! \E )21 * %2)2 - Js#)2 .(\gFN .()2, 1 1 .G .()2 l%2JN!** 1!)2 /)+* 3 %+%2%2 0)2 !*-%Z HE 6 -* %2. ) Js - *Nd*** *. )+* .] - 1!)+*. 1 1 *%+%4)+ 0! \E )21 1 %2)2 - Js#)2 DZ *.* ': E )+! PC%+E * .()2 )21!3 1!)2*1!% !)2*

ω = − Te ∂xn0(x) ky

e n0B(1 + mi Te

e2 B2 k2⊥

) . $ '(LOUe &

p 1 !)2 *1s#)+.$%21 .()2hV1!)+!*- )+* )2 1F!*- $ *.s - * $ '(LOdc & -%2)2 1E * !) %21F!*- & 1*HN %Z HE 67-* %2. )+*y@ 0)2 !*- 6 * 1!3!)2 Yf * )+*

!)& Bs %+%2(. )+ ?*HZω ≈ ω∗

HZ s )+ V1!)+!*- 1!- 1 2s#)+*.()+-UN )+0. .() E -* @!)21 * !)& !- 1 2s#)+*.()+-UN )+ .l %2JN !** $ '( 5Ue & *1!3!)2 <l10-)+%+

ω∗ = −Te ∂xn0 kye n0B

=Te kyeB L⊥

=%s cs kyL⊥

*.ω∗ = |v∗| ky ,

$ '(LOUO &E )+ . *)+!)2* ..()2**% C*-

L⊥g

1

L⊥= − 1

n0

∂ n0

∂x.

$ '(+99 &r1EhE *E )+ $ '( 59 & $ '(Le5 & *. $ '(LOUO & -)+%+ .** g0!)& Bs %+%2*

ω =ω∗(

1 + %2s k

2⊥

) . $ '(+9( &

5 $2-H$m|Y8) d$2979=$m( $8$&*(1)+*$&& t+

6 )+ .,V ** HE . !)& !*PC !*-%s k⊥ ∼ 1

)21 .()2MF!*- 1 *HZs 1 - !)+*- %21.()2 .() E -*!)21 w? *HZ

ω∗ A**N EhE .( .()2y_\s *.(*- ._!!)^Z )j@

TC1!-%2)2 *- . n 1!)+*!%+)+TC1! .()+*-*-yZ 1!*.$Z s )+ - JN **HZ)2 *%2)2 *- & )2 1 1 )+#.()2 , TC- )+ .bX **'s %2 g .* % !)21!!)2*11!!E j f * Bs !!%+)2 )21!V * *. .()2 1 %+)+** *1! *- \N ** E * 1 *.G$ * .()2X * E - *)+TC. 1 % 1\E 1 HZ s * .1h_! *.*1)+* )+* 1 )2 -.()2* *

∇n0N )+* 1!)2 1R@

)+*. )2 1!F!*- $ *. * !) %21F!*- & f ! *.* 1)+*N**4 67* ! .()2l * 1! *-kf * !)& s %+%2* .( .()2DV#.af JN!)2* . )+* !-!*..()2 YE )+!%21#.bl × p @ 1 s#)+*.()+-UN )+ $ '( 5dc & p )#)+*hf ! *.** *!) %21!F!*- Js - * 1)2 .()2 l %2JN2@!** *. X ** E )+/. -%2)2 *.l × p5@ 1 s#)+*.()+-UN )+/HE .()2 678i )Ê *. 6 )+*)Ê . 1 %2JN !!)21 * *!) %21 !HE )201Df %+%&g !p Js -*- )+%2. )+**/1)2 .( *.*" )+! %4%2JN!!)21 - % .*. , )+%2 * ] - * ** $ 1!)2 V)+%2.(*-/'( I &

y

x

φ(x, y, z0)

I& ( #/0$'&'()"H $'$'&'( /%/H))"H" -& ( # +/0)5/0 $0/"% 4# # # 7$0/0- /0# ) / </# 8 +/' "0)- / eφ =

&#+ $0/# $')<5"%#&'(5/0)" :! )4) /0$'& ( #D #$%/')) 5/'"%+-)61 2" :

×A 7 AB B# #" #&'() "# "%#( "0& (/0 $%/

- / ) "%2) ;1 )) )<)4))& ( /#

p ) )+*TCHE %+)2 * *!) %21!F!*- E * 1 %21 E )+*1 )2 .(%+)2 1 *%+%2* *.)+* *1 )2 .(%+)2 )2 !*- .( *.* " )+! %+* . aD D N'Z !**Z * kVE 3%+)+!.*a@ !- *. _\Z)2 * . 1#%2JN!!)21 *?%2. 1 )2 1V#.af JN!)2*1! Js -*- * !)+*- * * * )V *s 1* )+ )+* 1)2 -.()2* * $ )+*7.()2 1JE p )213)2% )+*7.() %2 0)2 !*- & / % 1\E .()2 )+* .*kp )2 *)2 .(!)+- % 1\E .()2 *. HE - JN ! 1 .G ..( )2 a@ 1!F!*- * *- -bs .*/s %2 1!)2 - JE CG .0)+- *Qp ! !*- * )+*

y@ 0)2 !*-

%21 !)& s %+%2* 1!)+*4 "$- * ./.() !)21 * N !)2* .wl %2JN !** %+- .)2 1!F!*-

n*HE )+!%+ #*. )+*1!** .()2 * !) %21F!*-

φ)+* .()2-%2)2 V1!)j@

!*- 1!!)2 !*-

M s#)2 ).* M %+%^s %+%2* *.~M R@BV%&fdo*2s %+%2* )21! $ )2 .()2k? *HZ .%2JN!1!!)21 */!)& Bs %+%2* )+*#)+* %+%2FG )2 @ *. *!) %21!F!*- *.*1!)2 s - * .hV#.() !)^Z )+TC1** HE $ '(LOdc & )+* 1/*. )+* 1!)2 *- 1F! )+*

y@

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t< # -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

"2CI2*"c, BI* , *1!3 !1 $ 1!)2 V1 *)+!'( ' & )21!0- - * .(

〈Γ 〉t = 〈n v 〉y,z,t .

$ '(+9' &X * .0 )2 ! *?!)2.1!%+%+*- !-)+ 1!)2 .E )+

Γ ∼

∑

k

kyXE

nk φ∗k

.

M)+*. )2 *. * !) % )+* 1 1 !-)+1!)2 *!hN )+* JN !)^f 0.() %2 , )+%2 *a@HG)2 1#)21! )+*QV )+%2.(*- '( 5yf *1 %+)2

x

y

∇n0

B

E # $0& (/ )" +/')/' $%/ "%*+ >' I *+ + )> )"%<# )H$' $0& ( "04)) :

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"0&'(5/0)> "02 )"( $ )"%<)5/ 8 +5/0"%) ))" $'& ( 4) +/')/' !!( $0 & () & (5/0)$0/"0 = "%+/$%/ $ & (5/0) $ & ( 7/')- "%)$0/0# /# en eφ !($'-$' @() ) $'&'( AB) /0" )" "' #" & (/' *)</')"9/') & (5/0 "02 ;1 ) D; & ()- /%/')

〈〉y,z

1 $'+ -<) <+*#</0 *#" "' $ +"%/ )5/0$0/0)(5/#

M)+%2(%+)+*6 .*/!- * .JE * *)2 bZ HE .() %2* , )+%2 *!*1!3 !0 ) 'p 1! )+*- - * )+* 1*2f 1 )2*- ..( .GQZ HE p )21!3)2% .()2 .() !)21 N !)2*. l %2JN!** *)2 E - - * )21!(1 g !#.()2 1 Z )+*JE .() %2* , )+%2 * !*1!3 !s#)2 )Ê %+- *.*V 1 *)+! !% C!Ws#)+. !

p ! ,s .*/* * 1!)21!)^f 0!)& s %+%2* . 4 .()2#3 %+%2%2]4)+!C )+-UN )+ .;l%2JN!**s#)+. .( .1 %2JN!1!)21 [ HE 1 1\Z $ .(!)+! - JN **HZ)2 * %2)2 *- &E )+! \gTN 1!)2 !)+-

∂ n

∂t= −v · ∇n0 +B∇‖0

(j‖

eB−n0 u‖

B

)+n0 K(φ) − Te

eK(n)

mi n0

B2

∂

∂t∇2

⊥φ = B∇‖0

j‖

B−TeK(n)

∂ A‖

∂t+me

e2 n0

∂ j‖

∂t=Tee n0

∇‖n0+Tee∇‖0

(n

n0− e φ

Te

)− η‖ j‖ .

5 $2-H$m|Y8) d$2979=$m( $8$&*(1)+*$&& tPt

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XBE ω ∼ η‖ .

\g )+* 1/\N ** E * .G$. XBE -)+*PC!)+%3 1)+!)^f . *)+ )21XBE ω > 0 1

. .G )+**( C*-%+)2 MF!*-$ !)& Bs %+%2 & )+*3 1!)+!)^f V*2s 1! 1!)+\Z *.1!)2 7.()2yMF!*- E )+DZ * HE *.%r )+)ÊhE s )+Df 1!TC\N )2 1!)21!!)^f )+TCη‖

g !Z )+*0 1*2f 1 )2*-wZ s#)21 * .#)2 @ *. *!) %21!F!*-1 .G .()2 )2 a@ 1!F!*- * * .JE %2JN!!)21 * * !) %f 1)+%+ E )+ !-)+ 1!)2 * $ '( 5( ' & )+* .() %2 , )+%2 *HG;s %2 1!)2 *1 %+)2 .( V )+%2.(*-$'( :f 1 * % CGXE _!-%2)2 Z V )+%2.(*-$'( 5 * 1!)2 ** .()2 .( .()2 " )+! % .() % * $HG *7*. )+***7- !)2 * , )+%2 * Js -*- * $V#.af JN !)2*7.)2 & )Ê ? %+HG <C *HE )+!% *)2 WE f %+%21!TC*.()+- (

x

y

∇n0

B

G= 5/0)"0$0& ()H4) ' E )/ )$')"5 /' /0+ # " -& ( !( $0#*)" 7$'& () 4 $'& ( # & (/0 7 +/0)5/0 $0/"0 6*+"# 13"% )" )& () )$0$0/0 * / /< +/')/' $%/ "% )" & (5/0)$0/"0 & (/ -)("- $0/'5/'

y 7 & (/' 13+)>+- / )"%) )) /')) - /H( ()"%#" &'(5/06-)(" )& () )6"0 & ( 7/' *#</0#"9/$ , ))"%& (5/0 = D &'()- /0/0) #"% /H$'& ( # +<)" "0 )" #&'()5/0"' $ +"%/#

)2WZ Hf )+* )+*.() %2 0)2 !*- !*1!3 !!)2!)2 Ds#)+./* * f 1!TC\N )+* .()2 10)2 a@!*-!*1!3 !!)2!6 )+ !1 )+*.5r )+s 1* %21.()2,VWE 3%+)+!.* . 1!)2 1!)j@*.*/!)& s %+%2* )ÊhE 5s )+ *4 E )+;s .* 1 .*kN %2)+*1\N%+)+- * 6 .* i(3 **a@!)2%+%*s 1*. X *1!)+%+)+TC*s %2 #.( .* )+*!-!*.(-.()2* * . % 1FE .()2 a@ \Ek-%+)2 HZ s .( .()2s#)+! % !!)+- V#.af JN !)2* .0)2 *- !!)2 *s .*4 ! !

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t # -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

∇n0 F g ∇n0

F $'& ( & ( :!"%<"% )" #.)( 7 #.+" 7 =$0/' / /G AB )/ $'& ( # $0& ( )"0 D 2$0$'<) / 2 )" ))" )& (5/0) D 2$0$'<) / $0+ $%/ )$')$ 5.$0/0) ) )$')() / )", "0#* /'/'+$0<"01 / $%/

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g*. f * . , !1\E )+! $

R = 0 & Z , !1 HG *1)+ )+* - !)2

F g ∼ g V* %2-Z HE p )21!3)2%. %2JN !!)21 *- % .**~? %^g11!)+-UN )+* )+* V )+%2.(*- '(qc

-)+ 1 )2 )+**p )2 )+*7.JE .!FN -.()2* .* * .()2 %+%2 . 1)2 -.()j@* * g !*)ÊhE )+*7.()2 -%2)2 0)2 !*-Qs )21!Ds#)2 .()2 JN !)^f hr *!!)& - %^N )2 1p )2 *. 1!)2 )+* )+*JE 1!)+%2* r1!*. E - JN ! f ! PC%+ 1 1!)2 ( . , @!1HG *1)+ !)21.()2 S . 1 % 1\E .(!TN 1 . r * !!)& - %^N *!- - *- !)2 1/.G 1!)2 ( -!*.mf *mE )^N1FN 3)21 *M s *HN *- * . 1 % 1\E .(!FN(1 $ )2 *., JE 3 ! & V1!1 )+*1!)+%+)+TC* *Bs#)2FN %+*Nd***4 /.1)+*1 %+)2JG *. )2.%2 6 -* %2.f * )+*** * HG *mE )+

1/R( C%+%+ $ 1!)2

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)+* )+*JE % 1\E !1.()2 1!)+%2* *. )+*1!)+%2* -)2**g .()2 67-* %2.(%+)+*)2*E )+)+* *.f !*.* 1!)+*. =Nd*** 1!)2 TCHE %+)2 -G 1FN%+)+- MF!*- * *Bs#)2FN %+*.*VE 3%+)+!. )Ê p )2 . 181 2s * 6 -* %2. 1 1TC\N 4)21!67* 1!3!)2 f *LH B B0J@X *1!)+%+)+TC*4)2 1Ds .* )+*V 1 *)+!'( :(LK - * )+%+%+1!!!)2!" )2 ) .* !)& s %+%2* g * )2 %2N%Wf ! *.*MF!*- * . )2 *., JE 3 !WZ N %2)+*1\N%+)+- *X *1)+%+)+TC*/s %2 )+** .() %2* , *1!3 ! f * , )+%2 **. " C\E 1*HN Z .* ? %+HG <C *Qf !1 *4 )2 1 V1 *)+!bg bV1!1 a@)+*1!)+%+)+TC *. .V1 *)+! '( :(LKWg p#%+%2 *)+*-@BX *1)+%+)+TC)+%2.* .()2# !*.(% - Z HE_1!TC*.(*)21. 10!*1!3 !\Z- *.*7 Z 11 1#)Ê p 6KW@ 6 .%+% $S 3)+%K &

5 $2-H$m|Y8) d$2979=$m( $8$&*(1)+*$&& t<

∇p0 ∇p0

Bϕ ∼ 1

R

F g

R , #+- /0"0$0& ( "%$0/0) )";A ++ 7 =$%/ / /! )) +<< $ 7 $ )$'& ( ) /'+"%+ )/%1 # 1 /@ "0 )" &'(5/0 - /

1/R 5/"%)

$' AB)"0)& ( )$ $%/ "%<) /%13))$ $0/' 3"%2 + )"%) AB)"0)& ( )$! $'/0+"%$ + $ /01 ) $0& ( & (6$0/ $0/' "0+/0 +

l )+* %+)+*,ADC !*-g .1 %2JE )21! * E )+s )+!- *.*h_)+*( *- *wEk-%+)2 4V )2 )21s#)2 ..15r)2% .( )+* * 1 *- . )21!3 1!)2*1!% !)2* .()2 %+)+**V *s 1* **4Z Nd***4'V%215p )21!3)2% ! E */.1,M(1!JE . 1!)21!!)^f *6 W@Bp %+%2 *)+*-@ 6 .*

∂ n

∂t= −v · ∇n0+B∇‖0

(j‖

eB−n0 u‖

B

)+ n0 K(φ) − Te

eK(n)

$ '(+9UK &

mi n0

B2

∂

∂t∇2

⊥φ = B∇‖0

j‖

B− TeK(n)

$ '(+9I &

∂ A‖

∂t+

me

e2 n0

∂ j‖

∂t=Tee n0

∇‖n0+Tee∇‖0

(n

n0− e φ

Te

)− η‖ j‖ ,

$ '(+95 &s ) * . %2JN!1!)21 M3 JZ ) %& %+% * 1! hs .* 1%+% $ . , \E

∂tA‖)+* .

.(!)+!* %2)2 *-ws#)+.A %+% - 1\Z & X * . 67-* 2(.( .a* E )^N $ 6 & )21 . % 1\E .(!TN - - * .( p = pi + pe

?!* -)+%+ .1Wf )+*( 3 %+%2%2 [ HE 1 1\Z

E‖ = η‖ j‖<s 1 .E )+ )+* !-

.G .()2yVWE 3%+)+!.*7. 1 %2JN !!)21 * * !) %21 11Wf )2% -FG 1)+*. %21 .()2 VE 3%+)+!.*. !FN(1!F!*- * $f -% %2)2 *- $ '(+95 & & Pl10-)+%+0.!

p

p0 e φ

Te

HZ s n

n0 e φ

Te.

$ '(+9: &)2 1p .()+*-*-/)21! %21;XH9;%7d?=2-'.10 JN** XBE HE * .()2 1 V ! )+/*. . 14_)2 %2.aE .%+%214s .* , JE 3 !1!F!*- * *)2

t # -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

! 1 .GyE )+Te =

*1! *.Ti = 0

.()2 !TN 1F!*- * .( )2 1!F!*- *1!- .(\gFN %s .* Nd***48 1 6 .%+% . %+)+** %2JN!1!)21 * 1!)21!!)^f * 6 W@p %+%2 *)+*-@ 6 .* *)ÊhE 0.** .()2)+*/ - JN **HZ)2 *? \E *4HM E )+,N**4E *'s#)2s )+ ** - 1 */E )+

p0 + p = (n0 + n)Te.()2 )2 -%2)2 *- %21 %2)2 *- g.* % 1\E .(!TN *1 * 1 !*E * .()2 - @1E 1! %2)2 *-wE )+0.Wl %2JN!**JE 3 !DE %+!)+3%+)^Z )2!?g .()2ms )+* p ! !*- * s .* , )2.%^N .()+* * f \s *. $ 1!)2 V * *-V + & *. Z h_)+*( *-~s .* MF!*- * )Ê p )2 . 1 1 s * 67-* %2. 1E )+

θ = 0 ! )2\E )+ f )+*( /1)2 .kS \gHEhE *- 13 K ( %+- *.

"$)21

sin θ ≈ 0 , cos θ ≈ 1 =⇒ K ≈ 2

r R0B0∂θ ≈

2

r0R0B0∂θ .

E )+ ! bE * .*4?%+% . Z %+)+*.(!)21 *AC !*-)+* . )2.%2DS \gHEhE *- 1R@ JN *)2 ;E )+! \gTN 1)2 !)+-,s .*42]CG;E * !*.()2 _1 !*- .E -*!)21 *?%2.(%+)+*)2*7HG $

q(r)=q0 & 1yN**E *E )+ )+%& . _ !) %2*!*1 \E !)2*

θ′ = θ − 1

q0ϕ , ϕ′ = ϕ+

r20q0R2

0

θ$ '(+9dc &

.*V 1.(!FN g0.()2 3 %+%2%2V %2)+!*-/)+* $V LO & f )+*( *4

∇‖0=

1

R0

(∂ϕ′ + O(ε)

).

$ '(+9Ue &6 )+ .()2 1;_)+*( *- C%+%+ * * .()20 )2 !*-Yf *

ϕ′ E )+ . )2 !*- . 167-* %2. 1Z 1EhE *46 )+ .* %+)+*!)21!)2!* %2)2 *- * *1! E * s#)2 .mMF!*- * )Ê % 1\E .(!TNHZ s )+* . % 1\E .()2 *. )Ê %2JN!!)21 * * !) % E )+ 1 2s wVWE 3%+)+!. 6 )+.JE ?!)2*1\Z -)+%+#* * g0.()2YMF!*- *

(φ

p

)=

(φ

p

)ei(krr+mθ−nϕ)+γ t =

(φ

p

)ei(krr+m′θ′−n′ϕ′)+γ t ,

E )+m′ = m+ O(ε2) , n′ = n− m

q0.

$ '(+9UO &6 )+

γ = i ωs#)+. .()2YV*2s 1! .0%+)+**X *1!)+%+)+TC JZ)2 * M \Z E * .()2 1*

V *1\Z )+* .()2 )+- * E \N )2!* %2)2 *- *7)+*1 ! PC%+ E *

−γ k2⊥ − B2

0

η‖0n0mi

k2‖ i

2B0

n0miR0kθ

ip0

B0L⊥

(1 − 2L⊥

R0

)kθ γ

(φ

p

)= 0 .

$ '(+9 &

5 $2-H$m|Y8) d$2979=$m( $8$&*(1)+*$&& t

)2! ) -)+%+#g0.()2 "$%+%2*HZ %2*

kθ =m′

r0, k2

⊥ = k2r + k2

θ , k‖ =n′

R0.

$ '(+ &6 )+

L⊥)21bs#)2 ..()2DN *1!*)+* !-!*.(-.()2* *% C*-

1

L⊥= − 1

n0

∂n0

∂r= − 1

p0

∂p0

∂r.

$ '(+' &g0.* ! * .() %2*mp )2 - JE )+* )2]1!*- 1! .()+*-*- %

γ2 + γ

B20

η‖0n0mi

k2‖

k2⊥

=2p0

n0miR0 L⊥

(1 − 2L⊥

R0

)k2θ

k2⊥

=1 − δ

τ2int

k2θ

k2⊥

.$ '(+7K &

)2! )<s#. )Ê %2\Z *M !)+!0.()2DVHNg\Z *-δ = 2L⊥/R0

* \Z 01=s#)2.()2 N @!)21!!)21 YV 1!1 HZ)+

τ

τ =

√2miR0 L⊥

Te

$ '(+JI &)+*- g !p ! E *4Z s )+*4_)+*( *- * 6 .*/.bMF!*- *'s %2 *!% *- .6 -* %2.(%+)+*)2* N *1!* 1!)+*. g .()2 %21

k‖ = 0-)+%+ 17! PC%+hE * %21h]1!*- f *

$ '(+7K & g0.()2YV *s 1*γ =

√1 − δ

τ

kθk⊥

.$ '(+5 &

)2 V 11 HZ)+ )21!8.E )+ .()2 N!)21!!)21 r )+1\N% 7E )+ . .()2%+)+**yV1!1 a@)+*1!)+%+)+TC* *s 1*4X * .()2 1JE ! * ? %+% sbg. %21 .()2wV*2s 1!E )+ )ÊhE / * 3 %2)2.%2*"$%+%2*HZ %2* )+*JE N *1!* * MaC!!)+-*- 1\s ! * !- - *1!! *4DX * . 3 2(1!)^N%+)21 *" )+\N %+)2 HN )+)21! 1 1 .G .()+*- .( N )+*!)21 )+*- 1 JN Hs %2 )+* .JE )2 ! *.? %+)2.aE .%+%#*)2 1 !)2 * s .*.()2 %+)+** V *s 1* gN%2)+* TCHE %+)2 MN%2* s#)2 . * HE *4 WE .()2 1*.M( f ! %+wZ \gFN(1!)2 !)+- * E * .()2 6-%+)2 HN )+)+*N'g*1!!%+)2 CE 3( *- )+* .10 ! %2)2 *- 11\(1!JE)+*HZ *s %2 11 %+)2JG%+)2 ( .()2Z HE 6 -* %2. 1*HN * E )^Nhs#)+\N'AC8@ l )+*HZ%+ )+* )2\Z s .* )+*mV 1 *)+!WK( '(+ *.QS 3)+% 5 13 *4p )213)2%21\s )21 %+)2JG 1!)2 )+*DZ 1TC\Z %+)2 CE 3( *- .! )+* *)+*.JE )+* .0)2 @HZ s !FN -%2)2 *- )+* , \E .? \E

D1 ∇2⊥p

*. )+* . * !) %+-%2)2 *- )+* , \ED2 ∇4

⊥φ..()2! s#)+.

?!* 1)2* .()2 )+* %2)2 *- $ '(+9 & ** ( *.* SkZ )2** . )211!)+3 !)2*-%2)2 -G1 .G 1!)2 E )+

D = D1 ∼ D2* )+- ]1!*- 1!3Z .( g#.()2DV*2s 1R@

* %+- *. 1!-)+

γ = γ0 −Dk2⊥

E )+γ0 =

√1 − δ

τ

kθk⊥

.$ '(+: &

X *V*%2 * *-7* .()2 )+* .* 6 .%+%+-%2)2 *- * . 1 Wp 6KW@ .kf \s *.* A\E )2 @!*- * $ 1!)2 QV1 *)+! K(LK( ' & N** E * )+* N!)21!!)21 ]C*-

ξ)+*(g * (

t # -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

s %2 1!)2 .()2 1*HN * E )^N .%V 1!1 )+*1!)+%+)+TC* *. .()2 . 1!3PC @! *4p#%+%2*)+*-@BX *1!)+%+)+TC*/1!3)2%+Ul* 1!3 *. .!%2)+!*- . 6 .%+%+-%2)j@ *- *4E )+!%21 . !)& !*PC !*- )21!

ξ)+* . FG *.(* *- f *

%s6 )+ .;l)+*(g !*-

. 1 *\E )2!*/)211!)+3 !)2*1FN kZ )2**D

=Dτ /ξ2

! PC%+;E * .**/.()2 %2)2 *- g.()2 *\E )2!YV*2s 1!

γ τ = γ0 τ

−D

(k⊥ ξ)2 E )+

γ0 τ =

√1 − δ

kθ ξ

k⊥ ξ.

$ '(+=c &X * V )+%2.(*-$'(LO )21! .()24V PC*-)+-UN )+ . *\E )2!* V*2s 1! f * . *\E )2!*3 %2)2.%2* "$%+%2*HZ % .!- 1!%+%+(6 )+,l )+*(g !*- .,N'g*1!!%+)2 * )211!)+3 !)2* ! PC%+ E ** * )Ê _!-%2)2 4Z $ '(+5 & * *( C*-%+)2 JE`V *1!!)2- g N%2)+*"$%+%2*HZ %2* )+*.R@%+)2 .(HZ )2!*-/.V*2s 1! g#-FG "$%+%2*HZ %2* )+* 3 %2)2.%20)2 !*- 16 8i()Ê HE )+* .()2 1JE '"2,'.1 D/O i !S*TFHC@h +- 'W* PC*- 1f E "$! . 8@.() %2* "$%+%2*HZ %

kr4

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

γ τ

kθ ξ

kr ξ = 0.12kr ξ = 0.24kr ξ = 0.36

>#"0 &'($'"0/') (<) / )" ++ ) ))4#()> AB)"0)& ( )"913+/0- /;)H1 2"; $ /0)"0) /0+)H*)" ) )/0) ! $' 76, #+- /0"0 "0- /')") )"0 )@$%/

δ=0.034

D

=0.81)$' /04 / +"%)>

uw$9=9 = a 4 *.(-H< $d =976$ D$89=6$2 p ! wE * * * MF!*- * E )+

k‖ 6= 0 %21 6 .* .()2 * !% *- .hE -*!)21 *

?%2.(%+)+*)2*wf!)+)2* 1 - 1%+ 1!)2 .()2 )+*/* %^ !)21 DM)+%+)+TC1!* 1! *- 1 2s#)2!)j@- 1) * .()2 1 M%+%2 ( .()2 )+%+)+!)2*11 !)& $ 'UK 'M )+DKUK & f \s#)2 1*)+* .%+%2s )+* * %^ !)21 *p **- * *. V1 PC\Z *- * 1 1g !%+)2 1 !)2 *1!)+*. /l101)2* )2 .()2Ds#)2 !)+- 1!*7 1!%+ )+*SDg\Zs#)2 .!- - *46 )+

k‖ 6= 0- ** .( .()2 ! %+%2%2.a* E )^N .()2 1!)21!!)^f )+TC )+* .()2)21!3 R@

1!)2*1!% !)2*E )+)+* $ 1!)2 %[ HE 1 1 1\Z 0M )+YK: & /V1#)+*V 1 PC\Z *- %+-

γ ∼ γ0 −Dk2⊥ − 1

η‖k2‖ .

$ '(+7e & 1 . .G/%+%2 6 .*Hs %2 %E )+ *.(%+)2 JE

k‖*!% *-/. 67-* %2.(%+)+*)2*f8@

!)+)2*41!)+%2 1!)+*. %21 1%2 yE )+k‖ = 0

1!3)2%2* - . .()2 6 .*E )+k‖ = 0

5 $2-H$m|Y8) d$2979=$m( $8$&*(1)+*$&& tP>

s %2 .()2 -%2)2 %+)^Z )+TC (s )21*s#)2 .()2E -*!)21 * ?%2.(%+)+*)2*$( .* !)2* %2*? %+HG <C * )

q0 = m/n.()2 * 1 )2.*. %+%2 )+*Qp JZ -/( .()2 % 1\E 1!)+%+)+TC

?!* \N ** yE * .GmE )+ )+* -FG * 1!)21!!)^f )+TC 6 .*~E )+ *.(%+)2 JEk‖)ÊhE 1TC\N #- ./CE 3( bs .*4Hl)+* / 1!)21!!)^f )+TC - bE )+ )+* -FG * % 1R@

E 1!FG)+-UN )+)+* p \gTN 1)2 !)+-E * * * .()2;E -*!)21 5_1 !*-10! PC%+E *%E )+ .] )+*!)21!)2!*-. 1

q@ %21

1

q=

1

q0− 1

q20

dq

dr

∣∣∣∣r=r0

(r − r0)

*. .b_ !) %2* !*1 \E !)2* $ '(+9dc & .*QV1.(!FN g0.* %+%2%+-.()2* *

∇‖0=

1

R0

(∂ϕ′ − R0

Ls r0(r − r0) ∂θ′ + O(ε)

).

M E )+Z)+- ** gr = r0

.167-* %2. )+*70)2 !*-f *ϕ′ <s,C *. g

r 6= r0.()2_1 !*- . g 1!-(.G .# %+%2%+-.()2*f * . .() %2*S.()+* PC*a@-)+- )21! E )+yN** .()2 .() %2 _ !) !)2* . ! * MF!*- * *)2 yE .( ?!)2\E .* 1 !)2 *Qs .*4<l10-)+%+0.

(φ

p

)=

(φ(r)

p(r)

)ei(m

′θ′−n′ϕ′)+γ t .$ '(+7O &

V 1 .Ds )+* * 1 *- g )+* 6 .<s %2 )r = r0

%2N%+)21!)2! )21! %+- %21]1!*-

φ(r) ∼ exp

[−(r − r0)

2

2σ2r

] E )+ σrξ

=(1 − δ)1/6

(kθ ξ)1/3.

$ '(+'9 &)2YVE 3%+)+!. .MF!*- %21 .()2?\E )+* HG 1 * %2 FN *HN \f /s %2 HEr = r0

!HE N *HZ* !!)2! )21! 6 )+σr/ξ

)21! )+ *\E )2!p )+ *- - *4 ?g/.()2*\E )2!YV*2s 1! -)+%+

γτint = [(1 − δ)kθξ]2/3 .

$ '(+'( &p ) .()2 1 **-4s#. .l JN )+*Z 1TC\Z %+)2 * )211!)+3 !)2*

D*)2 \gTN 1)2 a@

!)+-"$)+! )+*ms#. .()2 , 1 1- * \Z .G .()2 .() %2YV1. **-/. 6 .σrs 1*!%+)2 1N %2)+* %21)+ 3 %2)2.%2 V1. **- )21! %21

kθ |∂r|-)+%+1=s#)2 .()2 p .()+*a@

-*-kθ ξ 1

)+* . *)+!)2* . N !)21!!)21 *.]C*-

ξ$ K(+7O & .()2M !% C*- )+* . "$)21

ξ ∼ Ls ∼ 1/s)+*- \N **E *.G/.()2V *s 1!E )+ Z * HE *.WE -*!)21

_1 !*- *)ÊhE 'l )+* E -*!)21 _1 !*- s#)+\N 1!)2 /%21 1!)+%+)21!)2*. ( .()2V 1!1 )+*1!)+%+)+TC* 1

* hs#)2 ).* !)& Bs %+%2* Z)+-.()2bV * %^(1 .;V 11 HE .*.G *#.()2bV *s @1* )+ )+* 1 .()2** . % 1\E )+*!-!*..()2 HZ s . 1 % 1\E .(!TN 1YN )+*1l * R@1! *wf * %+)+**X *1)+%+)+TC* .()+*- .( 1!)21!)^f)+TC . 6 -* %2.aN \gHEhE *-Ek-%+)2 )21A . V* %^ 1 . 1 %+)+**._)2 %2.aE .%+%21 1%+%0** )Ê *PC 1!* V 1 *)+! - JZ)+-

# -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

s .*+s#)2 1!)2 7. 1DX *1!)+%+)+TC*$!- *. , )+%2 *!*1!3 !%f )+*( 1 PCR@Z* % CG

!+"' # (*" ( ( !" #%$ &)(

VE p )213)2%. !)& R@BV %&fdo*a@ "$%+%2*4s#)+. )2 .JE *1!!)2!2s#)2)+* f )+*( Dp a@* *- . 1 *E %2* , *13 !15E )+!%21 . 1 %+)+*!)21!)2!* %2)2 *- 11\(1!JE 1 f %+%^Z- * s R@.*kN**42V - 1 PC\Z , *1!3 !.() 1!)2*1\N kZ )2* * )2* )+*b_!-%2)2 1\Ek-%+)2 HN )+Z $.JE ** , *13 ! )+* *)2 !%+)+** , !%2*HZ * %^(1 .()2 .( 1f %+%21!TC*a@.()+- * HE !)21 %]1!*- ! %+*s#)+. )2 1 )21 *1/g .*$.( 1p#%+%2*)+*-@ 6 .* \f !- !( ** , *1!3 !Ek-%+)2 Z! @. E ! JE !)21 $.!%+)2 1 2s#)2!)+- Ds#)+. g .()2!-.2* B Q BRS: *S+*-/C.10 . 1.( wp#%+%2 *)+*-@BX *1)+%+)+TC*wf !1 * , )+%2 *!*1!3 !1 )Ê HE * .()2 15V! )+(5r)+f \s#)2 1*4

p ) . 1!)+%+)+** V 1 PC\Z *- . 1 , *1!3 !1 .( .* 1- * ***gXg ^] !0%7_[*1W!-/%aA*, !W - E *1f *$)+*JE .() 1!)^f * _! %+* . 1 , *1!3 !1 1 'dc )2 1N**E )+!%21 )+* 1 ) 1!)2*1\N kZ )2* * * .0% !)2* $ '(LK5 & 1 !)2 *Qs .*4

Γ(r) = −D⊥dn(r)

dr.

$ '(+'' &M E )+N**yE *yZ HEp )21!3)2%.1l !- *)21)+*p * *- . 1 , )+%2 *!*1!3 !1 .( .* 1*HN * ) 1)2*1\N kZ )2** 1.(\gTN *7s %2 E )21! )+* .4 ! * % 1\@E -)2* %21bN *1!**- *PC !Ws#)+. X * . %+)+** V * %^(10s %2 )+*7.*f ! !)+- *1V1 *)+!* f !- 1%+%+s#. )21!.1r)2% .()2 V*2s 1!* .X *1!)+%+)+TC*hZ **s %2 #.* , *1!3 !f !1 *4 )#!- * 1)2 .()2M3 JN !* .hV *s 1* )+* V PC*-)+-UN )+ f * .*YZ!Js )+%+)+- *"$%+%2*HZ %2*4 E )+0 bE * %21 )+*X *( \E !)2*Qg 0.()2?*HZ .MF!*- * - @s ***4 Ws#)+\N %+)2 *E %2 , *1!3 !

Γ 1!!)ÊhE 1!)2 .( %+%2 %+)+** *.

*)2 !%+)+** "$ 1%^s#)+\N*- * *. )+ * * 1!3 *.*mVWE 3%+)+!.* *. 1* JZ )2 @ *- *bs %2 7* .( .()2f %+%21TC*.()+- #*)2 !%+)+*1]1!*- . 1 %2)2 *- 11\(1!JE 1 *,s .* Nd***4p ). V1 PC\Z *- !)2* !)2! E * 1!)2 %21 * . *)2 !%+)+*8@*/* E )^N *. !)+*- .()2 1#)+*r1EhE * *- E )+ ** %+)+**wV*2s 1!*4A .* )21! !)+- *wp ! !*- *4g .1_)2 %2.aE .%+%-)+%+ g)+*bV1 PC\Z *-%f *FG *.(* *- *

∂n

∂t∼ γ n ∼ v · ∇n ∼ k2

⊥

Bn φ .

$ '(+'UK &?g.* ) 1!)2*1 *1\Z -)+%+

D⊥ ∼ Γ

/∂n0

∂r∼ Γ

L⊥

n0

$ '(+'I &

5 $2-H$ 9 7- & a * 35-/ 9=:;)P8

*. g0.* *)2 !%+)+**N *2f JN !)^f * , *1!3 !

Γ ∼ n v ∼ k⊥Bn φ ,

$ '(+'5 &1 .GkE * .* ) 1!)2*1\N kZ )2* *E )+

D⊥ ∼ k⊥Bn φ

L⊥

n0

$ '(+': &1 PC\Z*1N**4 6 )+ . !)& !*PC !*- g .()2 *)2 !%+)+* * E )^N $ 1!)2 $ '( 5I & +V a@1 *)+! '( I & s#)+. )+*4V ** HE wg .()2wVWE 3%+)+!.*-FG . ! * 6 .* - @E

∇⊥n ∼ ∇⊥n0 ⇐⇒ k⊥ n ∼ n0

L⊥⇐⇒ n ∼ n0

k⊥ L⊥.

$ '(+'dc &M \Z bE *QZ 1!$ '(+'UK & *. .** $ '(+'dc & )+* .()2V1 PC\Z *- $ '(+': & )+*1 ! PC%+ E *1 !

D⊥ ∼ γ

k2⊥

.$ '(+'Ue &

p )Ê 6 )ji )+*-@B]4*-! a@Bl 1!!)Ê Qs#)+. .** * .()2 )+*1!)+%21! 6 . %21 .()2/-FG V *a@s 1)Ê %+)+**M3 JN !!HE ! 17s#)2 )+ Z - /!)+- "$%+%2*HZ %

D⊥ ≈(γ

k2⊥

)

k .

$ '(+'UO &?!* )21 .()2V 1 PC\Z *- . 1 , *1!3 !1 .( .* 1- * *** 'H8I2SDLX\ ] !0%_[*0 !-/%,aA, BWbg%+)2 'Ue'UO

D ⊥ ≈

(γ

k2⊥

γ2

ω2k + γ2

)

k .

$ '(+7K9 & )2 )21!

ω.# %+)+% . N E 3%2 i* ? *HZ PM E )+#% 11* 1!)2 1 * * .bp 1!!)Êy@

E *- . %+)+** V*2s 1!* 1! V 1 PC\Z *- * g .* *E %2* ? %+HGE )+!%21. 10) 1!)2*1FN kZ )2** - Js#)+***4p Jf )Ê *PC 1*4V1 *)+! . 1!)+%+)+* , *1!3 ! . 1 %+)+** _)2 %2.aE .%+%215E )+R@%21V*2s 1!* *. 6 )ji )+*-@B]4*-! a@Bl 1!!)Ê .()21\N!)2!s#)+. 1) * .()2 1M%+%2 * )+* )+*( 4MN%2* 1 PC\Z *- g .* ) 1)2*1\N kZ )2** . 1 *E %2* , *13 !*- - *4;?g % 1FE 3 E

Te = 80!_

B = 2,

L⊥ = 0.03E s#)2 1!)2 Z p )Ê

, NE N , l , [ Z *.* 1!)+*. 17s#)2ρs = 7.2 · 10−4 E ! PC%+ E *QE )+

n ∼ ρsL⊥

n ; vE ∼ ρsL⊥

k⊥ T

eB17s#)2 .* % !)2**$ '(+'I & *. $ '(+'5 & .()2 FG *.(* *- . 1 i 3 !)Ê * %+%\E )+!%+**E %2* ) 1)2*1\N kZ )2**

D⊥ ∼ ρsL⊥

T

eB≈ 1

E 2

1 .

0 # -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

!

p ! s#)+. !* .1 %+)+*!)21!)2! _)2 %2.aE .%+% SZ! . * .()25S \gHEhE *- 1!\E

∂ n

∂t= −v · ∇n0+B∇‖0

(j‖

eB−n0 u‖

B

)

mi n0

B2

∂

∂t∇2

⊥φ = B∇‖0

j‖

B

∂ A‖

∂t+

me

e2 n0

∂ j‖

∂t=

Tee n0

∇‖n0+Tee∇‖0

(n

n0− e φ

Te

)− η‖ j‖

mi n0

∂ u‖

∂t= −Te∇‖n0 − Te

(∇‖0

+ ∇‖

)n ,

X *7% 1!)+*. .()2 )+1 .()21\N !)2!*M %+%^s %+%2*~E \N)2! -\g*E \N)2! , \E )Ê[ HE 1 *$ 1\Z )21!#g .1l *1! *f * 1- * *** S2"c, !I2* 'C*N* ! Q !N=Wkf * R@s !!%+)2 ': M )+c8I )2 1 , \E s#)+. g .()20 * E )^N 1 .1E 1 %2.hE *wE )+.* % 1\E .()2 * *. , JE 3 !* )+*p )2 ;f 1!FG )+* .**Z * HE *.N )+*!)21 l JN 0)+*/.3 %+%2%2*/ * E )^N .()2 )2*)2 \gFN(1!)2 !)+- s .* )+* %+%201!3)2%2*4 N** )+* *)2 Df 1 2s#)+*.*.l )+*HG .()2 1 1 , \E 1 ( .()2 *1! *- 1!- a@*)211 %21WX *.()^Z *- 1 *s .*.G E * .* g%+!)+-UN )+1! )2 7. 6 .%+%+-%2)2 *- *f !% 11* +7e 0M )+ 5( 'dc X * %+)2- ** )+*S E )+* !)2* 1 1!)21!!)^f * !)& s %+%2* *.QM R@BV%&fdo*a@ "$%+%2*f .()2hl × p @BV#.af JN!)2*$. )2 *. . .1 %+- *.*$ 1*2f 1 )2*-Z s#)j@1 *7)2 *. %2JN!!)21 JE * !) % )21!.()2 * E )^N .!)& s %+%2*7* *mE )+!%21 .3 %+%2%2*1M!E .()2 * .()2 * E )^N .%2JN!E -*!)21 *~M R@BV%&fdo*a@ "$%+%2* - @N 33 %+67* 1!3!)2 * *Qf *Y9?* <,%,FH <'IJ*C%cKO* ! ?g.1#)+* % *. -\g* )+*- !C!%M(1!JE s#. )+* , )2.%^N .()+* * . ?a@!)2*1\Z

f =∑

k,m,n

f ei(k r+mθ+nϕ−ω t)

- JE HE .()2 )21!3 1!)2*1!% !)2*QZ \E )+!%+*4/? !* s#.*QE )+bg%+)2 * MN%+)2!*a@- * .()2 %2)2 *- *7.()Ê *1!)2*1!%210*\E )2!HE 1!)2Dg 1!)2 !%+)2 Z 1 )+ *4

n→ n0 n ; φ→ Teeφ ; A‖ →

Te c

e v2; u‖ → c u‖ ; j‖ → e n0 c j‖ .

6 )+ .* )+%211 *f !- 1!%+%+*mVHNg\Z *- *

ω∗ =Te

eB0 r0 L⊥; β =

c2

v2; ω‖ = k‖ c ; ωC =

e2 n0 η‖

mi

k‖ =1

R0

(m

q− n

); x =

r

r0; µ = −%

2s

r20(−k2 + i k −m2)

5 $2-H$ 9 7- & a * 35-/ 9=:;)P8 <t

! PC%+ E * .()2 )21!3 1!)2*1!% !)2*

ω4 − mβ ω∗ − i µ ωCβ + me

miµ

ω3 −(

1 + µ

β + memiµ

+ 1

)ω2‖ ω

$ '(+7K( &

+

((1 − β)mω∗

β + memiµ

+mβ ω∗ − i µ ωC

β + memiµ

)ω2‖ ω +

ω4‖

β + memiµ

= 0 .

"$)+! )+*s#.* .()2f )+*( *.*V** HE *

ω‖ =c

q0R0; k = m ; x = 1

. 4µ = −%

2s

r20m (2m− i)

$ '(+7K' &- JE *.Z ]1!*-Wf * $ '(+7K( & s#. )+**HE !)21 !)+*5Z A%+%21!%+%2* 1!!)Êy@E *-/gWN E 3%2 i( %^ *E f \s *. LK9 <V 1.-FG*V *s 1!ys#. .**E )+!%21 6 )ji )+*-@B]8*-! a@Bl1!!)Ê $ '(+'UO & . , *1!3 !\N kZ )2* D⊥

)+*wV PC*-)+-UN )+,f *.*f 1 )2 .** % 1\E 3 E !* *V1 .*~f 1 )2 .** S E )+* !)2**. E \f!) !)2** 1) )2 %21p )21!3)2% )+*%V)+%2.(*- '(+9. , *1!3 !\N kZ )2* )+*V PC*-)+-UN )+f *7.l %2JN!**JE 3 ! *. . )+* !-!*.(-.()2**% C*- /.!- @1!%+%+a_\s *. s#.* .()2 , l , [ @ E !!)23 E E )+

R0 =1.75E

r0 =0.45E

17s#)2q(r0) = q0 = 3

6 *$\N **4.G ).*f \s *.* % 1\E 3 E !* .1 6 8@i )Ê HE . 1 , )+%2 * !*13 !1 )Ê p )2 f *

Te=40 . . . 100!_ *HZ ! * )21! -FG

.* .()2 , JE 3 ! s#)+. . 1!$- !)+*- ks#)+. . , *1!3 ! )2 1 1E \f !% (%+)2 ) _ !) !)2* . )2 %+)+!)^f ! %+*4 1kl !- *)21ws#. .** E )+.JE 1!%+)+*s )+* * 1! *-f !-%+)2 * )+* ..()2yM %+%^s %+%2* $ % mE R@N)2! , \E & *)2 E )+! \gFN(1!)2 !)+-Ds#.*4/l1bZ)+-* 1!)2 * JZ )2.* !)21 YS \f *4

10% CG 1!)2 1 %+)2JG *.G g0.* , )+%2 * !*13 ! )Ê p )2 . f R@s *.* % 1\E 3 E DM %+%^s %+%2*N )+* -G %+%2g#.()2%+)+* 2@* E )^N *. .* .E )+f !*.** , *1!3 !1!3)2%2*4 , *1!3 !s#)+. )2%21 *.( 7.()2 !)& R@BV %&fdo*a@ "$%+%2* $ V8" & 1!!)ÊhE "$)+! )+* % CG1!)2 1 . , JE 3 ! PC*-)+-UN )+ . 1 ) 1!)2*1\N kZ )2* * 1 ! %2)+*.G/ 11!)2 )8. V "@ * E )^N HE )+* )+* 1 % 1\E *.(3 PC*E * *a@.%+ s )+E * )+* .1WX ** . 1 % 1\E 1Wf .(!)+*- *. .()2 , JE 3 !-FG s#)+. . 1!ks *)+- 13)2%2* .()2 V " )+* %+%2

E .()2kV11- JN .()2 1Yl !- *)211hZ $ Js *s#. ) %+%2* E \f!) !)2@**7.( - 8.G .()2 6 .%+% ** HE *$. !)& !*PC !*- -Ug%+!)+- %+)2 *4 )2gs#.* .()2p .()+*-*- *

δ ≤ 0.05 ; β ≤ 0.05 ;

(ω)/Ωi ≤ 0.01*. XBE

(ω)/Ωi ≤ 0.01

E )+ .* %+- *.* E ! )2 *7)ÊhE g%+%+

B0 = 1 . . . 6,

; L⊥ = 0.02 . . . 0.1E

; q = 2 . . . 4 .

# -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 20 40 60 80 100 120 140 160 180 200

2

Te/

D⊥D ⊥

2

L⊥/

D⊥D⊥

0

0.05

0.1

0.15

0.2

0.25

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

2

Te/

D⊥

D⊥ 2

L⊥/

D⊥

D ⊥

!)"0 1 )" )$'&'(/'4 /0) $'+ $'<+5 4#)/') - ( <# / *+)" ) 7"'/0" B )" 5/0)"0"% "')/') 3

p ) .WV1\s % .)2 *.ml %2JN !** JE 3 !s#.%Z 1TC\Z %+)2 * .(- @ .G .()2 % 1\E 1!FG)+-UN )+1!)2 )+*yp )2 * *. 1 .Gg .()2 3 %+%2%2 W * E )^N)ÊhE * 7.()2 ? %+)2.** HE -Ug%+!)+- %+)24 )2g*)ÊhE DE *7* .G .()2%]C*- *1\N% . %+%2%2.a * E )^N s 1*!%+)2 -FG )21! %21 .()2yE )+!!%2 )2 "$-% C*- .Dl%2JN!**E )+0)+ * %2E 1FG *4

λe k‖1)2 1bs#.E )+ . p 1 TC*HN*-

λe k‖ =

√Te/me

νek‖ ≤ 0.5

HZ s 0.1

!)2 p ) )+*JE _!-%2)2 .D⊥

@BS \f * E )+ .* 1!%+* )+* * 1 *-4)+*. . , \E . N !)^f * X *1!)+%+)+TC* $ -\g* E \N)2! , \E & *)2 \gTN 1!)2 !)+-s#. 1%+%+ 1!)2 1 .G g .()2/- Js,C %+ E !-*HZ

λe k‖ ≤ 0.5.()2 FG

. 1 , *1!3 !\N kZ )2* * s 1*!%+)2 .( .()2 6 )+!* HE .()2 1 1 , \E 1 )+*HGs#)+. p )Ê 1!!*- * *HZ N !)+!)+HE

λe k‖ ≤ 0.1f 1 2s *. . l JN 0. 1#-\g** , \E 1(

.* , )+%2 *!*1!3 ! 1 .G .()2 g%+!)+-UN )+ .D? %+)2.** HE * +7e M )+ 5( 'dcg.()2 %+%2%2.a * E )^N - - *Qs X * , %+%2 '(+ )21)+* 1!)2 . , JE 3 !R@ )2 @BSE )+* !)2** Z 1 *)+* .g.1 ZJs )+%+)+- WMFG)+-UN )+1\N !)+!)+HE .()2 Z )+* , JE 3 !5Z - /!)+- E )+*)Ê %2 )2 *- * )21

λe k‖ ≤ 0.5g

q = 3

Te/!_

n/1019 E −3

59 ':dc '(+c5 '( :e( K(+OI I+99 I :'5 c '

λe k‖ ≤ 0.1g

q = 3

Te/!_

n/1019 E −3

9 9(LK'5 5

59 5(qc:dc 9(LK

) :!)</'"%+)/') )"'/0"%# 4# #( "0) -) & (/0#4)" :;"%1 2 #" D; ( 7-1 2" "0) . -< λe k‖ 1

" C %+QE * g .()2 .! *- - * , JE 3 ! )+*7*)2 .(!)+- )2 1 %21 1)27)+*, 4'(+ *- g !)211 g !.1 .Z .G . * , *13 !\N kZ )2* )ÊhE

5 $2-H$ 9 7- & a * 35-/ 9=:;)P8 0

E .( .* , \E . N !)^f * X *1!)+%+)+TC* )+*HGs#)+. )2 1 *!)+!)^f V 1\@1- 0HZ -% . % 1\E 1FG)+-UN )+ *./. 1 , *1!3 !\N kZ )2* * ! Z! . (8.* )2- ! **k_)+*( *- * $ '(+7K' & ). *HE !)21 * p *.(%+*- . 1 %+)+** 6 .%+%j@1\ 1JE 1 *. 1%+%+ *)2 g ! Js !s .*4 )2 )2f !- 1!%+%+ *1! *--)+ .()2 %+)+!)^f *mr1EhE * PC*- Z s#)21 * V "@ * E )^N *. , )+%2 * !*1!3 !s#)2 .X *YV)+%2.(*-'(+9 1!)2 E * )21!3)2%21\s )21 .()2;V PC*-)+-UN )+ . 1 , *1!3 !1f * .8)2 a@-.()2* *% C*- )2

D⊥@BS \f -)+81 4-s#)2 . .G)+* 1!)+%28)2 -.()2* )+*

1!TC\N %+)+*X *1)+%+)+TC0%+)2 ! *. )+*( %+- . 11* .0..( .()+*- , *1!3 ! 1!)2 ! /

!

)2,p#%+%2*)+*-UE .* 1)+\Z* )+*y?%+- . 67-* %2.aN \gHEhE *- )+* 3 %2)2.%2,_ !) !)2*4M)2 1)+*. %21 PC*-)+- f E " )+*HN %

θ8 hE HG g 1)2/.()2 %+)+*M)+%+)+TC1* %^(1

s %2 g8.()25V1!1 )+*1!)+%+)+TC* )+*%V1 *)+! '( 5( I1FN)^Z Z )2!s#.( .* - 1E *3 %2)2.%2*kp )2 /\s )+!;s .*4 6 )+

θ ∈ [0, 2π]% CG.()2 6 -* %2.aN \gHEhE *-ds %2

.( .*m[3 K )+* .()26 .%+%+-%2)2 *- *7)+*- $V LO & .()2?!)2\E .*QE )+0.*6 .*HZ %2*(m,n)

$f-% V*1\Z $ '(+7O & & E )+ )+ * (m±1,n)@BA \E .* N 33 %+*4dr

]1!*- . 1 )+*hV 1 *)+! '( 5( I ! * %+)+** 6 .%+%21\ 1JE 1 )21 .E )+ * %+- *.?!)2*1\Z g0.()2 ! * MF!*- *ms )+g *.

(φ

p

)=

(φ(r, θ)

p(r, θ)

)e−in

′ϕ′+γ t .$ '(+7KUK &

)2Ds )+ * %^ !)21 *1! *-/- 1%+ 1!)2 E ! JE !)21 HE *-)2 40M)2)21! * .JE A E * H ! ^WC B1%,9 '** ! i.10 .`VW B C !0%7d?=2-*'.10/ JN** *.$)+*LK(K'2KUK <Z h*.*42V%21 1!%+ .()2 1,p ! !*- * )+* *)2 .(!)+- 1! [.(* *-

(1/n)1! PC%+WE * %+)+*%V*2s 1!* *. )+* AC !*-/g.()2%VWE 3%+)+!.* .()2 Z! . 7%+%2.* -%2)2 * p !- (s )21*4 )2#)+- *!%+)2 p#%+%2 *)+*-UE .0)21! .()2VaG BC-0 ! )*8D/ .)+*a@Z%+**VE 3%+)+!.* $ 1!)2 V 4'(+ & )+* .()2 1JE ? %+% %+%2VE 3%+)+!.* -%2)2 -G1!)+*. 1)+\Z .()2hp#%+%2*)+*-UE .kN )+* .() %2 V PC*-)+-UN )+s 1 *)2 . " )+\N %+)2 HN )+* 1!3!)2 l )+*s )+g *. V* %^ 1E )+.0*PC 1!! / *[.(* *-

1/nLKI0K5PK:

%+)2 !0 *1 )2 .(%+)2 -G ?!)2\N kZ )2* *1 .GhE *4s#)+\N %+)2 f * )+* l )+*a@ 2g%+%2*.* 13 *N**4rX %+%+1!!!)2*s#. E )+.JE p 6K@ -EhE s %2 1 )Ê *PC 1*1S 3)+%f !- @1!%+%+;s#)+. .1 %+)+*!)21!)2!6 .%+%21\(1!JE * HE !)21 - % 1! )20 1!%+!)2*.* 6 .*/. 1%2JN!!)21 * * !) %21 1!)+*. )+*mV )+%2.(*- '(+ .!- 1!%+%+ r 1 * 1!)+*. .()2?!)2\N Ey@3 ***$)+* $

n=8 & @Bp#%+%2 *)+*-UE .+s %2 %21Dl )+* g%+%2*. . .!- 1!%+%2* .() %2* *!) %+3%2%f 1*.*Qs#)+. p %+%2 *)+*-UE .* s .*%s - * )+ -G * .() %2* *. 3 %2)2.%2*yV1. **- %21 ^ )EXY$D/* JZ)2 *al )+* 3 %2)2.%2 M *)+!# )

ϕ = 0Z)+- .()2S *!* . 1 %2JN !!)j@

1 * *!) %21 1%+!)2*. 1 )+- * *- *. * !.()2 3 %2)2.%2*V 1\E G . -%2 %2* 6 .*4 )2s 1* !%+)2 )+*1!)+%2 -)2* )21!#.!%+)2 ( .bV HG *1)+ . 1, !1 Z \N ***Hs .1)+*1 %+)2JG *. 6 -* %2. 1 2s C )21!

# -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

).()2 1 %+)+**~V * %^(1 * . % !)^f *1 )2 . .VE 3%+)+!.*HE 8i()Ê *a@)+* *. * s .* N** .()2V1%+s !7 %+)2)+- 1)+*. % CG 1!)2 ., *1!3 !s#)2 .0* g .()2V*2s 1!

γ*. .* 6 )ji )+*-@B]4*-! a@Bl 1!!)Ê 1 PCR@

Z*4

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.5

φm,8

q

m=19m=20

m=21

m=22

m=23

m=24

R

Torusachse

φm,8q

m=19m=20m=21m=22m=23m=24

"!"%+ )" $0)& ($ " /0# D +"%)"0-+5) # /0) )$ +/')/' $ eφ ))"+ ) (n=8) 7 6+5 )"% /- /0) >$0+)"0( / :!(52) " ++ )" & ( /0/H- / )8 +5/0"%) #" +/')/' /' #))"

n=24 7 6+5 =6 # # D) "%#"% )& (5-1 2" /. $0& (

: 7 , #+- /0"0- /n0 =6.5 · 1019

−3Te =30.8

- / <2 $0/0& ()" $'$0 /0+D

=0.8"%& ( 1 2("9/G

A .()2 1JE M!)&Z - .( .()24]1!*-7. 1 %+)+*!)21!)2!* _)2 %2.aE .%+%21 1%+% 1!)2 * *.*wA)2 !%+)+*!)+TC* Z - Js *. s .*2s %2 .()2!*.(% - #g!!%2* 15_! %+*)+%2.*4

, ( +"' ( '( + #%( " ( &* #%( "

l )+* l )+*1)2 )+*$.()2 *)2 !%+)+*/ * E )^N .YN%2)+*1\N%+)+- * MF!*- * !% .1S *a@Z3 . <'S* N*5fh i.0"c,.W, 76*7 7 s %2 1 )Ê %+- *.* V 1 *)+! f !- 1!%+%+%s#)+. X *V 1 *)+! '(qc 's#)+. .** .15r1!*.JN EhE */.5MaC!!)+-*-f *wMF!*- 1*)^f 1 .( .()2 *)2 !%+)+** Z 11 * . V *s 1!3 1 .0%+)+**X *1!)+%+)+TC* \N % C!

l )+* -6-%+)2 HN )+(.1 r1 EhE *1!3)2% . )+*HZ%+** 3 2(1!)^N%+)21 * Z 11)+*/)+*JE!)& Bs %+%2*HE .%+%HZ %f 1! * )2 .1S *HZ3 . <* 7*Lf Q.0"c,. , ,C6*, N6 )+)+%j@ .l *!-)2*)+%2 .)+*HZ%+** 1\N% *m?%2.% 11* 1!)2 *!-)2 ! %+*. Z 11*.~l *!-)2!*1! 1 1=s#)2 l *!-)2%+%2* *.l*!-)2 1*HN * . 1 ! *M 1!JE 1 *.*4 ))+* 11!)2! E * 1!)2 )+*/.()2 1JE S *HZ3011 %+)2JG%+)2 g .()2l *!-)2*a@)+%2 )+**! %+ )+* 1 1!!)ÊhE *1_%+HE *1 s %2 /.( $.()2 * E )^N .DN %2)+*1\N%+)+- *MF!*- * 1!!)ÊhE 1!)+*. ?g .1 )2 ! y_)2 %2.aE .%+% . *)2!E *7.()2H<'S* N:fh i.0"c,.W, 76*7 7SD ,+-0%21

U =1

2n0mi

(v2 + u2

‖

)+

B2

2µ0+

me

2 e2 n0j2‖ +

1

2n0 Te

n2

n20

.$ '(+7KI &

Q60 =& $2-H a 7$ * 97 7-a =)0 H$2 !;97 7-/ * +

l * ! %+* 1!)+*.7 )2 .()2hV *)+%2 .N )+*!)21 *~l *!-)2 JZ g-%+)2 . 1*HN *l × p @*. . 3 %+%2%2* * E )^N .()2wV* )+%2 .YE -*!)21 * ?%2.*!-)2 17s#)2 . V* )+%. 10 % 1\E .(!TN 1 )2! ) -)+%+

v2 =

∣∣∇⊥φ∣∣2

B2

*.B2 =

∣∣∇⊥A‖

∣∣2 . $ '(+7K5 &)2 3 %+%2%2yM!E .()2 1!F!*- )21!#- - * .( .1VE 3 1 1\Z $ '(Le' & 6 * )+* 11!)2!1)2 g0.()2DZ)+!%+)2 l * s#)2FN %+*- .0 )2*Ql *!-)2

∂E

∂t=

∫∂U

∂tdV ,

$ '(+7K: &s %2 $1 . _%+HE *)+* -!)2* .4l*!-)2 .()2 \f !- b?g )+* ! 1 % 1\E f %+HE * - WE *Qf * .l ! %+!*- . )2*? %+HN! !)2*1*!-)2 1. 4HE )+1!!)21!!)21 r )+\E )+!!%+*- 〈〉t

g )+*wX * \f%+% )+* . *)2 !%+)+** !!%2** 1 . 1M 1!JE 1 $ 1HV 1 *)+!'(qc ' & -)+%+

〈E〉t = 0 .$ '(+7Kdc &

5EYg11* 1!)2 /.()2n%+%2* *.4M *HN */. l*!-)20)+**! %+/)+* 1 _%+HE *1 . . 1 ! *1M)Ê % !)2*1! *- )21 )Ê Z)+!%+)2 * 6 )+!%N E 3 *1!)2*4l1 1) JE \N .G .Yp -!) ? %+HN! !)2*1*!-)2 )+- * !%+)2 .()2E<** NàG*7 ND0* %'.10/*yE )+*'V(8.()2 -!) )2 * 1 )2.(*-hf *QMF!*- * *.w? %+HN ! !)2**4s#)+.1!3PC )+* V 1 *)+! :(++ )+*- - *- *4p ) .Qp ! !*-.f * MF!*- * N** E *)+*Z)+!%+)2 1 6 )+!%g )+* 1!!)ÊhE 1 r )+!)+* \f%+%;f E 1E ! !- .%MF!*- *1!! )2* *. 1 )+*Y? %+HN ! !)2* ! %+* .*QZ)+!%+)2 16 )+!%f 1 s#)+*.<l )+*MF!*-/ 1! %21 )ÊhE 1)+*JE Z)+!- JE )+!%+* V* )+% *. )+*JE ? %+HN ! !)2*1*a@)+% ?g0.()2s )+YV1\s !!*-/. 1 V1.(!FN(1$ '(+7K: & s#)+. .()2 %+- *. % !)2* * \Z

1

2

∂∣∣∇⊥φ

∣∣2

∂t= ∇ ·

(φ∂∇⊥φ

∂t

)− φ

∇2⊥φ

∂t.

p ) )+*_%+HE *)+* -!)2*zf 1 2s#)+*.7* * . 1! , \E . * M )+1f *$ '(qc + & 8 %+%21 E * Z ]1*-7. 1 6 .%+%21F 1!JE 1 g .1 ! *- )2 $_@%+HE * & %21 *.( .()+*-*- **)ÊhE .G .()2wMF!*- * E *.f 1 s#)+*.*4 )2 1, \E )21! )+* )^f !- *HZ *.s#)+. ) ._%+HE *)+* -!)2* * 7.JE HG 1 * M(\Z)+*Qf 1 s#)+*.*.[ <C *\E ?g0.*ml *!-)2* )+% . 1 % 1FE .(!FN(10-)+%+

1

2n0 Te

∂

∂t

n2

n20

=Ten0

n∂n

∂t.

)2 WE \E *- . 1!!%+)2 * r )+%2)+!*- * .l *!-)2* )+%2wf * %2)2 *- $ '(+7K5 & %+-0* %2- E )+#1!)+*./.()2DV1.(\gFN )Ê`X * -*.*f * $ '(+7KUe & .( .()2 %2)2 *a@- * . 1_)2 %2.aE .%+%21 $ '(Le: & )21 $ '(LeUO & - - *467*%E %+!)+3%+)^Z )2! .()2 ZJs )+%+)+- * 1\N% *

# -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

MF!*- * E )+.Z - /!)+- * %2)2 *-4%21nE )+ . )2 -%2)2 *- $S* !)+*)+TC1R@

-%2)2 *- & φ E )+ ._!!)^Z )+TC1-%2)2 *- j‖ E )+.JE [ HE 1 * 1\Z *.u‖

E )+ .3 %+%2%2*mXE 3%21!)+% *HZ *. 1\Z 0.()2 * 1! *.*V 1.(\gFN )+* .*mX *-*.* )+*

∂E

∂t=

∫Ten0

n∂n

∂t− n0mi

B2φ∇2

⊥φ

∂t+ j‖

(∂A‖

∂t+

me

e2 n0

∂j‖

∂t

)+ n0mi u‖

∂u‖

∂tdV .

$ '(+7KUe &)2 - C!*X * -*.*)+%2 1!)+*. .** - - * .( .()2 , \E

−Ten0

n v · ∇n0 +Ten0 e

nB∇‖

j‖

B− Te n B∇‖

u‖

B+ Te nK(φ) − T 2

e

n0 enK(n) ,

−φ B∇‖

j‖

B+ Te φK(n) ,

Ten0 e

j‖∇‖n0 +Ten0 e

j‖∇‖n− j‖∇‖φ− η‖ j2‖ ,

−Te u‖∇‖n0 − Te u‖∇‖n . )'f 1 s#)+*.* .()2 *1! *.* [ <C *)+* -%22l *1f 1 s#)+*.* .()2 *R@-)2 ! %+*.* A)2 !%+)+*!)+TC*4 6 )+ .* f \s *.* *.( .()+*-*- * % CG/1!)2 .()2 1 )21!3)2%21Fs )21Z)+- * g

∫

V

mi n0

B2φ v · ∇∇2

⊥φ dV =$ '(+7KUO &

−mi n0

B2

∫

V

(∇2

⊥φ φ∇ · v + ∇2⊥φ v · ∇φ

)dV +

mi n0

B2

∫

V

∇ ·(

v φ∇2⊥φ)dV = 0 .

1 1!QX * -% f 1 2s#)+*. 1!*1ks - * *- *EhE * X *HN E 3 11)+)+%+)+TCφ∇ ·

v = 0 l1 1)* )+*HE % .( )+*- Js#)2 1* .G )+**! %+ . !)& !*PC !*- .

N E 3 11!)+%2 V* )+% . .() E -*!)21 * M!E .()2 / )+1 )+*$.*S \gHEhE *- 1!\E ** ! %+* )212r<s )+*1 -)+%+

v ·∇φ = 0E )+. *)+!)2* .5V#.af JN !)2* $ '( :dc & E )+$ '( 5dc & 1 Z s )+X * -%Ps#)+.ws#)2 .5Z )+*JE [ <C *\E .5s - * . *.( .()+*-*a@

- *f∣∣∂V

= 0f 1 2s#)+*.

)+*)+- . f !%+)2 ** !)+- *.X * -*.(\E % 11* 1)2 Z )^f !- *HZ*N E )+*)2*4,M -)+%+yZ p 8g .()2 MHEhE 1 .JE .(!)+!* * , \E E )+ .JE Z s )+*% * , \E

−B∇‖

n u‖

B= −B · ∇

n u‖

B= −∇ ·

(Bn u‖

B

).

1Y_1 s#)+*.* .()2 1 , \EhN E )+* !)2* * .Y_%+HE *)+*-!)2* . .G? %+HN ! !)2*1*!-)2%Z s#)21 * .()2 1* , \E * ! %+* %2)+<l )+*Qs 1%21)+!)+- Dl *!-)2 @1!1 .1\N% *1MF!*- *g .()2 1 , \E )21%21 Ek-%+)2 4 Ws .*7.()2 1

Q60 =& $2-H a 7$ * 97 7-a =)0 H$2 !;97 7-/ * <>

, \E %21aG*7 N,B<*** JZ)2 *l *!-)2 ! %+*. , *1! 1 1!)+*. Z s#)21 *7.JE Z s )+* * *.QZ s )+* - %+ * , \E 1=s#)2YZ s#)21 * .JE 1!* -\g***. .(!)+!* - %+ * , \E - - *4V * *. . *)+!)2* . 1kS \gHEhE *- 13 1 $ '( :5 & \N ** wE *.G$.()2 1 1%+1)+* )^f !- *HZ .1!%+%+

Te nK(φ) + Te φK(n) = TeK(n φ) = −∇ ·(

B

B2× ∇(n φ)

).

M E )+*. YZ s#)21 * .()2 1* S \gHEhE *- 1!\E * )+* l *!-)21!1 1!!=s,C *.s - *

nK(n) =1

2K(n2)

. %2\Z , \E 1%+1! 1 * *!-)2 ! %+*. )21! *. ) .w_%+HE *)+*-!)2*f 1 s#)+*.M %+)2JG%+)2 %2)+ *$* 1_1 2s#)+*.* %+%2 [ <C *\E * * .()2wn%+%2*$*.M *HN * . 1WM(1!JE 1bg!)+-

∂E

∂t=

∫−Ten0

n v · ∇n0 +Ten0 e

j‖ ∇‖n0 − Te u‖∇‖n0 − η‖ j2‖ dV .

$ '(+JI 9 &6 * JZ)2 * .()2 % !)2* $ '(+JI 9 & %21 aG*7 N' 4 . 1 _)2 %2.aE .%+%21 )2 R@1!* .() , \E %+)2 !* )+** 3 1!)+!)^f *p )+!- *. 1)+*. %21Wn%+%+\E Z f 1! *4 )2 )2 ? %+HN! !)2*1*!-)2 - % *- %21wg .* .()2* * . )+*!-!*..()2 )+* .1M 1!JE )2 1W_!- *- 1 )+- * 7.1WV*2s 1* f *X *1)+%+)+TC* )+*7)+*JE )+*a@ E - **7% 1\E Ps#)2 )+*mV1 *)+! '( 5 *. '( : - JZ)+-)2 3 %+%2%2* V* )+%2 $Z s )+*. .(!)+! , \E )+*$ '(+JI 9 & & .* . ))+** *- -* l *!-)2HG2s %2 * !% *-f *$- 1!F!* 67-* %2.(%+)+*)2* .( )+**$3 %+%2%2* !FN -.()2* * f !1 %s#)+. s,C *. . 1! , \E .()2WV#.af JN !)2* .( l × p @ " )+! % 1 )+( %2\Z , \E )21!*- !)^f . *)+ *.7)+%2. .()2hl *!-)2 1*HN . 1%M(1!JE 1 l .JE *1!!)2! .()2 .()211!)+3 !)^f " )+\N *- .03 %+%2%2* 1!)21!!)^f )+TC 1 S*HZ3 . )2*4? %+HN ! !)2*1*!-)2*. Z HE`p )213)2% ) . * 1! *- . 1Z s )2.()Ê *1!)2* %2* ?27 @h%2K`"2WNC !% c*` LKdc )+*JE %2JN !1!!)21 * *\E .%+%. 1b_)2 %2.aE .%+%21)+* .V! )+ LKUe %210) -*1!!)^NV *s *.(*-0l )+* 1g !%+)2 )21\@N11)2* . 1Yl *!-)2 1 \Z 1 g .1 .()2.()Ê *1)2* %2 %2JN !E -*!)21 M 1 %2.aE .%+% )+* .JE , JE 3 !1!F!*- *

T \gTN 1!)2 !)+- 1!)+*. a*. E *Z HE p )213)2%4)+* LKUO

)2)+* .()2 1JE`V 1 *)+! f !- 1!%+%+* !*.(% - * .()2** . )21\N 11!)2* . 1 )+* V1 *)+!:(qc 'yf !- 1!%+%+*ml*!-)2 1\Z 1#g .10Wp 6KW@ 6 .%+%

!

p ! E * !* .18*)2 !%+)+*b_)2 %2.aE .%+% $ '(Le: & )21 $ '(LeUO & 1 % CG 1!)2 %E )+!%21.?!)2\Z!%2-*-/ %+- *. 1\N ***46 * ! YZ p .()2 *)2 !%+)+* V#.af JN!)2* )+* . S * !)+*)+TC1!-%2)2 *-

dn

dt= . . . ⇐⇒ ∂n

∂t= −vE · ∇n+ · · · .

% # -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

X * , )2.%^N .()+* * \E %+)2!

∂n

∂t= − 1

rB0

(∂φ

∂r

∂

∂θ− ∂φ

∂θ

∂

∂r

)n+ · · · ,

* Bs#)2TN %+DE *Qs#)2Dg%+)2 * 3 %2)2.%2* *. )2.%2* 6 .*

n =∑

m,n

nm,n(r, t) ei(mθ−nφ) ,

1 .G)+* *)2 !%+)+* 1 ) *!) %+-%2)2 *- 11\(1!JE g .()25S.()+* *(r, t)

g!)+-%2)+?g.()2Z)+!%+)2 %l * s#)2FN %+*- )+* 6 .

(m,n)-)+%+0**

∂nm,n

∂t= − i

rB0

∑

m=m1+m2n=n1+n2

(m2 nm2,n2

∂φm1 ,n1

∂r−m1 φm1,n1

nm2,n2

∂r

)+ . . . .

)2 * E )^N )+* 6 . )21! %217 PC*-)+-f * Z s ) s )+* 6 .* *. .* * 8@E )^N +l1 *.%+ 1)2 7 )2 HE )+*9;S* !%NXYD0*C%cKO+*-/6* Q@ B'"$.10]4\Z !%+)2 $% 11* 1!)2 %+%26 .*2s 1%^s#)+\N *- * bXY$D/*"2* 8 Q.000* - * ** )+* 6 .* !!)+3 %( )+%2*4 )2 1)j@ 6 .*a@ "$ 1%^s#)+\N *- )21!bE )+

∂

∂t

( |nm,n|2 + |nm1,n1|2 + |nm2,n2

|22

)= 0

$ '(+JI &*!-)2 ! %+*. +7e *. .10 ..G .()2l*!-)2Ps %2 .( .()2 %+)+**X *18@)+%+)+TC* ) s#)+. 'g #.()2**.(%+)2 f)2%2* 6 .*HN 33%+*- * )+* f 1 )2 .* TCHE %+)2 MN%2*Qf !%+1 )4HE%f !)+%+Ws .*QN**4 % *-#.()2# )2Wl *!-)2)+* TCHE %+)2 Dp )2 's %2 DN%2)+* %21 )+*4X **-U !)2*1!.()j@1 1!)+*. 1s .* .()2 .!,f ! !1 *.* 3 2(1!)^N%+)21 * Z 110*)2 E .( .1? %+)2.'@ !)& Bs %+%2*a@ 6 .%+% 1 !)2 *4ap )41 -G * "$%+%2*HZ %2*Qs#)+. .()2Dl *!-)2 .( )211!)+3 !)2* $N )+*!)21 %l JN & - g !0*. )+* " C\E HE - Js *.%+p )p ! !*- . )2*? %+HN ! !)2*1*!-)2 \N **E * .G .()2 , \E yE )+ )+* R@-!*.(-.()2* /.()2 l *!-)2%+%2* )+* .JE ! * 6 .%+%21\(1!JE .1%+%2* *. .()2, \E ,s %2 )211)+3 !)2* 1 )+ * $ )Ê Id? 6 )21!/ 1 . , \E E )+

η‖ & %214l *!-)2 @1*HN *Z f 1! * 1!)+*. p ) , !%2*HZ1!)Ê % !)2** )2 .()2 3 %+%2%2 1)21!!)^f)+TC %21l *!-)2 1*HN *)2 1 HE .* V*1!!)2- * )2kl *!-)2 1 .* %+)+**X *18@)+%+)+TC*Z N E 3 *1!)2*4 )21! 1 *s *.()+- E )+!%21kZ 1TC\Z %+)2 /)211!)+3 !)2* .1- * ** * *, 46 7* ! )*e9? '' W, 7 (.1WM(1!JE Z 1!)+%+)21!)2*4<M !* .** - * -/)21\@1!)+3 !)2* )Ê M 1!JE f ! *.* )21!8*. .()2 %+)+**yX *1)+%+)+TC* -G - *-- Js .* 1)+*. HE .()2 *)2 !%+)+** !!%2* * Z 11 *HZ !)+ *+s#)+.$.()2hl*!-)2 HEYf !)+%+ *. )+N %2)+*1!*mMN%2* - g !Hl )+* =W7, .012'P+*-C B'L./ . 1MF!*- 1!*)^f 10HZ s .VE 3%+)+!.**Bs#)2FN %+*- 1\Z )+* )21 )+* * 1 .a* E )21 10%2)2 - Js#)2 E )+ *E %2JE, *1!3 !!)2 )21X * . V)+%2.(*- '(+' )21! .bMaC!!)+-*- 1\E *)21\E 10.( .()2 Z)+!%+)2 Dl * s#)2FN %+*-hf *)2 1!F!*- * f *1 %+)2 +?g B 3)21 , l , [b@ E s#. )2E )+.JE

Q60 =& $2-H a 7$ * 97 7-a =)0 H$2 !;97 7-/ *

rms−

Dic

hte

Zeit

0

2

4

6

8

10

12

14

16

18

0 50 100 150 200 250Zeit

rms−

Dic

hte

0

2

4

6

8

10

12

14

16

18

0 1000 2000 3000 4000 5000 6000 7000 8000

- # "0)$%/') /$0/ 4)) /'&'( :!/ &'< )$"%-$ 7 )"9/')$ )" & (5/0)$0/"0 p

〈en2〉V

1 2"B/. $'& ( " : 7 ;"0- /')") :! ) /0$'& ( "0 /%/ /'" /2µsec

p 6KW@ . )+* * *- .( - g !V ()+*JE N *1!* *7)2 )+*!-!*. s#.*7)2 1!F!*- * ( - 1\Z <s %2 1)+*;r( %+%21\f !)+%+*- - Js *** s .* *. .()2 * N \Z5r )+Z )+* i 3 ** !)2%+% *a@s 1*.*4X *1!)+%+)+TC g *4dV )+*JE 1!!)ÊhE * )2 1!F!*- 1*)^f . 11* W s 1*!%+)2 f */. f \s *.* )211!)+3 !)2* PC*- -)& */.()2WA )2 !%+)+*!)+TC* )+* *. .(HZ )2* .1WMF!*- 1!*)^f 4+[ \E %21bZ)+-1!)2 .()2 1#)+*? \E )+* 1 1 2s#)+*- 1 PXEZ s )+* p )+%2. )21 .y]8*-UZ)+\f !% ( .()2 1kM)Ê % !)2* Z 1 *4)2 \N **yE * .G* Bs 999r )+1 !)+!* $ ('E 1 & )+* * 1!)+%2 %2)2 - Js#)2 1FZ 1!*. !)2 s .* )21

M 2s !3*HN .()2 1V! )+ )21! .()2 *1! *- . 1 %+%2*)+*-@ .E )+*)2!* , )+%2 *!*1R@3 !1 *0 1*. p \gTN 1)2 !)+-*- )+* 1# i(!* )+*.(HZ )2!* 67-*1!F %2.1 s#)+. * * )Ê *PC 1!* S 3)+%.()2#*HE !)21 06 ! . f !- 1!%+%+UE )+. .()2 *)2 !%+)+* 2(1!)^N * 1! s#.

P # -H: 7$'&b B T"%$'&&$' ()+*$'&&5 *.-/ )0(1-/&$2 35$'&6$ 0 7-/ 97:5)P8

! !

XBE HE * .()2 1yV ! )+ 1%+% .1 *.1 )2 !3% 1\E . 1 , l , [b@ DlW@Bli(3 !)Ê *1)+* .()2.()Ê *1!)2* %2 E !!)2QE )+ )+%& . 1 p 6KW@ -EhE 1 ( 1)+* !!%2* 1_! %+* )+* * 1! s .*4 )2! ) 1!)2 7.1p 6KW@ -EhE )+* 1*."$)21 * ./)Ê , l , [ @Bli(3 !)Ê * .()2 " )+\N *- )+* 1 i(!* \Z-*767-*1!FR@ %2. 1 $ l & (.1 *.1 )2 !3% 1\E /* 1! Ds#)+. Pl)+*C *%+)2 1li(3 !)Ê *WE )+)+*JE l!- .()21 *$)^f ! $l & *.f )+*)+- * *7)+* . $!Mg. *HN )2 &1!!8.1 , [l@ MWV @Bli(3 !)Ê * :c p 6K@ .4. g .()2 * HE !)21 , !%2*HZ * %^(1 E )+ , [l@ MV@ E !!)2 * s#)2FN %+ks#. %+)2 ! .()2/6-%+)2 a@N )+ .()2#*)2 !%+)+* 2(1!)^N . 1! 1!!)21)2!* % 1\E *.1 )2 bf * , [l MWVZ 1!!.()2*4p Jf * *7( .()2YXBE 3%2JE *!)2!*- . , l , [ @ E !!)2 *. .()2 . 1l @ MF %2.1)+* .10 -EhE )+*- - *- *ms#)+. $S 3)+%I & 1%+% )+* .()2 1JE S 3)+%Z *PC 1.103 2@1!)^N%+)21 6 .%+%4. 1 p 6KW@ . 1bf !- 1!%+%+0*. .()21\N!)2!s .*4 1 -EhE *.%+ .()2 *)2 !%+)+* 2(1!)^N$. 1!)21!!)^f * p#%+%2*)+*-@ 6 .* %2JN2@!1!!)21 4 )2 6 .%+%+-%2)2 *- * 1%+%2* )+* * \E .%+% . 1yZ Hf .()21\N !)2!* _)2R@ %2.aE .%+%210.XBE 1!*4V1 *)+! K(+Ws#)+. Z *PC 1!#.()2WV*2s *.( \N )+0. %2JN !1!!)21 *AC !*-.()21\N!)2!M)2$)21! *- f !*.* E )+ . % 1\E 1FG)+-UN )+ s %2 .( .()2 % 1\E 8@3 E hs#)2 )2 *. , JE 3 ! 1!!)ÊhE hs#)+. 6 )+ )+%& /. 1 .!yf !- 1!%+%+*p %+%2 *)+*-3 E 1 % CG 1!)2 .5p )2 kE )+ %2JN!1!!)21 JE N*.kE )+ %2JN2@!E -*!)21 JE N . , !%2*HZ )+*-*HZ*4;X * V 1 *)+!wK( 's#)+. .** .1p 6KW@ 6 .%+% )+* .* HE * . 10)21! .()21\N !)2!*Q_)2 %2.aE .%+%210*. 1)+*JE %2JN2@!1!!)21 *~M3 JZ ) %+%& %+% )+*- .(*8JE 1 %+)2JG 1!)2 7)+* V1 *)+!K(LK .()2kMN)^Z Z )2!*-.h_!- *1\s )21 )#. *HE !)21 * ]1!*- . ) * !) %+-%2)2 *- * *4 )2 )Êp 6K * \Z kMN%+)2!*- .6 .%+%+-%2)2 *- *$17s#)2 .()2 *)+!)2*7.*.( .()+*a@-*- *Qs#)+. )2 f !- 1!%+%+

# -H: 7$'&bt -H9kuwvyxzt/4T|)P<8-/(~( **$Qx )+*$&&<&$'6+ <$

()&# ( "'( #), $! " " " # "#%( + (

V 1!- *.Qf E %+%+- JE )+***)2 !%+)+**m_)2 %2.aE .%+% $Id? 6 & s#)+. * * .()2YM %+%^s %j@%2*.a* E )^Nf !* % C11!)+- *. E * ! . )4.* %2JN !1!!)21 *M3 JZ ) %& %+%

d n

dt= −v · ∇n0 +B

(∇‖0

+∇‖

)( j‖

eB−n0 u‖

B

)+ n0 K(φ) − Te

eK(n)

$ K(+ &

mi n0

B2

d

dt∇2

⊥φ = B(∇‖0

+∇‖

) j‖B

− TeK(n)$ K( ' &

∂ A‖

∂t+

me

e2 n0

d j‖

dt=

Tee n0

∇‖n0 +Tee

(∇‖0

+∇‖

)( n

n0− e φ

Te

)− η‖ j‖ .

$ K(LK & E )+f 1 s#)+*. *)2 * .()2 r )+%2)+!*- . 1E -*!)21 *y_JN!3 * !) %21 *. .3 %+%2%2*M!E .()2 1*.!* .()2 .() %2*mS E 3 ** * . 3 %+%2%2*V %2)+!*-∇‖

's %2 .*gfh W7* . 1 )+* !!)+*1!)21 * 67-* %2. 1 1 )+ *4)2DA )2 !%+)+*!)+TC )21.E )+ * )+* .0%2*r )+%2)+!*-4f ! *.*

d

dt=

∂

∂t+ v · ∇ .

E .()2 V*2s *.( \N )+ . %2JN!1!!)21 * AC !*-Z $ Js !* % CG 1!)2 )Ê [ HEy@1 * 1\Z $ K(LK & %+- *.QV 1 PC\Z *- ! *4,M %+%2* %2JN !E -*!)21 ml JN N )+* %+%2 E 1!3)2%2*1 1%+%+ .()2FG *.(* *- .;_JN !3 *!) %+ )+!TC- s 1* R@%+)2 N %2)+* 1)+* %21 .()2 . 10%2JN !!)21 * *!) %21

|∂t A‖| |∇‖0φ| HZ s |ω A‖| |∇‖0

φ| , $ K( I &E )+

∂t ∼ ω )2 .()2* *s .* ** g .()2`Z!Js )+%+)+- * "$%+%2*HZ %2* - 1 PC\Z

6 )+ .JE VWE 3 1 * 1\Zws#)+. * * .1%_JN !3 * !) % g .* 3 %+%2%2* M!E 1!!)ÊhE

A‖ ≈ µ0

j‖

k2⊥

,

. 1)+*1)+1g .()2 VE )+3 % !)+TC1 .()+*-*- - 1 PC\Z s .*hN**46 )+ . )^f R@- *HZ )+ )+. 1E 1!E .()2 %N** * _!* % C11!)+-*- . 10 % !)21 !)2*11!!E 1.()2 3 %+%2%2bM!E .()2 g .()2 .() E -*!)21 bM!E .()2 #.l %2JN!** 1!- .(\gTNs .*4

∇‖j‖ ≈ ∇⊥j⊥ ∼ K(pe) )2! )s#.s#)2 . .()2YAC !*-kf *N%+*X ** - JE Ti=0

E )+0-)+%+

k‖ j‖ ∼ 1

B0Rk⊥ pe

HZ s A‖ ∼

µ0

B0R

1

k⊥ k‖pe .

$ K( 5 &?g.()2 ! W * E )^N *)ÊhE E * .()2 !)& s %+%2*a@ HZ s .() E -*!)21 ?*HZ

ω∗ ∼TeeB0

k⊥L⊥

$ K( : &

t D&$ 78)<9 7-a =976$ +:;$ 2-/&-/&&, * |Y&-H9=(1-H97 P$' P

%210B 3)21 yr )+1\N% *4<?!*1FN%+)2* .()2 ! *MF!*- 1E 3%+)+!.*7)Ê M)+**. !)& !*PC !*- $ 1)2 $ '( 55 & & E )+)+* *.

e φ

Te∼ n

n0=pep0.

$ K(qc &6 )+ . _)+*( *-

k‖ ∼ 1/(q R)1\Z E * .()2% !)2** $ K( 5 & )21 $ K(qc & )+* .()2Dp .()+*a@

-*- $ K( I & )+* *. ! PC%+ 1 %+)2JG%+)2

β = βe q2 R

L⊥ 1 .

$ K(Le &?g )+*$B 3)21 1 , l , [b@ % 1\E )21! )+* . *.1 )2

βe ≈ 10−4 1 .GQE )+q= 3

R=1.75

E *.L⊥ =0.03

E .()2 % !)2* $ K(Le & E )+ β≈0.05g%+%+)21! 6 )+. *)+!)2*

$ '( Ic & PC*- .JE * 7.1%2JN !1!!)21 Y_! %+*7. 1DM(1!JE 1 * * .67-* %2.'@1!TC\N f * . % 1FE .()2 *. .l%2JN!**JE 3 ! 4X * I 9 s#)+.PZ . - JZ)+- .G 1 * g 1 N %2)+* "$!%f *

βe ∼ 10−6 .()2 * E )^N.( %2JN !E -*!)21 ~l JN )+*HG 1)+* N**4,_)2%Ê )21 1 .()2G 2'2% 2"* , s %2 g .()2 V *s *.( \N )+ . %2JN !1!!)21 * AC !*- *1 )2. M)2 1!!)ÊhE .* -1W 2"2* D/*e.1 .0 )*14 HZ s .* -1W 2"2*`D/*PCN*!E .* N . , !%2*HZ )+* JZ -(.()2 % 1\E 1!FG)+-UN )+ g .1 )2 .()21\N!)2!Id? 6 HZ 1 PC\Z* 1!)2 . 1- * **EH ! ^$C !0'P*

ν %21 - )+-* \s#)2 1*4l 0)21!0. *)2!%21ν =

me

mi

q2R0 νecs

∼ n

T 2e

$ K(LO &*. )21 33 !!)2* %/Z )2 #*. HE - JN !33 !!)2* %PZ HE n .(.;l%2JN!**a@JE 3 ! 6 )+ 1)+- *.* "$!* . 1 .()Ê *1!)2*1!%21* E 1

ν %21E )+ -FG s .*.*$)2 *717s#)2 1!)+*HN *.* , JE 3 !*$1!)+- .()2 % 1\E 1FG)+-UN )+ 8@E )+ )+* 0- )+*l ! / *- .#3 %+%2%2* 1!)21!!)^f )+TC

η‖∼T−3/2

e

XBE *HZ %+%ν 1

s#)+.$.()2 * E )^N$11 %+)2JG%+)2 .( %2JN!1!)21 wp %+%2 *)+*-@6 .* 1!!)ÊhE s %2 )+* V1 *)+! '( :(LKf !- 1!%+%+hs .* 1!)+*. p ) )ÊhE hN%2)+*s .*.JE

ν )+*1! 1*. g ν 1 1!3)2%+.()2 * E )^N . !)& R@BV%&fdo*a@ "$%+%2* )+*

)ÊhE -FG %+%2 )2p JZ)2 **-Bp %+%2 *)+*-3 E 1!%+!)2!1.JE M( 2f R@ %+ .G g

ν ≥1.()2p %+%2 *)+*-@ * E )^N )Ê , !%2*HZ f ! %+* )+*/.*k_.!-!*.

!!)+! 4C *%+)2 ks#)2

β)21 /. MFG)+-UN )+13 E

ν %2 .()+-%+)2 )+*WMN%2* 1 PC\Z *-.()2 )+* 1!b[!)2* !)2!*-g .()2#g%+!)+-UN )+ . 1 %2JN!1!!)21 */*HZ %+%21 - * N**4

!)& R@BV%&fdo*a@ -)Ê $ %2JN !E -*!)21 & ν < 1- !)+*- MFG)+-UN )+

p#%+%2*)+*-@ -)Ê $ %2JN !1!!)21 & ν > 1 MFG)+-UN )+

JE CG$. 6 .%+%+-%2)2 *- * )+* V1 *)+! '( :( '71!)+*. .()2 !)& R@BV %&fdo*a@ "$%+%2* )+* %2JN2@!E -*!)21 1 PC*E *4 s % .()2 %2JN !E -*!)21 *ml JN )Ê A )2 .(!)+- 1!F@G)+-UN )+1!-)Ê )+** )ÊhE -FG *.l)+*HG ( .()2 * E )^N * HE * )+*.JE 1!)2Z HE


p )213)2%8.()2 *)2 R@ .() !)21 N !)2* .l %2JN !**Qf 1!TC\N *)21 . , )+%2 *!*1R@3 !;s )+ 1!!- *. .( /.()2l × p @ W * E )^N 1!)ÊhE *. . %2JN!1!)21 A! s#)+. . N *2f JN !)^f , *1!3 !

Γ = 〈n vr〉 +

n

B〈u‖ Br〉

$ K(+9 &)+* 1!4] )+*)2 .( .* 1!*.V* )+% 1!!)ÊhE rHE p )21!3)2%s#. )+* )+*wV! )+Z HE %2JN!E -*!)21 * 1- s 8@ " N*)j@ 6 .%+% $Id? 6 *M %+%^s %+%2* *.hS \gHEy@E *- 1!\E & E )+Dp ! !*- *g .()2 V *)+%2 . )2* ? %+HN! !)2*1*!-)2 - JZ)+-.G g -FG "$!Df *

β∼1*. / #.()2 , !%2*HZ )+** %2JN !1!!)21 * N@

(s )21! I XBE %+- *.* 1!)+*. )+* . , %+%2K(+Qf 1 )2 .*mMFG)+-UN )+*

ν *. .()2 Z - /!)+- *)2 * *. , JE 3 !* g#.()2 E !!)2 . 1 , l , [b@Bli(3 !)Ê * 10*./g#.()2. 1, [l@ MV@Bli(3 !)Ê *10*- - *4)2! ))21! )+* %+%2*4?C%+%2* .1 % 1\E

βeg

.*OZJs )+%+)+- * , NE N )2.* !)21 4

n/1019 m−3 Te/eVν

ν

' 99 9( 95 9( 9UeI 59 9( I 9( 555 I 9 9(qc5 5( 5 K: I: KUK(LK ' qc:( 5 K9(Le 5: '( 'UK

) ! $0$0/<) /') 4)#"9/!"%& ( )H -# $'+$0+$')HA +5+ 7 "0- /')"ν )"%<# )H$0/ G B:; )"%-)5/0)+-)/0"%B*+ :

!)"0)& (4) : ))"%)!)( ()"% A +5+</' * / / )" ! $ 7"0$'&'(& (/ 4)$0$0/#

XBE )+*%+)2FN7( )+* i(!*7)+*.(HZ )2! 1 6 -*1F %2. 1!%+%+ 1!)2 7.()2 ?- )+*2s#)2Js )+ .()2)21! !)+- * !%2-*- *Z HE N /. , !%2*HZ -Ug%+!)+-$%2)+ *4 X * )+*/N!%j@%2* V! )+%g .*~l)+*HG7. 1 1!!)21 * DlW@ MF %2. 1 ( .()2 !!%2* W * E )^N. 1W_)2 %2.aE .%+%21DN **%E )+ )+%& . (5V 1 *)+! '(qc + ! *.* l *!-)2 .() -*@1!!)^N7- JZ)+-%s .* .G )5V*2s 1* )+ . 1hMF %2. 1 )Ê 1!FG)+-UN )+1 %+% .()2%2JN!1!!)21 ADC !*-%Z !!) . 4Z)+- 1!)2 / )2 . .!%+)2 l )+*HG %2JN@!E -*!)21 # V "@BVWN !)^f)+TC )Ê p )2 /*)2 .(!)+- % 1FE 1!FG)+-UN )+. .1;_! %+*. 1 *E %2* , *13 !1 *!)+!)^f )+*HG )'V *s 1* )+ . 1 1!!)21 *MF %2. 1 .1 - * Qs#)2 .bE -*!)21 Y? % !.JE 6 -* %2. )Ê % 1\E )+* .() 8@%2 S E 3 ** 4f !%2)+ 1!%+%+/1!)2 1 .G . 3 %+%2%2 , )+%2 * !*1!3 ! $Z s )+, \E )+*/%2)2 *- $ K(+9 & & - - *2g .JE .( .()2bl × p @ * E )^N - !!)2 ** , *13 !f !* % C11!)+- %2)+ I '

t -H9ku4vyxtP4Tx )+*$&& 0

l1 % CG 1)2 1 %+)2JG *. 1!! %+* .G * .JE )21 )2! bl !% C!* gB 3)21 MFG)+-UN )+* . 1 *.1 )2 !3% 1\E 1's#)2 E * 1 ) , l , [ ., [l MWV *. . *E %2 , )+%2 *!*1!3 ! $ )V*2s 1* )+)+* 1 i(!** 1!!)21 * 6 -* %2. 1 .( .*1l × p @ , *13 ! . !)& R@s %+%2*.a * E )^N 1!)ÊhE )21+XBE p )2 MFG)+-UN )+Z p )

ν > 1.4

\s )21!/1!)2 . %2JN !1!!)21 M3 JZ ) %& %+% %21hZ Hf !% C11!)+- AC !*- $ E )+MF %2. &

A * .JE M s !3*HN .()2 1yV ! )+ . * 1! *-$. 1 , *1!3 !1 * hl )+*HG)+* 1 67-*1!F %2. 1 E )+ )+*JE %2JN!1!)21 * , !%2*HZ E .%+% )21! . Z 1TC\Z %+)2 l )+*HG . 1

ν @ "$! 1 ( .()2# ! * E )^N )+*#)+* 11* W?- 1!%+%+*-'X *kS 8@3)+%c 1%+% )2( * )+*HE % )+*- - *- *ms .*4

"# & +

)2 6 .%+%+-%2)2 *- *7. 10Wp 6KW@ -EhE 1! PC%+ E * * * 10.* )+*- *- 10#- @N **HZ)2 ** %2)2 *- * $ K(+ & )21 $ K(LK & s )0 )2 g .()2 1!3PCp ! !*- . 1l @ MF %2. 1 .()24f %+%21!TC*.()+- %+%2% %2)+!*-

∇‖=∇‖0+e∇‖

\gFN(1!)2 !)+- s#)+. 5M \Z E * .** .()2 3 %+%2%2YM!E .()2 ej‖

E )+ )+%& . 1b[ HE 1 * 1\Z 1 $ K(LK & )+* .()2DS *a@!)+* )+TC1!-%2)2 *- $ K(+ & *. .()2VWE )+3 % !)+TC1! .()+*-*-/HZ s /_!!)^Z )+TC1!-%2)2 *- $ K( ' &)+* 1/! PC%+E *7.()2 6 .%+%+-%2)2 *- * . 11- * ** * D B#' W, !I2* `' i.10%XYD/* ! i$ K(+ & *. $ K(+' & ;l1 )+%2./.()2l \s )+!*- . 1 %2JN !1!!)21 * 1- s 8@ " N*)j@6 .%+%21#HE .()2DZ 1TC\Z %+)2 ! *S \gHEhE *- 1 JN

dn

dt= −v · ∇n0 + χ‖ ∇2

‖(n0 + n)− 1

e η‖∇2

‖φ+ nK(φ)−Tee

K(n)$ K(+ &

mi n0

B2

d

dt∇

2⊥φ = e χ‖ ∇2

‖(n0 + n) − 1

η‖∇2

‖φ− TeK(n)$ K(+' &

A * . S 33%+*- . 6 .%+%+-%2)2 *- *.g .()2 *!-)2 ! %+*.* , *1! !\E .WS \gHEhE *-4*. S 33%+*- *. .()2 .E )+Wf !*.** l *!-)2/g11Yg .()2, \E . 3 %+%2%2* * E )^N 1!s %2 .( .()2 1!)21!!)^f )+TC .()211!)+3 !)^f * N 1!)+\Z $ 1!)2 WV 1 *)+! $ '(qc + & g l*!-)2!*1 1 & )2! )Ps#)+. . 3 %+%2%2 )211!)+3 8@!)2*1!3 E

χ‖%21 V HN'g\Z *-kf \s *.

χ‖ =Te

e2 n0 η‖.

$ K(+7K &)2 % E \N)2!* , \E 1)+*. )Ê p 6K@ -EhE *)2 f ! *.*4#)2 1 s \ZE \N)2!* , \E %+- * )+*- - * .()+JN#1 .*/5E \N )2!* %2)2 *- * )+*4V a@1 *)+! '( 5( I g8 1)21!!)^f 5p#%+%2 *)+*-UE .* )+* . 6 W@BADC !*- )2 6 )+!* HE . 18-\g*E \N)2!* , \E 1 )+**! %+ . 6 @BADC !*- % CG 1!)2 %+- *.\E G * -\g*.*4 1


p 6KW@ 6 .%+%s#. 1!3\g*-%+)2 Z * 1! *- f * !FN(1!F!*- * f \s *.s ) \gFN(1!)2 !)+- s#..G .()2#3 %+%2%2 " C\E .() 1!)2* )+*/. , JE 3 !!-%2)2 *-$ '(qc5 & $ %2\Z , \E & *)2 Wf !* % C11!)+-s .*N**1 !*mE * .*l )+*HGwE -*!)j@1 bMF!*- */ \gFN(1!)2 !)+- * s#)+%+% I2K dV* %2- .Z ws#)+. )+* .()2 1 V! )+ .()2WV ** HE - JE (.G .()2 3 %+%2%2 , )+%2 *.() 1!)2* )+* . )2 -%2)2 *- $ -\g* E \N )2! , \E)+* $ K(+ & & *)2 f !* % C11!)+-s .*N** IUI

p )4%+%2* , !%2*HZJ@ M)Ê % !)2** )21!# 1 .!%+)2 )+*WZ 1#C\Z %+)2 CE 3( *- )+* .1M@1!JE )+*HZ * .()2#1- * **:, 46 7 B )h9? '' W, 7 d[ *#.()2 1,N'g*1!!%+)2 #)211!)+3 !)2*sbg. .()2 .( .()2 1!)21!!)^f )+TC

η‖f ! *.* 3 %+%2%2 CE 3( *- *)2 1)2 * .8@

E )+.1M(1!JE )+* )+** - 1TC!!)+-**)2 !%+)+**1r1*. )+* !*N*** .JE 1!)2 %+)+*QX *1)+%+)+TC* * Bs#)2TN %+/ *4 )2 !!)j/Z )2%+%2 )211!)+3 !)2* g !*)ÊhE .()2 %+%2.WN )+*!)21 * l JN Ps %2 ( .*N%2)+*1!* TCHE %+)2 *MN%2* g .*V!*1!3 !.y? %+HN ! !)2*1*!-)2 .( " C\E \Z-*- 1!- *4 )+* )+*JE ? %+)2.aE .%+%0.()2 1l JN g .()2 1*HN * E )^N (MN%2*f *

ρs*)2 G s .*aN** E *Z p

E )+ )+%& f * .0P** 6+*-*e9; B#* , , s %2 .( .()2 1!3 JZ )2%+%2 " % . )Ê 3%2JE *a@!)2!* *HE !)21 * 6 ! . )+* )+*JE , !%2*HZ . 1!!)ÊhE s#)+. g )+*WZ 1TC\Z %+)2 l *!-)2 1*HN )Ê 1!)Ê %+)2!* M(1!JE 1!- *4PV%21 p )21!3)2%4g.()2 1_!- *1\s )21 1) . VW]?K8@ . I 5 <I :\s,C * 4)+* . .()2 !!)j/Z )2%+%2 )211!)+3 !)2*$11 %+)2JG%+)2 $.( $.1f \s *. * HE !)21 yM JE - %+)2 !s#)+. /V%+!* !)^f )21)2 .()2_\s *.(*- f *Z 1TC\Z %+)2 * , \E * * .* .()211!)+3 !)^f " )+\N *- )+* 1!h] )+*)2 (N %2)+* TCHE %+)2 MN%2* s#)+\N )2 14_!- *1\s )21*.hE * )Ê , b@ . Ic bs#)2 .;XBE Wp 6Ks,C %+E *Z HE p )21!3)2%4.()2 , \E

χ⊥∇2⊥n

*.ν∇4

⊥φ .$ K(+JI &

)2Z 1TC\Z %+)2 )211!)+3 !)2*Qs#)+. .**mE )+!%21 . E χ⊥

*.ν)+* .DM)Ê % !)2*

- 1!!

r1EhE *.E )+/.* )211!)+3 !)2*1!\E * $ K(+JI & )+%2.* .()2 1 2s \Z *. -\g*.E \N)2!*, \E )+* $ K(+ & *. $ K(+' & .** .1 6 .%+%21\(1!JE . 10p 6KW@ . 1

dn

dt= −vE · ∇n0 + χ‖∇2

‖(n0 + n) + n0 K(φ) + χ⊥∇2⊥n

$ K(+5 &mi n0

B2

d

dt∇

2⊥φ = − 1

η‖∇2

‖φ− TeK(n) + ν∇4⊥φ

$ K(+: & 1/p 6K@ 6 .%+%bE )+ .* Z s )1\N% *.?%2.!*

n*.

φ3TC1*!)2!/1!)2 . /%21

%2JN!1!!)21 1 *\E .%+% . 10%2JN!E -*!)21 *_)2 %2.aE .%+%21 A /.* )21 !)+- *4p ! !*- * Z HE )211!)+3 !)^f *4S 33%+*- 1\E .%+% \N ** ,E * * *

t t Q$ P9= <9=(1$2 =)+*$ />

.G/.()2 )+*HZ )+- S 33%+*- . )2.* %2)2 *- *7.( .()2%S \gHEhE *- 1!\E K %+-)2 .( .* .()2**7. 1 )2 )+* !-!*.1

n0HZ s . 1 % 1\E .(!FN(1

p0*. .

6 -* %2.aN\gHEhE *- f !1 * p#%+%2 *)+*-@BX *1!)+%+)+TC* !)+ * .()2 W * E )^N .MF!*- * *4X *1! 1*. 1!- * .()2 1!TC\N hs .*.* MF!*- * g )+** *2s a@1*.* .() E -*!)21 *M!E ..( .()2%V *s 1* )+ .67-* %2.aN\gHEhE *- )Ê, \E K(n)

.DZ s )+* %2)2 *- )+*ySE 3 11!)2* C ! ?%+-%+)2 $)21! .()2 .() E -*!)j@1 M!E .()2 *)2 E .()^f !- *HZ ) E .()2,n 1!)+*!%+)+TC . 1 % 1\E 1Z s ** 1! .()2/ % !)21 !)2*11!!E .()2 *. %21%?%+- )+%2.* 1!)2 .()2wl × p5@ " )+! % s %2 E )+ )+ %V#.af JN !)2*Z *)2 !%+)+** 1%+1!!- *)21!)2!* MaC!!)+-*- . 1yM 1!JE 1 g *4 .()24S \gHEhE *- 1!\E 4*. .** . V 1!-%2)2 ~Z s#)21 * .%N )+*!)21 * l *!-)2*. .#! \E )21 *l *!-)2 . 1 !TN 11!! 1 p 6KW@ 6 .%+%8 )+* %+%21 .()2 1!)^N .0 1)21!!)^f *p#%+%2*)+*-UE .*717s#)2 .()26 *)21\E * . 18.() %2* , )+%2 * !*1!3 !1.( .()2 l × p @ W * E )^N 6 )+ .JE p 6K@ -EhE % CG 1!)2 * .* !%2-*- * )Ê f ! !)+- * V1 *)+! )Ê p )2 MFG)+-UN )+.1#!!%2*_! %+* . 1 *.1 )2 !3% 1FE 11!.()2*4

#) # ( &

X * .()2 1JEV 1 *)+! s#)+. .1 ]1!*- 1Ff * . 1 p 6K@ -EhE 1 )+* .* s#)2 a@!)+- 1!* *HN* f !- 1!%+%+ A * .4V1\s % . 1 .()2.()Ê *1!)2* %2* *- )2 1)21! .()2 V 1\s % .?!)2\E .* *HZ %E )+.Z -!*.%+)2- *.*1?!)2\Z!%2-*- )+** 1 )2.*. 1S !)+!)+HE 5s %2 1/g .()2 1!3PCwf !- 1!%+%+* M)Ê % !)2** 17s %.()2 ! 2(1!)^N %21 .()2 *HZ)+! %+)2 )+*HG

?g .()2p **- . 1 !!%2* * , *1!3 !1ys,C %+ E * )Ê Wp 6KW@ -EhE 1. % 1\E *.1 )2 )+**$1 HE %2*1p )2 $)+* .() %2V1. * *-Qf * )+*)+- * r *a@!)Ê !*]C*- 1<p ! s#)+. * * .DE -*!)21 *1_1 !*- )+**! %+$. 1 *- )2 1 .()27" )+\N *- )+* i !** 6 -* %2.1!F!*-bs#)2 1!)2Z p .( .()2l @ MF %2.1!3%2* \Z-Ys#)+. XE p 6K s#)+. %21 *- )2 )+* .ag** , !1R@1 %2 ! s#)2 1!)2 )+* V )+%2.(*-1K(+ 1\N)^Z Z )2! )21! )2 .() %2 p )+ . 1 *a@- )21 1%+%+ - JE CG . !)& !*PC !*- )Ê p )2 . )+*!-!*.(-.()2* *% C*-

L⊥%+)2- *4 1 *- )2s#)+. 1 - Js,C %+ .G 1 .() % - * f *Z s ) !)2* %2* ? %+HG@<C *E )+

q

*.q

HE G0)21!r - E !!)21 *7 1!%+%+*-ws .* , )2.%^N .()+* *

(r, θ, ϕ)f \s *. )2 @

E !!)2 . 1 )+*1 %+)2JG *.* 67-* %2. 1 - .** g .()2[3 * )+* .()2/6 .%+%j@-%2)2 *- * $ K(+5 & *. $ K(+: & )+* $ 1)2 WV* *- V ' & .()2 g , NE N(1,E )+ -G JEV1!3 JN \f ! PC%+!*)21g%+)2 r<%+)+*.R@BADC !*- )+* 1Ys#)+. )Ê Wp 6K .()2 *%7d?=2-'.10f \s *.Us %2 0.()2 [3 * * )+*HE %Hf )+*( $ 1!)2 V* *-V I & ) %2)j@ * .()2 )2.%2*S \gHEhE *- 1 JN s )+! )+* \gTN 1)2 !)+-)2E -*!)21 W_1 !*-

ss#)+. g .1 .() %2

q@ % \gTN 1!)2 !)+-)2\Z %+)+*8@

!)21!)2! E * .1q@ % HE )+* 1!- 1! DE -*!)21 ? %+HG <C E )+

q0 !HE )

) # -H: 7$'&bt -H9kuwvyxzt/4T|)P<8-/(~( **$Qx )+*$&&<&$'6+ <$

s#)+. )Ê p 6K . S \s !1/q(r)

%+)+*!)21!)2!

1

q=

1

q0− 1

q20

dq

dr

∣∣∣∣r=r0

(r − r0) .$ K(+=c &

6 )+ .* *)+!)2** $ '(LKUO & -)+%+0.E )+

1

q=

1

q0− s

q0 r0(r − r0) =

1

q0− R0

Ls

(r − r0)

r0.

$ K(+7e &)2_1 !*-

ss#)+. ))+* M)Ê % !)2* )+**! %+ . 1 *- )21 %21wN *1!*

1!!- %2- )2YM3 JZ )j/N!)2* . 10)+*1 %+)2JG *.* 6 -* %2. 1 *. E * )+*QV* *-kV '(d

q q0 q

$B /'"0& (/' /0"0)-)$0+ )& ()) / + 5-) ! $0/'"% 7)4 $0/0)"%/ "% $%/ ) 2 +"0 $'$'&'( B#&'(H 13)"%# 4 & (

q0-$0& ( /G ))

)"1/q 7 >#"0$0#"%-$%/

q0-#$%/ "0)& (/0$*+ #" "0 ) & (# - /0/0 4) /'"% #

!

E .()2 6 .%+%+-%2)2 *- * * HE !)21 ~Z % 1* )21! 1 1!)+**f %+% .( )+* - )+-*4AR@E )2!*-7.()2 %2)2 *- * .()Ê *1!)2*1!%21 Z 1FN%+)2*4 1 .*_!)+% .G ) )+*- )+-**$" % .DA\E )2!*- 1\f 1 !)& .()2hV *HZ % . )Z s,C %2*.* E .(HZ )2!s#)+. <V %+%2 E Df * 3 2(1!)^N%+)21 *. - E !!)21 %2Jf*HZ 1!)+*. .**)+*s *)+- * .()Ê *1!)2*1!%21* S kZ )2* * $ )2 E & . .()Ê *1)2*1!%21* 6 .%+%j@-%2)2 *- * *! %+*4 \g )+* 1 * 1 )2.bE * 1!)2 / ) .,A\E )2!*- g TCHE %+)2 *.wZ)+!%+)2 YMN8@%2*s %2 B 3)21 g .()2 ! * Z 11 1)+*. XE p 6K@ -EhE s#)+. .()2 2(1!)^N$. 1!)21!)^f *p#%+%2*)+*-@ 6 .* *.%+ * 1 )2. E * 1!)2 g )+*r )+1\N%

τs %2 #)+* . FG *.(* *- . 3)21 * %+)+**kp#%+%2 *)+*-@BV *s 1*

%+)2-*. g )+*b]C*- *1\N% ξ8s %2 0.* TCHE %+)2 *wp )2 . 1*HN * * E )^N

G JE CG .0!)& !*PC !*- -)+%+0. ξ ≈ ρs

( ) *. .()2g p#%+%2*)+*-@V *s 1*$* 1 )2.*. 3 %+%2%2 1)21!!)^f)+TC *. .()2 M !% C*- )+* .()2Y]C*- *1\N% ξl)+*- *-

ξ =

√min0η‖ Ls√τ B0

; τ =

√R0L⊥√2 c

$ K(+7O &

t t Q$ P9= <9=(1$2 =)+*$ +

l1bs .* .()2 .()Ê *1!)2*1!%21*S .()+* *

x =r − r0ξ

; y =r0θ

ξ; z =

R0ϕ

Ls

$ K( '9 &f \s *.1 .G g .()2 ) * !) %23 * *. .1 .() %2

q@ % %+- *. % !)2@

** - %+*

∂r =1

ξ∂x ; ∂θ =

r0ξ∂y ; ∂ϕ =

R0

Ls∂z ;

1

q(x)≈ 1

q0− R0ξ

r0Lsx .

$ K( '( &)2A\E )2!*- * g .()2 3 1!)^N%+)21 *_ !) %2*s .* .**QE )+!%21

t −→ τ t

; n −→ n0ξ

L⊥n

; φ −→ B0ξ2

τφ

; A‖ −→ B0ξ2

Lsψ

$ K( '' &f !- *EhE *4P?g .()2 )211!)+3 !)2*1FN kZ )2** -)+%+# !*

χ‖ −→ L2s

τχ

‖ ; χ⊥ −→ ξ2

τχ

⊥ ; ν −→ ξ2

τ

min0

B20

ν

.$ K( 'UK &

6 )+ . 1 - Js,C %+*~A\E )2!*- gS .()+* * *._ !) %2* ! PC%+E * 1 $ K(+5 &*. $ K(+: & .()2 .()Ê *1!)2*1%21* p 6KW@ %2)2 *- * )+*M% a@ E !!)2

∂t n+φ, n

= −κ ∂y φ+ δ G(φ) + χ‖∇2

‖(−κx+ n) + χ⊥∇2⊥n

$ K( 'I &

∂t∇2⊥φ+

φ,∇2

⊥φ

= −∇2‖φ− G(n) + ν∇4

⊥φ$ K( '5 &

)2YM)+-*- . 1bE )+!!%2* )+* !-!*.(-.()2**7)21!0- - * .(

κ = −∂n0(x)

∂x= 1 ,

s E )+L−1⊥ = − 1

n0

∂n0(r)

∂r

$ K( ': &g%+%+,s#)+. ,X *. iwA g .()2*\E )2!* _ !) %2* s#. )2,Z 11*4]4 1! \N )+s -- % 11*4)2 *\E )2!*[3 * )+*mM% a@ E !!)2 1!)+*. - - * .(

f, g = ∂xf ∂yg − ∂yf ∂xg$ K( 'dc &

∇2⊥ = ∂2

x + ∂2y

$ K( 'Ue &G = sin(κyy) ∂x + cos(κyy) ∂y

$ K( 'UO &∇‖ = ∂z +

(κzκy q0

− x

)∂y

︸︷︷︸∇‖0

−ψ, · $ K(LK9 &

E )+ .*QV HN'g\Z *- *

κy =ξ

r0; κz =

LsR0

; δ =2L⊥

R0

17s#)2xq= m

n=κzκy

(1

q− 1

q0

).

$ K(LK( &

< # -H: 7$'&bt -H9kuwvyxzt/4T|)P<8-/(~( **$Qx )+*$&&<&$'6+ <$

.()2 )211*HN % EhE ψ, · )+* . %+%2% %2)+!*- $ K(LK9 & *. .()2 "$ 1%^s#)+\N *-. 1 *\E )2!*h_JN!3 * !) %21

ψE )+.* MF!*- * l )+*- *- )+* .1 6 .%+%21F 1!JE d[ *

V *s 1* )+ )+* 1kE -*!)21 * MF %2. 1wf 1 s#)+*. .()2 1 .() %2SE 3 **7. %+%2% %2)+!*- ∇‖

!

r0* HE !)21 *m]1*- . 1\N%+)2!* %2)2 *- * )+* .*mS .()+* *(x, y, z)

s#)+. )+*?!)2!)+ * *1\Z )+*

y@*.

z@ 0)2 !*- - JE

n(x, y, z, t) = n00(x, t) +∑

m,n>0

[nm,n(x, t) e

i (mκy y−nκz z) + ] $ K(LK' &

φ(x, y, z, t) = φ00(x, t) +∑

m,n>0

[φm,n(x, t) e

i (mκy y−nκz z) + ] $ K(LKUK &

6 )+!%21 .()2 1 1,V *1\Z 1 Z!C%+%+ .1 6 .%+%+-%2)2 *- 11\(1!JE )+* - JN 33 %+) * !) %+-%2)j@ *- */g .()2bN E 3%2 i*w?!)2\N kZ )2* *

nm,n(x, t)*.

φm,n(x, t))2 %2)2 *- *

1!)+*. 1E )+ )+* .() %2 0)2 !*-x*. )+* .kr )+

tZ % 1*48 ) )21!

m.()2 3 %2)2.%2

6 .*HZ %4*.n.()2 )2.%2 6 .*HZ %

?g )+* - Js C %+V *HZ %bf * .() %2* )+!!3*HN *nx

)+**! %+ . 1 *- )2 1s#)+. .()2bfh B !*%79; H*S*14c*C%NXU!-1D/hZ ]1!*-f \s *. )2 *HE !)21 ]1!*- .r )+1 !)+! - 1 )2 0g0.()2 *- 1!F! %+%2%2.a* E )^N $ ∇2

‖0

@ , \E & E )+0)+*JE BH8 Q 46 B%* +*"$%,d; ,+-1 iC% C*!<2-0* LK9 PM )+YeIc/s,C *. g0.()2Dg!)+- * , \E )+*]*8 i !%4 ,':, *S4W ,D W ';_['2<'S% +*-*%f \s *.Ws#)+. I2e A * .yV 1Fs % . .() %2* )+!!3*HN *HZ % 1!!)ÊhE .()2wV *HZ % .Yf R@s *.*?!)2\E .*

(m,n).()2 TCHE %+)2 V P1!*- ) )+*M)Ê % !)2*4 "$- *$.

6 -*!)21!)2!*- . 1 % 1\E 1-)+%+)+* . !)& !*PC !*- .()2y? %+JE .*PC !*- $ '( IUI & 1.G * 6 .* E )+k‖ ≈ 0

! Qs .*40)2 171!)+*. - . .()2s#)2 !)+- 1!* *.)+*1!)+%21!* 6 .* )+* .p#%+%2*)+*-@ * E )^N <?g01)2 -)+%+

k‖ =1

R0

(m

q− n

)≈ 0 .

$ K(LKI &XBE ! * *- )2bs .* .** * 0%+%26 .* 1 .JE ?!)2!HE \gTN2@1!)2 !)+-2s %2 .()2p .()+*-*- $ K(LKI & g%+%2*/HZ s ds %2 )+**! %+ . *.E )+* *.* 1)+!)2** .0!)2* %2*? %+HG <C * %2N%+)21!)2! 1!)+*. Pl1#-)+%+

q n ≤ m ≤ n q

E )+n = 0,∆n, 2∆n, . . . , n .

$ K(LK5 &M E )+)21! * * . )+!!3*HN *HZ %

nx

.()2 E 8i )Ê %2 , )2.%Ê .*HZ %n

1=s#)2#.()2)2.%2#6 .*1 !)+!s )+

∆nZ ys,C %2*4V )+%2.(*-YK( 'f *1 %+)2 * )+*HE %(.()2

V 1\s % .06 .* 1.JE ?!)2!HE

!

)2Z ]1!*-/*Bs *.()+- *7*.( .()+*-*- *s .* .( .()2 " % . 1 )2 )+* R@-!*.(-.()2* *

∇n0 1!!- %2- . .()2* . 1 .() %2* )2 3%21 .()2Wn%+%2.

t t Q$ P9= <9=(1$2 =)+*$ Pt

0

m

n

q n

q n

n & ()/0$'& ( "0$%/')- #" 6+5) $ ( $ ) D +"%)"0"06 5/"0 # "%#) , ") $' /H1 2" 4))( "%) D +"%)"0<+5 -4))5/0)ef−m,−n = ef∗m,n

6 /B #" ( /') "%$0/0# )"0 # 5" 6+5)4#() - /m,n ≥ 0

$0 ( /#

MF!*- * *.$. .E )+Yf !*.** , !%2*HZ )21!+E HG .(0- %s .* .G.()2 1 *)2 1 ! * kl *1! * . 1N *2f JN !)^f * , )+%2 *11 1;f %+%21TC*.()+- 1- -%+)2 *s#)+. *. .E )+0.1 *HE !)21 Yli 3 !)Ê * f \Z)+!)+- *.#)21!( bs#)+. .bZ)+!%+)2 *./TCHE %+)2 - JE )+!%+ )2 )+* !-!*.(-.()2*N *1!* 0- %+*4)2 1 - 1 )2 s#)2 %+-H?g .()2Z)+!%+)2 Yl *Bs#)2TN%+*- . 10)2 )+* !-!*.1 -)+%+

n0(x, t) = −κ (x− x ) + n00(x, t) .$ K(LK: &

.()2 -G 1FN%+)+- (0,0)

@BSE 3 ** . )2 1!F!*- )+*HG <C *- JE )+!%+ FG .1!%+%+!TC-0)+ 0"$! *( %+%21bZ l *Bs#)2TN%+*- . 10 %21

n0(x, t) )

?g0.1Z)+!%+)2 - JE )+!%+ )2 3% -)+%+ .**〈n0(x, t)〉t = −κ (x− x ) + 〈n00(x, t)〉t .

$ K(LKdc &6 )+0.* *.( .()+*-*- *

n00(x, t)|x

= n00(x, t)|x = 0$ K(LKUe &

! PC%+ E *QZ HE p )21!3)2%8)+* )2 3% <s#)2 1#)+*QV)+%2.(*-4K(LKyZ 1 * )21!?g .* ?%+% . 1%+1!\N *1!)21* * gFN's#)+\N*- .wMF!*- * $ - 1!!!)2 %+ 1 %0)+*V 4FK(LK & 1!- *$.()2 *.( .()+*-*- * $ K(LKUe & . g .G .Z)+!%+)2 $- JE )+!%+/)2 @-.()2*

∂x〈n0(x, t)〉t = −κ+ ∂x〈n00(x, t)〉t$ K(LKUO &

)Ê TCHE %+)2 *76 )+!%8.( .()2S*1!* κ 1!!- %2-0)21!

1

x −x

x ∫

x

∂x〈n0(x, t)〉t dx = −κ . $ K( I 9 & )+* .06 .%+%+-%2)2 *- $ K( 'I & * V %2)+!*- * . 1#)2 3%21 $ K(LK: & )+*- */s#)+.Z # 11* 1!)2 .()2 )+!%2Jf* S*1!*

κx f *f !* )+* )+* . 6 .%+%+-%2)j@

*-ws -- % 11*4


1 *- )2Ws#)+. HE .()2 1- * ** * G. H*S4 \s )+!)+* .** .()2 *.( @.()+*-*- * E %+)+*HN * *. * CHG * *. . 1 .() %2*.M)Ê % !)2*1!- )21 g%+%+s .*4

0

n0(x)

q q0 q x

x

n00 n00

〈φ2〉1/2

V

q0qq

xx 0

n0(x)

qq0

q

xxn00

n00

〈φ2〉1/2

V

q0q q x x

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E * * )+**QV11 *)+!1#.JE *.1 )2 !3% 1\E ! *. *)2 .()2 % 1\@E s *.as 1%^s#)+\N *- * )+* .()2 * 1! *- *4E )+)+* JZ )2 dN**hE * 1!)2 .()2# R@Z** %21bX 1%+)21 )2 * g0.1#1!)Ê %+)2! % 1\E f 1%+%2*4<r<s#)21 * .()2 1*mM )2 *s#)+. .()2 !!%2*/ * E )^N !\Z )+* 1 *1! *.* , )+%2 *11 1 ( ! %+*4X *7.()2 1*X 1%+)21 )2 *1s .*$%+%2 %+)+**7*. *)2 !%+)+**S33%+*- * f * 6 .*f ! )+*.! )+*.JE E * )+**! %+ . \Z**N )+* 6 .* %2N%+)21!)2!0*. * E )+ ! PC%+YE * )+* 1!)+- 1 *. % *- 1E 1%V( %+%2* . 6 .* E 3%+)+!.* )+**! %+. \Z** )21# )+*QZ .* WC*.!*Hs .()2 *.( .()+*-*- *

fm,n(x, t)∣∣∣x

= fm,n(x, t)∣∣∣x

= 0 m,n ≥ 0$ K( I &

g%+%+ s .*4X * V)+%2.(*-K(LK71)2 DE * %21Dp )213)2% .*_!% (#. 1Z)+!%+)2 $- JE )+!%+*$ %21 .%2JN!!)21 * *!) % HN! !)2**4<X **! %+ . \Z**'Z s#)21 *

x

*.q

1@s#)2yZ s#)21 *

q *.

x %+%2*$.()2yVWE 3%+)+!.*2s !/1!!)+- (,A%+%4 wE * 1!)2

%2\Z !%+)2 *0g0.()2 2(1!)^N )+**! %+ .X 1%+)21 )2 * %21yZ s#)21 * .* \Z**)+* 11!)2!)21! 1 s#)2 !)+- .G/.()2 *.( .()+*-*- *7)+**Ek-%+)2 1!- !)+*- *l JN (.()2 W * E )^N )Ê X ***$. 1 *- )21 1\g *4 1!)+*.7.()2hp )+*$. R@Z** - *2g- *. -GhZ 4s,C %2*1 .GyN )+* 1!\N * .()2* * )Ê`p )2 . )21\@TC*. )

q

*.q

( !*4E .()20.() %2V %2)+!*- . ?%2.

n*.

φ )

x

*.x

1!!)ÊhE * Z wNd*** -)+%+.()2p .()+*-*-

f−1 = −f1*.

fnx+1 = −fnx−1 . ) JZ)2 *

fi.*/"$! )+* 1 ?!)2\N kZ )2**

fm,n(x)* . .() %2* )+!!3 @

1!)+!)2*i = 0, . . . , nx

!% *HE !)21 r )+1 !)+! PC*-hf * . MTC\N . .()2* * E )21\@*.$4+p )1)+%2s .*.* .()2**$)21! .()2kM)+%+)+TC . 1 *HE !)21 *~_ *1

t t Q$ P9= <9=(1$2 =)+*$ <

* 0.( )+*D_!!)+*- !*- . 1Wr )+1 !)+!1 Z - Js,C !%2)21!*4P_* . bs,C* )Ê M)+**)+* 6 )+*)Ê )2!*-$. *HZ)+ .()2 \Z** Ek-%+)2 1! )+yZ s C %2*4p ) )+*N *1!* * .() %2*V P1*- *.$)+*_!)+!*- . \Z**s,C;Z . .()2r %8.#.() %2*7)+!!3*HN

nx + 1Z ! / *Hs 1 )+*l! / *-/. *HZ)+ g

.()2M)Ê % !)2* .5M mE HG E * )#. " %0. )+!!3*HN\Z % *. . 1 r )+R@1 !)+!1)+** 3!)Ê %2* 6 )+!%^s - g0)+*Ek-%+)2 1!0- !)+*- *.*.*4r1TC\Z %+)2 )21!0* .(Z * .G/.()2 6 .* E 3%+)+!.* g06 .* E *. . 1 *- )21 )

q

*.q

- JE CG )+ 6 .* )+ 1 *%+% - *- ( %+%2* Jf )+ "$! E *.A%+% - 1\Z 4s .*4 )2 6 .*)+N**.E *.f )+*( 4E )+%2)2 *- $ '(+'9 & 1 PC\Z* $ 1)2 V 1 *)+!'( 5( I & l !*- 1- JE CG 1!)2 g;f 1 )2 .* % 1\E 3 E )ÊhE )+* \Z**)j@Df *OZJs )+%21'58@K9 . *- )21)+ %2103!)Ê % \s#)2 1*4E .()2M)Ê % !)2* Z / -)+***s#)+. )+*DM!\f !)+%+*-hE )+ .* "$!*

fm,n(r, t0)E )+R@

%218)+* 1r( %+%21!- *1 \Z- "$- * .S33%+*- .86 .* *. . WEYf !)+%+*-f *m? %+HN! !)2*1*!-)2 )21.13PC *)2 !%+)+* !!%2* kr1!*. * PC*-)+-4f *.*M!! .()+*-*- *4?g .()2bs )+" %.,M)Ê % !)2*1!3 E 01) (4.()2 )21\N 11!)2* )+*wS 3)+% 5Df \s#)2 @1*4 r*PC 1! 1%+% * * .()2%XE 3%2JE * !)2!*- . DlW@ MF %2.(- E !!)2 )+* .*7p 6KW@ .Df !- 1!%+%+Ws .*4


#$

* Qs#)2 )Ê )+* !!)+*1!)21 *E -*!)21 *m? % !0 Js#)+\N )+* HG ! %+ . 10 % 1R@E 1 \Z- 1hE -*!)21 14MF %2. E )+

∂θA‖ 6= 0 .G7.1 )+*1 %+)2JG *. 1 E \E 8@

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# -H: 7$& + 7$2 $29 (-H< $d =976$9 PB$&*

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D⊥%21N *1!* Z -!*. *. 1!!.()2* .*

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& " ( & +

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q0*. )+ * A !* \f \Z !( * *. .E )+

.1mV( * .()2 1 * !*z? %+HG 1 % C ~Z f *% 11*4 l N** )+*f 1 )2 .** *D/*C"2 W.0W, 7* !!)2 * s .*5Z p N** )+*.K @ . )+*:'8@ MF\E .*( %2. )+*.(HZ )2! s .*4aX **! %+ .()2 1,V ! )+ s#)+. .()2 '8I@BS*a-!)2* ! )2 DlW@ M3%2*~s .* . ) 1E )+yM!E !!)2 * .G * . 1!)+!)2*

q0 =3)+* 3! 1* *HZ .067-* %2.1!F!*-kE )+0.* 6 .*HZ %2*

m=12*.

n=4* 1! )2 1s#)+. )2%21 1!!)21 ! <A * .()2 1 3! 1* *HZYs .* (!*. . 3 %2)2.%2* V 1. * *-$. M3%2*1!!TC*- )+* . , !1!)+***1)+ * s )+ 6 .* )Ê 3 %2)2.%2*M3 JN!!HE *- -0s %2 1!)2 7HE

m=12 !HE N *HZ*a@

!!)2*4?g .*$ ! * % 1\E )2 . .()2 1 .G )+*$0)2 !*- . 1N%2)+** % 1\E .()+1

rf E % 1\E Z* !!HE Z HE % 1\E *. )+* .1MF %2. *s,C 1*. .8@

) Z! . 1* *HZkE )+m = 12, 12 ± 1, 12 ± 2, . . .

)+* )+- * .() %24MF %2.af !)+%+*-Z -!*. %+)2- $ 1!)2 V )+%2.(*-4I ' & <A * .()2 1* !*.( \E *)21 * s .*7)Ê % 1\E %+%2 / * \E *)21 * * .*UZ!Js )+%21 Z HE MF %2. 1* * * ? %+HG@<C *3 1!)+!)2** *- - %21 Z p 8* . $

q = 11/4 & @ 1!)+!)2* .()2 6 .*(22, 8)

(44, 16)

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4 # )0 <8-a =)0 ;$ 3Y 3hu P>

n1 *)+!1 *E \N )2!'_ %+-bE * )21!3)2%21\s )21?%2.(%+)+*)2*as %2 ( !)2* %2*? %+HG <C *

q=m/nf !% ( * 14N * 1!)2 * - *

m3 %2)2.%2* *.

n)2.%2*

E % C( *$* )+ *71!3\g*-%+)2 * 1!G3*HN Z \gFN 6 )+ )+*V 1\s %f *E a@* .() %4*. 3 %2)2.%f !)+%+* M!!3*HN* g0.()2? %2.(%+)+*)2*2f %+-*- % CG1!)2 1* f )2%2* )2.%2* E % C( * .()2 , 3 %2-)2 . 1 _ NHEhE -* %2. 10)Ê`!!%+)2 *mp @)2 )+* .JE .1 % 1\E %2N%+)21)2! )21 *. ! s#)+.$ *- )2 & )+%2.*4rHE p )21!3)2% )21!g %f 1 )2 .*

q@ "$! .1 )+*Fo)+%2. )+* % \N .()+* *

(r, θ)g .* *- 1!F!* ?%+%)+*yV)+%2.(*-I+Z 1 *467* \N ** .G .()2 )+* .()2 1V! )+ ! 67-* %2.(- E !!)2YN )21 \E )+- ? %+HG <C * %+)2 !

I +&#" ) "%$0/0) )$ B<5- )/%1 # #$ : 7 , )+- /0"0 132" # 7)$0/"9/')6D 3 - /B<5)) / "913) # )) )$%/'"%+ *+

Ic =0.25<5 3 "0

$')(5H)" + "0 "%$0/0) $%/ ) D <5/0+" )$0/0"%#&'<5/ +"%)> "%2) 8 "%)$' "%<5 7"%)6)6AB) )" ; #"%4)+)>

M %+DE * ** .1 i !* 67-*1!F %2. )+*HZ 1 g !.()2 1* *HZ * .*? %+HG@<C * )+* . )+*Fo @ 1!%+%+*-Z WV1!)+%2.(*-4f * '* Q Hs#)2 1 )+*mV)+%2.(*-wI+.!- 1!%+%+#)21!Hr \N *** )21! .()2M!!HN !0. 1 '8I@ MF %2. 10.( .()2V 1)+%2.(*- . 8I@ '8I@*. 7K8I@BX *1%^N !*4Pp )8*s 1*.*MF %2.1!TC\N *Qs .* %+%2 \E @*)21 * 1!TC\N *- -1 .G.()2 ZJs )+%+)+- * 3! 1* *HZ*$ -G 1\N%+)+- X *1%+* & -FG s .* *.Qs )+ \E *)21 PZ p )

q = 23/8 .( mX *1%+)+%2.(*- 1!)2 ! s .*4

"$** 1!)2 .()2ws 1*.* X *1%+* * !k? %+HG <C * )+* .() %2 0)2 !*-7 \g a@* *.4g !% 33 *13!)2 bE * f * .M 1!!)21!)2!*- . 1#67-* %2. 1 ** % 11*1!)2 .()27)+*HZ%+** 67-* %2.(%+)+*)2* *)2 E )+* 1!!)ÊhE * ? %+HG <C Z .(**4)2h? %+HG <C * 1!)+*. ( - * *. i()21!!)2*$*)2 %E $ 1!)2 ~V 4RILK & 1l )+*1\Z* *./.()2WMTC\N . f[* D Q B 7*+*-1 W, B' N**.01bs#)+.kE )+ .JE 1- * *** )+!)^N =f2@ E

σ N !)21!)2!

σm =δm+1,n + δm,n

rq=(m+1)/n − rq=m/n.

$I+ & )2! ) )21

δ.()2 %+ .() %2 V1. * *- )+*e, B+-* '* &r<s ) * !

# -H: 7$& + 7$2 $29 (-H< $d =976$9 PB$&*

X *1%+* -)+*** 1!)2 Z g !% 33 *'s ** .()2YMHEhE . )2.*%+)+* -FG ,s#)+.%210.#.() %2V1!*. . X *1%Ê )+!%+3*HN $ (% G.1"c#- * ** & Ps ** %21

σ ≥ 1-)+%+ LK45I hE . )+!)^N 7f@ E .*$"$! yg !!!) 8. 1! 1!TC\N )21! .11! 1!!)21 $ !)21 & _! %+* . E -*!)21 *?%2.(%+)+*)2*4

( -/ *$29 uw$6$ P$ $d 79 1 *- 1!F!w_ NHEhE -* %2. ) , l , [ N** )+* . )Ê V* *-mV ' 1 !)2 @ **? \E *- - *ms .*4 )2! ) !TC-.0-G .()+1

R0 =1.75E N %2)+*

% 1\E .()+10 )q=3

!TC- *- !C r=0.45

E " )+%+% E * .1 , !%2*HZ f ! %+* . 11! 1!!)21!)2!* % 1\E 1 1!.()2*)21!0.0.() %2Yp )2 )+* .JE 1!)2 .()2YA !)+*1%+*E )+0.*QX *1%+* . 3! 1* *HZg !% 33 *E 1!* - )+-*. .!0. )+!)^N =f2@ E E -FG* )21!p ) . '8I@BS *a-!)2* . 1 , l , [ @Bli 3 !)Ê * 10)201!)2 . .() %2p )2 *)+* .JE .()2 8I@ 8'8I@ *. 7K8I@ 6 . *. )+ / * \E *)21 * %2N%+)21!)2!01!)+*. p ) .* )21! !)+- * p 6KW@ M)Ê % !)2**2s %2 .()2 , !%2*HZE )+ .7e:8@ MF %2.aN *a@-!)2* . 1 l !- .()21 * )^f !1 ) , [Wl MWV * 1! * I 5 s#./.1 *- )24Z s#)21 * . $

q = 2 & @ *. . $q = 3 & @B? %+HG <C *3 1!)+!)2* 1!!- %2- )2

*HZ <C hs )q0 = 2.5

$.1 i(3 **!)2%+%2kV ( %+%2* .YMF %2.aE .* Ey@3%+)+!.*f E * *. . 1 )210 )

q=3)+* 0)2 !*- . 10 % 1\E )+***Z

q=2- 1 Z! * *\Z * MF %2.1!TC\N * )+** )2JG *.* !- *- f * )+*JE 1! 1!!)j@1 * % 1\E )2 Z )+*JE *)2 #1! 1!!)21 *Qp )2 E )+

σ<1 1 1 1!- Js C %+

*- )2s %2 1 )Ê %+- *.* * * %21g&V * N JZ)2 *%s .* 1%+% )2 @ . .()2 6-%+)2 HN )+ .()2 1* )2JG *.* !- *-~Z 1!!.()2* *.g .()2 1/)+**.()+JN*w_!-%2)2 . M)Ê % !)2*1!- *)211 g , l , [ E )+ .**wf * , [l M V*HZ 1!%+%2*4 6 )+ .%2* , l , [ @ E !!)2 *. . 1 3 2(1!)^N%+)21 *$\g*.*Z f \s *.*.*/6 .* *HZ %

n )21 18Z! . ) B 3)21 * , l , [b@ % 1\E 3 E !*

.( .()2 .E )+ f !*.*/ *HZ)+ *)2 %Ek-%+)2 - Js 1* .* MF13%2*1!!E 1 Z 1\Z*.G.()2 1!%+!)2*.,_JN !3 *!) %^f !)+%+*-

A‖(r)$ 1)2 V1 *)+!I ' &Z )+* M 1!!)21!)2!*-%E )+

σ≥1g ! .()2 FG 0. 1* HE !)21 * r )+1 !)+!1HZ s

.()2WM)+%+)+TC . 1 * HE !)21 *w_ *1 .( .()2#" % .,MF %2.1!TC\N 0%+)Ê )+!)2!s#)+. PZ)2 * 1)2 7.1 )2DXFXFX5E )+1)+**

nx =256 )+!!)+* \f%+%2* $ \Z**7)+*a@

N%+1!)^f & %21%Z )+ 14HE )+*M)Ê % !)2* )+**! %+ f * )+*)+- * "$ * *HZ)+Z \Ek-%+)2 *4 / %2JE !)^N . *HZ)+ *. . " % . - E !!)21 * *.3 2(1!)^N%+)21 * E s#)2.aE 1!)2 .()2 )21\N11!)2* )+*S 3)+% 5(E )+* .!%+)2 .(HN!)2* . *HZ)+DZ !)2 *<s#. .1 Z Hf )+* . . @1!!!HN! . 1 p 6K 1! )Ê 3%2JE * !)2! *- )2 $ )2,XTXFX & Z /-FG * q @ "$!* )+* f 1 * *. !* )+* . .() %2*kp )+ %+)2!2M * 1!*.*hZ s )* *a@- )2as %2 HE .()2 $

q0 =3 & @ 1!)+!)2* !HE`N *HZ*!!)2! 1!)+*. ( 1 Z )2 XTXFX -%2)2 )+ &V * N HE 1 %+)2JG .1X *\f%+%

q ∈ [2.5; 3.5]s C *.$.1 1 HE %2U&V * N .1

X *\f%+%q∈ [2.75; 3.25]

. FN .()2 *HZ)+ .(!)21 E )+ . )+!!3*HN \Z %1!)+-5N** )+* )2 X .()2 *HZ)+ - - *g )2 XFXFX HE .* ? N4If !!)+*a@- !s .* $ .()2 \Z**1N *** s 1f \N %2)+*!Ys .* & 86 )+ .* s )+* !%2-*- * 1 S 3)+% 5s#.*/.** %+%2DM)Ê % !)2**4E )+ )2,X .( - g !l1

40 < 1(u4vyxtP <

* ! PC%+ .()2 '8I@ MF %2.aE .E )+ )+ * )2.*wA !* 8I *. 7K8I *. )2 . Z s ) p )2 E )+ * 1 )2 .(%+)2 JE )+!)^N 7f@ E Z HE )+** .* !% 33*- 1R@ )2 Z s#)21 * . 8I@ *. '8I@ 6 .*3 1!)+!)2*4E )+ .JE`M 1!!)^Z )+TC1\s !

σ−

UZ HE*.* .* !% 33*- 1! )2 hZ s#)21 * .* '8I@ *. 7K8I@ 6 .*3 1!)+!)2**hE )+ .JEM 1!!)^Z )+TC1\s !

σ+

( '*

18 i !* MF\E -* %2. ) , l , [ @ l N**E )+ )+%& .4)2.%2* S E 3 ** . 1_JN!3 * !) %21

A* %^!)21 1 !)2 *1s .*48 YV!!)^N % 5' %+)2 ! .Z .*

V *1\ZAϕ =

∑

m

Bc rc gm√1 + r cos θ/R0

(r

rc

)mcos(mθ − n0ϕ)

$I ' &E )+

gm = (−1)m+mcsin [(m−mc)∆θc/2]

mπ (m−mc),

$ILK &s ) )20.()2MTC\N . 1 E -*!)21 *MF %2. 1

Bc.( .*MF1!3%2*1!!E

Ic- - @

* )21!

Bc =2µ0 Ic n0

∆θc rc.

$I I & 1 MF %2. )21! (0. ( %2.1)+ %2N%+)21!)2! *.$.()2 3 %2)2.%2 V1. * *-

∆θc.

MF1!3%2* !TC- Bs 72

∆θc = 2 θc ≈2π

5.

$I 5 &)2 Z*!%2 MF\E .*HZ %

mc 1!)ÊhE 1!)2 7.( $.()2 )2.%2 MF\E .*HZ %

n0 =4*. .()2 3 %2)2.%2yM3%2* 1. * *-∆θc

mc =2π n0

∆θc.

$I : & "$! g , l , [ )21! 1E )+

mc≈20)2 1 )21! %+- *.\E G *kZ yf 1! *4 .()2

MF1!3%2* * . 1)+!)2*rc=0.5325

E * 1! )+*Z HE % 1\E )+* i(3 **!)2%+% ( %j@%2*. 1MF %2. )Êz_ NHE $ 1!)2 # hV4RI ' & 1 3 %2)2.%2bMF13 JN !!HEzN *HZ* !!)2!1!)2 /HE .* "$!

mc )2 1 1,M3 JN!!HE )21!- . 1N *HZ )+3)2!.G%E )+5p \gFN(1!)2 !)+-*-

.0 % 1\E N !)2* ( .1 _ N HE 1!F %2. )+**! %+ . 1# % 1FE 1# )q ≈ 3

E )+mc=20)+* 6 .

mc = 12*- -ks#)+. 1 MF %2. f !1 %21 )+*U ! ,' !* 6+*- '8I@

6 .* 1* *HZ )+**! %+ . 10 % 1\E 1 p 6K@ . *\Z )+* %2JN!1!)21 1 6 .%+% HM E )+ E HG )+*1%+1!\N *1)21!* p * *- . )+*!!)+*1!)21 * M!FE * %+%2*4<?g .()2%?\E % . 1 i(!**MF\E -*R@ %2. 1 )21! %21f *1f !* )+* .()2/ % 1\E N !)2* ( .()24MF!*-mE )+\Z \gFN(1!)2 !)+- *4 )2ghN**E * * )+* * %^!)21 *.ADC !*- 1!3Z .( )+* , *1 \E !)2* .6 .*HZ %2**.* 55 )2 1 , *1 \E !)2*

m −→ m+ (m β3/2β1)1/3

β1

*.mc =

mc

β1

H # -H: 7$& + 7$2 $29 (-H< $d =976$9 PB$&*

Gebiet I

−40

0

40

32.75 3.25

ψ em,4q

m = 11m = 12m = 13

ψ em,4

q

m = 10m = 11m = 12m = 13m = 14

Gebiet II

−50

0

50

2.5 3.5 3

ψ em,4

q

m = 11m = 12m = 13ψ em,4

q

m = 10m = 11m = 12m = 13m = 14

I 6 /0)*)"%/0) )$ 5/"%1 )$ )<5/0"%-$ $'&'() )& ()) /= "%# /0)"0) , ) /6=%= 3 :; 4) $0/0))" 1 )" eψ 7 D </'+ 0 )$ "0)& (/'), ) /'$0"' "%<5/ $0& ( )/0 *61 )6 -#"%$'&'() ) /0$'& ( "0 /%/ $G

* ! PC%+.()2 .()2 FG *

β1 =dθ

dϑ

∣∣∣∣θ=π

*.β3 =

d3θ

dϑ3

∣∣∣∣θ=π

,

s ). )+ 2s#)+*HN %β1

f E % 1\E -%2)2 - Js#)2 PC*- *. . _!% (g /.*)+* !!)+*1!)21 * 3 %2)2.%2* " )+*HN %

ϑ)Ê )2.%2* % 1\E *)2 %+)+* )21! 6 )+

m*.

mc1!)+*. .()2 )+* !!)+*1)21 *76 .*HZ %2* - JN **HZ)2 * )Ê Wp 6K )+*M% a@BAC !*-7 * \Z s#)+. 8)21 .()2 )+- , *1! \E !)2* *. .()2)2.%2 E !!)2 *)2 f \s *.( dl )+*#* %^!)21 !%2)+!*-g .()2,Z %+)+*.(!)21 ADC8@ !*- 1 )21!% *- *)2 bZ ,_g-*-Hr *4s C.G

β1)+*%+)+*D?*HN!)2*

f *θ)21! *. .G )2.%2wS !JN !!\E 4s#)2kZ p , \E E )+

β3f 1 2s#)+*.*s **

E * .()2%r< %+)+*.R@ .DM% *PC !*- f \s *. 1 %+- .()2V 1 PC\Z *-

m −→ m

β1

*.mc =

mc

β1

g .()2 , *1 E !)2*Z )+*!!)+*1!)21 * 6 .*HZ %2* 5: 10)+* .* p 6K )+*- yMF %2. )21!0.** )+*M% - E !!)2 .( .()2Y?\E %

A‖ =∑

em(−1) em+mc Bc rc e

“emβ1

rrc

−1”

β1sin [(m− mc)∆θc/(2β1)]

m π (m− mc)cos(m θ − 4ϕ)

$Iqc &- - *4<?g , l , [ )21!

β1 =0.6*. 1E )+

∆θc/(2β1)=π/3

l1 % CG 1!)2 wZ)+- * .G ** .1bMF1!3 JN!!HEAm,4

*)2 E 1!!E ))21! )2 1 M( a@f ! %+ )21s#)2 !)+-s ** Z HNg*( !)+-_!-%2)2 Z s#)21 * Wp 6KW@ M)Ê % !)2** *. %2JN!@E -*!)21 * , !%2*HZ1!)Ê % !)2** Z HE p )21!3)2%E )+ V]?K- JE s .* 1%+%2*4 1hMF!3 * !) % $Iqc & % CG 1!)2 ~E )+ )+%& .hMN%+)2!*- 1Ff 1 !)& * $ K( '9 & *. $ K( '' &*\E )2* *. )21 1 )+* .* Wp 6K )+*- .G .()2bM!*- .MF %2.1!TC\N #.( .()2l )+*- . 1M3%2*1!!E 1

Ic)+*N'V %+- *QN**4

X * V)+%2.(*- I '71!)+*. .()2 .() %2* %2 . 1 *\E )2!* MF!3 *!) %21!3 JN !!HE 1ψm,4gZ s )* 1 )2 .(%+)2 )+ *- )2kE )+(Z!Js )+%21

Ic = 1NaV - JZ)+-8)2 r % .

f \s *.*zMF\E .* . *)2! 1!)2 s#)2 .QE )+ $ K(LK5 & g .()2 p -*HZ *- . @ *- )2 1 *\E )2!MF %2.(3 *!) %

ψf 1 s#)+*. */.*CHG * C*.!* .

40 < 1(u4vyxtP at

\Z**4 1 1 .()2 1JE * !) % \f !- *. _ N HEhE -* %2. s#./ )+1 )+* . )+*a@Fo .1!%+%+*- .V)+%2.(*-I+ g .* *- 1!F!*?%+%E )+

Ic = 0NaV *. .* %2)2

- 1!F!*Q? %+%+E )+Ic=0.25

NaV M3%2*1!!E - JZ)+- ) )21!#.( .()2 -\g** 0)+*- .()2p -*HZ *- . 1 )21WX - JN **HZ)2 * V ms#)+. )8.()2 1 1!%+%+*- .()2 " )+\N*-. \Z** ( .1MF %2.(3 *!) % .!%+)2 4 )2 8I@ *.7.()2 7K8I@ MF %2.aE . *.* 1!)2 /- * * . 1!)+!)2* . 1 *- )2!*. 1 *.ks .* HG ! %+ . f *%21 )+* . \Z* 1!\N - ./CE 3( 0V1 .()2 1JE !*. 1!)2 E * )Ê )+*Fo @Bp )+%2.I+ .()2DZ - /!)+- *mX *1%+PC . 8I@ 6 . *.$7K8I@ 6 . * 0*- !C Z DC%& 1!- )+%2.2s C *./1)2 g .()2 \Z** )2 * JZ *- 1F!W? %+HG <C * R@1 *)+! !- *4)2 %+- *.* )2.* )+*Fo @ 1!%+%+*- * )+*V )+%2.(*-yILK%Z)+- * * *(.GyE )+ )+*JEM3%2*1!!E f *$ )21'N'V )+*%M 1!!)21!)2!*- . 1067-* %2. 1!)2 Ws#)+.

I +&#" 7 "%$0/0) )$ B<5- /%13))$ : 7 , )+- /'"% );)$' / 74)))" /'+5& ( $%/'$0#"% - /Ic = 1.2

< 3 # $0/'"0<)" /'+5& ( $%/'$0#"% - /Ic = 1.8

<5 3 !)"0( /0) $0& ( D )) & (+/'$0& ( $'-5" +& ( )"%"0)$0/0 )" 7 ! <5/0 )"! /'$0& ()=$0) 4))"0<) # +- / ) / ) $0$' )$0/0$0/0"0 <5/0" *+)" "0$ "%2& () =$')$%/'"%</'" ):;"0+5$')"%-)"%( /')> & (6( #" )"0)4)) "%2) $ #&'()) / =

)2! ) !- * .()2 "$! . 1 )+!)^N 7f@ E 1 g )+**QM!E f *Ic=1.2

NaV Bs σ− =1.0

*.σ+ = 1.2

$V)+%2.(*-ILK & +?g )+**7 / *~M3%2*1!!E f *Ic= 1.8

N'V! PC%+ E *

σ− =1.2*.

σ+ =1.5$V)+%2.(*-kILK &

# -H: 7$& + 7$2 $29 (-H< $d =976$9 PB$&*

!

XBE Wp 6KW@ -EhE % 11* 1!)2 /.()2#*)2 !%+)+** , \E 13 1 %+*1 .GYZ p* .()2 %+)+*p#%+%2*)+*-@ 2(1!)^N *1! s .* N**4,A . WE 3 1 . 1p 6K s#)2 1)2 )Ê %2\Z *S 3)+% 1 !)2 * s#.5N ** * .()2QAC !*- 1! \E %+*g .()2 %+)+** V*2s 1!*$ '(+=c & *. $ '(+'( & 17s#)2/g .()2 6 .* )+* $ '(+'9 &1S 3)+% '/.Z f \s *.s .* .()2 * -%2)2 * 3 1!)^N%+)21 *1p .()+*-*- *$1)j@E %+)2!*l !- *)211 $!r< %+)+*.\N\gHEhE *-5E )+ )211!)+3 !)2* *. * _1 !*- & %+-)2 E )+)+* *.hZ ~f !-%2)2 *4)2 , 1!!3 14s#.4E )+ . 3 .(HN !)2* f *l !- *)211* )+* , [l@ MV@ M)Ê % !)2* I - 1 %211*4 , [Wl@ MV $ , M & )+* Bs 10-FG , NE N %21 , l , [ )21!#*. HG .JE .()2MF %2.1!3%2*Z D?%2.!- .()21!)2!*- )Ê - *1\ZyZ , l , [ ( .DA )2 .(!)+- %2.1)+3 1!)+!)2*)2!s * $ , !1HG *1)+ & 4 1)2 $)+* .()+JN_!-%2)2 . , !%2*HZ1!)Ê a@% !)2** *4aX *4V1 *)+!#5(+s#)+. * *. .()2 1 M)Ê % !)2*1!- *)211 .!%+)2 - JE .G )0)+*JE .()+JN * _!-%2)2 . )2.* li 3 !)Ê * 1*.1/(.1wM3 JN!!HE. %+)+**~V*2s 1!* - %s .*~E HG WE )+* )+*1!)ÊhE *- )Ê , !a@%2*HZ f ! %+* ))+*- 1 %+ MF!*- ) )2.*yli 3 !)Ê * */ * Z yNd***)21!0.()2 " %8. ?!)2\E .* *HZ % *. .!!)j/Z )2%+%2* )211)+3 !)2*Qf * * 1 )2.*.p .!*-X *4V 1 *)+!05( 's#)+. )+* 3 2(1!)^N%+)21 1bS !)+!)+HE g .()2 1!!)2!*- . N'g*1!!%+)2 * )21\@1!)+3 !)2*hf !- 1!%+%+dV1 .()2 1JE \s,C 1! Z! . .1 %2JE f *hZ % *- * *HZ)+*. )+* .()2 1JE ?%+% )+* -G wV *HZ %;f *~?!)2\E .* )+* . M)Ê % !)2* \gFN(1!)2 !)+-s .*kE HG )2r1EhE * PC*- Z s#)21 * .5?!)2\E .* *HZ % . TCHE %+)2 *wV(@% 1!*- *. . !!)j/Z )2%+%2* )211)+3 !)2* s .* \N % C!2?!* s#)+. .()2?- 0 * Bs !)+*2s#)2Js )+)+* V1 *)2.* . 1 %+)+** 6 .*1!3 JN!!HE 1 Z bl )+*1!3 !*- f * *HZ)+.( .()2 V1\s %(. 6 .*HZ %(* !- - * . 1 1!!)2\N !)+!)+HE 1.()2 N !)2* . , !a@%2*HZ(4.1 DlW@ MF %2./ )+*HG'r p *Bs !!*-yE HGYZ HE )+*. 1! )+* _!-%2)2 1! %+%E )+f %+%21!TC*.()+- JE V*2s 1!*13 JN !!HE $E 8i )Ê %2wV *HZ %)+*1)+%2 6 .* & f ! *a@.* 1)+*4 .()2 6 .()j/N!)2* )+*)+- % 1\E 3 E s#)2Z p 6 -* %2.1!TC\N _1 !*- 17s#)2 )2 *. , JE 3 !8%+)2JG 1!)2 $)+*w[3!)Ê )2!*- . *HZ)+*

# -H: 7$&b )<;$8$' = 1*$29 +(1$976$' :;$2(1$' 0 79

!)2 *(HE .()2 1 * 1 *- Z \Ek-%+)2 *4)2l !- *)211Ws .* * *.wf * 3 @%2)2.%2* )2 1!3 JN !* f !- 1!%+%+ rHE V 1 %+11hs#)+. * )+*HE % .kl )+*HG )+*)211!)+3 !)2*1! / *- ( .()2V*2s 1!*1!3 JN !* *. .* *)2 !%+)+** , *1!3 !*a@- 1!3 *4X *V 1 *)+! 5(LKws .* .** .()2 gs )+ * 1! *- * E 1!*7- )+-**$ 8@E g )+*Wp 6KW@ M)Ê % !)2* )+* %+)21!!)21 , l , [b@ E !!)2 ( - %+)21!()2 )2hf !- 1!%+%+* * HE !)21 *.li(3 !)Ê * Z hV *.(*-$)+* 1 HZ -% NZ3%2 @ *HZ)+ *.$3 2(1!)^N%+)21 % 1)+)+%+)+TC 3!)Ê )2!* E 1\Z 1 )+%2.* .()2 !*.'@% - g0.()2 )+*S 3)+% : .()21\N !)2!* Wp 6KW@ M)Ê % !)2**4

+, ( + "

E .!%+)2 Z E * s#)2Qs#)2 !)+- .1 %+)+*V *s 1*1!3 JN !!HEE )+/. " %.)211!)+3 !)2*1!3 E *. .?!)2\E .* 1Fs % g.1 , !%2*HZ f ! %+* )21s#)+. )Ê %+- *.* )+* , [l@ MWV @ MZ* !)2 E )+0)+*JE , l , [b@ MZ* !)2wf !-%+)2 * ) .** .()2 % 1\E 3 E #)Ê s 1*!%+)2 * -%2)2 - Js,C %+ s#.*4( %2)2 HZ)+!)+- .()2* .()2 , [l@ MWV @ M)Ê % !)2* %21k_!-%2)2 Z HE *HZ %+%)+* I *. )21! * * .*)+* .Dl )+*%2)+!*- *- 1!3 ** , 11)+* 6-%+)2 HN )+ .1N !JN y?*HN !)2*)2*7. 1p 6KW@ -EhE 1* .JE )+*S 3)+%I 1 !)2 ** WE Z Qg !3\g( *4)2 )+* I * *.JE *- - ** l )+*- 3 E

κy*.

κz* 1!3 * - JE CG

.A\E )2!*- 1Ff 1 !)& * $S 3)+%K & .* %+- *.* 3 2(1!)^N%+)21 * *. - E !!)21 *FG *4

n = 5 · 1019 −3 , Te = 50 , B0 = 1 R0 = 2.5 , r0 = 0.35 , q0 = 2.5 , s = 2.5 , L⊥ = 0.03

χ

⊥ = 1.5 , ν

= 1.5 , χ

‖ = 0.5

n = 0, 6, 12, . . . , 30 , #mn = 96 , nx = 128

)2! )0!C%+%+ ( .G .wN%2)+* *HZ .()+1 * %+ 1 -G )21 s#)2 . 1TC %+)j@ "$!/ ) , [l MWV E )+

r0 = 0.70E )

q0;l *1$)21! .()2mMTC\N . 1 , )j@

.%& %2. 1yE )+B0 = 1

, s *)+- %21 %+ 1 -Gs#)2 .()2 )Ê %2* li(3 !)Ê *hg%+)2 *?%2.1!TC\N *48)2 1 6 .()j/N!)2*1s Z HE r )+!3*HN . *1! *-7( -!*.$. @ *HZ)+\N3 Z )+TC* . .E %+)+- * E 3 .!%+)2 5dc 67* f !-%2)2 /.()2 1 E )+ . E \s % )+* 'UK +M )+ ::( "$- * . *HZ)+\N3 Z )+TCs#. $.()2 1E \Z %8%+%2#6 .*

#mn- !)+*- - %+* *. 1 s#.* * .()2WZ , [l@ MWV

7e:8@ MF %2.aN *a-!)2* 1* * * 6 .* E )+∆n=6

\gFN(1!)2 !)+-0rbp **-. 1E \E .*HZ % 1)8(V * *- p f \s#)2 1*4X * V )+%2.(*-$5(+ \N ** yE * * * .() %2 %2 . )2 HN ! !)2** %21yl !- *)21./.( - g !* , [l@ MV@ M)Ê % !)2* E )+ .* )+- * E !*4 )2 1s#R@.*7.( $)+*kr )+R@ *.? %+HG <C *HE )+!!%+*- * 1 .*7 **1?%2.!* - Js ***s ) .()2hr )+\E )+!!%+*- g )+*mX * \f%+% )Ê *)2 !%+)+**4- 1TC!!)+-* , !%2*HZ Z 1!*.E )+ 59 r )+!3*HN * .( - g !s#. WV 1!*. .Dr )+\E JG3*HN !TC-0)+* R@s

80µ1 r )+*JE 1!!)ÊhE * r )+!3*HN 4s#. )+* . M)Ê % !)2* .1wl @ MF %2.

$2<&$6 ( k3huw |Yu " Tuw$B$8$' d-/&& P

)+* . 7e:8@BS *a-!)2*$)+*- 1 %+ )s#.*Z s )f 1 )2 .*hMF %2.1!TC\N *f \s *.s %2 )Ê % 1\E )+* .hAC 4f *

q0Z M 1!)^Z )+TC1\s !*.f *

σ = 1.1*.σ = 3.0

g !*4 )2 %2 E )+hlW@ MF!*-1s#.* g .1 -%2)2 r )+!)+*\f%+%\E )+!%+s#)2 .1 %8. 1#*- 1!F!*? %+%21

3.02.52.0

30

25

20

10

15

5

0

σ = 3.0σ = 1.1σ = 0.0

〈n2〉1/2

q

E !"%+ H)"& (5/0 </'/'+)> /0"0)$'+ 7 464)": 7 5/"0 *+ : / $'&'( - /

q = 3

"0)& (/') -)$B, ) /0$B=%=0= ; #"%4)+)$' ()"!&'(5/ "0)$%/') /#

6 * \N ** )+*~V )+%2.(*-75(+ .()2 .!%+)2 wV1*HN *- . )2 HN! !)2** )Ê p @)2 -G

q@ "$!/s .()2%MF %2.1!TC\N E 1!TC\N(1!* )21!)2 1 1l !- *)211!!)ÊhE WE )+

.JE *HZ %+% $? )+-;e )+* I & \f !- *.yg )+*4]4 .()+-%+)2 .()2 V1%+Bs !#. *R@E )2!* \E 1R@BVE 3%+)+!.*/1!)+*. Bs 1f 1 )2 .*4)2 1 -\g*.1!)2 .( .()2 " %)+*Bs 15N%2)+** !!)j/Z )2%+%2*/)211!)+3 !)2*/%21)Ê *HZ %+% I /E )+

χ⊥=ν=2 !s#.*

.()2*\E )2!* )211!)+3 !)2*1\s ! .( .* " C\E %2)+\N kZ )2**κ⊥i

- JE CG $ '(LKUK & *..()2X **f)21\N 1)+TCµ⊥i

(- JE CG +7K aE )+0.bZ 1#C\Z %+)2 *V** HE Ti=Te

1!!)ÊhE )23 %+%2%2 )211!)+3 !)2* !)2* !)2! 1!)2 7* .3 %+%2%2* " C\E %2)+ C )+-UN )+ .Wl%2JN!**κ‖e

+l)+* s 1YN%2)+* )211!)+3 !)2*s E )+χ⊥ = ν = 1.5

)+* . )2 .( - g !* M)j@E % !)2* */!)+-8HE )+** .()+JN *1_!-%2)2 ~E )+ . %+- *.* , l , [b@ M)Ê % !)2*~Z \Ek-%+)2 *4l1 1) * $ JE \N .G )+*$.()2 1* , [l@ MWV @ MZ* !)2* .()2 " )+\N *- .? %+HN! 8@!)2* (.* % 1FE )+* !-!*. *.(\gFN ks#. )2 1wl)+*1 TC*HN *- s#. )+* .1!3PC*V! )+ 5 4*)2 WE 0- JE rHE .()+JN *w_!-%2)2 s#.)+* , l , [ @ M)Ê % !)2*E )+bC *%+)2 * % 1\E @ *. )21\@1!)+3 !)2*1!3 E !*7.( - g !

n = 4 · 1019 −3 , Te = 50 , B0 = 2 R0 = 1.75 , r0 = 0.42 , q0 = 2.5 , s = 2 , L⊥ = 0.03

χ

⊥ = 1.5 , ν

= 1.5 , χ

‖ = 0.5

n = 0, 4, 8, . . . , 32 , #mn = 153 , nx = 256

)2! ) N ** * * #.()2%2 , l , [ @Bli(3 !)Ê *!- E !!)2YE )+ )+ JE N%2)+** *.-G * .()+1 \gTN 1!)2 !)+-bs .*4'p ) . " % . %+- *.* E bs#. *a@ %+%21 ( )+*yC *%+)2 1V1\s %f * ?!)2\E .* .(

n - s )E )+

∆n=4

# -H: 7$&b )<;$8$' = 1*$29 +(1$976$' :;$2(1$' 0 79

*( %+%2101# *HZ)+!-\g*.* * #.()2Z '8I@ DlW@ MF!*- 1* ** 6 .* )+* .()2M)Ê % !)2* )+* JZ- * s#.* . .()2 *HZ)+ .(!)21 ~E )+ . 1E \Z % %+%26 .*

#mn1!)+-

V %21 l !- *)210.()2 1 M)Ê % !)2* ! PC%+ E * .()2)+*V)+%2.(*- 5( ' - JZ)+-* .() %2* %2.)2 HN! !)2** $ & *. . 1 !!%2** , )+%2 *11 1 $ & +[2s %.()2Yf \s *a@.* E 0. 1 , l , [ @ MZ* !)21 )Ê s 1* !%+)2 *E )+ .* )+- * , M2@ E !*g )+*1!!)ÊhE * \N **%E * .G .()2 , !%2*HZ )+* .JE 1)Ê %+)2!* , l , [ @ % 1\E *.1(.()2 i(!* 67-* %2.1F!*-/-)2! %21 1#)+* I 8*. 5 8 Ws#.)2 HN ! !)2**$1!)+*HN * *)2 *. .? %+HG/1!)+-0)+* .-)2*QE )+ l !- .()j@1!)2!*- 1- *4

2.0 3.02.5

20

15

10

5

0

σ = 0.0σ = 1.1

〈n2〉1/2

q

〈Γ 〉q 3.02.0 2.5

σ = 0.0 25

20

15

10

5

0

σ = 1.1

〈n2〉1/2

q

〈Γ 〉

q

E ;"0+ !)" & (/0 <5/0 /0+)H #$ /'"%)/') D; $'$')$ # #" : 7H /'+- / $'+$%/') )& ( # $'$0 /0+$ "0- /')"% 6+5)4#() >0E 6 /)" (!*+

r0 = 42&)

q0 = 2.5 "0 2 )" @ G"%$')"% )$

q 7 !"%+ $ : 7, )+-)/0"% - /rq=3 =45

&) 6, ) /=%=%= $'& ()"%)$0/0# /#

)2 1 1 * 1 )2 .(%+)2 4_! %+* % CG 1!)2 E )+ )+%& . %+)+** V *s 1*1!3 JN !**. .4- ! ** 6 .* 1Fs %\N % C*4 )2\Z ! 5E * .18%+)+* M3 JN!!HE )+*V )+%2.(*- 5(LK(r 1 * )21! .()2;Z .*VZJs )+%+)+- * % 1\E 3 E !* . , NE N(1Z - /@!)+- %+)+*DV*2s 1! )+*wV PC*-)+-UN )+ f * . )2.%2* 6 .*HZ % 'X *wf)2%2! )21! .1Z 4V)+%2.(*- 5(+Z - /!)+- WV*2s 1!*13 JN !!HE . )2,f !- 1!%+%+* , M2@ M)Ê % !)2*E )+ .* )211!)+3 !)2**

χ⊥=ν=1.5Z 1 *4)2 S \f - /! Z HE , M2@ MZ* !)2 1

I;E )+ .* )211!)+3 !)2**χ⊥ = ν = 2

+?g , Ms#.EZ!Js )+%21 .1 )+ *- )2 XFXFX *\Z 1 .(*HN %+% E \N )2!M3 JN !!HE - /!yZ .* , l , [ @ E !*.E )+ *- )2hXFXTX 0s#)2 1)2 )+* . )+- * M)Ê % !)2*~E )+

χ⊥ = ν = 1.5f \s *.ys#.*4

X * 1 2s \Z 1)2 YE * .1 M3 JN!!HE . -%2)2 * , l , [ @ E *E )+ / )211!)+3 !)2*

χ⊥ = ν = 2 6 * \N ** .G )+* / "$! . *\E )2!* !!)j/Z )2%j@

%2* )211!)+3 !)2*13 E χ⊥

*.ν-!*.1#C\Z %+)2 Z N %2)+** V*2s 1!* g !8.

.()2 )211!)+3 !)2* 1)+%+)21!)2*.1s#)+\N XE ? %+% . )+- * , l , [b@ M)Ê % !)2*1!3 E g !* "$! f *

χ⊥ = ν = 21- .Z 8.G )+* *(!C*-%+)2 !!%2* 1y_! %+* 1@

! - ./CE 3( hs#.;M E )+yN ** 4N )+* * 1! *-$.( - g !hs .*s %2 E )+ .JE , M2@ *HZ %+% I .()+JN PC!f !-%+)2 *s .*Nd***4)2 g , M f \s *. @* )211!)+3 !)2** 1!)+*./g , l , [ @ E 1 %+)2 Z 4)2 1 *1 )2 . % CG1!)2 $*E )+ . " % .f 1 )2 .** CG 1\E G .(

R0*.

r0g .()2 )2.*

, NE N2@Bli(3 !)Ê * -\g*.*5s %2 .()2 *\E )2!*;E 8i()Ê %2*V*2s 1!* )Ê.!- 1!%+%+*1M3 JN !!HE g , M -FG Ws .* % CG%21 .()2 Z!*)+- * )Ê 1 s \Z E \N)2R@* , l , [b@ M3 JN!!HE

$2<&$6 ( k3huw |Yu " Tuw$B$8$' d-/&& a>

TS TEXTOR

TEXTOR

0

0.05

0.1

0.15

0.2

0 20 40 60 80 100 120 140 160

γ

n

q ∈ [2; 3]

q ∈ [2; 3]

q ∈ [2; 3]

q ∈ [2.75; 3.25]

q ∈ [2; 3]

q ∈ [2.75; 3.25]

χ⊥=ν=2.0

χ⊥=ν=1.5

E #"%$'&'(5/ #"!#"0) & ( $'"0/')- (<) /B*+-)"!/0+"0+# 6+5)4#(n "%#$%/') / $' & ( $'"0/') 132 " 7 : 7 , )+-)/0"% 132"/0#"%$'&'()& (

"%) /' )& ()) /0 1 2" /0#"%$'&'()& ( )"%/0 )";"%/0 4))# $0$' /0+> 8 "9*) "%)$ ))" )( *+ &'($'"0/') )"%/0)

γ (n)#"%$0/0) / B#&'("%& (6#"% A 7 )& (5)

1 ) "%)>

p )4)+*JE`_!-%2)2 . 1 , M2@ 1!%+1$V45(+ & E )+0.JE , l , [b@ 1%+$V 45( ' &f !-%2)2 E * )+* V )+%2.(*- 5(LK .* f )2%2! ! **zM3 JN !!HE 1\f !% (kE )+ .JE .(*a@N %+% *4 1 * 1 )2 .(%+)2 , !%2*HZ f ! %+* )+* .* l!- *)211*yE )+ )+*- 1 %+MF!*- )4*1*1!* -%2)2 * % 1\E 3 E !* % CG#1!)2 * *s#)2 %+- -\g*.*4(6 )+)+*6 .*s %f *

n =30\N ** E *.G )Ê , M2@ MZ* !)2wN \Z )+* .JE 678i )j@

E HE . 1WN E 3%2!*f)2%2!WE \N)2!* M3 JN !!HE 1 - 1 *)+!* ~s#. 1 )^G 1s#.*7g .()2 , !%2*HZ1!)Ê % !)2** *)2 %+%2 %+)+* )+*1!)+%2*$6 .*E )+

γ > 0 @

\gFN(1!)2 !)+- 6 )+ )+*)+* Bs -%2)2 -G *E 8i )Ê %2* , )2.%Ê .*HZ %f *n =32 ) . , l , [b@ M)Ê % !)2**.0.1 V1 *)2.* )Ê %+)+**M3 JN !!HE $ .(*HN %+%

S \f )+*QV 45(LK & .!%+)2 f 0.JE 678i()Ê HE 1!!HX * .0*)2 !%+)+** **-ks R@.* . ) , l , [ )Ê - *1 \Z,Z , M .( .()2 1 V1\s %.()2 )+*1!)+%21!* %+)+**p %+%2 *)+*-UE .* *)2 %E )+ )+* JZ- *4 )2 1 g !Z )+*JE ? %2s %2 1!)2 * * )+*. N !)2* . , !%2*HZ ( )+* i !* 1DMF %2. *)2 .1 % C-E .()2 1 V !-HE *!)2*hZ %2- *s#. .()2 -%2)2 , l , [b@ M)Ê % !)2*kE )+ )+* /@ */6 .*HZ % .( - g ! 1 .GYE )+ .JE V1 *)2.* . 1 M3 JN!!HE 15E )+

n =48Z HE )+*. 1!.()2%E 8i()Ê %2* %+)+** V*2s 1!*7)+* .*p ! !*- * * ! %+*s *4)2 1 * *-Ds 8Z! . )+* .JE )+* *- )2;XFXFX0E )+

q ∈ [2; 3]*.YE )+ )+* - @

*2g- *. -G * DlW@ MF %2.1!TC\N )+*%N )+* *- JE 11** *HZ)+E .( (g ! A * . .( .()2 6 .*s %8 .()+*-*ml! / *-4E HGZ 1#C\Z %+)2 * .0* HE !)j@1 kr )+1 !)+! ! %+)2 $- 1*HN s .*4. * * * E )+*1!)+%2 6 .*E )+ )+*7.()2p * *- )+* 11*4dV1.()2 1JE !*.ks#.0.()2 1M)Ê % !)2*wE )+.JE %+ 1)+* *- )2hXE )+

q ∈ [2.75; 3.25].( - g ! 1YZ - /!)+- /%+)+* V *s 1*a@

1!3 JN !!HE )21 .( .()20 %+%+% ] )+*)2)+*wV)+%2.(*- 5(LK .!- 1!%+%+V* .* 1 \E )+!%+*? %+HN ! !)2*1!3%2*7)+*mV)+%2.(*- 5( I/\N ** E * * *7.1V1!)+*HN * .)2 HN ! 8@!)2** ) )+*- 1 %+DlW@ MF!*- )2V1*HN *- )21!# ) .#- Js C %+* 1 2s *

) # -H: 7$&b )<;$8$' = 1*$29 +(1$976$' :;$2(1$' 0 79

M 1!!)^Z )+TC g .*/- 1E * * )2 s -%2)2 1!\N ( -!*./. N%2)+**TCHE %+)2 * V 1. **- HE .()2$

q0 =3 & @ 1)+!)2*4)2 1 1;l!- *)21 1!!)ÊhE 5E )+ .* , [l@

2.75 3.0 3.25 0

5

10

15

20

25σ = 0.0σ = 1.0

〈n2〉1/2

q

〈Γ 〉q 2.75 3.0 3.25

15

20

10

5

0

σ = 1.0σ = 0.0

〈n2〉1/2

q

〈Γ 〉

q

E IB !)"0( /') )" "0 ) ;"0+ , ) / =- #:;$'&'( /') #$ : 7 5/"%1 ))$G ) $%/& ( 1 2" : )$')<5-)" & (/0 <5/0 /0+) #"%<)5 "# )# )"$'& ( # $0)(5-)$ )& ()) /0$ $0/ ( /' 5/0+5& ( $0/04) / / ()4)6(+-+) *)"9/') /#

MWV @ 1!%+* %+)+!)^f~g )+* . 1 * * )Ê M)+** . C'"c,*.0`#"' 8.10f !-%2)2 )21!l1 1)8Z . 7 JE \N .G s )+g *. * !)+!)^f y_!-%2)2 kZ s#)21 * .*M)Ê % 8@!)2*1!- *)211* *)2 1!)+**2f %+% 1!)+*. +V %+%2)+* 1 * .1 g .()2 )+*V)+%2.(*- 5( I - JZ)+-* %25f \s *. r )+\E )+!% )Ê - 1!F!* ? %+% )214 )2g8*HZ )2 *. " C *.Yg%+)2 R@s )21 g .!!)+- r )+\E )+!% Z s#)21 * e9 *. 59mr )+!)+ *3*HN 4f \s *. s .*1!*.* )Ê )+- * ? %+% * '9 r )+!3*HNZ 4_g-*- V 1!*. .Qr )+!3*HN2@ !TC- '9 *\E )2!wr )+)+* )+*

τ WE .()2 I 99

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% 1\E 3 E !*Z ** */!)+-WE * s 1WE %21 , - %E )+ )+*JE E @.!**m] )+*ai@ $ K WZ <K')+R@ M 1!JE & "$)+p ! !*- * . )2,f !- 1!%+%+*_!-%2)2 1! * *- * !- * )+*1!- 1E #)+*1!!)ÊhE )+- 1p )+%2. .G 1)2 %+- *.\E G *Z 1EhE *( 11* % CG

)2 N !)2*$. 1!)Ê %+)2!* % 1\E * (#)+*$ i !* Z - 1 %+ 1hMF %2.)21! )+* )2.*kli 3 !)Ê * \N *a-!)2** 1 ;C *%+)2 4 JE * 1!3)2%+ 1N )+** 1 )2.*. %+%2 .()2 MF %2.1!3%2* ( . ( %2.1)+ $ ?;M & s#)2 ) , l , [8 . ( . A )2 .(!)+- %2.1)+ $]?;M & s#)2 ) , [l MWV%2N%+)21!)2! 1!)+*. bl * 1 )2.*. )214f )2%Ê s#)2Jf )2% )+*1!)+%2 6 .* g .()2M)Ê % !)2*mf \s *.s .*4

E )+ 1! ** .()2ws#)2 !)+- 4?- /)Ê HE *)2 yf)2%+%2)2 %+%2/)+*1!)+%2* 6 .*$N E 3%2! 1DM3 JN!!HE & )+* . * *- \gFN(1!)2 !)+-s .*E%g11* *. )+*s#)2Js )+ )+*f \Z)+!)+- 1S 33 * . 1WM3 JN!!HE 10.()2YM)Ê % !)2*1!- *)211 )+*HG )2 1?- - )+* YE )+ . 1!!)2!*-7. !!)j/Z )2%+%2* )211)+3 !)2*s %2 E G- *. g .()2wMTC\N . )+*1)+%2* 6 .* )21!X * .* * - JZ)+-* * *- * )21! .()2 !!)j/Z )2%+%2 )211)+3 8@!)2* )+*- 1!%+%+ys .*4p ) * / * "$!* f *

χ⊥*.

ν PC! E *

.1 !!%2* _! %+* -2C*HZ %+)2 * !*.*4<?!*#)21!#.()2Dl )+*1!%+%+*- . *\E )2!*3 %+%2%2*$)211!)+3 !)2*mE )+

χ‖ =0.5HE )+**? N IwZ mN %2)+* - Js,C %+ s ** 1!)2 1!)2 $*

. Z .* % 1\E 3 E !*Z - /!)+- * MFG)+-UN )+E )+ ?\E % $ K(+7K & !)2* !)2*71%+%

" = = 2$&&$Q979=:-a =)0 +

%+- *. V 1 *)+! *Bs ! .()2 1 ?- * 1\(1!JE !)21 *. -)+ )+* V *%2)+!*-Z 03 2(1!)^N%+)21 QN *1)21!* *l )+*1!%+%+*- . )211)+3 !)2*1!3 E

( + #)# $ "'( &

l1 1)* )+*HE % .* !)+**! .G .()2 !!)j/Z )2%+%2 )211!)+3 !)2* E )+ )+ * , \E * $ K(+JI &)+*- g !s .* E HG HE .()2 .()211!)+3 !)^f *0N )+*!)21 *l JN . 1% 1\E 1Z 7 \gTN2@1!)2 !)+- *+s %2 1!)2 (#)+*N %2)+**$HE 1\N% 1!3)2%2*4.()2 *)2 E .( .()2? %+)2.** HE 1 !)2 *Qs#)+. )2 .() %2* )+!!3*HN bs .*/)+* .*4M)Ê % !)2** 1!11 - Js C %+ .G .()2V P1!*-.M)Ê % !)2* )21# )+* QZ !)& 1\N% f * )+*)+- *

10−4 E -

∆r ∼ %s =⇒ ∆x ∼ %sξ.

$ 5(+ &".g.YE * )+* * -FG %VP1!*- E )+

∆r < %ss C %2*1hsbg.YE * )+* * 8@

E )^N/*1! * .()2 *)2 E E )+0.!)& !*PC !*- f )+* )21! 1 * .()2 *HE !)21 /6 ! . )+* - Js#)211 )211!)+3 !)2*1E )+ 1!)2 $!)+*- *~N**1%+%+ *ws )+ 1 " )211* .()2 !!)j/Z )2%+%2 )211)+3 !)2* ( )+* 6 )+*)Ê HE )+*- 1!%+%+ s R@.*81 .G .()2 *)2 !%+)+*MaC!!)+-*- 13 1 - ./1 * !)2 s .*~N**4 )2 1sbg..JE 1%+1!!-%+)2*.*_! %+* . 1 % 1\E 1#E - 1!*Qs .*4WE )+**l )+*.(!FNkf * .5Ek-%+)2 *w_ !) !)2* . *\E )2!* )211!)+3 !)2*1\s !WZ f \E )+!%+* 1).1#* HE !)21 mEk-%+)2 6 )+*)Ê HE .0)211!)+3 !)2* g0.()2 )+- * , 1! * *- *$*- - @ *4 )2 "$! g

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ν1)+*.7HE )+**1? N/98@54N%2)+*+s C *.$.()2 3 %+%2%2

)211!)+3 !)2* - JE CG7. % 1FE .()2 * *. , JE 3 !* "$! )Ê p )2 ~f *χ‖ ≈ 2** HE * 1%+%+

p ) .QM * .JE * HE !)21 * 6 )+*)Ê HE . )211!)+3 !)2* s#)+.5Z . *)2 .()21*HN V P1*- g .()2 ! * E )^N \gFN(1!)2 !)+-aA */.,V 1\s % .)+!!3*HN *. .? 1!!%2-*-Qf *

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k ⊥ =

n q(r)

r.

$ 5( ' & JE CG . !)& !*PC !*- 1%+%+ * * .()2 -FG7"$%+%2*HZ % )+* . FG *.(* *- .!)& 1\N% 1)+*%21

k ⊥ %s ∼ 1

$ 5(LK &- %+*4 (N %2)+**MN%2* %21

%sN )+* * E )^N/. ? %+HN ! !)2** 1!!R*.* N**

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γ (n ) ≈ 0g

k

⊥ %s ∼ 1 .$ 5( I &

< # -H: 7$&b )<;$8$' = 1*$29 +(1$976$' :;$2(1$' 0 79

M E )+ )21! .()2 TCHE %+)2 wVP1!*-Qg n

.()+JN E )+ . !!)j/Z )2%+%2* )211!)+3 !)2*1f R@N*g3( ./.()2 )211!)+3 !)2*7.*A %+%+3*HN )Ê )2.%2* V *s 1!*1!3 JN!!HE . *)2!$ 1!)2 V )+%2.(*- 5(LK & V1 . !%2-*- %+- .** .G .()27)211!)+3 !)2*1!3 E .!)+*- 1!%+%+ s .*E%g11*.G )

n ≈ r0q0 %s

$ 5( 5 &.()2/%+)+*V*2s 1!A %+% s#)+. )2! ) s#.

q0%21 *HZ 3 1!)+!)2* * \Z s )+%

1!)2 .10 *- )2.() %4HE .()2 1* "$!N *HZ* !!)2!?g .1 * ! , l , [b@ M)Ê % !)2*1! )21!3)2%4 )2JG .()2 1.G .()2)211!)+3 !)2* 1)+*- 1!%+%+s .*~EYgHG4.G .()2 V *s 1* )

n ≈ 300A %+%s .*48 1 678i )j@

E HE )+* .JE Z - /!)+- * M3 JN !!HE % C- .** )n= 80 − 100

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n =48(

n =300! / #1!)2 .()2f * . 6 .*HZ % PC*-)+- *HZ)+

)Ê )2X HE Bs .*? N '99 !

1 r)2% 4)+* f %+%21!TC*.()+- 1 %+)+* 1%V *s 1!*1!3 JN!!HE )+*$. M)Ê % !)2*~Z \gTN2@1!)2 !)+- * % CG1!)2 *WE )+ )+*s )+* 1!\N *7 .(HZ )2!*- . *HZ)+ !)2 *4 )2\Z )21! .()2%p *\Z *- . 1 1 HE %2* *- )21X,f *f !* )+*7 .!%+)2 4 1\@s )+*N**E )+ .* %+- *.* E \f TC*.!*- * - ! )+ s .*4uw$* =)< *$24x-H< $d 7B$'&*9 HP$ .(HZ )2!E * .1 )+*1 %+)2JG *. 67-* %2. 14f !-FG ! 1)2 $.()2 !)& !% C*- .

%s∼B−1

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B0 = 2, (

B0 = 1, E )+

%+)2! 1)2 /. "$! f *n

dX */)+*JE B 3)21 * % 1FE )Ê 1!FG)+-UN )+1-)Ê DE )+ν >1

)21!bZ HE p )213)2%n ≈400

6 )+0)+* 6 .*1 !)+!Bs )+ f *∆n=4

.(HZ )2! 1!)2 .**4E )+

n ≈200.()2 *HZ)+#HE )+**? N 5('?.!bE * s )+! )+* *)2 #)+*

1*HN TCHE %+)2 V P1!*-f *1 %s

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s 1/.( 1* wN *( \E E )+ . !)& !*PC !*-ys,C 1 .(HZ )2!#1!)2 wE )+

n ≈100.()2 *HZ)+

\E %21 HE )+** ? N 5(X *1!- 1E PC!wE * )+*/ .(HN !)2* . *HZ)+ HE)+**? N ''5 - Js ***4W 0+ 1*$20 0<$ l ! / WE * .()2% 1\E .()2 *. 1*HN -%2)2 HZ)+!)+- .()2l%2JN!**JE 3 ! 1 ! / E * .()2MFG)+-UN )+/. 1 % 1\E 1 *\E )2! 3 %+%2%2 )211!)+3 !)2*1!3 E )21hE )+

" = = 2$&&$Q979=:-a =)0 Pt

χ‖ ∼ T 2e /n0

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%s ∼√Te

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σ

m+1,m =2

√ψm+1 + 2

√ψm

xm+1 − xm

$ 5( : &6 )+0.* Wp 6KW@ MN%+)2!*- * ! PC%+E * .** .()2 % !)2*

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m+1,m ∼√

s

B0kθ

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.

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s=2(s= 3

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# -H: 7$&b )<;$8$' = 1*$29 +(1$976$' :;$2(1$' 0 79

%+%2 %+)+** )+*1!)+%2* 6 .* )21n = 92

g )+* M)Ê % !)2* E )+ - *2g- *. -G M 1!!)^Z )+TC \gFN(1!)2 !)+- s .* *. .()2 *HZ)+ g .()20 %+- *. *1! *- )Ê** HE *p )2 f * .()8)21bf)2#"$ * % -

n = 6.5 · 1019 −3 , Te = 30.8 , B0 = 1 R0 = 1.75 , r0 = 0.45 , q0 = 3 , s = 3 , L⊥ = 0.03

χ

⊥ = 0.81 , ν

= 0.81 , χ

‖ = 1.03

n = 0, 4, 8, . . . , 92 , #mn = 576 , nx = 54

X *V)+%2.(*- 5( 5 )21 .1Z - /!)+- /%+)+*kV *s 1!*1!3 JN!!HE Z 1 *48)2 )211)j@3 !)2*s#.1VZ1!!)2! .GyE )+ )+*#1*HN *VP1!*-yf *

2 %s)+*y@ 0)2 !*- *.

)+*YV P1*-f *1 %s

).* .() %2* )+!!3*HN 1!TC*.* .()2 %+)+** 6 .* )

n =94*. / 1!)+%+)21)2! 1!)+*.

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n = 12, 20, 32, 40, 60, 80*.~O'(8)2/ gTNs#)+\N *- . MF@

!*- * ( .* % 1\E )+*!-!*.s#. * * Qs#)2 . \gFN(1!)2 !)+-

−0.05

0

0.05

0.1

0.15

0.2

0.25

0 10 20 30 40 50 60 70 80 90 100

γ

n

χ⊥=ν=0.81

6E E @#"0)$ & ($0"'/0)$ )</'"% - / ))"" $'$0 /0+$')$%/')> ) (5.$'<$'& () 8 "% /')"%# )" "% 1 /'( #"% )2/# $-:!"%)$)" $0/ "%& (6 1"%) @ # 4)$ --) 13/ B# /0)" 5/0)6 5/ #"% /0)"%/#

p ) . V* %^ 10.5l !- *)211 * DlW@ MF %2. Z)+-1!)2 .()2 Z \s !*. N !)2*. !!%2* *? %+HN ! !)2**4 )2 1 !)+!Bs )21Wl! / *- . 6 .*HZ %2*4f *

n =12(I 9 ! .()2l! / *- .b? %+HN! !)2*1!*)^f 1 Z ? %+- )2 1 1!%+!)2! 1 . , R@1 .GyE #)+*1!)+%2 6 .* )Ê`p )2 . 1M3 JN!!HEhE 8i()Ê HE 1bE )+ )+* .()2DS %^N% 8@!)2**7)+* JZ- * s#.*4 l )+*s )+ 1Yl ! / * .6 .*HZ %2* %21 )+*1M)+- !*$.V *HZ %N %2)+*1\N%+)+- "$%+%2*HZ %2* yN )+* s 1*!%+)2 h_TC*.!*-E DE )+ 1!)2 . ** .()2 !!)j/Z )2%+%2 )211!)+3 !)2* g .*V!*1!3 ! . ? %+HN! !)2*1*!-)2 )0N%2)+* @*mMN%2* )ÊhE # JN !)^f s#)+. PM E )+ % - .()2D_\E !*- * .GyE * .( QS 33 *

" = = 2$&&$Q979=:-a =)0 <

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n(s )21* .( )+*

S \f E )+)+* *.f !*.*4

1e−07

1e−06

1e−05

1e−04

0.001

0.01

0.1

1

10

10 100 1000

〈|nmn|2〉

m

n = 92n = 80n = 60n = 40n = 32n = 20n = 12

E ++ )$ )<5/0"0 )" & (/'# /0)C5 "0/0 1 2" ) )$0/"9/') D :! *+"04)) /')$ 8 ) )</'"% )"D +"%)"0-+5) & (/ $'& ( "%& ( D )()" # # )/0))" 5H /0+ )-)"%< ")

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n =92(. .()23 1!)^N%+)21 %+)+TC0E 1!*4s#)2 .R@

-)+67* \N ** .G )+* %+%2* 1)2 * M3 JN!* )+* 1!\N _ !) !)2* .VWE 3%+)+!.*2s !)+* .* / 1!*)Ê ZJs )+%+)+- *4M3 JN!!HE f \N EhE *.* 6 .*HZ %2*kf ! *.* )21!_R@-%2)2 YE * Z p ) .* *1S \f * E )+

n = 32.()2 1 _ !) !)2*1E )+ .JE _! %+*

.1%+ * VWE 3%+)+!.* .(1\s ! 〈|nmn|2〉1 )+*JE`M3 JN !!HE E )+ 1!3PCbS 33*-

Z p .()2 % *1S \f *E )+n =80

1 s#)+.7.!%+)2 .G .()2 1 _ !) !)2** * .( .1f \Z)+!)+- V1 *)2.* . 1,M3 JN !!HE 1Z 1!*.bN EhE *4'M E )+)21! .()2 1 _ !) !)2* %21? %2 .( .1YV1 *)2.*~Z $)+*!3!)2*4 1!3PC%E * .1 M3 JN!!HE -*HZ %21OZ! E N %2)+*1\N%+)+- /"$%+%2*HZ %2* )+* .kM)Ê % !)2* \gFN(1!)2 !)+-hs .* . 1!- !)+*- 0)21!#.? %20 ) . p * *- .#!!%2* * FG *as#)2Z p . ? %+HN! 8@!)2*1!*)^f 1Hs %2 10.WMHEhE )2!*-wg %+%2 6 .*

(m,n) \f !- *4

% !)^f k? %2 ! * 1!)2 .** 1 . 1%+*1_ !) !)2*113 ** JZ- * (.* *HZ %+% $ 1 s \ZE \N )2! 1 M3 JN!!HE & .()^f)2.()2! .( .* 678i )Ê %^s ! .VE 3%+)+!.* .( ) .*/-FG* MN%2* $N %2)+*1!*/6 .*HZ %2* & 15S 33 */. 1%+)j@

# -H: 7$&b )<;$8$' = 1*$29 +(1$976$' :;$2(1$' 0 79

**M3 JN!!HE 1 $V45( 5 & s )+ f 1)+*JE 678i()Ê HE E )+n =12

g ! )21!3)2%21\s )21Z $)+*JE % !)^f *1? %2f *$Bs I 9 . )+*7.p !)+%+*-f * .() %2*$ %2* )+*)+* - Js#)2 * ? %21!HEhE !)2* \gFN(1!)2 !)+-ys .* EYgHGp )

n = 32 !TC-

.()2 1 ? %2 g .()2# / 1!*/6 .*HZ %2* * * 4K *. gn =60

* * 9( ' "$.( .()2 14_ !) !)2* )Ê l )+*HZ%+** \f !- !( *s#)+. N ** )+**! %+ .()2 1hVR@ )+ )21! *)2 - JN % C!4s .*4 V+C%+%+)+- )21! .G .()2_ !) !)2* * g .()2 / 1!**. *)2 .(!)+- 1!*

m@ "$!)+* 1!!)ÊhE *

n@ 6 .*HZ % ( !!)+! 1 1)+*. - * /.()26 @

.*s %2 )Ê *- )2 * .* WC*.!* %2N%+)21!)2! 1!)+*. )2 1 *.aE .* *)+* .* *-*HZ*.* \Z** N )+* 6 .*3 !!* E E )+ .** 1!)2 %+)+* .( .()2 67-* %2.aN \gHEhE *-yN 33 %+* Nd***42V / ). *)2 !%+)+** "$ 1%^s#)+\N *- )21.()2mr % . !!*\E .* )Ê - *1\ZZ 6 .*s %2 1!)2 )+* . 6 )+! . 1 @ *- )21 *.* - !)+*- E )+ks C )+* f TC*.! ? %+HN! !)2*1*!-)2!*1! * .* C*.!*f !*.* . 1 )+* . 6 )+! . 1 *- )21 *)2 f ! *.* )21l1Nd** 1)2 )2HE )+** !!)j/Z )2%+%2*7*. JN 0 *.%+*.)+* )+**Js **.yl )j@- *1 . 1yM)Ê % !)2*1 i 3 !)Ê * 1kE )+ 1)+** \Z** *.( .()+*-*- * 1)+** HE !)21 * 6 ! . )21!2V /)+*01!\N Wl ! / *- . )211!)+3 !)2*2s#)2 1!)2 Z p )+* I8 %+- .(HZ )2! .()2 1 1#*1 )2 .(%+)2 %_! %+* .06 .* * .* *- )21R@TC*.!* * *2s 1* !%+)2 4

)2* 1 )2.*.D?- #g .()2 1 V! )+)21!** )+*2s#)2Js )+#1!)2 .1;f \Z)+!)+- S 33 */. 1M3 JN !!HE 1 )+* )+*JE 1 1!!)21!)2!* 6 -* %2. 1Fs#)+\N s#. .()2/-%2)2 *a@1! *- * )+*HE %g )+*DlW@ MF %2.1!TC\N DE )+ )+* M 1!)^Z )+TCbf *

σ− =1.2*.σ+ =1.5

.( - g !Pl)+* *1 )2 .Z HE ? %2\f ! %+* )+*QV45( : s 0*)2 Z *4)2N HE %+)2*.*? %2#)Ê - 1F!*Q? %+% *. )+* V45( : g * .** )Ê _!-%2)2 ..() %2* %2 . )2 @ *. *!) % HN! !)2** 1=s#)2 . 1 !!%2* * , )+%2 *a@11 1%Z ! %+)2 * _TC*.!*- *Z! * " % f *

n QE * )+* .()2 1 V! )+

)+* 1!h] )+*)2 * )+* %+)+!)^f * V 11- *~s#)2D0*T8* ! +*-*G.0#b#- -0?D/* **C ND%

' Wh' 7+*- S* 9 a 9?% W'.10/* )+* 11)2! )21!0*.QE * .()2 1 1 -* *.Qf * .() %2* %2* \N ***4N**s#..()2 %+N !)2* )+*HZ%+* , !%2*HZ -FG * (8)+* )+*- @1 %+ 1MF %2. )+* V PC*-)+-UN )+,f * .5E 8i()Ê %2* 6 .*HZ % 1!!.()2! )2! )!- 1!)2 .G .()2 , !%2*HZ N !)2* ( .1 DlW@ MF %2.E )+ *s *)+- * 6 .* %21 E )+n ≤30−40

8fd%+%+)+- *1 )2 .(%+)2 Z .*h?C%+%2*yE )+n ≥60

1\ % UrHE p )21!3)2%(%+)2)Ê _!-%2)2 kE )+ .JE *- 1!F!* % 1FE )+*bV 1*HN *- . )2 HN ! !)2**/)Ê ? %+%E )+

n =60- - *2g #.JE ? %+%0E )+

n =92 1! *s ** .()2 1 - !)+*-g-)+- HE

)+*)+- Z* s#)2 4 )+*- - *4N **DE *wZ p )n =32

)+** %2)2 *QV *1!!)2- .)2 HN ! !)2**$)Ê - 1F!*m? %+% \N ***.1!)2 mf 1!TC\N WZ!s *)+- 6 .* )+*.*M)Ê % !)2** \gTN 1)2 !)+-s#.*4r1EhE *- G!-)+1!)2 .0)+*V)+%2.(*-/5( 5 !)+-hE \N)2!yr1EhE * *-

" = = 2$&&$Q979=:-a =)0

1 V1 *)2.* )+*/. ADC . 1bM3 JN !!HEhE 8i )Ê HE 1#. %+)+** V *s 1R@* $ ] )+*)2* )+*QV45( 5 & g !bZ *)2 N *( \E *l !- *)211* )Ê , R@%2*HZ f ! %+* )Ê _!-%2)2 Z HE ?%+%E )+N E 3%2!JE M3 JN !!HE 0s **$.1 % 1\E /1! 1!!)21!)2!Ds#)+. . .()2%S 33*- . 1M3 JN !!HE 1Z Dl )+*a@1!3 !*-1f * *HZ)+ *ks )+/- * -$ )+*/.JE V*2s 1!*1!3 JN !!HE- 1 * $ -\g*] )+*)2* )+*QV 45( 5 &

)2" % f *n =60

\s )21!01!)2 .E )+ g#.()2 )2 *1! * E %2101!)j@ *.

rHE V1 %+11#. *1! *- *Z 0!!)j/Z )2%+%2* )211!)+3 !)2* *. . Z - /!)+- * 6 @.*2s %(1%+%* )+*HE %.()2 M *1!)+!)^f )+TC . 1E .%+%+)2!* , !%2*HZ1\(1!JE 1 (.()2 1!!)2 @!*- . )211!)+3 !)2*1!3 E * Ys .*4r$.()2 1JE r<s FN s#. %+%2 )21! !)+- * E WN *1!* - %+* *. .()2 1*HN )211!)+3 !)2* .! ! / .G .()2 %+)+*8@* V*2s 1!* )+ * A %+%+3*HN )

n = 80HZ s

n = 60(s#)2 1*4 )2 1 )21! )+*

V )+%2.(*-/5(qc .!- 1!%+%+

−0.05

0

0.05

0.1

0.15

0.2

0.25

0 10 20 30 40 50 60 70 80 90 100

γ

n

χ⊥=ν=0.81χ⊥=ν=0.96χ⊥=ν=1.28

E F @#"%#$; & ( $'"0/')$ )</'"% 1 2" /')"%$'& ( # & ( $0$' 7/0+$ "'- /0)"# )" /

n =60 : k−1

⊥∼ 3.1%s

n = 80 : k−1

⊥∼ 2.3%s

n =92 : k−1

⊥∼ 2.0%s

?!*5s#.* **- *wE )+)211!)+3 !)2*1Ff!) !)2* .( - g ! )+* .** * !- - */.1!!)2!*- 1!- % 7.()2%E 8i()Ê %2 , )2.%Ê .*HZ % * * %+%2*$*.*7 E !*N *1!* - % 11*s#.X *1!- 1 E !- 71!)2 )Ê _!-%2)2 .? %+HN ! !)2*1!3%2 *. . 1!!%2* * ? %+11 1E )+0* 1 )2 .(%+)2 * )211!)+3 !)2*1\s !* %+- *.r1EhE * *-

-FG .()2 !!)j/Z )2%+%2 )211!)+3 !)2* )Ê *- 1!F!*m? %+%8)21!. 1! / 0)21 .#!!%2* , )+%2 *HG

)2 1-)+%+)+*1! 1*. )Ê E )+!!%2* p )2 $. 1 *- )21 )q=3

)2 1 1Z *PC 1!3 . i 1 )+**. 1!%+ s#)+. )+*V 1 *)+!:( 5kE )+ .CE 3( *-/.0-G 1FN%+)+- *

# -H: 7$&b )<;$8$' = 1*$29 +(1$976$' :;$2(1$' 0 79

3 %2)2.%2* !)2* . 10% 1\E 10 -\g*.0*. 1g !%+)2 \N% C!p ! DE * .*m_!-%2)2 )V *s 1* )+ )+*DlW@ MF!*- 1 !-)+ 1!)2 g.()2*)2 .(!)+- )211!)+3 !)2*$)+*$. 6 )+! . 1 *- )21 )

q= 3)+*h? %+HG ! / *- HE (

' s,C *. 1!)2 )Êz? %+%1!\N )211!)+3 !)2* )+*,? %+HG 1*HN*-f * *- !C 5 - - *a@g .JE *- 1F!* ? %+%!-)+dV .()2 1 PC*-5s#)2 .;E )+.JE l )+*HG .)211!)+3 !)2*( .()2 !)2* . 10 % 1FE 1 Z 1EhE *4M !* %21 N )+* *.* 3 1)^N%+)21 *.V!-HE * JN** 1)+*. 1%+%+ .()2 !!)j/Z )2%+%2)211!)+3 !)2* * .JE S !)+!)+HE . !)& !*PC !*-/HZ s - JE CG/..( 7.()2 % 1\E 3 8@E 1!!)ÊhE * TCHE %+)2 *mVP1!*- )+*Z1!!)2! s .*4

"'( & #%$ "'" (

6 )+ .* f *- - *- ** * 1! *- *Z !!)j/Z )2%+%2* )211!)+3 !)2* 6 .* 1\s %*. *HZ)+3!)Ê )2!*-7!-)+ 1!)2 .G .()2 * ( %+- *.* M)Ê % !)2*1!3 E g.()2 * 1! *- )+* 1 !!%2* * % 1\E 1 )+* , l , [ @ E !!)2 *. )+* '8I@ l @S *a-!)2* )Ê M)+** .Z)+!%+)2 * 67 \N )+E )+ .JE p 6KW@ -EhE E 1*- )+-*1!)+*.

n = 6.5 · 1019 −3 , Te = 30.8 , B0 = 1 , mi = 2mp

R0 = 1.75 , r0 = 0.45 , q0 = 3 , s = 3 , L⊥ = 0.03

χ

⊥ = 0.81 , ν

= 0.81 , χ

‖ = 1.03

n = 60, 80 92 , ∆n = 4 , nx = 54 , ν = 1.58

p ! kE * %+)21!!)21 )2 * )Ê *.1 )2 !3% 1\E . 1 , l , [b@Bli(3 !)Ê * 11 )21! .! .()24MFG)+-UN )+YE )+

ν = 0.04f )2% - !)+*- M)2 %+)2-E )+

n≈ 2 · 1019 E −3 *.

Te ≈ 100!_ )Ê p )2 )+* . .()2 !)& R@BV %&fdo*a@ "$%+%2*a@ * E )^N -G * l )+*HG ( .1

, !%2*HZ f ! %+* *)ÊhE $ 1!)2 S 3)+%K & ;M 71!)+*. V 11 - * g .1 , !%2*HZ f R@ %+* . 1 )2 1!)Ê %+)2!* % 1\E 1,Z *PC 1! * #1 2s bE )+ i(3 !)Ê *%+%2*Ql !- *)211*f * , l , [ Z f !-%2)2 *4 p )0.kV 1\s % . )2 *. . , JE 3 ! s#.Z HE )+*. 1! .( - .G/.1 % 1\E

βe<s %2 1g.1 % 1\E -%2)2 - Js#)2

)+* *1 )2.*.S **HZ ) )21! *- !C .*._! PC%+!*)211* f * , l , [ *- 3 G)21! " )2 !)+- )21!EZ! . .G .1 p 6KW@ 6 .%+%0.* )+* , l , [ .E )+*)2*.**)2 !%+)+** l × p @ , *1!3 !\E *)21\E 1 *! PC%+

BG #$

X * .()2 1JE S 3)+%Ps .* .()2M)Ê % !)2*1!- *)211. , !%2*HZ1!!.()2* f !- 1!%+%+ *a@1! %s#)+. )+* !!%2* 1 % 1\E )Ê 1FG)+-UN )+1!-)Ê )+* .JE .()2wp#%+%2*)+*-@VN !)^f)+TC . 1 % 1\E 1 .!%+)2 Z HE _1 )+* N EhE )2 N !)2* . , !%2*HZ ()+*# i !* )+*.(HZ )2!6 -* %2.1!F!*- $ Dl & *. .()2bV1\s#)+\N *- * ( .* *E %2*, *1!3 ! 1!)+*. .1WZ* !%2 , JE .()2 1DV! )+ )2 %+- *.*V1g !*- * 1%+%2* )Ê HE * . 1 %2JN !1!!)21 * Wp 6KW@ 6 .%+%21 .Z )+!- * )+*/ 11 l)+*1!)2 )+*.1YN E 3%+)^Z )2!Qr1EhE *1!3)2% *)2 !%+)+* Z 11/ ) V*2s 1* )+ )+* 1 1! 1!!)j@1!)2!* 67-* %2. 1WZ !% *- *4 )2\Z s#)+. .1 "$ 1%^s#)+\N *- 11 JE . !!%2*a@*/ Z 110g .1 p 6KW@ 6 .%+% *- JZ- * $ 1)2 WV)+%2.(*- :(+ & )2 s#)2 !)+- 1!* Z 11Z s#)21 * .* !!%2** MF!*- *0.* )+*!-!*.(3%2* *. . l @6 -* %2.1!F!*-Ds .* )2E )+!%21 )+*- JZ)2 * " )+\N *- 1!3( )+%2 3TC1* !)2!Ul )+*1\ 1JE !)21 1V *a@ *.V11 %+*mf 1 )2 .* "$ 1%^s#)+\N *- 1\E *)21FE *$)21!E )+.JE Wp 6KW@ .WEk-%+)2 *.4s#)+./)Ê %+- *.* .Z *\Z ag )+* *.#- % - !*)2 !%+)+* Z 11Z )21%+)2* *. - !**Z 1!.()2*4XBE 1*V 1 *)+! s .*Z *PC 1!/.()2 -!*.(%2- *.* , !%2*HZ 3Z 11 %2 s %2 * )+* i(!*$)+*.(HZ )2! 6 -* %2.1!F!*- )+* .JE )2 ! * 6 .%+%j@1\ 1JE s#)+\N *4 )2 3!- *)211 .%M)Ê % !)2**s .* )+*V1 *)+! :( '4Z *PC 1!)+*$ 1 )+ *. "$)21 f !- 1!%+%+ *. 1!3PC )+*~V 1 *)+! :( : )+* !3!)2! *.$)+* )+* 1E !)+%2. )+*- .(*Ps %2 1WE )+ .JE "$ 1%^s#)+\N*- 11 JE /)+*V)+%2.(*- :(+ .R@- 1!%+%+ )21! )2V 1 *)+! :(LK$)21/:( 57 )+* %+* .()2 .)+%+%+)2!* *1! *- * .)+*HZ%+** Z 11*./F ** .()26-%+)2 HN )+0)+* 1 11* _1!TC*.(*)211 1 . s 1* R@%+)2 *.r1EhE * PC*- V1 %+)2JG *. 1%+%2* )+* V1 *)+! :(qc * s )+V1!3 JN .l !- *)21!* 1 *- * *- 13 *s .*4

>) # -H: 7$'& 3 &$' 29 =*$ .( k$2P)+*9=$28 7$(x-H< $d 7$&*

5

88

7

3b

3a

6b6a

4

2

1

n, v

∂r n0(r) "! # %$ v00(r)

& ' ( ) * ( + ,- ' .( ) ) ' * )

& ' ( ) * ( + ,/1012 ) 0 ( ' 3 ' 415

"6 6 7

Γr = 〈n vr〉

89 ;:< $%=>?@

B

ABCDFE DHG8CG1E D1I1JKLMON G1KE P 〈n〉t

ABCDHE D@G8CQG1E D1I1JKLMON G1KE P 〈v〉t

)& ($0# "%< $'$0& () #"!/'"%)/')!"%+4)#$0$' 132"! $ A 7 6+5)$0.$0/0)6 $ : 75/"%1 ) )) / & (5/ 5" /' "0 #/0# D; <5/0 /0+) $0+)"0 & ( 5/0)"0"% "0+ #" & (5/0 )" , #$0& ( <) / )& ( ))"% & (/' G"%)" )& ($0# "%< - /-) D </' /0+) ) C5 $0$0/'/'$0& () /0#!)" / "%) $0/0)()> HD; <5/0 /0+) #"C5 $'$%/ /0$'&'( /')B #4)$ --)) "0 ) )& ()/'"0$ +"9/ > & (6<+*#</0 *#"D ) /G

" )+\N *- 1!3( )+% "$ 1%^s#)+\N *- 1!3Z 11 n%+%2 . , !%2*HZ' .(HN !)2* .( mN *2f JN !)^f *m? %+HGK M)+%+)21!)2!*-/.( r * % =s01K M)+%+)21!)2!*-/.( 6 * 7sI l \Z-*- .( J*%2.11! 115 .(HN !)2* .( 1!F! .()+*-*? %+HG: .(HN !)2*: _1!TC\N *-c .(HN !)2* .( M 1!!)21!)2!*-e l \Z-*-

# AB)) /' )" "%< $ 1 ) #" "%)4 "%+4))$'$' 5& ()

++ + $& +#)# & ( ' +

X *$.()2 1JE * \N3)+%N** E )+!%21 . p 6K@ 6 .%+%+-%2)2 *- * .1%r1EhE *1!3)2%. 1)2 )+*!-!*.1 *. . 1 1 2s#)+*.()+-UN )+1!3%21,E )+.*wMF!*- * .()21\N !)2!s R@

4 *&$2P$' *$3,&$ d:8) d$29=97$Q)< $ 0 <8$&* >+

.*4 )2 V11- *Ys .*yZ 1TC\Z %+)2 yE )+M)Ê % !)2*1!- *)211* )+%+%+1!!!)2! )2 )21FN11)2*.*)2 !%+)+** "$ 1%^s#)+\N *-QZ s#)21 * % 1\E )+*!-!*.$*.1MF!*- * *yV *a@s 1* )+)+* 1MF\E -* %2.1\Ek-%+)2 .* )+*( 1!* r- *- Z HE _1!TC*.(*)21. 1"$ 1%^s#)+\N *- 11 JE 1)+*QV)+%2.(*- :(+)2 )2f !- 1%+%+* * *- *s#.*E )+ .* )+*1S 3)+% 54E !)^f)2!*l)+*- 3 8@E !*7.( - g !8)2 - * yV *- %+%2%2Jf* *~M)Ê % !)2*1!3 E *.E * )Ê *PC 1!*~V 1 *)+! :( '( 6 )+ )+%& . 6 .%+%+-%2)2 *- *~N** * * - JZ)+-s R@.* .G .1 )2 ! "$ 1%21!3)2%;Z s#)21 *$.* !!%2* * ? %+HN! !)2**$*..JE % 1\E )+* !-!*. 1!)2 .( .()2E ! JE !)21 l )+- *1 0)+*HZ%+*#6 .%+%+\E -\g*.0*. *)2 * .()2D_!- f * % 1\E 3 E !* - JN *2g3( 0)21! 1)2* )2.()2 6 .%+%+-%2)2 *- */. 1 Wp 6K $ K( 'I & *. $ K( '5 & 1S 3)+%PK g .()2)2 @ *.$ * !) %21!F!*-7* )+*HE % )+*$)+ *\E )2!* ?\E ! ?g .()2 *)+!)2* .[3 * *. .()2p .!*-/. )+*HZ%+** , \E 1)8( .()2V1 *)+!K( ' *.QK(LK( 'yf \s#)2 1*4

∂t n+φ, n

= −κ ∂y φ+ δc G(φ) + χ‖∇2

‖(−κx+ n) + χ⊥∇2⊥n

$ :(+ &

∂t∇2⊥ φ+

φ,∇2

⊥ φ

= −∇2‖ φ− G(n) + ν∇4

⊥φ .$ :( ' &

6 )+κs#)+. .bE )+!!%2 )2 )+* !-!*.(-.()2* 1!!- %2-HX 1! E * * )+*01%+1!\N *a@

1!)21!* * ]1!*- . 1 %2)2 *- 11\ 1JE )+* 11!)2!)+*/.JE .()2 gTNs#)+\N *- . ? %+HN! 8@!)2** ( .()2 )+* !-!*.(3%2.)2 #*. . 1 %2JN !!)21 * *!) %21 \gFN(1!)2 !)+-s .* 1%+%2*81 )21! ).]1!*- . %2)2 *- * .()2yZ)+!%+)2 wl * s#)2FN %+*- ..() %2* )+*!-!*.(3%2 Z 1!!)ÊhE *4

!

" )2 )+1 ) .yp *.(%+*- . *.( .()+*-*- * \s C * $ 1V 1 *)+!hK(LK & 1\Z 1!)2 .1 )2 )+* !-!*.(3% 1 .JE 1f !- - ** )2 -.()2* * *. .

(0,0)@B?!)2\N E 3 ** .0)2 1!F!*- *Z 1EhE *

n0(x, t) = −κ (x− x ) + n00(x, t) ,$ :(LK &

s,C *. 1!)2 .1 * !) %+ )+* !-!*.(3%,f %+%21!TC*.()+- 1.]1!*-/. *!) %21!F@!*- * !-)+

φ0(x, t) = φ00(x, t) .$ :( I &

p )+%2.yE * .1h? %+HG <C *HE )+!% 〈〉y,z. )2.* %2)2 *- * $ :(+ & *. $ :( ' & )+*.JEE * %+%2 , \E g 0.()2

y@ *.

z@BS .()+* )+*-!)2!0*. !*#.()2V2s 1* )+ )+* 1

i !*7)+*.(HZ )2!* E -*!)21 * MF %2. 1 **)ÊhE 41 .G ∇‖ = ∇‖0

-)+%+ .**$! PC%+E * .()2 %2)2 *- * g0.()2Z)+!%+)2 Yl *Bs#)2FN %+*- . 1 )+* !-!*.1

∂t n00 = −∂x (〈n vx〉y,z − χ⊥ ∂x n00)$ :( 5 &

∂t v00 = −∂x (〈vy vx〉y,z − ν ∂x v00) .$ :( : &

>< # -H: 7$'& 3 &$' 29 =*$ .( k$2P)+*9=$28 7$(x-H< $d 7$&*

p ) ._!!)^Z )+TC1!-%2)2 *-Qs#. * 7)+*HE %Z 1TC\Z %+)2 g .()2 .() %2%S.()+* x)+* -!)2!

)2 .() %2 *. 3 %2)2.%2hSE 3 ** .Dl × p @ 1 s#)+*.()+-UN )+ !- * 1!)2 .( $.()23 !!)2%+%2*mV%2)+!*- * . * !) %21!F!*- *

vx = −∂y φ , vy = ∂x φ .$ :(qc &

?g.* $0,0 & @BV* )+% . 1 2s#)+*.()+-UN )+ \N ** WE *.G .()2 y @BV %2)+!*- f * φ00

A%+%!-)+ 1 .G .()2 )+*!-!*.(- 1 s#)+*.()+-UN )+

v00* 1 . 3 %2)2.%2* S E 3 **

1!

v00 = ∂x φ00 .$ :(Le &

Ws#)+.v00

%21GW ^ 7D ) WN , ,/ JZ)2 * " g.E * .()2 %2\Z YX * -!)2*. %2)2 *- :( 'Dg .()2

x@BS .()+* *)2 .( (g * 1 1!\g*. f Z .JE , \E .

%2)2 *- $ :( : & * )+*HE % .()2 3 !!)2%+%2WV %2)+!*- )+* .() %2 0)2 !*- 1 .GE * .**)+*l * s#)2FN %+*- 1!-%2)2 *- g0.()2 C*'+*-'.10bD/* ,D/ WNW, 7 ∂x v00

! )2%+)2 l !% C!*- . "$ 1%^s#)+\N*- 1FE *)21\E * Z s#)21 * !!%2** ? %+HN! !)2***. .* )+* !-!*.(3%2* 1%+% * * E )+ )+%& /. )2.* %2)2 *- * $ :( 5 & *. $ :( : & %+- *4r1! ! hE *$.* )+*( 1!*~? %+% )+*%M)Ê % !)2*4)+* .DE *$.* % 1\E )+*a@!-!*.Hi )2! % CG+XE M)+** .YMN%2*!** *- *.%+ 1 1!)2 7HE )+* !N *1!)21!* **-(.DE * )+* .()2 1JE ? %+%* .()2* E )^N .5N %2)+*1\N%+)+- * MF!*- * ! *. .()2 * E )^N .0-G 1\N%+)+- * )+* !-!*.(-FG * HG *Qf !% CG !!)+- * *a@- *mE )+N E 3%2!Hi )2!JE )+* !-!*.s#.*7 )+1)Ê HE * . * 1! *- *)+* V1 *)+! 5(+Qf !- 1%+%+ )2 %+- *.V )+%2.(*- :( '$1%+%.()2mf ! *.* M)+! !)2*)+%+%+1!!!)2*4

1

n, v

∂r n0(r)

v00 (r)

Γr = 〈n vr〉

B

! 0"# ( # ) " ) ( # + ,F' 3$ 4@) ' ( * 〈n〉t

! 0"# ( # ) " ) ( # + ,1' 3$ 4@) ' ( * 〈v〉t

)& ($0# "%< $'$0& () )"!/0"0 )5/0);"0+4))$'$0 1 2";) D # #$ 1 #$%/') ! $'( 5/0)"0"%$#

)2 V#.af JN !)2*7. 1)2 )+* !-!*.1 .( .()2%l × p @ 1 2s#)+*.()+-UN )+s %2 * 7.*l !% C!*- * . 1,Z s )+*QS 3)+%21 Z 1EhE * E )+#.* S \gHEhE *- 1!\E * .()2Dn%+%2g%+)+* X *1!)+%+)+TC* )+**! %+ . 1 6 .%+%21\(1!JE 1%+)2 ! )21! )2.l \Z-*- 1FE 8@*)21\E 1f * *)2 !%+)+** !!%2* * )2 @ *. * !) %21F!*- *4d?g .1%2)2 *- 1R@1\ 1JE $ :(+ & $ :( ' & $ :( 5 & *. $ :( : & 1%+% .()2 1* *7.( .* " )+\N *- 1!3( )+% 3TC1* !)2!s .*4

4 *&$2P$' *$3,&$ d:8) d$29=97$Q)< $ 0 <8$&* >Pt

V * .()2 1M%+%2 1) (.()2p .!*- .p -!) )2 HN )+*7 )+*- Js#)2 1*4Pp )21!% *- s f *.* !!%2* * W'.10/* .()2 ./s %2 1!)2 7%21W]1*-/.6 .%+%+-%2)2 *- *$!- @ *4 )2 1w]1!*- *

n(t)φ(t)

% 11* 1!)2 s )+! )+* )+* )+** Z)+ PC*-)+- * *. )+**Z)+!* PC*-)+- *V* )+% ( 1!3 %+*4 Z)+!* PC*-)+- V* )+% Hs %2 .( r )+\E )+!R@%+*- ! %+*s .*N**;s#)+. %21 -.2' B*NW, B+-*eFH8* B .w]1!*-$ JZ)2 * Z)+ PC*-)+- QV *)+% .()2 !!%2* * f Q.0"c,. , ,R* ! PC%+hE * .( M!N!)2* . 1 1!)21!)21 *V* )+%21 f * .*QMF!*- * 1%+1X *4V* *- s#)+. - JZ)+-(.G .()2WZ)+R@%+)2 - JE )+!%+* FG * 〈 n 〉t

*. 〈 φ 〉t%2 .()+-%+)2 7)+*YADC !*-/g.* 1!!)21 *V* )+%

.]1!*- . 10 )20 ! * , !%2*HZ E .%+%21\(1!JE 1.1%+%2*4 *1 )2.(*- f *MF!*- * *.Q? %+HN! !)2**N EhE . 1! %+ 1*.%p .a@!*-QZ s )+% ) i(3 !)Ê *%+%2*~V*.(* *- * Z 6 11!*-f * , !%2*HZ -FG * 1!1* .hZ)+ PC*-)+- mV *)+%# 1!!)ÊhE ks .*.N** 5Ue p ) *- 1!!*_!-%2)2 *Z s#)21 *4li 3 !)Ê * *.wM)Ê % !)2*4N** %21 * 5g .1_! %+*wf * ? %+HN! !)2**.()21\N!)2!s .*4E .()2 1JE M( 2f ! %+0 **- Z !- * )21! )+* )+- JE M JE . 1!)21!)21 V *a@)+% - 1*.! .!- 1!%+%+ /- 1E .() %2QN *f JN!)^f , )+%2 *HG

Γrs %2 .*

*E %2* , *1!3 ! .1%+%+ 1\Z 1!)2 1E )+ 1 )+*JE HN!)2*.* *. )+*JE 1)j@1!!)21 * V* )+%.MF!*- *

n*.

vZ 1EhE *4 " )+. ) **- *$.()2 gFN's#)+R@

N*- ( .* )+*!-!*.7 ! $ 1!)2 V )+%2.(*- :(LK & 1 1!)+*. .()2hM JE %2JE *? %+HN ! !)2**2? %+HG *. 1!)21!!)21 5V* )+%%21 )+*bl )+* )+ . !!%2**MF!*- *Z f 1! * $ - 1!!!)2 %+WE HE *- & p ! E *$* * %2)2 *- $ :( 5 & 81 \N ** YE *8.G .()2yZ)+!%+)2 wl * s#)2FN %+*-7.

(0,0)@ )2 JE . .( wZ s )0V* )+%2 1!!)ÊhE ,s#)+. $ M )+ & %2\Z V* )+% 1!%+%+)+* 1)+*;?\E

χ⊥ ∂2x n00

)+*#) 1!)2* .s C *. .1!bV* )+%.()20.() %2bV %2)+!*-. 1 HG <C *- JE )+!%+* N *f JN!)^f * ? %+11 1 )21! *. %21wl \Z-*- 1\E *)21\E 1 . 1

(0,0)@BV* )+%21f 1!*.*Qs .*N**4

?g .()2 M)Ê % !)2* E )+ 1!JE )+* !-!*. $V )+%2.(*- :( ' & N** . , \E E )+ .JEN *2f JN !)^f *.? %+HG g .()2l)+*- 1!!*- . 1 p 6K A%+%#- 1\Z hs .*4;V * .f !%2)+ *.* ) 1!)2*1!-%2)2 *-

∂t n00 − χ⊥ ∂2x n00 = 0

\N ** ,E * .G .()26 .#* / 1!!)ÊhE br )+ *. - - *JE N%2)+*JE`r( %+%21 *( *- 1R@s !5E )+ .* f ! *.** *.( .()+*-*- *4Z wA%+%- ./CE 3( ,s#)+. 2X * .*M)Ê % !)2**)21! .()2 1 1 * * 1s *)+- * r )+)+* )+*

τ.? %+% s,C *. Z p .()2 1 E . .

**- * )+*)+- *.! )21 1*. r )+)+* )+* HE G?g .1YZ)+!%+)2 - JE )+!%j@ )+*!-!*..()2 3%# . .()2 1 * %2)2 *- $ :(LK & 8.G .E )+ κ *(!C*-%+)2 f !- - * )+*!-!*.(-.()2* .0)2 ! %+* %2)+ $ 1)2 V)+%2.(*- :( I &

n0(x) = −κ (x− x ) .

"$*.E * 1!)2 $* * . %2)2 *- $ :( : & Z 41 !- * 1)2 ~C *%+)2 4M( f ! %+ ..()2 1 % !)2* .()2 -%2)2 E ! JE !)21 M!!HN! 1!)+\Z s#)2 %2)2 *- $ :( 5 & " C *.. %2\Z , \E s#)2 .!HE )+* CE 3( *- ./3 %2)2.%2* !)2*

v00 \f !!( % CG

> # -H: 7$'& 3 &$' 29 =*$ .( k$2P)+*9=$28 7$(x-H< $d 7$&*

1!)2 .#1! , \E %21 l \Z-*- 1FE *)21\E 1 -)& *4)2! ) *.%+0 1 1!)2 HE .()2.() %2 V%2)+!*- .1- * ** * 8 D2'N C.01 . 8 D2:,*# 0X * .D? %+)2.'@.a* E )^N )21 . J *%2.11!! 11 )+* !!%2* XE 3%21!!*1!3 ! .()+*- .( *)21!3 M!FE *- *4;l1 *.%+ 1!)2 HE .()2/1!)21!!)21 S!% !)2* Z s#)21 * . .() %2* *.3 %2)2.%2* 1 2s#)+*.()+-UN )+11F!*- 5UO :9(:( 2XE ? %+%2)+* 1 1*/ % 1\E )+*!-!*.1s#)+. .()2 1 , \E )Ê -EhE A %+% - 1\Z 1 .G s - * .#1 *V ./CE 3( *- *.$ :( I & .1)+* !-!*.(3 * !) %A %+%8%2)+M %+% * *$)+* 1%+1!\N *1)21!* k]1!*-E )+Df!) %2* )+* !-!*.(3%2* *s R@.* 1 1!)+*. .()2 , \E .yl \Z-*- 1\E *)21FE * .() %2wV%2)+!*-7. 1YN *f JN!)^f *? %+11 1 *. J *%2.11!! 11 )+*HZ HZ 1 %+*42?g.1 ! 6 .%+%+-%2)2 *- 11\(1!JE!- * 1!)2 .** "$ 1%^s#)+\N*- *0s#)2 1)2 )+*mV)+%2.(*- :(LK .( 7.()2 * )+%2 - @N **HZ)2 * 1!)+*. )+% ' 1\'E %+)21!)2! * * .*l )+*HG7. 1YN *f JN!)^f * ? %+11 1 〈n vr〉 ( .1)+* R@-!*..()2 3% A $.*f *- - *- ** !%2-*- *7g ! . .() %2y? %+HG s %2 1 .JE )2 a@-.()2* * . 1 )+*!-!*.1 - 1!3 )21!hs#)+. $ )+% & .Z .G .()2 $

0,0 & @BS E 3 ** . )2 1!F!*- * *2s,C 1!aA %2)2 *-$ :(LK & . .()2 1 )+* V * . 1)2 3%21n0(x)

l )+*~N%2)+* )2 -.()2* %+)2 !Z . 1s *)+- X *1!)+%+)+TCs %j@ s#)2 .!HE .()2yMTC\N .MF!*- *7*. . 1N *2f JN !)^f *? %+11 1 1!!)ÊhE .( s#)+. .** .()2 *./" )+\N*- . 1 .(HZ )2!*~? %+11 1 (#.1 )21 . )+* 1!)2 * Bs#)2TN %+ )2 3%N %2)+* *. 1 !;A )+* - Js#)211* r )+ !-)+ 1!)2 )2)+* *1!!)2*PCr1*. E * .*p -)+** . 10** 1!)2*PC* r1!*.1* 3!)21 JE l\E 11* - 1 PC\Z $ 1!)2 V* *- & 1 ! PC%+ E * .( .r )+\E )+R@%+*- g )+* Ek-%+)2 1! % *- 1 r )+!)+*\f%+%0)+* . 1!!)2*PC* 1 )+* - 1)2 )+*!-!*.(3% +s#)2 )+*QV)+%2.(*- :( I .!- 1!%+%+

3b

3a

4

2

1

n, v

∂r n0(r) !" # $&% v00(r)

' ( ) * + ) , -. ( /) * * ( + *

' ( ) * + ) , -0&132 * 1 ) ( 4 ( 5 6

7 7 8

Γr = 〈n vr〉

B

9: ;< = < >&;>3= <3?3@BACDFE >3A= G 〈n〉t

9: ;< = < >&;>3= < ?&@BAHCDFE >3A= G 〈v〉t

)& ( $') "0<5$0$'& () )"!/0"0 )5/0) !"%+4))$'$'- / *"% ) ; $'( 7/0)"0"% 2& < "0<5 )" /'"%)/') 5/"0 ) 1 $ & (/' 7 , )$'& ( 7<) /0$ "%+ "% 4)) $'$0)>

)2 l \Z-*- .43 %2)2.%2* !)2* .( .* J*%2.11!! 11R@ 6 *)21\E 1s#)+. .( )+% I73TC1*!)2! " )2 * - JZ)+-ys#)+. N** .()2 1/3 %2)2.%2M!FE *- )+* .() %20)2 !*- *1 )2 .(%+)2 1!\N *. * 1 )2 .(%+)2 - !)2 1)+* $_\Z)2 *s a@

3,&$ 0 7$43 8-/ 97:;)<= k($8<)+*9=$28 7$ x -H0 $2 7B$'&* ><

1% & E )+ !-)+ 1!)2 /)+*0.() %2V PC*-)+-UN )+ . _1 !*- ∂xv00(x) )2 .( .* J *%2.11!! 11 \Z- 3 %2)2.%2 !)2* s#)+. . a × % +*-*#, W`.10 .a × #-S @ - * ** )2 1 3 %2)2.%2M 1!FE *- )21 )Ê_!-%2)2 Z .* * 1 )2 .(%+)2 *

((m,n) 6= 0)@

MF\E .* . 1 2s#)+*.()+-UN )+-G 1\N%+)+- $(0,0)

@ 6 .0)Ê?!)2!HE & " )2 ! !)21 :' :UK *.7 i(3 !)Ê *%+%2kV! )+* :I :5Z)+- * s#)+\N .()2 1kM!FE *- 1!)+%+)21!)2*.( .()2 , !%2*HZ*.kf \E )+*.! ..( .* .() %2* , )+%2 *!*1!3 !( )+*b_R@E )+*.!*-7. !!%2* *.MF!*- * - s#)2 .yE )+ )+*yV ( 1!)+%+*-7. 1 )2 )+*a@!-!*.(-.()2* *$)+* $ )+% ' & 0M E )+W*.)+* 1!TC*.()+- 1*)2 !%+)+* 1 "$ 1%21!3)2%Z s#)21 * )2 )+* !-!*. !!%2**~MF!*- * *. M 1!FE *- 1!!Ps %2 1 )+*V )+%2.(*- :(LK .( .()2 )+%2 )21%I 3TC1* !)2!ys#)+. 1)21!!)21 4V* )+% .MF!*- *

n*.

v)210)+* .()2 1JE ?%+%4- - *2g .JE HN!)2*.*V *)+%f !* % C11!)+-@

N%2)+*4V ). M 1!!FE *- *1 )2.,E *kZ s#)21 * )+*JE Z)+!%+)2 - JE )+!%+*4V* )+%*. )+*JE Z)+!%+)2 HN !)2*.*V* )+% )2 1 ) *HZ )2!*-1s#)+. )+* V1 *)+!/:( 5( ' 1!3 *4

0

10

20

30

40

50

60

32.75 3.25

fester Hintergrundvariabler Hintergrund

n0(x)

q

〈n2〉1/2

V

τ 0

2

4

6

8

10

12

14

16

0 500 1000 1500 2000 2500 3000 3500 4000

variabler Hintergrundfester Hintergrund

n0(x)

q

〈n2〉1/2

V

τ

I ; )"%)& ()$ 4#) /0& ()- /0/0) /') & (/')(/0#"%"%"%+ $# ; !#"%)& ( )"B4)# / 7& ()6:;5/ & <5H)$"%-$ 7 )"9/')$ #" &'(5/0)$0/"0 )>1 )$0/0# *"% ) /0#"%"%>

? 1!\Z %+* %2)+.G )+* . 1E s#)+\N*- %+%2 Z 110 ))+*JE f!) 8@%2* % 1\E )+*!-!*. . )2 )+*!-!*. )+**! %+ . 1 %2Jf* * @ *- )21 - Ws#)+. $ 1)2 V 4:( I & *. 1!)2 ..( )Ê`_!-%2)2 Z .kM)+! !)2*~E )+ 1!JE )+* !-!*. .()2 !!%2*VN !)^f )+TC .(HZ )2!)2 1 0)+*D_\E )+*.!*- .WMF!*- 1!*)^f 1 Z b?%+- $ 1)2 V 4 :( I &

+ ( !*" #)$ &' ( * & # ( + " ( +

A .*1f ! )+*.* * 1 *- * . 1 g*( *S 3)+%21%Z %l )+*1%+%+*- . !!)j@/Z )2%+%2* )211!)+3 !)2* 17s#)2 Z kV1\s %0. % 1\E 3 E /1%+%2* ** .()2 l !- *)211. M)Ê % !)2** g %2 , l , [ @ E !!)2 *.4E )+ V*2s 1* )+ . 1# i !**$'8I@MF %2. 1,f !- 1!%+%+,s .*4( )<s .* )+* .()2 1JE 1!* , )+% .()2 311- *4g .1 _! %+* . , !%2*HZ *. . 1 .() %2* , )+%2 *!*1!3 !1( - %+)21! *. )Ê Z s )j@* , )+%$V1 *)+! :( : & E )+ .* l \N **!*)211* .YM)Ê % !)2** 1 .* * ( %+- *.* *\N3)+%+* )+* !3!)2!

> # -H: 7$'& 3 &$' 29 =*$ .( k$2P)+*9=$28 7$(x-H< $d 7$&*

p .!*- % 1FE 3 E p .!*- l )+*- Js !X **JE 3 !

Ti = 0!_ E 8i 6 .

n = 60l %2JN !** JE 3 !Te = 30.8

!_ 6 .* 1!*.∆n = 4

)2 n0 = 6.5 · 1019 E −3 . )+!!3*HN

nx + 1 = 54 + 16 -* %2.

B0 = 1.0, 3 % ("$%+%2*HZ %

κy = 0.0017G , !1!.()+1

R0 = 1.75E "$%+%2*HZ %

κz = 1S %2)+*.()+1r0 = 0.45

E )211!)+3 !)2*χN

‖ = 1.03 ! @B? %+HG <C

q0 = 3V!)211!)+3 !)2*

χN⊥ = 0.81_1 !*-

s = 3V!)211!)+3 !)2*

νN = 0.81% <MF %2.1!TC\N X

δB1/B0 = 2.0 · 10−4 MF1!E X )+*N'Vδψ = 1.2

% <MF %2.1!TC\N XFXδB2/B0 = 3.1 · 10−4 MF1!E XFX)+*N'V

δψ = 1.8

!)& !% C*- %s = 8.0 · 10−4 E p#%+%2 *)+*-% C*-

ξ = 7.9 · 10−4 EMFG)+-UN )+

ν = 1.58V 11 HZ)+

τ = 4.2 · 10−6 1 M 1!!)^Z )+TCWX

σ− = 1.0 , σ+ = 1.2 )2X

2.75n≤m≤3.25nM 1!!)^Z )+TCWXTXσ− = 1.2 , σ+ = 1.5

1E \E .*HZ %#mn = 256

) )"%$'& (/ )" *)" ) /') 5H /0+$ "'- /0#") )"9/' #" "0) /0 *# 5/"%1 )$0/ "%< $0/ % )$ 1 2"

q = 3 7 +$0 /'+ )"%)& ( /#

)2 , %+%2 :(+ -)+ )+* 1!)2 Dg .()2%f \s *.*$ % 1\E 3 E $ %+)+*HN(1 & 17s#)2.()2 .Z - /!)+- * *\E )2!*l )+*- 3 E $ 1 & .0p 6KW@ M)Ê % !)2**4

" 9 "%-/&,*$(&-a =)< 97*-a 7$ l1 1%+%** -\g*. s .*,s !HE )2 .()2 M)Ê % !)2*1!- *)211 g .* ? %+%DE )+n =60

f !- 1!%+%+ s .*( )4.JE .1M3 JN !!HE .#%+)+**QV*2s 1!* \g HZ)+!)+- ) )+*DE 8i )Ê %2*7)2.%2*$6 .*HZ %- 1 *)+!*1s#)+. $ 1!)2 yV 45( 5 & l1 1%+%+1!)2 .()2?- s !HE .()20 )+15f ! *.**QM)Ê % !)2*1.*E )+ .JE N E 3%2!* V*a@s 1*1!3 JN !!HE $

n =92 & *. )+*01*HN *QV P1*-hf * ∆⊥≈2 %s*)2 Z

V 1\s !!*-$ *- JZ- *s#.*4 )2 V* s ! %+)2- )+* .kABs *.()+-UN )+/)+* 1 -*r )+\E )+!%21 p ) .* V 1 *)2.* 1 *- * . 1 g*( *S 3)+%21ws#.* .()2 a@* *- * E )+yMF!*- *Z!Js )+%21 f *

τ = 5400)21

τ = 8000g .()2 *)2 .(!)+- QMF %2.1!TC\N

*. f *τ = 5400

)21τ = 7000

g .()2 MF %2.1!TC\N .( - g ! 14M3 )j@ !3% \Z -\g*.* E )+

∆τ = 20* %+%2 '9mr )+)+* )+* .()2wN E 3%2! .()2.()Ê *1)2* %2

]1!*- 1!)+*( \E !)2* - 1!3 )2 !Ys .*1N **4* 1!3!)2 .()2 1 7K9 HZ s e9Qr )+!3*HN2@*2g .()20- JE )+!%+,s .*4N**42X */. V)+%2.(*- :( 5 1!)+*. ** .()2Z)+!%+)2 *4_!% C( .WMF!*- 1!*)^f 1 . 1bN *f JN!)^f *m? %+11 1 $ %+)+*HN 1 & *. .0)2 1!F!*- * $ 1 & g.* ?%+%

n = 60Z 1 *4 67* \N ** .G ) )+*- 1 %+wMF!*- .()2wA)^f 1

Z)+!%+)2 1 1!\N4f!)+)2*4/XE ? %+%E )+n =92

f ! PC%+ 101!)2 mC *%+)2 $S 3)+% 5 &

3,&$ 0 7$43 8-/ 97:;)<= k($8<)+*9=$28 7$ x -H0 $2 7B$'&* >

E * )+*/.,V * %^(1 wZ)+!- JE )+!%+ FG */ ! (1!%+%+ 1)2 /.()2W?- ).*4f ! *.**Z)+!%+)2 *Q_ !) !)2** )+* .* * *- *QE )+0DlW@ MF!*- .1 r )+!)+*a@\f%+%HZ 6 )+!!%+*-f *

τ=5400)21

τ=8000- *2g- *. -G )21U[. *.1 1!- .(\gFN

S***7.()2l !- *)211 . %+)+!)^f * * 1! *- * . 1#g*( *S 3)+%21.( * % C*- * *- *E )+#-FG *r )+!)+* \f%+%2*Z 06 )+!!%+*-/ 1!TC!)+-bs .*E .()2w?- wZ * s !* s#.* .()2 **- * E )+ DlW@ MF!*-7*.

n :9

)21hZ )+*JE r )+!3*HN kf *τ = 14000

g *)2 .(!)+- MF1!!E 1TC\N *.τ = 12700

g yMF1!!E 1TC\N %f !% C*- ! E )+ ! PC%+DE * .** g .()2hr )+\E )+!!%+*- )+*1- 1E I2K9 HZ s <K:54r )+!3*HN<p .()+*-.( .()2Y] )Ê )+!)2!*- . *HZ)+Ws .()2 10g .*? %+%

n O' *)2 Ek-%+)2 4)2 **- E )+

n :9 *.mMF1!!E 1!TC\N

Ic=1.2NaV

*/!)+-f *τ=8000

)21τ=14000

*- !C 7K , - WE )+#)+*JE ] )+*ai@ $ K Z K')+R@ M 1!JE & 's C *. .()2M)Ê % !)2*E )+

Ic=1.8NaV f *

τ=7000)21

τ=12700Bs 'I

, - .! -%2)2 h]8( PC! E )+ JE MF %2. )n =92

Z * *R@%2)21!!*- 1 PC\Z *- 1\s )21 59 )21 :9 , - */!)+-

0

0.5

1

1.5

2

2.5

3

3.5

4

4000 6000 8000 10000 12000 14000

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

〈Γ〉V

τ

〈n2〉1/2

V

τ 0

1

2

3

4

5

6

4000 6000 8000 10000 12000 14000

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

〈Γ〉V

τ

〈n2〉1/2

V

τ

E ) /%*)"0 1 )$-*+-))- /0/0) /') <+5*)</' *) D $'$0)$G! ) /%*#"% 1 )"*+-))- /0/0) /') & (5/0)$0/"0 5/"0 "% 0 )$ #

τ = 5400))$'& ( /' /#

:!( # / )" +"%-)"%/0) ) / )/0" /1 τ = 4.2µ

$')&5

l )+* _!-%2)2 . Z)+!- JE )+!%+* l !- *)211 .hf !% C*- !* M)Ê % !)2*1!% C( QE )+ .*l !- *)211* .bN'g\Z* $

n =92 & @ M)Ê % !)2** 1!TC!)+- Z . .()2V11- .G 1)2 .1 %+)+!)^f ,_! %+* . , !%2*HZ# ) )+*- 1 %+JE MF %2. E )+

n =60*. .

X *HN( * HE . 1#)+*QV 1 *)+!5( ' f !- 1%+%+* % !)^f *m? %210*)2 s 1* !%+)2 C*.!

0

0.5

1

1.5

2

2.5

3 3.252.75

σ− = 1.2 , σ+ = 1.5σ− = 1.0 , σ+ = 1.2

σ = 0

σ=0

σ−=1.0, σ+=1.2

σ−=1.2, σ+=1.5

〈Γ 〉q

σ=0

σ−=1.0, σ+=1.2

σ−=1.2, σ+=1.5〈Γ 〉

q

nmax = 60

nmax = 92

0

0.5

1

1.5

2

2.5

3 3.252.75

σ− = 1.2 , σ+ = 1.5σ− = 1.0 , σ+ = 1.2

σ = 0

σ=0

σ−=1.0, σ+=1.2

σ−=1.2, σ+=1.5

〈Γ 〉

q

σ=0

σ−=1.0, σ+=1.2

σ−=1.2, σ+=1.5

〈Γ 〉q

nmax = 60

nmax = 92 B )$ !"%+ #$-/0"%)5/0) D $'$0)$-- /6 $'&'()"0() /0$'/')"9* 1 2" n 7 #&'( ) <2 "04))"0 1 4)# /H )$0/"9/') D )$ ;"0+ #$ /' "0 #/0#D $'$0#$ - /B6 $'& ( #"%() /0$'/')"9* 132 "

n 7 #&'( ) 3)"% 1 4#) / )$0/" 7/') D

)2\Z ! QE *V)+%2.(*- :( :( !- 1!%+%+ )21 .1 .() %2 % . 1 !!%2* *

> # -H: 7$'& 3 &$' 29 =*$ .( k$2P)+*9=$28 7$(x-H< $d 7$&*

? %+11 1 )* 1 )2 .(%+)2 *mMF1!E 1!TC\N * gn

O' $ & *. n :9 $ & )2! )s#.Z 1!0.()2 6 )+!!%+*-kg .()2 3 %2)2.%2 *. )2.%2YS.()+* .( - g ! * s#.*7.**Z Z! . 1!)+!)2*

r.()2Yf 1 )2 .** "$! .r )+!)+ %216 JG@

-FG . 1hZ)+!%+)2 - JE )+!%+* !!%2* * ? %+11 1Γ (r)

)+* !3!)2! 1 .G$.()2 *a@1 )2 .(%+)2 HN )+ . 6 JGUs ! $Z)+!%+)2 _ !) !)2** & %21 1ZJN !)^f .()+*- 1!)21!!)21 1!!*.p !*- 1! %2*- 1 *4s .*N ** *4)2 1Ws .* .( )+* HG@f !)+%+*- 1 !)2 *4 p )

n6 JGUs !*

xi . 4

(n − 1)S *!%+%Ê 11!*- *#1!)+*. .()2

1!!)21!!)21 *mADC !*- 1\s !Ys#)2 %+-0. *)2!

6 )+!%^s !x =

1

n

n∑

i=1

xi$ :(LO &

M*..s )2 *-s =

√n∑x2i − (

∑xi)

2

n (n− 1)

$ :(+9 &

6 JG*1!)2 ! )+ 1%+: u =

s√n

,% !)^f

: ε =u

x.

$ :(+ &rTZ! ./.() %2* 1!)+!)2* s#. .** ./6 )+!%^s ! . 6 JG-FG *. .()2 1%+6 JG*1!)2 ! )+ 1!!)ÊhE X *~V)+%2.(*- :( : )21! %21 - 1!!!)2 %+w] )+*)2 .1%E )+ .JE f R@s *.* !)+! HE !)21 * r )+\E )+!% - JE )+!%+ %Z $\N ***+s C *.$.()2 .( - @Z- **m] )+*)2* .1Z - /!)+- 6 JG*1!)2 ! )+1)+* \f%+%

x± uE \N)2*4

6 )+ .()2 1h_!- *1\s )21 )21! )Ê 1!!*- * 1!!)21!)21 *.M)+** * N )+*V11- 4g )+*_!!*1!)+*\f%+%#- ! *s .* )+* .JE .%s r )+\E )+!%^s ! .hr )+!)+ Z *.* s,C . )21hE * )2kZ *PC 1/* %+)+!)^f *_!-%2)2 * f * %2*E )+f ! *.*JE r )+\E )+!%^s ! )+* 11!)2!4 %+)2 ! .()2 1!%+%+*- . 1 6 JG*1)j@ ! )+1!)+*\f%+%21 )+* .* %2* .( 1 )+*0 DV 1 PC\Z *- . g)+*s#)2Js )+ %^s ! 1TC %+)2 )Ê HE * .0- Js C %+*r )+!)+* \f%+%2 * 1 )2.( 1!)+*. " C *. .;_!-%2)2 . 6 )+!%^s !!3%2 $ - 1!!!)2 %+D] )+*)2* & Z s#)21 * .*4V )+%2.(*a@- *7:( : *. :( :7.1_! %+*7. 1!!%2* * , *13 !1 ) i !* 67-* %2.1!F@!*- %21 **PC !*. *f TC*.! 1 )+** % CG *. 1E )+ .()2 V11- * . 1 S 3)+%21 5)+*1!- 1E 1!TC!)+-1)2 E * .** )Ê`? %+%2 . 1 N'g\Z*mr )+\E )+!%+*-)+* \f%+%21 $ & .G .()2 6 JG*1!)2 ! )+1!)+* \f%+%2 . %2WE )+ DlW@ MF!*- .! .!%+)2 )+ 1)+*.*./1- g !% 33 *4'X * .()2 1JE ?%+%2 %+)2JG 1!)2 *)2 ,E * s )+ *!)+!)^f 18@!)21!!)21 * 1 *- * * 1 )2.* )+*s#)2Js )+ )+* * 1 )2 .(%+)2 MF1!!E 1!TC\N (8.1 %^f ! %+*s#)+\N )+*- - */\N ** E * )+* V4:( : .()2.!%+)2 W_1 HE 8@%2!*-. *1!)2 ! )+1!)+* \f%+%2 .()+*- .( .()25V1\s %()+* 14-FG *kr )+!)+* \f%+%21?g .()2 * 1 )2 .(%+)2 *MF %2.1TC\N * 1)+*. .()2 %+*1!)2 ! )+1!)+* \f%+%2)ÊhE * .!%+)2 f *)+* *.Z * 1 )2.*4V %21s )+*w_!-%2)2 Z s#)21 */.* $

n O' & @8*. .* $

n :9 & @ **- * ! a@E *V)+%2.(*- :(qc )2 1!)+*. .()2 .() %2* %2 . !!%2* *7)2 HN ! !)2**

Z 1 *4V )2%N**E * .* *1 )2 . .kr )+\E )+!%+*- 1!)+* \f%+% C*- * *.. %+*1!)2 ! )+1!)+*\f%+%2 g/*1 )2 .(%+)2 MF %2.1!TC\N * \N *** s,C *.

3,&$ 0 7$43 8-/ 97:;)<= k($8<)+*9=$28 7$ x -H0 $2 7B$'&* >P>

.1 - *%+%2 %+)+!)^f m_! %+* . )2 HN ! !)2** )V *s 1* )+ . l @MF!*- -%2)2 %2)+ * . 1/! PC!*V** HE .G ) .f !% C*- !*$ * *-QE )+

n :9 .1

r )+\E )+!% *. .()2Z - /!)+- * %+*1!)2 ! )+1!)+* \f%+%2 - )+-*81!)+*. HE .() *HZ )2!V 11 - *Yg .1 - *%+%2 %^f ! %+* $ * PC*-)+-Df * .f \s *.* 6 .*HZ % &E )+ *1 )2 .(%+)2 * MF %2.1!TC\N * E * Z ~Nd***s .* g * ( %+- *. R@1!%+%+*- * *.()2 )+** r )+\E )+!% * *1!)2 ! )+1!)+* \f%+%2 - JZ)+-HE )+* 11 1!)2 .l !- *)211Z Qs *4

0

1

2

3

4

5

3 3.25

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75

〈n2〉1/2

q

〈n2〉1/2

q

nmax = 60

nmax = 92

0

1

2

3

4

5

3 3.25

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75

〈n2〉1/2

q〈n2〉1/2

q

nmax = 60

nmax = 92 F )$ !"%+ -)" "0-$ 7 & (/' <5/0 /0+ - / 6 $'& ( #"%() /0$'/')"9*-132 "

n 7 )& (5) 3<52"%4#)"% 134)) / )$0/"%/0) D )$!"%+ -)" "0-$ 7 & (/' <5/0 /0+ - /6 $0& ()"0( # /0$'/0#"9*;132 "

n 7 )& (5) )"% 1 4)) / )$%/ "9/')D

WP$ 9797$A .* _\g !%2-*- * Z kr )+\E )+!%+*-$. f ! *.* * 1%+%2* * * .()2 3 2@1!)^N%+)21 *~l!- *)211 .yM)Ê % !)2** $ 1 , %+%2 :(+ & 1!3 *~s .*s %2 / )2.( .()2 N!)2* . 1 !!%2**? %+11 1 )+*1V4:( : *. . )2 HN ! !)2** )+*V 4:(qc ( .()2 DlW@ MF!*- )+%+%+1!!)2! 1!)+*.

!!%2*Q? %+HG %2)+ )Ê .() %2* 6 )+!%0**PC !*. -%2)2 -Gs **.1 DlW@ MF %2. )+*- 1 %+ys#)+. . !C%+%+ ( 8.G )+* . 6 )+! . 1 *- )21 )Êp )2 $

q K & )Ê *- 1!F!* ? %+% $ 1 2s \Z1] )+*)2* )+*

V 4:( : & )+*_!!)+*- !*-4f ! *.* )21! .()2 1)2 )Ê ? %+%2 . $n

O' & @ * *- * /.!%+)2 ,Z)+- $V4:( : & 'p ) )+*- 1 %+ DlW@ MF!*-*. )2 )+*%2N%2Wl ! / *- . 1 , *1!3 !1 1!!'XBE .() %2*4p )2 /@ YMF %2.1!TC\N %*. )+* .!%+)2 hV 1*HN*- . 1!!%2* *1? %+11 1g .*.? %+% M 1!!)^Z )+TC $

σ+ ≈ 1.5 & 1!! $ -\g*] )+*)2* & s,C *.)Ê ? %+% )+* 1 s,C *.MF!*- $σ+ ≈ 1.2 & .()2 1 l JN%f !* % C11!)+- %2)+0M E )+WN** )+* 1!TC\N YV1*HN *- . 1 ? %+11 1#)+* .()2 1JE )2(

)+*l ! / *- .WMF %2.1!TC\N Z \gFN - g ! s .*4<X * . 6 )+! . 10 @ *- )21,f ! PC%+ 1 1!)2 :Z . /HE - JN ! )2)21! .()2#%2N%2Wl ! / *- . 1!!%2* * , *1!3 !10)Ê ?%+%4 / DMF %2.1!TC\N .!%+)2 - !)+*-

) # -H: 7$'& 3 &$' 29 =*$ .( k$2P)+*9=$28 7$(x-H< $d 7$&*

JE CG/. *1 )2.(*- Z s#)21 * .JE 1!)21!!)21 * *. .JE HN !)2*.*V* )+%.wMF!*- * * 1 )2.kE * )Ê .() %2* ? %+HG *( %+%21 .* V *)+% . 1 .*Z)+!%+)2 - JE )+!%+* FG * \f !- f * .JEV *)+% . 1 .*HN!)2*.* FG *- )+%2.s#)+. /l10-)+%+

Γ = 〈n vr〉θ,ϕ = Γ + Γ .$ :(+' &

)2! )4)21!0.()2? %+HG <C *HE )+!!%+*-/)+*? !)2.1!%+%+*-/- - * .(

〈f g 〉θ,ϕ =1

4π2

2π∫

0

2π∫

0

f g dθ dϕ =∑

m,n

fm,n(r, t) g∗m,n(r, t) .$ :(+7K &

1!)21!)21 V* )+%8)21!#- - * .(

Γ = 〈〈n〉t〈vr〉t〉θ,ϕ ,$ :(+JI &

s,C *. . HN!)2*.1V* )+% 1 .* )2 HN ! !)2**n′ = n − 〈n〉t

*. .* 1 s#)+*.()+-UN )+1 HN ! !)2**

v′= v − 〈v〉t *s#)+.

Γ = 〈n′v′r〉θ,ϕ .$ :(+5 &

HN!)2*.wV *)+%5s#)+. )2 %21 .P,.1 *.1 )* UC CN JZ)2 *s,C *. .- 1E , *13 !

Γ;s#)2 )21! )+* .()2 14V ! )+"CI2'"2, BI2 C WNW* - * **

s#)+. )2 )24E )+ .JEV3 1!!3 - Js C %+ *1 )2.(*-Z s#)21 * ? %+HN! !)2**

f ′*.

MF!*- *fs#)+. * * )+*rHN *( Ws#)2 . %+%2* - % 11*4Pl )+* )+*.!)+- * 1 )2.(*-

Z s#)21 *wMF!*- *.y? %+HN! !)2* !-)+ 1!)2 s )+! )+* .( .* -%2)+*.* , i p ).* ! * %2* )+* V )+%2.(*- :( : *.%+ 1/1)2 %21 HE .* - 1E *, *1!3 !HZ g-%+)2 . 1bZ)+!* PC*-)+- *V *)+%21 %21

〈Γ 〉t = 〈Γ〉t − 〈Γ〉t . 1!)2 $. .() %2 , *1!3 ! 1 . )2 @*. 1 s#)+*.()+-UN )+11!F!*- !-)++E HG1!)2 .1 1 !)2 *D_! %+* ( .()2 N!)2* .MF!*- 1!-FG * ( .()2 i(!* 678@-* %2.1!F!*-%Z \gFNg */% 11*4 )2\Z ! E * !* .()2V )+%2.(*- :(qc 4 )2 .1wr )+\E )+!%f * .JE - 1E * \E 1\@ )2 1!F!*- 13% - JZ- * s#. *.%+0 1 1)2 ) .()2 1JE % HE )+*3!)+*HZ )+3)2%+%4 i 3 !)Ê * %+%+Z -2C*-%+)2 , !%2*HZJ@-FG 67* \N **

X * 0)2 !*- -FG q@ "$!YN** )+* .!%+)2 %V1*HN *-/.)2 HN@

! !)2** 5s .*4 " C *. .()2bV1*HN *- )Ê p )2 .$q=3 & @ 1!)+!)2* g *1 )2 .(%+)2 MF %2.1!TC\N * -%2)2 1!\N )21!1 1 )+* .

V( %+% ).*/-FG*q@ "$!* . 1 *- )21 .!%+)2 E )+ .JE .!!)+- *

V*1!!)2- . DlW@ MF %2.1TC\N N !%+)2!Z 1)+*4

3,&$ 0 7$43 8-/ 97:;)<= k($8<)+*9=$28 7$ x -H0 $2 7B$'&* +

l )+* Bs 1 *.4M)2 s )21wg .1_! %+* . 1 !!%2* *~? %+11 1 *.$. )2 a@HN! !)2** - Js#)+** E *)+*.JE E * .()2 3 %2)2.%2* %2 .()2 1 FG * ! $ 1!)2 V 4:(Le & )2ys#.g .()2 .() %2 *. )2.%2QS.()+* - JE )+!%+ 17s#)2g 0.()2%r )+X *V )+%2.(*- :(Le \N **E * .*

(cos2(θ))@!_!% ( . 1 !!%2** , *1!3 !1 ( - @

!- *%g .* 3 %2)2.%2* " )+*HN % )2N EhE - *HZ N % .()2,l )+- *1 )+* 1 %+%2*)+*-@)+*1!)+%2* % 1\E 1YZ HE _1 )+*8)+*$.JE .()2hS \gHEhE *- 1)+*1!)+%+)+TC*s %2 ( .CHG * , !11)+ -FG 1!)+*. * .* l!% C!*- * . 1Z s )+*hS 3)+%21 .!8g )+** / * , *1!3 !01!- * %210( . , !1!)+***1)+6 )+ .* - Js C %+* % 1\E 3 E @!* $ 1!)2 , %+%2:(+ & *.5E * 1!)2 /- JE CG . 1 ν @ "$! 1 )Ê 1!FG)+-UN )+1! )2 . 10 % 1\E 1 $p %+%2*)+*-@ -)Ê &

0

0.5

1

1.5

2

2.5

3

−3 −2 −1 0 1 2 3

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

〈Γ 〉

θ

〈n2〉1/2

θ 0

1

2

3

4

5

−3 −2 −1 0 1 2 3

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

〈Γ 〉θ

〈n2〉1/2

θ

++ !"%+ #$ /'"%)/') D; $'$')$ )" "%-$ 7 &'(5/0 </' /0+ 3 )$';"0+ )$0/0)()$ ! <5/0)6

θ 7 &'(5/0

. )Ê ?%+% / DlW@ MF!*- 1!)2 DE * )2DN )+* _TC*.!*- . 1 , *13 !1s,C *. )Ê ? %+% 1 s,C MF!*- )+* %2)2 WV*1!!)2-hf * ( 9 )Ê *Qp )2 . 10 % 1\E !1 $ 3 1)+!)^f

θ@ "$! & Ws .*N**4V )Ê 3 %2)2.%2* % . \E 1R@ )2 HN ! !)2** $V4:(Le & % CG 1!)2 .Yp#%+%2@

*)+*- N \N ***4 .()2 %2)2 V1FEhE !!)2 . 1/%21 )Ê *- 1!F!* ? %+%$ 6 8i()Ê HE )21! Z * * , !11)+ )+* f 1 * & 1!)2 E * * * )Ê_!-%2)2 Z .*$- 1!F!*~?C%+%2* $ )2./1\EhE !!)21 HE

θ = 0 & Z HE )+** .()2 .!%+)2 4V 1*HN *-. )2 HN! !)2** )Ê ** , !1! )2 s,C *. )Ê %2)2 - 1!F!* ?%+% )+*l ! / *-Z $ * )21!4)2 1 1 %^f ! %+* N** .1 %2)2 1\'EhE !!)21 %^f ! %+*7 )Ê !!%2**? %+HG \N % C*4PX *1!- 1E 1!)2 E * )2 .G)+* / DlW@ MF!*-/.()2%MTC\N . )2 HN ! !)2**f !!)+*- ! 1 .() %2 % .%2JN !!)21 *7 * !) % HN ! !)2* )21!0)+*p )+%2. :(LOkZ 1 *4 )2 R@N ** E *7)+**7.!%+)2 *V*1!!)2- .? %+HN ! !)2** ) )+*- 1 %+yMF!*- )2 1- E )+ )+*l ! / *-. .() %2* 1 2s#)+*.()+-UN )+1 HN! !)2** )+* $V 4:(LO & 8X *1\@ 1*. )21! ) ..() %2* 1 2s#)+*.()+-UN )+1\N E 3 ** )+* .!%+)2 kV1*HN *-/)+*. 6 )+! . 1 *- )21 g .* *- 1!F!*Q? %+% ! .()2 ) )+*- 1 %+MF!*- *PC !*- 1Fs )21 ( - *s#)+. +r1EhE * E )+ .* )2 HN! !)2*13%2*E .()2 10.1 _! %+* . 10!!%2* *? %+11 1#)+*Qp )+%2. :( : 3% 1)+ %

< # -H: 7$'& 3 &$' 29 =*$ .( k$2P)+*9=$28 7$(x-H< $d 7$&*

0

5

10

15

20

25

30

3 3.25

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75

〈φ2〉1/2

q

〈v2

r〉1/2

q 0

0.2

0.4

0.6

0.8

1

3 3.25

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75

〈φ2〉1/2

q

〈v2

r〉1/2

q

;"0+ )"!"%-$ 7 +/0)5/0 </'/'+)> ;"0+ )$ "%-$ 7 )"%/0)$)""0 )6:

×A 7 , )$'& ( <) /0$'<+ +)5/0

X *1!- 1 E E .()2 .() %2 1 s#)+*.()+-UN )+1\N E 3 ** 7* )+**N %2)+**.V* )+%.l × p @ 1 s#)+*.()+-UN )+1 HN ! !)2**71s#)2)+*4V)+%2.(*- :(+9 g .* *- 1!F!*? %+%Z \N *** )21!("$- * |vθ|> |vr|

*. .*V1.(\gFN * $ :(qc & -)+%+ 1/ky > 1/kx

0

1

2

3

4

5

32.75 3.25

〈v2 〉1/2

〈v2

θ〉1/2

〈v2

r〉1/2

q

〈v2 〉1/2

q 0

1

2

3

4

5

3

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

3.252.75

〈v2 〉1/2

〈v2

θ〉1/2

〈v2

r〉1/2

q

〈v2 〉1/2

q

, ")*#"%)& ()" "' ) ++# &'(5/0$0<+ + #/0# #" :×A 7

, )$'& ( <) / ( #" "' )"%-$ 7 ;"0+ )$0/"%/0) D ;"0+ )""0-$ 7:×A 7 , )$0& ( <# /#

1 _! %+* . l × p5@ 1 s#)+*.()+-UN )+1 HN! !)2** )21! )+* 1! ] )+*)2.( .()2 3 %2)2.%20)2 !*- 1\N E 3 ** 1!!)ÊhE 8X * . *1 %+)2 *%_R@1!%+%+*- f *7!)2*.*1N %2)+*1\N%+)+- *7" )+! %+* $l ..()2 1 & E *7 1 %214E )+1!\N )+* 3 %2)2.%20)2 !*- %2*-)2!* " )+! %+*Z !*4X *p )+%2.7:(+9$\N ** E * .()2 " )+\N *- . DlW@ MF!*- ( .1 % .- 1E * l × p @B? %+HN! !)2**4V * .* 1!)+!)2**/. 1* * *4? %+HG <C *s )Ê *- 1!F!*? %+% .()21l × p @ 1 2s#)+*.()+-UN )+ $ )+*1!4.()2 3 %2)2.%2 @1 2s#)+*.()+-UN )+ & )+* %2N%2 1 6 )+*)Ê HE (s )21!*. )PMF!*- )+*#.!%+)2 l ! / *-/.? %+HN ! !)2** 1!!

l )+* s )+ )+* 11*YV13 JN .%M)Ê % !)2*1!- *)211 )21! .1D_! %+* . 1 1)j@1!!)21 *]1!*- 1*)+%21 )4)+*- 1 %+DlW@ MF!*-p ! hE * .()2 .() %2* %2 . 1 - 1E * N *2f JN !)^f * , *1!3 !1 )+* V )+%2.(*-:(+ *. f !-%2)2 0.()2 1 E )+ .* HN!)2*.*V* )+%2* )+*wV4:( :1 1!)2 ,E * .G. N *2f JN !)^f , *13 !0)+* .* - 1!F!*?C%+%2* * .* WC*.!* . 10 *- )210* *Bs 1-FG 0)21!

3,&$ 0 7$43 8-/ 97:;)<= k($8<)+*9=$28 7$ x -H0 $2 7B$'&* Pt

0

0.5

1

1.5

2

2.5

3

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

3.252.75

〈Γ〉

q

〈n2〉1/2

q 0

1

2

3

4

5

3 3.25

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75

〈Γ〉q

〈n2〉1/2

q

!"%+ 1 2" ) #$' /0) <+*)<5/0 *) D 3 1 2" "0-$ 7 & (/')$%/ "%) 3

A $.!%+)2 Z)+- 1!)2 )+*1V 4:(+ )Ê - *1 \ZkZ V 4:(qc 4)+*hl! / *- .)2 3%2 .( .()2/l @ MF!*-$* .* C*.!* - - *2g .JE *- 1!F!* ? %+% )2 1 )21\N 3 *HZyZ s#)21 *$.*1? %+HN ! !)2*1!3%2*$*.7.*~MF!*- 1!3%2*$)21- JE CG.kp ! !*- * )+* V 1 *)+!/:(++ .( .* 1!)21!!)21 *V* )+%0 .()+*-X * .V )+%2.(*- :(+')21 g .*k? %+% . * DlW@ MF!*- .1 % . 1)2 E 3%+)+!.*a@ .(1 $ 1 s \ZD] )+*)2 & ( - 1 %^g11%+ )+* .* HN !)2*.* $ % & *. .* 1!)21!!)j@1 *V *)+% $ & ( - !- *4 )2 - 1!!!)2 %+y] )+*)2 .()2* %21 *HZ%Z HE *- 1!F!*)2 3% XE *- 1!F!*? %+%8)21. 1!)21!!)21 YV* )+% 3N !)21 *)2 Df ! *.*4!%+)2 Z \N ***/)21!.()2 l ! / *- . 1 1)21!!)21 *wV *)+%21 )+*kZ .*/WC*.!*/. 1 *- )21 )2 1 )21!f !*.* E )+)+*JE %2N%2* 6 )+*)Ê HE * .$

q=3)@ 1!)+!)2*4'V *

.* 678i )Ê %21%+%2*mE . 1!)21!!)21 yV *)+% s K9 .- 1E * .(!)2!*)2 1!F!*- 1

0

2

4

6

8

10

12

14

16

18

3 3.25

turbulentquasistatisch

σ = 0

gesamt

2.75

σ− = 1.2 σ+ = 1.5

〈n2〉

q

〈φ2〉q

0

100

200

300

400

500

600

700

800

3 3.25

gesamtturbulent

quasistatischσ = 0

2.75

σ− = 1.2 σ+ = 1.5〈n2〉q

〈φ2〉

q

# G )"%)& ( )$ </' #"%)) C $0$0/'/'$0& () 5/0)$; #" &'(5/0)$0/"0 3 )" +/')/' $%/ "% @( H)" "0 ) /' )C"'/ "%+ B1 2" )D - />(+()"5/ " 713)$%/ "%<)$0/0"%& () /' >$%/') / 0 )$B))$%/ "9/') D " 5-) <) C5 $0$0/'/'$0& ()" /')>*+"%( # $0/#

p ! DE * ** .1 *!) %+3% )+*V 4:(+' 1 !C%+%+( .G/. 1)21!!)21 V *)+%.!s 1* !%+)2 - !)+*- )21!Bs * 58@9 . 1 - 1E * * !) % E 3%+)+!.* 8@.(1 "$)+! )+* % CG 1!)2 1N )+* .() %2kV PC*-)+-UN )+ HZ s _ !) !)2* . 1 1!)21!)21 *V *)+%21 \N ***4V )2 )21;N )+* 1!)21!!)21 V* )+%)Ê *- 1!F!* ? %+%Pf ! *.*s 1.()2_\E !*- * %2- .G .()2 1!)21!!)21 *V* )+%2 . 1 E !% 1!*- . 1 )2 ! * , !%2*HZ E .%+%21\ 1JE 10%+%2)+*)+- .( .1 )+*- 1 %+DlW@ MF %2. R@f !- !( *Qs .*4

# -H: 7$'& 3 &$' 29 =*$ .( k$2P)+*9=$28 7$(x-H< $d 7$&*

A .* )21! !)+- *Qp ! !*- * )21! .1 _! %+* . 1 .() %2* , *13 !R@3%21 )+MF!*- 1 1!\N .( .1,_! %+* . 1 1!)21!)21 *V* )+%21. )2 1!F!*- 1!!)ÊhE )+*- - * 1 )+* .()2 N !)2*/. 1 , *1!3 !1 )+*. AC . $

q=3)@ 1)+!)2*f !* HE %+)2 .( .1,_! %+* .# 1 s#)+*a@

.()+-UN )+11!F!*- * )+*HGZ 1)+*4

p %+)2TN E * (.()2 Z)+!%+)2 *_!% C( . 1hN *f JN!)^f * , *1!3 !1 *. . 1wMF!*- 1!*)j@f 1 . )2 Z \gFN $V4:( 5 & 1 )21!DN )+* )+*.!)+- 1V*1!)+- * .V 1!)+*HN * .A )^f 1YZ $1 *4 )2 1 1!!)ÊhE YE )+ .*$ %2* 1V4:(+yg )+*4. ).*~r )+R@f !% C( *~f %+HE *- JE )+!%+MF!*- 1!*)^f 1 )+*HN %+1!)^f /. 1 1!!)21 *~V* )+%21 ! s .*4 . %2)+# )20.()2?- DZ N% C*as !HE .()2DZ)+!%+)2 _ !) !)2* . 1 N *f JN2@!)^f *y? %+11 18*-%2)2 1TC\N )21!1 %2. .()2 i(!* 6 -* %2.1!F!*- )+*- 1 %+s#)+. )2 )2f !- 1!%+%+* M)Ê % !)2*1!- *)21101%+%2* * * )+* .* *PC 1*kV 1 *)+!* !)2 R@- *. * %^ 1)2!bs .*4 )+s#)+. .1r1EhE *1!3)2%8.0)+*HZ%+** !!%2** @Z 11 *. )+ N !)2* (0)+* i(!* 67-* %2.1!F!*- 1!!.()2!8" )2/1 * )+*Va@1 *)+!#:(+ \s C * ,s#. )21! 1;Ek-%+)2 )Ê Wp 6K . f 1 )2 .* , \E . 1 * R@1! * 6 .%+%21\ 1JE 18*a@*. 1\Z 1 %+*1.G )+* .JZ )2.()2!,p ! !*- )+*HZ%+* Z 11Ek-%+)2 )21!)2 1s#. g0.()2 %+- *.*l!% C!*- * - *\Z X * *\N3)+%8:( :Ys .* .** .()21 * ( - JZ=C %+*mM)Ê % !)2*1!- *)211 * p @\gFN(1!)2 !)+-*- . V1 *)+! :(LK )21:( 5 \N % C!0*.Z 1 EhE *- G

(* (' " + !" #%$ &)(

88

7

1

n, v

∂r n0(r)

v00 (r)

Γr = 〈n vr〉

B

! 0"# ( # ) " ) ( # + ,1' 3$ 4H) ' ( * 〈n〉t

! 0"# ( # ) " ) ( # + ,1' 3$ 4H) ' ( * 〈v〉t

#&'($') "0<5$0$'& ( # 1 2" )$')() / )" : 7 5/"0 )>1 #$%/'); $'(/0#"%"%> "%<5 )" )5/0#"%) /%13)$%/ "% 1D </0 /'+ # ) C $0$0/'/'$0& ()5/0) "0 )/0"%# 5/ /0"'&'(5/0 /G

N ! f " f "$#%" '&(*)+" , -+./ , 0-1324" '+65 , + 6 -78+ 29"& *:, 17; 31 , (*)<" '+ >= , &?-7! , 17@1+BAC(D)+*"&'EF , G17@HI") , , ,CJK, L** = 3 '7 M #N1 ,= 1&" O%&'1 1P +" 3Q 1P* *(*)+R> " '&>1+S3 >>T+(* , - +"U ,WV ' , 7 , 1+3. , -X+&Y* '+

! "#$&%')(*,+-%'./10# 2 43

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a` M 3 A , 17bB *:, 2 "& -132%"U# , -+. , 7!"HI" ' = E4"&(*)+Y (D)U31 , (D) " '+3 , " 6 G3 X+8 UJ &?H -) ' , 7 , 1+&I9653.&' 'L 31 , (*) 3 ;65 , +[B *:, 17 M , 78 =M G +7!D-H : +" (*)3 7 , *(*) "+" A , 17!*H : 78&' (*)!" " " '&'EW" * 7 , D , !" G , = , -1+(D) 7!"7!"+*" '7!H./"+ , " MO%T , + ) ,= , (*) cBL"P , EF1 , " L1H *.#D , "d5 , 78&" (D) H "e? , 7! = **"-1+ R = D(*) 93MLK 3 J , -HI , -1+>#% = "&'& 93M<;\7 , E4"+ M

gfhgfji k@lnmNopl1qjr#s)t$r \qjl t>l1uvt>wjxsnr4s6 " '+"H 29 " V ' , 7 , 1+ 29")& 1 3 ; = '&' * , "+UA , G17" '+I = R (*)+"+" (D) ") ' , 7 , 1+3. , -X+& *EF 3 , " '+* L"+" . & +-&"QB ,*: H 17yI9R = *(*) 93M<;8M<;zM6M 2 T) , 'H17! *:, "SOP-&'&PL1 " '+ , ) : )+ , " 1 ,= 1&" "UR 7 D 1+ +-H L1 ) : )+EA, "0O &'1 1P + 7 -1+ 'HU5 , 78&" (*)L1"y? , 7! = **"I R = *(*) 93MLK H 7 , ? = &"HV ' , 7 , 1+ M , !" H - 1 ' , R == '&317 93M<;| -)P-+ , 7 ,*: P! , "~ -)&"SAEW , S29T , 3 , 3 ?-&" JK, -X+&; 1 ,= 1&" " O &'1+** 1+ , 7 -1+2 T , 3 UL" &' (*)+?K EF (*G&'17S , (D) *:, 17 M

0

5

10

15

20

25

3 3.25

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75

〈Γ 〉

q

〈n2〉1/2

V

τ 9

10

11

12

13

14

15

16

17

2400 2600 2800 3000 3200 3400

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

〈Γ 〉q

〈n2〉1/2

V

τ

v|Eaa|||pEa|>4ELaO!ag|E|Eajj8 aEj¡j aa¢^£|g¤<¥4a¢¦§a|EJ|¨va¢^£ª©«¬E|¬bzz®¯&L¯n£|aO¬E|

` " )?K '+*(*)P-& " , @? ACB *:, 17 * ") H - " '+ *") , , ! R = *"G17 1 ,= 1&" "O%&'1+D* MV '7!"7!" = = (*) H - -H L" &' (*)+"c5 , &?-132 , (*) *:, 17 'H 6 "&N" ' 'HHI , 7 ,*: P! , EW , "+ 7 -1 = " ? , ) : ) 17U , B *:, 29"& D , ! MG , 29&'7!"+" R == '&317 93M<;° '+ 3 , 3 ?-&" JK, -X+& , 1 ,= 1&" " (*) EA 1+

2# %T/ b +-)g .! zg :#0g.z %T4)N (

J " ?-&_R+1 1P +" + , 7! "&'& MV , ' = " "UOD-&'&";" '+ "1 &' (*)+ R = *"G17 ,, HIA AC , 0L1 , !"+" = E4"&(*)+3 R = D"17U 7! *:, " 1 ,= 1&" "$O &'1+** ' ` '&93M<;| = "7 , T+" M

0

2

4

6

8

10

12

14

16

3 3.25

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75

〈n2〉1/2

q

〈φ2〉1/2

q 0

10

20

30

40

50

60

3 3.25

σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75

σ = 0

〈n2〉1/2

q

〈φ2〉1/2

q

v|Eaa|||EaE|#|La|av|v¯Ea |Ej§!EC|'¨va¢^£!E¦ !Ca¬_ E|n|'NE©L ||¦ !Ca¬®!ª

gfhgf [lnu$wt>uvtNw! #"1r'm%$s)t>r#wjq (D) "H = )+ , 7!D-7 " &ODP * (*) 1 , R>+ "7 , (D) *:, 17! 7 -1+ ' R == '&_A317 93M<;|8= 1 , H " '+"H "1 &' (*)+" R+ "7 Q 1P *(D)+"$R " '&Y '5 ,= '+317=, '7!" M G H 2 &'7!"+" ` '& 93M<;9 , !" H -I3 *""1 &' (*)+" R+ "7Y Q 1P* *(*)+"R> " '& M` *+ , -17!"+*(D)+" '&' (*)$ 3 , 3 ?-&R = )#D-78 '78!" YR+ "7! 2 T , 3 = " _A" B *:, 2 "& D , !"'& ( > D , ! , +pB *:, 2 "&I '

rA (D) 17 EF , I 7 78& M) ) , '! 7 A J , -HI ,§ME=

7 ,*: P! , -1+(D) , R+ "7 Q 1P D(*)+" RH.&' 1+" E4 , M

0

100

200

300

400

500

600

3 3.25

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75

〈n2〉

q

〈φ 2〉q

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

3 3.25

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75

〈n2〉q

〈φ 2〉

q

v|Eaa|||8EajE*jO¬aa¢^£Z|+ !aCa¢^£| 4 aaJ|n¨va¢^£O«¬O||,®g||Z|nN L ©«¬E| 4a¯||¬O«¬|| ¬«¬O8 a_¡z|¯ jO-*¬aa¢^£ ¯ v|Êa.¥TEj*|av¯|a ||/ !|C|¬¬

? *" -7!"HI , = +P 'H10 "7!"+D LIL1C" ? , 7! = **"$H -&'&"CAC(*)+*"&'EF , G17!. , AL**" I32> , -. "& 93MLK 1+ 93ML9NM 1 -1+(D); 'H 17! *:, " OP-&'&54"EW" '&F" ' Q 1P* *(*)+ ,R> " '& = " (D) 1+ J " ?-& 7 , )P-+"U M ` " 7 , ? = &"H V ' , 7 , 1+UEF , U ,Q 1P* _A *(*)+yW : *17!D- " '& )P-1. §D(*)&' (*)C31 , (D)C3 V ' , 7 , 1+3!H. +" " I

(0,0)A 6 G" M

7! = '& = *-2 , !" '+F@? ACB *:, 17 " '7!D(*)P-& M G 03 * , B 'H 1&? = '0 , 3 (0,0)

AO+1 , , !H./+" "&' '+ , = 7!'D-H.32 E4 , " I9R = *(*) @93M<;8M<;zME= EF , 1 ,Q 1P A *(*)+W : 17!D- " '& 31 , (*) 3

(m,n)A 6 " 7! = '& = 1+ LE4 ,:7 , +")H&' (D) 31 , (D) 3

7 , $P!-&' '7!" R> " '&[In= 4

1+n= 8

ME= EF 0 'CR == '&317 93M<; VSL1C*")+" M Y-)+ , E4" ,4Q 1P *(D)+ ` " , -7 'S" JK, -X+&" I9R ==M93M<;9NM -1+(D)U3 >. , D-78P- , 3 ?-&\B , 1 1 ,

! "#$&%')(*,+-%'./10# 25

-132 M (!G(*); 2 T , 3 ) , 7!629T) , Y 1+D U 1 , , Q 1P-&' 7 \5 , 78&" (*) LEF *(*)+" 7!EA *:, "H 1+;17! *:, "H OP-&'& 7 ` "&?-7 M

1e−05

1e−04

0.001

0.01

0.1

1

10

100

10 100

σ = 0

〈|nmn|2〉

m

〈|φmn|2〉m

1e−06

1e−05

1e−04

0.001

0.01

0.1

1

10

100

1000

10 100

σ = 0

〈|nmn|2〉m

〈|φmn|2〉

m

v|Eaa|||-Ea§J ||z¦O|¯ |Ca¢^£| 4 aa¬¨va¢^£ª©#||*>LaO«¬O||ba¯EE©¬«¬O a_ #a4¥TO|b|a%4¯|a ||/ !|C|j4|az¬a¬a|aJ |j£|

mapa¬ £E©

¯a¢^£Ej4E ¬||!O¬a

'T ),84#g).

O+&'7 H -0" R1+29T) , 17!" ' #=°j= * &ODP (D) 3 43 , " A , G17 , @? ACB *:, 17 '" '+"H -+*(D)P-1&' (*)+" ` '& 7 , ")+"'&- 'H -)HI" 6 G"&'& = , (*) " 7 , $P!*-&' '7!"B , 1 1 , " I &" '+ 6 G"L -)&" ME= EW"&(*)+U (D) -& 7 ,*: P! , I&8-& ? × ` A 0 D(*) EF '+3 '78!" ¬AEF ,= "& I5 , 'LIG , ? 3 M = "7 , " 2 " &?**" = 3 U3 ]> 7 " , #%" '&(*)+" )+ ,¬7 , A, 1329" = EW , "P(D)C" = *)+ , '7!" ` , (D) 17!" H 2 , " 'EW , "+ , O &'1 1P +*"+ , 78 , &' '+ , " G + = '&' D "S7!. " M * '+ 1 , " = , H , L17 , 1+"&' "7!"+" 6 A78+ 29"& . &78 7 , T.32 = M ) M '+ " . , (*)+"+S ') , , 6 G"L -)&";- ,WJ 8 , 4"EW" '&' '7!" O &'1TP$R4D(D)+ 'H 17! *:, " OP-&'& &-&' ,M B (D)P , 17 , 6 A78+ 29"&3&' ' " 29T) , L1 " '+"H R1+*" 'P-+ , &?-132 " 7 7 , )+ , "7 = "P(*) = , " O+"&3&' ' "I :, &' (*)S! ,, "&' ,§M6M )&' (*) EF >" '+0Y*(*)EF '+3 '78!" 7 , *(*)+ , 17:? × ` A A ,= "& "&78 , " , 7 , -132 =, (*)+" - = "&'& H - (D) 1 +SR>132 =, (*)+" 1+ R1+*" 'P-+ ,, " <P!" , 7 , $P!*-&' '7!" A ,= "& , 1 1 , "C31 , (*)C3 -1+*" 'P-+ , &?-132 "+" 6 -78+ 29"&3&' ' " 7 ,=EWG31 , (*)$+"1+I&" '+8-&' '7! , B , 1 1 , " " ")+" M ? X++ -&*;" '+ 2H 7 , " '&'17 7 O%&'1 1P +*"+ , 78 7 7 , $P!",B-&" ) ' L1 &" '+" B8-&" M * 2H 7 , " '&'17EF1 , S653. , 'HI" "&'& = " # ? B2 J >R -)P-+ , (*) ER+1 1P +" P(D)7!"EF *" MV , ! I" '+R = D"17 , (*) -H.&' 1+" = " K7 , $P!"$*" , (*) "CAC"&'&"L -)&"= = (D) E4 , " 9>9 M

R = D(*)&' |P!"+ &ODP (D) 2 )P-& " = +P " '+-? , 7!G3 , 17Y , 6 -78+ 29"& , 1A 1 , " '+IR = *"G17 , 1 ,= 1&" " O &'1 1P +"$L1 , O+&'7! )P I J 2 " '&pVS 'R ==M93M<;FNME= E4"&(*)+ H " '+"H R>+ "7S Q 1P* *(*)+" R " '& , (*) EA1+ J " ?-& *:, 17 7 ,= 1+" I J 2 " '&Y M6M G +7!D-H X++ H - !" '+"R>+ "7 , R = 2 -&'& 7!D-H "C! 7 " 7 " # , -+. , = EJD-) , "+C , 1 , A= 1&" R> " '&N # , -+*./ , = ' M

2 %T/ b +-)g .! zg :#0g.z %T4)N (

"!#$

KR17!"HI , 03 *2> , -. "& , (*) * (*) -132/3 , 4 '+3 , " A , G17Y , 8? *AB *:, 17 -132+" # , -+*./ ,M D = ") -1+N , R = R (D) 17 (*) ") ' , 7 , 1+ 31 , (*)3 Y8? *ACB *:, 17S-&'&" ';1+ = "EF , " '+L1+§D L&' (*)+ B = '&' , 17 , #1 ,= 1&"L M 1 , (*)" '+" Q 1P- 7 " 5 , 78&" (*)H " 1& " & L " 2 , 8-. "&Y7!"EF ' H -3 Z? , !" = +P 3 R = R (*)17U (*) ". , -X+& -&'&" ' 31 , (*) 3 @? ACB *:, 17 (D) ,>V -1. HI(D)P- H 1+29T , + R = '!" , (*) ER+1 1P +" M

% 041 z))2>H 3 L1 7 , ")+" = 0+ EW"+3 '7 , +"1 3 ?K EF (D&'17!78&" (D) 17 I 93ML°NM 29T , " (D) ") ' , 7 , 1+BL1 = , (*) " Mb` " YR EWD")+" " '+B *:, 2 "&UEF , 3 * 1H "#N , H

⟨χ‖ ∇2

‖(−κx+ n)⟩y,z

I 93M<;9NM

HI3 X+L ,= EW"&(*)+ , = " N , O &'1TP$R4D(*)+"H "&'17; ,6 G"&'&'78&" (*) 17I 93M<;zM " ") M 65 , + 6 -78+ 29"& *:, 177!") T = , 3 J **G&?-HHI , ';" J , -&'&"&? = &" 17!*. , A , H " ' I9 ")+ 6X+ I F3MLF'&NME= ] -. "& FNM &

∇‖ = ∂z +C(x) ∂y −ψ, ·,H , R = T , L17

C(x) :=

(κzκy q0

− x

).

-H 29&'7 29T , " #% , H 93M<;9 3 "&? ⟨χ‖ ∇2

‖(−κx+ n)⟩y,z

=I 93M<; V M

χ‖

⟨∂z[∇‖(−κx+ n)

]+ C(x) ∂y

[∇‖(−κx+ n)

)] −ψ,(∇‖(−κx+ n)

)⟩y,z

.

A D-) , "+ = " K , G "7 , $T = , 3 0O &'1TP$R4D(*)+I3 0 , " = " "#N , HI 7 , *(D) EF '+" =78 '& 2 T , " R>1+* , 1+(DUH , B *:, 17

χ‖

⟨−ψ,(∇‖(−κx+ n)

)⟩y,z

I 93M<; Y M

= −χ‖

⟨∂x ψ ∂y

[∇‖(−κx+ n)

]⟩

y,z+ χ‖

⟨∂y ψ ∂x

[∇‖(−κx+ n)

]⟩

y,z

= −χ‖

⟨∂x ψ ∂y

[∇‖(−κx+ n)

]⟩

y,z+ χ‖ ∂x

⟨∂y ψ∇‖(−κx+ n)

⟩

y,z

−χ‖

⟨∂x

(∂y ψ

)∇‖(−κx+ n)

⟩y,z

= −χ‖

⟨∂y

[∂x ψ∇‖(−κx+ n)

]⟩y,z

+ χ‖ ∂x

⟨∂y ψ∇‖(−κx+ n)

⟩y,z

= ∂x χ‖

⟨∂y ψ∇‖(−κx+ n)

⟩y,z

.

-H , )#D-& H -U29T , 3 L" &' (*)+ ?K EF (*G&'17U> (D) ") ' , 7 , 1+

∂t n00 = −∂x (〈n vx〉y,z − χ⊥ ∂x n00 + ΓδB)I 93M<;)(NM

8! v "4 ) v 25

H , R = GT , L17

ΓδB := −χ‖

⟨∂y ψ∇‖(−κx+ n)

⟩

y,z.

I 93MLK'&NM

G EF "EW" * (*)C3 * , *:, 17! = 3 '7 #N" '&(*)+"SR+1TPΓδB

= EW"&(D)+ , 31 , (*) 3 , 3 ?-&:]>HyA./+" S 7! *:, " 6 -78+ 29"&0" ") = -1323 ?K EF (*G&'17 ,

(0,0)A (D) "!HyA

./+" -1+EF , = - = " " '+ , -P-& *(D)+" 2 , 1+(D) 17 1 , H ,b7 , " '32 (*)+"+"R>P-)HI = +P>+ + , H , pB *:, 2 "&3. " ?-&

ψ-1+**(D)&' |P8&' (*) H ,

(0,0)A (*) "!H. A

+" EW(D)+*"&'EF , = = 7!*(*)#D L E4 , " I9R>)P-7? M<;zM6M O T , " '+ 7 &'& D-+3 '7! ` , (D) 17 , AC(*)+*"&'EF , G17 3 Y 1HI , *(D)+ R1+EW , 17 LEF '7!"+ M

A D-) , "+ , *:, 17! = 3 '7 = .P , -&'&"&U#%" '&(*)+"SR+1TPΓδB

' 0 &" (D) 17,I 93M<;)(NM 'H 17!EA *:, " OP-&'& (*) p7 , )P-+"S I 7 78& M R = *(D) J93M<;8M<;zME= EF , , = " (*)17!" H 29 "HJ &?H -) ' , 7 , 1+ 1+ = " /R E4*")+" B *:, 29"&4 'H ` 6cF A ) G>T = &' (D)+ , EW" * 'SA , 1&'& 7!* L = D EF Y-1+(*) = " " 7 , 7! "&'& "[? , 7! = *D" & L " R = *(*) 93MLF , OP-&'&E ,M

gfnfji wjm l1s ulnu w #")t>r "-wjs)t>r'm ml-s1mNxJq? KEW , " 1 4G(*) (*) 17!" = , (D) = 'I"+" " ' (*) ") ' , 7 , 1+ = , T+(*3 (*)SA '7 EF , = , 1 , , 5 , ? 31 , (*) " *:, 17! = 3 '7 " O%&'1TP

ΓδB

, @? ACB *:, 171 , &' "7 MGV , 29T , H 1TP , = " = . , G(*)+"+ I

ΓδB

M A #N , H. , 7 , -HH ' , S" '7!D(*)P-& EW , " M

5

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n, v

) %"*' + ",*+-. ' / 0" !1 (32 -∂r n0(r)

v00 (r)

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Γr = 〈n vr〉

- 6%+ ' - / 0+-.,78.:9 ; <3 = ! *

B

>&?&@!A B A C @6C B A D E1' FGIH C ' B J 〈n〉t

>&?&@+A B A C @6C B A D E1' FGIH C ' B J 〈v〉t

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QSRT AUR , G1WV7 WT , Q@?QACBYX :, 1WV7[Z-132 WT\V]Q^R(*)YX_T")WRVOX_T , 7 , 1WV+ = E`T"&(*)aTRV&BG(D)aT"HbZ 93M<; Y 31 , (*)WT\V J 2cT\R'& ° , T". , D_T\VOX,RT , X EdR , = efVWVYX_TY31 , (*) gRT_TFOZ-&'& = T\R.WRT"&hR_e&RT , XF1WVYX_T , 1+(*)YX EiT , WT\V MQSRTkjST(*)WV 1WV7 T\VEF1 , WT\VEdRTWT , HlRXYWT\V]RVgRT_T"H ]mZ-.WRX_T"&KT = &R(*)aT\V J Z , Z-HnT\X_T , V$WT , #oZA= T"&'&T 93M<; 1WV+ WT\V We , XpZV7 T"7 T = T\VaT\VqQ@?QACBYX :, 2cT"&,X D , T\V 31 , (*)7 T629T) , X MGMV R == R'&31WV7 93M<;)( ZrT , T\VWVOX HbZV WT\V sT\RX 7 T , &Z-1324WT ,, HIAMQ^R(*)YX_T,X :, 1WV7 & ` T\R)?tRV+D(*)Z-&X_T\VWT , BYX :, 1WV7 uZVWVHbZVC)WRT , R'H OZ-&'& WT ,XvZ , T\VdBYX :, 2wT"&WTmT\RVaTIWT"1WX&R(*)aT R = vT\V1WV7 WT

2#2 %T/ b +-)g .! zg :#0g.z %T4)N (

QSR(D)OX_T\VWR 7 T Z-1+^T , T\VWVaT\V M ]ZV = T Z(D)OX_T = aZP)WRT , EdRTWT , 1H WT , R+1 X1WRT , T\V+WTY1WV+ Z-1+(*) WT ,Q 1Z,R_XvZX,R*(*)aT RpVYX_T\R'& R'HHBYX :, 1WV7!,VWR 7 T Z-1]T\VOX)Z-&X_T\V _RV+ M

9

10

11

12

13

14

15

16

2000 2500 3000 3500 4000

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

〈n2〉1/2

V

τ

〈n0〉q

10

15

20

25

30

35

40

45

3

σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75 3.25

σ = 0

〈n2〉1/2

V

τ

〈n0〉

q

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ΓδB

WT , Z , X^Z-1+EdR , X = aZP T\RV R = R Z(D)aT\V WT^Q^R(*)YX_T")WRVOX_T , A7 , 1WV+kR'H ` T , T\R(D) = T_efV+WT , 0,XvZ , T , BYX :, 2wT"&,X D , T;,XvZX,XXaV+WT\X M Q^RT )ZX0L1 , Oe&'7 T = aZP,R(*)CgRTkX1 ,= 1&T\VYX_T\V O &'1YX1ZX,RefVaT\VCWT , Q^R(D)OX_T1WV+WT J efX_T\VOX,RZ-& 2 T , 7 , e$P T

qA A T , X_T = = T\R

WT\VaT\V WT , ) )WR , R' e 7 A J Z , Z-HnT\X_T , RV T\X E`Z K'& 7 ,*: P T , R_XZ-&R'H ` T , T\R(D) G&T\RVaT , T ,qA A T , X_T =

WT"1WX&R(*)Z = _T\V T\V M Q^RT R,X ZV WT\V , ZgRZ-&T\V JK, e-X+&T\V WT , R == R'&31WV7 93MLK'& G&Z , L1 T , T\VWVaT\V M

0

2

4

6

8

10

12

14

16

3 3.25

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75

〈n2〉1/2

q

〈φ 2〉1/2

q 0

10

20

30

40

50

60

3 3.25

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75

〈n2〉1/2

q

〈φ 2〉1/2

q

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ΓδBI A R , G1WV7!.32cT\R'& °NM WT"1WX&R(*) EdR , M G8V$R == R'&31WV7

93MLKS; ,RVgRT , Z gRZ-&T\V Q^R(*)YX_TER+1YX1ZX,RefV+. , e-X+&T 29T , WT\V$OZ-&'&WT , )ae)aT\V Q@?QACBYX :, 1WV7qR'H5T , 78&T\R(*) HlRX 1WV e)WVaTdI

ΓδBM A AUR , G1WV7$L1 _T")aT\V M A RT;HbZV )WRT , ,RT")OX = ,RV T\VgRT , HI¬A

A T , X_Tme)WVaTWT\V ,X :, 1WV7! = TgRV7uX_T\V O &'1TPLEdR*(D)aT\V K'& 1WV &0JK, eL T\VOX^Z == EpD-) , T\V _RT HlRXWT , I

ΓδBM A A R , 1WV7 LEdRD(*)aT\V F'& 1WV °'& JK, eL T\VOX Z = ,RV T\V = 4,T VZ(*) , Z gRZ-&T ,4J e8,RX,RefV M Q Z-HlRX

HbZ(*)OXgRT>L1+ DXL&R(*)aT R = vT\V1WV7WT , Q^R(*)YX_TER+1YX1ZX,RefVaT\V Z-13297 , 1WVWT , Q^R(*)OX_T"7 , Z gRT\VYXvZ = AR Z(*) 1WV7S&TgR'78&R(*) ;)& Z-1+ M

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2

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8

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σ− = 1.2 σ+ = 1.5

σ = 0ΓδB ΓδB

〈n 2〉1/2

q

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( TWeG(*) = T"&T"7 T\VgRT 7 e , 7 T,X_T"&'&X_T\VBYR'H 1&ZX,RefV_T , 7 T = VWR_TST\RVWT"1WX,R'7 = aZP gRTb5T , A, RV7 T , 1WV7 WTmQ^R(*)YX_T"7 , Z gRT\VOX_T\V 31 , (*) WT\V,X :, 2cT"& = TgRV7uX_T\V = .Z , Z-&'&T"&T\V # T\R'&_A(*)aT\VSR+1TP]VWR(*)YX WT , V Z-1.WXHnT(*)ZVWRH 1kR,X = WT , L1 , R = _T\VG1WV7WT , X1 ,= 1&T\VSAX_T\VCO &'1YX1ZX,RefVaT\V29T) , X = _efVWT , V aZPgRT BYX_eG(*)Z,X,R_RT , 1WV7 WT , OaT"&3&RVWRT\V gRTV Z-1.WX1 , Z(*)aTR,X M Q^RT RVgR , T X_T A R , G1WV7 WT , Q@?QACBYX :, 1WV7 Z-132 WT\V X1 ,= 1SA&T\VOX_T\VCO &'1TPUT = T , gRT0R = _T\V1WV7 WT Q^R(*)YX_T")WRVOX_T , 7 , 1WV3. , e-X+& 31 , (*) WT\V,X : A, 1WV7 = TgRV7uX_T\V O &'1TP I J 2wT\R'&T ° 1WV ;zM .WRT"&XpV 1 , T\RVaTG&T\RVaT , TmjSe&'&T M

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- 6%& ' - / 0&-.,78.:9 ; <" = + *

B

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v00M Aj]peH.hefVaT\VOX_T T = T , X , Z-7 T\V = E`T"&(*)aT , _efV,XX_T\R'&'E`T\R_TSRVgRT>R = R Z(*)1WV7WTiQSR(D)OX_TEA

)WRVOX_T , 7 , 1WViZ = 7 T"&T\RX_T\XKEè , WT\VEpD , T M R& ` T"&T"7gRT_T , R , 781HnT\VOXvZX,RefV uZVWV HbZV gRTR == R'&_A31WV7 93MLK>F ZV_T")aT\V'& V RT , R,XiR'H &RV T\V;# T\R'& gRTL T\RX&R(*)aTZ?tVOX EdR(G&'1WV7WT ,

φ00Aqe WT 29T , WT\V

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M 1WV2cT,X_T"H Q^R(*)YX_T"7 , Z gRT\VOX_T\V M 1]T , T\VWVaT\V]R,X = aZP[e)WVaT 2 , T\RT\V]Q^R(*)OX_T")WRVYX_T , 7 , 1WV aZ R 7 T Z-1WT V RVOX_T , 7 , 1WV3.hefX_T\VOX,RZ-&1H T\RV 5SRT"& 2IZ(*)aTW7 T"7 T\V T = T , WT"H R 7 T Z-1WT efV,R,X_T\VYX_T\VSOZ-&'&ZV EpD(D),X M Q Z-HlRX 7 T ,= 1WVWT\VqR,X gRT ? , L T"1781WV7 T\RVaT , _T") , 7 , e$P T\V .he&efRaZ-&T\VqjSefXvZX,RefV M G H, T(*)OX_T\V # T\R'& 7 efVCR ==M93MLK>F T , T\VWVYX HbZV aZYL T\RX&R(*)aT:5T , )Z-&X_T\V WT efV 7 T X,R 7 T\V O &'1vT & RX 2wT,X_T"H Q^R(D)OX_T")WRVOX_T , 7 , 1WV R,X gRT L T\RX&R(D)aT5Z , RZX,RefV,X D , T , 1WVaZ R 7 T Z-1WT"1WX&R(*)) : )aT ,M = Eie)& V 1WVUHnT") , .se&efRaZ-&T BYX ,*: H 1WV7IL1 , BYXvZ = R'&R,RT , 1WV7WT , #1 ,= 1&T\VL 7 e , )ZVWT\VR,X = R,X gRT X1 ,= 1&T\VOX_T R X,R 7 RX DXkRV7 TvZ-H XEiT_T\VYX&R(*) 7 ,*: P T , Z-&kR'H OZ-&'&TUHlRX 7 Z , RZ = &T"HV RVYX_T , 7 , 1WV M

Q^RTSHbZ(*)OX vT") , WT"1WX&R(D) = aZP gRTqVWR(D)OX&RVaT Z , T\V A T(D)_T"&'EdR , 1WV7 T\VBHlRX WT ,Q^R(*)YX_TI J 2wT\R'&T ; 1WV K RV R ==ML93MLFNM WT , Eie)& EdR(*)OX,R'7_X_TB T"& = _X , T"781&RT , 1WV7HnTEA(*)ZVWR*H 1>2 T , gRTY#N1 ,= 1&T\VL RVWT"H = T\X , Z(*)YX_T\X_T\V qe WT"&'&, _X_T"H R,X M

O%T , gRT )WRT ,Y= T\X , Z(*)YX_T\X_T 2pVYX_T , 1+(*)1WV7 = T\RV)Z-&X_T\X gRTmZ = T , WT\V&5e , X_T\R'& = aZPgRT?T X_T =EiT"&(*)aT HlRX WT , .se&efRaZ-&T\V jSefXvZX,RefV 1WV WT , BYX :, 1WV7 T\RV)aT , 7 T")aT\V = V 1WV Z-17 T". , D-7uX_T , L1T , T\VWVaT\V ,RV M2>H gRT)WRT , 7 T629T) , X_TrQ^RG1,RefVT = T , gRTrQ@?QA AUR , G1WV7qZ-132 aZ 0 T*(*)EdRVgR'7 T\RX. , e-X+&= TvT , L1 7 T , ,X_T")aT\V = uZVWV HbZV ,R(*) L1WV#D(*),X EdRTWT , WT ,>J efX_T\VYX,RZ-&')WRVOX_T , 7 , 1WVWT\VYX EdR( &'1WV7= LE M WT , I

v00M A 0 &T\R(*)1WV7 I 93ML9NM L1EiT\VWT\V M )WV&R(D)CEdRTnZ-1+(*) = T\RWT , qe WT"&'&'78&T\R(D) 1WV7WT ,

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−⟨∇2

‖φ⟩y,z

I 93MLKS;zM

= T\R WT , O%&'1TP$R4D(D)aT\VHlRX,X_T"&'1WV7 WT , qe WT"&'&'78&T\R(D) 1WV7 I 93MLKNM6M ?tRVaT L1I 93M<; Y M ZVZ-&e7 T jpT(D)WV 1WV72 T) , XFL1 WT , ?tVOX EdR(G&'1WV778&T\R(*)1WV7WT , .se&efRaZ-&T\V V RVYX_T , 7 , 1WV,X ,*: H 1WV7

∂t v00 = −∂x (〈vy vx〉y,z − ν ∂x v00) +AδB

I 93MLK>KNM

HlRXpT\RVaT"H ,X :, 1WV7 = TgRV7uX_T\V& 1 ZXL X_T , H

AδB := −⟨∂yψ∇‖φ

⟩y,z

.I 93MLK>FNM

` T\R T\RVaT , ZVZ-& X,R*(*)aT\V ` T\X , Z(D)OX1WV7 WT ,` T*(D)&T"1WVWR'781WV7XI 93MLK>FNM 7 T"EdRVWVOX HbZV V1 , 1WVOX_T ,WT , RSVWVZ-)HnT = aZPaZ BYX :, 2cT"&3.hefX_T\VOX,RZ-&^Z-1*(*)&RT|P8&R(D) HlRXWT , I

0,0M Aqe WT WT J efX_T\VYX,R_A

Z-& E`T(*)_T"&'EdR , X = gRT*? , T\VWVYX,VWR = aZP 4:T\VaT , R>1 , 1+(nVaT"7OZX,R 7 WT6XaVWRX`R,X = 1WVrT , _eHlRX`T\RVaT'D-H.32cT\VWT4A R , 1WV7mZ-132 gRTJ? , L T"1781WV7 WT , .se&efRaZ-&T\V BYX ,*: H 1WV7 31 , (*)WT\VkjpT\ Vae&,X , T)ZX*I M RpV)ZV7 ? MLKNM6M O%T , T\RVaT Z-&'&'7 T"HnT\RV78T&X,R'7 TR>1 Z-7 Tè)WVaT EiT\RX_T , T RSVWVZ-)HnT\V fZVWVEdRTEAWT , 1H V 1 , gRT 1HnT , R = T"H T)OX>EiT , WT\V MQSRT AUR , G1WV7 WTT\RV7 T*(*)Z-&X_T\X_T\V #oT , H0I 93MLK>FNM EdR , R'H BG(D)aT"HbZ 93MLK>K 31 , (*) aZ 5e , )ZVSAWT\V_T\RV WT ,>J 2wT\R'&T 9 Z01WV 9-=;, T". , DvT\VOX,RT , X M

0

50

100

150

200

250

0 1000 2000 3000 4000 5000 6000 7000 8000

n00 n00

〈φ 2

00〉1/2

V

τ

n00

n00

〈Γ〉V

τ−5

0

5

10

15

20

25

30

35

0 1000 2000 3000 4000 5000 6000 7000 8000

n00

n00

〈φ 2

00〉1/2

V

τ

n00n00

〈Γ〉V

τ

Kv|Eaa|||pE .E *jO¬aa¢^£8EjoP COa¬||¬ O£! jE|#a ¬O||z La«¬O|| ®||*|j aa¢^£|* OLCa¬@Ej¦¬¬ j¦ j\ Na|®!a¯ ||¬«¬ a ¬O£!|||¯e||8|a¢^£¬£||||¯ ¨va¢^££|ajO¬O||

gf gfji dwjs)t>r'm ml-sbxt>r#s)t>wjuq*l1siUum>tNww'wjr#qjqjr ¬l1s RXWT\V0T = &R(*)aT\V J Z , Z-HnT\X_T , VI&RT62wT , XgRT BYR'H 1&ZX,RefVWT R'H BG(D)aT"HbZ 93MLK>K= T*(D) , RT = T\VaT\V BL TEAVZ , Re aZ RV R == R'&31WV7 93MLK Z aZ , 7 T,X_T"&'&X_TJ5T , )Z-&X_T\VWT ,KV RVYX_T , 7 , 1WV3.sefX_T\VYX,RZ-&,X :, 1WV7 M ` T\R?tRV*(*)Z-&X_T\V WT ,\7 T , *(*)WRTWT\V ,XvZ , T\V Q@?QACBYX :, 2wT"&WT , ,RV XaZ I

φ00M A R 7 T Z-1 , Z_X,R*(*) Z =

1WV 1WVYX_T , &RT"7uXmT65 X , T"HnT\VL T\RX&R(D)aT\V~BG(*)E`ZVG1WV7 T\V M jpT(D)OX = RV R ==M93MLK8== R,X gRT L17 T") : A, R'7 TZ?tVaT , 7uRTWT , .he&efRaZ-&T\VBYX ,*: H 1WV77 T*(D) EdRVgR'7 T\RX aZ , 7 T,X_T"&'&X M ?,SR,XSZ-1+(D)]ZVWV#D-)aT , VL1 T , T\VWVaT\V = aZPgRTYL T\RX&R(D)aT\V&BG(D) E`ZVG1WV7 T\V HlRX WT , BYX :, 2wT"&,X D , T L1WVaT")HnT\V M

2#2Y %T/ b +-)g .! zg :#0g.z %T4)N (

0

50

100

150

200

250

0 2000 4000 6000 8000 10000 12000 14000

σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

σ = 0

〈φ 2

00〉1/2

V

τ

12 〈v 2

00〉V

τ 0

10

20

30

40

50

60

70

80

90

100

0 2000 4000 6000 8000 10000 12000 14000

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

〈φ 2

00〉1/2

V

τ

12 〈v 2

00〉V

τ

Kv|Eaa|||1E .|¡z aa¢^£|g¤ <¥#a¢^¦ a||n|#a ¬O||z La ®4|Eb|¦§aEa¢£|¤|¬a| ¬a¬a!ap#Ca¬ ®S$a|¬¢^£| ªj¨¤ ¨4©P «¬O|||

RpV)ZV WT , sT\RX_T\RV)aT\RX_T\V = T\R'H 5T , &Z-132 WT , Iφ00

M ACBYX :, 1WV7dI9R ==M93MLK>FNM T , T\VWVYX HbZV = aZP= R L1 T\RVaT"H J 1WV X 7 efV T\X E Z ;)&'&'& sT\RX_T\RV)aT\RX_T\V T\RV ,X_T\R'&T , RpV,X,RT"7 ,XvZX,XXaVWT\X M Q^RTR,XiR'H0 T"7 T\VvZXLYL1H BSDX,X,R'781WV7 7 T , )Z-&X_T\VWT , O &'1 X1ZX,RefVaT\V T\RVaT1H T\X E`ZlT\RVaT\V;OZ X_e , K'& &ODVSA7 T , T@sT\RX aZ-1aT ,M ? = T\V_e EdR , WT"1WX&R(*) = aZP gRT_T-5T , )Z-&X_T\V+veEie)&sR'H6BL T\VZ , Re HlRXW2cT_X_T"HQSR(D)OX_T")WRVYX_T , 7 , 1WV Z-& Z-1+(D) R'H OZ-&'& WT , efV,R,X_T\VOX_T\V V RVOX_T , 7 , 1WV = T")ZV3&'1WV7 I*(*)E`Z , L TWRVWRT M-7 e , )ZVWT\V R,X M qZV+,RT")OX = 4,T>) : )aT , aZ R 7 T Z-1 WT ,FJ efX_T\VOX,RZ-&')WRVOX_T , 7 , 1WV_X :, 1WV7 R_X =WT,X_e&ODV7 T , R,XpZ-1+(D)WT , sT\RX , Z-1H =+= RpT\RVaT VaT"1aT>,XvZX,RefV#D , TyBSDX,X,R'781WV7rT ,, T\R(D)OXpR_X M Q Z-HlRX7 T ,= 1WVWT\V R,X Z-1+(*)T\RVaT.#DX_T , T BSDX,X,R'781WV7WTZI

n00M A R 7 T Z-1dR'H 1WV7 T,X :, X_T\V efV_R,X_T\VOX_T\V

OZ-&'& MQ Z47 TvZ-H X_T^X1 ,= 1&T\VOX_T BY _X_T"H = T6XaVWT\X _R(*) 4:TWeG(*) V 1 , aZVWV[RVrT\RVaT , EdR , G&R(*) VaT"1aT\V+_XvZAX,RefV#D , T\V J )Z_T = _e-2wT , V]Z-&'&T R+1 X1WRT , T\VWT\V 0 ,*: P T\V T\RV*(*)&RT|P8&R(D) WT , V RVOX_T , 7 , 1WV 7 Z , RZX,ReAVaT\V 4,T"EiT\R'& R') , T VaT"1aT BYXvZX,RefVZ , RX DX 7 T62 1WVWT\V )Z = T\V ; Y M Q^RT R,XpZ-1+(D) WT , 0 , 1WV = E`Z , 1H= T\R Z-&'&T\V BYR'H 1&ZX,RefVaT\V gRT&BYX :, 1WV7 T , _XIL1 4,T\VaT"H , T(D)OX7.#DX_T\V sT\RX.1WV X[T\RV7 T*(*)Z-&X_T\XEF1 , WT = ZVWT"H ,R(D)aZ5 eH.&T\X,X_T .)Y _R fZ-&R*(D)aT #1 ,= 1&T\VL _ ,X_T"H Z-17 T\V*(*)aT\RV&R(D)RV BYXvZAX,RefVZ , RX DX = T62 1WVWT\V )ZX M

?,0,X_T"&'&X ,R(D) V 1WV gRT;O , Z-7 T = RVEdRT"EiT\RXbT\RVaT qe gR XfZX,RefV WT , Z , X,R X+L RT"&'&T\V Q^R_R'.ZX,RefV RVWT , 5e , X,R'L RX DX*78&T\R(*) 1WV7 I 93MLKNM )WR'& 2 , T\R(D) 1WV?_RVWV 7 e&'& R,X Mp` T\X , Z(*)OX_T\X HbZV WT\V@#oT , H WT ,TWV,X&R(D)aT\VQ\D-H.3291WV7

ν∇4⊥φ ∼ ν k4

⊥φk ,I 93MLKM

_e T , T\VWVYXHbZV = aZP gRTpQ\D-H.3291WV7 = T_efVWT , ,XvZ , bZ-132 G&T\RVfZ-&R'7 T\BYX :, 1WV7 T\VUHlRXW7 , e$P T\Ve WT\VL Z-)&T\VEdR , YX M G ) , T AUR , G1WV7UT = T ,

k4r

Z-132 gRT .he&efRaZ-&TljpefXvZX,RefV R,X 7 T , 78&T\R(*)*EiT\R_T7 T , RV7 M : 78&R(*)SEpD , TSV 1WV[RV WT , 0 &T\R(D) 1WV7I 93MLK>KNM 2 T , gRT V RVOX_T , 7 , 1WV,X ,*: H 1WV7 = EiT"&(D)aT^R'Hj ` F Q 7 T\X , T\VWVOX 7 efVWT\Vqe WT"&'&'78&T\R(D) 1WV7 T\V$2 T , gRTI

m,nM A RpVYX_T\R'&T = T")ZVWT"&XYEdR , = T\RVaT

ZVWT , T Q\D-H.32 1WV7 efV_XvZVOX_TpT\RVL1 = Z-1aT\V = L Ma` M T\RVaTF1H T\RVaT\V = T_X,R'HH X_T\V O Z YX_e ,αT , ) : )OX_T

QSR,R'.ZX,RefVν

ν00 := α ν .?tRVaT,X D , T , T Q\D-H.32 1WV7WT , .he&efRaZ-&T\V jpefXvZX,RefV;)#DX,X_TWT\V 5e , X_T\R'& = aZPaZ V RVOX_T , 7 , 1WV A.sefX_T\VYX,RZ-& *(D)WVaT"&'&T , RVgRT BSDX,X,R'781WV7.)Z_TvT\RVaT R 7 T Z-1mT\RVYX , RX,X 1WV T = T , gRT VZ(*) WT\V= R)aT , R'7 T\V ? , T\VWVYX,VWR_T\V gRT L T\RX&R(*)aT\V&BG(D) E ZV1WV7 T\VqR'H 7 T,X :, X_T\V OZ-&'& WT"1WX&R(*) 7 T , RV7 T ,EpD , T\V M W T\XL X_T , T)#DX,X_T WT\V T\Vae , HnT\Ve5e , X_T\R'&dT\RV )ae)aT qZP ZV jpT(*)aT\VL T\RXnT\RV*.Z , T\V L1 : VWVaT\V = aZnRV 4,T\VaT"H OZ-&'& T\RV 781WX_T _XvZX,R,X,R*(*)aT\sT\RXHlRX,X_T"&$*(*)aefVqVZ(*) EiT_T\VYX&R(*) EiT\VWR'7 T ,

O3&8! v "4 ) $. f.80' v z% 242S3

sT\RX*(*) , RX,X_T\V T ,, T\R(*)OX EpD , T M Q^RTG WT T WT , ?K5 X , Z 'D-H.3291WV7 29T , gRTI0,0

M Aj]peH.hefVaT\VOX_TUEdR , Z-1+(*) = T\RV 1HnT , R*(D)aT\V 2SVOX_T , *1+(*) 1WV7 T\VWT , ;/ 2 8 ZV7 T"EiT\VWT\X 9'( MA D-) , T\V EdRT = R)aT , gRT Z , X,R X+L RT"&'&T Q^R,R'.ZX,RefV

νVZ(*) WT"H .)O ,R uZ-&R*(*)aT\V R .hT X WT

_T\V , T(D)OX_T\VUR>1SR : *1WV7 7 T , H : 7 T\V T\RV7 T,X_T"&'&X EdR , cI,RT")aT R = *(*)WVWRX,X °3MLKNME= EpD , T gRTFEdR(D)OX,R'7 TO , Z-7 T^V 1WV = EdRT gRT@ 1vZXL gR,R'.ZX,RefV;HlRX WT"H J Z , Z-HnT\X_T ,

αT\RV!4 1_X,RT , X4E`T , WT\V _e&'& MGV RT , L1

: VWVOX_T T\RV5T , 78&T\R(D) WT , 5e , X,R'L RX DX

w = ∇2⊥φ

HlRXφ 6= φ00

I 93MLK>°NM

1WVWT , BG(*)aT , ,X ,*: H 1WV7∂2x φ00

Z-&p] , RX_T , R'1H )WR'& 2 , T\R(D)+vT\RV M B e-2wT , VgRT5e , X,R'L RX DX = LE M AUR , A= T"&,X D , T WT , G&T\RVaT , T\VdBYX , 1 X1 , T\V7 ,*: P T , R,XmZ-& gRT:BG(*)aT , ,X ,*: H 1WV7 WT , 7 , e$PuZ-&R'7 T\VC.heA&efRaZ-&T\VrjpefXvZX,RefV = fZVWV HbZV 7 efVbT\RVaT"H BY _X_T"H Z-17 T")aT\V = RVWT"H gRTpX1 ,= 1&T\VOX_T R X,R 7 RX DXVWR(*)OX7DVL&R(*) 31 , (*)gRTZBG(D)aT , ,X ,*: H 1WV7 , T31L RT , X>EdR , V MQ1 , (D) aZ85e , )ZVWT\V_T\RVWT , T65 X_T , VaT\V]Z-7uVaT\X2cT"&,X :, 1WV7U7 T,XvZ-&X_T\X,R(D) aZ\ 1vZ-HHnT\VSA.WRT"& 7 efV

ν001WV

AδB

RVWT , BYX ,*: H 1WV778&T\R(D) 1WV7&I 93MLK>KNM 4,TWeG(*)@ eH.&R'L RT , X &RpV7 T\VaeHHnT\VgRT Q^R,R'.ZX,RefV EpD , TqZ-132 gRT e = T\V 7 e , 7 T*(D)&Z-7 T\VaT A T\RvT,4 1_X,RT , X = _eaZP gRT BYX D , TWT ,BYX ,*: H 1WV7 7 T , *(D)aT , 1WV7 gRT@5e , X,R'L RX DX1WVOX_T , X , RhX = _e : VWVOX_T = T\RT\RVaT , = T,X,R'HH X_T\VcBYX :, 2cT"& A,X D , TkT\RVaT31 , (D) WT\V# T , H

AδB

T , L T"17uX_T = T\VOX7 T"7 T\VC7 T , R(D)OX_T\X_T J e&efRaZ-&,X ,*: H 1WV7 29T , T\RVaT,X D , T , T BYX ,*: H 1WV7 7 T , *(D)aT , 1WV7_e , 7 T\V = _eaZP _R(*) e = R'7 T85T , )#D-&X,VWR_T 1H T") , X_T\V M Q^RTiR_X4:TWeG(*) , T\RVaTQB.sTG1&ZX,RefV M?tRVaTFL1 DXL&R(D)rT\RV7 T = Z-1WX_T^QSR,R'.ZX,RefV29T , aZ V RVYX_T , 7 , 1WV3.sefX_T\VYX,RZ-&hRVIL1TWV32cX,R'7 T\V R ,= T\R_AX_T\V = TaZ , 2T\RVaT , 7 , TWV3&R(*)aT\V5e , 1WVYX_T , 1+(*)1WV7;1WVOX_T ,@` T , T+( ,R(*)YX,R'781WV7WTST65gX_T , VaT\VBYX :, A2cT"&WT MPV RT , L1 : VWVOX_TgRTmT65G.hT , R'HnT\VOX_T"&'&TqT1WV7WT , .se&efRaZ-&T\V]jpefXvZX,RefV 1WVWT aZ-HlRX7 T ,= 1WVWT\VaT\V , Z gRZ-&T\VrT"&T X , R*(D)aT\V OaT"&WT = EdRT,RTL1H ` T\R.WRT"&aRV V & 31 , (*)7 T629T) , X EF1 , WT =)WR'& 2 , T\R(D)@_T\RV M ( TWe(D) 7uR'&XW)WRT , L1 = T Z(*)YX_T\V = aZP = T\R WT , T653.sT , R'HnT\VYX_T"&'&T\V qT1WV7 WT ,WJ &ZAHbZ , efXvZX,RefV RVC#b? #^j gRTljSZVD(*)WR(*)YX_T hT X_T I J &ZHbZA A ZV A A T(*)_T"&'EdR , G1WV7 T\V M T = T ,gRT RH = R'.he&Z , RX DX = TgRV781WV7aZ , Z gRZ-&TqT"&T X , R*(D)aT OT"& HlRX = T,X,R'HHnT\V 1WV T\RVaT 5T , A78&T\R(*) = Z , T\RX 7 efV jST1&XvZX_T\V T , *(D) EiT , T\V M

gf gf r'us\qjx l1siJx)snuq\qx% Z(D) gRTvT\VQ5e , T = T , &T"781WV7 T\V0uZVWVkV 1WVgRT WT,XvZ = R'&R,RT , T\VWT = RVgR , T X_TAUR , G1WV7WT , Q8?Q*ABYX :, 1WV7rZ-132WT\V X1 ,= 1&T\VOX_T\V # , ZV.he , X R'H T\RVL T"&VaT\VqT ,*:, X_T , X EiT , WT\V MQSRT LEiT\RX_T RVgR , T X_TSAUR , G1WV7 WT , Q@?QACBYX :, 1WV7 Z-132 WT\V # , ZV.he , X XaVWT\XkVZ(*),BG(*)aT"HbZ93MLK>K T = T , gRT qe gR XfZX,RefV WT qT ZVSReE 1WVWT , sefVZ-&_Re E ,XvZX,X8I J 2wT\R'&T 9 1WV FNM6M AURTdRVR = *(D)WVWRX,X 93M<;8M<; *(*)aefV T , &OD-1WX_T , X = )ZX gRT .he&efRaZ-&T 0 T*(D) EdRVgR'7 T\RX,X ,*: H 1WV7

v00(r)31 , (D)

R') , Tb5T , *(*)aT , 1WV7T\RVaT ,XvZ = R'&R_RT , T\VWTFAUR , G1WV7 Z-132 gRTdX1 ,= 1&T\VYX_T\VSO &'1 X1ZX,RefVaT\V M Q^RTeAgR X+L RT , 1WV7 WT 0 T*(*)EdRVgR'7 T\RX. , e-X+& 31 , (*) gRT BYX :, 1WV7cI J 2cT\R'&T 9 Z01WV 9-=4M4= T"EdR , X V 1WVT\RV RSV,X_T\R'7 T\V WT ,SJ efX_T\VYX,RZ-&_R+1YX1ZX,RefVaT\V =b7 T ,= 1WVWT\V HlRX T\RVaT"H WT"1WX&R(*)aT\V@RSV,X,RT"7 WTX1 ,= 1&T\VOX_T\V # , ZV.he , X RVWT , RX,X_TWTpjST(*)aT\V7 T = RT\X = EdRT RV R == R'&31WV7 93MLK>° L1 _T")aT\V M2>H gRT_T@5T , )Z-&X_T\VL1vZ-HHnT\VHlRX ` R'& 93MLK L1 7 T , ,X_T")aT\V = R_X^T,RVWV 7 e&'& = T\R WT , .se&efR_AaZ-&T\V V RVOX_T , 7 , 1WV_X ,*: H 1WV7 LEdRD(*)aT\VWT"H L T\RXvZ = )#DV7uR'7 T\VcsefVZ-&_ReE ZVOX_T\R'& 1WV WT"H L T\RX1SAVZ = )#DV7uR'7 T\V T ZVSRe E`ZVYX_T\R'&%L1U1WVOX_T , D(*)aT\RWT\V M Q T\VqT ZVSReE@7 T"EdRVWVYX HbZV T\RV32wZ(D)31 , (*)

2#2 %T/ b +-)g .! zg :#0g.z %T4)N (

sT\RXHlRX,X&'1WV7WT ,YV RVYX_T , 7 , 1WV37 T*(*)EdRVgR'7 T\RXv00M O T , aZ , Z gRZ-&T JK, e-X+& gRT_T L T\RX1WVZ = A

)#DV7uR'7 T\V RSVOX_T\R'& 7uR'&X aZVWVv00(x) = 〈v00〉y,z,t .

I 93MLK>9NMG H Z-&'&'7 T"HnT\RVaT\VEdR , = T\RWT , Q^RG1,RefV 7 efV .se&efRaZ-&T , jpefXvZX,RefV e WT , BG(D)aT , ,X ,*: H 1WV7 7 TEAVZ-1 T = T , gRT_T\V RpVYX_T\R'&N7 T. , eG(*)aT\V M

0

5

10

15

20

25

3 3.25

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75

〈Γ 〉

q

〈φ 2〉1/2

q 0

20

40

60

80

100

3 3.25

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75

〈Γ 〉q

〈φ 2〉1/2

q

Kv|Eaa|||E . E #|La#O!a4EjE|Eajj1 Na| ®|E| O¯©>jL !|¦!Ca¬| ®!¯ 'a|¬¢^£! ªv¨¤ ¨4©P «¬O|||

Q Z , Z gRZ-&T J , e-X+& WT qT ZVSReE R_XoRVR == R'&31WV7 93MLK>9 Z 29T , WT\V01WV7 T,X :, X_T\VIOZ-&'&aZ-&"*(*)E`Z , AL TZ] 1 ,¬7 T aZ , 7 T_X_T"&'&X M Q T , , Z gRZ-&T 5T , &Z-132 gRTvT JK, e-X+& = EiT"&(D)aT , RV gRT_T"H OZ-&'& T\RVaT HbZ 5gR_AHbZ-&T*5T , *(D)aT , 1WV7

∂x v00R'HHlRX,X&T , T\V ` T , T\R(*) WTjpT(D)aT\V7 T = RT\X_T = T\RV)Z-&X_T\X = 2 T) , XV 1WVSL1

WT , WT"1WX&R(*)aT\V R = _T\VG1WV7 WT X1 ,= 1&T\VYX_T\V O &'1_T`RV WT , 1WVHlRX,X_T"& = Z , T\V D-)aT WT , Iq=3

M AJ e,RX,RefV I ` R'& 93MLK>°NM6M

−15

−10

−5

0

5

10

15

3

σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

3.252.75

σ = 0

〈v00〉

q

〈v 〉1/2

q 0

2

4

6

8

10

12

14

3 3.25

σ = 0σ− = 1.0 , σ+ = 1.2σ− = 1.2 , σ+ = 1.5

2.75

〈v00〉q

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v2ZF = (v00 − 〈v00〉t)2

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⟨(v00 − 〈v00〉t)2

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.I 93MLK Y M

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0

0.5

1

1.5

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〈Γ 〉

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AδBAδB

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κ = 0.388 : n = 13 , γ τ = 0.04 =⇒ D⊥ = 0.19

2

_T

κ = 1 : n = 25 , γ τ = 0.21 =⇒ D⊥ = 1.80

2

_TQ Z RX`R,X gRTST,X_T $XWV RX

κ=0.388 T\RVaT\V Z X_e L° efV WT XvZX R"aT\V

) vZVhe,X eYT RT\VYX_T\V T\VOX %wT,VOX ` T\R RVaT Z_T\V ) vZVhe,X,XgRT\V ZX ZVnZsT aZ"_T ,X_eOZVWR_RT,X_T V `ZvZX_T\VsT X OZ VWR"YX T %20!WV RVaT Z_TqT\Xae WT\V ` T,X,R WV efV ` Z e efVWRV V `ZvZX_T\VdRTmRV WVWRX,X \ % 0_T\V hT,X_T\V%IZ T\RVaT WV R"aT\V sT X dRT aZ RX

κ=1 _efX_T J T Q T>_e Z T$X X_T ) vZVse,XmR_X aZVWV T\RVaT\V Z X_e rR T! T\R" _T ,X efV_R,X_T\VOX sT_TWVaT\X_T\V ) vZVse,X iT,X !_e QSRT ZYX WTWX R"

aZ gRT

V T,RaT YR ZX,RefV WV gRT0vT ,X efV,R,X_T\VYX_T ` T_TWV WVWTmVWR"OX RVaT Z_T\V QS VZ R T %ce WT R" R_X

_e%wT,V ZV ZV T\RVaT T\VZaT_T\V vZ T0hTWT\V ZVae Z T\V ) vZVse,X

RVOX_T_T,RT,XSR,X RT T\V _T\R`Vae Z aZvZV T,RVWVaT,X

aZgRT VOZsT T\RVaT ) vZVhe,X e T RT\VOX_T\V

D⊥gRT VWVZ T hT\RVZ X_T\X

aZ WT ZVae Z T ) vZVse,X

WT ,R"40hTWT\V TvZ X_T\V

jpTaT\V sT_T\R" T_T\RuVaT\X efV gR ,RM T ) ZvZ YX_T R,X Q^RT_T VWVZ TqR,X 4,TWe UVWR"OX

dRV T\V


!"# $&%&' ( )*%&,+ - )

"MV ZWRX_T ' WT#sT_T\RX gRT/. TWTWXWV WT Z Z,X$ uR T\RX % 0WT\V10 ZvZ X_TWT ) T\V T WVOX Q^RT Z Z_X$ uR T\RXT\VOXaT\RWT\X0sTgRT32 0 X,R T\RXWTtT TYX_e_XvZX,RaT\V4 aTWV

dRT _RT R j5. Q T`T\VWT\X#dR Q ZmT ,R" hT\R WT\V We,X,R T\V6. T\XvZYXWV T\V

gRT Z RXvZX,RM TtRVae gV WV WT T$ X,VWR_T RX,X_T WT.dZ e efVWRV!ZvZ T\X_T ZVWT XR,X`T\RVaT aZ10hT'WRVZ TaT\VWT ZvZ T\X_T,XgRT Z% !_T\R" "MVsT_efVWT_T uZVWV7_e WT

V iT\VWVhT_T\R" WTj7. Q _e!vZ T\V TT\RV T!_T\V X iT WT\V "8VgRT_T ZWRX_T iT WT\V aZaT _T\R `T\RX_T_TYR ZX,RefVaT\V RX WVOX_TWRT R"aT Z Z,X$ uR T\RX e T,X_T X

iT "aTbT,X_T\V WV%cX 0sTWT\VtRV WT YX$ uR T\RX hT\RT\RVaT ,X_eZ,X,R,RT,X_T\VX

T\VOX_T\V Z Z ThT\V WV iT\RX_T\V WT\V V iT\VWVsT_T\R" WTSj7. Q Z RT_T\V , T\RuX>,R"

aZ[RVWVaTZ WTqe WT !_T\V T\V gRT TWRTWT\VaT\V X T\VYX_T\V qTZVWR

T\V`T "aT aZ T_T dRWV gRZ!vZ WT ZWRX_T hT,RThT\V iT WT\V

Z hT\R WVYX_TWRT R"aT YX$ uR T\RXdRV[R_T dRWVkTZ X_T\V T\RsT\V Q Z dR RX,X_T V T,RaTYR ZX,RefV hT TuX

aZ gRT TiefVWVaT\VaT\V TWVWR_TkRV ZWRX_T T\RVaT

!WV T T\VWT tR T\VZ %X WT qe WT ! T\R" WV T\V ,RV WV Z RXvZX,RM[Z % 0_T'WVOX_T WRT R"aT Z ZZvZ T\X_T&WV TVWT,X& T\RhT\V

98 : ** ( $ & !*%" *% /; 5# $

RXiZVvefV,X_T\V ! T\R"aT\V ZvZ T\X_T,V ' WT\V _T\R iT\RX_T_T YR ZX,RefV %wT RX WVOX_TWRT R"aT Z Z,X$ uR T\RX T(%20,X

iT "aT WT\V<.dZ eYefVWRV!ZvZ T\X_T

ν = ZvZ X_T,R,RT,X iT WT\V fZVWV?>

' T\VZ,Re@>ν = = 1.00 ⇐⇒ n0 = 5.5 · 1019 −3 ; Te = 36.0

T

T\VZ,Re@>ν = = 0.74 ⇐⇒ n0 = 5.0 · 1019 −3 ; Te = 40.0

T

T\VZ,Re@>ν = = 0.40 ⇐⇒ n0 = 4.0 · 1019 −3 ; Te = 50.0

T

Q ZhT\R' WT aZ Z ZhT\XvZβe

iT "aT ,R" WT\V Z Z WVaZ ]ZuVaT\X %cT sT,X,R X

efV,XvZVYX T Z_T\V QSRT YR ZX,RefVZvZ T\X_T aVWT\X ZV RV VZV

QSRT^Q^R,RZX,RefVZvZ T\X_T#' WT\V dRTWT RX WT $WV,RX_T,R T WVWRX,X

a | s uv m|W~ |W~u s v s u |x~ !ku

T\RV ,X,RT,X_e aZ,R" aZ RV WR WV ' T T\RuX_T`Vae RT,X_T V `ZvZX_T\VsT X

%20 WVYX_TWRT R"aT YX$ uR T\RX_T\VTuRWX

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 10 20 30 40 50 60 70 80 90

γ τ

n

ν = 1.00ν = 0.74ν = 0.40

K4||a_P "!$#%&'(#Na()*#% K+",-)*.0/1$#2)3!$ 5476#8" #212.0/| sP!$9¬*: !2(;=<vsP> !2#2&?1$*)*@476#AEP§&B|C)3!D1 *47E,E#%>

n =52 &F)*.0/"!G

qZV T T\VWVOXWTWX R"aZ RXnZWVaT T\VWTYX$ uR T\RXgRT RVaT Z_T"MV_XvZWR RXX T,RV T

dR `T\VWR T . Z e efVWRVOZ X,RMYRXXlR Y ,X_T eZVWT\V R_X Q^RTkT ,XkVae Z gRT 4^Z T\V TWVWT gR T\V,RefV e_T\V.dZ eYefVWRV!ZvZ T\X_T

ν = "MV T,X_T RVWRT dR gRT_T TZ X_T\V RV WT\V qe WT ! T\R" WV T\V gRT RX ,RV T\VWTYX$ uR T\RXtR T !$ T iT WT\VWT

Vae RT,X_T ZvZ T gR,RZX,RefV

χN

‖

hT!10WVWT\X "MV VZV-aVWT\X ZV gRT Q^R,RZX,RefVZvZ T\X_T

χN⊥ νN

WVχN

‖ `T "aT VZWT7e TaT\ViT\R_T

RV gRT_T sT\RX'dRV T\V RX WT\V O ,R uZ RaT\V ZvZ T\X_T,V efV%we _T\RV _e T\V H dRTWTSjpTaT\V T\RXST\RVZ_T\V

' WT\V gRT e WT\V Z T\V RX

n =52hT!_T\V X WV

aZ RVaT Z_T V ZvZX_T\VhT X `T\RX T\V WRVOX_T vT\RVaT qZ 5 R Z TWVWRX,X_T\V

" YR ZX,RefV Z% eWVaT Q Q YX$WV RXν = = 0.40

TaT\VgRT YX$WV T\VWTQ^R"YX_T WV WT efX_T\VOX,RZ rVZWT RVaT Z_T\V V ZZvT RV T\RVaT dRV !WV RXpT\RVaTpT\RV R T\V efV,XvZVOX_T\V _T aT\V 0sT I T\RX R"aT X T\VOX_T XZX,RefVaT\V ,RV ve RXrVWR"OX eZVWT\V Q Z RX hT%aVWT\X7,R" gRT,2_T\V TWT V iT\VWVsT_T\R" WT j7. Q % 0q) T\V ,XgRT\V R YX$ uR T\RXsT_T\R"dRaT\V

ν = = 0.40WV

ν = = 0.74

"MV WR WV _RTOX ZV]V WV gRT se efRaZ T\V _e T WTSX T\VYX_T\V _T _e dRT gRTWT QSR"OX_T XZX,RefVaT\V Q^RT_T&hT T T\V Z% !WV WT5,RV T\VWT\V T,X_ThT\R

θ≈0 T T\V 0hT

WT\VYRT T\RVaT_T\VZhT#! T\R" T\RhT\VWT\V T,X_T\V[R ) e,RVWVaT\V sT_T\R"

aZ ,R" RX _RV

T\VWT YX$ uR T\RX gRT .dZ e efVWRVOZ YX,RM RXX WT Z Z _T RT,X

.`T\RhT\RVaT T! T\R"WT Z T\V RXν = = 1.58

WVν = = 1.0

TuRWXgRTtVaTuRTgRZuVae,X,R TsT\V%IZ % 0_T0sTWT\V ,R".VWT,VWT\VgR,RZX,RM T\V 0 ZvZ X_TWT Z Z Q^RT _efX ZYRT,X_T\V tVaTuRT_T\V T\VQ 0 ' WV Q 0 VaT T\V sT\R ,RV T\VWT YX$ uR T\RX R T$ X,VWR aT X_T Q " T\R"OX

aZ7_RT T T,X _ehe,X,RefVZ -,RV T\VWT\V jST,R,X,RM RXX

% |W s | u s v m|W~|W~u s !v s u

0

0.5

1

1.5

2

2.5

3

−3 −2 −1 0 1 2 3

〈Γ 〉

θ

〈n2〉1/2

θν = 1.58ν = 1.00ν = 0.74

ν = 1.58ν = 1.00ν = 0.74

0

1

2

3

4

5

−3 −2 −1 0 1 2 3

〈Γ 〉θ

〈n2〉1/2

θ

ν = 1.58ν = 1.00ν = 0.74

ν = 1.58ν = 1.00ν = 0.74

Kv|Ea D¬D¬=)#%D!476#( !2#O| "!$ ) - 476#E| <v.0/"!$="!2=)3!D( >j!2 #212.0/| |a.0/(#WP!$9¬ !("- #

ν = 1%!$D , | # )aDDE")*! 8§ ! 3! 12! 8"D#2/)* F 12!2D5 j(##12!-(#!p a.0/ "!$#2)*1$>D# !G

η‖_RV "8V WR WV R,X sT ZOX_T\V

aZgRT _e WX `T,X_TWT)oT TlRV Z

WV [VWR"YX gR_TYX RX_T\RVZVWT T! R"aT\V iT WT\V7 $fVWVaT\VaZ ,RT :T`T\R RX`R_T

T $,R T\V 2 TvZ X_T\VaTuRTEVae RT,X è WT\V ,RV Q ZaTR_XT hT\R WT . T\XvZYXWVgRT

_T tVaTuRTgRZ!vZ T,RVWV e gRT .`T\X T WT T\RV T VaT\VtVaTuRT ZVYX_T\R TnRV jpT ZX,RefV aT X_T Q " _T\X T\V

1e−04

0.001

0.01

0.1

1

ν = 1.58

∑

ν = 1.0

∑ 1e−04

0.001

0.01

0.1

1

ν = 1.58

∑

ν = 1.0

∑

Kv|Ea #%¬=61$12S& 1%!$9# !$ >)a -C)3!D &% , aa( p #%& < K #2 "! &F)*>=)*/&? 1 |121=)3! 8" <#"3:Mp #2&?1& !1 #C)*12&F)*12!29¬ !

ν = )*b>G)*.$/!2=) | ! ")3!2 ?)*4/#2#% , aa +(1$)*&B!2 #2¬ eE D#%& #%!1B5(12/=)a5|WK41$D¬! , #%!2|.0/"!E|#2 "!8" #%¬a.0/ , #% 9

QSRT T,RV TWV WT YX$ uR T\RX dR X ,R"nRVWTj5. Q <fZ RTWV %20gRT RVOZsTZvZ T\X_T sTvefVWTdRV WT Z WT Vae RT,X_T\V

ZvZ T T\VQ^R_RZX,RefV eYT RT\VYX_T\V

χN

‖

ZdRT gRT I Z T\V iT,X_T&% 0 gRTYR ZX,RefVaT\V RV VZV T\R T\V Q ZaT ,RTYX ZV WT\VTYX

WT YX$ uR T\RXJ TVWTWV RV sTvefVWTSZVZV WTSQ 0 ) T RV WT\V

χN

‖

Z e %wZ X_epT\RV TOX "MV T Z X T T\VWVOX ZV ,TWe T\RVaT Z RXvZX,RM T\V HpVYX_TWRTWTkRVWT\V tVaTuRT 0_T\V WTY ,X_T '%20 WVYX_TWRT R"aT YX$ uR T\RX_T\V Q^RT WTWX_T\X aZvZ% WRV

aZaZ ) T\V T

Z X_T\V RV _T\RVaT 0 ZvZ X_TWV TVWT,X T\RWX QSRT . T\XvZYXWV WT vZ gRZ T\V _e TWT × . 2 T dRVgR T\RX WVkR_T vZ gRZ T\V e he VaT\VOX_TRV WR WV $ WVYX_T ZaT,X gRT_T vZ T . T\RWRT iT\RvTT T\VWVOX ZV RV $ Z

aZ gRT YX X WT !_e fZ R T\V(e,X,R T WV TVWT,X T\RWXiT\VWV ZV gRT YX$ uR T\RX

fZ,RRT,X.,RTaTkZ Q^R,RefV WT = ' & Z% T\RX_T' ) "MV $ ,RTYX ZV _T!WX WT\V RV ZWRX_T e T,X_T X_T\VtRV WT he efRaZ T\V jSefXvZX,RefV

iT "aT T\RVaTYXvZ

! | s uv m|W~ |W~u s v s u |x~ !ku

WR R,RTWV WTvZ gRZ T\V Q^ VZ R %20,X sT\R gRT_T sT X hT eZYX_T\X ZV T\RVaTTVWTWVWT ) T\V_T Z YX,RefV Z% WT\V !_e fZ R T\V qT ZVe

iT\VWV gRT YX$ uR T\RX

fZ,RRT,X dR

0

1

2

3

4

5

3

〈v2 〉1/2

q

ν = 1.58ν = 1.00ν = 0.74

〈v2

r〉1/2

q

ν = 1.58ν = 1.00ν = 0.74

0

0.2

0.4

0.6

0.8

1

3

〈v2 〉1/2

q

ν = 1.58ν = 1.00ν = 0.74

〈v2

r〉1/2

q

ν = 1.58ν = 1.00ν = 0.74

Kv|EaB E)*|C)#%D!- # × :"+(12.0/",4| ! ) ?(##$)*EC)( + 12.0/,#Ej !®

<v-"!$ #%1$.0/| |a.$/ `P"!$9¬ !2(F476/#%( (? #O tKv", .0/ n@ # nEDD#%|=)3!2 1$3)a #21$D=) | 2 , aa # D #% ! (n # >)*#> 1# /9#O(#%D!1 =)*#2 1%!$a !1 A :#(# EsP"!$9¬ ! 1%!$DB)*1$ /"!2(#- #%1$.0/"! =)*1 #2D| 1$ 10)*& !S *!A1.0/? #2 ( =) |P!2#2"!2#A # NQ#O>jC)*"! 8E ! 3!A@ # = #>D¬D¬=) P!$#%9&@ B # E .0/(jE !21$& !%!2¬.0/ 1

QSRTST\RV R TSjST Z YX,RefV[Z% gRT TVWT,X_T YX$ uR T\RX hT,X_TYXiRV WTTVWTWV WT .dZ e e VWRVOZ YX,RM RXX

gRT dRTWT RXmT\RVaT ZVOX,RXvZX,RM T\VTVWTWV WT ) T\R "aT\VOXvZVhe,X

TWVWT\V R,X WT efV TYX,RM T\Vt) vZVhe,X WT WRT e T,X_T X_T\VYR ZX,RefVaT\V ' WTdRTWT WT ) vZVse,X e T RT\VOXhT,X,R X ]ZV T$ X'% 0 gRTWV T,X$,X_T\V T >

ν = = 1.58 : D⊥ = 0.67

2

_T

ν = = 1.00 : D⊥ = 0.30

2

_T

ν = = 0.74 : D⊥ = 0.23

2

_T .

QSRTvT T,RV TWV WT ZVae Z T\V ) vZVse,X e T RT\VOX_T\V hT\R ,RV T\VWTYX$ uR T\RX,X,R X Z RXvZX,RM _T !WX RX WT\V HSVOX_T WV_T TWVWR_T\V RV $ 70hT_T\RV Q e,X efVWVOX_T RX pR %wT WT ) j 0è WT WT .`T_T\R" WT YX$ uR T\RX Vae `T\RX_T T\RVaT_T\V T,X_T\V TiT\R X_T,X`T WT\V

ve aZaZVWV Z WT*tRV WTkT T X_e ZuVaT\X,RaT\V Q QS VZ R Z%

WT\V) vZVse,X ,R"YXZ ' WT Q T sTOZVdRaT\V WT Z e efVWRVWe RVWRT,X_T\Vt) vZV se,X_TuR T WV WT WTkQ ,X,R XkRV T\X `Z RX WT YX$ uR T\RXsT_T\R" 0sT_T\RV

hT\R

WT ZV RX WT j5. Q T\RVaT) T\V T WVYX_T aT\V fZVWV-ν = ≈ 0.4

0gRT ) T\V ,R ZX,RefVaT\V RXν = = 1.0

WVν = = 0.74

' WTkV WV gRT URWV WTQQ YX$ %wT RXpT\RVaT T\V,X_e efV

Ic = 1.3 Z%gRT ) T\V WVYX_T YX

"MV WR WV C T T\VWVOX ZVgRTljpT Z X,RefVWT efV T X,RM T\V ) T\R "aT\VOXvZVhe,XWVWTQSR"OX_T XZX,RefVaT\V R Z T WT YX$ uR T\RX

ν = = 1.0 Q^RT_T WVYX_TaT\RWT\V ,R" Z RXvZX,RM

VWR"OX efV WT\V TWVWR_T\V WT/ ê ,X$ uR T\RX %IZ ν = = 1.58 Z ZWRX_T e è

% |W s | u s v m|W~|W~u s !v s u

gRT WT,XvZWR R,RT_T\VWTURWVWTSQQ YX$ %wT Z% WT\V ) vZVhe,XpRVWT RX,X_T WTpjST aT\V TWRT\X gRT jST RTWV WTqT ZVe dRTpZ WT V,X,RT WT Z,R_XvZX,RaT\V $WVvZVYX_T\R ,X,R T\V 0sT_T\RV SRT T! T\R"aT ZV . T\RWRT WR WV ' Z RX WR WV C

0

0.5

1

1.5

2

32.75 3.25

σ = 0quasistatisch

turbulentgesamt

σ− = 1.0 σ+ = 1.3

〈Γ〉

q

〈n2〉q

σ− = 1.0 σ+ = 1.3

0

2

4

6

8

10

12

14

16

32.75 3.25

σ = 0quasistatisch

turbulentgesamt

〈Γ〉q

〈n2〉

q

Kv|Ea E)*|C) #2D| 1 p #$)*1$>D#%!21 ) @ #F<v.0/"!2(1%!$9#% 0 >12 / 1%!? # 5 #E< <E:P!29#2& >)a

ν = = 1.00

V WT\V vZ gRZ T\Vt _e T\VWTqT ZVe WV WT I efVZ e pRV WR WV \ T T\VWVOX ZVgRT@aefV hTfZVWVYX_T URWVWTkQ Q YX$WV dRTWT

gRT YX,RM RXX WT I efVZ e

T $aT\V WVWT\VqT ZVe _T RT_T\V YX$ WV_e sRV \ Z>,RTYX ZVlV WVT,X Z

aZ gRT jpT RTWV WT qT ZVe WThe efRaZ T\VjSefXvZX,RefV VWR"OX^V ST\RVaT

T\RV%wZaT % WV]R,XdRT7,RT . RV VZV T\RVaT[ZVZ X,RaT 4 aTWV

gR WX,RT,X#dR _efVWT,V RV gRT_T Z R_X gRTvT % WVnT\RVaT e Q Q YX$ %cT [R

Z Z T TuX_T[jSefXvZX,RefV-0hT Z T,X`T "aT !$ T R,XkZ WT10WV! R"aT qT ZVe

Q ZaT T,X ,R"gRT jpefXvZX,RefV,R"OXWV WT Z Z R sT\XvZOX_T\X_T\V]jpTaT\V TWRT\X QSRTvT TWVWR&$X,X_T ZV RX WT Z,R RVaT Z_T\V VZ _T Z VZV VWR"YX^TZ X_T\V $fVWVaT\V WV WVOX_T,X_T\R"OXnVae T\RV Z gRT14êfX `T\VgR T\RXnT\RVaT V T,RaT\V $WVWTVWR"OX RVaT Z_T\V ) T\V e WT

−5

−4

−3

−2

−1

0

1

2

3

4

5

2.75 3.253

σ = 0σ− = 1.0 , σ+ = 1.3

〈v00〉

q

〈v2 〉1/2

q 0

1

2

3

4

5

32.75 3.25

σ− = 1.0 , σ+ = 1.3σ = 0

〈v00〉q

〈v2 〉1/2

q

Kv|Ea ()* =D(, D=) DG,E1g& >)aν = = 1.00

& ! D/`P!$9# 47j; $ )a !0)3! 8/=)3! 1.$/ 6E # & P"!$9¬ !21$#%(¬&?F& !

ν = = 1.58&KvE |.$/!21 * #%!( (D.0/ #%:

!&F)*F )*/=)* 1 G)*=DG, ) =) 5| P!29#%? + #2D*!$)3!D # (*! =, .0/=)*1 D# .0/ # P!$#%9&@?& /#%( 8"(#%&F)*

.`T\XvZOX_T\X ZV V WV WT\V Z RX Vae T\RVaT_TtYX$ uR T\RX RV WR WV _e T\RuX

,R" T\RVaT vZ,X,RaT _T\VWVWT ) T\R "aT\VOXvZVhe,XbR . T_T\R" aeaTYX$ %wT ,X T VZVWT Q^R"YX_T TZ X_T\V RV ]T T\VWVYX ZV

aZ gRT RX T\RVaT vT!_e T\V

V,X,RTWT Z,R,XvZX,RaT\V VOX_T\R TWVWT\V[R,X Q T Z,R,XvZX,RaT VYX_T\R R,X iT_T\VYX R"

| s uv m|W~ |W~u s v s u |x~ !ku

!$ TSZ WT X T\VYX_T VOX_T\R WT YX$WV Q^RT'' WT WRaT hT\R: T\RVaT^ZVWT_T\V YR ZX,RefV RVgRT_T hT\RX'sT eZOX_T\X e _euX RV gRT_T Z gRTrQ Q YX$WV

`T "aTrVZ WT\V*. T\XvZYXWV T\V RV WVWRX,X

gRT tVaTuRT_T\V T WTijST,R,X,RM RXX0hTWT\V ZvZ T T\V[Q^R,RZX,RefV,X_T χ‖∇2

‖(n0 + n)RV WT Q^R"OX_T! T\R" WV T,X X

aZ % 0

aZaZ e WT , ,X_T WT j7. Q 0hT gRTvT\V

T[ZV _T\RVaT V `T\VZ T\RX!_T\V T0,X$ uXZWT T\RVaT) T\V T&WVOX_T OX iT

WT\V fZVWV

0

0.5

1

1.5

2

32.75 3.25

σ = 0quasistatisch

turbulentgesamt

σ− = 1.0 σ+ = 1.3

〈Γ〉

q

〈n2〉q

0

2

4

6

8

10

12

14

16

18

32.75 3.25

σ = 0quasistatisch

gesamtturbulent

σ− = 1.0 σ+ = 1.3

〈Γ〉q

〈n2〉

q

Kv|Ea E)*|C) #2D| 1 p #$)*12>D#%!21 ) 5 # <v.0/"!$ 1%!$9#% 0 > 12(/(Z12!5 # 5 #< <:P!$9#%& >)a

ν = = 0.74

"MV T Z XWRT T V gRTYR ZX,RefV_T TWVWR_TWT\V gZ TZ X dRWTaZ RX0,RV T\VWT

Z Z,X$ uR T\RX gRT7.dZ e efVWRVOZ YX,RM RXX _T RT,X dR WV0_e RX iT\VWR T ) T\VWV) vZVhe,XST\V,X_TaT\V fZVWV

Q& gRT T(%20,X_TYX$ uR T\RXJuZ,RZX,RefV ' WT T T\RuXaZ ,R" aZ RV

ZWRX_T T WX_T,X_T*) T\V TZ X_T\V hT\R WV V iTvT\VaT\RX WT^Q Q YX$ %wT WT Z RXvZX,RMmVWR"YX TVWT,X 4 gRT YX TWT RVaT Z_T\V . Z e efVWRVuRV ,XvZWR RXX#dR gRT TVWTWVWT Z Z,X$ uR T\RX'sT T\RV uX Q ZaT_T X,RT,XaZ RVgRT_T sT\RX hT eZYX_T\X_T) T\V TZ X_T\V[Z WT ZvZ X_T,R,X,RaT\VtR T\VZ %X WT e WT ! T\R" WV T\V WV R,X Z RXvZX,RM WVZ$VuR efVWT Z WT Z ZZvZ T\X_T

$& ! %7 $&%

"MV WT e RT T\VWT\V sT\RX dR aZkX T\VYX_T TZ X_T\V WT . Z e efVWRV qe WT\VaTJ e T%cT\VaT\V ) T\R "aT\VYXvZVhe,X`T\RVaTjSZVWR"OX Z ZRV ) )^j 2 T e T\X,RT hT\R V iT_T\VaT\RX WT,XvZX,RaT\V Q Q YX$ %cT WT WVOX_T OX SRT ' WT WT j5. Q 0è WTZV gRT ) )^j 2 T e T\X,RTmZV TZuX'WV T\RV e WT %20 aZ QQ YX$ %wT [RV WT

m/n=12/4

efV !vZX,RefV R T T\VOX,RT,X Q^RTmj7. Q e WT ! T\R" WV T\V hTZVWT VgRT _T,R,X,R T\V.dZ eYefVWRV qe WT\V RV T TYX_e_XvZX,RaT 4 aTWV TWRTWT\VaT j5. Q YR ZX,RefVaT\V T\RVaTtT e gR_RT,X_T\V Z ZtR ê ,X$ uR T\RX_TuR T,RV T(%20,X#è WT\V

aZ I vZ T\VWRT WT`VWR"OX RVaT Z_T\V T_T dRWV_e

T_T dRaT\VrQSR"OX_TT T X,RaT oefX_T\VYX,RZ WV WTQ Q RV RT,X_T\V ZuVaT\X,RaT\V YX$

WV T,XV R" ZaT\V QSRT Q^R,RefV WTlj5. Q e WT ! T\R" WV T\V Z kT\RV HSVOX_T e WT WT T T X_e ZuVaT\X,R aT\V6^RT %cT e WT ZOXWTWX R"

aZ WT ZVae Z T ) T\R "aT\VYXvZVhe,XkZ sT\R0 R

aT\V Q Q YX$ %wT ,X T\V ZWX R"gRT × . Q^YVZ R hT,X,R X R_X Q ZV T,RaT 5 sT,R T\VYX

T\RVaT ) )^j jpTWV WV RX T\RVaT ,R RT,X_T\V)^j

H j jST(%cT_T\V%IZ RX VZaT ! T\R"aT\V Z ZZvZ T\X_T,V T! T\R"aT\VTuRWX

aZ

gRT jpT Z X,RefVWT ) vZVhe,XiZ% T\RVaT T 5 X_T,VaT qZuVaT\X %wT ,X$WVVWR"YXaZ efV[Z$VuR R_XeaZYX$ %wT Z% WT Se%wT _T\RX_TdRT sT\R) )^j e WTSZ% WT74SRT,R%wT _T\RX_T dRT

sT\R )^j H j efZ R_RT,X R,X RX T\RVaT _ ,X_T ZX,RaT\V HpVYX_T WV tRV _X,RTWV WTtZ,X,R RT T\VnQ^R,RZX,RefV fZVWV T T\RuX iT WT\V

aZgRT _R RT,X_T Z ZX T\V _e ie R WV T,X$,X_T\V Z Z mZ

sT\R e gR,RTWV)vTnT aV R" Z% gRT Z WT0vT\V _TOX_T\VUQ % WVZvZ T\X_T_T ZuRT,X RXWT\V e T,X_T X_T\V TWVWR_T\V dR WTWX R"

aZ gRTqZ,X,R RT T Q^R,RZX,R

efV T\RVaT `T\RWTWX,R T jpe T+WRT X > I T\RVaT\V % 0,X _RT hT_efVWT hT\R#!_e T\V _T\V _TOX_T\V T T\V Z T\V hT\V $fX,RuX_T\V WXvZVse,X efV XZX,RefV_T\VaTuRT WV ZX aZ RXkT\RVaT,XvZWR R,RT_T\VWT RWV Z% gRT) T\V

ZVWT_T\V ,TWe UT $OX gRT Q^R,RZX,RefV

WT\V) T\R "aT\VOXvZVhe,X RV jpTuRefVaT\VRV WT\VaT\V _RT gRT TaTWV WT T ZVe _T RT,X

Q^T\VWV gRT Q^R,RZX,RefV VWR XnRVWT.e,X,R RXX! T\R" WV Z%gRT se efRaZ T jpefXvZX,RefV WT pRVYX_T!WV Z Z tRV Q^RT TaTWVgRT_T pRVYX_T!WV,X$ WVrR,X T\RV `T_T\VOX R"aT Q % WV TZVWR % 0 WT\V]ZVae Z T\V ) T\R "aT\VYXvZVhe,X QSRT HpVYX_T WV T\V T\R T\V `T\RX_TWRV

aZ gRT^VWR"OX RVaT Z_T\V T_T dRWV T\V RXWT Q^R"OX_TWRVYX_T!WVWT

Y | s uv ~ |- v? 8|W~~

dR"OX,R_X_T T ,X_T! RTWV TZVWR %20gRT) T\VSeWVaT V `T_T\VaT\RXiT\RVaT YX$ ZuVaT\X %cT WTSRVWT hT\XvZOX_T\X_T\Vqe WT R,X .`T\RaT\RV TZ X_T\X_T Q Q YX$ %wT uX,,R"lT\RVaTdQ T,XvZWR R,RTWV WT ) T\V nT\RVaTjpT X,RefVWT jpefXvZX,RefVJ TaTWV hT eZOX_T\V Q^RT_T jpT X,RefV7fZVWV T\RVaT\VT\RVaT Z,R RVaT Z_T Q % WVbT\VOX_X_TaT\V WV ZVWT_T\V T\RVaT,X$WVsTgRVuX_T jSe XvZX,RefV aTJ e T%wT\V iT WT\V 4SThT\V gRT_Te gR fZX,RefV WT pRVYX_T!WV T dRVgR T\RX R,X gRT Z WVWT QSR"OX_T WRVOX_T!WV_e aZ Q Q YX$ %cT T\RVTsT\V_elT\VYXaT\RWT\VWT _e T vT\V WV efV X T\VYX_T\V XZX,RefVaT\V Q^RT_T Z WV e X gRT tVOX,X_T WV WT,X$ %cT hTgRVuX_T\V ) T\R "aT\V _T ,XvZVWT

iT "aT gRT vZ gRZ T]ZuVaT\X %wT e se

VaT\VOX_T aTJ e T%wT\V dR ,&dR RVgRT_T sT\RX ZhT Z T T\RuXaZ gRT YX_eZ,X,R

,RTWVWTqZuVaT\X %wT RVWRT\V Z T\RV7aefV T\RVaT\V ThT\V_e !_e T\V RV bZ% gRT^X T\VOX_T\V XZX,RefVaT\V ZX Q^RTaT RVWRT\V,X_eZ,X,R_RTWV hTdR X gRT _T\VWVWTpQ^R"YX_T XZX,RefVaT\V WV[T TYX,RaT\VtoefX_T\VYX,RZ YXZX,RefVaT\V

iT "aTmT 5 sT,R T\VYX_T V! R"aT2$

T\V _RVWVqR,X efVWT\V ,X$WVsTgRVuX_T\V SRVOX_T!WV_e e gR fZX,RefVaT\V WT QSR"OX_T

WV 2 T dRVgR T\RX& WVOX_TaT\RWT\V QSRTYX T WT X T\VYX_T\V) T\R "aT\V _T& T\RWX R RX,X_T &0hT aZ'hT\XvZOX_T\X_T Z Z e T\V sT\R V iT_T\VaT\RXWTlQ Q YX$WV VZaT WV TVWT,X Q^RTR,X gRTe T WT hT Z TWV efV YXZX,RefV_T RTWVYX_e Z,X,R,RTWV WV _X$WVsTgRVuX_TQSR"OX_T_e Z Z WV

RX XZX,RefVJ T,XWV gRT T,RV TWV WT'!_e uZ

R T\V se efRaZ T\VqjpefXvZX,RefV , dR iT\RX_T T T\RuX

aZ gRT ,XvZX,RaT Q Q YX$WV hT\R !$ T iT WT\VWT\V YX$ %cT

,X T\V T\RVaT\V V,X,RT WT Z,R,XvZX,RaT\V VYX_T\R RVWT\V Q^R"YX_T WV oefX_T\VYX,RZ ,X$WV T\VsTdR X VZV WTe WT\VhTYX_T\V dR hT TuX

aZ WT Z,R,XvZX,RaT VOX_T\R e,VaT

R" RVWT\V !_e fZ R T\V YXYX_T\VWT Z ZpT\VOXZ X_T\VqR,X % gRT_T T\RvTuX ,R"gRT YXYX WT qZuVaT\X_X$ %cT WT WT X T\VOX_T\V Z ZnZ% "MVgRT_T hT\RX efVWVYX_T aZ RX% 0 T\RV ) )^j Z Z sT TuX&iT WT\V

aZ WT TvZ X_T

efV T X,RM T ) T\R "aT\V `T "aT WT\V ZVae Z T\V ) T\R "aT\VYXvZVhe,X aZ,X_T X

R e T\V

RX,X_T sT\R]ZuVaT\X %cT WT e gR,RTWV ! T\R" !_e T\RWXZhT ! T\R" T\RX,R T\RVaT aZ

QQ T sT,X,R X_T T RT X_T R"aT YXYX,RTWVWT) T\R "aT\V _TmT_T\R"OX iT

WT\V fZVWV Q gRT Z,RZX,RefV WT Z Z,X$ uR T\RX7 efVWVOX_TgRT V `T\VWV!_T\V T WT[j7. Q _e!vZ sT_X,R X iT WT\V QSRTvT2&_T\V T RTuX % 0 ) )^j YR ZX,RefVaT\V dRaT\V

n = 5 · 1019 −3 Te = 40T WV

n = 4 · 1019 −3 Te = 50T YRT ZOX@,R"aZ

sT TZaZ sT\RiT\RVaT hT,X,R X_T\V T\RVaT\V YX$ uR T\RX gRTnX T\VOX_T YX,RM RXX RV WT

YR ZX,RefV V R"t T dRVWT\XaZ gRT X_T\RhT\VWT RVaT Z_T "8V,XvZWR RXX WT/.dZ eYefVWRV e

WT\V RX ,RV T\VWTYX$ uR T\RX T\RVaTdR QSRT ehT\V gR WX,RT,X_T\V TWVWRvThT 0! R"WTdZVae Z T\Vt)oT\R "aT\VOXvZVse,X T\RsT\V Z RXvZX,RM[Z TZ X_T\V

iT\VWV ZVgRT Z Z,X$ uR T\RX Z,RZX,RefV WT Q^R"OX_T WV WT

) T hTvZX TVWT,X Q Z RX dR T T\RuXaZaZkRV gRTvT hT\RXWVYX_T YX_T Z R

XvZX,RM T) T\V TZ X_T\VUZ WT ZvZ X_T,R,X,RaT\V R T\VZ %cX WT qe WT ! T\R" WV T\V%ce uX

- -)*'&%& ! %&'& )*<!"$&%

"MV gRT_T VZV dR aZ !WVWT RT T\VWT e e gRVZX_T\V, ,X_T WV gRT sT\V WX X_TqZ uVaT\X %cT gX_ehe euRT WV$,X Z T T\RV WV aZVWVmRV WT hT RT T\V 4 aTWV WT Z 2 T e T\X,RT e T,X_T X QSRT sT\XvZOX_T\X_T]ZuVaT\X %cT gX_ehe euRT# TOX RV gRTiQ T%aVWRX,RefVaT\V WT TWRTWT\VaT\V sTvZX_e_T\V RXST\RV

8 +( ( *!#(( " * $

2 T TsT\V7_T\RT\RVaT RV T,X,RTZ_T ) vZV%ce ZX,RefVR(u1, u2, u3)

RX WT\V e e gRVZX_T\Vu1 = r

u2 = θWV

u3 = ϕ_e dRT WT\V) vZV %we ZX,RefV! T\R" WV T\V

x = (R0 + r cos θ) cosϕ , y = r sin θ , z = (R0 + r cos θ) sinϕ , '

iehT\RR %ce T\VWT\VgRT 0WVR = R0 + r cos θ

T`T\VWT\X dR R0

R,X WT# T,X WT!_e T\V) evZ gR

3_T\V

rgRT eYe gRVZX_T WT T\RVaT\V Z ZvZ gRsT T\R"WVaT\X

Q Z RXST$ X ZV gRT e fZ,RZVYX_T\V .dZ,RJ T X_e_T\VZ 12 / 2! 1; 1 = = /k4 / 2 T\VZVWVOX >

er =∂R

∂r= (cos θ cosϕ , sin θ , cos θ sinϕ)

eθ =∂R

∂θ= (−r sin θ cos θ , r cos θ , −r sin θ sinϕ)

eϕ =∂R

∂ϕ= (−R sinϕ , 0 , R cosϕ)

QSRT O ,R uZ RaT\V tRVaT\RXJ T X_e_T\VZ 2W/m =12 /2! / 2 /k4 / 2 T\VZVWVOX

,RV WT%

VWRT,X

ei :=ei

|ei|=∂R

∂ui

/∣∣∣∣∂R

∂ui

∣∣∣∣ =ei√|gii|

RXi = r, θ, ϕ

Y o|x ! s | uv |W~uv $

WV RX WT\V e sefVaT\VYX_T\V WT./ = /2 / 2=4 =

grr = 1 grθ = 0 grϕ = 0

gθr = 0 gθθ = r2 gθϕ = 0 . $

gϕr = 0 gϕθ = 0 gϕϕ = R2

Q ZlV gRT ^ZWX gRZ efVZ e hefVaT\VOX_T\V WT T\X,RaT\V)oT\V_e^T\RVaT\V T,X WV! T\R",4 ZVWVaT T\V

ZVWT XT ,R" sT\RWT\V )oe_efRaZ eYe gRVZX_T\V T\RVre,Xae efVZ T eYe gRVZX_T\V

, _X_T 4 aT_T*tRV T aT\RX_T\V mZ T T\RVaT\V<. T_T\RWVt efV RVWR T\V eYe gRVZ X_T\V, _X_T T\V WV )oT\V_evZVZ _R7aVWT\X ZV RV

$ * $ & $"! $ :* ($

Q ZpT\RV RT T\VWTqZuVaT\X %wT WT Z Z R,X' T ThT\V

Br = 0 , Bθ =B0R0

q(r)R2, Bϕ =

B0R0

R2.

C Q ZhT\R R,X

B0WT7. T\XvZWT ]ZuVaT\X %wT WTSZV WTYX_T T WT'!_e T\V Z ZvZ gR

R0

RX WT\V Y _R fZ RaT\V e sefVaT\VYX_T\V

Bi =√|gii|Bi \

T$ X ZV

B2 = B20R2

0

R2

(1 +

r2

q2(r)R2

).

QSRTpRV WVWRX,X $ hT\V WX X_T\V sTvZX_e_T\V VaT T\V sT\R!)oe_efRaZ eYe gRVZX_T\V %we T\VWT e fZ,RZVYX_T Q Z,X_T WV ZV?>

∇ = er ∂r + eθ1

r2∂θ + eϕ

1

R2∂ϕ

v · ∇ =RBϕ

r B2

(∂rφ ∂θ − ∂θφ ∂r

)+r Bθ

RB2

(∂ϕφ ∂r − ∂rφ ∂ϕ

)

K =RBϕ

r B4

(∂θB

2 ∂r − ∂rB2 ∂θ)

+r Bθ

RB4

(∂rB

2 ∂ϕ − ∂ϕB2 ∂r)

∇‖0=

Bθ

B∂θ +

Bϕ

B∂ϕ

∇‖ =RBϕ

r B2

(∂θA‖ ∂r − ∂rA‖ ∂θ

)+r Bθ

RB2

(∂rA‖ ∂ϕ − ∂ϕA‖ ∂r

)

ε 1 v Y

∇2⊥ = ∂2

r +

(1

r2−(Bθ

B

)2)∂2θ +

(1

R2−(Bϕ

B

)2)∂2ϕ +

(1

r+

cos θ

R

)∂r

−sin θ

r R∂θ −Bθ ∂θ

(Bθ

B2

)∂θ −Bθ ∂θ

(Bϕ

B2

)∂ϕ − 2

Bθ Bϕ

B2∂θ ∂ϕ

pRThT\R ' WT gRT 0WV∂ui

% 0 gRT Z,X,RT T T\RXWVSVZ WT-:T`T\R R T\V eYe gRVZX_T

∂/∂uihT\V WX X

ε 1

I! #"%$'&)(*$' *$'+ $' *$'&)+-,/.10*23!+!465!$' 87 $' 9.:(*;1 *$'+<&)+=>"#&) *55-&?$ 4 2@$' !+!4A5!$CBD1EFGH'I KJ/L H NMPO HQERTS1U M I@V JJ H J 4W$'XA.1092(Y>[Z\.1BK=8B 7 $']T( $' 2 :^ (+!&?B

A&?B(85!$ +!&?$' (_. ^ B @;1(&?$'+(_.:@B`4 *; $'X ) ; @B 9.15-&)@B

R0!+@5a] ^ $'&)+@$'X ) ;1 @B* 9.15-&)@B

a>A = R0/a

>cb 0 `dc$'( 9.10*2(*$'(*$feg$' (*$5!$' ` 9.15-&h. ^ $'+i;T;1 *5-&)+j.:(*$

r 5!$ +!&?$' (K5-! *0*2lk ^ .1BXA.:dc$' *$'&?0*2ad!mC"n>joi$C092@$'+!4W$'d!&?$'( 5- p@0*]T(8B&?092q5-&?$CB*$ 4`r 2@$Qs !+!4f&)+<5!$' ioi$ ^ .:(&?;1+<,tp! $'&)+<&)+3uW$' *B*$CBi=`B 7 $']3(uW$' 2 r:^ (+!&?B8.:@B >

ε :=r

R0 1 . v => Ww

bxp! ùT&?$ ^ $yx;1]1.:XA.:]-BYz"#&?$.:@092a,tp! Ky#|Ky o_z&?B(`5-&?$CBK$'&)+@$d! 9.:@0*23dj.: *$~!+@5ap!d ^ &?0*2@$ 4`r 2@$Qs !+!4@>+f5-&?$CB*$' 4Kr 2@$' !+!4n"%$' *5!$'++! %+@;3092A=8@B*5- p@0*]W$i&)+A+ ^)^ (*$' *5-+!+!4εdc$' p@0*]-B&?0*23(&)4(Yz. ^ B*; O(ε) = 1

> H +(*$' . $' p@09]TB&?092(&)4!+!45!$' l.:4+@$'(,$ ^ 5-]W;1X 7 ;1+@$'+3(*$'+ v =>CWwnuW$' *$'&)+-,/.10*2@$'+B&?0*25@.:X~&)(`. ^)^ $ 87 $' 9.:(*;1 *5@.: *B*(*$ ^)^ !+!4W$'+qmC∇ = er ∂r + eθ

1

r2∂θ + eϕ

1

R20

∂ϕ

v · ∇ =1

r B0

(∂rφ ∂θ − ∂θφ ∂r

)

K =2 sin θ

R0B0∂r +

2 cos θ

r R0B0∂θ

∇‖0=

1

q(r)R0∂θ +

1

R0∂ϕ

∇‖ =1

r B0

(∂θA‖ ∂r − ∂rA‖ ∂θ

)

∇2⊥ = ∂2

r +1

r∂r +

1

r2∂2θ

v =>Ww

Z8&?$ v ε 1ws 4Kr 2@$' !+!4$'+3(*B 7 &?0*23(K5!$'Xidc$' 4.:+!4AuW;1+a$'&)+@$' i(*;1 *;1&?5@. ^ $'+ 2 $C;1Xf$'( &?$mCq$'&)+@$' mC ^ &)+@5- &?B*092@$'+ 2 $C;1Xf$'( &?$>xW$C5!;T0*2d ^ $'&)dc$'+2!&?$' "%$'&)(*$' 2!&)+(*;1 *;1&?5@. ^ $6K p!X~X!+!4WB*$Q[$']3(*$dc$Qs p@0*]-B&?0*23(&)4(YzW".1BB&?0*2f5-! *092A5!$'+~+!&?0*23(uW$' *B*0*23"#&)+@5!$'+@5!$'+K p!X~X!+!4WB*; 7 $' 9.:(*;1 K .:@B*5- p@0*]T(Y>

$! #" $&%('*) + -,/. 021*Z8&?$43jU6587 O:9 S:R-H'E<;TI3DB(*$ ^)^ ($'&)+@$6"%$'&)(*$' *$=$' *$'&)+-,/.10*2!+!4g5!$' 2 $C;1Xf$'( &?$5@.: Y>x=8@B4W$'2@$'+@5uW;1+5!$' v ε 1

ws 4Kr 2@$' !+!4 d$'( 9.1092(*$'(KXA.:+q+3!+l+! K$'&)+@$'+lB9$'2! 8B*092!XA. ^ $'+a 9.15-&h. ^ $'+ . $' *$'&?0*2l!X$'&)+@$\d$CB*(&)X~X_(*$_k ;B&)(&?;1+r02@$' !X X~&)(

r = r0 ± ∆rz!B*;~5@.>

r0∆r

1 v =>6?@Ww4& ^ (Y> Z\.:X~&)( ^hr >( B&?0*25-&?$mC ^ &)+@5- &?B*0*2@$*AT0*2j. ^ $ v oi$C0*2@$'+!4W$'d!&?$'(w . ^ B +j.:2@$'mCCB3j.15!$' ,D1 X~&)4\.:-[.1BsB*$'+ v l.:+B(*$ ^)^ $nB*&?0*2q$'&)+@$'+l.:-,t4W$CB90*2!+!&)((*$'+@$'+l!+@5.:@B*$'&)+j.:+@5!$' `4W$' *; ^)^ (*$'+ I ^ &)+@5!$' &)+!46uW;1 Y> wQ>bxp! 85-&?$ 87 $' 9.:(*;1 *$'+q$' 4&)d!(iB&?0*2qB*0*2 ^ &?$E> ^ &?092

∇ = er ∂r + eθ1

r20∂θ + eϕ

1

R20

∂ϕ

v · ∇ =1

r0B0

(∂rφ ∂θ − ∂θφ ∂r

)

K =2 sin θ

R0B0∂r +

2 cos θ

r0R0B0∂θ

∇‖0=

1

q(r)R0∂θ +

1

R0∂ϕ

∇‖ =1

r0B0

(∂θA‖ ∂r − ∂rA‖ ∂θ

)

∇2⊥ = ∂2

r +1

r20∂2θ .

v =>6?F? w

=i@092 2!&?$' d ^ $'&)dc$'+ 5-&?$lK p!X~X_!+!4WB*$Q[$']3(*$gd$' p@09]TB&?092(&)4(Y>Z`&?$CB*$q4W$C;1Xf$'( &?B*092@$ 4`r 2@$' !+!4"#&) *5&)X o . HG1Z uW$' "%$'+@5!$'(Y>

$ ' !") -$ )*!") - ' ! %

Z8&?$=8@B".:2 ^ 5!$' nbj;1! &?$' Xf;35!$'+g"#&) *5lX~&)(n5!$' \e .:2 ^ 5!$CBno8$C0*2@$'+!4W$'d!&?$'(*BK,$CB(4W$ ^ $'4(Y>8&?$' ,tp! B(*$'2@$'+<&)X o . HG1Z 5- *$'& uW$' *B*0*2!&?$C5!$'+@$ 2 $'d!&?$'(*$\mC! i=8@B" .:2 ^ >@B#4& ^ (q n ≤ m ≤ n q X~&)(

n = 0,∆n, 2∆n, . . . , n , v . >6? w"%;1dc$'&[email protected]_o8$C0*2@$'+!4W$'d!&?$'( v ;12!+@$6k-[$' mY;1+@$'+w\5-! *0*25-&?$~eg$' (*$q !+@5 q ,$CB(4W$ ^ $'4(_&?B*(v B&?$'2@$_.:@0*2 v G-> G WwiwQ>jBX>65@.:+!+q5-&?$_XA.T&)XA. ^ $y ;1 *;1&?5@. ^ Xf;T5!$'+!m .:2 ^ n B*; "#&?$_5-&?$n4.:+!m'sm .:2 ^ &)4W$/AT0*2! &)(("$'&)(*$

∆n,$CB(4W$ ^ $'4("%$' *5!$'+ >WZ\.:dc$'&-X_>

n $'&)+4.:+!mCm .:2 ^ &)4W$CB =8&?$ ^ ,/.10*2@$CBuW;1+∆n

B9$'&)+ >

0

m

n

q n

q n

n !vE|a#"+%$.0/ &F)3!1$.0/ < )*#%12!2aF # D =)*12,-)*/| )*1+ & =D#O #2#2)*&> = 12!(? 1 - .0/ | !21 D/'& # D 0

4 .1092-,P; ^ 4W$'+@56B&)+@5~,Pp! . ^)^ $X D14 ^ &?092@$'+o8$C0*2@$'+!4W$'d!&?$'(*$`5-&?$ . $' *$C0*23!+!4WB,P;1 Xf$ ^ +5!$' 2 $CB9.:X_(Xf;:s5!$'+!m .:2 ^#mn

.:+!4W$'4W$'dc$'+ z "%$ ^ 092@$A,Pp! 5-&?$A+3!Xf$' &?B*0*2@$ X 7 ^ $'Xf$'+(&?$' !+!4 4W$Q,P!+@5!$'+"$' *5!$'+]W;1+!+(*$'+!+@5<"%$ ^ 0*2@$$'+(*B*092@$'&?5!$'+@5mC! =8d@B*092 r (mC!+!4AuW;1+o8$C0*2@$'+!mY$'&)(*$'+dc$'&)( 9.:4W$'+<] D1+!+@$'+ >ZK.:dc$'& d$'"#&) ]T(5-&?$=8+"$'+@5-!+!4 5!$CB~+(*$'4W$' s 87 $' 9.:(*;1 (int)

.:-,8$'&)+@$ *$C$ ^)^ $ Ic.:2 ^ [email protected]=8d-sB*0*2!+@$'&?5!$'+5!$' %ZK$'mC&)XA. ^ B(*$ ^)^ $'+ +j.10925!$'X i;1X~XA.;12!+@$ò!+@5!$'+ zB9;_5@.>XA.:+$'&)+@$i+j.:(p! ^ &?0*2@$Ic.:2 ^ $' 2 r:^ (Y>ZK$' ~a;35- ^ ;:s 87 $' 9.:(*;1

(%)d$'"#&) ]T(~5-&?$ . $' *$C092!+!+!45!$CB4.:+!mCm .:2 ^ &)4W$'+o8$CB(*B$'&)+@$' iZ`&)u3&?B&?;1+X~&)(imC"%$'& 4.:+!mY$'+ Ic.:2 ^ $'+ >

C@ 2 8

2.75n ≤ m ≤ 3.25n

&)(i5!$' =8d!]Tp! mC!+!4N

′

mn =(n

∆n+ 1)(n

4+ 1)

!+@5<5!$' ibj. ^)^ !+3(*$' *B*0*2@$'&?5-!+!4,Pp! 5-&?$ni;1 *$']T(! (*$' Xf$∆i

(n ∆n

)%4 =

0;T5!$'

3 =⇒ ∆1 = ∆3 =n ∆n

− (int)( n

4∆n

)

1;T5!$'

2 =⇒

∆1 =n ∆n

− (int)( n

4∆n

)− 0.5

∆3 =n ∆n

− (int)( n

4∆n

)+ 0.5

B*; "#&?$∆2 = (int)

( n 2∆n

+ 0.8)

$' 2 r:^ (iXA.:+<5-&?$ 2 $CB.:X_(m .:2 ^ 5!$' id$' p@09]TB&?092(&)4(*$'+lq;T5!$'+<5-! *0*2

#mn =

(int)(N′

mn) ,"$'+!+

∆n%4 = 0

(int)(N′

mn − ∆1) ,"$'+!+

∆n%4 = 1

(int)(N′

mn − ∆2) ,"$'+!+

∆n%4 = 2

(int)(N′

mn − ∆3) ,"$'+!+

∆n%4 = 3

v . > Ww

2.5n ≤ m ≤ 3.5n

N′′

mn =(n

∆n+ 1)(n

2+ 1)

; ∆2 = (int)( n

2∆n+ 0.8

)

2 $CB.:X_(Xf;T5!$'+!m .:2 ^#mn =

(int)(N′′

mn) ,"$'+!+

∆n%2 = 0

(int)(N′′

mn − ∆2) ,B*;1+@B*( v . > GWw

2n ≤ m ≤ 3n

2 $CB9.:X_(Xf;T5!$'+!m .:2 ^#mn =

(n ∆n

+ 1)(n

2+ 1). v . > $3w

$ ) %

+5-&?$CB9$'X =8+!2j.:+!4 B*; ^)^ 5-&?$ . $'4 & [Bd!& ^ 5-!+!4 E;5 J V JQM 5 M V J Radc$'4 p!+@5!$'(A"$' *5!$'+ >% B(~XA.:+ .:+$'&)+@$' n=i+j. ^ -B*$5!$' nB(9.:(&?B*092@$'+ DB!+!4<5!$CB\2!&?$' Kdc$'2j.:+@5!$ ^ (*$'+ y ! d! ^ $'+!mCXf;T5!$ ^)^ B_&)+3(*$' *$CB*B&?$' (Yz"#&?$$CBid$'&?B 7 &?$ ^ B*"%$'&?B*$d$'&)X A3(@5-&)!XuW;1+ J'M 5 M VPFIS1E*HQI V E7*HQU I5!$' `b. ^)^ &?B( zjB*;A&?B(imCadc$Qs p@0*]-B&?0*23(&)4W$'+ zW5@.>8&)X . ^)^ 4W$'Xf$'&)+@$'[email protected]$'&)(X~&)((*$ ^ 5!$' A3((D1 !+!4WB ^ DB!+!4W$'+~+!&?0*23(5!$' B*(9.:(&?B*0*2@$'+ DB!+!4A5!$CBa;T5!$ ^)^ BTB*(*$'XfB$'+(*B 7 &?092(Y>Z`&?$CB#d$'4 p!+@5!$'(8B&?0*2<"#&?$\,P; ^ 4( >2 $'4W$'d$'+qB*$'& [email protected] 2`^ $'&?092!+!4WB*B-B(*$'X

∂t f = N(f ,f) + L(f) , v 0 >6? w"%$ ^ 0*2@$CBa$'&)+ y ! d! ^ $'+!mCXf;35!$ ^)^ B-B(*$'X uW$' *$'&)+-,.1092(l5@.: *B(*$ ^)^ $'+ B*; ^)^ >Z\.: &)+ $'+3(2j. ^ (*$'+ &?B(<$'&)++!&?0*2( ^ &)+@$Y.: *$' Yz*B3j.15- 9.:(&?B*092@$' 87 $' 9.:(*;1 N

!+@5 $'&)+ ^ &)+@$Y.: *$' 87 $' 9.:(*;1 Lz"%$ ^ 092@$ad$ ^ &?$'d!&)4W$ r !X ^ &?0*2@$\=8d ^ $'&)(!+!4W$'+ $'+3(2j. ^ (*$'+ >-Z8&?$ DB!+!4

fB*$'(mY$8B*&?0*2mC@B9.:X~Xf$'+<.:@B$'&)+@$'XmY$'&)(!+j.:d-s2 r +!4&)4W$'+!+@5$'&)+@$'X mY$'&)(9.:d!2 r +!4&)4W$'+l=i+3(*$'& ^

f = f + f . v 0 > WwZ`$' #mY$'&)( ^ &?092u.: &)&?$' *$'+@5!$_=8+(*$'& ^ "#&) *5b ^ !]T(j.:(&?;1+<4W$'+j.:+!+3(i!+@5<$CB#4W$ ^ (*$0 = 〈f〉t

!+@5f = 〈f〉t , v 0 > GWw"%;1dc$'& 5!$' #mY$'&)(!+j.:d!2 r +!4&)4W$_=8+(*$'& ^ .:@B$'&)+@$' I$'&)(X~&)(( ^ !+!4A5!$' DB!+!4f4W$'"%;1+!+@$'+"#&) *5[z-mW> . >5-! *0*2

〈f〉t :=1

T

t2∫

t1

f dtX~&)(

T = t2 − t1 . v 0 > $3w=i@092$'&)+@$\dc$ ^ &?$'d!&)4W$_.:+@5!$' *$nZK$@+!&)(&?;1+q5!$CB I$'&)(X~&)((*$ ^ B8&?B(5!$'+!]Tdj.: Y>=i@B 2K^ $'&?0*2!+!4 v 0 >6? w!+@5 v 0 > Ww%,; ^ 4(#!+!X~&)((*$ ^ dj.:

∂tf = N(f , f) + N(f , f) + N(f , f) + N(f , f) + L(f) + L(f) . v 0 >CWwA3@0*2(8XA.:+<+!++j.10*2q5!$' B*(9.:(&?B*0*2@$'+ DB!+!4fz@B9;X_>A5-&?$CB*$_5-&?$ 2`^ $'&?0*23!+!4

0 = N(f ,f) + L(f) v 0 >Ww= N(f , f) + N(f , f ) + N(f , f) + N(f , f) + L(f) + L(f) v 0 >w

8 2 0$' ,tp ^)^ $'+ > 4 !+B9$'&T5-&?$e &?5!$' *B 7 @0*2@B9.:+!+j.:2!Xf$z5@.>

f = f5-&?$B*(9.:(&?B*0*2@$ DB!+!4\5!$' 2`^ $'&?092!+!4

v 0 >Ww&?B(Y>@Z\.:+!+X_>5-&?$ 2`^ $'&?0*23!+!40 = N(f , f) + L(f) v 0 >Ww

$' ,tp ^)^ (iB9$'&)+ >@=i@B#5!$'X =$' 4 ^ $'&?0*2uW;1+ v 0 >Ww!+@5 v 0 >w $' ]W$'+!+3(iXA.:+ z@5@.>A5@.:+!+N(f , f) + N(f , f) + N(f , f) + L(f ) = 0 v 0 >Ww

4W$ ^ (*$'+X_p>(*$>Z8! *092 I$'&)(X~&)((*$ ^ !+!4q5!$' \! *B 7 p!+!4 ^ &?0*2@$'+ 2K^ $'&?0*2!+!4 v 0 >CWwK$' 2 r:^ (\XA.:+gmC@B9.:X~Xf$'+ X~&)(5!$'+Z`$@+!&)(&?;1+@$'+ v 0 > GWw#5-&?$\o8$ ^ .:(&?;1+〈 ∂tf 〉t = 〈 N(f , f ) 〉t + 〈 N (f , f) 〉t + 〈 N (f , f) 〉t + 〈 N (f , f) 〉t + 〈 L(f) 〉t + 〈 L(f) 〉t

= N(f , f) + 〈 N (f , f) 〉t + L(f) . v 0 >6?@WwZK.: 9.:@B,P; ^ 4(#X~&)(85!$' iZK$@+!&)(&?;1+q5!$CB I$'&)(X~&)((*$ ^ B v 0 > $3w 5!$' I@B9.:X~Xf$'+!2j.:+!4

f t2 − f t1T

− 〈 N(f , f)〉t = N(f , f) + L(f) . v 0 >6?F? w&)+ =$' 4 ^ $'&?0*2fX~&)(5!$' =i+!+j.:2!Xf$ v 0 >Ww mD14W$+!+,tp! dc$ ^ &?$'d!&)4W$ I$'&)(&)+3(*$' u1. ^)^ $

T5-&?$b@;1 *5!$' !+!4

f t2 − f t1T

!= 〈 N (f , f) 〉t v 0 >6? Ww

+j.10*2B*&?0*2 > X . ^)^ 4W$'Xf$'&)+@$'+ &?B( $C5!;T0*2 +!&?0*23($' *B&?092( ^ &?092 zx5@.>,Pp! n$'&)+ B 7 $'mC&?$ ^)^ $CB I$'&)(&)+3(*$' u1. ^)^T5-&?$n&)(( ^ !+!46$'&)+@$CB#+!&?0*23( ^ &)+@$Y.: *$'+ 87 $' 9.:(*;1 *B&)+<$'&)+@$'+ ^ &)+@$Y.: *$'+qZ8& [$' *$'+!mY$'+j.:@B*5- @09]<p!d$' s,tp!2! ("$' *5!$'+].:+!+ >

+5!$' 2!&?$' uW;1 ^ &?$'4W$'+@5!$'+ =i dc$'&)(_"#&) *5 $C5!;T0*2 .:@B*B*092 ^ &?$E> ^ &?0*2 5-&?$'I$'&)(X~&)((*$ ^ 5!$@+!&)(&?;1+ v 0 > $3wuW$' "%$'+@5!$'(Yz5-&?$<&)+ &)2! *$' f5-&?B]3 *$'(*$'+ b@;1 X &)+ 5-&?$<ZK$@+!&)(&?;1+5!$CBf.: &)(2!Xf$'(&?B*092@$'+ I$'&)(X~&)((*$ ^ Bp!d$' 4W$'23(YzB*; "#&?$_5-&?$kx;1&?B*B9;1+!] ^ .:X~Xf$' v G-> wi. ^ Bi+!&?0*23( ^ &)+@$Y.: *$ 87 $' 9.:(&?;1+ >bxp! 7*HQU VH7'V DH I$'&)(s&)+(*$' u. ^)^ $ ^hr >(iB&?0925@.:+!+5-&?$\b@;1 *5!$' !+!4 v 0 >6? Ww+!&?092(iXf$'2! i2j. ^ (*$'+ >ZK.:X~&)(#&?B*(4W$'mY$'&)4(Yz!5@.>f5-&?$`mY$'&)( ^ &?092<4W$'X~&)((*$ ^ (*$'+ 2 (D >W$'+ 〈 n 〉t!+@5 〈 φ 〉t

^ $C5-&)4 ^ &?092$'&)+@$ 4Kr 2@$Qs !+!4f,Pp! 5!$'+qB(9.:(&?B*092@$'+=8+(*$'& ^ 5!$' DB!+!465!$CB2!&?$' idc$'( 9.10*2(*$'(*$'+ly ! d! ^ $'+!mCXf;T5!$ ^)^ B-B(*$'XfB5@.: *B(*$ ^)^ $'+g] D1+!+@$'+ >[ZK$CB"%$'4W$'+ "$' *5!$'+ 5-&?$CB*$mY$'&)(4W$'X~&)((*$ ^ (*$'+ 2 (D >W$'+ . ^ B&B3j.1B&?B(9.:(&?B90*2@$~=8+-s(*$'& ^ $5!$' DB!+!4W$'+n!+@5

φdc$'mY$'&?0*2!+@$'(Y>e &?$n4!(85-&?$CB9$ 4Kr 2@$' !+!4A,Pp! i5-&?$_B*(9.:(&?B*0*2@$ DB!+!4&?B(Yzc] D1+!+(*$A.:+ .:+@5!$' *$' A3(*$ ^)^ $~!+(*$' *B@092(!+@5ld$'"$' (*$'(n"%$' *5!$'+ z mW> . >[5-! *092$'&)+@$'+ 2 (D >W$'+-suW$' 4 ^ $'&?0*2l5!$' y $' Xf$nuW;1+ v 0 >Ww!+@5 v 0 >Ww%,Pp! $'&)+d$CB*(&)X~X_(*$CB I$'&)(&)+(*$' u. ^)^ >

-

=i+<5-&?$CB*$' A3(*$ ^)^ $nB*; ^)^ .:-,5!$'+ $'4 & A3(9.:(&?;1+j.: &)( r (`$'&)+!4W$'4.:+!4W$'+q"$' *5!$'+ > A3(9.:(&?;1+j.: &)( r (8"#&) *54 !+@5!B r (m ^ &?0*2 uW;1 9.:@B4W$CB*$'(mC(Yz#B*;1dj. ^ 5 XA.:+ ,Pp! 5-&?$ =i+j. ^ -B*$l5!$' A3&)X_ ^ .:(&?;1+@B9$' 4W$'d!+!&?B*B*$ $'& s+@$$'&)(X~&)(( ^ !+!46p!d$' #$'&)+dc$CB(&)X~X_(*$CBi+3(*$' u1. ^)^ 5!$' #4W$CB r ((&)4(*$'+ A3((D1 !+!4WB*+!&)uW$Y.:@$'+("#&?09] ^ !+!45-! *0*2-,Pp!2! (Y>2AT;~] D1+!+3(*$\mW>K>@5!$' $'&)(uW$' ^ .:-,uW;1+=8d!d!& ^ 5-!+!4 -> d<2@$' 9.:+!4W$'mY;14W$'+q"%$' *5!$'+ z@[email protected]$'&)(&)+(*$' u. ^)^ mC! i *B(*$ ^)^ !+!46$'&)+@$CB# 9.15-&h. ^ $'+ab ^ !]3(j.:(&?;1+@B 7 *; ^ B8mC<5!$@+!&?$' *$'+ z@dc$'&?B 7 &?$ ^ Bs"%$'&?B*$t ∈ [1000 τ ; 4000 τ ]

> ZK.:X~&)(A"#&) *5 . ^ B*; 5!$' $'4 & 3 M 5 M VPF1I251EV M S M X~&)(f5!$'X ; 7 (&?B*092@$'+&)+@5- @0*]4 ^ $'&?0*2!4W$CB*$'(mC(Yz5@.>lB*&?0*2 [email protected] A3TB*(*$'X .:d $'&)+@$'X$'&)( 7 !+!]T(~5!$' $'&)( *$'&)2@$qDH J S MM V D M HQV)D M >8dqB&?0*[email protected] A3-B(*$'X "#&) ] ^ &?0*2q&)+q$'&)+@$'X 4W$CB r ((&)4(*$'+@B(9.:+@5d$@+@5!$'(Yzj]1.:+!+l.:-,t4 !+@55!$' $'+@5 ^ &?092@$'+ r +!4W$l5!$' uW;1 2j.:+@5!$'+@$'+$'&)( *$'&)2@$'+ +!&?$'XA. ^ B$'+(*B*092!&?$C5!$'+ "%$' *5!$'+ >#ZK$' $Qs;1dj.10*2(*$' uW;1+ B*&)X_ ^ &?$' (*$'+$'&)( *$'&)2@$'+2j.:($CB5@.:2@$' d$'&B9$'&)+@$'+ 4W$'d!+!&?B*B9$'+ &)X~Xf$' +! _X~&)(3 M V R L EF87*HQIHQVtIHQE E ;TI"!YDH J 5$# M R-HQV M mC(!+ >Z`$'+!+@;T0*2 &?B( $CB6X D14 ^ &?0*2 5-&?$ A3(9.:(&?;1+j.: &)( r (<$'&)+@$' %$'&)( *$'&)2@$l+ r 2@$' mC Bj.:+3(& @mC&?$' *$'+ > i&?$' smC].:+!+XA.:+5!$'+ B*;14W$'+j.:+!+3(*$'+'&I)(:HQE J VF1I J'M H JQM 2@$' 9.:+!mC&?$'2@$'+ >Z`&?$CB*$CBB(9.:(&?B(&?B*092@$=$' ,.:2! *$'+d$'+3!(mC(65-&?$+*8 ^)^ 2 7 ;1(2@$CB9$z5@.>g&)+5!$'X d$'( 9.1092(*$'(*$'+ +3(*$' u1. ^)^ B(9.:(&?B(&?B*092@$ A3(9.:(&?;1+j.: &)( r (uW;1 ^ &?$'4(Yz !+@5 ^ &?$Q,$' (_5@.:+!+gX~&)( 8& ^ ,P$~5!$' 3 M ;,!3HQI MPOMPO$- HQE M HQV U ;TI3D. $/$'&)+ =$' ( 9.:@$'+@B&)+(*$' u. ^)^ z&)+!+@$' 2j. ^ d5!$CB*B*$'+5-&?$*i ^)^ 23 7 ;1(2@$CB*$IjV R M 587D3HQU)HQRI M mCg"%$' *5!$'+d! 9.:@0*2(Y> Z\.:d$'&" r 2 ^ (5!$' $C;1dj.1092(*$' [email protected] A3&)4+!& @].:+!mC+!&)uW$Y.: [email protected]&?$0 *$'&)(*$5!$CB =$' ( 9.:@$'+@B&)+(*$' u. ^)^ B%d$CB(&)X~X(YzB*$ ^ d@B*(.:@BY>!BB*$'&[dc$'(*;1+(Yzj5@.>f5-&?$\B(9.:(&?B(&?B*092<]W;1 *$']3(*$_+3(*$' 7 *$'(9.:(&?;1+q5!$CB*=$' ,/.:2! *$'+@B#+!&?$'XA. ^ Bi$'&)+@$=i+!+j.:2!Xf$f5!$' 1*i ^)^ 23 7 ;1(2@$CB*$f$' ^ .:!d!(Yz "%;1X~&)(n"#&?$C5!$' !X 5-&?$~XA.:(2@$'XA.:(&?B*0*2@$328+dc$'"%$'&?Bdj.: s]W$'&)(uW;1+ A3(9.:(&?;1+j.: &)( r (`$'&)+!2@$' i4W$'2(Y>eg$'&)(*$' 2!&)+ ^ &?$'4(5!$' ~=8+"$'+@5-!+!4 5!$CB =$' ,.:2! *$'+@B~5-&?$ J'M 5 M V J'M V J R-H54xI 587R-S1I3D1V)D761HQV M . ^)^ $' ~dc$Qs( 9.10*2(*$'(*$'+ q$E> 7 !+!]T(*$AmC!4 !+@5!$>8 $'& 5!$'X ;1d$'+d$'( 9.1092(*$'(*$'++(*$' u. ^)^ 5!$' 9$'&)( *$'&)2@$.:@B=id!d!& ^ 5-!+!4-> d]1.:+!+ XA.:+5@.:+!+mW>`>X~&)( i& ^ ,$5!$CB;: 5 M R<#V MM HQU>=%H'E M (:H'E@?5:RE*HQI J .:@BA. ^)^ $'+uW$' ,tp!4dj.: *$'+'$'&)( 7 !+!]T(*$'+$'&)+@$ ; 7 (&)X~&?$' (*$Ac.:2 ^ uW;1+ &)((*$ ^ "$' (*$'+ d!& ^ 5!$'+ z"%$ ^ 092@$<B&?092"#&?$-,. ^)^ B4 (D >W$'+~uW$' 2j. ^ (*$'+~!+@5&)X B(9.:(&?B(&?B90*2@$'+A3&)+!+@$!+j.:d!2 r +!4&)4_B&)+@5[>1= ^)^ $uW$' ,Pp!4dj.: *$'+f&)(s(*$ ^ "%$' (*$"%$' *5!$'+g5@.:+!+ 5!$'X +uW$' *B&?;1+@B(*$CB*(\!+(*$' mY;14W$'+ > Z\.:X~&)(K]1.:+!+aXA.:+ B-B(*$'XA.:(&?B*0*2 [email protected]$'( 9.1092(*$'(*$f+(*$' u. ^)^ .:@B\$'&)+@$'X A3&)X_ ^ .:(&?;1+@B ^ .:-,#&)+ 5-&?$ =$' 4.:+!4W$'+!2@$'&)(*B &?092(!+!4l$' "%$'&)(*$' + z5-&?$8;1X_d!&)+j.:(&?;1+ 5!$' Kdc$'&?5!$'+H=$' ,.:2! *$'+g$' +@$'!(n.:+"$'+@5!$'+g!+@5lB*;B!]3mY$CB9B&)u<5!$'+B$'&)( 7 !+!]3(@+@5!$'+ zT.:+A5!$'X+j.10*25!$'+64W$'" r 2 ^ (*$'+ A3&)4+!& @].:+!mC]T &)(*$' &?$'+ 5-&?$ A3(9.:(&?;1+j.: &)( r ( +j.10*2 5!$' DC8I@U65 ;E? OL R 5 J H%d$'4&)+!+3(Y>!=i-,x5-&?$CB9$8eg$'&?B9$82 r ((*$`XA.:+5-&?$`4 (D >(X D14 ^ &?0*2@$n=8+!m .:2 ^ uW;1+$'&)(*B*0*2! &)((*$'+<4W$Qs

0 8 ,t!+@5!$'+ z"%$ ^ 0*2@$iXA.:+~,Pp! 5-&?$# *B(*$ ^)^ !+!4uW;1+Ak *; ^ $'+6;T5!$' A 7 $']3( *$'+A&)+A5!$' y ! d! ^ $'+!m .:+j. ^ -B*$5-! *0*23$'&)(X~&)(( ^ !+!4uW$' "%$'+@5!$'+A].:+!+ >ZK$' +3uW$' *B&?;1+@B(*$CB(!+@[email protected]#.:(*092!X~&)((*$ ^ "%$' (uW$' ,/.:2! *$'+"%$' *5!$'+&)+.1 /x5-&?B]T!(&?$' (Y>Z`$'+!+@;T0*2d ^ $'&)d!(.:@092<5@.:X~&)( 5-&?$&A3(9.:(&?;1+j.: &)( r (i$'&)+@$' $'&)( *$'&)2@$`,Pp! 5!$'+<.:+j. ^ -B&?$' *$'+@5!$'+%$C;1d-s.10*2(*$' $'&)+@$fB!d$']3(&)uW$ K (D >W$>AT;B*; ^)^ &)+!+@$' 2j. ^ d 5-&?$CB*$' n=8 d$'&)(n5-&?$ A3(9.:(&?;1+j.: &)( r (5-! *0*25-&?$; 7 (&?B*0*2@$ $'"$' (!+!4q5!$' CA3((D1 !+!4WB*$'+3("#&?0*] ^ !+!4W$'+ $' ,P; ^ 4W$'+ >[+ B3j. ^ &)(9.:(&)uW$'X A3&)+!+@$~B(*$'2(5-&?$F LM V J R-H 3 M 5 M VPFI 51EQV M S MtJ 51I 51U J H .:+@5!$' *$'+B*(9.:(&?B(&?B*0*2@$'+ =$' ,.:2! *$'+&)++!&?0*2(*B8+j.10*2 >=id@B*092 ^ &?$E>W$'+@5B*$'&c+@;T0*2<dc$'Xf$' ]T(Yz!5@.>fB&?0*2<4W$' 9.15!$\5-&?$_;1d$'+.:+!4W$CB 7 *;T0*2@$'+@$'+4=$' ,.:2! *$'+5@.:mC$'&)4+@$'+ z 5!$'+ 2i+3(*$' *B*0*2!&?$C5 mC"#&?B*0*2@$'+5!$'X B(9.:(&?B*0*2@$'+=i+3(*$'& ^ !+@5 5!$'X Bj.1B&?B*(9.:(&?B*0*2@$'+ =8+(*$'& ^5!$' A3((D1 !+!4WB ^ DB!+!4W$'+ Bj.:+3(&)(9.:(&)u.:-,PmC ^ DB*$'+ v BY>3=8d@B*092!+!&)((#wQz3&)+@5!$'X $'&)+=$' ( 9.:@$'+@B*&)+(*$' su1. ^)^ ,tp! 5!$'+".:2! *$'+$'&)(X~&)((*$ ^ "%$' (i$'&)+@$' Z\.:(*$'+! *$'&)2@$`4W$Q,t!+@5!$'+ "#&) *5[>TZ`$' ".:2! *$ $'&)(X~&)((*$ ^ s"%$' (K$'+(*B 7 &?092(K5@.:+!+q5!$' 8B*(9.:(&?B*0*2@$'+ DB!+!465!$CB8y ! d! ^ $'+!mYB-B(*$'XfBY>b p! i5-&?$CB*$ $' *$C092!+!+!4X_>.:dc$' K5-&?$B(9.:(&?B*(&?B*0*2@$928+j.:d!2 r +!4&)4]W$'&)(_. ^)^ $' nq$E>"$' (*$~B*;Y"#&?$B(9.:(&?B*(&?B*0*2@$ A3(9.:(&?;1+j.: &)( r (4W$Q,P;1 *5!$' (8"%$' *5!$'+ >

8

=i+5-&?$CB*$' A3(*$ ^)^ $nB9; ^)^ 5!$' &)+@>A5!$CB#$'T(*$' +&)+@5-!mC&?$' (*$'+aa.:4+@$'(*B*((D1 ,P$ ^ 5!$CBi.:-,5!$'+<k ^ .1BXA.Ns2!&)+(*$' 4 !+@5g&)+ .:+j. ^ T(&?B*092@$' _b@;1 X 5-&?B]T!(&?$' (\"%$' *5!$'+ > Z`&?$CB\&?B*(`2!&?$' \+3! n&)+ $'&)+!4W$CB*0*2! r +!]T(*$' b@;1 X X D14 ^ &?0*2 >l.:+dc$'+!(mC(~5@.:dc$'&#$'&)+@$B3j.1B& ^ &)+@$Y.: *$%* r 2@$' !+!4@z 5[> 2 > XA.:+dc$'( 9.10*2(*$'(f5-&?$eg$C0*2@B*$ ^ "#&) ]T!+!4<5!$CB A3((D1 ,P$ ^ 5 7 ;1(*$'+(&h. ^ BψX~&)(K5!$'

(0,0)sq;T5!$~5!$' \Z`&?0*2(*$!+@5l5!$CB\$ ^ $']T( & sB*0*2@$'+k ;1(*$'+(&h. ^ B >

2i+3(*$' *B@0*23(x"#&) *5n5!$' x&)+_5!$' Z`&?0*23(*$'2!&)+(*$' 4 !+@5-4 ^ $'&?092!+@4 v ->6?YWwcB((D1 !+!4WBdc$C5-&)+!4(*$b ^ > v -> F@WwΓδB := −χ‖

⟨∂y ψ∇‖(−κx+ n)

⟩y,z

v >6? wB*;Y"#&?$n5-&?$_B((D1 !+!4WBdc$C5-&)+!4(*$ $CB*0*2 ^ $'!+!&)4!+!4 v -> FGWw&)+5!$' ^ $'&?0*23!+!4A,Pp! 5-&?$ 7 ; ^ ;1&?5@. ^ $ i&)+-s(*$' 4 !+@5-4W$CB*0*23"#&)+@5-&)4]W$'&)( v -> Ww

AδB := −⟨∂yψ∇‖φ

⟩y,z

. v > WwZ8&?$_2!&?$' 8$' 2j. ^ (*$'+@$'+l=id@B90*2 r (mC!+!4W$'+a,tp!2! *$'+q+! `(*$'& ^ "%$'&?B*$mC!X dc$C;1dj.10*23(*$'(*$'+H=$' 2j. ^ (*$'+l5!$CB+!Xf$' &?B*092uW; ^)^ B( r +@5-&)4q4W$ ^ DB(*$'+a;T5!$ ^)^ BTB*(*$'XfB\d$'&%&)+@B*0*2j. ^ (*$'+5!$' _ZKZ8s A3((D1 !+!4@> ZK.:X~&)("#&) *5<+@;T0*2q$'&)+!XA. ^ 5-&?$ *`;1("$'+@5-&)4]W$'&)(K$'&)+@$' 8 *$'&)++3!Xf$' &?B*0*2@$'+ DB!+!45!$CB+!&?092( ^ &)+@$Y.: *$'+ A33sB(*$'XfB#dc$']3 r ,P(&)4(Y> 1c 0C19/0#c$'&c5!$' #=i@B*"%$' (!+!4~5!$CB B((D1 !+!4WBdc$C5-&)+!4(*$'+<b ^ @B9B*$CB%$' 2 r:^ ( XA.:+ 4W$'X r >~5!$' 0 7 $' 9.:(*;1 *5!$@+!& s(&?;1+@$'+q.:+j. ^ ;14fmC v ->6? w 5!$'+q=i@B*5- @09]

ΓδB = −χ‖

⟨∂y ψ

(∂z (−κx+ n) + C(x) ∂y (−κx+ n) −

ψ, (−κx+ n)

)⟩y,z

. v > GWw$'( 9.1092(*$'(\XA.:+a+! \5-&?$eg$C0*2@B*$ ^ "#&) ]T!+!4X~&)(`5!$'

(0,0)s8;1X 7 ;1+@$'+3(*$z B*;5@.>

n ≡ n004& ^ (Yz5@.:+!+$' 2 r:^ (XA.:+<"$'&)(*$'

# 8

ΓδB ≈ −χ‖

⟨−∂y ψ

(∂x ψ ∂y (−κx+ n00) − ∂y ψ ∂x (−κx+ n00)

)⟩y,z

= −χ‖

⟨∂y ψ ∂y ψ ∂x (−κx+ n00)

⟩y,z

= −χ‖

⟨∂y ψ ∂y ψ ∂x n0(x, t)

⟩

y,z= −χ‖

⟨∂y ψ ∂y ψ

⟩

y,z∂x n0(x, t)

= −χ‖ ∂x n0(x, t)

2πκy∫

0

2πκz∫

0

∑

m,n

∑

m′,n′

∂y ψm,n ei (mκyy−nκzz) ∂y ψm′,n′ ei (m

′κyy−n′κzz) dy dz

= −χ‖

∑

m,n

m2 κ2y

∣∣∣ψm,n(x)∣∣∣2

(−κ+ ∂x n00(x, t)) ,

"%;1dc$'&@,Pp! [email protected] A3((D1 ,P$ ^ 5 7 ;1(*$'+(&h. ^ &)+f5-&?$CB*$' =8 dc$'&)(4W$'X r >_n. 7 &)(*$ ^ \+3! 5-&?$a;35!$'+n = 4

!+@5m = 11, 12, 13

dc$'( 9.10*23(*$'(~"%$' *5!$'+ >Z`&?$ A3!X~Xf$p!dc$' 5-&?$b@;1! &?$' ]W;1X 7 ;1+@$'+(*$'+&?B*( 7 ;B&)(&)u5!$@+!&)(i!+@5<B*; ^)^ . ^ Bα2(x) :=

∑

m,n

m2 κ2y

∣∣∣ψm,n(x)∣∣∣2 v > 3w

.:d!4W$']3p! mC("%$' *5!$'+ > ZK.:X~&)(_,P; ^ 4(n,tp! _5-&?$fmY$'&)( ^ &?0*2@$+("#&?0*] ^ !+!4l5!$CBZ`&?0*2(*$ 7 *; ^ B v ->6?YWwn5-&?$ ^ $'&?0*23!+!4

∂t n00(x, t) = −∂x(Γ − χ⊥ ∂x n00(x, t) − χ‖ α

2(x) (−κ + ∂x n00(x, t))). v >CWw

2iX $'(" .1B p!d$' 5-&?$loi; ^)^ $a5!$CB ^ $'(mC(*$'+ yx$' XfB &)+ ,.:2! !+!4d! &)+!4W$'+ mC ] D1+!+@$'+ "#&) *5 . ^ B"%$'&)(*$' *$ =$' *$'&)+-,.1092!+!4+! %5-&?$iB(9.:(&?B*0*2@$ DB!+!45-&?B]3!(&?$' (!+@5~p!d$' %5-&?$# 9.15-&h. ^ $i8;3;1 *5-&)+j.:(*$x&)+(*$'4 &?$' (

0 =

x∫

x ∂x(Γ − χ⊥ ∂x n00(x, t) − χ‖ α

2(x) (−κ + ∂x n00(x, t)))dx . v >Ww

B4W$ ^ (*$'+5-&?$n&)X o G1Z8s ;T5!$KuW;1 2j.:+@5!$'+@$'+ao`.:+@5-d$C5-&)+!4!+!4W$'+Γ |x = 0 , α2(x)

∣∣x = 0 , ∂x n00|x = −s , v >w

"%;1dc$'&s7 ;B&)(&)u ]W;1+@B(9.:+(~&?B(!+@55@.:X~&)(~$'&)+@$ +@$'4.:(&)uW$ A3(*$'&)4!+!4g5!$CB v n00

wsk *; ^ BA.:+ 5!$'+o8.:+@5 7 !+!]T(*$'+f5!$CB o8$C0*2@$'+!4W$'d!&?$'(*B4W$'" r 2! ^ $'&?B(*$'(&?B(YzN".1BmC@B.:X~Xf$'+~X~&)(5!$'X X~&)(( ^ $' *$'+fZ`&?0*2-s(*$'4 9.15-&?$'+(*$'+ −κx mC$'&)+@$'X Z`&?0*2(*$ 7 *; ^ "#&?$K&)+<=id!d!& ^ 5-!+!4 v -> .Ww,Pp!2! (Y>!Z`&?$\;1d!&)4W$'+<o8.:+@5Tsd$C5-&)+!4!+!4W$'+q4W$ ^ (*$'+q. ^)^ $CB9.:X_(K.:@0*2<dc$'&x = x >!Z\.:X~&)(#,; ^ 4(

Γ − χ⊥ ∂x n00 − χ‖ α2 (−κ+ ∂x n00) − χ⊥ s = 0 ,

Γ + χ‖ α2 κ− χ⊥ s = (χ⊥ + χ‖ α

2) ∂x n00 ,

∂x n00 =Γ + χ‖ α

2 κ− χ⊥ s

χ⊥ + χ‖ α2.

8 2 =i@B~5!$' fy .:(*B9.10*2@$z5@.>g5!$' ~ 9.15-&h. ^ $ K 9.15-&?$'+(A5!$'

(0,0)sZ8&?092(*$CB((D1 !+!4&)X 9.15-&h. ^ $'+q&)((*$ ûW$' *B*0*23"#&)+@5!$'(Yz

〈∂x n00(x)〉x = 0 , v >Wwd$CB(&)X~X(iB&?0925-&?$no8.:+@5-dc$C5-&)+!4!+!4~,Pp! 5-&?$CA3(*$'&)4!+!4s =

〈Γ 〉x + χ‖

⟨α2(x)

⟩xκ

χ⊥. v >Ww

ZK.:X~&)(8$' 2 r:^ (XA.:+B*092 ^ &?$E> ^ &?0*2q5!$'+ @B9.:X~Xf$'+!2j.:+!4∂x n00 =

Γ − 〈Γ 〉 + χ‖

(α2 −

⟨α2⟩)κ

χ⊥ + χ‖ α2. v >6?@Ww

&)(5!$' =8+!+j.:2!Xf$z@5@.>[email protected]#mY$'&)( ^ &?0924W$'X~&)((*$ ^ (*$\ 9.15-&h. ^ $nk *; ^ 5!$' v n00ws A3((D1 !+!4@z@"$ ^ 0*2@$CB&)+5!$'+HA3&)X_ ^ .:(&?;1+@B9$' 4W$'d!+!&?B*B*$'+ld$'( 9.1092(*$'(K"#&) *5[z.:+!+ r 2@$' +@5a5!$' 8;1dc$'+<dc$'( 9.10*23(*$'(*$'+lB*(9.:(&?B*0*2@$'+

DB!+!4 v >6?@Ww8$'+(*B 7 &?092(Yz ^hr >(\B&?092 5-&?$5-&) *$']T(*$e &) ]T!+!45!$' `ZKZ8s A3((D1 !+!4q.:-,%5!$'+lZ`&?0*2-s(*$'2!&)+(*$' 4 !+@5a5-&?B]3!(&?$' *$'+ X bj. ^)^ ;12!+@$ A3((D1 ,$ ^ 5l4& ^ (α2 = 0

zB9;5@.>5-&?$ A3(*$'&)4!+!4<5!$' v n00ws A3((D1 !+!4+3! \&)+g5!$'X 9.Ns5-&h. ^ $'+ $' *$'&?092g5!$CB\oi$C0*2@$'+!4W$'d!&?$'(*$CB 7 ;B&)(&)u&?B(Yz&)+a5!$'X 5!$' $'"$'& ^ &)4W$eg$' (n5!$CB`(! d! ^ $'+(*$'+b ^ @B9B*$CB4 (D >W$' &?B*( . ^ B#B*$'&)+]W;1+@B(9.:+3(*$' $'&)(X~&)((*$ ^ "%$' (Y>jZ`&?$CB&?B(2!&?$' &)X 7 23TB*&)]1. ^ &?B*092 *$ ^ $'u1.:+-s(*$'+ K$'d!&?$'(mC"#&?B*092@$'+q5!$'+k-c$' mY;1+@$'+a5!$' ibj. ^)^ !+@5.:@092&)X .:+!4 *$'+!mY$'+@5!$'+a=i+-,/.:+!4WBdc$' *$'&?0*25!$' ik-[$' mY;1+@$'+ z@&)+<5!$'X 5-&?$nk *; ^ "%$' (*$_B*(*$'(&)46.:d-,. ^)^ $'+ v B&?$'2@$_.:@0*2q=id@B*092!+!&)(( G-> G-> 3wQ>jq&)(

∂x n0(x) = −κ+ ∂x n00(x) v >6?F? w,tp!2! (f5-&?$CB~mC 5!$' ~dc$' *$'&)(*Bf5-&?B]T!(&?$' (*$'+=8d@B*$'+!]T!+!4g5!$CBfZ`&?0*2(*$'4 9.15-&?$'+3(*$'+ d!mC"n>mC! f=id.Ns0*2!+!45!$CB# 9.15-&h. ^ $'+qZ`&?092(*$'2!&)+(*$' 4 !+@5 7 *; ^ B v B&?$'2@$n=8d!d > -> .-z@B*0*23".: mY$K! uW$ wQ>bxp! 5!$'+ bj. ^)^ $'&)+@$CBn,P$CB*(*$'+gZ8&?092(*$'2!&)+3(*$' 4 !+@5!BYz&)+ 5!$'X ( *;1(mY5!$'X 5-&?$f5-&) *$']T(*$~e &) ]T!+!4q5!$CBB((D1 !+!4WBdc$C5-&)+!4(*$'+gb ^ @B9B*$CBΓδB

dc$' p@0*]-B&?092(&)4(_"#&) *5 v B&?$'2@$~=8d@B*092!+!&)(( -> @>6? wQzc"#&) *5a5!$' \(! sd! ^ $'+(*$<b ^ > &)+ 5!$' ^ $'&?092!+!4 v ->6?YWwd!mC"n> v >6?@Ww *8 ^)^ 4W$CB*$'(mC(Y> 28+(*$' $' p@0*]-B&?0*23(&)4!+!45!$CB/A3((D1 7 ;1(*$'+3(&h. ^ 7 *; ^ B v B&?$'2@$\=8d!d > @> Ww$' 2 r:^ (XA.:+<5@.:+!+&)X *$C092(*$'+ $' *$'&?0*2q5!$CB 7 2-B&)]1.Ns^ &?B*0*2@$'+ oi$C092@$'+!4W$'d!&?$'(*BYzc"%;α2 > 〈α2〉 4& ^ (Yzc$'&)+@$=id.1092!+!4<5!$CB\Z`&?0*23(*$ 7 *; ^ BYzc" r 2! *$'+@5l&)X^ &)+!]W$'+3 $' *$'&?0*2 !+@5A&)+5!$'+k-c$' mY;1+@$'+ zT";

α2 < 〈α2〉 4& ^ (YzT$'&)+@$i=8-,PB(*$'& ^ !+!4$'&)+( &)((Y>-Z`&?$CB*$CB 4W$'d!+!&?B"#&) *5<5-! *092q=8d!d!& ^ 5-!+!4 ->6?Y:d<dc$CB( r (&)4(Y>X B*$ ^ d@B(]W;1+@B&?B*(*$'+(*$'+ bj. ^)^ 4& ^ ( X~&)(5!$' oi$ ^ .:(&?;1+ v >6?@Ww65-&?$ai;1Xd!&)+j.:(&?;1+ 5-&?$CB*$' d$'&?5!$'+=i@B*B.:4W$'+ >!X $'&?B 7 &?$ ^ $' " .: (*$'(KXA.:+q&)X $' *$'&?0*2lX~&)(Γ > 〈Γ 〉 !+@5 α2 > 〈α2〉 $'&)+@$=id.1092!+!45!$CBiZ`&?0*23(*$ 7 *; ^ BYz" r 2! *$'+@5a&)X $' *$'&?092qX~&)(Γ > 〈Γ 〉 !+@5 α2 < 〈α2〉 $'+@$=id.1092!+!4uW$' X~&)+@5!$' (8B*$'&)+"#&) *5[>@Z`&?$_.:+j. ^ T(&?B*092@$'+q 4W$'d!+!&?B*B9$n5!$' 2!&?$' X D14 ^ &?092@$'+qbj. ^)^ !+-s(*$' *B*0*2@$'&?5-!+!4W$'+ ^ .1B*B*$'+B*&?0*2a.:dc$' +!&?092(`Xf$'2! 8B*;~5-& [$' *$'+!mC&?$' (8&)+q5!$'+HA3&)X_ ^ .:(&?;1+@B9$' 4W$'d!+!&?B*B*$'+"#&?$C5!$' @+@5!$'+ >

+ 1* 1* -1 0C19/0#c+l.:+j. ^ ;14W$' èg$'&?B9$_mCl=id@B90*2!+!&)((\>6?$' 2 r:^ (8XA.:+q,Pp! `5-&?$B((D1 !+!4WBdc$C5-&)+!4(*$ $CB90*2 ^ $'!+!&)4!+!4v -> FGWw`&)+g5!$' n+("#&?0*] ^ !+!4WB4 ^ $'&?0*23!+!4 5!$' 7 ; ^ ;1&?5@. ^ $'+ i&)+3(*$' 4 !+@5- *;1(9.:(&?;1+ v -> Ww`5!$'+ =i@Bs5- @0*]

AδB =⟨∂y ψ

(∂z φ+ C(x) ∂y φ−

ψ, φ

)⟩y,z

. v >6? Ww

# 8 $'( 9.1092(*$'(%XA.:+~"#&?$C5!$' !X &)+ Bj.1B& ^ &)+@$Y.: *$' * r 2@$' !+!4\+! 5-&?$eg$C092@B*$ ^ "#&) ]3!+!45!$CB A3((D1 ,P$ ^ 5!BX~&)(i5!$'

(0,0)s8;1X 7 ;1+@$'+3(*$~5!$CBk ;1(*$'+(&h. ^ BYzB*;~$' 2 r:^ (XA.:+<X~&)(

φ ≡ φ005-&?$\oi$ ^ .:(&?;1+

AδB ≈⟨∂y ψ

(∂xψ ∂y φ00 − ∂y ψ ∂x φ00

)⟩y,z

= −⟨∂y ψ ∂y ψ ∂x φ00

⟩y,z

= −

2πκy∫

0

2πκz∫

0

∑

m,n

∑

m′,n′

∂y ψm,n ei (mκyy−nκzz) ∂y ψm′,n′ ei (m

′κyy−n′κzz) v00 dy dz

= −∑

m,n

m2 κ2y

∣∣∣ψm,n(x)∣∣∣2v00(x, t) .

&)(K5!$' 8=8d!]3p! mC!+!4 v > 3wi$' 2 r:^ (8XA.:+l5@.:+!+l5-&?$n4W$'+ r 2@$' (*$ ^ $'&?0*23!+!4,tp! `5-&?$ 8&)+(*$' 4 !+@5Ts4W$CB*0*23"#&)+@5-&)4]W$'&)(∂t v00 = −∂x (〈vy vx〉y,z − ν ∂x v00) − α2 v00 . v >6?GWw&)(5!$' A"%$'&)(*$' *$'+ =8+!+j.:2!Xf$z%5@.>5-&?$o8$'3+@; ^ 5!B*B 7 .:+!+!+!4mC@B9.:X~Xf$'+ X~&)(5!$' .: (& @mC&?$ ^)^ $'+Z8&?B9B& 7 .:(&?;1+6$'&)+@$#mY$'&)( ^ &?092A]W;1+@B(9.:+3(*$8@$ ^)^ $

Q,tp! 5-&?$ioi;1(9.:(&?;1+

v005@.: *B*(*$ ^)^ ( v B&?$'2@$=8d@B*0*2!+!&)((->6?>6? wQz@"#&) *5<5-&?$ ^ $'&?092!+!4 v >6?GWw $'&)+@$nZ`& [$' *$'+(&h. ^ 4 ^ $'&?092!+!4uW;1X y 7

v = Q− α2 v . v >6? 3wZ8&?$nmC!4W$'2D1 &)4W$ DB!+!4AX~&)(i5!$' =i+-,/.:+!4WBdc$C5-&)+!4!+!4

v0 ≡ v(t = t0)^ .:!(*$'(

v =Q

α2−(Q

α2− v0

)e−α

2(t− t0) . v >6?>WwZK.:X~&)(\"#&) *5 5-&?$~.:+-, r +!4 ^ &?0*2@$ 7 ; ^ ;1&?5@. ^ $~oi;1(9.:(&?;1+

v0X~&)(\$'&)+!4W$CB*0*2j. ^ (*$'(*$'X A3((D1 ,P$ ^ 5 v α2 > 0

wd$'&,P;1 (*B*092! *$'&)(*$'+@5!$' $'&)( v t > t0wK$' 7 ;1+@$'+(&?$ ^)^ .:d!4W$C5 r X 7 ,P(Yz d!&?BB&?$fB&?092 &) 4W$'+@5-" .:+!+5!$'Xeg$' (

Q/α2 .:+!+ r 2@$' (Y>Z8&?$~.:dc5 r X 7 ,P$'+@5!$e &) ]T!+!45!$' \ZKZ8s A3((D1 !+!4<]W;1+!+(*$&)+ 5!$'+ A3&)X_ ^ .:(&?;1+@$'+gdc$CB( r (&)4(K"%$' s5!$'+ v B&?$'2@$KmW>K>!=id!d!& ^ 5-!+!4 -> G .WwQ>!$C5!;3092 ^ &?$Q,P$' (5-&?$CB9$\=id@B*092 r (mC!+!4f+!&?0*23([email protected]#uW; ^)^ B( r +@5-&)4W$%& ^ 5[>xe &?$A$'&)+@$32i+3(*$' *B@0*23!+!4 &)+ . $/mY$'&)4(Yzx].:+!+5-&?$$'-(*$' +&)+@5-!mC&?$' (*$a.:4+@$'(,$ ^ 5!B((D1 !+!4$'d$'+-,/. ^)^ B`p!d$' \5!$'+ly $' X v -> FGWw8$'&)+@$o8;1(9.:(&?;1+l2@$' uW;1 -,P$'+ > 28X5-&?$CB`mCamY$'&)4W$'+ zc"#! *5!$5-&?$oi$'T+@; ^ 5!B*B 7 .:+!+!+!4f.:@B4W$CB90*2j. ^ (*$'(Yz!. ^ B*;_5!$' k ;1(*$'+(&h. ^ 2!&)+(*$' 4 !+@5]3p!+@B*( ^ &?0*2 T&?$' (Yz3" r 2! *$'+@55-&?$ndc$'&?5!$'+ ^ $'(mC(*$'+qyx$' Xf$5!$' ^ $'&?0*23!+!4 v -> Ww#"%$'&)(*$' 2!&)+a&)+q5!$' A3&)X ^ .:(&?;1+ld$' p@09]TB&?092(&)4("#! *5!$'+ >


X,; ^ 4W$'+@5!$'+B&)+@5 5-&?$K"#&?0*23(&)4WB(*$'+HA3&)X ^ .:(&?;1+@B 7 .: 9.:Xf$'(*$' 8uW;1+<5- *$'&[!+3(*$' *B*0*2!&?$C5 ^ &?092@$'+ oi$C0*2-s+!+!4W$'+.:+!4W$'4W$'dc$'+ z!5-&?$8mC! #k ^ .1B*XA.1B((D >&)4]W$'&)(*B!+(*$' *B*@0*2!+!45!$CB n. 7 &)(*$ ^ B _uW$' "%$'+@5!$'("#! s5!$'+ >Z`&?$k.: 9.:Xf$'(*$' _5!$CB\5- &)((*$'+ A3mY$'+j.: &?;B`,tp!2! (*$'+ mCl]W$'&)+@$'+g5-&?B]T!(9.:d ^ $'+ 4W$'d!+!&?B*B*$'+ z[[email protected]!;1 (85-&?$y ! d! ^ $'+!mf.:-,t4 !+@5a5!$' i+!&?$C5- &)4W$'+ A3((D >&)4]W$'&)(K!+@55!$CB82@;12@$'+χ ‖ s/eg$' (*$CBiuW; ^)^ ]W;1XsXf$'+q.:d!4W$C5 r X 7 ,t(i"#! *5!$>

$C5!$'!(!+!4 k ^ .1BXA. 7 .: 9.:Xf$'(*$' $C5!$'!(!+!4 &)+!4.:dc$'"%$' ( ;1+@$'+(*$'X 7 $' 9.:(!

Ti = 0$ = XA.[>!(*;1 Y>@a;35!$

n = 52 ^ $']T( *;1+@$'+(*$'X 7 $' 9.:(! Te = 36.0

$#= q;T5!$'+j.:d@B(9.:+@5∆n = 4Z`&?0*23(*$

n0 = 5.5 · 1019 X −3 9.15[> K&)((*$' 7 !+!]3(*$nx + 1 = 50 + 1l.:4+@$'(,P$ ^ 5

B0 = 1.0y 7 ; ^ >@eg$ ^)^ $'+!m .:2 ^

κy = 0.0015` *; >W$' 8y ;1 @B 9.15-&)@B

R0 = 1.75X (*;1 Y>!eg$ ^)^ $'+!m .:2 ^

κz = 1 ^ $'&)+@$' io`.15-&)@Br0 = 0.45

X Z8&?B9B& 7 .:(&?;1+χN

‖ = 1.64o8$Q,> sb ^ > r 0*2@$q0 = 3

=8 (Y>@Z`&?B*B& 7 .:(&?;1+χN⊥ = 0.905=$' *B*092@$' !+!4

s = 3=i (Y>jZ8&?B9B& 7 .:(&?;1+

νN = 0.905 *$ ^ >A3((D1 ,$ ^ 5!B( r ]W$δB/B0 = 2.7 · 10−4 A3((D1 *B( *;1X &)+]T=

δψ = 1.3

Z` & ,P( ^hr +!4W$%s = 8.7 · 10−4 X #. ^)^ ;T;1+!&)+!4 ^hr +!4W$

ξ = 6.8 · 10−4 XA3((D >&)4]W$'&)(ν = 1.00

=i@B(9.:@B90*2!mY$'&)(τ = 3.9 · 10−6 B*$C0A3(*;T0*2j.1B(&)mC&)( r (

σ− = 1.0 , σ+ = 1.3K$'d!&?$'(i

2.75 n≤m≤3.25nK$CB.:X_(Xf;T5!$'+!m .:2 ^

#mn = 196

0H 2- 2 2

$C5!$'!(!+!4 k ^ .1BXA. 7 .: 9.:Xf$'(*$' $C5!$'!(!+!4 &)+!4.:dc$'"%$' ( ;1+@$'+(*$'X 7 $' 9.:(!

Ti = 0$ = XA.[>-(*;1 Y>@a;35!$

n = 52 ^ $']T( *;1+@$'+(*$'X 7 $' 9.:(! Te = 40.0

$#= q;T5!$'+j.:d@B(9.:+@5∆n = 4Z`&?0*23(*$

n0 = 5.0 · 1019 X −3 9.15[> `&)((*$' 7 !+!]3(*$nx + 1 = 48 + 1l.:4+@$'(,P$ ^ 5

B0 = 1.0y 7 ; ^ >-eg$ ^)^ $'+!m .:2 ^

κy = 0.0014` *; >W$' 8y ;1 @B 9.15-&)@B

R0 = 1.75X (*;1 Y>!eg$ ^)^ $'+!m .:2 ^

κz = 1 ^ $'&)+@$' ò8.15-&)@Br0 = 0.45

X Z`&?B*B& 7 .:(&?;1+χN

‖ = 2.2o8$Q,> sb ^ > r 0*2@$q0 = 3

=8 (Y>!Z`&?B*B& 7 .:(&?;1+χN⊥ = 0.938=$' *B*092@$' !+!4

s = 3=8 (Y>!Z8&?B9B& 7 .:(&?;1+

νN = 0.938 *$ ^ > A3((D1 ,$ ^ 5!B( r ]W$δB/B0 = 2.6 · 10−4 A3((D1 *B*( *;1X &)+<]-=

δψ = 1.3

Z` & ,P( ^hr +!4W$%s = 9.2 · 10−4 X #. ^)^ ;T;1+!&)+!4 ^hr +!4W$

ξ = 6.2 · 10−4 XA3((D >&)4]W$'&)(ν = 0.74

=i@B*(9.:@B*0*2!mY$'&)(τ = 3.7 · 10−6 B*$C0A3(*;T0*2j.1B(&)mC&)( r (

σ− = 1.0 , σ+ = 1.3K$'d!&?$'(i

2.75n≤m≤3.25n\$CB9.:X_(Xf;T5!$'+!m .:2 ^

#mn = 196

$C5!$'!(!+!4 k ^ .1B*XA. 7 .: 9.:Xf$'(*$' $C5!$'!(!+!4 &)+!4.:d$'"$' (;1+@$'+3(*$'X 7 $' 9.:(!

Ti = 0$#= XA.c>-(*;1 Y>jq;T5!$

n = 52 ^ $']T( *;1+@$'+(*$'X 7 $' 9.:(! Te = 50.0

$ = a;T5!$'+j.:d@B(9.:+@5∆n = 4Z`&?0*2(*$

n0 = 4.0 · 1019 X −3 9.15[> `&)((*$' 7 !+!]T(*$nx + 1 = 44 + 1a.:4+@$'(,$ ^ 5

B0 = 1.0y 7 ; ^ >-eg$ ^)^ $'+!m .:2 ^

κy = 0.0011` *; >W$' iyx;1 @B 9.15-&)@B

R0 = 1.75X (*;1 Y>-eg$ ^)^ $'+!m .:2 ^

κz = 1 ^ $'&)+@$' io8.15-&)@Br0 = 0.45

X Z`&?B*B*& 7 .:(&?;1+χN

‖ = 4.2oi$Q,> sb ^ > r 0*2@$q0 = 3

=i (Y>!Z`&?B*B*& 7 .:(&?;1+χN⊥ = 1.05=$' *B*0*2@$' !+!4

s = 3=i (Y>!Z`&?B*B& 7 .:(&?;1+

νN = 1.05

Z` & ,t( ^hr +!4W$%s = 1.0 · 10−3 X #. ^)^ ;3;1+!&)+!4 ^hr +!4W$

ξ = 5.0 · 10−4 XA3((D >&)4]W$'&)(ν = 0.40

=8@B(9.:@B*092!mY$'&)(τ = 3.3 · 10−6 B9$C0

K$'d!&?$'(2.75 n≤m≤3.25n

K$CB9.:X(Xf;35!$'+!m .:2 ^#mn = 196


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Turbulenter Plasmatransport im ergodisierten Magnetfeld · -H: 7$ & D C HG * % 1 \E )2 kf .( !)+*- *4 67 -* %2. ! PC %+.( .()2 i ( !*)+* .(HZ)2 ! M

Turbulenter Plasmatransport im ergodisierten Magnetfeld · -H: 7$ & D C HG * % 1 \E )2 kf .( !)+- 4 67 -* %2. ! PC %+.( .()2 i ( !)+ .(HZ)2 ! M