Taylor-Couette turbulence

Taylor-Couette turbulence

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lT iQ MQr- 7Q` i?2 ?B;? _2vMQH/b MmK#2`b U> 104V- i?2 7Q+mb ?b #22M QM /2@ b+`B#BM; i?2 ;HQ#H UBX2X iQ`[m2V ~Qr T`QT2`iB2b- M/ KQbiHv 7Q` i?2 +b2 r?2`2 i?2 Qmi2` +vHBM/2` Bb biiBQM`vX AM i?Bb i?2bBb r2 rBHH 7Q+mb QM HQ+H [mMiBiB2b M/ i?2B` +QMM2+iBQM iQ ;HQ#H [mMiBiB2b- MQi QMHv 7Q` i?2 +b2 Q7 Tm`2 BMM2` +vHBM/2` `QiiBQM- #mi HbQ 7Q` i?2 +b2 r?2`2 #Qi? +vHBM/2`b `2 `QiiBM;X AM +?Ti2` R UT;2 8V i?2 ;HQ#H i`MbTQ`i Bb +?`+i2`Bx2/ #v }M/BM; i?2 b+HBM; 2tTQM2Mi Q7 i?2 i`MbTQ`i Lmω∝ hγX AM +?Ti2` k UT;2 R8V i?2 7Q+mb Bb QM i?2 HQ+H ~mt Jω BMbB/2 i?2 ~Qr M/ r2 rBHH /2}M2 M/ K2bm`2 i?2 ǵrBM/Ƕ p2HQ+Biv Q7 i?2 ~QrX 6QHHQrBM; i?2 MHQ;v /2b+`B#2/ BM _27X (R)- i?2 #QmM/@ `v Hv2` T`QT2`iB2b `2 K2bm`2/ M/ +QKT`2/ iQ _" ~Qr (k)- b22 +?Ti2` j UT;2 k8VX AM +?Ti2` 9 UT;2 jdV i?2 TQbbB#BHBiv Q7 KmHiBTH2 ~Qr bii2b- 2p2M 7Q` p2`v ?B;? h U_2V- Bb 2t2KTHB}2/ #v bBKmHiM2Qmb ;HQ#H M/ HQ+H K2bm`2@ K2MibX AM +?Ti2` 8 UT;2 9NV i?2 HQ+H ~Qr Bb MHvx2/ rBi? biiBbiB+H iQQHb, T`Q##BHBiv /2MbBiv 7mM+iBQMb US.6bV- S.6b Q7 p2HQ+Biv BM+`2K2Mib- bi`m+im`2 7mM+iBQMb- M/ bT2+i`H MHvbBbX AM +?Ti2` e UT;2 edV r2 T2`7Q`K p2HQ+Biv T`Q}H2 K2bm`2K2Mib- M/ r2 rBHH ?p2  HQQF i i?2 #B@KQ/HBiv Q7 i?2 ~QrX AM +?Ti2` d UT;2 RyNV r2 rBHH ?p2  HQQF i i?2 η /2T2M/2M+2 Q7 i?2 ~Qr 7Q` i?2 ;HQ#H M/ HQ+H [mMiBiB2bX h?2 2tT2`BK2MiH pHm2b `2 ++QKTMB2/ #v MmK2`B+H `2bmHib #v _Q/QH7Q PbiBHH JƦMB+QX *?Ti2` 3 UT;2 R9dV 2tTHBMb ?Qr Hb2` .QTTH2` M2KQK2i`v K2bm`2K2Mib +M #2 T2`7Q`K2/ BMbB/2 i?2 hvHQ`@*Qm2ii2 /2pB+2- /2bTBi2 Bib +m`p2/ Qmi2` bm`7+2X HH i?2 +QM+HmbBQMb +M #2 7QmM/ QM T;2 ReRX



5PSRVF TDBMJOH JO UVSCVMFOU 5BZMPS$PVFUUF øPX

XJUI DP BOE DPVOUFSSPUBUJOH DZMJOEFST

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q2 MHvx2 i?2 ;HQ#H i`MbTQ`i T`QT2`iB2b Q7 im`#mH2Mi hvHQ`@*Qm2ii2 ~Qr BM i?2 bi`QM;Hv im`#mH2Mi `2;BK2 7Q` BM/2T2M/2MiHv `QiiBM; Qmi2` M/ BMM2` +vHBM/2`-`2+?BM; _2vMQH/b MmK#2`b Q7 i?2 BMM2` M/ Qmi2` +vHBM/2` Q7 _2i = 2Ǧ 106 M/

_2o = ±1.4 Ǧ 106- `2bT2+iBp2HvX 6Q` HH _2i- _2o- i?2 /BK2MbBQMH2bb iQ`[m2 G

b+H2b b  7mM+iBQM Q7 i?2 hvHQ` MmK#2` h Ur?B+? Bb T`QTQ`iBQMH iQ i?2 b[m`2 Q7 i?2 /Bz2`2M+2 #2ir22M i?2 M;mH` p2HQ+BiB2b Q7 i?2 BMM2` M/ Qmi2` +vHBM/2`V rBi?  mMBp2`bH 2z2+iBp2 b+HBM; Hr G ∝ h0.88_{- +Q``2bTQM/BM; iQ Lm}

ω ∝ h0.38

7Q` i?2 Lmbb2Hi MmK#2` +?`+i2`BxBM; i?2 pQ`iB+Biv i`MbTQ`i #2ir22M i?2 BMM2` M/ Qmi2` +vHBM/2`X h?2 i`MbTQ`i Bb KQbi 2{+B2Mi 7Q` i?2 +QmMi2`@`QiiBM; +b2 HQM; i?2 /B;QMH BM T?b2 bT+2 rBi? ωo ≈ −0.4ωiX

◦_{Sm#HBb?2/ b, .2MMBb SXJX pM :BHb- aM/2` :X >mBbKM- :2`i@qBK "`m;;2`i- *?Q}

amM-M/ .2iH27 GQ?b2- hQ`[m2 b+HBM; BM im`#mH2Mi hvHQ`@*Qm2ii2 ~Qr rBi? +Q@ amM-M/ +QmMi2`@`QiiBM;

+vHBM/2`b- S?vbX _2pX G2iX Rye- yk98yk UkyRRVX h?Bb Tm#HB+iBQM rb b2H2+i2/ b 1/BiQ`bǶ am;@

;2biBQM - b2H2+i2/ 7Q`  bvMQTbBb BM S?vbB+b dz>2i M/ irBbi Q7 im`#mH2Mi ~QrbǴ- M/ rb +Qp2`2/ #v S?vbB+b hQ/v (j)X h?2 2tT2`BK2Mib `2 T2`7Q`K2/ #v >mBbKM- amM- M/ pM :BHb- MHvbBb Bb /QM2 #v >mBbKM- amM- M/ pM :BHb- M/ i?2 KMmb+`BTi Bb r`Bii2M #v GQ?b2X .Bb+mbbBQM Q7 i?2 `2bmHib M/ T`QQ7`2/BM; Q7 i?2 KMmb+`BTi #v 2p2`vQM2X

e *>Sh1_ RX hP_Zl1 AL h*

 *OUSPEVDUJPO

:HQ#H i`MbTQ`i T`QT2`iB2b Q7 im`#mH2Mi ~Qrb `2 Q7 T`BK2 BKTQ`iM+2 7Q` KMv TTHB+iBQMb Q7 ~mB/ /vMKB+b- #mi HbQ 7Q`  7mM/K2MiH mM/2`biM/@ BM;- b i?2v `2~2+i i?2 BMi2`THv #2ir22M #QmM/`v Hv2` M/ #mHFX h?2 irQ +MQMB+H bvbi2Kb mb2/ iQ MHvx2 i?2 i`MbTQ`i T`QT2`iB2b BM +HQb2/ im`#m@ H2Mi bvbi2Kb `2 _vH2B;?@"ûM`/ U_"V +QMp2+iBQM M/ hvHQ`@*Qm2ii2 Uh*V ~Qr- M/ i?2v `2 +QM+2TiBQMHHv +HQb2Hv `2Hi2/ (9- 8- R)X AM _" ~Qr- ?2i UBM /BK2MbBQMH2bb 7Q`K i?2 Lmbb2Hi MmK#2`V Bb i`MbTQ`i2/ 7`QK i?2 ?Qi #Qi@ iQK THi2 iQ i?2 +QH/ iQT THi2 (e- d)- r?2`2b BM h* ~Qr M;mH` p2HQ+Biv Bb i`MbTQ`i2/ 7`QK i?2 BMM2` iQ i?2 Qmi2` +vHBM/2` U7Q` ωi > ωoVX AM MHQ;v iQ _" ~Qr- 1+F?`/i 2i HX (R) B/2MiB}2/- 7`QK i?2 mM/2`HvBM; LpB2`@aiQF2b

2[miBQMb-Jω= r3(_⟨urω⟩A,t− ν∂r⟨ω⟩A,t) URXRV b `2H2pMi +QMb2`p2/ i`MbTQ`i [mMiBiv- `2T`2b2MiBM; i?2 ~mt Q7 M;mH` p2@ HQ+Biv 7`QK i?2 BMM2` iQ i?2 Qmi2` +vHBM/2`X >2`2 urUuφV Bb i?2 `/BH Ux@ BKmi?HV p2HQ+Biv- ω = uφ/r i?2 M;mH` p2HQ+Biv- M/ ⟨. . . ⟩A,t /2MQi2b p@ 2`;BM; Qp2` iBK2 M/ M `2 rBi? +QMbiMi r 7`QK i?2 tBbX Jω _{Bb K/2} /BK2MbBQMH2bb rBi? Bib pHm2 Jω

HK = 2νr2ir2o(ωi− ωo)/(ro2− ri2) 7Q` i?2 HKBM` +b2- ;BpBM;  dzLmbb2Hi MmK#2`Ǵ b /BK2MbBQMH2bb i`MbTQ`i

[mMiBiv-Lmω = Jω/J_HKω , URXkV

r?2`2 ri,oM/ ωi,o/2MQi2 i?2 `/Bmb M/ i?2 M;mH` p2HQ+Biv Q7 i?2 BMM2` M/ Qmi2` +vHBM/2`- `2bT2+iBp2Hv- ν Bb i?2 FBM2KiB+ pBb+QbBiv Q7 i?2 ~mB/X Lmω Bb +HQb2Hv +QMM2+i2/ iQ i?2 iQ`[m2 τ i?i Bb M2+2bb`v iQ F22T i?2 BMM2` +vHBM/2` `QiiBM; i +QMbiMi M;mH` p2HQ+Biv- Q` BM /BK2MbBQMH2bb 7Q`K iQ

G = τ

2πℓρ~mB/ν2 =Lmω J_HKω

ν2 =LmωGHK, URXjV

r?2`2 ℓ Bb i?2 ?2B;?i Q7 i?2 +vHBM/2` M/ ρ~mB/ i?2 /2MbBiv Q7 i?2 ~mB/X u2i MQi?2` Q7i2M mb2/ TQbbB#BHBiv iQ `2T`2b2Mi i?2 /i Bb i?2 7`B+iBQM +Q2{+B2Mi cf = ((1− η)2/π)G/_22i (3)X

6Q` _" ~Qr- i?2 b+HBM; T`QT2`iB2b Q7 i?2 Lmbb2Hi MmK#2` BM i?2 7mHHv im`#mH2Mi `2;BK2 UBX2X 7Q` p2`v H`;2 _vH2B;? MmK#2`b- bv _ ≥ 1010V ?p2 `2+2Bp2/ i`2K2M/Qmb ii2MiBQM BM i?2 Hbi /2+/2 M/ p`BQmb ?2i ~mt K2bm`2K2Mib

RXRX ALh_P.l*hAPL d ?p2 #22M T2`7Q`K2/c ;BM- b22 i?2 `2pB2r `iB+H2 (e)X AM +QMi`bi- h* ~Qr BM i?2 7mHHv im`#mH2Mi `2;BK2 ;Qi Km+? H2bb ii2MiBQM- rBi? i?2 QMHv 2t+2TiBQM #2BM; i?2 h2tb 2tT2`BK2Mi #v arBMM2v- Gi?`QT- M/ +QrQ`F2`b (N- 3- Ry- RR)X AM i?i 2tT2`BK2Mi  _2vMQH/b MmK#2` _2i = 106 Q7 i?2 BMM2` +vHBM/2` rb `2+?2/ UrBi? i?2 Qmi2` +vHBM/2` i `2biV M/ M 2z2+iBp2 TQr2` Hr Q7 G ∝ _2γ i rBi? γ ≈ 1.6Ĝ1.85 rb /2i2+i2/ (N- 3) BM i?2 im`#mH2Mi `2;BK2- i?Qm;? i?2 b+HBM; T`QT2`iB2b `2 MQi T`iB+mH`Hv ;QQ/ M/ bi`B+iHv bT2FBM; γ /2T2M/b QM _2i- BX2X i?2`2 Bb MQ Tm`2 b+HBM;X AM/22/- BM `27bX (Rk- Rj- R) r2 ?p2 `;m2/ i?i i?2`2 b?QmH/ #2  bKQQi? i`MbBiBQM 7`QK γ = 3/2 7Q` i?2 bKHH _2vMQH/b MmK#2` Q7  #QmM/`v Hv2` /QKBMi2/ ~Qr iQ γ = 2 7Q` i?2 H`;2` _2vMQH/b MmK#2` Q7  ~Qr /QKBMi2/ #v i?2 im`#mH2Mi #mHFX hm`#mH2Mi h* 2tT2`BK2Mib ;2i 2p2M KQ`2 b+`+2 7Q` h* ~Qr rBi? BMM2` M/ Qmi2` +vHBM/2` `QiiBM; BM/2T2M/2MiHvX q2 `2 QMHv r`2 Q7 i?2 q2M/i 2tT2`BK2Mib BM i?2 RNjyb (R9)- `2+?BM; _2i ≈ 105M/ _2o ≈ ±105- M/ i?2 `2+2Mi QM2b #v _p2H2i

2i HX (R8)- `2+?BM; _2i ≈ 5 Ǧ 104 M/ _2o = 2Ǧ 104X h?2 ?Bi?2`iQ 2tTHQ`2/ T?b2 /B;`K Q7 h* ~Qr rBi? BM/2T2M/2MiHv `QiiBM; +vHBM/2`b Bb b?QrM BM };X RXRX M Hi2`MiBp2 `2T`2b2MiiBQM Q7 i?2 T?b2 /B;`K Bb ;Bp2M #v .m#`mHH2 2i HX (Re)- r?Q BMi`Q/m+2b  b?2` _2vMQH/b MmK#2` M/  `QiiBQM MmK#2` b Hi2`MiBp2 `2T`2b2MiiBQM Q7 i?2 T?b2 bT+2- b22 #2HQrX

AM i?2 T?b2 /B;`K };X RXR r2 ?p2 HbQ //2/ i?2 _2i- _2o MmK#2`b r?B+? r2 2tTHQ`2/ rBi? Qm` M2rHv +QMbi`m+i2/ hr2Mi2 im`#mH2Mi h* 7+BHBiv Uh3 *V-r?B+? r2 ?p2 /2b+`B#2/ BM ;`2i /2iBH BM `27X (kR)X AM b?Q`i- i i?Bb 7+BHBiv i?2 +vHBM/2`b `2 ℓ = y.Nd K ?B;?c i?2 BMM2` +vHBM/2` ?b  `/Bmb Q7 ri =ky +K-M/ i?2 Qmi2` +vHBM/2` Q7 ro = kd.N +K- M/ i?2 KtBKH BMM2` M/ Qmi2` M;mH` p2HQ+Biv `2 ωi/(2π) = ky >x M/ ωo/(2π) = ±Ry >x- `2bT2+iBp2Hv-+Q``2bTQM/BM; iQ _2i = riωid/ν = 2Ǧ 106 M/ _2o = roωod/ν = ±1.4 Ǧ 106 -rBi? d = ro− riX h?2 bvbi2K Bb 7mHHv i2KT2`im`2 +QMi`QHH2/ i?`Qm;? +QQHBM; Q7 i?2 mTT2` M/ HQr2` THi2X h?2 iQ`[m2 Bb K2bm`2/ i i?2 KB//H2 T`i Q7 i?2 BMM2` +vHBM/2` UbBKBH`Hv b /QM2 BM (N)V #v HQ/ +2HHb BK#2//2/ BMbB/2 i?2 BMM2` +vHBM/2` M/ MQi #v K2bm`BM; i?2 iQ`[m2 i?`Qm;? i?2 b2HBM;X PM2 Q7 i?2 ;QHb r2 rMi iQ +?B2p2 rBi? i?Bb M2r 7+BHBiv Bb iQ 2tTHQ`2 i?2 U_2i -_2o- η = ri/roV T`K2i2` bT+2- i?mb 2Mi2`BM; i2`` BM+Q;MBi- M/ K2bm`2 i?2 iQ`[m2 UBX2X- i`MbTQ`i Q7 M;mH` p2HQ+BivěLmω BM /BK2MbBQMH2bb 7Q`Kě Q`- ;BM 2tT`2bb2/ /Bz2`2MiHv- i?2 Qp2`HH /`;V M/ i?2 BMi2`MH _2vMQH/b MmK#2` Q7 i?2 ~QrX

3 *>Sh1_ RX hP_Zl1 AL h*

6B;m`2 RXR, UV 1tTHQ`2/ T?b2 bT+2 U_2o- _2iV Q7 h* ~Qr rBi? BM/2T2M/2MiHv `QiiBM; BMM2` M/ Qmi2` +vHBM/2`X _B;?i Q7 i?2 ?Q`BxQMiH tBb i?2 +vHBM/2`b `2 +Q@`QiiBM;- iQ i?2 H27i Q7 Bi i?2v `2 +QmMi2``QiiBM;- M/  HQ;@HQ; `2T`2@ b2MiiBQM ?b #22M +?Qb2MX h?2 /b?2/ HBM2b `2 1bb2` M/ :`QbbKMMǶb (Rd) 2biBKi2 7Q` i?2 QMb2i Q7 im`#mH2M+2 rBi? η = 0.71X h?2 KMv /i TQBMib BM i?2 bKHH _2vMQH/b MmK#2` `2;BK2 Q7 Tii2`M 7Q`KiBQM M/ bTiBH i2KTQ`H +?Qb Ub22 2X;X (R3- RN- ky)V ?p2 MQi #22M BM+Hm/2/ BM i?Bb T?b2 /B;`KX Pm` /i TQBMib Q7 i?Bb Tm#HB+iBQM `2 i?2 #H+F /BKQM/bX U#V Pm` /i TQBMib BM i?2 T?b2 /B;`K QM  HBM2` b+H2X U+V Pm` /i TQBMib BM i?2 T?b2 bT+2 Uh- aVc MQi2 i?i h HbQ /2T2M/b QM aX U/V Pm` /i TQBMib BM i?2 T?b2 bT+2 Uh- RΩV Ub22 2[X URXdVVX

RXkX _1alGha L. LGuaAa N

 3FTVMUT BOE "OBMZTJT

AM i?Bb +?Ti2` r2 rBHH 7Q+mb QM i?2 `2[mB`2/ iQ`[m2 7Q` 7mHHv /2p2HQT2/ im`@ #mH2Mi ~Qr U_2i- _2o > 105V- r?2`2 η = 0.716 rBi? BM/2T2M/2MiHv `QiiBM; BMM2` M/ Qmi2` +vHBM/2`- r?B+? ?Bi?2`iQ ?b MQi #22M 2tTHQ`2/X h?2 2tKBM2/ T`K2i2` bT+2 BM i?Bb +?Ti2` Bb b?QrM BM i?2 bT+2 Q7 U_2i- _2oV BM };X R#-M/ Uh- aV BM };X R+- R#-M/ Uh- RΩV BM };X R/ iQ #2 2tTHBM2/ #2HQrX q2 rBHH MQi //`2bb i?2 [m2biBQM r?2i?2` Tm`2 b+HBM; Hrb 2tBbi, 6B`bi- i?2 2tTHQ`2/ _2vMQH/b MmK#2` `M;2 Bb iQQ b?Q`i iQ Mbr2` i?Bb [m2biBQM- M/ b2+QM/- i?2 2`HB2` rQ`F (N- 3- Ry- Rk- Rj- R) ;Bp2b Qp2`r?2HKBM; 2tT2`BK2MiH M/ i?2Q`2i@ B+H 2pB/2M+2 i?i i?2`2 `2 MQ Tm`2 b+HBM; Hrb 2p2M mT iQ _2vMQH/b MmK#2`b Q7 106<sub>X aQ HH b+HBM; 2tTQM2Mib BM i?Bb +?Ti2` ?p2 iQ #2 `2/ b 2z2+iBp2</sub> b+HBM; HrbX 105 <sub>10</sub>6 109 1010 1011 Re<sub>i</sub> G –2 –1 0 0 1 2 Reo x106 x106 Re<sub>i</sub> 6B;m`2 RXk, h?2 /BK2MbBQMH2bb iQ`[m2 G(_2i) 7Q` +QmMi2``QiiBM; h* ~Qr 7Q` 7Qm` /Bz2`2Mi }t2/ pHm2b Q7 _2o/105 = −15- −8- −5- M/ 0 UiQT iQ #QiiQK /i b2ibV- b22 BMb2i 7Q` i?2 T`Q#2/ `2 Q7 i?2 T`K2i2` bT+2X

Pm` `2bmHib 7Q` i?2 +QmMi2``QiiBM; +b2 7Q` i?2 /BK2MbBQMH2bb iQ`[m2 G b 7mM+iBQM Q7 _2i 7Q` }t2/ _2o `2 b?QrM BM };X RXkX PM2 BKK2/Bi2Hv b22b i?i +QmMi2``QiiBQM 2M?M+2b i?2 iQ`[m2 UM/ i?mb i?2 Qp2`HH /`;V- #mi i?i 7Q` ;2M2`H _2o ̸= 0 i?2 2z2+iBp2 TQr2` Hr G ∝ _21.76i - i?i ?QH/b BM i?2 +b2 Q7 BMM2` +vHBM/2` `QiiBM; QMHv- ;2ib HQbic BM 7+i- i?2`2 Bb 2p2M MQ 2z2+iBp2 TQr2` Hr i HHX

Ry *>Sh1_ RX hP_Zl1 AL h* bvbi2K\ h?2 MHvbBb Q7 1+F?`/i 2i HX (R) M/ i?2 MHQ;v Q7 i?2 h* bvbi2K iQ i?2 _" bvbi2K bm;;2bi iQ #2ii2` THQi Lmω b 7mM+iBQM Q7 i?2 hvHQ` MmK#2`

h = 1 4σd 2_(r i+ ro)2(ωi− ωo)2ν−2, URX9V r?2`2 σ = (((1 + η)/2)/√η)4_{- BX2X HQM; i?2 /B;QMHb} ωo=−aωi URX8V

BM i?2 T`K2i2` bT+2 };X RXR#X AM/22/- 1+F?`/i 2i HX (R) /2`Bp2/ 7`QK i?2 mM/2`HvBM; LpB2`@aiQF2b 2[miBQM i?2 2t+i `2HiBQM

ϵw = ϵ− ϵHK = ν3d−4σ−2h(Lmω− 1) URXeV 7Q` i?2 2t+2bb FBM2iB+ 2M2`;v /BbbBTiBQM `i2 ϵw UBX2X- iQiH FBM2iB+ 2M2`;v /BbbBTiBQM `i2 ϵ KBMmb i?2 FBM2iB+ 2M2`;v /BbbBTiBQM `i2 BM i?2 HKBM` +b2 ϵHK = 4νr2_ir2_o(ri + ro)−2d−2(ωi − ωo)2X AM 2[X URXeV σ +M #2 BMi2`T`2i2/ b  U;2QK2i`B+V S`M/iH MmK#2`- M/ h M/ Lmω `2 i?2 2t+i h* MHQ;b iQ i?2 _vH2B;? M/ Lmbb2Hi MmK#2` BM _" ~QrX HQM; i?2 /B;QMH- 2[X URX8V BM T`K2i2` bT+2- QM2 ?b h = 1

4σd2(ri+ ro)2(1 + a)2ωi2ν−2- M/ i?2 r2HH@ bim/B2/ (e) 2z2+iBp2 b+HBM; Hr Lm ∝ _γ˜ <sub>7Q` _" ~Qr UrBi? ˜γ ≈ 0.31 (kk- e)V</sub> MQr rQmH/ +Q``2bTQM/ iQ M 2z2+iBp2 b+HBM; Hr Lmω ∝ hγ 7Q` h* ~QrX Lmω pbX T a Bb b?QrM BM };X RXj 7Q` p`BQmb a- BX2X HQM; p`BQmb bi`B;?i HBM2b i?`Qm;? i?2 Q`B;BM Q7 i?2 T`K2i2` bT+2 };X RXR#X M mMBp2`bH- BX2X a@BM/2T2M/2Mi- 2z2+iBp2 b+HBM; Lmω ∝ hγ rBi? γ ≈ 0.38 Bb +H2`Hv `2p2H2/X h?Bb +Q``2bTQM/b iQ  b+HBM; Q7 G ∝ _21.78

i 7Q` i?2 /BK2MbBQMH2bb iQ`[m2 HQM; i?2 bi`B;?i HBM2b 2[X URX8V BM i?2 T`K2i2` bT+2 };X RXR- iQ cf ∝ _2−0.24i 7Q` i?2 /`; +Q2{+B2Mi- M/ iQ G ∝ h0.88_{X h?2 +QKT2Mbi2/ THQib Lm}

ω/h0.38BM

};X RXj# /2KQMbi`i2 i?2 [mHBiv Q7 i?2 2z2+iBp2 b+HBM; M/ BM //BiBQM b?Qr i?2 a@/2T2M/2M+2 Q7 i?2 T`27+iQ` Q7 i?2 b+HBM; HrX

h?i a@/2T2M/2M+2 Q7 i?2 T`27+iQ` Lmω/h0.38 Bb THQii2/ BM };m`2 RX9X Ai b?Qrb  T`QMQmM+2/ KtBKmK `QmM/ a = 0.4- BX2X 7Q` i?2 KQ/2`i2Hv +QmM@ i2``QiiBM; +b2- bB;MHBM; i?2 KQbi 2{+B2Mi M;mH` p2HQ+Biv i`MbTQ`i 7`QK i?2 BMM2` iQ i?2 Qmi2` +vHBM/2` i i?i pHm2X q2 K2MiBQM i?i Bi Bb Q#pBQmb i?i i?Bb +m`p2 ?b  KtBKmK- b BM #Qi? HBKBiBM; +b2b a → ±∞ UQMHv `QiiBM; Q7 i?2 Qmi2` +vHBM/2`V i?2 ~Qr Bb HKBM` M/ Lmω = 1- #mi Bi Bb BM@ i2`2biBM; iQ MQi2 i?i i?2 KtBKmK /Q2b MQi Q++m` 7Q` i?2 KQbi T`QMQmM+2/ +QmMi2`@`QiiBM; +b2 ωo = −ωi UQ` a = 1VX *QKT`2/ iQ i?2 +b2 Q7 Tm`2 BMM2` +vHBM/2` `QiiBQM Ua = 0V- i a = 0.4 i?2 M;mH` p2HQ+Biv i`MbTQ`i 7`QK BMM2` iQ Qmi2` +vHBM/2` Bb 2M?M+2/ #v KQ`2 i?M kyWX

RXkX _1alGha L. LGuaAa RR 1011 1012 1013 102 103 Ta (a) (b) Nu Z 1011 1012 1013 2 4 6 8 10 12x 10 !" Ta Nu Z Ta !# $" %

6B;m`2 RXj, UV Lmω pbX h 7Q` p`BQmb ac b22 };X RXR# 7Q` i?2 HQ+iBQM Q7 i?2 /i BM T`K2i2` bT+2X  mMBp2`bH 2z2+iBp2 b+HBM; Lmω ∝ h0.38 Bb `2p2H2/X h?2 +QKT2Mbi2/ THQib Lmω/h0.38 BM U#V b?Qr i?2 [mHBiv Q7 i?2 2z2+iBp2 b+HBM; M/ i?2 a@/2T2M/2Mi T`27+iQ` Q7 i?2 b+HBM; HrX

Rk *>Sh1_ RX hP_Zl1 AL h* !"#$2 " "#$ 4 6 8 %" 12&'%" !( a Nu Z Ta !" #( ) : a mp lit ud e 1 %#$ 2 *#$

6B;m`2 RX9, S`27+iQ` Q7 i?2 2z2+iBp2 b+HBM; Hr Lmω ∝ h0.38 Ub?QrM BM };X RXjV b 7mM+iBQM Q7 a = −ωo/ωiX h?2 BMb2i b?Qrb i?2 2z2+iBp2 2tTQM2Mib γ r?B+? `2bmHib 7`QK M BM/BpB/mH }i Q7 i?2 b+HBM; Hr Lmω ∝ hγX

h?2 T`K2i2` a = −ωo/ωi Bb +QMM2+i2/ iQ i?2 bQ@+HH2/ `QiiBQM MmK#2` RΩ= (1− η)(_2i+_2o)/(η_2o− _2i) URXdV BMi`Q/m+2/ #v .m#`mHH2 2i HX (Re) M/ mb2/ #v _p2H2i 2i HX (R8)- MK2Hv RΩ = (1− η)η−1(a− η)(a + 1)−1X q2 HbQ THQi Qm` /i TQBMib BM i?2 T?b2 bT+2 Q7 Uh- RΩV b b?QrM BM };X R/X h?2 QTiBKH pHm2 a ≈ 0.4 r2 7QmM/ 7Q` i?2 i`MbTQ`i T`QT2`iB2b Q7 i?2 bvbi2K +Q``2bTQM/b iQ RΩ≈ −0.09X AM i?Bb +?Ti2` r2 T`272` a b +QKT`2/ iQ RΩ b i?2 bB;M Q7 a BKK2/Bi2Hv bB;MHb r?2i?2` i?2 bvbi2K Bb +Q`QiiBM; Q` +QmMi2``QiiBM;X

 $PODMVTJPO

AM +QM+HmbBQM- r2 ?p2 2tTHQ`2/ i?2 i2`` BM+Q;MBi Q7 7mHHv /2p2HQT2/ im`@ #mH2Mi h* ~Qr rBi? BM/2T2M/2MiHv `QiiBM; BMM2` M/ Qmi2` +vHBM/2`- #2vQM/ _2vMQH/b MmK#2`b Q7 106<sub>- }M/BM; M mMBp2`bH 2z2+iBp2 b+HBM; Hr G ∝ h</sub>0.88 -+Q``2bTQM/BM; iQ Lmω ∝ h0.38- 7Q` HH U}t2/V a = −ωo/ωi- rBi? QTiBKH i`Mb@ TQ`i [mMiBiB2b i a ≈ 0.4X Ai Bb `2K`F#H2 i?i i?2 2z2+iBp2 b+HBM; 2tTQM2Mi 0.382t+iHv `2b2K#H2b i?2 MHQ;Qmb 2z2+iBp2 b+HBM; 2tTQM2Mi BM Lm ∝ _0.38 BM _" +QMp2+iBQM BM i?2 mHiBKi2 `2;BK2 Q7 i?2`KH +QMp2+iBQM (kj- k9)- `2~2+i@ BM; i?2 MHQ;v #2ir22M h* M/ _" ~Qr HbQ BM i?2 bi`QM;Hv im`#mH2Mi `2;BK2X

RXjX *PL*GlaAPL Rj

h?2 M2ti bi2Tb rBHH #2 iQ 7m`i?2` 2ti2M/ i?2 T`K2i2` bT+2 };X RXR iQr`/b ?B;?2` `/Bmb `iBQb η iQ b22 r?2i?2` i?2 Q#b2`p2/ mMBp2`bHBiv +``B2b QM iQr`/b M 2p2M H`;2` T`K2i2` `M;2- M/ iQ HbQ K2bm`2 i?2 hvHQ`@ _2vMQH/b MmK#2` M/ i?2 rBM/ _2vMQH/b MmK#2`b Q7 i?2 BMi2`MH ~Qr- r?B+? `2 +HQb2Hv +QMM2+i2/ iQ Lmω M/ 7Q` r?B+? i?2Q`2iB+H T`2/B+iBQMb 2tBbi (R)X qBi? bm+? K2bm`2K2Mib M/ +?`+i2`BxiBQMb Q7 i?2 ~Qr bi`m+im`2b r2 rBHH HbQ #2 #H2 iQ +?2+F r?2i?2` i?2b2 `2 `2~2+i2/ BM i?2 Qp2`HH i`MbTQ`i T`QT@ 2`iB2bX



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h?2 ~Qr bi`m+im`2 Q7 bi`QM;Hv im`#mH2Mi hvHQ`@*Qm2ii2 ~Qr rBi? _2vMQH/b MmK@ #2`b mT iQ _2i = 2Ǧ 106 Q7 i?2 BMM2` +vHBM/2` Bb 2tT2`BK2MiHHv 2tKBM2/ rBi?

?B;?@bT22/ T`iB+H2 BK;2 p2HQ+BK2i`v USAoVX h?2 rBM/ _2vMQH/b MmK#2`b _2w

Q7 i?2 im`#mH2Mi hvHQ`@pQ`i2t ~Qr Bb 7QmM/ iQ b+H2 b _2w ∝ h1/2- 2t+iHv b

T`2/B+i2/ (k) 7Q` i?2 mHiBKi2 im`#mH2M+2 `2;BK2- BM r?B+? i?2 #QmM/`v Hv2`b `2 im`#mH2MiX h?2 /BK2MbBQMH2bb M;mH` p2HQ+Biv ~mt ?b M 2z2+iBp2 b+HBM; Q7 Lmω∝ h0.38- HbQ BM +Q``2bTQM/2M+2 rBi? im`#mH2M+2 BM i?2 mHiBKi2 `2;BK2X h?2

b+HBM; Q7 LmωBb +QM}`K2/ #v HQ+H M;mH` p2HQ+Biv ~mt K2bm`2K2Mib 2ti`+i2/

7`QK ?B;?@bT22/ SAo K2bm`2K2Mib, i?Qm;? i?2 ~mt b?Qrb ?m;2 ~m+imiBQMb- Bib bTiBH M/ i2KTQ`H p2`;2 MB+2Hv ;`22b rBi? i?2 `2bmHi 7`QK i?2 ;HQ#H iQ`[m2 K2bm`2K2MibX

◦_{Sm#HBb?2/ b, aM/2` :X >mBbKM- .2MMBb SXJX pM :BHb- aB2;7`B2/ :`QbbKM- *?Q}

amM-M/ .2iH27 GQ?b2- lHiBKi2 hm`#mH2Mi hvHQ`@*Qm2ii2 6HQr- S?vbX _2pX G2iX Ry3- yk98yR UkyRkVX h?Bb Tm#HB+iBQM rb b2H2+i2/ b 1/BiQ`bǶ am;;2biBQM - M/ rb 72im`2/ BM  S?vbB+b oB2rTQBMi dzh?2 hrBMb Q7 hm`#mH2M+2 _2b2`+?Ǵ (k8)X 1tT2`BK2Mib- MHvbBb M/ r`BiBM; `2 /QM2 #v >mBbKM- bmT2`pBbBQM #v amM- M/ GQ?b2X S`QQ7`2/ #v 2p2`vQM2X

Re *>Sh1_ kX lGhAJh1 h*

 *OUSPEVDUJPO

h?2 hvHQ`@*Qm2ii2 Uh*V bvbi2K Bb QM2 Q7 i?2 7mM/K2MiH ;2QK2i`B2b +QM@ +2Bp2/ BM Q`/2` iQ i2bi i?2Q`B2b BM ~mB/ /vMKB+bX 6HmB/ Bb +QM}M2/ #2ir22M irQ +QtBH- /Bz2`2MiBHHv `QiiBM; +vHBM/2`bX h?2 bvbi2K ?b #22M mb2/ iQ K2bm`2 pBb+QbBiv- bim/v ?v/`Q/vMKB+ BMbi#BHBiB2b- Tii2`M 7Q`KiBQM- M/ i?2 ~Qr rb 7QmM/ iQ ?p2  p2`v `B+? T?b2 /B;`K (R3)X AM i?2 7mHHv im`@ #mH2Mi `2;BK2- i?2 7Q+mb mT iQ MQr ?b #22M QM ;HQ#H i`MbTQ`i [mMiBiB2b (N- 3- Ry- ke- kd)- r?B+? +M #2 +QMM2+i2/ iQ i?2 iQ`[m2 τ- r?B+? Bb M2+2bb`v iQ F22T i?2 BMM2` +vHBM/2` `QiiBM; i +QMbiMi M;mH` p2HQ+BivX AM `27X (R) i?2 MHQ;v #2ir22M i?2 M;mH` p2HQ+Biv ~mt BM h* im`#mH2M+2 M/ i?2 ?2i ~mt BM _vH2B;?@"ûM`/ U_"- b22 `27X (e)V ~Qr rb rQ`F2/ Qmi- bm;;2biBM; iQ 2tT`2bb i?2 7Q`K2` BM i2`Kb Q7 i?2 Lmbb2Hi MmK#2`- Lmω- r?B+? BM `27X (ke) rb 7QmM/ iQ ?p2 M 2z2+iBp2 b+HBM; Lmω ∝ h0.38 rBi? i?2 hvHQ` MmK#2` Ui?2 MHQ; iQ i?2 _vH2B;? MmK#2` _ BM _" ~QrVX am+? 2z2+iBp2 b+HBM; Lm ∝ _0.38 <sub>+?`+i2`Bx2b i?2 bQ@+HH2/ mHiBKi2 b+HBM; `2;BK2 BM _" ~Qr</sub> (kj- k9- k3)X 6QHHQrBM; i?2b2 TT2`b- :`QbbKMM M/ GQ?b2 (k) ?p2 BMi2`@ T`2i2/ i?Bb b+HBM; b bB;Mim`2 Q7 im`#mH2Mi #QmM/`v Hv2`bX h?2v /2`Bp2/ Lm ∝ _1/2_{× HQ;@+Q``2+iBQMb U_"V M/ Lm}

ω ∝ h1/2× HQ;@+Q``2+iBQMb Uh*VX h?2 HQ;@+Q``2+iBQMb BKTHv i?2 2z2+iBp2 b+HBM; Hr 2tTQM2Mi Q7 0.38X h?2v HbQ K/2  T`2/B+iBQM 7Q` i?2 ++QKTMvBM; b+HBM; Q7 i?2 rBM/ _2vMQH/b MmK#2` _2w- MK2Hv

_2w∝ _1/2 M/

_2w∝ h1/2 UkXRV

7Q` _" M/ h* im`#mH2M+2- `2bT2+iBp2HvX >2`2 i?2 HQ;`Bi?KB+ +Q``2+iBQMb `2@ K`F#Hv +M+2H Qmi- BM +QMi`bi iQ r?i E`B+?MM ?/ T`2/B+i2/ (kN) 2`HB2`-MK2Hv

_2w ∝ _1/2(HQ; _)−1/2 M/

_2w ∝ h1/2(HQ; h)−1/2, UkXkV

r?B+? H2/b iQ M 2z2+iBp2 b+HBM; 2tTQM2Mi Q7 #Qmi yX9d BM i?2 `2H2pMi im`#mH2Mi `2;BK2X AM Q`/2` iQ p2`B7v i?2 BMi2`T`2iiBQM Q7 `27X (k) M/ iQ +?2+F

kXkX 1sS1_AJ1Lh Rd i?2 T`2/B+iBQM UkXRV- HQ+H ~Qr K2bm`2K2Mib `2 `2[mB`2/ iQ 2ti`+i i?2 rBM/ _2vMQH/b MmK#2` _2wX >Qr2p2`- r?i ?TT2Mb HQ+HHv- BMbB/2 i?2 h* ~Qr-?b mT iQ MQr QMHv #22M bim/B2/ 7Q` `2HiBp2Hv HQr _2vMQH/b MmK#2`b _2 < 105 -M/ ?b #22M `2bi`B+i2/ iQ ~Qr T`Q}H2b -M/ bBM;H2@TQBMi biiBbiB+b (R9- jy- RN-jR- jk- jj- N- Ry- j9- j8- je- R8)X

AM i?Bb +?Ti2` r2 bmTTHv HQ+H ~Qr K2bm`2K2Mib 7`QK ?B;?@bT22/ T`iB+H2 BK;2 p2HQ+BK2i`v USAoV i bi`QM;Hv im`#mH2Mi h* ~QrX 6`QK i?2b2 r2 rBHH p2`B7v i?i BM/22/ _2w ∝ h1/2X AM //BiBQM- 7`QK i?2 SAo K2bm`2K2Mib r2 `2 #H2 iQ HbQ 2ti`+i HQ+H M;mH` p2HQ+Biv ~mt2bX h?2b2 `2 7QmM/ iQ bi`QM;Hv ~m+imi2 BM iBK2- #mi r?2M p2`;2/ xBKmi?HHv- `/BHHv- M/ BM iBK2- 7Q` i?2 HQr2` h b?Qr  bHB;?i tBH /2T2M/2M+2- r?B+? r2 BMi2`T`2i b `2KBMBb+2M+2 Q7 i?2 im`#mH2Mi hvHQ` pQ`iB+2b- M/ r?B+? M2`Hv pMBb?2b 7Q` i?2 H`;2bi h r2 +?B2p2X

 &YQFSJNFOU

h?2 TT`imb mb2/ 7Q` i?2 2tT2`BK2Mib ?b M BMM2` +vHBM/2` rBi?  `/Bmb Q7 ri = y.kyy K-  i`MbT`2Mi Qmi2` +vHBM/2` rBi? M BMM2`@`/Bmb Q7 ro = y.kdN K- `2bmHiBM; BM  ;T@rB/i? Q7 d = ro− ri =y.ydN K M/  `/Bmb `iBQ η = ri/ro = 0.716X h?2 ?2B;?i Bb L = y.Nkd K BKTHvBM; M bT2+i `iBQ Q7 Γ = L/(ro− ri) = 11.7X JQ`2 /2iBHb `2;`/BM; i?2 2tT2`BK2MiH 7+BHBiv +M #2 7QmM/ BM `27X (kR)X >2`2 r2 7Q+mb QM i?2 +b2 Q7 BMM2` +vHBM/2` `QiiBQM M/ }t2/ Qmi2` +vHBM/2`X h?2 HQ+H p2HQ+Biv Bb K2bm`2/ mbBM; SAoX q2 miBHBx2 i?2 pB2rBM; TQ`ib BM i?2 iQT THi2 Q7 i?2 TT`imb iQ HQQF i i?2 ~Qr 7`QK i?2 iQTX h?2 ~Qr Bb BHHmKBMi2/ 7`QK i?2 bB/2 mbBM;  TmHb2/ L/@uG6 Hb2` (jd)-+`2iBM;  ?Q`BxQMiH Hb2` b?22iX h?2 rQ`FBM; ~mB/ Uri2`V Bb b22/2/ rBi? ky µK TQHvKB/2 b22/BM; T`iB+H2b- M/ Bb `2+Q`/2/ mbBM;  ?B;? bT22/ +K2` (j3)X h?2 SAo bvbi2K Bb QT2`i2/ BM /Qm#H2@7`K2 KQ/2 r?B+? HHQrb mb iQ ?p2  ∆t 7` bKHH2` i?M 1/f- r?2`2 f Bb i?2 7`K2 `i2X h?2 SAo K2bm`2K2Mib ;Bp2 mb /B`2+i ++2bb iQ #Qi? i?2 M;mH` p2HQ+Biv ω(θ, r, z, t) = uθ(θ, r, z, t)/r M/ i?2 `/BH p2HQ+Biv ur(θ, r, z, t)- bBKmHiM2QmbHvX

6`QK i?2 Hii2` r2 2ti`+i i?2 rBM/ _2vMQH/b MmK#2` b _2w := ur,σ d/ν-r?2`2 ur,σ Bb i?2 biM/`/ /2pBiBQM Q7 i?2 `/BH p2HQ+BivX AM };X kXR _2w Bb b?QrM b  7mM+iBQM Q7 i?2 hvHQ` MmK#2`

h = 1

4σ(ro− ri)

2_(r

R3 *>Sh1_ kX lGhAJh1 h* 109 1010 1011 1012 1013 1014 103 104 105 Ta Re w 109 ₁₀11 ₁₀13 0.030 0.035 0.040 Ta Re w Ta ! 1 !2 Rew" 0.0424Ta0.495 # 0.010

6B;m`2 kXR, _2w pbX hX h?2 /i 7`QK `2T2i2/ 2tT2`BK2Mib i KB/@?2B;?i `2 THQii2/ b b2T`i2 U#Hm2V /Qib- b?QrBM; i?2 [mHBiv Q7 i?2 `2T`Q/m+B#BH@ Biv M/ i?2 biiBbiB+H biiBQM`Biv Q7 i?2 K2bm`2K2MibX q2 ?p2 p2`;2/ xBKmi?HHv- Qp2` iBK2- M/ BM i?2 #mHF ~Qr Uy.kj K ≤ r ≤ y.k8 KVX h?2 bi`B;?i HBM2 Bb i?2 #2bi }i _2w = 0.0424h0.495±0.010M/ i?2 U`2/V /b?2/ HBM2 Bb i?2 E`B+?MM T`2/B+iBQM (kN) 2[X UkXkVX h?2 BMb2i b?Qrb i?2 +QKT2Mbi2/ THQi _2w/h1/2 pb hX h?2 ?Q`BxQMiH U;`22MV HBM2 Bb i?2 T`2/B+iBQM UkXRV Q7 `27X (k)X

AM `27bX (R- ke) h ?/ #22M bm;;2bi2/ b KQbi TT`QT`Bi2 BM/2T2M/2Mi p`B@ #H2 Q7 i?2 h* bvbi2K BM Q`/2` iQ rQ`F Qmi i?2 MHQ;v rBi? _"X >2`2 σ = #(1 + η)/(2√η)$4 +M #2 BMi2`T`2i2/ b  ;2QK2i`B+ dzS`M/iH MmK#2`Ǵ (R)- ωi,o Bb i?2 M;mH` p2HQ+Biv Q7 i?2 BMM2` M/ Qmi2` +vHBM/2`- `2bT2+iBp2Hv-M/ ν Bb i?2 FBM2KiB+ pBb+QbBivX LQi2 i?i h ∝ (ωi − ωo)2, r?BH2 _ BM _" +QMp2+iBQM Bb T`QTQ`iBQMH iQ i?2 i2KT2`im`2 /Bz2`2M+2 iBK2b i?2 ;Bp2M ;`pBiv 7Q`+2- h BM h* ~Qr Bb T`QTQ`iBQMH iQ i?2 M;mH` p2HQ+Biv /Bz2`@ 2M+2 ωi− ωo iBK2b i?2 +2Mi`B7m;H 7Q`+2- r?B+? Bib2H7 Bb HbQ T`QTQ`iBQMH iQ ωi − ωo- BKTHvBM; i?2 b[m`2@/2T2M/2M+2X h?2`27Q`2- #v /2}MBiBQM- i?2 irQ +QMi`QH T`K2i2`b _2i U`272`BM; iQ i?2 BKTQb2/ xBKmi?H p2HQ+BivV M/ h `2 +QMM2+i2/ #v _2i ∼ h1/2- #mi bm+?  i`BpBH `2HiBQM Q7 +Qm`b2 /Q2b MQi 2tBbi #2ir22M i?2 rBM/ _2vMQH/b MmK#2` _2w M/ h Ur?B+? Bb  `2bTQMb2 Q7 i?2 bvbi2Kb M/ `272`b iQ i?2 `/BH p2HQ+BivVX

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6B;X kXR `2p2Hb  +H2` b+HBM; Q7 i?2 rBM/ _2vMQH/b MmK#2` rBi? i?2 hvHQ` MmK#2`- MK2Hv _2w ∝ h0.495±0.010- r?B+? Bb +QMbBbi2Mi rBi? i?2 T`2/B+iBQM (k) _2w∝ h1/27Q` i?2 mHiBKi2 h* `2;BK2- #mi BM+QMbBbi2Mi rBi? E`B+?MMǶb 2`HB2` T`2/B+iBQM UkXkV Q7  b+HBM; 2tTQM2Mi 1/2 rBi? HQ;`Bi?KB+ +Q``2+iBQMb (kN)X 6Q` +QKT`BbBQM- r2 BM+Hm/2/ i?Bb `2HiBQM BMiQ };X kXR- r?B+? +H2`Hv Bb BM+QMbBbi2Mi rBi? i?2 2tT2`BK2MiH /iX q2 bi`2bb i?i i?2 +M+2HHiBQM Q7 i?2 HQ;@+Q``2+iBQM 7Q` _2w b bm;;2bi2/ BM (k) Bb ?B;?Hv MQM@i`BpBH M/ i?i BM _" ~Qr BM i?2 MQM@mHiBKi2 `2;BK2b i?2 rBM/ _2vMQH/b MmK#2` b+H2b b _2w ∼ _0.44 (jN)- T`QMQmM+2/Hv /Bz2`2Mi i?M i?2 1/2 2tTQM2Mi r2 }M/ ?2`2 BM i?2 mHiBKi2 `2;BK2X PMHv p2`v `2+2MiHv i?2 rBM/ _2vMQH/b MmK#2` b+HBM; BM mHiBKi2 _" ~Qr +QmH/ #2 K2bm`2/- HbQ }M/BM; _2w ∼ _1/2 (9y) b T`2/B+i2/ BM `27X (k)X

ky *>Sh1_ kX lGhAJh1 h* !Nu Ω "Θ,r ,t # 325 Ωi 0.10 0.14 0.18 0.22 $ 0.22 $ 0.20 $ 0.18 $ 0.16 $ 0.14 $ 0.12 $ 0.10 $ 0.08 x #m $ y# m$ a $ 30 000 $ 20 000 $ 10 000 0 10 000 20 000 30 000 Nu Ω 0.0 0.2 0.4 0.6 0.8 1.0 0 100 200 300 400 500 r' !Nu Ω " Θ,t b !! !! !! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! !! ! !! !! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! !! !! !! ! ! ! !! !! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! ! ! !! !! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! ! ! !! ! ! ! ! !! !! !! ! !! !! ! ! ! ! ! ! ! !! !! !! ! ! !! ! ! ! !! ! !! !! !! !! !! ! ! !! !! ! ! ! ! !! ! ! ! !! ! !! ! ! ! ! ! !! !! ! !! !! !! ! ! ! !! !! ! ! ! !! !! !! ! ! !! ! !! !! !! ! ! ! !! ! ! !! ! ! !! !! ! ! ! ! ! ! !! !! ! ! !! !! !! !! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! !! ! !! !! !! !! ! ! ! !! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! !! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! !! ! !! !! !! !! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! !! !! !! ! ! ! ! !! !! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! !! !! !! ! ! !! ! ! ! !! !! !! ! !! ! ! !! ! ! ! ! !! !! ! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !! ! !! ! ! !! ! !! ! !! !! !! !! ! !! ! ! ! ! ! !! ! !! !! ! ! !! ! ! ! !! !! ! ! ! !! !! !! ! ! !! ! ! ! ! ! ! ! ! !! !! ! ! ! !! !! ! ! ! !! ! ! !! !! !! ! ! !! ! !! !! ! !! !! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! !! ! ! ! !! ! !! ! ! !! !! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! ! ! ! !! !! ! ! ! ! ! ! ! !! !! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! !! !! !! !! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! ! !! ! ! ! !! !! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! !! !! !! ! ! !! !! ! ! ! ! ! ! ! !! !! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! !! ! ! !! ! !! !! ! ! ! ! !! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! ! !! ! ! ! ! ! ! !! !! ! !! ! !! ! ! ! ! ! !! !! !! !! ! ! ! ! ! ! ! 0.0 0.2 0.4 0.6 0.8 1.0 10 20 50 100 200 500 1000 r' !Nu Ω " Θ,t c 6B;m`2 kXk, UV aMTb?Qi Q7 i?2 BMbiM iM2Qmb +QM p2+iBp 2 M;mH` p2HQ +Bi v ~mt-K2bm`2/ i z = L /2 -7Q` h =1 .5 Ǧ 10 12 X h?2 (r, θ) @THM2 M/ iBK2 p 2`;2/ ~mt Bb 7QmM/ iQ #2 2[mH iQ ⟨Lm ω ⟩θ,r ,t = 325 X U#V GQ +H MQ`KHBx2/ +QM p2+iBp 2 ω @~mt b 7mM+iBQMb Q7 r ′ =( r− ri )/( ro − ri ) 7Q` e ?2B;? ib p`vBM; #2i r 22M 0 .5 ≤ z /L ≤ 0 .73 -7Q` h =1 .5 Ǧ 10 12 X h?2 #H+ F bQHB/ HBM2 Bb i?2 p 2`;2 Q7 i?2 Rk 2tT 2`BK2M ib U⟨ Lm ω ⟩θ,r ,z ,t V-r?B+ ? Bb p2`v +HQb2 iQ i?2 2tT 2+i 2/ pHm2 326 ± 6 7`QK ;HQ#H iQ`[m2 K2bm`2K2M ib (ke )X U+V GQ +H MQ`KHBx2/ +QM p2+iBp 2 M;mH` p2HQ +Bi v ~mt pbX `/BH TQbBiBQM r ′ 7Q` p`BQmb `QiiBQM `i2b-K2bm`2/ i z = L /2 X 6`QK #QiiQK iQ iQT r 2 ? p2 `2T 2i2/ 2tT 2`BK2M ib 7Q` h =3 .8 Ǧ 10 9 -1 .5 Ǧ 10 10 -6 .2 Ǧ 10 10 -3 .8 Ǧ 10 11 -1 .5 Ǧ 10 12 -M/ 6 .2 Ǧ 10 12 X h?2 /b?2/ HBM2b 7Q` i?2 i?`22 ?B;?2bi h `2T`2b2M i i?2 Lm ω pHm2 /2`Bp 2/ 7`QK ;HQ#H iQ`[m2 K2bm`2K2M ib (ke )X

kXjX LGuaAa L. _1alGha kR L2ti- b i?2 SAo K2bm`2K2Mib ;Bp2 mb #Qi? i?2 M;mH` p2HQ+Biv M/ i?2 `/BH p2HQ+Biv Uω(θ, r, z, t) M/ ur(θ, r, z, t) `2bT2+iBp2HvV- r2 +M /B`2+iHv +H+mHi2 i?2 UiQiHV M;mH` p2HQ+Biv ~mt U+QMp2+iBp2 + KQH2+mH`V

Jω(θ, r, z, t) := r3(urω− ν∂rω) , UkX9V r?B+? Bb K/2 /BK2MbBQMH2bb rBi? Bib pHm2 7Q` i?2 HKBM` BM}MBi2 bT2+i `iBQ +b2- Jω

HK = 2νr2ir2o(ωi− ωo)/(ro2− ri2)- ;BpBM; (R) i?2 HQ+H dzLmbb2Hi MmK#2`Ǵ Lmω(θ, r, z, t) = Jω(θ, r, z, t)/JHKω .

AM/22/- b b?QrM BM `27X (R)- i?2 M;mH` p2HQ+Biv Bb i?2 `2H2pMi [mMiBiv i`Mb@ TQ`i2/ 7`QK i?2 BMM2` iQ i?2 Qmi2` +vHBM/2`- b Bib ~mt UkX9V Bb `/BHHv +QM@ b2`p2/- QM+2 Bi Bb p2`;2/ xBKmi?HHv- tBHHv- M/ Qp2`

iBK2-d dr⟨J

ω_{(θ, r, z, t)⟩}

θ,z,t= 0.

AM i?2 im`#mH2Mi `2;BK2 i?2 +QMp2+iBp2 i2`K Bb i?2 KDQ` +QMi`B#miQ` iQ i?2 ~mt BM i?2 #mHF (9R)X

AM };X kXk r2 b?Qr  bMTb?Qi Q7 Lm(θ, r) i KB/@?2B;?i z = L/2 7Q` h = 1.5Ǧ 1012X h?2 [mMiBiv b?Qrb ?m;2 ~m+imiBQMb- `M;BM; 7`QK +105 iQ −105 M/ #2vQM/- r?2`2b i?2 p2`;2 ⟨Lmω(θ, r, t)⟩<sub>θ,r,t</sub> = 325 Bb p2`v +HQb2 iQ i?2 pHm2 Lm;HQ#

ω = 326± 6 Q#iBM2/ 7`QK ;HQ#H iQ`[m2 K2bm`2K2Mib (ke)X h?2 HQ+H ~mt +M i?mb #2 KQ`2 i?M ±300 iBK2b b H`;2 b i?2 K2M ~mtX G`;2 ~m+imiBQMb ?p2 HbQ #22M `2TQ`i2/ 7Q` i?2 HQ+H ?2i@~mt BM _" ~Qr (9k)-#mi BM i?i +b2 i?2 H`;2bi ~m+imiBQMb r2`2 QMHv k8 iBK2b H`;2` i?M i?2 K2M ~mtX

7i2` xBKmi?H M/ iBK2 p2`;2b- ⟨Lmω(θ, r, t)⟩_θ,t- i?2 ~m+imiBQMb M2`Hv pMBb?- b22 };X kXk# U`2p2HBM; bQK2 `/BH M/ ?2B;?i /2T2M/2M+2 7Q` }t2/ h = 1.5 Ǧ 1012- T`2bmK#Hv `2KBMBb2Mi Q7 i?2 hvHQ` pQ`iB+2bV M/ };X kXk+-r?2`2 r2 b?Qr i?2 HQ+H M;mH` p2HQ+Biv ~mt r′<sub>@T`Q}H2b 7Q` `QiiBQM `i2b 7`QK</sub> ωi/(2π) = y.8 >x iQ ky >x- +Q``2bTQM/BM; iQ h = 3.8 Ǧ 109 iQ 6.2 Ǧ 1012X 1+? T`Q}H2 Bb #b2/ QM xBKmi?H p2`;BM;- `/BH #BMMBM;- M/ p2`;BM; Qp2` jkyy 7`K2b U+Q``2bTQM/BM; iQ k8Xe `QiiBQMb 7Q` i?2 i?`22 HQr2bi `QiiBQM `i2b- M/ jk- e9- M/ Rk3 `QiiBQMb 7Q` i?2 7bi2bi `QiiBQMb `i2bVX 6Q` 2+? `QiiBQM `i2 `2T2i2/ 2tT2`BK2Mib ?p2 #22M T2`7Q`K2/ M/ i?2 T`Q}H2b `2 `2T`Q/m+B#H2X PMHv BM QM2 +b2 i?2 im`#mH2Mi hvHQ` pQ`i2t ~Qr b22Kb iQ #2 BM  /Bz2`2Mi bii2UbVX 6`QK };X kXk+ r2 +QM+Hm/2 i?i i?2 bT`2/ BM i?2 `2T2i2/ 2tT2`BK2Mib /2+`2b2b rBi? BM+`2bBM; h- 7Q` r?B+? i?2 hvHQ` pQ`i2t bi`m+@ im`2 rBHH #2 KQ`2 M/ KQ`2 rb?2/ QmiX AM //BiBQM- 7Q` BM+`2bBM; h- MQi

kk *>Sh1_ kX lGhAJh1 h* QMHv /Q r2 K2bm`2 /m`BM; KQ`2 `2pQHmiBQMb- #mi HbQ i?2 i`Mbp2`b2 p2HQ+Biv BM+`2b2b- #Qi? BKT`QpBM; i?2 biiBbiB+bX h?2 /b?2/ HBM2b BM };X kXk+ +Q``2@ bTQM/ iQ i?2 K2bm`2/ ;HQ#H i`MbTQ`i 7Q` i?2 i?`22 ?B;?2bi `QiiBQM `i2bc i?2b2 pHm2b r2`2 Q#iBM2/ 7`QK i?2 iQ`[m2 K2bm`2K2Mib (ke) M/ b?Qr H@ `2/v ;QQ/ ;`22K2Mi rBi? Qm` HQ+H K2bm`2K2MibX

M //BiBQMH tBH p2`;2 Bb M2+2bb`v iQ Q#iBM i?2 2t+i `2HiBQM #2ir22M Lmω M/ i?2 ;HQ#H iQ`[m2 τ `2[mB`2/ iQ /`Bp2 i?2 BMM2` +vHBM/2` i +QMbiMi p2HQ+Biv

(R)-τ = 2πLρJ_HKω _⟨Lmω⟩_θ,z,t. UkX8V

Ai Bb i?2 H+F Q7 bm{+B2Mi tBH p2`;BM;- r?B+? ++QmMib 7Q` i?2 bKHH /2@ pBiBQMb #2ir22M ⟨Lmω(θ, z, t)⟩θ,r,t M/ Lm;HQ#ω X AM/22/- /m2 iQ i?2 hvHQ`@ pQ`i2t bi`m+im`2 Q7 i?2 h* ~Qr QM2 rQmH/ 2tT2+i bQK2 tBH /2T2M/2M+2 Q7 ⟨Lmω(θ, r, z, t)⟩_θ,r,t- r?B+? b?QmH/ #2+QK2 r2F2` rBi? BM+`2bBM; /2;`22 Q7 im`#mH2M+2 M/ i?mb BM+`2bBM; h- Dmbi b };X kXk+ bm;;2bibX h?Bb TB+im`2 Bb +QM}`K2/ BM };m`2 kXjX >2`2 r2 T`2b2Mi HQ+H K2bm`2K2Mib Q7 i?2 +QMp2+iBp2 M;mH` p2HQ+Biv ~mt 7Q` p`vBM; `QiiBQM `i2b- `2bmHiBM; BM  hvHQ` MmK#2` `M;2 Q7 3.8 Ǧ 109_{Ĝ6.2 Ǧ 10}12<sub>X 6Q` 2+? hvHQ` MmK#2` r2 T2`7Q`K2/ KmHiBTH2</sub> 2tT2`BK2Mib M/ K2bm`2/ i?2 Lmω i`MbTQ`i i KB/@?2B;?iX h?2 #Hm2 TQBMib `2 `2bmHib Q#iBM2/ 7`QK SAo K2bm`2K2Mib i KB/@?2B;?i- r?2`2 i?2 H2M;i? Q7 i?2 #`b BM/B+i2 i?2 2``Q` Q#iBM2/ 7`QK i?2 `2T2i2/ 2tT2`BK2MibX h?2 ;`22M M/ Q`M;2 TQBMib `2 `2T2i2/ K2bm`2K2Mib i z = L/2 + d/2 M/ z = L/2 + d- `2bT2+iBp2HvX M 2z2+iBp2 b+HBM; Lmω ∝ h0.45±0.04 Bb `2p2H2/ 7Q` i?2 #Hm2 /i TQBMib- r?BH2  b+HBM; Q7 Lmω ∝ h0.39±0.08 Bb `2p2H2/ 7Q` i?2 Q`M;2 /i TQBMibX

Ai Bb `2K`F#H2 ?Qr i?2 ~Qr T`QpB/2b M;mH` p2HQ+Biv i`MbTQ`i 7`QK i?2 BMM2` iQ i?2 Qmi2` +vHBM/2`- BM bTBi2 Q7 i?2 ~m+imiBp2 Mim`2- r?B+? `2 b22M BM };m`2 kXkX AM };X kX9 r2 T`QpB/2  biiBbiB+H MHvbBb Q7 i?2b2 ~m+imiBQMb, q?BH2 i?2 T`Q##BHBiv /Bbi`B#miBQM 7mM+iBQMb US.6bV Q7 i?2 M;mH` p2HQ+Biv U};X kX9V M/ i?2 `/BH p2HQ+Biv U};X kX9#V `2 M2`Hv bvKK2i`B+- i?2 S.6 Q7 i?2B` T`Q/m+i r3_u

rω ∝ Lmω U};X kX9+V Bb +H2`Hv TQbBiBp2Hv bF2r2/X AM/22/- i?2 +`Qbb@+Q``2HiBQM +Q2{+B2Mi Q7 ur M/ ω U};X kX9/V Bb `2HiBp2Hv H`;2X

q2 MQi2 i?i i?MFb iQ i?2 SAo K2bm`2K2Mib Q7 i?2 7mHH p2HQ+Biv }2H/- i?2 2ti`+iBQM Q7 i?2 HQ+H M;mH` p2HQ+Biv ~mt Lmω(θ, r, z, t) ∝ ωur Bb 2bB2` BM h* b +QKT`2/ iQ i?2 MHQ; i2KT2`im`2 ~mt Lm(⃗x, t) ∝ Tuz BM _" ~Qr, BM Q`/2` iQ Q#iBM i?Bb Hii2` [mMiBiv HQ+HHv- QM2 ?b iQ K2bm`2 i?2 i2KT2`im`2 T M/ i?2 p2HQ+Biv bBKmHiM2QmbHvX "2+mb2  ?B;?@T`2+BbBQM }2H/ K2bm`2K2Mi Q7 i?2 i2KT2`im`2 Bb T`2b2MiHv MQi TQbbB#H2 M/ i?mb MQi pBH#H2- i?2 #2bi QM2 +M /Q 7Q` _" ~Qr Bb iQ K2bm`2 Lm(⃗x, t) TQBMi #v TQBMi (9k- 9j) Q` mb2

kX9X *PL*GlaAPL kj !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! !! !! !! ! !! ! !!! !! !!!!!!!!!!!!!! !! !!!!!!!!!! !!!!!!!! !!!!!!!!!!!!!!!!!!!!! ! !! ! !!! !!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 109 1010 1011 1012 1013 1014 10 100 1000 20 50 200 500 Ta !Nu Ω "Θ,r ,t

6B;m`2 kXj, GQ+H +QMp2+iBp2 M;mH` p2HQ+Biv ~mt b  7mM+iBQM Q7 hvHQ` MmK#2`X h?2 #Hm2 /Qib `2 `2bmHib Q#iBM2/ 7`QK SAo K2bm`2K2Mib M/ b?Qr  b+HBM; Q7 Lmω ∝ h0.45±0.04X h?2 ;`22M M/ Q`M;2 /Qib `2 `2T2i2/ K2bm`2K2Mib i  ?2B;?i Q7 z = L/2+d/2 M/ z = L/2+d- `2bT2+iBp2HvX h?2 #H+F /i TQBMib `2 Q#iBM2/ 7`QK ;HQ#H iQ`[m2 K2bm`2K2Mib M/ b?Qr  b+HBM; i?i Bb H2bb bi22T, Lmω∝ h0.38X h?2 /b?2/ ;`22M HBM2 Bb Q#iBM2/ #v Ki+?BM; irQ HQ;@Hv2`b (3)- M/ ?b  b+HBM; 2tTQM2Mi Q7 0.37 i h = 109 -M/ 0.41 i h = 1013X h?2 `2/ HBM2 Bb 7`QK i?2 im`#mH2Mi #QmM/`v Hv2` i?2Q`v Q7 `27X (k)X Ai ?b  b+HBM; 2tTQM2Mi Q7 0.43 `QmM/ h = 109 _{M/ 0.44} `QmM/ h = 1013<sub>X .`F `2/ /i TQBMib `2 Q#iBM2/ #v K2Mb Q7 ;HQ#H</sub> iQ`[m2 K2bm`2K2Mib (Ry)X

M BMbi`mK2Mi2/ i`+2` (99)X

 $PODMVTJPO

AM +QM+HmbBQM- 7`QK ?B;?@bT22/ SAo K2bm`2K2Mib r2 ?p2 7QmM/ i?2 rBM/ _2vMQH/b MmK#2` BM bi`QM;Hv im`#mH2Mi h* ~Qr iQ b+H2 b _2w ∝ h0.495±0.010 -BM ++Q`/M+2 rBi? i?2 i?2Q`v Q7 `27X (k) M/ -BM +QM~B+i rBi? E`B+?MMǶb (kN) T`2/B+iBQM UkXkVX AM //BiBQM- r2 2ti`+i2/ i?2 HQ+H M;mH` p2HQ+Biv ~mt M/ 7QmM/ i?i Lmω ∝ hγ rBi? γ ∼ 0.39 − 0.45 /2T2M/BM; QM i?2 tBH TQbBiBQM M/ +QMbBbi2Mi rBi? 2`HB2` ;HQ#H iQ`[m2 K2bm`2K2Mib (ke- kd)X 6Q` BM+`2bBM; h-  bKHH tBH /2T2M/2M+2 Q7 Lmω Bb 7/BM; rv- `2~2+iBM; i?2 /2+`2bBM;

k9 *>Sh1_ kX lGhAJh1 h* !6 !4 !2 0 2 4 6 !5 !4 !3 !2 !1 0 Ω# Log 10 PDF !Ω # " a !6 !4 !2 0 2 4 6 !5 !4 !3 !2 !1 0 u#r Log 10 PDF !ur # " b !6 !4 !2 0 2 4 6 !5 !4 !3 !2 !1 0 NuΩ # Log 10 PDF !Nu Ω # " c 0. 0.1 0.2 0.3 0.4 0.5 0.00 0.05 0.10 0.15 0.20 ΤΩi# !2 Π" $ur # ! Ω # %! Τ " d

6B;m`2 kX9, _2bmHib Q7 i?`22 2tT2`BK2Mi rBi? p`vBM; `QiiBQM `i2 `2bmHi@ BM; BM h = 3.8 Ǧ 1011- 1.5 Ǧ 1012- M/ 6.2 Ǧ 1012- +QHQ`2/ BM `2/- ;`22M-M/ #Hm2- `2bT2+iBp2HvX HH i?2 /i b?QrM Bb p2`;2/ Qp2` i?2 `2;BQM y.kj K ≤ r ≤ y.k8 K- M/ K2bm`2/ i KB/@?2B;?iX HH [mMiBiB2b rBi? iBH/2b `2 biM/`/Bx2/ Ub?B7i2/ M/ b+H2/ bm+? b iQ ?p2 x2`Q K2M M/ mMBi p`B@ M+2VX UV S.6 Q7 i?2 biM/`/Bx2/ M;mH` p2HQ+BivX U#V S.6 Q7 i?2 biM@ /`/Bx2/ `/BH p2HQ+BivX U+V aiM/`/Bx2/ MQ`KHBx2/ HQ+H +QMp2+iBp2 M;mH` p2HQ+Biv ~mt S.6X U/V *`Qbb@+Q``2HiBQM +Q2{+B2Mi Q7 i?2 M;mH` p2HQ+Biv M/ i?2 `/BH p2HQ+Biv- i?2 /BK2MbBQMH2bb /2+vBM; iBK2 UBM MmK#2` Q7 `QiiBQMbV Bb 7QmM/ iQ #2 0.07X h?2 +Q``2bTQM/BM; H2M;i? b+H2 +M #2 7QmM/ #v KmHiBTHvBM; i?Bb MmK#2` rBi? i?2 +B`+mK72`2M+2 Q7 i?2 BMM2` +vHBM/2` ;BpBM; δ = 33 KK-r?B+? Bb Q7 i?2 bK2 Q`/2` Q7 K;MBim/2 b i?2 ;T rB/i? d = 3y KKX

BKTQ`iM+2 Q7 i?2 hvHQ` pQ`iB+2bX h?2 M2ti bi2T rBHH #2 iQ T`QpB/2 7mHH p2HQ+Biv M/ M;mH` p2HQ+Biv T`Q}H2 K2bm`2K2Mib- BM+Hm/BM; i?Qb2 BM i?2 #QmM/`v Hv2`b- M/ iQ 2ti2M/ i?2 T`2b2Mi K2bm`2K2Mib iQ i?2 +QmMi2`@`QiiBM; +b2 M/ Qi?2` `/BB `iBQb η- BM Q`/2` iQ 7m`i?2` i?2Q`2iB+HHv mM/2`biM/ i?2 HQ@ +H ~Qr Q`;MBxiBQM M/ i?2 BMi2`THv #2ir22M #mHF M/ #QmM/`v Hv2`b BM im`#mH2Mi h* ~QrX  7m`i?2` ?B;?Hv BMi2`2biBM; bmTTQ`i 7Q` i?2 T`2b2Mi2/ B/2 Q7 i?2 +HQb2 +Q``2bTQM/2M+2 #2ir22M i?2 h* M;mH` p2HQ+Biv i`MbTQ`i BM i?2 bim/B2/ h@`M;2 rBi? i?2 mHiBKi2 `M;2 Q7 _" i?2`KH +QMp2+iBQM Bb iQ B/2MiB7v i?2 QMb2i Q7 i?Bb mHiBKi2 `M;2 r?2M BM+`2bBM; hc ?2`2 r2 2tT2+i  +?M;2 Q7 i?2 Lmω b+HBM; 2tTQM2Mi M/ HbQ  i`MbBiBQMH +?M;2 BM i?2 rB/i?b M/ T`Q}H2b Q7 i?2 "GbX



-PHBSJUINJD CPVOEBSZ MBZFST JO TUSPOH

5BZMPS$PVFUUF UVSCVMFODF

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q2 T`QpB/2 /B`2+i K2bm`2K2Mib Q7 i?2 #QmM/`v Hv2` T`QT2`iB2b BM ?B;?Hv im`@ #mH2Mi hvHQ`@*Qm2ii2 ~Qr mT iQ _2 = 2 Ǧ 106Uh = 6.2 Ǧ 1012V mbBM; ?B;?@

`2bQHmiBQM T`iB+H2 BK;2 p2HQ+BK2i`v M/ T`iB+H2 i`+FBM; p2HQ+BK2i`vc SAo M/ Sho- `2bT2+iBp2HvX q2 }M/ i?i i?2 K2M xBKmi?H p2HQ+Biv T`Q}H2 i i?2 BMM2` M/ Qmi2` +vHBM/2` +M #2 }ii2/ #v i?2 pQM E€`K€M HQ; Hr u+ ₌ 1

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h?2 pQM E€`K€M +QMbiMi κ Bb 7QmM/ iQ /2T2M/ QM i?2 /`BpBM; bi`2M;i? h M/ 7Q` H`;2 h bvKTiQiB+HHv TT`Q+?2b κ ≈ 0.40X h?2 p`BM+2 T`Q}H2b Q7 i?2 HQ+H xBKmi?H p2HQ+Biv ?p2  mMBp2`bH T2F `QmM/ y+_{≈ 12 M/ +QHHTb2 r?2M}

`2b+H2/ rBi? i?2 /`BpBM; p2HQ+Biv UM/ MQi rBi? i?2 7`B+iBQM p2HQ+BivV- /BbTHvBM;  HQ;@/2T2M/2M+2 Q7 y+ <sub>b HbQ 7QmM/ 7Q` +?MM2H M/ TBT2 ~QrbX</sub>

◦_{Sm#HBb?2/ b, aM/2` :X >mBbKM- ap2M a+?`MQrbFB- *?`BbiBM *B2`TF- *?`BbiBM CX}

E ?H2`-.2iH27 GQ?b2- M/ *?Q amM GQ;`Bi?KB+ #QmM/`v Hv2`b BM bi`QM; hvHQ`@*Qm2ii2 im`#mH2M+2-S?vbX _2pX G2iX RRy- ke98yR UkyRjVX 1tT2`BK2Mib T2`7Q`K2/ #v >mBbKM- a+?`MQrbFB- M/ *B2`TFX o2+iQ` +H+mHiBQM M/ T`2HBKBM`v MHvbBb /QM2 #v a+?`MQrbFB M/ *B2`TFX 6m`i?2` MHvbBb- THQiiBM;- M/ r`BiBM; Q7 i?2 KMmb+`BTi /QM2 #v >mBbKMX amT2`pBbBQM #v E ?H2`- GQ?b2-M/ amMX S`QQ7`2/ #v 2p2`vQM2X

ke *>Sh1_ jX GP: "G AL h*

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hvHQ`@*Qm2ii2 Uh*V ~Qr Bb QM2 Q7 i?2 T`/B;KiB+H ~Qrb BM T?vbB+b Q7 ~m@ B/b M2ti iQ _vH2B;?@"ûM`/ +QMp2+iBQM- +?MM2H- ~i THi2- M/ TBT2 ~QrX Ai +QMbBbib Q7 irQ `QiiBM; +QtBH +vHBM/2`b b?2`BM;  ~mB/ BM #2ir22M i?2 +vHBM/2`b- b22 };X jXRX 6Q` QMHv BMM2` +vHBM/2` `QiiBQM i?2 _2vMQH/b MmK#2` _2 = ωiri(ro− ri)/ν [mMiB}2b i?2 /`BpBM; Q7 i?Bb bvbi2KX Ai Bb +HQb2Hv +QM@ M2+i2/ iQ i?2 hvHQ` MmK#2` h = 1

4(

1+η

2√η)4(ro− ri)2(ri+ ro)2ω2i/ν2- i?2 `iBQ Q7 +2Mi`B7m;H 7Q`+2b iQ pBb+Qmb 7Q`+2bX >2`2- ω = uθ/r Bb i?2 M;mH` p2HQ+Biv +QKTQM2Mi- r i?2 `/Bmb- ν i?2 FBM2KiB+ pBb+QbBiv- η = ri/ro i?2 `/Bmb `iBQ-M/ i `iBQ-M/ o bm#b+`BTib /2MQi2 [mMiBiB2b `2Hi2/ iQ i?2 BMM2` `iBQ-M/ Qmi2` +vHBM@ /2`- `2bT2+iBp2HvX 6Q` BM+`2bBM; _2 i?2 bvbi2K Bb }`bi /QKBMi2/ #v +Q?2`2Mi bi`m+im`2b (R3) r?Qb2 H2M;i? b+H2 Bb Q7 bBKBH` bBx2 b i?2 ;T rB/i?X 6Q` 7m`@ i?2` BM+`2bBM; _2 im`#mH2M+2 /2p2HQTb BM i?2 #mHF i H2M;i? b+H2b #2ir22M i?2 BMi2;`H M/ i?2 EQHKQ;Q`Qp b+H2 r?BH2 i?2 #QmM/`v Hv2`b `2 biBHH Q7 HKBM` ivT2X h?Bb `2;BK2- BM r?B+? i?2 ~Qr ?b  im`#mH2Mi #mHF M/ i?2 #QmM/`v Hv2`b `2 Q7 S`M/iH@"HbBmb ivT2- Bb +HH2/ i?2 +HbbB+H `2;BK2 (98)X "v 7m`i?2` BM+`2bBM; _2 i?2 bvbi2K 2Mi2`b i?2 bQ@+HH2/ mHiBKi2 im`#mH2Mi bii2 BM r?B+? HbQ i?2 #QmM/`v Hv2`b ?p2 im`M2/ im`#mH2Mi (kN- k- 9e- 9d)X LQi2 i?i pQ`iB+H bi`m+im`2b TT2` i i?2 }`bi i`MbBiBQM #Qp2  +`BiB+H h (93) M/ +QMiBMm2 iQ T2`bBbi rBi?BM i?2 mHiBKi2 `2;BK2 (Ry- 9N)X "b2/ QM ;HQ#H i`MbTQ`i K2bm`2K2Mib- i?2 mHiBKi2 `2;BK2 Q7 im`#mH2M+2 b2ib BM i _2UhV ≈ 104₍₁₀8₎(N- Ry- 9R)X

h?2 +Q2tBbi2M+2 Q7  HKBM`@ivT2 #QmM/`v Hv2` M/ im`#mH2Mi #mHF BM +HbbB@ +H im`#mH2Mi _vH2B;? "ûM`/ U_"V +QMp2+iBQM ?b #22M r2HH 2bi#HBb?2/ 7`QK MmK2`Qmb 2tT2`BK2MiH M/ MmK2`B+H BMp2biB;iBQMb (8y- e- d- 8R)X h?Bb Bb HbQ i?2 +b2 7Q` i?2 +HbbB+H `2;BK2 BM h* ~Qr (R9- jR- 8k- Ry- 8j) M/ i?2 i`MbB@ iBQM `2;BK2 iQ mHiBKi2 h* ~Qr (89- 88- 8e)X _2+2MiHv-  /B`2+i K2bm`2K2Mi (8d) Q7 i?2 K2M i2KT2`im`2 T`Q}H2 +HQb2 iQ i?2 rHH BM i?2 mHiBKi2 _" bii2 `2p2H2/ HQ;`Bi?KB+ #2?pBQ` BM i?2 mHiBKi2 `2;BK2X 6Q` TBT2- ~i THi2- M/ +?MM2H ~Qrb MmK2`Qmb 2tT2`BK2Mib ?p2 `2p2H2/ i?2 2tBbi2M+2 Q7  HQ; Hv2` 7Q` i?2 p2HQ+Biv M/ Bib p`BM+2 b22 2X;X (83- 8N- ey- eR- ek)X

L2Bi?2` 7Q` h* MQ` 7Q` _" i?2`2 ?/ #22M Mv /B`2+i M/ bvbi2KiB+H K2@ bm`2K2Mi Q7 i?2 p2HQ+Biv #QmM/`v Hv2` BM i?2 ?B;?Hv im`#mH2Mi mHiBKi2 bii2 /m2 iQ i?2 2tT2`BK2MiH /B{+mHiB2bX bbmKBM;  HQ;`Bi?KB+ p2HQ+Biv T`Q}H2 BM i?2 #QmM/`v Hv2`b 7Q` ?B;?Hv im`#mH2Mi h* ~Qr- M/ Ki+?BM; i?2 K2M p2HQ+BiB2b i KB/;T- Gi?`QT 2i HX (N- ej) Q#iBM2/  /2T2M/2M+2 Q7 i?2

jXRX ALh_P.l*hAPL kd ;HQ#H iQ`[m2 M/ i?2 _2- r?B+? ;`22b r2HH rBi? i?2 iQ`[m2 K2bm`2K2Mib BM i?2 mHiBKi2 im`#mH2M+2 `2;BK2 (N- ej- e9)X PMHv `2+2MiHv /B`2+i K2bm`2K2Mib QM #QmM/`v Hv2`b r2`2 +QM/m+i2/ #v pM >Qmi M/ Eix (89) 7Q` h mT iQ 2Ǧ 109- r?2`2 i?2v 7Q+mb2/ QM i?2 2z2+i Q7 +QmMi2``QiiBQM M/ 7QmM/ i?i i?2 pQM E€`K€M +QMbiMi /2T2M/b QM i?2 M;mH` p2HQ+Biv `iBQ ωo/ωiX AM i?Bb +?Ti2` r2 `2TQ`i i?2 /B`2+i bvbi2KiB+H 2tT2`BK2MiH BMp2biB;iBQM Q7 i?2 #QmM/`v Hv2` T`QT2`iB2b 7Q` p2`v ?B;? h 7`QK h = 9.9 Ǧ 108 <sub>iQ</sub> h = 6.2 Ǧ 1012 <sub>mbBM; ?B;? `2bQHmiBQM Sho M/ SAo (e8- ee- ed) rBi? M</sub> mMT`2+2/2Mi2/ bTiBH `2bQHmiBQM /QrM iQ ≈ Ry µKX q2 7Q+mb QM i?2 +b2 Q7 BMM2` +vHBM/2` `QiiBQM- M/ 2tKBM2 i?2 #QmM/`v Hv2` T`QT2`iB2b b  7mM+iBQM Q7 _2 UhV BM i?2 mHiBKi2 im`#mH2Mi h* `2;BK2X

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Ai rb Ki?2KiB+HHv 7QmM/ (R) 7`QK i?2 LpB2`@aiQF2b M/ +QMiBMmBiv 2[m@ iBQMb i?i Jω = r3(⟨urω⟩_z,θ,t− ν∂r⟨ω⟩_z,θ,t) Bb bi`B+iHv +QMb2`p2/ BM h* ~QrX >2`2 ⟨X⟩z,θ,t `2T`2b2Mib tBH- xBKmi?H- M/ iBK2 p2`;BM; Q7 X M/ ur Bb i?2 `/BH p2HQ+BivX h?Bb i`MbTQ`i [mMiBiv Bb BM/2T2M/2Mi Q7 rc Mv ~mt ;Q@ BM; i?`Qm;? M BK;BM`v +vHBM/2` rBi? `/Bmb r HbQ ;Q2b i?`Qm;? Mv Qi?2`

k3 *>Sh1_ jX GP: "G AL h* BK;BM`v +vHBM/2`- Q` Ki?2KiB+HHv ∂rJω = 0X h?Bb ~mt +M #2 K2bm`2/ HQ+HHv (e3- e9) #mi HbQ ;HQ#HHv (N- kd- ke- eN) #v K2bm`BM; i?2 iQ`[m2 M22/2/ iQ bmbiBM +QMbiMi p2HQ+Biv Q7 i?2 +vHBM/2`bX h?2 iQ`[m2 T Bb `2Hi2/ iQ i?2 /BK2MbBQMH2bb iQ`[m2 G M/ iQ Jω b 7QHHQrb,

G = T

2πℓρν2 = Jω

ν2 UjXRV

r?2`2 ρ Bb i?2 /2MbBiv Q7 i?2 ~mB/- M/ ℓ i?2 ?2B;?i Q7 i?2 +vHBM/2`bX q2 +M 7m`i?2` `2Hi2 i?2b2 [mMiBiB2b iQ i?2 rHH b?2` bi`2bb τw- i?2 7`B+iBQM p2HQ+Biv uτ- M/ i?2 pBb+Qmb H2M;i? b+H2 δν i i?2 BMM2` +vHBM/2` rHH,

τw,i= T

2πr2_iℓ UjXkV

uτ,i= %

τw,i/ρ UjXjV

δν,i= ν/uτ,i . UjX9V

LQi2 i?i b Jω Bb +QMb2`p2/ `/BHHv- Bi Bb i?2 bK2 i #Qi? +vHBM/2`b- M/ mbBM; 2[X UjXRV- HbQ i?2 iQ`[m2 T i #Qi? +vHBM/2`b Bb i?2 bK2X *QMb2[m2MiHv-τw- uτ- M/ δν `2 /Bz2`2Mi i i?2 BMM2` M/ Qmi2` +vHBM/2`- M/ i?2 7QHHQrBM; `2HiBQMb ?QH/,

τw,i/τw,o = 1/η2 UjX8V

uτ,i/uτ,o= 1/η UjXeV

δν,i/δν,o= η UjXdV

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i = y.kyy K-  i`Mb@ T`2Mi Qmi2` +vHBM/2` rBi? BMM2` `/Bmb ro =y.kdN K- ;BpBM;  `/Bmb `iBQ Q7 η = 0.716X h?2 +vHBM/2`b ?p2  ?2B;?i Q7 ℓ = y.Nkd K- `2bmHiBM; BM M bT2+i `iBQ Q7 Γ = ℓ/(ro− ri) = 11.7X JQ`2 /2iBHb +M #2 7QmM/ BM `27X (kR)X 6Q` i?2 SAo M/ Sho K2bm`2K2Mib- i?2 rQ`FBM; ~mB/ Uri2`V Bb b22/2/ rBi? ~mQ`2b@ +2Mi TQHvK2` T`iB+H2b (dy)X lbBM;  Hb2` (dR) r2 +`2i2  ?Q`BxQMiH HB;?i b?22i Q7 `Qm;?Hv 8yy µK i?B+FM2bb 7Q` BHHmKBMiBQMX h?2 ~Qr Bb i?2M BK;2/ 7`QK i?2 iQT Ub22 };X jXRV mbBM;  ?B;?@`2bQHmiBQM +K2` (dk) rBi? H`;2 /vMKB+H `M;2X

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0.00 0.01 0.02 0.4 0.6 0.8 1.0

a

0.98 0.99 1.00 0.0 0.2 0.4

b

2.5 % 104!9.9 % 108" 5.0 % 104!3.9 % 109" 1.0 % 105!1.5 % 1010" 2.0 % 105!6.1 % 1010" 5.0 % 105!3.8 % 1011" 9.8 % 105!1.5 % 1012" 2.0 % 106!6.2 % 1012"

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2.6 ! 10 2!1.0 ! 10 5" 1.1 ! 10 3!2.0 ! 10 6" 2.6 ! 10 3!1.0 ! 10 7" 5.7 ! 10 3!5.0 ! 10 7" 1.0 ! 10 5!1.5 ! 10 10 " 2.0 ! 10 5!6.1 ! 10 10 " 5.0 ! 10 5!3.8 ! 10 11 " 9.8 ! 10 5!1.5 ! 10 12 " 2.0 ! 10 6!6.2 ! 10 12" u " # y " Κ# 0.40 B# 5.2 10 0 10 1 10 2 10 3 10 4 10 5 10 6 0 5 10 15 20 25

y

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Outer

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6B;m`2 jXj, UV xBK mi?H p2HQ +Bi v T`Q}H2 M2` i?2 BMM2` +vHBM/2` Ur ′ ∈ [0 ,1/2] V 7Q` p`vBM; _2 (h )X y + Bb i?2 /BbiM+2 7`QK i?2 BMM2` +vHBM/2` BM mMBib Q7 i?2 pBb+Qmb H2M;i? b+H2 δν X u + Bb /2}M2/ b # u (ri ) − u (r ) $ /u τ -r?2`2 u (r ) Bb i?2 xBK mi?H +QKT QM2M i Q7 i?2 p2HQ +Bi v-M/ u (ri ) Bb i?2 xBK mi?H p2HQ +Bi v Q7 i?2 BMM2` +vHBM/2`X U#V xBK mi?H p2HQ +Bi v T`Q}H2 M2` i?2 Qmi2` +vHBM/2` r ′ ∈ [1/2 ,1] X y + = # ro − r $ /δ ν Bb i?2 /BbiM+2 7`QK i?2 Qmi2` +vHBM/2` BM r HH mMBibX 6Q` i?2 +b2 Q7 i?2 Qmi2` #QmM/`v H v2`-i?2 p2HQ +Bi v Bb b+H2/ b u + = u (r )/ uτ X "Qi? };m`2b BM+Hm/2 i?2 HQ;`Bi?KB+ H r Q7 i?2 r HH u + = 1 HM κ y + + B #v pQM E€`K€M-rBi? i?2 ivTB+H pHm2b Q7 κ =0 .40 M/ B =5 .2 -i?2 pBb+Qmb bm#H v2` u + = y + -M/ BM ;` v i?2 }iiBM; /QKBM y + ∈ [50 ,600] X 1+ ? /i@b2i Bb ++QKTMB2/ #v M ``Q r rBi? +TiBQM BM/B+iBM; _2 (h )X h?BM bQHB/ HBM2b `2 .La `2bmHib (88 )X

jXkX 1sS1_AJ1Lha L. _1alGha jR 6Q` 2+? `QiiBQMH p2HQ+Biv- 104 BK;2 TB`b r2`2 +[mB`2/ i  `2+Q`/BM; 7`2@ [m2M+v Q7 Ry >xX h?2 K2M p2HQ+Biv /Bbi`B#miBQM rb +QKTmi2/ mbBM; bBM;H2@ TBt2H 2Mb2K#H2 +Q``2HiBQMX h?Bb i2+?MB[m2 (dj- e8) H2/b iQ  }MH `2bQHmiBQM Q7 ≈ R8y µK M/ BM Qp2` 8yy BM/2T2M/2Mi /iTQBMib BM i?2 3y KK ;TX h?2 biM/`/ /2pBiBQM rb /B`2+iHv +QKTmi2/ 7`QK i?2 p2HQ+Biv T`Q##BHBiv /2M@ bBiv r?B+? rb 2ti`+i2/ 7`QK i?2 b?T2 Q7 i?2 +Q``2HiBQM 7mM+iBQM-b /Bb+mbb2/ BM `27X (ed)X h?Bb T`Q+2/m`2 2Mbm`2b i?i HH im`#mH2Mi b+H2b `2 BM@ +Hm/2/ BM i?2 biM/`/ /2pBiBQMX AM +QMi`bi iQ biM/`/ SAo MHvbBb- r?2`2 QMHv bTiBHHv HQr@Tbb }Hi2`2/ `2bmHib `2 +?B2p2/- ?2`2 HbQ i?2 +QMi`B#miBQM Q7 i?2 bKHH b+H2 ~m+imiBQMb `2 T`QT2`Hv +QMbB/2`2/X AM Q`/2` iQ `2bQHp2 i?2 M2` rHH@`2;BQM i i?2 BMM2` +vHBM/2`-  KB+`Qb+QT2 (d9) rb KQmMi2/ BM 7`QMi Q7 i?2 +K2`X qBi? i?Bb b2imT-  b+HBM; 7+iQ` Q7 ≈ Ry µK/Tt rb +?B2p2/X hQ KtBKBx2 i?2 bTiBH `2bQHmiBQM i?2 M2`@rHH `2;BQM rb 2pHmi2/ rBi? Sho K2i?Q/b- r?B+? Bb #2bi bmBi2/ 7Q` i?Bb Tm`TQb2 (ee)X

6B;X jXk b?Qrb HH i?2 K2bm`2/ KB/@?2B;?i T`Q}H2bc }p2 +Qp2`BM; i?2 7mHH ;T-M/ irQ +Qp2`BM; Dmbi i?2 `2;BQM M2` i?2 BMM2` +vHBM/2`X b 2tT2+i2/- i?2 T`Q}H2b /Q MQi +QM7Q`K iQ i?2 HKBM` MQMpQ`iB+H T`Q}H2- M/ i?2 #mHF ?b  Km+? b?HHQr2` bHQT2 /m2 iQ im`#mH2Mi KBtBM; b i?2 #mHF ~Qr BM Qm` T`K2i2` `2;BK2 Bb 7mHHv im`#mH2Mi (Ry- e9)X b  +QKT`BbQM- r2 BM+Hm/2/ i?2 T`Q}H2 7Q` QM2 Q7 i?2 }`bi pQ`iB+H ~Qrb- _2 = 260 (88)X 6Q` i?2 HKBM` tBHHv BM/2T2M/2Mi bQHmiBQM- i?2 +QMp2+iBp2 T`i Q7 Jω Bb x2`Qc Mv +QMp2+iBQM BM i?2 #mHF Q7 i?2 bvbi2K i?2`27Q`2 /2+`2b2b i?2 ∂rω i2`K BM i?2 #mHF- `2bmHiBM; BM  b?HHQr2` M;mH` p2HQ+Biv ;`/B2MiX 6`QK i?2 BMb2ib r2 +M +H2`Hv b22 i?i 7Q` BM+`2bBM; _2(h) i?2 #QmM/`v Hv2`b #2+QK2 bi22T2`- M/ BM/22/ i?2 M;mH` p2HQ+Biv T`Q}H2 ?b i?2M iQ #2+QK2 bi22T2` b ur BM Jω Bb x2`Q i i?2 rHH- M/ r2 `2 QMHv H27i rBi? i?2 i2`K ∂rωX q2 MQr bTHBi Qm` /i BM irQ T`ib, r′ <sub>∈ [0, 1/2] UBMM2` #QmM/`v Hv2`V M/ r</sub>′ <sub>∈ [1/2, 1] UQmi2` #QmM/`v</sub> Hv2`V M/ MQ`KHBx2 p2HQ+BiB2b rBi? i?2 TT`QT`Bi2 uτ U2[bX UjXjV M/ UjXeVV M/ /BbiM+2b rBi? i?2 `2bT2+iBp2 δν U2[bX UjX9V M/ UjXdVV- b22 };X jXjX q2 mb2/ ;HQ#H iQ`[m2 K2bm`2K2Mib (Ry- ke) iQ }M/ uτ M/ δνX q2 bTiBHHv `2bQHp2/ i?2 pBb+Qmb bm#Hv2` (y+ _{< 5)} 7Q` i?2 HQr2bi _2X Ai Bb FMQrM i?i BM i?Bb pBb+Qmb bm#Hv2` i?2 p2HQ+Biv T`Q}H2 7QHHQrb u+ <sub>= y</sub>+<sub>X h?Bb +QM+2Ti HbQ</sub> TTHB2b iQ h* ~Qr- b bmTTQ`i2/ #v MmK2`B+H bBKmHiBQMb i HQr _2UhV Ub22 2X;X `27X (88)VX Ai b?QmH/ #2 MQi2/ i?i ;HQ#H iQ`[m2 K2bm`2K2Mib T`QpB/2 M p2`;2 iQ`[m2 T - r?BH2 i?2 HQ+H iQ`[m2 /2T2M/b QM ?2B;?i U7QHHQrBM; i?2 H`;2@b+H2 hvHQ` pQ`i2t bi`m+im`2VX aQ r?BH2 2[X UjXkV ?QH/b 7Q` i?2 p2`;2 τw,i- Bi KB;?i ?p2 M tBH /2T2M/2M+2X >Qr2p2`- Bi ?b #22M 7QmM/ (9R) i?i i?2 xBKmi?H p2HQ+Biv i KB/ ;T QMHv ?b  r2F tBH /2T2M/2M+2 Q7 i?2

jk *>Sh1_ jX GP: "G AL h* Q`/2` Q7 ∼ 1W 7Q` a = 0X hQ MQ`KHBx2 i?2 p2HQ+Biv T`Q}H2b- r2 mb2/ i?2 ;HQ#HHv K2bm`2/ iQ`[m2 Ub QMHv i?Bb [mMiBiv Bb pBH#H2 7Q` HH _2(h) rBi? bm{+B2Mi T`2+BbBQMV M/ i?mb i?2 p2`;2 τw,ic i?Bb +mb2b i?2 BKT2`72+i Ki+?BM; Q7 u+_{= y}+X

AM ~i im`#mH2Mi #QmM/`v Hv2` ~Qrb- i y+ <sub>> 50 i?2 2z2+ib Q7 pBb+QbBiv</sub> /BKBMBb?X 6m`i?2`KQ`2- b bm;;2bi2/ #v S`M/iH M/ pQM E€`K€M (d8)- BM i?Bb HBKBi i?2 p2HQ+Biv T`Q}H2 +QMp2`;2b iQ,

u+= 1 κHM y

+_{+ B, UjX3V}

rBi? κ i?2 pQM E€`K€M +QMbiMi M/ B i?2 HQ;`Bi?KB+ BMi2`+2TiX q2 rBHH MQr TTHv i?Bb +QM+2Ti iQ h* ~QrX 1! ! ! ! ! ! ! _!! ! ! ! ! ! ! !! " " _"" " " " " "" "" 107 108 109 1010 1011 1012 1013 0.0 0.5 1.0 1.5 104 ₁₀5 ₁₀6 0.40

Ta

Re

!

"

Κ inner BL Κ outer BL 600 ∆Ν,i!ri

6B;m`2 jX9, h?2 T`K2i2` κ Q#iBM2/ #v }iiBM; u+ ₌ 1

κHM y++ B 7Q` y+ ∈ [50, 600] M2` i?2 BMM2` M/ Qmi2` +vHBM/2`X h?2 +QHQ` b+?2K2 Bb B/2MiB+H iQ };m`2b jXk M/ jXj- i?2 #H+F bvK#QHb `2 .La `2bmHib (88)X h?2 bQHB/ +QHQ`2/ HBM2 b?Qrb i?2 `iBQ Q7 i?2 mTT2` 2/;2 600δν Q7 i?2 }iiBM; `2;BK2 iQ i?2 `/Bmb Q7 +m`pim`2 Q7 i?2 +vHBM/2`- [mMiB7vBM; i?2 `2HiBp2 BM~m2M+2 Q7 i?2 +m`pim`2X LQi2 i?i δν,i/ri = δν,o/roX

PmibB/2 i?2 pBb+Qmb rHH `2;BQM y+ <sub>> 50UBMbB/2 i?2 Qmi2` Hv2`V };X jXj b?Qrb</sub> i?2 2tBbi2M+2 Q7  HQ;@Hv2` BM i?2 mHiBKi2 h* `2;BK2 Uh ! 109<sub>V- r?B+? Bb BM</sub> b?`T +QMi`bi iQ i?2 HKBM` #QmM/`v Hv2`b 7QmM/ BM i?2 .La bBKmHiBQMb (88) UBM/B+i2/ #v i?BM HBM2bV BM i?2 +HbbB+H im`#mH2Mi `2;BK2X Pm` T`Q}H2b `2 }ii2/ iQ 2[X UjX3V Qp2` i?2 BMi2`pH y+ <sub>∈ [50, 600] Ub22 i?2 b?/2/ `2 BM };X</sub> jXjVX q2 ?p2 +?Qb2M y+ _{= 50} b HQr2` 2/;2 Q7 i?2 BMi2`pH b Bi Bb i?2 bi`i Q7 i?2 Qmi2` `2;BQM- M/ i?2 mTT2` 2/;2 y+ _{= 600 +Q``2bTQM/b iQ #Qmi KB/}

jXkX 1sS1_AJ1Lha L. _1alGha jj ;T U(ri+ ro)/2V 7Q` Qm` HQr2bi _2(h) M/ i?2 2/;2 Q7 i?2 HQ;@`2;BK2 7Q` Qm` ?B;?2bi _2(h)- b22 };m`2 jXj- #mi r2 bi`2bb i?i i?2 pHm2b 7Q` κ M/ B QMHv r2FHv /2T2M/ QM i?2 2t+i 2ti2MbBQM Q7 i?2 }iiBM; BMi2`pH- b H`2/v 7QmM/ 7Q` TBT2 ~Qr (eR)X h?2 pHm2b 7Q` κ `2 b?QrM BM };X jX9X h?2v `2 /Bz2`2Mi 7Q` i?2 BMM2` M/ Qmi2` #QmM/`v Hv2` M/ /2T2M/ QM _2(h)X 6Q` BM+`2bBM; _2(h)- r2 b22 i?i i?2 pHm2b 7Q` i?2 BMM2` #QmM/`v Hv2` TT`Q+?  pHm2 Q7 κ ≈ 0.40- +HQb2 iQ i?2 FMQrM +HbbB+H pHm2 Q7 κ = 0.40 (ey- ek)- `2+2MiHv bvbi2KiB+HHv 2tKBM2/ #v J`mbB+ 2i HX (eR)X

6Q` i?2 bKHH2` _2(h) i?2 pHm2 7Q` κ Bb H`;2` b i?2 HQ;@Hv2` Bb MQi v2i 7mHHv /2p2HQT2/- b +H2`Hv b22M 7`QK };X jXjX >Qr2p2`- HbQ ;2QK2i`B+ 2z2+ib-MK2Hv i?2 +m`pim`2 Q7 i?2 +vHBM/2`- Kv +QMi`B#mi2 iQ i?Bb /2pBiBQMX hQ [mMiB7v i?Bb 2z2+i- r2 THQi i?2 `iBQ Q7 i?2 Qmi2` 2/;2 U600δνV Q7 i?2 }iiBM; `2;BK2 iQ i?2 `/Bmb Q7 +m`pim`2 Q7 i?2 +vHBM/2`- b22 };X jX9- 7`QK r?B+? r2 +QM+Hm/2 i?i 7Q` ?B;? _2 > 106 <sub>i?2 2z2+i Q7 i?2 +m`pim`2 #2+QK2b H2bb</sub> i?M 1W- BX2X M2;HB;B#H2X 6BMHHv r2 MQi2 i?i /m2 iQ  TQbbB#H2 bHB;?i ?2B;?i /2T2M/2M+2 Q7 τw,i +mb2/ #v i?2 hvHQ` pQ`iB+2b 2p2M 7Q` i?Bb p2`v im`#mH2Mi ~Qr- M/ i?2 `2bmHiBM; BKT2`72+i Ki+?BM; Q7 u+_{= y}+_{- i?2 pHm2b Q7 κ M/ B} +QmH/ biBHH p`v bHB;?iHv rBi? ?2B;?iX

AM //BiBQM iQ SAo- HbQ ?B;? `2bQHmiBQM Sho K2bm`2K2Mib M/ MHvbBb ?p2 #22M T2`7Q`K2/X 6Q` i?2b2 K2bm`2K2Mi r2 xQQK2/ BMiQ i?2 `2 M2` i?2 BMM2` +vHBM/2` mbBM;  HQM;@/BbiM+2 KB+`Qb+QT2 iQ Q#iBM  b+HBM; 7+iQ` Q7 ≈ Ry µK/TtX h?2 bTiBH `2bQHmiBQM Q7 Sho /2T2M/b QM i?2 MmK#2` Q7 BK;2b-M/ +M i?mb #2 #2ii2` i?M i?2 TBt2H ;`B/ bT+BM; T`QD2+i2/ BM iQ T?vbB+H bT+2 (e8)X q2 2ti`+i i?2 p`BM+2 σ2_(u

θ) 7`QK i?2 b?T2 Q7 i?2 T`Q##BHBiv /2MbBiv 7mM+iBQM Q7 i?2 +Q``2HiBQM 7mM+iBQM (ed)- b22 };X jX8X q2 MQ`KHBx2 σ(uθ)2rBi? i?2 7`B+iBQM p2HQ+Biv uτ Ub22 };X jX8V M/ rBi? i?2 /`BpBM; p2HQ+Biv u(ri)Ub22 };X jX8#VX 6Q` #Qi? +m`p2b i?2 KtBKmK Q7 σ(uθ)2Bb `QmM/ y+= 12-r?B+? Bb `2K`F#Hv bBKBH` iQ i?2 pHm2b Q#iBM2/ BM TBT2 M/ +?MM2H ~Qrb Ub22 2X;X (8N- ey)VX AM //BiBQM- Bi +M #2 Q#b2`p2/ i?i i?2 +QHHTb2 Q7 i?2 /i Bb #2ii2` r?2M r2 MQ`KHBx2 σ(uθ)rBi? i?2 /`BpBM; p2HQ+Biv `i?2` i?M i?2 b?2` p2HQ+Biv uτX PTTQb2/ iQ i?2 T`Q}H2b b?QrM BM };X jXj BM i?2 Qmi2` Hv2` Uy+ > 50V i?2 p`BM+2 T`Q}H2b mMBp2`bHHv +QHHTb2 BMbB/2 i?2 #mz2` Hv2` U5 < y+ <sub>< 30V M/ b22K MQi iQ /2T2M/ QM _2(h)- pQ`iB+H</sub> bi`m+im`2-M/ bi`2KrBb2 +m`pim`2X b b?QrM BM };X jX8- r2 }i i?2 /i 7Q` y+<sub>> 50</sub> UQmi2` Hv2`V rBi? (σ(uθ)/uτ)2 = B1− A1HM y+ěi?2 HQ; Hr 7Q` i?2 p2HQ+Biv p`BM+2 (ek)X h?2 +Q``2bTQM/BM; }iiBM; T`K2i2`b `2 BM/B+i2/ BM };X jX8X _2K`F#Hv- i?2 bHQT2 A1Up`vBM; 7`QK 1.26 iQ 1.45V Bb +QKT`#H2 rBi? pHm2b 7QmM/ UA1 ≈ 1.25V BM ?B;? _2 #QmM/`v Hv2` ~Qrb (ek)X

j9 *>Sh1_ jX GP: "G AL h* 10 100 1000 2 5 20 50 200 500 0.000 0.002 0.004 0.006 0.008 0.010 2 5 12 20 50 100 200 5001000 y! !Σ !uΘ "# u !ri "" 2

b

0 2 4 6 8 10 !Σ !uΘ "# uΤ " 2 2.5% 104 !9.9% 108" 5.0% 104 !3.9% 109" 1.0% 105 !1.5% 1010 " 2.0% 105 !6.1% 1010 " 5.0% 105 !3.8% 1011 " 9.8% 105 !1.5% 1012 " 1.5% 106 !3.5% 1012 "

a

1.40 9.0 1.26 9.3 1.34 11.0 1.36 11.8 1.45 13.1 A1 B1

6B;m`2 jX8, UV h?2 p`BM+2 Q7 i?2 HQ+H xBKmi?H p2HQ+Biv Bb T`2b2Mi2/ b  7mM+iBQM Q7 `/BH /BbiM+2 7`QK i?2 BMM2` +vHBM/2`X h?2 p2HQ+Biv Bb K/2 /BK2MbBQMH2bb mbBM; i?2 7`B+iBQM p2HQ+Biv uτ- M/ i?2 /BbiM+2 Bb BM rHH@mMBib y+ ₌#_r−r

i$/δνX U#V aK2 b BM };X  #mi i?2 p2HQ+Biv ?b #22M `2b+H2/ mbBM; i?2 /`BpBM; p2HQ+Biv u(ri)X *Q``2bTQM/BM; +QHQ`b BM };bX  M/ # +Q``2bTQM/ iQ i?2 bK2 /i@b2i- M/ `2 +QMbBbi2Mi rBi? T`2pBQmb };m`2bX h?2 _2(h) `2 BM/B+i2/ rBi? ``Qrb BM };m`2 X 6Q` y+ _{> 50 i?2 }p2 ?B;?2bi h +b2b} `2 }ii2/ rBi? (σ(uθ)/uτ)2 = B1− A1HM y+X h?2 }iiBM; T`K2i2`b `2 HBbi2/ BMbB/2 };X  7Q` BM+`2bBM; h M/ +QHQ`2/ ++Q`/BM;HvX

jXjX alJJ_u j8

 4VNNBSZ

hQ bmKK`Bx2- r2 T2`7Q`K2/ /B`2+i K2bm`2K2Mib Q7 i?2 p2HQ+Biv #QmM/`v Hv2` T`Q}H2b BM ?B;?Hv im`#mH2Mi h* ~Qr mT iQ h = 6.2 Ǧ 1012 <sub>M/ 7QmM/</sub> i?2 2K2`;2M+2 Q7  HQ; Hv2`- b i?2Q`2iB+HHv T`QTQb2/ BM `27bX (kN- k)X h?2 }ii2/ pQM E€`K€M +QMbiMi κ rb 7QmM/ iQ TT`Q+? i?2 +HbbB+H pHm2 Q7 0.40 7Q` H`;2 2MQm;? hX 6m`i?2`KQ`2- r2 7QmM/ i?i i?2 T2F BM σ(uθ) mMBp2`bHHv +QHHTb2b `QmM/ y+= 12- M/ i?i i?2 ?2B;?i Q7 i?2 T2F Bb 7QmM/ iQ +QHHTb2 #2ii2` r?2M b+H2/ rBi? i?2 /`BpBM; p2HQ+Biv b +QKT`2/ iQ i?2 7`B+iBQM p2HQ+BivX GbiHv- i?2 p`BM+2 T`Q}H2b /2TB+i2/  HQ; /2T2M/2M+2 7Q` y+ > 50X



.VMUJQMF TUBUFT JO IJHIMZ UVSCVMFOU 5BZMPS$PVFUUF

øPX

◦

h?2 m#B[mBiv Q7 im`#mH2Mi ~Qrb BM Mim`2 M/ i2+?MQHQ;v KF2b Bi Q7 miKQbi BK@ TQ`iM+2 iQ 7mM/K2MiHHv mM/2`biM/ im`#mH2M+2X EQHKQ;Q`QpǶb RN9R T`/B;K bm;;2bib i?i 7Q` bi`QM;Hv im`#mH2Mi ~Qrb rBi? KMv /2;`22b Q7 7`22/QK M/ Bib H`;2 ~m+imiBQMb- i?2`2 rQmH/ QMHv #2 QM2 im`#mH2Mi bii2 b i?2 H`;2 ~m+im@ iBQMb rQmH/ 2tTHQ`2 i?2 2MiB`2 ?B;?2` /BK2MbBQMH T?b2 bT+2X >2`2 r2 `2TQ`i i?2 }`bi +QM+HmbBp2 2pB/2M+2 Q7 KmHiBTH2 im`#mH2Mi bii2b 7Q` H`;2 _2vMQH/b MmK#2` _2 = O(106<sub>)UhvHQ` MmK#2` h = O(10</sub>12<sub>)V hvHQ`@*Qm2ii2 ~Qr BM i?2 `2;BK2 Q7</sub>

mHiBKi2 im`#mH2M+2- #v T`Q#BM; i?2 T?b2 bT+2 bTMM2/ #v i?2 `QiiBQM `i2b Q7 i?2 BMM2` M/ Qmi2` +vHBM/2`X h?2 KMB72biiBQM Q7 KmHiBTH2 im`#mH2Mi bii2b Bb 2t@ 2KTHB}2/ #v T`QpB/BM; +QK#BM2/ ;HQ#H iQ`[m2@ M/ HQ+H@p2HQ+Biv K2bm`2K2MibX h?Bb `2bmHi p2`B}2b i?2 MQiBQM i?i #B7m`+iBQMb +M Q++m` BM ?B;?@/BK2MbBQMH ~Qrb Ui?i Bb- p2`v H`;2 _2V M/ [m2biBQMb EQHKQ;Q`QpǶb T`/B;KX

◦_{Sm#HBb?2/ b, aM/2` :X >mBbKM- _Q2HM/ *XX pM /2` o22M- *?Q amM- M/ .2iH27}

GQ?b2-JmHiBTH2 bii2b BM ?B;?Hv im`#mH2Mi hvHQ`@*Qm2ii2 ~Qr- LiX *QKKmMX 8- UkyR9VX

1tT2`BK2Mib-M/ MHvbBb T2`7Q`K2/ #v >mBbKM 1tT2`BK2Mib-M/ pM /2` o22MX SHQiiBM; 1tT2`BK2Mib-M/ r`BiBM; Q7 i?2 KMmb+`BTi /QM2 #v >mBbKMX amT2`pBbBQM #v amM M/ GQ?b2- M/ T`QQ7`2/ #v 2p2`vQM2X