The basic elements of the magnetotail of the magnetosphere (Figure 9.1) are

Transcription

1 Chaptr 9 Quit Magntotail Th baic lmnt of th magntotail of th magntophr (Figur 9.1) ar Mantl (currnt): Rgion of opn fild with high dnity magntohath plama. Lob: Low dnity, trong magntic fild rgion. Enrgy i tord in th lob magntic fild. Plama ht: Rgion of highr dnity and highr thrmal prur clo to th quatorial plan. Th plama β O(1). Currnt ht or nutral ht: rgion of th cro tail currnt gnrating th lob fild. Hr th magntic fild i rathr wak thu th nam nutral ht. Fild alignd currnt: Birkland currnt originat from Earthward boundary of th currnt ht and clo in th ionophr. Th magntotail i th rgion of th magntophr whr nrgy i tord. Th tail play an important rol in magntophric ubtorm. Hr th proc of torag and rla of nrgy in th tail dtrmin much of th magntophric dynamic and i important for th coupling to th ionophr and all rlatd pac wathr ffct. 9.1 Magntotail modl Magntic fild modl Thr ar numbr of modl which dcrib th magntic fild of th magntophr [Tygannko, 1990; Tygannko and Strn, 1996; Tygannko, 000]. Th modl u a uitabl t of ba function and thn fit th cofficint of th function from larg databa of atllit obrvation. Th modl can aily b paramtrizd for variou IMF condition, olar wind prur, dipol tilt tc. It i important to undrtand that th modl ar not quilibrium modl. Thy do not provid any plama data and in gnral it i not poibl fit an iotropic prur ditribution to gnrat quilibria jut from th magntic fild modl. 137

2 CHAPTER 9. QUIET MAGNETOTAIL 138 Intrplantary magntic fild Z Tail (mantl) currnt Plama mantl Magntopau Tail lob Plama ht Fild-alignd currnt Nutral ht currnt Y X Plamaphr Ring currnt Solar wind Magntopau currnt Figur 9.1: Sktch of th magntophr with variou lmnt of th magntotail and th magntophric currnt ytm. Magntic fild modl hav many diffrnt application. For intanc thy can provid a rfrnc for paccraft obrvation. Th modl ar alo ud frquntly to carry out tt particl computation to tudy mchanim which form ditribution function or to dtrmin how particl can ntr into crtain rgion of th magntophr. Howvr, om caution i ncary particularly for thi u. To comput particl trajctori both magntic and lctric fild ar rquird. Th lattr ar oftn aumd for intanc a a contant in th cro tail dirction th. A contant lctric fild, howvr, impli B/ t = 0 and thrfor impli tationary convction in th magntotail. It i highly qutionabl whthr uch a tady tat convction xit. Thi point i dicud latr in thi ction. Whil th magntic fild modl ar not quilibrium modl thy can b ud to obtain typical proprti of th magntic fild uch a th magntic flux tub volum (Figur 9.). Thi can for intanc provid inight into magntophric convction Equilibrium Configuration Th magntotail i urpriingly tabl for long priod of tim. Typically convction i mall and th tail configuration i wll dcribd by quilibrium olution. Analytic quilibria ar availabl for th ction of th magntotail whr th variation along th magntotail (or in th cro-tail dirction y) i mall compard to th variation prpndicular to th currnt ht (wakly two- or thr-dimnional) (.g., Birn t al., 1975; Schindlr, 1975; Birn t al., 1977; Schindlr, 1979a). Bcau of th variation in x th olution ar applicabl only at ufficint ditanc ( 10R E ) from th Earth. Th quilibria can b contructd a fully kintic olution. In th MHD approximation thy olv th tatic MHD quation.

3 CHAPTER 9. QUIET MAGNETOTAIL 139 Figur 9.: Illutration of th magntic fild and om aociatd proprti uch a prur, magntic flux tub volum, and ntropy on magntic fild lin. A impl xampl for thi cla of analytic olution i th following olution which rprnt a two-dimnional modification of th claic Harri ht configuration [Harri, 196]. Kintic quilibria can b contructd by auming th ditribution function a a function of th contant of motion. Spcifically for th magntotail th Harri olution can b drivd by auming t = 0 and y = 0 Such that th contant of motion for th particl pci ar H = m v / + q φ and P y = m v y + q A y whr φand A y ar th lctric potntial and th y componnt of th vctor potntial. Any function of th contant of motion f (r, v) = F (H, P y ) olv th colliionl Boltzmann quation. To obtain/pcify ditribution function that ar in local thrmodynamic quilibrium on can choo F (H, P y ) = c xp ( α H + β P y )

4 CHAPTER 9. QUIET MAGNETOTAIL 140 whr α and β nd to b pcifid to obtain th rquird ditribution function. Th xponnt W = α H + β P y can b r-writtn a [ m ( ) W = α v x + vy + vz + q φ β ] (m v y + q A y ) α = α m ( vx + v y β ) + v z m β + q φ β q A y α α α Thi illutrat that α = 1/ (k B T ) to obtain a Maxwll ditribution (local thrmodynamic quilibrium). Furthr, β /α hould b intrprtd a a contant vlocity that hift th ditribution in th v y dirction. Not alo that φ = φ (r) and A y = A y (r) which nd to b dtrmind through th olution of Maxwll quation (Poion quation and Ampr law). Dfining w = β /α and ĉ = c xp (α m w /) (not that all trm in th xponntial in ĉ ar contant uch that w can jut r-dfin th normalization contant c. Th complt ditribution function for pci can now b writtn a { m ( f (r, v) = ĉ xp v k B T x + (v y w ) ) } { + vz xp q } (φ (r) w A y (r)) k B T Not that th firt xponntial dpnd only on vlocity coordinat and th cond dpnd only on patial coordinat through φ and A y. To obtain charg and currnt dniti w nd to intgrat th ditribution function ovr vlocity pac. Th intgral for th dnity alo dtrmin th normalization cofficint ĉ. Lt u abbrviat th cond xponntial function with { h (φ (r), A y (r)) = xp q } (φ (r) w A y (r)) k B T Normalization: with th dfinition n (r) = n 0 h (φ, A y ) and n (r) = = ĉ h (φ, A y ) = ĉ ( πkb T m th normalization contant i f (r, v) d 3 v { m ( xp v k B T x + (v y w ) ) } + vz d 3 v ) 3/ xp { q k B T (φ (r) w A y (r)) ĉ = n 0 ( m πk B T Charg dnity ρ c and currnt dnity j = vf (r, v) d 3 v for pci ar now obtaind a ) 3/ }

5 CHAPTER 9. QUIET MAGNETOTAIL 141 ρ c = q n 0 h (φ, A y ) j = q n 0 w h (φ, A y ) y Not that currnt dnity in th x and z dirction i 0 bcau of th ymmtry. th currnt dnity along y can b formally computd from th abov intgral or it can b argud that th bulk vlocity mut b w bcau th ditribution function i a Maxwllian (ymmtric) hiftd by th th vlocity w in th v y dirction. Auming a plama of ingl chargd ion and lctron yild for th total charg and th currnt dnity { { ρ c = n 0i xp } { }} (φ w i A y ) n 0 xp (φ w A y ) k B T i k B T { { j y = n 0i w i xp } { }} (φ w i A y ) n 0 w xp (φ w A y ) k B T i k B T With th ourc w nd to olv th Poion quation and Ampr law. Sinc w k a olution on cal much largr than th Dby lngth w can aum a nutral plama. Hr i cutomary to aum ithr quai-nutrality ρ c = 0 or xact nutrality φ = 0. Sinc th aumption of quainutrality lad to baically th am rult but rquir a mor complicatd tratmnt lt u tart dirctly with xact nutrality φ = 0. Th condition ρ c = 0 yild thi quation i olvd by { } { wi n 0i xp A y n 0 xp w } A y = 0 k B T i k B T n 0 = n 0i = n 0 w = T T i w i Subtitution in th quation for currnt dnity and uing B = A = µ 0 j yild A y = λ xp {A y /κ} with λ = µ 0 n 0 w i ( 1 + T T i ) and κ = kb T i / (w i ). Th abov quation i calld th Grad-Shafranov quation. Thr ar variou analytic olution to thi quation. Not that th pcific form of th currnt dnity i du to th choic of th ditribution function. Not, howvr, that our choic only prmit two-dimnional olution. To driv th Harri ht lt u conidr a on-dimnional olution of th Grad-Shafranov quation. d A y dx = λ xp (A y /κ)

6 CHAPTER 9. QUIET MAGNETOTAIL 14 Multiplying thi quation with da y /dx and r-arranging yild d 1 ( ) day + λ dx dx κ xp (A y/κ) = 0 (9.1) Not that B z = da y /dx uch that th abov xprion can alo b writtn a or 1 Bz + λ µ 0 µ 0 κ xp (A y/κ) = cont ( ) day + λ dx κ xp (A y/κ) = cont (9.) which i th quation for total prur balanc whr Bz/µ 0 i th magntic and λ xp (A µ 0 κ y/κ) th thrmal plama prur. W can now intgrat (9.) to obtain th olution for A y a A y = A 0 ln coh x L Rducing th lngth cal L and A 0 to typical plama proprti yild L = λ i v thi w i A 0 = k BT i w i B 0 = gn (w i ) Ti T i + T µ 0 (p i0 + p 0 ) with p 0 = n 0 k B T and v thi = k B T i /m i. Th rlation how that th magntic fild amplitud B 0 cal with th quar root of th thrmal prur. Th width L of th Harri ht i proportional to th ion inrtial lngth and cal with th ratio of ion thrmal vlocity v thi to drift vlocity w i. Th magntic fild, numbr dnity, currnt dnity, and prur ar B z = B 0 tanh x L n = n 0 coh x L j y = j 0 coh x L p = p 0 coh x L with j 0 = B 0 /µ 0 and p 0 = B 0/µ 0. Th magntic fild, dnity, prur, and currnt dnity ditribution ar hown in Figur 9.3. Th Harri ht quilibrium conit of a ht of currnt that parat two rgion of oppoitly dirctd magntic fild of qual magnitud. Th magntic fild prur outid of th currnt ht i balancd by th thrmal prur inid th currnt ht.

7 CHAPTER 9. QUIET MAGNETOTAIL 143 Figur 9.3: Illutration of th Harri magntic fild, dnity, prur and currnt dnity configuration. Aymptotically dnity, prur, and currnt dnity go to zro at incraing ditanc from th currnt ht. Th configuration i clarly highly idalizd and thr ar othr analytic olution of th Grad Shafranov quation. Th importanc of th Harri quilibrium i that it contitut an xact analytic olution of th Vlaov quation. It ha xtnivly bn ud to xamin tability proprti and linar intabiliti of currnt ht uch a th magntotail or th magntopau currnt. Howvr, it hould b notd that quilibria ar uually air to obtain in a fluid or MHD plama. To illutrat baic proprti lt u rprnt th magntic fild through B = A z (x, y) z = A z (x, y) z (9.3) From thi form w can driv th currnt dnity in trm of th vctor potntial j = 1 µ 0 A z z + 1 µ 0 B z z uch that th z componnt of th currnt dnity i j z = 1 µ 0 A z Subtituting thi rprntation of B and j into th tatic forc balanc quation in MHD dmontrat that th prur mut b alo a function of A z. Not that thi i intuitivly clar bcau prur mut b contant on magntic fild lin in an quilibrium. Including a nonzro B z i alo poibl and it turn out that B z i alo a function of A z. Summarizing all of th rult and combining thm with Ampr law yild

8 CHAPTER 9. QUIET MAGNETOTAIL 144 A z = µ 0 d da z p(a z ) with p(a z ) = p(a z ) + B z(a z ) Thi i alo th Grad Shafranov quation, howvr now with a trm on th right id that i a gnral function of A z whra thi trm wa pcifid in our kintic tratmnt through th aumption of local thrmodynamic quilibrium. Th abov quation can alo b olvd in two-dimnion. A common tratmnt particularly for th magntotail i to aum that any variation in th dirction down th tail i much mallr than th variation (gradint) prpndicular to th quatorial plan. On cla of uch olution can b xprd through a modification of th Harri ht olution: µ 0 A y = A c ln coh(z/l(ɛx)) + f(ɛx) B x = A y / z = B 0 (ɛx) tanh(z/l(ɛx)) B z = A y / x Hr l(ɛx) i a gnral function which can b ud to match ithr th prur or magntic fild variation along th magntotail. Figur 9.4 how a configuration with a ralitic prur variation along th tail. Th Earthward boundary i locatd at approximatly 15 R E. Thi cla of analytic olution i particularly uful for a numbr of tudi uch a th tability of particular configuration, prur aniotropy in th tail, and a initial configuration for numrical imulation. In addition to analytic quilibrium olution thr ar numrical quilibrium olution for th magntotail. Th rulting configuration ar not ubjct to th contraint of analytic quilibria and can b xtndd much clor to th Earth. Thy provid inight into th mchanim which cau fildalignd currnt and th tructur at th arthward dg of th plama ht. Thr ar two baic approach on of which olv th MHD quilibrium quation dirctly uing numrical itration mthod. Th othr approach tart from a magntic fild modl with a good gu for th plama and prur ditribution. A numrical rlaxation mthod i thn ud to obtain an quilibrium configuration. 9. Convction in th Magntotail During priod of outhward IMF on can attmpt to approximat convction with a contant crotail componnt of th lctric fild. Conidring a cut in th noon midnight mridian it i poibl to valuat th vlocity prpndicular to th magntic fild uing th E B drift (Figur 3.10). For a contant lctric fild in th poitiv y dirction (which hould b qual to th dayid rconnction rat of 10 3 V/m) on obtain convction of lob fild lin (0 to 40 nt) toward th plama ht of a fw 10 km/. In th plama ht convction i dirctd toward th Earth with rlativly larg vlociti of a fw 100 km/ bcau of th wak magntic fild in th nutral ht (fw nt). Th contant lctric fild impli B/ t = 0. Uing long tim avrag (hour) th magntic fild may

9 CHAPTER 9. QUIET MAGNETOTAIL z Plama Vlocity and Currnt Dnity = Dnity z 10 tim = x 0.5 Figur 9.4: Magntic fild and currnt dnity (top) and plama dnity of a two-dimnional tail quilibrium. indd not chang much locally uch that thi pattrn dcrib corrctly th avrag convction ovr ufficintly long priod of tim. Howvr, thr ar important contraint to convction in th plama ht which nd to b conidrd. Whil tationary convction provid om inight into magntophric convction thr ar important contraint which prohibit tationary convction for mot tim. Uing th MHD quation it i ay to how that th quantity pρ γ i conrvd, i.., d (pρ γ ) /dt = 0. Hr pρ γ i a maur of ntropy. By intgrating ovr th lngth of a fild lin and dfining th diffrntial flux tub volum a V = d/b on can find two additional conrvd quantiti which ar th numbr of particl N and th pcific ntropy S on fild lin. N = ρdl B S = pv γ

10 CHAPTER 9. QUIET MAGNETOTAIL 146 Strictly th conrvation law apply only in idal MHD and if thr i no lo of particl or kintic nrgy into th ionophr. Howvr, for typical application non MHD ffct and/or lo in th ionophr ar ngligibl. Th conrvation law imply that if a fild lin i convctd from 40 R E to 10 R E th numbr of particl and th ntropy rmain contant. Howvr, th fild lin volum i much largr for a typical fild lin originating at 40 R E thn it i at 10 R E. Uing th xampl of th Tygannko modl (Figur 9.) th flux tub volum chang by about two ordr of magnitud which impli that th prur ha to incra by mor th 10 3 to maintain a contant ntropy during convction. Not that th local magntic fild do not chang for tady convction. Howvr, thi would yild an ntirly unralitic prur clo at 10 R E uch that convction for thi fild modl cannot b tationary [Schindlr, 1979b; Erickon and Wolf, 1980; Birn and Schindlr, 1983]. Not that a ralitic prur ditribution, which yild approximatly an quilibrium in th noon midnight mridian, yild an ntropy that vari by two ordr of magnitud (Figur 9.) thrby xcluding tady tat convction. Both dnity and pcific ntropy ditribution from atllit obrvation ar not conitnt with tady tat convction from th mid-tail (30 to 40 R E ) to th nar Earth rgion. Svral modl dcrib a quai-tatic (low) volution of th magntotail (.g., [Schindlr, 1979b; Schindlr and Birn, 198]) and dmontrat that th lctric fild (or convction) i hildd from th plama ht.