The spontaneous magnetic field direction in an anisotropic MHD dynamo

The phenomenon of magnetic field generation in an astrophysical plasma in the frame of developed magnetohydrodynamic (MHD) turbulence is considered. The functional quantum field renormalization group approach is applied to helical anisotropic MHD developed turbulence which is stabilized by the self-...

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Datum:	2012
Hauptverfasser:	Shakhov, B.A., Jurcisin, M., Jurcisinova, E., Stehlik, M.
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Sprache:	English
Veröffentlicht:	Головна астрономічна обсерваторія НАН України 2012
Schriftenreihe:	Кинематика и физика небесных тел
Schlagworte:	Космическая физика
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Zitieren:	The spontaneous magnetic field direction in an anisotropic MHD dynamo / B.A. Shakhov, M. Jurcisin, E. Jurcisinova, M. Stehlik // Кинематика и физика небесных тел. — 2012. — Т. 28, № 5. — С. 27-36. — Бібліогр.: 37 назв. — англ.

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spelling	irk-123456789-773692015-02-28T03:01:43Z The spontaneous magnetic field direction in an anisotropic MHD dynamo Shakhov, B.A. Jurcisin, M. Jurcisinova, E. Stehlik, M. Космическая физика The phenomenon of magnetic field generation in an astrophysical plasma in the frame of developed magnetohydrodynamic (MHD) turbulence is considered. The functional quantum field renormalization group approach is applied to helical anisotropic MHD developed turbulence which is stabilized by the self-generated homogeneous magnetic field. The purpose of the study is to calculate the value as well as direction of the magnetic field in the stochastic dynamo model. The generated magnetic field is determined by ignoring divergent rotor part of Green function of the magnetic field. It is shown that the magnetic field direction is connected with unique existing vector n describing the anisotropic turbulence forcing. Розглядається явище генерації магнітних полів в астрофiзичнiй плазмi у межах моделi розвиненої МГД-турбулентностi. Застосовується квантовопольовий пiдхiд ренормалiзацiйної групи до розвиненої гiротропно-анiзотропної турбулентностi, яка стабiлiзуєть ся самогенерованим магнiтним полем. Метою цього дослiдження є обчислення як значення, так i напрямку магнiтного поля у стохастичнiй моделі динамо. Генероване магнiтне поле визначається при знехтуваннi дивергентної роторної частини функцiї Грiна магнiтного поля. Показано, що напрямок магнiтного поля пов'язаний з наявністю вектора n, який описує анiзотропне накачування енергiї у турбулентнiсть. Рассматривается явление генерации магнитных полей в астрофизической плазме в рамках модели развитой МГД-турбулентности. Применяется квантовополевой подход ренормализационной группы к развитой гиротропно-анизотропной турбулентности, которая стабилизируется самогенерируемым однородным магнитным полем. Целью этого исследования является вычисление как значения, так и направления этого магнитного поля в стохастической модели динамо. Генерируемое магнитное поле определяется при пренебрежении дивергентной роторной частью функции Грина магнитного поля. Показано, что направление магнитного поля связано с наличием вектора n, описывающего анизотропное накачивание энергии в турбулентность. 2012 Article The spontaneous magnetic field direction in an anisotropic MHD dynamo / B.A. Shakhov, M. Jurcisin, E. Jurcisinova, M. Stehlik // Кинематика и физика небесных тел. — 2012. — Т. 28, № 5. — С. 27-36. — Бібліогр.: 37 назв. — англ. 0233-7665 http://dspace.nbuv.gov.ua/handle/123456789/77369 52-337 en Кинематика и физика небесных тел Головна астрономічна обсерваторія НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Космическая физика Космическая физика
spellingShingle	Космическая физика Космическая физика Shakhov, B.A. Jurcisin, M. Jurcisinova, E. Stehlik, M. The spontaneous magnetic field direction in an anisotropic MHD dynamo Кинематика и физика небесных тел
description	The phenomenon of magnetic field generation in an astrophysical plasma in the frame of developed magnetohydrodynamic (MHD) turbulence is considered. The functional quantum field renormalization group approach is applied to helical anisotropic MHD developed turbulence which is stabilized by the self-generated homogeneous magnetic field. The purpose of the study is to calculate the value as well as direction of the magnetic field in the stochastic dynamo model. The generated magnetic field is determined by ignoring divergent rotor part of Green function of the magnetic field. It is shown that the magnetic field direction is connected with unique existing vector n describing the anisotropic turbulence forcing.
format	Article
author	Shakhov, B.A. Jurcisin, M. Jurcisinova, E. Stehlik, M.
author_facet	Shakhov, B.A. Jurcisin, M. Jurcisinova, E. Stehlik, M.
author_sort	Shakhov, B.A.
title	The spontaneous magnetic field direction in an anisotropic MHD dynamo
title_short	The spontaneous magnetic field direction in an anisotropic MHD dynamo
title_full	The spontaneous magnetic field direction in an anisotropic MHD dynamo
title_fullStr	The spontaneous magnetic field direction in an anisotropic MHD dynamo
title_full_unstemmed	The spontaneous magnetic field direction in an anisotropic MHD dynamo
title_sort	spontaneous magnetic field direction in an anisotropic mhd dynamo
publisher	Головна астрономічна обсерваторія НАН України
publishDate	2012
topic_facet	Космическая физика
url	http://dspace.nbuv.gov.ua/handle/123456789/77369
citation_txt	The spontaneous magnetic field direction in an anisotropic MHD dynamo / B.A. Shakhov, M. Jurcisin, E. Jurcisinova, M. Stehlik // Кинематика и физика небесных тел. — 2012. — Т. 28, № 5. — С. 27-36. — Бібліогр.: 37 назв. — англ.
series	Кинематика и физика небесных тел
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first_indexed	2025-07-06T01:40:31Z
last_indexed	2025-07-06T01:40:31Z
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fulltext	ÊÎÑÌÈ×ÅÑÊÀß ÔÈÇÈÊÀ UDC 52-337 B. A. Shakhov1, M. Jurcisin2, E. Jurcisinova2, M. Stehlik2 1Main As tro nom i cal Ob ser va tory of NASU, Zabolotnoho Str. 27 Kyiv, Ukraine, 03680 2In sti tute of Ex per i men tal Phys ics of SAS, 04001 Kosice, Watsonova 47, Slovakia stehlik@saske.sk The spon ta ne ous mag netic field di rec tion in an anisotropic MHD dy namo The phe nom e non of mag netic field gen er a tion in an as tro phys i cal plasma in the frame of de vel oped magnetohydrodynamic (MHD) tur bu lence is con - sid ered. The func tional quan tum field renormalization group ap proach is ap plied to he li cal anisotropic MHD de vel oped tur bu lence which is sta bi - lized by the self-gen er ated ho mo ge neous mag netic field. The pur pose of the study is to cal cu late the value as well as di rec tion of the mag netic field in the sto chas tic dy namo model. The gen er ated mag netic field is de ter mined by ig nor ing di ver gent ro tor part of Green func tion of the mag netic field. It is shown that the mag netic field di rec tion is con nected with unique ex ist ing vec tor n de scrib ing the anisotropic tur bu lence forc ing. ÍÀÏÐßÌÎÊ ÑÏÎÍÒÀÍÍÎÃÎ ÌÀÃÍIÒÍÎÃÎ ÏÎËß Ó ÀÍIÇÎÒÐÎÏ - ÍÎÌÓ ÌÃÄ-ÄÈÍÀÌÎ, Øàõîâ Á. Î., Þð÷èøèí Ì., Þð÷èøèíîâà Å., Ñòåãëiê Ì. — Ðîçãëÿäàºòüñÿ ÿâèùå ãåíåðàöi¿ ìàãíiòíèõ ïîëiâ â àñòðî - ôiçè÷íié ïëàçìi ó ìåæàõ ìîäåëi ðîçâèíåíî¿ ÌÃÄ-òóðáóëåíòíîñòi. Çàñòîñîâóºòüñÿ êâàíòîâîïîëüîâèé ïiäõiä ðåíîðìàëiçàöiéíî¿ ãðóïè äî ðîçâèíåíî¿ ãiðîòðîïíî-àíiçîòðîïíî¿ òóðáóëåíòíîñòi, ÿêà ñòàáiëiçó - ºòü ñÿ ñàìîãåíåðîâàíèì ìàãíiòíèì ïîëåì. Ìåòîþ öüîãî äîñëiäæåííÿ º îá÷èñëåííÿ ÿê çíà÷åííÿ, òàê i íàïðÿìêó ìàãíiòíîãî ïîëÿ ó ñòîõàñ - òè÷íié ìîäåëi äèíàìî. Ãåíåðîâàíå ìàãíiòíå ïîëå âèçíà÷àºòüñÿ ïðè çíåõòóâàííi äèâåðãåíòíî¿ ðîòîðíî¿ ÷àñòèíè ôóíêöi¿ Ãðiíà ìàãíiò íî - ãî ïîëÿ. Ïîêàçàíî, ùî íàïðÿìîê ìàãíiòíîãî ïîëÿ ïîâ'ÿçàíèé ç íàÿâí³ñ - òþ âåêòîðà n, ÿêèé îïèñóº àíiçîòðîïíå íàêà÷óâàííÿ åíåðãi¿ ó òóð áó - ëåíò íiñòü. ÍÀÏÐÀÂËÅÍÈÅ ÑÏÎÍÒÀÍÍÎÃÎ ÌÀÃÍÈÒÍÎÃÎ ÏÎËß Â ÀÍÈÇÎ - ÒÐÎÏ ÍÎÌ ÌÃÄ-ÄÈÍÀÌÎ, Øàõîâ Á. À., Þð÷èøèí Ì., Þð÷èøèíî - âà Å., Ñòåãëèê Ì. — Ðàññìàòðèâàåòñÿ ÿâëåíèå ãåíåðàöèè ìàãíèòíûõ 27 ÊÈÍÅÌÀÒÈÊÀ È ÔÈÇÈÊÀ ÍÅÁÅÑÍÛÕ ÒÅË òîì 28 ¹ 5 2012 © B. A. SHAKHOV, M. JURCISIN, E. JURCISINOVA, M. STEHLIK, 2012 28 B. A. SHAKHOV et al. ïîëåé â àñòðîôèçè÷åñêîé ïëàçìå â ðàìêàõ ìîäåëè ðàçâèòîé ÌÃÄ-òóð - áóëåíòíîñòè. Ïðèìåíÿåòñÿ êâàíòîâîïîëåâîé ïîäõîä ðåíîðìàëèçà öè - îí íîé ãðóïïû ê ðàçâèòîé ãèðîòðîïíî-àíèçîòðîïíîé òóðáó ëåíò- íî ñ òè, êîòîðàÿ ñòàáèëèçèðóåòñÿ ñàìîãåíåðèðóåìûì îäíîðîäíûì ìàã íèò íûì ïîëåì. Öåëüþ ýòîãî èññëåäîâàíèÿ ÿâëÿåòñÿ âû÷èñëåíèå êàê çíà÷åíèÿ, òàê è íàïðàâëåíèÿ ýòîãî ìàãíèòíîãî ïîëÿ â ñòîõàñ - òè÷åñêîé ìîäåëè äèíàìî. Ãåíåðèðóåìîå ìàãíèòíîå ïîëå îïðåäåëÿ åò - ñÿ ïðè ïðåíåáðåæåíèè äèâåðãåíòíîé ðîòîðíîé ÷àñòüþ ôóíêöèè Ãðè íà ìàãíèòíîãî ïîëÿ. Ïîêàçàíî, ÷òî íàïðàâëåíèå ìàãíèòíîãî ïîëÿ ñâÿ çàíî ñ íàëè÷èåì âåêòîðà n, îïèñûâàþùåãî àíèçîòðîïíîå íàêà÷è - âàíèå ýíåðãèè â òóðáóëåíòíîñòü. IN TRO DUC TION Magnetohydrodynamic (MHD) tur bu lent dy namo still at tracts large at ten - tion due to many ap pli ca tions in both the as tro phys i cal and lab o ra tory plas - mas. De spite an enor mous ef fort, now a days the re lated prob lems and ques - tions re main de fi ciently ex plored. That is ques tion, how does the tur bu lence am plify and sus tain mag netic fields (MFs), what is the spec trum and struc - ture of this field at var i ous scales in as tro phys ics as well as in a lab o ra tory. Many au thors at tempt to jus tify the ba sic treat ment of sta ble re gimes of fully de vel oped MHD tur bu lence in var i ous ap proaches: ki netic or mag - netic driv ing the he li cal or non-he li cal dy namo. An other even tual ap proach is based on the self-con sis tent non lin ear set of MHD equa tions, the Navier-Stokes equa tion in clud ing the Lo rentz force to gether with the in - duc tion equa tion. Now a days the mean field he li cal dy namo is used to be ap plied in as tro - phys i cal ro ta tors [22, 26, 28]. It in volves ini tially weak large-scale field am - pli fied by strong he li cal ve loc ity fluc tu a tions. In this un der stand ing the large-scale field is am pli fied and sus tained on scales sig nif i cantly larger than the scale of the driv ing tur bu lence, and the source of E = aB (a is known as the dy namo co ef fi cient) typ i cally is the ki netic helicity, Hk º v v× rot . Cal cu la tions of the field spec tra are ob vi ously per formed in the case of iso tro pic MHD tur bu lence. Re cently, anisotropic con tri bu tions from mean ve loc ity flows have been con sid ered as ad di tional con tri bu tions to the elec tro mo tive force driv ing the ve loc ity driven he li cal dy namo [4, 31]. Gen er ally, the MHD tur bu lence is now a days un der stood to be in her - ently anisotropic. A num ber of im por tant phys i cal pro cesses in anisotropic MHD tur bu lence was clar i fied in [2, 17, 23, 24] where the con cept of crit i - cal bal ance was used to de ter mine the ra tio of the di men sions of tur bu lent ed dies in the di rec tions par al lel and per pen dic u lar to the lo cal MF. As was em pha sized in [6], the un der stand ing of lab o ra tory and/or as tro - phys i cal dy namo de pends on dom i nant con tri bu tion (role) of mag netic fields, or al ter na tively, typ i cal flows (known as the ki ne matic dy namo). In clas si cal un der stand ing of non lin ear as tro phys i cal dy namo the ory one sep a - rates “small-scale nonhelical dy na mos” in which mag netic en ergy is am pli - fied by ran dom force-line stretch ing on scales ap prox i mately less to that of the tur bu lent forc ing [16, 20, 21]. Con trary to “large-scale he li cal dy na - mos”, the he li cal hy dro dy namic tur bu lence pro duces MF on scales larger than that of the in put tur bu lent forc ing (this is one we will con sider here) [1, 5, 7, 22, 26, 28, 29]. This type of dy namo is of ten used to ex plain MFs of the so lar and stel lar co ro nae as well as co ro nae above ac cre tion discs [14,15]. Nu mer i cal stud ies of the ef fect of a reg u lar MF on a-dy namo ac tion, due to helicity driven MHD tur bu lence, are re ported in [27] for unite Prandtl num - ber. They found that the ap par ent ef fect of the dc-mag netic field is to sup - press the dy namo ac tion. Re cently sev eral pa pers deal with a spe cial flow pat tern, so-called “Rob erts flow”, which con cerns the low-di men sional dy - namo mod el ing [12, 32, 37]. This model can be use ful for the ex am i na tion of lab o ra tory ex per i ments. They at tempt to solve nu mer i cally the dy namo prob lem in the Fourie rep re sen ta tion with small dis crete modes. Note that the ki netic helicity is not nec es sary for a mod i fied ver sion of the Ro b erts- type dy namo [30]. In the he li cal MHD an ad di tional prob lem arises: the in sta bil i ties in - duce the ex po nen tial in crease of the mag netic fluc tu a tions in the large scales range (see in [11], for ex am ple). The elim i na tion of this in sta bil ity leads to for ma tion of a large-scale mag netic field known as the tur bu lent dy - namo. Re moval of the in sta bil ity in quan tum field for mu la tion of he li cal MHD can be achieved by means of a nice and very well-known spon ta ne - ous sym me try break ing mech a nism fol lowed by the cre ation of a ho mo ge - neous sta tion ary mag netic field. The prob lem was an a lyzed in [3] in the spe - cial case of iso tro pic MHD tur bu lence. In deed, the ba sic phys i cal prob lem aris ing in the MHD tur bu lence sta bil ity re gime is that aris ing “ro tor in sta - bil i ties” in MHD tur bu lence needs the cre ation of a large-scale mag netic field, but then the MHD tur bu lence does not to be iso tro pic. For ex am ple, the AlfvJnic tur bu lence (weak forc ing when the tur bu lence ex ci ta tions are small-am pli tude dis tur bances prop a gat ing along the MF) is man i festly anisotropic [33]. The sta bi li za tion of an instable sys tem through the ap pear ance of a spon ta ne ous mean field is a typ i cal ef fect in field-the o retic mod els of var i - ous phe nom ena. A typ i cal ex am ple is the ap pear ance of spon ta ne ous mag - ne ti za tion be low the crit i cal tem per a ture Tc in ferromagnetics. There, the stan dard tech ni cal pro ce dure can be used. How ever, in the pres ent prob lem this tech nique is not ap pli ca ble (as was dis cussed in [3]). Here, we in tro duce a non zero mean mag netic field al ready in or i gin prob lem to sta bi lize the tur - bu lent sys tem. In [10, 13] the dy namo co ef fi cient a was de rived us ing the quasilinear MHD tur bu lent the ory. They found that in the limit of large ki netic Re and mag netic Rem , the a-co ef fi cient may be not small if the cor re la tion time of ve loc ity field and mag netic field are shorter than the eddy turn over time of the MHD tur bu lence and a de pends in ten sively on the mag netic Prandtl num ber. The op po site limit of small mag netic and fluid Reynolds num bers was con sid ered in [35]. In this case there is no small-scale dy namo ac tion and so the small-scale mag netic field is solely due to shred ding the large- 29 THE SPONTANEOUS MAGNETIC FIELD DIRECTION scale mag netic field. In gen eral a has a ten sor form, a ij [22], which was also used in nu mer i cal sim u la tions [8, 9]. They showed rather fluc tu at ing value of the a-ef fect ten sor in all the com po nents. Thus the gen eral case of an ani so tropy pres ence is to be ex am ined. Here we will use the func tional quan tum field renormalization group (RG) ap proach [1, 36]. It as signs a field ac tion to the sto chas tic prob lem and makes pos si ble to use el e gant and very well de vel oped RG pro ce dure in quan tum field the ory to in ves ti gate in fra red as ymp totic re gimes of a sto - chas tic sys tem. This the o retic-field RG ap proach con cerns to (de scribes) sta tion ary state of gen er ated MF un der equi lib rium in the sense when the state is the so lu tion of self-con sis tent non lin ear MHD equa tions (in a he li cal me dium) with the he li cal in jec tion of en ergy. (In the RG ter mi nol ogy the sta ble re gime is de scribed by the fixed point of RG-equa tions.) Con tin u ous in jec tion of helicity typ i cally leads to a quasi-steady dy nam i cal equi lib rium with self-gen er ated MF as a sta bi liz ing fac tor [18, 34]. But in the iso tro pic de vel oped MHD tur bu lence it is not pos si ble to determine a di rec tion of gen er ated ho mo ge neous MF. This ques tion is the goal of the pres ent pa per. We use the the o retic-field RG ap proach to he li cal anisotropic MHD tur bu - lence which is sta bi lized by the self-gen er ated MF. Its di rec tion is con - nected with unique ex ist ing vec tor n de scrib ing the anisotropic forc ing (see be low) . THE MODEL FOR MU LA TION The in ter ac tion of elec tri cally neu tral con duc tive tur bu lent in com press ible fluid (with the unit mag netic per me abil ity) with the mag netic field in the case of anisotropically driven MHD tur bu lence is de scribed by the MHD equa tions for the fluc tu at ing part of the fields: $ ( ) ( ) ~ Lv t A v vº -¶ + + Ñ + Ñ + + =v v v v b b f f fnD , (1) $ ( ) ( ) ~ Lb t A b bº -¶ + ¢ + Ñ - Ñ + + =b b v b b v f fn D 0. The first equa tion is the well-known Navier — Stokes equa tion for the transversal ve loc ity field v x( , )t with the ad di tional non lin ear con tri bu tion of the Lo rentz force (the lon gi tu di nal con tri bu tion is as cribed to pres sure p). The sec ond equa tion for mag netic field b x( , )t (in the Alfven units b = B / 4pr, r is a fluid den sity) fol lows from the Maxwell equa tions for con - tin u ous me dium. The mag netic dif fu sion co ef fi cient n¢ is con nected with the co ef fi cient of mo lec u lar vis cos ity by re la tion n¢ = un with dimensionless mag netic Prandtl num ber u-1 . The terms f A i are re lated to uni ax ial ani so - tropy and they have the fol low ing form [2]: f n v n nv n n nvA v = Ñ + Ñ + Ñ +n c c c[ ( ) ( ) ( ) ( )]1 2 2 2 3 2 + Ñ + Ñ + Ñ + Ñl l l l1 2 2 3 4 2b n nb n n n b nb n n nb( )( ) ( ) ( )( ) ( )( )b , f n b n nv n n nvA b = ¢ Ñ + Ñ + Ñn t t t[ ( ) ( ) ( ) ( )]1 2 2 2 3 2 . (2) 30 B. A. SHAKHOV et al. The pa ram e ters c i and t i char ac ter ize the weight of the in di vid ual struc tures in Eqs (2), and the unit vec tor n spec i fies the di rec tion of the ani - so tropy axis. The large-scale ran dom force per unit mass f cor re sponds to a ki netic en ergy dop ing and it is as sumed to have Gaussi an sta tis tics de fined by the av er ages: f i = 0, f t f ti j( , ) ( , )x x1 1 2 2 = D t tij ( , )x x1 2 1 2- - . (3) The two-point cor re la tion ten sor D t t d D iij ij( , ) ( ) ( ) ~ ( )exp( )x k k k x= ×òd p2 3 (4) is con ve nient to parametrize in the fol low ing way: ~ ( ) [( ) ( ) ( )D gu k P R Qij k ij ij ik k k= + + +-20 12 3 1 2 1 2 2p n a x a re j ] (5) as lin ear com bi na tion of both ten sor and pseudotensor (in our he li cal case). Here e ³ 0 is dimensionless pa ram e ter of the model; its phys i cal value is e = 2 which cor re sponds to the Kolmogorov en ergy pump ing from in fra-red re gion of the small k. The value e = 0 cor re sponds to a log a rith mic per tur ba - tion the ory for a cal cu la tion of Green func tions when g, which plays the role of a bare cou pling con stant of the model, be comes dimensionless [36]. The prob lem of the con tin u a tion from e = 0 to the phys i cal val ues was dis cussed in [1]. The (3´3)-ma tri ces Pij , Rij , and Qij are the trans verse pro jec tors. Their ex plicit forms are de fined by the re la tions (in the wave-num ber space): P k k k ij ij i j ( )k = -d 2 , R P n n Pij ir r s sj( ) ( ) ( )k k k= , (6) Q k k ij ijm m( )k = ie , where e ijm is the Levi — Civita pseudotensor and x is given by the equa tion x = k n / k× . The ten sor ~ Dij , given by Eq. (5), is the most gen eral form with re spect to the con di tion of incompressibility of the sys tem un der con sid er - ation and con tains two dimensionless free pa ram e ters a 1 and a 2 . The pos i - tive ness of the correlator ten sor Dij leads to re stric tions on the above pa ram - e ters, namely, a 1 > –1 and a 2 > –1 . It is men tioned in [1, 3] that the sys tem (1) (with zeroth both ~ f ) is un sta - ble and in QFT for mu la tion the di ver gence pro por tional to the UV cut off L can ap pear in the Green func tion b b¢ . It ac quires the form of b b¢ rot L. The L — UV di ver gence gen er ates the in sta bil ity of the the ory, that causes an ex po nen tial growth in time of the cor re spond ing re sponse func tion. There - fore, its di rect in ser tion into the ac tion is not al lowed and we have to find an ef fec tive way to elim i nate it. So, one must con sider the new vac uum state with a non-van ish ing mean value of b º C ¹ 0. [1]. In QFT the ap pear ance of non-zero vac uum value of field is as so ci ated with spon ta ne ous sym me try break ing [34]. The value of spon ta ne ous mean field is de ter mined from re - 31 THE SPONTANEOUS MAGNETIC FIELD DIRECTION quire ment of min i mum of po ten tial en ergy at the tree level. The L-di ver - gence can be elim i nated by means of the shift of b by the value of spon ta ne - ous ho mo ge neous mag netic field, namely b( )x ® b( )x + C. This shift in the sys tem (1) with zeroth both ~ f pro duces the new non-van ish ing terms ~ ( ) ( )( ) ( )( )f C b C n nb n n Cbv = Ñ + Ñ + Ñ +l l1 2 2 + Ñ + Ñl l3 4 2n C nb n nC n nb( )( ) ( )( )( ), ~ ( )f C vb = Ñ with new pa ram e ters l i (i = 1, ..., 4) . THE RENORMALIZATION The com plete unrenormalized ac tion of the anisotropically forced MHD tur bu lence [18, 19] with the shifted MF (b b C® + ) [3] can be ob tained us - ing ~ f and ~ f in the sto chas tic MHD equa tions (1). Us ing the standart for mal - ism, the sto chas tic prob lem (1) with correlator (4) can be trans formed into the field the o ret i cal model of fields F = {v, b, v¢, b¢} where v¢, b¢ are the aux - il iary in com press ible fields with the ac tion (see [1, 36], for de tails): S D L Lv b( ) $ $F = ¢ ¢+ ¢ + ¢ 1 2 v v v b . (8) Here af ter in the sim i lar ex pres sion the in te gra tion over cor re spond ing vari ables (x, t in this case) and the traces over the vec tor in di ces are im plied. As it is usual in QFT, the ac tion (8) is con sid ered to be unrenormalized with the bare con stants marked by the sub script “0”. The ba sic ob jects of the study are the Green func tions of the fields F (i. e., the cor re la tion func tions and re sponse func tions in the ter mi nol ogy of the orig i nal prob lem (1)). They can be de ter mined as func tional de riv a tives with re spect to ex ter nal sources A = {Av , Ab , Av ¢, Ab¢} of the gen er at ing func tional G A( ) = Dò +F Fexp[ ( ) ]S + AF], i. e., they are the func tional av er aged val ues of the cor re spond ing num ber of the fields F with a weight exp[ ( )]S F . In the Fourie rep re sen ta tion one ob tains: S v D v v V v v v W b b b U b vi ij j i ijl j l i ijl j l i ijl j l= ¢ ¢ + ¢ - ¢ - ¢ 1 2 ~ + + ¢ - - + - - + 1 2 1 2 2 3 2v n n n vi i j in jj [( ) ( )]iw k kc x k c c x + - - - + - - ¢ + 1 2 1 2 2 3 2v n n n n vi i j i j j[( ) ( )]iw k kc x k c c x + ¢ - - + - - + 1 2 1 2 2 3 2b u u u n n n n bi i j i j j[( ) ( )]iw k kt x k t t x 32 B. A. SHAKHOV et al. + - - - + - - ¢ + 1 2 1 2 2 3 2b u u u n n bi in in jj j [( ) ( )]iw k kt x k t t x + ¢ + + + + 1 2 2 23 4 1 2v n n C n n C bi n i j i j i j j[ ( ) ( )]i i ig gl l g x l x l x + + - - + - + ¢ 1 2 2 23 4 1 2b n n n C C n vi n i j i j i j[ ( ) ( )]i i ig gl l g x l x l x j + + ¢ + - ¢ 1 2 1 2 b v v bi j i j[ ] [ ]i ig g where k n= k 2 , g = ( )Ck , g n = ( )Cn and the ver texes are: V k kijl j il l ij= +i( )d d , U k kijl j il l ij= -i( )d d , W V qijl ijl= + , i( ( ) ( )l x d d l x d l l1 2 3 42 2k n n n n k n nj il l ij k i jl i j l jk l + + + + + k n n ni j lx ). In ver sion of the qua dratic part of (9) leads to the Green func tions (prop - a ga tors). In gen eral, the Green func tions vv , vb , bv , bb are ex pected to be lin ear com bi na tions of the pro jec tors Pij , Rij º P n n Pir r s sj , P C n Pir r s sj , P n C Pir r s sj , P C C Pir r s sj , Qij , Q n n Pir r s sj , Q C n Pir r s sj , Q n C Pir r s sj , Q C C Pir r s sj , P n Qir r sjs , P n C Qir r s sj , P C n Qir r s sj , and P C C Qir r s sj . They all are nec es sary to clos ing the group of pro jec tors. An a log i cally, the Green func tions v v¢ , v b¢ , b v¢ , b b¢ are ex pected to be lin ear com bi na tions of the first five above pro jec tors. In the case of the weak ani so tropy limit one can re main the terms of only the first or der with re spect to all pa ram e ters and one can leave out all the terms con tain ing the pa ram e ters c 3 , t 2 , t 3 , and all l i , be - cause these van ish in the fixed RNG points [2]. As a re sult, one ob tains: v v a P a R a Q a Q n n P a P n n Qi j ij ij ij ir r s sj ir r s sj= + + + +1 2 6 7 11 , b v b P b R b Q b Q n n P b P n n Qi j ij ij ij ir r s sj ir r s sj= + + + +1 2 6 7 11 , v b c P c R c Q c Q n n P c P n n Qi j ij ij ij ir r s sj ir r s sj= + + + +1 2 6 7 11 , b b d P d R d Q d Q n n P d P n n Qi j ij ij ij ir r s sj ir r s sj= + + + +1 2 6 7 11 , v v x P x Ri j ij ij¢ = +1 2 , b v y P y Ri j ij ij¢ = +1 2 , v b z P z Ri j ij ij¢ = +1 2 , b b t P t Ri j ij ij¢ = +1 2 . The co ef fi cients in the set (10) have rather com pli cated form and they lin early de pend on five small pa ram e ters c1 , c 2 , t1 , a 1 , a 2 . All above Green func tions must be cal cu lated for the gyrotropic MHD renormalization, namely, their di ver gent parts of pos sess ing poles ~ e -1 in e = 0. These parts are needed for the ory renormalization and they are pro por tional to the sec - ond or der in mo men tum unit. This ques tion is not dis cussed here. In the dy - namo the ory the lin ear part of the Green func tion b bi j plays a cru cial role [3]. 33 THE SPONTANEOUS MAGNETIC FIELD DIRECTION CON DI TION FOR THE HO MO GE NEOUS MAG NETIC FIELD The cal cu la tion of the Feynman di a gram set b bi j , be side the sec ond-or der part, gives also the lin ear part (in mo men tum unit) which cor re sponds to L — UV di ver gent ro tor part and it must be van ished. The cal cu la tion can be per formed only un der the con di tion that the ho mo ge neous mag netic field C di rec tion is par al lel to the ani so tropy di rec tion n. It yields a strong re quire - ment for pos si ble C di rec tion, C = n\| \|C . Its ab so lute value fol lows from the con di tion of the ro tor part van ish ing, namely, in the case of the weak ani so - tropy limit one ob tains: b b k k gu u u u ui j ijl l¢ » + - + - +ie p r p n 12 1 10 1 3 8 2( ) { ( )( \| \| )C L + + -[ ( )\| \| ]5 1 3 16 1p n tu u uC L + + - +[ ( )\| \| ( ) ] }5 5 3 16 2 1p n cu u uC L . (11) As a re sult, the re quire ment of the van ish ing of this term yields the value of spon ta ne ous field C n= + + + + + + 4 45 1 30 1 5 9 13 91 1 u u u u u Ln p t c ( ) [ ( ) ( ) ( ) ]. (12) Note that in iso tro pic tur bu lence the pres ence of ar bi trary small gyrotropy r in Eq. (5) leads to the mean field C gen er a tion [3], but its di rec - tion can not be de ter mined (the di rec tion is spon ta ne ous). In con trary, the pres ence of a small ani so tropy of en ergy forc ing into the he li cal tur bu lent sys tem leads to the de ter mined di rec tion of C, C \|\| n. In the limit of zero ani - so tropy (t1 = c1 = 0), Eq. (12) holds only as sca lar equa tion and \| \|C ac quires the known value C = 8 3uLn p/ [3]. CON CLU SIONS In the pa per, the phe nom e non of the mag netic field gen er a tion in as tro phys - i cal en vi ron ment is stud ied. The sta tis ti cal dy namo model is con structed for he li cal weakly anisotropic full de vel oped MHD tur bu lence in the frame - work of the quan tum field renormalization group ap proach. It is shown that the renormalization leads to aris ing of ul tra-vi o let di ver gence in ro tor part of the mag netic field Green func tion b bi j¢ which must be elim i nated from the the ory. The weak he li cal anisotropic tur bu lence is then sta bi lized by the self-gen er ated mag netic field C whose value is cal cu lated. The di rec tion of the mag netic field is con nected with unique ex ist ing vec tor n de scrib ing the anisotropic tur bu lence forc ing, C n\|\| . Note that the renormalized Green func tion is ob tained which is fi nite as L ® ¥ for mally, as it is usual in the field the ory. But in real prob lems a nat u - ral cut off re ally ex ists. In the de vel oped tur bu lence the Kolmogorov dissipative length lD = L-1 plays the role of a min i mal scale. This length can 34 B. A. 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The spontaneous magnetic field direction in an anisotropic MHD dynamo

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