Magnetic rope structures in the electromagnetic interchange mode

Transcription

1 PHYSICS OF PLASMAS VOLUME 9 NUMBER 7 JULY 00 Magnetc rope structures n the electromagnetc nterchange mode J Vranješ a) and M Y anaka Natonal Insttute for Fuson Scence 3-6 Orosh-cho ok Cty Japan M Kono Faculty of Polcy Studes Chuo Unversty Hachoj okyo Japan Receved 6 February 00; accepted 10 Aprl 00 Nonlnear electromagnetc nterchange mode n a magnetzed plasma wth the spatally dependent ntensty of the magnetc feld s studed he equlbrum condtons n such a system requre the exstence of a spatally nonunform on drft and the effects of such a drft on the nonlnear evoluton of unstable modes are nvestgated he exstence of spatally localzed structures s demonstrated for the densty electrc and magnetc feld In the case of the magnetc feld these structures have a twsted or rope form 00 Amercan Insttute of Physcs DOI: / I INRODUCION he nterchange or flute mode and the nstablty related to that mode are one of the most dangerous n magnetzed plasmas Physcally t s analogous to the Raylegh aylor nstablty n ordnary fluds as t leads to the nterchange of layers wth dfferent denstes that are kept by the magnetc feld 1 he most unstable stuaton s certanly when a perturbaton propagates strctly perpendcular to the magnetc feld lnes snce n that case lght electrons are unable to shortcrcut the regons of perturbed potental However n realty the effects of parallel motons should usually be taken nto account eg when the magnetc shear s present n the system or when the curvature effects are spatally dependent In the latter case the electromagnetc perturbatons and the related nstablty are termed as balloonng modes localzed n the regons of unfavorable curvature he most smple model for the study of such modes ncludes a parallel dynamcs of electrons and two-dmensonal e strctly perpendcular moton of ons Yet even n that case there are new effects that nclude bendng of the magnetc feld lnes whch s stablzng as t ncreases the magnetc energy 1 he system s descrbed by equatons that are bascally nonlnear he unstable mode wll grow untl the stage when the nonlnearty can no longer be neglected It turns out that the nonlnear terms are of the vector-product or the Posson bracket type whch s on the other hand known to be responsble for the creaton of specfc types of structures lke vortces that can be double 3 and trple 4 9 as well as n the form of vortex chans 10 In laboratory experments wth rotatng fluds the trpolar vortex develops from a perturbed monopole 4 It turns out to be a remarkably stable structure survvng many rotatons of the system and consstng of a rotatng central vortex wth two lateral vortces wth opposte vortcty A nce example of a trpolar structure n nature can B 0 x 1x e z 1 hs knd of magnetc feld nonunformty s typcal for the tokamak plasma confguraton where n that case the parama Author to whom correspondence should be addressed On leave from Center for Plasma Astrophyscs Celestjnenlaan 00B 3001 Leuven Belgum Permanent address: Insttute of Physcs PO Box 57 Yu Belgrade Yugoslava Electronc mal: jvranjes@yahoocom be found n Ref 5 In plasma systems the frst analytcal soluton of that type was predcted n Ref 6 As for the expermental verfcaton of the trpolar vortex n plasmas recently t has been obtaned as a standng electrostatc global structure whch develops due to nonlnear effects n a plasma produced by the effect of electron cyclotron resonance 11 In plasmas n most cases a trpolar vortex appears due to the presence of some nonunformty n the system eg n the presence of a nonunform flow 7 or current 8 snce mathematcally that spatal dependence necessarly nvolves hgher Bessel harmoncs n descrbng possble solutons that are localzed n space In smple electrostatc problems t s created by the nonlnear effects n the presence of the flow and just carred by the flow 7 9 In the present study we nvestgate the nonlnear behavor of the electromagnetc nterchange mode n a nonunform system where the densty s nonunform and there s a nonunform on drft n the system caused by a specfc nonunformty of the magnetc feld he lnear and nonlnear behavor of the mode wll be dscussed In Sec II we gve the model and derve the basc set of three closed nonlnear equatons descrbng perturbatons n the electron on flud In Sec III we shortly dscuss the lnear stablty of the mode and n Sec IV present a possble strongly nonlnear soluton whch may represent a saturated state of the lnearly hghly unstable mode At the end we gve a summary II MODEL AND DERIVAIONS We start from a model of a nonunform quasneutral plasma consstng of electrons and ons wth densty gradents n the drecton of the x axs placed n an external magnetc feld whch s orented n the drecton of z axs wth a spatally dependent ntensty B X/00/9(7)/954/5/$ Amercan Insttute of Physcs Downloaded 19 Jun 00 to Redstrbuton subject to AIP lcense or copyrght see

2 Phys Plasmas Vol 9 No 7 July 00 Magnetc rope structures n electromagnetc nterchange 955 eter represents the rato between small and large rad and the x term s replaced by a cos term whch descrbes the polodal dependence of the magnetc feld A smlar geometry and nonunformty can also be found n the magnetc arcades on the Sun as well as on the day-sde magnetosphere whch s n the stage of bouncng back after beng compressed by the solar wnd In such a stuaton the magnetospherc plasma s subject to an effectve gravty whch s actually orented opposte to the drecton of the g term caused by the planet In the model whch we use the densty gradent s orented along the x axs as well as the g term whch can represent a real external force or the effectve one caused by the plasma streamng along the feld lnes herefore n the equlbrum state the system wll develop an on drft perpendcular to both the magnetc feld lnes and the gravty term e n the y drecton As the on moton s nvolved we shall nvestgate some low frequency perturbatons compared to the on gyrofrequency eb /m that develop n the system In the lmt of low- plasma the perturbatons of the electrc and magnetc feld expressed n terms of correspondng potental are gven by E 1 A z e t z B 1 e z A z We use the momentum equaton for electrons and ons n the form m n v v t v q n A z e t z v B m n g p Assumng cold ons t s seen that n the absence of electrc felds the equlbrum state of ons s descrbed by ev 0 B 0 m g 0 yeldng the on drft velocty whch can be wrtten n a few approprate forms that wll be used n the followng text v 0 gm 1xe eb y 1xe y 1 e B z x 6 he meanng of the ntroduced quanttes s obvous A smlar equaton can be wrtten for electrons however due to ther small mass compared to ons ther g drft wll be neglected 1 though they can n prncple be subject to a damagnetc drft moton due to the pressure gradent Makng a vector product of Eq 4 we can wrte a recurrent formula for the total perpendcular velocty for ons We keep small lnear and nonlnear terms up to the second order assumng a weak nonunformty n the equlbrum he perturbatons propagate almost perpendcular to the magnetc feld k z k so that the on moton s assumed to be two dmensonal hen for the ons we obtan v 1 B 0 x e z 1 B e z 1 B Smlarly for electrons we have t 1 B e z v e 1 B 0 x e z 1 0 v B ez B en 0 B 0 x e z n e 8 Further we use the charge conservaton equaton j z j z 0 j z en 0 v ez1 en 1 v ez1 j 1 en 0 v 1 v e1 n 1 v 0 Usng B 0 j and Eqs 7 9 we obtan the frst necessary equaton 1 B t 1 e B z 1 0 en 0 z 1 B e z A z A z 1 B e z n Here the densty perturbaton s normalzed to n 0 and the electron thermal effects are neglected From the electron parallel momentum equaton n the same lmts of small parameters we obtan e t 1B z A z z 11 he electron contnuty n 1 t n 0xn 1 v e1 0 s the thrd equaton necessary to close the system Usng Eqs 7 and 8 ts transformaton s straghtforward yeldng e t 1B z n 1 ln n 0 cx 1 0 en 0 z 1 B e z A z A z 0 1 Here the prme denotes the dervatve n the x drecton the densty s normalzed to n 0 and ce/gm Downloaded 19 Jun 00 to Redstrbuton subject to AIP lcense or copyrght see

3 956 Phys Plasmas Vol 9 No 7 July 00 Vranješ anaka and Kono III LINEAR ANALYSIS Lnearzng Eqs 10 1 on the condton of small densty gradents and for perturbatons of the form fˆ(x)exptk y y where fˆ(x) denotes the x-dependent ampltude one can obtan the followng dfferental equaton for the ampltude of the electrostatc potental: x k y 1x xˆ x0 1 x k y x1 k z c a 1x n 0 n 0 k y 1x 1 k z c a 13 c a B 0 n 0 m Here the Alfvén velocty c a naturally appears due to the fnte electron response along the magnetc feld lnes Neglectng spatal dependence of the on drft and settng k x 0 from Eq 13 one obtans 3 k y n 0 n 0 k z c a k y k z c a 0 14 he dsperson relaton 14 s dentcal to the one derved n Ref 3 for an unstable electromagnetc balloonng mode In the lmt k z 0 one obtans a standard dsperson equaton for the flute mode 1 that s unstable provded that the drecton of the g vector opposes the drecton of densty gradent k y 4 n 0 15 gn 0 In a partcularly smple case when the change of the drft velocty s such that t balances the densty gradent n 0 /n 0 from Eq 13 one obtans the unstable mode wth the x-dependent ampltude whch changes as expk y x We wll come agan to that pont later on IV NONLINEAR SOLUIONS Due to the presence of the external force g the system s prone to the development of nstablty In the nonlocal case whch s the subject of the study the spatal dependence of the on drft can modfy the behavor of the unstable mode At a certan stage of the growng mode the nonlnear terms may become mportant and may determne the drecton to whch the nstablty wll develop Double vortex as a possble state of the developed nstablty s shown to be possble n the case wthout the spatal dependence of the drft velocty 3 Here we wll show that the spatally dependent drft s crucal for another type of soluton whch conssts of monopolar and quadrupolar parts resemblng the trple vortex known to exst n rotatng fluds 45 and predcted to exst 6 9 and observed 11 n plasmas as well hus we search for nonlnear solutons that are statonary n a reference frame travelng wth veloctes u y and u z along the y and z axs respectvely We can wrte /te z u y x /ye z x /ze z u y x/u z Here we normalze veloctes u yz to the earler ntroduced and rewrte Eqs 10 1 n the dmensonless form as z e z A z t e z c a A z L e z n t 1B 0 e z A z z t e z n 1 ln n 0 x c a L z e z A z A z 0 Here the followng normalzaton s ntroduced: t L L t B L B L A z A z LB and L1/ On condton of localzed solutons Eq 17 can be ntegrated yeldng A z 1 u z 19 Usng 19 we can wrte Eqs 16 and 18 n the followng way: e z u y x c a u z e z u y x L e z n e z u y x n 1 ln n 0 x c a e L u z u y x 0 z 1 We proceed by searchng for solutons that satsfy the followng set of equatons: a 1 a x n 1 b 1 b x 3 Downloaded 19 Jun 00 to Redstrbuton subject to AIP lcense or copyrght see

4 Phys Plasmas Vol 9 No 7 July 00 Magnetc rope structures n electromagnetc nterchange 957 e we use a procedure smlar to Refs 3 and 1 wth the dfference due to the x term whch wll be crucal for a new type of solutons and n order to have spatally localzed solutons t turns out to be necessary to cancel out the spatal dependence of the on drft Here we ntroduce the constants a 1 b 1 that should be determned from the correspondng boundary condtons whch the solutons are subject to Settng expressons and 3 nto Eqs 0 and 1 we obtan the followng two condtons for solutons satsfyng Eqs and 3: a 1 a a c a V L b 1 0 u z xa 1 u y a 1 c a u y a 1 V L b u z b a c a x n 0x xu n y b 1 0 L u z c a u y a L u z We search for localzed solutons n space and for that purpose t s convenent to dvde space by a crcle of the radus r 0 and search for solutons outsde and nsde of that crcle ndependently and match them smoothly at the border defned by rr 0 he constants a 1 b 1 ntroduced n Eqs and 3 are allowed to have dfferent values outsde marked by the superscrpt o and nsde marked by the superscrpt of the crcle For spatally localzed solutons we have a o 0 b o 0 and from Eq 4 we obtan a o 1 L b o V Introducng the notaton a o 1 from Eq we readly obtan a soluton whch can be wrtten n terms of the zeroth and second Bessel harmoncs and for the cyllndrcal coordnates r and n the form o r 0 K 0 rk rcos 7 Here K 0 are the modfed Bessel functons and 0 are the constants of ntegraton whch should be determned from the boundary condtons at rr 0 Havng found solutons for o t s then straghtforward to obtan n o 1 (r) and A o z (r) From Eq 4 we fnd also that t s satsfed for the dmensonless velocty u z gven by u z c a 8 Consequently the proposed soluton has a translatory speed along the magnetc lnes equal to the Alfvén velocty he condton 5 must be satsfed as well Note that 1 Usng Eq 6 we fnd out that n dmensonless form n 0 /n 0 1 he proposed nonlnear soluton represents a possble saturated state of the lnearly growng mode for the case when the spatal change of the drft velocty s gven by Eq 6 and densty by n 0 e x As for the nner solutons we use Eqs and 3 Introducng notaton a 1 for the potental we can wrte a x a wth the soluton r a 4 a r 0 J 0 r a r J rcos 30 Here J 0 are the Bessel functons and 0 are new constants of ntegraton he second harmoncs enter as a drect consequence of the lnearly varyng on drft velocty e through the term x r (1cos )/ For that reason the second harmoncs are kept for the outsde soluton as well he soluton for A z s easy to fnd as well as densty n 1 whch then becomes n 1 r b 1 a b 1 a 4 a From the condton 4 we fnd b 1 L and from 5 wefnd r 0 J 0 r b r r b 1 J r b rcos 31 3 b a 33 L he other constants of ntegraton should be found from the contnuty condtons at rr 0 hese condtons nclude the contnuty of the potental (r) whch accordng to Eqs 19 3 and mples the contnuty of A z n 1 and respectvely Physcal condtons at rr 0 also requre the contnuty of the gradent (r) mplyng the contnuty of (r)/r ogether these condtons yeld four equatons so obvously some of the constants wll reman free to choose Solutons gven by Eqs 7 30 and 31 consst of monopolar and quadrupolar parts however the contour plots of such mathematcal objects for some approprate values of constants of ntegratons yeld trpolar vortces consstng of a central vortex wth two lateral ones wth an opposte polarty n vortcty 7 9 herefore n referrng to such a structure we use the term trpolar vortex V SUMMARY As ponted out n the begnnng the present study should be applcable to varous stuatons n laboratory and space Downloaded 19 Jun 00 to Redstrbuton subject to AIP lcense or copyrght see

5 958 Phys Plasmas Vol 9 No 7 July 00 Vranješ anaka and Kono plasmas that are subject to some external force or that are n the state of flow along curved magnetc feld lnes For clarty the model used n the descrpton s relatvely smple and n order to be used n some realstc stuatons perhaps should be mproved by addng some new phenomena lke the fnte on Larmor radus effects known to act n such a way to stablze the nterchange mode due to the on damagnetc drft However the purpose of the present study s to show as clearly as possble the second order effects that appear as a result of the spatally dependent ntensty of the magnetc feld causng on the other hand the spatally dependent on drft whch necessarly exsts n the system n order to have a proper eqlbrum 1 Snce the effects whch we study are hgher order and the nstablty s present n the system the nonlnear effects can no longer be omtted herefore we derve a set of three coupled nonlnear equatons descrbng perturbatons of the densty and electrostatc and vector potentals n a quasneutral and low plasma he nonlneartes turn out to be of the vector-product type allowng for the ntegraton of the parallel electron momentum equaton o solve the other two equatons we use the ansatzes 3 smlar to some ansatzes that can be found n the lterature wth the dfference n the x terms We show the possblty for the formaton of electromagnetc trpolar vortces n the system assocated wth the local condensaton and rarefcaton of the densty he solutons are shown to be possble for a specfc case when the gradents of the logarthm of the equlbrum densty and the on drft are equal but wth the opposte sgn he magnetc feld lnes are assumed as straght n the equlbrum he perturbaton of the magnetc feld s expressed through the A z component of the vector potental only e there s no compresson of the magnetc feld As A z turns out to be proportonal to the perturbed electrostatc potental clearly due to Eq 3 the perturbed magnetc lnes wll concde wth the stream lnes of partcles nsde the vortex he total magnetc feld wll therefore have a small spatally dependent descrbed by the spatal profle of the trpolar vortex perpendcular component e locally t wll have a twsted or rope structure hs means that n the doman of the vortex the magnetc feld lnes are not straght but twsted n the form of spatally dependent helces and the wndng n the core vortex s opposte compared to lateral vortces ACKNOWLEDGMENS hs study was performed durng the vst of one of the authors JV to the Natonal Insttute for Fuson Scence ok Japan he scentfc atmosphere and excellent workng condtons at the Insttute are gratefully acknowledged 1 F F Chen Introducton to Plasma Physcs and Controlled Fuson nd ed Plenum New York 1990 pp J Weland Collectve Modes n Inhomogeneous Plasma Insttute of Physcs Brstol 000 pp R Bharuthram and P K Shukla J Plasma Phys G J F van Hejst and R C Kloosterzel Nature London R D Pngree and B Le Cann J Geophys Res I O Pogutse Sov J Plasma Phys J Vranješ Planet Space Sc J Vranješ G Marć and P K Shukla Phys Lett A J Vranješ D Petrovć and P K Shukla Phys Lett A J Vranješ Phys Rev E A Okamoto Ishhara K Nagaoka S Yoshmura and M Y anaka J Plasma Fuson Res V D Larchev and G K Reznk Polym News Downloaded 19 Jun 00 to Redstrbuton subject to AIP lcense or copyrght see