PHYSICS OF PLASMAS 14, 10906 007 Reduced magnetohydrodynamc equatons wth coupled Alfvén and sound wave dynamcs R. E. Denton and B. Rogers Department of Physcs and Astronomy, Dartmouth College, Hanover, New Hampshre 03755, USA W. Lotko Thayer School of Engneerng, Dartmouth College, Hanover, New Hampshre 03755, USA Receved 14 June 007; accepted 4 August 007; publshed onlne 31 October 007 A set of reduced magnetohydrodynamc equatons s descrbed that s approprate for the smulaton of auroral Alfvén waves usng curvlnear coordnates. These equatons nclude the parallel electrc feld, ponderomotve force, damagnetc drft effects, and gravty and rotaton, but do not nclude the fast mode dynamcs. The equatons are formulated for multple speces, but quasneutralty s explctly mantaned. The equatons have an exact conserved energy. 007 Amercan Insttute of Physcs. DOI: 10.1063/1.786060 I. INTRODUCTION Observatons from polar orbtng spacecraft suggest that auroral phenomena such as parallel electrc felds, accelerated electrons, and uplfted onospherc ons are caused by Alfvén waves wth dsperson resultng from the fnte temperature and/or electron nerta of the plasma. 1 3 Numercal smulatons of these phenomena have been based manly on two-flud equatons, ncludng n addton to dsperson the perpendcular dsplacement current 4 6 and the damagnetc current and the Alfvén ponderomotve force. 7 10 The dsplacement current supplants the Alfvén wave polarzaton current n regons of low plasma densty, whereas damagnetc currents are nduced by flute perturbatons n the drecton of the wave electrc drft. The ponderomotve force nonlnearly couples Alfvén waves and on sound waves when the wave ampltude and wave frequency become relatvely large. All of these effects appear to operate n the topsde auroral onosphere and low-alttude magnetosphere. Our purpose here s to derve a set of equatons approprate for smulatng auroral Alfvén waves that s more complete than those used prevously, and for whch there s an exact conserved energy. When we say that the energy s conserved, we mean that the energy relaton of the system may be expressed n a conservatve form, E = S. t 1 The energy densty and energy flux S are gven n Sec. III below. A conservatve form for the energy equaton s especally useful for evaluatng power flow through the system and for descrbng the parttonng and converson of energy between knetc, thermal, electromagnetc, and gravtatonal forms. As n prevous analyss, the equatons to be derved here nclude parallel electrc felds due to the fnte on Larmor radus, electron nerta, and electrcal resstvty, damagnetc drft effects, the perpendcular dsplacement current, and the Alfvén wave ponderomotve force, 8 whch s proportonal to B E /t n the reduced flud model. The ponderomotve force s mportant n drvng parallel flows n the auroral regon. 11 Multple speces e.g., H +,He +, and O +, curvlnear geometry, gravtaton, and rotaton are also allowed. The equatons are global, and dervatves on the background magnetc feld are retaned. In addton to the usual magnetohydrodynamcs MHD orderngs, /t c and /L 1 n terms of the on gyrofrequency c and on gyroradus, we further assume that /t /L c wheren the polarzaton and damagnetc drfts are comparable n magntude and weak n comparson wth the domnant EB drft. The analyss s also restrcted to a class of dsturbances n whch the magnetc compressblty s weak, and the parallel component of the magnetc perturbaton may be neglected. For such dsturbances, Faraday s law mples that the perpendcular velocty perturbaton satsfes v /t whle the solenodal condton on B mples that the perpendcular and parallel length scales satsfy L L. Magnetc ncompressblty does not restrct the gradent v, so order unty densty perturbatons are allowed. These approxmatons are sutable for descrbng ultralow frequency electromagnetc power flows from hghalttude magnetospherc dynamos to low-alttude auroral regons when the transverse length scale of the electromagnetc perturbatons s suffcently small. Sngly onzed oxygen s the domnant plasma consttuent of the topsde onosphere, and ts low-alttude, magnetc feld-algned dynamcs are strongly constraned by gravty and ambpolar electrc felds; thus we have ncluded these effects n the model equatons. The topsde onospherc plasma s also weakly collsonal, so we also nclude on-electron collsons n the descrpton of the parallel moton and the parallel thermal conductvty as gven by Bragnsk. 1 However, the model does not explctly nclude on-neutral collsons, whch are mportant n the E regon and bottomsde F regon of the onosphere, 13.e., the low-alttude end of the physcal doman of nterest Although the derved equatons are not solved n ths paper, low-alttude boundary condtons for auroral applcatons typcally nclude the effects of the conductng onosphere. Smple treatments of the onosphere nclude a per- 1070-664X/007/1410/10906/7/$3.00 14, 10906-1 007 Amercan Insttute of Physcs Downloaded 31 Oct 007 to 19.170.67.4. Redstrbuton subject to AIP lcense or copyrght, see http://pop.ap.org/pop/copyrght.jsp
10906- Denton, Rogers, and Lotko Phys. Plasmas 14, 10906 007 fectly conductng sphercal surface to represent the alttudnally thn E regon of the onosphere, or a fnte, unform constant conductng surface domnated by on-neutral collsons and characterzed by an ansotropc heghtntegrated conductvty, 13 combned wth current contnuty. Ether type of boundary condtons promotes standng or nterferng wave structures; the fnte conductvty addtonally ntroduces energy dsspaton n the Alfvén wave dynamcs. Streltsov and Lotko 7 relax the constant conductvty assumpton by ncludng a tme-dependent equaton for contnuty of the heght-ntegrated on densty and, therefore, the heghtntegrated conductvty, ncludng sources and losses of onzaton. Ths electrodynamcally actve boundary condton leads to the nterestng phenomenon of spontaneously generated Alfvén waves va a negatve-energy feedback nstablty. 14 Because Alfvén and on sound waves are magnetcally guded, the lateral boundary condtons transverse to the gude feld do not affect the dynamcs of dsturbances ntally localzed away from the lateral boundares. Nevertheless, the lateral boundary condtons must be consstent wth the model equatons f artfcal dynamcs are to be avoded. 15 Due to the magnetc and plasma nhomogenety, the length scales L and L of ntal dsturbances vary contnuously along the magnetc feld. Magnetc gudance of shear Alfvén and on sound dsturbances mples that the perpendcular scale length L scales as the dstance between neghborng feld lnes, whch vares approxmately as B 1/ 0 n terms of the geomagnetc gude feld. For applcatons to Alfvén waves, L scales locally as the effectve parallel wavelength V A / f, where V A s the local Alfvén speed and f s the wave frequency. On auroral feld lnes, V A vares from a low value of about 1000 km/s n the outer magnetosphere 16 and near the F-regon peak n the onosphere, to a hgh value that can exceed the speed of lght n the very tenuous lowalttude magnetosphere. 17 Relevant frequences n the MHD range span f 0.1 Hz. Thus for f 0.1 Hz, L /L wll reman small along the entre flux tube when the transverse length scale of the dsturbance s 1000 km n the outer magnetosphere. Ths transverse length scale maps along the flux tube to a value of about 30 km n the onosphere. The dervaton of the equatons gven n the next sectons follows the analyss of Zeler et al. Sec. II of Ref. 18, whch, n turn, was based on the equatons of Bragnsk. 1 For other reduced Bragnsk equatons, see the references lsted n Ref. 18. We have wrtten Zeler et al. s equatons n such a way as to allow for a curved background magnetc feld. Also, the equatons of Zeler et al. are not practcally solvable as wrtten snce the polarzaton drft velocty ncludes a tme dervatve of the polarzaton drft velocty, leadng to a recursve term. In our formulaton, only the lowest-order velocty appears n the polarzaton drft, and the polarzaton drft appears only n the current contnuty vortcty equaton 5. The neglect of the polarzaton drft n the contnuty equaton 16 means that the densty, and hence the nonlnear Alfvén speed, wll not be correctly descrbed for Alfvén waves, but for Alfvén waves the densty perturbaton wll be small. The contnuty equaton does nclude the parallel velocty, so large densty perturbatons can possbly arse from the parallel dynamcs. In addton to terms from Zeler et al., we have ncluded electron nerta, as well as gravty, rotatonal forces, and the perpendcular dsplacement current. Electron nerta s ncluded n the equatons of Zeler, 19 though not n a manner that exactly conserves energy. These equatons are smlar to those recently publshed by Brzard. 0 Hs equatons are derved from a drft-flud Lagrangan, but our equatons are more easly nterpreted and nclude addtonal effects such as rotaton, gravty, resstvty, and parallel thermal conducton. II. EQUATIONS A. Coordnate system and electromagnetc felds Dependng of the applcaton, the equatons derved below can be smplfed by settng the resstvty, the parallel thermal conducton e, the gravtaton constant G, or the rotatonal angular frequency r equal to zero. Further smplfcatons can be obtaned by droppng terms enclosed n the subscrpted square brackets that appear n the varous reduced equatons. For example, f all terms enclosed by j wth the same subscrpt j are dropped, the conservatve form of the energy equaton derved n Sec. III wll be preserved. The j=1 terms nclude fnte Larmor radus modfcatons to the polarzaton drft; 19 the j= term comes from the on heat flux Eq..3 wth Eq..14 from Ref. 1; and the j=3 terms come from the so-called thermal force Eqs..e and.3e wth Eq..9 from Ref. 1. We assume that the background magnetc feld B 0 les n a merdonal plane necessary to defne orthogonal coordnates, and that B 0 =0. That s, there s no current assocated wth the equlbrum feld. Ths restrcton can be relaxed, but n order to defne orthogonal coordnates t s necessary that b 0 J 0 =0, where b 0 =B 0 /B 0 and J 0 =c/4 B 0. 1 A dpole magnetc feld s an example of a feld satsfyng these condtons. Let s be the parallel coordnate equal to the dstance along a reference feld lne t wll not n general be exactly equal to the dstance s along adjacent feld lnes. Let B = B 0 + A s, where B =As the perpendcular symbol s relatve to b 0. Let b 0 = B 0 B 0, 0 = b 0, b = b 0 + B B 0 = b 0 + b, = b. 3a 3b 3c 3d Note then that b= b b= 1+B /B 0 1. But B /B 0 s assumed to be small. Downloaded 31 Oct 007 to 19.170.67.4. Redstrbuton subject to AIP lcense or copyrght, see http://pop.ap.org/pop/copyrght.jsp
10906-3 Reduced magnetohydrodynamc equatons Phys. Plasmas 14, 10906 007 Let E = 1 A c t s = b 1 A 0 c t 0 s. 4 Note then that B=0 and Faraday s law holds exactly. Note that s=b 0 on the reference feld lne, where 0 s=1, and that ths wll be approxmately true on adjacent feld lnes. The current s J = c 4 B, 5 and the parallel b 0 component from Eq. s J 0 = b 0 J = c 4 b 0 A s = c 4 A 0 s + 0 s A, where b 0 b 0 s the curvature vector. B. Veloctes and convectve dervatves We defne several forms for the on velocty that are used n dfferent equatons, v 0 = v b + v E, v 1 = v b + v E + v d, v = v b + v E + v d + v G + v pol, n order of ncreasng accuracy, where 6 7a 7b 7c v E = ce b 0 B 0, 8a v d = cb 0 p q n B 0, 8b v G = v g + v c, F G = F g + F c, 8c 8d v g = cf g b 0 q B 0, 8e F g = m g, g = M EG R 8f 1 rr sn, 8g v c = cf c b 0 q B 0, 8h F c = m r v 1, and the on polarzaton drft s 8 where v pol = b 0 d c v 1 0 + b 0 n m c p b 0 1 + 0 n m cb p b 0 v E c c v 0 p v, 9 c E1 d = t + v 0 = t + v b + v E, 10 and where c =q B 0 /m c. Agan, terms n square brackets are not necessary for energy conservaton. Our plan s to have multple on speces, so the subscrpt serves as a speces ndex. In Eq. 8, v E s the EB velocty, v d s the on damagnetc drft, F g and v g are the gravtatonal force and gravtatonal drft velocty, respectvely, ncludng the pseudo-gravtatonal effect of rotaton, M E s the mass of the Earth, G s the unversal gravtaton constant, R s the geocentrc dstance, r s the rotaton frequency vector wth the drecton of the axs of rotaton, s the polar angle from the rotaton axs, and F c and v c are the Corols force and Corols drft velocty, respectvely. For a detaled explanaton of the on gyrovscous terms n Eq. 9 and other equatons, see the references lsted n Ref. 18. The electron velocty s where v e0 = v e b + v E, v e1 = v e b + v E + v de, v eq = q n v 1 J en e, v e = q n v J 0 en e, and the electron damagnetc drft velocty s v de = cb 0 p e /en e B 0. 11a 11b 11c 1 13 The electron densty s n e = q n /e 14 assumng quasneutralty, and we defne d e = t + v e q. 15 Note that the perpendcular current appears only n v eq. The use of J n v eq s necessary for d e / Eq. 15, whch appears n Eq. 4. Then the convectve dervatve n Eq. 4 s consstent wth our defnton of n e n Eq. 14, allowng us to use Eq. 39 to derve energy conservaton. Downloaded 31 Oct 007 to 19.170.67.4. Redstrbuton subject to AIP lcense or copyrght, see http://pop.ap.org/pop/copyrght.jsp
10906-4 Denton, Rogers, and Lotko Phys. Plasmas 14, 10906 007 C. Contnuty and pressure equatons The on contnuty equaton for each speces s n t + n v 1 =0. The on pressure equaton s 3 d p + 5 p v 0 + p v G 16 + 5 c e b 0 B 0 p T =0, 17 where T = p /n. The electron pressure equaton s 3 d e p e + 3 p e v eq + p e v e0 J 0 + 0.71T e bn e v e 3 = b e T e, 18 where s the parallel resstvty, T e = p e /n e, and e s the electron thermal conductvty. If the term n square brackets marked wth subscrpt 3 s kept, the correspondng term n the electron momentum equaton should also be kept. We have assumed that resstve heatng domnantly heats the electrons, whch s approxmately true because of ther lower mass. D. Momentum equatons The on parallel momentum equaton s derved from Ref. 18, m n t + v v = p P + q n E J 0 b 0 + v B c + n F G, 19 where v ndcates the full on velocty, and the dvergence of the pressure tensor ncludng fnte Larmor radus effects s P = m n v d v + p b c v + p b v c + b p v, 0 c 1 where here only s perpendcular to b. Note that Eq. 0 was used already for v pol, approxmatng b by b 0 and v by v 0 n the last two terms on the rght sde of Eq. 0. In order to make further progress, we approxmate the frst v n the convectve term on the left sde of Eq. 19 by v 1, and the second v n the same term by v 0. There s then a cancellaton nvolvng the frst term on the rght sde of Eqs. 0. 19 We also use v 0 for v n the frst two terms on the rght sde of Eq. 0. The tme dervatve actng on the velocty yelds the polarzaton drft, so we are excludng here the part of the polarzaton drft wth the tme dervatve actng on the damagnetc drft velocty. Ths could be added wth some ncrease n complexty. Dottng Eq. 19 wth b, we get m n b d v 0 We can expand b d v 0 = p p b b 0 c v 0 + q n E J 0 + n b F G. 1 = b d v + v b d b 0 + b b t + b v 0 b + b d v E. The b d b 0 / term s equal to b v +v E b 0, where =b 0 b 0 s the curvature of the background magnetc feld. The b b /t term can be rewrtten as c/b 0 b E usng Faraday s law, B/t=B /t= c E. Then we can rewrte Eq. 1 as m n b d v = m n v b v + v E b 0 c B 0 b E + b v 0 b m n b d v E p p b b 0 c v 0 + q n E J 0 + n b F G. 3 The m n b d v E / term ncludes the so-called ponderomotve force E /tb. The electron parallel momentum equaton s smpler, snce we can neglect most of the fnte Larmor radus FLR terms as well as gravty and the ponderomotve terms. m e n e d e v e = p e en e E J 0 0.71n e T e 3, 4 where ce =eb 0 /m e c, and the term n brackets must be kept f the correspondng term s kept n the electron pressure equaton. E. Current contnuty equaton Fnally, the current contnuty vortcty equaton s q n v en e v e1 + 1 E =0, 4 t 5 where we have ncluded the dsplacement current n the perpendcular electrc feld. The parallel dsplacement current dvded by the perpendcular dsplacement current E /L /E /L E /E L /L. Both E /E and L /L are small as expermentally observed, so the parallel Downloaded 31 Oct 007 to 19.170.67.4. Redstrbuton subject to AIP lcense or copyrght, see http://pop.ap.org/pop/copyrght.jsp
10906-5 Reduced magnetohydrodynamc equatons Phys. Plasmas 14, 10906 007 dsplacement current should be neglgble compared to the perpendcular dsplacement current. Futhermore, keepng the parallel dsplacement current b 0 E /t would lead to undesrable hgh-frequency plasma wave oscllatons along the magnetc feld. F. Summary of key equatons Ion contnuty equaton 16 n t + n v 1 =0, 6 wth the defnton n Eq. 7b. Electron densty from the quasneutralty equaton 14 n e = q n /e. 7 Ion pressure equaton 17 3 d p + 5 p v 0 + p v G + 5 c e b 0 B 0 p T =0, 8 wth defntons n Eqs. 10, 7a, and 8c. Electron pressure equaton 18 3 d e p e + 3 p e v eq + p e v e0 J 0 + 0.71T e bn e v e 3 = b e T e, 9 wth defntons n Eqs. 15, 11c, 11a, and 6. Ion momentum equaton 3 m n b d v = m n v b v + v E b 0 c B 0 b E + b v 0 b m n b d v E p p b b 0 c v 0 + q n E J 0 + n b F G, 30 wth defntons n Eqs. 10, 8a, 7a, and 6. Electron momentum equaton m e n e d e v e = p e en e E J 0 0.71n e T e 3. Current contnuty vortcty equaton 5 q n v en e v e1 + 1 E =0, 4 t wth defntons n Eqs. 7c and 11b. 31 3 III. ENERGY CONSERVATION The dervaton of energy conservaton s smlar to that of Zeler et al. 18 To derve an energy, we frst multply the on parallel momentum equaton 1 by v, the electron momentum equaton 4 by v e, and add them together. For multple on speces, there s an mpled sum over the ndex, + m n v b d v 0 + p v b b 0 d e + n e 1 m ev e + v p + v e p e c v 0 + m n v b g +m n v b r v 1 J 0 E J 0 + 0.71n e v e T e 3 =0. 33 We then add the on pressure equaton 17 and the electron pressure equaton 18 together, t 3 p + e 3 p v 0 p v e0 p e + p v G J 0 + 5 c e b 0 B 0 p T + 0.71T e bn e v e 3 = 5 p v 0 + p e 3 v e q + v e0 b e T e. Usng Faraday s law, t B 8 1 c A 0sJ 0 t 34 = A 1 4 0s A 35 t. Then we multply the current contnuty equaton by, t E J 8 0 b + v E m n d v 0 + m n g + r v 1 + p b 0 c v 0 + p + p e = J 0 b + q n v d + v pol + v G en e v de + 1 E 4 t p c v E b 0 v E b 0 v E v E 1. 36 Now we make use of several useful denttes the volume ntegrated forms of the on equatons are gven n Ref. 18, p b 0 c f = m n v d f p f b 0 c, 37 Downloaded 31 Oct 007 to 19.170.67.4. Redstrbuton subject to AIP lcense or copyrght, see http://pop.ap.org/pop/copyrght.jsp
10906-6 Denton, Rogers, and Lotko Phys. Plasmas 14, 10906 007 n d + v d f = n f + v 1 n f, 38 t usng the contnuty equaton 16, and d e n e f = n ef + v eq n e f, 39 t usng the effectve electron contnuty equaton mpled by quasneutralty Eq. 14. Addng Eqs. 33 36, we fnd Eq. 1 wth E = m n v b + v E + m en e v e + 3 p + p e + 1 8 B + E + m n G, and the energy flux S s 40 S = 1 m n v 0 v 1 + 1 m en e v e v eq + + 5 p v 0 + p v G + p b 0 c v 0 + p v E b 0 v E b 0 v E v E c 1 5 c b 0 p T + p ev e0 + 3 eb 0 v e q b e T e + 0.71n e T e v e b 3 1 4 0s A t A + J 0 b + q n v d + v pol + v G en e v de + 1 E. 4 t 41 From Eq. 40, we can see that the energy densty s composed of terms representng the knetc energy, the thermal energy, the electromagnetc feld energy, and the gravtatonal potental energy. If the boundary condtons are perodc or such that S nˆ =0, where nˆ s the boundary normal, the total energy ntegrated over the smulaton volume wll be conserved. If the boundary condton s not perodc and S nˆ 0, an accurate evaluaton of the smulaton energy wll have to take account of the flux of energy through the boundary. There are a number of possble varatons n the equatons that can be consdered. A possble smplfcaton s to neglect the convectve tme dervatve of b 0 n Eq. 1 leadng to a term proportonal to the curvature by separatng the terms actng on b 0 v from those actng on b v +v e. There are also some generalzatons leadng to more complexty. The b 0 terms can be generalzed to b wthout any addtonal couplng of equatons because we can use Faraday s law to evaluate partal tme dervatves actng on B. The polarzaton drft can be extended to nclude dv d /. Then the p v term n the on pressure equaton must nclude the polarzaton drft, and the energy densty wll nclude a term 1/m n v b+v e +v d. Another straghtforward modfcaton would be to make the pressure ansotropc usng equatons based on those of Chew, Goldberger, and Low see Ref. 0. The effects of on cyclotron waves could be ncluded usng the approach of Denton and Lyon. 3 However, correct treatment of the parallel dynamcs ncludng Landau dampng would requre a gyroflud approach. 4 a v v v t + a v t + a A v A t = S v, 4 where a v v, a v, and a v A are felds, and S v collects all the explct terms. The term wth /t comes from the ponderomotve force. The term wth A/t comes from E. The electron momentum equaton has the form a ve J J 0 t A + a ve A t + a ve v v t = S ve. 43 Here J 0 A/t s the largest term from v e /t. The sum wth v /t also comes from v e /t because of the defnton of v e n terms of J 0 Eq. 4. Ths relaton s used n several steps dervng energy conservaton, for nstance summng the E terms from the momenta equatons to get J 0 E J 0 n Eq. 33. The term wth A/t comes from E. The vortcty equaton s of the form a vor t + a vor v v t = S vor. 44 Here the /t term comes from the dv e / part of the polarzaton drft and the dsplacement current collected together nto one term, and the v /t term comes from the dv b / part of the polarzaton drft. The equatons can be solved by substtutng v /t from Eq. 4 nto Eqs. 43 and 44, yeldng two coupled equatons for A and. Havng the solutons for A and, one can substtute back nto Eq. 4 to evolve v. IV. SOLVING THE EQUATIONS The on momentum equaton for each speces has the form V. LINEAR DISPERSION RELATION For a two-component plasma n a homogeneous magnetc feld, the lnear dsperson relaton of our equatons s Downloaded 31 Oct 007 to 19.170.67.4. Redstrbuton subject to AIP lcense or copyrght, see http://pop.ap.org/pop/copyrght.jsp
10906-7 Reduced magnetohydrodynamc equatons Phys. Plasmas 14, 10906 007 where 1 V 1+ V1+ m tot c + m V K V e + m K + e K =0, 45 V /k V A, 46a c K k p, 46b c V A c, m m e m, 46c 46d s 5 s 6 = s, wth = 5 3, 46e tot e +, 46f where V A B 0 / 4m n s the Alfvén speed, p 4n q /m s the on plasma frequency, and s =8p s /B 0 s the plasma beta for speces s = or e for ons or electrons. As can be seen from Eqs. 46a and 46b, V s the normalzed, and K s the normalzed k. The frst lne of Eq. 45 neglectng the terms proportonal to K yelds the Alfvén wave V1 and the sound wave V tot, the normalzed sound speed squared solutons. Terms n the dsperson relaton result from the product of two terms. The m V K term n Eq. 45 k c/ pe comes from the combnaton of electron nerta and on nerta. The V e + m K term V e K f e m, and V e Kk se, where se =c se / c and c se = Te /m s the on sound speed based on T e. Ths term comes from the combnaton of the electron parallel pressure gradent and on nerta. The last term comes from the combnaton of the electron parallel pressure gradent and the on parallel pressure gradent. Equaton 45 s nearly dentcal to the correspondng expresson for the equatons of Ref. 0. In that case, the c term should be replaced by m + c. VI. SUMMARY Startng from the equatons of Zeler et al., 18 we have derved a set of equatons for the smulaton of coupled Alfvén and sound waves along magnetospherc feld lnes. These equatons nclude parallel electrc felds, the ponderomotve force, and the effects of feld lne curvature, rotaton, and gravty. We have shown that there s a conserved energy gven approprate boundary condtons, and we have gven the energy flux. We dscussed how the equatons can be solved n a numercal code. We have also gven the lnear dsperson relaton for a homogeneous plasma, whch shows that there are Alfven and sound wave solutons coupled by fnte kc/ p. ACKNOWLEDGMENTS Work at Dartmouth was supported by NASA Grants No. NNG05GJ70G and No. NNG04GEG and by NSF Grants No. ATM-063740 and No. ATM-038694. We thank Alan Brzard and Anatoly Streltsov for helpful dscussons. 1 W. Lotko, A. V. Streltsov, and C. W. Carlson, Geophys. Res. Lett. 5, 4449 1998. C. C. Chaston, J. W. Bonnell, C. W. Carlson, J. P. McFadden, R. E. Ergun, R. J. Strangeway, and E. J. Lund, J. Geophys. Res. 109, A0405 004. 3 C. C. Chaston, n Magnetospherc ULF Waves: Synthess and New Drectons, Geophyscal Monograph Seres 169, edted by K. Takahash, P. J. Ch, R. E. Denton, and R. L. Lysak Amercan Geophyscal Unon, Washngton, D.C., 006, p.39. 4 A. V. Streltsov and W. Lotko, J. Geophys. Res. 104, 4411 1999. 5 A. V. Streltsov and W. Lotko, J. Geophys. Res. 108, 189 003. 6 A. V. Streltsov and W. Lotko, J. Geophys. Res. 110, A0803 005. 7 A. V. Streltsov and W. Lotko, Couplng between densty structures, electromagnetc waves and onospherc feedback n the auroral zone, J. Geophys. Res. submtted for publcaton. 8 P. Frycz, R. Rankn, J. C. Samson, and V. T. Tkhonchuk, Phys. Plasmas 5, 3565 1998. 9 J. Y. Lu, R. Rankn, R. Marchand, and V. T. Tkhonchuk, Geophys. Res. Lett. 30, 1540003. 10 R. Rankn, J. Y. Lu, R. Marchand, and E. F. Donovan, Phys. Plasmas 11, 168 004. 11 X. L. L and M. Temern, Ponderomotve effects on on-acceleraton n the auroral-zone, Geophys. Res. Lett. 0, 131993. 1 S. I. Bragnsk, n Revews of Plasma Physcs, edted by M. A. Leontovch Consultants Bureau, New York, 1965, Vol. I. 13 M. C. Kelley and R. A. Heels, The Earth s Ionosphere: Plasma Physcs and Electrodynamcs, Internatonal Geophyscs Academc Press, San Dego, 1989, Vol 43. 14 R. L. Lysak and Y. Song, Alfvén wave dynamcs n reduced magnetohydrodynamcs, J. Geophys. Res. 107, A800. 15 R. E. Denton, B. Rogers, W. Lotko, and A. V. Streltsov, Phys. Plasmas submtted for publcaton. 16 T. E. Moore, D. L. Gallagher, J. L. Horwtz, and R. H. Comfort, Geophys. Res. Lett. 14, 1007 1987. 17 J. P. McFadden, C. W. Carlson, R. E. Ergun, D. M. Klumpar, and E. Mbus, J. Geophys. Res. 104, 14671 1999. 18 A. Zeler, J. F. Drake, and B. Rogers, Phys. Plasmas 4, 1341997. 19 A. Zeler, Max-Planck-Insttut fur Plasmaphysk IPP Report 1998. 0 A. J. Brzard, Phys. Plasmas 1, 0930 005. 1 A. Salat and J. A. Tatarons, J. Geophys. Res. 105, 13055 000. G. F. Chew, M. L. Goldberger, and F. E. Low, Proc. R. Soc. London, Ser. A 36, 111956. 3 R. E. Denton and J. G. Lyon, J. Geophys. Res. 105, 7545 000. 4 G. W. Hammett and F. W. Perkns, Phys. Rev. Lett. 64, 3019 1990. Downloaded 31 Oct 007 to 19.170.67.4. Redstrbuton subject to AIP lcense or copyrght, see http://pop.ap.org/pop/copyrght.jsp