More than mass proportional heating of heavy ions by supercritical collisionless shocks in the solar corona
aa r X i v : . [ a s t r o - ph . S R ] J un More than mass proportional heating of heavy ions bysupercritical collisionless shocks in the solar corona
Gaetano Zimbardo
Physics Department, University of Calabria, Arcavacata di Rende, Italy (Dated: June 3, 2018)
Abstract
We propose a new model for explaining the observations of more than mass proportional heatingof heavy ions in the polar solar corona. We point out that a large number of small scale intermit-tent shock waves can be present in the solar corona. The energization mechanism is, essentially,the ion reflection off supercritical quasi-perpendicular collisionless shocks in the corona and thesubsequent acceleration by the motional electric field E = − (1 /c ) V × B . The acceleration dueto E is perpendicular to the magnetic field, in agreement with observations, and is more thanmass proportional with respect to protons, because the heavy ion orbit is mostly upstream of thequasi-perpendicular shock foot. The observed temperature ratios between O ions and protonsin the polar corona, and between α particles and protons in the solar wind are easily recovered. K and more is oneof the outstanding problems of solar physics. Beside the high temperatures, Soho/UVCSobservations have shown that heavy ions in polar corona, like O and Mg , are heatedmore than protons, and that heavy ion heating is more than mass proportional; further,the perpendicular temperatures T ⊥ are much larger than parallel temperatures T k [1, 2, 3].As a consequence of magnetic mirroring in the diverging magnetic field of coronal holes,heavy ions are observed to be faster than protons in the solar wind [2, 4]. In addition,the collisional coupling with protons up to 1.32 R ⊙ indicates that the Mg heating hasto be faster than minutes [5]. These observations give stringent contraints on the coronalheating mechanism. Ion cyclotron heating has been considered since long (e.g., [3, 6, 7, 8]),but some details are not yet fully understood. The comprehension of coronal heating is ofgeneral physical interest, as the sun serves both as a huge plasma laboratory and as a modelfor a large class of stars.Shock waves are considered to be common in the chromosphere/transition region andin the corona (e.g., [9, 10, 11, 12, 13]). For instance, photospheric convection leads to theemergence of small magnetic loops, which lead to magnetic reconnection with the networkmagnetic field; small scale plasma jets are formed in the reconnection regions, and fastshocks can form when jets encounter the ambient plasma [10, 11, 14]. Indeed, recent X-rayHinode and UV Stereo observations have shown that many more plasma jets are presentin the polar corona than previously thought [15, 16, 17]. Therefore, a large number ofsmall scale, intermittent shocks can form in the reconnection regions and propagate towardthe high altitude corona. Recent numerical simulations show that bursty, time dependentreconnection in solar flares can eject many plasmoids and create oblique shocks [18]. Inthe high corona, magnetic reconnection happens when current sheets form because of theevolving coronal structures [19], while large scale shocks propagate in the corona becausesolar flares and of the emergence of coronal mass ejections [20, 21]. Such shocks are detectedas type II radio bursts [22]. In the case of solar flares, the associated reconnection outflowtermination shocks can be so strong as to accelerate electrons to 100 keV energies in afraction of a second [14, 23], and in some cases both the upper and the lower terminationshocks are identified in radio observations [24].In the low β , nearly collisionless corona, a shock wave is formed when a superAlfv´enicplasma flow having velocity V > V A collides with the ambient coronal plasma. Here, the2lasma β is given by β = 8 πp/B , where p is the total plasma pressure, B is the magneticfield magnitude, V is the plasma velocity upstream of the shock, and V A = B/ √ πρ isthe Alfv´en velocity, with ρ the mass density. The Alfv´enic Mach number is defined as M A = V /V A . We notice that although the typical Alfv´en velocity in the corona, of theorder of 1000 km/s, is larger than the observed jet velocities of 200–800 km/s [15, 25],the Alfv´en velocity in the reconnection region can be much lower, since B is weaker there.Indeed, the reconnection regions are characterized by current sheets, magnetic field reversals,and magnetic quasi-neutral lines. For instance, Tsuneta and Naito [14] argue that an obliquefast shock is naturally formed below the reconnection site in the corona, see their Figure 1.The plasma velocity in the reconnection outflow region between the slow shocks is of theorder of V A in the inflow region, that is much larger than V A in the outflow region, and thisleads to the formation of shocks (e.g., [9, 14, 20]).Previously, shock heating of coronal heavy ions was considered by Lee and Wu [11],but mostly in connection with subcritical shocks. Here, we propose that the more thanmass proportional heating of heavy ions in polar coronal holes is due to ion reflection atsupercritical quasi-perpendicular shocks and to the ion acceleration by the V × B electric fieldin the shock frame. In this connection, we notice that more than mass proportional heatingof α particles and O has been observed in the solar wind by the Ulysses spacecraft, between2.7 and 5.1 AU, downstream of interplanetary shocks, most of which were supercritical [26](see also Ref. [27]).It is well known both from laboratory [28, 29] and from spacecraft experiments (e.g., [30,31, 32]) that above a critical Mach number M ∗ A ≃ . θ Bn between the shock normal (pointing in the upstream direction)and the upstream magnetic field is larger than about 45 ◦ , the reflected ions reenter the shockafter gyrating in the upstream magnetic field. Such shocks are termed quasi-perpendicular.Conversely, for θ Bn < ◦ , the reflected ions propagare upstream, forming the ion foreshockwhich characterizes the quasi-parallel shocks. The critical Mach number M ∗ A can decreasebelow 1.5 for oblique shocks in a warm plasma [34], so that ion reflection is a relativelycommon process. Ion reflection can be considered to be the main dissipation mechanism bywhich collisionless shocks convert the flow directed energy into heat, while the electrons are3 IG. 1: Schematic of the magnetic field profile of a supercritical quasi-perpendicular collisionlessshock. The main features like the magnetic foot, the ramp, and the magnetic overshoot areindicated. heated much less (typically, one tenth of proton heating) [31].For the solar corona, we consider a quasi-perpendicular supercritical collisionless shock,and we assume a simple one dimensional shock structure. The upstream quantities areindicated by the subscript 1, and the downstream quantities by the subscript 2. We adoptthe Normal Incidence Frame (NIF) of reference, in which the shock is at rest, the upstreamplasma velocity is along the x axis and perpendicular to the shock surface, V = ( V x , , xz plane, B = ( B x , , B z ), so that the motionalelectric field E = − V × B /c is in the y direction, E y = V x B z /c . We further assume that B z ≫ B x ( θ Bn ≃ ◦ ), in order to simplify the discussion. An order-of-magnitude estimateof the energy gained by ions after reflection at the shock can be obtained by approximatingthe reflected ion trajectory with a circle of radius r L , with r L the ion Larmor radius, inthe upstream magnetic field. Assuming specular reflection [30, 31, 35], on average the ionvelocity at the reflection point is perpendicular to the shock and along the x axis. Keepingin mind that ion reflection gives rise to a non adiabatic displacement in the y direction, thework W done by the electric field is W = q i E y ∆ y, (1)where ∆ y ∼ r L . For specularly reflected ions, the Larmor radius has to be evaluated withthe upstream flow speed (neglecting the thermal velocity of the incoming ion distribution),i.e., v ⊥ ≃ | V x | , so that W = q i E y × r L = 2 q i V x B z c m i V x cq i B z , (2)4hich yields W = 2 m i V x . This estimate shows that the energy gain is mass proportional.A more detailed calculation yields a more precise result, and shows that heavy ion heatingis more than mass proportional. In order to do this, we remind that a distinctive featureof quasi-perpendicular collisionless shocks is the formation of a “foot” in the magnetic fieldprofile in front of the main magnetic ramp, the latter culminating in the magnetic overshoot,beyond which the downstream values are gradually attained [29, 32, 36], see Figure 1. Thefoot is due to the population of reflected and gyrating protons, which causes an increase inthe plasma density, and, because of the magnetized electrons, in the magnetic field strength[29]. Even if the solar corona composition encompasses several ion species, the foot extentin the upstream direction is determined by the proton gyroradius, since protons are themajor species. We define B foot = (1 + b ) B z , with b depending on the ion reflection rate; b can be estimated to be of the order 0.5–1 for typical shocks in the heliosphere [31, 32, 36].Direct observations in space show that very strong fluctuation levels are found in associationwith collisionless shocks. However, in what follows we will neglect fluctuations and we willconsider only the average quantities, in order to set the stage. Taking into account thefact that the orbit in crossed electric and magnetic fields is a trochoid, we start from theequations of the particle trajectory. We assume the magnetic field to be along the z axis,and set the origin of coordinates at the point of ion reflection, with t = 0 (e.g., Ref. [37]): x ( t ) = − v ⊥ Ω sin(Ω t ) + cE y B t (3) y ( t ) = v ⊥ Ω [1 − cos(Ω t )] (4)where Ω = q i B/m i c , and B the local magnetic field. The corresponding particle velocity is v x ( t ) = − v ⊥ cos(Ω t ) + cE y B (5) v y ( t ) = v ⊥ sin(Ω t ) . (6)Specular ion reflection implies that at t = 0 the ion velocity v x is opposite to the incomingplasma velocity, v x ( t = 0) = − V x , whence v ⊥ = V x + cE y B foot = V x + V x B z B foot = V x b b . (7)The reflected ions meet again the shock surface, at x = 0, at a later time t >
0, correspond-ing to v ⊥ Ω sin(Ω t ) = cE y B t . (8)5 IG. 2: (Color online) Projection in the xy plane of the trajectories of hydrogen and oxygen ionsreflected at the shock ramp. The motional electric field is also indicated. As in Figure 1, thevertical dashed lines separate the main magnetic field regions, such as the upstream region, thefoot, the ramp, and the downstream region. Upon inserting the values of v ⊥ and of E y in the above equation we obtainsin(Ω t ) = Ω t b , (9)whose numerical inversion yields Ω t = 2 . b = 1, Ω t = 2 . b = 0 .
5, andΩ t = 1 . b = 0 (see below). At this time the particle will have moved in the y direction by an amount given by∆ y ( t ) = v ⊥ Ω [1 − cos(Ω t )] = m i V x cq i B z [1 − cos(Ω t )] 2 + b (1 + b ) . (10)This displacement in the y direction determines the energy gained by reflected ions duringthe gyromotion in the field E y : W = q i E y ∆ y = m i V x [1 − cos(Ω t )] 2 + b (1 + b ) = 2(2 + b )(1 + b ) [1 − cos(Ω t )] 12 m i V x (11)Taking into account the fact that protons move in the foot magnetic field B foot , we canassume that b ≃ . − cos(Ω t ) = 1 . b = 1 (or 1.52652 for b = 0 . W p ≃ × . × ( 12 m p V x ) . (12)6n the other hand, for heavy ions like O most of the trajectory is upstream of the foot,see Fig. 2, in the unperturbed plasma where B ≃ B z . Then we can set b = 0 with goodapproximation, and obtain 1 − cos(Ω t ) = 1 . W heavy ≃ × . × ( 12 m i V x ) . (13)Here we can see that, with respect to protons, heating is more than mass proportional. For b = 1, the ratio of the heavy ion energy gain over the proton energy gain is W heavy /W p ≃ . × m i /m p , while for b = 0 . W heavy /W p ≃ . × m i /m p . Varying the value of b between 0.5 and 1 yields an O temperature about 25–34 times larger than the protontemperature ( m O ≃ m p ), in good agreement with Soho/UVCS observations which give T O /T p = 27–37 [5]. Also, assuming a typical value of b = 0 .
5, we can easily recover thetemperature ratios observed in the solar wind for helium ( m α ≃ m p ), where T α /T p ≃ b = 1, T α /T p ≃ E y which is perpendicular to the magnetic field by definition. This allowsto understand the observed strong temperature anisotropy with T ⊥ ≫ T k . In addition, asingle shock encounter is required to accelerate the ions, and the acceleration time is on thescale of the ion gyroperiod, so that the heating mechanism is very fast, as required by theobservations in Ref. [5].Typically, the reflection rate for protons is found from numerical simulations to be 20–30%. The energy gained by reflected ions is distributed to the transmitted ions by waveparticle interactions [31], so that we can assume for the bulk of protons a heating rate about1/4 of W p . Let us define the heating efficiency, η , for protons, as the ratio of the energy gainof both reflected and trasmitted ions over the upstream thermal energy m p v th : η = W p / m p v th ≃ . m p V x )4 × m p v th = 0 . M s (14)where M s = V x /v th is the sonic Mach number. In the low β corona, the thermal speed ismuch less than the Alfv´en speed, so that M s ≫ M A . However, in the reconnection outflowregion the magnetic field is weaker than in the ambient corona, so that for a first estimatewe assume that the sonic Mach number is of the same order of the Alfv´en Mach number. For7nstance, assuming that M s = 7, we can see that the efficiency for protons equals η ≃ K to coronal temperatures of the order of 10 K. Asa general trend, we can say that the stronger the magnetic field in the reconnection inflowregion, the larger the plasma velocity and the Mach numbers in the reconnection outflowregion, and the larger the heating efficiency. On the other hand, several shock crossings maybe required for the high altitude corona to reach the observed temperatures, and we noticethat large scale shocks associated with coronal mass ejections and type II radio bursts canpropagate all the way into polar corona.Clearly, the present model has to be further developed, since a wide range of differentMach numbers, plasma β , shock normal angles θ Bn , fluctuation levels, and heating ratioscan be envisaged in the corona. In addition, collisional coupling with protons can decreasethe obtained temperature ratios, as may be the case of Mg . Further, multiple reflectedions at a single shock can also be envisaged [43], a phenomenon which would increase theheating efficiency.Our model leads to the prediction that a fraction of heavy ions comparable to the protonfraction is also reflected at quasi-perpendicular shocks. In collisionless shocks, an electro-static potential barrier ∆ φ ≃ m p ( V x − V ) / e arises, which slows down the incoming ions[29, 39, 44]. In simple discussions of ion reflection, ions are expected to undergo specularreflection if their kinetic energy in the shock frame is less than the potential energy barrier q i ∆ φ : however, for heavy ions this is usually found only for a tiny fraction of the upstreamvelocity distribution. Nevertheless, experimental evidence of α particle reflection at Earth’squasi-perpendicular bow shock has been reported by in Ref. [40], while evidence of α par-ticle specular reflection off the quasi-parallel bow shock was reported in Ref. [41]. On theother hand, laboratory experiments show that increasing M A , the number of reflected ionsincreases while the potential jump decreases [29], contrary to expectations if the potentialjump would be the only cause of ion reflection. This shows that ions are not simply reflectedby the average potential jump across the shock, and that also the magnetic foot and thefluctuating electric and magnetic overshoots play a role for ion reflection [30, 35, 39, 42]. In-deed, cross shock electric fields measured by the Polar spacecraft at Earth’s bow shock showthat the potential ∆ φ is strongly spiky and fluctuating [44]. Further, quasi-perpendicularshocks also exhibit cyclic reformation, which implies time-depending electric and magnetic8vershoots and reflection rate [33, 45]. Indeed, the Cluster spacecraft have recently shownthat ion reflection is highly unsteady [45], so that the strong variations in the shock structurecan also induce heavy ion reflection.In conclusion, we have considered the heavy ion energization due to the ion reflectionoff quasi-perpendicular shocks. Fast, supercritical shocks are assumed to form because ofreconnection of small scale magnetic loops at the base of coronal holes, like those associatedwith polar coronal jets, and because of the merging of magnetic structures in the highercorona. The energy stored in the coronal magnetic field is transformed to bulk kineticenergy by reconnection, and into heat and heavy ion heating by the quasi-perpendicularshocks which form in the reconnection outflow region. Our model can explain both coronalheating and the more than mass proportional heavy ion heating observed by Soho/UVCS.In addition, this heating mechanism is strictly perpendicular to the magnetic field and it isvery fast (a single shock encounter is needed); most heating goes into the ions, with electronsundergoing an energy gain which is about an order of magnitude smaller than that of protons[31, 32]. Further, the strongly anisotropic heating with T ⊥ ≫ T k can give rise to efficiention cyclotron emission; this phenomenon is actually observed downstream of the Earth’sbow shock. Indeed, recent Stereo, Helios, and Venus Express data show that ion cyclotronwaves are probably present in the corona [46]. Therefore, quasi-perpendicular collisionlessshocks can be the source of ion cyclotron waves in the corona, too. These waves later on canheat locally the solar wind by cyclotron resonance dissipation, as suggested by a number ofobservations.This research was partially supported by the Italian INAF and the Italian Space Agency,contract ASI n. I/015/07/0 “Esplorazione del Sistema Solare”. [1] J.L. Kohl et al., Solar Phys. 175, 613 (1997).[2] J.L. Kohl et al., Astrophys. J., 501, L127 (1998).[3] S. R. Cranmer, Field, G. B., and Kohl, J. L., Astrophys. J., 518, 937 (1999).[4] E. Marsch and C.-Y. Tu, J. Geophys. Res., 106, 227 (2001).[5] R. Esser et al., Astrophys. J., 510, L63 (1999).[6] E. Marsch, Goertz, C. K., Richter, K., J. Geophys. Res., 87, 5030 (1982).
7] P. A. Isenberg and Hollweg, J. V., J. Geophys. Res., 88, 3923 (1983).[8] J. V. Hollweg and P. A. Isenberg, J. Geophys. Res., 107(A7), 1147, doi:10.1029/2001JA000270(2002).[9] T. Yokoyama and Shibata, K., Nature, 375, 42 (1995).[10] T. Yokoyama and Shibata, K., Publ. Astron. Soc. Japan, 48, 353 (1996).[11] L.C. Lee, and B.H. Wu, Astrophys. J., 535, 1014–1026, 2000.[12] M. Ryutova and Tarbell, Th., Phys. Rev. Lett., 90, 191101 (2003).[13] A. Vecchio, G. Cauzzi, and K. P. Reardon, Astron. Astrophys., 494, 269 (2009).[14] S. Tsuneta, and T. Naito, Astrophys. J., 495, L67 (1998).[15] J.W. Cirtain et al., Science 318, 1580, doi: 10.1126/science.1147050 (2007).[16] S. Patsourakos, Pariat, E., Vourlidas, A., Antiochos, S. K., Wuelser, J. P., Astrophys. J., 680,L73 (2008).[17] G. Nistic`o, V. Bothmer, S. Patsourakos, and G. Zimbardo, Solar Phys., in press (2009).[18] S. Tanuma, and K. Shibata, Publ. Astron. Soc. Japan, 59, L1–L5 (2007).[19] L. Fisk, J. Geophys. Res., 108, 1157 (2003).[20] M.J. Aschwanden,
Physics of the Solar Corona: An Introduction,
Springer, Berlin (2005).[21] I. Ballai, Erdelyi, R., and Pinter, B., Astrophys. J., 633, L145 (2005).[22] G. S. Nelson, and D. Melrose, in Solar Radiophysics, ed. D. J. McLean, and N. R. Labrum(Cambridge: Cambridge University Press), 333 (1985).[23] A. Warmuth, G. Mann, and H. Aurass, ˚A, 494, 677 (2009).[24] H. Aurass and G. Mann, Astrophys. J., 615, 526 (2004).[25] F. Moreno-Insertis, K. Galsgaard, and I. Ugarte-Urra, Astrophys. J., 673, L211–L214, (2008).[26] D. Berdichevsky, J. Geiss, G. Gloeckler, and U. Mall, J. Geophys. Res., 102, 2623 (1997).[27] K. E. Korreck, T.H. Zurbuchen, S. T. Lepri, and J.M. Raines, Astrophys. J., 659, 773 (2007)[28] J. W. M. Paul, L. S. Holmes, M. J. Parkinson, and J. Sheffield, Nature, 208, 133 (1965).[29] P.E. Phillips and A.E. Robson, Phys. Rev. Lett., 29, 154 (1972).[30] S.D. Bale et al., Space Sci. Rev., 118, 161, DOI: 10.1007/s11214-005-3827-0 (2005).[31] J. Gosling and Robson, A.E., in
Collisionless Shocks in the Heliosphere: Reviews of CurrentResearch , edited by R. G. Stone, B. T. Tsurutani, Geophysical Monograph Series, AGU, 35,Washington D.C. (1985).[32] J. D. Scudder et al., J. Geophys. Res., 91, 11,019 (1986).
33] K.B. Quest, J. Geophys. Res., 91, 8805 (1986).[34] J.P. Edmiston and C.F. Kennel, J. Plasma Phys., 32, 428 (1984).[35] K. Meziane et al., Annales Geophys., 22, 2325, (2004).[36] L. F. Burlaga et al., Nature, 454, 7200, 75 (2008).[37] L.D. Landau and E.M. Lifshitz,
Field Theory, Course of Theoretical Physics, eq. (22,6), Perg-amon, New York, (1983).[38] J.C. Kasper, Lazarus, A.J., and Gary, S.P., Phys. Rev. Lett., 101, 261103 (2008).[39] M. M. Leroy, D. Winske, C. C. Goodrich, C. S. Wu, and K. Papadopoulos, J. Geophys. Res.,87, 5081 (1982).[40] M. Scholer, Ipavich, F. M., Gloeckler, G., J. Geophys. Res., 86, 4374 (1981).[41] S. A. Fuselier, Thomsen, M. F., Ipavich, F. M., Schmidt, W. K. H., J. Geophys. Res., 100,17107 (1995).[42] K. Meziane et al., J. Geophys. Res., 109, A05107, doi: 10.1029/2003JA010374 (2004).[43] A. S. Lipatov and G. P. Zank, Phys. Rev. Lett., 82, 3609 (1999).[44] S.D. Bale and F.S. Mozer, Phys. Rev. Lett., 98, 205001 (2007).[45] V. V. Lobzin et al., Geophys. Res. Lett., 34, L05107, doi:10.1029/2006GL029095 (2007).[46] L.K. Jian, C.T. Russell, R. J. Strangeway, et al., Evolution of ion cyclotron waves in the solarwind from 0.3 to 1 AU, paper presented at the Solar Wind 12 conference, 21-26 June 2009,St. Malo, France.eq. (22,6), Perg-amon, New York, (1983).[38] J.C. Kasper, Lazarus, A.J., and Gary, S.P., Phys. Rev. Lett., 101, 261103 (2008).[39] M. M. Leroy, D. Winske, C. C. Goodrich, C. S. Wu, and K. Papadopoulos, J. Geophys. Res.,87, 5081 (1982).[40] M. Scholer, Ipavich, F. M., Gloeckler, G., J. Geophys. Res., 86, 4374 (1981).[41] S. A. Fuselier, Thomsen, M. F., Ipavich, F. M., Schmidt, W. K. H., J. Geophys. Res., 100,17107 (1995).[42] K. Meziane et al., J. Geophys. Res., 109, A05107, doi: 10.1029/2003JA010374 (2004).[43] A. S. Lipatov and G. P. Zank, Phys. Rev. Lett., 82, 3609 (1999).[44] S.D. Bale and F.S. Mozer, Phys. Rev. Lett., 98, 205001 (2007).[45] V. V. Lobzin et al., Geophys. Res. Lett., 34, L05107, doi:10.1029/2006GL029095 (2007).[46] L.K. Jian, C.T. Russell, R. J. Strangeway, et al., Evolution of ion cyclotron waves in the solarwind from 0.3 to 1 AU, paper presented at the Solar Wind 12 conference, 21-26 June 2009,St. Malo, France.