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	 "body":"\nSH 4.1 .07\n\nMultiple Magnetic Field-Shock Crossings and Particle\n\nAcceleration at Quasi-perpendicular Shocks\n\nD. Ruffolo and J. Sukonthachat\nDepartment ofPhysics, Chulalongkorn University, Bangkok 10330, THAILAND\n\nThe importance of accounting for diffusion perpendicular to the mean magnetic field during quasi-perpendicular\nshock acceleration is well documented. Here we note that perpendicular diffusion is typically envisioned as\ndue to the random walk of field lines, with particle guiding centers closely tied to and diffusing back and forth\nalong the field. A turbulent magnetic field line can cross and recross the shock, like a sawtooth edge. In this\n'sawtooth mechanism,' if there are N magnetic field-shock crossings that are separated by distances L >\nthe scattering mean free path, a particle diffusing along the field line will cross the shock an average of N\ntimes before escaping. This could increase the total shock-drift distance and energization of particles. We\nhave verified that multiple field-shock crossings do occur for reasonable values of (8B/B the shock,\nand have measured the distribution of N, and L for simulated random magnetic fields. For the special\ncase of the solar wind termination shock, this mechanism may help explain the observationally inferred drift\nof anomalous cosmic rays (ACR) over much of the distance from the solar equator to the poles or vice-versa.\n\n1 Introduction:\n\nWhen particles are largely tied to a given magnetic field line, for a single field-shock crossing the\nacceleration rate can greatly increase as the field-shock normal angle, approaches 900 i.e., for a nearly\nperpendicular shock (Jokipii 1987). The energization of particles can be viewed (in the fixed frame) as mainly\ndue to the shock-drift mechanism (Schatzman 1963), in which particles drift along the electric field while\nencountering the shock. Particle diffusion perpendicular to the mean magnetic field direction has been shown\nto play an important role (e.g., Jokipii 1987; Jokipii, Kóta, & Giacalone 1993; Jones, Jokipii, & Baring 1998).\nGiacalone, Jokipii, and Kóta (1994) and Ellison, Baring, and Jones (1995) have performed Monte Carlo (MC)\nsimulations for this situation that include ad hoc diffusion perpendicular to the magnetic field, and the latter\nauthors found that while the acceleration rate rose with the injection rate declined.\n\nIn theoretical models of particle transport in turbulent magnetic fields such as those found in the solar\nsystem (e.g., Bieber & Matthaeus 1997), particle diffusion perpendicular to the mean magnetic field direction is\nmainly ascribed to a random walk of the magnetic field, i.e., particle guiding centers are basically tied to field\nlines, which themselves wander perpendicular to the mean field. Note that a turbulent magnetic field line can\ncross and recross the shock, like a sawtooth edge (Figure 1). In this work, we consider the implications of\nthis concept of perpendicular diffusion, and we identify a mechanism, which we term the 'sawtooth\nmechanism,' that can greatly enhance the particles' total energization and shock drift distance, which could give an\nimproved physical explanation of the observationally inferred drifts (Cummings, Stone, & Webber 1985) and\nassociated energy cutoffs (Mewaldt et al. 1996) of ACR.\n\nOur proposed mechanism involves similar basic physics to that contained in MC calculations with ad hoc\nperpendicular diffusion (e.g., Giacalone, Jokipii, & Kóta 1994; Ellison, Jones, & Baring 1999). Our goal is\nto elucidate the key physical mechanisms of energetic particle acceleration at nearly perpendicular shocks,\nwhich would also underly the MC results. Here we do not address the issue of injection. The mechanism we\nconsider is appropriate when the particle speed is fast relative to the convection speed; for slow particles or a\nfast convection speed, a more important mechanism might be that involving multiple reflection in collapsing\nmagnetic traps (Decker 1990, 1993).\n© University of Utah • Provided by the NASA Astrophysics Data System\n\n2 Multiple Magnetic Field-Shock Crossings:\n\nIn order to determine the characteristics of\nmagnetic field-shock crossings, we computationally\ngenerated random magnetic fields for a specified\npower spectrum matrix using inverse Fourier\ntransforms and a random phase. Figure la shows an\nexample of such a random magnetic field line for\nslab turbulence with P _ P _ C(1 + x/z U\nk (öB _ 0.05, and a correlation\nlength of 16L where L is the grid size for the\ninverse Fourier transform. The mean magnetic field\nis in the horizontal direction, and the shock plane\n(solid line) is slightly tilted relative to the mean\nmagnetic field; note the greatly expanded vertical axis.\nActually, the magnetic field line should be verti- z/L\ncally compressed downstream of the shock, which b\nfor clarity is not shown in Figure la; that would (absorbing)\nnot affect the number or characteristics of\nfieldshock crossings studied in the present work. This\n. . shock\nis an example of how one does find multiple\nfieldshock crossings for reasonable values of the tur- (reflecting)\nbulent energy. In fact, stronger turbulence, with\n(öB 0. 1 , is expected to be generated by a Figure 1 : a) A magnetic sample field line that crosses a\nquasi-perpendicular shock, according to the hybrid shock (diagonal line) multiple times. Note the greatly\nsimulations of Liewer, Rath, & Goldstein (1995). expanded vertical scale. b) Schematic of the above\n\nFigure 2 shows examples of the statistics we can 'sawtooth' magnetic field, and boundary conditions for\ncollect regarding field-shock crossings, for the same the random walk of particles along B.\ntype of turbulence. The calculation of the upstream field-shock angle, , takes into account the orientation\nof B in three dimensions. It is noteworthy that even when taking magnetic field irregularities into account, the\ndistribution of field-shock angles has peaks near 900 , indicating that particle-shock encounters can potentially\nyield a large amount of shock-drift acceleration. As shown by Jokipii (1982), the ratio of energy gain to\nq/M the change in potential energy when drifting along the electric field, is approximately unity for such\nhigh angles. The distribution of L, the distance between consecutive crossings, helps determine the number of\ncrossings which are sufficiently far apart for the sawtooth mechanism to take effect (see §3). These calculations\nwill be extended to consider 2D and 2D + slab (three-dimensional) turbulence models as well.\n\n3 Sawtooth Mechanism:\n\nLet us first consider the random walk of particles along the random magnetic field by ignoring the\npossibility of reflection when approaching the shock from upstream. Referring to Figure 1, in this framework we view\nthe acceleration process in terms of discrete episodes of diffusive shock acceleration (which includes shock\ndrift acceleration) when the particle encounters a field-shock crossing. In addition to the correlation lengths,\nother relevant length scales for a given particle species and energy include the gyroradius rg and the scattering\nmean free path AH. Since the particle motion follows a sort of average of B over a gyroradius, field-shock\ncrossings closer together than rg should be grouped together so that the particle interaction in that region is\nconsidered to constitute a single particle-shock encounter.\n\nNext, field-shock crossings spaced farther than rg but closer than will generally be traversed in sequence;\nN such crossings can then yield an N-fold enhancement in shock acceleration. We refer to this as a linear\n(degrees)\n\n© University of Utah • Provided by the NASA Astrophysics Data System\n\n150 6000\n\n100 4000\n\n50 2000\n\n0 0\n\nU 30 60 90 U 10 20 30\n\nN L/(100 z\n\nFigure 2: Histograms of a) the number of field-shock crossings, b) the upstream angle between the field and\nthe shock normal, and c) the distance between crossings for 1000 simulated turbulent magnetic fields.\n\nNow consider only field-shock crossings or groups of field-shock crossings that are spaced farther apart\nthan A For this purpose, the magnetic field in Figure la can be conceptually simplified as in Figure lb.\n(Again, for simplicity we have not indicated the refraction of magnetic field lines.) Between two crossings, a\nparticle's motion is randomized, and there is an equal probability of either moving forward to the next crossing\nor returning to the previous crossing.\n\nThis then becomes a classic random walk problem and is amenable to mathematical analysis\n(Chandrasekhar 1943). Starting from upstream of the shock on the left hand side, let ri be the number of times a\nparticle encounters the shock, and let rn indicate the regions between field-shock crossings from left (rn - 0)\nto right (711) - N). To represent the ultimate return of particles upstream of the first field-shock crossing,\ndue to convection, we place a reflecting barrier at ri-i - 0, and to represent escape downstream, we use the\nconservative assumption of absorption at rn - N. The probability of escape after ri shock encounters can be\nshown to be\n\nP(n) _ • (( + - . ((n (1)\n\nF::o,7n F::O,rn\n\nwhere Ti) + N is even and rn - (2j + 1)N. This probability sums to 1 (when summing over all ri   N\nsuch that ri + N is even), and the mean value of ri is N The mean number of shock encounters before\nescape should actually be even larger if we consider that a large fraction of particles approaching a shock from\nupstream (87% for a strong shock with a compression ratio of 4) should be reflected backup upstream, which\ngives some probability of trapping between two adjacent field-shock crossings.\n\nTherefore, even with conservative assumptions the sawtooth model predicts a quadratic enhancement by\na factor of N the shock drift and total energization of particles, where N is the number of field-shock\ncrossings spaced farther than A. Presumably this enhancement is occurring in MC simulations of particle\nacceleration at nearly perpendicular shocks. In practice the total energy gain of particles will also be limited\nby the lateral extent of the shock, and the convection of field lines past the shock. In fact, for the case of the\nsolar wind termination shock, observations of anomalous cosmic rays can be understood in terms of particle\ndrift at the shock over a large fraction of the distance from the heliospheric equator to the poles or vice-versa\n(Cummings, Stone, & Webber 1985), with even an indication of a spectral break corresponding to particles\nthat traverse that entire distance (Mewaldt et al. 1996). Such observations point to successful shock-drift\nacceleration up to the limit of the size of the termination shock.\n\nFurther work will aim to clarify the effect of the level of turbulence on the acceleration of energetic charged\nparticles at nearly perpendicular shocks. We hope that this framework will also prove useful for assessing a\npossible species dependence of the acceleration efficiency.\nThe authors wish to acknowledge useful discussions with John Bieber, Cliff Lopate, Bill Matthaeus, and\nGary Zank. This work was partially supported by a Basic Research Grant from the Thailand Research Fund.\nDR is grateful to the Bartol Research Institute of the University of Delaware for their hospitality while part of\nthis work was carried out, and for support during that time from NASA grant NAG 5-8 134.\n\nBieber, J. W., & Matthaeus, W. H. 1997, ApJ, 485, 655\nChandrasekhar, 5. 1943, Rev. Mod. Phys., 15, 1\nCummings, A. C., Stone, E. C., & Webber, W. R. 1985, Proc. 19th ICRC (La Jolla, 1985), 5, 163\nDecker, R. B. 1990, J. Geophys. Res., 95, 11993\nDecker, R. B. 1993, J. Geophys. Res., 98, 33\nEllison, D. C., Baring, M. G., and Jones, F. C. 1995, ApJ, 453, 873\nEllison, D. C., Jones, F. C., and Baring, M. G. 1999, ApJ, 512, 403\nGiacalone, J., Jokipii, J. R., & Kóta, J. 1994, J. Geophys. Res., 99, 19351\nJokipii, J. R. 1982, ApJ, 255, 716\nJokipii, J. R. 1987, ApJ, 313, 842\nJokipii, J. R., Kóta, J., & Giacalone, J. 1993, Geophys. Res. Lett., 20, 1759\nJones, F. C., Jokipii, J. R., & Giacalone, J. 1998, ApJ, 509, 238\nLiewer, P. C., Rath, S., & Goldstein, B. E. 1995, J. Geophys. Res., 100, 19809\nMewaldt, R. A., Selesnick, R. S., Cummings, J. R., Stone, E. C., and von Rosenvinge, T. T. 1996, ApJ, 466,\nSchatzman, E. 1963, Ann. d'Astrophys., 26, 234\n\n",
	 "pubyear_facet":1999,
	 "bibcode":"1999ICRC....7..476R",
	 "abstract":"The importance of accounting for diffusion perpendicular to the mean magnetic field during quasi-perpendicular shock acceleration is well documented. Here we note that perpendicular diffusion is typically envisioned as due to the random walk of field lines, with particle guiding centers closely tied to and diffusing back and forth along the field. A turbulent magnetic field line can cross and recross the shock, like a sawtooth edge. In this \"sawtooth mechanism,\" if there are Æ magnetic field-shock crossings that are separated by distances Ä , ¾ the scattering mean free path, a particle diffusing along the field line will cross the shock an average of Æ times before escaping. This could increase the total shock-drift distance and energization of particles. We near the shock, have verified that multiple field-shock crossings do occur for reasonable values of Æ and have measured the distribution of Æ , Ò, and Ä for simulated random magnetic fields. For the special case of the solar wind termination shock, this mechanism may help explain the observationally inferred drift of anomalous cosmic rays (ACR) over much of the distance from the solar equator to the poles or vice-versa.",
	 "ft_source":"http://adsabs.harvard.edu/full/1999ICRC....7..476R",
	 "id":"514126",
	 "title":[
	  "Multiple Magnetic Field-Shock Crossings and Particle Acceleration at Quasi-perpendicular Shocks"],
	 "author_facet":["Ruffolo, D"],
	 "database":["AST"],
	 "keyword_facet":[""]}]
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