arXiv:1501.06509v1 [astro-ph.HE] 26 Jan 2015 o.Nt .Ato.Soc. Astron. R. Not. Mon. aav Lingam Manasvi mechanism unified Dynamo the Dynamo-Reverse via outflows astrophysical Modeling 2Jn 2021 June 12 † ⋆ Pino Dal Gouveia (2010). Rodr´ıguez de Beskin & (2003); (2005); Mirabel Blandford Chernin & (1999); & Lyutikov Livio Masson (1999); (1985); (1998); Lada Ferrari (1984); foun (1993); Perley is & systems astrophysical and Bridle in in jets outflows to of comprehe refer one discussion A sive shall the “outflows”. as we than cousins diffuse Letter, smaller (relatively) much their this jets are In exhibits that above. Sun ones mentioned our albeit addition, their flares, In in and evolution. stars of in often stages detected emanations, been final also diffuse have outflow, “outflows”, termed narrow highly and a suggests Objects collimated “jets” word Protoplanetary Stellar the and Although wider Young (PPNe). systems Nebulae low-mass much binary X-ray as a massive such in (YSOs), systems observed of micro- been array (AGNs), have Nuclei non- versions Smaller-scale Galactic (GRBs). relativistic Bursts observed Active Ray Gamma been of and have quasars context jets myriad the Relativistic in in co velocity. luminosity, appear and in variations mation they significant universal: exhibiting forms, are jets Astrophysical INTRODUCTION 1 c 1 nttt o uinSuis h nvriyo ea,Aust Texas, of University The Studies, Fusion for Institute -al [email protected] E-mail: -al [email protected] E-mail: 04RAS 2014 ie h iest fterocrec n the and occurrence their of diversity the Given 000 1 ⋆ – 21)Pitd1 ue22 M L (MN 2021 June 12 Printed (2014) 1–5 , n wds .Mahajan M. Swadesh and cls h ag cl lve ahnumber Alfv´en ion Mach scale the microscop scale, large of length The categorization and intrinsic scales. normalization an proper Dy-R introduces the The for o also lowing evolution It context. vorticity. the astrophysical unifies fluid which broad and magnetohydrodynamics, amb a Hall an via in from derived fields explored magnetic is and capable outflows reservoir, mechanism, scale large (Dy-RDy) generating Dynamo ously Dynamo-Reverse unified The ABSTRACT e words: Key ees yao(D)i h rnia aietto famagnetica be p may a outflow have RDy of efficient that the microscopic an outflows manifestation while that astrophysical a principal conjectured observed kinetic, to is the It mostly is tied reservoir. is is field, turbulent be (RDy) turbulence magnetic to dynamo ambient scale sca shown reverse large short the is ambient the when the field producing mode of preferentially magnetic characteristics (Dy), the the namo to reflects that field length flow the of eeaino atce tr:wns uflw aais jets galaxies: – outflows winds, stars: – particles of celeration mgeoyrdnmc)MD–pams–mtos nltcl–ac – analytical methods: – plasmas – MHD (magnetohydrodynamics) lli- n X772 USA 78712, TX in, n- d nnurnsa rss(odec esnge 1992; comes Reisenegger & recognition (Goldreich This evolution 2004). crusts field Zweibel & star Arras magnetic Cumming, discs neutron and mag- protostellar 2001) in as 2011), Terquem such Shang & & scenarios, (Balbus Krasnopolsky Shang several Li, & sys- shown Li in 2011; (Krasnopolsky, been astrophysical formation role star has and crucial in effect braking netic a Hall effect decade, the play Hall last recognized; to been the the has in of tems However, ex- importance model. highly the fluid and simplest plasma the plored model, (MHD) hydrodynamic these of origin the explain mag- to the by attempts mediated their outflows. all 2014), in al. field, et 2001; netic Colgate Livio 2006; & al. Ogilvie et 1991; Li Contopoulos mechanisms & other 1999), Berk and Blandford (Lovelace, 2003) & Li (Lynden-Bell Sub- Krasnopolsky, towers 1999; outflows. magnetic mecha- al. these magneto-centrifugal et (Shu of the nism and of introduced generation importance works the 1970s the sequent in elusive highlight fields did the K¨onigl 1979; magnetic proven & 1982) Blandford Payne in 1977; mechanism & has Blandford Znajek & works underlying Lovelace Blandford outflows 1976; 1976; Ruzmaikin ex- broad seminal & (Bisnovatyi-Kogan these 1980s outflows a early several power these for although can quest that that the parameters hibit, different vastly 1 oto h rcdn nlssue h da magneto- ideal the used analyses preceding the of Most † A T E tl l v2.2) file style X M A M ≫ A enn h eaie“abundance” relative the defining , 1. ersror h dy- The reservoir. le h oreo many of source the cadmacroscopic and ic ymcaimis mechanism Dy etmicroscopic ient antcfield magnetic f kndph al- depth, skin l dominated lly h dominant the fsimultane- of roducing scale - 2 M. Lingam and S.M. Mahajan along with the realization that MHD may be an inad- magnetic field: equate framework for generating the observed outflows ∂b (Bisnovatyi-Kogan & Lovelace 2001). = [(v α0 b) b] , ∂t ∇× − ∇× × In order to explore the simultaneous dynamics of mag- 2 ∂v v netic fields and outflows, one must resort to models more = v ( v)+( b) b p + , (1) ∂t × ∇× ∇× × −∇  2  encompassing than MHD. As an illustration, we will ana- lyze incompressible Hall magnetohydrodynamics (HMHD), wherein α0 = λi0/R0, with λi0 representing the ion skin the simplest “beyond MHD” model that captures one cru- depth, which serves as the intrinsic length scale character- cial two-fluid effect: the Hall current, expressing differential istic of HMHD. R0 represents a characteristic scale length electron-ion motion, introduces a fundamental scale length, to be specified shortly hereafter. In (1), the magnetic field the ion skin depth (ideal MHD is scale-free) that provides is normalized to an arbitrary representative magnetic field a convenient fiduciary length in terms of which one can de- B0, the velocity to the corresponding Alfven velocity VA0, fine ‘large scale’ and ‘small scale’ quantities. HMHD assigns the length scale to R0 and the timescale to R0/VA. Subse- assigns equal weights to, and thereby unifies, the magnetic quently, we choose R0 to be the ion skin depth, as we are fields and flows in a natural manner, as demonstrated in interested in the underlying microscopic physics. Thus, we Section 2. Qualitatively, this can be understood as a conse- observe that λi0 will no longer appear explicitly, i.e, α0 = 1. quence of one of the HMHD equations involving a simultane- With such a choice of units, one may re-express (1) as a pair ous evolution of the magnetic field B and the fluid vorticity of vorticity-like equations (Mahajan & Yoshida 1998): v, resulting in the inexorable linking of these two quan- Ω ∇× ∂ j tities. (vj Ωj ) = 0, ∂t −∇× × In this Letter, we explore what was termed as the where the vorticities Ω and the associated velocities v are ‘reverse dynamo’ mechanism; derived and developed in j j Mahajan et al. (2005). The mechanism employs HMHD, and Ω1 = B, v1 = v B, exhibits potential universality in generating non-relativistic −∇× Ω2 = B + v, V2 = v. outflows. The original moniker ‘reverse dynamo’ consti- ∇× tutes an inadequate (and incomplete) description of the The structure of the above equations is radically different more encompassing process. Though dynamo based ap- from that of ideal MHD, and is also indicative why B and proaches have been around since the 1970s (Lovelace 1976), v ( v to be precise) are generated in tandem. ∇× with several important developments in the subsequent Notice that the governing equation for the canonical decades (Bhattacharjee & Hameiri 1986; Brandenburg et al. vorticity Ω2 treats the magnetic field and the vorticity on 1995; Vishniac & Cho 2001; Blackman & Field 2002; par. There are two conserved helicities in the system: the 3 A B Brandenburg 2005; Brandenburg & Subramanian 2005; magnetic helicity D d x and the canonical helicity 3 · Ebrahimi & Bhattacharjee 2014), the Dy-RDy approach is d x (A + v) (R v + B); the latter is essentially the D · ∇× conceptually unique as it incorporates the following features: ionR helicity reflecting the (finite) ion inertia. Unlike in ideal MHD, the cross helicity, d3x v B, is no longer conserved. D · The HMHD is the governing model from the outset (the The velocity and magnetic fields are decomposed into • R Hall term is not appended to an essentially MHD calcu- the equilibrium seed fields (v0 and b0) and the fluctuations; lation). Consequently the magnetic field and the flows are the latter, in turn, comprising of the macroscopic (U and H) treated on an entirely equal footing. The “dynamo” desig- and microscopic (˜v and b˜) components: nates the creation of long-scale magnetic fields from a short b H b˜ b scale (turbulent) velocity field, and the “reverse dynamo” = + + 0, denotes the complementary/opposite phenomenon of gener- v = U + ˜v + v0. (2) ating long-scale flows from a short scale (turbulent) mag- It is important to realize that the fluctuations are comprised netic field. The HMHD based theory is a unified Dy-RDy of both macroscopic and microscopic components. The for- theory; both flows and magnetic fields emerge simultane- mer yield non-zero expressions upon carrying out a suit- ously from a given source of short scale energy be it kinetic able ensemble or spatial averaging, whilst linear combina- or magnetic. tions of the latter yield no contributions. Over a microscopic Unlike MHD, the HMHD has an intrinsic length scale. • scale, we also assume that the large scale fluctuations do not Thus it becomes possible to characterize long and short vary significantly, i.e. that their derivatives vanish. We refer scales in a well-defined way. This will prove to be crucial the reader to Mininni, G´omez & Mahajan (2002, 2003a,b); in estimating the length scale relevant to, for example, the Mahajan et al. (2005) for a detailed discussion of the closure generated outflows. scheme, and its accompanying assumptions. The equilibrium fields are the energy reservoir for driv- ing the fluctuations. A comment regarding the length scales is in order. The only intrinsic scale is defined by the normal- 2 DY-RDY: A SYNOPSIS izing length, the ion skin depth. Any macroscopic length scale of interest is way bigger than λi0, but there is some We begin with a short recapitulation of the formalism de- latitude in what could be called the microscopic length. The rived in Mahajan et al. (2005). Assuming a constant den- latter (microscopic length) is taken to be any length that is sity (purely for analytic simplicity) and equal temperatures on the order of, or smaller than, the ion skin depth λi0 (or of for the electrons and ions (p = pi + pe 2nT ), the HMHD order unity and less in normalized units). The equilibrium ≈ yields the following evolution equations for the flow and the fields are small scale (microscopic), have been produced by

c 2014 RAS, MNRAS 000, 1–5 Unified Dynamo mechanism 3 some microscopic process, undergone saturation, and have scale lengths, namely Λ−1 and ζ−1. The helicity measures, built up an energy reservoir that will drive the large scale in turn, reflect the initial state of the system, in particu- (macroscopic and observable) and small scale (microscopic) lar, the ratio of the ambient kinetic energy to the ambient fluctuations. magnetic energy. No matter what their ratio, the flows and Given that one has little information about the equi- the magnetic fields are generated together; the HMHD is a librium fields, one makes the most ‘natural’ choice in the theory of the unified Dynamo-Reverse Dynamo (Dy-RDy) context of Hall MHD; the double Beltrami equilibria, de- process. scribed in Mahajan & Yoshida (1998). The double Beltrami Simple algebra reveals that for a d 1, the short ∼ ≪ equilibria are the solutions of scale reservoir is mostly magnetic, v0 ab0 b0 leading ∼ ≪ b0 to strong (compared to magnetic fields) macroscopic flows + b0 = v0, b0 + v0 = dv0, (3) U ΛH H. This limit pertains only to those systems a ∇× ∇× ∼ ≫ where the observed flows are super-Alfv´enic. The standard which, along with the Bernoulli condition 2 dynamo scenario pertains to the opposite case: a d 1, v ∼ ≫ p + 2 = 0 constitute the stationary solutions of (1). implying the microscopic relation v0 ab0 b0, and the ∇ ∼ ≫ The constants a and d, set by the helicities of the system macroscopic relation H U. ≫ (Mahajan et al. 2005), define two (dimensionless) inverse This deterministic relationship between the energy mix 1 −1 −1 2 in the short scale reservoir, and the consequent long scale scale lengths λ± = d a (d + a ) 4 . Ap- 2  − ± −  observables is one of the major results of the paper. The the-
q propriate tuning of a and d can lead to vastly separated ory has provided a probe into the earlier era microphysics; roots, λ+ Λ and λ− ζ; the former (latter) represents measuring the long range fields and flows yields information ≡ ≡ the microscopic (macroscopic) inverse scale length. Such a about the source that caused them. choice is physically meaningful - the astrophysical systems Let us explore a little further the regime where RDy are macroscopic, but their underlying physics may have a part of Dy-RDy will be dominant. We will examine the rele- substantial microscopic component. In fact, we could as- vant relation U ΛH in physical units. Since H and U are ∼ sume that the ambient fields are entirely microscopic; this normalized by B0 and VA0 = B0/√4πmin (with constant is again a rational assumption since the underlying reser- n), we find that voir that generates observable (macroscopic) phenomena is, U¯ U¯ indeed, presumed to be microscopic. Λ A. (7) ∼ H/¯ √4πmin ∼ ¯A ∼M With these assumptions and a great deal of algebra V (Mahajan et al. 2005), one derives the final equations for Hence, the inverse (dimensionless) microscopic scale length the macroscopic fluctuations: Λ turns out to be approximately equal to the large scale H¨ H U¨ U H Alfv´en Mach Number A, where U¯ and ¯A are, respec- = r ( ) , = (s q ) (4) M V − ∇× ∇× − tively, the (dimensional) large scale flow and Alfv´en veloci- where r, s and q, given by ties. Equation (7) captures the very essence of this analysis, 2 2 b b − − as it directly ties together the microscopic and macroscopic q = Λ2 0 , r = Λ 0 1 Λa 1 a 2 , 6 3 − − physics. This is also seen in (6), as it relates q/(s + r) (mi- 2
croscopic) with U/H (macroscopic). In our subsequent dis- b0 −1 2 s = Λ Λ+ a 1 , (5) cussion, we drop the overbars and note that all quantities 6 − h
i on the RHS of (7) are large scale and dimensional. 2 are determined by b0, representing the normalized ambient The reverse dynamo mechanism, favored by the order- magnetic energy, and by the scale lengths set by the mi- ing a d 1 (Λ 1), will operate exactly in those regimes ∼ ≪ ≫ croscopic helicities; the entire macroscopic dynamics is con- where the observed A is much greater than unity. Equiv- M trolled by the ambient microscopic dynamics. By manipu- alently, if the observed large scale flows are highly super- lating the equations in (4), we find that Alfv´enic, then it is quite likely that they are generated via q the reverse dynamo mechanism. The characteristic Mach U = H. (6) s + r number A of a given outflow is an index of the relative M Thus, by solving the first equation in (4), we can fully de- efficiency of the RDy and Dy mechanisms, which in turn, is termine H and U. The details of the solution can be found a mirror of the constitution of the ambient state - what mix- in Mahajan et al. (2005). ture of magnetic to kinetic turbulence it is endowed with. Let us now use (7) to probe the range of densities and magnetic fields for which the RDy is likely to be important. Choosing Λ & 10 (which loosely satisfies Λ 1) and differ- 3 DY-RDY IN ASTROPHYSICAL SYSTEMS ≫ ent values of U, we can calculate A, and the corresponding V Equation 6 is the most potent and revealing statement of the values of H and n. The results are presented in Fig. 1. For Unified dynamo - the simultaneous generation of large scale the current non-relativistic treatment, we keep U bounded magnetic fields and flows out of a short-scale reservoir of below 104 km/s. Each of the curves in Fig. 1 corresponds to kinetic and magnetic energy. The dominant dynamo H a fixed value of U, and allows us to probe the ranges of n | |≫ U or the dominant reverse dynamo U H state are and B for which the reverse dynamo may be operational. In | | | | ≫ | | simply the two limiting cases of the same mechanism. the regions lying above and on the curve, RDy will dominate What mix of H and U will finally emerge depends and generate large scale outflows, while in the regions below | | | | upon the coefficients q, s,and r, which are in turn deter- the curves, the RDy will be sub-dominant. mined by a and d. The latter duo are also manifest in the In Table 1, the computed Alfv´en Mach numbers are

c 2014 RAS, MNRAS 000, 1–5 4 M. Lingam and S.M. Mahajan

Table 1. The Alfv´en Mach number for astrophysical outflows

3 System ρg/cm
H(G) U(cm/s) MA Reference GRBs 1016 1015 1010 103 Beskin (2010) Microquasars 1016 1010 1010 108 Beskin (2010) Radio pulsars 1015 1012 1010 106 Beskin (2010) YSOs 10 103 107 105 Beskin (2010) PPNe 1 102 107 105 Huggins (2007); Asensio Ramos et al. (2014)

The table illustrates the values of the Alfv´en Mach number MA for astrophysical systems where jets are observed. In the case of Protoplanetary Nebulae (PPNe), the outflows are modeled as originating from the central star, which possesses specifications akin to the Sun. An upper bound on the large scale magnetic field in PPNe was obtained in Asensio Ramos et al. (2014), and the typical jet velocities in PPNe were taken from Huggins (2007). The rest of the parameters are presented in Table 1 of Beskin (2010).

n the chromosphere where the RDy has been shown to play a 1040 key role (Mahajan et al. 2002). 1032 Lastly, we note that the HMHD is a theory of unified

1024 Dy-RDy mechanism - the general theory is just as valid in scenarios where A . 1. Even in this sub-Alfv´enic regime, 1016 M the flows, along with the dominant magnetic fields, will con- 108 tinue to amplify as long as there is turbulent energy to drive

1 them. Both fields grow at the rate determined by the dis- persion relation, ω2 = r k (Mahajan et al. 2005) with r 10-8 6 9 12H − | | 0.001 1 1000 10 10 10 given by (5). One expects the outflow level to be boosted over time, albeit in a sub-Alfv´enic setting, until it is com- Figure 1. The figure depicts the permitted values of n and H patible with observations. for the reverse dynamo mechanism. For all the curves, Λ = 10 has been chosen. The black, red, green, blue and orange curves correspond to the cases with U = 1 , 10 , 102 , 103 , 104 km/s re- 4 DISCUSSION AND CONCLUSION spectively. Note that this is a log-log plot. In this Letter, we have explored the unified Dynamo-Reverse Dynamo mechanism within an HMHD model. The RDy displayed for a variety of astrophysical systems. The com- (Dy), operates preferentially when the magnetic component putations should be accepted with the following caveats: (the kinetic component) accounts for the bulk of the short scale energy reservoir. The end product of dominant RDy The magnetic fields (H) in the large scale outflows can- • (Dy) is a super Alfv´enic (sub Alfv´enic) outflow, where the not be easily measured, and hence we have chosen to use the Alfv´en Mach number obeys A 1 ( A 1). Conse- magnetic fields present in the ‘engine’ (driving the outflows) M ≫ M ≪ quently, if the observed large scale outflows were to satisfy as a substitute. However, since most of the values of A M A 1 ( A 1), they may indicate the operation of a are very high, it is reasonable to suppose that reducing the M ≫ M ≪ reverse dynamo (dynamo) mechanism. A knowledge of A magnetic field by several orders of magnitude may still plant M as per this model can serve as a probe of the “content” of the system firmly in the regime where the reverse dynamo the underlying ambient reservoir. mechanism determines the outcome. The unified Dy-RDy mechanism is characterized by a The magnetic fields in both the ‘engine’ and the out- • striking reciprocity between the micro and the macro scales: flows are both very complex, and the Alfv´en Mach number the ambient (microscopic) Alfv´en Mach number, denoted by serves only as a simplified criterion for evaluating the va- −1 µA, precisely equals ( A) - the inverse Alfv´en Mach lidity of the reverse dynamo mechanism. In addition, it is M M number of the large scale (macroscopic) outflow. known that the winds in GRBs, microquasars and pulsars We have also found that the reverse dynamo mechanism are relativistic, and a fully consistent treatment must take may operate in a wide variety of astrophysical objects whose such effects into account. We plan to present a relativistic ambient density n and the large scale magnetic fields H lie in version in a forthcoming paper. the ranges delineated in Fig. 1 and Table 1. We do not claim The Alfv´en Mach numbers, displayed in Table 1, are all that the RDy mechanism is the sole source of super-Alfv´enic very high - much greater than unity. The simplest criterion outflows. Instead, we emphasize the importance of the large that such outflows may originate in a RDy mechanism is scale Alfv´en Mach number A as a means of gauging the M amply satisfied. It must be borne in mind that this does not composition of the ambient magnetic and kinetic energies of necessarily imply the operation of the RDy mechanism. Nev- the reservoir. However, our analysis is not exhaustive; some ertheless, this simple criterion can help us probe the nature of the inherent limitations were outlined earlier. of the underlying source that drives large scale outflows and We must stress that the Hall term, which introduces an magnetic fields. An example in contrast is the solar wind, a intrinsic micro-scale, and thereby opens up the possibility of highly sub-Alfv´enic outflow; the RDy is not likely to be the physics at disparate macros and micro scales, is an essential primary driver. However, we note that there are regions in enabler of the unified Dy-RDy theory. This step, however,

c 2014 RAS, MNRAS 000, 1–5 Unified Dynamo mechanism 5 represents only the tip of the iceberg as Hall MHD, itself, Krasnopolsky, R., Li, Z.-Y., & Blandford, R. 1994, ApJ, is the simplest of the extended MHD models. A natural ex- 526, 631 tension of the formalism presented in this paper involves Krasnopolsky, R., Li, Z.-Y., & Shang, H. 2011, ApJ, 733, the construction of an extended MHD and/or two-fluid uni- 54 fied Dy-RDy that can capture the non-ideal MHD effects Lada, C. J. 1985, ARA&A, 23, 267 in a more direct manner. To understand relativistic flows, Li, H., Lapenta, G., Finn, J. M., Li, S., & Colgate, S. A. we must resort to relativistic MHD (Lichnerowicz 1967) or 2006, ApJ, 643, 92 the coupled relativistic magnetofluid model (Mahajan 2003). Li, Z.-Y., Krasnopolsky, R., & Shang, H. 2011, ApJ, 738, For incompressible Hall MHD, we hypothesize that the re- 180 sults are likely to remain similar under v γv in the rel- Lichnerowicz, A. 1967, Relativistic Hydrodynamics and → ativistic regime. Lastly, we note that the assumption of ho- Magnetohydrodynamics (New York: Benjamin) mogeneity can also be relaxed without difficulty, and the en- Livio, M. 1999, PhR, 311, 225 suing model is still easily solvable via computational means. Lovelace, R. V. E. 1976, Nature, 262, 649 The unified Dy-RDy mechanism presented herein Lovelace, R. V. E., Berk, H. L., & Contopoulos, J. 1991, (and/or one of its several suggested extensions) constitutes a ApJ, 379, 696 very viable candidate, as well as a strong indicator, for gen- Lynden-Bell, D. 2003, MNRAS, 341, 1360 erating large scale outflows in a wide range of astrophysical Lyutikov, M., & Blandford, R. 2003, systems. A more detailed picture of the model’s strengths arXiv:astro-ph/0312347 and limitations are likely to emerge only via further numer- Mahajan, S. M. 2003, Phys. Rev. Lett., 90, 035001 ical simulations. Mahajan, S. M., & Yoshida, Z. 1998, Phys. Rev. Lett., 81, 4863 Mahajan, S. M., Nikol’skaya, K. I., Shatashvili, N. L., & ACKNOWLEDGMENTS Yoshida, Z. 2002, ApJL, 576, L161 Mininni, P. D., G´omez, D. O., & Mahajan, S. M. 2002, SMM’s research was supported by the U.S. Dept. of En- ApJL, 567, L81 ergy Grant DE-FG02-04ER-54742. ML’s research was sup- Mininni, P. D., G´omez, D. O., & Mahajan, S. M. 2003a, ported by the U.S. Dept. of Energy Contract DE-FG05- ApJ, 584, 1120 80ET-53088. Mininni, P. D., G´omez, D. O., & Mahajan, S. M. 2003b, ApJ, 587, 472 Mahajan, S. M., Shatashvili, N. L., Mikeladze, S. V., & REFERENCES Sigua, K. I. 2005, ApJ, 634, 419 Masson, C. R., & Chernin, L. M. 1993, ApJ, 414, 230 Asensio Ramos, A., Mart´ınez Gonz´alez, M. J., Manso Mirabel, I. F., & Rodr´ıguez, L. F. 1999, ARA&A, 37, 409 Sainz, R., Corradi R. L. M., & Leone, F. 2014, ApJ, 787, Ogilvie, G. I., & Livio, M. 2001, ApJ, 553, 158 111 Shu, F., Najita, J., Ostriker, E., Wilkin, F., Ruden, S. & Balbus, S. A., & Terquem, C. 2001, ApJ, 552, 235 Lizano, S. 1994, ApJ, 429, 781 Beskin, V. S. 2010, PhyU, 53, 1199 Vishniac, T. E., & Cho, J. 2001, ApJ, 550, 752 Bhattacharjee, A. & Hameiri, E. 1986, Phys. Rev. Lett., 56, 206 Bisnovatyi-Kogan, G. S., & Ruzmaikin, A. A. 1976, Ap&SS, 42, 401 Bisnovatyi-Kogan, G. S., & Lovelace, R. V. E. 2001, NewA Rev., 45, 663 Blackman, E. G., & Field, G. B. 2002, Phys. Rev. Lett., 89, 265007 Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433 Blandford, R. D., & K¨onigl, A. 1979, ApJ, 232, 34 Blandford, R. D., & Payne, D. G. 1982, MNRAS, 199, 883 Brandenburg, A. 2005, ApJ, 625, 539 Brandenburg, A., Nordlund, A., Stein, R. F., & Torkelsson, U. 2005, ApJ, 446, 741 Brandenburg, A., & Subramanian, K. 2005, PhR, 417, 1 Bridle, A. H., & Perley, R. A. 1984, ARA&A, 22, 319 Colgate, S. A., Fowler, K. T., Li, H., & Pino, J. 2014, ApJ, 789, 144 Cumming, A., Arras, P., & Zweibel, E. 2004, ApJ, 609, 999 de Gouveia Dal Pino, E. M. 2005, Adv. Space Res., 35, 908 Ebrahimi, F., & Bhattacharjee, A. 2014, Phys. Rev. Lett., 112, 125003 Ferrari, A. 1998, ARA&A, 36, 539 Goldreich, P., & Reisenegger, A. 1992, ApJ, 395, 250 Huggins, P. J. 2007, ApJ, 663, 342

c 2014 RAS, MNRAS 000, 1–5