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Modeling Astrophysical Outflows Via the Unified Dynamo-Reverse

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Modeling Astrophysical Outflows Via the Unified Dynamo-Reverse Mon. Not. R. Astron. Soc. 000, 1–5 (2014) Printed 12 June 2021 (MN LATEX style file v2.2) Modeling astrophysical outflows via the unified Dynamo-Reverse Dynamo mechanism Manasvi Lingam1⋆ and Swadesh M. Mahajan1† 1Institute for Fusion Studies, The University of Texas, Austin, TX 78712, USA 12 June 2021 ABSTRACT The unified Dynamo-Reverse Dynamo (Dy-RDy) mechanism, capable of simultane- ously generating large scale outflows and magnetic fields from an ambient microscopic reservoir, is explored in a broad astrophysical context. The Dy-RDy mechanism is derived via Hall magnetohydrodynamics, which unifies the evolution of magnetic field and fluid vorticity. It also introduces an intrinsic length scale, the ion skin depth, al- lowing for the proper normalization and categorization of microscopic and macroscopic scales. The large scale Alfv´en Mach number MA, defining the relative “abundance” of the flow field to the magnetic field is shown to be tied to a microscopic scale length that reflects the characteristics of the ambient short scale reservoir. The dy- namo (Dy), preferentially producing the large scale magnetic field, is the dominant mode when the ambient turbulence is mostly kinetic, while the outflow producing reverse dynamo (RDy) is the principal manifestation of a magnetically dominated turbulent reservoir. It is conjectured that an efficient RDy may be the source of many observed astrophysical outflows that have MA ≫ 1. Key words: (magnetohydrodynamics) MHD – plasmas – methods: analytical – ac- celeration of particles – stars: winds, outflows – galaxies: jets 1 INTRODUCTION vastly different parameters that these outflows ex- hibit, the quest for a broad underlying mechanism Astrophysical jets are universal: they appear in myriad that can power these outflows has proven elusive forms, exhibiting significant variations in luminosity, colli- although several seminal works in the 1970s and mation and velocity. Relativistic jets have been observed early 1980s (Bisnovatyi-Kogan & Ruzmaikin 1976; Lovelace in the context of Active Galactic Nuclei (AGNs), micro- 1976; Blandford & Znajek 1977; Blandford & K¨onigl 1979; quasars and Gamma Ray Bursts (GRBs). Smaller-scale non- arXiv:1501.06509v1 [astro-ph.HE] 26 Jan 2015 Blandford & Payne 1982) did highlight the importance of relativistic versions have been observed in a much wider magnetic fields in the generation of these outflows. Sub- array of systems such as low-mass Young Stellar Objects sequent works introduced the magneto-centrifugal mecha- (YSOs), massive X-ray binary systems and Protoplanetary nism (Shu et al. 1999; Krasnopolsky, Li & Blandford 1999), Nebulae (PPNe). Although the word “jets” suggests a highly magnetic towers (Lynden-Bell 2003) and other mechanisms collimated and narrow outflow, diffuse emanations, often (Lovelace, Berk & Contopoulos 1991; Ogilvie & Livio 2001; termed “outflows”, have also been detected in stars in their Li et al. 2006; Colgate et al. 2014), all mediated by the mag- final stages of evolution. In addition, our Sun exhibits jets netic field, in their attempts to explain the origin of these and flares, albeit ones that are much smaller than the one outflows. mentioned above. In this Letter, we shall refer to jets and their (relatively) diffuse cousins as “outflows”. A comprehen- Most of the preceding analyses used the ideal magneto- sive discussion of outflows in astrophysical systems is found hydrodynamic (MHD) model, the simplest and highly ex- in Bridle & Perley (1984); Lada (1985); Masson & Chernin plored plasma fluid model. However, in the last decade, (1993); Ferrari (1998); Livio (1999); Mirabel & Rodr´ıguez the importance of the Hall effect in astrophysical sys- (1999); Lyutikov & Blandford (2003); de Gouveia Dal Pino tems has been recognized; the Hall effect has been shown (2005); Beskin (2010). to play a crucial role in several scenarios, such as mag- Given the diversity of their occurrence and the netic braking and star formation (Krasnopolsky, Li & Shang 2011; Li, Krasnopolsky & Shang 2011), protostellar discs (Balbus & Terquem 2001) and magnetic field evolution ⋆ E-mail: [email protected] in neutron star crusts (Goldreich & Reisenegger 1992; † E-mail: [email protected] Cumming, Arras & Zweibel 2004). This recognition comes c 2014 RAS 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: Rion 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).
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