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Reconnection Diffusion in Turbulent Fluids and Its Implications for Star Formation

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Abstract

Astrophysical fluids are turbulent a fact which changes the dynamics of many key processes, including magnetic reconnection. Fast reconnection of magnetic field in turbulent fluids allows the field to change its topology and connections. As a result, the traditional concept of magnetic fields being frozen into the plasma is no longer applicable. Plasma associated with a given magnetic field line at one instant is distributed along a different set of magnetic field lines at the next instant. This diffusion of plasmas and magnetic field is enabled by reconnection and therefore is termed “reconnection diffusion”. The astrophysical implications of this concept include heat transfer in plasmas, advection of heavy elements in interstellar medium, magnetic field generation etc. However, the most dramatic implications of the concept are related to the star formation process. The reason is that magnetic fields are dynamically important for most of the stages of star formation. The existing theory of star formation has been developed ignoring the possibility of reconnection diffusion. Instead, it appeals to the decoupling of mass and magnetic field arising from neutrals drifting in respect to ions entrained on magnetic field lines, i.e. through the process that is termed “ambipolar diffusion”. The predictions of ambipolar diffusion and reconnection diffusion are very different. For instance, if the ionization of media is high, ambipolar diffusion predicts that the coupling of mass and magnetic field is nearly perfect. At the same time, reconnection diffusion is independent of the ionization but depends on the scale of the turbulent eddies and on the turbulent velocities. In the paper we explain the physics of reconnection diffusion both from macroscopic and microscopic points of view, i.e. appealing to the reconnection of flux tubes and to the diffusion of magnetic field lines. We make use of the Lazarian and Vishniac (Astrophys. J. 517:700, 1999) theory of magnetic reconnection and show that this theory is applicable to the partially ionized gas. We quantify the reconnection diffusion rate both for weak and strong MHD turbulence and address the problem of reconnection diffusion acting together with ambipolar diffusion. In addition, we provide a criterion for correctly representing the magnetic diffusivity in simulations of star formation. We discuss the intimate relation between the processes of reconnection diffusion, field wandering and turbulent mixing of a magnetized media and show that the role of the plasma effects is limited to “breaking up lines” on small scales and does not affect the rate of reconnection diffusion. We address the existing observational results and demonstrate how reconnection diffusion can explain the puzzles presented by observations, in particular, the observed higher magnetization of cloud cores in comparison with the magnetization of envelopes. We also outline a possible set of observational tests of the reconnection diffusion concept and discuss how the application of the new concept changes our understanding of star formation and its numerical modeling. Finally, we outline the differences of the process of reconnection diffusion and the process of accumulation of matter along magnetic field lines that is frequently invoked to explain the results of numerical simulations.

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Notes

  1. This makes the problem of the initial or seed magnetic field, that for a long time has worried researchers, rather trivial. Very weak magnetic fields, e.g. generated by Bierman battery (see Lazarian 1992) can be amplified fast in a turbulent plasmas.

  2. In supersonic flows compressibility effects induce deviations from the equipartition.

  3. Reynolds number \(\mathit{Re}\equiv L_{f}V/\nu=(V/L_{f})/(\nu/L^{2}_{f})\) which is the ratio of an eddy turnover rate \(\tau^{-1}_{eddy}=V/L_{f}\) and the viscous dissipation rate \(\tau_{dis}^{-1}=\eta/L^{2}_{f}\). Therefore large Re correspond to negligible viscous dissipation of large eddies over the cascading time τ casc which is equal to τ eddy in Kolmogorov turbulence.

  4. The description in terms of interacting wavepackets or modes is also possible with the corresponding wavevectors tending to get more and more perpendicular to the magnetic field as the cascade develops.

  5. For compressible MHD turbulence simulations in Beresnyak et al. (2005) and Kowal et al. (2007) demonstrated that the density spectrum becomes more shallow and isotropic as the Mach number increases.

  6. Recent work by Beresnyak and Lazarian (2009a, 2010) shows that present day numerical simulations are unable to reveal the actual inertial range of MHD turbulence making the discussions of the discrepancies of the numerically measured spectrum and the GS95 predictions rather premature. In addition, new higher resolution simulations by Beresnyak (2011) reveal the predicted −5/3 spectral slope.

  7. We would like to stress that we do not deal with the properties of MHD turbulence with imbalance, i.e. when the energy flow from one direction exceeds the flow from the opposite direction. Such turbulence has different properties. At the moment there exist several theories of imbalanced turbulence (see Lithwick and Goldreich 2001, Beresnyak and Lazarian 2008; Chandran 2008; Perez and Boldyrev 2009). Among those theories only Beresnyak and Lazarian (2008) seem to agree with the numerical testing (see Beresnyak and Lazarian 2010). In compressible interstellar medium back scattering of Alfven waves is expected to decrease the imbalance of turbulence making GS95 a good approximation.

  8. Formal mathematical arguments on how and why the frozen-in condition fails may be found in Eyink (2011) (see also Eyink 2007, 2009).

  9. Figure 2 presents a cross section of the 3D reconnection layer. A shared component of magnetic field is present in the generic 3D configurations of reconnecting magnetic flux tubes.

  10. As discussed in LV99 and in more details in ELV11 the magnetic field wandering, turbulence and magnetic reconnection are very tightly related concepts. Without magnetic reconnection, properties of magnetic turbulence and magnetic field wandering would be very different. For instance, in the absence of fast reconnection, the formation of magnetic knots arising if magnetic fields were not able to reconnect would destroy the self-similar cascade of Alfvenic turbulence. The rates predicted by LV99 are the rates required to make Goldreich-Sridhar model of turbulence self-consistent.

  11. The Richarson diffusion was reported in case of magnetized fluids in Eyink et al. (2013).

  12. The model in LV99 is three dimensional, and it is not clear to what extend it can be applied to 2D turbulence (see discussion in ELV11 and references therein). However, the cases of pure 2D reconnection and 2D turbulence are of little practical importance.

  13. It is shown in LV99 that if the reconnection is calculated assuming that the bottle neck is due to Ohmic resistivity, the reconnection rate gets much larger than the Alfven speed. This is due to the effect of 3D turbulent magnetic fluxes getting in contact over many independent patches. The total cumulative rate of reconnection “cutting magnetic field lines” becomes very large and it is the outflow of magnetized plasma from the reconnection region that enters to limit the overall reconnection speed. Naturally, increasing the local reconnection speed of magnetic patches either due to plasma effects or, alternatively, numerical effects in computer simulations does not increase the reconnection speed of the large scale turbulent magnetic fluxes.

  14. We expect the effect of field wandering to play the crucial role for the reconnection with non-zero cross-helicity. This wandering should be determined using the theory of strong imbalanced MHD turbulence, for instance, by one in Beresnyak and Lazarian (2008) if higher resolution and longer averaging future testings confirm it. We also expect that similarly as a successful model of imbalanced MHD turbulence must produce GS95 scaling for the case of zero cross helicity, the reconnection in flows with imbalanced turbulence should converge to LV99 predictions for the zero cross helicity. For very high cross helicity field wandering may be small and other, e.g. plasma effects, become important.

  15. The latter was a sort of Holy Grail for many researchers studying reconnection.

  16. In the case of a dynamically unimportant field the magnetic dissipation and reconnection happens on the scales of the Ohmic diffusion scale and the effects of magnetic field on the turbulent cascade are negligible. However, turbulent motions transfer an appreciable portion of the cascading energy into magnetic energy (see Cho et al. 2009; also Sect. 3.1). As a result, the state of intensive turbulence with negligible magnetic field is short-lived.

  17. If magnetic reconnection is slow then, as was claimed by Don Cox (private communication), the interstellar medium should behave not like a fluid, but more like felt or Jello.

  18. One should be careful with the application of Eq. (26), however. When the dynamics of magnetic fields is studied at scales less than the injection scale, the magnetic fields obey the Richardson diffusion law and superdiffusion law δx∼t 3/2 is applicable and the diffusion gets faster with the increase of scale. The use of Eq. (26) assumes that the cloud is not changed by turbulence and therefore we always study the diffusion at the same scale.

  19. It is important to stress that fast reconnection does not change the helicity of the magnetic flux and therefore it does not solve the problem of helicity in astrophysical dynamos (see Vishniac and Cho 2001). Therefore the simulations where researchers use enhanced many orders of magnitude resistivity in an attempt to mimic effects of turbulence smoothing on magnetic field are in error. However, the transport of magnetic flux and smoothing that does not change helicity are important ingredients of any mean field dynamo that reconnection diffusion takes care of.

  20. There is also expansion of the spot arising from the Lyapunov deviation of the flow lines as we discuss in Sect. 6.1.

  21. The difference of the reconnection diffusion and ambipolar diffusion is that the former is associated with turbulent motions of matter. Therefore, too intensive turbulence may upset the virial balance and instead of magnetic field diffusion, a dispersion of the entire cloud can take place.

  22. Superdiffusion in terms of the accelerated divergence of magnetic field lines changes a lot of physics for the propagation and acceleration of cosmic rays (Lazarian and Yan 2013).

  23. Richardson diffusion presents an example of superdiffusion, i.e. diffusion process for which l 2∼t β, β>1. The important consequences of Richardson diffusion have been studied for heat transfer and cosmic ray propagation (see Lazarian 2006; Yan and Lazarian 2008).

  24. The Kolmogorov velocity field is Holder continuous, i.e. |v(r 1)−v(r 2)|≤C|r 1−r 2|1/3.

  25. This is not only similarity in terms of the spectrum. Cho et al. (2003) showed that in the local system of reference the intermittency of turbulence is also similar to the hydrodynamic one.

  26. This regime induces perpendicular “superdiffusion” in terms of cosmic rays and other charged particles streaming along magnetic field lines.

  27. Incidentally, in terms of reconnection, Eq. (36) expresses the thickness of outflow denoted as Δ in Fig. 2. Substituting this value in Eq. (2) one recovers the LV99 reconnection rate, i.e. Eq. (9).

  28. One can also claim that spontaneous stochasticity of magnetic field in turbulent fluids is an underlying process that governs magnetic reconnection (ELV11). As we mentioned earlier, fast reconnection makes MHD turbulence theory self-consistent.

  29. In fact, Lazarian et al. (2004) show that the turbulence in partially ionized gas demonstrates several regimes, including the intermittent “resurrection” of turbulence cascade at scales less that the ambipolar damping scale. These effects make magnetic fields stochastic on scales much less that the naively estimated l min,∥ that enters Eq. (34).

  30. To see the effect the authors had to adopt the Hall term much larger than its value for the adopted parameters of the media.

  31. Incidentally, this can explain the formation of density fluctuations on scales of thousands of Astronomical Units, that are observed in the ISM.

  32. The magnetic Reynolds number, which is the ratio of the magnetic field decay time to the eddy turnover time, is defined using the injection velocity v l as a characteristic speed instead of the Alfvén speed V A , which is taken in the Lundquist number.

  33. In comparison the models with enhanced resistivity operate with the resistivities that are not motivated by the known physics.

  34. For highly supersonic turbulence shock formation may somewhat alter the picture and the correlation of magnetic field and density is being observed in supersonic simulations (Burkhart et al. 2009). However, the bulk of the turbulence in warm diffuse media for which the observations are applicable is subsonic (see Burkhart et al. 2011).

  35. The issue of the metalicity dependence is also discussed in Meurer et al. (2009), where the dependence of the Initial Mass function on the galactic surface brightness was detected. This can be the effect of shielding of gas (see Krumholz et al. 2008) rather than the change in the ambipolar diffusion rates.

  36. The predicted spectrum without taking the backreaction of the accelerated particles is N(E)dE∼E −5/2 dE. Considerations in Drake et al. (2006) suggest that the spectrum of the particles can get shallower if the backreaction is taken into account.

  37. Modeling in Kowal et al. (2011b) showed that the acceleration in 2D and 3D proceed at a different rate, which questions the applicability of results obtained in 2D simulations (see Drake et al. 2010) to the actual astrophysical systems.

  38. The idea that cosmic ray can diffuse perpendicular to magnetic field lines due to magnetic field wandering is not new. However, the fact that this wandering is intrinsically connected with fast reconnection induced by turbulence has not been realized until quite recently (see LV99, Eyink et al. 2011).

  39. A similar process takes place in the case of molecular diffusivity in turbulent hydrodynamic flows. The result for the latter flows is well known: in turbulent regime molecular diffusivity is irrelevant for the turbulent transport. Indeed, in the case of high microscopic diffusivity, the turbulence provides mixing down to a scale l 1 at which the microscopic diffusivity both, suppresses the cascade and ensures efficient diffusivity of the contaminant. In the case of low microscopic diffusivity, turbulent mixing happens down to a scale l 2≪l 1, which ensures that even low microscopic diffusivity is sufficient to provide efficient diffusion. In both cases the total effective diffusivity of the contaminant is given by the product of the turbulent injection scale and the turbulent velocity.

  40. A possible point of confusion is related to the difference of the physical scales involved. If one associates the scale of the reconnection with the thickness of the Sweet–Parker layer, then, indeed, the ambipolar diffusion scale is much larger and therefore the reconnection scale gets irrelevant. However, within the LV99 model of reconnection, the scale of reconnection is associated with the scale of magnetic field wandering. The corresponding scale depends on the turbulent velocity and is not small.

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Acknowledgements

I acknowledge grantthe NSF grain ATM 1212096, Vilas Associate Award as well as the support 1098 from the NSF Center for Magnetic Self-Organization. Stimulating environment provided by Humboldt Award at the Universities of Cologne and Bochum, as well as a Fellowship at the International Institute of Physics (Brazil) is acknowledged. A productive exchange on star formation with Bruce Elmegreen was particularly valuable. I am grateful to Elisabeth Gouveia dal Pino, Reinaldo Santos-Lima, Dick Crutcher, Chris Mckee, Greg Eyink, Ethan Vishniac for stimulating discussions on various aspects of the problem. Exchanges with Julian Krolik on the reconnection diffusion around black holes and with Ellen Zweibel on the role of ambipolar diffusion are acknowledged. We thank the anonymous referee for useful input and Blakesley Burkhart for reading the manuscript.

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    A. Lazarian

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Note by the Editor: This paper was meant to be part of the topical volume on Microphysics of Cosmic Plasmas (Volume 178, Numbers 2–4).

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Lazarian, A. Reconnection Diffusion in Turbulent Fluids and Its Implications for Star Formation. Space Sci Rev 181, 1–59 (2014). https://doi.org/10.1007/s11214-013-0031-5

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