We study the effects of turbulence on magnetic reconnection using 3D numerical simulations. This is the first attempt to test a model of fast magnetic reconnection in the presence of weak turbulence proposed by Lazarian & Vishniac (1999). This model predicts that weak turbulence, generically present in most of astrophysical systems, enhances the rate of reconnection by reducing the transverse scale for reconnection events and by allowing many independent flux reconnection events to occur simultaneously. As a result the reconnection speed becomes independent of Ohmic resistivity and is determined by the magnetic field wandering induced by turbulence. To quantify the reconnection speed we use both an intuitive definition, i.e. the speed of the reconnected flux inflow, as well as a more sophisticated definition based on a formally derived analytical expression. Our results confirm the predictions of the Lazarian & Vishniac model. In particular, we find that Vrec Pinj^(1/2), as predicted by the model. The dependence on the injection scale for some of our models is a bit weaker than expected, i.e. l^(3/4), compared to the predicted linear dependence on the injection scale, which may require some refinement of the model or may be due to the effects like finite size of the excitation region. The reconnection speed was found to depend on the expected rate of magnetic field wandering and not on the magnitude of the guide field. In our models, we see no dependence on the guide field when its strength is comparable to the reconnected component. More importantly, while in the absence of turbulence we successfully reproduce the Sweet-Parker scaling of reconnection, in the presence of turbulence we do not observe any dependence on Ohmic resistivity, confirming that our reconnection is fast.
Deep Dive into Numerical Tests of Fast Reconnection in Weakly Stochastic Magnetic Fields.
We study the effects of turbulence on magnetic reconnection using 3D numerical simulations. This is the first attempt to test a model of fast magnetic reconnection in the presence of weak turbulence proposed by Lazarian & Vishniac (1999). This model predicts that weak turbulence, generically present in most of astrophysical systems, enhances the rate of reconnection by reducing the transverse scale for reconnection events and by allowing many independent flux reconnection events to occur simultaneously. As a result the reconnection speed becomes independent of Ohmic resistivity and is determined by the magnetic field wandering induced by turbulence. To quantify the reconnection speed we use both an intuitive definition, i.e. the speed of the reconnected flux inflow, as well as a more sophisticated definition based on a formally derived analytical expression. Our results confirm the predictions of the Lazarian & Vishniac model. In particular, we find that Vrec Pinj^(1/2), as predicted b
Magnetic fields play a key role in astrophysical processes such as star formation, the transport and acceleration of cosmic rays, accretion disks, solar phenomena, etc. (Crutcher 1999;Beck 2002;Schlickeiser 2004;Elmegreen & Scalo 2004). Typically magnetic diffusion is very slow on astrophysical scales, so to a good approximation we can treat magnetic fields as being purely advected with the flow, which is frequently referred to in the literature as the "frozen in" condition for the plasma (see Moffat 1978).
Do we expect the frozen in condition to be violated in typical astrophysical conditions? The answer to this question is a qualified yes. Indeed, most astrophysical flows are chaotic; when adjacent parcels of fluids do not move in the same direction, the frozen-in magnetic fields become tangled. This can be easily visualized by viewing magnetic fields lines as threads moving with the fluid. When the distance between magnetic bundles of different direction becomes small, the finite resistivity of fluids starts to be important. In a generic situation of 3D flows, the bundles of magnetic fields come into contact with their neighbors at an angle of the order unity. Over the small scales at which fluid resistivity is important, magnetic field lines change their topology, or reconnect. However, once magnetic field energies become large, bending magnetic fields on small scales requires energies much larger than the turbulent energies on that scale. The magnetic field lines become stiff, even in the presence of strong turbulence, and the scales associated with contact between regions with very different magnetic fields become large.
What is the resulting reconnection speed? This is a vital question for many areas of astrophysics (see Biskamp 2000;Priest & Forbes 2000;Bhattacharjee 2004;Zweibel & Yamada 2009). A first guess could be that magnetic reconnection is generically slow in astrophysical circumstances. There is a large disparity in the scales involved. The scale of the magnetic flux bundle is astronomically large. The microphysical scale over which the Ohmic dissipation is important is relatively small. In this case the crossing magnetic bundles will create ubiquitous unresolved intersections or “knots” in the fluid. Magnetic tension that arises from those intersections as the magnetic bundles press against each other should dramatically change the properties of magnetized fluids1 . This would be devastating news for most numerical MHD simulations, as numerical diffusion in the simulations is high and magnetic bundles in simulations easily reconnect.
To understand the difference between astrophysical reconnection and the one in numerical simulations, one should recall that the dimensionless combination that controls the reconnection rate is the Lundquist number2 , defined as S = LV A /η, where L is the length of the reconnection layer (see Figure 1, upper panel), V A is the Alfvén velocity, and η is Ohmic diffusivity. Because of huge astrophysical sizes L involved, the astrophysical Lundquist numbers are huge, e.g. for the ISM they are about 10 16 , while present-day MHD simulations correspond to S < 10 4 . As the numerical efforts scale as L 4
x , where L x is the size of the box, it is not feasible at present and will not be feasible in the foreseeable future to have simulations with the sufficiently high Lundquist numbers. Incidentally, this also presents a problem for numerical simulations of magnetic reconnection unless one has theoretically-derived scaling relations to test. Even with the limited resolution, numerical simulations are a good tool to study scaling relations, the point that has been proved by successful numerical studies of MHD turbulence.
Due to huge values of astrophysical Lundquist numbers, any rate of reconnection that depends on S is extremely slow3 . Fast reconnection is reconnection that does not depend on resistivity.
What are the conditions that can make magnetic reconnection fast? Does it rely on special initial or boundary conditions for the flow, or does it require particular plasma effects? These are burning astrophysical questions, which, for example, define the extent that we can rely on numerical simulations of magnetized fluids as models of astrophysical phenomena. To understand processes of magnetic field generation associated with dynamos, the dynamics of the interstellar medium and accretion disks, or other related phenomena, it is important to understand magnetic reconnection. In most cases, astrophysical reconnection is difficult to observe, with the notable exception of solar flares (see Sturrock 1966;Masuda et al. 1994) and gamma ray bursts (see Galama et al. 1998). This sometimes creates an illusion that the importance of reconnection is limited to those selected phenomena.
A famous model of magnetic reconnection was suggested by Parker (1957) and Sweet (1958). Unfortunately, this model, usually referred to as Sweet-Parker reconnection, provides very slow
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