Editing Magnetic reconnection (section)

==Theoretical descriptions==

===Slow reconnection: Sweet–Parker model<!--'Sweet–Parker model' and 'Sweet-Parker model' redirect here-->===
The first theoretical framework of magnetic reconnection was established by [[Peter Alan Sweet|Peter Sweet]] and [[Eugene Parker]] at a conference in 1956. Sweet pointed out that by pushing two plasmas with oppositely directed magnetic fields together, resistive diffusion is able to occur on a length scale much shorter than a typical equilibrium length scale.<ref>Sweet, P. A., The Neutral Point Theory of Solar Flares, in IAU Symposium 6, Electromagnetic Phenomena in Cosmical Physics, ed. B. Lehnert (Dordrecht: Kluwer), 123, 1958</ref> Parker was in attendance at this conference and developed scaling relations for this model during his return travel.<ref>{{Cite journal|last=Parker|first=E. N.|date=December 1957|title=Sweet's mechanism for merging magnetic fields in conducting fluids|journal=Journal of Geophysical Research|volume=62|issue=4|pages=509–520|doi=10.1029/JZ062i004p00509|bibcode=1957JGR....62..509P}}</ref>

The '''Sweet–Parker model'''<!--boldface per WP:R#PLA--> describes time-independent magnetic reconnection in the resistive MHD framework when the reconnecting magnetic fields are antiparallel (oppositely directed) and effects related to viscosity and compressibility are unimportant.  The initial velocity is simply an <math>E\times B</math> velocity, so
<math display="block">E_y = v_\text{in} B_\text{in}</math>

where <math>E_y</math> is the out-of-plane electric field, <math>v_\text{in}</math> is the characteristic inflow velocity, and <math>B_\text{in}</math> is the characteristic upstream magnetic field strength.  By neglecting displacement current, the low-frequency Ampere's law, <math>\mathbf{J} = \frac{1}{\mu_0}\nabla\times\mathbf{B}</math>, gives the relation
<math display="block">J_y \sim \frac{B_\text{in}}{\mu_0\delta},</math>

where <math>\delta</math> is the current sheet half-thickness.  This relation uses that the magnetic field reverses over a distance of <math>\sim2\delta</math>.  By matching the ideal electric field outside of the layer with the resistive electric field <math>\mathbf{E} = \frac{1}{\sigma}\mathbf{J}</math> inside the layer (using [[Ohm's law]]), we find that
<math display="block">v_\text{in} = \frac{E_y}{B_\text{in}} \sim \frac{1}{\mu_0\sigma\delta} = \frac{\eta}{\delta},</math>

where <math>\eta</math> is the [[magnetic diffusivity]].  When the inflow density is comparable to the outflow density, conservation of mass yields the relationship
<math display="block">v_\text{in}L \sim v_\text{out}\delta, </math>

where <math>L</math> is the half-length of the current sheet and <math>v_\text{out}</math> is the outflow velocity.  The left and right hand sides of the above relation represent the mass flux into the layer and out of the layer, respectively.  Equating the upstream magnetic pressure with the downstream [[dynamic pressure]] gives
<math display="block">\frac{B_\text{in}^2}{2\mu_0} \sim \frac{\rho v_\text{out}^2}{2}</math>

where <math>\rho</math> is the mass density of the plasma.  Solving for the outflow velocity then gives
<math display="block">v_\text{out} \sim \frac{B_\text{in}}{\sqrt{\mu_0\rho}} \equiv v_A</math>

where <math>v_A</math> is the [[Alfvén wave|Alfvén velocity]].  With the above relations, the dimensionless reconnection rate <math>R</math> can then be written in two forms, the first in terms of <math>(\eta, \delta, v_A)</math> using the result earlier derived from Ohm's law, the second in terms of <math>(\delta, L)</math> from the conservation of mass as
<math display="block">R = \frac{v_\text{in}}{v_\text{out}} \sim \frac{\eta}{v_A\delta} \sim \frac{\delta}{L}.</math>

Since the dimensionless [[Lundquist number]] <math>S</math> is given by
<math display="block">S \equiv \frac{Lv_A}{\eta},</math>

the two different expressions of <math>R</math> are multiplied by each other and then square-rooted, giving a simple relation between the reconnection rate <math>R</math> and the Lundquist number <math>S</math>
<math display="block">R ~ \sim \sqrt{\frac{\eta}{v_A L}} = \frac{1}{S^\frac{1}{2}}.</math>

Sweet–Parker reconnection allows for reconnection rates much faster than global diffusion, but is not able to explain the fast reconnection rates observed in solar flares, the Earth's magnetosphere, and laboratory plasmas.  Additionally, Sweet–Parker reconnection neglects three-dimensional effects, collisionless physics, time-dependent effects, viscosity, compressibility, and downstream pressure.  Numerical simulations of two-dimensional magnetic reconnection typically show agreement with this model.<ref>{{Cite journal|last=Biskamp|first=D.|date=1986|title=Magnetic reconnection via current sheets | journal=Physics of Fluids| language= en| volume= 29| issue= 5| pages = 1520–1531| doi= 10.1063/1.865670| issn = 0031-9171 | bibcode=1986PhFl...29.1520B}}</ref>  Results from the Magnetic Reconnection Experiment (MRX) of collisional reconnection show agreement with a generalized Sweet–Parker model which incorporates compressibility, downstream pressure and anomalous resistivity.<ref>{{Cite journal|last1=Ji|first1=Hantao| last2=Yamada|first2=Masaaki| last3=Hsu|first3=Scott| last4=Kulsrud|first4=Russell|last5=Carter|first5=Troy|last6=Zaharia|first6=Sorin|date=26 April 1999|title=Magnetic reconnection with Sweet-Parker characteristics in two-dimensional laboratory plasmas|journal=Physics of Plasmas| language=en| volume=6|issue=5|pages=1743–1750|doi=10.1063/1.873432|issn=1070-664X|bibcode=1999PhPl....6.1743J|url=https://digital.library.unt.edu/ark:/67531/metadc681346/}}</ref><ref>{{Cite journal| last1=Ji|first1=Hantao| last2=Yamada|first2=Masaaki| last3=Hsu|first3=Scott|last4=Kulsrud|first4=Russell|date=1998|title=Experimental Test of the Sweet-Parker Model of Magnetic Reconnection|journal=Physical Review Letters| volume=80| issue=15| pages=3256–3259| doi=10.1103/PhysRevLett.80.3256| bibcode=1998PhRvL..80.3256J|url=https://digital.library.unt.edu/ark:/67531/metadc675424/}}</ref>

===Fast reconnection: Petschek model===

The fundamental reason that Petschek reconnection is faster than Parker-Sweet is that it broadens the outflow region and thereby removes some of the limitation caused by the build up in plasma pressure. The inflow velocity, and thus the reconnection rate, can only be very small if the outflow region is narrow.  In 1964, Harry Petschek proposed a mechanism where the inflow and outflow regions are separated by stationary slow mode shocks that stand in the inflows.<ref>Petschek, H. E., Magnetic Field Annihilation, in The Physics of Solar Flares, Proceedings of the AAS-NASA Symposium held 28–30 October 1963 at the Goddard Space Flight Center, Greenbelt, MD, p. 425, 1964</ref>  The aspect ratio of the diffusion region is then of order unity and the maximum reconnection rate becomes
<math display="block">\frac{v_\text{in}}{v_A} \approx \frac{\pi}{8 \ln S}.</math>

This expression allows for fast reconnection and is almost independent of the Lundquist number. Theory and numerical simulations show that most of the actions of the shocks that were proposed by Petschek can be carried out by [[Alfvén waves]] and in particular rotational discontinuities (RDs). In cases of asymmetric plasma densities on the two sides of the current sheet (as at Earth's dayside magnetopause) the Alfvén wave that propagates into the inflow on higher-density side (in the case of the magnetopause the denser magnetosheath) has a lower propagation speed and so the field rotation increasingly becomes at that RD as the field line propagates away from the reconnection site: hence the magnetopause current sheet becomes increasingly concentrated in the outer, slower, RD. 

Simulations of resistive MHD reconnection with uniform resistivity showed the development of elongated current sheets in agreement with the Sweet–Parker model rather than the Petschek model.  When a localized anomalously large resistivity is used, however, Petschek reconnection can be realized in resistive MHD simulations.  Because the use of an anomalous resistivity is only appropriate when the particle mean free path is large compared to the reconnection layer, it is likely that other collisionless effects become important before Petschek reconnection can be realized.

===Anomalous resistivity and Bohm diffusion===
{{see also|Spitzer resistivity#Disagreements with observation}}

In the Sweet–Parker model, the common assumption is that the [[magnetic diffusivity]] is constant. This can be estimated using the equation of motion for an electron with mass <math>m</math> and electric charge <math>e</math>:

<math display="block">{d{\mathbf{v}} \over dt} = {e \over m}\mathbf{E} - \nu\mathbf{v},</math>

where <math>\nu</math> is the collision frequency. Since in the steady state, <math>d{\mathbf{v}}/dt = 0</math>, then the above equation along with the definition of electric current, <math>{\mathbf{J}} = en{\mathbf{v}}</math>, where <math>n </math> is the electron number density, yields

<math display="block">\eta = \nu{c^2 \over \omega_{pi}^2}.</math>

Nevertheless, if the drift velocity of electrons exceeds the thermal velocity of plasma, a steady state cannot be achieved and magnetic diffusivity should be much larger than what is given in the above. This is called anomalous resistivity, <math>\eta_\text{anom}</math>, which can enhance the reconnection rate in the Sweet–Parker model by a factor of <math>\eta_\text{anom}/\eta</math>.

Another proposed mechanism is known as the Bohm diffusion across the magnetic field. This replaces the Ohmic resistivity with <math>v_A^2 (mc/eB)</math>, however, its effect, similar to the anomalous resistivity, is still too small compared with the observations.<ref>{{cite book|last1=Parker|first1=E. G.|title=Cosmical Magnetic Fields|date=1979|publisher=[[Oxford University Press]]|location=Oxford}}</ref>

===Stochastic reconnection===

In stochastic reconnection,<ref>{{cite journal|author1-link=Alexandre Lazarian|last1=Lazarian|first1=Alex|last2=Vishniac|first2=Ethan|title=Reconnection in a Weakly Stochastic Field|journal=The Astrophysical Journal|date=1999|volume=517|issue=2|pages=700–718|doi=10.1086/307233|arxiv = astro-ph/9811037 |bibcode = 1999ApJ...517..700L |s2cid=119349364}}</ref> magnetic field has a small scale random component arising because of turbulence.<ref>{{cite journal|last1=Jafari|first1=Amir|last2=Vishniac|first2=Ethan|title=Topology and stochasticity of turbulent magnetic fields |journal=Physical Review E|date=2019|volume=100|issue=1|pages=013201|doi=10.1103/PhysRevE.100.013201|pmid=31499931|bibcode=2019PhRvE.100a3201J|s2cid=199120046}}</ref> For the turbulent flow in the reconnection region, a model for magnetohydrodynamic turbulence should be used such as the model developed by Goldreich and Sridhar in 1995.<ref>{{cite journal|last1=Goldreich|first1=P.|last2=Sridhar|first2=S.|title=Toward a theory of interstellar turbulence. 2: Strong Alfvenic turbulence|journal=The Astrophysical Journal|date=1995|volume=438|page=763|doi=10.1086/175121|bibcode = 1995ApJ...438..763G |url=https://authors.library.caltech.edu/38003/}}</ref> This stochastic model is independent of small scale physics such as resistive effects and depends only on turbulent effects.<ref>{{cite journal|last1=Jafari|first1=Amir|last2=Vishniac|first2=Ethan|last3=Kowal|first3=Grzegorz|last4=Lazarian|first4=Alex|title=Stochastic Reconnection for Large Magnetic Prandtl Numbers|journal=The Astrophysical Journal|date=2018|volume=860|issue=2|pages=52|doi=10.3847/1538-4357/aac517|bibcode=2018ApJ...860...52J|s2cid=126072383|doi-access=free}}</ref> Roughly speaking, in stochastic model, turbulence brings initially distant magnetic field lines to small separations where they can reconnect locally (Sweet-Parker type reconnection) and separate again due to turbulent super-linear diffusion (Richardson diffusion <ref>{{cite journal|last1=Jafari|first1=Amir|last2=Vishniac|first2=Ethan|title=Magnetic stochasticity and diffusion|journal=Physical Review E|date=2019|volume=100|issue=4|pages=043205|doi=10.1103/PhysRevE.100.043205|pmid=31770890|arxiv=1908.06474|bibcode=2019PhRvE.100d3205J|s2cid=201070540}}</ref>). For a current sheet of the length <math>L </math>, the upper limit for reconnection velocity is given by

<math display="block">v = v_\text{turb} \; \operatorname{min}\left[\left( {L \over l} \right)^\frac{1}{2}, \left( {l \over L} \right)^\frac{1}{2} \right],</math>

where <math>v_\text{turb} = v_l^2/v_A</math>. Here <math>l</math>, and <math>v_l</math> are turbulence injection length scale and velocity respectively and <math>v_A </math> is the Alfvén velocity. This model has been successfully tested by numerical simulations.<ref>{{cite journal|last1=Kowal|first1=G.|title=Numerical Tests of Fast Reconnection in Weakly Stochastic Magnetic Fields|last2=Lazarian|first2=A.|last3=Vishniac|first3=E.|last4=Otmianowska-Mazur|first4=K.|journal=The Astrophysical Journal|year=2009|volume=700|issue=1|pages=63–85|doi=10.1088/0004-637X/700/1/63|arxiv = 0903.2052 |bibcode = 2009ApJ...700...63K |s2cid=4671422}}</ref><ref>{{cite journal| last1=Kowal|first1=G| last2=Lazarian|first2=A.| last3=Vishniac|first3=E.| last4=Otmianowska-Mazur|first4=K.| title=Reconnection studies under different types of turbulence driving| journal=Nonlinear Processes in Geophysics|date=2012|volume=19|issue=2|pages=297–314|doi=10.5194/npg-19-297-2012| arxiv = 1203.2971 |bibcode = 2012NPGeo..19..297K |s2cid=53390559|doi-access=free}}</ref>

===Non-MHD process: Collisionless reconnection===

On length scales shorter than the ion inertial length <math>c / \omega_{pi}</math> (where <math>\omega_{pi} \equiv \sqrt{\frac{n_i Z^2 e^2}{\epsilon_0 m_i}}</math> is the ion plasma frequency), [[ion]]s decouple from electrons and the magnetic field becomes frozen into the electron fluid rather than the bulk plasma.  On these scales, the [[Hall effect]] becomes important.  Two-fluid simulations show the formation of an X-point geometry rather than the double Y-point geometry characteristic of resistive reconnection.  The [[electron]]s are then accelerated to very high speeds by [[Electromagnetic electron wave|Whistler waves]].  Because the ions can move through a wider "bottleneck" near the current layer and because the electrons are moving much faster in Hall MHD than in [[Magnetohydrodynamics|standard MHD]], reconnection may proceed more quickly.  Two-fluid/collisionless reconnection is particularly important in the Earth's magnetosphere.