Editing Magnetic reconnection (section)

===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>