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Phase Diagram for Magnetic Reconnection in Heliophysical,

Astrophysical and Laboratory Plasmas

Hantao Ji1and William Daughton2

1Center for Magnetic Self-Organization,

Princeton Plasma Physics Laboratory,

Princeton University, Princeton, New Jersey 08543

2Los Alamos National Laboratory, Los Alamos, New Mexico 87545

(Dated: September 6, 2011)

Abstract

Recent progress in understanding the physics of magnetic reconnection is conveniently summa-

rized in terms of a phase diagram which organizes the essential dynamics for a wide variety of

applications in heliophysics, laboratory and astrophysics. The two key dimensionless parameters

are the Lundquist number and the macrosopic system size in units of the ion sound gyroradius.

In addition to the conventional single X-line collisional and collisionless phases, multiple X-line

reconnection phases arise due to the presence of the plasmoid instability either in collisional and

collisionless current sheets. In particular, there exists a unique phase termed “multiple X-line

hybrid phase” where a hierarchy of collisional islands or plasmoids is terminated by a collision-

less current sheet, resulting in a rapid coupling between the macroscopic and kinetic scales and a

mixture of collisional and collisionless dynamics. The new phases involving multiple X-lines and

collisionless physics may be important for the emerging applications of magnetic reconnection to

accelerate charged particles beyond their thermal speeds. A large number of heliophysical and

astrophysical plasmas are surveyed and grouped in the phase diagram: Earth’s magnetosphere,

solar plasmas (chromosphere, corona, wind and tachocline), galactic plasmas (molecular clouds,

interstellar media, accretion disks and their coronae, Crab nebula, Sgr A*, gamma ray bursts,

magnetars), extragalactic plasmas (Active Galactic Nuclei disks and their coronae, galaxy clus-

ters, radio lobes, and extragalactic jets). Significance of laboratory experiments, including a next

generation reconnection experiment, is also discussed.

PACS numbers:

1

arXiv:1109.0756v1 [astro-ph.IM] 4 Sep 2011

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I.COLLISIONAL AND COLLISIONLESS RECONNECTION

It has been a long held view that magnetic reconnection is primarily characterized by

plasma collisionality. This is evidenced by the common uses of the resistive magnetohy-

drodynamic (MHD) models, which is parameterized solely by the dimensionless Lundquist

number,

S ≡µ0LCSVA

η

, (1)

as a starting point of the discussion for magnetic reconnection. In Eq.(1), LCSis the half

length of the reconnecting current sheet, and can be taken as LCS = ?L where L is the

plasma size and 0 ≤ ? ≤ 1/2 (the choices of ? are discussed in Sec.VI.A.). VAis the Alfv´ en

velocity based on the reconnecting magnetic field component and η is the plasma resistivity

due to Coulomb collisions. The well-known Sweet-Parker model1,2predicts reconnection

rates as an explicit function of S,

VR

VA

=

1

√S,

(2)

where VR is the reconnection inflow speed. When collisions are sufficiently infrequent or

S is sufficiently large, physics beyond resistive MHD becomes crucial3, leading to a fast

reconnection rate nearly independent of S. A large body of the work in the past decades,

therefore, has focused on reconnection either in collisional or collisionless limit as summarized

by recent reviews4,5.

The collisional MHD description provides a good description of magnetic reconnection

for plasmas in which all the resistive layers remain larger than the relevant ion kinetic scale.

For example, without a guide field (i.e. anti-parallel reconnection), the transition between

collisional and collisionless reconnection occurs6–10when the current sheet half thickness

predicted by the Sweet-Parker model approaches

δSP≡LCS

√S

= di

(3)

where di≡ c/ωpiis the ion skin depth. By properly varying both L (and thus LCS) and

η (through changing e.g. electron temperature), S and dican be kept constant while the

relative magnitude of δSP to di can be reversed, leading to dramatic differences in the

structure of the reconnection layer along with clear changes in the magnitude and scaling of

the reconnection rate. This qualitative change can be characterized by the effective plasma

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size which is defined by

λ ≡L

di

(4)

so that the second equality in Eq.(3) can be written as

S = ?2λ2. (5)

In the case of a finite guide field, the transition occurs11–14when δSP = ρs where ρs ≡

?(Ti+ Te)mi/qiBT is ion sound gyroradius, Teand Tiare electron and ion temperatures,

BTis the total magnetic field including both the reconnecting and guide components, and mi

and qiare the ion mass and charge. In the case of anti-parallel reconnection with upstream

plasma βup? 1, ρswill be equal to diby the virtue of the force balance across the current

sheet, if the reconnecting magnetic field and the temperatures at the current sheet center

are used to calculate ρs. (We note that when βup? 1, the transition scale for δSP is less

clear since ρs(??βupdi) is separated from di.) Therefore, the boundary between collisional

and collisionless reconnection is defined by Eq.(5) regardless of the presence of a guide field

when the definition of plasma effective size is modified to

λ ≡L

ρs. (6)

Thus, the collisional and collisionless reconnection phases are distinguished in the parameter

space of (λ,S). This is illustrated as the black line in the phase diagram in Fig. 1 assuming

? = 1/2. We note that the term “collisionless reconnection” is used in this paper for the

reconnection process dominated by the effects beyond collisional MHD, such as two-fluid

effects, ion and electron kinetic effects. Among these, the electron kinetic effects should

become important in a similar parameter space defined by the black line as shall be discussed

in Sec.V.

II. SINGLE AND MULTIPLE X-LINE COLLISIONAL RECONNECTION

For plasmas larger than those specified by Eq.(5), it would appear that the collisional

MHD description might be valid despite the large S. It shall become clear later in Sec.III.B.,

however, collisional models are not sufficient for describing reconnection in these regimes.

Discussions on collisional reconnection have been long dominated by debates between the

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log (S)

log (λ)

0

5

10

15

0

2

4

6

8 1210

NGRX NGRX

MRX MRX

MagnetosphereMagnetosphere

SolarSolar

Corona Corona

SolarSolar

Tachocline Tachocline

SolarSolar

ChromosphereChromosphere

Single X-line collisionless

Multiple X-line

collisionless

Multiple X-line

hybrid

Multiple X-line

collisional

Single X-line collisional

S=Sc

S S= λ = λ

2 2

√ √S S

c c

S=λ /4

2

λ=λc

FIG. 1: A phase diagram for magnetic reconnection in two dimensions. If either S or the nor-

malized size, λ, is small, reconnection with a single X-line occurs in collisional or in collisionless

phases. When both S and λ are sufficiently large, three new multiple X-line phases appear with

magnetic islands. The dynamics of new current sheets between these islands are determined ei-

ther by collisional physics or by collisionless physics (see Sec.III and Sec.IV.). The conditions for

electron runaway are shown as red lines (see Sec. V). The locations for reconnection in Earth’s

magnetosphere, solar corona, solar chromosphere, and solar tachocline are also shown. The existing

experiments, such as Magnetic Reconnection Experiment (MRX), do not have accesses to these

new phases. A next generation reconnection experiment (NGRX) is required for such accesses to

these new phases directly relevant to reconnection in heliophysical and astrophysical plasmas.

Sweet-Parker model1,2and the Petschek model15, both of which, however, are unsatisfac-

tory. The Sweet-Parker model has been verified numerically16,17and experimentally18at

relatively small values of S, but it predicts reconnection rates too slow to be consistent

with observations of larger S plasmas. On the other hand, the Petschek model, invoking

slow-mode shocks, predicts rates consistent with observations but it requires a localized re-

sistivity enhancement in simulations19–22and has not yet been verified experimentally. The

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origin of the localized resistivity enhancement is hypothesized to be kinetic in nature, but

the underlying mechanisms still remain illusive. While signatures of slow-mode shocks have

been reported in the Earth’s distant magnetotail23, large-scale hybrid (kinetic ions and fluid

electrons) simulations have revealed significant discrepancies in the expected structure of

the discontinuities due to the strong ion temperature anisotropy that is naturally generated

in these configurations24,25.

There is a growing body of work that suggests it may be necessary to move beyond these

steady-state models in order to understand the dynamics of magnetic reconnection in large-

scale collisional plasmas. In particular, for sufficiently high Lundquist numbers, resistive

MHD simulations feature highly elongated layers which breakup into muliple X-lines sepa-

rated by magnetic islands (or plasmoids)16,26–30. These multiple-X line models are inherently

time dependent, often generating impulsive reconnection consistent with observations such

as Flux Transfer Events (FTE)31. The plasmoid-like structures are also observed in Earth’s

distant magnetotail during substorms32and in the current sheet during solar Coronal Mass

Ejections (CME)33. Although the multiple-X line models were also applied to explain these

observed plasmoids in the magnetotail34–36or on the solar surface37–39, they did not receive

much attention until recent theory40and numerical simulations41–47offered detailed predic-

tions concerning the break-up of Sweet-Parker layers to the so-called plasmoid instability,

which produces numerous secondary magnetic islands. Although the time-averaged rates

can be still different45,48,49depending on the detailed divisions within (see Sec. III below),

all of them are definitely much faster than the Sweet-Parker rate [Eq.(2)]. Thus, the multiple

X-line reconnection, associated with the plasmoid instability, constitutes a new reconnection

phase within the collisional reconnection regime. Recent MHD simulations indicate that the

critical Lundquist number, Sc, for the onset of the plasmoid instability is approximately

Sc∼ 104, (7)

shown by the green line in Fig. 1. However, the precise value of Scprobably depends on

the level of pre-existing fluctuations in the plasma30,44,50,51. We note that the green line is

necessarily stopped at the low λ end defined by Eq.(5) due to the invalidity of MHD models

in the collisionless phase.

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