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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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2L1

2δ1

2δ2

2nd Level

3rd Level

N1− islands

N2− islands

S1= (L1/δ1)2

S2= (L2/δ2)2

2L2= 2L1/N1

1st Level

FIG. 2: In the regimes of high Lundquist number and large plasma size, the plasmoid instability

gives rise to a hierarchy of interacting current sheets and islands. The above sketch gives the

notation used here for describing this hierarchy.

III. MULTIPLE X-LINE COLLISIONAL AND COLLISIONLESS RECONNEC-

TION

Until quite recently, the boundary between collisional and collisionless reconnection was

thought to be given by Eq.(5). This may not be true anymore when the current sheet is

unstable to the plasmoid instability, forming thinner current sheets which may be further

subject to new plasmoid instability leading to yet thinner current sheets in a hierarchical

fashion as proposed by Shibata and Tanuma39. Eventually, these new current sheets can

approach the ion kinetic scales triggering collisionless reconnection as recently demonstrated

by full kinetic simulations with a Fokker-Planck treatment of Coulomb collisions44. There-

fore, the multiple X-line phase can be further divided into a phase involving only collisional

physics (i.e. purely resistive MHD) and a phase involving both collisional and collisionless

physics, which we denote as the multiple X-line hybrid phase.

The boundary line in the (λ,S) space between these two sub-phases depends on the de-

tailed physics of unstable current sheets. The main uncertainty originates from the question

of how many islands, on the average, remain within the unstable current sheet at any given

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time. According to linear analytic theory40,52, the number of secondary islands scales as

N ∼

?S

Sc

?α

, (8)

where α = 3/8. This linear prediction has been carefully verified in simulations designed to

study the initial breakup of the Sweet-Parker layers43,51. However, nonlinearly many more

islands are observed in the simulations44,45,51,53corresponding to scaling parameters in the

range α = 0.6 → 1. A possible explanation for this discrepancy is that, in these nonlinear

simulations, the break-up of the original Sweet-Parker layer leads to new current sheets

between the islands which are also unstable to the same plasmoid instability as illustrated

in Fig. 2, and the islands on more than one level in the hierarchy were counted (see below

for more discussions).

At the present time, it appears that there are only two ways to terminate the downward

progression in this hierarchy: (1) either the local Lundquist number of the new current

sheets falls below the critical value for the plasmoid instability or (2) the new current sheets

approach the ion ρsscale where collisionless effects dominate. One can make quantitative

predictions regarding these two possible outcomes with just a few simple assumptions.

A.Multiple X-line collisional reconnection

As illustrated in Fig. 2, we start by defining the half-length and thickness of the top

level Sweet-Parker by L1 ≡ LCS and δ1 ≡ δSP [Eq.(3)], corresponding to a macroscopic

Lundquist number of S1≡ S [Eq.(1)]: S1= (L1/δ1)2. At this top level, the development

of the plasmoid instability gives rise to N1= (S1/Sc)αislands, which breaks the original

layer into new sheets with length L2= L1/N1. We assume these new layers are governed by

the Sweet-Parker scaling relationships and are also susceptible to the plasmoid instability

in the same manner. Then, the number of islands generated within the second level of the

hierarchy is N2= (S2/Sc)αwhere S2= (L2/δ2)2is the Lundquist number of the new sheets,

assuming the same reconnection magnetic field strength upstream. Therefore, the jth level

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quantities in the hierarchy are related to the (j-1)th level quantities by

Lj =

Lj−1

Nj−1

Sj−1

Nj−1

Lj−1

?Sj−1

(9)

Sj = (10)

δj≡

Lj

?Sj

=

1

?Nj−1

=

δj−1

?Nj−1

, (11)

where the number of islands in the jth level, Nj, is given by a recursion relation,

?Sj

= N(1−α)

j−1

Nj =

Sc

?α

=

?Sj−1

Sc

?α?

= ... = N(1−α)j−1

Sj

Sj−1

?α

= N(1−α)2

j−2

1

. (12)

If we terminate the hierarchy at the jth level, then the total number of islands in the system

Njcorresponds to the product of the islands in all the levels

Nj ≡ N1N2N3... Nj

= N1N(1−α)

1

N(1−α)2

1

?S1

... N(1−α)j−1

1

≡ Nβj

1 =

Sc

?αβj

(13)

where

βj≡

j−1

?

n=0

(1 − α)n=1 − (1 − α)j

α

. (14)

Note that as the hierarchy becomes deeper j ? 1 it converges towards β∞= α−1. We can

now conveniently express the scaling for the total number of islands up through jth level in

terms of the global Lundquist number

Nj=

?S1

Sc

?1−(1−α)j

,(15)

while the Lundquist number of the new current sheets at the jth level is

Sj =

Sj−1

Nj−1

=

Sj−2

Nj−1Nj−2

= ...

=

S1

Nj−1Nj−2...N1

=

S1

Nj−1

= S(1−α)j−1

1

S1−(1−α)j−1

c

.(16)

Notice that for α < 1 this result implies Sj> Scfor any finite level in the hierarchy, which

implies the new levels are always unstable to the plasmoid instability! Strangely enough,

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this implies the hierarchy has infinite depth for α < 1, and only terminates in the limit

j → ∞ where N∞= S1/Scand S∞= Sc. However, our basic scaling assumption for the

number of islands at each level Nj= (Sj/Sc)αbecomes invalid when Njdecreases towards

unity. To make a reasonable estimate for the number of levels in the hierarchy, one should

consider a cutoff Nj∼ Nminbelow which continuous scaling arguments are meaningless. For

example, by setting Nj≥ Nminin Eq. (12) as a cutoff, the maximum level in the hierarchy

is

jmax= 1 +ln(ln(Nmin)) − ln(αln(S1/Sc))

ln(1 − α)

, (17)

which diverges logarithmically as Nmin→ 1. Picking some reasonable cutoff Nmin∼ 2 and

assuming α = 3/8 will terminate the hierarchy fairly quickly (jmax= 2 → 8) for most any

conceivable Lundquist numbers (see Sec. VI).

Therefore, the scaling of the total number of islands in the hierarchy [Eq. (15)] depends

on the maximum number of levels, jmax[Eq. (17)], and the island number scaling power

index from one level to the next level, α. Applying this estimate to the reported numerical

MHD simulations51yields jmax∼ 3, using α = 3/8. This leads to the predicted scaling of

Njmax∼ (S1/Sc)0.76. However, the linear scaling of α = 3/8 does not necessarily apply in the

hierarchy model where the nonlinear evolution of islands at one level is required to generate

new current sheets for the islands at the next level. Using α = 0.8, for example, leads to

jmax∼ 2 and Njmax∼ (S1/Sc)0.96. These scalings are not very far from the reported linear

scaling of S∼1given the large uncertainties that still exist (see Fig. 5 in Ref. 51). As the

hiearchy becomes increasingly deep, the precise value of α no longer matters and the result

approaches the linear scaling of S as evident from Eq.(15), consistent with earlier heuristic

arguments44,49,51.

B.Transition to multiple X-line collisionless reconnection

There is a second way to terminate the downward progression in the hierarchy before

reaching the maximum level estimated by Eq. (17). As discussed in Sec. I, this occurs when

the thickness of a current sheet at a level j (≥ 1), given by Eq. (11), approaches the ion

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log (S)

5

4

3

2

1

1

6

7

23

4

log (λ)

MRXMRX

NGRXNGRX

Single X-line collisional

Multiple X-line

collisional

Multiple X-line

hybrid

Multiple X-line

collisionless

Single X-line

collisionless

S=λ /4

2

λ=λc

S S= λ = λ

2 2

√ √S S

c c

S=Sc

FIG. 3: The phase diagram in a smaller parameter space to show dotted and dashed lines better

(see texts in Sec. III). Other symbols are same as in Fig.1.

sound radius to trigger collisionless reconnection:

δj =

δj−1

?Nj−1

?Nj−1Nj−2...N1

=

δj−2

?Nj−1Nj−2

= ...

=

δ1

=

δ1

?Nj−1

= ρs.(18)

Using Eq. (15), this can be expressed in terms of the critical global Lundquist number,

S = S1, as a function of plasma size, λ, as

S = S(1−c)/2

c

(?λ)1+c, (19)

where

c =

(1 − α)j−1

2 − (1 − α)j−1

which vanishes in the limits of α = 1 or j = ∞ for 0 < α < 1. The blue line in Fig. 1

shows the transition boundary in the (λ,S) space from multiple X-line collisional phase to

multiple X-line hybrid phase in these limiting cases,

S =

?

Sc?λ (20)

where Sc= 104and ? = 1/2, consistent with the previous heuristic arguments51,54.

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For more realistic scenarios with α < 1 and a finite number of levels, the appxoimate

transition boundary can also be estimated. For any given S(> Sc), the deepest level, jmax, in

the hierarchy is given by Eq. (17). Then using this jmax, the maximum λ for current sheets

at the deepest level in the hiearchy to remain collisional is determined by Eq. (19). Examples

for α = 3/8 and α = 0.8 are shown in Fig. 3 as a dotted line and a dashed line, respectively.

The steps in these lines correspond to the increases in jmax, but overall they are only slightly

above the limiting cases of c = 0. Lastly, we note that the physics across this boundary

is vastly different: on the collisional side the reconnection is completely determined by

collisional MHD physics while on the hybrid side, both collisional and collsionless physics is

important. This is consistent with the boundary determined by electron runaway conditions

which also indicate the change in the required physics (see Sec.V later). The reconnection

rate, on the other hand, is given by the Sweet-Parker rate of S−1/2

c

∼ 0.01 on the collisional

side while on the hybrid side, the reconnection is faster, but not by a large amount, at the

collisionless rate of 0.01 − 0.1.

IV.SINGLE AND MULTIPLE X-LINE COLLISIONLESS RECONNECTION

The last part in our phase diagram concerns the fact that the single X-line current sheet

in the collisionless phase [defined by Eq.(5)] may be also subject to secondary collisionless

tearing instability. Unlike the MHD counterpart, we are not aware of any analytic work in

this area although there have been numerical demonstrations55–57and some observational

evidence58. In the collisionless limit, sufficiently large kinetic simulations suggest55that the

critical size for the secondary island formation in the extended current sheet as a result of

nonlinear evolution is,

λ = λc∼ 50, (21)

which is shown as the orange line in Fig. 1.

In principle, multiple X-line collisionless reconnection can also occur in the hybrid phase

if the effective plasma size is larger than λcat the hierarchy level when the current sheet

thickness reaches ρs. But it turns out that the condition for such transition is almost identical

to Eq. (20) as shown below. The effective plasma size at the jth level in the hierarchy, λj,

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is given by

λj=Lj

ρs

=L1

ρs

1

Nj−1

?S

= ?λ

?Sc

S

?1−(1−α)j−1

. (22)

Equating this to λcyields

?λ

λc

=

Sc

?1−(1−α)j−1

(23)

which reduces to S = (Sc?/λc)λ in the limiting cases of α = 1 or j = ∞. This condition

is different from Eq. (20) by only a factor of 2 when Sc = 104and λc = 50. Thus, the

parameter space for the multiple X-line collisionless reconnection is very limited within the

hybrid phase.

V. ELECTRON RUNAWAY CONDITION

In real plasmas, the whole concept of collisional resistivity (and thus Lundquist number) is

restricted to parameter regimes in which the reconnection electric field is small in comparison

to the Dreicer runaway limit59given by

ED≡

√meTeνei

e

, (24)

where νeiis electron-ion collision frequency. To put the phase diagram into better perspec-

tives with regard to its previous version60, it is important to understand where runaway

conditions are unavoidable and how these boundaries correlate with kinetic-scale transitions

already discussed. For a Sweet-Parker reconnecting current sheet, the reconnection electric

field is given by

ER= ηj =meνei

e2n

BR

µ0δSP

(25)

where η is classical resistivity, j is the peak current density, and BRis the reconnecting field

component. Using the relation for the plasma β

β = 2

?ρs

di

?2

(26)

where Te= Tiis assumed, we have

ER

ED

=2√2

β

BR

BT

?me

mi

ρs

δSP.(27)

For the single X-line reconnection phase, δSP is given by Eq.(3), and ER/ED= 1 leads to

?BT

S =

β2

128BR

?2mi

meλ2

(28)

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where BT is the total magnetic field. Assuming β = 0.01, BT/BR= 10, mi/me= 1836, the

above equation becomes

S = 0.14λ2

(29)

which is below the boundary line defined by Eq.(5) but within only a factor of 2 when

? = 1/2. Therefore, the electron runaway condition is well coincident with the boundary

line between collisional and collisionless phases.

For the multiple X-line reconnection phase, we can use Eq.(18) with α = 1 or j = ∞ for

simplicity, yielding

β

8√2

BR

Using the same example parameters as above, we have

S =

BT

?mi

me

?

Scλ.(30)

S = 0.38

?

Scλ (31)

which is again within a factor of 2 from Eq.(20) when c = 0 and ? = 1/2. Red lines in

Fig. 1 illustrate the boundaries for electron runaway conditions, which separate collisional

and collisionless reconnection phases for both single X-line and multiple X-line geometries.

The significance of the red lines in the phase diagram is that they separate the regime where

reconnection can be described by collisional physics alone from the regime where collisionless

physics is required. The alignment of red lines with either the black line or blue line is

consistent with the transitions from collisional reconnection to collisionless reconnection,

regardless whether it takes the form of the single X-line or multiple X-lines.

Besides MHD models and fully kinetic models, Hall MHD models and hybrid models (fluid

electrons and kinetic ions) are often used to study the transitions between collisional and

collisionless phases3,6,7,9,10,12,13,48,54,61. As demonstrated by the comparative studies between

different models3, Hall MHD models and hybrid models can capture the qualitative bound-

aries between these two phases. However, the coincidence of electron runaway conditions

with the transition boundaries between collisional reconnection and collisionless reconnec-

tion raises questions on the suitability of these fluid models when they are used to study

detailed dynamics near the transitions7,13,48,54,61. The detailed electron kinetic dynamics

become important in these regimes but are not yet accurately treated in fluid models. In

particular, the transition into the multiple X-line hybrid phase unavoidably leads to runaway

electric fields (E > ED) as illustrated by the red line in Fig. 1. Fully kinetic simulations

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including the collision operator have demonstrated62,63that the mechanism breaking the

frozen-in condition changes rapidly across this transition, from ordinary resistivity in the

sub-Dreicer collisional limit (E ? ED) to off-diagonal terms in the electron pressure tensor

for the runaway regime. Once this transition to runway electric fields has occurred, it is

unlikely to be reversed as suggested by Hall MHD models7,54until fast reconnection has

depleted the available flux. Indeed, large-scale collisional kinetic simulations44have demon-

strated that resulting electron layers in this runaway regime can become highly extended

and are unstable to secondary magnetic islands in a manner similar to previous collisionless

simulations55,64. Further insights emerge when we divide S by λ, yielding

S

λ= ?µ0ρsVA

η

, (32)

which is simply the Lundquist number based on ρs. It has been suggested that there exists

a critical value of S/λ ∼ 50 where the dynamics can revert from Hall dynamics (kinetic)

back into the Sweet-Parker regime7,54. However, notice that Eq.(31) implies that electron

runaway will occur for S/λ > 40 beyond which simple resistivity models are known to

break down. It remains an outstanding challenge to properly model this dynamics within

two-fluid approaches, but comparative studies between these different models should be

useful to provide guidance on reliable two-fluid models which can be practically used for the

detailed investigations of the phase diagram at large S and λ values.

VI.DISCUSSION

While the simple S −λ diagram conveniently summarizes much of the present knowledge

regarding the dynamics of magnetic reconnection, there are a variety of other factors that

may significantly influence reconnection which have not yet been discussed. For example,

we have largely avoided the onset question (i.e. how reconnection gets started) which is

likely very different for the various parameter regimes. In particular, the detailed properties

of tearing instabilities in various regions of parameter space may play some role in the onset

phase of reconnection, but this subject is beyond the focus of the present paper. As another

example, the structure of the large-scale initial condition can also influence the reconnection

dynamics, including features such as asymmetric current layers and velocity shear. There

has been some work on both of these issues, but many uncertainties remain. Below we

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specifically discuss external drive dependence and three-dimensional effects, followed by the

discussions on the locations of various plasmas in the phase diagram.

A. Dependence of external drive

One might consider that the reconnection process should depend on the external drive

or on how much free energy available in the system for reconnection. This is especially

relevant when reconnection is modeled by a local box around the diffusion region. In real

applications, the boundary condition may significantly influence the reconnection process

within. However, we point out that the definition of the Lundquist number given by Eq.(1)

already takes into account of the available free energy in the system: the half length of the

reconnecting current, LCS, which is taken as a fraction of the system size, L, is used. If the

system is completely relaxed without free energy for reconnection, then LCS= 0 even L can

be very large. Having LCSon the order of L implies that the available free energy is near its

maximum. One can imagine a time evolution when the system is driven from its completely

relaxed state with LCS= 0 to LCS? L/2 to reach a maximum S, and the dependence on

the drive is actually reflected in the magnitude of S already. If the free energy is less than

its maximum in a given system, S should be less than its maximum value even L is still

same.

B. Influence of realistic three-dimensional dynamics

The ideas leading to the phase diagram in Fig. 1 are largely based on two-dimensional

(2D) models and simulations of reconnection. At present time, very little is known regarding

how reconnection will proceed in large three-dimensional (3D) systems - either in fluid or

kinetic parameter regimes. To begin with, the whole idea of magnetic islands relies upon a

high degree of symmetry, which can be achieved in laboratory plasmas (and 2D simulations)

but is unlikely to occur in space and astrophysical plasmas. Instead, extended flux ropes

are the natural 3D extension of magnetic islands, and the manner in which these can form

and interact is much more complicated than in 2D models. This is true of both the primary

flux ropes which may form due to tearing instabilities, and also secondary flux ropes which

can form in the new current sheets that arise from the nonlinear evolution of the primary

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Je

Jo

0

4

2

Ly= 70di

flux ropes

Lx= 70di

flux ropes

x

z

FIG. 4: Results from a kinetic simulation65of guide field reconnection showing the formation

and interaction of flux ropes as illustrated by an isosurface of the particle density colored by

the magnitude of the current density along with sample magnetic field lines (yellow). Simulation

parameters are mi/me = 100 with the initial guide field equal to the reconnecting field. The

domain size is 70di× 70di× 35dicorresponding to 2048 × 2048 × 1024 cells and ∼ 1012particles.

flux ropes. The advent of petascale computers over the last few years is permitting 3D

kinetic simulations65to explore these ideas for guide field reconnection geometries, where

tearing modes can be localized at resonant surfaces across the initial current sheet. These

initial simulations together with Vlasov theory have demonstrated that the spectrum of

oblique tearing modes within ion-scale layers is simpler than previously thought66, but the

resulting flux rope dynamics is still quite rich. Furthermore, the nonlinear development

of the primary flux ropes produces intense electron-scale current sheets near the active x-

lines and along the separatrices. As illustrated in Fig. 4, in 3D simulations these layers are

unstable to the formation of secondary flux ropes over a broad range of oblique angles. The

continual formation and interaction of these flux ropes gives rise to a turbulent evolution

that is significantly different than 2D models. However, the full implication of these results

will take years to sort out; researchers are just beginning to scratch the surface.

In addition to these fundamental issues associated with island (flux rope) formation, there

16

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are a wide range of other processes to consider in 3D which may potentially influence the dy-

namics of magnetic reconnection. These processes include the lower-hybrid drift instability,

driven by the strong diamagnetic currents, streaming instabilities, modes driven by either

electron or ion velocity shear, and a range of kinetic instabilities driven by temperature

anisotropy. Even in MHD, influence of a pre-exiting turbulence on reconnection remains an

outstanding issue in both 2D50and 3D67studies. Huge challenges remain in understanding

the role these various process play in reconnection, and how they might change the phase

diagram in Fig. 1. One of the most long-standing ideas is that instabilities may modify

the dissipation physics within electron-scale regions. However, there are other possibilities

to consider including non-linear couplings between electron and ion-scale features, or the

possibility that these instabilities may seed the formation of new flux ropes.

C. Reconnection in heliophysical, astrophysical and laboratory plasmas

Despite these rather serious caveats discussed in the previous section, it is interesting to

place plasmas from laboratory, heliophysics and astrophysics in the phase diagram. In Fig. 1,

some heliophysical and laboratory example plasmas are shown. In this section, results from

a more extensive survey of astrophysical plasmas are summarized in Table I and Fig. 5 with

references from which typical plasma parameters were taken. In general, these parameters

are associated with large uncertainties due to limited measurements available from these dis-

tant plasmas and crude models used for the estimation. Extreme astrophysical conditions92,

such as special relativity and radiation, are not taken into account here since these effects

on collisionless plasmoid instabilities is just beginning to be explored numerically101,102. On

the log-scales as in Fig. 5, however, even an order of magnitude of the uncertainty does not

change the location of these plasmas by much in the phase diagram.

The cases shown in the phase diagram can be roughly grouped into three groups. The first

group includes high temperature fusion plasmas and Earth’s magnetospheric plasmas. These

plasmas are completely in the collisionless phases, either with a single X-line or multiple X-

lines, depending on whether the plasma effective sizes, λ, are larger than the critical λc. The

plasma for the Sgr A* flares may also belong to this group.

The second group of plasmas cluster along the black line separating multiple X-line col-

lisionless phases and multiple X-line hybrid phase. It spans over huge ranges from solar

17

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TABLE I: Key parameters of various plasmas from laboratory, heliophysics and astrophysics. Un-

less explicitly stated, assumptions are (1) ? = 1/2, (2) the reconnecting field is 1/10 of total

magnetic field, BR= 0.1BT, (3) equal electron and ion temperatures, Te= Ti, and (4) ions are

protons. We note that there are opinions that the plasmas in Crab pulsar wind and radio lobes

are nonthermal so that temperature may not be a good description68,69. There are some labora-

tory experiments which are not listed: flux rope experiments70–72with S ∼ λ ∼ 101, and plasma

merging experiments73,74with S = 102− 103and λ = 101− 102.

location plasmasize(m)Te(eV)ne(m−3) BT(Telsa)SλNotes

MRX75

0.8 101 × 1019

1.5 × 1018

5 × 1025

9 × 1018

1 × 1020

1 × 1020

3 × 1019

1 × 107

3 × 105

7 × 106

1 × 1015

1 × 1017

1 × 1029

6 × 108

0.13 × 1031.5 × 102? = 1/4, Ti= Te/2, BR= 0.3BT

3 × 102

2 × 101

3 × 1066.2 × 101Ti= 350eV, D+, BR= 0.05BT

1 × 1082.3 × 102Ti= 36keV, D+, BR= 0.01BT

6 × 108

2 × 1051.2 × 103? = 1/4, Ti= Te/2, BR= 0.2BT

5 × 10−86 × 1013

2 × 10−84 × 10151.3 × 103BR= BT, Ti= 4.2keV, (p.233)

7 × 10−93 × 1012

2 × 10−21 × 1013

2 × 10−2

11 × 109

2 × 10−5

included82, Mg+

VTF14

0.4 25 0.0444 × 100

1 × 101

? = 1/4, Ti= 5eV, Ar+

Laser Plasma76

2 × 10−4

1.0

103

100Al+13, BR= BT

LabMST77

1.3 × 103

1.3 × 104

2 × 104

15

0.5

TFTR78

0.9 5.6

ITER79

4 5.35 × 102

D+, BR= 0.01BT

NGRX80

1.60.5

Magnetopause81

6 × 107

6 × 108

2 × 1010

1 × 107

1 × 107

1 × 107

9 × 109

3009 × 102

BR= BT, (p.267)

Magnetotail81

600

SolarSolar Wind81

102 × 105

4 × 107

3 × 108

5 × 1010

1 × 109

(p.92)

System Solar Corona81

200(p.79)

Solar Chromosphere82

0.51 × 108

neutral particle effects are weak82

Solar Tachocline83,84

200

Protostellar Disks85

3 × 10−2

8 × 103

L=2h(R=1AU), e-n collisions

X-ray Binary Disks86,87

4 × 104

751 × 1027

363 × 107

9 × 108

M = 10M?, L = 2h(R = 102RS),

α = 10−2, ˙M = 1016g/s

X-rayBinaryDisk

Coronae88

3 × 104

5 × 105

1 × 1024

1 × 104

1 × 1016

9 × 107

M= 10M?, R = RS, Ti

(mp/me)Te, ηComptonincluded88

=

Galaxy Crab Nebula Flares89–911 × 1014

Gamma Ray Bursts92

130 106

10−7

5 × 10202 × 1011pair plasma, T from B2

6 × 10172 × 1016pair plasma

6 × 10165 × 1017pair plasma, SGR 1806-20

2 × 1024

2 × 10−91 × 10117 × 1012neutral particle effects included82,

HCO+

R/2µ0= 2nT

104

3 × 105

5 × 105

7 × 106

10−3

2 × 1035

1041

4 × 109

2 × 1011

10−3

Magnetar Flares92,93

104

Sgr A* Flares94,95

2 × 1011

3 × 1016

1013

5 × 108

L = 2R = 20RS

Molecular Clouds96,97

109

Interstellar Media96,97

5 × 1019

2 × 1011

1 105

5 × 10−102 × 10201 × 1014L=magnetic field scale height

0.52 × 10131 × 1014M = 108M?, L = 2h(R = 102RS),

α = 10−2, ˙M = 1026g/s

AGN Disks86,87,98

248 × 1023

Extra-

galactic Radio Lobes69

AGN Disk Coronae88

3 × 1011

5 × 105

1 × 1017

41023

3 × 1011M= 108M?, R = RS, Ti =

(mp/me)Te, ηComptonincluded88

3 × 1019

3 × 1019

6 × 1018

10015 × 10−102 × 10258 × 1012

10−7

6 × 10291 × 10143C 303

2 × 10−92 × 10256 × 1011A1835

Extragalactic Jets99

104

3 × 101

4 × 104

Galaxy Clusters100

5 × 103

18

Page 19

log (λ)

log (S)

MRX MRX

NGRXNGRX

0

5

10

15

0

5

10

15

20

25

30

VTFVTF

Laser Laser

MSTMST

TFTR TFTR

ITER ITER

MagnetotailMagnetotail

MagnetopauseMagnetopause

SolarSolar

WindWind

SolarSolar

CoronaCorona

X-ray BinaryX-ray Binary

Disk CoronaDisk Corona

Crab NebulaCrab Nebula

FlareFlare

Sgr A*Sgr A*

FlareFlare

AGN DiskAGN Disk

CoronaCorona

GalaxyGalaxy

ClusterClusterRadio

Radio

LobeLobe

ExtragalacticExtragalactic

JetJet

Multiple X-line

InterstellarInterstellar

MediumMedium

γ γ-ray Burst-ray Burst

MagnetarMagnetar

AGNAGN

DiskDisk

MolecularMolecular

CloudCloud

SolarSolar

TachoclineTachocline

ProtostellarProtostellar

DiskDisk

Single X-line collisional

Solar ChromosphereSolar Chromosphere

X-ray Binary DiskX-ray Binary Disk

Multiple X-line

collisionless

Multiple X-line

collisional

hybrid

Single X-line collisionless

λ=λc

S=λ /4

2

S S= λ = λ

2 2

√ √S S

c c

S=Sc

FIG. 5: Various laboratory, heliophysical and astrophysical plasmas, in which magnetic reconnec-

tion is believed to occur, are shown in the phase diagram. Other symbols are same as in Fig.1. See

the text and Table I for details.

corona, accretion disk coronae, Crab nebula flare, to galaxy clusters, radio lobes, and extra-

galactic jets. When S and λ are both small, this same line separates single X-line collisional

phase and collisionless phase. With a single X line, the reconnection in the collisional phase

was known to be much slower than in the collisionless phase. The plasma collisionality was

argued103to regulate itself so that the plasma always stay near the marginal collisionality,

based on the reasoning that fast reconnection should effectively release magnetic energy

evaporating nearby dense neutral gases (such as in the solar chromosphere) to increase den-

sity and collisionality until reconnection slows to a collisional rate. An alternative model was

also proposed104based on self-regulation of electron temperature to maintain marginal colli-

sionality through a similar but different reasoning: higher temperature lowers collisionality

19

Page 20

and fastens reconnection, and thus depletes quickly available magnetic energy and eventu-

ally slows reconnection and cools the plasma while lower temperature increases collisionality

and slows reconnection, and thus accumulate magnetic energy and eventually trigger faster

reconnection and heat the plasma.

However, at large S and λ values for all plasmas in the second group, the marginality

black line now separates multiple X-line collisionless phases and multiple X-line hybrid phase

in the phase diagram. Now there is numerical evidence44,48that the reconnection rates in

multiple X-line hybrid phase are as fast as the single X-line collisionless rate, consistent with

the theoretical argument49that the global reconnection rate is determined by a dominant

reconnection site in the island hierarchy which should be collisionless in the hybrid phase.

Therefore, the self regulation arguments for the collisionality mentioned above do not seem to

hold at the large S and λ phases, but much work still remains to be done. The accumulation

of energetic particle populations is suggested57as another player in the self regulation process

of reconnection rate in the multiple X-line collisionless phase. Energetic particle populations

should be regulated also by finite collisions in the hybrid phase, but detailed dynamical

processes need to be investigated.

The third and final group of plasmas shown in Fig. 5 occupy much of the multiple X-

line collisional phase: accretion disk interiors, solar chromosphere and tachocline, molecular

clouds, gamma ray bursts and magnetar flares. It could be argued that they form a line

slightly below but along the boundary (blue line) between the multiple X-line collisional and

hybrid phases. It is conceivable that the self-regulation arguments for collisionality103,104

could be applied here since collisional reconnection dominates at the deepest level of the

hierarchy on the one side of the boundary while collisionless reconnection dominates on the

other side. In fact, it has been suggested through Hall MHD simulations48that reconnection

in the multiple X-line collisional phase is much slower than that in the hybrid phase although

it is much faster than the single X-line collisional (Sweet-Parker) rate. However, the recon-

nection rate is not so different: collisionless rates are around 0.1 and while the collisional

rates are around 1/√Sc∼ 0.01 at the deepest level of the hierarchy. A key question here

is what determines the overall reconnection rate in a hierarchy of islands and whether it is

indeed dominated by the reconnection process at the deepest level49or by the reconnection

process at all levels in an integrated way.

There are two special cases which do not belong to either of the above three groups:

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protostellar disks and interstellar media. Protostellar disks have lowest S among the objects

we surveyed and are located between the single X-line and multiple X-line collisional phase.

Interstellar media are right in the middle of the hybrid phase, and probably both collisional

MHD physics and collisionless physics are important in charactering reconnection processes

there as a part of the galactic dynamo.

Lastly, we note that currently there are no laboratory experiments which can be used to

study all of these new phases of magnetic reconnection. Laboratory experiments have been

playing important roles in the reconnection research: confirming some leading theoretical

or numerical models such as Sweet-Parker18and collisionless reconnection models105while

challenging others such as Petschek model; benchmarking state-of-the-art numerical simu-

lations63,106,107; discovering 3D phenomena108–111; studying flux rope dynamics71,72, to name

a few. As mentioned above, the main research tool on physics of new reconnection phases

is numerical simulations using either full particle, Hall MHD, or resistivity MHD codes.

Existing experiments, such as MRX, do not have accesses to these new phases which are

important for the emerging themes of particle acceleration by magnetic reconnection56,57.

While numerical simulations, coupled closely with analytic theory, will continue to be a

major player at this front, a next generation reconnection experiment (NGRX) based on

the MRX concept is considered as a candidate for such a laboratory experiment80. The

parameter ranges for both MRX and NGRX are also indicated in the phase diagram for

their coverages.

Acknowledgements

HJ acknowledges support from the U.S. Department of Energy’s Office of Science - Fu-

sion Energy Sciences Program, and the Princeton Plasma Physics Laboratory’s Laboratory

Directed Research and Development Program. WD acknowledges support from the U.S.

Department of Energy through the Los Alamos National Laboratory’s Laboratory Directed

Research and Development Program. Simulation in Fig. 4 was performed on Kraken with

an allocation of advanced computing resources provided by the National Science Founda-

tion at the National Institute for Computational Sciences. HJ appreciates suggestions on

parameters of various astrophysical plasmas and their references by Jeremy Goodman, Hui

Li, Alex Schekochihin, Farhad Zadeh, and Ellen Zweibel. We greatly appreciate critical

21

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feedbacks from Amitava Bhattacharjee, Ellen Zweibel, an anonymous referee, and espe-

cially from Dmitri Uzdensky who read our manuscript carefully and provided a long list of

constructive comments. We are also grateful to Masaaki Yamada and Stewart Prager for

valuable discussions.

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