Quantitative, Comprehensive, Analytical Model for Magnetic Reconnectionin Hall Magnetohydrodynamics

Andrei N. Simakov and L. Chacon

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA(Received 11 April 2008; published 3 September 2008)

Dissipation-independent, or ‘‘fast’’, magnetic reconnection has been observed computationally in Hall

magnetohydrodynamics (MHD) and predicted analytically in electron MHD. However, a quantitative

analytical theory of reconnection valid for arbitrary ion inertial lengths, di, has been lacking and is

proposed here for the first time. The theory describes a two-dimensional reconnection diffusion region,

provides expressions for reconnection rates, and derives a formal criterion for fast reconnection in terms of

dissipation parameters and di. It also confirms the electron MHD prediction that both open and elongated

diffusion regions allow fast reconnection, and reveals strong dependence of the reconnection rates on di.

DOI: 10.1103/PhysRevLett.101.105003 PACS numbers: 52.35.Vd, 52.30.Cv

Extremely fast time scales observed for magnetic recon-nection phenomena are consistent with reconnection ratesbeing independent of collisional dissipation coefficients.Such ‘‘fast’’ reconnection rates have been observed com-putationally in fluid [Hall magnetohydrodynamics (MHD)]and particle simulations [1]. However, a comprehensiveanalytical theory capable of explaining them has beenlacking. Hall MHD is arguably the simplest fluid descrip-tion of a magnetized plasma that features fast reconnection[1]. An important physical parameter in Hall MHD recon-nection is the ratio of the ion inertial length, di, to themagnetic field dissipation layer thickness, �. Existingquantitative analytical models have only concentrated onlimiting regimes of Hall MHD, such as resistive MHD(di � �) [2] and electron MHD [3] (di � �). None isvalid in general. A quantitative analytical model valid inall regimes of Hall MHD (arbitrary di) is essential toexplain observations, and to describe transitions betweenthese different regimes.

Here we present, for the first time, a simple analyticaltheory of magnetic reconnection in all di regimes. Thetheory describes the reconnection diffusion region in twodimensions (2D). It considers steady-state reconnectionwithout a guide field, and takes into account both plasmaresistivity and hyper-resistivity (electron viscosity). Finiteguide field effects can be straightforwardly added [4].

The theory recovers resistive MHD (Sweet-Parker [2])and electron MHD results [3] in the appropriate limits, andis valid everywhere in between. It yields the diffusionregion aspect ratio and the reconnection rate as functionsof inertial and dissipation parameters. In particular, forgiven dissipation, it predicts the di threshold for whichreconnection rates become dissipation-independent. Wehave validated the theoretical predictions with nonlinearsimulations of the magnetic island coalescence problem[5]. Our theory confirms a number of long-standing em-pirical results, such as the possibility of fast reconnectionfor moderate di (di * �) in Hall MHD [1], and the inde-

pendence of the reconnection rate of the dissipationmechanism [6]. It also resolves some outstanding contro-versies in Hall MHD reconnection. In particular, it con-firms our electron MHD findings [3] that both open [1] andelongated [7] X-point configurations can result in fastreconnection, and that, contrary to previous claims [8],the reconnection rate can exhibit a strong dependence ondi [9,10].Incompressible Hall MHD model.—Assuming that

plasma pressure exceeds magnetic pressure, so that plasmadensity, n, is constant in space and time, and normalizing tothe Alfven speed and an arbitrary length, L, there resultsthe incompressible Hall MHD model [11]

@t ~Bþ ~r� ð ~B� ~VeÞ ¼ �� ~r� ð ~r� ~BÞþ �H

~r� ð ~r�r2 ~BÞ; (1)

@t ~V þ ~V � ~r ~Vþ ~rp ¼ ~B � ~r ~B; ~r � ~V ¼ 0: (2)

Here, ~B is the magnetic field, ~Ve � ~V � di ~J is the electron

flow velocity (with ~V the ion flow velocity and ~J ¼ ~r� ~Bthe total current), and p is the sum of magnetic and plasmakinetic pressures, which is eliminated by employing the~r � ~V ¼ 0 constraint. The ion inertial length is di �c=ð!piLÞ, with c the speed of light, !pi � ð4�ne2=MÞ1=2the ion plasma frequency, e the unit electric charge, andMthe ion mass. Finally, � and �H are the normalized resis-tivity and hyper-resistivity, respectively.This study concentrates on the reconnection region and

does not consider a particular macroscopic physical systemsupplying magnetic flux. A fully closed dynamical descrip-tion can be obtained by coupling our model equations witha suitable macroscopic driver [5]. We assume the 2Dreconnection geometry shown in Fig. 1, with the ignorabledirection along the z axis. The electron diffusion region isapproximated with the smaller green rectangle of (normal-ized) dimensions � and w, and the ion inertial region with

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the larger gray rectangle of dimensions h � � and � � w.We assume that � corresponds to an external length scale.We also assume that h � maxðdi; �Þ, since no electron-iondecoupling is expected to occur for di < � [12,13]. For thesame reason, we take w ¼ � whenever h ¼ �.

The reconnection region is conveniently described withdiscrete electron and ion flow and magnetic field variables[3,5], as shown in Fig. 1. Electrons and ions enter and exitthe ion inertial region with approximately equal velocities:Vouty � Vout

e;y and Voutx � Vout

e;x . The species decouple within

the ion inertial region, creating in-plane currents and aquadrupolar magnetic field Bz. The ion flow is negligiblewithin the electron diffusion region: V in

y � Vine;y and V

inx �

V ine;x.

Equations for the discrete variables can be obtained fromEqs. (1) and (2). Notice, that their z components predictthat Bz is coupled with the ion flow in the z direction.However, based on the numerical results for Hall MHDreconnection [13], Vz is expected to be small, Vz � Ve;z,

and will be neglected. Time-derivative terms in Eqs. (1)and (2) are normally found small at and around the time ofthe local maximum of the reconnection rate [5], and asteady-state analysis is appropriate in such situations.

Discretizing the x, y, and z components of Eq. (1) atðx; yÞ ¼ ð0; �=2Þ, ðw=2; 0Þ, and ðw=2; �=2Þ, respectively,and employing @x 2=w, @y 2=� gives [3]

�Binx V

ine;y

�¼ �eff

�Biny

�w� Bin

x

�2

�;

Biny V

ine;x

w¼ �eff

�Binx

�w� Bin

y

w2

�;

di

�Binx

wþ Bin

y

�

��Biny

w� Bin

x

�

�¼ �Bz�eff

�1

�2þ 1

w2

�;

(3)

with �eff � �þ��Hð��2 þ w�2Þ. For simplicity we ne-glect all numerical factors ofOð1Þ. The numerical factor�describes relative importance of resistive and hyper-resistive dissipation and is of order �2r2Jz=Jz, as evalu-ated at ðx; yÞ ¼ ð�=2; 0Þ. Assuming Jz / exp½�ð2x=�Þ� �

ð2y=wÞ� gives � ¼ 4�. Here, we employ � ¼ 16, asestimated from numerical simulations.Discretizing the x and y components of Eq. (1) in the ion

inertial region at ðx; yÞ ¼ ð0; h=2Þ and ð�=2; 0Þ, respec-tively, and ignoring dissipation, gives equations for Bout

x

and Bouty :

Voute;y B

outx � V in

e;yBinx

h� �¼ Vout

e;x Bouty � Vin

e;xBiny

�� w¼ 0: (4)

Equation (2) is employed next to obtain equations for

discrete flow quantities. Using ~Ve ¼ ~V � di~r� ~B, and

assuming that the ion flow velocity decreases linearlyfrom the boundary towards the center of the ion inertialregion gives

V ine;x ¼ V in

x þ diBz

�¼ Vout

x

w

�þ diBz

�;

Vine;y ¼ V in

y þ diBz

w¼ Vout

y

�

hþ diBz

w:

(5)

According to numerical simulations [13], Bz is approxi-mately constant along magnetic separatrix everywhere in-side the ion inertial region (but outside the electrondiffusion region). Therefore, Bzðw=2; �=2Þ �Bzð�=2; h=2Þ. Then, electron and ion flows at the ioninertial region boundaries are related as

Voute;x ¼ Vout

x þ diBz

h; Vout

e;y ¼ Vouty þ diBz

�: (6)

Finally, we follow Ref. [5] to obtain equations for the ionflow components Vout

x and Vouty from the x component of

Eq. (2) and incompressibility condition ~r � ~V ¼ 0, respec-tively:

h

�

ðVouty Þ2 þ ðBout

x Þ2 � ðVoutx Þ2 � ðBout

y Þ22

þ ðBouty Þ2 � h2

�2ðBout

x Þ2 ¼ 0; �Vouty ¼ hVout

x : (7)

Discrete Eqs. (3)–(7), together with the definition of h,comprise a system of 12 equations for 14 unknowns: h, �,w, Bin

x , Biny , B

outx , Bout

y , Bz, Vine;x, V

ine;y, V

oute;x , V

oute;y , V

outx , and

Vouty .

Diffusion region aspect ratio.—An important parametercharacterizing the reconnection region is the aspect ratio ofthe electron diffusion region, � � �=w. The reconnectionrate, Ez ¼ �ð�� �Hr2ÞJz ¼ ½�þ��Hð��2 þ w�2Þ�ðBin

x =�� Biny =wÞ, is given in terms of � by

Ez� � Ez

ðBinx Þ2

¼ 2fð�Þ2�1� �2

�

�; (8)

where fð�Þ � ½ ffiffiffi2

pwSð�Þ�1=2. The effective Lundquist

number Sð�Þ is defined as S�1ð�Þ � S�1� þ S�1

H ð1þ��2Þ, where S� � ffiffiffi

2p

Binx =� and SH � ffiffiffi

2p

Binx w

2=ð��HÞare the resistive and the hyper-resistive Lundquist num-

FIG. 1 (color online). Reconnection region geometry.

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bers, respectively. We assume that collisional dissipation issmall, so that corresponding Lundquist numbers are large:S�, SH, Sð�Þ � 1. It is clear from Eq. (8) that electron

diffusion regions with �2 � 1 are expected to result inlarger Ez� and faster magnetic energy release than thosewith � 1. Therefore, we will concentrate on the limit�2 � 1 (but allow � < 1) in the remainder of this work, byapproximating 1� �2 � 1þ �2 � 1.

An equation for � can be obtained from Eqs. (3)–(7). It isconvenient to employ normalized quantities �� � �=fð�Þand d� � di=½fð�Þw to write

�

�1� d2��2�

2

�2 þ 1� ð1þ d2�Þ�2�

2¼ 0: (9)

Here, � � h�=ð�wÞ � 1 is a measure of two-fluid sepa-ration, with h�=ð�wÞ the ratio of ion inertial over electrondiffusion areas. Equation (9) describes � in all regimes ofinterest. The magnitude of � (but not its exact value)clearly determines the asymptotic solution of Eq. (9). Weexpect to have � � 1 (and the first term to be negligible)in the resistive MHD limit and � � 1 (and the last twoterms to be negligible) in the electron MHD limit. It isconvenient to consider these two limits before discussingthe general solution of Eq. (9).

Resistive MHD limit.—Assuming � � 1 (and w � �)in Eq. (9) gives

�� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

1þ d2�

s) � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

p

�Sð�Þ þ ffiffiffi2

pd2i Sð�Þ2

vuut : (10)

The standard resistive MHD result [2] is obtained by

letting di ! 0. For negligible hyper-resistivity, ��½�S�=

ffiffiffi2

p �1=2�1 and the reconnection region is al-

ways fairly elongated. For negligible resistivity, � �½�SH=

ffiffiffi2

p �1=4. Reconnection is slow in both cases, Ez� �� � 1.

Electron MHD limit.—Assuming � � 1 in Eq. (9) gives

�� �ffiffiffi2

pd�

) � � 1

diSð�Þ ; (11)

which recovers the electron MHD result obtained inRef. [3]. In particular, for negligible hyper-resistivity, � �ðdiS�Þ�1 and the electron diffusion region is more elon-

gated than in resistive MHD. For negligible resistivity, � �ðdiSHÞ�1=3. Since SH / w2, depending on the macroscopicdriver, this permits electron diffusion regions which areeither elongated, � � w �, or have near-unity aspectratios, � w � �. The reconnection rate is not explicitly

dependent on dissipation coefficients, Ez� �ffiffiffi2

pdi=w, and

is therefore fast. In particular, the reconnection rate isformally independent of the dissipation mechanism, andfeatures a strong dependence on di. Moreover, Ez� does notdepend on �, i.e., both square and elongated electrondiffusion regions allow fast reconnection.

Hall MHD solution.—Expressions (10) and (11) for ��are valid for � � 1 and � � 1, respectively. However, wenote that Eq. (10) smoothly transitions into Eq. (11) ford� � 1. Consequently, it appears reasonable to approxi-mate � for all di with Eq. (10) and the reconnection ratewith

Ez� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2fð�Þ2ð1þ d2�Þ

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

pwSð�Þ þ

2d2iw2

s: (12)

Results (10) and (12) do not involve �, and reduce to thecorrect resistive and electron MHD limits. Moreover, weshall show that Eqs. (10) and (12) are in excellent agree-ment with nonlinear numerical simulations in all regimesof interest.Equation (12) predicts that ‘‘slow’’ (Sweet-Parker) re-

connection occurs for d� & 1, while ‘‘fast’’ (dissipation-independent) reconnection occurs for d� * 3. The naturalthreshold between slow and fast reconnection is d� ¼ 1.Using Eq. (10), this can be written as di ¼ dic � w�c, with

�c satisfying �c ¼ fð�cÞ ¼ ½ ffiffiffi2

pwSð�cÞ�1=2. Then, recon-

nection is slow for di & dic and fast for di * 3dic. Inpractice, one can estimate dic from dissipation coefficientsand w (for which the appropriate equilibrium length scale� can be used) and therefore determine a priori whether agiven di will produce slow or fast reconnection. Thethreshold estimate d� * 3 is consistent with the intuitivecriterion [12,13] that Hall effects are relevant when di > �.Numerical validation of the theory.—We proceed to

validate predictions of Eqs. (10) and (12) by comparingthem with full 2D numerical simulations [14] of the mag-netic island coalescence instability [5]. Values of �, w, Ez,Binx , and B

iny are obtained from the simulation at the time of

FIG. 2 (color online). Ratio of numerical results to analyticalpredictions for � versus normalized di. h (red) correspond todi ¼ 3� 10�4, j (red) to di ¼ 10�3, d (brown) to di ¼ 2�10�3,� (brown) to di ¼ 4� 10�3,w (black) to di ¼ 7� 10�3,e (green) to di ¼ 8� 10�3,r (green) to di ¼ 10�2,b (yellow)to di ¼ 2� 10�2, m (blue) to di ¼ 3� 10�2, 4 (blue) to di ¼9� 10�2, . (blue) to di ¼ 3� 10�1, and 5 (blue) to di ¼ 1.

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maximum Ez. Parameters di, �, and �H are varied overwide ranges to describe situations corresponding to elec-tron, Hall, and resistive MHD regimes: 3� 10�4 di 1, 0 � 10�3, and 0 �H 10�7. Situations wheneither resistive or hyper-resistive effects dominate arestudied, as well as those where both are comparable.Thirty-six nonlinear simulations are considered in total.

The ‘‘raw’’ simulation results for � are found to varybetween � � 0:012 (corresponding to di ¼ 3� 10�4) and� � 0:53 (corresponding to di ¼ 1). Somewhat unexpect-edly, but in agreement with Eq. (10), values of � 0:3–0:4are found in the resistive MHD regime for hyper-resistivitydominated cases, which do not feature fast reconnection.

Equation (10) provides theoretical predictions �th for theaspect ratio. Ratios of numerical results over the theoreticalpredictions, �=�th, are shown versus d� in Fig. 2, anddemonstrate excellent agreement between the theory andthe numerical experiments. The results are coded accord-ing to the value of di, with the same symbols correspondingto the same di but to different values of � and/or �H. Thefact that the average value for �=�th is slightly above unitycan be attributed to our neglect of numerical factors ofOð1Þ in the discrete equations.

Figure 3 shows maximum reconnection rates Ez versusdi obtained from the numerical simulations. One can seethat Ez is small and independent of di for di & 10�2

(resistive MHD regime), but becomes much larger andproportional to di for di * 0:1 (electron MHD regime).The scaling Ez / di is shown with a solid black line. Thisagrees with our prediction Ez ¼ diðBin

x Þ2=w provided thatw and Bin

x are independent of di (the case for the islandcoalescence instability). We note that, depending on themacroscopic driver, situations might exist where w and Bin

x

scale with di, and hence Ez / d�i , � � 1 [9,10].For all di, we find that ratios of the numerical over the

theoretical predictions (12) for normalized reconnectionrates are 0:5< Ez�=Eth

z� < 2:3, i.e., our analytical theoryunder- or over-predicts the numerical reconnection rate by

no more than a factor of 2. This is remarkable given the1:2� 103 spread in numerical Ez� values.In conclusion, we have obtained, for the first time, a

simple analytical theory for magnetic reconnection in all diregimes of Hall MHD. Our theory recovers the well-knownslow Sweet-Parker reconnection in resistive MHD. It con-firms long-standing numerical results by predicting fastreconnection and its independence of the dissipationmechanism in Hall and electron [3] MHD. The theoryalso describes the transition between different regimes,and is thoroughly benchmarked with 2D numerical simu-lations. In addition, it resolves a number of existing con-troversies. In particular, the theory confirms our earlierelectron MHD findings [3] that fast reconnection is pos-sible for both open and elongated X-point configurations,and demonstrates that the reconnection rate can have astrong scaling with di. The theory provides simple bench-marks for existing and future fluid codes, and allows quickestimates of the reconnection rate and of certain parame-ters of the microscopic reconnection region from experi-mental data. More importantly, our approach provides aflexible, extendable framework for the study of magneticreconnection in a fluid context. Many additional physicaleffects, such as electron inertia, ion viscosity, guidingmagnetic field, and perhaps certain kinetic effects, can beconsidered, and their influence on magnetic reconnectionassessed.This research was supported by a Laboratory Directed

Research and Development grant at Los Alamos NationalLaboratory under contract No. DE-AC52-06NA-25396.

[1] J. Birn et al., J. Geophys. Res. 106, 3715 (2001) andreferences therein.

[2] E. N. Parker, J. Geophys. Res. 62, 509 (1957).[3] L. Chacon, A. N. Simakov, and A. Zocco, Phys. Rev. Lett.

99, 235001 (2007).[4] L. Chacon, A.N. Simakov, V. S. Lukin, and A. Zocco,

Phys. Rev. Lett. 101, 025003 (2008).[5] A. N. Simakov, L. Chacon, and D.A. Knoll, Phys. Plasmas

13, 082103 (2006) and references therein.[6] M.A. Shay and J. F. Drake, Geophys. Res. Lett. 25, 3759

(1998).[7] W. Daughton, J. Scudder, and H. Karimabadi, Phys.

Plasmas 13, 072101 (2006).[8] M.A. Shay, J. F. Drake, B. N. Rogers, and R. E. Denton,

Geophys. Res. Lett. 26, 2163 (1999).[9] X. G. Wang, A. Bhattacharjee, and Z.W. Ma, Phys. Rev.

Lett. 87, 265003 (2001).[10] R. Fitzpatrick, Phys. Plasmas 11, 937 (2004).[11] D. Biskamp, Magnetic Reconnection in Plasmas

(Cambridge Univ. Press, New York, 2000).[12] I. J. D. Craig and P. G. Watson, Sol. Phys. 214, 131 (2003).[13] D. A. Knoll and L. Chacon, Phys. Rev. Lett. 96, 135001

(2006).[14] L. Chacon and D.A. Knoll, J. Comput. Phys. 188, 573

(2003).

FIG. 3 (color online). Numerical reconnection rates Ez. Solidblack line corresponds to Ez / di.

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