A saddle-node bifurcation model of magnetic reconnection onset

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1 PHYSICS OF PLASMAS 17, A saddle-node bifurcati model of magnetic recnecti set P. A. Cassak, 1 M. A. Shay, 2 and J. F. Drake 3 1 Department of Physics, West Virginia University, Morgantown, West Virginia 26506, USA 2 Department of Physics and Astromy, University of Delaware, Newark, Delaware 19716, USA 3 IREAP, University of Maryland, College Park, Maryland 20742, USA Received 12 February 2010; accepted 5 May 2010; published line 11 June 2010 It was recently shown that magnetic recnecti exhibits bistability, where the Sweet Parker collisial and Hall collisiless recnecti solutis are both attainable for the same set of system parameters. Here, a dynamical model based saddle-node bifurcatis is presented which reproduces the slow to fast transiti. It is argued that the properties of the dynamical model are a result of the Hall effect and the dispersive physics associated with it. Evidence from resistive two-fluid and Hall magnetohydrodynamics simulatis are presented that show that the time evoluti agrees with the dynamical model, the outflow speed is correlated with the dispersive physics due to the Hall effect, and bistability persists in the absence of electr inertia American Institute of Physics. doi: / I. INTRODUCTION The nlinear dynamics of magnetic recnecti, the process which is thought to enable rapid magnetic energy release in solar eruptis and fusi devices, has lg been a topic of interest. 1,2 Recently, it was shown that recnecti is bistable for a wide range of Lundquist numbers, and that bistability leads to hysteresis-like behavior. 3,4 The two stable states are the collisial Sweet Parker soluti 5,6 which is very slow and the collisiless Hall soluti 7 9 which is fast. A third soluti, which is unstable to small perturbatis and lies between the two stable solutis, was predicted and identified numerically. 10 A csequence of bistability is that the transiti from the collisial to collisiless states is catastrophic, which prompted the suggesti that this is the underlying physics of abrupt set seen in observatis. 3,11 13 A number of recent studies addressed the nlinear dynamics of recnecti using Sweet Parker-type scaling analyses. Flux pileup and time dependence were incorporated into the Sweet Parker theory. 14 A similar approach was employed for recnecti within electr magnetohydrodynamics EMHD without electr inertia, 15 pair plasmas, 16 Hall-MHD without electr inertia, two-fluid Hall- MHD with electr inertia, 22 and EMHD with electr inertia. 23 These analyses employed a resistivity or hyperresistivity electr viscosity to break the frozen-in cditi, though two recent studies used off-diagal pressure tensor terms in scaling analyses within the two-fluid 24 and pair plasma 25 models. A comm result is that resistive recnecti has elgated layers, but hyper-resistive recnecti has either elgated slow or x-type fast layers depending macroscopic driving. Note that many authors use fast to mean weakly dependent dissipati mechanism, whereas we use the more restrictive definiti of weakly dependent dissipati mechanism and system size. Bistability was identified in some of these results. Since bistability occurred in the scaling analyses when electr inertia was present but not otherwise, 17,19,23 it was suggested 23 that electr inertia causes bistability. Previous simulatis 3,4 were ambiguous this point because both the Hall effect and electr inertia were employed simultaneously. This presents an unresolved questi about the physical cause of bistability and hysteresis. In this paper, we present a physical model showing that bistability and hysteresis are caused by the Hall effect. First, we motivate that the dynamics can be described as a saddlenode bifurcati 26 which brings steady-state solutis into and out of existence as a ctrol parameter varies. Then, we argue that the saddle-node bifurcati is a result of the dispersive effects introduced by the Hall term and present a physical explanati of why this is the case. Finally, we present evidence from two-fluid and Hall-MHD numerical simulatis that such a model is borne out numerically. In particular, we show that the time evoluti of the current sheet displays the properties of a saddle-node bifurcati. We show that the outflow speed during the transiti is closely coupled to the dispersive character of the phase speed of waves in Hall-MHD. 7,27 We then use Hall-MHD simulatis without electr inertia to demstrate that the Hall effect ctrols the observed dynamics rather than electr inertia. We cclude with a discussi of implicatis of the theory and discuss how inclusi of secdary islands in Sweet Parker recnecti affects the results. II. THEORY A summary of recnecti dynamics as learned in Refs. 3 and 10 is sketched in Fig. 1 a. The normalized recnecti rate E =ce/c A B x is plotted as a functi of collisiality parameter = c 2 /4 c A d i, where E is the recnecti electric field, B x is the recnecting field immediately upstream of the current layer, is the Spitzer resistivity, c A is the Alfvén speed based B x, and d i is the i inertial length. We define x as the outflow directi, y as the inflow directi, and z as the out-of-plane directi. At high, ly the Sweet Parker soluti exists. At low, ly the X/2010/17 6 /062105/7/$ , American Institute of Physics

2 Cassak, Shay, and Drake Phys. Plasmas 17, physics of the standard Sweet Parker soluti. In this soluti, cvecti outside the layer balances diffusi within it. If cvecti exceeds diffusi, the layer becomes thinner; if diffusi exceeds cvecti, the layer becomes broader. This leads e to posit an evoluti equati for of the form 28 d dt = v in + c2 4, 1 where v in is the inflow speed. Formally, this can be derived from a scaling analysis applied to the out-of-plane z compent of the resistive MHD Ohm s law E z + v B = J z. 2 c z FIG. 1. a Schematic summary of the nlinear dynamics of magnetic recnecti found in Refs. 3 and 10. The dashed line shows a predicted unstable steady-state recnecti soluti. b Bifurcati model of recnecti which reproduces the behavior in panel a. Shown is a schematic phase portrait for. As ctrol parameter decreases, the curve moves down. Heavy dots denote steady-state solutis. Hall soluti exists. At intermediate, both exist and the system is bistable. The dashed line correspds to the unstable soluti found in Ref. 10. In trying to understand the cause of the observed transiti dynamics, we sketch the simplest dynamical model which reproduces the observed behavior. A bifurcati model is plotted schematically in Fig. 1 b as a phase portrait a variable plotted against its time derivative for various values of a ctrol parameter. We identify the dynamical variable as the half-thickness of the current sheet and the ctrol parameter as the collisiality parameter. We show later why this choice of variables and parameters was made. We now motivate that the dynamics described by Fig. 1 b reproduces the observed dynamics. Steady-state equilibria fixed points occur where d /dt=0 and are marked by heavy dots. The uppermost curve is for large collisiality, with a single stable steady-state soluti at relatively high, correspding to the Sweet Parker soluti. As decreases, the curve moves down and two fixed points e stable, e unstable are borne in a saddle-node bifurcati. 26 The stable fixed point at small correspds to the Hall soluti. For a range of, there are two distinct stable steady-state solutis as shown in the middle curve, i.e., there is bistability. The existence of the unstable soluti 10 between the two stable solutis appears naturally in this model. As ctinues to decrease, the curve moves lower, and the unstable and Sweet Parker fixed points approach each other, coalesce, and disappear in a secd saddle-node bifurcati. For low collisiality, the ly fixed point is the Hall soluti, as is shown in the lowest curve in Fig. 1 b. Therefore, this simple dynamical picture reproduces the complicated observed dynamics. While Fig. 1 b is potentially a valid descripti, it does not explain why such a descripti should arise. Before arguing that the key physics is the Hall effect, we review the Using J z cb x /4 from Ampère s law, v B z v in B x, and E z 1/c A z / t with A z B x and assuming that B x is relatively cstant and being careful with minus signs, e arrives at Eq. 1. This result is similar to that derived independently in Ref. 14. Using ctinuity, e finds v in c A /L SP, where L SP is the length of the Sweet Parker current layer in the outflow directi. This leaves d dt = c A L SP + c2 4. As a check, the fixed point where d /dt=0 is indeed the standard Sweet Parker soluti, = c 2 L SP /4 c A 1/2. The phase space portrait starts at positive infinity for small and decreases motically to negative infinity for large. Therefore, there is ly e steady soluti in resistive MHD recnecti, the Sweet Parker soluti. In particular, the robust dynamics observed in Refs. 3 and 10 is not captured. The Sweet Parker soluti is stable to small perturbatis in. This is because a small decrease in the layer leads to a decrease in cvecti and an increase in diffusi, so the layer s respse is to broaden, which opposes the perturbati. With this interpretati of Sweet Parker recnecti in mind, csider the dynamics described in Fig. 1 b. It is known that Hall recnecti is marked by the decoupling of is from the magnetic field and electrs at a length scale of the thermal or inertial i gyroradius. 29 The current sheet thickness is electr scales, 30 and while resistive effects may play a small role, the bulk of the recnecti electric field is balanced by portis of the off-diagal pressure tensor. 31 The unstable soluti exists at a current sheet width that is intermediate between electr and i length scales. 10 As a csequence, is have decoupled from the field, but the electrs have not. In this regi, electr inertia effects are negligible, which means that the ly possible physical mechanism allowing a balance of the recnecti electric field is for electr cvecti i cvecti plus the Hall term to balance resistive diffusi. The mechanism for Ohm s law balance is plotted schematically in Fig. 2 for the a Sweet Parker, b unstable, and c Hall solutis. Red lines denote oppositely directed magnetic fields, blue lines are i flow, and black lines are electr flow. The blue box 3

3 A saddle-node bifurcati model Phys. Plasmas 17, FIG. 2. Color Schematic of the governing physics for the a Sweet Parker, b unstable, and c Hall recnecti solutis. Red lines are oppositely directed magnetic field lines, blue lines denote i flow, and the black lines denote electr flow. The purple box denotes a resistive diffusi regi, while the black box denotes a diffusi regi dominated by kinetic effects. denotes i gyroscales. The shaded boxes denote the dissipati regi defined as where the frozen-in cditi is violated, with purple for resistive diffusi and black for kinetic effects such as pressure tensor terms. The recnecti rate of the unstable soluti can readily be computed in the limit where the i inflow is small compared to the electr inflow. Balancing the Hall term and the resistive term in the out-of-plane z compent of the generalized Ohm s law gives J y B x /nec J z. Using J y cb z /4 L and J z cb x /4 from Ampere s law gives E, 4 where E =ce/b up c Aup /L is the normalized recnecti rate, L is the length of the dissipati regi in the outflow directi, and we have assumed that B z B x. This predicti is in good agreement with the simulatis in Ref. 10. Note that E scales like as opposed to 1/2 for Sweet Parker, but is independent of the system size L SP, so is actually faster than the Sweet Parker rate, as is seen in Fig. 1 a. This scaling analysis has been previously carried out, 15,17 19,22,23 but it has not been appreciated that this soluti correspds to an unstable, and thus physically unrealizable, soluti. The fact that the unstable soluti is a balance between electr cvecti and resistive diffusi has profound implicatis about recnecti dynamics. As in the Sweet Parker model, cvecti tends to reduce the thickness of the layer, while diffusi broadens it. For the unstable soluti to be unstable, it must be true that compressing the layer leads to an increase in cvecti that overcomes the increase in diffusi, so runaway toward smaller length scales occurs. The runaway process stops ly when additial physics, such as off-diagal elements of the pressure tensor, become important at electr scales, which is the Hall soluti. In two-fluid simulatis of Hall recnecti, the runaway process is often stopped using an explicit high order dissipati term such as hyperviscosity or through numerical dissipati because off-diagal pressure tensor terms are absent from the model. The physical cause of this runaway lies in the Hall effect. Recnecti outflow is driven by the straightening of newly recnected field lines, which at subi gyroradius scales is the whistler wave. The whistler wave is dispersive, so is faster at smaller scales. Up squeezing the layer, the outflow speed increases. 7,32 By ctinuity, the increase in outflow speed leads to an increase in inflow speed. This increase in cvecti due to Hall physics is sufficient to overpower diffusi and lead to runaway. We cclude that the Hall effect is crucial to enabling the bistability and hysteresis. For bistability, in terms of the phase portrait in Fig. 1 b, the Hall effect increases the inflow speed above where it would be in the absence of the Hall effect, which from Eq. 1 makes d /dt less negative between i and electr length scales. This leads to the bump in the phase portrait which gives rise to saddle-node bifurcatis as is varied. For hysteresis, the resistive term balances the recnecti electric field during Sweet Parker recnecti. Once a transiti to Hall recnecti is made, the recnecti electric field is orders of magnitude faster, so the resistive term cannot play much of a role. Changing by a small amount after Hall recnecti starts has no effect the system, which is the physical cause of hysteresis. Therefore, the Hall effect and the dispersive behavior introduced by it are instrumental in leading to the rich dynamics observed in previous simulatis. III. NUMERICAL SIMULATIONS The bifurcati model in terms of saddle-node bifurcatis makes predictis that can be directly tested with numerical simulatis. In this secti, we present three pieces of evidence that support the model in the previous secti. In what follows, magnetic fields, densities, velocities, lengths, electric fields, and resistivities are normalized to B 0, n 0, the Alfvén speed c A0 =B 0 / 4 m i n 0 1/2, the i inertial length d i = m i c 2 /4 n 0 e 2 1/2, E 0 =c A0 B 0 /c, and 0 =4 c A0 d i /c 2. A. Time evoluti of the dissipati regi thickness Figure 1 b makes a predicti about the time evoluti of the thickness of the dissipati regi. We can test this predicti using results from the two-fluid numerical simulatis presented in Ref. 3. The simulatis are described fully in Ref. 3, but we relay the salient details here. We employ the double Harris sheet cfigurati with doubly periodic boundary cditis in a system with size of d i with grid scale of 0.1d i and initial Harris sheet width of 2d i using the two-fluid code F3D. 33 The electr mass is m e =m i /25 and the fourth order diffusi coefficient hyperviscosity is Cvergence tests have been performed. In these simulatis, Sweet Parker recnecti was attained with a resistivity of =0.015 despite the presence of the Hall term. Then, the resistivity was lowered below the threshold value. This creates a system that is

4 Cassak, Shay, and Drake Phys. Plasmas 17, (a) (b) (c) t FIG. 3. Simulati data of the dissipati regi thickness vs time t for simulatis from Ref. 3 in which a transiti from Sweet Parker to Hall recnecti is made when is lowered from to a 0.003, b 0.007, and c out of equilibrium which will transiti to Hall recnecti, so its time evoluti can be studied. During the transiti, the thickness of the layer defined as the half-width at half-maximum of the out-of-plane current density J z is measured as a functi of time. The behavior of versus time appears in Fig. 3 for runs with the lowered resistivities =0.003, 0.007, and 0.009, seen as a collapse from i 1 to electr d e = length scales. The resultant phase portraits of are shown in Fig. 4, with the solid black, blue, and red lines for the =0.003, 0.007, and simulatis, respectively. Clearly, they qualitatively resemble the small curve in Fig. 1 b, which supports the saddle-node bifurcati picture of the dynamics. For completeness, the dot-dashed line is for a simulati in which a transiti from Hall to Sweet Parker recnecti occurs when the resistivity is suddenly increased to d/dt FIG. 4. Color Phase portrait for cstructed from the data in Fig. 3. The solid lines are for =0.003 black, blue, and red. The results are similar to Fig. 1 a. The dot-dashed line is for a transiti from Hall to Sweet Parker recnecti when the resistivity is increased to The dashed line is for a run with zero electr mass, showing that the behavior persists. v e,out / B e,up As expected, increases in time until the system reaches the Sweet Parker soluti. A quantitative comparis with Eq. 3 is not feasible because the upstream magnetic field changes in time, an effect not included in the present analysis. Such effects are csidered in Ref. 14. B. Relati of outflow speed and Hall-MHD wave phase speed It was argued in Sec. II that the cause of bistability is the nature of the dispersive behavior of Hall physics. In particular, it was argued that the outflow is driven by the wavelike character of the newly recnected field lines. Here, we show evidence that the outflow speed during a transiti from Sweet Parker to Hall recnecti is correlated with the phase speed of the whistler wave in two-fluid theory. The phase speed v phase of a two-fluid wave with wavenumber k parallel to the equilibrium field is 33,34 2 v phase = 2 k 2 = c 2 A D 1+ k2 2 2 d i d i 2D + k2 D Simulati data Whistler Disp. Rel. FIG. 5. Color Electr outflow speed v e,out normalized to the magnetic field upstream of the electr layer B e,up as a functi of current layer half-thickness during the transiti from Sweet Parker to Hall recnecti from Ref. 10. The dashed line gives the phase speed v phase of Hall-MHD waves from Eq. 5 using k 1/. + k4 4 d i 4D 2, where D=1+k 2 d 2 e and d e is the electr inertial scale. This gives Alfvén waves at small k, whistler waves for intermediate k, and electr cyclotr waves for large k. Results are shown in Fig. 5 for the simulati in which a transiti from Sweet Parker to Hall recnecti occurred when was lowered from to During this transiti, we measure the electr outflow speed v e,out the maximum of electr velocity in the outflow directi in a cut through the X line, the half-width at half-maximum of the out-of-plane current density J z, and the magnetic field B e,up at the electr layer the magnetic field upstream of the X line. The solid blue line is v e,out normalized to B e,up as a functi of during the transiti. On the same plot, the phase speed of the two-fluid wave from Eq. 5 is plotted as the red dashed line, using k 1/. Clearly, the outflow speed is well described by the phase speed of the two-fluid wave throughout the transiti from Sweet Parker to Hall 5

5 A saddle-node bifurcati model Phys. Plasmas 17, E η off η Hall off Hall t/t f FIG. 6. Color Demstrati of bistability in the absence of electr inertia. Recnecti rate E vs time t normalized to the final time t f of the simulati for the runs described in the text. The vertical dashed lines show when the added effects were enabled. The horiztal line gives the predicted Sweet Parker rate for these parameters. recnecti, spanning MHD length scales down to electr length scales. Using a more precise value of k /2 Ref. 33 merely shifts the theoretical curve but does not change that the two curves follow each other. This provides evidence that the outflow speed increases with thinner layers within i gyroradius scales due to the Hall effect. C. Dependence of bistability electr inertia To determine whether bistability occurs as a result of the Hall effect or electr inertia, we perform new simulatis similar to those in Ref. 3 using the massively parallel code F3D, 33 but without electr inertia. Instead of electr inertia, we use a fourth order hyper-resistivity with coefficient D 4 = in all of the equatis. This value of D 4 is four times larger than in Ref. 3. This term regularizes the whistler mode at small scales in the absence of electr inertia. Even with the larger D 4, a time step 20 times smaller than in Ref. 3 is required to ensure stability. The two-dimensial periodic domain has size L x L y = with grid scale of 0.1. The initial equilibrium is two Harris sheets in a double tearing mode cfigurati with initial current sheet thickness of 1.2. Initially, is are statiary and pressure is balanced with a nuniform density n asymptoting to 1 far from the sheet. There is no initial out-of-plane guide magnetic field. The plasma is isothermal with cstant and uniform temperature T=1. There is no viscosity. Initial random perturbatis the magnetic field of amplitude break symmetry so secdary magnetic islands are ejected. The resistivity =0.012 is cstant and uniform when used. This value of ensures that we are in the bistable range, but also outside of the range of ctinual secdary island formati. We address the formati of secdary islands in Sec. IV. Recnecti is initiated using a coherent field perturbati with amplitude of A simulati is begun with nzero but the Hall term turned off. The recnecti rate E, measured as the time rate of change in magnetic flux between the X and O line, is (a) (b) y y x FIG. 7. Color Out-of-plane current density J z for the a fast and b slow stable solutis in Fig. 6 in the absence of electr inertia. For clarity, ly a porti of the computatial domain is plotted with an altered aspect ratio and the color table in a has been stretched. plotted as the red solid line in Fig. 6 as a functi of time t relative to the final time of the simulatis t f =1776. As expected, E agrees with the Sweet Parker predicti E SP /L SP 1/ , shown as the horiztal solid line. Here, L SP is the half-length of the current sheet in the outflow directi; L SP L x /4 since the full length takes up half the periodic domain. At t=1200, the Hall term is enabled. The lower dashed line shows E, revealing that recnecti remains slow. A secd series of simulatis is initiated with the Hall effect enabled, but =0. The recnecti is fast, as seen in the blue solid line in Fig. 6, where t f = At t=538.75, a resistivity of =0.012 is enabled. The upper dashed line shows E. While it does drop slightly because the layer broadens slightly, it clearly remains much faster than E SP. In summary, the two dashed curves differ ly in their time history, but are in different cfiguratis, which demstrate bistability even in the absence of electr inertia. E η: Hall: off off t FIG. 8. Color Demstrati of hysteresislike behavior in the absence of electr inertia. Plotted is the recnecti rate E as a functi of time t. The system changes states when is changed from to 0 and back.

6 Cassak, Shay, and Drake Phys. Plasmas 17, The out-of-plane current density J z is plotted in Fig. 7 with magnetic field lines superimposed for the Sweet Parker and Hall recnecti solutis. The dramatic difference between the states is clearly seen. Note that the Sweet Parker cfigurati exhibits a secdary island, which can induce a transiti to fast recnecti. 35,36 However, despite the island, fast recnecti does not occur. While hysteretic behavior seems to follow directly from bistability, it is worth cfirming with simulatis see Fig. 8 for E versus t. The run is started in the slow recnecti regime using = At t=1500, is zeroed and the recnecti becomes fast. The resistivity is then returned to its original value, and recnecti remains fast. Thus, removing and replacing leads to different physical states, which is hysteretic. Since the ly nresistive MHD effect is the Hall effect, it is the cause of bistability and hysteresis. For completeness, we show the phase portrait for the evoluti of the thickness for the simulati with zero electr mass as the dashed line in Fig. 4, which shows d /dt versus. Even in the absence of electr inertia, e sees the bump in the plot which is a key signature of the saddle-node bifurcati. This behavior is further evidence that the Hall effect drives the dynamics. IV. DISCUSSION In summary, we propose a model describing the nlinear dynamics of magnetic recnecti using saddle-node bifurcatis. This model is able to reproduce previously observed phenomena, such as hysteresis, bistability, and the existence of an unstable mode. 3,10 We argue that the unstable mode is a balance between electr cvecti i cvecti plus the Hall term and resistive diffusi. As a result, we provide a physical mechanism demstrating that the dispersive behavior of the Hall effect causes instability of the mode and show why the dynamics of recnecti is described as a saddle-node bifurcati. Results of two-fluid and Hall-MHD simulatis are presented which are csistent with new predictis of this model. We show that the time evoluti of the thickness of the layer follows the predicted behavior. We show that the outflow speed during a transiti from Sweet Parker to Hall recnecti is well described by the phase speed of the two-fluid wave. Finally, we show that the bistability and hysteresis ctinue to occur in a system without electr inertia, which implies that the Hall effect is the relevant physics for achieving bistability and hysteresis. These results counter the claim 23 that the cause of the hysteresis is electr inertia. It is outside the scope of this study to determine why previous analyses attain different results, but it should be addressed in future work. In particular, it has been emphasized 20,21 that the models do not selfcsistently predict the length of the diffusi regi, which makes them incomplete. As mentied in Sec. II, some previous authors studied Hall recnecti with balancing the recnecti electric field. 15,17 19,22,23 The present results reveal that this state correspds to the unstable soluti, so this regime is physically unrealizable. The present results do not address recnecti in pair plasmas. Such a system has no Hall effect, but has other additial physics due to the equal inertia of the two species. 16 The present study uses parameters where secdary islands do not play an important role Lundquist numbers below However, for large Lundquist numbers, secdary islands qualitatively change Sweet Parker recnecti. 37 Their role remains under study. 36,38 41 One result that is clear is that secdary islands lead to smaller length scales in the outflow directi, which causes the layers to become thinner in the inflow directi where they induce the set of collisiless effects soer than in the absence of secdary islands. 35,36 However, collisiless recnecti still sets catastrophically when the thinnest part of the Sweet Parker layer reaches i gyroscales. 36 Thus, in making compariss to physical systems, it is the current sheet thickness including secdary islands that should be used to compare to kinetic scales to see when transitis arise rather than the global Sweet Parker scale lengths. 36 ACKNOWLEDGMENTS The authors have learned that a simulati study similar to that in Sec. III C is being carried out by B. P. Sullivan et al. see Ref. 42. The authors acknowledge valuable cversatis with J. D. Huba. Computatis used resources at the Natial Energy Research Scientific Computing Center. The authors acknowledge support by grants NSF Grant No. PHY P.A.C., NSF Grant No. ATM M.A.S., NASA Grant No. NNX08AM37G M.A.S., and NSF Grant No. PHY J.F.D.. 1 X. Wang and A. Bhattacharjee, Phys. Rev. Lett. 70, Z. W. Ma and A. Bhattacharjee, Geophys. Res. Lett. 23, 1673, doi: /96gl P. A. Cassak, M. A. Shay, and J. F. Drake, Phys. Rev. Lett. 95, P. A. Cassak, J. F. Drake, and M. A. Shay, Phys. Plasmas 14, P. A. Sweet, in Electromagnetic Phenomena in Cosmical Physics, edited by B. Lehnert Cambridge University Press, New York, 1958, p E. N. Parker, J. Geophys. Res. 62, 509, doi: /jz062i004p M. E. Mandt, R. E. Dent, and J. F. Drake, Geophys. Res. Lett. 21, 73, doi: /93gl M. A. Shay, J. F. Drake, B. N. Rogers, and R. E. Dent, Geophys. Res. Lett. 26, 2163, doi: /1999gl J. Birn, J. F. Drake, M. A. Shay, B. N. Rogers, R. E. Dent, M. Hesse, M. Kuznetsova, Z. W. Ma, A. Bhattacharjee, A. Otto, and P. L. Pritchett, J. Geophys. Res. 106, 3715, doi: /1999ja P. A. Cassak, J. F. Drake, M. A. Shay, and B. Eckhardt, Phys. Rev. Lett. 98, P. A. Cassak, J. F. Drake, and M. A. Shay, Astrophys. J. Lett. 644, L D. A. Uzdensky, Astrophys. J. Lett. 671, P. A. Cassak, D. J. Mullan, and M. A. Shay, Astrophys. J. Lett. 676, L A. N. Simakov, L. Chacón, and D. Knoll, Phys. Plasmas 13, L. Chacón, A. Simakov, and A. Zocco, Phys. Rev. Lett. 99, L. Chacón, A. Simakov, V. S. Lukin, and A. Zocco, Phys. Rev. Lett. 101, A. N. Simakov and L. Chacón, Phys. Rev. Lett. 101,

7 A saddle-node bifurcati model Phys. Plasmas 17, A. N. Simakov and L. Chacón, Phys. Plasmas 16, L. M. Malyshkin, Phys. Rev. Lett. 101, D. A. Uzdensky, Phys. Plasmas 16, B. P. Sullivan, A. Bhattacharjee, and Y.-M. Huang, Phys. Plasmas 16, L. M. Malyshkin, Phys. Rev. Lett. 103, A. Zocco, L. Chacón, and A. N. Sinakov, Phys. Plasmas 16, D. Tsiklauri, Phys. Plasmas 15, M. Hesse, S. Zenitani, M. Kuznetsova, and A. Klimas, Phys. Plasmas 16, S. H. Strogatz, Nlinear Dynamics and Chaos Addis-Wesley, Reading, B. N. Rogers, R. E. Dent, J. F. Drake, and M. A. Shay, Phys. Rev. Lett. 87, P. A. Cassak, Ph.D. thesis, University of Maryland, B. U. Ö. Snerup, in Solar System Plasma Physics, edited by L. J. Lanzerotti, C. F. Kennel, and E. N. Parker North Halland, Amsterdam, 1979, Vol. 3, p M. Hesse, K. Schindler, J. Birn, and M. Kuznetsova, Phys. Plasmas 6, M. Hesse, M. Kuznetsova, and M. Hoshino, Geophys. Res. Lett. 29, 1563, doi: /2001gl M. A. Shay and J. F. Drake, Geophys. Res. Lett. 25, 3759, doi: / 1998GL M. A. Shay, J. F. Drake, M. Swisdak, and B. N. Rogers, Phys. Plasmas 11, X. Wang, A. Bhattacharjee, and Z. W. Ma, J. Geophys. Res. 105, 27633, doi: /1999ja K. Shibata and S. Tanuma, Earth, Planets Space 53, W. Daught, V. Roytershteyn, B. J. Albright, H. Karimabadi, L. Yin, and K. J. Bowers, Phys. Rev. Lett. 103, D. Biskamp, Phys. Fluids 29, N. F. Loureiro, A. A. Schekochihin, and S. C. Cowley, Phys. Plasmas 14, A. Bhattacharjee, Y.-M. Huang, H. Yang, and B. Rogers, Phys. Plasmas 16, P. A. Cassak, M. A. Shay, and J. F. Drake, Phys. Plasmas 16, P. A. Cassak and J. F. Drake, Astrophys. J. Lett. 707, L B. P. Sullivan, Y.-M. Huang, and A. Bhattacharjee, On the questi of hysteresis in Hall MHD recnecti, Phys. Plasmas submitted.