triggering of magnetic reconnection in the current sheet...

Astron. Astrophys. 326, 1252–1258 (1997) ASTRONOMYAND

ASTROPHYSICS

Triggering of magnetic reconnection in the current sheetby shock wavesD. Odstrcil and M. Karlicky

Astronomical Institute, 251 65 Ondrejov, Czech Republic

Received 14 March 1997 / Accepted 27 May 1997

Abstract. Interaction of a shock wave with a current sheet is in-vestigated by the two-dimensional numerical magnetohydrody-namic model. In accordance with the solar coronal conditions,the ratio of the thermal to magnetic pressures of 0.1 and theshock Alfven Mach number of about 1.25 are used. It is foundthat the shock wave initiates magnetic reconnection process inthe current sheet. Further, it is found that the post-shock situa-tion rather than the shock compression is a cause of the magneticreconnection. Interaction of the shock with the current sheet re-sults in formation of two shocks that propagate away from thecurrent sheet on opposite sides followed by a rarefaction wavesregion. This dynamic situation causes the current sheet to be-come gradually thinner and the magnetic reconnection processis initiated probably due to the tearing-mode instability. Pre-sented results support the idea that the solar flare can be trig-gered by the shock wave from a distant flare.

Key words: Magnetohydrodynamics (MHD) – shock waves –methods: numerical – Sun: flares – Sun: magnetic fields

1. Introduction

Magnetic reconnection is a process where the magnetic fieldchanges its topology and the magnetic field energy is convertedinto kinetic and thermal energies. This local process can playseveral different roles in solar flare phenomena (see the latestresults in Bentley & Mariska 1996). It is believed (for a reviewsee Svestka 1981) that some flares, which represent the recon-nection process, can be initiated by distant flares. The presentsituation in the investigation of the so called sympathetic flareswas recently reviewed and new observational evidence was pre-sented by Bumba & Klvana (1993). Also, there are cases whenso called Moreton waves destabilize distant filaments (Smith &Harvey 1971). Moreover, the flare cannot start immediately infull flare volume; the flare process is probably spreading from aninitially very small volume. All these phenomena need agents

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which are generated in the first flare (or in the first small volumeof the flare) and which trigger the distant flare (or the reconnec-tion in bigger volume of the same flare). Two type of agentswere suggested: (a) superthermal particle beams and (b) shockwaves (e.g. Norman & Smith 1978).

From the theoretical point of view, the scheme of the flarespreading or the triggering of the distant flare by electron beamswas more or less described (Norman & Smith 1978, Karlicky& Jungwirth 1989). Namely, electron beams generate high-frequency waves, which increase the anomalous resistivity ina flare electric current system and thus the reconnection can beinitiated (see also Papadopoulos & Coffey 1974). On the otherhand, physical aspects of the triggering and spreading of flarereconnection processes by shock waves are not known.

Numerical magnetohydrodynamic (MHD) studies in this di-rection were initiated in our previous paper (Odstrcil & Karlicky1997), where the response of the current sheet to parallel shocks(i.e. shocks that propagate along the current sheet) was inves-tigated. This problem is similar to the interaction of interplan-etary (fast-mode MHD) shocks with the heliospheric currentsheet investigated earlier (Odstrcil et al. 1996). However, weinvestigated the interaction process in more detail and, in addi-tion to the fast-mode MHD shock, studied switch-on and slow-mode MHD shocks. We confirmed a distortion of the shockfront and magnetic field lines at the current sheet for all typesof shocks. Because the electric current density in the currentsheet increases, an anomalous resistivity can occur and triggerthe magnetic reconnection process. The current density reachesvery large values in the special case of the switch-on shock butthe propagation of such shocks along the current sheets has alow probability of occurring in real situations.

Therefore, in this paper, we devote our attention to the per-pendicular interaction of shocks with the current sheet. A shockis generated by an “explosion” of a hot plasma tube outside thecurrent sheet and the subsequent interaction of the shock withthe current sheet is followed by numerical simulation. The pa-rameters of the problem are chosen to specify a situation withmoderate shock strength and low plasma β (ratio of thermal tomagnetic pressure) that corresponds to conditions in the solarcorona.

D. Odstrcil & M. Karlicky: Triggering of magnetic reconnection in the current sheet by shock waves 1253

2. Numerical model

The time-dependent resistive MHD approximation is used. Thismodel, written in the conservative form, consists of the follow-ing system of equations:

∂

∂t(ρ) + ∇ · (ρV ) = 0 , (1)

∂

∂t(ρV ) + ∇ · (ρV V ) = −∇(P ) + ∇ ·

(BB

µ

), (2)

∂

∂t(E) + ∇ · (EV ) = −∇ · (PV ) + ∇ ·

(BB

µ· V)

+∇ · (ηJ ×B/µ) , (3)

∂

∂t(B) = ∇× (V ×B) −∇× (ηJ ) , (4)

where ρ is the mass density,V is the flow velocity,P is the totalpressure (sum of the thermal and magnetic pressures),B is themagnetic field induction, E is the total energy density (sum ofthe thermal, kinetic and magnetic energy densities), J is theelectric current density, µ is the magnetic permeability, and η isthe electric resistivity. The equation of state is

p = RρT , (5)

where R is the gas constant and T is the temperature.The actual variables are normalized by the following charac-

teristic quantities: magnetic field by B?, density by ρ?, velocityby the Alfven velocity V ?

A = B?/(µ0ρ?)1/2, length by the half-width of the current sheetL, and time by the Alfven transit timetA = L/V ?

A . Pressure, temperature, current density, and resistiv-ity are then normalized by B2

?/µ0, B2?/(µ0RHρ?), B?/(µ0L),

µ0LV?A , respectively. Finally, the ratio of specific heats, γ, is

5/3, the magnetic permeability is normalized by the permeabil-ity of free space, µ0, and the gas constant is normalized by thegas constant of ideal hydrogen plasma, RH .

The resistivity is assumed to be a function of the currentdensity

η =

{α(|J | − Jc), if |J | > Jc ;0, otherwise,

(6)

where α = 0.05 and Jc = 1.1. Thus it is assumed that abovesome critical value of the current density, Jc, the plasma insta-bility takes place which leads to the anomalous resistivity. Sucha dependence of the resistivity also allows a feedback mecha-nism that could increase the resistivity when the current sheetbecomes thinner and the current density increases (Scholer &Roth 1989). Hayashi & Sato (1978) found that the forced re-connection process depends only weakly on the resistivity coef-ficients and that there is no essential difference between resultsobtained with different critical current densities, Jc. Similarly,Min at al. (1985) found that the reconnection rate does not de-pend on resistivity, but the resistivity influences the onset timeof reconnection.

The explicit finite-difference YDFCT algorithm (Odstrcil1993) is used by a space-splitting technique on a non-staggerednumerical grid. The algorithm can resolve shocks without need-ing an artificial diffusion to stabilize the numerical solution(Toth and Odstrcil 1996). Terms with the resistivity, η, in theMHD model are treated explicitly as “source” terms in the dif-ference schemes. Numerical stability condition restricts valueof the time step, ∆t, by

∆t < ACmin

(∆x

|Vx| + Cx,

∆y

|Vy| + Cy,

(∆x)2

2η,

(∆y)2

2η

)(7)

where Cx and Cy are components of the fast-mode MHD wavespeed and AC = 0.8 is used.

3. Initial conditions

We assume a two-dimensional model for the interaction of theshock with the current sheet and neither a plasma flow nor amagnetic field in the z direction. All quantities depend on thex and y coordinates and the electric current has a z componentonly. The initial state consists of a current sheet near a hot plasmatube and these structures are surrounded by a uniform ambientstate. We shall use an ambient plasma characterized by the massdensity ρ0 = 1, temperature T0 = 0.05, flow velocity V0 = 0,and magnetic field B0 = 1. This corresponds to the followingcharacteristic parameters: plasma β = 0.1 and the Alfven speedVA = 1.

A shock is generated by an “explosion” of the hot plasma att = 0. The hot plasma tube has a radius of 1 and is centered at x= -10 and y = 0. The temperature within this tube is 100 timesthe surrounding ambient values.

The current sheet, a region separating antiparallel magneticfield, is assumed in the y-direction and centered at x = 0. Theinitial configuration of the magnetic field is specified as follows:

B0y(x) =

{B0x/L, if |x| ≤ L;B0x/|x|, otherwise,

(8)

where L = 1 is the half-width of the current sheet. This linearmagnetic field gives a constant current density across the currentsheet. The current sheet is assumed to be initially isothermalwith a constant total pressure. Because the magnetic pressuredecreases within the current sheet to zero at its axis, the thermalpressure increases. The density is then modified to adjust thethermal pressure in order to obtain a constant total pressure.This causes the mass density to increase up to 11 at the currentsheet axis for our case with β = 0.1.

The computational domain represents a region with theequidistant difference mesh with the ∆x = ∆y = 0.2. Thus,10 mesh points are within the current sheet width. All bound-aries are assumed to be free-flow and they are well away fromthe interaction region of interest.

4. Results

Fig. 1 shows the global view on the interaction of the shockwave with the current sheet. This process is expressed here as

1254 D. Odstrcil & M. Karlicky: Triggering of magnetic reconnection in the current sheet by shock waves

Fig. 1. Shock wave interaction with the current sheet (global view) expressed as an evolution of the mass density at t = 10, 20 and 30 tA, wheretA is the Alfven transit time.

an evolution of the mass density within x ∈ (-40,20) and y ∈(-40,40) (in units of L, the current sheet half-width) at t = 10,20, and 30 (in units of tA, the Alfven transit time). Note that themass density values are shown within the subscale of the actualrange of their minimum and maximum values to see details.

The hot plasma tube expands and generates MHD waves thatcan steepen into a shock wave. Velocities of MHD waves dependon the orientation of their propagation direction with respect tomagnetic field lines. This causes a non-symmetric configurationof the resulting shock structure. The fast-mode MHD shockis relatively weak and it is the strongest in a direction acrossthe magnetic field lines. The fast-mode MHD waves propagateslower along the magnetic field lines and they do not steepeninto a shock in our case. The slow-mode MHD shock gives muchlarger density compression and propagates along the magneticfield lines. Only the fast-mode MHD shock can reach the currentsheet, in our configuration, and interact with it. In this paper, weshall use the term shock for the fast-mode MHD shock.

The shock wave front that propagates from its initiation re-gion to the right reaches the current sheet at about t = 6. Theinteraction is perpendicular initially and part of the shock is re-flected (back to the left direction) while another part of the shockpasses through the current sheet and continues in its propagation(to the right direction). Both of these shocks are weaker than theoriginal shock. The reflected shock is seen at t = 10 betweenthe initiation region and the current sheet. This shock interactsand passes through the slow-mode MHD shock structure at latertimes. Part of the original shock that passed through the currentsheet is seen at t = 20 and 30. The current sheet is heavy for

β = 0.1 and thus deflection of its position caused by the shockimpact is very small.

Now, let us see in more details what happens in the currentsheet during the interaction. Figs. 2-5 present the distribution ofthe total pressure (sum of the thermal and magnetic pressure),flow velocity, magnetic field, and current density, respectively,at five times: t = 10, 20, 60, 80, and 100. Here, you can see howthe situation at the current sheet is changed. At t = 10 and 20,plasma velocities correspond to the propagating and reflectingshocks. Large changes of the total pressure and flow veloci-ties occur but the magnetic field lines remain unchanged afterpassage of the shock. Increased values of the current densityin the current sheet are caused by superposition of the originalvalues with the current densities generated at the shock front.These increased values cause anomalous electric resistivity inthe current sheet, however, the reconnection process is not ini-tiated immediately for our case of moderate shock strength inlow β plasma. Later on (t = 60 and 80) the velocity pattern typ-ical for the fast magnetic field reconnection is formed (Priest &Forbes, 1986). This is confirmed by the magnetic field structureand current densities at the current sheet. The plasma is suckedinto the low-pressure region along the magnetic field. Then theplasma enters the interaction region and propagates in oppo-site directions with velocities accelerated to values comparableto ambient Alfven speed. The magnetic field island in the re-connection region is formed (t = 100) due to the tearing-modeinstability in a long current sheet. The distribution of currentdensity (Fig. 5) exhibits a current ridge extending outward fromthe diffusion region. These current ridges are characteristic for


Fig. 2. Shock wave interaction with the current sheet expressed as an evolution of the total presure at t = 10, 20, 60, 80, and 100 tA, where tAis the Alfven transit time.

Fig. 3. Shock wave interaction with the current sheet expressed as an evolution of the velocity vectors at t = 10, 20, 60, 80, and 100 tA, wheretA is the Alfven transit time. The arrow at the bottom corresponds to value of the ambient Alfven velocity, V 0

A.


Fig. 4. Shock wave interaction with the current sheet expressed as an evolution of the magnetic field lines at t = 10, 20, 60, 80, and 100 tA,where tA is the Alfven transit time.

Fig. 5. Shock wave interaction with the current sheet expressed as an evolution of the electric current density at t = 10, 20, 60, 80, and 100 tA,where tA is the Alfven transit time.


Fig. 6. Evolution of the flow velocity at positions x = -2 (dashed line)and x = 2 (solid line) for y = 0. Initial value of the velocity is shownas dotted line. Periods when the velocities are directed towards eachother is indicated by grey filling.

Fig. 7. Evolution of the total pressure at positions x = -2 (dashed line)and x = 2 (solid line) for y = 0. Initial value of the total pressure isshown as dotted line. Periods when the total pressure is less than itsambient value at both positions is indicated by grey filling.

the slow shocks attached to the X point (e.g. Hayashi & Sato1978, Scholer & Roth 1987).

Figs. 6 and 7 show evolution of the flow velocity and totalpressure, respectively, at positions before (x = -2) and behind(x = 2) the current sheet. The shock front arrives at x = -2 atapproximately t = 5. The local shock velocity, computed usingthe travel time between the positions x = -3 and x = -2, is 1.25.Because the ambient Alfven velocity, VA, is 1, the shock Mach

number, just ahead of the interaction region, has the same value.The shock wave has a blast-wave character and both the veloc-ity and total pressure profiles have a sharp peak followed by arapid decrease. The shock interacts with the current sheet andpartially reflects from and partially propagates through the cur-rent sheet (centered at x = 0). It arrives at x = 2 at approximatelyt = 11 and its amplitude is about three times smaller than it wasat x = -2. This effect is mostly caused by interaction with theheavy current sheet though the shock also gradually weakens asit propagates through the ambient medium. The reflected shockis much weaker and it arrives back at x = -2 also at approxi-mately t = 11 (it propagates shorter distance). It can be seen inFigs. 6 and 7 as a small riple in the time-dependent profiles. Astwo shocks propagate away from the current sheet rarefactionregions with backward flows are formed. The situation with col-limated mass flows causes compression of the current sheet thatbecomes thinner and unstable to the tearing instability. This ini-tiates the reconnection process as has been seen also in Figs. 2-5.Once the reconnection process has been initiated, the suckingof plasma into low-pressure region continues faster as shown inFigs. 6 and 7.

5. Conclusions

We have used a two-dimensional resistive MHD model to studythe interaction of a shock with a current sheet. The parametersused for this problem correspond to parameters of the solarcorona except for the large resistivity. Thus, the plasma β (ratioof the thermal to magnetic pressures) is 0.1 and the shock AlfvenMach number is about 1.25.

We have found that the shock wave initiates magnetic recon-nection process in the current sheet. Further, we have found thatthe post-shock situation, rather than the shock compression, isa cause of the magnetic reconnection. The shock compressionheats up the plasma and increases current densities in the currentsheet. However, the effect is of a short duration and the systemdoes not have time to response. The shock passes through thecurrent sheet without any change in the global topology of mag-netic field lines. On the other hand, the dynamic situation afterthe passage and reflection of the shock has much longer effect.These two shocks propagate away from the current sheet on op-posite sides followed by a rarefaction waves region. The currentsheet gradually becomes thinner and the magnetic reconnectionprocess is initiated, probably, due to the tearing-mode instability.A characteristic feature of this process is the pressure decreasein the reconnection region and the plasma flow sucked into thisregion along the magnetic field lines. Plasma flow is acceleratedto velocities of the order of the Alfven speed and a magneticisland is formed at the position of the initial X-point.

For comparison, see the reconnection which was initiatedby the local resistivity enhancement (Karlicky 1988): this re-sistivity locally heated the current sheet, the pressure increasedand the expansion pressure wave was formed. Then, similarlyas in the present case the surrounding plasma was sucked intothe X-point of the reconnection.


We note that another mechanism for triggering magneticreconnection can exist for parallel interaction of the shock withthe current sheet as described in our previous paper (Odstrcil andKarlicky 1997). Mutual interaction between the MHD shocksand the current sheet causes distortion of the shock front andmagnetic field lines. This generates additional electric currentwithin the current sheet and an anomalous resistivity can occurand trigger the magnetic reconnection process. Large currentdensities are generated especially in the case of the switch-onshock, however, propagation of such shocks along the currentsheets have a low probability of occurring in real situations.

Our results support the idea that a solar flare can be triggeredby the shock wave from a distant flare. Thus, so called sympa-thetic flares could be explained. We note that the interaction ofshocks waves with the current sheet seems to be an efficient trig-gering mechanism for magnetic reconnection. If we compare thetriggering of the reconnection by shocks and by superthermalparticle beams, we can see that while beams influence inter-nal conditions – anomalous resistivity, the shocks generate thereconnection through external plasma velocity flows. Becauseshock fronts are much larger structures than cross sections ofparticle beams, they have a larger probability of interacting withcoronal current sheets. Further, we expect that the results pre-sented here could be used also for the interpretation of anotherphenomena observed in the solar atmosphere, e.g. brightening ofsome X-ray features or sudden disappearance of prominences.The disturbances may originate from nearby active regions orsolar flares. Finally, we note that magnetic field reconnection,except sudden release of thermal energy, causes also accelera-tion of particles. Therefore, some radio bursts which represent aradio manifestation of the particle acceleration could be initiatedby shocks.

More studies are necessary to determine which shock waveand current sheet parameters are necessary to initiate the mag-netic reconnection, especially, to investigate lower limit of theshock strength and influence of the electric resistivity.

Acknowledgements. This work was supported by Key Projects K1-003-601, K1-043-601, and Grant No. A3003707 from the Academy ofSciences of the Czech Republic.

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