ASTRONOMY AND Modelling explosive events in …aa.springer.de/papers/9351002/2300721.pdfAstron. Astrophys. 351, 721–732 (1999) ASTRONOMY AND ASTROPHYSICS Modelling explosive events

Astron. Astrophys. 351, 721–732 (1999) ASTRONOMYAND

ASTROPHYSICS

Modelling explosive events in the solar atmosphere

L.M. Sarro 1, R. Erdelyi2, J.G. Doyle3, and M.E. Perez3

1 Laboratorio de Astrofısica Espacial y Fısica Fundamental (LAEFF) INTA, P.O. Box 50727, 28080 Madrid, Spain ([email protected])2 Department of Applied Mathematics, University of Sheffield, Hicks Building, Western Bank, S3 7RH, England, UK

([email protected])3 Armagh Observatory, College Hill, Armagh, BT61 9DG, Ireland (jgd,[email protected])

Received 4 June 1999 / Accepted 30 August 1999

Abstract. High–resolution ultraviolet (UV) spectra of the outersolar atmosphere show transient brightenings often referred toas explosive events. These are localized regions of small spatialextent that show sudden enhancements in the intensities of linesformed between 20,000 and 200,000 K, accompanied by strongnon-gaussian profiles.

The present work is an attempt to extract observational con-sequences from computational simulations of the dynamic re-sponse of a coronal loop to energy perturbations. Explosiveevents are simulated in semi-circular magnetic flux tubes. Ther-mal energy perturbations drive flows along the flux tube giv-ing rise to thermodynamic phenomena. The temporal evolu-tion of the thermodynamic state of the loop is converted intoC iv λ 1548.2A line profiles in (non)-equilibrium ionization.Time dependent carbon ion populations are obtained in the non-equilibrium conditions derived from the thermodynamic vari-ables by means of an adaptive grid code. Most important, de-partures from ionization equilibrium are assessed for the firsttime under conditions such as those encountered in explosiveevents.

Key words: Sun: atmosphere – Sun: particle emission – Sun:UV radiation

1. Introduction

High resolution ultra-violet (UV) spectra taken with theNRL High Resolution Telescope and Spectrograph (HRTS)(Brueckner & Bartoe 1983) show transient enhancements in thewings of lines formed in the solar transition region result-ing in highly non-Gaussian line profiles. Similar observationswere carried out with the UVSP instrument on boardSMMwhich detected line broadenings in ‘hotter lines’ associated withneutral lines in magnetic bipoles (see e.g. Porter et al. 1987);and finally with the instrument SUMER on boardSOHO inJune–July 1996 (see e.g. Doschek 1997, Perez et al. 1999 andChae et al. 1998). The most complete statistical description ofthe explosive events can be found in Brueckner & Bartoe (1983)and Dere et al. (1989). The main conclusions derived from thesestudies can be summarized as follows:

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– the average size of the events along the slit is≈ 2 arcsec(1500 km) as seen in the Civ resonance lines.

– the observed velocities range between±250 kms−1 and±50 kms−1

– a particular explosive event may present different time his-tories with distinct activity episodes which makes it diffi-cult to assign time scales to these phenomena. The lifetimedistribution of observed explosive events ranges between20–200 s with an average value of≈ 40 s and a peak in thedistribution around20 s

– the maximum line-of-sight velocity of explosive events onthe disk is independent of latitude, suggesting an isotropicdistribution of velocities at the explosion site.

The strongest explosive events have been associated withthe He i dark points which coincide with X-ray brightpoints in the corona (Porter et al. 1987). X-ray bright pointsare the observational signature of the hot tops of magneticbipoles made up of smaller loops (see e.g. Krieger et al. 1971or Sheeley & Golub 1979).Hα observations and magne-tograms suggest a connection with emerging magnetic flux(Brueckner et al. 1988). In a recent paper Chae et al. (1998)conclude, based on a sample of 163 explosive events observedsimultaneously with SUMER on boardSOHOand with the BigBear Solar Observatory video-magnetograph, that this type ofphenomena occur preferentially in regions of mixed polarityand not in the interior of strong flux concentrations, and that themajority of explosive events are associated with photosphericmagnetic flux cancellation.

The ultimate origin of the input energy that drives the flowof material detected in the explosive events has not been es-tablished yet. Consequently, computational efforts concentratedinitially on the simulation of the response of the solar atmo-sphere to a sudden release of energy (see e.g. Sterling et al. 1993or Sterling et al. 1991) regardless of the nature of the energy re-lease itself.

Only recently, has the physical reason for the energy re-lease, and in particular, magnetic reconnection as the mostplausible mechanism behind these phenomena (Parker 1988,Cargill 1994), began to be included in the simulations. This hasbeen accomplished by means of magnetohydrodynamic codesthat simulate magnetic reconnection episodes in the solar at-

722 L.M. Sarro et al.: Modelling explosive events in the solar atmosphere

mosphere (e.g. Karpen et al. 1995 or Jin et al. 1996) althoughunderad hocassumptions that neglect the effect of radiativelosses and heat conduction.

In this paper we attempt to assess the validity of the inter-pretation of the explosive events as the response of the solaratmosphere to an energy release of unknown origin placed be-low the transition region by extracting spectral information ofthe Civ λ1548.2A line from numerical simulations. This re-quires the calculation of the time-dependent ion populations ofcarbon along the loop given that the dynamical situation is suchthat, as discussed in Sect. 3, the hypothesis of ionization equi-librium is no longer acceptable. Therefore, ionization balancecalculations have to be carried out in order to synthesize theradiative output of the loop in a given spectral line.

In Sect. 2 we describe the numerical simulation of explo-sive events and in Sect. 3 the extraction of observational con-sequences from the explored parameter space. In Sect. 4 andSect. 5 we present and discuss the main conclusions that can bedrawn from the present study.

2. Hydrodynamical simulations

2.1. Physical layout

Explosive events are simulated in a one-dimensional semi-circular rigid magnetic flux tube of total length 13,000 km. Forplasma where the plasma-beta,β = Pg/Pmag is smaller thanunity, the assumption of a rigid loop is commonly used.

The equations governing the structure and evolution of theloop are (see Mariska 1992):

∂ρ

∂t+

∂(ρ · v)∂s

= 0, (1)

∂(ρ · v)∂t

+∂(ρ · v2)

∂s= −ρ · g(s) − ∂P

∂s, (2)

∂E

∂t+

∂

∂s

[(E + P ) · v − κ · ∂T

∂s

]= −ρ · v · g(s) − L + S (3)

where

E =12

· ρ · v2 +P

γ − 1, (4)

is the sum of kinetic and internal energy,s the distance along theloop,ρ,v andT are the plasma density, velocity and temperaturerespectively,P the pressure,g(s) the gravity,γ the ratio ofspecific heats,κ the Spitzer’s classical conductivity,L denotesthe radiative loss function andS denotes the volume heatingrate.

These equations have to be complemented with the perfectgas law (see e.g. Mariska 1992)

P =ρ

µmpKbT, (5)

whereµ is the mean atomic weight andmp is the proton mass.We have used a constant value of 0.61 forµ corresponding to a

totally ionized gas composed of a 90% of hydrogen and a 10%helium. The implications of this assumption for the computationof the ionization states of the loop are discussed in Sect. 3.

Gravity forces are included taking into account the rigidityof the loop so that,g(s) = g0 cos α whereα denotes the anglebetween the tangent to the loop at points, and the gravity vectordownwards, andg0 = g |s=0= 2.7 102 m s−2.

For the radiative loss function,L, we use the analyti-cal expression given by Rosner et al. (1978) modified as inSterling et al. (1991) while for the input heating rate,S, we takea constant value per unit volume of3.6 10−4 erg cm−3 s−1 thatyields the desired loop length.

A stable initial solution is readily obtained by dropping thetime derivatives in Eqs. (1) – (3) and imposing the existenceof two model chromospheres 1,500 km thick at both ends ofthe loop. This is accomplished by imposing the exact balancebetween heating and radiative losses. Ats=1,500 km, we use aRunge-Kutta algorithm with adaptive step-size to integrate thecomplete set of Eqs. (1)–(3).

This initial solution is perturbed introducing an additionalheating term localized below one of the transition regions ofthe initial solution. The exact functional form of the energysupplement is given by

∆H = H0 · exp(−α · t) · exp(

− (x − x0)2

β2

)(6)

that represents an exponential decay in time and a Gaussiandistribution of energy in space. In Eq. (6),H0 is the maximumamount of energy injected at the onset of the explosion and

α = − ln 0.1∆t

(7)

and

β =

√M2

− ln 1/2. (8)

are the parameters controlling the shape of the temporal andspatial distributions.α is such that 90% of the total energy isdeposited in the first∆t seconds of the simulation whereasβis related to the Gaussian distribution throughM , the FWHM.x0 represents the point of energy deposition around which thespatial distribution is centered.

H0 is obtained from

ET =H0σβ

√π

α(9)

once the total energy to be deposited,ET, is chosen. This totalenergy depends onσ, the cross section of the loop. Given thesymmetries adopted, the value ofσ will not affect the result ofthe simulations and, therefore, in what follows we will useET/σ(erg cm−2) as the free parameter regulating the magnitude ofthe energy input. In practice, onlyx0, ∆t andET/σ are variedin order to explore the parameter space, whileM is fixed andequal to 30 km. Table 1 shows the region of the parameter spaceexplored in the simulations.

L.M. Sarro et al.: Modelling explosive events in the solar atmosphere 723

Table 1. Regions of parameter space explored in the simulations.x0

is the space coordinate of the explosion expressed in Mm,ET/σ is thetotal energy injected in erg cm−2, and∆t is the parameter governingthe rate at which this energy is deposited.

∆t = 60 sx0 ET/σ = 2.5 × 109 ET/σ = 6.25 × 109

1.0 ×1.3 × ×1.45 × ×

∆t = 300 sx0 ET/σ = 2.5 × 109 ET/σ = 6.25 × 109

1.0 ×1.3 × ×1.45 × ×

∆t = 600 sx0 ET/σ = 2.5 × 109 ET/σ = 6.25 × 109

1.0 ×1.3 × ×1.45 × ×

This exploration of the parameter space was designed itera-tively based on the emerging line profiles. The aim was to find alimited number of simulations and associated line profiles withproperties similar to those characterizing the explosive events.Therefore, it is not meant to be complete. It has been proved (seee.g. Shibata et al. 1982 or Sterling et al. 1993) that a thermody-namic behaviour qualitatively different from that discussed be-low is possible under different sets of parameters not coveredin Table 1.

Eqs. (1) – (3) are solved using the Fortran 90 code EMMAD(De Sterck et al. 1998) based on high resolution shock captur-ing schemes and an approximate Riemann solver. EMMADis a general code for the numerical solution of the full MHDequations on structured non-cartesian grids. The discretizationis carried out using finite volume Godunov-type techniques. Thecode achieves second order accuracy in space coordinates us-ing limited linear reconstruction (min-mod limiter) and secondorder accuracy in time using a 4-stage Runge-Kutta integrator.We use a fixed grid spacing corresponding to 13 km per gridcell. The code is implemented with solid wall boundary condi-tions. This choice can potentially affect the emergent line pro-files if waves reflected at the boundaries reach the Civ emittingregions. The choice of a total integration time of 60 secondsensures that the perturbations caused by the explosion eithernever reach the boundaries during the computations or, if theydo so, it occurs at the end of the integration, thus leaving unaf-fected the regions where Civ is emitted. On both foot-points,the temperature and the pressure are fixed at 10,000 K and 2.1dyn cm−2 respectively. This correponds to a height between1380 and 1515 km above the layer withτ5000=1 in models C orD of Vernazza et al. (1981). These boundary conditions resultin a total gas pressure equal to 0.11 dyn cm−2 at the base ofthe transition region. Again, this value corresponds to the in-termediate model 2 of Fontenla et al. (1990). According to the

Fig. 1. Time evolution betweent=0 s andt=120 s of the temperaturedistribution in the loop derived from Eqs. 1–3 for the parameter setET/σ = 2.5 109 erg cm−2, x0 = 1.3 Mm and∆t=300 s. Thex axisrepresents the normalized spacial coordinate along the loop, and thedifferent plots, which correspond to time increments of 10 s have beenshifted upward bylog T=1 dex.

authors, their models (all differing in the amounts of mechanicalheating and total heat flux at the base) may be representative ofa distribution of loops.

2.2. Results

In this section, we describe the dynamical response of our loopmodel to different energy perturbations. First, we give an over-all description of results obtained for the set of parametersE/σ = 2.5 109 erg cm−2, x0 = 1.3 Mm and∆t = 300 s,hereafter central simulation, and then we discuss the differenceswith respect to this case produced by changes in the drivingparameters. Plots shown in this section are normalized in thespatial coordinate so that the loop starts ats = −1 (x=0 km)and ends ats=+1 (x=13,000 km). Thus,x=1.3 Mm correspondsto s=–0.8.

2.2.1. Central simulation

The qualitative description of the evolution followed by the loopafter the onset of the energy deposition is in rough agreementwith previous studies (Sterling et al. 1991). A common featurefor any given set of parameters is that, during the first few sec-onds of the simulation, most of the energy injected goes intointernal energy, thus increasing the temperature and pressureat the explosion site. In Fig. 1 it is readily seen how, at the ex-plosion site, the plasma is heated up to coronal temperaturesas it expands due to the increased pressure. As a consequenceof the temperature rise, new narrow regions at transition zonetemperatures are created at both sides of the evacuated region.At the same time, the expansion produces two oppositely di-rected plasma flows centered roughly in the middle of the twonew transition zones (see Fig. 2). Therefore, these new transi-tion zones move at opposite velocities thus contributing to the


Fig. 2.Time evolution betweent=0 s andt=120 s of the velocity distri-bution in the loop for the same energy perturbation as in Fig. 1. In thiscase, successive plots are shifted upwards by an amount equivalent to50 km s−1.

integrated line profile with the corresponding Doppler shift. Theplasma flows –which during the first4 s are of the same order ofmagnitude despite the effect of the asymmetric gravity force–will soon be characterized by different flow speeds due to theoriginal asymmetry in the physical conditions on both sides ofthe evacuated region: while the up-flowing stream encounters amedium of decreasing density, on the opposite side, hydrostaticequilibrium governs the run of density with depth and thus, themedium consists of an increasingly denser chromosphere thatprevents the development of fast down-flows towards the foot-point of the loop.

At t ≈5.0 s the up-flowing material reaches the position ofthe transition zone originally placed 200 km above the explo-sion site and, as a result of this interaction, two pressure wavesdevelop. It is at this point that the asymmetry between the op-positely directed flows becomes more dramatic. The two waveshave different propagation speeds. One is a slow pressure wavethat propagates with the same speed of the cool dense plasmoidmade up of the chromospheric material originally situated be-tween the explosion site and the original corona and the secondis a fast pressure wave of smaller magnitude that accelerates,compresses and heats the plasma while it develops into a shock.In Figs. 1–3 and for the sake of clarity we have only shown thecomputed solution at 10 s intervals. Therefore, this process isnot shown in detail, but the resulting pressure waves can be seenin the solution att=10 s.

After this, the cold dense plasmoid begins to travel across theloop towards the apex at velocities of the order of 100 km s−1.This plasmoid is separated from the coronal plasma by two tran-sition regions moving with the same approximate speed anddirection, one corresponding to the original transition zone ofthe initial loop, the other created during the explosion. At thispoint, the loop has already reached a temperature slightly lessthan log Te = 6.1 at the explosion site. This is the maximumtemperature in the loop at that stage and will not change sig-nificantly during the rest of the simulation.log Te = 6.1 can

Fig. 3.Time evolution betweent=0 s andt=120 s of the pressure distri-bution in the loop for the same energy perturbation as in Fig. 1. In thiscase, successive plots are shifted upwards by an amount equivalent to1 dyn cm−2.

thus be taken as the maximum temperature reached by the loopduring the simulation.

The subsequent evolution of the loop can be divided intothree stages: during the first stage, both the pressure shock andthe plasmoid travel along the loop and cross the loop apex; inthe second, the preceding shock wave approaches the secondchromosphere placed on the loop-leg opposite to the explosionsite and compresses the plasma in between. Finally, in the thirdstage it is the cool plasmoid that reaches the second chromo-sphere giving rise to a complex mixture of pressure and densitywaves. Shortly before the end of the simulation, a reverse shockis easily discernible travelling back to the loop apex and decel-erating the previously accelerated plasma. Though not shownin the plots, we also find two density condensations, the firstcorresponding to the original plasmoid and the second to theenhanced density produced by the compression of the chromo-sphere at the far end of the loop by the shock wave. At the pointwhere the pressure increase (associated with this second densitycondensation) approaches the boundary of the loop we end thesimulation for this parameter set.

2.2.2. Exploration of the parameter space

In the simulation described above, we followed the loop evolu-tion during the first 120 s. This cannot be achieved for the wholeexploration of the parameter space due to the computational re-sources involved. It will be shown in the next section that it ismainly in the first stages of the evolution that the changes inthe emergent line profiles occur. For both reasons, we limit theduration of the computations to the first 60 seconds after theonset of the energy injection. The differences with respect tothe evolution described above, due to changes in the parametersET/σ, ∆t andx0 can be summarized as follows.

– Changes in the total energy injected.As expected, there are a number of thermodynamic parame-ters that scale with the total energy deposited at the explosion


site such as the pressure maxima for a given time after theonset of the energy deposition, the maximum and minimumtemperatures attained in the loop and in the plasmoid, or themaximum velocities developed.Examining the solution to Eqs. 1–3 at intervals of 0.5 s, weobserve that the pressure at the explosion site reaches amaximum value around 1.5 dyn cm−2 beforet=1 s and re-mains roughly constant during the first 10 seconds of theintegration. A change in the total energy injected will risethis value up to 2.5 dyn cm−2 in the case ofET/σ =6.25 109 erg cm−2.The maximum temperature of the loop is attained in allour simulations at the explosion site. This maximum tem-perature reaches a value oflog T = 6.25 for ET/σ =6.25 109 erg cm−2 while the minimum temperature of theplasmoid is of the order oflog T = 4.8 before the interac-tion with the original transition region. The influence of thischange in the minimum temperature in the evolution of thecool plasmoid will be discussed in detail in the context ofchanges in the energy injection rate.The maximum positive velocity in the loop shows a strongercorrelation with the value ofET/σ (varying between 75and 100 km s−1) than that of the minimum negative velocityassociated with the down-flowing plasma which only variesbetween 25 and 30 km s−1.Finally, another significant change in the loop response tothe energy injection is the absolute magnitude of the dis-continuity associated with the shock wave, increasing as thetotal energy deposition per unit area is increased.

– Changes in the energy injection rate.The most conspicuous consequence of increasing the energyinjection rate is, as could be expected, an acceleration of theprocesses described above. The fact that most of the energydeposition occurs during a shorter interval results in muchhigher pressures and temperatures at the explosion site andalso in the ejected plasmoid; most important, it will alsodrive faster outflows from the explosion site. In particular,the velocity in the up-flowing part of the loop can reachvalues close to 175 km s−1 for ∆t=60 s.It is interesting to stress that, as noted bySterling et al. (1991), the minimum temperature ofthe plasmoid is going to affect the line radiation thatemerges from the loop. Given that heat conduction has astrong dependence on temperature (T 5/2), those simula-tions giving rise to high coronal temperatures will producehigher heat fluxes towards the cool core of the plasmoid. Ifheat conduction is sufficiently high the plasmoid can reachtemperatures above the Civ formation temperature. Inthese cases, the Civ emission originated in the transitionzones originally separating the plasmoid from the corona,vanishes at some point during the simulations.

– Changes in the energy injection site.From a qualitative point of view, the most remarkable in-fluence of a variation in the explosion site coordinate is rel-ative to the total amount of mass contained in the ejectedplasmoid. This total mass can be roughly estimated as the

chromospheric material found betweenx0 and the origi-nal transition region. This estimate is not very precise in thesense that the coronal plasma surrounding this cool plasmoidwill drive evaporation through heat conduction as describedabove. In the simulations withx0 = 1.45 and∆t=60 s or∆t=300 s, we find that heat conduction is enough to bringthe plasmoid into thermal equilibrium with the surroundingcorona. Even in the simulation with∆t=60 s,x0 = 1.3 andET/σ = 6.25 109 erg cm−2 we findevaporating gas plugs,in the terminology used by Sterling et al. (1991). In fact,the phenomenology associated with the heating of the plas-moids in these simulations is remarkably similar in all ther-modynamic variables (temperature, pressure, density andvelocity) to the one described in Sterling et al. (1991) forthe simulation characterized byE0=100 erg cm−3 s−1 de-spite the different total energies and the fact that we use anexponential decay to model the energy perturbation whileSterling et al. (1991) used a one-half period of the sine func-tion. We therefore conclude that the thermalization of theplasmoids found in our simulations is the same phenomenonas the disappearing gas plugs of Sterling et al. (1991) inan intermediate regime betweenE0=20 erg cm−3 s−1 andE0=100 erg cm−3 s−1. For a comparison between the dis-appearing gas plugs and the evaporation in flare models werefer to the discussion included in Sterling et al. (1991).In any case, it is observed that the total mass increases asx0diminishes and approaches the foot-point of the loop. As aconsequence, explosions taking place far below the transi-tion zone will produce massive plasmoids moving slowly.Also, the time interval after the onset of the energy injec-tion (during which the velocity distribution remains roughlyequal for the up-flowing and down-flowing plasma) is thetime elapsed until the up-flow reaches the original transitionregion as it is only when the plasma encounters low densitycoronal regions that velocities of the order of 100 km s−1

are attained.

3. Observational consequences

As already mentioned in the introduction, the aim of this paperis to confront computational simulations of the hydrodynamicresponse of a coronal loop to energy perturbations such as thosedescribed above with direct observations of the solar transitionregion dynamic phenomena, in particular with explosive events.In order to do so, it is necessary to convert the outcome fromthe computational simulations into emergent spectral profiles oflines formed mainly at transition region temperatures. In par-ticular, in this work we have selected Civ λ1548.2 A as thetransition region line to synthesize. In order to compute theemergent line profiles, we use the equations that link the ther-modynamic variables with the distribution of emergent energyas a function of wavelength (see e.g. Spadaro et al. 1990).

It is evident that the amount of energy emitted by the loopwill strongly depend on the amount of Civ present in the loop.Therefore, we are confronted with the necessity of computingion populations for carbon as a function of position along the


loop and time, in conditions far from ionization equilibrium, asdiscussed below. This is discussed in Sect. 3.1 and the resultingpopulations introduced in the spectral equations in Sect. 3.2.

3.1. Non-equilibrium ionization

In most of the diagnostics used to infer the thermodynamic prop-erties of the solar atmosphere it is usually assumed that theplasma is in a state of ionization balance. This is a valid approx-imation when any temperature change takes place on time scaleslonger than the ionization and recombination time scales of theplasma, so that it has enough time to readjust the relative ionpopulations to the new temperature. Nevertheless, the pictureof the solar transition region that emerges from the UV obser-vations made withSOHOis that of an ever changing dynamicinterface that experience sudden episodes of activity character-ized by velocities of the order of, or greater than 100 km s−1

such as the explosive events mentioned in the introduction. Fur-thermore, any mass flow through the steep temperature gradientof the transition region will undergo rapid heating or coolingdue to the interaction with the surrounding plasma. This meansthat it is necessary to refine many of the diagnostics used so farin the Sun in order to take into account such deviations.

In the case of our simulations, the plasma properties changeat such speeds (especially during the first few seconds of inte-gration, when most of the injected energy goes into heat) that itis necessary to consider possible deviations from the ionizationequilibrium populations in order to obtain reliable descriptionsof the spectral evolution emerging from any simulation.

Up until now, the study of the departures from ionizationbalance focused on their effect on stationary flows with veloci-ties of the order of 10 km s−1 (see e.g. Spadaro et al. 1995 andreferences therein) in the context of siphon flows along loops.In this work, we pursue further the study of these deviations byextending the calculations to non-stationary flows. In order tocalculate the ion populations along the loop for a given time wehave to integrate the ionization equations,

∂Ni

∂t+

∂(Ni · v)∂s

= Ne(Ni+1αi+1 + Ni−1Si−1 − Ni(αi + Si)) (10)

whereαi andSi are the recombination and ionization coeffi-cients respectively of the ionization stagei, andNi is the num-ber density of ioni. In our case, we have selected carbon as thespecies whose ion populations we intend to calculate, the rea-son being the conspicuity with which explosive events appearin the resonance lines of Civ at 1548 and 1550A. This makes iteasy to compare our detailed predictions about the time evolu-tion of the observational signatures of the events with real dataobtained in a parallel observing programme that makes use ofthe SUMER instrument on boardSOHO. Another set of rou-tines has been developed to perform similar studies with Oxy-gen resonance lines. In both cases we have followed Arnaud &Rothenflug (1985) to compute the recombination and ionizationcoefficients. According to Arnaud & Rothenflug (1985), chargetransfer reaction rates far less than10−9 can be neglected. Thus,

the contribution of the charge transfer ionization of Cii withHe ii or the charge transfer recombination of Cii, C iii and Cvwith H i and Hei will not greatly affect the computations. Forsimplicity, we have not included these terms in the ionizationequations. The charge transfer recombination rates of Civ withH i and Hei are not clearly less than10−9 cm3 s−1, but, as dis-cussed by Arnaud & Rothenflug (1985), these neutral speciesdo not exist at the temperature of maximum abundance of Civ.Thus, the ionization state of hydrogen and helium does not affectthe solution of the ionization equations and we can consistentlyuse, as explained in Sect. 2, a constant value of the mean atomicweightµ.

The equations are solved using the values ofNe andTe pro-vided by the hydrodynamical simulations. In an ideal case, thewhole set of hydrodynamic and ionization balance equationswould be solved simultaneously and the radiative loss functionupdated at each time step to take into account new values of theion populations. Given the lack of detailed knowledge aboutthe exact shape of the radiative loss function, we have preferredto keep both sets of equations separated in order to take fulladvantage of two different numerical schemes particularly suit-able for each set. While the hydrodynamic equations are bestintegrated by means of a shock capturing scheme such as theone mentioned above, adaptive grid algorithms are essential ifa moving ionization front is expected in the solution.

In our model loop, the initial transition region (betweenT=20,000 K andT=200,000 K) spans a distance of about250 km (20 grid cells). The corresponding temperature gradientimplies that the full width at half maximum of the region withsignificant presence of a given ion, say Cii, is of the order of65 km. This corresponds to 5 grid cells in the computational do-main used for the hydrodynamic simulations, something clearlyinsufficient for the kind of distribution it is meant to represent.Furthermore, the appearance of new moving transition regionswith associated ionization fronts demands extremely high spa-tial resolution everywhere in the integration domain. The use inthe hydrodynamical simulations of a grid that conveys sufficientgrid points in regions of steep gradients in the population den-sities makes the CPU time for running EMMAD prohibitive.On the other hand, most of the loop is occupied only by Ci (thechromosphere) and Cv (the corona) and it is only in the transi-tion regions that the ion populations exhibit such steep spatialgradients. As mentioned above, the solution to this problem isthe use of an adaptive grid algorithm that places grid points in theregions of high spatial variations. We have adapted an algorithmbased on the method of lines (MOVGR, Blom & Zegeling 1994)to our particular equations. The ionization and recombinationcoefficients adopted from Arnaud & Rothenflug (1985) wereslightly modified in order to obtain smooth functions of time,given that their original formulation introduced spurious oscil-lations around the matching points of the parameterizations.

Figs. 4–6 show five snapshots each, of the fractional ion pop-ulations of carbon in selected regions around the four transitionzones, note that the value of the Civ ion fraction is vanishinglysmall everywhere else in the loop, present in the central simula-tion shown in Fig. 1. Dashed lines represent equilibrium values


Fig. 4.Civ ionic fraction att=0.0 s (leftmost column of panels),t=1.0 s,t=2.0 s,t=3.0 s andt=4.0 s (rightmost column of panels) in the threetransition regions present in the central simulation. Solid lines representthe normalized ion populations out of equilibrium computed with theadaptive grid code and dashed lines correspond to the values derivedunder the assumption of ionization equilibrium. Note that thex axisscale that corresponds to the normalized arc length changes from plotto plot in order to follow the position of the ionization fronts.

Fig. 5. C iv ion fraction in the three portions of the loop at transitionregion temperatures att=5.0 s (leftmost column of panels),t=6.0 s,t=7.0 s,t=8.0 s andt=9.0 s (rightmost column of panels) in the centralsimulation. For further information, see caption of Fig. 4

while solid lines correspond to the populations obtained withthe adaptive grid code.

It is evident from the plots that, at least for the energy injec-tion parameters explored in this paper, strong deviations fromthe equilibrium values of the ion populations can be expected.This is particularly true for the first few seconds of the inte-gration, when the carbon component of the plasma, initially inthe predominant form of neutral carbon, attains temperatures of

Fig. 6. C iv ion fraction in the three portions of the loop at transitionregion temperatures att=10.0 s (leftmost column of panels),t=20.0 s,t=30.0 s,t=40.0 s andt=50.0 s (rightmost column of panels) in thecentral simulation. For further information, see caption of Fig. 4

the order oflog Te=5.5. It is then when the delay between thetemperature rise and the ionization processes due to the finite-ness of the ionization rates gives rise to the most significantdepartures from the equilibrium values. We see, then, how dueto this delay of the ionization process with respect to the tem-perature rise, the relative population of any ionization stage willbe initially deficient with respect to the equilibrium values butwill subsequently increase even when the electron temperatureis above the formation temperature of the ionization stage underconsideration. At some point, this trend reverses and a higherionization stage takes over.

In our simulation, att=1.0 s the plasma at the explosionsite has almost reached the Civ formation temperature but thenon-equilibrium populations is roughly half the equilibrium one(Fig. 4, top panel). On the other hand, one second later the com-puted ion population is more than twice the maximum relativepopulation in equilibrium at the injection point, where no Civshould be present due to the high temperatures. The relativedelay between the temperature rise and the ionization processincreases as the electron density decreases due to the expan-sion. It is also worth noting the departures from the equilibriumpopulation in the pre-existing transition region on the energy-injection-side of the loop due to the mass flows produced by theexpansion of the plasma at the explosion site. This is most crit-ical when the up-flowing stream interacts with this pre-existingtransition region giving rise to fast flows and pressure waveswhich subsequently develop into shocks. At this point (t=5–6 s),the temperature gradient gets even steeper reaching values sim-ilar to those attained in the newly created transition regions.Thus, the equilibrium distribution of Civ becomes narrowerwhile the computed one evolves slowly towards equilibriumvalues and finally reaches a steady distribution characterized byvalues of the maximum relative population of Civ of less thanhalf the equilibrium values (see Fig. 6). This non-equilibrium


curve, due to the effect of the flows, is also somewhat widerthan the equilibrium distribution.

We postpone the detailed analysis of the evolution of thefractional ion populations with respect to the equilibrium val-ues to a subsequent paper devoted entirely to the study of theevolution of the ionization state of several species in a loopsubject to this kind of energy perturbations.

3.2. Line synthesis

Once the ion populations are computed, the emissivity of a givenemission line per unit interval of wavelength in an optically thin,collisionally excited resonance line can be obtained from

Eλ ∝ hc

λ

Ωω

N1

Nion

Nion

Nelem

Nelem

NHNHNe

exp −WKbT√T

φ(λ) (11)

whereh is the Planck constant,c is the speed of light,Ω isthe collisional strength,ω is the statistical weight of the lowerlevel, N1/Nion is the ratio of ions responsible for the emis-sion in the ground state relative to the total number of ions perunit volume,Nion/Nelem is the relative population of the ion,Nelem/NH is the element abundance,NH is the proton density,Ne is the electron density,W is the energy difference betweenthe upper and lower levels,Kb is the Boltzmann constant,T isthe temperature, and

φ(λ) =exp−( (λ−λ0−λs)

∆λ0)2

∆λ0√

π(12)

(see e.g. Spadaro et al. 1990). In the definition ofφ(λ), λ0 is therest wavelength of the line,λs = (λ0

c )vp is the Doppler shiftcorresponding to the velocity of the plasma projected on the lineof sight (vp), ∆λ0 is the Doppler width of the line

∆λ0 =λ0

c

√2 KbT

mi(13)

andmi is the mass of the ion. Given a distribution of emissivitiesalong the loop, the total intensity can be calculated as

Iλ =

se∫0

Eλds (14)

wherese is the total length of the loop.

4. Results

Figs. 7–9 show the results of these calculations as applied tothe 1548A resonance line of Civ. The profiles represent theemission integrated along the whole loop length for a structure atlatitude0 and placed at the solar meridian and normalized to theintensity emitted by the whole loop before any energy injectiontakes place. The plots are constructed from individual line pro-files computed every 0.5 seconds and any structure below thistime resolution will be due to perspective effects. In the horizon-tal plane,xandy represent wavelength in velocity units and time

in seconds. Positive velocities represent blue-shift whereas neg-ative velocities should be interpreted as red-shifts. Changes inperspective were necessary to provide visibility of as many pro-file components as possible. Hereafter and for the sake of claritywe shall refer to the energy inputET/σ = 2.5 109 erg cm−2

as the low energy case, andET/σ = 6.25 109 erg cm−2 as thehigh energy case.

4.1. ∆t = 600 s

In general, the radiative output of the simulations, shown inFig. 7, can be described as a very rapid and short-lived en-hancement of the line intensity followed by the appearance ofDoppler shifted components. In the low energy cases (ET/σ =2.5 109 erg cm−2), a blue shifted component is clearly visiblewhose maximum Doppler shift increases as the energy deposi-tion is placed closer to the original transition region (1,500 kmabove the foot-point). In these low energy cases, the maximumred-shift is attained during the first 5 seconds of the simula-tions and diminishes thereafter. The blue-shift, on the contrary,increases with time, reaches the maximum shift and remainsroughly constant until the end of the simulations except for thecase withx0 = 1.45 where the higher velocity attained impliesthat the up-flowing material is close to the loop apex at the endof the simulation. At the loop apex, the radial velocity vanishesand therefore the blue-shifted component tends to return to therest wavelength as the material responsible for this emissionapproaches this point. It is also noticeable the decrease in max-imum relative intensity whenx0 increases. This is due to thefact that the density decreases exponentially with height in thehydrostatic atmosphere that characterizes the initial loop. Thus,there is less available material to ionize during the explosion.

In the high energy cases (ET/σ = 5 109 erg cm−2), thereis a marked difference between the simulations withx0=1.3 andx0=1.45. In the first case, the blue component initially increasesin Doppler shift (and decreases in intensity) until it reaches amaximum of about 70 km s−1. Then, it returns back to the restwavelength (due to the curvature of the loop) thus merging withthe red-shifted component. In the latter case the evolution takesplace on a much faster time scale. The blue component has avery short lifetime and disappears from the line profiles due tothe rapid heating of the plasma up to temperatures well above theC iv optimum formation temperature. Originally, the core of thecold plasmoid maintains temperatures below the Civ formationtemperature (TC IV) and is thus separated from the surroundingcorona by a Civ emitting transition region. Given that, initially,this plasmoid is ascending along the loop-leg, this is the origin ofthe blue-shift in the profiles. Whenever heat conduction acrossthe transition layers rises the temperature of the plasmoid aboveTC IV the emission from this part of the loop vanishes. Theblue component actually disappears before showing hints of anydecreasing trend in the wavelength shift. The speed at which thisprocess occurs for a given amount energy depends mainly onthe amount of cool material in the plasmoid so that the moremassive the plasmoid, the longer it takes to heat it above theC iv formation temperature.


Fig. 7. C iv line profiles normalized to themaximum intensity emitted by the quiescentloop. These line profiles were obtained forthe set of simulations characterized by∆tparameter equal to 600 s. From left to rightand from top to bottom, the top row of plotscorresponds toET/σ = 2.5 109 erg cm−2

andx0 = 1.0, x0 = 1.3 andx0 = 1.45.The last two plots in the bottom row cor-respond toET/σ = 6.25 109 erg cm−2

with x0 = 1.3 andx0 = 1.45. Thex axisshows wavelength in velocity units (km s−1)and they axis corresponds to time in sec-onds. Positive velocities represent blue-shiftwhereas negative velocities should be inter-preted as red-shifts. Changes in perspectiveare necessary in order to facilitate the followup of the different profile components.

Fig. 8.C iv line profiles obtained for the setof simulations characterized by∆t equal to300 s. From left to right and from top tobottom, the top row of plots correspondsto ET/σ = 2.5 109 erg cm−2 andx0 =1.0, x0 = 1.3 and x0 = 1.45. The lasttwo plots in the bottom row correspond toET/σ = 6.25109 erg cm−2 with x0 = 1.3andx0 = 1.45.

The red component shows a similar behaviour as in the for-mer case where it reaches maximum shift right after the explo-sion and subsequently returns to the rest wavelength.

For this energy injection rate, the maximum blue-shift canreach values of the order of 100 km s−1 in the most extremecases while the maximum redshift is always of the order of20–25 km s−1. These values, nevertheless, depend on the posi-tioning angles of the loop on the solar disk.

4.2. ∆t = 300 s

As the energy injection rate decreases we find that the mostevident change with respect to the description of the simulations

with ∆t = 600 s is the increase in the relative intensity andin the rate of change of the different profile components. Thecorresponding line profiles are shown in Fig. 8.

Again, for the low energy cases the intensity increasesabruptly while the line profile splits into two components.Then, the up-flowing component decreases in intensity whilethe Doppler shift reaches a maximum and subsequently dimin-ishes. In the simulation withx0 = 1.3 the blue component hasnot yet returned to the rest wavelength whent=55 s, while in thecase withx0 = 1.45 the plasma is heated up to temperatureswhere no Civ can be present and, thus, the blue componentdisappears from the profiles.


Fig. 9. C iv line profiles obtained for the set of simula-tions characterized by∆t equal to 60 s. From left to rightand from top to bottom, the top row of plots correspondsto ET/σ = 2.5 109 erg cm−2 with x0 = 1.3 andx0 = 1.45. The last two plots in the bottom row corre-spond toET/σ = 6.25 109 erg cm−2 with x0 = 1.3and x0 = 1.45. The missing plot at the upper left cor-ner of the Fig. corresponds to the parameter setx0=1.0,ET/σ = 2.5 109 erg cm−2. These values lead to insta-bilities in the solution of the ionization balance equations asdescribed in the text.

In the high energy case withx0 = 1.3, the initially up-flowing plasma reaches the loop apex well before the end of thesimulation and thereafter flows downwards towards the foot-point opposite to the explosion site. This is observed in the lineprofiles as a displacement of the component initially associ-ated to up-flowing material from blue-shifts to red-shifts. Atthe point where its wavelength coincides with the characteris-tic wavelength of the down-flowing material both componentsmerge thus increasing the relative intensity. In the case withx0 = 1.45 we see again the effect of the increased conductiv-ity at high temperatures. In this simulation, the plasma reachestemperatures where the conductive energy transport becomesso efficient that the heating of the cool plasmoid above the Civformation temperature occurs before the end of the simulation.Nevertheless, though not visible in Civ, the plasmoid crossesthe loop apex and travels downwards towards the foot-point op-posite to the explosion site. It is result of the compression thattakes place when this plasmoid approaches the chromosphereat the far end of the loop in the initial setup that gives rise tothe new burst of activity in the red wing of the line att ≈ 45svisible in the simulation withx0 = 1.45 and in all cases with∆t = 60 s.

Again, the maximum blue-shifts are of the order of100 km s−1. On the contrary, in these simulations the cross-ing of the cool plasmoid past the loop apex and its arrival tothe ‘second’ chromosphere increase the maximum redshift upto values of the order of 60–70 km s−1.

4.3. ∆t = 60 s

In this section there is a missing plot corresponding to the pa-rameter setx0=1.0, ET/σ = 2.5 109 erg cm−2. This isdue to the instabilities appearing in the solution of the ioniza-tion balance equations when the descending ionization front

reaches the boundary of the loop closest to the energy injec-tion site. Given the fast energy injection rate and the proximityof the explosion site to this boundary whenx0=1.0, this oc-curs well before the end of the hydrodynamic simulation. Therest of the line profiles are gathered in Fig. 9. In it, we see howa faster energy injection produces more intense line profiles,larger Doppler shifts, and faster evolution. In this extreme case,the blue wing of the profiles obtained forx0=1.45 vanishes im-mediately after the onset of the energy deposition resulting innet red-shifts throughout the simulation. This also applies in thecase ofET/σ = 6.25 109 erg cm−2 andx0=1.3, except that inthis case the blue wing is present for the last 10–15 s with inten-sities higher than those characterizing the red component. Thecase withET/σ = 2.5 109 erg cm−2 andx0=1.3, on the otherhand, shows a well developed blue wing whose presence in theprofiles can be traced even after its merging with the red-shiftedcomponent.

In all four cases, the arrival of the cool plasmoid can bedetected as a new episode of activity in the red part of the lineprofiles, and again, its strength depends mainly on the mass ofthe plasmoid and on its speed.

5. Discussion

In general, the observed trends can be summarized as follows:soon after the onset of the energy injection the line profile expe-riences a sudden increase in intensity and splits into two compo-nents with opposite wavelength shifts. The red-shifted compo-nent is produced by plasma moving in an ever increasing-densitymedium and therefore we interpret the monotonic decrease inthe wavelength shift of this component as the manifestation ofthis deceleration process. The blue-shifted component, on theother hand, travels upward along the loop length surrounded bylow density coronal plasma. The blue-shift will correspond to


Fig. 10.C iv line profiles obtained for the setof simulations characterized byET/σ =6.25 109 erg cm−2, ∆t equal to 300 s andx0=1.3 assuming different angular coordi-nates of the loop. (See the middle plot in theupper row of Fig. 8 for a comparison withthe same simulation placed at Sun centre.)

the net effect of two different factors: the natural hydrodynamicevolution of the velocity along the semicircular geometry pre-scribed in the simulations and the projection of this velocityalong the line of sight. The intensity of each component willdepend on the number of Civ ions at each velocity and thiswill, in turn, depend on the relative maximum and the width ofthe Civ ion population out of equilibrium and on the density inthe region of emission.

Two main conclusions can be drawn from the comparisonof the Civ ion populations in and out of equilibrium. First, thedetailed calculations of the populations out of equilibrium repre-sent a major change in the expected spectral output only duringthe first stages of the explosion, when the temperature changesrapidly and fast flows cross temperature gradients typical ofthe transition region. Once the plasmoid and the surroundingcorona move at comparable velocities the only differences inthe populations are due to the temperature changes caused byheat conduction towards the cool core of the plasmoid, and theseare in the sense of reducing the emissivity on the Civ lines. Theimportance of these effects in the intermediate and last stages ofthe evolution can be expected to increase in those simulationswhere rapid evaporation of the gas plug occurs. The second con-sequence is related to the differences occurring during the firststages of the explosion. These are such that they enhance theemissivity of the Civ lines when the emitting plasma is at rest ormoves at low velocities. Furthermore, due to the delay in the ion-ization processes with respect to the temperature changes, thisoccurs at temperatures above the Civ formation temperature,as explained in Sect. 3.1, giving rise to larger thermal widthsof the lines than would be expected if the lines were formed attheir equilibrium temperature. The result therefore, is a wide andbright component centred or close to the rest wavelength whichis observed in some explosive events (see e.g. Perez et al. 1999).

As mentioned before, all the plots shown in Figs. 7–9 havebeen computed assuming that the loop is located at Sun centreand aligned with the equator. A full exploration of the (pre-dictable) behaviour of the line profiles as functions of the po-sition of the loop (given by its latitude, longitude and rotationangle with respect to the parallel circle that passes below itsapex) is beyond the scope of this paper. Nevertheless we haveincluded three examples of the dependence of the line profileson the positioning angles for one of the simulations shown in

Fig. 8. These examples are shown in Fig. 10 where the latitudeis measured as usual and the longitude is zero at Sun centreand−π/2 in the western limb. The inclination angle representsthe clockwise measured angle between the parallel circle thatpasses beneath the loop apex and the loop itself.

Fig. 10 shows how different positions of the loop on the solardisk can result in very different phenomenologies for the sameunderlying processes. The original asymmetry between red andblue shifts observed in the exploration of the parameter spacewhen the loop was placed at Sun centre –in the sense that blue-shifts are always higher in absolute value than red-shifts– is thusless significant as it depends very strongly on the three position-ing angles. In this context, a possible correlation between the lat-itude and the maximum line-of-sight velocity in our simulationsneeds some discussion. In the work by Dere et al. (1989), ob-servations of explosive events along a solar radius (and thus, forroughly equal longitude angles) showed no such correlations.In their study, they only considered the maximum blue shiftvelocities of explosive events. In a one-dimensional semicircu-lar geometry, the resulting line profiles will necessarily show alatitudinal dependence of the maximum line-of-sight velocity.The different projection factors that apply at different positionsalong the loop, the possibility of different rotation angles and theslowly varying velocity that characterizes the transition regionssurrounding the cool plasmoid after their passing of the loopapex, will tend to attenuate this dependence. But in any case, itis clear that a different geometry in which the flow axis can berandomly oriented independent of the position on the solar diskwould better reproduce the observations of Dere et al. (1989).Magnetic reconnection in twisted magnetic flux tubes bundledtogether would be a good example of this.

One of the most prominent results derived from these sim-ulations is the fact that the Civ emission coming out from theloop after the explosion can outshine by more than two ordersof magnitude the emission from the quiet loop. In general, thehigher the total energy injected, the injection rate or the depth inthe chromosphere of the injection point, the brighter the loop.In order to compare our results with observations of explosiveevents in the Sun, we first have to take into account that the plotsshown in Sect. 4 are normalized to the maximum intensity in theline profile emitted by the quiescent loop and only represent ra-diation coming from the perturbed loop. If the cross section of


Fig. 11. Time evolution of the emission of an ensemble of 9 quies-cent loops and one loop undergoing an energy injection ofET/σ =2.5 109 erg cm−2 with ∆t =300 s and placed atx0=1.3.

the loop is roughly equal to the field of view of our spectrographit is then possible to compare directly its observations with theline profiles shown in Figs. 7–10 of this paper. If, on the otherhand, one assumes that the perturbed loop belongs to an ensem-ble that occupies a given filling factor of the observed volume,then it is necessary to add to the synthetic line profiles a newcomponent at the rest wavelength produced by the other, qui-escent loops. We show, as an example, a line profile computedassuming that the field of view covers 10 loops with the samecharacteristics as the one we perturb in the simulations. This isincluded in Fig. 11.

The intensity is normalized to the intensity emitted by tenquiescent loops. The assumptions made to construct the lineprofiles are based on the model by Dere et al. (1987) in whichthe transition region is made up of filamentary structures withcross sections of the order of 10 km. In this case, the line profilesproduced by the volume limited by the spectrograph slit do notshow such dramatic enhancements as those present in plots 7–9.

Regarding morphology, we see how in this exploration ofthe parameter space some basic phenomenological characteris-tics of explosive events arise in a natural manner. We find howvarying the energy deposition parameter and the positioningangles we can obtain line profiles very similar to most of theobservational examples provided by Dere et al. (1989) whichcomprise line profiles with just one component (the so called‘plasma jets’) such as example 3b and very broadened profileswith unresolved components such as examples 2 and 5; and,finally, one explosive event with three separated components ofsimilar intensity and Doppler shifts of the order of±100 km s−1

(shown in example 1 of the paper).It is evident that, under the simplifying assumptions made

in this work, only partial agreement with observations can beachieved. Specially because no physical mechanism responsiblefor the energy injection has been put forward. But the tools de-veloped here for the analysis of the ionization fronts in the solar

atmosphere should allow us to explore more detailed scenariossuch us two dimensional magnetic reconnection.

Acknowledgements.The authors would like to thank the anonymousreferee for the critical reading of the manuscript along with many usefulcomments and suggestions on how to improve the paper. L.M.S. andR.E would like to thank Hans De Sterck for his help during the adap-tation of EMMA D to the particular problem we were dealing with.R.E. thanks M. Keray for patient encouragement. R.E. also thanksPPARC in U.K. for financial support. L.M.S. and R.E are very gratefulto LAEFF and Armagh Observatory for providing them a very kindhospitality during their visits. This work was partially financed by theSpanish DGICYT under grant number PB-941275. Research at Ar-magh Observatory is grant-aided by the Department of Education forN. Ireland while partial support for software and hardware is providedby the STARLINK Project which is funded by the UK PPARC. Thiswork was supported by PPARC grant GR/K43315.

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ASTRONOMY AND Modelling explosive events in …aa.springer.de/papers/9351002/2300721.pdfAstron. Astrophys. 351, 721–732 (1999) ASTRONOMY AND ASTROPHYSICS Modelling explosive events