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Solar plasma theoretical models for STEREO and Solar-B
Advances in Space Research 36 (2005) 1561–1571 www.elsevier.com/locate/asr
Solar plasma theoretical models for STEREO and Solar-B M.L. Khodachenko *, H.O. Rucker Space Research Institute, Austrian Academy of Sciences, Schmiedlstraße 6, A-8042 Graz, Austria Received 5 October 2004; received in revised form 22 December 2004; accepted 22 December 2004
Abstract An overview of several theoretical models addressed to the dynamical and energetic processes in the solar plasmas and coronal magnetic loops, and related to the phenomena of solar flares, CMEs, coronal loop interaction and oscillations is given. The presented models can be organized into two basic groups. The theoretical concept of one group of the models, called as the inductive models, employs the ideas of possible inductive and ponderomotoric interaction between the electric currents confined within the current-carrying magnetic loops moving relative to each other. The models of another group, so-called non-adiabatic MHD models, are applied to describe the MHD response of plasma in the low solar atmosphere to a changing current system of a flaring magnetic tube, which contains an injected beam of fast non-thermal electrons. A detailed information about the structure of coronal loops, their dynamics and global magnetic environment in solar active regions is crucial for all the considered models and it is expected that STEREO and Solar-B observations will be useful for testing and further improvement of these models. 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Sun:magnetic fields; Sun:flares; Sun:oscillations; Sun:corona
1. Introduction The solar corona has a highly dynamic and complex structure. It consists of a large number of constantly evolving loops and filaments, which interact with each other and are closely associated with the local magnetic field. The non-stationary character of solar plasma-magnetic structures appears in various forms of the coronal magnetic loops dynamics (grow motions, oscillations, meandering and twisting). Energetic phenomena, related to these types of magnetic activity, range from tiny transient brightenings (micro-flares) and jets to large, activeregion-sized flares and CMEs. Understanding of physical mechanisms in the background of the solar dynamic * Corresponding author. Tel.: +43 316 4120661; fax: +43 316 41206090. E-mail address: [email protected] (M.L. Khodachenko).
and energetic phenomena and their consequences in the near-Sun space is the main task of the modern solar physics. In this paper, we present briefly several theoretical models addressed to the dynamical and energetic processes in the solar plasma-magnetic structures, and related to the phenomena of solar flares, CMEs, coronal loop interaction and oscillations. The presented models can be separated on two general groups: (1) Inductive models of magnetic loops, which deal with the processes of global dynamics and interaction of coronal loops (Khodachenko et al., 2003) and (2) non-adiabatic MHD models dealing with the internal dynamics of plasma and fields in a particular magnetic tube (Khodachenko and Rucker, 2004). The data on the 3D structure and dynamics of coronal loops and their participation in the solar flaring events, expected from STEREO and SolarB, will provide the necessary information for testing and further development of the presented models and underlying concepts.
0273-1177/$30 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2004.12.056
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2. Inductive models of interacting current-carrying magnetic loops Complex dynamics of the solar coronal magnetic loops together with action of possible under-photospheric dynamo mechanisms cause majority of the loops to be very likely as the current-carrying ones. However, because of difficulties in measurement of a 3D magnetic field structure in the corona with a sufficiently high resolution, the presence of electric currents in the loops and their structure are not yet confirmed observationally. At the same time, the existing magnetometric measurements, which allow to determine a vertical component of rot~ B and hence to identify the vertical component of a current flowing from below the photosphere into the corona, provide a strong support in favor of the current-carrying loops hypothesis (Moreton and Severny, 1968; Hagyard, 1989; Gary and Demoulin, 1995; Leka et al., 1996; Hardy et al., 1998). Moving relative each other current-carrying magnetic loops should interact via the magnetic field and currents (Melrose, 1995). The simplest way to take account of the inductive and ponderomotoric interaction between the electric currents confined within the current-carrying magnetic loops moving relative to each other consists in application of the equivalent electric circuit models of the loops. In these models, each loop is considered as an equivalent electric circuit with variable parameters (resistance, capacitance and inductive coefficients) which depend on shape, scale, position of the loop with respect to other loops, as well as on the plasma parameters and value of the total longitudinal current in the magnetic tube (Khodachenko and Zaitsev, 1998; Khodachenko et al., 2003). The main attention in our research is paid to modelling of (1) the loop–loop flaring interaction, including as well the processes of electric current build up in the loops; (2) acceleration of solar Coronal Mass Ejections (CMEs); (3) oscillations of the loops in active regions. The equation for the electric current I in the coronal circuit of a separate, but not isolated from surroundings, thin (Rloop r0) magnetic loop has the following form (Khodachenko et al., 2003): IR ¼ U 0
1 d 1 dWext ðLIÞ ; c2 dt c dt 8R
ð1Þ
where L ¼ 4pRloop ðln rloop 74Þ is the inductance of the 0 2Rloop loop; R ¼ r2 rðT Þ is the resistance of the loop caused by a fi0 nite, depending on temperature conductivity of plasma r(T) = 9 · 106T3/2; U0 is a drop of potential between the loops footpoints, and Wext = B^Sloop is an external magnetic flux through the circuit of the loop. Here Rloop and Sloop are the main radius of the loop and the area covered by the loop, respectively. The term (1/c)(dWext/dt) in Eq. (1) is the external inductive electromotive force (EMF) eind. For the multiple loop systems, eind appears as EMF of mutual inductance eind ¼ c12 dtd ðM ij I j Þ; i 6¼ j, i
where i,j are the loop numbers, and Mij, mutual inductances. Characteristic time of the current change tc = L/(Rc2) in the coronal electric circuit is very large (104 years). Thus, the dynamics of current is defined by the motion of loops (emergence, submergence, etc.), which results in the change of inductive coefficients on the time-scales M L sL ¼ dL=dt and sM ij ¼ dM ijij=dt. It should be noted that the considered here equivalent electric circuit models are of course the idealization of real coronal magnetic loops. They usually involve a simplified geometry assumptions and ignore the fact that changes of the magnetic field propagate in plasma at the Alfve´n speed. This means that the circuit equations correctly describe temporal evolution of the currents in a system of coronal magnetic loops only on a time-scale longer than the Alfve´n propagation time. The actual spatial distribution of electric current in a loop is approximated by the current homogeneously distributed over the loop cross-section. At this stage, the stability of the loop is not considered anymore, but a certain care about not overcoming the kink instability threshold values for the magnetic tube twist is taken. On the other hand, the electromagnetic inductive coupling of current-carrying loops can play a significant role in stabilizing their possible kink instabilities. From the geometric point of view, the kink-type deformation of a loop results in a change of the magnetic loop inductance. In particular, for a spiral kink with one full twist, the disturbed value of inductance will be defined by the following expression: 2 4 7 113pRloop d d Lkink ¼ L þ L þ0 ; 16 Rloop Rloop 48 ð2Þ where L is the inductance of the undisturbed magnetic loop and d is the radius of the spiral disturbance, i.e., the distortion of the initial radius Rloop of the loop. For the typical scales of solar coronal magnetic loops, Eq. (2) gives the increase of inductance of the kink-disturbed magnetic loop, which in its turn corresponds to the increase of magnetic energy of currents stored in the loop. Moreover, the influence of the inductively connected neighboring loops will also strongly affect the energy change in the system during the deformation of a particular loop. Both these factors can result that kink deformation of a strongly twisted, carrying a high and dense current magnetic loop, in some cases will be still energetically not optimal, and thus this deformation will not take place. 2.1. Loop–loop flaring interaction It follows from observations that a large number of flares occurs in the regions where a new magnetic loop emerges and interacts with the existing loops (Nishio
M.L. Khodachenko, H.O. Rucker / Advances in Space Research 36 (2005) 1561–1571
et al., 1997). Such events are called as interacting flare loops. Here, we present further development of the idea of the flaring loop–loop inductive interaction, first suggested by Melrose (1997) and later applied by Aschwanden et al. (1999a) for interpretation of observations. The easiest case of two interacting loops is considered (see Fig. 1). In addition to Melrose (1997) and Aschwanden et al. (1999a), we note that relative motion of the loops creates significant inductive electromotive forces in their electric circuits which appear as a powerful source for changing of the currents in the loops. Change of the currents disturbs the initial thermal equilibrium of the loops and results in a change of the plasma temperature, which in its turn influences the resistivity of the circuit and the radiative energy losses. Therefore, each of the rising magnetic loops (Fig. 1) in our model is described by two equations (Khodachenko and Zaitsev, 1998;Khodachenko et al., 2003): (1) the equation for the electric circuit of the loop 1 X _ _ ij I i Ri ¼ U 0i 2 M ij I j þ I j M ð3Þ c j and (2) the energy equation 1c 2 j2 ni QðT i Þ i H i ; T_ i ¼ 2ni k B rðT i Þ
ð4Þ
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where Mij = Mji, Mii ” Li, indices i, j indicate the loop number (1 or 2), and the dot denotes the time derivative. In Eq. (4), T i ; ni and ji ¼ I i =ðpr20i Þ are the temperature, density of plasma and the current density in the ith loop, respectively; c = 5/3 is the adiabatic constant, Q(Ti), the radiative loss function for the optically thin emission (Cox and Tucker, 1969; Rosner et al., 1978; Peres et al., 1982), and Hi is a stationary background heating introduced to provide thermal equilibrium in the initial steady state. Within the thin torus approximation, applied for the loops, we use for the mutual inductance coefficients Mij = Mji in Eq. (3) the approximate expression derived by Melrose (1997) and modified by Aschwanden et al. (1999a) " # 1=2 Rloop i Rloop j cos uij ; M ij ¼ 8 Li Lj ð5Þ ðRloop i þ Rloop i Þ2 þ d 2ij where Li, Lj are the inductances of the loops, Rloop i, Rloop j, their major radii, dij, the distance between the centers of the loops tori, and uij, the angle between the normal vectors to the loops planes. A linear increase in time of the major radii of the loops Rloop i ¼ R0loop i þ vi t; i ¼ 1; 2 is considered, and the initial steady state and thermal equilibrium are assumed. One of the loops is taken to be initially current-free
Fig. 1. (a) Loop–loop flaring interaction and (b) approximation of the flaring domain by a pair of rising inductively connected current-carrying loops.
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Fig. 2. Dynamics of currents and plasma temperature in the parallel (u12 = 0) loops 1 (a) and 2 (b) in dependence on the distance d12 (Khodachenko et al., 2003). ðR0loop 1 ¼ 2 109 cm; R0loop2 ¼ 108 cm; r01 ¼ r02 ¼ 5 107 cm; v1 ¼ 5 105 cm=s; v2 ¼ 106 cm=s; n1 ¼ n2 ¼ 109 cm3 ; T 1 ðt ¼ 0Þ ¼ T 2 ðt ¼ 0Þ ¼ 106 K; I 10 ¼ 1011 A; I 20 ¼ 0Þ.
(I20 = 0). The numerical solution (Khodachenko et al., 2003) of Eqs. (3) and (4) for both loops reveals the main features of the current dynamics in the loops (Fig. 2):
– Quick build up of the current in the initially currentfree loop; – Fast significant increase of the current value and plasma temperature in when the loops become to be of the same size (a flare); – The most efficient interaction of the parallel loops (u12 = 0). The inductive electric field produced in the loop can be roughly estimated as E ¼ I 2i =ððpr20i Þ2 rðT i ÞÞ. When the value of this field exceeds the critical value Ec 0.2(4pne3Le)/kBT, where Le is Coulomb logarithm, the main part of electrons appears to be in a run-away regime and can be accelerated by the inductive electric field. According to Fig. 3 which presents the dynamics of E and Ec in the loop 1 (the ‘‘slow’’ loop), the situation E > Ec takes place for t > 2000 s. The estimations show that inductive field can accelerate electrons in the loop up to energies We[eV] = q[e] Æ E[V/m] Æ lloop[m] = 100 1000 eV (Khodachenko et al., 2003). 2.2. CMEs as rising magnetic loops. Acceleration of CMEs
Fig. 3. Dynamics of the inductive electric field E and critical field Ec in the loop 1 (for the case u12 = 0) (Khodachenko et al., 2003).
Another aspect of inductive models consists in a possibility to describe the effects of the dynamic coupling (due to the force ½~ j~ B=c) of the large-scale coronal electric currents. These can influence the global dynamics of current-carrying magnetic loops, such as the loop
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(a)
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(b)
(c) Fig. 4. (a) Inductively interacting multiple current-carrying loops; (b) simplified representation of the multi-loop system and (c) the lifting force acting on the central loop as a function of the lateral loops height.
rising, oscillations, change of inclination. Considering the loops as electric circuits with flowing currents, one can find a generalized ponderomotive force of their interaction Fi ¼
oU ; oxi
ð6Þ
1 X M ij I i I j 2c2 i;j
ð7Þ
where U ¼
is a potential force function of the system of currents and xi is a generalized coordinate. Note that when the motion of a dynamic system is described by the generalized coordinates in terms of generalized forces Fi and momenta Pi the equations of motion are still P_ i ¼ F i . But only in Cartesian coordinates, the generalized momenta are the components of usual mechanical momentum vectors mvi. In general case, the Pi not necessarily reduce to products of mass and velocity and therefore the dimension of the generalized force Fi could be different from the one of the usual mechanical force (Landau and Lifshitz, 1960, p. 16). The advantage of the potential force function approach is that it simplifies significantly the analysis of the ponderomotoric interaction of currents. It allows one to avoid the complex integration of forces, acting on each separate element of a current. The motion of a rising current-carrying loop can be described as a result of the competition between the lifting force, Flift = oU/oRloop and magnetic stress force
2
2
n , where U is the potential force F m ¼ oso ðB8pÞs0 þ B4p Rloop function of a system (Eq. (7)), s and s0 are the coordinate and the unit vector along the field line, respectively, and n is the normal vector to the field line. Fm decreases when the Rloop increases. And in the case of Flift > Fm, the acceleration of the loop by the ponderomotive forces takes place (Fig. 4(a)). The MHD simulations in Wu et al. (1997) find that a flux rope with sufficient magnetic energy density will erupt. The source of this energy comes from the inductive and ponderomotoric interaction of currents in the filament and the system of underlying loops. In the model calculations, we consider a symmetric group consisting of three current-carrying loops (Fig. 4(b)). The bigger loop with the radius Rloop 2 stays vertically (H2 = 0) in the center of the group, whereas the lateral loops, located at the distance d on both sides from the central loop, are inclined at some angle H1 = H3 to the vertical direction. All the loops are assumed to be the current-carrying ones (with currents Ii, i = 1, 2, 3). To simplify calculation of the inductive coefficients, we approximate each loop by a rectangular structure characterized by height hi, length bi, and thickness 2r0i, where i = 1, 2, 3. For the simplicity reasons, the model loops are assumed to have a quadratic form (hi = bi), and the lateral loops are taken to be of the same size with the height h1 = h3 ” h. Fig. 4(c) shows the lifting force applied to the central loop as a function of the height of the lateral loops calculated for the following model parameters: Rloop 2 = 5 · 108 cm,
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r01 = r02 = r03 = 5 · 107 cm, H1 = H3 = p/4, H2 = 0, d = 5 · 109 cm, I1 = I3 = 0.5 · 1010 A, I2 = 1010 A. The increase of the lifting force during the increase of the lateral loops height can be clearly seen in Fig. 4(c). This means that under the certain conditions the rising motion of the lateral loops can result in a situation when the ponderomotoric lifting force exceeds the value of the magnetic stress force and the central loop starts to be accelerated. 2.3. Oscillations of current-carrying magnetic loops Oscillations of the loops are traditionally explained as standing, or propagating MHD waves. The transverse oscillations of coronal loops have been interpreted as kink fast magnetoacoustic modes (Aschwanden et al., 1999b; Nakariakov et al., 1999). Besides, the longitudinal waves (or propagating EUV disturbances) were discovered by Berghmans and
Clette (1999) and interpreted as slow magnetoacoustic waves (Nakariakov et al., 2000; De Moortel et al., 2000). In this section, we demonstrate a possibility of transverse oscillations of the loops which are caused by the ponderomotoric interaction of currents in groups of inductively coupled current-carrying loops. In that sense, the presented alternative mechanism is not traditional MHD in nature. The forces responsible for the loop oscillations here have essentially non-local character and appear as an interaction of electric currents running in different spatial domains. As for the MHD waves propagating along the loop, they also can take place in our case, but as a kind of a secondary effect, arising during the loop motion under the action of the ponderomotive forces. It is natural to expect that the whole large loop will move not as a rigid object, but having local transverse deformations, which can appear as the origin of MHD waves travelling along the loop.
Fig. 5. Temporal evolution of the mutual potential force function U123(H, t) (Khodachenko et al., 2003).
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We consider the modelling system of the current-carrying loops, shown on Fig. 4(b), assuming that the lateral loops are inclined at the angle H1 = H3 = p/4 and grow linearly in time: hi(t) = h0i + ai t, where i = 1, 3, h01 = h03 = 5 · 108 cm, and a1 = a3 = 1.5 · 105 cm s1. The lateral loops are assumed to be initially current free, i.e., I1(t = 0) = I3(t = 0) = 0, whereas the initial current in the central loop is taken as I2(t = 0) = 104 A. Other parameters of the model are taken as the following: h02 = 3 · 109 cm, d = 5 · 108 cm, r0i = 5 · 107 cm, Ti = 106 K, (i = 1, 2, 3). Based on the numerical solution of the set of Eqs. (3) and (4), we study the temporal evolution of the mutual potential force function of the system U 123 ðH; tÞ ¼ U ðH; tÞ þ
3 3 1 X 1 X 2 L I ¼ M ij I i I j ; i i 2c2 i 2c2 i;j;i6¼j
which describes the dynamic interaction of the loops and depends on their mutual orientation, i.e., on the variable angle H2 ” H, characterizing the inclination of the central loop (Khodachenko et al., 2003). The general potential force function U(H, t) is defined by Eq. (7). Formation of the dip in U123(H,t) during the system evolution (see Fig. 5) means the possibility of oscillations of the central loop near its vertical position. The oscillations can take place in the case when the system is disturbed by a sufficient external energy Eext input from the neighbouring flaring events and shocks propagating in the coronal plasma. To estimate the main characteristics of the oscillatory regime of the central loop, we assume that the amplitudes are small enough and the oscillation periods are not very long, i.e., Posc < scurr, where scurr is the characteristic time of the current variation in the system. This allows one to neglect the influence of a changing orientation of the oscillating central loop on the global currents dynamics in the system and use the quasi-stationary current approximation. On the other hand, in order to apply the electric circuit model for the current-carrying loops, Posc should be larger than the Alfve´n propagation time sA 1 to 10 s. The period of oscillations in the system with a dipped potential profile U123(H, t*), can be calculated by the formula pffiffiffiffiffiffiffiffi Z h2 dH P osc ¼ 2I0 ; ð8Þ 1=2 h1 ðE ext U 123 ðH; t ÞÞ
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where I0 is the inertia momentum of the loop. To be specific, we take here a particular form of the mutual potential force function U123(H, t*) (see Fig. 5) realized at the moment t = t* 16,000 s, corresponding to the largest dip. The reflecting points h1 and h2 are defined by the condition Eext = U123(H, t*). The amplitude D, angular amplitude dH and velocity v of the top point of the oscillating loop are determined by D ¼ h2 sin h2 ;
dH ¼
h2 h1 ; 2
v ¼ 2h2
h2 h1 : P osc
ð9Þ
Table 1 presents the main characteristics of the oscillatory regime of the loop, calculated by Eqs. (8) and (9) for different values of a disturbing energy input. The results obtained are rather close to the observed quantities (Aschwanden et al., 1999b). The observed decay of oscillations of solar magnetic loops which is usually explained by damping of a corresponding MHD mode (Nakariakov et al., 1999), can also be interpreted within the framework of the inductive model, taking into account the ponderomotoric interaction of the loops, without any appeal to some energy dissipation mechanism. For example, oscillations of the central loop in our modelling case will gradually vanish if after the initialization of the loop oscillations due to the input into the system of a certain energy E0 the potential function of the system U123(H, t) evolves so that after some time t 0 the minimum U123(H = 0, t = t 0 ) of the potential dip where the loop oscillates becomes to be above the level of the energy input E0. Oscillations of the loop during such evolution will gradually vanish. By this, not only the amplitude, but also the period of the vanishing oscillations will decrease (see Eq. (8) and Fig. 5). In some cases of the observed by TRACE oscillations of loops, presented by Aschwanden et al. (2002), the tendency of a changing period of oscillations during their decay probably takes place.
3. Non-adiabatic MHD models The models of this group describe the internal dynamics of plasma in a magnetic tube. They are based on the self-similar solution of the MHD equations, describing plasma flows with a homogeneous deformation. The processes of Joule heating, radiative energy losses, thermoconductivity and viscosity are taken into
Table 1 Amplitude, velocity and period of oscillations of the coronal loop, calculated for different values of the disturbing energy Eext Eext (erg)
1.38 · 1015
1.46 · 1015
1.55 · 1015
1.63 · 1015
1.72 · 1015
1.80 · 1015
dH () D (km) Posc (s) v (km s1)
2.72 1420 194 29.3
4.40 2300 247 37.2
6.09 3180 292 43.6
7.78 4050 331 49.2
9.46 4930 366 54.1
11.1 5800 398 58.6
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Fig. 6. Scheme of a flaring event and disturbance of magnetic tube by the reverse current.
account (Khodachenko, 1996a,b; Khodachenko and Rucker, 2004). In particular, the MHD response of plasma in a magnetic tube to a change of the electric current system structure caused by the injection into the tube of a beam of fast non-thermal electrons during a flare is considered. This study is addressed to a special aspect of the process of an energetic particle beam injection and propagation in a magnetic tube not yet considered. When a beam is injected into the magnetic tube, the initial (pre-flaring) current system is disturbed due to appearance of the so-called return current jr.c. = jb, where jb is the beam current. Changes of the current density (dj = jr.c.) in the magnetic tube disturb its thermodynamic equilibrium as well as the force balance (see Fig. 6). This results in a complex dynamic behaviour of the whole plasma-magnetic structure, which influences radiation of the system in X-rays and microwaves (Khodachenko and Rucker, 2004). These emissions appear additionally to the beam generated radio bursts. Besides, due to the temporal changes of the plasma parameters (density and temperature) in the disturbed magnetic tube, the features of the local radio source itself are changing. The impulsive character of the beam injection causes two stages in the evolution of the magnetic tube. (1)
During the first stage, characterized by the presence of a beam, the preliminary equilibrium of the magnetic tube is disturbed and complex MHD reaction of the system takes place in the region of the beam propagation. (2) During the second stage, when the injection of the beam is already over, the disturbed plasma-magnetic system gradually evolves to a new equilibrium state. The beam of fast electrons plays in this scheme a role of only a disturbing factor. It is important to note that the energy which is released during the MHD evolution and heating of the magnetic tube comes not from the energy of the beam, but is stored in the initial pre-flaring magnetic field. The destabilization of the magnetic tube by a beam is effective if duration of the beam sb is at least of the order of magnitude of the time s, characterizing the temperature variation in response to a changed current density, or, in other words, the reaction time of the system to a change of the Joule heating term. From the energy equation, the reaction time s can be estimated as (Khodachenko and Rucker, 2004): s
5=2 2nk B T rðT Þ 10 nT 37:26 10 : ðc 1Þ j2b j2b
ð10Þ
In the case of a 20 keV electron beam, Eq. (10) yields 5=2 s20 keV 2:3 1010 nTn2 , where nb is the electron density b
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in a beam. For the initial (pre-flaring) density n = 1010 cm3 and a 20 keV beam with nb/n = 104 the initial temperature 105 K gives s20keV 7.3 s, but for the temperature 106 K we have s20keV = 2300 s. The last seems to be too large for duration of an electron beam, whereas the disturbing effect of more possible shorter beams will be insufficient. At the same time, for the considered parameters of plasma and beam the mean free path of fast electrons 2 !2 31 2 e 5 ke20 keV ¼ 44pn 2 me ðV e20 keV Þ 2
ðT e20 keV Þ ; ð11Þ n is about ·1011 cm, which is much greater than the atmospheric gravity scale and the typical scales of the considered fragment of the flaring magnetic loop. Thus, fast electrons of the beam do not interact collisionally with the plasma and there is no any collisional energy deposit from the beam to the plasma here. Therefore, the considered mechanism of plasma disturbing and heating works on the heights where the direct collisional energy deposition from the non-thermal beam to the background plasma Emslie, 1983 is still not efficient. The optimal range of heights is probably located in the lower corona or upper chromosphere. The dynamics of a magnetic tube disturbed by a beam of fast electrons is described by the model based on the self-similar MHD solutions: 2:85 104
a_ V r ðr; tÞ ¼ r; a Br ðr; tÞ ¼ 0;
b_ V z ðz; tÞ ¼ z; b
V u ðr; tÞ ¼ 0; Bu ðr; tÞ ¼ Bu0 ðtÞ
r ; R
ð12Þ
Bz ðr; tÞ ¼ Bz0 ðtÞ; ð13Þ
qðr; tÞ ¼ qðtÞ; pðr; z; tÞ ¼ p0 ðtÞ p1 ðtÞ
ð14Þ r 2
p2 ðtÞ
z 2
; R L r 2 z 2 T 2 ðtÞ ; T ðr; z; tÞ ¼ T 0 ðtÞ T 1 ðtÞ R L
ð15Þ
where R and L are transverse and longitudinal scales of the considered fragment of a flaring magnetic tube; R* is the external transverse scale of the whole magnetic structure (r [ R R*) (see Fig. 6); a(t),b(t) are dimensionless time-dependent components of the deformation tensor, characterizing the degree of plasma compression in the magnetic tube. Substitution of the self-similar solution (12)–(15) to the MHD equations set and further grouping of terms proportional 0th, 1st, and 2nd power of r and z yields a set of ordinary differential equations for a(t), b(t), Bu0(t), Bz0(t), q(t), T0(t), T1(t) and T2(t), which is much more convenient for the analysis than
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the initial set of MHD equations. Assuming the small 2 2 quantities: n1 ¼ TT 10 ðRr Þ 1, n2 ¼ TT 20 ðLz Þ 1, we present a any function of the type T (r, t) in the energy equation as T a ðr; tÞ T a0 ðtÞð1 a n1 a n2 Þ (Khodachenko, 1996a,b). 2 The Joule heating term ð~ j ~ jb Þ =rðT Þ in the energy equation includes as well the return current caused by the propagating beam of non-thermal electrons. In this paper, we illustrate briefly the main consequences of such modelling, whereas more details regarding the model itself and used approaches can be found in (Khodachenko, 1996a,b; Khodachenko and Rucker, 2004). The MHD response of the magnetic tube onto a beam is defined by (a) initial equilibrium parameters of plasma (T0(t = 0) = T00, T1(t = 0) = T10, T2(t = 0) = T20, n00 = q (t = 0)/mi); (b) current density of the beam (d = jb/j(t = 0)); (c) characteristic scales of the model (R, L, R*). Three general types of dynamics can be distinguished: (1) compressional regime; (2) quasi-periodic (pulsating) regime and (3) decompressional regime. All the dynamical regimes result from the competition between the pressure gradient and Ampere forces in the plasma, whereas the temperature change is determined by heating and cooling mechanisms acting differently on different dynamical stages of the magnetic tube evolution. Fig. 7 shows the examples of dynamics of the plasma parameters in the magnetic tube during the tube disturbance by a beam of fast electrons (Fig. 7(a)), and during its consequent relaxation to a new equilibrium state (Fig. 7(b)). In Fig. 7(b), plasma velocities V r ðr ¼ _ RÞ ¼ aa_ VR0 and vz ðz ¼ LÞ ¼ bb VL0 are normalized to V 0A ¼ Bu0 ðt¼0Þ ð4pqðt¼0ÞÞ
A
1=2
A
R . R
Thus, the disturbances of the current system produced by a beam of fast electrons propagating in the magnetic tube result in the dynamic processes, which in their turn, will influence radiation of the system in X-rays and microwaves, appeared additionally to the beam generated radio bursts. By this, due to the temporal changes of the plasma parameters (n, T) in the disturbed magnetic tube, the features of the local radio source itself are changing. This may explain a variety of observational data (Khodachenko and Rucker, 2004), which cannot be easily interpreted in terms of the traditional flaring scheme operating only with the collisional beam-plasma mechanism of the energy deposition into the system. Pulsating regimes can be analog of flaring events with precursors, or the quasi-periodically modulated flares. These pulsations correspond to the global sausage mode which has recently been identified in the radio observations with spatial resolution (Nakariakov et al., 2003). A rough estimation of the period of pulses during the quasi-oscillatory relaxation of the disturbed magnetic tube can be obtained within the assumption of adiabaticity. Khodachenko and Rucker (2004) provide the following formula for estimation
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Fig. 7. (a) Dynamics of q(t) and T0(t) in the tube with T10/T00 = 102, R = 3 · 107 cm, b(t) ” 1, T2(t) = 0, during propagation of the beam with d = 105 for different values of T00: (I) 3 · 105 K; (II) 1.8 · 105 K; (III) 105 K, and n00: (1) 109 cm3; (2) 1010 cm3; (3) 1011 cm3. (b) Oscillatory relaxation of the disturbed magnetic tube with n00 = 1.5 · 1011 cm3, T10/T00 = T20/T00 = 102, R = 3 · 107 cm, L = 3 · R, R* = 3 · 108 cm in dependence on the T00: (1) 107 K; (2) 6 · 106 K and Bu0(t = 0): (I) 55 G; (II) 45 G (Khodachenko and Rucker, 2004).
of the period of the small-amplitude adiabatic compressional oscillations of the magnetic loop fragment: 2psM a P ad ¼ pffiffiffiffiffiffiffiffiffiffiadffi ; c1
ð16Þ
where 1=ð2c2Þ B n00 T 00 ; b ¼ B 8pk sm ¼ R=V 0A ; aad ¼ ðb TT 1000 Þ 2 2 , and c u0 ðt¼0Þ ðR=R Þ is adiabatic constant. In particular, for the parameters of the model used for calculations in Figs. 7b I(1), II(1), Eq. (16) yields Pad = 54.5 s and Pad = 90.3 s for Bu0(t = 0) = 55 G and Bu0(t = 0) = 45 G, respectively. Of a special interest appear the cases where the regime of the compressional cooling is realized (Figs. 7a I(2); I(3); II(2)). In these cases, the conditions of cold and dense photosphere-like plasma are created in a higher (chromospheric or low coronal) levels, and a direct, collisional heating of plasma by the beam becomes possible there. This results in a shift of the location region of the source of hard X-ray bursts towards the higher than the typical photospheric levels. The self-similar modelling presented
in this section, allows to define the main possible dynamic regimes of the disturbed magnetic tube, which depend strongly on the parameters of plasma and scales of event. More detailed quantitative study of the effect requires an extensive numerical MHD simulations.
4. Conclusion Addressed to the basic dynamic and energetic processes in the solar coronal magnetic loops, the models presented here, can be applied for interpretation of the phenomena of CMEs, flares, oscillations of loops, acceleration of charged particles, etc. They are closely related to the main scientific goals of STEREO and Solar-B missions, such as explanation of the energetic processes in groups of interacting magnetic loops; explanation of oscillations of the loops and magnetic arcades; description of acceleration of CMEs and generation of accelerated particles on different stages of CME formation and
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propagation; explanation of the temporal features of the flaring electromagnetic emissions and location of the radiating sources; description of the plasma parameters dynamics in the solar filamentary material. An information about the 3D structure, mutual relation, radiative features, and global dynamics of the coronal loops and CMEs, as well as the detailed data on the associated magnetic fields, currents, and parameters of plasma are crucial for the considered models. We expect that high resolution 3D data from STEREO combined with the Solar-B vector magnetograms will appear as a good base for practical application and further development of the presented models.
Acknowledgments This work was supported by the Austrian ‘‘Fonds zur Fo¨rderung der wissenschaftlichen Forschung’’ (Project ¨ AD-RFBR Scientific and Technical P16919-N08), the O Collaboration Program (Project No. I.21/04), and the ¨ AD-’’Acciones Integradas’’ Program (Project No. 11/ O 2005).
References Aschwanden, M.J., Kosugi, T., Hanaoka, Y., Nishio, M., Melrose, D.B. Quadrupolar magnetic reconnection in solar flares. I. Threedimensional geometry inferred from Yohkoh observations. Astrophys. J. 526, 1026, 1999a. Aschwanden, M.J., Fletcher, L., Schrijver, C.J., Alexander, D. Coronal loop oscillations observed with the transition region and coronal explorer. Astrophys. J. 520, 880, 1999b. Aschwanden, M.J., DePontieu, B., Schrijver, C.J., Title, A. Transverse oscillations in coronal loops observed with TRACE: II. Measurements of geometric and physical parameters. Solar Phys. 206, 99, 2002. Berghmans, D., Clette, F. Active region EUV transient brightenings – First results by EIT of SOHO JOP80. Solar Phys. 186, 207, 1999. Cox, D.P., Tucker, W.H. Ionization equilibrium and radiative cooling of a low-density plasma. Astrophys. J. 157, 1157, 1969. Emslie, A.G. Energetic electrons as an energy transport mechanism in solar flares. Solar Phys. 86, 133, 1983. Gary, G.A., Demoulin, P. Reduction, analysis, and properties of electric current systems in solar active regions. Astrophys. J. 445, 982, 1995.
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Hagyard, M.J. Observed nonpotential magnetic fields and the inferred flow of electric currents at a location of repeated flaring. Solar Phys. 115, 107, 1989. Hardy, S.J., Melrose, D.B., Hudson, H.S. Observational tests of a double loop model for solar flares. Publ. Astron. Soc. Aust. 15, 318, 1998. Khodachenko, M.L. A dynamic model of a solar magnetic tube. Astron. Rep. 40, 252, 1996a. Khodachenko, M.L. Modeling the dynamics of partially ionized plasma in solar magnetic tubes. Astron. Rep. 40, 273, 1996b. Khodachenko, M.L., Zaitsev, V.V. Modeling energetic processes in solar active regions taking account of interaction between magnetic loops. Astron. Rep. 42, 265, 1998. Khodachenko, M.L., Rucker, H.O. MHD effects triggered by beams of fast particles in magnetic tubes and their possible relation to plasma heating during solar flares. Astrophys. Space Sci. 289, 111, 2004. Khodachenko, M.L., Haerendel, G., Rucker, H.O. Inductive electromagnetic effects in solar current-carrying magnetic loops. Astron. Astrophys. 401, 721, 2003. Landau, L.D., Lifshitz, E.M. Mechanics. Pergamon Press, Oxford, 1960. Leka, K.D., Canfield, R.C., McClymont, A.N., van Driel-Gesztelyi, L. Evidence for current-carrying Emerging flux. Astrophys. J. 462, 547, 1996. Melrose, D.B. Current paths in the corona and energy release in solar flares. Astrophys. J. 451, 391, 1995. Melrose, D.B. A solar flare model based on magnetic reconnection between current-carrying loops. Astrophys. J. 486, 521, 1997. De Moortel, I., Ireland, J., Walsh, R.W. Observation of oscillations in coronal loops. Astron. Astrophys. 355, L23, 2000. Moreton, G.E., Severny, A.B. Magnetic fields and flares in the region CMP 20 September 1963. Solar Phys. 3, 282, 1968. Nakariakov, V.M., Ofman, L., Deluca, E.E., Roberts, B., Davila, J.M. TRACE observation of damped coronal loop oscillations: Implications for coronal heating. Science 285, 862, 1999. Nakariakov, V.M., Verwichte, E., Berghmans, D., Robbrecht, E. Slow magnetoacoustic waves in coronal loops. Astron. Astrophys. 362, 1151, 2000. Nakariakov, V.M., Melnikov, V.F., Reznikova, V.E. Global sausage modes of coronal loops. Astron. Astrophys. 412, L7, 2003. Nishio, M., Yaji, K., Kosugi, T., Nakajima, H., Sakurai, T. Magnetic field configuration in impulsive solar flares inferred from coaligned microwave/X-ray images. Astrophys. J. 489, 976, 1997. Peres, G., Rosner, R., Serio, S., Vaiana, G.S. Coronal closed structures. IV – Hydrodynamical stability and response to heating perturbations. Astrophys. J. 252, 791, 1982. Rosner, R., Tucker, W.H., Vaiana, G.S. Dynamics of the quiescent solar corona. Astrophys. J. 220, 643, 1978. Wu, S.T., Guo, W.P., Dryer, M. Dynamical evolution of a coronal streamer – flux rope system – II. A self-consistent non-planar magnetohydrodynamic simulation. Solar Phys. 170, 265, 1997.
Solar plasma theoretical models for STEREO and Solar-B M.L. Khodachenko *, H.O. Rucker Space Research Institute, Austrian Academy of Sciences, Schmiedlstraße 6, A-8042 Graz, Austria Received 5 October 2004; received in revised form 22 December 2004; accepted 22 December 2004
Abstract An overview of several theoretical models addressed to the dynamical and energetic processes in the solar plasmas and coronal magnetic loops, and related to the phenomena of solar flares, CMEs, coronal loop interaction and oscillations is given. The presented models can be organized into two basic groups. The theoretical concept of one group of the models, called as the inductive models, employs the ideas of possible inductive and ponderomotoric interaction between the electric currents confined within the current-carrying magnetic loops moving relative to each other. The models of another group, so-called non-adiabatic MHD models, are applied to describe the MHD response of plasma in the low solar atmosphere to a changing current system of a flaring magnetic tube, which contains an injected beam of fast non-thermal electrons. A detailed information about the structure of coronal loops, their dynamics and global magnetic environment in solar active regions is crucial for all the considered models and it is expected that STEREO and Solar-B observations will be useful for testing and further improvement of these models. 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Sun:magnetic fields; Sun:flares; Sun:oscillations; Sun:corona
1. Introduction The solar corona has a highly dynamic and complex structure. It consists of a large number of constantly evolving loops and filaments, which interact with each other and are closely associated with the local magnetic field. The non-stationary character of solar plasma-magnetic structures appears in various forms of the coronal magnetic loops dynamics (grow motions, oscillations, meandering and twisting). Energetic phenomena, related to these types of magnetic activity, range from tiny transient brightenings (micro-flares) and jets to large, activeregion-sized flares and CMEs. Understanding of physical mechanisms in the background of the solar dynamic * Corresponding author. Tel.: +43 316 4120661; fax: +43 316 41206090. E-mail address: [email protected] (M.L. Khodachenko).
and energetic phenomena and their consequences in the near-Sun space is the main task of the modern solar physics. In this paper, we present briefly several theoretical models addressed to the dynamical and energetic processes in the solar plasma-magnetic structures, and related to the phenomena of solar flares, CMEs, coronal loop interaction and oscillations. The presented models can be separated on two general groups: (1) Inductive models of magnetic loops, which deal with the processes of global dynamics and interaction of coronal loops (Khodachenko et al., 2003) and (2) non-adiabatic MHD models dealing with the internal dynamics of plasma and fields in a particular magnetic tube (Khodachenko and Rucker, 2004). The data on the 3D structure and dynamics of coronal loops and their participation in the solar flaring events, expected from STEREO and SolarB, will provide the necessary information for testing and further development of the presented models and underlying concepts.
0273-1177/$30 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2004.12.056
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2. Inductive models of interacting current-carrying magnetic loops Complex dynamics of the solar coronal magnetic loops together with action of possible under-photospheric dynamo mechanisms cause majority of the loops to be very likely as the current-carrying ones. However, because of difficulties in measurement of a 3D magnetic field structure in the corona with a sufficiently high resolution, the presence of electric currents in the loops and their structure are not yet confirmed observationally. At the same time, the existing magnetometric measurements, which allow to determine a vertical component of rot~ B and hence to identify the vertical component of a current flowing from below the photosphere into the corona, provide a strong support in favor of the current-carrying loops hypothesis (Moreton and Severny, 1968; Hagyard, 1989; Gary and Demoulin, 1995; Leka et al., 1996; Hardy et al., 1998). Moving relative each other current-carrying magnetic loops should interact via the magnetic field and currents (Melrose, 1995). The simplest way to take account of the inductive and ponderomotoric interaction between the electric currents confined within the current-carrying magnetic loops moving relative to each other consists in application of the equivalent electric circuit models of the loops. In these models, each loop is considered as an equivalent electric circuit with variable parameters (resistance, capacitance and inductive coefficients) which depend on shape, scale, position of the loop with respect to other loops, as well as on the plasma parameters and value of the total longitudinal current in the magnetic tube (Khodachenko and Zaitsev, 1998; Khodachenko et al., 2003). The main attention in our research is paid to modelling of (1) the loop–loop flaring interaction, including as well the processes of electric current build up in the loops; (2) acceleration of solar Coronal Mass Ejections (CMEs); (3) oscillations of the loops in active regions. The equation for the electric current I in the coronal circuit of a separate, but not isolated from surroundings, thin (Rloop r0) magnetic loop has the following form (Khodachenko et al., 2003): IR ¼ U 0
1 d 1 dWext ðLIÞ ; c2 dt c dt 8R
ð1Þ
where L ¼ 4pRloop ðln rloop 74Þ is the inductance of the 0 2Rloop loop; R ¼ r2 rðT Þ is the resistance of the loop caused by a fi0 nite, depending on temperature conductivity of plasma r(T) = 9 · 106T3/2; U0 is a drop of potential between the loops footpoints, and Wext = B^Sloop is an external magnetic flux through the circuit of the loop. Here Rloop and Sloop are the main radius of the loop and the area covered by the loop, respectively. The term (1/c)(dWext/dt) in Eq. (1) is the external inductive electromotive force (EMF) eind. For the multiple loop systems, eind appears as EMF of mutual inductance eind ¼ c12 dtd ðM ij I j Þ; i 6¼ j, i
where i,j are the loop numbers, and Mij, mutual inductances. Characteristic time of the current change tc = L/(Rc2) in the coronal electric circuit is very large (104 years). Thus, the dynamics of current is defined by the motion of loops (emergence, submergence, etc.), which results in the change of inductive coefficients on the time-scales M L sL ¼ dL=dt and sM ij ¼ dM ijij=dt. It should be noted that the considered here equivalent electric circuit models are of course the idealization of real coronal magnetic loops. They usually involve a simplified geometry assumptions and ignore the fact that changes of the magnetic field propagate in plasma at the Alfve´n speed. This means that the circuit equations correctly describe temporal evolution of the currents in a system of coronal magnetic loops only on a time-scale longer than the Alfve´n propagation time. The actual spatial distribution of electric current in a loop is approximated by the current homogeneously distributed over the loop cross-section. At this stage, the stability of the loop is not considered anymore, but a certain care about not overcoming the kink instability threshold values for the magnetic tube twist is taken. On the other hand, the electromagnetic inductive coupling of current-carrying loops can play a significant role in stabilizing their possible kink instabilities. From the geometric point of view, the kink-type deformation of a loop results in a change of the magnetic loop inductance. In particular, for a spiral kink with one full twist, the disturbed value of inductance will be defined by the following expression: 2 4 7 113pRloop d d Lkink ¼ L þ L þ0 ; 16 Rloop Rloop 48 ð2Þ where L is the inductance of the undisturbed magnetic loop and d is the radius of the spiral disturbance, i.e., the distortion of the initial radius Rloop of the loop. For the typical scales of solar coronal magnetic loops, Eq. (2) gives the increase of inductance of the kink-disturbed magnetic loop, which in its turn corresponds to the increase of magnetic energy of currents stored in the loop. Moreover, the influence of the inductively connected neighboring loops will also strongly affect the energy change in the system during the deformation of a particular loop. Both these factors can result that kink deformation of a strongly twisted, carrying a high and dense current magnetic loop, in some cases will be still energetically not optimal, and thus this deformation will not take place. 2.1. Loop–loop flaring interaction It follows from observations that a large number of flares occurs in the regions where a new magnetic loop emerges and interacts with the existing loops (Nishio
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et al., 1997). Such events are called as interacting flare loops. Here, we present further development of the idea of the flaring loop–loop inductive interaction, first suggested by Melrose (1997) and later applied by Aschwanden et al. (1999a) for interpretation of observations. The easiest case of two interacting loops is considered (see Fig. 1). In addition to Melrose (1997) and Aschwanden et al. (1999a), we note that relative motion of the loops creates significant inductive electromotive forces in their electric circuits which appear as a powerful source for changing of the currents in the loops. Change of the currents disturbs the initial thermal equilibrium of the loops and results in a change of the plasma temperature, which in its turn influences the resistivity of the circuit and the radiative energy losses. Therefore, each of the rising magnetic loops (Fig. 1) in our model is described by two equations (Khodachenko and Zaitsev, 1998;Khodachenko et al., 2003): (1) the equation for the electric circuit of the loop 1 X _ _ ij I i Ri ¼ U 0i 2 M ij I j þ I j M ð3Þ c j and (2) the energy equation 1c 2 j2 ni QðT i Þ i H i ; T_ i ¼ 2ni k B rðT i Þ
ð4Þ
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where Mij = Mji, Mii ” Li, indices i, j indicate the loop number (1 or 2), and the dot denotes the time derivative. In Eq. (4), T i ; ni and ji ¼ I i =ðpr20i Þ are the temperature, density of plasma and the current density in the ith loop, respectively; c = 5/3 is the adiabatic constant, Q(Ti), the radiative loss function for the optically thin emission (Cox and Tucker, 1969; Rosner et al., 1978; Peres et al., 1982), and Hi is a stationary background heating introduced to provide thermal equilibrium in the initial steady state. Within the thin torus approximation, applied for the loops, we use for the mutual inductance coefficients Mij = Mji in Eq. (3) the approximate expression derived by Melrose (1997) and modified by Aschwanden et al. (1999a) " # 1=2 Rloop i Rloop j cos uij ; M ij ¼ 8 Li Lj ð5Þ ðRloop i þ Rloop i Þ2 þ d 2ij where Li, Lj are the inductances of the loops, Rloop i, Rloop j, their major radii, dij, the distance between the centers of the loops tori, and uij, the angle between the normal vectors to the loops planes. A linear increase in time of the major radii of the loops Rloop i ¼ R0loop i þ vi t; i ¼ 1; 2 is considered, and the initial steady state and thermal equilibrium are assumed. One of the loops is taken to be initially current-free
Fig. 1. (a) Loop–loop flaring interaction and (b) approximation of the flaring domain by a pair of rising inductively connected current-carrying loops.
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Fig. 2. Dynamics of currents and plasma temperature in the parallel (u12 = 0) loops 1 (a) and 2 (b) in dependence on the distance d12 (Khodachenko et al., 2003). ðR0loop 1 ¼ 2 109 cm; R0loop2 ¼ 108 cm; r01 ¼ r02 ¼ 5 107 cm; v1 ¼ 5 105 cm=s; v2 ¼ 106 cm=s; n1 ¼ n2 ¼ 109 cm3 ; T 1 ðt ¼ 0Þ ¼ T 2 ðt ¼ 0Þ ¼ 106 K; I 10 ¼ 1011 A; I 20 ¼ 0Þ.
(I20 = 0). The numerical solution (Khodachenko et al., 2003) of Eqs. (3) and (4) for both loops reveals the main features of the current dynamics in the loops (Fig. 2):
– Quick build up of the current in the initially currentfree loop; – Fast significant increase of the current value and plasma temperature in when the loops become to be of the same size (a flare); – The most efficient interaction of the parallel loops (u12 = 0). The inductive electric field produced in the loop can be roughly estimated as E ¼ I 2i =ððpr20i Þ2 rðT i ÞÞ. When the value of this field exceeds the critical value Ec 0.2(4pne3Le)/kBT, where Le is Coulomb logarithm, the main part of electrons appears to be in a run-away regime and can be accelerated by the inductive electric field. According to Fig. 3 which presents the dynamics of E and Ec in the loop 1 (the ‘‘slow’’ loop), the situation E > Ec takes place for t > 2000 s. The estimations show that inductive field can accelerate electrons in the loop up to energies We[eV] = q[e] Æ E[V/m] Æ lloop[m] = 100 1000 eV (Khodachenko et al., 2003). 2.2. CMEs as rising magnetic loops. Acceleration of CMEs
Fig. 3. Dynamics of the inductive electric field E and critical field Ec in the loop 1 (for the case u12 = 0) (Khodachenko et al., 2003).
Another aspect of inductive models consists in a possibility to describe the effects of the dynamic coupling (due to the force ½~ j~ B=c) of the large-scale coronal electric currents. These can influence the global dynamics of current-carrying magnetic loops, such as the loop
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(a)
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(b)
(c) Fig. 4. (a) Inductively interacting multiple current-carrying loops; (b) simplified representation of the multi-loop system and (c) the lifting force acting on the central loop as a function of the lateral loops height.
rising, oscillations, change of inclination. Considering the loops as electric circuits with flowing currents, one can find a generalized ponderomotive force of their interaction Fi ¼
oU ; oxi
ð6Þ
1 X M ij I i I j 2c2 i;j
ð7Þ
where U ¼
is a potential force function of the system of currents and xi is a generalized coordinate. Note that when the motion of a dynamic system is described by the generalized coordinates in terms of generalized forces Fi and momenta Pi the equations of motion are still P_ i ¼ F i . But only in Cartesian coordinates, the generalized momenta are the components of usual mechanical momentum vectors mvi. In general case, the Pi not necessarily reduce to products of mass and velocity and therefore the dimension of the generalized force Fi could be different from the one of the usual mechanical force (Landau and Lifshitz, 1960, p. 16). The advantage of the potential force function approach is that it simplifies significantly the analysis of the ponderomotoric interaction of currents. It allows one to avoid the complex integration of forces, acting on each separate element of a current. The motion of a rising current-carrying loop can be described as a result of the competition between the lifting force, Flift = oU/oRloop and magnetic stress force
2
2
n , where U is the potential force F m ¼ oso ðB8pÞs0 þ B4p Rloop function of a system (Eq. (7)), s and s0 are the coordinate and the unit vector along the field line, respectively, and n is the normal vector to the field line. Fm decreases when the Rloop increases. And in the case of Flift > Fm, the acceleration of the loop by the ponderomotive forces takes place (Fig. 4(a)). The MHD simulations in Wu et al. (1997) find that a flux rope with sufficient magnetic energy density will erupt. The source of this energy comes from the inductive and ponderomotoric interaction of currents in the filament and the system of underlying loops. In the model calculations, we consider a symmetric group consisting of three current-carrying loops (Fig. 4(b)). The bigger loop with the radius Rloop 2 stays vertically (H2 = 0) in the center of the group, whereas the lateral loops, located at the distance d on both sides from the central loop, are inclined at some angle H1 = H3 to the vertical direction. All the loops are assumed to be the current-carrying ones (with currents Ii, i = 1, 2, 3). To simplify calculation of the inductive coefficients, we approximate each loop by a rectangular structure characterized by height hi, length bi, and thickness 2r0i, where i = 1, 2, 3. For the simplicity reasons, the model loops are assumed to have a quadratic form (hi = bi), and the lateral loops are taken to be of the same size with the height h1 = h3 ” h. Fig. 4(c) shows the lifting force applied to the central loop as a function of the height of the lateral loops calculated for the following model parameters: Rloop 2 = 5 · 108 cm,
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r01 = r02 = r03 = 5 · 107 cm, H1 = H3 = p/4, H2 = 0, d = 5 · 109 cm, I1 = I3 = 0.5 · 1010 A, I2 = 1010 A. The increase of the lifting force during the increase of the lateral loops height can be clearly seen in Fig. 4(c). This means that under the certain conditions the rising motion of the lateral loops can result in a situation when the ponderomotoric lifting force exceeds the value of the magnetic stress force and the central loop starts to be accelerated. 2.3. Oscillations of current-carrying magnetic loops Oscillations of the loops are traditionally explained as standing, or propagating MHD waves. The transverse oscillations of coronal loops have been interpreted as kink fast magnetoacoustic modes (Aschwanden et al., 1999b; Nakariakov et al., 1999). Besides, the longitudinal waves (or propagating EUV disturbances) were discovered by Berghmans and
Clette (1999) and interpreted as slow magnetoacoustic waves (Nakariakov et al., 2000; De Moortel et al., 2000). In this section, we demonstrate a possibility of transverse oscillations of the loops which are caused by the ponderomotoric interaction of currents in groups of inductively coupled current-carrying loops. In that sense, the presented alternative mechanism is not traditional MHD in nature. The forces responsible for the loop oscillations here have essentially non-local character and appear as an interaction of electric currents running in different spatial domains. As for the MHD waves propagating along the loop, they also can take place in our case, but as a kind of a secondary effect, arising during the loop motion under the action of the ponderomotive forces. It is natural to expect that the whole large loop will move not as a rigid object, but having local transverse deformations, which can appear as the origin of MHD waves travelling along the loop.
Fig. 5. Temporal evolution of the mutual potential force function U123(H, t) (Khodachenko et al., 2003).
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We consider the modelling system of the current-carrying loops, shown on Fig. 4(b), assuming that the lateral loops are inclined at the angle H1 = H3 = p/4 and grow linearly in time: hi(t) = h0i + ai t, where i = 1, 3, h01 = h03 = 5 · 108 cm, and a1 = a3 = 1.5 · 105 cm s1. The lateral loops are assumed to be initially current free, i.e., I1(t = 0) = I3(t = 0) = 0, whereas the initial current in the central loop is taken as I2(t = 0) = 104 A. Other parameters of the model are taken as the following: h02 = 3 · 109 cm, d = 5 · 108 cm, r0i = 5 · 107 cm, Ti = 106 K, (i = 1, 2, 3). Based on the numerical solution of the set of Eqs. (3) and (4), we study the temporal evolution of the mutual potential force function of the system U 123 ðH; tÞ ¼ U ðH; tÞ þ
3 3 1 X 1 X 2 L I ¼ M ij I i I j ; i i 2c2 i 2c2 i;j;i6¼j
which describes the dynamic interaction of the loops and depends on their mutual orientation, i.e., on the variable angle H2 ” H, characterizing the inclination of the central loop (Khodachenko et al., 2003). The general potential force function U(H, t) is defined by Eq. (7). Formation of the dip in U123(H,t) during the system evolution (see Fig. 5) means the possibility of oscillations of the central loop near its vertical position. The oscillations can take place in the case when the system is disturbed by a sufficient external energy Eext input from the neighbouring flaring events and shocks propagating in the coronal plasma. To estimate the main characteristics of the oscillatory regime of the central loop, we assume that the amplitudes are small enough and the oscillation periods are not very long, i.e., Posc < scurr, where scurr is the characteristic time of the current variation in the system. This allows one to neglect the influence of a changing orientation of the oscillating central loop on the global currents dynamics in the system and use the quasi-stationary current approximation. On the other hand, in order to apply the electric circuit model for the current-carrying loops, Posc should be larger than the Alfve´n propagation time sA 1 to 10 s. The period of oscillations in the system with a dipped potential profile U123(H, t*), can be calculated by the formula pffiffiffiffiffiffiffiffi Z h2 dH P osc ¼ 2I0 ; ð8Þ 1=2 h1 ðE ext U 123 ðH; t ÞÞ
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where I0 is the inertia momentum of the loop. To be specific, we take here a particular form of the mutual potential force function U123(H, t*) (see Fig. 5) realized at the moment t = t* 16,000 s, corresponding to the largest dip. The reflecting points h1 and h2 are defined by the condition Eext = U123(H, t*). The amplitude D, angular amplitude dH and velocity v of the top point of the oscillating loop are determined by D ¼ h2 sin h2 ;
dH ¼
h2 h1 ; 2
v ¼ 2h2
h2 h1 : P osc
ð9Þ
Table 1 presents the main characteristics of the oscillatory regime of the loop, calculated by Eqs. (8) and (9) for different values of a disturbing energy input. The results obtained are rather close to the observed quantities (Aschwanden et al., 1999b). The observed decay of oscillations of solar magnetic loops which is usually explained by damping of a corresponding MHD mode (Nakariakov et al., 1999), can also be interpreted within the framework of the inductive model, taking into account the ponderomotoric interaction of the loops, without any appeal to some energy dissipation mechanism. For example, oscillations of the central loop in our modelling case will gradually vanish if after the initialization of the loop oscillations due to the input into the system of a certain energy E0 the potential function of the system U123(H, t) evolves so that after some time t 0 the minimum U123(H = 0, t = t 0 ) of the potential dip where the loop oscillates becomes to be above the level of the energy input E0. Oscillations of the loop during such evolution will gradually vanish. By this, not only the amplitude, but also the period of the vanishing oscillations will decrease (see Eq. (8) and Fig. 5). In some cases of the observed by TRACE oscillations of loops, presented by Aschwanden et al. (2002), the tendency of a changing period of oscillations during their decay probably takes place.
3. Non-adiabatic MHD models The models of this group describe the internal dynamics of plasma in a magnetic tube. They are based on the self-similar solution of the MHD equations, describing plasma flows with a homogeneous deformation. The processes of Joule heating, radiative energy losses, thermoconductivity and viscosity are taken into
Table 1 Amplitude, velocity and period of oscillations of the coronal loop, calculated for different values of the disturbing energy Eext Eext (erg)
1.38 · 1015
1.46 · 1015
1.55 · 1015
1.63 · 1015
1.72 · 1015
1.80 · 1015
dH () D (km) Posc (s) v (km s1)
2.72 1420 194 29.3
4.40 2300 247 37.2
6.09 3180 292 43.6
7.78 4050 331 49.2
9.46 4930 366 54.1
11.1 5800 398 58.6
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Fig. 6. Scheme of a flaring event and disturbance of magnetic tube by the reverse current.
account (Khodachenko, 1996a,b; Khodachenko and Rucker, 2004). In particular, the MHD response of plasma in a magnetic tube to a change of the electric current system structure caused by the injection into the tube of a beam of fast non-thermal electrons during a flare is considered. This study is addressed to a special aspect of the process of an energetic particle beam injection and propagation in a magnetic tube not yet considered. When a beam is injected into the magnetic tube, the initial (pre-flaring) current system is disturbed due to appearance of the so-called return current jr.c. = jb, where jb is the beam current. Changes of the current density (dj = jr.c.) in the magnetic tube disturb its thermodynamic equilibrium as well as the force balance (see Fig. 6). This results in a complex dynamic behaviour of the whole plasma-magnetic structure, which influences radiation of the system in X-rays and microwaves (Khodachenko and Rucker, 2004). These emissions appear additionally to the beam generated radio bursts. Besides, due to the temporal changes of the plasma parameters (density and temperature) in the disturbed magnetic tube, the features of the local radio source itself are changing. The impulsive character of the beam injection causes two stages in the evolution of the magnetic tube. (1)
During the first stage, characterized by the presence of a beam, the preliminary equilibrium of the magnetic tube is disturbed and complex MHD reaction of the system takes place in the region of the beam propagation. (2) During the second stage, when the injection of the beam is already over, the disturbed plasma-magnetic system gradually evolves to a new equilibrium state. The beam of fast electrons plays in this scheme a role of only a disturbing factor. It is important to note that the energy which is released during the MHD evolution and heating of the magnetic tube comes not from the energy of the beam, but is stored in the initial pre-flaring magnetic field. The destabilization of the magnetic tube by a beam is effective if duration of the beam sb is at least of the order of magnitude of the time s, characterizing the temperature variation in response to a changed current density, or, in other words, the reaction time of the system to a change of the Joule heating term. From the energy equation, the reaction time s can be estimated as (Khodachenko and Rucker, 2004): s
5=2 2nk B T rðT Þ 10 nT 37:26 10 : ðc 1Þ j2b j2b
ð10Þ
In the case of a 20 keV electron beam, Eq. (10) yields 5=2 s20 keV 2:3 1010 nTn2 , where nb is the electron density b
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in a beam. For the initial (pre-flaring) density n = 1010 cm3 and a 20 keV beam with nb/n = 104 the initial temperature 105 K gives s20keV 7.3 s, but for the temperature 106 K we have s20keV = 2300 s. The last seems to be too large for duration of an electron beam, whereas the disturbing effect of more possible shorter beams will be insufficient. At the same time, for the considered parameters of plasma and beam the mean free path of fast electrons 2 !2 31 2 e 5 ke20 keV ¼ 44pn 2 me ðV e20 keV Þ 2
ðT e20 keV Þ ; ð11Þ n is about ·1011 cm, which is much greater than the atmospheric gravity scale and the typical scales of the considered fragment of the flaring magnetic loop. Thus, fast electrons of the beam do not interact collisionally with the plasma and there is no any collisional energy deposit from the beam to the plasma here. Therefore, the considered mechanism of plasma disturbing and heating works on the heights where the direct collisional energy deposition from the non-thermal beam to the background plasma Emslie, 1983 is still not efficient. The optimal range of heights is probably located in the lower corona or upper chromosphere. The dynamics of a magnetic tube disturbed by a beam of fast electrons is described by the model based on the self-similar MHD solutions: 2:85 104
a_ V r ðr; tÞ ¼ r; a Br ðr; tÞ ¼ 0;
b_ V z ðz; tÞ ¼ z; b
V u ðr; tÞ ¼ 0; Bu ðr; tÞ ¼ Bu0 ðtÞ
r ; R
ð12Þ
Bz ðr; tÞ ¼ Bz0 ðtÞ; ð13Þ
qðr; tÞ ¼ qðtÞ; pðr; z; tÞ ¼ p0 ðtÞ p1 ðtÞ
ð14Þ r 2
p2 ðtÞ
z 2
; R L r 2 z 2 T 2 ðtÞ ; T ðr; z; tÞ ¼ T 0 ðtÞ T 1 ðtÞ R L
ð15Þ
where R and L are transverse and longitudinal scales of the considered fragment of a flaring magnetic tube; R* is the external transverse scale of the whole magnetic structure (r [ R R*) (see Fig. 6); a(t),b(t) are dimensionless time-dependent components of the deformation tensor, characterizing the degree of plasma compression in the magnetic tube. Substitution of the self-similar solution (12)–(15) to the MHD equations set and further grouping of terms proportional 0th, 1st, and 2nd power of r and z yields a set of ordinary differential equations for a(t), b(t), Bu0(t), Bz0(t), q(t), T0(t), T1(t) and T2(t), which is much more convenient for the analysis than
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the initial set of MHD equations. Assuming the small 2 2 quantities: n1 ¼ TT 10 ðRr Þ 1, n2 ¼ TT 20 ðLz Þ 1, we present a any function of the type T (r, t) in the energy equation as T a ðr; tÞ T a0 ðtÞð1 a n1 a n2 Þ (Khodachenko, 1996a,b). 2 The Joule heating term ð~ j ~ jb Þ =rðT Þ in the energy equation includes as well the return current caused by the propagating beam of non-thermal electrons. In this paper, we illustrate briefly the main consequences of such modelling, whereas more details regarding the model itself and used approaches can be found in (Khodachenko, 1996a,b; Khodachenko and Rucker, 2004). The MHD response of the magnetic tube onto a beam is defined by (a) initial equilibrium parameters of plasma (T0(t = 0) = T00, T1(t = 0) = T10, T2(t = 0) = T20, n00 = q (t = 0)/mi); (b) current density of the beam (d = jb/j(t = 0)); (c) characteristic scales of the model (R, L, R*). Three general types of dynamics can be distinguished: (1) compressional regime; (2) quasi-periodic (pulsating) regime and (3) decompressional regime. All the dynamical regimes result from the competition between the pressure gradient and Ampere forces in the plasma, whereas the temperature change is determined by heating and cooling mechanisms acting differently on different dynamical stages of the magnetic tube evolution. Fig. 7 shows the examples of dynamics of the plasma parameters in the magnetic tube during the tube disturbance by a beam of fast electrons (Fig. 7(a)), and during its consequent relaxation to a new equilibrium state (Fig. 7(b)). In Fig. 7(b), plasma velocities V r ðr ¼ _ RÞ ¼ aa_ VR0 and vz ðz ¼ LÞ ¼ bb VL0 are normalized to V 0A ¼ Bu0 ðt¼0Þ ð4pqðt¼0ÞÞ
A
1=2
A
R . R
Thus, the disturbances of the current system produced by a beam of fast electrons propagating in the magnetic tube result in the dynamic processes, which in their turn, will influence radiation of the system in X-rays and microwaves, appeared additionally to the beam generated radio bursts. By this, due to the temporal changes of the plasma parameters (n, T) in the disturbed magnetic tube, the features of the local radio source itself are changing. This may explain a variety of observational data (Khodachenko and Rucker, 2004), which cannot be easily interpreted in terms of the traditional flaring scheme operating only with the collisional beam-plasma mechanism of the energy deposition into the system. Pulsating regimes can be analog of flaring events with precursors, or the quasi-periodically modulated flares. These pulsations correspond to the global sausage mode which has recently been identified in the radio observations with spatial resolution (Nakariakov et al., 2003). A rough estimation of the period of pulses during the quasi-oscillatory relaxation of the disturbed magnetic tube can be obtained within the assumption of adiabaticity. Khodachenko and Rucker (2004) provide the following formula for estimation
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Fig. 7. (a) Dynamics of q(t) and T0(t) in the tube with T10/T00 = 102, R = 3 · 107 cm, b(t) ” 1, T2(t) = 0, during propagation of the beam with d = 105 for different values of T00: (I) 3 · 105 K; (II) 1.8 · 105 K; (III) 105 K, and n00: (1) 109 cm3; (2) 1010 cm3; (3) 1011 cm3. (b) Oscillatory relaxation of the disturbed magnetic tube with n00 = 1.5 · 1011 cm3, T10/T00 = T20/T00 = 102, R = 3 · 107 cm, L = 3 · R, R* = 3 · 108 cm in dependence on the T00: (1) 107 K; (2) 6 · 106 K and Bu0(t = 0): (I) 55 G; (II) 45 G (Khodachenko and Rucker, 2004).
of the period of the small-amplitude adiabatic compressional oscillations of the magnetic loop fragment: 2psM a P ad ¼ pffiffiffiffiffiffiffiffiffiffiadffi ; c1
ð16Þ
where 1=ð2c2Þ B n00 T 00 ; b ¼ B 8pk sm ¼ R=V 0A ; aad ¼ ðb TT 1000 Þ 2 2 , and c u0 ðt¼0Þ ðR=R Þ is adiabatic constant. In particular, for the parameters of the model used for calculations in Figs. 7b I(1), II(1), Eq. (16) yields Pad = 54.5 s and Pad = 90.3 s for Bu0(t = 0) = 55 G and Bu0(t = 0) = 45 G, respectively. Of a special interest appear the cases where the regime of the compressional cooling is realized (Figs. 7a I(2); I(3); II(2)). In these cases, the conditions of cold and dense photosphere-like plasma are created in a higher (chromospheric or low coronal) levels, and a direct, collisional heating of plasma by the beam becomes possible there. This results in a shift of the location region of the source of hard X-ray bursts towards the higher than the typical photospheric levels. The self-similar modelling presented
in this section, allows to define the main possible dynamic regimes of the disturbed magnetic tube, which depend strongly on the parameters of plasma and scales of event. More detailed quantitative study of the effect requires an extensive numerical MHD simulations.
4. Conclusion Addressed to the basic dynamic and energetic processes in the solar coronal magnetic loops, the models presented here, can be applied for interpretation of the phenomena of CMEs, flares, oscillations of loops, acceleration of charged particles, etc. They are closely related to the main scientific goals of STEREO and Solar-B missions, such as explanation of the energetic processes in groups of interacting magnetic loops; explanation of oscillations of the loops and magnetic arcades; description of acceleration of CMEs and generation of accelerated particles on different stages of CME formation and
M.L. Khodachenko, H.O. Rucker / Advances in Space Research 36 (2005) 1561–1571
propagation; explanation of the temporal features of the flaring electromagnetic emissions and location of the radiating sources; description of the plasma parameters dynamics in the solar filamentary material. An information about the 3D structure, mutual relation, radiative features, and global dynamics of the coronal loops and CMEs, as well as the detailed data on the associated magnetic fields, currents, and parameters of plasma are crucial for the considered models. We expect that high resolution 3D data from STEREO combined with the Solar-B vector magnetograms will appear as a good base for practical application and further development of the presented models.
Acknowledgments This work was supported by the Austrian ‘‘Fonds zur Fo¨rderung der wissenschaftlichen Forschung’’ (Project ¨ AD-RFBR Scientific and Technical P16919-N08), the O Collaboration Program (Project No. I.21/04), and the ¨ AD-’’Acciones Integradas’’ Program (Project No. 11/ O 2005).
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