On MHD Modeling of Coronal Mass Ejections

Transcription

1 NUMERICAL MODELING OF SPACE PLASMA FLOWS / ASTRONUM-2007 ASP Conference Series, Vol. 385, c 2008 Nikolai V. Pogorelov, Edouard Audit, and Gary P. Zank, eds. On MHD Modeling of Coronal Mass Ejections Horst Fichtner, Andreas Kopp Institut für Theoretische Physik IV, Ruhr-Universität Bochum, Universität Str. 250, Bochum D-44780, Germany Jens Kleimann Max-Planck-Institut für Sonnensystemforschung, Katlenburg-Lindau, Germany Rainer Grauer Institut für Theoretische Physik I, Ruhr-Universität Bochum, Germany Abstract. We give at first a brief overview of the motivation for magnetohydrodynamic simulations of coronal mass ejections that can be classified as principal, local, and global, and discuss some of the present problems with the modeling. Besides the initiation, acceleration, evolution, and interactions of CMEs with the background solar wind, with each other and with planetary magnetospheres, we identify the need for large-scale MHD and multi-fluid models that explicitly incorporate effects of kinetic processes occurring on micro- or meso-scales. Concentrating on the acceleration and heating of the solar wind and CMEs by plasma waves, we describe an alternative route to this goal. Subsequently, we present CWENO-based local CME simulations and relate them to observations with the ACE spacecraft near 1 AU. 1. Why are CMEs interesting? The study of coronal mass ejections (CMEs), especially by numerical means, has been tremendously intensified during recent years for several reasons, see, e.g., the reviews by Aschwanden et al. (2006) and Forbes et al. (2006). First, CMEs are amongst the main mediators of the influence of the Sun on the inner heliosphere, particularly on the Earth and its environment, where they significantly codetermine the space weather conditions (Scherer et al. 2005; Bothmer & Daglis 2006; Schwenn 2006). Second, as a consequence of the shocks driven by CMEs, they serve as particle accelerators that do not only contribute to space weather effects, but can be used to study acceleration processes in detail (Reames 1999; Mewaldt et al. 2005; Li et al. 2005). Third, with the recently launched two-spacecraft mission STEREO (Kaiser 2005), for the first time the full three-dimensional structure of CMEs can be observed both remotely and in-situ, for first results see, e.g., Howard & SECCHI Team (2007). And, fourth, the MHD modeling of CMEs provides an excellent testbed for numerical codes, as is manifest with numerous approaches documented in the literature. These approaches can be grouped into principal (analytical or symmetry-assuming) modeling (e.g., Titov & Démoulin 1999; Roussev et al. 2003; Schmidt & Cargill 2003; Jacobs et al. 2005), local modeling limited to a few tens of solar radii (e.g., 151

2 152 Fichtner et al. Mikić & Linker 1994), and global modeling covering the whole range from the solar surface out to 1 AU and beyond (e.g., Manchester et al. 2004; Odstrcil et al. 2005; Tóth et al. 2005). Despite these intensified efforts and activity regarding the study of CMEs, there remains a number of unsolved problems as well as modeling deficiencies, as described in the following section. We single out the problem of plasma acceleration and heating due to waves that most likely has an influence on the dynamics of the background solar wind and of the CMEs themselves. After we have described a way to incorporate the relevant micro- or meso-scale kinetic physics of wave propagation into a macroscopic MHD model, we present first results obtained with a recently developed code (Kleimann et al. 2004) based on the also rather new CWENO algorithm (Kurganov & Levy 2000) that is designed for an optimized treatment of shocks. 2. Open problems in the modeling of CMEs There are still unanswered questions regarding several aspects of CMEs as such. For example, it is unknown whether they are initiated by flux cancelation or injection or an as yet unidentified process. Also, their initial and interplanetary acceleration, propagation and evolution in size and shape is by far not fully understood in all detail, and neither is their interaction with the background solar wind, with other CMEs, and with planetary magnetospheres (e.g., Ip & Kopp 2002). Besides problems directly related to CMEs, there exist also difficulties related to the incorporation of fundamental physical processes into large-scale CME models like magnetic reconnection, which is expected to play a role in coronal heating and eruptions, or the influence of plasma waves on the dynamics of the solar wind and of CMEs. A principal difficulty is the gap between relevant scales (see Fig. 1): the kinetic physics is to be described on length scales shorter than 10 3 m, while the MHD interactions take place on scales longer than 10 6 m. The task is, therefore, to bridge this gap. This can be achieved purely numerically by applying the method of adaptive mesh refinement (AMR) as is described by Tóth et al. (2005). The AMR method is, however, conceptually non-trivial computationally demanding and, thus, applicable only, if a suitable, highly sophisticated computing environment is available. An alternative is a two-step procedure in which a first step, consisting of a coupling of the kinetic and MHD regimes by analytical methods, is followed by a numerical integration of the MHD equations. The latter are supplemented by additional terms and, possibly, additional equations, both quantifying the effects of the kinetic processes on the micro- and meso-scales characterized by (Forbes et al. 2006), e.g., the inertial length or the gyro radius. In the following section we discuss one example of how the effect of the plasma wave dynamics can be included in a large-scale MHD model. 3. Plasma acceleration and heating due to waves Many MHD models of CMEs lack an explicit formulation of the non-thermal heating of the background solar wind. One approach dating back to the work

3 MHD Modeling of Coronal Mass Ejections 153 Figure 1. Time and length scales associated with CMEs as they propagate outwards through the solar system. The shaded regions indicate the ranges covered by numerical codes, and the curved lines indicate the leading and trailing edges of a CME as it propagates outwards through the heliosphere. Taken from Forbes et al. (2006). by Hartle & Barnes (1970) employs ad-hoc heating functions (e.g., Groth et al. 2000; Manchester et al. 2004) } ( (r R)2 Q(r) = ρ q exp { σ 2 T 0 γ p ) ρ that can actually cool the plasma at radii where depending on the values of the pressure p, mass density ρ and polytropic index γ the actual temperature drops below a prescribed target temperature T 0. Here, q and σ are constants, and R is the solar radius. Another also at least three-decade old idea (Fahr et al. 1976) invokes variable polytropic indices γ = γ(r) (e.g., Lugaz et al. 2007). And somewhat younger phenomenological heating functions explicitly based on wave effects introduced by Hollweg (1986) are still discussed in recent literature (e.g., Usmanov et al. 2000). None of these formulations intends to incorporate the plasma waves self-consistently. This might not be necessary for those studies that concentrate on interplanetary propagation of CMEs, i.e. on the distances greater than some 30 solar radii, but it is required for modeling the region closer to the Sun, where the initiation and acceleration of CMEs takes place. We have recently attempted the incorporation of wave effects (Laitinen et al. 2003; Vainio et al. 2003) for the case of the solar wind background plasma. To this end we extended the approach by Tu (1987) (see also Hu et al. (1999)) to explicitly describe the wave dynamics. Using wave power spectra P(r, f), where (1)

4 154 Fichtner et al. f is the frequency, the wave pressure p w (r) = 1 8π 0 P(r,f)df allows to compute the plasma acceleration a w = 1 n p w (with number density n). The plasma heating can be computed from Q p = F K(r, f H ) 4π (u + v A ) f H P(r, f H ) 4π where u and v A are the solar wind and the Alfvén velocity, respectively. The heating due to wave cascading into the dissipation region is described by the first term (for explicit forms of F K see Vainio et al. (2003)) and the effect of the changing dissipation frequency f H (r) by the second. The quantities F K and f H (r) are the central quantity connecting the kinetic with the MHD scales. Following an idea by Vainio & Laitinen (2001), equating the wave damping length (u + v A )/(2γ w ) to the distance from the Sun s surface r R, where γ w is the damping rate, an analytical expression can be derived: f H (r) Ω p 2π α f(r) (see Laitinen et al. (2003)). In difference to earlier work, the auxiliary quantity α f and, thus, the dissipation frequency f H is a function of heliocentric distance. Completing the approach by supplementing the basic equations (where p e,p denote the thermal electron and proton pressure, j the current density, B the magnetic field, Φ grav the gravitational potential, ν ep the collision frequency and η the (scalar) resistivity) ρ (ρu) p p p e B = (ρu) (3) = (ρu u) (p e + p p + p w ) + j B ρ Φ grav (4) = (p p u) 2 3 [p p ( u) Q p ] ν ep (p p p e ) (5) = (p e u) 2 3 (2) [ pe ( u) ηj 2] ν ep (p e p p ) (6) = (ub Bu) + η B η j (7) with an equation for the wave transport ([ ] ) P 3 = 2 u + v A P (u ) P F K f allows to model the solar wind as well as CMEs on large scales and, nonetheless, to include the effects of the wave dynamics at the same time. These equations are mutually coupled via the wave power P, the wave pressure p w, as well as the solar wind and Alfvén velocity u and v A, respectively. This approach has been validated by the results for the simplified case of a radially expanding steady-state slow and fast solar wind, which are in fairly good agreement with the available data (Laitinen et al. 2003). In the next section we discuss first results of our subsequent CME modeling. (8)

5 MHD Modeling of Coronal Mass Ejections 155 Figure 2. Selected snapshots from a simulation of two interacting CMEs expanding from the Sun (marked in black at the lower left), shown as cuts in the x z-plane (with z along the rotation axis of the Sun and x being the direction of propagation of the CMEs) through the velocity field (color-coded in units of the sound speed), have been superimposed with field lines of the solar magnetic field. The normalized simulation times are given above the individual images, the length scale is one solar radius. 4. First results from a new CME model In a next step towards an improved CME model we have solved Eqs. (3) to (7) with our CWENO-based code (Kleimann et al. 2004) for various time-dependent, fully three-dimensional MHD cases, not yet considering any two-fluid effects. For the first test cases described here, we have not integrated the wave transport Eq. (8), but rather used γ = 1 for the polytropic index, i.e. assumed constant temperature T = 10 6 K. A typical case of two expanding CMEs is shown in Fig. 2 displaying a time series of the color-coded, normalized velocity field and the magnetic field structure in the meridional plane y = 0. In the first and second frame, both CMEs are still present in the simulation volume; note the structures indicated by the kinks in the magnetic field lines, which could result in shocks for the non-isothermal case. The CME that was launched later is more massive and faster than the earlier one so that their relative distance is decreasing and they start to collide. In the third frame, while still continuing to accelerate, the faster CME also reaches the volume boundary, thereby dragging the field lines outwards and deforming them almost radially. In the final frame both CMEs have completely left the simulation volume and the system has relaxed into a state similar to the initial quiet conditions. Figure 3 gives a comparison of another of our (isothermal) simulation results with observations. Despite the rather different heliocentric distances, the overall similarity of the computed time profiles and the actual observations is remark-

6 156 Fichtner et al. Figure 3. Left: time profile of magnetic field strength, particle number density, plasma temperature, and flow velocity of the near-earth solar wind as measured in situ by the ACE spacecraft in September 1998 during the passage of a magnetic cloud. The vertical lines denote the cloud s delimitation from the preceding shock front. Taken from Burlaga et al. (2001). Right: time profiles for the magnetic field component B z, number density n, and velocity v at selected solar radii r i along a single CME s trajectory. The minus sign of B z compensates for the magnetic field s North-South polarity (which has B z < 0 in the ecliptic plane z = 0). Time is given in hours after the CME onset. B z and n have been scaled with (r i /R) 2 and (r i /R) 3, respectively, such that similar curves can each be shown in a single viewgraph. able. Particularly the sharp rise and slow decay of the velocity s maxima is clearly discernible in both cases. Also the sharp, almost needle-shaped peaks of the magnetic field profiles exhibit a strong similarity. The notable differences in the total duration of passage (about one day for the magnetic cloud opposed to about one hour in the simulations) can easily be accounted for by the very different site of the observations. The cloud had much more time to expand from a presumably rather compact object to its full length of up to 1 AU. These findings underline the importance of an as detailed as possible modeling of CMEs close to the Sun: it appears that after a few solar radii many of their characteristics have already developed and are mainly scaled during the further propagation out into the heliosphere. 5. Résumé We have discussed the need for CME studies that incorporate micro- and mesoscale kinetic effects into large- or macro-scale MHD models. At the example of the acceleration and heating of the solar wind and CMEs by plasma waves, we have described a way to achieve such a combination. While the first modeling results are still based on rather simplifying assumptions, we have demonstrated nevertheless the usefulness of the approach by showing their remarkable principal similarity to observations in interplanetary space. Subsequent investigations will comprise more complex cases and the explicit inclusion of the wave dynamics.

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