Comparison of force-free flux rope models with observations of magnetic clouds

Transcription

1 Advances in Space Research xxx (2004) xxx xxx Comparison of force-free flux rope models with observations of magnetic clouds M. Vandas a, *, E.P. Romashets b, S. Watari c, A. Geranios d, E. Antoniadou d, O. Zacharopoulou d a Astronomical Institute, Academy of Sciences, Boční II 1401, Praha 4, Czech Republic b Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation, Russian Academy of Sciences, Troitsk, Moscow Region , Russia c National Institute of Information and Communications Technology, Nukuikita, Koganei, Tokyo , Japan d Department of Physics, University of Athens, Panepistimioupoli-Kouponia, Athens 15771, Greece Received 7 September 2004; received in revised form 22 November 2004; accepted 24 November 2004 Abstract Recently, we have found a force-free solution inside an elliptic cylinder for magnetic cloud models. For a diminishing oblateness, the solution tends to the widely used Lundquist constant-alpha force-free solution inside a circular cylinder. The solution may include effects of a flux rope expansion. A comparison of this new solution and the Lundquist solution with magnetic cloud observations is done for magnetic clouds with flat magnetic field magnitude profiles. In such cases, oblateness improves fits of magnetic field magnitude profiles in clouds significantly. Ó 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Magnetic clouds; Solar wind; Interplanetary magnetic field 1. Introduction Magnetic clouds are large interplanetary flux ropes propagating in the solar wind, which were ejected from the Sun; their legs are probably still connected to the Sun when the clouds are observed at 1 AU. Their signatures in solar wind observations were defined by Klein and Burlaga (1982) as regions in the solar wind with (i) an enhanced magnetic field magnitude, (ii) a smooth rotation of the magnetic field vector through a large angle, and (iii) a decreased proton temperature. Magnetic clouds are low-b objects and therefore they are often modelled by force-free magnetic fields. The simplest one is a constant-a axially symmetric force-free field in a (circular) cylinder, the so called Lundquist solution * Corresponding author. address: vandas@ig.cas.cz (M. Vandas). (Lundquist, 1950). It was successfully used by Burlaga (1988) and then by many others to fit observed magnetic field profiles in magnetic clouds. These fits yield basic flux rope parameters like their axis orientations, chirality, magnetic field strength, and diameter. In reality, magnetic clouds will not be circular, but oblate, as various studies indicate (e.g., from stand-off distances of their bow shocks (Russell and Mulligan, 2002), or from MHD simulations (Vandas et al., 2002; Odstrcil et al., 2002)). Oblate shapes of flux ropes were integrated into non force-free models first (Mulligan and Russell, 2001; Hidalgo et al., 2002). Vandas and Romashets (2003) generalized the Lundquist solution for a case of an elliptic cylinder. Vandas et al. (2005) presented the first fits of magnetic cloud profiles by this (static) solution. The present paper is an extension of the former paper, it incorporates temporal effects (cloud expansion) and discusses more cases /$30 Ó 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. doi: /j.asr

2 2 M. Vandas et al. / Advances in Space Research xxx (2004) xxx xxx 2. Models Two models are used and presented for comparisons, the Lundquist one and the constant-a force-free field in an elliptic cylinder. The latter one becomes the former one when the oblateness disappears (and the cylinder becomes circular), but this fact is not obvious from analytic expressions, so formulas for both solutions are given below. The Lundquist solution (Lundquist, 1950) reads B r ¼ 0; ð1þ B u ¼ B 0 J 1 ðarþ; ð2þ B Z ¼ B 0 J 0 ðarþ; ð3þ where J 0 and J 1 are the Bessel functions; a = const.; B 0 is the magnetic field strength; r, u, and Z are cylindrical coordinates, related to Cartesian ones by common expressions x ¼ r cos u; ð4þ y ¼ r sin u; ð5þ z ¼ Z: ð6þ The flux rope boundary is set at the place where B Z = 0; then a = 2.41/r 0, where r 0 is the flux rope radius and the number 2.41 (approximately) stands for the first root of J 0. A constant-a force-free solution in an elliptic cylinder (Vandas and Romashets, 2003) reads 1 ob Z B u ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ac cosh 2 u cos 2 v ov ; ð7þ B v ¼ p ac 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh 2 u cos 2 v ob Z ou ; ð8þ ceh 0 ðu; e=32þce 0 ðv; e=32þ B Z ¼ B 0 ce 2 0 ð0; e=32þ ; ð9þ where u, v, and Z are elliptic cylindrical coordinates, related to Cartesian ones by x ¼ c cosh u cos v; y ¼ c sinh u sin v; ð10þ ð11þ z ¼ Z; ð12þ p where c ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 b 2, u 2 Æ0,1), and v 2 Æ0,2p). The generating ellipse with semimajor axis a and semiminor axis b has u = u 0 = const., coshu 0 = a/c. The parameter e =(ac) 2 ; a = const. is related to the size and oblateness of the flux rope (B Z component is zero on the elliptic flux rope boundary, defined by a and b, similarly to the convention introduced for the Lundquist solution). B 0 determines the magnetic field strength; ce 0 and ceh 0 are ordinary and modified even Mathieu functions of zero order. The field configurations (1) (3) and (7) (9) are static. It is known that magnetic clouds are expanding. Modifications of the solution (1) (3) in case of a radial flux rope expansion was first carried out by Osherovich et al. (1993). This solution described an expanding flux rope only for the polytropic plasma index c < 1 and was not force-free. It was used by Marubashi (1997) to fit magnetic field and velocity profiles in magnetic clouds. Shimazu and Vandas (2002) derived a slightly different solution, which described a force-free field of an expanding flux rope for an arbitrary polytropic index (a similar result was obtained by Berdichevsky et al., 2003). In an asymptotic limit, the field configuration is described by (1) (3), but a and B 0 are time dependent in a way a a! ð1 þ t=t 0 Þ ; B 0! B 0 ð1 þ t=t 0 Þ ; ð13þ 2 where t is time and t 0 is a parameter related to the expansion rate. A model flux rope is radially expanding with velocity V r ¼ r : ð14þ t þ t 0 It is assumed that the same holds for an oblate flux rope and the same changes are made in (7) (9). The expansion law (14) ensures that an oblate flux rope conserves its oblateness (i.e., a/b) during expansion. 3. Comparison of the models with observations Theoretical magnetic field magnitude profiles (Vandas and Romashets, 2003; Vandas et al., 2005) show that oblate magnetic clouds have flat profiles when the clouds are crossed parallel to their b axis (preferential crossing because b is roughly oriented along the cloudõs bulk velocity). Therefore we selected for our analysis magnetic cloud observations during the period with pronounced magnetic field vector rotations and flat B profiles, where oblateness presumably plays a role. Data were taken from the omni data base at the World Data Center A (WDC-A-R&S). Boundaries of magnetic clouds are determined using identifications already published and/or adjusted from the magnetic field components and plasma (mainly proton temperature) behaviour. The position of the boundaries determines two times, t 1 and t 2, when a spacecraft entered and exited a cloud. These times are kept fixed during a fitting procedure. The intermediate variance direction of magnetic field vectors (Lepping et al., 1990) is used for an estimate of the axis orientation. A slope of the total velocity profile serves for the first evaluation of t 0. A non-linear least-square (PowellÕs) method

3 M. Vandas et al. / Advances in Space Research xxx (2004) xxx xxx 3 (Press et al., 1992) determines cloud parameters from observed magnetic field and total velocity profiles. The method searches a minimum of the expression D ¼ 1 X N jb obs NB obs i B mod i jþ w av i¼1 X N NV obs av i¼1 jv obs i V mod i j; ð15þ where N is a number of observed (hourly) values within the time interval Æt 1,t 2 æ, ÔobsÕ and ÔmodÕ refer to the observed and model values, respectively, ÔavÕ is the average value within the time interval, and w is a weight (usually 0 or 1). Both measured and model values are spatially and temporarily dependent. It is assumed that a flux rope moves with a constant velocity V obs av in the x direction (in GSE) and radially expands according to (14). Model parameters are determined in several steps (in which more and more parameters are switched on in searching procedures) and final fits are obtained with w =1in(15). We shall present several examples of fits, which also compare the models (1) (3) and (7) (9). The former model is referred as ÔcircularÕ and the latter one as ÔoblateÕ in figures and in Table 1, which summarizes our results. Fig. 1 shows the magnetic cloud of 30 October It is #10 from the Lepping et al. (1990) list. The intermediate variance direction gives the axis orientation: h 0 = 77, / 0 = 200. The h 0 is the inclination of the axis to the ecliptic plane, / 0 is the axis azimuthal angle in the ecliptic plane. So the flux rope is highly inclined. The fit with the Lundquist solution yields: h 0 = 74, / 0 = 208, r 0 = 0.11 AU, p = The r 0 is the radius of the flux rope at the first encounter (at time t 1 ). The impact factor p is the closest distance of the spacecraft to the rope axis divided by r 0. Its sign determines on which side of the axis the spacecraft crossed, it is the sign of y (in GSE coordinates) of the crossing point of the Table 1 Magnetic cloud model parameters Case Fit begin 78/10/ /02/ /10/ /01/10-05 end 78/10/ /02/ /10/ /01/11-02 h 0 [ ] i c o / 0 [ ] i c o V obs av ½km s 1 Š r 0 [AU] c o p c o t 0 [h] h B 0 [nt] c o a/b o w 0 [ ] o D c o The rows ÔbeginÕ and ÔendÕ determine the boundary of the clouds in the form YY/MM/DD-HH (UT). The column ÔfitÕ indicates which model is used (i axis orientation from the intermediate variance direction, c model with a circular cross section, o model with an elliptic cross section). The other symbols are described in the text. It was used w =0 for D calculations in order to have a net comparison of magnetic field fits. Fig. 1. Magnetic cloud of 30 October Plotted are magnetic field magnitude and components (in GSE), solar wind velocity, density, and proton temperature. The vertical lines indicate estimated cloud boundaries. Model values are displayed in thick lines, a fit with the Lundquist solution (labelled ÔcircularÕ) is on the left, a fit with the elliptic solution (labelled ÔoblateÕ) is on the right.

4 4 M. Vandas et al. / Advances in Space Research xxx (2004) xxx xxx rope axis with the ecliptic plane, the y is assumed to be constant. The flux rope has a negative chirality h (see Table 1), that is, it is left-handed. This model yields a nearly central crossing (small p). The geometry of the crossing is shown in Fig. 2. The fit with the elliptic solution yields: h 0 = 79, / 0 = 325, r 0 = 0.11 AU, p = 1.4, a/b =4,w = 11. The oblateness a/b is determined only roughly, because its changes are not very sensitive during fitting. We define r 0 = b at the first encounter (semiminor axis at time t 1 ). The w angle is the azimuthal angle in the ecliptic plane of the b axis (for w =0 a projection of the b axis to the ecliptic plane is oriented along the background flow, it is an expected orientation). The geometry of the crossing is shown in Fig. 3. The latter model improves the fit of the magnetic field magnitude profile. Fig. 4 shows the magnetic cloud of 7 February It is #15 from the Lepping et al. (1990) list. A large field increase at the trailing edge of the cloud was excluded during fits. It is accompanied by jumps in the other plasma parameters and caused by a shock penetrating the cloud. The intermediate variance direction gives the axis orientation: h 0 =71, / 0 = 164. The fit with the Lundquist solution yields: h 0 =20, / 0 = 176, r 0 = AU, p = This fit gives a different axis orientation than the intermediate variance direction is, a low inclined flux rope instead of a highly inclined one, which yields a very small radius of the rope. The fit with the elliptic solution produces h 0 =77, / 0 = 171, r 0 = 0.19 AU, p = 0.73, a/b = 1.2, w =42. This model improves the fit of the magnetic field and velocity profiles significantly and yields the axis orientation close to the intermediate variance direction. Fig. 5 displays the magnetic cloud of 18 October It was analyzed by Lepping et al. (1997). It has a very flat profile. Vandas et al. (2005) presented Fig. 2. Geometry of the spacecraft crossing through the model magnetic cloud of 30 October 1978 described by the Lundquist solution, plotted in the flux rope rest frame with the rope axis perpendicular to the figure plane. A projection of the spacecraft trajectory through the cloud is the displayed line, the direction of the spacecraft motion is indicated by the arrow. The smaller circle is the cloud boundary at the spacecraft first encounter, the larger one is the boundary at the exit. Three bullets along the spacecraft trajectory subsequently mark positions of the first encounter, the closest approach, and the exit. Fig. 3. Geometry of the spacecraft crossing through the model magnetic cloud of 30 October 1978 described by the elliptic solution. Layout is the same as in Fig. 2. In addition, the orientations of the rope major (a) and minor (b) axes at the first encounter are shown by the dashed lines. At the exit, the positions of these axes are the same, only the axes are longer. The minor axis is not parallel to the spacecraft trajectory (which has an opposite direction than the background flow) because w 6¼ 0. Fig. 4. Magnetic cloud of 7 February Layout is the same as in Fig. 1.

5 M. Vandas et al. / Advances in Space Research xxx (2004) xxx xxx 5 Fig. 5. Magnetic cloud of 18 October Layout is the same as in Fig. 1. Fig. 6. Magnetic cloud of 10 January Layout is the same as in Fig. 1. a comparison of fits by the Lundquist and elliptic static models. As in the previous case, a large field increase at the trailing edge of the cloud was excluded during fits from a similar reason. The intermediate variance direction gives the axis orientation: h 0 = 14, / 0 = 289. The fit with the Lundquist solution yields h 0 = 16, / 0 = 292, r 0 = 0.12 AU, p = This model gives a nearly central crossing. The fit with the elliptic solution produces h 0 = 14, / 0 = 300, r 0 = 0.12 AU, p = 0.37, a/b =6, w = 3. The latter model improves the fit of the magnetic field magnitude profile significantly. Fig. 6 shows the magnetic cloud of 10 January It was analyzed by Burlaga et al. (1998) and Ivanov et al. (2004). Vandas et al. (2005) presented a comparison of fits by static models. It also has a very flat profile and is similar to the previous case. Again, a large field increase at the trailing edge of the cloud was excluded during fits; it is supposed to be a remnant of a prominence (Burlaga et al., 1998). The intermediate variance direction gives the axis orientation: h 0 = 18, / 0 = 245. The fit with the Lundquist solution yields: h 0 = 16, / 0 = 236, r 0 = AU, p = This model also yields a nearly central crossing. The fit with the elliptic solution produces: h 0 = 15, / 0 = 226, r 0 = AU, p = 0.60, a/b =4, w = 1. As in the previous case, the latter model improves the fit of the magnetic field magnitude profile significantly. 4. Conclusions A comparison of magnetic field and plasma observations of magnetic clouds with linear force-free models were presented. The models were a constant-a force-free field in a circular cylinder (Lundquist solution) and its generalization, a constant-a force-free field in an elliptic cylinder. These static models were modified to include temporal effects (flux rope expansion). It was demonstrated that oblateness in the model flux rope improves field magnitude fits of clouds with flat magnetic field

6 6 M. Vandas et al. / Advances in Space Research xxx (2004) xxx xxx magnitude profiles significantly. Rotation of the magnetic field vectors in clouds is described comparably well by the models. Observed velocity magnitude profiles are fitted quite well by the models. Acknowledgements The observational data were provided by the WWW services of NSSDC (WDC-A-R&S); we acknowledge PIs who provided data to the omni data set, namely from the ISEE 1, ISEE 3, and Wind missions. This work was supported by INTAS grant and RFBR grant , and by projects S from AV ČR and ME501 from MŠMT ČR, and by grant 205/ 03/0953 from GA ČR. We also acknowledge support from the bilateral Czech-Greek agreement on collaboration in science and technology. References Berdichevsky, D.B., Lepping, R.P., Farrugia, C.J. On geometric considerations of the evolution of magnetic flux ropes. Phys. Rev. E 67, , Burlaga, L.F. Magnetic clouds and force-free fields with constant alpha. J. Geophys. Res. 93, , Burlaga, L., Fitzenreiter, R., Lepping, R., Ogilvie, K., Szabo, A., Lazarus, A., Steinberg, J., Gloeckler, G., Howard, R., Michels, D., Farrugia, C., Lin, R.P., Larson, D.E. A magnetic cloud containing prominence material: January J. Geophys. Res. 103, , Hidalgo, M.A., Nieves-Chinchilla, T., Cid, C. Elliptical cross-section model for the magnetic topology of magnetic clouds. Geophys. Res. Lett. 29 (13), , Ivanov, K.G., Bothmer, V., Cargill, P.J., Kharshiladze, A.F., Romashets, E.P., Veselovsky, I.S. Slow dynamics of photospheric regions of the open magnetic field of the Sun, solar activity phenomena, substructure of the interplanetary medium, and near- Earth disturbances in the beginning of the 23rd cycle: The 1996 to February 1997 events. Int. J. Geomag. Aeronomy 4, Klein, L.W., Burlaga, L.F. Interplanetary magnetic clouds at 1 AU. J. Geophys. Res. 87, , Lepping, R.P., Jones, J.A., Burlaga, L.F. Magnetic field structure of interplanetary magnetic clouds at 1 AU. J. Geophys. Res. 95, 11,957 11,965, Lepping, R.P., Burlaga, L.F., Szabo, A., Ogilvie, K.W., Mish, W.H., Vassiliadis, D., Lazarus, A.J., Steinberg, J.T., Farrugia, C.J., Janoo, L., Mariani, F. The Wind magnetic cloud and events of October 18 20, 1995: Interplanetary properties and as triggers for geomagnetic activity. J. Geophys. Res. 102, 14,049 14,063, Lundquist, S. Magnetohydrostatic fields. Ark. Fys. 2, , Marubashi, K. Interplanetary magnetic flux ropes and solar filaments. In: Crooker, N., Joselyn, J., Feyman, J. (Eds.), Coronal Mass Ejections, Geophys. Monogr. Ser., vol. 99. AGU, Washington, DC, pp , Mulligan, T., Russell, C.T. Multispacecraft modeling of the flux rope structure of interplanetary coronal mass ejections: cylindrically symmetric versus nonsymmetric topologies. J. Geophys. Res. 106, 10,581 10,596, Odstrcil, D., Linker, J.A., Lionello, R., Mikic, Z., Riley, P., Pizzo, V.J., Luhmann, J.G. Merging of coronal and heliospheric numerical two-dimensional MHD models. J. Geophys. Res. 107 (A12), 2002, SSH Osherovich, V.A., Farrugia, C.J., Burlaga, L.F. Dynamics of aging magnetic clouds. Adv. Space Res. 13 (6), 57 62, Press, W.H., Flannery, B.P., Teukolsky S.A., Vetterling, W.T. Numerical Recipes in Pascal. The Art of Scientific Computing. Cambridge University Press, New York, p. 331, Russell, C.T., Mulligan, T. The true dimensions of interplanetary coronal mass ejections. Adv. Space Res. 29 (3), , Shimazu, H., Vandas, M. A self-similar solution for expanding cylindrical flux ropes for any polytropic index value. Earth Planets Space 54, , Vandas, M., Romashets, E.P. Force-free field with constant alpha in an oblate cylinder: a generalization of the Lundquist solution. Astron. Astrophys. 398, , Vandas, M., Odstrčil, D., Watari, S. Three dimensional MHD simulation of a loop-like magnetic cloud in the solar wind. J. Geophys. Res. 107 (A9), SSH , Vandas, M., Romashets, E., Watari, S. Magnetic clouds of oblate shapes. Planet. Space Sci. 53, 19 24, 2005.