JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A07104, doi: /2007ja012814, 2008

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi: /2007ja012814, 2008 MHD simulations of the global solar corona around the Halloween event in 2003 using the synchronic frame format of the solar photospheric magnetic field Keiji Hayashi, 1 Xue Pu Zhao, 1 and Yang Liu 1 Received 13 September 2007; revised 7 February 2008; accepted 17 March 2008; published 19 July [1] We performed two time-relaxation magnetohydrodynamics (MHD) simulations of the solar corona: one uses the boundary map representing the solar surface magnetic field distribution before the Halloween event in 2003, and the other uses map representing the postevent distribution. The aims of this study are to test a new concept of a solar surface magnetic field map capable of representing a particular time of interest and to examine the coronal responses to the solar photospheric magnetic field changes occurring over a few days. We used a new mapping scheme named synchronic frame that can include the longitudinal shift caused by the solar differential rotation and the solar surface variations occurring at the time of interest. These two time-relaxation MHD simulations using the two maps are separately performed to numerically obtain the quasi steady states of the solar corona before and after the Halloween event. Comparisons of the simulated coronal magnetic field structures to the SOHO/EIT measurements show that the combinations of our mapping method and simulation model reproduce the changes of the coronal structures well. We also find that the consequences of solar surface variations can be seen in the plasma quantities in the solar corona. These results show the capability and importance of the solar surface magnetic field mapping scheme for better reconstruction of global coronal structures, parts of which are sensitive to the solar surface magnetic field variations. Citation: Hayashi, K., X. P. Zhao, and Y. Liu (2008), MHD simulations of the global solar corona around the Halloween event in 2003 using the synchronic frame format of the solar photospheric magnetic field, J. Geophys. Res., 113,, doi: /2007ja Introduction [2] The solar surface magnetic field is one of major factors that determine solar coronal structures on a global scale, such as the position, size, and shape of the coronal holes and the coronal streamers. Therefore, a map of the solar surface magnetic field has long been used to give the boundary constraints in various models, such as the potential field model [Schatten et al., 1969] and the timedependent magnetohydrodynamics (MHD) simulations. The potential field model assumes the vacuum limit so that the system can be described with one Laplace equation. The calculation of the potential field model is much faster than MHD simulations and can provide a possible reconstruction of the coronal magnetic field; therefore, this model has long been used as a standard scheme. In many MHD simulation studies of the solar corona [e.g., Han et al., 1988; Linker et al., 1990; Usmanov and Dryer, 1995; Groth et al., 2000; Riley et al., 2001; Usmanov et al., 2005; Hayashi et al., 2006; Mikić etal., 2007], the simulation domain has the 1 W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, California, USA. Copyright 2008 by the American Geophysical Union /08/2007JA boundary at the base of the solar corona so that the measurement-based or synthetic data of the solar photospheric magnetic field can be applied to specify the magnetic situation to be simulated. In most cases, the potential field is first used to give the initial values of the threedimensional magnetic field, then the nonlinear interactions between the plasma and magnetic field in the sub-alfvénic solar corona and the super-alfvénic interplanetary space can be numerically treated. In these major approaches, the solar surface magnetic field map covering the entire solar surface is an essential input. [3] The solar surface measurements are generally limited to the Earth side hemisphere. Therefore, making the entire solar surface map needs a process to combine data taken at different times. The concept of a synoptic chart was proposed by Bumba and Howard [1965] to make the entire solar map by collecting the solar magnetic field measurements taken at different times. The synoptic map assumes that the solar surface magnetic field in a global scale does not change significantly during one Carrington rotation period and that the solar differential rotation can be neglected. These assumptions allow us to treat the meridional bins sampled at different times evenly and determine the coronal structures stable during one Carrington rotation period. In turn, the constructed synoptic chart does not 1of13

2 contain information specifying the particular time in the Carrington rotation period it represents. Therefore the results of the MHD simulations using the solar surface magnetic field in the synoptic chart format are unable to represent the solar coronal features at a particular time. [4] In some cases, we need to determine the global coronal structures at a particular time. For example, a recent study on the propagation of the coronal mass ejection (CME) by Liu and Hayashi [2006] uses a traditional synoptic chart to numerically reproduce the global coronal magnetic field structures as the background of CME propagation. The synoptic chart is sufficiently useful to discuss the general tendency of the CME propagation. However, the mapping scheme to specify the solar surface magnetic field at a particular region and a particular time of interest would be helpful to discuss a particular event. [5] To improve solar surface mapping, Zhao et al. [1997] proposed a modified version of a synoptic chart, named a synoptic frame, in which the solar surface magnetic field measurement data taken at the time of interest replaces the corresponding part of the synoptic chart. This partially replaced synoptic chart is used to calculate the potentialfield source-surface model, and it was demonstrated the solar coronal magnetic field structures at the time of interest could be constructed better than with the synoptic chart. [6] Recently, Zhao et al. (X. P. Zhao et al., The effect of the differential rotation of photospheric magnetic features on the synoptic frames of the photospheric magnetic flux, submitted to Journal of Geophysical Research, 2008)] proposed a synchronic frame, a modified version of the synoptic chart and the synoptic frame formats, to better reproduce the magnetic field structures at both particular time and particular solar longitude. In the concept of the synchronic frame, first the synoptic chart is prepared so that the central longitude will correspond to the time of interest. Second, the positions of the synoptic chart data are longitudinally shifted in accordance with the solar differential rotation so that the data point at the chosen central longitude does not move [Zhao et al., 2004]. The synoptic chart is thus modified so that the regions around the longitude of interest are better represented than in the original synoptic chart. Finally, the synchronic frame is made by replacing the data around the central longitude with the corresponding measurement data taken at the time of interest. [7] The map showing only the differential rotation correction can be regarded as representing the conditions just before the time of interest, and the map corrected in accordance with the solar differential rotation and replaced with the data at the time of interest can be regarded as representing the conditions at and after the time of interest. Therefore, by simulating the solar corona using the two maps, we can obtain the solar coronal responses to the solar surface changes that occurred around time of interest. By comparing coronal observations such as SOHO/EIT, we can check the ability of the combinations of the mapping method and the coronal model to reconstruct the coronal features. [8] Because the solar magnetic field measurement can be done best for the meridional bin on the solar disc, it is the best case if the solar longitude of interest is near the solar meridional bin. We chose the Halloween event in 28 October 2003, because the X-class flare event occurred near the center of the solar observation disc and there were substantially large solar surface magnetic field variations. Because of the position as well as the size of the event, this event represents a good case for seeing solar coronal variations and testing the concept of the synchronic frame. [9] The most orthodox approach for obtaining the coronal responses to the surface magnetic field variations is to perform time-dependent MHD simulation with a timevarying boundary magnetic field. This simulation approach would be capable of coupling the measured magnetic field with the associated plasma flow, determining the full MHD conditions on the solar surface and in the solar corona, and helping estimate the flux of the magnetic energy and helicity to the corona though the photosphere. However, in practice, it is generally difficult to prepare the timevarying boundary MHD variables when the temporal variation of the normal component of boundary surface magnetic field is given. That is, the solution must satisfy the sinusoidal condition [Yeh and Dryer, ¼r ~V ~B n ; ð1þ where subscription n stands for the normal direction to the boundary surface (in this study, the radial direction). Equation (1) is identical to the induction equation along the normal direction. When the left-hand t B n is given, though, this equation is under-determined. In this kind of problem, the shortage of information describing the physical quantities and their temporal evolutions is generally inevitable; the measurements are limited only to the solar photosphere, or an observable solar atmospheric layer, while the three-dimensional data including the radial gradients are needed to determine the temporal evolution of a magnetic field satisfying the divergence-free condition. Therefore, models and assumptions have been proposed to compensate for the lack of information in physics. [10] The probably first effort to calculate the plasma flow from the solar photospheric magnetic field measurement data was done by Levine and Nakagawa [1974]. They successfully derived the horizontal plasma flow from the measurement of the solar surface magnetic field variations at a sunspot region but failed at the extreme and saddle points of B n because of the assumption of the zero vertical flow. Recently, many sophisticated methods coupling the induction equation with the measurement sequence of the solar surface magnetic field have been proposed [e.g., Kusano et al., 2002; Longcope, 2004; Welsch et al., 2004; Schuck, 2006]. By incorporating the measurement sequence of the normal and/or tangential components of the magnetic field and giving reasonable physical and mathematical constraints, these methods have been proved to be quite powerful in determining the surface plasma flow and magnetic helicity flux, and thus the consequent magnetic energy in the solar corona. However, these methods are originally designed to analyze a small area with strong magnetic fields like sunspot regions, and it will be difficult to apply these methods to the entire solar surface. For example, these methods may not work for calculating the plasma motion at the quiet sun with small and relatively weak fields and at the high heliographic latitude regions with low quality of measurement. In addition, it is difficult 2of13

3 to calculate the plasma motion on the entire solar surface such that fully satisfies physical conservation laws. [11] Because these difficulties may not be soon solved, we have chosen a simple approach: we performed two MHD simulations to obtain the steady corona with the fixed normal component of the boundary magnetic field, using two data maps that represent time periods before and after the Halloween event. One data map is the synoptic chart modified in accordance with the solar differential rotation, and the other is the synchronic frame. This choice is reasonable as long as only the steady states of the solar corona are discussed. [12] The simulation method is described in section 2, and the boundary magnetic field maps are shown in section 3. In section 4, the simulation results and comparisons with the SOHO/EIT measurements are shown, and section 5 contains the discussion. 2. MHD Simulation Code [13] The basic equations governing the system are the set of the ideal MHD equations in the @t ¼ r % ~V ¼ r P g þ B2 I þ % ~V ~V 1 8p 4p ~ B~B h þ % ~g þ W ~ ~r ~ W þ 2 ~V ~ ¼ r ¼ r E þp g þ B2 ~V 1 8p 4p ~ B ~V ~B h þ % ~V ~g þ ~ W ~r W ~ i ; ð5þ where %, ~V, ~B, P g, E, ~r, t, ~g, g are mass density, velocity of plasma flow viewed in the rotating frame, magnetic field vector, gas pressure, energy density E = %v 2 /2 + P g /(g 1) + B 2 /2, position vector originating at the center of the Sun, time, solar gravitational force ~g = GM/r 3 ~r, and the specific heat ratio, respectively. The 3 by 3 unit tensor is written as I, and ~V ~V, ~B ~B, ~V ~B, and ~B ~V are dyadic products. The MHD simulation assumes the solar rigid rotation so that the steady relaxed state of the solar corona will be obtained for each magnetic field data. The sidereal angular velocity of the rigid solar rotation in the MHD simulation, j ~ Wj, is taken to be 2p/25.3 radian/d (or 14.2 /d). Note that the input boundary magnetic field maps will include the effect of the solar differential rotation. [14] In this MHD simulation model, the solar wind heating and acceleration are represented by the specific heat ratio g that is assumed to be This choice is reasonable because of our focus on the global coronal ð2þ ð3þ ð4þ responses to the surface magnetic field variations. To obtain detailed solar coronal structures, such as the temperature structures at the active regions, realistic physical processes such as the Alfvén-wave decay, the electron heat conduction, and the radiation thermal loss at the solar transition region and the lower corona must be included [e.g., Mok et al., 2005]. We did not attempt to obtain such smallscale structures in this work, because extending the study requires additional computational resources and techniques to refine the computation grids handling the steep vertical gradient of plasma variables and to handle the very small time-increment steps that result from physical processes varying more quickly than the hyperbolic MHD waves. [15] The simulation code we have developed [Hayashi, 2005] is based on the concepts of the total variation diminishing (TVD), the monotonic upstream scheme for conservation laws (MUSCL), and the finite volume method (FVM), and is modified so that the steep gradients and wide ranges of quantities of MHD variables in the solar corona and solar wind can be properly treated. The numerical grids are constructed in the spherical coordinate system, and the computational region is between the two spheres. The radius of the inner boundary sphere is chosen to be 1.01 solar radii, so that the simulation can treat the plasma already heated up to the order of 1 MK at the solar transition regions and the boundary magnetic field values can be well specified with the photospheric measurement map. The radius of the outer sphere is chosen to be 50 solar radii at which the solar wind is always super- Alfvénic. The radius from 1.01 to 50 solar radii is divided with 144 nonuniform cells. The latitude and longitude are uniformly divided by 128 and 256 cells, respectively, thus Dq = Df = p/128. [16] Without any modification, the size of numerical cells (/ Df sinq) is very small near the solar rotational axis, and the computations will be severely slow due to the Courant- Friedrichs-Lewy (CFL) condition on the time increment step. To speed up the calculation, the MHD variables allocated at cells near the solar rotation axis are averaged along the longitude every time increment step. This averaging is done for every k 0 cells, where number k 0 is the largest of the integer power of 2, satisfying k 0 Dr/DfR s sinq. The longitudinal differencing in updating data at the cell address k uses the cell data at k ±2k 0, k ± k 0 and k, while the differencing with respect to the other directions is done normally. Because the difference method is based on the FVM, the conservative variables, such as mass, momentum, and energy, are preserved through this longitudinal differencing and averaging process. [17] The outer boundary sphere is set at the heliocentric distance of the super-alfvénic region where all information needed to update the boundary variables come from the computation domain. In contrast, for the inner sub-alfvénic boundary sphere where the radial component of the magnetic field will be specified, the boundary treatments based on the concept of the method of characteristics is used so that only the physically meaningful information can be used to update the boundary variables. In our simulation model, the concept of the projected normal characteristic method [e.g., Nakagawa et al., 1987; Wu and Wang, 1987] is applied so that the boundary solutions will be consistent 3of13

4 with the hyperbolic system of the governing MHD equations. The projected normal characteristic method reasonably works to minimize the physical inconsistency and reduce the unphysical vibrations near the sub-alfvénic solar surface boundary. In most cases, this method gives three constraints regarding three outgoing MHD waves, and in turn allows us to give five constraints. In this paper, where the time-relaxed solar corona will be obtained, the radial component of the solar surface boundary magnetic field is assumed to be fixed all through the simulation. The fixed B r requires two additional constraints so that the sinusoidal condition will be satisfied [Yeh and Dryer, 1985]. In our test simulations [Hayashi, 2005], we examined some choices of the remaining arbitrary constraints. In the present study, we chose the simplest among those we tested and found reasonable; that is, the surface density and temperature are fixed at the initial values of 2.0e8/cc and 1.46 MK when the surface plasma flows outward (at the coronal hole base), and the plasma variables can vary (generally increase) keeping the specific enthalpy when the plasma is stagnant (in the closed-field regions). [18] We performed a time-relaxation simulation of the solar corona for 40 h in real timescale starting with the initial values of the analytical solution of the spherically symmetric plasma flow [Parker, 1958] and the potential field magnetic field. In this study, the source surface of the potential field is chosen at 50 solar radii, so that almost all initial magnetic field lines are closed, and whether the field lines are closed or open to the interplanetary space is not determined at the initial stage. The scalar potential of field was computed by solving the time-relaxation of the Laplace t Y = r 2 y, instead of the spherical harmonic polynomial expansion, so that the derived field will fully match the specified boundary map and the information contained in the given boundary map will be preserved. By solving the timerelaxation of the scalar magnetic potential with respect to unphysical time t, we do not have to choose the truncation term of the spherical harmonics polynomial and can avoid the ring-like unphysical vibration of the magnetic field values around the large magnetic signals corresponding to the sunspots. [19] The system of the global solar corona will be sufficiently relaxed after the time-relaxation simulation for 40 h in real-timescale. It must be mentioned that the obtained relaxed solution is not exactly steady. For example, residual plasma motions remain in the closed-field region where the plasma flow speed is expected to be completely zero. We suspect that a main cause of the residuals is the Kelvin-Helmholtz type instability at the boundary between the coronal hole and the stagnant closed-field region where the horizontal gradient of the radial component of plasma flow is large. Such instability in a spatial scale comparable to or probably smaller than the numerical cell size can generate weak but ceaseless numerical fluctuations. In the simulated three-dimensional solar corona for this study, the speed of the residual plasma flow in the closed-field regions is less than 0.05 km/s, or 1/2000 of the local sound speed or Alfvén speed. This number is larger than those of twodimensional cases in which the size of the numerical cell can be much smaller. Because the residual plasma motion in the three-dimensional cases is still sufficiently small, we regard the obtained numerical time-relaxed state as the steady state. 3. Boundary Magnetic Field Maps [20] The synchronic frame, an improved version of the synoptic chart, will be described in detail by Zhao et al. (submitted manuscript, 2008). Therefore, we will briefly mention the mapping processes. [21] Figure 1 shows the boundary magnetic field data used in this simulation study to demonstrate our mapping process: Figure 1a the synoptic chart (hereafter, abbreviated SC), Figure 1c the synoptic chart accounting the solar differential rotation (DR), and Figure 1d the synchronic frame (SF). In order to take the solar differential rotation of the surface magnetic field into account, we use the solar rotation as the function of the heliographic latitude derived by Snodgrass [1983]. For reference, Figure 1b shows the synoptic frame, an early version of the synchronic frame, made by replacing the SC map with the data at the time of interest. [22] To construct SF map, first, the SC is made with the central Carrington longitude of 294 that approximately corresponds to the time of the Halloween X-flare event. We used the SOHO/MDI magnetograph data made from 14 October (corresponding to the Carrington longitude of 114, and Carrington rotation number of 2008) to 11 November 2003 (corresponding to the longitude of 114, and the rotation number of 2009) to make this SC map. Second, the SC is longitudinally shifted in accordance with the solar differential rotation rate to form the DR map (Figure 1c). Because the period of the rotation at the high latitude is longer than the Carrington rotation period, the extra 3-d data are used to cover the whole longitude at the solar high-latitude regions. We regard the DR map as the map best representing the entire solar surface magnetic field distribution before the Halloween event. By replacing the central part of the DR map with the data taken after the Halloween event on 28 October, we can finally construct the SF map (Figure 1d) which we regard as best representing the entire surface magnetic field after the Halloween event. Because of the longitudinal shift in accordance with the solar differential rotation and the replacement of the data around the central longitude, the longitude in the DR and SF map differ from that of the Carrington longitude in the SC. With the bottom label heliographic longitude in Figure 1, we mean to emphasize that the DR and SF represent the entire solar surface at a particular time of interest. [23] The SF is made by replacing the central part of the SF map with the measurement data taken at the time of interest. Regarding this updating or replacing process, there are two points to mention. The first one is the size of the replacement, because the quality of the data near the solar limb is not good owing to the inclination of the magnetic vector, the limb s darkening, and the systematic increase of noise. We used the data at the longitude range from 45 Wto 45 E in the disc measurement for this replacement. The second point is that generally the photospheric magnetic field directing outward and inward is not balanced over the replaced part. The solar magnetic field measurement data inevitably contains observational errors, and we cannot correctly estimate the amount of unbalanced magnetic flux 4of13

5 Figure 1. Solar photospheric magnetic field maps, (a) the synoptic chart (SC), (b) the synoptic frame, (c) the SC map modified in accordance with the differential rotation (DR), and (d) the synchronic frame (SF). The white lines in the four plots show the polarity inversion lines where the radial component of magnetic field is zero. Note that the heliographic longitude in the DR and SF map specifies the longitude at a particular time while the Carrington longitude in the SC map does not differentiate the longitude and the time. and determine the locations of the overestimated parts. Therefore, to make the whole solar surface magnetic flux balanced, we are only permitted to make a simple choice: to correct the magnetic field strength over the entire map or within the replaced part. We chose the former because it will change the magnetic field values evenly all over the surface, and therefore the effects of such surplus magnetic flux in the replaced part will be minimized. In the present case, as shown in Figure 1, the changes outside the replaced part around the central longitude are very small; the largest displacement of the surface polarity inversion lines outside the replaced part is less than 2, comparable to or smaller than the angular size of the computational cell. In general, as shown by Zhao et al. [1999], the changes of the potential field global corona due to this correction are small enough. [24] Figure 2 shows the base of the coronal hole in the simulated corona using the SC (Plot (a) of Figure 1) and DR (Plot (c) of Figure 1) maps. While the corrections of the solar differential rotation will be largest at the east limb (90 E from the central longitude), large differences are not seen between the two cases (see Plots (b) and (e) of Figure 2). Therefore, the discussion in the following sections will use only the mapping format DR and SF. [25] In this study, the DR and SF maps were smoothed by Gaussian with half-width of 2.5 arc degrees so that data gaps will be filled, the smoothness of the boundary map will match the size of the numerical grids, and the computational stability is obtained. The information about the smallscale field structures are lost in this smoothing procedure. However, the information about the solar differential rotations included in the synchronic frame format will not be lost. Therefore, we think that the smoothing procedures will not significantly affect our study in which we simulate the large-scale coronal structures. The radial method [Wang and Sheeley, 1992] is used to convert the observed line-of-sight component of magnetic field to the radial component. 4. Simulation Results [26] The MHD simulations obtaining the quasi steady states matching the specified boundary magnetic field maps DR and SF were separately performed with the identical MHD code. Therefore, it is a reasonable expectation that the differences between the two simulation results will be best seen in the coronal magnetic field structures. Such simulated differences must have counterparts in the real corona, if the combination of the solar surface magnetic field mapping scheme and the MHD model worked reasonably well. The coronal hole is chosen for the comparison, because whether the simulated coronal field lines are closed or open to the interplanetary space can be numerically determined, and the observational coronal holes will be clearly identified. 5of13

6 Figure 2. Base of open-field lines in the solar corona simulated with the SC and DR maps. Plots in the upper and lower row are obtained from the simulation using the SC map and the DR map, respectively. The plots in the right column are viewed from the central longitude that corresponds to the Earth s position on 28 October The plots in the left column are viewed from the opposite side of the Earth. The difference between the simulations with SC and DR must be most significant when viewed from 90 E from the Earth (middle column), though it is not in the simulated case. [27] In Figure 3, the coronal bases of the open- and closed-field region are drawn with darker and brighter gray, respectively. Figure 2 has been made in the same format. The solar inclination angle, B-zero angle, is taken into account so that the plots can be compared to the corresponding SOHO/EIT images shown in Figure 4. Because the concept of the source surface is no longer applicable to the MHD simulation results [e.g., Riley et Figure 3. Base of open-field lines in the solar corona simulated with the DR and SF maps. Plots in the upper and lower row are obtained from the simulation using the DR map and the SF map, respectively. The intervals of the longitude of the viewpoints are 14, synthesizing three consecutive daily observations from the Earth over the Halloween events (27, 28, and 29 October 2003). 6of13

7 Figure 4. SOHO/EIT images made on 27, 28, and 29 October The disc images measured at 195A are plotted in the upper row, and that measured at 304 A are plotted in the lower. al., 2006], the numerical coronal holes are determined here by whether or not the field lines starting from the inner boundary sphere at 1.01 solar radii can reach the super- Alfvénic flow regions, which are typically at 5 solar radii in this simulation study. It was confirmed that all simulated field lines once reaching the super-alfvénic regions extend outward to the interplanetary space and never return the inner boundary sphere. In Figure 4, the reversed gray scale of EIT images taken at 195 and 304 angstrom are used to show the coronal density and temperature distribution tendencies. [28] Figure 5 assembles Plots (a) and (f) in Figure 3 and the corresponding plots in Figure 4, for comparison with the simulations and the observations. Two points are notable: the first is that a southward bulge (marked with circles) from the existing ring-shaped coronal hole appears in the simulation of the post-halloween corona with the SF map. While the ring-shaped coronal hole is not clearly seen in the EIT measurement, the southward extension of the new coronal hole is clearly observed. The second in Figure 5 is the changes in size of another coronal hole (marked with rectangles). The coronal hole extending from northeast to southwest in EIT images shrank after the Halloween event. In the simulation result of the post-halloween corona, this part of the coronal hole disappeared completely. It can be said that our model simulation reasonably reproduced the decrease in the size of this coronal hole, although it was not perfect. [29] Both the expanding and shrinking coronal holes observed by the SOHO/EIT might be the consequences of the coronal process in which the global corona seeks the lower-energy and thus more stable state that matches the new solar surface magnetic field distribution. We do not mean that the observed coronal variations were caused only by the surface magnetic field variations. However, it may be reasonable to think that the concept of the synchronic frame capable of including the surface magnetic field changes observable from the Earth (near the central meridional longitude) provided the simulation model with a boundary magnetic field condition reasonably well, to better reproduce the coronal magnetic field. [30] Figure 6 shows the solar wind flow speeds on the 2.5 solar radii sphere obtained by the MHD simulations. The distributions of the simulated flow speed in the open-field regions (marked with X) change significantly, especially in the northern hemisphere. The slower wind regions correspond to the unipolar boundary layers [Zhao and Webb, 2003] that locate above a pair of the small closed-field regions and divide one large open-field region into two, originating from different coronal holes with the same magnetic field polarity. In contrast, the shape of the slowest flow regions that correspond to the coronal streamer (or the heliospheric current sheet in the interplanetary space) did not largely change. The obtained difference in the sensitivity of the flow speed to the changes of the surface magnetic field can be understood by considering that the plasma flow speed in the coronal hole must be strongly affected by the shape of the small-scale magnetic structures at the lower corona. The small-scale coronal magnetic field structures at the lower corona are sensitive to the small-scale short-term surface variations, while the large-scale closed-field structures are determined by the large-scale surface magnetic field, and are thus less sensitive to the small-scale shortterm solar surface variations. [31] Figure 7 shows the simulated solar wind speed at 2.5 and 5.0 solar radii to demonstrate the radial evolution of the coronal plasma flow. The similar shapes of the slowest-wind belt are seen at 2.5 and 5.0 solar radii with both DR and SF maps, which means that the large-scale coronal structures at the trans-alfvénic regime are mostly determined by the inner coronal structures and less sensitive to solar surface variations. In contrast, the topology of the slower wind 7of13

8 Figure 5. Comparison of the simulated corona with the EIT observations. The plots in the left (right) column show the simulated and observed corona viewed from the Earth on 27 (29) October 2003, 1 d before (after) the Halloween event. region in the open-field region at 5.0 solar radii obtained with the DR map (Plot (c)) is significantly different from that at 2.5 solar radii (Plot (a)). The distributions of the slower wind obtained with SF map (Plots (b) and (d)) resemble each other. The radial evolutions are mostly consequences of the balance of the kinematic, thermal, and magnetic pressure along the tangential directions [e.g., Usmanov et al., 2000]. The solar wind simulated with the SF map reached the balanced state at the lower corona, while the heliocentric distance of 2.5 solar radii was not enough for the solar wind simulated with the DR map. It is obvious that the differences are caused by the different coronal structures that result from the differing solar surface magnetic field distribution. [32] Figure 8 depicts the coronal magnetic field structures simulated with DR and SF maps. Plots (a) and (b) show the closed-field lines obtained with DR and SF maps, respectively, and plots (c) and (d) enlarge a unipolar boundary layer part. The field lines are drawn as the tubes, size changes so that the magnetic flux tube (FTE) expansion can be expressed. In plots (c) and (d), we added the openfield lines that pass the points {r =2.5R s, N9.856 q N67.5, f = E23.9 } at which the flux tube expansion factor, the number density, the radial component of the magnetic field, and plasma flow are sampled. Those variables are plotted in Figure 9. In Figure 8, it is clearly seen that the portion of the streamer belt in the southern hemisphere (marked by dashed circles) shrank, while the smallscale closed-field region below the unipolar boundary layer in the northern hemisphere (marked by dot-dashed circles) expanded. The structures near the part of the limb distant from the replaced central part were affected less, but there are some minor changes because of the global changes of the magnetic flux balance. Such changes in the magnetic field structures are obtained by the simulations using DR and SF maps. [33] Focusing on the plasma quantities, we sampled the simulated variables at the unipolar boundary layer, because the MHD variables around the unipolar boundary layer are sensitive to the solar surface magnetic field distributions. Thus the layer will give a good example. In Plot (a) of Figure 9, we can see that the flux tube expansion factors obtained with SF map are generally larger than that with DR map. The larger expansions are due to the shrinkage of coronal holes at the middle latitude (marked with dashed rectangle in Plot (a) of Figure 5) and at the polar region. Because the plasma flowing toward the unipolar boundary layer undergoes rapid expansion, the density of the SF case is lower than the DR case overall (see Plot (b)). The rapid expansion also causes the relatively large (about 5 km/s in this simulation study) horizontal component of plasma flow, and the plasma flow originating from two different coronal holes merge at the unipolar boundary layer region to form a density enhancement (marked with vertical lines). In Plot (c), the radial component of plasma flow obtained with the SF map is slower than that with DR map, and there is a small enhancement at the same location of the density enhancement. Again, this is caused by the merging of the two plasma streams. In Plot (d), we also see the small drop of the magnetic field strength in the case with the SF map. We think the lack of enhancements in the DR case is due to the height of the closed-field region below the unipolar boundary layer; the two streams encounter before they are accelerated to gain enough speed. [34] We obtain the values and spatial variations by the nonlinear MHD simulations, and find the differences by comparing the simulation results using the DR and SF maps. We do not intend to apply the virtues of using DR and SF maps to the small-scale coronal structures such as active regions. The active regions and other small-scale coronal structures are out of our present scope. However, we think that the small structures of the global coronal features, for example, of about 2 or 3 in angular size, can be well simulated by using the DR and SF maps. [35] We examine the coronal magnetic energy variations in the open- and closed-field regions. Table 1 shows the coronal magnetic field energy and the volume of the openand closed-field regions simulated with the DR and SF maps. The summations are done for the two different longitudinal ranges; one is same as that of the replaced part of the SF map (from 45 E to45 W, 90 in total), and the other is for the latitude 45 wider to both sides (from 90 E to 90 W; all Earth-side hemisphere). The ranges of the radius and latitude are from 1.02 and 2.47 solar radii, and from 0 to 8of13

9 Figure 6. Radial component of simulated plasma flow at 2.5 solar radii. From the left the viewpoints are set at the Earth s position on 27, 28, and 29 October The plots in the upper (lower) row show the plasma flow speed simulated with the DR (SF) map. The darkest parts are the stagnant region corresponding to the coronal streamer belt region, or the magnetically neutral line which is the coronal origin of the heliospheric current sheet. 180 degrees, respectively. A notable point in Table 1 is that the closed field regions acquired more magnetic energy than did the open-field. In the region from 45 E to 45 W, more than 80 percent of the magnetic energy increase was distributed to the closed field regions, and in the volume from 90 E to 90 W, the magnetic field energy in the openfield even decreased. The volume of the closed-field just above the replaced part (from 45 E to 45 E) decreased, meaning that the closed magnetic field structures in this part became much more compact. The total decrease is due to the slight but global shrinkage of the bipolar closed-field structures at the streamer belt in the southern hemisphere, which was not compensated by the volume increase at the unipolar boundary layer demonstrated in the enlarged plots (c) and (d) of Figure 8. We think the increase of the coronal closed-field energy examined in this study may show an aspect of the complexity of the coronal dynamics. We do not intend to say that the intensified coronal closed-field means Figure 7. Radial component of simulated plasma flow at 2.5 (upper row) and 5.0 solar radii (lower row). The viewpoints are fixed and same as in Plots (b) and (e) of Figure 6. 9of13

10 Figure 8. Field lines of the simulated solar corona. The closed-field lines at r 2.5 R s simulated with the DR and SF maps are drawn in Plots (a) and (b), respectively. The viewpoints are identical to Plots (a) and (b) in Figure 7. The field lines are drawn as the tubes. For simplicity, the intersection of the tube is circle, and the diameter of the circles is set to be inversely proportional to the square root of the ratio of the magnetic field strength to that at the foot point on the coronal base. In Plots (a) and (b), the dashed circles and the dot-dashed circles show the shrunk closed-field region and the expanded closed-field region, respectively. Plots (c) and (d) enlarge the portion marked by dashed rectangles in Plots (a) and (b), respectively. In Plots (c) and (d), the open-field lines that pass the points {r =2.5R s, f = +23.9, N 9.85 q N 67.5 } are added. The gray of the open-field tubes shows the simulated density relative to the reference value of the Parker solution at each height, and the brighter gray represents higher relative density. Note that the three coronal holes visible in this Figure are of same (negative) polarity. that global corona is more likely to causes eruptive event because the coronal instability can be determined by the small-scale conditions and by the history of the entire corona as well. However, the estimation of the magnetic energy and the volume of the open- and closed-field regions shown in Table 1 would be useful for analysis of the complexity of the coronal eruptive events. The global coronal closed field can be the cradle of the coronal eruptive events holding the energy potentially involved in the future event, and can be the obstacle decelerating the coronal eruption at the early phase. 5. Discussion [36] In this study, we performed the MHD simulations to obtain the steady state of the global solar corona using boundary magnetic field maps constructed with a new format. The first one (DR) is the synoptic chart modified in accordance with the solar differential rotation, and the other one (SF) is the DR map including, in addition, the measurement just after the Halloween event. The former represents the solar magnetic field map before the Halloween event, and the latter shows the post-halloween map. The MHD simulations using these maps are separately performed, and the consequent MHD steady states of the solar corona are examined to see how the coronal structures vary in response to the changes of the surface magnetic field on the Halloween event, 28 October [37] The open-field regions in the simulated corona are calculated as the proxies of the global solar coronal structures and compared to the coronal holes measured by SOHO/EIT. Several similarities are obtained. The changes 10 of 13

11 Figure 9. Solar coronal quantities at a unipolar boundary layer of the simulated corona. The MHD data are sampled at points, {r =2.5R s, f = +23.9, N 10 q N 70 }, one end of the open-field lines drawn in Plots (c) and (d) of Figure 8. The vertical lines show the approximate position of the unipolar boundary layer where the coronal plasma originating from two coronal holes encounter; the plasma plotted in the left (right) part originates from the coronal hole at the northern polar region (low latitude region). of the coronal hole were numerically traced by using the two magnetic field maps. [38] The differences of the solar surface magnetic field can significantly affect the structures of the plasma flow speed in the open-field regions at upper corona. The difference of the plasma flow is caused by changes in the small-scale closed-field structures at the lower corona. In contrast, the slowest wind belt that corresponds to the coronal streamer and the heliospheric current sheet did not change so much as the open-field regions, meaning that the Table 1. Magnetic Energy and Volume in the Open- and Closed-Field Regions at the Central Longitude Region a E45 f W45 E90 f W90 P DVB 2 P /2 Magnetic Energy in P DVB 2 /2 Magnetic erg DV Volume in 3 Rs Energy in P erg DV Volume in 3 Rs Open Closed Open Closed Open Closed Open Closed DR SF diff a The summations are done for 1.02 R s r 2.47 R s and 0 q 180. The longitudinal ranges of the summed volume are written in Carrington coordinate. The longitudinal ranges of the summed volume in the table are centered at the 294 of the heliographic longitude that corresponds to the view point of plots of Figures 7 and of 13

12 large-scale closed coronal magnetic structures are determined mostly by the large-scale surface magnetic field. Therefore, the traditional synoptic chart is good for the determination of the global magnetic field polarity in the solar corona and interplanetary space. We have to be cautious, however, when predicting the plasma quantities from the solar surface magnetic field data. [39] As a whole, the MHD-simulated solar wind speed depends on the coronal magnetic field structures at the lower corona and thus the solar surface magnetic field: the fast wind comes from the near the center of the coronal hole and the slow wind comes the outermost part. This tendency is similar to the relationship between the flux tube expansion factor and flow speed [Wang and Sheeley, 1990; Arge et al., 2003]. In terms of quantity, though, the simulation used in this study could not produce the large contrast of the solar wind flow speed, typically from 300 km/s to 700 km/s at the Earth, because our model does not include realistic descriptions of the solar wind heating and acceleration processes, such as the Alfvén wave decay. The polytropic choice was made so that the simulated situations would be simpler and the coronal responses to the solar surface changes can be clearly seen. We will include the heating and acceleration processes in future studies. [40] We obtained the coronal slower wind regions in the unipolar open-field region. In the coronal measurement, the features of this kind must be faint, because they locate in the open-field region; thus the plasma density is not as high as that of the streamers. In addition, the slower wind regions tend to be ephemeral because of their sensitivity to the surface magnetic field variations. Therefore, we need more investigation to see the coronal features in the measurements. Because the simulated slower wind regions in the unipolar open-field regions at 2.5 and 5.0 solar radii tend to have larger horizontal width than the stagnant streamer regions, we think that the interplanetary high-density broad regions without the magnetic sector boundary [e.g., Crooker et al., 2004; Neugebauer et al., 2004] would be their interplanetary counterparts. The measurement and analysis of the interplanetary radio scintillation (IPS) could find the interplanetary signature of the slower wind regions because the ground-based IPS measurement has the ability to make constant observations of solar wind speed in the wide range of interplanetary space, including the high heliographic latitude regions [e.g., Jackson et al., 1998; Kojima et al., 1998; Ohmi et al., 2003; Hayashi et al., 2003]. As for the coronal observation, the tomography analysis [e.g., Frazin and Janzen, 2002] may be pointed out as an approach capable of detecting the long-lasting parts of the faint coronal structures. [41] The solar surface magnetic field measurements are today limited in only the Earth-Sun direction. Therefore, we have to collect the data made at different instances to make the synoptic chart and other versions of the solar surface maps covering all longitudes. In the standard synoptic chart, there is the 27- or 28-d difference in the measurement time between the left- and rightmost parts. If there is no solar surface evolutions or differential rotation, these data must be identical and thus the map will be seamless at the longitudinal overlap. The large differences between the data at two meridional bins measured at different times can cause inaccurate estimations of the solar coronal magnetic field. In the potential-field source-surface model, however, the error due to the data discontinuity rapidly decreases with respect to the distance. Therefore, that inaccuracy at the overlapping bin will not be significant in this work because our interests are limited to and focused on the coronal regions around the central longitude in the maps. However, we should be cautious when using the solar surface map to extrapolate the solar wind variables at the distant interplanetary space, especially when beyond 2 AU, because the estimation of the solar wind properties at such a distance can be affected by the data discontinuity. To avoid such problems, we may have to shift the longitude of the solar surface map (or the date of the measurement) so that the coronal origin of the solar wind at the position of interest will be at the central longitude of the longitudinally shifted map. In addition, the magnetic field near the solar polar region needs data corrections to adjust the limb darkening and other effects, including geometric uncertainty in calculating the radial component. These difficulties would be solved in future, for example, with the simultaneous measurements by the space probes from various directions, although the efforts to improve solar surface mapping, the MHD, and other models of the solar corona and solar wind must be continued. [42] Acknowledgments. The data set of the SOHO/EIT and MDI were used in this study. The SOHO is an international project between NASA and ESA. The simulations shown in this paper were performed at the Columbia supercomputer system at the NASA Ames Research Center. The authors were supported in part by the NASA/MDI project under grant NNG05GH14G, and the NSF/CISM project under grant NSF [43] Amitava Bhatarjee thanks the reviewers for their assistance in evaluating this paper. References Arge, C. N., D. Odstrcil, V. J. 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