The solar corona is sub-Alfvénic, and the time-relaxation simulation with time-dependent MHD code is probably a unique method to solve the nonlinear MHD equations and obtain the general three-dimensional solution of the solar corona. To specify the time of interest and to obtain realistic solution, the measurement-based data is used as the boundary value. After giving three-dimensional initial value matching the boundary data, we can start the time-relaxation simulation to obtain the sub/trans-Alfvénic solar corona. It must be mentioned that there remain lots of assumptions in model descriptions. These assumptions would be in future replaced with measurement-based and theoretical descriptions.
We have developed the three-dimensional magnetohydrodynamic (MHD) simulation code based on the concept of MUSCL (of 3rd-order, van Leer-type) and TVD (with linearized Riemann solver) strategy for these simulation model [c.f. Hayashi, 2005 ; Hayashi et al., 2006]. As the quick version, the Lax-Friedrichs method is used for provisional studies.
The boundary treatment on the inner boundary surface is challenge : the boundary is sub-Alfvénic, thus the numerical inconsistency will arise unless the adequate treatment is imposed there. The projected normal characteristic method [Nakagawa, 1980; Wu et al., 1983 ; Han et al., 1988] provides the boundary treatment satisfying both the basic MHD equations and the imposed boundary constraints. We developed modified versions of this method for our purposes [Hayashi, 2005].
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The closed field lines in the solar corona simulated with the measurement-based magnetic field data:
The colors on the sphere represent the polarity of the measurement-based magnetic field data imposed as the boundary value; the blue and red represent the positive (outward) and negative (toward the Sun) magnetic field polarity, respectively. Field lines at viewer's side are not drawn so that field lines inside closed field regions can be well seen. An mpeg movie (2.9MB) is now available. |
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The field lines in the solar corona and the density at the base of corona:
The contrasts in density and other plasma parameters can be retrieved by using the boundary treatment based on the projected normal characteristic method [c.f. Hayashi, ApJS 2005]. |
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This VRML (Virtual Reality Modeling Language; gzipped 33kB) shows the field lines, and another one (68kB) does both field lines and magnetically neutral surfaces. |
The MHD codes can be applied for the regions distant from the Sun. Time-dependent codes are generally used, and the radial-increment (non-time dependent) version of MHD code [c.f. Hayashi,K. et al., 2003 JGR] can be used when the interest is on the steady state of the super-Alfvénic solar wind in the inner heliosphere at r < 5 AU or so. The radial-increment method is inevitably more numerically diffusive (or erroneous) than the time-dependent version. But, it runs much faster and thus needs less computational resources, and yields reasonably accurate solutions. These features help do the model calculations needing iteratively calculation of the solar-wind MHD variables. I have developed both radius-increment and time-dependent versions.
To start the simulations, we always have to set the boundary values. One good set of the boundary solar wind variables is the MHD solution obtained by the sub/trans-Alfvénic MHD simulation.
The MHD version and other versions of computer assisted tomography (CAT) analysis for the interplanetary scintillation (IPS) measurement [c.f. Jackson,V.B. et al., 1998 JGR ; Kojima,M. et al., 1998 JGR ; Hayashi,K. et al., 2003 JGR] can provide the "measurement-based" boundary variables of solar wind in the inner heliosphere.
In the outer part of solar wind, the effects of the solar rotation becomes dominant, and the fast and slow solar wind starting from different solar regions interact each other. The MHD simulations can solve this nonlinear interaction process and theoretically extend the solar wind solution outward up to several or tens AUs. It must be noted that the pickup ion (interstellar-origin materials ionized by solar (E)UV, mostly proton) and electron should be taken into account for better simulations of interplanetary space beyond 4 or 5 AUs [e.g. Usmanov and Goldstein, 2006].
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Cross-section view of a three-dimensional simulated solar wind structure
on the solar equatorial plane (up to 2000 solar radii) :
This plot shows the formation of steep gradients or discontinuities at the co-rotating interaction regions (CIR) between faster wind (drawn with green) and slower one (yellow and orange) in the simulated spiral structure of the solar wind. |
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Magnetically neutral region where magnetic field changes direction (toward the Sun or outward),
or heliospheric current sheet (HCS), in the co-rotating spiral structure of the solar wind
from 30 solar radii to 2030 solar radii (∼9.4AU):
This (rose-like) spiral surface is the contour surface of Br = 0 (the blue surface is meant the other side of the sheet) that well corresponds to the HCS. |