Solar MHD: Theory and Observations – a High Spatial Resolution Perspectiv
ASP Conference Series, Vol. .., 2006
J. Leibacher, H. Uitenbroek, & R. F. Stein
Holistic MHD-Simulation from the Convection Zone to
the Chromosphere
Werner Schaffenberger, Sven Wedemeyer-Böhm, Oskar Steiner
Kiepenheuer-Institut für Sonnenphysik, Schöneckstrasse 6, D-79105
Freiburg, Germany
Bernd Freytag
Dep. for Astron. and Space Phys., Uppsala University, Box 515, 75120
Uppsala, Sweden; Los Alamos Nat. Lab., Los Alamos, NM 87545, USA;
Dep. of Phys. and Astron., MSU, East Lansing, MI 48824-2320, USA
Abstract.
A three-dimensional magnetohydrodynamic simulation of the integral layers from the convection zone to the chromosphere has been carried
out. The simulation represents magnetoconvection in a quiet network-cell interior. The following preliminary new results are obtained: The chromospheric
magnetic field is very dynamic with a continuous rearrangement of magnetic flux
on a time scale of less than 1 min. Rapidly moving magnetic filaments (rarely
exceeding 40 G) form in the compression zone downstream and along propagating shock fronts that are present throughout the chromosphere. The magnetic
filaments rapidly move, form, and dissolve with the shock waves. Flux concentrations strongly expand through the photosphere into a more homogeneous,
space filling chromospheric field. “Canopy fields” form on a granular scale above
largely field-free granule centers leading to a mesh-work of current sheets in a
height range between approximately 400 to 900 km.
1.
The Numerical Method
The simulations were made with the CO 5BOLD-code1 , which was recently extended to include magnetic fields. The code solves the coupled system of the
equations of compressible magnetohydrodynamics in an external gravity field
and non-local, radiative transfer. The code features a realistic equation of state,
including partial ionization. Radiative transfer, hydrodynamics, and additional
turbulent diffusion are treated separately using operator splitting. Here, we
briefly describe the MHD extension of CO 5BOLD.
The magnetohydrodynamic equations are solved on a static, non-equidistant
cartesian grid. The multidimensional problem is reduced to 1-D problems by
dimensional splitting. Each of these 1-D problems is solved with a Godunovtype finite-volume scheme using an approximate Riemann solver modified for a
general equation of state and gravity. The extension to second order in space
and time is done using linear reconstruction for the primitive variables ρ, v, B,
p, and ρ followed by a Hancock predictor step (Toro 1999).
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www.astro.uu.se/˜bf/co5bold main.html
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1.1.
Schaffenberger et al.
The Riemann solver
The original hydrodynamics step of CO 5BOLD uses a Riemann solver of Roe
type. For MHD, we decided to use a HLL-solver (Harten, Lax & van Leer 1983)
instead, because the Roe solver does not guarantee the positivity of the density
and the pressure. This problem, which may occur also for hydrodynamics, gets
worse for MHD: If the thermal energy is much smaller than the total energy,
small errors in the total energy lead to large errors of the thermal energy, which
may turn the pressure to negative values, especially in regions of strong magnetic
field and low plasma beta. Values of β ≈ 10 −4 are common in the chromospheric
regions of the simulation.
In case of hydrodynamics, the HLL approximative Riemann solver ensures
positivity of density and pressure provided that the exact solution of the Riemann problem has positive density and pressure (Einfeldt et al. 1991). For
MHD, this is true only if the normal component of the magnetic field has no
jump, which is the case for 1-D problems. For multidimensional problems, jumps
in the normal component of the magnetic field are unavoidable when using cell
centered magnetic fields, even if the divergence-free condition is fulfilled in a
discrete sense. It was shown by Janhunen (2000) that allowing for magnetic
monopoles and taking into account their contribution to the Lorentz-force, an
additional source term occurs only in the induction equation. Using a special
discretization of this source term, the HLL-solver for the MHD equations is
positive (Janhunen 2000).
1.2.
The constrained-transport step
A special requirement of MHD-calculations is the enforcement of the divergencefree condition for the magnetic field. Violating this condition can lead to unphysical forces which can degrade the solution (Brackbill & Barnes 1980).
After performing the 1-D substeps, the updated magnetic field is not divergence-free, even if it was before the update. Therefore, a special step for updating the magnetic field is required. In our scheme, we use the flux-interpolated
constrained transport of Balsara & Spicer (1999). This method uses a staggered grid, where the magnetic fields are stored at the cell boundaries whereas
the remaining hydrodynamic variables are cell centered. For the 1-D updates,
the magnetic field at the cell centers is computed from the values of the cell
boundaries by arithmetic averaging. After the 1-D substeps are performed, the
updated values of the cell-centered magnetic field are discarded. Instead, the
boundary located magnetic field values are updated using the fluxes from the 1D substeps in such a way, that a discrete version of the divergence-free condition
is fulfilled.
2.
First Simulations
A first three-dimensional magnetohydrodynamic simulation of the integral layers
from the upper convection zone to the middle chromosphere was carried out. The
computational domain extends over a height range of 2800 km of which 1400 km
reach below the average height of optical depth unity and 1400 km above it. The
horizontal dimensions are 4800 km × 4800 km. With 120 3 grid cells, the spatial
Holistic MHD-Simulation
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resolution in the horizontal direction is 40 km, while in the vertical direction
it is 20 km throughout the photosphere and chromosphere increasing to 50 km
through the convection-zone layer. The lateral boundary conditions are periodic,
whereas the lower boundary is “open” in the sense that the fluid can freely flow
in and out of the computational domain under the condition of vanishing total
mass flux. The specific entropy of the inflowing mass is fixed to a prescribed
value so as to yield solar radiative flux at the upper boundary. The upper
boundary is “closed” so far, i.e., reflecting boundaries are applied to the vertical
velocity, while stress-free conditions are in effect for the horizontal velocities,
viz., dvx,y /dz = 0. The vertical derivative of the specific internal energy is zero.
The MHD simulation starts with a homogeneous, vertical, unipolar magnetic field of a flux density of 10 G superposed on a previously computed, relaxed
model of thermal convection. This flux density ought to mimic magnetoconvection in a network-cell interior. The magnetic field is constrained to have
vanishing horizontal components at the top and bottom boundary but lines of
force can freely move in the horizontal direction, allowing for flux concentrations
to extend right to the boundaries. Although this condition is quite stringent,
especially at the top boundary, it still allows the magnetic field to freely expand
with height through the photospheric layers into the more or less homogeneous
chromospheric field, different from conventional simulations that extend to a
height of typically 600 km only.
2.1.
Dynamic chromospheric magnetic field
Subsequent to superposition of the magnetic field, flux expulsion from the granule centers takes place. Within less than 5 minutes, the magnetic field concentrates in narrow sheets and small knots near the surface of optical depth unity
with flux densities up to approximately 1 kG. Occasionally, these magnetic flux
concentrations extend down to the bottom boundary at a depth of 1400 km
but more often, they disperse again at a depth of less than 1000 km. Figure 1
(bottom) shows the logarithm of the absolute magnetic flux density in three
horizontal sections through the computational domain at a given time instant.
The left panel refers to a depth of 1210 km close to the bottom boundary, the
middle panel to a height of 60 km above the average height of optical depth unity,
where one can see that horizontal flows have expelled the field from the granule
centers. The right panel refers to a height of 1300 km, which is located in the
chromospheric layers, close to the upper boundary. There, the magnetic field is
very dynamic with a continuous rearrangement of magnetic flux on a time scale of
less than 1 min, much shorter than in the photospheric or convection-zone layers.
The field has a strength between 2 and 19 G in this snapshot. Different from
the surface magnetic field, it is more homogeneous and virtually fills the entire
space so that the magnetic filling factor in the top layers is close to unity. There
seems to be no direct correlation between chromospheric flux concentrations and
the small-scale field concentrations in the photosphere. The former rather have
a slight tendency to be located in between the flux concentrations at the surface.
The rightmost panel of Fig. 1 (bottom) shows the emergent intensity. Several granules are visible. Comparing it with the panel near the τ c = 1 level, one
readily sees that the magnetic field is concentrated in intergranular lanes and
at lane vertices. The field concentrations do not show a corresponding inten-
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Schaffenberger et al.
Figure 1. Top: Snapshot of a vertical section through the three-dimensional
computational domain showing the logarithm of the absolute field strength
and velocity vectors projected on the vertical plane (arrows). The dashed
curve shows optical depth unity for vertical lines of sight, the black contour
correspond to β = 1. Bottom: Horizontal sections showing log |B|. Left:
Bottom layer at a depth of 1180 km. Middle: Layer 90 km above optical
depth τc = 1. Right: Top, chromospheric layer in a height of 1310 km.
Rightmost: Emergent visible continuum intensity.
sity signal because the magnetic flux is too weak to form a significant “Wilson
depression” (as can be seen from the top panel of Fig. 1) so that no radiative
channeling can take place.
Figure 1 (top) shows log |B| trough a vertical section of the computational
domain indicated by the dashed line in the bottom panels (see also Schaffenberger et al. (2006) for other cross sections). Overplotted are arrows indicating the velocity field. The dashed curve corresponds to optical depth unity
and the black curve to equipartition between thermal and magnetic pressure,
β = 1. Highly dynamic transient filaments of magnetic field are a ubiquitous
phenomenon in the chromosphere and are also present in this snapshot, e.g.,
near x = 1300 km. They typically form in the compression zone downstream
and along propagating shock fronts to where the magnetic field gets swept by
the supersonic plasma flow. These magnetic filaments that have a field strength
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Figure 2. Logarithmic current density, log |j|, in a vertical cross section
(top panel) and in four horizontal cross sections in a depth of 1180 km below,
and at heights of 90 km, 610 km, and 1310 km above the average height of
optical depth unity from left to right, respectively. The arrows in the top
panel indicate the magnetic field strength and direction. The dashed line
indicates the position of the vertical section.
rarely exceeding 40 G, rapidly move, form, and dissolve with the shock fronts.
The surface of β = 1 separates the region of highly dynamic magnetic fields from
the more slowly evolving field of high beta plasma below it.
2.2.
Chromospheric current sheets
A very common phenomenon in the present simulation run (also visible in Fig. 1,
top) is the formation of a small-scale magnetic “canopy” field that extends in a
more or less horizontal direction over expanding granules and in between photospheric flux concentrations. The formation of such canopy fields proceeds by the
action of the expanding flow above granule centers. This flow transports “shells”
of horizontal magnetic field to the upper photosphere and lower chromosphere,
where layers of different field directions may come close together, leading to a
complicated meshwork of current sheets in a height range from approximately
400 to 900 km.
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Schaffenberger et al.
This meshwork can be seen in Fig. 2 (top), which shows the logarithmic current density, log |j|, together with arrows indicating the magnetic field strength
and direction in the same vertical section as of Fig. 1. The bottom panel of
Fig. 2 shows from left to right log |j| in four horizontal cross sections in a depth
of 1180 km below, and at heights of 90 km, 610 km, and 1310 km above the
mean height of the surface of optical depth unity. Higher up in the chromosphere
(rightmost panel), thin current sheets form along shock fronts like in the lower
left corner near x = 1.3 Mm.
3.
Conclusions
The hydrodynamic code CO5BOLD has been extended to magnetohydrodynamics using a second order accurate HLL Riemann solver for computing the
numerical fluxes and a constrained transport for divergence-free transport of the
magnetic field.
A first three-dimensional magnetohydrodynamic simulation of the integral
layers from the convection zone to the chromosphere has been carried out. The
simulation represents magnetoconvection in a quiet network-cell interior. It reveals a chromospheric magnetic field that is very dynamic. There, the magnetic
field is continuously rearranged on a time scale of less than 1 min. Rapidly
moving, transient magnetic filaments form in the compression zone downstream
and along propagating shock fronts. They move, form, and dissolve with the
shock fronts and carry thin current sheets.
The surface of β = 1 separates the region of highly dynamic magnetic fields
from the more slowly evolving field of high beta plasma below it. This rather
corrugated surface is at a height of around 1000 km and strongly varies in time.
The magnetic field gets almost completely expelled from the photospheric
layers of granule interiors, resulting in a horizontally directed but continuously
changing small-scale “canopy” field that overlays these magnetic “voids”. This
“canopy” field consists of diversely directed magnetic layers, leading to a complicated meshwork of current sheets in a height range from approximately 400
to 900 km above optical depth unity.
Acknowledgments. WS was supported by the Austrian Fonds zur Förderung der Wissenschaften through the Erwin Schrödinger Stipendium, project
J2211. SW acknowledges support by the Deutsche Forschungsgemeinschaft
(DFG), project Ste 615/5.
References
Balsara, D. S., & Spicer, D. S. 1999, J. Comput. Phys., 149, 270
Brackbill, J. U., & Barnes, D. C. 1980, J. Comput. Phys., 35, 426
Einfeldt, B., Munz C. D., Roe, P. L., & Sjögreen, B. 1991, J. Comput. Phys., 92, 273
Harten, A., Lax P. D., & van Leer, B. 1983, SIAM Review, 25, 35
Janhunen P. 2000, J. Comput. Phys., 160, 649
Toro, E. F. 1999, Riemann Solvers and Numerical Methods for Fluid Dynamics (BerlinHeidelberg: Springer-Verlag)
Schaffenberger, W., Wedemeyer-Böhm, S., Steiner, O., & Freytag, B., 2006, in ESA
Publ. SP-596, Chromospheric and Coronal Magnetic Fields, ed. B. Fleck &
D. Danesy