Solving the system of radiation magnetohydrodynamics for solar physical simulations in 3d [Elektronische Ressource] / Andreas Dedner

"Andreasdedner"

Sujets

• Astronomie

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• Publié par: albert-ludwigs-universitat_freiburg
• Publié le: 01 janvier 2003
• Nombre de lectures: 3
• Langue: English
• Poids de l'ouvrage: 32 Mo

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Albert-Ludwigs-Universit¨at Freiburg i. Br.
Fakult¨at fur¨ Mathematik und Physik
Solving the System of Radiation
Magnetohydrodynamics
for solar physical simulations in 3d
Andreas Dedner
Dissertation zur Erlangung des Doktorgrades der Fakult¨at fur¨ Mathematik und
Physik der Albert-Ludwigs-Universitat¨ Freiburg im Breisgau
Betreuer: Prof. Dr. Dietmar Kroner¨
Abteilung fur¨ Angewandte Mathematik
Freiburg im Breisgau, April 2003Dekan : Prof. Dr. Rolf Schneider
Referenten : Prof. Dr. Dietmar Kr¨oner
: Prof. Dr. Gerald Warnecke, Universit¨at Magdeburg
Datum der Promotion : 22. September 2003
Picture on title page:
simulation of a circular slip stream. In the purely hydrodynamic setting the interface is unstable with
respect to perturbations (Kelvin–Helmholtz instability). By means of a sufficiently strong magnetic
field that is tangential to the interface, this instability can be suppressed. For the simulation shown
here, the magnetic field is not yet strong enough so that the development of the Kelvin–Helmholtz
instabilities can be clearly seen. The large picture shows the density of the fluid. The two smaller
pictures show the locally adapted grid (top) and the third component of the magnetic field (bottom)
in a small section of the domain in the vicinity of the interface.Abstract
In this study we present a finite–volume scheme for solving the equa-
tions of radiation magnetohydrodynamics in two and three space di-
mensions. Among other applications this system is used to model the
plasmainthesolarconvectionzoneandinthesolarphotosphere. Itis
a non–linear system of balance laws derived from the Euler equations
of gas dynamics and the Maxwell equations; the energy transport
through radiation is also included in the model. The starting point of
our presentation is a standard explicit first and second order finite–
volume scheme on both structured and unstructured grids. We first
study the convergence of a finite–volume scheme applied to a scalar
model problem for the full system of radiation magnetohydrodynam-
ics. We then present modifications of the base scheme. These make
itpossibletoapproximatethesystemofmagnetohydrodynamicswith
anarbitraryequationofstate; theyreduceerrorsduetoaviolationof
the divergence constraint on the magnetic field, and they lead to an
improved accuracy in the approximation of solution near an equilib-
rium state. These modifications significantly increase the robustness
oftheschemeandareessentialforanaccuratesimulationofprocesses
in the solar atmosphere. For simulations in the solar photosphere, we
have to take the radiation intensity into account. A scheme for solv-
ing the radiation transport equation is a further focus of this study.
We present both analytical results and numerical tests, comparing
our scheme with some standard schemes found in the literature. We
conclude our presentation with a study of the parallelization strategy
for distributed memory computers that we use in our 3d code.iiIntroduction
Numerical simulations have become an important tool for studying many different
physical and technical problems. Ranging from the formation of galaxies to weather
forecasts to the design for parts of complex machinery, the applications are numerous.
On the one hand, numerical simulations serve as a tool for the verification of physical
theories deduced from observation; on the other hand, they play an important role
in reducing development cost in manufacturing. Although the range of applications
is extremely broad, the methods used for solving problems numerically have many
features in common. This is due to the fact that the physical models used have similar
properties. For example, fluid flow in the atmosphere of stars or in car engines can be
modeled by very similar systems of equations and can be simulated using very similar
numerical methods.
In this study we investigate numerical schemes that can be used to simulate the evo-
lution of a compressible fluid. The governing system of partial differential equations
is based on the Euler equations of gas dynamics. Over the last centuries, this system
has been the focus of both analytical and numerical studies. A large number of dif-
ferent schemes have been developed and tested using this system. One very successful
approach turned out to be the finite–volume framework, and many different schemes
basedonthisapproachhavebeenpresented. Thesamemethodshavealsobeenapplied
to different extensions of the basic system of Euler equations, including, for example,
reactive flow and magnetohydrodynamics. The latter will be the main focus of our
study.
Solar physical applications
The material presented in the following is part of a project financed by the Deutsche
Forschungsgemeinschaft(DFG)aimedatderivingandanalyticallyjustifyingnumerical
methods for studying fluid flow in the solar atmosphere. The development of many of
the methods is a direct consequence of the interaction between members of our group
here in Freiburg (Dietmar Kroner,¨ Christian Rohde, Matthias Wesenberg, and myself)
and solar physicists (Manfred Schuss¨ ler and Peter Vollm¨oller from the Max–Planck
Institute for Aeronomie in Kattlenburg–Lindau), whose ideas greatly influenced our
work. Many of the problems discussed here occur only if the methods are applied
not to academic test cases, but to realistic settings. Therefore, the discussions with
the solar physicists and their help in developing and testing the numerical methods
influenced the direction in which our work progressed.
Although a variation in solar activity has a strong impact on life here on earth, a
thorough understanding of the physical processes behind these phenomena is still the
subject of research all over the world:
iiiiv INTRODUCTION
http://image.gsfc.nasa.gov/poetry/storm0/black1.html http://sec.noaa.gov/SWN/index.html
Storms are usually responsible for the losses of Early records of sunspots indicate that the Sun
electricity we endure, but did you know that went through a period of inactivity in the late
”storms” as far away as the sun are capable 17thcentury. Veryfewsunspotswereseenonthe
of knocking out large areas of electric service? Sun from about 1645 to 1715. [...] This period
Amazingly, the sun is capable of not only dis- of solar inactivity also corresponds to a climatic
rupting electrical power, but also short wave period called the ”Little Ice Age” when rivers
radio, television and telegraph signals, naviga- that are normally ice-free froze and snow fields
tional equipment (GPS and LORAN), defense remained year-round at lower altitudes. There
(military) early warning radar systems, the cli- is evidence that the Sun has had similar peri-
mate,andcanevenknockoutourcommunication ods of inactivity in the more distant past. The
satellites in space. connection between solar activity and terrestrial
http://image.gsfc.nasa.gov/poetry/storm0/black1.html climate is an area of on-going research.
http://sec.noaa.gov/SWN/index.html
Since the possibilities for direct observation of physical processes below the solar sur-
face are limited, numerical simulations play an important role in obtaining a clearer
understanding of solar phenomena. A further example of a solar phenomena not yet
fully understood is the eleven year cycle in which the number of sun spots on the solar
surfaceincreaseanddecrease. Onedifficultyisthatthefilamentsattheboundaryofthe
5sun spots are made up of magnetic fluxtubes that are formed about 2·10 kilometers
below the solar surface in the lower convection zone of the sun. In this region direct
observation is hardly possible so that the formation and evolution of the fluxtubes has
to be studied by means of numerical simulations. Although the presentation here is far
more general and such solar phenomena are not the immediate focus, the application
of our method to problems in solar physics has been a constant motivation.
Mathematical model
Themathematicalmodelconsistsofasystemofbalancelawscombiningtheequationsof
magnetohydrodynamics (MHD) and the radiation transport (RT) equation. The MHD
equations are a non–linear system of eight conservation laws; the energy transport
through radiation leads to an additional source term that is non–local in space.
The MHD equations describe the evolution of an electrically conductive plasma in the
presence of magnetic fields and combine the Euler equations of gas dynamics and theINTRODUCTION v
Maxwellequations. Thelatteralsointroduceaconstraintequationonthedivergenceof
the magnetic field. In the solar atmosphere the force of gravity plays an important role
and is included in our model via source terms. To perform the simulations, we have to
prescribe suitable initial conditions for the fluid. These often consist of a perturbation
of a stratified and static background atmosphere. One intrinsic problem of simulations
with this type of initial data is the size of the computational domain. The setting
allows for no physical boundaries, and the construction of suitable artificial boundary
conditions that can be used in numerical simulations is no easy undertaking.
The main difficulty in approximating the radiation field is the high dimensionality
of the problem and the propagation speed of the radiation, which is several orders
of magnitude above the speed of the fluid. In our model we deal with the second