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*Astron. Astrophys. 342, 300-310 (1999)*
## 2. Physical models and governing equations
Three different physical models of increasing complexity are
considered in this paper, namely, a single magnetic interface, a
single magnetic slab and two adjacent magnetic slabs. In the *single
interface model* , the background medium is separated by an
interface into two regions with different strengths of the magnetic
field; in a special case, the field can be zero in one of these
regions. To satisfy the pressure balance across the interface, the gas
pressure must be higher in the region where the magnetic field is
weaker. It is assumed that the interface is located along the y-axis
of a cartesian (x,y) coordinate system. A more detailed description of
this model is given in Sect. 4, where it is used to verify the
developed numerical code.
In the *single slab model* , two magnetic interfaces are
introduced in the background medium to form a magnetic slab. The
interfaces are located symmetrically with respect to the y-axis, so
that the slab extends along that axis. The slab thickness is a free
parameter in this approach and the external medium is assumed to be
field-free. More details are given in Sect. 5, where the model is used
to investigate the behavior of nonlinear MHD surface and body
waves.
Finally, in the *two adjacent magnetic slabs model* , four
interfaces are introduced in the background medium to form two slabs
that are located parallel to the y-axis. Both slabs have the same
thickness and the same strength of the magnetic field. The external
medium is again assumed to be field-free. More physical details of
this model are given in Sect. 6, where the excitation of magnetic slab
waves by external acoustic waves is discussed.
A mathematical description of the considered physical models is
given by the set of ideal and two-dimensional MHD equations. After
neglecting gravity, the set can be written in its conservative and
dimensionless form (see Huang 1995) as follows
In this set, all variables are dimensionless and defined as
follows: ,
, ,
,
and , where the superscript "d"
refers to a dimensional quantity, and
is the gas density,
is the velocity, *p* is the gas
pressure, is the magnetic field
strength, *t* is the time and
is the position vector. The quantities
,
and represent the reference density,
magnetic field strength and sound speed, respectively, and
is the time scale. In addition,
and
are the components of the velocity,
and
are the components of the magnetic
field strength, is the ratio of
specific heats, and is the ratio of
gas pressure to magnetic pressure: .
Initially there are no waves or fluid motions in the systems. The
velocity perturbations (see Sects. 5 and 6) are introduced in the
models at . Then, the unknowns
, ,
, ,
and *p* are computed as a
function of *x*, *y* and *t*. The numerical procedure
adopted to solve the governing MHD equations is described in the
following section.
© European Southern Observatory (ESO) 1999
Online publication: December 22, 1998
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