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Astron. Astrophys. 342, 300-310 (1999)

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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

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

In this set, all variables are dimensionless and defined as follows: [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA], where the superscript "d" refers to a dimensional quantity, and [FORMULA] is the gas density, [FORMULA] is the velocity, p is the gas pressure, [FORMULA] is the magnetic field strength, t is the time and [FORMULA] is the position vector. The quantities [FORMULA], [FORMULA] and [FORMULA] represent the reference density, magnetic field strength and sound speed, respectively, and [FORMULA] is the time scale. In addition, [FORMULA] and [FORMULA] are the components of the velocity, [FORMULA] and [FORMULA] are the components of the magnetic field strength, [FORMULA] is the ratio of specific heats, and [FORMULA] is the ratio of gas pressure to magnetic pressure: [FORMULA]. 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 [FORMULA]. Then, the unknowns [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] 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.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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