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Astron. Astrophys. 357, 1073-1085 (2000)

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2. Basic equations

The wave electromagnetic fields obey Maxwell's equations

[EQUATION]

[EQUATION]

[EQUATION]

where the current density, [FORMULA][FORMULA], and the charge density, [FORMULA]s[FORMULA], have to be calculated using a suitable mathematical model of the plasma. The most popular models are based on the ideal MHD equations, the two-fluid MHD equations, and kinetic Vlasov equations.

As discussed in the Introduction, we use the mathematical model of two-fluid MHD in order to take into account some important linear and nonlinear effects in the low-frequency short-scale AWs. In two-fluid MHD the electron and ion fluids are allowed to move in separate ways, but are coupled by collective electromagnetic fields and by the electron-ion friction force. The equations of motion for the electrons and ions are:

[EQUATION]

[EQUATION]

where [FORMULA] is the ion/electron pressure, [FORMULA] (but we do not require [FORMULA]). The parallel friction force, [FORMULA], responsible for the (parallel) resistivity along [FORMULA], is

[EQUATION]

where [FORMULA] [FORMULA] is the electron collision frequency.

Note that we have dropped the perpendicular friction force, responsible for the perpendicular resistivity, because its contribution in the wave dissipation is [FORMULA] times smaller than contribution of the parallel resistivity ([FORMULA] and [FORMULA] are the parallel and perpendicular wavelengths).

The particles number densities obey the continuity equations

[EQUATION]

where the subscript s denotes particles species, [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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