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Astron. Astrophys. 337, 887-896 (1998)

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3. Basic equations

We start from the equations of three-fluid plasma magnetohydrodynamics for electrons, ions, and neutral atoms. Let us introduce the bulk plasma velocity [FORMULA] in the laboratory reference frame

[EQUATION]

where k = e, i, a, and v k are the velocities of the species in the laboratory reference system. Denote the velocities of the species in the reference frame connected with bulk plasma (diffusion velocities) as [FORMULA]. When the diffusion velocities, as well as their derivatives, are small compared with the velocity and acceleration of the bulk plasma (this allows us to consider [FORMULA]dt), and the dependence of the friction force due to collisions vs. velocity is linear, the equation of motion for k-th component can be written as

[EQUATION]

Here [FORMULA] = [FORMULA] + [FORMULA] is the electric field in the reference frame moving with the bulk plasma, [FORMULA] = [FORMULA], [FORMULA] =[FORMULA] for k = i, e, a, and [FORMULA] is the momentum losses of the k-type particles, due to collisions with particles of l-types. After summing all three equations (1) for each k = e, i, a, and taking the relations

[EQUATION]

into account one can get the equation of motion for the bulk plasma

[EQUATION]

where [FORMULA] = [FORMULA] + [FORMULA] + [FORMULA] is the density of partially ionized plasma and [FORMULA] = [FORMULA] + [FORMULA] + [FORMULA].

Excluding from Eq. (1) velocities [FORMULA], after neglecting the terms as small as [FORMULA] in comparison with the units, one can obtain the generalized Ohm's law

[EQUATION]

Here [FORMULA] = [FORMULA]/([FORMULA]([FORMULA]+[FORMULA])) is conductivity, and F = [FORMULA]/[FORMULA] is the relative density of neutrals. Eqs. (2) and (3) together with the Maxwell equations and the mass conservation law

[EQUATION]

describe self-consistently the behavior of the plasma and the electromagnetic fields.

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© European Southern Observatory (ESO) 1998

Online publication: August 27, 1998
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