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Astron. Astrophys. 327, 377-387 (1997)

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2. Equilibrium and governing equations

Consider a current sheet in equilibrium modelled using a Harris profile (Harris 1962). The one-dimensional magnetic field is given by

[EQUATION]

representing a continuous change of the magnitude of the field from [FORMULA] at large positive [FORMULA] to [FORMULA] at large negative [FORMULA] and passing through zero at [FORMULA] =0. Equilibrium demands that the total pressure (plasma plus magnetic) is uniform:

[EQUATION]

yielding a plasma pressure [FORMULA] given by

[EQUATION]

given that [FORMULA] (cold plasma) as [FORMULA]. The plasma density is arbitrary; we assume a density profile [FORMULA] of the form (cf. Epstein 1930)

[EQUATION]

declining from [FORMULA] at [FORMULA] to [FORMULA] as [FORMULA]. Through the ideal gas law [FORMULA], the temperature [FORMULA] is non-isothermal. The implied sound speed [FORMULA] varies from [FORMULA] at the centre of the current sheet to zero in the far environment of the sheet, consistent with our assumption of a cold plasma as [FORMULA]. Specifically, the square of the sound speed [FORMULA] is given by

[EQUATION]

The square of the Alfvén speed [FORMULA] is given by

[EQUATION]

where [FORMULA]. Thus the Alfvén speed [FORMULA] declines from [FORMULA] in the far environment of the current sheet [FORMULA] to zero at the centre of the sheet. Fig. 1 gives plots of these two speeds, together with the tube speed [FORMULA] [FORMULA] and the fast speed [FORMULA], using [FORMULA]. The equilibrium condition (6) may be written in the alternative form

[FIGURE] Fig. 1. A plot of the Alfvén [FORMULA], sound [FORMULA], fast [FORMULA] and tube [FORMULA] speeds (normalised against [FORMULA]) in a neutral current sheet with uniform density and [FORMULA] =5/3. The region of low Alfvén speed about [FORMULA] acts as a duct for magnetoacoustic waves. Note that the fast speed is approximately constant throughout the whole domain. The sound and tube speeds have maxima of 0.9129 [FORMULA] and 0.4771 [FORMULA], respectively. These speeds are important in determining the nature of the magnetoacoustic waves in a current sheet.

[EQUATION]

The Alfvén and sound speed profiles alongwith the density profile satisfies the equilibrium condition (11).

Consider the linearised equations of ideal magnetohydrodynamics, assuming gravity is negligible. The wave equation for plasma motions,

[EQUATION]

in a non-uniform magnetic field [FORMULA] where the equilibrium parameters (density and pressure) are dependent upon x is given by (Roberts 1981a)

[EQUATION]

where

[EQUATION]

Here [FORMULA] is the velocity component normal to the magnetic field, [FORMULA] is the frequency and [FORMULA] is the longitudinal wavenumber along the sheet. The velocity parallel to the magnetic field is given by

[EQUATION]

We consider only motions that are independent of the y -coordinate so that propagation is in the xz -plane.

The perturbed gas [FORMULA], magnetic [FORMULA] and total [FORMULA] (gas and magnetic) pressure perturbations are given by

[EQUATION]

whilst the magnetic tension force [FORMULA] takes the form

[EQUATION]

where [FORMULA] is the perturbed magnetic field and

[EQUATION]

To investigate the nature of the modes we reduce Eq. (13) to the canonical form. Setting [FORMULA] reduces the governing wave equation to

[EQUATION]

with [FORMULA] given by Eq. (3) and [FORMULA] = [FORMULA], with f given by Eq. (14). Here the dash denotes a derivative with respect to x. For a uniform medium, [FORMULA] is constant [FORMULA] ; body and surface modes are determined by positive and negative [FORMULA], respectively. However, for a medium which is continuously structured, we define body and surface modes when [FORMULA] attains positive or negative values, respectively, across the whole sheet [FORMULA]. In addition, hybrid modes occur when [FORMULA] possesses both positive and negative [FORMULA] across the width of the sheet.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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