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
representing a continuous change of the magnitude of the field from at large positive to at large negative and passing through zero at =0. Equilibrium demands that the total pressure (plasma plus magnetic) is uniform:
yielding a plasma pressure given by
given that (cold plasma) as . The plasma density is arbitrary; we assume a density profile of the form (cf. Epstein 1930)
declining from at to as . Through the ideal gas law , the temperature is non-isothermal. The implied sound speed varies from 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 . Specifically, the square of the sound speed is given by
The square of the Alfvén speed is given by
where . Thus the Alfvén speed declines from in the far environment of the current sheet to zero at the centre of the sheet. Fig. 1 gives plots of these two speeds, together with the tube speed and the fast speed , using . The equilibrium condition (6) may be written in the alternative form
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,
in a non-uniform magnetic field where the equilibrium parameters (density and pressure) are dependent upon x is given by (Roberts 1981a)
Here is the velocity component normal to the magnetic field, is the frequency and is the longitudinal wavenumber along the sheet. The velocity parallel to the magnetic field is given by
We consider only motions that are independent of the y -coordinate so that propagation is in the xz -plane.
The perturbed gas , magnetic and total (gas and magnetic) pressure perturbations are given by
whilst the magnetic tension force takes the form
where is the perturbed magnetic field and
To investigate the nature of the modes we reduce Eq. (13) to the canonical form. Setting reduces the governing wave equation to
with given by Eq. (3) and = , with f given by Eq. (14). Here the dash denotes a derivative with respect to x. For a uniform medium, is constant ; body and surface modes are determined by positive and negative , respectively. However, for a medium which is continuously structured, we define body and surface modes when attains positive or negative values, respectively, across the whole sheet . In addition, hybrid modes occur when possesses both positive and negative across the width of the sheet.
© European Southern Observatory (ESO) 1997
Online publication: April 8, 1998