## 2. Equilibrium and governing equationsConsider 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 where 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 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
=
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