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

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1. Introduction

Neutral sheets may guide waves. Ducted waves occur in regions of low Alfvén speed (see Edwin & Roberts 1982; Roberts, Edwin & Benz 1984; Murawski &; Roberts 1993, b; Nakariakov & Roberts 1995; Smith, Roberts & Oliver 1997). Therefore we may expect wave energy to be localised where the magnetic field is weak, as in a neutral sheet, or where plasma density is high, as within a coronal loop. Hence coronal streamers, which contain both closed dense loops and current sheets, are expected to provide effective wave ducts. The investigation of magnetoacoustic wave propagation in inhomogeneous photospheric flux tubes and coronal loops has been comprehensively studied (see the reviews by Roberts 1985, 1991, 1992 and Edwin 1991, 1992). A variety of waves may be present in the solar atmosphere - body or surface, fast or slow - their existence and properties depending upon the relative orderings of the Alfvén and sound speeds.

Direct observations of current sheets in the solar atmosphere are not common. Eddy (1973), analysing results from the 1922 eclipse, suggests the existence of a coronal neutral sheet which separates regions of opposite magnetic polarity. The width and length of the sheet were found to be 9 arc-seconds and 4 solar radii respectively. Neutral sheets are expected to be present above the helmet structures in coronal streamers and between coronal loop systems of opposite polarity. Current sheets may be formed in one of three ways (Priest 1982): the interaction of topologically distinct regions, the loss of equilibrium of a force-free field, and X-point collapse. In addition, in the Earth's magnetosphere, the nightside central plasma sheet and dayside magnetopause both contain current sheets. We show here that such sheets may support magnetoacoustic waves.

Current sheets are important structures which may guide waves; the damping of these waves may be an important mechanism for coronal heating (see, for example, Cramer 1994; Tirry,[FORMULA]ade[FORMULA] & Goossens 1997). The study of magnetoacoustic waves in current sheets may also provide valuable insight into the origin of certain frequently reported oscillations in the corona (Aschwanden 1987; Tsubaki 1988) and the Earth's magnetosphere (see Anderson 1994 and references therein). Evidence of wave-like motions in solar current sheets is limited, although Aurass & Kliem (1992) associate a Type IV radio burst with activity in a large sheet. These authors suggest that the formation of current filaments by the tearing mode instability may create propagating magnetoacoustic waves. In addition, there have been many recent observations of short period coronal oscillations (Correia & Kaufmann 1987; Zhao et al. 1990; Ru[FORMULA]in & Minarovjech 1994; Zlobec et al. 1992), with periods ranging from 0.3 to 12 seconds, which may be associated with oscillating current sheets. There have also been reports of current sheet oscillations in the Earth's magnetotail. Periodicities in the range 3.5 to 5 minutes in the electron flux were reported by Montgomery (1968). Mihalov, Sonett & Colburn (1970) found periods ranging from 0.5 to 15 minutes with peaks at 100 seconds, 4.5 minutes and 8 minutes. Recently, Bauer, Baumjohann & Treumann (1995a) and Bauer et al. (1995b) have reported 1-2 minute oscillations in the vicinity of the current sheet where the perturbed gas and magnetic pressures were out of phase.

Several authors have investigated wave propagation in neutral sheets (see the review article by Cramer 1995). Hopcraft & Smith (1985) solved the governing wave equation for a Harris sheet analytically through a small parameter expansion. Although dispersion curves for fast magnetoacoustic waves were presented, the validity of the expansion near the edge of the sheet is uncertain. Further work by Hopcraft & Smith (1986) solved the governing wave equation numerically, although no dispersion curves were given. The magnetic field and pressure profiles used by Hopcraft & Smith (1986) were discontinuous at the sheet boundary. Although the sheet was in equilibrium an additional current at the edge of the sheet was generated which may have affected the results. Using the Harris sheet model, Seboldt (1990) investigated the singular solutions of the governing wave equation. This analysis showed that perturbations may be subject to phase mixing. Discrete eigenmodes were also briefly discussed, although no detailed analysis of the modes was given. Cramer (1994) examined the Alfvén and slow resonances of surface waves in a current sheet. The sheet was modelled as a field with an arbitrary change of direction through a narrow transition region. Cramer found that for certain angles of propagation surface waves were non-existent; two surface waves were present for some angles of propagation (compare with Roberts 1981a). The recent investigation by Tirry, [FORMULA]ade[FORMULA] & Goossens (1997) explored the damping of surface waves in a Harris profile through the Alfvén resonance; they found only two surface modes existed (the kink and sausage), with the sausage mode undergoing more damping than the kink.

A simple model of a current sheet was analysed by Edwin, Roberts & Hughes (1986) and Edwin (1992) to explain the generation of Pi2 pulsations in the plasma sheet. These oscillations are quasi-periodic pulsations with periods ranging between 40 and 150 seconds, typically lasting for a few cycles (Southwood & Stuart 1980; Singer et al. 1985). The sheet was modelled as an unbounded hot plasma slab, with a narrow field-free region, between the anti-parallel fields of the magnetotail. In the magnetic region the cold plasma approximation was used. Both fundamental kink body and sausage surface modes were found. In the slender sheet approximation, their phase speeds tended towards the exterior Alfvén and tube speeds, respectively. The fundamental kink body mode transforms to a surface wave when its phase speed falls below the constant sound speed within the sheet. Taking the velocity component [FORMULA] normal to the equilibrium magnetic field and the total pressure perturbation both continuous across the sheet boundary at x = [FORMULA] leads to the dispersion relations (assuming [FORMULA] as [FORMULA]) (Edwin et al. 1986; Edwin 1992)


for the surface [FORMULA] modes, and


for the body [FORMULA] waves. Here


the subscript i equals o in the field-free region and e in the magnetic region. The `tan' and `tanh' solutions refer to the sausage mode, whereas the `cot' and `coth' terms relate to kink modes. The frequency and longitudinal wavenumber are [FORMULA] and [FORMULA], respectively, whilst the sound and Alfvén speeds are denoted by [FORMULA] and [FORMULA]. [FORMULA] is the ratio of specific heats and R is the ratio of exterior Alfvén speed to interior sound speed. The dispersion diagram is discussed later (see Fig. 3b).

The dimensionless phase speed of both sausage and kink surface waves, [FORMULA], tends to the same limiting value at short wavelengths [FORMULA], given by (Edwin et al. 1986)


(There is a spurious solution to Eq. (4) which does not satisfy the original dispersion relation.) In addition, sausage and kink body modes were found above wavenumber cut-offs. The phase speed of these body waves lie between the interior sound speed [FORMULA] and the exterior Alfvén speed [FORMULA]. Edwin et al. (1986) and Edwin (1992) suggested the long period magnetospheric oscillations (see, for example, Anderson 1994) may be due to surface waves, whereas the shorter period Pi2 pulsations may be body waves (see also McKenzie 1970 for a similar model). Impulsively generated waves in a plasma sheet were found to consist of a well defined wave packet consisting of a periodic phase, followed by quasi-periodic phase and then a decay phase, in good agreement with observations. However, an important aspect neglected in this model was the field reversal region. It is the purpose of this paper to consider propagation in a structured current sheet.

In this paper we investigate in detail the eigenmodes supported by a current sheet. Such an investigation may provide important seismic information about the corona and magnetosphere. We show that a current sheet supports both kink and sausage oscillations. In the sausage mode the current sheet pulsates like a blood vessel, with the central axis remaining undisturbed. In the kink mode the central axis moves back and forth during the wave motion. In a simple slab, modes may also be characterised as either body (oscillatory in the interior) or surface (hyperbolic in the interior); see Roberts (1981a,b,1985,1991,1992), Edwin & Roberts (1982) and Edwin (1991, 1992) for a full discussion. We show that in a continuously structured medium, three types of mode can exist: body, surface and hybrid. The nature of the mode is governed by the phase speed and wavenumber. Hybrid modes contain elements of both body and surface waves. We restrict attention to modes which are evanescent outside the current sheet and therefore ignore any wave leakage (see, for example, Cally 1986).

The component of velocity transverse to the sheet can have nodes in two spatial directions, across the sheet width or along the sheet length. Modes with the least number of nodes across the sheet are referred to as fundamental modes. Traversing the sheet from one boundary to the other, the velocity in the fundamental kink mode does not change sign whereas the fundamental sausage mode changes sign once (at the central axis). Overtones of the kink and sausage modes have a greater number of nodes in the direction perpendicular to the current sheet axis.

For the neutral sheet in the Earth's magnetotail, widths of 3-6 [FORMULA] ([FORMULA] km) have been used by Patel (1968), McKenzie (1970) and Edwin et al. (1986) in theoretical models. Hopcraft & Smith (1985) used a range from 1000 km to 50,000 km, whilst Seboldt (1990) used a sheet thickness of 30,000 km. For our calculations we use current sheet widths of [FORMULA] km for solar applications and [FORMULA] for the Earth's magnetotail. The constant Alfvén speed in the corona is taken to be 1000 km s-1, whilst in the magnetosphere a value of 500 km s-1 is considered (Mihalov et al. 1970). In this paper we use the standard ideal magnetohydrodynamic (MHD) equations. The mean free path in the solar corona (for temperatures 106 K and number densities 109 cm-3) is of the order of 50 km whereas the electron gyro-radius is 20 cm for a field of 1G (Spitzer 1962). Hence, we consider only current sheets with widths greatly in excess of 50 km to ensure the validity of MHD. Ideal MHD may only be applied with caution in the Earth's magnetosphere. A discussion on the validity of the use of MHD in the magnetosphere may be found in Schindler and Birn (1978, 1986) and Birn, Hesse and Schindler (1996). The authors suggest that a MHD approach may be justified because the plasma pressure is approximately isotropic within the plasma sheet. In addition, Birn et al. (1996) compare the results of simulations of the magnetotail using MHD, hybrid and particle codes. The results are qualitively similar in each case, suggesting MHD is a good approximation to the more rigourous particle treatment.

The format of the paper is as follows. In Sect. 2 the equilibrium and governing wave equation are discussed. In Sect. 3 the numerical procedure for solving the wave equation is described. In Sect. 4 we explain the properties of magnetoacoustic waves in current sheets and in Sect. 5 our conclusions and applications are made.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998