5. Conclusion and applications
5.1. Summary of results
We have seen that a current sheet may support magnetoacoustic guided waves. The waves are trapped by the inhomogeneity in magnetic field and/or plasma density. Modes may be excited by, say, solar flares, magnetospheric substorms or reconnection. Once excited, the waves may propagate along a current sheet and be the source of reported oscillations in the solar corona and Earth's magnetosphere.
A current sheet supports waves which may be body, surface or hybrid. An examination of the driving forces and perturbed pressures shows that no purely fast or slow modes exist in the structure. Observed oscillations have properties of both types of mode. A sausage surface wave exists for all values of the wavenumber; in the long wavelength limit the phase speed approaches the maximum value of the tube speed in the current sheet. A fundamental kink wave also exists in a current sheet. For long wavelengths (with phase speeds approaching the maximum value of the Alfvén speed in the sheet) the wave is characterised as a body mode ( for all in the inhomogeneous region). As the phase speed decreases, attains both positive and negative solutions; this is a hybrid mode. For lower phase speeds the mode changes nature and becomes a surface wave ( for all in the structured region). For short wavelength oscillations the phase speed of the fundamental kink and sausage surface waves is constant, tending to approximately the value given in the simple slab model of Edwin et al. (1986) [see Eq. (4)]. Harmonics also exist above wavenumber cut-offs; these may be body or hybrid depending upon the phase speed. The dispersion curves of a pair of kink and sausage modes merge when their phase speeds fall below than the maximum sound speed within the sheet (the fundamental sausage interleaves with the first harmonic kink, first harmonic sausage with second kink and so on). The phase speeds of each pair merge after the phase speed passes through the maximum sound speed. For a uniform density profile, a minimum in the group velocity exists only for the fundamental kink mode, whereas for the Epstein profile minima also exist for other modes. The results are similar to those obtained by Edwin et al. (1986). The main difference is that, due to the continuous structuring, body modes transform into hybrid modes for all harmonics. In the slab model of Edwin et al. body modes were trapped in the phase speed range between the sound speed in the field-free region and the exterior sound speed. In addition, the fundamental sausage mode, which in our case tends to the for , approaches zero as ; in the slab model the tube speed in the field-free region is zero.
5.2. Application to observations
We now apply our results to some observations. For convenience, we use the case of uniform density. For solar applications we consider a constant Alfvén speed in the exterior of the sheet of =1000 km s-1 and a sheet width equal to 1000 km. For magnetospheric applications, an Alfvén speed of 500 km s-1 is used with a sheet width of (=6400 km). The periods determined using our model are given in Table 1. The fundamental surface mode periods are estimated using a value of , whilst the periods for the other modes are estimated at the onset of the modes (i.e. at the cut-off value of ).
Table 1. Estimated periods of ducted waves in current sheets
For the coronal case periods lie in the range 0.5 to 12 seconds. Interestingly, periods of this order are frequently reported (see Aschwanden 1987 for a review of radio and X-ray observations). In particular Zlobec et al. (1992) report a 11.4 second oscillation two hours after a solar flare; we suggest this may be due to surface waves in a current sheet. There are many recent reports of short period oscillations, which may be due to magnetoacoustic waves in current sheets. Pasachoff & Landman (1984) and Pasachoff & Ladd (1987) detected intensity variations of the green coronal line with periods between 0.5 and 4 seconds. Short oscillations are also abundantly reported in radio and X-ray emission. For example, Correia & Kauffman (1987) report oscillations of 0.3 seconds in hard X-rays whilst Zhao et al. (1990) give periods of 1.4 and 1.6 seconds in microwaves; Aschwanden (1994) and Karlický & Jiika (1995) give further examples of short period coronal oscillations.
For the Earth's magnetosphere, our calculations give periods in the range 8 seconds to 2.5 minutes, for the various modes. Therefore the 1-2 minute magnetic field fluctuations of the neutral sheet observed by Bauer et al. (1995a,b) may indeed be due to waves propagating in a neutral sheet. Magnetic variations and boundary motions on time scales from a few minutes to several hours have been explained as "flapping" motions of the tail ( et al. 1970; Hruka & Hrukova 1970; Hones et al. 1971; Russell 1972). Longer periods may be due to surface waves with , or for circumstances with lower Alfvén speeds and/or wider sheets.
It is of interest to consider impulsively generated waves. Suppose that at an impulse (such as due to reconnection or an instability) occurs at a location . This impulse is composed of all frequencies. Then the wave observable at large distances from the initial impulse evolves as follows (Roberts, Edwin & Benz 1984) on the basis of the group velocity curve. The event begins with a periodic phase starting at a time and consisting of low frequency, low amplitude waves; this corresponds to the left-hand branch of the group velocity curve (Fig. 6). The frequency and amplitude of the waves in the periodic phase grow until a time , where is the constant group velocity at high frequency. This is the onset of the quasi-periodic phase. A train of high frequency waves from the right-hand branch of the group velocity curve are superimposed on the low frequency waves from the left-handside. The amplitude of this phase, in an impulsively excited disturbance, is strongly enhanced due to the superposition of high and low frequency waves (Pekeris 1948; see also Ewing, Jardetzky & Press 1957). The oscillations are quasi-periodic; the amplitude varies inversely proportional to the slope of the group velocity curve. During this phase, the frequency of the high frequency low and high frequency waves continues to decrease, whilst those of the low frequency waves continue to increase. This occurs up to a time , where is the minimum of the group velocity, when the two frequencies take the same value. This marks the onset of the decay phase. The disturbance then consists of a single frequency , and the plasma continues to oscillate with this frequency although its amplitude decays rapidly.
The durations of the periodic and quasi periodic phases are given by (Roberts et al. 1984)
As a numerical example, consider the case of uniform density with the results displayed in Fig. 6. The minimum group velocity is =0.617 occurring at a frequency of =1.166 . At high frequency, the group velocity approaches =0.675 . For solar applications, we assume an observational height h =105 km above the impulse and an Alfvén speed of 1000 km s-1; then the periodic phase starts 100 seconds after the impulse and lasts for 48 seconds. The quasi-periodic phase has a duration of 13.9 seconds with a minimum period of 2.7 seconds at the end of the phase. For the magnetosphere consider 50 and an Alfvén speed of 500 km s-1. Then the periodic phase begins about 10.5 minutes after the impulse and lasts for 5 minutes. The quasi-periodic phase has a duration of 89 seconds. The decay phase then follows. The periodic phase is relatively low in amplitude and so may not be observable because of noise or poor resolution. In which case, our illustration gives a wavepacket of about 14 seconds duration with a minimum period of 2.7 seconds observable in a coronal current sheet. In the magnetospheric sheet, the observable quasi-periodic phase would last for about 89 seconds, with a minimum period of 34 seconds. Of course, this illustration is for a single pulse initiated at the centre of the sheet. More complicated time signatures may be expected for multiple impulses or pulses that occur away from the centre of the sheet (see the numerical simulations by Murawski & Roberts 1993a,b for the case of a coronal loop). In our equilibrium a pulse will generate the disturbances described plus also a z -component of velocity ( and are coupled). Therefore more complicated temporal signatures may result from this coupling.
© European Southern Observatory (ESO) 1997
Online publication: April 8, 1998