4.1. Uniform density
We begin by looking at the various modes which may propagate in a current sheet. The function arising in Eq. (21) is strongly dependent upon the mode of oscillation (wavenumber and phase speed) and also the position along the sheet. We find that three types of mode may exist in a continuously structured sheet, compared with two in a simple slab. Modes of oscillation may have or across the inhomogeneous region; these modes are classified as body or surface, respectively. We note, however, that these modes are different from those in a uniform medium, where is constant. In addition, we also find modes where takes both positive and negative values across the sheet; we call these hybrid modes. Fig. 2 shows an example of a fundamental sausage hybrid mode (solid line) along with the x -component of velocity (dashed line). Firstly, varies across the whole domain, being positive in some regions and negative in others. Also, the velocity has maxima where , whereas wave motions are smaller where . The eigenfunction is characterised by regions of different signs of ; a hybrid mode exists. For harmonics, the wave motions are more oscillatory across the sheet, in which case oscillations can occur in regions of both positive and negative . For all modes, we find for , and solutions are evanescent outside the current sheet.
Fig. 3a shows the dispersion curves obtained for the case of constant density, = . Kink and sausage modes are shown as dashed and solid lines, respectively. The horizontal dotted line shows the position of the maximum sound speed, . Above this line only body and hybrid modes are found. For phase speeds close to the maximum Alfvén speed, body modes arise ( for all in the structured region). As the phase speed decreases the body modes transform into hybrid modes ( attains both positive and negative values). The dot-dashed line shows the maximum value of the tube speed . Below this value only continuum solutions exist (Rae & Roberts 1982 and references therein; see also Seboldt 1990; Poedts & Goossens 1991; Cramer 1994). Between these two lines both surface and hybrid waves arise. Trapped modes exist only in the phase speed range
The fundamental sausage surface wave (lowest curve in Fig. 3a) is the only mode which exists as a surface wave ( for all ) for all wavenumbers. At long wavelengths its phase speed falls to the maximum tube speed in the current sheet. As increases the phase speed increases, tending to a constant value () for short wavelengths. It is interesting to note that this limiting value at short wavelengths is close to the value given by the slab model of Edwin et al. (1986). If we set in Eq. (4), as the ratio of maximum Alfvén speed to maximum sound speed, we obtain a phase velocity of 0.689 for , which compares favourably with the numerical results for the Harris sheet.
The fundamental kink wave exists for all wavenumbers, although it is a surface wave only for above a threshold. Fig. 4 illustrates the eigenfunction for the fundamental kink mode. For low , the kink wave starts off as a body wave and most of the velocity amplitude is concentrated at the centre of the sheet (see Fig. 4a). As the phase speed decreases the kink mode transforms to a hybrid mode (where attains both positive and negative values). As the phase speed falls below the maximum value of the sound speed the mode transforms into a surface wave, where across the entire sheet (see Fig. 4b, where a small "dip" can be observed in the eigenfunction in the centre of the sheet). For larger , this dip becomes more pronounced (Fig. 4c) so ultimately, for very short wavelengths, there are two peaks in the velocity concentrated near the edge () of the sheet (Fig. 4d). The phase speeds of the kink and sausage waves approach the same limiting value for short wavelengths.
In a similar way to the work by Edwin & Roberts (1982), magnetoacoustic modes also exist above wavenumber thresholds (Fig. 3a). At the wavenumber threshold the phase velocity equals the exterior Alfvén speed and the mode is classified as body. As their phase speeds decrease, attains both positive and negative values and a hybrid mode forms. The dispersion curve for the fundamental sausage mode, occurring at a critical wavenumber of , interleaves with the first harmonic kink mode when the phase speed falls below the maximum sound speed in the sheet. In addition, the first harmonic sausage mode interleaves with the second harmonic kink mode , and so on. Notice that each merging pair contains a higher overtone kink mode than the sausage. When the "dip" appears in the eigenfunction for the kink mode (Fig. 4) the number of maxima is the same across the sheet for both kink and sausage modes (although the parity about is different).
Due to the continuous structuring of sound and Alfvén speed profiles in the current sheet, the phase speeds of these modes do not tend to the maximum value of sound speed, as in the case studied by Edwin et al. (1986). Instead the phase speeds fall below ; notice how the phase speeds (and therefore frequencies) of these pairs tend to the same value in Fig. 3 after passing through the maximum value of the sound speed. The phase speed tends to an asymptote which is slightly different for each pair of modes. Each pair tends to a slightly higher phase velocity than the previous pair; for the first pair this is 0.69. The value of this asymptote is approximately given by the slab model of short wavelength oscillations in Edwin et al. (1986); see Eq. (4). For short wavelength perturbations we see that the velocity is concentrated about the sheet boundary, whereas for longer wavelengths oscillations across the entire sheet may be expected.
The x and z -components of the velocity are shown in Fig. 5 for a first harmonic kink hybrid wave. This mode is characterised in by two large peaks at , with ("body") plus two smaller peaks at with ("surface") (cf. Fig 4). The z -component of velocity shows similar characteristics to but is of opposite parity about . The velocity in the inner part of the sheet is approximately zero with wave motions concentrated near the edge of the sheet. The amplitudes of the velocity components are similar; this is to be expected since the maximum values of the sound and Alfvén speeds are similar. The eigenfunction of the associated sausage hybrid wave (the mode that transformed from the fundamental sausage body) is very similar, the main difference being the parity of the two solutions about the current sheet centre. The oscillation frequencies approach the same value as increases (Fig. 3a).
Fig. 6 shows the group velocity curve for the fundamental kink mode. The minimum group velocity is =0.617 , occurring at a frequency . These values are of importance in the theory of impulsively generated waves (see Roberts, Edwin & Benz 1984, and below). However no minimum in the group velocity exists for the other modes in the uniform density current sheet. The absence of a minimum in the group velocity also arises in the work of Nakariakov & Roberts (1995). Considering coronal loops modelled by different density profiles, they found that the Epstein profile was a special case: it had no minimum, whereas other density profiles possessed minima.
We now examine the driving forces (tension and gradients of gas and magnetic pressure) and the perturbed pressures. This gives some insight into what forces are important in driving wave motions, and it affords a comparison with the results of a uniform field giving us an indication of whether the modes are fast or slow. The linearised MHD equations (leading to the governing wave equation 13) may be written in the form
where and are the gas and magnetic pressure perturbations, is the total pressure perturbation, and and are the components of the tension. We first recall the results for a uniform medium, i.e. one in which B, p, (and therefore and ) have no x -variation in the region under consideration. From Eqs. (16)-(18) we see that if the magnetic pressure is out of phase with both the gas and total pressure perturbation; this is a feature of the slow mode. Moreover for this mode, the x -projections of the tension force and velocity are out of phase whereas the z -components are out of phase. For , the total pressure perturbation, gas pressure and magnetic pressure are all in phase; this is a property of the fast mode. In addition, for a fast mode in a uniform medium, is out of phase with whilst the z -components of velocity and tension are in phase.
The formulae for the total pressure perturbation and the tension force is the same in both uniform and non-uniform media . However, the perturbed gas and magnetic pressures are altered by the non-uniformity in pressure and magnetic field; see Eqs. (16) and (17). The results show that the magnetoacoustic modes of a current sheet case are more difficult to classify than those of a uniform medium, with the non-uniformity playing an important role.
Firstly we consider the x -projection of the driving forces (Fig. 7a). In the centre of the current sheet the gradients of plasma and magnetic pressures are clearly out of phase with ; this is a characteristic of a slow mode. Notice also the gradients of gas and magnetic pressures are in anti-phase at , where the amplitudes of the driving forces are approximately the same. In the outer part of the current sheet , the gradient of plasma pressure tends to zero rapidly. In addition, in this region the gradient of magnetic pressure is greater than the gradient of gas pressure; this is a signature of a fast mode. Thus, as we move away from the centre of the sheet, the mode changes from being slow in character to being fast. This is not surprising since the plasma pressure is maximum (magnetic field minimum) at whereas the situation reverses as we increase . In other words we are moving from a high plasma beta regime (where we expect slow waves to dominate) to a low beta region, where fast waves are likely to propagate. Therefore, it is not possible to classify the mode as globally slow or fast; the mode is locally slow in the inner region fast in the outer parts of the structure. Furthermore, unlike the uniform case, due to the non-uniformity in Alfvén speed the x -projection of tension is not out of phase with the velocity. The Alfvén speed is low in the vicinity of the centre of the sheet and therefore tension forces are small. Similar results are also found for the z -component. The gradients of plasma and magnetic pressures are out of phase around the centre of the sheet. As we move away from the centre the gas pressure perturbations become less dominant than magnetic pressure perturbations. In the outer part of the configuration the magnetic pressure becomes greater than the plasma pressure.
These points are reinforced by examining the plasma pressure, magnetic pressure and the total perturbed pressure (Fig. 7b). Notice that the plasma and total pressures are out of phase with the magnetic pressure around . However, as we increase the perturbed plasma pressure decays rapidly and the magnetic pressure and the total pressure are in phase for . Thus for the fundamental modes the oscillations within the sheet are essentially slow, whereas the evanescent decaying velocity is predominantly fast.
By contrast, for the harmonics of the modes we find that the oscillations within the sheet may possess both fast and slow mode properties. To illustrate this we show the perturbed forces in Fig. 8 for the first overtone sausage hybrid mode with and . Notice that in the inner part of the current sheet the perturbed magnetic pressure is small (since B is small). Consequently the plasma and total pressure perturbations are almost equal for . This suggests that the mode is slow. However, the perturbations in the outer part of the sheet are different in character. At the total, plasma and magnetic pressures are almost in phase. Thus both slow - and fast -type oscillations occur simultaneously in the sheet for the same velocity component. For the pressure term is small and the evanescent decaying velocity has the character of the fast mode.
Therefore, in contrast with a uniform medium, there are no pure fast or slow modes in a current sheet. This is due to the non-uniformity in pressure and magnetic field; the plasma beta ranges from infinity at to zero for . Modes do not possess all the characteristics of slow or fast modes, although we can identify properties of both types in the different regions.
Similar results are found for all modes of oscillation. The tension force in the central part of the sheet is very small, and tension plays an important role only as the Alfvén speed approaches its maximum value. We always see that in the high plasma beta part of the sheet the plasma and magnetic pressures are in anti-phase, with the plasma pressure dominating. In the exterior of the sheet the magnetic pressure is greater than the plasma pressure. For all modes considered we see both slow and fast mode characteristics.
It is interesting to note that in the observations of the neutral sheet oscillations by Bauer et al. (1995a,b) the plasma and magnetic pressures were in anti-phase, in agreement with our theoretical results.
4.2. Epstein density profile
When the plasma density inside the current sheet is no longer equal to the density in the far environment of the sheet, so that , the profile (8) is the Epstein one. The results for an Epstein profile are similar to those in the uniform density case and so we summarise the main results and illustrate the main differences. We take the plasma density at the centre of the current sheet to be five times larger than the constant density away from the current sheet ().
The curves in the dispersion diagram are similar to the uniform density case; see Fig. 9. Both kink and sausage, body, surface and hybrid modes may exist. A current sheet therefore supports a rich spectrum of modes. The sausage surface mode, which again exists for all , has a limiting phase speed of the tube speed in the slender current sheet approximation. For short wavelengths the phase speed attains a constant value of 0.38 . An application of Eq. (4) gives a phase speed of 0.40 . The fundamental kink mode originates as a body mode . As the phase speed decreases the body mode transforms into a hybrid mode. When the phase speed passes through the maximum value of the sound speed the mode changes character and becomes a surface wave. The phase velocity of this mode tends to the same value as the sausage surface wave. In addition, body and hybrid waves exist above wavenumber cut-offs: the fundamental sausage mode mode has a threshold of =0.63, first kink harmonic 1.10, first sausage harmonic 1.65 and the second kink =1.96. These cut-offs are lower than in the case of uniform density. These body modes, as in the uniform density case, transform into hybrid waves when their phase speed decreases. Again, a sausage and a kink mode interleave when their phase speed falls below the maximum sound speed within the current sheet. Each kink and sausage mode pair tends to a slightly higher value than the previous pair. We see, however, that for the Epstein density profile, the phase speed only falls below the maximum sound speed level for much higher values of the wavenumber than the case of uniform density. For example, for the fundamental body sausage mode the transition occurs at whilst for uniform density case it is .
The driving forces show a similar form to the uniform density case. Due to the large value of for the phase speed to fall below the maximum sound speed, we find the group velocities possess minima for all modes except the sausage surface. The value of is approximately the same for all cases considered; namely 0.316 for the surface kink mode and approximately 0.356 for all the other modes considered. The minimum frequency equals 0.462 and 2.774 for the fundamental and first harmonic kink modes, and 1.672 and 3.846 for the fundamental and first harmonic body sausage modes. Therefore impulsively generated waves will show a similar temporal form to that analysed by Roberts et al. (1984). For coronal loops Nakariakov & Roberts (1995) found minima in the group velocity for a wide variety of density profiles. However, the Epstein profile was an exception where the minimum moved out to infinity.
© European Southern Observatory (ESO) 1997
Online publication: April 8, 1998