Coronal Waves and Oscillations

2.1 MHD modes of a straight cylinder

An important elementary building block of this theory is dispersion relations for modes of a magnetic cylinder (see Figure 2). Magnetic cylinders are believed to model well such common coronal structures as coronal loops, various filaments, polar plumes, etc. Consider a straight cylindrical magnetic flux tube of radius a filled with a uniform plasma of density ρ₀ and pressure p₀ within which is a magnetic field B₀e _z ; the tube is confined to r < a by an external magnetic field B_ee _z embedded in a uniform plasma of density ρ_e and pressure p_e (here we neglect the effects of twist and steady flows). The very existence of such a plasma configuration requires the balance of the total pressure P_tot, which is the sum of the plasma and magnetic pressures, between the two media at the boundary. For the equilibrium state, this condition is

$${p_0} + {{B_0^2} \over {2{\mu _0}}} = {p_{\rm{e}}} + {{B_{\rm{e}}^2} \over {2{\mu _0}}}.$$

(1)

Similarly, a magnetic slab may be introduced in much the same way, having a width 2a.

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In the internal and external media, the sound speeds are C_s0 and C_se, the Alfvén speeds are C_A0 and C_Ae, and the tube speeds are C_T0 and C_Te, respectively. Relations between those characteristic speeds determine properties of MHD modes guided by the tube.

Figure 3 shows a typical dispersion diagram of a coronal loop. This figure generalises the dispersion plot given by Edwin and Roberts (1983) to the inclusion of higher-m modes. In the Figure 3, all the possible modes correspond to body modes, which have oscillatory behaviour inside and evanescent behaviour outside. This is in contrast with surface modes, which have evanescent behaviour in both media. Phase speeds of MHD modes guided by the tube can have values in two bands: either between C_A0 and C_Ae (provided C_A0 < C_Ae) and between C_T0 and C_s0. The wave modes in these two bands have been named, respectively, fast and slow, in analogy with the types of magnetoacoustic wave modes present in a homogeneous medium. The fast modes are highly dispersive. In the long wavelength limit, the phase speed of all but sausage fast modes tends to the so-called kink speed

$${C_{\rm{K}}} = {\left({{{B_0^2/{\mu _0} + B_{\rm{e}}^2/{\mu _0}} \over {{\rho _0} + {\rho _{\rm{e}}}}}} \right)^{1/2}} = {\left({{{{\rho _0}C_{{\rm{A}}0}^2 + {\rho _{\rm{e}}}C_{{\rm{Ae}}}^2} \over {{\rho _0} + {\rho _{\rm{e}}}}}} \right)^{1/2}},$$

(8)

which corresponds to the density weighed average Alfvén speed. The sausage mode approaches a cut-off at the external Alfvén speed. Trapped sausage modes do not exist at longer wavelengths. But mode solutions can be found if the condition of mode localisation is relaxed, i.e., if waves are allowed to radiate into the external medium. Such wave modes are called leaky modes and have complex eigenfrequencies. To include leaky modes, dispersion relation (7) is modified such that the modified Bessel functions K _m (x) are replaced by Hankel functions. Details of the leaky solutions may be found in Zaitsev and Stepanov (1975); Cally (1986); Stenuit et al. (1998). In Figure 3, leaky waves would be situated outside the regions of existence of the trapped waves. In particular, the sausage mode continues in the region over the horizontal asymptote C_Ae but its frequency is complex there.

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In closed fields of coronal active regions, the longitudinal wave number k _z of standing modes is usually prescribed by the line-tying boundary conditions at the photosphere. Modes with the lowest wave numbers are called global or fundamental.

The magnetoacoustic modes (with an important exception) are collectively supported by the plasma environment, i.e., the wave mode acts across neighbouring magnetic field lines and across transverse plasma inhomogeneities. Alfén waves, though, are locally supported. They have phase and group velocities, with magnitudes equal to the local Alfvén speed, which are directed along the magnetic field. This means that an Alfén wave propagates along its local magnetic field line without interaction with neighbouring field lines. This particular property allows for the existence of continua of eigenfrequencies and which will be discussed in Section 2.2.

In a cylinder model, Alfvén waves are torsional waves that twist tube. In the case of a straight cylinder these modes are incompressible, however in a slightly twisted cylinder they are accompanied by perturbations of plasma density (Zhugzhda and Nakariakov, 1999). In the slab geometry, torsional waves perturb the magnetic field and generate perturbations of plasma velocity in the direction perpendicular to magnetic field and to the direction of the inhomogeneity. Alfvén waves are very weakly dissipative. This means they can propagate very long distances and deposit energy and momentum far from their source. Concerning the generation of Alfvén waves, they can easily be excited by various dynamical perturbations of magnetic field lines. This makes Alfvén waves a promising tool for heating and diagnostics of coronal magnetic structures.

Movies visualising the structure of the MHD modes in a magnetic cylinder are available in Resource 1. Figure 4 shows the transverse and longitudinal density and velocity structure of a magnetic cylinder perturbed by a fundamental fast kink oscillation.

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2.2 MHD continua

When the group speed of wave modes is directed along the magnetic field, then such modes on neighbouring field lines do not interact with each other. They may oscillate locally with their own eigenfrequency without disturbing the rest of the medium. If the equilibrium parameters of the plasma vary continuously across the magnetic field, then the eigenfrequency of such modes may also vary continuously. This gives rise to continuous intervals of eigenfrequencies, i.e., continua, in frequency space. This situation, though, may change by including resistivity or non-MHD effects (see, e.g., Appert et al., 1986).

In ideal MHD, there exist two continua: the Alfvén continuum and the slow (or cusp) continuum (see Goedbloed, 1983; Goedbloed and Poedts, 2004). Equations (2, 3) are also valid for linear waves in a coronal loop modelled as a plasma cylinder if the equilibrium quantities vary continuously in the radial direction (see, e.g., Sakurai et al., 1991a). The continua are in this case characterised by the frequencies C_A(r)∣k _z ∣ and C_T(r)∣k _z ∣. There are two effects associated with wave modes from these continua: resonant absorption and phase-mixing.

2.2.1 Resonant absorption

If the loop supports a collective (a mode) wave, either driven externally or set up initially, which has a frequency ω that falls within one of the two continua, a resonance is set up at the location(s) where \(\omega^{2}=C_{\rm A}^{2}(r)k_{z}^{2}\) or \(\omega^{2}=C_{\rm T}^{2}(r)k_{z}^{2}\) (where Equation (4) becomes zero). The physical location where the resonance occurs is called the resonance layer and in the plasma cylinder model it corresponds to a radial shell (see Figure 5). From now on we shall consider the case of a resonance in the Alfvén continuum only. The locally excited resonant wave mode is a torsional Alfvén wave and its amplitude peaks in the resonance layer where the perturbation develops large gradients. Dissipation has to be taken into account to prevent the perturbation to diverge. The amplitude of the resonant mode scales as Re^1/3 at the resonant layer, where Re is the shear viscous and/or the magnetic Reynolds number (Kappraff and Tataronis, 1977). In the solar corona this number is much larger than unity (Re ∼ 10¹⁴). Since non-resonant modes dissipate with an amplitude proportional to Re, it is clear that the resonantly excited modes experience enhanced dissipation. This also implies that the resonant absorption process is inherently nonlinear, since the amplitude of the resonant mode can not grow to the values implied by the Re^1/3 scaling (Ofman et al., 1994; Ofman and Davila, 1995). Thus, wave energy is extracted secularly through the resonance from the collective wave to the benefit of a local wave mode (mode conversion), which then is dissipated in an enhanced manner. This mechanism of wave heating is called resonant absorption (Goedbloed, 1983, and references therein) and has been put forward in the context of the coronal heating problem (Ionson, 1978) and in sunspot seismology for explaining the loss of acoustic power in sunspots (see, e.g., Sakurai et al., 1991b; Bogdan, 2000, and references therein).

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There are inherently two time scales involved. Firstly, there is the damping time scale of the mode conversion from the collective to the local mode, which generally is independent of dissipation. From the point of view of linear theory and classical, theoretical values of dissipation coefficients, it is generally much shorter than the second time scale, which is linked to the dissipative damping of the small-scale perturbations of the local mode in the resonance layer.

Note that for sausage wave modes, where m=0, the equations describing the magnetoacoustic and torsional waves (i.e., Equations (2, 3)) are decoupled so that mode conversion and absorption through the Alfvén resonance cannot take place (the slow resonance can still operate; see, e.g., Erdélyi, 1997). For the slab geometry, this corresponds to propagation parallel to the magnetic field, i.e., k _y = 0.

To study resonant absorption the following procedure is often undertaken. Whilst outside of the resonant layer the ideal MHD equations can be applied, inside the resonant layer dissipative effects have, in principle, to be taken into account to ensure that the solution remains regular. But unless one is interested in the details of the solution in the resonant layer, this can be avoided. The thickness of the resonant layer, δ, is proportional to (ℓL²Re)^−1/3, where ℓ is the length scale over which the Alfvén speed varies around the resonance. In the solar atmospheric context δ can be assumed to be small. The solution in the resonant layer is replaced by a jump relation, which is based upon a Taylor series expansion of the ideal MHD equations around the resonance (Kappraff and Tataronis, 1977; Ionson, 1978; Hollweg, 1987; Hollweg and Yang, 1988; Sakurai et al., 1991a). It is assumed that the quantities that are conserved across the layer in ideal MHD will remain so in weakly dissipative MHD.

Consider a loop model where the internal and external media are homogeneous, except for a loop edge layer of width ℓ where the density varies monotonically from ρ₀ to ρ_e (see Figure 5). A global wave mode is present, which has a frequency ω which matches the Alfvén frequency at C_A(r = r_A)∣k _z ∣ within the Alfvén continuum of the edge layer (see Figure 6). In the internal and external media the solutions are calculated using, e.g., Equation (6). Those solutions contain arbitrary integration constants, which are related to each other with use of certain jump relations at the edge layer. The width of the edge layer, ℓ, is considered to be thin, i.e., ℓ ≪ a, but wider than the resonant layer, i.e., δ > ℓ. The jump relations are constructed as follows. Inside the edge layer, the system of Equations (2) is Taylor expanded around the resonance using the small parameter s ≡ r − r_A (∣s∣ ≪ 1). Thus, a second order differential equation for δP_tot/s is derived, which is of the form

$$\left[ {{{{d^2}} \over {d{s^2}}} + {1 \over s}{d \over {ds}} - \left({{1 \over {{s^2}}} + {{{m^2}} \over {r_{\rm{A}}^2}}} \right)} \right]\left({{{\delta {P_{{\rm{tot}}}}} \over s}} \right) = 0.$$

(9)

It has solutions in the form of modified Bessel functions. The total pressure perturbation is described as

$$\delta {P_{{\rm{tot}}}} = a\;s\;{I_1}\left({{{\vert m\vert s} \over {{r_{\rm{A}}}}}} \right) + b\;s\;{K_1}\left({{{\vert m\vert s} \over {{r_{\rm{A}}}}}} \right) \approx b{{{r_{\rm{A}}}} \over {\vert m\vert}},$$

(10)

which is approximately constant across the layer (Hollweg, 1987; Sakurai et al., 1991a). Since for m=0 no resonant coupling can occur, the Equation (10) does not apply for that case. Therefore, the jump relation for the total pressure perturbation is simply [δP_tot] = 0, where the square brackets denote the difference between the solutions in the right and left limits of s tending to zero, respectively. Note that, when the loop is twisted, the total pressure is no longer a conserved quantity (Sakurai et al., 1991a). Similarly, the solution for the radial displacement perturbation can be found to be approximately

$${\xi _r} \approx b{{\vert m\vert} \over {{r_{\rm{A}}}}}{1 \over {{\rho _0}({r_{\rm{A}}})\Delta}}\ln \left({{{\vert m\vert s} \over {2{r_{\rm{A}}}}}} \right) + {\rm{constant,}}$$

(11)

where \(\Delta=d(\omega^{2}-C_{\rm A}^{2}k_{z}^{2})/dr_{r=r_{\rm A}}\). ξ _r depends on s through a logarithmic term, which diverges for s tending to zero. The jump relation for ξ _r is

$$[{\xi _r}] = - i\pi {{{m^2}} \over {r_{\rm{A}}^2}}{{{\rm{sign}}\omega} \over {{\rho _0}({r_{\rm{A}}})\vert \Delta \vert}}\delta {P_{{\rm{tot}}}}({r_{\rm{A}}}),$$

(12)

for which the relation [ln(s)] = −iπ sign(ω) sign(Δ) has been used. To match the internal and external solutions, the usual jump conditions of continuity of total pressure and radial displacement are now replaced by the above derived jump relations. Depending on the condition that is imposed at r → ∞, the trapped and/or leaky eigenmodes of the system may be studied (involving a dispersion relation) or the reflection/absorption problem of an externally driven wave that interacts with the loop.

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By scanning through the frequency of the collective wave, the wave absorption as a function of ω is studied (see, e.g., the numerical simulations by Poedts et al., 1989). The fractional absorption spectrum often shows well-defined maxima where the absorption reaches 100%. The spatial structure of the excited wave mode at those maxima shows a combination of localised and global behaviour. It is a global wave (e.g., discrete fast eigenmode) with a frequency that lies in the Alfvén continuum and is, therefore, locally coupled to an Alfvén wave. These types of wave modes are known as quasi-modes and they are natural wave modes of the dissipative and inhomogeneous system (Balet et al., 1982; Steinolfson and Davila, 1993; Ofman et al., 1994; Ofman and Davila, 1995; Tirry and Goossens, 1996). Therefore, it is easily understood why maximum absorption occurs when driving at the frequency of a quasi-mode. Also, the presence of steady flows can significantly change the efficiency of resonant absorption (Erdélyi, 1998).

Furthermore, in the absence of flow, these modes are damped (Poedts et al., 1990; Ofman et al., 1994; Wright and Rickard, 1995). From the point of view of the global nature of the mode, the damping is primarily a conversion of energy from the collective to the local. This damping rate has been calculated for various geometries (see, e.g., Lee and Roberts, 1986; Hollweg, 1987; Goossens et al., 1992). Ruderman and Roberts (2002) calculated the damping time, τ, of a global kink wave in a long, thin loop (a ≪ L) in the limits of weak dissipation (Re ≫ 1) and zero plasma-β:

$$\tau = {{2a\vert \Delta \vert {{({\rho _0} + {\rho _{\rm{e}}})}^3}} \over {\pi \rho ({r_{\rm{A}}})\vert {C_{\rm{K}}}{k_z}{\vert ^3}{{({\rho _0} - {\rho _{\rm{e}}})}^2}}}.$$

(13)

This time scale is generally much shorter than the time scale of the dissipative damping of the small-scale perturbations of the local mode in the resonance layer. In Section 3.4 this aspect of rapid mode conversion will be explored further within the context of the observed rapid damping of transverse loop oscillations (Roberts, 2000; Ruderman and Roberts, 2002; Goossens et al., 2002a; Van Doorsselaere et al., 2004).

Ofman et al. (1994) and Ofman and Davila (1995) studied numerically nonlinear resonant absorption and found that the large shear velocities produced at the resonance layer are subject to Kelvin-Helmholtz instabilities. The velocity amplitudes derived from linear theory are much larger than the observed velocities from nonthermal broadening of coronal emission lines. This discrepancy may be explained by a turbulent enhancement of dissipation parameters due to the instabilities. Additional complexity is brought by the effects of boundary conditions in the longitudinal direction, e.g., Beliën et al. (1999) examined numerically the effect of the transition region and chromosphere on the resonant absorption in coronal loops. They found that the nonlinear energy transfer from the Alfvén waves to slow magnetoacoustic waves in the lower atmosphere can much diminish the absorption efficiency compared with models of line-tied loops without a lower atmosphere. The driver they considered was, though, monoperiodic and this study should be extended to include more realistic drivers. Furthermore, the heating of the resonance layer would spread due to thermal conduction and heat the lower atmosphere. The resulting chromospheric evaporation enhances the loop density at the resonance layer and, hence, shifts the Alfvén frequency away from resonance, as well as change the quasi-mode frequencies (see, e.g., the discussion in Ofman and Davila, 1995). Ofman et al. (1998a) considered a broad band Alfvén wave driver (see also DeGroof and Goossens, 2002), and coupling to the chromosphere of the loop density with the use of a quasi-static equilibrium scaling law. They found that the heating is concentrated in multiple resonance layers, rather than in the single layer of previous models, and that these layers drift throughout the loop to heat the entire volume. These properties are in much better agreement with coronal observations that imply multithreaded loop structure.

2.2.2 Alfvén wave phase mixing

Consider again a structure with a continuous inhomogeneity profile across the magnetic field. Instead of an initial collective mode, this time on each field line an Alfvén wave is excited. This wave oscillates independently from its neighbours, with a frequency that lies in the Alfvén continuum. For simplicity a Cartesian geometry is chosen where the magnetic field is in the z-direction and the inhomogeneity is in the x-direction. The Alfvén waves, which are polarised in the y-direction, are described by the wave equation (Heyvaerts and Priest, 1983)

$$\left({{{{\partial ^2}} \over {\partial {t^2}}} - C_{\rm{A}}^2(x){{{\partial ^2}} \over {\partial {z^2}}}} \right){V_y} = 0,$$

(14)

with the solution

$${V_y} = \Psi (x)f(z \mp {C_{\rm{A}}}(x)t),$$

(15)

where f(z) and Ψ(x) are functions prescribed by the initial profile of the wave. The sign in the argument of this function corresponds to a wave propagating in the positive or negative direction of the z-axis, respectively. When Ψ(x) = const, the wave is plane.

Equation (15) shows that the Alfvén waves propagate on different magnetic surfaces, corresponding to different values of x, with different phase speeds equal to the local Alfvén speed C_A(x). If the wave is initially plane in the x direction, it gets gradually inclined. This leads to generation of very small transverse (in the direction of the inhomogeneity) spatial scales. These high transverse gradients, in the presence of finite viscosity or resistivity, which leads to the appearance of a ν∂/∂t ∇²V _y term on the right hand side of Equation (14), are subject to efficient dissipation. Here, ν is the coefficient of viscosity and/or resistivity, small enough so that dissipation may be considered weak. This is the effect of Alfvén wave phase mixing, suggested by Heyvaerts and Priest (1983) as a possible mechanism for heating of open coronal structures.

In the developed stage of phase mixing (when ∂/∂x ≫ ∂/∂z) the Alfvénic perturbations of different magnetic surfaces become uncorrelated with each other, the perturbations decay according to the law (Heyvaerts and Priest, 1983)

$${V_y}(z) \propto {V_y}(0)\exp \left\{{- {{\nu k_z^2} \over {6C_{\rm{A}}^2(x)}}{{\left[ {{{d{C_{\rm{A}}}(x)} \over {dx}}} \right]}^2}{z^3}} \right\},$$

(16)

for propagating harmonic Alfvén waves, and

$${V_y}(t) \propto {V_y}(0)\exp \left\{{- {{\nu k_z^2} \over 6}{{\left[ {{{d{C_{\rm{A}}}(x)} \over {dx}}} \right]}^2}{t^3}} \right\},$$

(17)

for standing harmonic Alfvén waves. The decay time due to phase-mixing is proportional to Re^1/3, similar to the dissipative decay of Alfvén waves due to resonant absorption. Figure 7 shows the dissipative decay of propagating Alfvén waves in a 1D plasma inhomogeneity with Alfvén speed profiles of different steepness, compared with the dissipative decay of Alfvén waves in a homogeneous medium (solid curve).

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If instead of a monochromatic wave in the longitudinal direction a localised Alfvén pulse is considered, Heyvaerts and Priest’s expression for the exponential decay (17) should be replaced by the power law

$${V_y} \propto {\left[ {\nu {{\left({{{d\;{C_{\rm{A}}}(x)} \over {dx}}} \right)}^2}{t^3}} \right]^{- 1/2}};$$

(18)

see Hood et al. (2002) and Tsiklauri et al. (2003) for two methods of derivation and comparison with results of numerical modelling.

In the nonlinear regime, growing transverse gradients induce oblique fast magnetoacoustic waves (Nakariakov et al., 1997). Consequently, phase mixing regions may be characterised by the presence of compressible perturbations.

2.3 Zero plasma-β density profiles

The limiting case of zero plasma-β (where β is the ratio of the gas pressure to the magnetic pressure), when the plasma pressure force is neglected in comparison with the Lorentz force, and there is a one-dimensional profile of the plasma density, embedded in the constant and parallel magnetic field, the eigenvalue problem is similar to the quantum mechanical problem of the interaction of a particle with a nonuniform potential. In this section we restrict our attention to the Cartesian geometry only. The governing equations, in this case, allow for exact analytical solutions. We consider wave propagation along the magnetic field (i.e., k _y = 0) to avoid the Alfvén resonance. Nakariakov and Roberts (1995b) established that the qualitative dispersive properties depend weakly upon the specific profile of the density. Consider a coronal loop as a magnetic slab with a smooth density profile, given by the profile function

$${\rho _0} = {\rho _{\max}}{\rm{sec}}{{\rm{h}}^{\rm{2}}}\left({{x \over a}} \right) + {\rho _\infty},$$

(19)

where ρ_max, ρ_∞ and a are constant. Here, the parameter ρ_max is the density at the centre of the inhomogeneity, ρ_∞ is the density at x = ∞ and a is a parameter governing the inhomogeneity width. This inhomogeneity, plotted in Figure 8, is called the symmetric Epstein profile (see, e.g., Nakariakov and Roberts, 1995b). With an inhomogeneity of this form exact analytical solutions can be obtained. The plasma is inhomogeneous across the straight and uniform magnetic field B₀ = B₀ẑ. In the zero plasma-β limit considered, the equilibrium total pressure balance is identically fulfilled.

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Dispersion relations (23) and (24) and solutions (21) are a convenient tool for the study of the effect of the transverse profile on wave properties. In particular, Figure 9 demonstrates that the sausage mode group speed is affected by the steepness of the profile quite significantly. One of the observational manifestations of this effect is the shape of the fast wave trains formed by the dispersive evolution of initially broad band perturbations. Applications of this theory are discussed in Section 7.

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2.4 Effects of twisting

It is clear that twisting of the magnetic field in the cylinder will lead to linear coupling of various MHD modes. However, modification of dispersion relation (7) by the twisting is not well understood.

Bennett et al. (1999) considered the collective MHD modes of a straight uniformly twisted magnetic cylinder in the incompressible limit, and concluded that the twist leads to the appearance of body modes with phase speeds about the “longitudinal” Alfvén speed (calculated with the use of the longitudinal component of the field only). As well as in the untwisted case, there is a surface mode with the phase speed about the kink speed.

An alternative approach to the modelling of twisted and curved coronal loops was suggested by Cargill et al. (1994), which allowed the authors to take into account the hoop force — the feature missing from the straight cylinder model. The force is connected with both the loop twist and the curvature. The presence of the new restoring force was shown to give rise to a new oscillation mode manifested as the periodic change of the loop major radius and the loop density, whose frequency could be independent of the loop length. Oscillations of the loop minor radius were also found. The oscillation frequencies obtained were significantly different from the frequencies of straight cylinder eigenmodes. This approach certainly requires attention and further development.

Also, oscillations of current carrying loops can be described in terms of the LCR-model, see Section 3.5.

3.1 TRACE observations

On 14 July 1998, the imaging telescope on board the Transition Region and Coronal Explorer (TRACE) registered, in both 171 Å and 195 Å lines, spatially resolved decaying oscillating displacements of coronal loops in the active region AR 8270 (Aschwanden et al., 1999; Nakariakov et al., 1999; Schrijver et al., 1999). These oscillations happened shortly after a solar flare and, most probably, were generated by the flare. The mechanism of the excitation remains hidden, but it can be connected with a blast wave generated in the flare epicentre. Some of the loops seem to be more responsive to the oscillation than others and it could probably be connected with the magnetic topology of the active regions (Schrijver and Brown, 2000). Oscillations of different loops were not synchronised in phase. The highest amplitude was seen near the loop apices.

Since this discovery, the loop oscillations were subject to an extensive observational study and the results are summarised in Schrijver et al. (2002) and Aschwanden et al. (2002). In particular, it was found that the kink oscillations do not always have a simple form of a global (or principle) mode and there can be higher spatial harmonics observed.

Anyway, the oscillation examined by Nakariakov et al. (1999) may be considered as a typical example of kink oscillations of coronal loops. The analysis of the loop displacement shows that the oscillation is almost harmonic with the period of about P = 256 s (the frequency about 4 mHz). Figure 10 shows the temporal evolution of the displacement at the loop apex. About three periods of oscillation were observed. Displacement amplitudes are several Mm for the distance between the loop footpoints estimated to be about 2L/π = 83 Mm. The displacement amplitude is several times larger than the loop cross-section radius, which was observed to be about 2a = 1 Mm. The oscillation shows evidence of strong damping. Simultaneously, similar quasi-periodic oscillations were observed in several other loops at the distance of several Mm to 60–70 Mm from the flare epicentre (Aschwanden et al., 1999). All these observational findings suggested the oscillations, at least observed in this event, to be interpreted as a kink global standing mode of the loop.

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According to the theory of MHD modes of a magnetic cylinder, discussed in Section 2, the fast kink magnetoacoustic modes of a magnetic cylinder do not have dispersive cut-offs and exist for all wavenumbers. In all cases, the wavelength of the observed kink oscillations is much longer than the loop cross-section diameter (e.g., the width of the oscillating loop observed on 14 July 1998 is about 1 Mm, while the loop length may be estimated as 261 Mm for the distance between the footpoints of about 83 Mm; Nakariakov and Ofman, 2001). In this limit, the phase speed of fast kink modes waves approaches the kink speed C_K given by Equation (8).

Terradas and Ofman (2004) pointed out intensity variations localised at the tops of some large-amplitude oscillating loops observed with TRACE in the flaring event on 14 July 1998. As no noticeable changes of the plasma temperature were found at those regions, the intensity variations were interpreted as density variations, approximately in the range 14%–52%. The amplitude could possibly be even higher if the filling factor was less than 1. In the analysed loop, the projected maximum amplitude of the oscillations was about 90 km s⁻¹. In the loops oscillating with smaller amplitudes this effect is not detected, indicating its nonlinear nature.

Another possible polarisation of kink oscillations is when the loop moves in the same plane and its apex displaces horizontally. This mode has not been identified yet.

3.2 Non-TRACE observations

Kink modes of coronal loops — often erroneously called Alfvén waves (erroneously, because the true Alfvén waves guided by coronal loops are the torsional modes, see Section 8) — can also be observed through modulation of the broadband gyrosynchrotron emission by the periodic variation of the local magnetic field in flaring loops and through periodic variations of the Doppler shift.

However, confident identification of the kink mode requires observations with high spatial resolution. The pixel size should be smaller than the wave length of the mode. The first spatially resolved detection of microwave quasi-periodic pulsations, which could be associated with the fast kink mode, was performed by Asai et al. (2001) with the Nobeyama Radioheliograph (see Figure 11). The oscillation period was 6. 6 s. They determined the number density in the loop to be 4.5 × 10¹⁶ m⁻³ from filter ratios of soft X-ray images taken by SXT. The loop length was 16 Mm. The microwave pulsations had a less weaker modulated counterpart in hard X-ray emission observed by the HXT on Yohkoh. The modulation of the hard X-ray emission by the kink wave can be connected with modulation of the electron acceleration by the kink oscillation of the flaring loop, or with interaction of the flaring loop with another loop which performs kink oscillations. In both cases the reconnecting magnetic field is periodically fed to the reconnection site by the kink oscillation. As the thickness of the reconnection site is believed to be very small, even weak kink oscillations can produce the required modulation.

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When the line of sight has a significant component parallel to the plane of the oscillations, the kink mode can be detected with spectral instruments through the periodically modulated Doppler shift. For example, 300 s, 80 s, and 43 s periodicities were found by Koutchmy et al. (1983) in the Doppler shift of the green coronal line. In this event, no prominent intensity variations were observed. These oscillations were interpreted as standing kink waves by Roberts et al. (1984).

3.3 Determination of coronal magnetic fields

A practical formula for the magnetic field determination by the observables is

$${B_0} \approx 1.02 \times {10^{- 12}}{{d\sqrt {\mu {n_0}} \sqrt {1 + {n_{\rm{e}}}/{n_0}}} \over P},$$

(32)

where the magnetic field B₀ is in G, the distance between the footpoints d is in m, the number density in the loop n₀ is in m⁻³, and the oscillation period P is in s; μ is the effective particle mass with respect to the proton mass. In the solar corona, because of the presence of heavier elements, μ = 1.27. Applying this formula, Nakariakov and Ofman (2001) estimated the magnetic field in an oscillating loop observed on the 14 July 1998, as 13 ± 9 G (see Figure 12, where the number density is measured in cm⁻3). This error bar can be significantly reduced by improving the determination of the density in the loop and by better statistics.

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A similar estimation for the field strength (about 15 G) was obtained by Roberts et al. (1984) from the observations of Koutchmy et al. (1983) discussed in Section 3.2. However, in contrast with the TRACE observations, the lack of the direct observability of the oscillating loop did not make the interpretation of the oscillations in terms of the kink modes absolutely secure.

Asai et al. (2001) observed microwave quasi-periodic pulsations with a periodicity of 6.6 s, which are associated with a global kink oscillation. Using Equation (32) and assuming n_e/n₀ = 0. 1, we find the loop to have a magnetic field strength of 400 G. This value is consistent with a magnetic field extrapolation (see Asai et al., 2001, which found a magnetic field strength of 300 G). For an alternative interpretation of this observation in terms of the global fast sausage mode, see Section 4.

3.4 Decay of the oscillations

In addition, Equation (34) may give unrealistically short decay times comparable with the oscillation period. It is not clear how the phenomenon of resonant absorption can happen in those cases when the decay time is comparable with the oscillation period and, consequently, the oscillation is not harmonic, making the resonance impossible. In any case, resonant absorption remains a plausible interpretation of the decay.

A different approach to the problem is based on the leakage of wave energy from the loop. Cally (1986, 2003) showed that the fast kink mode is almost identical to a fast leaky kink mode. In the long wavelength limit both modes propagate at the kink speed. In the limit of the plasma-β tending to zero, the amplitude of the leaky kink mode decreases as it radiates into the coronal environment with a decay time (Cally, 2003, note a factor 2 error in his expression)

$${\tau \over P} = {2 \over {{\pi ^4}}}{\left({{a \over L}} \right)^{- 2}}{\left({{{{\rho _0} + {\rho _e}} \over {{\rho _0} - {\rho _e}}}} \right)^2},$$

(36)

which, unless the loop width 2a is considerably larger than the observed loop emission width, is too long to explain the observed decay, especially for large loops (see Figure 13). For example for a loop of length 200 Mm, a needs to be in the range of 10–20 Mm.

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Ofman and Aschwanden (2002) suggested that observationally determined scaling laws, connecting the decay times with oscillation periods and lengths of oscillating loops, may provide some information allowing us to distinguish between the interpretations discussed above. Indeed different decay mechanisms give different scaling laws: coronal leakage (τ ∝ L²P), footpoint leakage (τ ∝ LP), resonant absorption (τ ∝ P), and phase-mixing (τ ∝ P^2/3). By using the fact that the period is proportional to the loop length, one can be eliminated in favour of the other.

Figure 13 shows the observationally determined decay times as a function of wavelength and period and uses measurements from Nakariakov et al. (1999), Aschwanden et al. (2002), Wang and Solanki (2004), and Verwichte et al. (2004), which adds two more measurements compared with the study of Ofman and Aschwanden (2002). We have to bear in mind, though, that the measurement from Wang and Solanki (2004) refers to a vertically polarised oscillation and that the measurement from Verwichte et al. (2004) is an averaged value from seven damped kink oscillations in a prominence driven loop arcade. The data points are fitted by power laws which have the dependencies τ ∝ λ^0.70±0.32 and τ ∝ P^1.12±0.36. The dependence on the wavelength seems to favour phase-mixing, without excluding resonant absorption, and the dependence on period seems to favour resonant absorption but is also consistent with the modified scaling law for phase-mixing. In any case, more examples are needed as the data are too scattered.

Verwichte et al. (2004) studied TRACE observations of 15 April 2001 of kink oscillations in a post-flare loop arcade. The loops of this arcade were made to oscillate by the actions of a nearby prominence eruption. The oscillation signatures of nine loops were determined as a function of distance along each loop. They found that the displacement amplitude decreases with distance from the loop top as expected for a fundamental mode. However, two of the loops showed two, simultaneous oscillation modes with the longest period, roughly twice the shortest period. Also, the displacement amplitude of the shortest period oscillation increases with distance from the loop top, which indicates that it is an harmonic mode. The measured decay times of the loop oscillations shows the same dependence on the oscillation period as for previous observations, but with a clear bias towards longer decay times. There are several possible explanations for this bias. Firstly, this study considered an oscillating arcade of post-flare loops, while previous observations are concerned with more isolated active region loops. The decay times may be influenced by the different structuring of post-flare loops or by interactions between neighbouring loops in the arcade. Secondly, the arcade oscillation is driven by a nearby prominence eruption that may have given multiple impulses to the arcade. If a time of 5–10 min is subtracted from all the decay times, then they fall in the range of decay times of the earlier studies. The relationship between the decay time and the period, though, would then be much steeper.

Further development of this discussion on the decay of kink oscillations is connected with the improvement of the observational statistics and high-resolution 3D numerical modelling of oscillating loops. In particular, an important issue is whether the loop cross-section remains unperturbed during kink oscillations. Also, the question of the excitation of these modes, connected with the observational fact that the kink oscillations are a rather rare phenomenon, is still open.

3.5 Alternative mechanisms

The idea that a coronal loop is twisted and then carries an electric current gave rise to several alternative mechanisms for the loop oscillations.

Khodachenko et al. (2003) applied the idea of inductive interaction of electric currents in a group of neighbouring loops to an alternative interpretation of kink oscillations, suggesting that they are caused by the ponderomotoric interaction of currents in groups of inductively coupled current-carrying loops. More specifically, the ponderomotoric interaction of current-carrying magnetic loops can lead to the oscillatory change of the loops inclination. The efficiency of coupling, the period of oscillations and the decay time are connected with mutual inductance of different loops in the active region analysed. Also, it was pointed out that the interaction of the oscillating loop with neighbouring loops can lead to strong damping of the oscillations.

We would like to stress that the periods of loop oscillations described by LCR-contour models should be longer than the Alfvén travel time along the loop, which determines the response time of the current system, so P > L/C_A0.

5.1 Global acoustic mode

The SOHO/SUMER spectral instrument has recently discovered quasi-periodic oscillations of intensity and a Doppler shift in the coronal emission lines Fe xix and Fe xxi (Kliem et al., 2002; Wang et al., 2002, 2003b,a, and references therein). Figure 14 gives an illustration of such oscillations. These spectral lines are associated with a temperature of about 6 MK, corresponding to the sound speed of about 370 km s⁻¹. The observed periods are in the range 7–31 min, with decay times 5.7–36.8 min, and show an initial large Doppler shift pulse with peak velocities up to 200 km s⁻¹. The intensity fluctuation lags the Doppler shifts by 1/4 period. In a statistical study of 54 oscillation cases, Wang et al. (2003a) found that except for a few cases, the presence of definite periodicity in intensity fluctuations is not certain. Moreover, oscillations are not seen in other emission lines observed simultaneously with the oscillating lines. In the initial stage of all 54 cases analysed by Wang et al. (2003a), there is a rapid increase in the line intensity and a large Doppler shift, indicating that the oscillations are excited impulsively.

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There are still several open questions in both the theory and the observations: how are the oscillations triggered and excited; why are intensity oscillations not always seen, whether the occurrence rate of oscillations is temperature dependent (in major cases in Fe xix, i.e., in hot plasma), what is the role of non-adiabatic effects (e.g., thermal instability)?

Also, it is not clear how the SUMER oscillations are related with other coronal oscillations, observed in the radio (e.g., Aschwanden, 2003) and X-ray bands.

5.2 Second standing harmonics

Recently, Nakariakov et al. (2004b) modelled the evolution of a coronal loop in response to an impulsive energy release and demonstrated that the second standing acoustic harmonics appears as a natural response of the loop to an impulsive energy deposition. Modelling the loop as a 1D hydrodynamic system with nonlinearity, radiative damping, thermal conduction and accounting for possible chromospheric up and down flows, it was demonstrated that the second harmonics is a common feature of the loop evolution. Figure 15 shows typical time curves of the density and temperature at the loop apex. The quasi-periodic behaviour is clearly seen in the apex density curve, which is consistent with the mode structure

$${V_x}(s,t) = A\cos \left({{{2\pi {C_{\rm{s}}}} \over L}t} \right)\sin \left({{{2\pi} \over L}s} \right),$$

(50)

$$\rho (s,t) = - {{A{\rho _0}} \over {{C_{\rm{s}}}}}\left({{{2\pi {C_{\rm{s}}}} \over L}t} \right)\cos \left({{{2\pi} \over L}s} \right),$$

(51)

where A is the wave amplitude. (The paper of Nakariakov et al., 2004b, contains a misprint, the factor of two is missing from Equations (3) and (4)). The density perturbations have a maximum near the loop apex, while longitudinal velocity perturbations have there a node.

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The second standing acoustic mode may be responsible for quasi-periodic pulsation with periods in the range 10–300 s which are often observed in flare light curves in radio and X-ray bands. The SUMER oscillations mentioned above are likely to be associated with some other excitation mechanism, as only a small fraction of SUMER oscillations are observed in association with solar flares (Wang et al., 2003a). Traditionally, the acoustic wave interpretation has been excluded as the waves of these periodicities were supposed to be highly dissipative in the hot plasma of flaring loops. However, the numerical simulations performed by Nakariakov et al. (2004b), as well as the recently gained abundant observational evidence of the presence of acoustic waves in the solar corona, suggest that the observed periodicities can be associated with this mode. The decayless character of these oscillations may be explained in terms of auto-oscillations: By the competition of the oscillation energy losing by dissipation and the energy deposition to the oscillation, e.g., through thermal over-stability.

6.1 Observational results

One of the most wide-spread examples of the coronal wave activity is the slow propagating intensity disturbances observed with imaging telescopes in both open and closed coronal magnetic structures. The standard detection technique is the use of the stroboscopic method: the emission intensities along a chosen path, taken in different instants of time are laid side-by-side to form a time-distance map. Diagonal stripes on these maps exhibit disturbances which change their position in time and, consequently, propagate along the path. This method allows the determination of periods (or wavelengths), relative amplitudes and projected propagation speeds. Usually these waves are observed propagating along the assumed coronal magnetic structures and, thus, along the magnetic field. Their speeds are usually much slower than the expected coronal Alfvén speed which leads to their interpretation as longitudinally propagating slow magnetoacoustic (aka acoustic or longitudinal) waves.

The first observational detection of longitudinal waves came from analysing the polarised brightness (density) fluctuations. The fluctuations with periods of about 9 min, were detected in coronal holes at a height of about 1.9 R_⊙ by Ofman et al. (1997, 1998b) using the white light channel of SOHO/UVCS. Developing this study and analysing several other UVSC data sequences with a 75–125 s cadence time, Ofman et al. (2000b) determined the fluctuation periods in the range of 7–10 min. The propagation speeds of the fluctuations indicated values in the range of 160–260 km s⁻¹ at 2 R_⊙. This value of the speed is slightly below the estimated value of the sound speed. Possibly, a similar phenomenon was observed by Marsh et al. (2002), who detected quasi-periodic variability of coronal EUV emission lines covering a temperature range of log T_e = 5.3–6.1 K with SOHO/CDS. Statistically significant periods were found within the range 100–900 s and 1500 s, also short wavepackets with periods of the order 50–100 s with durations of 2–5 cycles were reported.

DeForest and Gurman (1998), using EIT 171 Å data confirmed this discovery: Outwardly propagating perturbations of the intensity were observed at distances of 1.01–1.2 R_⊙, gathered in quasi-periodic groups of 3–10 periods, with periods of about 10–15 min, The projected speeds are about 75–150 km s⁻¹ and the relative amplitude (in density) was about 2–4 %.

A similar phenomenon was observed near the footpoints of coronal loops, with EIT (Berghmans and Clette, 1999) and TRACE (Nightingale et al., 1999; De Moortel et al., 2000, 2002a,b) imagers as near-isothermal EUV intensity disturbances, which start near the loop footpoints and propagate along the loops at the apparent speed lower than the sound speed. Multi-wavelength observations have been performed by Robbrecht et al. (2001) combining TRACE 171 Å and EIT 195 Å and by King et al. (2003) with TRACE 171 Å and 195 Å. Robbrecht et al. (2001) found the projected propagation speeds to vary roughly between 65 and 150 km s⁻¹ for both instruments, which is close to and below the expected sound speed in the coronal loops, respectively. King et al. (2003) pointed out the high correlation of the disturbances observed in the different bandpasses. Also, recently, a coordinated observation of this phenomenon with SOHO/CDS and TRACE instruments has been carried out Marsh et al. (2003). A propagating oscillation with a period of about 300 s, observed by TRACE in the 171 Å bandpass, was also observed in He i, O v and Mg ix emission lines with CDS, corresponding to the chromospheric, transition region, and coronal temperatures, respectively. This is consistent with about 5 min oscillations observed by O’Shea et al. (2001) with CDS in both velocity and intensity time series associated with the coronal lines Mg ix and Fe xvi, as well as in O v.

A comprehensive overview of observational properties of the longitudinal oscillations, based upon the analysis of 38 examples, is given in De Moortel et al. (2002a), and a more recent one in Nakariakov (2003). The properties of propagating EUV disturbances may be summarised as follows: the projected propagation speed is 35–165 km s⁻¹; the amplitudes are always less than 10 % in intensity (less than 5 % in density); the disturbances are quasi-periodic with the periods about 140–420 s. In most cases, only upwards propagating disturbances have been detected (from the footpoints to the apex of the loop). Sometimes, the waves can be present for several consecutive hours with, more or less, constant period. It is possible that the disturbances with shorter, about 3 min, periods are situated above sunspot regions, whereas disturbances propagating along the loops which are not associated with sunspots have longer periodicity, of about 5 min (De Moortel et al., 2002c). However, King et al. (2003) showed that both 3 min and 5 min perturbations can coexist in the same coronal structure, at least in the analysed example, so the question still remains open.

The propagation direction and speed, together with the fact that the observed waves are compressible suggest their interpretation as slow magnetoacoustic waves. Slow waves of the observed periodicities (shorter than 20 min) can propagate without reflection in the 1.0–2.0 MK corona, as the acoustic cut-off period is about 70 min. According to this interpretation, the waves propagate at about the sound speed in the loop. The observed speed of the waves is reduced by line-of-sight projection.

Waves were also recently detected in the Doppler velocity data in the 1–3 mHz and 5–7 mHz ranges by Sakurai et al. (2002). The line intensity and line width did not show clear oscillations, but their phase relationship with the Doppler velocity indicates propagating waves rather than standing waves. These waves were interpreted as superposition of propagating slow and Alfvén waves.

6.2 Theoretical modelling

Let us illustrate the derivation of Equation (52) in a specific situation. Following Tsiklauri and Nakariakov (2001), we consider a semi-circular loop of the curvature radius R _L , with the inclination angle α (measured from the normal to the solar surface) and a non-zero offset of circular loop centre from the coronal base line, Z₀ (i.e., distance from the circle’s centre to the solar surface). The loop cross-section is taken to be constant. The loop is filled with a gravitationally stratified magnetised plasma of constant temperature. The sketch of the model is shown in Figure 16.

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Solutions of Equation (52) are in a satisfactory agreement with the observed evolution of the wave amplitude: For example, Figure 17 demonstrates the comparison of the observed and theoretically predicted growth of the longitudinal wave amplitude with height. Also, full MHD 2D numerical modelling of these waves gives similar results (see Ofman et al., 1999, 2000a). The comparison of the observed and modelled evolution scenarios give the possibility of estimating, in particular, the efficiency of non-adiabatic processes, such as the competition of heating and radiative losses.

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The energy carried and deposited by the observed waves is certainly insufficient for heating of the loop. However, Tsiklauri and Nakariakov (2001) have shown that sufficiently broadband slow magnetoacoustic waves, consistent with currently available observations in the low frequency part of the spectrum, can provide the rate of heat deposition sufficient to heat the loop. In this scenario, the heat would be deposited near the loop footpoints which agrees with observationally determined positioning of the heat source.

6.3 Propagating slow waves as a tool for coronal seismology

According to the previous sections, slow magnetoacoustic waves follow magnetic field lines and propagate at the local sound speed. Because of the high coronal thermal conduction along the field, the equilibrium temperature does not experience significant changes in this direction, at least at the distances comparable with the detection distance of the slow waves. As the sound speed is proportional to the temperature squared, the propagation speed should be constant along the structure, and the speed measured observationally would contain information about the temperature and about the value of the adiabatic index γ. However, such a measurement is impossible at the moment, because the observations with the single line of sight provide us with the apparent speed only, which is affected by the projection effect. Possibly, future 3D observations of the propagating slow waves with STEREO will make such an estimation possible.

Another interesting possibility arises when the waves are observed simultaneously in different bandpasses which correspond to different temperatures of the emitting plasma. First such observation was done by Robbrecht et al. (2001) with SOHO/EIT 195 Å filter and TRACE 171 Å filter. It was found that wave propagating along the same coronal structure was detected in both bandpasses. The big difference in spatial resolution of EIT and TRACE imagers (2.5″ vs 0.5″) did not allow to study the correlation in detail. Later on, King et al. (2003) analysed slow waves observed in 171 Å and 195 Å with TRACE, confirming the correlation (see Figure 18). Figure 19 shows the evolution of the correlation coefficient of the perturbations observed in the different bandpasses with distance along the loop. There is a systematic decrease of the correlation coefficient which may be caused by the different propagating speed of the disturbances observed in different bandpasses. This may be interpreted as an indication to the sub-resolution structuring of the analysed active region: The observed loop can be either a bundle of magnetic threads of varying temperature, or have a transverse temperature profile. The further development of this study seems to be promising.

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7.1 Propagating fast waves in coronal loops

Wave lengths of propagating waves should be much shorter than the size of the structure guiding the wave. The shorter wave lengths correspond to shorter periods of the waves, requiring high cadence observational tools. In particular, the typical cadence time of TRACE and EIT EUV imagers is 20–30 s, which gives the minimal period of detected waves to be about 2–3 min. For a typical coronal Alfvén speed of about 1000 km s⁻¹, at least, this corresponds to wavelengths longer than 120–180 Mm, which is comparable with the typical size of a coronal loop. Thus, the EUV imagers cannot normally be used for the detection of propagating fast waves guided by coronal loops. On the other hand, time resolution of a second or better can be achieved with ground based coronographs or radioheliographs.

Recently, Williams et al. (2001, 2002) and Katsiyannis et al. (2003) reported the observational discovery of rapidly propagating compressible wave trains in coronal loops with the SECIS instrument during a full solar eclipse. As the observed speed was estimated at about 2100 km s⁻¹, these waves were interpreted as fast magnetoacoustic modes. The waves were observed to have a quasi-periodic pattern with a mean period of about 6 s. Cooper et al. (2003) found an encouraging agreement between the observed evolution of the wave amplitude along the loop with the theoretical prediction made for kink and sausage modes (see Figure 20). Unfortunately, the high uncertainty in the measurement of the amplitude does not allow the authors to distinguish between the kink and sausage modes in this observation.

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The main difference of these works from a number of previous papers analysing eclipse observations is the application of the methods of the time-distance (stroboscopic) plot (Williams et al., 2001, 2002) and of the wavelet transform (Katsiyannis et al., 2003). Those techniques are certainly more preferable in the situation when the waves have modulation or form wave trains. According to dispersion relation (7), fast waves are highly dispersive when their wave length is comparable or longer than the radius of the loop. It is well known that, in a dispersive medium, impulsively generated (or broadband because of another reason) waves evolve into a quasi-periodic wave train with a pronounced period modulation. In the coronal context, it was pointed out by Edwin and Roberts (1983) and Roberts et al. (1983, 1984) that periodicity of fast magnetoacoustic modes in coronal loops is not necessarily connected with the wave source, but can be created by the dispersive evolution of an impulsively generated signal. Studying the dispersive evolution, Roberts et al. (1984) analytically predicted that the development of the propagating sausage pulse forms a characteristic quasi-periodic wave train with three distinct phases. Such evolution scenario is determined by the presence of minimum in the group speed dependence upon the wave number. That analysis was restricted to the case of the slab with sharp boundaries and to sausage modes only. The initial stage of the pulse evolution was numerically modelled by Murawski and Roberts (1993, 1994), Murawski et al. (1998), and was found to be consistent with the analytical prediction.

Nakariakov et al. (2004a) numerically simulated the developed stage of the dispersive evolution of a fast wave train in a smooth straight slab of a low plasma-β plasma. It was found that development of an impulsively generated pulse leads to formation of a quasi-periodic wave train with the mean wavelength comparable with the slab width. In agreement with the analytical theory (Roberts et al., 1984), the wave train has a pronounced period modulation which was demonstrated with the wavelet transform technique (see Figure 21). In particular, it is found that the dispersive evolution of fast wave trains leads to the appearance of characteristic “tadpole” wavelet signatures. Similar signatures were found in the wavelet analysis of SECIS data (Katsiyannis et al., 2003), strengthening the interpretation of the SECIS waves as the fast magnetoacoustic wave trains.

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This mechanism may also be responsible for the formation of quasi-periodic pulsations with the periods of about a second, observed in association with flares. However, the main property of this effect, which must be taken into account in interpretation, is the period modulation.

7.2 Propagating fast kink waves in open structures

Recently, Verwichte et al. (2005) presented for the first time a direct observational study of propagating fast waves in an open magnetic structure, i.e., observations of a long duration flaring event using TRACE 195 Å (a movie of this event can be seen in Resource 2). Because of the presence of an Fe xxiv emission line in the instrumental bandpass, which is sensitive to temperatures of around 20 MK, the hot supra-arcade above the post-flare loop arcade was also visible. The supra-arcade is an open magnetic structure containing plume-like rays. In the particular event analysed, dark, tadpole-like structures appeared in the lanes between the rays. They were density depletions that moved sunwards, decelerating from speeds above 500 km s⁻¹ to less than 100 km s⁻¹. We shall adopt the name tadpole to describe such structures.

Tadpoles were first reported by McKenzie and Hudson (1999) using the Yohkoh Soft X-ray Telescope. Their physical nature is not fully understood and various models have been put forward. Here we shall focus on one particular behavioural feature of tadpoles, namely, after the passage of the tadpole head, the boundaries of the tail regions with the neighbouring rays oscillate transversely. In other words, one can say that the tadpole wiggles its tail. Figure 22 shows slices of the data set analysed by Verwichte et al. (2005). The transverse motions of the tadpole-ray boundaries are clearly visible as wave packets of 3–4 oscillation periods, following the tadpole heads. From this data cube the transverse displacement of four edges, which each may contain multiple tadpole events, have been extracted. Figure 23 illustrates the typical characteristics of these wave packets. A wave packet propagates sunwards, decelerating with phase speeds in the ranges 200–700 km s⁻¹ and 90–200 km s⁻¹ at heights above the post-flare loop footpoints of 90 and 60 Mm, respectively. Simultaneously, the displacement amplitude, which is of the order of several Mm, decreases. The nature of this decrease is not obvious. Besides dissipation, vertical structuring of the supra-arcade may be responsible. The wave periods lie in the range of 90–220 s. The equivalent wavelengths are of the order of 20–40 Mm, which is much shorter than for the standing, fast kink oscillations in coronal loops.

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Because of their clear transverse signatures, the tadpole waves are interpreted as fast magnetoacoustic kink waves. They are guided by the vertical, ray-tadpole structure. An MHD model with slab geometry may be used to characterise the structure (Roberts, 1981b). A slab is preferred over a cylindrical model because little is known about the extent of the structure along the line of sight. Because the density is lower in the tadpole than in the ray, the observed waves may be surface modes. Fast surface modes have a phase speed below the minimal Alfvén speed, i.e., the ray Alfvén speed. If we assume that this speed lies plausibly in the range of 500–1000 km s⁻¹, then there is quite a difference with the observed phase speeds, especially at lower heights. This discrepancy may be explained by considering the geometric configuration of the angles between wave propagation, tadpole and ray magnetic field directions and/or by the presence of upflows, possibly connected with the strong Doppler blue shifts reported by SUMER (Innes et al., 2003). In the last case, these waves may be generated by negative energy mechanisms (see, e.g., Ruderman et al., 1996; Joarder et al., 1997), which lead to amplification of waves by nonuniform steady flows.

The period (phase speed) of consecutive tadpole wave packets passing the same neighbouring ray show a linear increase (decrease) with time. This trend can be explained by a density increase in the ray. Using the simple assumptions \(I-I_{0}\sim n_{\rm e}^{2}\sim V_{{\rm A},{\rm ray}}^{-4}\) and V_ph ∼ V_A,ray, where V_A,ray is the Alfvén speed in the ray, we find the relations I − I₀ ∼ P⁴ and \(I-I_{0}\sim V_{\rm ph}^{-4}\). I₀ is an arbitrarily chosen constant background intensity. A fit to the observed intensity profile yields the dependencies I − I₀ ∼ P^2.8±2.0 and \(I-I_{0}\sim V_{\rm ph}^{-3.0\pm 1.5}\). The difference with the theoretical power law indices lies within the error. Theoretical modelling of excitation and propagation of kink waves in supra-arcades has not been fulfilled.