Figure 1:
A scheme of the method of MHD coronal seismology. 

Figure 2:
A magnetic flux tube of radius embedded in a magnetised plasma. 

Figure 3:
Dispersion diagram showing the real phase speed solutions of dispersion relation (7) for MHD waves in a magnetic cylinder as a function of the dimensionless parameter . The typical speeds in the internal and external media are shown relative to the internal sound speed: , , and . The solid, dotted, dashed and dashdotted curves correspond to solutions with the azimuthal wave number equal to , , and , respectively. The torsional Alfvén wave mode solution is shown as a solid line at . 

Figure 4:
Transverse and longitudinal density perturbation (shown as intensity) and velocity structure (shown as a vector field) of a homogeneous magnetic cylinder model of a coronal loop, perturbed by a fundamental fast kink oscillation (from Resource 1). 

Figure 5:
A magnetic flux tube of radius embedded in a magnetised plasma with a thin edge layer of width where the density varies monotonically. 

Figure 6:
Radial profile of the Alfvén frequency in a loop with a thin edge layer of width where the density varies monotonically. A resonance occurs where the global wave mode frequency matches locally the Alfvén frequency inside the edge layer. 

Figure 7:
Comparison of the dissipative decay of Alfvén waves in 1D plasma inhomogeneities with Alfvén speed profiles of different steepness. The solid curve shows the dissipative decay of Alfvén waves in an homogeneous plasma. Phasemixing is therefore absent and the damping time is proportional to . The other curves show the dissipative decay in a 1D plasma inhomogeneity with a nonzero gradient in the Alfvén speed. Phasemixing, therefore, occurs and the damping times are proportional to . 

Figure 8:
Symmetric Epstein density profiles of the form with the solid, dashed, dotted and dashdotted curves corresponding to values of , , , and (effectively ). The case of corresponds to Equation (19). 

Figure 9:
Comparison of group speeds of a propagating sausage mode guided by a 1D plasma inhomogeneity with the Alfvén speed contrast ratio of , with the step function profile (the dashed line), and with the symmetric Epstein profile (the solid line). The group speed is normalised to the minimum value of the Alfvén speed, which is equal to the Alfvén speed inside the slab with the step function profile, represented by the horizontal solid line. 

Figure 10:
The temporal evolution of the loop displacement as an average coordinate of the loop position for four neighbouring, perpendicular cuts through the loop apex (diamonds), with error bars ( pixels), starting at 13:13:51 UT on 14 July 1998. The solid curve is a best fit of the function with , , and , corresponding to a period and efolding decay time of and , respectively (from Nakariakov et al., 1999). 

Figure 11:
Light curves of the second burst (scaled arbitrarily). From top to bottom: Radio brightness temperature observed at by NoRH (solid line) and hard Xray count rate measured in the M2 band (; dashdotted line) and M1 band (; dotted line) of Yohkoh/HXT. The vertical lines show the peak times of the microwave emission (from Asai et al., 2001). 

Figure 12:
The magnetic field inside a coronal loop as function of plasma density inside the loop, determined by Equation (32). The external to internal density ratio is 0.1. The solid curve corresponds to the central value of the kink speed (for the event of the 4 July 1999), and the dashed curves correspond to the upper and the lower possible values of the speed. The vertical dotted lines give the limits of the loop density estimation using TRACE 171 Å and 195 Å images. The distance between the loop footpoints is estimated as (from Nakariakov and Ofman, 2001). 

Figure 13:
Observationally determined loop oscillation decay times as a function of wavelength (left) and period (right) from: Nakariakov et al. (1999) and Aschwanden et al. (2002) (), Wang and Solanki (2004) (), and Verwichte et al. (2004) (). The solid lines are a best fit to the observations, corresponding to and . The dashed and dashdotted lines are the best fits using the models of resonant absorption and phasemixing, respectively. The vertically and diagonally shaded region correspond to theoretical decay times due footpoint leakage of Alfvén waves (Hollweg, 1984) and coronal leakage of a fast leaky kink mode (Cally, 2003), respectively, for a range of realistic coronal values for , , and . 

Figure 14:
Doppler oscillation events in the Fe XIX line observed with the SUMER instrument on 9 March 2001. a) Doppler shift time series. The redshift is represented with the bright colour, and the blueshift with the dark colour. b) Average time profiles of Doppler shifts along cuts AC and BD. The thick solid curves are the best fit functions of the form . c) Lineintegrated intensity time series. d) Average time profiles of lineintegrated intensities along cuts AC and BD. For a clear comparison, the intensity profile for BD has been stretched by a factor of . e) Line width (measured Gaussian width) time series. f) Average time profiles of line width along cuts AC and BD (from Wang et al., 2003a). 

Figure 15:
A typical response of a 1D loop to the flaring heat deposition near the apex. The density curve demonstrates pronounced quasiperiodic pulsations associated with the second standing acoustic harmonics. 

Figure 16:
The sketch of the coronal loop model suggested by Tsiklauri and Nakariakov (2001). A coronal loop is considered as a magnetic field line with density and gravitational acceleration varying along the axis of the cylinder. 

Figure 17:
The comparison of the theoretically predicted (the continuous curve) and observed amplitudes of the upwardly propagating EUV disturbances observed in polar plumes (from Ofman et al., 1999). 

Figure 18:
Simultaneous timedistance plots of propagating EUV disturbances observed by TRACE in the 171 Å and 195 Å bandpasses along a slit. The distance along the slit is shown in opposite directions to demonstrate that the disturbances observed in different bandpasses form a “fishbone” structure (from King et al., 2003). 

Figure 19:
Six upper panels (loops A  F): Evolution of correlation coefficients of propagating disturbances observed simultaneously in 171 Å and 195 Å bandpasses with the distance along six different slits. The solid lines show the correlation of unfiltered data while the dashed lines show the correlation of the signals after subtraction of slower variation. The dotted lines are the bestfitted straight lines. Two lower panels: Evolution of correlation coefficients of simulated signals. The left panel shows the correlation for the same angle with the line of sight, but for different temperatures , , and and , , and for the solid, dotted and the dashed line, respectively. The right panel shows varying angle and , , and for the solid, dotted and dashed lines, respectively, while keeping the temperature the same. The dashdotted lines in both panels are the correlation of the simulated signals with amplitude noise added (from King et al., 2003). 

Figure 20:
A plot of the maximum amplitude at the most frequent scale of the points, analysed by Williams et al. (2001), against their loop position. The axis error bars are the uncertainty in position and the axis error bars have been calculated statistically by taking the square root of the number of photons and normalising with respect to time. The figure also includes theoretically calculated dependences for kink (solid) and sausage (dashed) modes with the parameters (from Cooper et al., 2003). 

Figure 21:
Numerical simulation of an impulsively generated fast magnetoacoustic wave train propagating along a coronal loop with a density contrast ratio of and profile steepness power index equal to . Upper panel: The characteristic time signature of the wave train at , where is the loop semiwidth, from the source point. The vertical lines show the pulse arrival time if the density was uniform, the dotted line using the external density, and the dashed line using the density at the centre of the structure. Lower panel: Wavelet transform analysis of the signal, demonstrating the characteristic “tadpole” wavelet signature (from Nakariakov et al., 2004a). 

Figure 22:
Left: TRACE 195 Å field of view on 21 April 2002 at 01:49:57 UT. The square gives the location of the subfield used in the data analysis. The subfield data cube has spatial coordinates and , which represent height and horizontal coordinates, respectively. Middle: Slice of the subfield data cube for fixed vertical coordinate , as a function of and (dashed line in left panel). The oscillatory motions of the tadpoles are clearly visible. The tadpole heads are indicated with dashed lines. The tadpole edges analysed in Verwichte et al. (2005) are marked with letters. Right: Slice of the subfield data cube averaged over , as a function of and (between the dashdotted lines in left panel). This range of corresponds to the location of edge C (see middle panel). The solid lines indicate the location of tadpole heads (from Verwichte et al., 2005). 

Figure 23:
Characterisation of the transverse displacement of a wave packet in edge B (see middle panel in Figure 22). Left: Transverse displacement as a function of and height . The wave packet is visible as quasiperiodic sets of ridges. Righttop: Relative displacement as a function of time for . The thick line is a fitted cosine function. Rightmiddle: Phase speed as a function of height . The thick, solid line is an inverse linear fit. Rightbottom: Displacement amplitude as a function of height . The thick, solid line is a fitted exponential curve and the dashed line represents the minimum amplitude that can be resolved (from Verwichte et al., 2005). 
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