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4 Sausage Oscillations of Coronal Loops

The fast magnetoacoustic sausage mode (m = 0) is another type of localised, modified fast magnetoacoustic wave. It is associated with perturbations of the loop cross-section and plasma concentration This mode is mainly transverse and the perturbations of plasma velocity in the radial direction are stronger than perturbations along the field. According to Figure 3View Image, the phase speed of this mode is in the range between the Alfvén speed inside and outside the loop. This mode has a long wavelength cutoff (Roberts et al., 1984Jump To The Next Citation Point),
[ ]1/2 (C2s0 +-C2A0)(C2Ae--C2T0)- kzca = j0 (C2 - C2 )(C2 - C2 ) , (43) Ae A0 Ae s0
where j0 ~~ 2.40 is the first zero of the Bessel function J0(x). For a magnetic slab with the step function and Epstein density profiles, this value is
----------- C2 { p/2 (step function) kzca = V~ -2--A0--2-- V~ -- , (44) CAe - C A0 2 (Epstein function)

For k-- > kzc the mode approaches the cut-off, the phase speed, Cph, which is equal to w/kz, tends to C Ae from below, and in the short wave length limit, for k --> oo, C ph tends to C A0 from above.

The period PGSM of the global sausage mode of a coronal loop is determined by the loop length L,

2L PGSM = ----, (45) Cph
where Cph is the specific phase speed of the sausage mode corresponding to the wave number kz = p/L, C < C < C A0 ph Ae. The length of the loop L should be smaller than p/k zc to satisfy the condition k > kc. For a strong density contrast inside and outside the loop, the period of the sausage mode satisfies the condition
-2pa-- 2.62a- PGSM &lt; j0CA0 ~~ CA0 , (46)
as the longest possible period of the global sausage mode is achieved when k = kzc. We would like to emphasise that Equation (46View Equation) is an inequality, and that the actual resonant frequency is determined by Equation (45View Equation), provided Equation (46View Equation) is satisfied. Combining Equations (46View Equation) and (45View Equation), we conclude that the necessary condition for the existence of the global sausage mode is
V~ --- L p CAe r0 2a- &lt; 2j- C--- ~~ 0.65 r-, (47) 0 A0 e
so the loop should be sufficiently thick and dense (e.g., in the case of a flaring loop).

Nakariakov et al. (2003) demonstrated the applicability of the correct estimation for the global sausage mode period (Equation (45View Equation)) by interpreting high spatial and temporal resolution observations of 14 -17 s oscillations of coronal loops, performed with the Nobeyama radioheliograph. For the analysed flare, it was found that the time profiles of the microwave emission at 17 and 34 GHz exhibit synchronous quasi-periodical variations of the intensity in different parts of the corresponding flaring loop. The length of the flaring loop is estimated as L = 25 Mm and its width at half intensity at 34 GHz as about 6 Mm. These estimations are confirmed by Yohkoh/SXT images taken on the late phase of the flare. The distribution of the spectral density in the interval 14 -17 s along the loop showed the peak of oscillation amplitude near the loop apex and depression at the loop legs, consistent with the structure theoretically predicted for a global (fundamental) mode. Estimation of the period of this mode, according to Equation (47View Equation), gives the resonant period in the observed range. Also, for the loop considered, the sausage mode cut-off value kzca is about 0.25- 0.28. Thus, the longest theoretically possible wavelength c of the trapped sausage mode of the considered loop is c ~~ (22 -25)a. Consequently, as the observed loop radius is about 1/8 of its length, this loop could indeed support the global sausage mode.

Observations in the radioband and in X-rays often show also shorter periodicities, in particular in the range 0.5 -10 s (see, e.g., Aschwanden, 19872003Jump To The Next Citation Point). These oscillations are also traditionally associated with sausage (or radial) modes (see, e.g., Zaitsev and Stepanov, 20021989). It was suggested that the energetic particles produced by a flare are somehow modulated by the sausage oscillations of the flaring loop, localised near the top of the loop. The period of this oscillations is supposed to be given by the fast magnetoacoustic wave travel time across the loop, in other words as the ratio of the loop diameter and the fast magnetoacoustic speed (C2A0 + C2s0)1/2. However, it is not clear what determines the longitudinal length of the oscillation and why it does not propagate along the loop. If the longitudinal wave length is prescribed by the length of the loop, the sausage mode wave number is lower than the cut-off value and the mode is leaking, which is in contradiction with observed high quality of the short period oscillations (see also Aschwanden et al., 2004). The last difficulty can be overcome if there is some mechanism continuously feeding the oscillations or if the leakage is negligible. Also, quasi-periodic pulsations of shorter periods (0.5 -10 s) may be associated with sausage modes of higher spatial harmonics (Roberts et al., 1984Jump To The Next Citation Point), if the longitudinal wave length is shorter than the loop length. However, usually the short period oscillations are observed as a single high quality peak in the periodogram, and it is not clear why only this particular harmonics is excited. The role of ballooning modes has not been established yet.

Earlier, we discussed the microwave quasi-periodic pulsations observed by Asai et al. (2001Jump To The Next Citation Point) in the context of a global kink mode. The spatial resolution of the radio data is not sufficient to actually observe the spatial loop displacements. Therefore, can this pulsation also be attributed to a global sausage mode? A sausage mode can more naturally explain the modulation of X-ray emission as it is compressive. Asai et al. (2001Jump To The Next Citation Point) estimated the loop width to be 6 Mm. Condition (47View Equation) for the existence of a sausage normal mode restricts the density ratio to be r0/re > 17, which is reasonable. If we take r0/re = 20, then the external Alfvén speed is approximately C ~~ 2L/P = 4850 km s-1 Ae. The internal Alfvén speed is then found to be 1/2 -1 CA0 ~~ (re/r0) CAe = 1080 km s. Using the value of the loop density determined by Asai et al. (2001Jump To The Next Citation Point), a value for the magnetic field strength of 120 G follows. This value is three times smaller than Asai et al. (2001) obtained from a magnetic field extrapolation. From this point of view the global kink mode seems to be the most likely explanation for the microwave pulsations, but the global sausage mode cannot be dismissed outright.

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