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.

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 $x$ and $y$ , which represent height and horizontal coordinates, respectively. Middle: Slice of the subfield data cube for fixed vertical coordinate $x = 45.6 Mm$ , as a function of $t$ and $y$ (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 $y = 32.8 -36.4 Mm$ , as a function of $t$ and $x$ (between the dash-dotted lines in left panel). This range of $y$ corresponds to the location of edge C (see middle panel). The solid lines indicate the location of tadpole heads (from Verwichte et al., 2005

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.

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 $t$ and height $x$ . The wave packet is visible as quasi-periodic sets of ridges. Right-top: Relative displacement $dqy$ as a function of time for $x = 45.3 Mm$ . The thick line is a fitted cosine function. Right-middle: Phase speed $Vph$ as a function of height $x$ . The thick, solid line is an inverse linear fit. Right-bottom: Displacement amplitude $A$ as a function of height $x$ . 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 ).

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 $-1 500 -1000 km s$ , 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 $2 -4 I - I0 ~ ne ~ VA,ray$ and $Vph ~ VA,ray$ , where $VA,ray$ is the Alfvén speed in the ray, we find the relations $I - I0 ~ P 4$ and $I - I0 ~ Vp-h4$ . $I0$ 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 $-3.0± 1.5 I - I0 ~ Vph$ . 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.