4.6 MHD waves in coronal loops

Besides fast-mode MHD waves, which can be subdivided into kink-mode (asymmetric, see Section 4.5) and sausage-mode (symmetric) types, there are also slow-mode MHD waves, which propagate with acoustic (sound) speed. The sound speed can be expressed in terms of the electron temperature T e (assuming the coronal approximation Te = Ti, although Ti > Te according to some observations),
( )1∕2 ∘ --- c = γpth- = 1.66 × 104 Te- [cm s−1], (43 ) s ρ μ
where γ = 5∕3 is the adiabatic index, pth = 2nekBTe the thermal pressure, ρ = niμmp the mass density, and μ = 1.27 the mean molecular weight. However, if a sound wave propagates at an angle of 𝜃 to the line-of-sight, the observed speed scales with the sine function of the line-of-sight angle, i.e., v = c sin ϕ obs s. This projection angle ϕ can be determined by stereoscopic triangulation of the loop structure that guides the sound wave propagating in longitudinal direction.
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Figure 38: Derived 3D geometry of the wave propagation along the coronal (fan) loop, observed on 2008 Jan 10 with STEREO/A and B (bottom panels) The top panels show the projected geometry of the propagation vector with the view rotated by 90° seen from solar north (from Marsh et al., 2009Jump To The Next Citation Point).
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Figure 39: Time-slice plots of the integrated intensity along the (fan) loops for STEREO/B (left) and A (right). The abscissa indicates the observed distance long the loop, perpendicular to the line-of-sight from each spacecraft (from Marsh et al., 2009Jump To The Next Citation Point).

Such a measurement has been conducted for slow-mode (acoustic) waves observed with STEREO/A and B on 2008 Jan 10 (Marsh et al., 2009). The propagating waves were observed in a fan-like loop structure emanating from one magnetic pole of a bipolar active region (Figure 38View Image). A time-slice plot of the EUV flux along the fans as a function of time is shown in Figure 39View Image, which confirms the quasi-periodic wave propagation (diagonal pattern), for which a velocity of −1 vA = 62.9 ± 0.1 km s was measured with STEREO/A, and −1 vB = 104.1 ± 0.2 km s with STEREO/B, respectively. The wave period was found to be P ≈ 12 min. Stereoscopic triangulation of the fans yielded an inclination angle of 𝜃 = 37∘ ± 6∘ to the local vertical. 3D projections of the triangulated directions are also shown in Figure 38View Image (top panels). Correcting the observed wave speed for the corresponding line-of-sight angle yields a corrected speed of −1 cA = 132 ± 9 km s and −1 cB = 132 ± 11 km s, which represents the first measurement of the true coronal longitudinal slow-mode speed cs in 3D. Inserting this true sound speed into Eq. (43View Equation), a temperature of Te = 0.84 ± 0.15 MK is inferred, which is close to the peak sensitivity of the used 171 Å passband. The temperature structure of the same fan loop was also investigated with EIS/Hinode and a mean temperature of Te = 0.89 ± 0.09 MK was obtained, in excellent agreement with the EUVI measurements, which confirms that the slow-mode phase speed is identical to the sound speed (Marsh and Walsh, 2009).

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