What makes solar coronal waves wavy?
The intrinsically filamentary nature of the coronal plasma plays a crucial role in the dynamics of magnetohydrodynamic (MHD) waves. In particular, the waves could be guided along the plasma inhomogeneities. Due to this waveguiding effect, coronal MHD waves are known to be subject to effective dispersion manifested through the dependence of the wave speed on the wavenumber (frequency), which is most pronounced for fast magnetoacoustic (MA) waves. Similarly to a mass-start in road cycling, where all cyclists begin the race at the same time and position, an impulsively excited ensemble of fast MA harmonics, initially localised in space and time, disperses along the waveguide and forms a quasi-periodic fast-propagating perturbation of the local plasma parameters (see Fig. 1, for example). Such rapidly propagating trains of fast waves, guided by field-aligned plasma non-uniformities, are confidently observed both near the base and at higher heights of the Sun’s corona .
We determine characteristic signatures of fully developed fast wave trains in the time domain, obtained from theoretical modelling of fast MA waves in a coronal plasma non-uniformity. We demonstrate the link between the time history of a fast wave train and parameters of the hosting waveguide, which gives important and sometimes unique seismological information about the cross-field structuring of the coronal plasma.
From fast magnetoacoustic tadpoles to boomerangs
We model fast MA waves in a low-β plasma slab stretched along the z-axis in the linear regime (e.g., ). In the model, the direction of the equilibrium magnetic field coincides with the slab axis, and the cross-field profile of the plasma density (in the x-direction) is given by a continuous function with varying steepness. Thus, the waveguide is characterised by two free parameters, its depth (i.e. ratio of the plasma densities or Alfven speeds inside and outside) and steepness (i.e. smooth or sharp boundary), which determine the dynamics of fast MA waves in this model.
For smooth transverse density profiles, the group speed of fast MA waves varies with the parallel wavenumber monotonically between the Alfven speeds inside and outside the waveguide, CAi and CAe; and it has a well-pronounced minimum for steeper waveguide boundaries (see Fig. 2). The former means that among the impulsively excited ensemble of fast MA waves in a waveguide with sufficiently diffuse boundaries, all guided parallel harmonics will propagate at their own group speeds (between CAi and CAe), eventually forming a quasi-periodic pattern seen in the Morlet wavelet power spectrum as a tadpole with a narrow-band tail and broader-band head . In steeper waveguides, in contrast, there will be a relatively narrow interval of parallel harmonics propagating at the highest group speeds CAi < Vgr < CAe (a quasi-periodic phase I in Fig. 1); pairs of harmonics with distinctly different wavelengths travelling at the same group speed Vgrmin < Vgr < CAi (a multi-periodic or a peloton phase II in Fig. 1); and a single parallel harmonic trailing behind all other guided harmonics at Vgrmin (Airy phase III in Fig. 1). Such a structure of the wave train is seen as a boomerang shape in the wavelet spectrum with two well-pronounced arms at shorter and longer periods (bottom panel in Fig. 1). The revealed multi-phase structure of fast MA wave trains is consistent with the suggestion of , based on the analogy with Pekeris waves in an ocean layer.
The high time resolution traditionally available in the radio band seems to be most suitable for the detection of these boomerang-shaped fast MA wave trains in observations.
For example,  observed signatures of a fast wave train in the decimetric radio emission at 973-1025 MHz, with a frequency drift of Δf/Δt = 8.7 MHz/s towards lower frequencies which could be interpreted as an upward propagation of the wave train at the speed about 870 km/s (see Fig. 3). The shape of the wavelet spectrum of this wave train is seen to change from a tadpole at higher frequencies (lower heights) with a narrow-band tail around 81-s oscillation period and a broader-band head to a boomerang at lower frequencies (higher heights) with two well-pronounced arms around 81 s and 30-40 s. These observational properties are consistent with the theoretical scenario described in our work for the development of a fast MA wave train in a waveguide with a steep transverse density profile.
Prospects for coronal seismology
A synergy of observations of such boomerang-shaped wave trains with theory offers a unique possibility for probing simultaneously the plasma waveguide depth (Alfven speed ratio CAe/CAi) and steepness. According to the model, one can estimate CAe/CAi from the observed duration of phase I (see Fig. 1). Likewise, the duration of peloton phase II gives the ratio CAi/Vgrmin which is sensitive to the waveguide steepness. Thus, the time history of fast-propagating quasi-periodic wave trains carries important information about the transverse structuring of the coronal plasma which opens up interesting perspectives for coronal seismology, using high-resolution and high-sensitivity observations from existing (e.g. AIA/SDO, LOFAR) and upcoming (e.g. SKA, METIS/SO, ASPIICS/Proba-3) instruments.
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