The heating of the solar chromosphere and corona to the observed high temperatures, imply the presence of ongoing heating that balances the strong radiative and thermal conduction losses expected in the solar atmosphere. It has been theorized for decades that the required heating mechanisms of the chromospheric and coronal parts of the active regions, quiet-Sun, and coronal holes are associated with the solar magnetic fields. However, the exact physical process that transport and dissipate the magnetic energy which ultimately leads to the solar plasma heating are not yet fully understood. The current understanding of coronal heating relies on two main mechanism: reconnection and MHD waves that may have various degrees of importance in different coronal regions. In this review we focus on recent advances in our understanding of MHD wave heating mechanisms. First, we focus on giving an overview of observational results, where we show that different wave modes have been discovered in the corona in the last decade, many of which are associated with a significant energy flux, either generated in situ or pumped from the lower solar atmosphere. Afterwards, we summarise the recent findings of numerical modelling of waves, motivated by the observational results. Despite the advances, only 3D MHD models with Alfvén wave heating in an unstructured corona can explain the observed coronal temperatures compatible with the quiet Sun, while 3D MHD wave heating models including cross-field density structuring are not yet able to account for the heating of coronal loops in active regions to their observed temperature.

Kink oscillations of coronal loops, i.e., standing kink waves, is one of the most studied dynamic phenomena in the solar corona. The oscillations are excited by impulsive energy releases, such as low coronal eruptions. Typical periods of the oscillations are from a few to several minutes, and are found to increase linearly with the increase in the major radius of the oscillating loops. It clearly demonstrates that kink oscillations are natural modes of the loops, and can be described as standing fast magnetoacoustic waves with the wavelength determined by the length of the loop. Kink oscillations are observed in two different regimes. In the rapidly decaying regime, the apparent displacement amplitude reaches several minor radii of the loop. The damping time which is about several oscillation periods decreases with the increase in the oscillation amplitude, suggesting a nonlinear nature of the damping. In the decayless regime, the amplitudes are smaller than a minor radius, and the driver is still debated. The review summarises major findings obtained during the last decade, and covers both observational and theoretical results. Observational results include creation and analysis of comprehensive catalogues of the oscillation events, and detection of kink oscillations with imaging and spectral instruments in the EUV and microwave bands. Theoretical results include various approaches to modelling in terms of the magnetohydrodynamic wave theory. Properties of kink oscillations are found to depend on parameters of the oscillating loop, such as the magnetic twist, stratification, steady flows, temperature variations and so on, which make kink oscillations a natural probe of these parameters by the method of magnetohydrodynamic seismology.

Aims. We investigate the nonlinear phenomena accompanying long-wavelength torsional waves in solar and stellar coronae. Methods. The second order thin flux-tube approximation is used to determine perturbations of a straight untwisted and non-rotating magnetic flux-tube, nonlinearly induced by long-wavelength axisymmetric magnetohydrodynamic waves of small, but finite amplitude. Results. Propagating torsional waves induce compressible perturbations oscillating with double the frequency of the torsional waves. In contrast with plane shear Alfvén waves, the amplitude of compressible perturbations is independent of the plasma-β and is proportional to the torsional wave amplitude squared. Standing torsional waves induce compressible perturbations of two kinds, that grow with the characteristic time inversely proportional to the sound speed, and that oscillate at double the frequency of the inducing torsional wave. The growing density perturbation saturates at the level, inversely proportional to the sound speed.

Characterized by cyclic axisymmetric perturbations to both the magnetic and fluid parameters, magnetohydrodynamic fast sausage modes (FSMs) have proven useful for solar coronal seismology given their strong dispersion. This review starts by summarizing the dispersive properties of the FSMs in the canonical configuration where the equilibrium quantities are transversely structured in a step fashion. With this preparation we then review the recent theoretical studies on coronal FSMs, showing that the canonical dispersion features have been better understood physically, and further exploited seismologically. In addition, we show that departures from the canonical equilibrium configuration have led to qualitatively different dispersion features, thereby substantially broadening the range of observations that FSMs can be invoked to account for. We also summarize the advances in forward modeling studies, emphasizing the intricacies in interpreting observed oscillatory signals in terms of FSMs. All these advances notwithstanding,

Aims. The theoretical model for magnetohydrodynamic (MHD) modes guided by a field-aligned plasma cylinder with a steady flow is adapted to interpret transverse waves observed in solar coronal hot jets, discovered with Hinode/XRT in terms of fast magnetoacoustic kink modes. Methods. Dispersion relations for linear magnetoacoustic perturbations of a plasma jet of constant cross-section surrounded by static magnetised plasma are used to determine the phase and group speeds of guided transverse waves and their relationship with the physical parameters of the jet and the background plasma. The structure of the perturbations in the macroscopic parameters of the plasma inside and outside the jet, and the phase relations between them are also established. Results. We obtained a convenient expansion for the long wave-length limit of the phase and group speeds and have shown that transverse waves observed in soft-X-ray solar coronal jets are adequately described in terms of fast magnetoacoustic kink modes by a magnetic cylinder model, which includes the effect of a steady flow. In the observationally determined range of parameters, the waves are not found to be subject to either the Kelvin-Helmholtz instability or the negative energy wave instability, and hence they are likely to be excited at the source of the jet.

Aims. We investigate the interaction of nonlinear fast magnetoacoustic waves with a magnetic null point in connection with the triggering of solar flares. Methods. We model the propagation of fast, initially axisymmetric waves towards a two-dimensional isothermal magnetic null point in terms of ideal magnetohydrodynamic equations. The numerical simulations are carried out with the Lagrangian remap code Lare2D. Results. Dynamics of initially axisymmetric fast pulses of small amplitude is found to be consistent with a linear analytical solution proposed earlier. The increase in the amplitude leads to the nonlinear acceleration of the compression pulse and deceleration of the rarefaction pulse and hence the distortion of the wave front. The pulse experiences nonlinear steepening in the radial direction either on the leading or the back slopes for the compression and rarefaction pulses, respectively. This effect is most pronounced in the directions perpendicular to the field. Hence, the nonlinear evolution of the fast pulse depends on the polar angle. The nonlinear steepening generates the sharp spikes of the electric current density. As in the uniform medium, the position of the shock formation also depends on the initial width of the pulse. Only sufficiently smooth and low-amplitude initial pulses can reach the vicinity of the null point, create there current density spikes, and initiate magnetic reconnection by seeding anomalous electrical resistivity. Steeper and higher amplitude initial pulses overturn at larger distance from the null point, and cannot trigger reconnection.

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Aims. We study the efficiency of the energy transfer to shorter scales in the field-aligned direction -the parallel nonlinear cascadethat accompanies the propagation of torsional Alfvén waves along open magnetic fields in the solar and stellar coronae, and compare it with the same effects for the shear Alfvén wave. The evolution of the torsional Alfvén wave is caused by the back reaction of nonlinearly induced compressive perturbations on the Alfvén wave. Methods. The evolution of upwardly propagating torsional Alfvén waves is considered in terms of the second-order thin flux-tube approximation in a straight untwisted and non-rotating magnetic flux-tube. The Cohen-Kulsrud equation for weakly nonlinear torsional waves is derived. In the model, the effect of the cubic nonlinearity on the propagation of long-wavelength axisymmetric torsional waves is compared with the similar effect that accompanies the propagation of plane linearly-polarised (shear) Alfvén waves of small amplitude.Results. The solution to the Cohen-Kulsrud type equation for torsional waves shows that their evolution is independent of the plasma-β, which is in contrast to the shear Alfvén wave. In a finite-β plasma, the nonlinear evolution of torsional Alfvén waves is slower and the parallel nonlinear cascade is less efficient than those of shear Alfvén waves. These results have important implications for the analysis of possible heating of the plasma and its acceleration in the upper layers of solar and stellar coronae. In particular, onedimensional models of coronal heating and wave acceleration, which use shear Alfvén waves instead of torsional Alfvén waves, over-estimate the efficiency of these processes.