2.1 Alfvén Waves

One set of two algebraic equations describes perturbations of the field and plasma flows in the direction perpendicular to both $B$ and $k$. By setting the determinant of the set of equations equal to zero, one obtains the dispersion relation for Alfvén waves:

where $v_{\text{A}}\equiv B_0/({\mu }_0{\rho }_0{)}^{1/2}$ is the Alfvén speed. Waves of this kind were predicted by Alfvén (1942). An Alfvén wave possesses the following characteristics:

As this process is caused by the magnetic elasticity (which is proportional to the magnetic field strength $B_0$) and the plasma inertia, which is proportional to its mass density ${\rho }_0$, the Alfvén speed is determined by $B_0$ and ${\rho }_0$.

The main properties of Alfvén waves are as follows:

In both linearly and elliptically polarized Alfvén waves, the absolute value of the magnetic field varies in time. In circularly polarized Alfvén waves, the absolute value remains constant, while its direction varies in time and space.

A convenient way to describe Alfvén waves is in terms of Elsässer variables,

where the different signs correspond to waves propagating in the positive and negative directions of the field (Elsasser, 1950). In terms of Elsässer variables, the MHD equations take a symmetric form

where $C_{\text{A}}$ is a vector with a magnitude equal to the Alfvén speed and a direction parallel to the magnetic field, and the right-hand sides contain dissipative and compressive terms. Equations 3 provide the basis for the study of MHD turbulence (e.g., Bruno & Carbone, 2013).

2.2 Magnetoacoustic Waves

The second partial set that has six remaining algebraic equations links perturbations of the magnetic field and the velocity vector in the plane formed by the magnetic field and the wave vector, and also perturbations of the density and gas pressure with each other. The dispersion relation in this case is

where $c_{\text{s}}=(\gamma p_0/{\rho }_0{)}^{1/2}$ is the sound speed, $\alpha$ is the angle between the wave vector and the magnetic field, and $\gamma$ is the adiabatic index. These waves are called magnetoacoustic, as they are essentially compressive, since they perturb the plasma density. It can be shown with the use of the continuity equation that in magnetoacoustic waves generally, $\nabla \mathrm{.}v\ne 0$. Equation 4 is bi-quadratic with respect to $\omega$ and $k$, and consequently has two pairs of roots for ${\omega }^2$ or $k^2$. Solving it with respect to ${\omega }^2$, one obtains

As the argument of the square root is always positive, and as the right-hand side is always positive, Equation 5 has two pairs of roots that describe oscillatory motions. The positive sign on the right-hand side corresponds to the fast magnetoacoustic wave, and the negative sign to the slow magnetoacoustic wave.

Physical processes that drive magnetoacoustic waves are associated with gradients of the gas pressure and magnetic pressure, as well as the magnetic tension. For illustration, consider a plane fast magnetoacoustic wave propagating across the field. In regions of compression of the plasma, because of the frozen-in effect, there is also an increase in the absolute value of the magnetic field. Hence, there are gradients of both gas and magnetic pressures, directed outward from the compressed regions, driving the plasma (with the frozen-in magnetic field) outward toward regions of rarified plasma density and decreased field. The force that attempts to restore the equilibrium is thus the gradient of the total (gas plus magnetic) pressure. Because of the finite inertia of the plasma, the plasma overshoots the equilibrium and creates new regions of enhanced and decreased gas and magnetic pressure. Thus, gradients of the total pressure occur, moving the plasma, and so on.

The main properties of magnetoacoustic waves are as follows:

In all other directions, the phase speeds of the fast and slow waves are greater or lower than the Alfvén speed, respectively.

2.3 Entropy Waves

The set of linearized ideal MHD equations has eight scalar equations and hence, in general, the dispersion relation polynomial is of the eighth order. Two roots correspond to Alfvén waves and four other roots to the fast and slow waves. The two remaining roots, which in ideal MHD are simply ${\omega }^2=0$, are called entropy waves. In an ideal plasma, the entropy mode is characterized by perturbations of thermodynamical properties of the plasma, frozen in the plasma flow. They become more interesting when nonideal effects are included. The entropy waves are non-propagating but could be advected by the plasma flow.

2.4 Friedrichs Diagrams

One of the essential features of MHD waves is the anisotropy determined by the angle $\alpha$ between the equilibrium field direction and the wave vector. A convenient way to illustrate the dependence of the phase and group speeds on the angle is to show it in a polar plot, called a Friedrichs diagram (see Figure 1). For a given angle $\alpha$, the value of the speed is determined by the distance from the origin to the crossing of the ray in that direction and the curve corresponding to the mode of interest.

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Figure 1. Polar plots showing the phase speeds (left panel) and group speeds (right panel) of MHD waves in a plasma with $\beta <1$: Alfvén (green), and slow (blue) and fast (red) magnetoacoustic waves—Friedrichs diagrams.

In ideal MHD, in a uniform medium, MHD waves of all kinds are dispersionless (i.e., their phase and group speeds are independent of the frequency or wavelength). This property has important implications for nonlinear MHD wave phenomena (see “5. Nonlinear Effects”).

4.1 Phase Mixing of Alfvén Waves

An intrinsic feature of Alfvén waves is their propagation strictly along the magnetic field. Consider an Alfvén wave in a plasma penetrated by a uniform magnetic field directed along $e_z$, with the Alfvén speed $v_{\text{A}}(x)$ varying smoothly in the perpendicular direction. Let the wave be linearly polarized in the $y$ direction (i.e., a shear Alfvén wave). In ideal MHD, the wave evolution is governed by the nonuniform wave equation,

with solutions $v_y=\text{Ψ}(x)f(z\mp v_{\text{A}}(x)t)$, where $f(z)$ and $\text{Ψ}(x)$ are smooth functions prescribed by the initial profile of the wave. The sign in the argument of this function corresponds to a wave propagating in the positive or negative $z$ direction, respectively. This solution implies that different magnetic surfaces $x=\text{const}$ support independent shear Alfvén waves that propagate along the field, at the local Alfvén speed. For each magnetic surface, the perturbation keeps its shape in the $z$ direction. Because of the Alfvén speed nonuniformity, the wave front in the $x$ direction becomes deformed, leading to the formation of smaller and smaller spatial scales in that direction. The perturbations of the neighboring magnetic surfaces $x=\text{const}$ become gradually uncorrelated with each other, leading to the phenomenon of Alfvén wave phase mixing (see Figure 3). This process could be considered as a perpendicular cascade, but its nature is purely linear. In cylindrical geometry, torsional Alfvén waves are subject to phase mixing too if the radial profile of the Alfvén speed varies smoothly.

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Figure 3. Effect of Alfvén wave phase mixing due to a nonuniformity of the Alfvén speed (in the $x$ direction), localized near $x=0$, when uniform magnetic field is in the $z$ direction. The left and right panels show the initial and developed profiles of the wave. The perturbation is normalized to the amplitude and the spatial units are arbitrary.

The cascade leads to a secular decrease in the perpendicular spatial scale of the wave and the onset of various physical effects associated with short wavelengths. In particular, it dramatically increases the effectiveness of short wavelength dissipation connected with, for example, finite viscosity or electrical resistivity. In the developed stage of phase mixing, which is defined by the condition $\partial /\partial x\gg \partial /\partial z$, the Alfvén wave decays as

where $\nu$ is the coefficient of viscosity and/or resistivity (Heyvaerts & Priest, 1983). The damping $\exp (-z^3)$ is much faster than exponential, with a characteristic time determined by the steepness of the Alfvén speed profile. Even more rapid damping occurs when the Alfvén speed nonuniformity is also along the field. For example, the damping could occur as $\exp [-\exp (z)]$ (Ruderman et al., 1998).

4.2 Alfvén Continuum and Resonant Absorption

Another important effect connected with a smooth nonuniformity of the plasma in a direction across the field is the Alfvén continuum. Following Chen (2008), consider a zero-$\beta$ plasma with density ${\rho }_0(x)$ nonuniform in the $x$ direction, across the field $B_0=B_0{e}_z$. Let the plasma be confined between two perfectly conducting plates at $z=0$ and $L$. Taking the plasma displacement in the $x$direction to be ${\xi }_x={\widehat{\xi }}_x\sin (n\pi z/L)$, with a positive integer parallel wave number $k_z=n$, one obtains for oblique linear harmonic perturbations

where ${ϵ}_{\text{An}}(x)={\omega }^2/v_{\text{A}}^2(x)-{(n\pi /L)}^2$, and $k_y$ is the wave number in the $y$ direction. Equation 15 has a singularity if the coefficient in front of the highest derivative becomes zero, ${ϵ}_{\text{An}}=0$, constituting an Alfvén continuum in the plasma. For a given frequency $\omega ={\omega }_0$, a resonance occurs at a certain surface $x=x_0$ determined by the condition $C_A{(x)}^2{(n\pi /L)}^2={\omega }_0^2$. The surface $x=x_0$ is called the Alfvén resonance layer, determined by the wave frequency ${\omega }_0$. The singularity results in a piling up of the wave energy carried by a magnetoacoustic wave at the Alfvén resonance layer. The energy goes from the collective magnetoacoustic oscillations to Alfvénic oscillations that are highly localized near $x=x_0$. It leads to apparent collisionless resonant absorption of magnetoacoustic oscillations as their frequency becomes complex. The effect of resonant absorption of magnetoacoustic oscillations is thus characterized by the transfer of their energy to Alfvén waves localized in the vicinity of the resonant layer. Accounting for finite-$\beta$ effects could lead to the appearance of a lower frequency continuum, associated with the variation of the tube speed $C_{\text{T}}(x)$ across the field. This continuum is usually referred to as the cusp continuum.

This effect received great attention in cylindrical geometry in the context of resonant absorption of kink oscillations of solar coronal loops (see “6.1 Kink Oscillations”). It has been shown (see Ruderman & Roberts, 2002; Goossens et al., 2006, for a comprehensive discussion) that the ratio of the damping time ${\tau }_{\text{D}}$ to the oscillation period $P$, determined by the imaginary and real parts of the frequency, respectively, is

where $l$ is the width of the resonant layer, determined by the steepness of the smooth radial profile of $v_{\text{A}}$, and other notations are the same as used in “3. Magnetohydrodynamic Modes of a Plasma Cylinder”. An important feature of resonant absorption is the linear scaling of the damping time with oscillation period.

5.1 Weakly Nonlinear Magnetoacoustic Waves

Just like regular sound waves, magnetoacoustic waves are characterized by a quadratic nonlinearity. The evolution of a plane magnetoacoustic wave that propagates along the $z$ axis with the angle $\alpha$ to a uniform magnetic field in a uniform plasma is governed by the inviscid Burgers equation,

where $U$ is a physical quantity perturbed by the wave, $C$ is the speed of a magnetoacoustic wave, determined by the dispersion relation (Equation 4), and the nonlinear coefficient ${ϵ}_{\text{B}}$ is determined by the angle $\alpha$, and the Alfvén and sound speeds,

This equation becomes invalid for the fast wave in the $\alpha \to 0$ limit, when it degenerates into the Alfvén wave. It may also include a linear term describing the effect of weak dissipation, for example, connected with volume viscosity, electrical resistivity, and thermal conductivity, or their combination, which will contain the second derivative of $U$ with respect to $z$, and also another linear term describing the misbalance between the radiative losses and heating of the plasma. The Burgers equation describes an energy cascade to small wavelengths in the direction of the wave vector, which leads to wave steepening and the formation of shocks and associated enhanced dissipation. In addition, the cascade brings the energy to progressively smaller scales where non-MHD dispersive effects (e.g., connected with Hall current or electron inertia) come into play, introducing short-wavelength dispersion. The basic physical process responsible for the nonlinear evolution is nonlinear resonant coupling of two identical spectral harmonics, resulting in the excitation of another harmonic with double the initial frequency and wave number. Schematically this processes could be illustrated as $\omega +\omega =2\omega$, $k+k=2k$. In the absence of dispersion, the excited harmonic $(2\omega \mathrm{,2}k)$ belongs to the same dispersion curve as the original harmonic $(\omega \mathrm{,}k)$ and causes a subsequent frequency and wave number doubling.

In a plasma cylinder or another plasma nonuniformity, the parallel cascade could be counteracted by dispersion, as the induced perturbations with $2\omega$ and $2k$ do not belong to the dispersion curve anymore. In fast modes, dispersion could fully suppress the wave steepening. In slow modes, it could lead to the appearance of stationary waves that propagate without evolution when the effects of nonlinearity and dispersion are in balance with each other. In the latter case, Equation 18 obtains an additional, weakly dispersive term that is determined by the dispersion of the mode of interest. For example, weakly nonlinear, long-wavelength surface sausage slow waves of a plasma cylinder, which are weakly dispersive, are described by the Leibovich–Roberts equation,

where the coefficients ${ϵ}_{\text{B}}^{\text{LR}}$, ${ϵ}_{\text{D}}^{\text{LR}}$, and $\lambda$ are determined by the characteristic speeds and the density (Roberts, 1985). Numerical investigation of Equation 19 demonstrated the existence of solitary wave solutions.

In highly dispersive fast modes of a perpendicular plasma nonuniformity, the parallel cascade is not pronounced, but they can effectively interact nonlinearly with themselves, or self-interact. This effect occurs on both quadratic and cubic nonlinearities and could be schematically illustrated as $\omega +\omega -\omega =\omega$, $k+k-k=k$. In other words, the nonlinearly induced perturbation with $2\omega$ and $2k$ immediately interacts with the perturbation with $\omega$ and $-k$, returning the energy to the original harmonic. Thus, the wave spectrum remains narrow, and the self-interaction causes evolution of the envelope of the narrowband wave train. The complex function $A(z\mathrm{,}t)$ describing the envelope is governed by the nonlinear Schrödinger equation,

where $V_{\text{gr}}$ is the group speed, the coefficient ${ϵ}_{\delta }^{\text{NSE}}=(1/2)({\text{d}}^2\omega /\text{d}{k}^2)$ accounts for the dispersion in the vicinity of the “filling” frequency of the narrowband wave train, and the asterisk denotes a complex conjugate. The coefficient ${ϵ}_{\text{N}}^{\text{NSE}}$ is determined by the profile of the perpendicular profile of the waveguiding nonuniformity and the perpendicular mode number (Nakariakov et al., 1997; Mikhalyaev & Ruderman, 2015). There are several important physical phenomena described by Equation 20, such as modulational instability and rogue waves.

5.2 Weakly Nonlinear Alfvén Waves

Weakly nonlinear elliptically polarized Alfvén waves propagating in a uniform plasma in the $z$ direction along a straight magnetic field $B_0$ are governed by the set of coupled Cohen–Kulsrud equations,

with the nonlinear coefficient ${ϵ}_{\text{CK}}=[4B_0^2{v}_{\text{A}}(v_{\text{A}}^2-c_{\text{s}}^2{)]}^{-1}$ (Cohen & Kulsrud, 1974). In the case of a shear Alfvén wave (e.g., polarized in the $x$ direction), the set of equations (Equation 21) reduces to a single Cohen–Kulsrud equation with respect to $B_x$ and with $B_y=0$. The same simplification occurs for a torsional wave that perturbs $B_{\theta }$ in the cylindrical geometry.

The physics of the nonlinear evolution of an Alfvén wave, described by Equation 21, is based on the action of the ponderomotive force. This force induces plasma flows along the gradient of the total magnetic pressure. Thus, the perturbation of the absolute value of the field by the wave drives parallel plasma flows and hence modifies the local plasma density. Therefore, the local Alfvén speed becomes modified, back-reacting, in turn, on the wave. As in the parallel direction, a perturbation of the density propagates at the sound speed, the equation describes effectively the interaction of Alfvén and sound waves.

According to Equation 21, elliptically or linearly polarized weakly nonlinear Alfvén waves evolve into a circularly polarized rotational structure, and a fast shock ahead of the rotation, a rotational discontinuity. The thickness of the rotational structure depends on the initial ellipticity of the wave. However, the proper description of the discontinuity requires accounting for short-wavelength physical effects, missing from ideal MHD. As circularly polarized Alfvén waves do not perturb the absolute value of the field, the ponderomotive force does not appear in those waves, and they are not subject to the processes described by Equation 21.

The coefficient ${ϵ}_{\text{CK}}$ is sensitive to the difference between the Alfvén and sound speeds (or the plasma $\beta$). When the Alfvén and sound speeds are equal to each other, Equation 21 is not valid. Interestingly, this constraint is softened in the case of nonlinear long-wavelength torsional Alfvén waves in a plasma cylinder. Locally, torsional Alfvén waves are linearly polarized and hence nonlinearly induce compressive flows of the plasma along the field. In a cylinder, the parallel plasma flows associated with the slow waves propagate at the tube speed that is always lower than both sound and Alfvén speeds. In the case of long-wavelength torsional waves in a plasma cylinder, the expression for the coefficient ${ϵ}_{\text{CK}}$ is modified. The denominator of ${ϵ}_{\text{CK}}$ has $C_{\text{T}}$ instead of $c_{\text{s}}$, and hence never becomes zero.

5.3 Multi-wave Resonant Interactions

The co-existence of different wave modes suggests the possibility of their nonlinear coupling. The most effective nonlinear interaction occurs when certain resonant conditions between their frequencies and wave vectors are fulfilled. In the case of a three-wave interaction, these resonant conditions are

where the indices label different interacting waves. The complex amplitudes $U_{\mathrm{1,2,3}}$ of the interacting waves are governed by the set of equations

where the integer indices label different interacting waves, and the nonlinear coefficients ${ϵ}_{\text{N}}^{\left\{\mathrm{1,2,3}\right\}}$ are determined by the properties of the waves. Because of energy conservation, a decrease in the amplitude of some waves is accompanied with an increase in the others. In particular, in the case of parallel propagation, two counter-propagating Alfvén waves (even circularly polarized) can nonlinearly interact with a sound wave (Sagdeev & Galeev, 1969). This phenomenon is responsible for a parametric decay process in which a finite amplitude Alfvén wave decays into an acoustic wave and a backward propagating Alfvén wave. The latter can be “taken” from noise. This process is believed to play a key role in plasma turbulence.

Similarly, in a plasma waveguide, modes of different azimuthal wave numbers $m$ can nonlinearly interact with each other. In particular, a sausage wave can decay into two counter-propagating kink waves. An interesting effect appears when one of the interacting waves is of negative energy, which can occur in a waveguide with a shear parallel flow (e.g., Joarder et al., 1997). In this case, amplitudes of all three interacting waves grow explosively in time, as ${(t_{\infty }-t)}^{-1}$, where $t_{\infty }$ is a constant. The energy comes from the mechanism that supports the flow.

6.1 Kink Oscillations

Coronal loops are dense structures forming coronal active regions. In the EUV band, they are seen as bright curved threads that link regions of opposite magnetic polarity on the surface of the Sun (see Figure 4, left panel). Loops are believed to be magnetic flux tubes filled with a hot dense plasma, and hence they are considered as tracers of the magnetic geometry. Transverse oscillations of coronal loops, usually called kink oscillations, are one of the most intensively studied wave phenomena in the corona. The oscillations are transverse, almost harmonic displacements of the loop, resembling oscillations of a guitar string. Usually, the maximum displacement amplitude is detected near the loop top, with the nodes at the chromospheric footpoints. Typical oscillation periods are several minutes. The oscillations are seen in two regimes, low-amplitude decayless and large-amplitude, rapidly decaying (see Figure 4). In this article, the decaying regime is concentrated on. Decaying kink oscillations are usually excited by a mechanical displacement of a loop from equilibrium by a low coronal eruption and are well seen in high-resolution EUV movies of the corona. However, in some cases, the excitation of kink oscillations is linked with other dynamic processes, such as colliding transient plasma flows or shear flow instabilities. In almost all the cases when it was possible to determine the oscillation polarization, it was polarized parallel to the solar surface (i.e., horizontal polarization).

Parameters of the oscillations are usually determined by the time–distance map technique. In an EUV imaging data cube of the active region of interest, one selects a narrow slit directed across an oscillating segment of the loop (see Figure 4, left panel). In each observational frame, the distribution of EUV intensity along the loop is measured. The intensities along the slit in different frames are stacked next to each other, creating a time–distance map. The periodic almost-harmonic transverse displacements of the loop are readily seen in the map (see Figure 4, middle panel). The time–distance oscillatory pattern is used to estimate the instantaneous period and apparent amplitude, and also other parameters of the oscillations such as the damping time. Typical oscillation amplitudes are from a few Mm to a few tens of Mm, which is much shorter than the length of the oscillating loop. The measured amplitude should be considered as a lower limit, since it is projected on the plane of the sky. Taking slits at different locations along the loop allows a determination of the spatial structure of the oscillation. In the vast majority of cases, kink oscillations appear to be standing, fundamental modes, while sometimes higher parallel harmonics are detected as well. Usually, the apparent width of the oscillating loop remains constant during the whole observed oscillation. Thus, the observations suggest that transverse oscillations of coronal loops could be interpreted as standing kink modes of the loop (see the discussion in “3. Magnetohydrodynamic Modes of a Plasma Cylinder”). Theoretical analysis performed by Van Doorsselaere et al. (2004) demonstrated the applicability of the results obtained for a straight cylinder to a curved loop.

Catalogues of kink oscillation events (e.g., Goddard et al., 2016) allow for the determination of correlations between different oscillation parameters. In particular, the oscillation period is found to scale linearly with the length of the oscillating loop (see Figure 4, right panel). This finding strengthens the interpretation of the kink oscillation as a natural standing oscillation of the loop. Indeed, as the wavelength of the fundamental mode is double the length $L$ of the oscillating loop and the phase speed of the kink mode is, in the long-wavelength limit, about the kink speed (Equation 11), the oscillation period is about $2L/C_{\text{K}}$. The linear scaling of the oscillation period with the loop length gives an estimation of the average kink speed, of about $1,300\text{km}{\text{s}}^{-1}$. The scattering of the kink speed values in individual cases should be attributed to the variation of the magnetic field, density, and the density contrast ratio in individual loops.

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Figure 5. Left panel: Image of a solar coronal active region, taken in the EUV band. A slit along a loop is indicated with a black bar. Right panel: The time–distance map made for the slit, covering about 10 cycles of the propagating EUV disturbances.

Adapted from Yuan and Nakariakov (2012).

The damping time of kink oscillations is found to scale linearly with the oscillation period (e.g., Ofman & Aschwanden, 2002; Goddard et al., 2016), indicating a possible association of the damping with the effect of resonant absorption described by Equation 16. In this scenario, kink oscillations are linearly coupled with azimuthal motions that are highly localized in the vicinity of the resonant layer. As kink oscillations involve motions of the whole body of the loop (i.e., are collective), they are observed as the transverse displacements of the loop. In contrast, the induced torsional motions do not perturb the loop boundary and hence remain invisible to imaging instruments. However, they could be picked up by high-resolution spectrometers. A more elaborate theory of resonant absorption suggests that the kink oscillation damping could occur as a combination of the Gaussian and exponential damping profiles (Hood et al., 2013). The shear flows excited between neighboring phase-mixed torsional waves induced in the resonant layer by a decaying kink wave may lead to its destabilization because of the Kelvin–Helmholtz instability (e.g., Terradas et al., 2008). Numerical simulations show that this process may lead to the plasma fragmentation and appearance of field-aligned fine structuring (e.g., Antolin et al., 2017).

6.2 Longitudinal Waves

Another ubiquitous coronal wave phenomenon is slow magnetoacoustic waves observed in plasma fan structures above sunspots and in coronal holes as propagating quasi-monochromatic disturbances of the EUV emission (see, e.g., de Moortel, 2009; Banerjee et al., 2011). The standard detection technique is also based on the construction of time–distance maps, but this time with the slit taken along the waveguiding structure (see Figure 5). Time–distance maps show the presence of almost periodic variations of the emission intensity, which form characteristic diagonal lanes. It indicates that the emission intensity disturbances propagate along the apparent direction of the magnetic field at a constant speed. The phase speed is determined by the gradient of the diagonal lanes in the time–distance map. The propagation speed ranges from a few to several tens of $\text{km}{\text{s}}^{-1}$and is always directed upward. The estimated values of the speed are lower than the sound speed in the plasma, estimated by its temperature. The observed apparent speed is lower than the actual value because of the effect of the line-of-sight projection. Usually, the apparent phase speed of the propagating EUV intensity disturbances remains constant during the whole duration of the detection and along the whole slit. Typical periods range from a few to several minutes. In some cases, up to several tens of oscillation cycles are detected. The period remains stable during the whole observation. Typical values of the EUV intensity variations are several percent of the background value. As in the regime of optically thin emission, the intensity is proportional to the density squared; the associated relative density perturbations are about the intensity variation divided by two. The waves are detected without any association with dynamic processes in the solar atmosphere such as flares and coronal mass ejections.

The consistency of the phase speed with the sound speed, parallel propagation, and the amplitude indicates the slow magnetoacoustic nature of the propagating EUV intensity disturbances. The stable oscillation period additionally strengthens this interpretation and is likely to be associated with physical conditions in the vicinity of the wave excitation location. In particular, it could be determined by the acoustic cutoff frequency in the chromosphere (De Moortel & Nakariakov, 2012). In this scenario, the narrowband signal is formed by the response of the stratified atmosphere to a broadband buffeting, for example, by ubiquitous granulation flows.

Slow magnetoacoustic waves are also observed in coronal loops in a standing form (e.g., Wang, 2011). The periods of standing slow oscillations, referred to as SUMER oscillations after the instrument used in their first detection, are typically longer than that of the propagating slow waves, ranging from several minutes to a few tens of minutes. The relative amplitude may exceed 10%. The perturbation of the plasma density is seen phase-shifted with respect to the parallel flow speed by a quarter of the oscillation period, which indicates the standing nature of SUMER oscillations. The oscillations are subject to strong damping, with the exponential damping time found to be about the oscillation period. The damping time scales linearly with the oscillation period. In contrast with propagating slow waves, SUMER oscillations seem to be excited by some impulsive processes. SUMER oscillations are interpreted as fundamental longitudinal harmonics of coronal loops. Similarly to kink oscillations, the waves excited in the coronal part of the loop and traveling downward get reflected at the steep gradients of the characteristic speed at the loop’s footpoints. In contrast with usual sound waves, slow magnetoacoustic waves propagate almost along the field. Hence both incident and reflected waves follow the same path. The superposition of the waves propagating in opposite directions forms a standing wave pattern.

6.3 MHD Seismology

Just as for any wave, MHD waves carry information about the medium of propagation and hence can be used for its remote diagnostics. The method of MHD seismology is extensively used in solar atmospheric studies. MHD spectroscopy and magnetoseismology are similar techniques used in laboratory (e.g., Fasoli et al., 2002) and magnetospheric plasmas (e.g., Chi et al., 2009), respectively. In the context of the solar corona, MHD seismology was theoretically proposed by Uchida (1970), but its full-scale implementation became possible with the first unequivocal detections of oscillations and waves in coronal plasma structures in the late 1990s.

Kink oscillations of coronal loops provide estimations of the Alfvén speed and, if the density is known, also of the magnetic field strength in the oscillating loop. The procedure for this estimation is as follows: The length of the loop and oscillation period are determined observationally. Also, the structure of the transverse displacement along the loop indicates that the oscillation corresponds to the fundamental parallel harmonic. Thus, the phase speed of the kink wave is estimated as the ratio of its wavelength (double the loop length for the fundamental harmonic) and the oscillation period. Using dispersion relation (Equation 9), it is assumed that the phase speed is roughly the kink speed. The kink speed (Equation 11) is determined by the Alfvén speed inside the loop and the density contrast inside and outside it. If the latter parameter can be measured by the ratio of the emission intensities, one obtains an estimate of the Alfvén speed. With the use of an independent estimate of plasma density in the oscillating loop, the value of the magnetic field can be deduced (Nakariakov & Ofman, 2001). More elaborate seismological techniques utilize additional information; in particular, the ratio of the periods of the fundamental and second parallel harmonics, mode structure along the loop, and the time when the Gaussian decay regime changes to an exponential one. These techniques provide estimates of the equilibrium density scale height (Andries et al., 2009), the expansion of the coronal loop with height (Verth & Erdélyi, 2008), and the transverse profile of the density in the loop (Pascoe et al., 2016).

Longitudinal waves have a promising seismological potential as well. Wang et al. (2007) measured the difference between the observed phase speed, taken to be the tube speed (Equation 6), and the estimated sound speed, and used it to estimate the magnetic field in the loop. Van Doorsselaere et al. (2011) used the relationship between relative density and temperature perturbations to measure the effective adiabatic index. Wang et al. (2015) established seismologically the suppression of parallel thermal conductivity, and a possible enhancement of the effective compressive viscosity, from the phase shift between the density and temperature perturbations. The confinement of slow waves to the local direction of the magnetic field allows the field geometry to be traced (e.g., above a sunspot) (Jess et al., 2016).