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date: 27 October 2020

# Magnetohydrodynamic Waves

• V.M. NakariakovV.M. NakariakovPhysics, University of Warwick

### Summary

Magnetohydrodynamic (MHD) waves represent one of the macroscopic processes responsible for the transfer of the energy and information in plasmas. The existence of MHD waves is due to the elastic and compressible nature of the plasma, and by the effect of the frozen-in magnetic field. Basic properties of MHD waves are examined in the ideal MHD approximation, including effects of plasma nonuniformity and nonlinearity. In a uniform medium, there are four types of MHD wave or mode: the incompressive Alfvén wave, compressive fast and slow magnetoacoustic waves, and non-propagating entropy waves. MHD waves are essentially anisotropic, with the properties highly dependent on the direction of the wave vector with respect to the equilibrium magnetic field. All of these waves are dispersionless. A nonuniformity of the plasma may act as an MHD waveguide, which is exemplified by a field-aligned plasma cylinder that has a number of dispersive MHD modes with different properties. In addition, a smooth nonuniformity of the Alfvén speed across the field leads to mode coupling, the appearance of the Alfvén continuum, and Alfvén wave phase mixing. Interaction and self-interaction of weakly nonlinear MHD waves are discussed in terms of evolutionary equations. Applications of MHD wave theory are illustrated by kink and longitudinal waves in the corona of the Sun.

### Subjects

• Astronomy and Astrophysics
• Plasma Physics

### 1. Introduction

The theory of long-wavelength large-scale magnetohydrodynamic (MHD) plasma motions, based upon the assumption that the medium behaves as a single, electrically conducting fluid, is one of the most successful theories of modern physics. Indeed, MHD (Priest, 2019) describes physical phenomena from nanoscales, in solids and semiconductors, to galactic and extragalactic scales, that is, spatial scales from $10−12m$ to $1022m$.

The vast majority of natural and laboratory plasmas can be considered as ideal or almost ideal (i.e., with values of the magnetic and viscous Reynolds numbers). An intrinsic property of MHD motions in ideal plasmas is the effect of the frozen-in magnetic field. Perpendicular motions of the plasma are accompanied by the deformation of the magnetic field. Hence, three intrinsic forces of MHD, the gradients of the magnetic and gas pressures, and the magnetic tension force, make the plasma a compressible and elastic medium. Therefore, a localized perturbation of the field or plasma parameters tends to spread over space in the form of MHD waves. Properties of these waves are determined in general by magnetic, inertial, and thermodynamical properties of the plasma, as well as by the angle between the wave vector and the magnetic field direction. MHD wave processes are studied in the whole domain of applicability of MHD, from semiconductors (e.g., Baynham & Boardman, 1970) to galactic arms (e.g., Fan & Lou, 1996).

In this article, low-frequency MHD waves in an ideal plasma are considered, neglecting effects of the dissipation connected with finite thermal conduction, viscosity, resistivity, radiation, and so on, as well as high-frequency effects such as electron inertia, Hall physics, finite gyroradius, and electron pressure; effects of temperature anisotropy and partial ionization; and also kinetic effects. All these generalizations of MHD wave theory are comprehensively covered in specialized research literature. Ideal MHD wave theory is illustrated here by MHD wave phenomena observed in the corona of the Sun. Modern solar coronal observational tools allow for the simultaneous resolution of the wavelength and oscillation period in the projected plane of space perpendicular to the line of sight (known as the plane of the sky) and time domains, respectively, and hence make the corona a unique natural laboratory for the study of MHD waves. Terminology used in the solar coronal physics, and introduced in this article, may be different in other applications. Moreover, different plasma environments (e.g., the plasma of controlled fusion and natural plasmas of the Earth’s magnetosphere and solar corona) have very different values of transport coefficients. However, the basic physical phenomena associated with MHD waves are universal. These universal phenomena intrinsic to MHD waves and independent of the specific plasma environment include wave anisotropy, appearance of waveguide effects, wave dispersion, Alfvén continuum and linear coupling of different modes, effects of finite amplitude, and associated MHD instabilities, although the presence of these depend more strongly on the environment.

### 2. Magnetohydrodynamic Waves in a Uniform Medium

Consider weak magnetohydrodynamic (MHD) perturbations of a uniform plasma with gas pressure $p0=const$, density $ρ0=const$, and temperature $T0=const$, penetrated by a straight and uniform magnetic field $B0=const$. Let the field be in the $xz$-plane. Consider perturbations of this equilibrium in the form of a plane wave propagating in the $z$ direction, that is, with the wave vector $k$ directed along the $z$ axis. Thus, the wave fronts are planes parallel to the $xy$ plane. The wave vector has an angle $α$ to the equilibrium magnetic field $B0$. Assume that the perturbations are harmonic, that is, proportional to $exp(ikz−iωt)$. Linearizing the ideal MHD equations (see Priest, 2019) with respect to this equilibrium, and using the assumptions just described, one obtains a set of eight homogeneous algebraic equations. This set actually splits into two independent subsets.

#### 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:

$Display mathematics$(1)

where $vA≡B0/(μ0ρ0)1/2$ is the Alfvén speed. Waves of this kind were predicted by Alfvén (1942). An Alfvén wave possesses the following characteristics:

it can be considered as a transverse oscillation of a “heavy magnetic string”;

a localized transverse deformation of the field leads to the appearance of a magnetic tension force, which tends to restore the equilibrium (i.e., to make the field straight);

when the field is returning back toward the equilibrium configuration, it drags the plasma with it, since the magnetic field is “frozen” in the plasma;

during the return of the field toward equilibrium, the plasma gains kinetic energy (as the field line is “heavy” because of the frozen-in condition);

because of the inertia (the kinetic energy), the plasma overshoots the equilibrium, and the field line becomes deformed in a direction opposite to the initial deformation until the magnetic tension force stops the deformation (when all the kinetic energy has been converted into magnetic potential energy) and then the plasma motion is revered, so that the field and plasma are returned back toward the equilibrium;

the plasma again returns back to the equilibrium position with the kinetic energy, and the process repeats; and

this perturbation propagates along the magnetic field at the Alfvén speed.

As this process is caused by the magnetic elasticity (which is proportional to the magnetic field strength $B0$) and the plasma inertia, which is proportional to its mass density $ρ0$, the Alfvén speed is determined by $B0$ and $ρ0$.

The main properties of Alfvén waves are as follows:

Alfvén waves are transverse: The plasma is displaced in a direction perpendicular to $k$. Thus, like any transverse wave, Alfvén waves can be polarized linearly, elliptically, or circularly.

In the linear regime, Alfvén waves are essentially incompressive: They do not modify the density of the plasma, which results in $∇.v=0$.

The group velocity ($Vgr$) is always parallel to the magnetic field, but the phase velocity

($Vph$) can be oblique to the field. It implies that the wave vector and the phase velocity may be non-parallel to the group velocity and the magnetic field direction. To illustrate this, consider a guitar with identical strings parallel to each other, separated by a vacuum. One may produce a localized transverse perturbation of those strings, which would develop in a form of transverse waves propagating along the strings. The group speed is always along the direction of the string, since it is the direction in which the energy and information are transferred by the wave. But the “wave front” produced by such a perturbation could be oblique to the direction of the strings, for example, if the individual strings are pitched at different distances from the guitar’s nut. In this case, the phase speed of the excited propagating perturbation is oblique to the strings, and hence the group and phase velocities are not parallel to each other.

The absolute value of the group velocity equals the Alfvén speed, $vA$.

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,

$Display mathematics$(2)

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

$Display mathematics$(3)

where $CA$ 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

$Display mathematics$(4)

where $cs=(γp0/ρ0)1/2$ is the sound speed, $α$ is the angle between the wave vector and the magnetic field, and $γ$ 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, $∇.v≠0$. Equation 4 is bi-quadratic with respect to $ω$ and $k$, and consequently has two pairs of roots for $ω2$ or $k2$. Solving it with respect to $ω2$, one obtains

$Display mathematics$(5)

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:

Magnetoacoustic waves are, in general, neither transverse nor longitudinal: The induced plasma flow has a component along the wave vector.

The waves are essentially compressive: They always perturb the density of the plasma.

Magnetoacoustic waves can, in general, propagate in all directions with respect to the magnetic field, while their properties depend strongly upon the angle and the plasma parameter $β$, which is the ratio of the plasma pressure ($p0$) to the magnetic pressure ($B02/(2μ)$) and so is proportional to the ratio of the squares of the sound and Alfvén speeds.

The fast wave propagates in a direction perpendicular to the field at the phase and group velocities $Vph$ and $Vgr$ with absolute values equal to the fast speed, $CF≡(vA2+cs2)1/2$. In the perpendicular direction, the phase speed of a slow wave goes to zero, while its parallel group speed tends to the tube (or cusp) speed,

$Display mathematics$

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

In the $β<1$ case, the fast wave cannot propagate along the field if $k∥B0$, the fast wave, becomes incompressive and purely transverse, and hence degenerates into the Alfvén wave. In the $β>1$ case, the parallel fast wave is the acoustic wave, propagating at the sound speed. In the $β<1$ case, the parallel slow wave propagates along the field at the speed $cs$ and does not perturb the magnetic field, and hence degenerates to the usual sound wave propagating along the field. For all values of $β$, the slow wave cannot propagate across the field.

For the oblique slow wave, the density and the absolute value of the magnetic field are perturbed in anti-phase, while in the fast wave they are in phase.

In the zero-$β$ limit, which describes well a number of important natural and laboratory plasmas, the slow wave ceases to exist. The fast wave propagates at the Alfvén speed in all directions (while in the parallel propagation case, it degenerates into an Alfvén wave). Because of that, in the zero-$β$ limit, the fast magnetoacoustic wave is often referred to as a compressional Alfvén wave.

In the $β<1$ case, the largest perturbation of the plasma in all fast waves and the Alfvén wave is transverse to the field.

#### 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 $ω2=0$, 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 $α$ 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 $α$, 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.

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”).

### 3. Magnetohydrodynamic Modes of a Plasma Cylinder

In the majority of natural and laboratory equilibria, plasmas are found to have nonuniform configurations of density, gas pressure, and magnetic field. The nonuniformity of the plasma dramatically complicates the dynamics of magnetohydrodynamic (MHD) waves. Here, the effect of plasma structuring on MHD waves is illustrated by the important case of a plasma cylinder (or a straight magnetic flux tube), which is an elementary building block for various plasma systems. For example, a plasma cylinder models a segment of a plasma loop in a solar coronal active region, a sunspot, a plume in a coronal hole, or a prominence fibril.

Consider a straight cylinder of radius $a$, filled with a static uniform plasma of density $ρ0$ and pressure $p0$. An untwisted uniform magnetic field is directed along the axis of the cylinder, $B0ez$. The cylinder is surrounded by a plasma with the uniform density $ρe$ and pressure $pe$. The magnetic field ($Beez$) outside the cylinder is directed along the cylinder’s axis too. Parameters of the plasma inside and outside the cylinder are linked with each other by the total pressure balance condition,

$Display mathematics$(7)

In terms of the characteristic wave speeds, condition (7) determines the density ratio

$Display mathematics$(8)

where $Cs0$ and $Cse$ are the sound speeds inside and outside the cylinder, respectively, and $CA0$ and $CAe$ are the internal and external Alfvén speeds, respectively.

Consider small perturbations of this equilibrium. In the direction along the axis and in the azimuthal direction, one can make a Fourier transform, introducing a parallel wave number $kz$ and an integer azimuthal wave number $m$. The latter parameter determines the azimuthal symmetry of the perturbations, which strongly affects the wave properties. Axisymmetric perturbations, with $m=0$, are usually called sausage modes. Perturbations with $m=1$ are kink modes. They displace the axis of the cylinder, and hence, depending on the polarization, can be linearly, elliptically, or circularly polarized. Perturbations with $m>1$ are called fluting or ballooning modes. Distortion of the cylinder by modes with $m=0$, $1$, and $2$ is shown in Figure 2.

Dispersion relations for the modes and phase relations between the perturbations of different physical quantities in these modes could be obtained by solving linearized MHD equations inside and outside the cylinder. As the medium is not uniform in the radial direction, one cannot make a Fourier transform in that direction. Therefore, the perturbations are described by ordinary differential equations of Bessel type with respect to the radial coordinate $r$. In the external medium, the perturbations should decrease with $r$ faster than $1/r$. Inside the cylinder, the perturbations must not tend to infinity at $r=0$. Those conditions determine the choice of appropriate Bessel functions of order $m$, that is, $Km(x)$ outside the cylinder and $Im(x)$ inside it.

External and internal solutions are matched with each other at the cylinder’s boundary by the conditions of continuity of the radial displacement and of total pressure, giving the following dispersion relations for magnetoacoustic modes of a plasma cylinder:

$Display mathematics$(9)

where $κ0$ and $κe$ are the effective radial wave numbers defined as

$Display mathematics$(10)

with the index $α=0$ or $e$ for the internal or external medium, respectively; and $CTα$ are internal and external tube speeds. The prime denotes a derivative of the corresponding Bessel function with respect to its argument. In the context of the solar atmosphere, dispersion relations (9) were first derived and studied in the seminal papers of Zajtsev and Stepanov (1975) and Edwin and Roberts (1983).

For the modes that are evanescent outside the cylinder, called trapped modes, the condition $κe2>0$ has to be fulfilled. It corresponds to the threshold for total internal reflection at the cylinder’s boundary. For $κe2<0$, the modes become leaky and propagate away from the cylinder. Their radial structure outside the cylinder is given by Hankel functions, $Hm(x)$, instead of the MacDonald functions $Km(x)$. In this regime, the cylinder operates as a magnetoacoustic antenna. Inside the cylinder, the value of $κ02$ could, in general, be either positive or negative. In the former case, the internal solutions are given by modified Bessel functions $Im(x)$, which monotonically decrease from the boundary to the axis. Those modes are known as surface modes. In the latter case, $κ02<0$, the internal solution is given by Bessel functions $Jm(x)$. In this regime, there exist body modes that have an oscillatory structure in the radial direction. The number of radial oscillations in a body mode determines the radial mode number $nR$.

The presence of a characteristic spatial scale, the radius of the cylinder $a$, produces wave dispersion. In particular, the phase and group speeds depend on the ratio of the parallel wavelength and the radius $a$ and the corresponding frequency of the mode of interest. This effect is most pronounced for perturbations with parallel wavelengths approximately equal to $a$. As the cylinder acts as a waveguide, globally both the phase and group speeds of the MHD modes are directed along its axis. However, locally the waves are oblique. In particular, this effect allows the fast wave to propagate along the magnetic field. The existence and properties of the modes are determined by the specific equilibrium. For example, a coronal plasma loop has trapped and leaky body waves if the external Alfvén speed is greater than internal Alfvén speed, but it does not have surface modes.

The waves of different azimuthal symmetry (i.e., corresponding to different values of $m$) have rather different physical properties. The lowest $nR$ modes (except the $m=0$ mode) are trapped for all values of $kz$. In the long wavelength limit, $kza→0$, their phase and group speeds tend to the kink speed,

$Display mathematics$(11)

Specific values of $κe2$ and $κ02$ are determined by the ratios of the phase speed to the Alfvén and sound speeds. An example of the dispersion plot for trapped MHD modes of a typical plasma loop of the solar corona, with $Cse, is shown in Figure 2. The trapped modes appear in two intervals of phase speed, $[CT0,Cs0]$ and $[CA0,CAe]$, which correspond to fast and slow bands, respectively. Fast waves are mainly characterized by radial flows. Perturbations of the magnetic and gas pressures are in phase in those waves. Fast modes experience stronger dispersion than slow modes. Trapped sausage modes do not exist at longer wavelengths, for $kz, with a cutoff wave number

$Display mathematics$(12)

where $j0,s$ are the $s$-th zeros of the Bessel function $J0(x)$. At the cutoff, the phase speed approaches the external Alfvén speed. For longer wavelengths, sausage modes become leaky. The frequency of a leaky mode is complex and its amplitude inside the cylinder exponentially decreases in time. This effect is known as radiative damping of the waves. All other fast magnetoacoustic modes with $nR>1$ have long wavelength cutoffs as well.

Properties of slow modes of different $m$ are not much different from each other, and hence those modes are all considered to be of the same, longitudinal, kind. Slow modes are characterized by the domination of parallel flows. In the long wavelength limit, the longitudinal modes are essentially oblique, as their perpendicular wavelength, which is about the radius of the cylinder, becomes much smaller than the parallel wavelength. In this limit, the phase and group speeds of longitudinal waves are about the tube speed $CT0$. Perturbations of the magnetic and gas pressures are in anti-phase.

In addition, the plasma cylinder supports torsional waves described by the dispersion relation $ω2=CA02kz2$. These are characterized by the propagating azimuthal twisting and untwisting of the magnetic field, accompanied by alternate rotational azimuthal plasma flows. The radial structure of torsional waves is arbitrary, determined by the initial perturbation only. Torsional waves are dispersionless and incompressive and hence are of Alfvénic nature. Torsional waves do not perturb the boundary of the cylinder, as the magnetic tension and centrifugal forces in them are opposite and exactly equal to each other.

All magnetoacoustic modes are collective, that is, they perturb the whole plasma cylinder and its surroundings, and their properties are determined by parameters of the plasma both inside and outside the cylinder, even if the plasma parameters vary smoothly across the cylinder. In other words, magnetoacoustic modes carry information about the plasma parameter in the whole cross section of the waveguide. In addition, perturbations associated with a certain magnetoacoustic mode propagate at the same phase and group speeds everywhere. In contrast, torsional waves are not collective, as they perturb the plasma at the magnetic surface that hosts them only. In an ideal plasma, torsional waves situated at the neighboring magnetic surfaces are fully uncorrelated. Likewise, properties of torsional waves are determined by the properties of the hosting magnetic surface only. In particular, the propagation speeds of torsional waves localized at different radii from the axis of the cylinder are different if the Alfvén speed varies with radius.

Often, the plasma cylinder has a finite length. For example, plasma loops in solar coronal active regions begin and end at the so-called footpoints at the photosphere. At the footpoints, the magnetic field is anchored in the dense plasma, and hence they can be considered as rigid wall boundaries for MHD waves. In this case, the waves become standing and the parallel wave numbers $kz$ become discrete. The modes with the lowest values of $kz$ are called fundamental or global modes. In general, standing MHD modes of a loop are characterized by a triplet of discrete wave numbers, $kz$, $m$, and $nR$.

### 4. Effects of the Smooth Perpendicular Plasma Nonuniformity

#### 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 $ez$, with the Alfvén speed $vA(x)$ 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,

$Display mathematics$(13)

with solutions $vy=Ψ(x)f(z∓vA(x)t)$, where $f(z)$ and $Ψ(x)$ 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 $x=const$ 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 $x=const$ 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.

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 $∂/∂x≫∂/∂z$, the Alfvén wave decays as

$Display mathematics$(14)

where $ν$ is the coefficient of viscosity and/or resistivity (Heyvaerts & Priest, 1983). The damping $exp(−z3)$ 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-$β$ plasma with density $ρ0(x)$ nonuniform in the $x$ direction, across the field $B0=B0ez$. 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 $ξx=ξ^xsin(nπz/L)$, with a positive integer parallel wave number $kz=n$, one obtains for oblique linear harmonic perturbations

$Display mathematics$(15)

where $ϵAn(x)=ω2/vA2(x)−(nπ/L)2$, and $ky$ is the wave number in the $y$ direction. Equation 15 has a singularity if the coefficient in front of the highest derivative becomes zero, $ϵAn=0$, constituting an Alfvén continuum in the plasma. For a given frequency $ω=ω0$, a resonance occurs at a certain surface $x=x0$ determined by the condition $CA(x)2(nπ/L)2=ω02$. The surface $x=x0$ is called the Alfvén resonance layer, determined by the wave frequency $ω0$. 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 $x=x0$. 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-$β$ effects could lead to the appearance of a lower frequency continuum, associated with the variation of the tube speed $CT(x)$ 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 $τD$ to the oscillation period $P$, determined by the imaginary and real parts of the frequency, respectively, is

$Display mathematics$(16)

where $l$ is the width of the resonant layer, determined by the steepness of the smooth radial profile of $vA$, 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. Nonlinear Effects

When amplitudes of magnetohydrodynamic (MHD) perturbations are not negligible in comparison with their equilibrium values, nonlinear effects come into play, leading to nonlinear coupling of modes of different kinds and the transfer of energy across the wave spectrum. A convenient method for the description of nonlinear effects is to restrict attention to weakly nonlinear phenomena, considering weak nonlinear corrections to linear MHD solutions. This technique simplifies the study to the analysis of a reduced, evolutionary equation in which different physical phenomena affecting the wave are described by different terms. Evolutionary equations are universal (i.e., they appear in completely different branches of science) and allow for an effective knowledge transfer between different research fields.

#### 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 $α$ to a uniform magnetic field in a uniform plasma is governed by the inviscid Burgers equation,

$Display mathematics$(17)

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 $ϵB$ is determined by the angle $α$, and the Alfvén and sound speeds,

$Display mathematics$(18)

This equation becomes invalid for the fast wave in the $α→0$ 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 $ω+ω=2ω$, $k+k=2k$. In the absence of dispersion, the excited harmonic $(2ω,2k)$ belongs to the same dispersion curve as the original harmonic $(ω,k)$ 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 $2ω$ 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,

$Display mathematics$(19)

where the coefficients $ϵBLR$, $ϵDLR$, and $λ$ 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 $ω+ω−ω=ω$, $k+k−k=k$. In other words, the nonlinearly induced perturbation with $2ω$ and $2k$ immediately interacts with the perturbation with $ω$ 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 $A(z,t)$ describing the envelope is governed by the nonlinear Schrödinger equation,

$Display mathematics$(20)

where $Vgr$ is the group speed, the coefficient $ϵδNSE=(1/2)(d2ω/dk2)$ 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 $ϵNNSE$ 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 $B0$ are governed by the set of coupled Cohen–Kulsrud equations,

$Display mathematics$(21)

with the nonlinear coefficient $ϵCK=[4B02vA(vA2−cs2)]−1$ (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 $Bx$ and with $By=0$. The same simplification occurs for a torsional wave that perturbs $Bθ$ 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 $ϵCK$ is sensitive to the difference between the Alfvén and sound speeds (or the plasma $β$). 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 $ϵCK$ is modified. The denominator of $ϵCK$ has $CT$ instead of $cs$, 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

$Display mathematics$(22)

where the indices label different interacting waves. The complex amplitudes $U1,2,3$ of the interacting waves are governed by the set of equations

$Display mathematics$(23)

where the integer indices label different interacting waves, and the nonlinear coefficients $ϵN{1,2,3}$ 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 $(t∞−t)−1$, where $t∞$ is a constant. The energy comes from the mechanism that supports the flow.

### 6. Observations of Magnetohydrodynamic Waves in the Solar Corona

This section presents several spectacular examples of observational detections of magnetohydrodynamic (MHD) waves, resolved in both time and space, in the corona of the Sun. Typical time periods of coronal MHD waves are a few minutes, and typical wavelengths are from tens to hundreds of Mm, which allows for their confident study with spaceborne extreme ultraviolet (EUV) and soft-X-ray imagers and ground-based radioheliographs. A possible role of MHD waves in the heating of the solar coronal plasma is addressed in Klimchuk (2019).

#### 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 $2L/CK$. The linear scaling of the oscillation period with the loop length gives an estimation of the average kink speed, of about $1,300kms−1$. 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.

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 $kms−1$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).

### 7. Conclusions

The elasticity and compressibility of a plasma in the presence of a frozen-in external magnetic field lead to the occurrence of natural oscillatory motions around an equilibrium. Long-period, large-wavelength perturbations that satisfy the magnetohydrodynamic (MHD) conditions are MHD waves. In ideal MHD in a uniform plasma, MHD waves are classified as incompressive Alfvén waves, compressive fast and slow magnetoacoustic waves, and non-propagating entropy waves. Properties of these waves depend strongly on the Alfvén and sound speeds, and also on the angle between the wave vector and the equilibrium magnetic field. In a uniform plasma, in ideal MHD, all MHD waves are dispersionless.

The dynamics of MHD waves is strongly affected by nonuniformity of the plasma. In particular, it leads to the occurrence of a waveguiding effect. An MHD waveguide can support a number of modes whose properties depend on the structure of the perturbation, such as the azimuthal wave number in the case of a plasma cylinder. Some of those modes are highly dispersive. In addition, a smooth perpendicular nonuniformity of the Alfvén speed causes Alfvén wave phase mixing, occurrence of the Alfvén continuum, and linear coupling of magnetoacoustic and Alfvén waves.

Effects of finite amplitude lead to the nonlinear interaction of MHD waves of different kinds as well as wave self-interaction. The behavior of weakly nonlinear MHD waves is discussed in terms of evolutionary equations. Applications of MHD wave theory are illustrated by kink and longitudinal waves in plasma loops in the solar corona. The combination of state-of-the-art observations and MHD wave theory allows for the remote diagnostics of plasma parameters by the method of MHD seismology.

The discussed wave phenomena are intrinsic to MHD waves in various environments, but the terminology and the implications of these phenomena could, of course, be specific. Comprehensive reviews of MHD wave phenomena in other environments can be found elsewhere, in particular, in chromospheric magnetic flux tubes (Verth & Jess, 2016); the solar wind (Ofman, 2016; Cranmer, 2019); Earth’s inner magnetosphere (Takahashi, 2016) and geomagnetic tail (Leonovich et al., 2016); the interstellar medium (Ptuskin et al., 2006); and laboratory plasmas (Chen & Zonca, 2016). Promising future research avenues could be opened up by the effective transfer of the knowledge between different plasma environments. In particular, there is fruitful cross-fertilization between magnetospheric and fusion plasmas (Chen, 2008) and between magnetospheric and coronal plasmas (Nakariakov et al., 2016).

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