Most textbooks on Magnetohydrodynamics (MHD) and plasma physics contain at least an elementary discussion of MHD waves in a uniform plasma of infinite extent (see e.g., Thompson, 1964; Mestel and Weiss, 1974; Goedbloed, 1983; Goossens, 2003; Goedbloed and Poedts, 2004; Walker, 2004). It is shown that the MHD waves are either Alfvén waves or slow/fast magneto-sonic waves. The Alfvén waves are incompressible and propagate parallel vorticity. They do not have a parallel component of displacement and are driven by magnetic tension only. The magneto-sonic waves are compressible and have a parallel component of displacement. They do not propagate parallel vorticity and are driven by pressure and magnetic tension. In non-uniform plasmas the situation can be very different. The clear division between Alfvén waves and magneto-sonic waves is no longer present. The MHD waves have mixed properties in non-uniform plasmas. Mixed properties mean that the general rule is that MHD waves propagate both parallel vorticity as in classic Alfvén waves and compression as in classic magneto-sonic waves. This behavior causes exciting wave physics. For instance, the phenomenon of MHD waves with mixed properties can lead to damping, with relevance in explaining the attenuation observed in coronal and prominence oscillations and discussed by Goossens et al. (2002a, 2011), Terradas et al. (2006), Arregui et al. (2008), Pascoe et al. (2010, 2011), among many others. The use of the information on wave damping has also been found useful to perform solar coronal seismology (see e.g., Goossens et al., 2002a, 2008; Arregui et al., 2007; Goossens, 2008). The mixed properties arise because in an inhomogeneous plasma the Eulerian perturbation of total pressure couples with the dynamics of the motion (Hasegawa and Uberoi, 1982). Mathematically this is translated into the fact that the differential equations for the radial component of the Lagrangian displacement ξ_r and the Eulerian perturbation of total pressure P′ are coupled to algebraic equations for compression ∇·ξ→, the parallel and perpendicular projections of the Lagrangian displacement ξ_∥, ξ_⊥, and vorticity ∇×ξ→ ¹. The coupling of the equations is due to the coupling functions C_A and C_S which were introduced by Sakurai et al. (1991b) in their study of resonant absorption. The relevance of the coupling functions goes beyond resonant absorption. The spatial behavior of the coupling functions and of the local Alfvén frequency ω_A and local cusp frequency ω_C determine the spatial behavior of the various components of velocity and vorticity and of compression. The simultaneous presence of compression and vorticity is hard to avoid.

Goossens et al. (2009) investigated the forces that drive these waves and found that the magnetic tension force always dominates the pressure force for the kink mode. In addition, they showed that compression is small in the particular case of thin tubes. Hence, these waves do not have the typical properties of fast magneto-sonic waves and behave more as Alfvén waves. Goossens et al. (2011) reconsidered these waves in their section on quasi-modes and decided to call them surface Alfvén waves. In the present paper, we continue the theoretical investigation of the nature of the waves. In section 2, we describe pure Alfvén and pure magneto-acoustic waves in a uniform plasma of infinite extent, by analysing their eigenfrequencies, eigenfunctions, vorticity and compression. In section 3, the analysis is generalized to MHD waves in non-uniform plasmas, which propagate both compression and parallel vorticity at the same time. This leads to new expressions for the components of vorticity that are derived for axi-symmetric/non-axi-symmetric motions in a non-uniform 1-dimensional cylindrical plasma. In section 4, we show that resonant Alfvén /slow waves are characterized by strong shear in the perpendicular/parallel component of displacement with large values of the parallel/perpendicular component of vorticity. This strong shear causes violent KH-instabilities (Terradas et al., 2008; Antolin et al., 2018) that accelerate the damping of the MHD waves and facilitate heating of plasma (Antolin et al., 2015; Arregui, 2015; Terradas and Arregui, 2018).

The properties of MHD waves in a uniform plasma of infinite extent are often used to characterize MHD waves in general. For a uniform plasma of infinite extent the MHD waves can be subdivided into two classes with distinct properties. The first class contains the magneto-sonic waves. They are compressive but do not propagate parallel vorticity. The second class contains the Alfvén waves. Alfvén waves propagate parallel vorticity and are incompressible. The equilibrium quantities are constant. The constant magnetic field

is used to define the direction of the z- axis of a Cartesian system of coordinates. The equilibrium density and pressure are constant

In what follows ξ→ is the Lagrangian displacement. In the present subsection the background is static and uniform. As a consequence solutions can be obtained in the form of plane harmonic waves and ξ→ is written

Here ξ^→ is the constant amplitude of ξ→, k→=kx1→x+ky1→y+kz1→z is the wave vector, and ω is the frequency of the wave. In what follows the hat on ξ→ will be dropped. Since the constant magnetic field defines a preferred direction a clever choice of dependent wave variables is X, Y, Z defined as

X, Y, Z are dimensionless quantities and allow us to obtain an elegant version of the governing equations. In terms of these variables the equations for linear ideal MHD waves can be written as

v_A, v_S are the Alfvén velocity and the velocity of sound. They are defined by

ω_A is the local Alfvén frequency. It is defined as

In a uniform plasma v_A, v_S, ω_A are constant. In a non-uniform plasma these quantities depend on position.

The system (5) consists of two uncoupled subsets of equations. The first subset is the third equation for the variable Z. The second subset contains the wave variables ξ_z and Y. The first type of MHD waves are characterized by

They are the classic Alfvén waves. The eigenfrequencies associated with the Alfvén waves (8) are infinitely degenerate as they only depend on the parallel component of the wave vector k→. Alfvén waves do not cause compression and have no component of the displacement parallel to the magnetic field. They are the only waves that propagate parallel vorticity in a uniform plasma of infinite extent. The only restoring force is the magnetic tension force. Note also that Alfvén waves in a uniform plasma of infinite extent exist for any wave vector k→=(kx,ky,kz).

The displacement ξ→ for Alfvén waves is

For k_y = 0 we obtain the popular result ξ→A=ξy1→y. These y- independent Alfvén waves are a special case. In the cylindrical case k_y = 0 and k_y ≠ 0 correspond to respectively axisymmetric waves with m = 0 and to non-axisymmetric waves with m ≠ 0 with m the azimuthal wave number. For a wave vector with both horizontal components of the wave vector different from zero both horizontal components of the displacement vector are non-zero. Let us now keep k_y ≠ 0, k_z ≠ 0 and mimic a situation with non-uniformity in the x- direction and a resonant condition where lim k_x → +∞ so that ∣k_y ∣ ≪ ∣k_x ∣, ∣k_z ∣ ≪ ∣k_x ∣. Find then

The motion in the Alfvén wave is predominantly in the y- direction and rapidly varying in the x- direction. The displacement (10) is not y- independent because of the factor exp(ik_yy) with k_y ≠ 0. The ≈ sign means that the two components (ξ_x, ξ_y) are non-zero but ξ_y is far larger in absolute value than ξ_x. The two components are needed to satisfy the incompressibility condition.

For a general wave vector k→=(kx,ky,kz)t the three components of vorticity ∇×ξ→ are non-zero. In addition to the parallel component (∇×ξ→)z also the components in planes normal to B→0 are non-zero:

For our later discussion on resonant Alfvén waves it is instructive to look at the components of vorticity ∇×ξ→ under conditions that mimic resonant behavior, i.e., when ∣k_y ∣ ≪ ∣k_x ∣, ∣k_z ∣ ≪ ∣k_x ∣ and find that

Hence

so that

Here also the ≈ sign means that the three components (∇×ξ→) are non-zero but the parallel component is far larger in absolute value than the two horizontal components.

The second class of MHD waves corresponds to

They are the magneto-sonic waves. They cause compression but do not propagate parallel vorticity. However, they cause horizontal vorticity. Their displacement has a component parallel to the magnetic field that is driven by the magnetic pressure force. The dispersion relation is

The well-known solutions for the eigenfrequencies are

k2=kx2+ky2+kz2, ω_C, and v_C are the cusp frequency, and the cusp velocity.

In Equation (15) “sl” corresponds to the minus sign, and “f” to the plus sign. The corresponding waves are the slow and fast magneto-sonic waves. The frequencies of the magneto-sonic waves depend on the three components (k_x, k_y, k_z) of the wave vector k→. They depend in the same way on k_x and k_y because of isotropy in the planes normal to B→0. It is instructive to consider the variation of ωsl,f2 as function of k_x for fixed values of (k_y, k_z). The cut-off frequencies ω_I, ω_II are defined as

Also

The cut-off frequencies ω_I, ω_II and the characteristic frequencies ω_A, ω_C obey the sequence of inequalities

Hence the spectrum of linear motions of a uniform plasma of infinite extent can be divided in a slow subspectrum ]ω_C, ω_I], a degenerate Alfvén point spectrum ω_A and a fast subspectrum [ω_II, +∞[. The first equality in (18) means that ω_C is an accumulation point of the slow subspectrum.

The magneto-sonic waves are driven by tension and pressure forces and cause variations in density and pressure and horizontal vorticity.

The solutions for the eigenfunctions are

or equivalently,

The popular view is that the horizontal motion (ξ_x, ξ_y) is the dominant motion for fast waves while the parallel motion ξ_z is the dominant motion for slow waves. In order to point out that this is not the general rule, ξ_x is used as the measuring unit in (20) and ξ_z in (21). It is straightforward to show that in general the parallel component in (20) is not small compared to the horizontal components, and similarly that the horizontal components in (21) are not per se much smaller than the parallel component. However, for strong magnetic fields, i.e., v_A ≫ v_S it can be shown that

The popular view corresponds to the limiting case of a strong field.

The parallel component of vorticity (∇×ξ→)z=iZ is of course zero. However, the horizontal components are non-zero

For k_y = 0 the expressions (20) for the displacement ξ→ and (23) for vorticity ∇×ξ→ can be simplified to

Keep k_y ≠ 0, k_z ≠ 0 and finite and mimic a situation with non-uniformity in the x- direction and a turning point where k_x = 0 and find

In summary for a uniform plasma of infinite extent the division is clear. The equation for vorticity is uncoupled from the equations for compression and parallel displacement. The waves have either parallel vorticity and no compression and no parallel displacement, these are the Alfvén waves, or they have compression and parallel displacement and no parallel vorticity, they are magneto-sonic waves. There are no waves with compression and parallel vorticity at the same time. There is no mixing of properties.

For a pressureless plasma with

the solutions for the magnetosonic waves are

In this situation there are no slow waves and the fast magneto-sonic waves have no parallel motions. The parallel motions are driven by the gradient of plasma pressure and here plasma pressure vanishes by assumption. The absence of slow waves and of parallel motions is a general result for a pressureless plasma. In what follows, no particular attention will be devoted to pressureless plasmas. The equations for MHD waves for a pressureless plasma are easily obtained by putting vS2=0 in the general equations.

The aim of the present section is to show that MHD waves in a non-uniform plasma have mixed properties. In general they propagate compression and parallel vorticity at the same time. The phenomenon of mixed properties follows from the fact that the equations that describe the linear motions are coupled, unlike for the case of a uniform plasma of infinite extent. In particular the focus is on MHD waves on 1-D cylindrical plasma columns. The equilibrium model is a straight cylindrical plasma column of radius R in static equilibrium. In what follows we use cylindrical coordinates r, φ, z. The magnetic field has both an axial and an azimuthal component

The equilibrium density ρ₀(r), equilibrium pressure p₀(r), and the components of the equilibrium magnetic field B_z,0(r), B_φ,0(r) are functions of r or constant. The equilibrium quantities satisfy the equation of static equilibrium

In a nonuniform plasma vS2, vA2, ωA2, and ωC2 are functions of position. In what follows f′ and δf denote respectively the Eulerian and Lagrangian variation of a quantity f. In linear theory they are related as

with f₀ the equilibrium value of f. In the following equations P′=p′+B0→·B′→/μ is the Eulerian perturbation of total pressure; p′ is the Eulerian perturbation of plasma pressure. ξ→ is the Lagrangian displacement.

We use the mixed field line / magnetic surface triad (b→,n→,π→) defined by Goedbloed et al. (2010) in their Equations (17.23). In the present case of a straight cylindrical flux tube with the equilibrium magnetic field B0→ defined in the Equation (28)

with 1→∥,1→⊥ the unit vectors in the magnetic surfaces respectively parallel and perpendicular to the magnetic field lines.

ξ_r is the radial component of Lagrangian displacement and ξ_∥, ξ_⊥ are the projections of the Lagrangian displacement in the magnetic surfaces parallel and perpendicular to the magnetic field lines:

Since the equilibrium quantities are independent of φ and z the wave variables can be put proportional to the exponential factor exp[i(mφ + k_zz)] with m, k_z the azimuthal and axial wave numbers, m is an integer. For example, for the Lagrangian displacement we write

ξ^→(r) is the radially varying amplitude of ξ→. In what follows the hat on ξ→ and on the other wave variables will be omitted. It is convenient to introduce the wave vector k→=(0,m/r,kz).

The linear MHD waves can be described by two ordinary differential equations for ξ_r and P′ (see e.g., Appert et al., 1974; Sakurai et al., 1991b; Goossens et al., 1992, 1995)

The coefficient functions D, C₁, C₂, C₃ are given by

v_A and v_S are the Alfvén speed and the speed of sound as before and are defined in Equation (6). In a non-uniform plasma they are functions of position. The quantities f_B and g_B are defined as

The frequencies ω_A and ω_C are the local Alfvén frequency and the local cusp frequency as before. They are defined for the planar case in Equation (7). Here in the cylindrical case their squares are defined as

Note that ω_A and ω_C are functions of position. For a given set of wave numbers (m, k_z) ω_A and ω_C map out two ranges of frequencies known as the Alfvén continuum and the cusp continuum. The frequencies ω_I, ω_II are defined as

They are the cylindrical analogues of the Cartesian cut-off frequencies defined in (17). Here they are not cut-off frequencies but rather frequencies that restrict the Sturmian or anti-Sturmian behavior of the spectrum as explained by Goedbloed (1975, 1983).

To emphasize that parallel motions are solely driven by the gradient plasma pressure force, the parallel component of the equation of motion is written as

δp is the Lagrangian variation of plasma pressure.

For the discussion of the mixed properties it is necessary to look at the wave variables ξ⊥,ξ∥,∇·ξ→ and (∇×ξ→). They are given by expressions in ξ_r and P′ and their derivatives. Algebraic expressions for ξ⊥,ξ∥,∇·ξ→ can be found in e.g., Sakurai et al. (1991b)

The coupling functions are defined as (see e.g., Sakurai et al., 1991b)

They are linear combinations of P′ and ξ_r. The coefficients of ξ_r in C_A and C_S vanish when the equilibrium magnetic field is straight B_φ,0 = 0. C_A depends on the azimuthal wave number m and the longitudinal wave number k_z. C_S on the other hand is independent of the wave numbers (m, k_z). The coupling functions play an essential role for the mixing properties of MHD waves and for resonant absorption. They are called coupling functions for the good reason that they couple the differential equations (33) for ξ_r and P′ to the expressions for all of the remaining wave variables ξ⊥,ξ∥,∇·ξ→,(∇×ξ→). First they couple the differential equations for ξ_r and P′ to the algebraic equations (39) for ξ⊥,ξ∥,∇·ξ→. When C_A ≠ 0 the first equation of (39) implies that ξ_⊥ ≠ 0. Similarly when C_S ≠ 0 the second and third equation of (39) imply that ∇·ξ→≠0. When in addition to C_S ≠ 0 also v_S ≠ 0 it follows that ξ_∥ ≠ 0.

Let us now consider (∇×ξ→). In section 2 it was pointed out that a division of linear waves can be based on compression, parallel displacement and parallel vorticity. A characterization based on the components (ξ_x, ξ_y, ξ_z) is in general not possible. When we move from Cartesian geometry to cylindrical geometry the horizontal components (ξ_x, ξ_y) are replaced by the components (ξ_r, ξ_⊥) in the planes normal to B→0 and ξ_z is replaced by the component ξ_∥ parallel to the equilibrium magnetic field. For a uniform plasma of infinite extent, the MHD waves could be divided into incompressible waves that propagate parallel vorticity, i.e., the Alfvén waves and waves that propagate compression and have a parallel displacement component i.e., the magneto-sonic waves. In what follows it will be shown that for a non-uniform plasma MHD waves propagate both compression and parallel vorticity and have non-zero radial, perpendicular and parallel components of displacement and vorticity. To the best of our knowledge expressions for the components of (∇×ξ→) are not available in the literature. They are

Expressions for P_⊥, P_∥, Q_⊥, Q_∥ are

The Equations (41) show that the components of (∇×ξ→) can be expressed in terms of (ξ_r, ξ_⊥, ξ_∥). Since ξ_⊥, ξ_∥ are expressed in terms of ξ_r and P′ it follows that also the components of (∇×ξ→) can be expressed in terms of ξ_r and P′. When (ξ_r, ξ_⊥, ξ_∥) are non-zero, the components of vorticity are in general also non-zero. All of the wave variables are coupled. The MHD waves have mixed properties, they propagate both compression and parallel vorticity and have non-zero radial, perpendicular and parallel components of displacement and vorticity. In general all wave variables are non-zero. A situation in which a subset of the wave variables is not coupled to the other wave variables is an exception. Such a situation will appear for axi-symmetric motions in the presence of a straight field. The clear division into Alfvén waves and magneto-sonic waves that exists for a uniform plasma of infinite extent does not any longer hold.

Hence in general for linear MHD waves on a non-uniform plasma

Let us consider the special case of axi-symmetric motions with m = 0. The expressions for f_B, g_B, C_A, C_S can be simplified to

For a twisted magnetic field with both a longitudinal component B_z,0 and a non-zero azimuthal component B_φ,0, the coupling functions C_A and C_S are non-zero. This implies that the preceding analysis on mixed properties also applies to axi-symmetric motions. The axi-symmetric motions propagate vorticity and compression. The situation is different when the magnetic field is straight.

Since C_A and C_S are functions of position the coupling of the equations depends on position and so does the strength of the mixing of the wave properties. For example a wave can start off as a predominantly fast wave, change into a wave that has both fast and Alfvén properties and turn into a predominantly Alfvénic wave. MHD waves have mixed properties and have different appearances in different parts of the plasma because of the inhomogeneity of the plasma. This phenomenon was discussed by Goossens et al. (2002b), Goossens (2008), and Goossens et al. (2011, 2012, 2014). Waves with mixed properties are also referred to as coupled MHD waves (Pascoe et al., 2010, 2011). This is a rather strange name as it seems to suggest that there are two or more waves involved.

Let us now focus on MHD waves in the presence of a straight field. For a straight field (B_φ,0 = 0) the magnetic surfaces are cylinders: r = constant, and the φ- and z- directions are the directions in the magnetic surfaces respectively perpendicular and parallel to the magnetic field lines. The r- direction is normal to the magnetic surfaces. Hence ξ_r is associated with motions normal or across magnetic surfaces; ξ_∥ = ξ_z are motions along the magnetic field lines and ξ_⊥ = ξ_φ are motions in the magnetic surfaces perpendicular to the magnetic field lines. For a straight field the expressions for f_B, g_B, C_A, C_S are simplified to

The coupling functions C_A, C_S only contain P′. The coefficients of ξ_r in C_A and C_S vanish when B_φ,0 = 0. Hence the coupling of the waves variables is solely due to P′ as will become clear in what follows. As far as the wave numbers (m, k_z) are concerned, C_A no longer depends on k_z, only on m.

The differential equations (33) for ξ_r and P′ and the algebraic equations for ξ⊥,ξ∥,∇·ξ→ (39) are now

Use now (41) and note that for a straight field

to obtain

Equations (45) and (46) govern the MHD waves on a non-uniform straight cylindrical plasma column with a straight magnetic field. There is a natural subdivision between, respectively, axi-symmetric motions with m = 0 and non-axisymmetric motions with m ≠ 0. The reason being that the equation for ξ_φ for m = 0 is decoupled from the remaining equations. Let us first focus on axi-symmetric motions with m = 0.

The equation for ξ_⊥ = ξ_φ is decoupled from the remaining equations

Equation (48) can be satisfied in two ways. First of all by choosing

The second choice is

The solutions given in (49) and (50) correspond respectively to the axi-symmetric Alfvén waves and the sausage magneto-sonic waves. The axi-symmetric MHD waves are decoupled in sausage magneto-sonic waves and axi-symmetric Alfvén waves. The solutions for the axi-symmetric magneto-sonic waves are

The solutions for the axi-symmetric Alfvén waves are

For an axi-symmetric non-uniform 1-dimensional cylindrical plasma this is the only case where pure Alfvén waves show up in the analysis. Each magnetic surface oscillates with its own local Alfvén frequency. In a twisted magnetic field, C_A ≠ 0 for m = 0 so that the equations are coupled and the corresponding MHD waves have mixed magneto-acoustic and Alfvén properties. Also C_S ≠ 0 for any azimuthal wave number m. The absence of pure Alfvén waves in a non-uniform 1-D cylindrical plasma for azimuthal wave numbers m ≠ 0 is in stark contrast to the situation for a magnetic flux tube with piece wise constant density and magnetic field. Spruit (1982) showed that solutions with ∇·v→=0 exist for any m. H. Spruit correctly identified these solutions as Alfvén waves. Flow patterns for Alfvén waves with m = 0 and m = 1 are shown on Figure 1 in Spruit (1982). In addition to the Alfvén waves there are compressive waves. The fact that pure non-axisymmetric Alfvén waves do not exist in a non-uniform straight plasma cylinder is an illustration of how the non-uniformity produces waves with mixed properties.

Let us now turn back to the non-axisymmetric MHD waves with m ≠ 0. Actually the analysis also holds for axi-symmetric MHD waves with ξ_φ = 0. Excluded from the analysis are the axi-symmetric Alfvén waves defined in (52). The Equation (46) can be rewritten as

Note that the expressions for the components of vorticity for axi-symmetric magneto-sonic waves can be obtained from (53) by putting m = 0.

Here all wave variables are coupled and all wave variables are non-zero. In case of a straight field, it is the non-zero Eulerian perturbation of total pressure P′≠0 that produces MHD waves with mixed properties reminiscent of Alfvén waves and magneto-sonic waves. See also the comments by Hasegawa and Uberoi (1982) in their Chapter 3 on MHD waves in an inhomogeneous medium.

Special interest goes to the components of ∇ × ξ. It is obvious that (∇ × ξ)_r ≠ 0 irrespective if the equilibrium is uniform or not. The same is true for (∇ × ξ)_φ. The second term is always non-zero. The first term is non-zero for a non-uniform equilibrium and for a piece-wise constant density model the derivative results in a delta-function contribution. The parallel component (∇ × ξ)_z is non-zero for a non-uniform equilibrium with

different from zero. In a fully non-uniform equilibrium this condition is satisfied everywhere. In a piece-wise constant density model the derivative results in a delta-function contribution.

Let us try to understand the cause of the vorticity. The equilibrium model is a 1-D straight cylinder with the equilibrium quantities functions of the radial distance r to the axis. There is no baroclinic source of vorticity since the iso-surfaces of density and pressure coincide. Equations (41) combined with the expressions for P_⊥, P_∥, Q_⊥, Q_∥ in principle contain all the information. They are rather complicated and do not allow a straightforward interpretation. Physical insight can be gained by considering the case of a straight field. For a straight field the equation of motion in the horizontal planes follows from the 2nd and 3rd equations of Equation (45).

ξ→h is the displacement in horizontal planes and ∇_h is the gradient operator in horizonal planes

The left hand side of Equation (55) is mass density times acceleration. The first term in the right hand side of Equation (55) is the horizontal gradient total pressure force; the second term is mass density times the magnetic tension force

Hence