The presence of the internal spatial scale, the radius of the tube , brings wave dispersion. The standard derivation of linear dispersion relations is based upon linearisation of MHD equations around the equilibrium. The following system of first order differential equations and algebraic equation governs the behaviour of linear perturbations of the form (Sakurai et al., 1991a):
and where and are the perturbation displacements in the radial and azimuthal direction, respectively. The quantity is defined as and plays the role of the transverse wave number and is defined as In each medium separately, the system of Equations (2, 3) can be reduced to the equation where . The first term of Equation (6) represents torsional Alfvén wave solutions with . The second term is a Bessellike equation that describes magnetoacoustic wave modes. External and internal solutions of this equation have to be matched by the use of jump conditions: the continuity of total pressure and the normal velocity (see, e.g., Roberts, 1981a,b). Furthermore, a condition of mode localisation is applied, requiring that the wave energy should decline with a lateral distance from the structure (tube or slab). In the presence of a steady flow, the condition of continuity of normal velocity is replaced by continuity of the transverse displacement (see, e.g., Nakariakov and Roberts, 1995a). Applying the boundary conditions to the solutions of the Bessel equation leads to the dispersion relation for magnetoacoustic waves in a magnetic flux tube (Edwin and Roberts, 1983; see also Roberts and Nakariakov, 2003 and references therein) and are modified Bessel functions of order , and the prime denotes the derivative of a function or with respect to argument . The functions and are the transverse wave numbers in the external and internal media, respectively, which are obtained from Equation (5) by the substitution of the appropriate characteristic speeds. For modes that are confined to the tube (evanescent outside, for ), the condition has to be fulfilled. In the equation, it is assumed that . The integer determines the azimuthal modal structure: waves with are called sausage modes, waves with are kink modes, waves with higher are sometimes referred to as flute or ballooning modes. The existence and properties of the modes are determined by the equilibrium physical quantities. In particular, a coronal loop or a filament can trap MHD waves if the external Alfvén speed is greater than internal.

In closed fields of coronal active regions, the longitudinal wave number of standing modes is usually prescribed by the linetying boundary conditions at the photosphere. Modes with the lowest wave numbers are called global or fundamental.
The magnetoacoustic modes (with an important exception) are collectively supported by the plasma environment, i.e., the wave mode acts across neighbouring magnetic field lines and across transverse plasma inhomogeneities. Alfén waves, though, are locally supported. They have phase and group velocities, with magnitudes equal to the local Alfvén speed, which are directed along the magnetic field. This means that an Alfén wave propagates along its local magnetic field line without interaction with neighbouring field lines. This particular property allows for the existence of continua of eigenfrequencies and which will be discussed in Section 2.2.
In a cylinder model, Alfvén waves are torsional waves that twist tube. In the case of a straight cylinder these modes are incompressible, however in a slightly twisted cylinder they are accompanied by perturbations of plasma density (Zhugzhda and Nakariakov, 1999). In the slab geometry, torsional waves perturb the magnetic field and generate perturbations of plasma velocity in the direction perpendicular to magnetic field and to the direction of the inhomogeneity. Alfvén waves are very weakly dissipative. This means they can propagate very long distances and deposit energy and momentum far from their source. Concerning the generation of Alfvén waves, they can easily be excited by various dynamical perturbations of magnetic field lines. This makes Alfvén waves a promising tool for heating and diagnostics of coronal magnetic structures. Movies visualising the structure of the MHD modes in a magnetic cylinder are available in Resource 1. Figure 4 shows the transverse and longitudinal density and velocity structure of a magnetic cylinder perturbed by a fundamental fast kink oscillation.

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