2.1 MHD modes of a straight cylinder

An important elementary building block of this theory is dispersion relations for modes of a magnetic cylinder (see Figure 2

). Magnetic cylinders are believed to model well such common coronal structures as coronal loops, various filaments, polar plumes, etc. Consider a straight cylindrical magnetic flux tube of radius $a$ filled with a uniform plasma of density $r0$ and pressure $p0$ within which is a magnetic field $B0ez$ ; the tube is confined to $r < a$ by an external magnetic field $Beez$ embedded in a uniform plasma of density $re$ and pressure $pe$ (here we neglect the effects of twist and steady flows). The very existence of such a plasma configuration requires the balance of the total pressure $Ptot$ , which is the sum of the plasma and magnetic pressures, between the two media at the boundary. For the equilibrium state, this condition is

Similarly, a magnetic slab may be introduced in much the same way, having a width $2a$ .

Figure 2:

A magnetic flux tube of radius $a$ embedded in a magnetised plasma.

In the internal and external media, the sound speeds are $Cs0$ and $Cse$ , the Alfvén speeds are $CA0$ and $CAe$ , and the tube speeds are $CT0$ and $CTe$ , respectively. Relations between those characteristic speeds determine properties of MHD modes guided by the tube.

The presence of the internal spatial scale, the radius of the tube $a$ , 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 $dPtot(r) exp[i(kzz + mf - wt)]$ (Sakurai et al., 1991a ):

d ( )( ) ( m2 ) D ---(rqr) = C2A + C2s w2- C2Tk2z k2 + --2 rdPtot, dr r -d- ( 2 2 2) dr (dPtot) = r0 w - C Akz qr, (2)

and

where $qr$ and $qf$ are the perturbation displacements in the radial and azimuthal direction, respectively. The quantity $D$ is defined as

and $k$ 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 $a = 0,e$ . The first term of Equation (6

) represents torsional Alfvén wave solutions with $w = ± CAakz$ . The second term is a Bessel-like 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)

$Im(x)$ and $Km(x)$ are modified Bessel functions of order $m$ , and the prime denotes the derivative of a function $Im(x)$ or $Km(x)$ with respect to argument $x$ . The functions $ke$ and $k0$ 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 $r > a$ ), the condition $2 ke > 0$ has to be fulfilled. In the equation, it is assumed that $ke,0 > 0$ . The integer $m$ determines the azimuthal modal structure: waves with $m = 0$ are called sausage modes, waves with $m = 1$ are kink modes, waves with higher $m$ 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.

Figure 3:

Dispersion diagram showing the real phase speed solutions of dispersion relation (7

) for MHD waves in a magnetic cylinder as a function of the dimensionless parameter $kza$ . The typical speeds in the internal and external media are shown relative to the internal sound speed: $C = 2C A0 s0$ , $C = 5C Ae s0$ , and $C = 0.5C se s0$ . The solid, dotted, dashed and dash-dotted curves correspond to solutions with the azimuthal wave number $m$ equal to $0$ , $1$ , $2$ and $3$ , respectively. The torsional Alfvén wave mode solution is shown as a solid line at $w/kz = CA0$ .

Figure 3

shows a typical dispersion diagram of a coronal loop. This figure generalises the dispersion plot given by Edwin and Roberts (1983

) to the inclusion of higher- $m$ modes. In the Figure 3

, all the possible modes correspond to body modes, which have oscillatory behaviour inside and evanescent behaviour outside. This is in contrast with surface modes, which have evanescent behaviour in both media. Phase speeds of MHD modes guided by the tube can have values in two bands: either between $CA0$ and $CAe$ (provided $CA0 < CAe$ ) and between $CT0$ and $Cs0$ . The wave modes in these two bands have been named, respectively, fast and slow, in analogy with the types of magnetoacoustic wave modes present in a homogeneous medium. The fast modes are highly dispersive. In the long wavelength limit, the phase speed of all but sausage fast modes tends to the so-called kink speed

which corresponds to the density weighed average Alfvén speed. The sausage mode approaches a cut-off at the external Alfvén speed. Trapped sausage modes do not exist at longer wavelengths. But mode solutions can be found if the condition of mode localisation is relaxed, i.e., if waves are allowed to radiate into the external medium. Such wave modes are called leaky modes and have complex eigenfrequencies. To include leaky modes, dispersion relation (7

) is modified such that the modified Bessel functions $Km(x)$ are replaced by Hankel functions. Details of the leaky solutions may be found in Zaitsev and Stepanov (1975 ); Cally (1986

); Stenuit et al. (1998 ). In Figure 3

, leaky waves would be situated outside the regions of existence of the trapped waves. In particular, the sausage mode continues in the region over the horizontal asymptote $CAe$ but its frequency is complex there.

In closed fields of coronal active regions, the longitudinal wave number $kz$ of standing modes is usually prescribed by the line-tying 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.

Figure 4:

Transverse and longitudinal density perturbation (shown as intensity) and velocity structure (shown as a vector field) of a homogeneous magnetic cylinder model of a coronal loop, perturbed by a fundamental fast kink oscillation (from Resource 1).