2.2 MHD continua

When the group speed of wave modes is directed along the magnetic field, then such modes on neighbouring field lines do not interact with each other. They may oscillate locally with their own eigenfrequency without disturbing the rest of the medium. If the equilibrium parameters of the plasma vary continuously across the magnetic field, then the eigenfrequency of such modes may also vary continuously. This gives rise to continuous intervals of eigenfrequencies, i.e., continua, in frequency space. This situation, though, may change by including resistivity or non-MHD effects (see, e.g., Appert et al., 1986 ). In ideal MHD, there exist two continua: the Alfvén continuum and the slow (or cusp) continuum (see Goedbloed, 1983

; Goedbloed and Poedts, 2004 ). Equations (2

, 3

) are also valid for linear waves in a coronal loop modelled as a plasma cylinder if the equilibrium quantities vary continuously in the radial direction (see, e.g., Sakurai et al., 1991a

). The continua are in this case characterised by the frequencies $CA(r) |kz|$ and $CT(r)|kz|$ . There are two effects associated with wave modes from these continua: resonant absorption and phase-mixing.

2.2.1 Resonant absorption

If the loop supports a collective (a mode) wave, either driven externally or set up initially, which has a frequency $w$ that falls within one of the two continua, a resonance is set up at the location(s) where $w2 = C2A(r)k2z$ or $w2 = C2T(r)k2z$ (where Equation (4

) becomes zero). The physical location where the resonance occurs is called the resonance layer and in the plasma cylinder model it corresponds to a radial shell (see Figure 5

). From now on we shall consider the case of a resonance in the Alfvén continuum only. The locally excited resonant wave mode is a torsional Alfvén wave and its amplitude peaks in the resonance layer where the perturbation develops large gradients. Dissipation has to be taken into account to prevent the perturbation to diverge. The amplitude of the resonant mode scales as $Re1/3$ at the resonant layer, where $Re$ is the shear viscous and/or the magnetic Reynolds number (Kappraff and Tataronis, 1977

). In the solar corona this number is much larger than unity ( $14 Re ~ 10$ ). Since non-resonant modes dissipate with an amplitude proportional to Re, it is clear that the resonantly excited modes experience enhanced dissipation. This also implies that the resonant absorption process is inherently nonlinear, since the amplitude of the resonant mode can not grow to the values implied by the $Re1/3$ scaling (Ofman et al., 1994

; Ofman and Davila, 1995

). Thus, wave energy is extracted secularly through the resonance from the collective wave to the benefit of a local wave mode (mode conversion), which then is dissipated in an enhanced manner. This mechanism of wave heating is called resonant absorption (Goedbloed, 1983 , and references therein) and has been put forward in the context of the coronal heating problem (Ionson, 1978

) and in sunspot seismology for explaining the loss of acoustic power in sunspots (see, e.g., Sakurai et al., 1991b ; Bogdan, 2000 , and references therein).

Figure 5:

A magnetic flux tube of radius $a$ embedded in a magnetised plasma with a thin edge layer of width $l$ where the density varies monotonically.

There are inherently two time scales involved. Firstly, there is the damping time scale of the mode conversion from the collective to the local mode, which generally is independent of dissipation. From the point of view of linear theory and classical, theoretical values of dissipation coefficients, it is generally much shorter than the second time scale, which is linked to the dissipative damping of the small-scale perturbations of the local mode in the resonance layer.

Note that for sausage wave modes, where $m$ =0, the equations describing the magnetoacoustic and torsional waves (i.e., Equations (2 , 3 )) are decoupled so that mode conversion and absorption through the Alfvén resonance cannot take place (the slow resonance can still operate; see, e.g., Erdélyi, 1997 ). For the slab geometry,this corresponds to propagation parallel to the magnetic field, i.e., $k = 0 y$ .

To study resonant absorption the following procedure is often undertaken. Whilst outside of the resonant layer the ideal MHD equations can be applied, inside the resonant layer dissipative effects have, in principle, to be taken into account to ensure that the solution remains regular. But unless one is interested in the details of the solution in the resonant layer, this can be avoided. The thickness of the resonant layer, $d$ , is proportional to $(lL2Re) -1/3$ , where $l$ is the length scale over which the Alfvén speed varies around the resonance. In the solar atmospheric context $d$ can be assumed to be small. The solution in the resonant layer is replaced by a jump relation, which is based upon a Taylor series expansion of the ideal MHD equations around the resonance (Kappraff and Tataronis, 1977 ; Ionson, 1978 ; Hollweg, 1987 ; Hollweg and Yang, 1988 ; Sakurai et al., 1991a ). It is assumed that the quantities that are conserved across the layer in ideal MHD will remain so in weakly dissipative MHD.

Figure 6:

Radial profile of the Alfvén frequency in a loop with a thin edge layer of width $l$ where the density varies monotonically. A resonance occurs where the global wave mode frequency $w$ matches locally the Alfvén frequency $CA(rA)kz$ inside the edge layer.

Consider a loop model where the internal and external media are homogeneous, except for a loop edge layer of width $l$ where the density varies monotonically from $r0$ to $re$ (see Figure 5

). A global wave mode is present, which has a frequency $w$ which matches the Alfvén frequency at $CA(r = rA)|kz|$ within the Alfvén continuum of the edge layer (see Figure 6

). In the internal and external media the solutions are calculated using, e.g., Equation (6

). Those solutions contain arbitrary integration constants, which are related to each other with use of certain jump relations at the edge layer. The width of the edge layer, $l$ , is considered to be thin, i.e., $l« a$ , but wider than the resonant layer, i.e., $d > l$ . The jump relations are constructed as follows. Inside the edge layer, the system of Equations (2

) is Taylor expanded around the resonance using the small parameter $s =_ r - rA$ ( $|s|« 1$ ). Thus, a second order differential equation for $dP /s tot$ is derived, which is of the form

It has solutions in the form of modified Bessel functions. The total pressure perturbation is described as

which is approximately constant across the layer (Hollweg, 1987

; Sakurai et al., 1991a

). Since for $m$ =0 no resonant coupling can occur, the Equation (10

) does not apply for that case. Therefore, the jump relation for the total pressure perturbation is simply $[dPtot] = 0$ , where the square brackets denote the difference between the solutions in the right and left limits of $s$ tending to zero, respectively. Note that, when the loop is twisted, the total pressure is no longer a conserved quantity (Sakurai et al., 1991a ). Similarly, the solution for the radial displacement perturbation can be found to be approximately

where $D = d(w2- C2Ak2z)/dr|r=rA$ . $qr$ depends on $s$ through a logarithmic term, which diverges for $s$ tending to zero. The jump relation for $qr$ is

for which the relation $[ln(s)] = - ip sign(w) sign(D)$ has been used. To match the internal and external solutions, the usual jump conditions of continuity of total pressure and radial displacement are now replaced by the above derived jump relations. Depending on the condition that is imposed at $r --> oo$ , the trapped and/or leaky eigenmodes of the system may be studied (involving a dispersion relation) or the reflection/absorption problem of an externally driven wave that interacts with the loop.

By scanning through the frequency of the collective wave, the wave absorption as a function of $w$ is studied (see, e.g., the numerical simulations by Poedts et al., 1989 ). The fractional absorption spectrum often shows well-defined maxima where the absorption reaches 100%. The spatial structure of the excited wave mode at those maxima shows a combination of localised and global behaviour. It is a global wave (e.g., discrete fast eigenmode) with a frequency that lies in the Alfvén continuum and is, therefore, locally coupled to an Alfvén wave. These types of wave modes are known as quasi-modes and they are natural wave modes of the dissipative and inhomogeneous system (Balet et al., 1982 ; Steinolfson and Davila, 1993 ; Ofman et al., 1994 ; Ofman and Davila, 1995 ; Tirry and Goossens, 1996 ). Therefore, it is easily understood why maximum absorption occurs when driving at the frequency of a quasi-mode. Also, the presence of steady flows can significantly change the efficiency of resonant absorption (Erdélyi, 1998 ).

Furthermore, in the absence of flow, these modes are damped (Poedts et al., 1990 ; Ofman et al., 1994 ; Wright and Rickard, 1995 ). From the point of view of the global nature of the mode, the damping is primarily a conversion of energy from the collective to the local. This damping rate has been calculated for various geometries (see, e.g., Lee and Roberts, 1986 ; Hollweg, 1987 ; Goossens et al., 1992 ). Ruderman and Roberts (2002 ) calculated the damping time, $t$ , of a global kink wave in a long, thin loop ( $a « L$ ) in the limits of weak dissipation ( $Re » 1$ ) and zero plasma- $b$ :

This time scale is generally much shorter than the time scale of the dissipative damping of the small-scale perturbations of the local mode in the resonance layer. In Section 3.4 this aspect of rapid mode conversion will be explored further within the context of the observed rapid damping of transverse loop oscillations (Roberts, 2000

; Ruderman and Roberts, 2002

; Goossens et al., 2002a

; Van Doorsselaere et al., 2004

Ofman et al. (1994 ) and Ofman and Davila (1995 ) studied numerically nonlinear resonant absorption and found that the large shear velocities produced at the resonance layer are subject to Kelvin-Helmholtz instabilities. The velocity amplitudes derived from linear theory are much larger than the observed velocities from nonthermal broadening of coronal emission lines. This discrepancy may be explained by a turbulent enhancement of dissipation parameters due to the instabilities. Additional complexity is brought by the effects of boundary conditions in the longitudinal direction, e.g., Beliën et al. (1999 ) examined numerically the effect of the transition region and chromosphere on the resonant absorption in coronal loops. They found that the nonlinear energy transfer from the Alfvén waves to slow magnetoacoustic waves in the lower atmosphere can much diminish the absorption efficiency compared with models of line-tied loops without a lower atmosphere. The driver they considered was, though, monoperiodic and this study should be extended to include more realistic drivers. Furthermore, the heating of the resonance layer would spread due to thermal conduction and heat the lower atmosphere. The resulting chromospheric evaporation enhances the loop density at the resonance layer and, hence, shifts the Alfvén frequency away from resonance, as well as change the quasi-mode frequencies (see, e.g., the discussion in Ofman and Davila, 1995 ). Ofman et al. (1998a ) considered a broad band Alfvén wave driver (see also DeGroof and Goossens, 2002 ), and coupling to the chromosphere of the loop density with the use of a quasi-static equilibrium scaling law. They found that the heating is concentrated in multiple resonance layers, rather than in the single layer of previous models, and that these layers drift throughout the loop to heat the entire volume. These properties are in much better agreement with coronal observations that imply multithreaded loop structure.

2.2.2 Alfvén wave phase mixing

Consider again a structure with a continuous inhomogeneity profile across the magnetic field. Instead of an initial collective mode, this time on each field line an Alfvén wave is excited. This wave oscillates independently from its neighbours, with a frequency that lies in the Alfvén continuum. For simplicity a Cartesian geometry is chosen where the magnetic field is in the $z$ -direction and the inhomogeneity is in the $x$ -direction. The Alfvén waves, which are polarised in the $y$ -direction, are described by the wave equation (Heyvaerts and Priest, 1983

)

with the solution

where $f(z)$ and $Y(x)$ are 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 direction of the $z$ -axis, respectively. When $Y(x) = const$ , the wave is plane.

Figure 7:

Comparison of the dissipative decay of Alfvén waves in 1D plasma inhomogeneities with Alfvén speed profiles of different steepness. The solid curve shows the dissipative decay of Alfvén waves in an homogeneous plasma. Phase-mixing is therefore absent and the damping time is proportional to $Re$ . The other curves show the dissipative decay in a 1D plasma inhomogeneity with a nonzero gradient in the Alfvén speed. Phase-mixing, therefore, occurs and the damping times are proportional to $Re1/3$ .

Equation (15

) shows that the Alfvén waves propagate on different magnetic surfaces, corresponding to different values of $x$ , with different phase speeds equal to the local Alfvén speed $CA(x)$ . If the wave is initially plane in the $x$ direction, it gets gradually inclined. This leads to generation of very small transverse (in the direction of the inhomogeneity) spatial scales. These high transverse gradients, in the presence of finite viscosity or resistivity, which leads to the appearance of a $n @/@t \~/ 2Vy$ term on the right hand side of Equation (14

), are subject to efficient dissipation. Here, $n$ is the coefficient of viscosity and/or resistivity, small enough so that dissipation may be considered weak. This is the effect of Alfvén wave phase mixing, suggested by Heyvaerts and Priest (1983

) as a possible mechanism for heating of open coronal structures.

In the developed stage of phase mixing (when $@/@x » @/@z$ ) the Alfvénic perturbations of different magnetic surfaces become uncorrelated with each other, the perturbations decay according to the law (Heyvaerts and Priest, 1983 )

for propagating harmonic Alfvén waves, and

for standing harmonic Alfvén waves. The decay time due to phase-mixing is proportional to $Re1/3$ , similar to the dissipative decay of Alfvén waves due to resonant absorption. Figure 7

shows the dissipative decay of propagating Alfvén waves in a 1D plasma inhomogeneity with Alfvén speed profiles of different steepness, compared with the dissipative decay of Alfvén waves in a homogeneous medium (solid curve).

If instead of a monochromatic wave in the longitudinal direction a localised Alfvén pulse is considered, Heyvaerts and Priest’s expression for the exponential decay (17 ) should be replaced by the power law

see Hood et al. (2002 ) and Tsiklauri et al. (2003 ) for two methods of derivation and comparison with results of numerical modelling.

In the nonlinear regime, growing transverse gradients induce oblique fast magnetoacoustic waves (Nakariakov et al., 1997 ). Consequently, phase mixing regions may be characterised by the presence of compressible perturbations.