The numerical code described above has been applied to study the heating of a solar coronal loop by the resonant absorption of an incident wave with a given amplitude and given wave numbers m and n (see Sect. 5). The results are presented in this section. In Sect. 4.1, the main linear MHD results on time scales and localisation of the heating are reproduced as a check and brief reminder. In Sect. 4.2, the variation of the background magnetic field is demonstrated. Next, nonlinear mode coupling is shown to occur in the resonant layer in Sect. 4.3. Finally, in Sect. 4.4the implications of this nonlinear mode coupling for the resonant aborption heating mechanism are discussed.
4.1. Main linear MHD results
The main linear MHD results on resonant absorption were reconstructed as part of the testing phase of the semi-implicit predictor-corrector scheme. For these simulations only one mode has been updated in each time step (as there is no nonlinear mode coupling in the circular cross-section cylindrical coronal loop model considered here) and the background has been fixed, i.e., the -`mode' is not updated but kept constant as specified in Eqs. (1)-(5).
The main linear MHD results were confirmed. When the system is driven at a frequency in the range of the ideal MHD Alfvén continuum, as time evolves the plasma response localizes in an ever diminishing layer around the resonance point, i.e. the point where the local Alfvén frequency matches the frequency of the external driver. Due to the finite electric conductivity () this localization stops after a finite time and the system attains a stationary state in which all quantities oscillate harmonically with constant amplitude. This happens when the energy supplied by the external source is exactly balanced by the Ohmic dissipation in the resonant layer. The width of the resonance layer(s) scales as , as was already known from linear MHD calculations (see e.g. Kappraff & Tataronis 1977 , Poedts et al 1989a, 1989b). Also, the time scale to reach the stationary state is proportional to , in linear MHD. This can be seen in Fig. 1 where the Ohmic dissipation rate is plotted versus time for different values of the plasma resistivity. It is clear that the Ohmic dissipation rate is small in the beginning of the simulation when all profiles are still smooth and the Ohmic terms do not play a role in the MHD equations. After some time, however, the phase-mixing process has created strong gradients and the resistive terms then become important. As a result, the Ohmic dissipation rate starts to grow. When the stationary state is reached, this grow stops again and the Ohmic dissipation rate is constant in time. The initial time interval in which the Ohmic dissipation rate increases depends clearly on the resistivity. But Fig. 1 also demonstrates a remarkable feature of resonant absorption, viz. that the Ohmic dissipation rate becomes independent of plasma resistivity in the limit . Indeed, in Fig. 1 it can be seen that it takes longer to reach the stationary state for than for but the Ohmic dissipation rate is almost exactly the same for both cases. Hence, the main linear results on resonant absorption are confirmed by these test runs for the nonlinear code.
4.2. Variation of the `background' field
One of the advantages of the used spectral discretization is that the `background' can be separated from the rest of the plasma response by virtue of the fact that it is given by the `mode'. In linear MHD simulations of coronal loop heating by the absorption of incident waves, the plasma response to the external driving is assumed to consist of a small perturbation of a fixed, usually static, `background equilibrium'. In nonlinear MHD simulations, the `background equilibrium' varies in two ways, for two different reasons, and on two different time scales and length scales. First, as mentioned above, the dissipation of the background magnetic field generates flow (and heating!). However, this change of the `equilibrium' takes place on a long length scale and on a long time scale, viz. the resistive diffusion time scale . Second, the build-up and heating of the resonant layer changes the background equilibrium locally. This variation of the background takes place on a shorter time scale, viz. the time scale of resonant absorption, which is proportional to . The plasma reacts to the heating process by trying to stop it or, at least, to reduce its rate. The most effective way to accomplish this is to modify the background magnetic field and, hence, the profile of the local Alfvén frequency such that the resonance becomes wider.
In the simulations we did, both changes of the background can be observed. In Fig. 2 this is illustrated by plotting the average profile of the local Alfvén frequency over one driving period, together with the initial -profile (. The average is made for driving period number 140, i.e. after a long time so that the effect of the global diffusion of the background magnetic field on the local Alfvén frequency is also visible. Since the plasma resistivity is the largest in the outer layers of the coronal loop, this effect is the largest there. In addition to this global variation, the local heating in the resonant layer around also affects the local Alfvén frequency there. In the resonant layer, the -profile oscillates around the plotted average building up a `plateau' at the driving frequency . As a result, the resonant layer widens and, hence, the gradients diminish and the heating process becomes less effective. Clearly, this reduction of the effectiveness of the heating process only works in this case of monoperiodic driving and can not work when the loop is driven by a continuous spectrum of incident waves as would be the case on the sun.
The generated `background flow' is dominantly in the tangential directions ( and z) and is the largest in the resonant layer and near the plasma edge.
4.3. Nonlinear mode coupling
Due to the spectral discretization in the tangential directions, the nonlinearity of the dynamics is translated into a coupling of the Fourier modes taken into account. The coupling of the Fourier modes is not to be confused with linear mode coupling which occurs as a result of e.g. geometrical effects or variation of equilibrium quantities in the direction that is discretised by means of Fourier modes or line-tying effects. Such linear mode coupling does not occur in the present simulations where a straight axisymmetric cylinder is considered with a circular cross-section and effects of line-tying have not been considered. Hence, the observed coupling of the Fourier modes is a pure nonlinear effect. This nonlinear mode coupling occurs mainly in the resonant layer because the amplitudes of the fields are the largest there. In the present section, we want to show that the nonlinear mode coupling is i) selective, ii) restricted to the resonant layer(s), and iii) dependent on the amplitude of the incident waves and the plasma resistivity.
4.3.1. Selective mode coupling
In contrast to the linear mode coupling which is the strongest between `nearest neighbours', the nonlinear mode coupling is more selective. Consider a wave with a given frequency and with given wave numbers and . When this wave is incident on a coronal loop it excites mainly the mode of oscillation charaterized by the same mode numbers. However, in addition, other modes of plasma oscillation are excited by this incident wave. One way to measure the strength of this nonlinear mode coupling is to calculate the amount of kinetic and magnetic energy in each of the modes at a specific point in time. Our simulations show that, as expected, the `overtones' of the basic mode are excited most effectively. In other words, modes with the same helicity as the basic mode, i.e. with the same -ratio, absorb most of the energy of the incident wave. The energy absorbed by the other nearby modes, with a different helicity, is several orders of magnitude lower than the energy absorbed by the overtones. The reason for this is the fact that the excited Alfvén waves are incompressible waves which do not involve a density perturbation. Since we do not update the density in the present simulations, the overtones are in fact the only modes that can be excited nonlinearly. Due to numerical `noise', however, some of the supplied energy couples to the other modes too. This is demonstrated by the following experiment which was done to check the code and measure the numerical `noise'. Consider a coronal loop with initial equilibrium profiles given by Eqs. (1)-(5) and a resistivity profile given by Eq. (6) with and excited by an incident wave with wave numbers and amplitude at the loop surface. In the simulation of the plasma response to this excitation the following Fourier modes are taken into account:
plus, of course, the Fourier modes with the corresponding mode numbers of opposite sign (the characters refer to the labels in Fig. 3. The modes indicated with bold characters should absorb all the energy, viz. the `basic' mode, which is the one that is excited directly, the `background', and the `overtones' with the same ratio as the basic mode. In order to check the numerical accuracy, a few modes with different helicity are considered as well. The amount of energy contained in these modes should remain zero. In Fig. 3 the kinetic (a) and magnetic (b) energy distribution are shown versus time. Of course, values below have no physical meaning (but the standard plotting routine used here plots them anyway).
Clearly, most of the kinetic energy is absorbed by the (`e'-) mode which is the driven mode. During the time interval considered in the present simulation, which comprises 20 driving periods, i.e. , the overtones , , , and (`j', `o', `t', and `y', respectively) absorb roughly about 1-10% (each) of the energy absorbed by the `basic' mode. The kinetic energy contained in the other modes, with a different helicity, however, remains at least eight orders of magnitude lower than the kinetic energy of the basic mode, which means that the accuracy of the calculation is all right. Notice that the background () also absorbs some kinetic energy related to the flow generated in the equilibrium, as mentioned above. The magnetic energy of the background of course dominates the magnetic energy contained in the incident wave which, again, is only contained in the basic mode and in its overtones. However, although these modes absorb only little energy they might play an important role in the heating process because the small-scale modes might affect the effective viscosity and resistivity felt by the large-scale modes. But it is extremely expensive (CPU time and computer memory) to take all these modes along in a three-dimensional simulation. Therefore, in the remainder of this paper, the simulations are done with only the overtones of the basic mode and the effect of the small-scale modes on the effective viscosity and resistivity is not investigated here.
4.3.2. Small plasma resistivity
The strength of the mode coupling discussed above is a measure for the nonlinearity of the dynamics. The mode coupling is restricted to the resonant layer that builds up around the idealy singular plasma layer. This is illustrated in Fig. 4a where the contribution of the different Fourier modes to the plasma pressure is plotted versus the loop radius. Outside the resonant layer around the amplitudes of the overtones are almost zero, indicating that the dynamics is linear there. But inside the resonant layer the amplitudes of the overtones become comparable to the amplitude of the basic mode and the overtones contribute substantially to the dynamics of this resonant layer, which is therefore nonlinear. The nonlinearity of the dynamics of the resonant layer depends on the magnetic Reynolds number. This is illustrated by comparing Fig. 4 to Fig. 6 which is the result of exactly the same simulation but with an eight times larger magnetic Reynolds number. From Fig. 6 it is clear that the dynamics in the resonant layer can become very nonlinear, even for relatively small values of the amplitudes of the incident waves, which makes realistic coronal loop heating simulations very expensive. As a matter of fact, for the very small -values of interest (), the narrow resonance layers require very small grid spacing and, hence, very small time steps and, in addition, many modes have to be taken into account in order to resolve the nonlinear mode coupling.
Fig. 5 displays a snapshot of the longitudinal component of the current density after 80 driving periods. The initial parabolic current density profile is still clearly visible in this case but the perturbation is nonlinear. In fact, after 80 driving periods, the perturbation on the current density is already very localized and the amplitude in the resonant layer is more than two times larger than the initial current density there, in spite of the relatively low driving amplitude at , viz. .
For smaller values of , the current density is even more localized, as shown Fig. 7 where a snapshot of the current density is displayed for after 40 driving periods and for the same driving amplitude as in Fig. 5. The snapshot is taken earlier (after 40 ) and the phase-mixing process is not finished yet, which explains the many narrow current layers observed in this snapshot.
4.3.3. Large amplitude waves
Clearly, the amplitude of the plasma response is also determined by the amplitude of the incident waves. The waves observed in the solar corona have amplitudes of up to km/s (Hollweg & Yang 1988), i.e. 1 to 2% of the typical coronal value of the Alfvén velocity ( km/s). In the previous sections we have seen that driving amplitudes of the order of 0.1% of the background Alfvén velocity already yield very nonlinear behaviour in the resonant layers, due to the extremely low plasma resistivity in the hot coronal loops.
Fig. 8 displays a snapshot of the longitudinal current density for a driving amplitude of 0.3% of the background Alfvén velocity and a relatively high plasma resistivity, viz. . It is clear that the nonlinear effects are very dominant in this case and the resonant layer does not stay nicely at the same radial position anymore. The variation of the background and the nonlinear mode coupling are very substantial in this case. As a matter of fact, the modal kinetic energy distribution shows that the amount of energy in the `overtones' becomes comparable to the amount of energy in the excited mode and the plasma behaviour looks very much like MHD turbulence in this large amplitude case. As a result, the heat deposition is spread over the whole loop volume. It is to be noted, however, that this run was performed with a resolution of and the 24 modes may be insufficient to resolve the nonlinear mode coupling in this case. The above discussion on the heat deposition brings us to the important issue of the effects of the nonlinearity in the resonant layers on the efficiency of the heating mechanism and the localization of the heat deposition. These points are discussed in the next section.
4.4. Effects on resonant absorption
In Sect. 4.2we have seen that the `background', i.e. the -mode, changes in two different ways, on two different time scales. First, there is a global change of the background magnetic field due to the magnetic diffusion. This is a slow change on a time scale . Due to the choice of the -profile (6), the magnetic diffusion is much faster near the loop boundary than in the loop center. The second change of the background is much more local and due to the resonant dissipation of the supplied wave energy. It occurs on a much shorter time-scale and the variation of the background magnetic field is localized in a relatively thin layer around the ideal resonance position. The changes of the background magnetic field influence both the efficiency of the heating process and the localization of the heat deposition. As a matter of fact, any change of the local Alfvén frequency changes the resonance positions and, hence, the locations where the heat is deposited. Moreover, the linear MHD studies have shown that the efficiency of the heating process depends on the gradients of the background fields. The time scale for resonant heating, e.g., scales with . The average effect of the background variation, as discussed in Sect. 4.2, therefore leads to less efficient heating as compared to linear MHD simulations.
On the other hand, the variation in time of the local Alfvén frequency results in a continuous shifting of the resonance position(s) when the external driving frequency is kept constant. As a result, the heat deposition profile changes in time and, for very nonlinear cases, this means that the heat deposition is spread over the whole loop volume instead of in a narrow resonant layer, as would be concluded from linear MHD simulations. For weakly nonlinear cases too, the `flattening' of the local Alfvén frequency profile near the resonance layer leads to wider resonance layers and less localized heat deposition profiles.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998