3. Three-dimensional nonlinear simulations
In this section, the problems of the strongly localized plasma dynamics and of the widely disparate MHD time scales and the numerical methods applied to overcome these numerical problems are addressed briefly. More details on these numerical techniques are given in Appendix A.
3.1. Strongly localized solutions
The linearized ideal MHD equations contain mobile singularities which give rise to continuous parts in the MHD spectrum of oscillation frequencies of a coronal loop. When such a coronal loop is excited side-ways at a frequency within the range of the continuous spectrum, ideal resonances occur inside the plasma at the flux surfaces where the local Alfvén frequency matches the frequency of the external driver, i.e. the incident wave. The non-zero plasma resistivity removes the singularity from the MHD equations but for the relevant, very small, values of in the solar corona, the resonant behaviour is clearly present. As a result, the plasma solutions become very localized in the radial direction. To simulate such behaviour accurately on a computer requires local expansion functions in this direction. In our code, we used finite differences on two staggered meshes in the radial direction and a spectral discretization (Fourier modes) in the - and z -direction in which such strong localizations do not occur.
3.2. Widely disparate time scales
Widely disparate time scales are another problem when simulating the three-dimensional nonlinear dynamical behaviour of an elongated coronal loop on a computer. The ideal MHD spectrum consists of three subspectra: the fast magnetosonic subspectrum and the Alfvén and slow magnetosonic subspectra with the mentioned continuous parts. In cylindrical geometry, the time scale associated with the compressional fast magnetosonic wave, also called the compressional time scale, is measured by the transit time of the fast magnetosonic wave over the plasma radius a, , with the phase velocity of the fast magnetosonic wave. For the low plasmas in solar coronal loops, and, hence, ( is the ratio of the plasma pressure and the magnetic pressure, ). Shear Alfvén waves, on the other hand, propagate mainly along the magnetic field lines and, hence, the time scale related to these waves is measured by the transit time of the shear Alfvén wave over the length of the coronal loop L: . In the large aspect ratio loops that are observed in the corona of the sun, the time scales and differ substantially. As a matter of fact, in the cylindrical model that is used in the present paper, the fast magnetosonic time scale is much shorter than the shear Alfvén time scale: for .
When the heating of coronal loops is studied, the time scale of resistive diffusion of magnetic fields, , is also of interest. In coronal plasmas, where the magnetic Reynolds number is typically , the diffusion time scale of the background field is much longer that both and . In the resonant absorption mechanism, however, plasma heating takes place on a shorter time scale than as a result of the short length scales created by the resonances. The width of the resonant layer scales as and, hence, the time scale of resonant absorption . Clearly, is also longer than the Alfvén time scale. Hence, in the elongated coronal loops we have
The fast magnetosonic waves play the important role of `energy-carrier': They are responsible for the transport of the energy from the source, across the magnetic surfaces, to the resonant layer(s).
3.3. Numerical scheme
The widely disparate time scales (12) make three-dimensional nonlinear time dependent MHD calculations very expensive because of the time-step limitation for conventional explicit numerical schemes. This limitation follows from the Courant-Friedrichs-Lewy (CFL) stability condition imposed by the compressional fast magnetosonic motion. Indeed, explicit numerical schemes are stable only if the time step is smaller than or
where is the smallest mesh interval in the radial direction. The high radial resolution () required to resolve the resonant layers that occur and the small fast magnetosonic time scale () make this CFL condition extremely restrictive for numerical simulations of coronal loop heating which occurs on a much longer time scale (see Eq. (12)).
The results presented in the next section are obtained by means of a semi-implicit predictor-corrector scheme. In this method, the CFL restriction on the fast magnetosonic waves is avoided by treating these waves, and only these, implicitely. Some more details on the spatial discretization and on the semi-implicit algorithm we used can be found in Appendix A. This method was applied to the numerical simulation of time dependent MHD phenomena for the first time by Harned & Kerner 1985. Since then, it has been used by many authors to simulate the MHD behaviour of both laboratory plasmas (tokamaks, stellerators, etc.) and solar plasmas (see e.g. Harned & Kerner 1986 , Harned & Schnack 1986 , Schnack et al. 1987 , Lerbinger & Luciani 1991, etc.).
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998