| The recently developed numerical technology for a hyperbolic system is reviewed. The techniques which have yielded a fairly robust performance especially when the discontinuity is encountered in a system have three major ingredients: (1) integral hyperbolic conservation laws; (2) Riemann solvers and (3) smart interpolation procedures. Codes based on these techniques have been built up and implemented. To study hydrodynamic waves in the lower solar atmosphere, we first rediscuss the linear adiabatic wave theory, permitting a complex frequency. The linear solution of an initial-boundary value problem consists of two parts, a normal oscillation with an imposed complex frequency and a transient oscillation with the nominal cutoff frequency. Under the regime of the linear waves, waves with all real frequencies, either above or below the nominal acoustic cutoff frequency, could propagate in a gravitationally stratified atmosphere and transport mechanical energy out to the upper atmospheric layers. The wave functions, group velocities, phase relation, and energy fluxes are fully presented. With the aid of the numerical techniques, the nonlinear behavior of the hydrodynamic waves are studied further. The computation results shows the features of (1) non-linear wave propagation with all real frequencies, (2) the velocity power spectral peak around the nominal acoustic cutoff frequency, (3) the velocity power plateau between 1-2 nominal acoustic cutoff frequency band in the power spectra, (4) adiabatically heating atmosphere up to 20000{dollar}spcirc{dollar} and (5) adiabatically cooling atmosphere down to 3000{dollar}spcirc,{dollar} all of which might be matched by some solar chromosphere observations. Finally, a 2-D ideal isothermal MHD study by the recent numerical technology has revealed various missing aspects of the magneto-convective instability. A full phenomenological and morphological comparison between the computational features in the dynamic evolution of Parker instability and the observational features of solar atmospheric activities raises a speculation/possibility: there might be a non-Babcock type dynamo mechanism which would accommodate both the solar magnetic cycle evolution and detailed evolution of most major solar magnetic features within one single package. |
