Quasi-stationary and fast transient processes connected with powerful lightning discharges and large-scale thunderstorm systems are analyzed. The main physical ideas serving as the foundation for sprite and sprite-producing cloud modeling are discussed with simple examples. Special attention is paid to the adequate description of the field sources and appropriate set of boundary conditions. The features of electron heating and ionization in the middle atmosphere are briefly discussed. The importance of the global circuit for the modeling of sprites and sprite-producing clouds is recognized. Along with the setting of boundary conditions, its role is connected with the importance of large thunderstorm complexes, including mesoscale convective systems (MCSs). It was shown recently that MCS stratiform regions make an especially large current contribution to the global circuit, serving either as an effective generator or as a discharger of the circuit depending on the polarity, magnitude and thickness of the MCS layers. On the other hand, stratiform regions of MCS are characterized by an enhanced rate of positive flashes, which are known to correlate with sprites. In the case of MCS the big narrow layers, generated near the 0°C isotherm serve as the source of electric charges for positive CG flashes. We suggest a model of a positive charge layer near the 0°C isotherm, based on the hypothesis that the melting-charging mechanism plays a principal role in the formation of the layer. We illustrate how microphysical considerations result in electric currents for the use in modeling of the global circuit and discharge processes. Further we address two aspects of the global electric circuit conception, particularly important from the viewpoint of sprites and sprite-producing cloud research. First is a classical aspect of the global circuit as the quasi-stationary current contour supported by the operation of thunderstorm generators over the globe. Another aspect is connected to the energy deposition and dissipation into the circuit, treated as an open dissipative system. Simple energetic estimates of sprite occurrence are presented. Nonlinear aspects of modeling throughout the chapter are emphasized.
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