The nonlinear dynamics of the free surface of an ideal dielectric liquid in a strong electric field is studied. The equation for the evolution of surface electrohydrodynamic waves is derived in the approximation of small surface-slope angles. It is established that the equation can be solved for liquids with sufficiently high values of the permittivity. This makes it possible to describe the interaction of the counter-propagating waves.

We present and analyze characteristics of the runaway electron flow in a high-voltage (the voltage rise rate of up to 1.5 MV/ns) air-filled electrode gap with a strongly nonuniform electric field. It is demonstrated that such a flow contains a high-energy electron component of duration not more than 10 ps. According to numerical simulations, runaway electron generation/termination is governed by impact ionization of the gas near the cathode and switching on/off a critical (sufficient for electrons to run away) electric field at the boundary of the expanding cathode plasma. The corresponding characteristic time estimated to be 2–3 ps is defined by the ionization rate at a critical field.

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The hydrodynamic evolution of the surface of a liquid metal in the presence of an electric field is investigated using both analytical and numerical techniques. It is established that a free liquid surface with axial symmetry evolves with time into a cone-like shape, with the cone angle equal to Taylor's static value of 98.6°. The mechanism behind such interesting flow behaviour is that the system of electrohydrodynamic (EHD) equations has a self-similar asymptotic solution that generalizes Taylor's static result. The asymptotic solutions are found and the time-dependent behaviours of basic physical quantities (electric field strength, fluid velocity and surface curvature) near the singularity are established. The results and the analytical and numerical techniques used are thought to be useful in the development of time-dependent models of operating liquid metal ion sources and EHD sprayers.