SUMMARY
The review highlights the main achievements in the theory of the vortex dynamo in rotating media. Various models of a vortex dynamo in rotating media are discussed: a homogeneous viscous fluid, a temperature-stratified fluid, a moist atmosphere, and a stratified nanofluid. The main analytical and numerical results for these models obtained by the method of multiscale asymptotic expansions are presented. The main attention is paid to models with a non-spiral external force. For a rotating moist atmosphere, characteristic estimates of the spatial and temporal scales of the generated vortex structures are obtained. New effects of the vortex dynamo in a rotating stratified nanofluid, which arise due to thermophoresis and Brownian motion of nanoparticles, are shown. The results of the analysis of the nonlinear equations of the vortex dynamo in the stationary regime are presented in the form of spiral kinks, periodic nonlinear waves, and solitons.