  A new class of exact solutions of hydrodynamic equations for an
incompressible fluid (gas) at the presence of a bulk sink and uprising vertical
flows of matter is considered. The acceleration of the rotation velocity of
classical non-stationary vortices is conditioned by the joint action of the
convective and Coriolis hydrodynamic forces (accelerations), which appear due
to the converging radial flows of the matter in the region of a bulk sink. It
is shown that there exist velocity profiles that nullify viscous terms in the
Navier-Stokes equations and represent a vortex structure with a "rigid-body"
rotation of its core and converging radial flows.
  The concept of non-stationary vortices in open systems is applied to
description of origination of power atmospheric vortices (whirlwinds, tornados,
and typhoons). In the classical hydrodynamics a favorable condition for the
origination and existence of such vortices is the exact nullification of the
terms, which describe kinematic viscosity of an incompressible fluid. Such
flows have the minimal rate of energy dissipation that corresponds to the
"minimum entropy production principle", and therefore may relatively easily
appear in favorable natural conditions.
