  We study vortex dynamics in the solar atmosphere by employing and deriving
the analytical evolution equations of two vortex identification criteria. The
two criteria used are vorticity and the swirling strength. Vorticity can be
biased in the presence of shear flows, but its dynamical equation is well
known; the swirling strength is a more precise criterion for the identification
of vortical flows, but its evolution equation is not known yet. Therefore, we
explore the possibility of deriving a dynamical equation for the swirling
strength. We then apply the two equations to analyze radiative MHD simulations
of the solar atmosphere produced with the CO5BOLD code. We present a detailed
review of the swirling strength criterion and the mathematical derivation of
its evolution equation. This equation did not exist in the literature before
and it constitutes a novel tool that is suitable for the analysis of a wide
range of problems in (magneto-)hydrodynamics. By applying this equation to
numerical models, we find that hydrodynamical and magnetic baroclinicities are
the driving physical processes responsible for vortex generation in the
convection zone and the photosphere. Higher up in the chromosphere, the
magnetic terms alone dominate. Moreover, we find that the swirling strength is
produced at small scales in a chaotic fashion, especially inside magnetic flux
concentrations. The swirling strength represents an appropriate criterion for
the identification of vortices in turbulent flows, such as those in the solar
atmosphere. Moreover, its evolution equation, which is derived in this paper,
is pivotal for obtaining precise information about the dynamics of these
vortices and the physical mechanisms responsible for their production and
evolution. Since this equation is available, the swirling strength is now the
ideal quantity to study the dynamics of vortices in (magneto-)hydrodynamics.
