  We investigate the evolution and origin of small-scale chromospheric swirls
by analyzing numerical simulations of the quiet solar atmosphere, using the
radiative magnetohydrodynamic code CO$5$BOLD. We are interested in finding
their relation with magnetic field perturbations and in the processes driving
their evolution. For the analysis, the swirling strength criterion and its
evolution equation are applied in order to identify vortical motions and to
study their dynamics. We introduce a new criterion, the magnetic swirling
strength, which allows us to recognize torsional perturbations in the magnetic
field. We find a strong correlation between swirling strength and magnetic
swirling strength, in particular in intense magnetic flux concentrations, which
suggests a tight relation between vortical motions and torsional magnetic field
perturbations. Furthermore, we find that swirls propagate upward with the local
Alfv\'en speed as unidirectional swirls, in the form of pulses, driven by
magnetic tension forces alone. In the photosphere and low chromosphere, the
rotation of the plasma co-occurs with a twist in the upwardly directed magnetic
field that is in the opposite direction of the plasma flow. All together, these
are characteristics of torsional Alfv\'en waves. We also find indications of an
imbalance between the hydrodynamic and magnetohydrodynamic baroclinic effects
being at the origin of the swirls. At the base of the chromosphere, we find a
net upwardly directed Poynting flux, which is mostly associated with large and
complex swirling structures that we interpret as the superposition of various
small-scale vortices. We conclude that the ubiquitous swirling events observed
in simulations are tightly correlated with perturbations of the magnetic field.
At photospheric and chromospheric levels, they form Alfv\'en pulses that
propagate upward and may contribute to chromospheric heating.
