  The evolution of electromagnetic and thermodynamic fields in a non-ideal
fluid are studied in the framework of ultrarelativistic transverse
magnetohydrodynamics (MHD), which is essentially characterized by electric and
magnetic fields being transverse to the fluid velocity. Extending the method of
self-similar solutions of relativistic hydrodynamics to the case of
non-conserved charges, the differential equations of non-ideal transverse MHD
are solved, and two novel sets of self-similar solutions are derived. The first
set turns out to be a boost-invariant and exact solution, which is
characterized by non-rotating electric and magnetic fields. The second set is a
non-boost-invariant solution, which is characterized by rotating electric and
magnetic fields. The rotation occurs with increasing rapidity $\eta$, as the
angular velocity is defined by
$\omega_{0}\equiv\frac{\partial\zeta}{\partial\eta}=\frac{\partial\phi}{\partial\eta}$,
with $\zeta$ and $\phi$ being the angles of electric and magnetic vectors with
respect to a certain axis in the local rest frame of the fluid. For both sets
of solutions, the electric and magnetic fields are either parallel or
anti-parallel to each other. Performing a complete numerical analysis, the
effects of finite electric conductivity as well as electric and magnetic
susceptibilities of the medium on the evolution of rotating and non-rotating
MHD solutions are explored, and the interplay between the angular velocity
$\omega_{0}$ and these quantities is scrutinized. The lifetime of
electromagnetic fields and the evolution of the temperature of the
electromagnetized fluid are shown to be affected by $\omega_{0}$.
