  The secular variation of the geomagnetic field suggests that there are
anticyclonic polar vortices in the Earth's core. Under the influence of a
magnetic field, the polar azimuthal flow is thought to be produced by one or
more coherent upwellings within the tangent cylinder, offset from the rotation
axis. In this study, convection within the tangent cylinder in rapidly rotating
dynamos is investigated through the analysis of forced magnetic waves. The
first part of the study investigates the evolution of an isolated buoyancy
disturbance in an unstably stratified rotating fluid subjected to an axial
magnetic field. It is shown that the axial flow intensity of the slow
Magnetic-Archimedean-Coriolis (MAC) waves becomes comparable to that of the
fast MAC waves when $|\omega_M/\omega_C| \sim 0.1$, where $\omega_M$ and
$\omega_C$ are the Alfv\'en wave and inertial wave frequencies respectively. In
spherical shell dynamo simulations, the isolated upwellings within the tangent
cylinder are shown to originate from the localized excitation of slow MAC waves
in the dipole-dominated regime. Axial flow measurements in turn reveal the
approximate parity between the slow and fast wave intensities in this regime,
which corresponds to the existence of strong polar vortices in the Earth's
core. To obtain the observed peak azimuthal motions of
$0.6$--$0.9^\circ$yr$^{-1}$, the Rayleigh number in the low-inertia geodynamo
must be $\sim 10^3$ times the Rayleigh number for the onset of nonmagnetic
convection. However, if the forcing is so strong as to cause polarity
reversals, the field within the tangent cylinder decays away, and the
convection takes the form of an ensemble of plumes supported entirely by the
fast waves of frequency $\omega \sim \omega_C$. The resulting weak polar
circulation is comparable to that obtained in nonmagnetic convection.
