  Considerable progress in experimental studies of atomic gases in a toroidal
geometry has opened up novel prospects for the investigation of fundamental
properties of superfluid states and creation of new configurations for
atomtronic circuits. In particular, atomic Bose-Einstein condensates loaded in
a dual-ring trap suggest a possibility to consider the tunneling dynamics
between coupled condensates with different angular momenta. Accordingly, we
address the tunneling in a pair of coaxial ring-shaped condensates separated by
a horizontal potential barrier. A weak-coupling truncated (finite-mode)
Galerkin model and direct numerical simulations of the underlying
three-dimensional Gross-Pitaevskii equation are used for the analysis of
tunneling superflows driven by an initial imbalance in atomic populations of
the rings. The superflows through the Bose-Josephson junction are strongly
affected by persistent currents which are present in the rings. Josephson
oscillations of the population imbalance and angular momenta in the coupled
rings are obtained for co-rotating states and non-rotating ones. On the other
hand, the azimuthal structure of the tunneling flow implies formation of
Josephson vortices (fluxons) with zero net current through the junction for
hybrid states, built of counter-rotating persistent currents in the coupled
rings.
