  [Abridged] Tides may play an important role in determining the observed
distributions of mass, orbital period, and eccentricity of the extrasolar
planets. In addition, tidal interactions between giant planets in the solar
system and their moons are thought to be responsible for the orbital migration
of the satellites, leading to their capture into resonant configurations. We
treat the underlying fluid dynamical problem with the aim of determining the
efficiency of tidal dissipation in gaseous giant planets. In cases of interest,
the tidal forcing frequencies are comparable to the spin frequency of the
planet but small compared to its dynamical frequency. We therefore study the
linearized response of a slowly and possibly differentially rotating planet to
low-frequency tidal forcing. Convective regions of the planet support inertial
waves, while any radiative regions support generalized Hough waves. We present
illustrative numerical calculations of the tidal dissipation rate and argue
that inertial waves provide a natural avenue for efficient tidal dissipation in
most cases of interest. The resulting value of Q depends in a highly erratic
way on the forcing frequency, but we provide evidence that the relevant
frequency-averaged dissipation rate may be asymptotically independent of the
viscosity in the limit of small Ekman number. In short-period extrasolar
planets, if the stellar irradiation of the planet leads to the formation of a
radiative outer layer that supports generalized Hough modes, the tidal
dissipation rate can be enhanced through the excitation and damping of these
waves. These dissipative mechanisms offer a promising explanation of the
historical evolution and current state of the Galilean satellites as well as
the observed circularization of the orbits of short-period extrasolar planets.
