  We present the results of a variational calculation of the frequencies of the
low-lying Landau two-fluid hydrodynamic modes in a trapped Fermi superfluid gas
at unitarity. Landau's two-fluid hydrodynamics is expected to be the correct
theory of Fermi superfluids at finite temperatures close to unitarity, where
strong interactions give rise to collisional hydrodynamics. Two-fluid
hydrodynamics predicts the existence of in-phase modes in which the superfluid
and normal fluid components oscillate together, as well as out-of-phase modes
where the two components move against each other. We prove that at unitarity,
the dipole and breathing in-phase modes are locally isentropic. Their
frequencies are independent of temperature and are the same above and below the
superfluid transition. The out-of-phase modes, in contrast, are strongly
dependent on temperature and hence, can be used to test the thermodynamic
properties and superfluid density of a Fermi gas at unitarity. We give
numerical results for the frequencies of these new modes as function of
temperature in an isotropic trap at unitarity.
