Two-fluid hydrodynamics for a trapped weakly interacting Bose gas

We derive coupled equations of motion for the condensate (superfluid) and noncondensate (normal fluid) degrees of freedom in a trapped Bose gas at finite temperatures. Our results are based on the Hartree-Fock-Popov approximation for the time-dependent condensate wave function, and thermodynamic local equilibrium for the noncondensate atoms. In the special case of a uniform weakly interacting gas, our hydrodynamic equations are shown to be consistent with the two-fluid equations of Landau. The collective modes in a parabolically trapped Bose gas include the analog of the out-of-phase second-sound mode in superfluid ${}^4\mathrm{He}$.