| Rotating superfluid helium contains normal and superfluid components, the latter exhibits irrotational flow except on discrete vortex lines which are arranged in a dense two-dimensional lattice, and result in a velocity pattern which varies rapidly in a microscopic scale and is thus inconvenient to use in a hydrodynamic description of the system. Instead Bekarevich and Khalatnikov (BK) introduced a macroscopic description using averaged velocity fields in the hydrodynamic equations. However, the BK hydrodynamics does not adequately include the elastic properties of the vortex lattice and cannot predict the Tkachenko shear waves that this lattice supports. In this thesis, a coarse-grained hydrodynamics is developed from the exact description of Tkachenko. To account for the dynamics of the vortex lattice, the macroscopic vortex displacement field is treated as an independent degree of freedom. The conserved energy is written in terms of the coarse-grained normal fluid, superfluid, and vortex velocities and includes an elastic energy associated with deformations of the vortex lattice. Equations of motion consistent with the conservation of energy, entropy and vorticity and containing mutual friction terms arising from microscopic interactions between normal fluid excitations and the vortex lines are derived. When the vortex velocity is eliminated from the damping terms, this system of equations becomes essentially that of BK with added elastic terms in the momentum stress tensor and energy current. The dispersion relation and damping of the first and second sound modes and the two transverse modes sustained by the system are investigated. It is shown that mutual friction mixes the transverse modes of the normal and superfluid components and damps the transverse mode associated with the relative velocity of these components, making this wave evanescent in the plane perpendicular to the rotation axis. The wave associated with transverse motion of the total mass current is a generalized Tkachenko mode, whose dispersion relation reduces to that derived by Tkachenko wave when the wavevector lies in this plane. |
