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We construct the hydrodynamic theory of coherent collective motion("flocking") at a solid-liquid interface. The polar order parameter andconcentration of a collection of "active" (self-propelled) particles at aplanar interface between a passive, isotropic bulk fluid and a solid surfaceare dynamically coupled to the bulk fluid. We find that such systems arestable, and have long-range orientational order, over a wide range ofparameters. When stable, these systems exhibit "giant number fluctuations",i.e., large fluctuations of the number of active particles in a fixed largearea. Specifically, these number fluctuations grow as the $3/4$th power of themean number within the area. Stable systems also exhibit anomalously rapiddiffusion of tagged particles suspended in the passive fluid along anydirections in a plane parallel to the solid-liquid interface, whereas thediffusivity along the direction perpendicular to the plane is non-anomalous. Inother parameter regimes, the system becomes unstable.

Hydrodynamic theory of flocking at a solid-liquid interface: long range order and giant number fluctuations