  We develop a formalism to describe extensions of existing axion haloscope
designs to those that possess directional sensitivity to incoming dark matter
axion velocities. The effects are measurable if experiments are designed to
have dimensions that approach the typical coherence length for the local axion
field. With directional sensitivity, axion detection experiments would have a
greatly enhanced potential to probe the local dark matter velocity
distribution. We develop our formalism generally, but apply it to specific
experimental designs, namely resonant cavities and dielectric disk haloscopes.
We demonstrate that these experiments are capable of measuring the daily
modulation of the dark matter signal and using it to reconstruct the
three-dimensional velocity distribution. This allows one to measure the Solar
peculiar velocity, probe the anisotropy of the dark matter velocity ellipsoid
and identify cold substructures such as the recently discovered streams near to
Earth. Directional experiments can also identify features over much shorter
timescales, potentially facilitating the mapping of debris from axion
miniclusters.
