  The non-relativistic covariant framework for fields is extended to
investigate fields and fluids on scale covariant curved backgrounds. The scale
covariant Newton-Cartan background is constructed using the localization of
spacetime symmetries of non-relativistic fields in flat space. Following this,
we provide a Weyl covariant formalism which can be used to study scale
invariant fluids. By considering ideal fluids as an example, we describe its
thermodynamic and hydrodynamic properties and explicitly demonstrate that it
satisfies the local second law of thermodynamics. As a further application, we
consider the low energy description of Hall fluids. Specifically, we find that
the gauge fields for scale transformations lead to corrections of the Wen-Zee
and Berry phase terms contained in the effective action.
