  This paper considers a family of non-diffusive active scalar equations where
a viscosity type parameter enters the equations via the constitutive law that
relates the drift velocity with the scalar field. The resulting operator is
smooth when the viscosity is present but singular when the viscosity is zero.
We obtain Gevrey-class local well-posedness results and convergence of
solutions as the viscosity vanishes. We apply our results to two examples that
are derived from physical systems: firstly a model for magnetostrophic
turbulence in the Earth's fluid core and secondly flow in a porous media with
an "effective viscosity".
