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We show that literature results claimed for the magnetic field dependence ofthe longitudinal conductivity in anomalous first-order hydrodynamics are framedependent at this derivative order. In particular, we focus on$(3+1)$-dimensional hydrodynamics in the presence of a constant ${\calO}(\partial)$ magnetic field with a $U(1)$ chiral anomaly and demonstrate that,for constitutive relations up to and including order one in derivatives, theanomaly drops out of the longitudinal conductivity. In particular, magneticfield dependent terms that were previously found in the literature only enterthe non-zero frequency thermoelectric conductivities through explicitly framedependent pieces indicating that they are not physical. This issue can beavoided entirely by incorporating the magnetic field into the fluid'sequilibrium state.

Leading order magnetic field dependence of conductivities in anomalous hydrodynamics