  Existing hydrodynamic models of charged fluids consider any external electric
field acting on the fluid as either first order in the hydrodynamic derivative
expansion and completely arbitrary or zeroth order but constrained by the
fluid's chemical potential. This is in tension with experiments on charged
fluids, where the electric field is both zeroth order and completely arbitrary.
In this work, we take the first step at resolving this conundrum by introducing
a new class of hydrodynamic stationary states, including an arbitrary zeroth
order electric field, upon which hydrodynamics can be built. We achieve this by
first writing down the hydrostatic constitutive relations for a boost-agnostic
charged fluid up to first order in derivatives. Then we introduce suitable
energy and momentum relaxation terms to balance the influence of the electric
field on the fluid. This analysis leads to a new hydrostatic constraint on the
spatial fluid velocity, which can be used to define our class of states. This
constraint generalizes to the realm of hydrodynamics a similar constraint on
the velocity found in the Drude model of electronic transport. Our class of
states exhibits non-trivial thermo-electric transport even at ideal order,
since it hosts non-zero DC electric and heat currents. We derive the explicit
form of the corresponding conductivities and show they depend non-linearly on
the electric field.
