  Magnetohydrodynamics is a theory of long-lived, gapless excitations in
plasmas. It was argued from the point of view of fluid with higher-form
symmetry that magnetohydrodynamics remains a consistent, non-dissipative theory
even in the limit where temperature is negligible compared to the magnetic
field. In this limit, leading-order corrections to the ideal
magnetohydrodynamics arise at the second order in the gradient expansion of
relevant fields, not at the first order as in the standard hydrodynamic theory
of dissipative fluids and plasmas. In this paper, we classify the
non-dissipative second-order transport by constructing the appropriate
non-linear effective action. We find that the theory has eleven independent
charge and parity invariant transport coefficients for which we derive a set of
Kubo formulae. The relation between hydrodynamics with higher-form symmetry and
the theory of force-free electrodynamics, which has recently been shown to
correspond to the zero-temperature limit of the ideal magnetohydrodynamics, as
well as simple astrophysical applications are also discussed.
