  In the presence of a strong electric field perpendicular to the magnetic
field, the electron cross-field (E$\times$B) flow relative to the unmagnetized
ions can cause the Electron Cyclotron Drift Instability (ECDI) due to
resonances of the ion acoustic mode and the electron cyclotron harmonics. This
occurs in collisionless shock ramps in space, and in $\rm{E \times B}$
discharge devices such as Hall thrusters. ECDI can induce an electron flow
parallel to the background E field at a speed greatly exceeding predictions by
classical collision theory. Such anomalous transport may lead to particle
thermalization at space shocks, and may cause unfavorable plasma flows towards
the walls of E$\times$B devices. The development of ECDI and anomalous
transport is often considered fully-kinetic. In this work, however, we
demonstrate that a reduced variant of this instability, and more importantly,
the associated anomalous transport, can be treated self-consistently in a
two-fluid framework without any collision. By treating electrons and ions on an
equal footing, the free energy allows the growth of an anomalous electron flow
parallel to the background E field. We first present linear analyses of the
instability in the two-fluid 5- and 10-moment models, and compare them against
the fully-kinetic theory. At lower temperatures, the two-fluid fastest-growing
mode is in good agreement with the kinetic result. Also, by including more
($>=10$) moments, secondary (and possibly higher) unstable branches can be
recovered. The dependence of the instability on various parameters is also
explored. We then carry out direct numerical simulations of the cross-field
setup using the 5-moment model. The growth of the instability and the anomalous
transport is confirmed. Finally, 5-moment and Vlasov simulations using
identical parameters in the lower-temperature regime are performed, showing
reasonable agreement.
