High-frequency (ω ⪆ ωci) electrostatic instabilities driven by an electron-ion relative drift velocity across an external magnetic field are discussed for an inhomogeneous plasma. The presence of a density gradient perpendicular to the magnetic field has been found to greatly reduce critical velocities for the onset of several instabilities such as lower hybrid oscillations and ion cyclotron harmonic waves. It has been also shown that the lower hybrid instability can be caused by some positive dissipation mechanism such as Landau damping due to thermal ions and a finite electron collision rate. Some applications have been made to turbulent heating experiments in plasmas in which cross-field currents are present.