A detailed analysis is given of the development of the electrostatic turbulence excited by a cross‐field current in a high density(&ohgr;p e ≫ &OHgr;e )collisionless plasma, as obtained from one‐ and two‐dimensional computer simulations. The drift velocity required for instability is found to be only slightly smaller than the critical drift for the unmagnetic two‐stream instability in contrast to the electron cyclotron drift instability. In one dimension the system remains self‐similar in the turbulent phase,< E˜2 > /8&pgr;nTe ≃ const, while the electron thermal velocity and the effective collision frequency increase linearly with time,&ngr;* ∝ &OHgr;et, in agreement with the theory of coherent trapped‐electron heating by Forslund, Morse, and Nielson. The switch‐off drift velocity, however, is&ngr;d ≈ 2cs, and hence the thickness of a magnetic sheath would be&Dgr; ∼ c/&ohgr;piinstead of&Dgr; ∼ c/&OHgr;e. In two dimensions two projections are investigated, thej‐Bplane and the plane perpendicular toB, the latter being the more relevant. The heating is much less efficient than in one dimension,&ngr; * ≲ 5 × 10−3 &ohgr;pe. It is found that&ngr;* ∝ ⟨E˜2⟩/8&pgr;nT, indicating stochastic dissipation. The gross features of the two‐dimensional behavior correspond to the picture of ion‐sound instability with electron runaway prevented by gyration.