A weakly ionized plasma with gradients in the ionization density under the influence of the crossed electric and magnetic fields is subject to a unique type of plasma instability called the cross-field or E×B instability. This paper examines by means of a linear analysis the relevance of such an instability mechanism to the formation of the field aligned columns of enhanced ionization or striations in the artificially generated plasma clouds in the upper atmosphere. A Gaussian model for the electron density distribution in the cloud and the electric fields that have been derived from the ion and neutral cloud drift motions have been used in computing the characteristic parameters, which are the minimum wavelength λmin and the e-folding growth time τ for the instability. A brief comparison has shown good agreement between the computed instability model and the observed striation formation in one of the high latitude Barium releases conducted in Alaska during March 1969 (test hemlock of Secede III series). Some general results that show the dependence of the instability parameters on the height of the release and cloud dimension are also presented.