Magnetic field observations in the dayside ionosphere of Venus revealed the existence of magnetic flux ropes. The general properties of these small-scale magnetic field structures can be explained by a theory of magnetic fluctuations excited by random magnetohydrodynamic flows of ionospheric plasma. The local spatial distribution of the magnetic field is random: the field is concentrated inside flux tubes separated by regions with weak fields. A mechanism of amplification of magnetic fluctuations in the presence of zero mean field, proposed by Zeldovich, is applied to the nonlinear theory of flux rope formation by means of a nonlinear equation derived from the induction equation; the nonlinearity is associated with the Hall effect. The equation describes the evolution of the correlation function of the magnetic field and resembles the Schr¿dinger equation except for a variable mass and the absence of the imaginary unit in the time-derivative term. In the limit of large Reynolds number the formulation is amenable to treatment by a modified WKB method. On the basis of this theory it is possible to explain why flux ropes are not observed if there is a strong regular large-scale magnetic field caused by the lowering of the ionopause. The theory predicts correctly the cross section of the flux ropes in the ionosphere of Venus and the maximum value of the magnetic field inside the flux rope. ¿ American Geophysical Union 1994