The aim of this paper is to model for the first time the variation of field and flow parameters in the magnetosheath during Earth passage of an interplanetary magnetic cloud. Under typical Solar wind conditions, magnetohydrodynamic (MHD) effects on the flow of plasma in the terrestrial magnetosheath are important only in a layer adjacent to the magnetopause which is a few thousand kilometers thick (“depletion layer” or “magnetic barrier”). During the passage of an interplanetary magnetic cloud, however, conditions upstream of the bow shock depart strongly from the norm. In this case, interplanetary parameters vary slowly over a wide range of values. Values of the upstream Alfvén Mach number are much lower than those otherwise sampled (∼3 versus 8–10). Together with the magnetic shear across the magnetopause, this parameter plays a central role in determining the structure of the magnetosheath close to the magnetopause. As a consequence of sustained low values of the upstream Alfvén Mach number, the magnetic field exerts a strong influence on the flow over a very substantial fraction of the magnetosheath throughout the duration of cloud passage, i.e., for a time period of the order of 1–2 days. We apply an algorithm to integrate the ideal MHD equations, using a boundary layer technique, and compute the variations of field and flow parameters along the stagnation streamline. We choose as our example the magnetic cloud which passed Earth on January 14–15, 1988. The interaction of this cloud with the magnetosphere, as regards the resulting ionospheric flow patterns and the substorm activity, has been the subject of various investigations. Using information from these studies, we obtain results on the magnetosheath when the magnetopause is modeled, first as a tangential discontinuity and then as a rotational discontinuity. Our results are in good general agreement with recent observations on the behavior of field and flow quantities in the magnetosheath region adjacent to the magnetopause. In addition, we predict the existence of a magnetic barrier when the upstream Alfvén Mach number is low, irrespective of the magnetic shear across the magnetopause.