A two-dimensional, time dependent, numerical model of the atmospheric boundary layer is formulated and used to investigate various characteristics of air motion when a change occurs in the roughness of the underlying surface. The model assumes a dry adiabatic atmosphere and adiabatic changes of state, no dynamic pressure effects due to flow accelerations and no vertical displacement of the zero-velocity level at the lower boundary when a discontinuity in surface roughness is encountered. Numerical evaluation of all terms in the resulting equations shows little change in the computed fields of velocity if an incompressible atmosphere is assumed and horizontal mixing is neglected. A computational stability analysis is made which provides the relationship between time-step and grid spacing necessary to insure convergence of the numerical solution. For horizontally homogeneous conditions, comparison of the numerical solutions with various analytical solutions shows that the errors inherent in the finite difference formulation can be reduced to any desirable level by reducing the grid spacing, particularly near the lower boundary. Computed quantities include the two-dimensional distributions of horizontal velocity, vertical velocity and eddy shearing stress. Increasing surface roughness causes a decrease in the horizontal velocity near the lower boundary. This results in upward vertical motion. For a prevailing geostrophic wind speed of 10 m/sec and roughness parameters varying from 1 cm through 10 cm, maximum vertical motion on the order of 5 cm/sec is found over the roughness discontinuity at heights ranging from 50 to 100 meters. Vertical velocities in excess of 0.1 cm/sec are found several meters upstream from the discontinuity and persist for several. hundred meters downstream for the cases considered. The computations also show that a nearly logarithmic wind profile with associated constant shearing stress becomes established throughout a layer which increases in depth with distance downwind from the roughness discontinuity. The ratio of depth to downwind distance of this layer lies between 1/50 and 1/100. Reestablishment of the logarithmic profile does not imply equilibrium flow, however. It is found that a fetch on the order of several kilometers is necessary before the surface eddy stress differs by less than ten percent from the final. equilibrium values. Excellent quantitative' agreement is found with Panofsky and Townsend (1964), who employed data of Kutzbach (1961) to show that their theory agreed with observations. Because of the inability of the numerical model to consider zero-plane displacement and non-neutral atmospheric stability, both of which are probably significant and certainly present in the observations available from the University of Wisconsin, no further quantitative comparisons were made. However, the general features of the computed velocity and stress distributions agree very well with results of Stearns and Lettau (1963), hence there is at least good qualitative agreement with observations. Suggestions are made for generalizing and improving the numerical model.