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Abstract Realistic computational simulations in different oceanic basins reveal prevalent prograde mean flows (in the direction of topographic Rossby wave propagation along isobaths; aka topostrophy) on topographic slopes in the deep ocean, consistent with the barotropic theory of eddy-driven mean flows. Attention is focused on the western Mediterranean Sea with strong currents and steep topography. These prograde mean currents induce an opposing bottom drag stress and thus a turbulent boundary layer mean flow in the downhill direction, evidenced by a near-bottom negative mean vertical velocity. The slope-normal profile of diapycnal buoyancy mixing results in downslope mean advection near the bottom (a tendency to locally increase the mean buoyancy) and upslope buoyancy mixing (a tendency to decrease buoyancy) with associated buoyancy fluxes across the mean isopycnal surfaces (diapycnal downwelling). In the upper part of the boundary layer and nearby interior, the diapycnal turbulent buoyancy flux divergence reverses sign (diapycnal upwelling), with upward Eulerian mean buoyancy advection across isopycnal surfaces. These near-slope tendencies abate with further distance from the boundary. An along-isobath mean momentum balance shows an advective acceleration and a bottom-drag retardation of the prograde flow. The eddy buoyancy advection is significant near the slope, and the associated eddy potential energy conversion is negative, consistent with mean vertical shear flow generation for the eddies. This cross-isobath flow structure differs from previous proposals, and a new one-dimensional model is constructed for a topostrophic, stratified, slope bottom boundary layer. The broader issue of the return pathways of the global thermohaline circulation remains open, but the abyssal slope region is likely to play a dominant role.