The interaction of a barotropic flow with topography generates baroclinic motion which exerts a stress on the barotropic flow. Here, explicit solutions are calculated for the spatial-mean flow (i.e. the barotropic tide) resulting from a spatially-uniform but time-varying body force (i.e. astronomical forcing) acting over rough topography. This approach of prescribing the force contrasts with that of previous authors who have prescribed the barotropic flow. It is found that the topographic stress, and thus the impact on the spatial-mean flow, depends on the nature of the baroclinic motion that is generated. Two types of stress are identified: (i) a ‘wave drag’ force associated with propagating wave motion, which extracts energy from the spatial-mean flow; (ii) a topographic ‘spring’ force associated with standing motion at the seafloor, including bottom-trapped internal tides and propagating low-mode internal tides, which significantly damps the time-mean kinetic energy of the spatial-mean flow, but extracts no energy in the time-mean. The topographic spring force is shown to be analogous to the force exerted by a mechanical spring in a forced-dissipative harmonic oscillator. Expressions for the topographic stresses appropriate for implementation as baroclinic drag parameterisations in global models are presented.

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