A note on compressible dynamics in a barotropic ocean

In the ocean, the barotropic mode governs tides, tsunamis, fast basin-scale adjustment, and provides the primary pathway through which atmospheric pressure and wind forcing are transmitted into the ocean interior. Although current observing systems have reached millimeter-level precision, the theoretical formulation of the motions they reveal is most often based on the Boussinesq approximation, where seawater is treated as incompressible and the depth-integrated continuity equation expresses volume conservation. In a compressible fluid, pressure-induced density changes cause the water column to expand and contract: changes in column mass are therefore not represented solely by free-surface displacement. Here we derive a depth-integrated continuity equation for the barotropic mode that retains the effects of compressibility. It contains two additional terms: one depends on the time derivative of surface pressure, the other modifies the relation between free-surface height and transport divergence, thus leading to a depth-dependent correction to the long-wave phase speed. Comparisons with deep-ocean bottom-pressure measurements suggest that these terms can account for the observed attenuation between sea-surface height and bottom-pressure signals. Moreover, for a spatially uniform but time-varying surface pressure, the new equation generates sea-level gradients over variable bathymetry producing barotropic transports. This mechanism is absent in incompressible formulations, as illustrated by an idealized numerical experiment. The obtained correction can be implemented in existing Boussinesq ocean models without altering the momentum equations or the hydrostatic structure. Thus, time-varying pressure forces barotropic currents over variable bathymetry, and a simple correction that reconciles exact mass conservation with the observed sea-level–bottom-pressure relation is discovered. Plain Language Summary The barotropic mode of the ocean controls tides, tsunamis, and how the ocean rapidly responds to atmospheric pressure and winds. Current models mostly use the Boussinesq approximation, which treats seawater as incompressible and ignores how pressure compresses the water column. In this study we derive a depth integrated equation that includes compressibility. The new equation adds a term proportional to the time derivative of surface pressure and modifies the link between sea level changes and horizontal transport. It explains why sea level variations observed by satellites are systematically smaller than those inferred from bottom pressure and shows that a spatially uniform but time varying surface pressure can generate sea level gradients and hence ocean currents over variable seafloor topography—a mechanism absent in current models. We demonstrate this mechanism explicitly through an idealized barotropic numerical experiment. The correction is local, easy to implement, and reconciles exact mass conservation with observations. Whether these components of the depth integrated barotropic mass balance play a significant role in realistic ocean general circulation model simulations remains to be investigated.