Baroclinically driven estuarine flow: A perturbation approach

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The residual circulation and density in a shallow estuary of constant depth and width and flowing at high Rayleigh number (Ra) and small internal Froude number (Fr), based on riverine flow (U∞) and the horizontal density variation, are studied. Criteria under which direct tidal effects on the residual flow can be neglected are established. The effects of varying the bottom boundary condition on the velocity field from free slip to no slip are examined using a linear bottom stress boundary condition. The governing equations are nondimensionalized, expanded in a power series in the small parameter, and the zeroth‐ and first‐order equations governing the density field are solved; these solutions yield the zeroth‐ and first‐order results for the velocity field. The equation governing the density field at lowest order is nonlinear and is solved numerically. The results show that for sufficiently large Ra, the mean transport of salt is balanced by baroclinic advection near the mouth and by horizontal turbulent transport farther upstream. For these large Ra the horizontal structure as well as the vertical structure of the flow is strongly affected by the bottom boundary condition with the salinity disturbances determined under the free‐slip condition extending twice as far upstream as the no‐slip results. For smaller Ra, though still large compared to 1, the mean transport is balanced by horizontal mixing throughout the system, and the bottom boundary condition does not affect the horizontal structure. The penetration distance of oceanic waters, with the internal Fr fixed, increases as Ra increases above 24, reaches a maximum upstream penetration distance of Kh/U∞ when Ra reaches 65, and decreases as Ra increases further. The ratio of entrained oceanic waters to riverine inflow is inversely proportional to the internal Fr to the 2/3 power for very large Ra and becomes proportional to Ra as the latter decreases.