3D analytical solution to the tidally induced Lagrangian residual current equations in a narrow bay

The 3D first-order Lagrangian residual velocity (LRV) equation is established, and its analytical solution is obtained in a narrow bay. The results show clearly the 3D structure of the first-order LRV. When the exponential bottom profile is assumed, the upper half layer of the water flows in through the deep channel from the open boundary directly to the head of the bay. Then the water will return to the area surrounding the lower half of the inflow area. The downwelling area is located mainly at the deep channel, while the upwelling area occupies both sides of the bay. The inter-tidal water transport, obtained by integrating the 3D first-order LRV through the water column, has a pattern similar to the previous study in which the 2D depth-averaged Lagrangian residual current equations were solved. The inter-tidal water transport is used to analyze the water exchange, and it is found that the water exchange at different cross sections increases smoothly with the distance between the cross sections and the head of the bay until about one wavelength. It is also found that the pattern of the breadth-averaged Lagrangian residual current varies with the length of the bay if a non-flat bottom profile is used. The depth-integrated LRV and the breadth-averaged LRV are mainly determined by the different terms of the tidal body force, with the former determined by the bottom friction related term and the latter by the eddy viscosity related term. When the bay is longer than one wavelength, different results in the outer bay can be observed.