Tidal Response of a Bay
A lagoon or bay connected to the ocean by an inlet has a tide that is smaller than that of the ocean due to the flow restriction imposed by the small cross-sectional area of the inlet. In addition, energy losses, due to friction and entrance and exit losses, serve to reduce the tide range of the bay. Finally, the bay tide also lags that of the ocean due to the time required to fill the lagoon.
This tide-calculating applet uses Keulegan's equation to solve for the bay tide. Keulegan showed that only a single parameter (the repletion coefficient K ) described the ocean/bay tidal system. The repletion coefficient is a dimensionless number involving the tide range in the ocean, the length and cross-sectional area of the inlet, and the planform area of the bay. Losses are also included (here, entrance and exit losses were taken as 0.2 and 1.0, respectively and the Darcy-Weisbach friction factor is 0.001).
The plotted results show the ocean and bay tides. Notice that the bay tide is smaller and lags the ocean tide. The larger the inlet geometry and the smaller the bay, the less the lag and the greater the bay tide range. The output data include the repletion coefficient corresponding to the input data, the phase lag (in degrees) of the bay tide, and the ratio of the bay tide to the ocean tide (denoted as response).
Also notice that the ocean and bay tides have the exact same elevation, when the tide in the bay reverses. The match means that there is no water level difference between ocean and bay.
Keulegan's method utilizes two equations: the first is a conservation of mass equation that relates the flow of water through the inlet to the change of water level within the bay. He assumed that the bay was small enough that tidal propagation times across the bay were small. The second equation relates the flow in the inlet to the water level difference between the ocean tide and the bay tide. The larger the difference the greater the flow within the inlet. Keulegan combined these equations into one nonlinear equation.
The procedure to obtain the solution above involves numerically integrating the single ODE using a Runge-Kutta 4th order method and examining the results after 3 tidal cyles (to reach steady state). The nonlinear equation introduces higher harmonics into the bay that that do not exist in the ocean tide (taken to be a sinusoidal semi-diurnal tide--occurring twice per day).
Keulegan, G.H., "Tidal Flow in Entrances,'' U.S. Army Corps of Engineers, Committee on Tidal Hydraulics, Tech. Bull. 14, Vicksburg, 1967.
Comments: Robert Dalrymple
Center for Applied Coastal Research
University of Delaware, Newark DE 19716