A two‐dimensional analytic tidal model for a narrow estuary of arbitrary lateral depth variation: The intratidal motion

An innovative method is introduced to solve a two-dimensional, depth-averaged analytic model for narrow estuaries or tidal channels with arbitrary lateral depth variations. The solution is valid if the lateral variation of the amplitude of tidal elevation (|Δa|) is small, i.e., |Δa| ≪ a, where a is the amplitude of the tidal elevation. This assumption is supported by a 60-day observation of elevation in the James River Estuary using pressure sensors at both sides of a cross section of the estuary. The error introduced by the solution is of the order of |Δa|/a, which has a maximum of ∼5% in the James River Estuary. The propagation of the tidal wave (elevation) is therefore essentially one-dimensional (along the estuary), regardless of the depth distribution, whereas tidal velocity has a strong transverse shear and is three-dimensional in general. Dozens of depth functions in six groups of various forms are used to calculate the solution. The tidal velocity is highly correlated with the bathymetry. The largest amplitude of the along-channel velocity is in the deepest water. The phase of the along-channel velocity in the shallow water leads that in deep water, causing a delay in time of flood or ebb in the deep water. The transverse velocity is generally small in the middle of a channel but reaches its maximum over the edges of bottom slopes. The depth function has a significant effect on the ellipticity and the sense of rotation of the tidal ellipses. By fitting the observed phase of semidiurnal tide in the James River Estuary to the phase of the momentum equation, we have obtained optimal values of the drag coefficient: 1.5 × 10−3 and 1.8 × 10−3 for the spring and neap tides, respectively. Then we apply these values of the drag coefficient and the model to the James River Estuary using the real bathymetry. Results show remarkable agreement between the observations and the model along the transects for both spring and neap tides. The cross-channel phase difference of the along-channel velocity between the channel and the shoal is found to be ∼1 hour, a value consistent with that from the model. The model-estimated lateral variation of elevation is 2.5% of the tidal amplitude, which is slightly smaller than the observed value.