## Nonlinear effects of a non-uniform bathymetry on the tidal response within a Helmholtz basin

##### Summary

The tidal response of an almost-enclosed basin, connected to an open tidal sea by a narrow strait, is examined. Such a basin is a Helmholtz resonator. A particular focus lies on nonlinear phenomena and in particular the effect of a non-uniform bathymetry (bottom profile) on tidal motion in the strait and bay. An idealised, process-based model for a nonlinear tidal Helmholtz basin is constructed, expressing the state of the system into a single variable ordinary differential equation for the excess volume of water in the basin. From its solutions, sea level variations within the basin and tidal currents through the strait can be extracted. These solutions have certain characteristics. Firstly, within the solution for the water level within the basin, the phase of each of the overtides (higher harmonics) is strongly determined by the phase of the component of the response which has the same frequency as the main external tidal forcing component (the main response). In other words, the phases of the overtides are ‘enslaved’ to the phase of the main response. Secondly, the squared amplitude of each of the overtides is related to the amplitude of the main response by a polynomial relation that is one order higher than the order of overtide. For example, for the first overtide, this polynomial is of order two. Thirdly, due to a non-uniform bathymetry, the phase of the first overtide is shifted by a factor π compared to the main response, leading to a vertical asymmetry in the tidal wave, while other nonlinear effects lead to different types of asymmetry in the tidal curve. Finally, the solution of the model in a low-friction and low-forcing limit suggests that multiple different responses in the water level and current velocity are possible for the same tidal forcing. It is also possible for the system to switch between the possible equilibria of the response. When this happens, it makes the tidal response chaotic. To test these model findings, water level and velocity time series from four different tidal basins, Strangford Lough, Marsdiep, San Francisco Bay and Tampa Bay, are examined. To quantify the asymmetry in the tidal response caused by the shift of the phase of the first overtide, several parameters are defined. These parameters are either based on the phases and amplitudes of different tidal components or on the difference in height and duration between high and low tide. To find amplitudes and phases of tidal components as a function of time, a wavelet analysis is used. The observations confirm that the phases of the overtides are enslaved to that of the main response. Namely, correlation coefficients for the first overtide inside the basins range between 0.935 and 0.968. Furthermore, in all basins, the correlation increases when going further inside the basin. For two of the four basins, the Marsdiep basin and San Francisco Bay, observations indicate that the tidal response is vertically asymmetric. Namely, in the Marsdiep basin the tidal response is strongly asymmetric and in the San Francisco Bay, there is a shift towards vertical asymmetry when going from outside to inside the bay. This suggests that for these two basins, the non-uniform bathymetry may be a main source of nonlinearity. For the other two basins, the Strangford Lough and Tampa Bay, the opposite is true, indicating that the non-uniform bathymetry is not a main source of nonlinearity. Observations suggest that there is a polynomial relation between the squares of the amplitudes of the overtides and the amplitude of the main response of one order higher than the order of overtide. Namely, generally the data corresponds well with the suggested polynomial relation, correlating with up to R^2 = 0.959 and increasing in correlation when going further inside a basin. However, some other local effect can also be observed, changing the coefficients of the polynomial through time. This appears to happen in the San Francisco Bay, where the correlation for the amplitude of the first overtide is only R^2 = 0.278. From a wavelet analysis, no evidence of multiple equilibria and no clear evidence for chaotic tides is observed for any of the tidal basins. This is most likely because a low-friction limit is not present.