## 4 Discussion
The time scale necessary to bring the system back to its original state could be calculated as [11]:  
\[\mathrm{T_{r}}=U\!I\!I/\!g^{\prime}H \tag{3}\]  
where \(H\) was the depth of the estuary and \(U\) was the longitudinal velocity generated during the relaxation process, calculated in turn as:  
\[U\ =\ \sqrt{g^{\prime}H} \tag{4}\]Given that \(L\) was small, the order of magnitude was of just a few hours. Specifically, for Fangar Bay, T\({}_{\rm r}\) ranges of five to nine hours implied very short relaxation periods in comparison to larger domains. A visual inspection suggested a time lag between wind decay and the apparition of estuarine circulation in Fangar Bay of this order of magnitude (see Figure 8). The water renewal has a relevant influence on water quality within the bay. A simple assessment of the residence time may be computed dividing the volume of water by the flow rate that comes out of the bay (calculated from the velocities in Figure 12). In these cases, we obtain residence times of 4 d for SIMU 2 and 4, and 1 day for SIMU 1 and 5 and respectively. The rest of the simulations are in the same order of magnitude. From the observed change flow (Figures 8 and 9) the average residence time is estimated in 4 d. The study carried out by [PERSON] et al. [42] compute the residence times in the bay in more detail and some possible modifications to help in the bay management. Empirical orthogonal functions (EOF) estimated for the alongshore water current directions from the entire vertical profile were used to identify the modal variability in the water current measurements (Figure 13a-d). EOFs allow multivariable data to be decomposed into its main components. Along-axis EOF analysis depicted the prevalence of the first component in both periods: 58/76% (Mouth/Bay) in FANGAR-I and 80/90% (Mouth/Bay) in FANGAR-II. During weak wind or calm periods (i.e. most of the time in FANGAR-I), the main components of the EOF analysis showed crossing-zero (suggesting stratification) with baroclinic conditions (see Figure 13a-c). However, during periods of intense winds, as was true of FANGAR-II, the first component revealed a mixing situation, seen especially at the inner bay B point, with barotropic flows. Rotational effects in small bays and estuaries were expected to be small. Using the geometry of Fangar Bay, the Kelvin number was estimated, which considers the characteristic width of the estuary and the internal radius of deformation [43]:  
\[K_{e}=B/R_{in} \tag{5}\]  
where \(R_{in}\) was equal to \((g^{\prime}H)^{(1/2)}f\), \(g^{\prime}=\) reduced gravity, \(H=\) channel depth and \(f=\) Coriolis force. Rotational effects play a secondary role on the basin circulation when \(K^{e}<1\), and they can be considered negligible when \(K^{e}<<1\). In the case of Fangar Bay, which is only 2 km wide and 4 m deep, \(K^{e}=0.39\) (\(<<\)1) suggesting that the Coriolis effect is of little importance in terms of water circulation. This was supported by a comparison of SIMU 1 and SIMU 8 (Figure 11a,h), where the differences in water currents were negligible. Questions remain open from observational analysis, in particular the hydrodynamic response during downwind and the spatial variability of the currents. From a numerical point of view, numerical modeling effort may include the bottom roughness sensitivity test, longitudinal and vertical density  
Figure 13: Empirical orthogonal functions (EOF) analysis for low-frequency filtered data during FANGAR-I **(a,b)** and FANGAR-II **(c,d)** along-shore currents. The legend shows the corresponding percentage of the explained variability. variations, sea-level oscillations, effect density gradients and baroclinic process due to river discharge sources. The numerical model implemented here is focused on solving the bathymetric effect which affect the wind-driven water circulation due to its shallowness under idealized perspective. So, realistic model implementation is a challenge to evaluate some remaining questions addressed before.