Supporting Information for River-enhanced non-linear overtide variations in river estuaries

a. State Key Lab of Estuarine and Coastal Research, East China Normal University, Shanghai 200241, China; b. Department of Hydraulic Engineering, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft 2600GA, the Netherlands; c. Marine and Coastal Systems Department, Deltares, Delft 2629 HV, the Netherlands; d. School of Ocean and Earth Sciences, University of Southampton, Southampton, UK; e. School of Marine Science, Sun Yat-Sen University, Guangzhou, China

is prescribed by constant values of 0, 10,000, 30,000, 60,000 m 3 /s, symbolized 216 as Q0, Q1, Q3, and Q6 scenarios, respectively, to facilitate harmonic analysis 217 with a stationary assumption. A dimensionless parameter, defined as the ratio 218 of river discharge to tide-averaged mean discharge (i.e., tidal prism divided by 219 tidal period) at the mouth section (R2T ratio), is estimated to be 0, 0.5, 2.6, and 220 42, which can be classified into tide-dominant, low, medium, and very high 221 river discharge circumstances, respectively (see section 3.3). The size of the 222 schematized estuary and the forcing conditions are characterized for a large 223 river estuary and in this case key dimensions from the Changjiang River 224 Estuary are used.

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To obtain a suitable bottom profile for the tidal model, we first run a 226 https://doi.org/10.5194/hess-2021-75 Preprint. Discussion started: 26 February 2021 c Author(s) 2021. CC BY 4.0 License. morphodynamic simulation based on the above-mentioned model outline, with 227 an M 2 tide and a river discharge seasonally varying between 10,000 and 228 60,000 m 3 /s as the boundary forcing conditions, as that in Guo et al. (2016).

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The long-term morphodynamic simulation starts from an initial sloping bed with  Past studies using similar 1D representation of tidal estuaries confirm the 242 capture of leading-order dynamic processes (Friedrichs and Aubrey, 1994;243 Lanzoni and Seminara, 1998). But it is noteworthy that the 1D model excludes  force and density variations are neglected (Dronkers, 1964), as follows, where u is velocity, η is water height above mean sea level, h is water depth 260 below mean sea level, g is gravitational acceleration (9.8 m 2 /s), and C is a 261 Cheźy friction coefficient prescribed as 65 m 1/2 /s uniformly.

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As the bed level is prescribed as an equilibrium profile, the water depth h is 263 constant. In the presence of a river discharge, the water level height is 264 composed of two parts, namely a mean water height related to river flow η 0 , 265 and a tide-induced water level oscillation, in case of the presence of M 2 tide only, in which η M2 is the surface amplitude of 268 M 2 , and ω is the frequency of M 2 , and k is tidal wave number. Similarly, the 269 current is composed of a mean current and a tidal component, in which u 0 is the mean current velocity, u M2 is the velocity amplitude of M 2 , θ 272 is the phase difference between tidal surface wave and tidal currents.

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Three nonlinear terms are identified in the tidal wave equations, namely 274 the discharge gradient term in the continuity equation, and the advection and 275 quadratic friction terms in the momentum equation: The bottom friction term is approximately expanded into a bottom shear 280 stress term and a term considering depth variations, as the two terms on the 281 right hand of Eq. (7), respectively, according to Godin and Martinez (1994), 282 given the tidal amplitude to water depth ratio (|η|/h) is generally smaller than Equation (9) suggests that the self-interaction of M 2 tide through the depth 301 variation term generates even-multiple frequency harmonics, e.g., M 4 and M 8 .

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Similar decomposition analysis for the advection and discharge gradient term 303 suggests the generation of even-frequency overtide as well (Parker, 1984;304 Wang et al., 1999). Following similar logic, when two components such as M 2 305 and S 2 tides are prescribed, compound tides with frequencies the sums (e.g.,

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MS 4 ) or differences (e.g., MSf) of the prescribed constituents are generated.

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The main focus of this study is devoted to M 4 overtide, given it is the first 308 overtide of M 2 and of profound importance for tidal asymmetry. further illustrate the river impact on the incoming tides (see Figure S1 in the SI).   amplified and distorted to some degree (Bonneton et al., 2015). Tidal bores are 578 less likely to occur in river estuaries because of river-enhanced damping, 579 although deformation is enhanced. The interaction between tidally-averaged 580 mean current and quarter-diurnal overtide current may contribute to net water 581 transport (Alebregtse and de Swart, 2016). Tide-averaged sediment transport 582 induced by tidal asymmetry related to M 2 -M 4 interaction plays a profound role 583 in controlling sediment import or export and resultant infilling or empty of 584 estuaries (Postma, 1961;Guo et al., 2014). It is noteworthy that the horizontal 585 velocity of the quarter-diurnal tide may exhibit more spatial variations than its 586 surface amplitude, owing to interaction with estuarine morphology and

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Although we have argued that channel convergence will not fundamentally 590 change the model results and main findings, the potential impact of the 591 simplified model setting in this study still mandates careful evaluation when 592 applying them to actual estuaries. For instance, regional narrowing and 593 shallowness in geometry and morphology is expected to induce variations in 594