Influence of asymmetric tidal mixing on sediment dynamics in a partially mixed estuary
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Abstract: To investigate the influence of asymmetric tidal mixing (ATM) on sediment dynamics in tidal estuaries, we developed a vertically one-dimensional idealized analytical model, in which the M2 tidal flow, residual flow and suspended sediment concentration (SSC) are described. Model solutions are obtained in terms of tidally-averaged, and tidally-varying components (M2 and M4) of both hydrodynamics and sediment dynamics. The effect of ATM was considered with a time-varying eddy viscosity and time-varying eddy diffusivity of SSC. For the first time, an analytical solution for SSC variation driven by varying diffusivity could be derived. The model was applied to York River Estuary, where higher (or lower) eddy diffusivity was observed during flood (or ebb) in a previous study. The model results agreed well with the observation in both hydrodynamics and sediment dynamics. The vertical sediment distribution under the influence of ATM was analyzed in terms of the phase lag of the M2 component of SSC relative to tidal flow. The phase lag increases significantly in estuaries with typical ATM (higher diffusivity during flood and lower diffusivity during ebb) for the case of seaward-directed net bottom shear stress (e.g., strong river discharge). In contrary, the phase lag is reduced by ATM, if the tidally-averaged bottom shear stress is landward (e.g., strong horizontal density gradient). The dynamics of sediment transport was analyzed as a function of ATM phase lag to identify the time of highest sediment diffusivity, as well as a function of the residual flow, to evaluate the relative importance of seaward and landward residual flows. In estuaries with relative strong fresh water discharge or weak tidal forcing (in case of flood season or neap tide), the near bottom SSC could be higher during ebb than during flood, since the bottom shear stress is higher during ebb due to seaward residual flow. However, landward net sediment transport can be expected in these estuaries in case of a typical ATM, because higher diffusivity causes higher SSC and landward transport during the flood period, while both SSC and seaward transport could be lower during ebb. On the contrary, seaward sediment transport can be expected in estuaries with landward tidally mean bottom shear stress in case of a reverse ATM, where sediment diffusivity is higher during the ebb.
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Figure 1. Site map of the York River Estuary showing the location of field observations. A sketch map of the model geometry attached in the top-right corner, where
$ z=\eta $ is the elevation of the water surface and$ z=-H $ is the bed. The horizontal dashed line indicates the average sea level (z = 0), while the solid line shows surface elevation during a tidal period of length P.
Figure 2. Model results of the vertical-temporal structure of total flow velocity (u = u0+u2) (a) and of the vertical structure of the residual components (b) based on the parameters of an observation point in the York River Estuary (Table 1). The blue line shows the flow component driven by the longitudinal density gradient (
$ {u}_{0\rho } $ ), the green line is contribution of asymmetric tidal mixing ($ {u}_{0a} $ ), and the red line is the contribution from fresh water discharge ($ {u}_{0q} $ ). The solid black line shows the corresponding modeled mean velocity and the dashed line shows the observed mean velocity from Scully and Friedrichs (2003).
Figure 3. Model results (a) and observations (b, taken from Scully and Friedrichs (2003)) for the vertical-temporal structure of suspended sediment concentration (SSC) over two M2 tidal periods (~ 25 h). The line plots in panel b compares temporal variation of near bottom SSC (z = 7 m) from observations and model results. The dashed horizontal lines in panel a mark the depth range with observational data shown in panel b. The labels “F” and “E” in panel a and b denote “flood” and “ebb”, respectively. Panels c, d and e show individual components of M2 component SSC induced by bottom shear stress (c2t), M2 component SSC induced by asymmetry of diffusivity (c2a), and M4 component SSC induced by bottom shear stress (c4t), respectively. Panel f shows the vertical distribution of SSC amplitude of each component.
Figure 4. Vertical structure of sediment flux and its components based on the parameters for a point in the York River Estuary.
$ {F}_{0a} $ ,$ {F}_{0\rho } $ and$ {F}_{0q} $ represent residual sediment flux related to asymmetry tidal eddy viscosity, horizontal density gradient, and river discharge.$ {F}_{2t} $ and$ {F}_{2a} $ represent tidal sediment flux related to bottom shear stress varying in M2 frequency and asymmetry tidal diffusivity. The dashed black line (YR99) shows the observed sediment flux cited from Scully and Friedrichs (2003).
Figure 5. Vertical-temporal structure of the asymmetric tidal mixing (ATM)-induced SSC (c2a) during a single M2 tidal period for different phase angles between tidal flow and sediment diffusivity (
$ \theta $ ). Panels a, b, c and d represent$\theta =0,\theta =0.5\pi , $ $ \theta =\pi \;\mathrm{a}\mathrm{n}\mathrm{d}\;\theta =1.5\pi$ , respectively. The remaining parameters are shown in Table 1.
Figure 6. Vertical structure of phase lag of suspended sediment concentration in response to diffusivity phase
$ \theta $ in case of:$\partial \rho /\partial x=0,{U}_{0}=0.15\;\mathrm{m}/{\mathrm{s}}$ (scenario z (i)) (a) and$\partial \rho /\partial x=1\times {10}^{-3}\;\mathrm{k}\mathrm{g}/{\mathrm{m}}^{4},{U}_{0}=0$ (scenario (ii)) (b). The black line represents the analytical result of Yu et al. (2011). The gray line shows the model results of c2t, i.e., without consideration of asymmetric tidal mixing. The remaining 4 colored lines represent the result of$ \phi \left(z\right) $ with consideration of c2a for four$ \theta $ values ($ 0, 0.5\pi ,\pi , 1.5\pi $ ).
Figure 7. Vertical structure of M2 suspended sediment concentration (SSC) component (c2) over a tidal period under four different values of phase angle of tidally varying sediment diffusivity (
$ \theta $ ) in case of scenario (i),$\partial \rho /\partial x=0,{U}_{0}=0.15\;{\rm{m}}/{\rm{s }}$ . Panels a, b, c and d show results for$\theta =0,\;\theta =0.5\pi ,\; \theta =\pi \;\mathrm{a}\mathrm{n}\mathrm{d}\;\theta =1.5\pi$ , respectively.
Figure 8. Vertical structure of M2 suspended sediment concentration (SSC) component (c2) over a tidal period under four different values of phase angle of tidally varying sediment diffusivity (
$ \theta $ ) in case of scenario (ii),$ \partial \rho /\partial x=1\times {10}^{-3}\; {\rm{kg}}/{{\rm{m}}}^{4},\;{U}_{0}=0 $ . Panels a, b, c and d show results for$ \theta =0, \; \theta =0.5\pi ,\;\theta =\pi \;\mathrm{and}\;\theta =1.5\pi $ , respectively.
Figure 9. Tidally mean near bottom shear stress
$ \left\langle{{\tau }_{b}}\right\rangle $ (a) and sediment flux (b–i) as a function of$ \theta $ and$ \gamma =\left[{u}_{0\rho }\right]/\left[{u}_{0q}\right] $ . The dashed white line indicates a ratio$ \gamma =1 $ . The solid white line indicates the value$ \gamma $ for which the average shear stress$ \left\langle{{\tau }_{b}}\right\rangle=0 $ . The solid red line in panel b–i indicates values of$ \gamma $ for which the sediment flux is zero. The remaining boundary conditions can be found in Table 1.Table 1. Assumptions of the one-dimensional analytical model
Assumption Explanation $ \dfrac{\eta }{H}=O\left(\varepsilon\right) $ water level fluctuation is an order of magnitude smaller than water depth $ \left(\dfrac{U_0}{U_2},\dfrac{U_d}{U_2},\dfrac{U_q}{U_2}\right)=O\left(\varepsilon\right) $ residual currents are an order of magnitude smaller than tidal currents $ \left(\dfrac{A}{\omega H^2},\dfrac{K}{\omega H^2}\right)=O\left(1\right) $ frictional force can affect the whole water column $ \left(\dfrac{A_2}{A_0},\dfrac{K_2}{K_0}\right)=O\left(\varepsilon\right) $ the fluctuating component of eddy viscosity (diffusivity) is an order of magnitude smaller than the mean component $ \dfrac{w_s}{\omega H}=O\left(1\right) $ length scale of sediment settling during a tidal cycle is comparable to water depth Note: Here, $ \left({U}_{0},{U}_{2}\right) $ represent typical values of the mean velocity and M2 tidal velocity, respectively. The parameters ${U}_{d}=g{H}^{3} \Big(\dfrac{\partial \rho }{\partial x}\Big)/ \left(48\rho A\right)$ and $ {U}_{q}=2q/\left(3H\right) $ are the typical velocity scales for residual currents driven by an along-estuary density gradient and fresh water discharge. The parameter $ \varepsilon \ll 1 $ indicates the magnitude of the difference to higher-order physical components. 
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Table 2. Definition of symbols used in the model description
Symbol Definition Symbol Definition $ \omega $ M2 tidal frequency $ {w}_{s} $ settling velocity $ P $ M2 tidal period $ c $ SSC $ Ri $ Richardson number $ {c}_{0} $ tidally mean SSC (dominant order) $ s $ partially slip parameter $ {c}_{4} $ M4 component SSC (dominant order) $ \partial {\rho }/\partial {x} $ along-estuary density gradient $ {c}_{2} $ M2 component SSC (first order) $ {{q}}_{0} $ fresh water discharge $ {c}_{2t} $ bottom shear stress induced $ {c}_{2} $ $ {{q}}_{2} $ M2 tidal discharge $ {c}_{2a} $ ATM induced $ {c}_{2} $ $ {u} $ along-estuary velocity $ {c}_{a} $ reference concentration $ {{u}}_{2} $ M2 tidal velocity (dominant order) ($ A $, $ K $) eddy viscosity and eddy diffusivity $ {{u}}_{0} $ tidally mean velocity (first order) ($ {A}_{0} $, $ {K}_{0} $) tidally mean component of A and K $ {{u}}_{0{a}} $ ATM induced u0 ($ {A}_{2} $, $ {K}_{2} $) M2 component of A and K $ {{u}}_{0{\rho }} $ $ \partial \rho /\partial x $ induced $ {u}_{0} $ ($ \left|{A}_{2}\right| $, $ \left|{K}_{2}\right| $) amplitude of A2 and K2 $ {{u}}_{0{q}} $ fresh water discharge induced u0 $ \theta $ phase of A2 or K2 Note: ATM: asymmetric tidal mixing; SSC: suspended sediment concentration.
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Table 3. Model input parameter values representative for a point in the York River Estuary, USA during April 1999 (referenced from Scully and Friedrichs (2003, 2007))
Quantity Symbol Value Water depth H 7.0 m Amplitude of tidal velocity U2 0.6 m/s Depth mean velocity U0 0.045 m/s Angular M2 tidal frequency $ \omega $ 1.4 × 10−4 s−1 Gravitational acceleration g 9.81 m/s2 Reference density $ {\rho }_{0} $ 1 020 kg/m3 *Bed to surface density difference $ \Delta \rho $ 2.0 kg/m3 *Along-channel density gradient $ \partial \rho /\partial x $ 8.6 × 10−4 kg/m4 *Threshold bed shear stress $ {\tau }_{c} $ 0.002 kg/(m·s2) *Settling velocity ws 0.6 mm/s *Ratio of $ \left|{K}_{2}\right|/{K}_{0} $ or $ \left|{A}_{2}\right|/{A}_{0} $ $ \alpha $ 0.3 *Phase lag of $ {K}_{2} $ or $ {A}_{2} $ $ \theta $ $ 1.4\pi $ Note: The stars mark the parameters deduced from the above references and were modified during model validation.
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