Effects of bottom slope, flocculation and hindered settling on the coupled dynamics of currents and suspended sediment in highly turbid estuaries, a simple model

Abstract

This study aims at gaining basic understanding about two specific phenomena that are observed in the highly turbid estuaries tidal Ouse, Yangtze and Ems, i.e. (1) the accumulation of suspended matter in the deeper parts of the estuaries and (2) the relatively high values of turbidity near the surface in the area of the turbidity maximum. A semi-analytical model is analysed to verify the hypothesis that these phenomena result from bottom slope-induced turbidity currents and from hindered settling, respectively. The model governs the dynamics of residual flow, driven by fresh water discharge, salinity gradients and turbidity gradients. It further uses the condition of morphodynamic equilibrium (no divergence of net sediment transport) to compute the residual sediment concentration. New aspects are that depth variations on flow and mixing processes, as well as flocculation and hindered settling of sediment, are explicitly accounted for. Tides act as a source of mixing and erosion of sediment only, thus processes like tidal pumping are not considered. Model results show that the estuarine turbidity maximum (ETM) shifts in the down-slope direction, compared to the case of a constant depth. Slope-induced turbidity currents, which are directed down-slope near the bottom and up-slope near the surface, are responsible for this shift, thereby confirming the first part of the hypothesis above. The down-slope shift of the ETM is reduced by currents resulting from gradients in depth-dependent mixing, which counteract turbidity currents, but which are always weaker. Including flocculation and hindered settling yields increased surface sediment concentrations in the area of the turbidity maximum, compared to the situation of a constant settling velocity, thereby supporting the second part of the hypothesis. Sensitivity experiments reveal that the conclusions are not sensitive to the values of the model parameters.

Appendix: Expressions

Description of the model parameters values for the model. All model parameters are functions of the scaled vertical coordinate ζ, the Péclet number Pe _v depends on horizontal coordinate x (see Eq. (16)) and the scaled depth $\tilde{h}$ which is defined as h(x)/H.

$$ m_1 = (\tilde{h}^3-9\zeta^2\tilde{h}-8\zeta^3), $$

(23)

$$ m_2 = \frac{12G_1}{Pe_v^{4}\tilde{h}^3}\mathrm{exp}\Big(-Pe_v\big(\zeta+\tilde{h}\big)\Big), $$

(24)

$$ m_{3} = -m_{2}Pe_{v}, $$

(25)

$$ m_{4} = \Bigg(\frac{24G_2}{Pe_v^5\tilde{h}^3}\mathrm{exp}\Big(-Pe_v\big(\zeta+\tilde{h}\big)\Big)-m_2\tilde{h}\Bigg)Pe_v, $$

(26)

in which

$$ \begin{array}{rll} G_1&=&4\tilde{h}^3Pe_v \\ &&+\,\Big(-6\zeta^2+6\tilde{h}^2+\tilde{h}^2Pe_v^2(\zeta+\tilde{h})(3\zeta+\tilde{h})\Big) \\ &&{\kern12pt}\times\,\mathrm{exp}\Big(Pe_v\zeta\left.\right)\Big) \\ &&+\Big(6\zeta^2-6\tilde{h}^2-6\zeta^2\tilde{h}Pe_v+2\tilde{h}^3Pe_v\Big) \\ &&{\kern12pt}\times\,\mathrm{exp}\Big(Pe_v\big(\zeta+\tilde{h}\big)\Big),\\ G_2&=&6\tilde{h}^3Pe_v+2\zeta Pe_v^2\tilde{h}^3 \\ &&+\,\Big(\tilde{h}^4Pe_v^2+4\zeta\tilde{h}^3Pe_v^2+12\tilde{h}^2 \\ &&+\,3\zeta^2Pe_v^2\tilde{h}^2-12\zeta^2\Big)\mathrm{exp}\Big(Pe_v\zeta\left.\right)\Big) \\ &&+\,\Big(-12\zeta^2\tilde{h}Pe_v-\tilde{h}^4Pe_v^2+6\tilde{h}^3Pe_v \\ &&-\,12\tilde{h}^2+3\tilde{h}^2\zeta^2Pe_v^2+12\zeta^2\Big)\mathrm{exp}\Big(Pe_v\big(\zeta+\tilde{h}\big)\Big). \end{array} $$

$$ \begin{array}{rll} T_s&=&\frac{1}{Pe_v^4}\Big\{\Big(-48+Pe_v^3\tilde{h}^3-18Pe_v\tilde{h}\Big)\\ &&{\kern18pt}-\times\,\mathrm{exp}\big(Pe_v\tilde{h}\big)+48-30Pe_v\tilde{h}\\ &&{\kern18pt}+\,6Pe_v^2\tilde{h}^2\Big\}, \end{array} $$

(27)

$$ T_t = \frac{-12H_1}{Pe_v^{7}\tilde{h}^3}\mathrm{exp}\big(-2Pe_v\tilde{h}\big), $$

(28)

$$ T_h = -T_tPe_v, $$

(29)

$$ T_{p} = \Bigg(T_t\tilde{h}-\frac{-12H_2}{Pe_v^{8}\tilde{h}^3}\mathrm{exp}\big(-2Pe_v\tilde{h}\big)\Bigg)Pe_v, $$

(30)

$$ \begin{array}{rll} T_Q &=& \frac{-2}{Pe_v^3\tilde{h}^3}\Big\{(-1+\frac{1}{2}Pe_v^2\tilde{h}^2)\mathrm{exp}\big(-Pe_v\tilde{h}\big)\\ &&+\,1-\tilde{h}Pe_v\Big\}, \end{array} $$

(31)

$$ T_{K_h} = \frac{1-\mathrm{exp}\big(-Pe_v\tilde{h}\big)}{Pe_v}, $$

(32)

$$ T_{hK_h} = \mathrm{exp}\big(-Pe_v\tilde{h}\big)-1, $$

(33)

$$ \begin{array}{rll} T_{pK_h} &=& \Bigg(T_{hK_h}\tilde{h}-\frac{\mathrm{exp}\big(-Pe_v\tilde{h}\big)+Pe_v\tilde{h}-1}{Pe_v^2}\Bigg) \\ &&\times\,Pe_v, \end{array} $$

(34)

in which

$$ \begin{array}{rll} H_1&=&12-\tilde{h}^2Pe_v^2\Big(12+\tilde{h}Pe_v\big(6+\tilde{h}Pe_v\big)\Big) \\ &&+\,\Big(-24+4\tilde{h}Pe_v\big(6+3\tilde{h}Pe_v-\tilde{h}^2Pe_v^2\big)\Big) \\ &&\times\,\mathrm{exp}\Big(Pe_v\tilde{h}\Big)\\ &&+\,\Big(12-2\tilde{h}Pe_v\big(12-6\tilde{h}Pe_v+\tilde{h}^2Pe_v^2\big)\Big) \\ &&\times\,\mathrm{exp}\Big(2Pe_v\tilde{h}\Big),\\ H_2&=&48-\tilde{h}^2Pe_v^2\big(36+15\tilde{h}Pe_v+2\tilde{h}^2Pe_v^2\big) \\ &&+\,\Big(2\tilde{h}Pe_v\big(48+\tilde{h}Pe_v\big(-6+\tilde{h}Pe_v\big) \\ &&\times\,\big(-2+\tilde{h}Pe_v\big)\big)-96\Big)\times\mathrm{exp}\Big(Pe_v\tilde{h}\Big) \\ &&-\,\Big(\tilde{h}Pe_v\big(4-\tilde{h}Pe_v\big)\big(24+\tilde{h}Pe_v \\ &&{\kern12pt}\times\big(-9+2\tilde{h}Pe_v\big)\big)+48\Big)\times \mathrm{exp}\Big(2Pe_v\tilde{h}\Big). \end{array} $$