Fine sediment transport by tidal asymmetry in the high-concentrated Ems River: indications for a regime shift in response to channel deepening

Abstract

This paper describes an analysis of the observed up-river transport of fine sediments in the Ems River, Germany/Netherlands, using a 1DV POINT MODEL, accounting for turbulence-induced flocculation and sediment-induced buoyancy destruction. From this analysis, it is inferred that the net up-river transport is mainly due to an asymmetry in vertical mixing, often referred to as internal tidal asymmetry. It is argued that the large stratification observed during ebb should be attributed to a profound interaction between turbulence-induced flocculation and sediment-induced buoyancy destruction, as a result of which the river became an efficient trap for fine suspended sediment. Moreover, an asymmetry in flocculation processes was found, such that during flood relative large flocs are transported at relative large flow velocity high in the water column, whereas during ebb, the larger flocs are transported at smaller velocities close to the bed—this asymmetry contributes to the large trapping mentioned above. The internal tidal asymmetry and asymmetry in flocculation processes are both driven by the pronounced asymmetry in flow velocities, with flood velocities almost twice the ebb values. It is further argued that this efficient trapping is the result of a continuous deepening of the river, and occurs when concentrations in the river become typically a few hundred mg/l; this was the case during the 1990 survey analyzed in this paper. We also speculate that a second regime shift did occur in the river when fluid mud layers become so thick that net transport rates are directly related to the asymmetry in flow velocity itself, probably still in conjunction with internal asymmetry as well. This would yield an efficient mechanism to transport large amounts of fine sediment far up-river, as currently observed.

Appendix 1—The 1DV POINT MODEL

The 1DV equation for horizontal momentum reads:

$$ \frac{{\partial u}}{{\partial t}} + \frac{1}{\rho }\frac{{\partial p}}{{\partial x}} = \frac{\partial }{{\partial z}}\left[ {\left( {\nu + {\nu_T}} \right)\frac{{\partial u}}{{\partial z}}} \right] $$

(9)

The pressure term in (9) is adjusted to maintain a given, time-dependent depth-averaged velocity:

$$ \frac{1}{\rho }\frac{{\partial p}}{{\partial x}} = - \frac{{{\tau_{\rm{b}}}}}{{\rho h}} + \frac{{U(t) - {U_0}(t)}}{{{T_{\rm{rel}}}}}{, }\quad U(t) = \frac{1}{h}\int_{{z_{\rm{bc}}}}^{{Z_{\rm{s}}}} {u\left( {z\prime, t} \right)} {\hbox{d}}z\prime $$

(10)

The standard k–ε model, i.e. with the common coefficients with sediment-induced buoyancy destruction reads:

$$ \frac{{\partial k}}{{\partial t}} = \frac{\partial }{{\partial z}}\left\{ {\left( {\nu + {\nu_T}} \right)\frac{{\partial k}}{{\partial z}}} \right\} + {\nu_T}{\left( {\frac{{\partial u}}{{\partial z}}} \right)^2} + \frac{g}{\rho }{\Gamma_T}\frac{{\partial \rho }}{{\partial z}} - \varepsilon $$

(11)

$$ \frac{{\partial \varepsilon }}{{\partial t}} = \frac{\partial }{{\partial z}}\left\{ {\left( {\nu + \frac{{{\nu_T}}}{{{\sigma_\varepsilon }}}} \right)\frac{{\partial \varepsilon }}{{\partial z}}} \right\} + {c_{1\varepsilon }}\frac{\varepsilon }{k}{\nu_T}{\left( {\frac{{\partial u}}{{\partial z}}} \right)^2}{ + }\left( {1 - {c_{3\varepsilon }}} \right)\frac{g}{\rho }{\Gamma_T}\frac{\varepsilon }{k}\frac{{\partial \rho }}{{\partial z}} - {c_{2\varepsilon }}\frac{{{\varepsilon^2}}}{k} $$

(12)

The advection–diffusion equation for the mass concentration of suspended sediment reads:

$$ \frac{{\partial {c^{(i)}}}}{{\partial t}} - \frac{\partial }{{\partial z}}\left\{ {w_{\rm{s}}^{(i)}{c^{(i)}}} \right\} - \frac{\partial }{{\partial z}}\left\{ {\left( {D_{\rm{s}}^{(i)} + \Gamma_T^{(i)}} \right)\frac{{\partial {c^{(i)}}}}{{\partial z}}} \right\} = 0 $$

(13)

Water density and suspended sediment concentration are related through the equation of state:

$$ \rho \left( {{c^{(i)}}} \right) = {\rho_{\rm{w}}} + \sum\limits_{(i)} {\left\{ {\left( {1 - \frac{{{\rho_{\rm{w}}}}}{{\rho_{\rm{s}}^{(i)}}}} \right){c^{(i)}}} \right\}} $$

(14)

The advection–diffusion equation for the number concentration of suspended flocs accounts for turbulence-induced flocculation through an aggregation term (1st term in RHS (15)) and a floc breakup term (2nd term in RHS (15)).

$$ \begin{array}{*{20}{c}} {\frac{{\partial N}}{{\partial t}} + \frac{\partial }{{\partial z}}\left( {\frac{{\left( {1 - {\phi_*}} \right)\left( {1 - {\phi_{\rm{p}}}} \right)}}{{\left( {1 + 2.5{\phi_{\rm{f}}}} \right)}}{w_{{\rm{s,r}}}}N} \right) - \frac{\partial }{{\partial z}}\left( {{\Gamma_t}\frac{{\partial N}}{{\partial z}}} \right) = } \hfill \\{ - k_{\rm{A}}^\prime k_N^3\left( {1 - {\phi_*}} \right)G{c^{\tfrac{3}{{{n_f}}}}}{N^{\tfrac{{2{n_f} - 3}}{{{n_f}}}}} + k_{\rm{B}}^\prime k_N^{2q}{G^{q + 1}}{{\left( {{k_N}{c^{\tfrac{1}{{{n_f}}}}}{N^{\tfrac{{ - 1}}{{{n_f}}}}} - {D_{\rm{p}}}} \right)}^p}{c^{\tfrac{{2q}}{{{n_f}}}}}{N^{\tfrac{{{n_f} - 2q}}{{{n_f}}}}}} \hfill \\\end{array} $$

(15)

The settling velocity of single flocs in still water and its effective value by hindered settling are given by:

$$ {w_{{\rm{s,r}}}} = \frac{\alpha }{{18\beta }}\frac{{\left( {{\rho_{\rm{s}}} - {\rho_{\rm{w}}}} \right)g}}{\mu }D_{\rm{p}}^{3 - {n_f}}\frac{{D_{\rm{f}}^{{n_f} - 1}}}{{1 + 0.15Re_{\rm{f}}^{0.687}}} $$

(16)

$$ {w_{{\rm{s,eff}}}} = \frac{{{{\left( {1 - {\varphi_f}} \right)}^2}\left( {1 - {\varphi_{\rm{s}}}} \right)}}{{1 + 2.5{\varphi_f}}}{w_{{\rm{s,r}}}} $$

(17)

These equations are completed by geometrical relations between number concentration, mass concentration, volumetric concentration and floc size:

$$ N = \frac{1}{{{f_{\rm{s}}}}}\frac{c}{{{\rho_{\rm{s}}}}}D_{\rm{p}}^{{n_f} - 3}D_{\rm{f}}^{ - {n_f}} $$

(18)

$$ {\phi_{\rm{f}}} = \left( {\frac{{{\rho_{\rm{s}}} - {\rho_{\rm{w}}}}}{{{\rho_{\rm{f}}} - {\rho_{\rm{w}}}}}} \right)\frac{c}{{{\rho_{\rm{s}}}}} = \frac{c}{{{\rho_{\rm{s}}}}}{\left[ {\frac{{{D_{\rm{f}}}}}{{{D_{\rm{p}}}}}} \right]^{3 - {n_f}}} $$

(19)

This system of equations is closed with a set of boundary conditions, which, however, are not presented in this paper; the reader is referred to Winterwerp (2002). In these equations, the following symbols are used:

c ⁽ⁱ⁾ :

sediment concentration by mass for fraction (i)

D _f :

diameter of mud flocs

D _p :

diameter of primary particles

D _s :

molecular diffusion coefficient for sediment

G :

shear rate parameter; $G = \sqrt {{\left( {\varepsilon /\nu } \right)}}$

k :

turbulent kinetic energy

k _A :

flocculation parameter

k _B :

floc breakup parameter

N :

number concentration of the mud flocs

n _f :

fractal dimension

p :

pressure

q :

empirical coefficient; q = 0.5

t :

time

T _rel :

relaxation time

U :

actual computed depth-averaged flow velocity

U ₀ :

prescribed depth-averaged flow velocity

u :

horizontal flow velocity, positive in x-direction

z _bc :

apparent roughness height

w _s :

effective settling velocity

w _s,r :

settling velocity of individual particle

x :

horizontal coordinate

z :

vertical coordinate

Γ _T :

eddy diffusivity

ε :

dissipation rate per unit mass

ρ :

bulk density of water–sediment mixture

ν :

molecular viscosity

ν _T :

eddy viscosity

ρ _f :

floc density

ρ _s :

density of the sediment

ρ _w :

density of the water due to salinity only

ν _T :

eddy viscosity

σ _T :

turbulent Prandtl-Schmidt number

τ _b :

bed shear stress

ϕ :

volume concentration

ϕ _* :

min {1, ϕ}