Modeling the impact of wind and waves on suspended particulate matter fluxes in the East Frisian Wadden Sea (southern North Sea)

Abstract

Suspended particulate matter (SPM) fluxes and dynamics are investigated in the East Frisian Wadden Sea using a coupled modeling system based on a hydrodynamical model [the General Estuarine Transport Model (GETM)], a third-generation wave model [Simulating Waves Nearshore (SWAN)], and a SPM module attached to GETM. Sedimentological observations document that, over longer time periods, finer sediment fractions disappear from the Wadden Sea Region. In order to understand this phenomenon, a series of numerical scenarios were formulated to discriminate possible influences such as tidal currents, wind-enhanced currents, and wind-generated surface waves. Starting with a simple tidal forcing, the considered scenarios are designed to increase the realism step by step to include moderate and strong winds and waves and, finally, to encompass the full effects of one of the strongest storm surges affecting the region in the last hundred years (Storm Britta in November 2006). The results presented here indicate that moderate weather conditions with wind speeds up to 7.5 m/s and small waves lead to a net import of SPM into the East Frisian Wadden Sea. Waves play only a negligible role during these conditions. However, for stronger wind conditions with speeds above 13 m/s, wind-generated surface waves have a significant impact on SPM dynamics. Under storm conditions, the numerical results demonstrate that sediments are eroded in front of the barrier islands by enhanced wave action and are transported into the back-barrier basins by the currents. Furthermore, sediment erosion due to waves is significantly enhanced on the tidal flats. Finally, fine sediments are flushed out of the tidal basins due to the combined effect of strong erosion by wind-generated waves and a longer residence time in the water column because of their smaller settling velocities compared to coarser sediments.

Appendix

1.1 SPM transport module

The SPM transport module used in this study is described in more detail. For each sediment fraction c ⁱ, the usual diffusion–advection equation can be written as

$$\begin{array}{l}\partial_t c^i + \partial_x(u c^i) + \partial_x(v c^i) + \partial_z((w - w_s^i)c^i) = \\ \qquad \partial_x (A_h \partial_x c^i) + \partial_y(A_h \partial_y c^i) + \partial_z(A_v \partial_z c^i)\end{array}$$

(5)

Here, u, v, and w denote current speeds in the three cartesian directions; $w^i_s$ denotes the settling velocity of the i ^th particle fraction; and A _v and A _h denote the vertical and horizontal eddy diffusivity.

While the horizontal diffusivity A _h was set constant, the vertical diffusivity A _v was a combination of three components:

$$A_v(z) = A_v^t(z) + A_v^{W}(z) + A_v^b$$

(6)

$A_v^t(z)$ accounted for the vertical turbulent diffusion induced by the current field and was calculated by the General Ocean Turbulence Model (GOTM, Burchard et al. 1999), which is part of the GETM model. $A_v^b$ denotes the constant background diffusivity, which, in contrast to the turbulent diffusivity $A_v^t(z)$, was also present when there was no motion of the water column. In this model study, a value of $A_v^b = 1.0 \cdot 10^{-6}$ m²/s was used.

To include the effect of surface gravity waves on the vertical diffusion of SPM, a parameterization proposed by Pleskachevsky et al. (2005) was implemented, which is based on the wave induced-orbital velocity field:

$$A_v^{W}(z) = (k\cdot H_{\rm sig})^2 \cdot U_w(z)^2 \cdot T_{\rm peak} $$

(7)

Here, H _sig denotes the significant wave height, T _peak the peak period of the underlying wave spectrum, k the corresponding wave number according to the dispersion relation of surface gravity waves (a formula to calculate the wave number is given below in Eq. 18), and U _w (z) the amplitude of the wave-induced orbital velocity, which is given by:

$$ U_w(z) = \frac{\pi \, H_{\rm sig}}{T_{\rm peak}} \frac{cosh(k\,(h+z))}{sinh(k\,h)}, $$

(8)

where h denotes the water depth measured downward from the surface (the bottom level is z = − h).

The settling velocity of the i ^th particle fraction was calculated after a formula from Soulsby (1997), which is valid for noncohesive irregular sand grains (Table 3):

$$\begin{array}{rll}w^i_s &= & \frac{\nu}{d} \left \{ \sqrt{10.36^2 + 1.049 \, D_s^3} - 10.36 \right\} \\ D_s & = & \left[\frac{g}{\nu^2} \left(\frac{\rho_s}{\rho_W} -1 \right) \right]^{1/3} \, d\end{array}$$

(9)

Here, g denotes the acceleration due to gravity, ρ _s and ρ _W the sediment and water density in kilograms per cubic meter, ν the kinematic viscosity in square meters per second, and d the sediment diameter in meters (see left graph in Fig. 18).

Table 3 Settling velocities at 4°C and critical shear stress for the different sediment fractions after Eqs. 9 and 13

Fig. 18

Left: settling velocity of sand grains as a function of grain diameter after Soulsby (1997) (see Eq. 9). Right: critical shear stress as a function of grain diameter after Soulsby and Whitehouse (1997) (see Eq. 13)

The temperature dependency of the kinematic viscosity ν was calculated by:

$$ \nu = \frac{1}{\rho_W} \eta \, exp \left( \frac{b}{T+273.0} \right),$$

(10)

with temperature T in ^o C, η = 1.9909 · 10^− 6 kg/(ms), and b = 1.8284 · 10³ K.

In the vertical direction, sediment only entered and left the water column through the bottom boundary layer, for which the following sediment flux condition was used:

$$ \left(A_v \partial_z c^i + w^i_s c^i \right)_{z = -h} = F^i_{\rm ero} - F^i_{\rm dep} $$

(11)

Here, F _ero and F _dep denote the sediment fluxes at the bottom due to erosion and deposition.

According to Einstein and Krone (1962), the deposition flux was calculated with:

$$ F^i_{\rm dep} = \begin{cases} w^i_s \cdot c^i_{z=-h} \left( \frac{\tau_B}{\tau^i_c} -1 \right) & \tau_B < \tau^i_c \\ 0 & \tau_B > \tau^i_c \end{cases}$$

(12)

Here, τ _B denotes the bottom shear stress and $\tau^i_c$ a critical shear stress, which had to be reached by the fluid in order to erode sediment from the bottom into the water column. If the bottom shear stress was below that value, suspended sediment settled to the bottom. Soulsby and Whitehouse (1997) provided an empirical formula for that critical shear stress, which is given by (Table 3):

$$\begin{array}{rll} \tau_c & = & g \, d \, (\rho_S - \rho_W) \, \theta_c \\ \theta_c &:=& \frac{0.3}{1+1.2\, D_s} + 0.055 \, \left(1- exp(-0.02\, D_s) \right)\end{array}$$

(13)

In Fig. 18, the critical shear stress is depicted for two different water temperatures. It is evident from this figure that the critical shear stresses used in this study are of the same order as those reported by Austen and Witte (2000) for sandy sediments in the Wadden Sea region. However, these values do not account for muddy and cohesive sediments, which have higher critical shear stresses of about 0.5–1.0 N/m². This effect will be implemented in a future study.

Furthermore, to determine the critical shear stress for a particular sediment fraction, the bottom sediments were modeled as separate pools that did not interact with each other. Therefore, the different sediment fractions were eroded and deposited separately from each other without accounting for mean grain size and mean critical shear stress at a specific location.

The erosion flux was calculated according to the formula of Partheniades (1965):

$$ F^i_{\rm ero} = \begin{cases} M_e \left( \frac{\tau_B}{\tau^i_c} -1 \right) & \tau_B > \tau^i_c \\ 0 & \tau_B < \tau^i_c \end{cases}$$

(14)

M _e is a free parameter that can be adapted to measurements of sediment concentration. For their studies in the East Frisian Wadden Sea, Stanev et al. (2006) used a value of $M_e=3.7 \cdot 10^ {-6} \mbox { kg}/(\mbox{m}^2 \mbox{s})$.

Total bottom shear stress responsible for sediment erosion and deposition was modeled as a nonlinear combination of a current-induced component τ _current and a wave-induced component τ _wave (Soulsby 1997). This approach was also used by Pleskachevsky et al. (2005) and Gayer et al. (2006):

$$\begin{array}{rll}\tau_B & = & \left[ \left( \tau_m + \tau_{\rm wave} \cos(\phi) \right)^2 + \left( \tau_{\rm wave} \sin(\phi)\right)^2 \right]^{1/2} \\ \tau_m &:=& \tau_{\rm current} \, \left[ 1 +1.2 \, \left( \frac{\tau_{\rm wave}}{\tau_{\rm wave}+ \tau_{\rm current}}\right)^{3.2} \right]\end{array}$$

(15)

Here, ϕ denotes the angle between the direction of wave propagation and current flow.

The current-induced bottom shear stress was calculated in GETM via an iterative procedure with a prescribed constant bottom roughness length z ₀ (see more details in Burchard and Bolding (2002)).

The wave-induced component was calculated after an approach proposed by Soulsby (1997). This approach begins with a general relation for bottom shear stress due to surface gravity waves:

$$\tau_{\rm wave} = \frac{1}{2} \rho_W f_w U_w^2$$

(16)

Here, f _w denotes the wave friction factor, ρ _W the water density, and U _w the horizontal wave orbital velocity near the bottom just above the wave boundary layer. The bottom velocity was approximated by:

$$ U_w = \frac{\pi H_{\rm sig}}{T_{\rm peak} sinh(k\, h )}, $$

(17)

which is the bottom velocity expression of Airy-waves. Here, h denotes the water depth, H _sig the wave height, T _peak the wave period, and k the corresponding wave number.

In general, the wave number is related to the peak period by the dispersion relation of surface gravity waves:

$$ \frac{2 \pi}{T_{\rm peak}} = g\,k\,~tanh(k\,h) $$

However, as it is not straightforward to obtain the wave number k from this dispersion relation if only T _peak is known, a simple estimation provided by Soulsby (1997) was used in this study:

$$ k =\left\{ \begin{array}{ll} \displaystyle\frac{1}{h} \left(\!\frac{4 \pi^2 h}{T_{\rm peak}^2 g}\!\right)^{1/2} \left(\!1 \!+\! 0.2 \cdot \frac{4 \pi^2 h}{T_{\rm peak}^2 g}\! \right)& \!, ~ for ~ \displaystyle\frac{4 \pi^2 h}{T_{\rm peak}^2 g} \leq 1 \\[10pt] \displaystyle\frac{1}{h} \frac{4 \pi^2 h}{T_{\rm peak}^2 g} \left[\!1\!+\! 0.2 \cdot exp\left(2\!-\!2\cdot \frac{4 \pi^2 h}{T_{\rm peak}^2 g} \right)\!\right]& \!, ~ for ~ \displaystyle\frac{4 \pi^2 h}{T_{\rm peak}^2 g} > 1 \end{array}\right.$$

(18)

The wave friction factor f _w was calculated for f _ws in the case of smooth beds or f _wr in the case of rough beds by:

$$ f_{wr} = 1.39 \left( \frac{A}{z_0}\right)^{-0.52}, ~~~ f_{ws} = B\,R_W^{-N} $$

(19)

The bottom roughness was calculated by z ₀ = d ₅₀/12 and the semiorbital excursion A was given by A = U _w T _peak / (2 π). The parameters of the smooth bed friction factor f _ws were dependent on the turbulence of the motion, which could be determined from the wave Reynolds number R _w  = U _w A / ν. According to this number, the parameters N and B were calculated by:

$$B = 2; ~ N = 0.5 \qquad\text{if}\; R_w \leq 5 \cdot 10^{5} ~ \text{(laminar)}$$

(20)

$$B = 0.0521; ~ N = 0.187 \; \text{if}\; R_w > 5 \cdot 10^{5} ~ \text{(smooth turbulent)}$$

(21)

Finally, if sediments were deposited at or eroded from the bottom, the mass of the available sediment pools changed. Each bottom pool $P_B^i$ was changed according to the following equation:

$$\partial_t P_B^i = F^i_{\rm dep} - F^i_{\rm ero}$$

(22)