Modeling the response of shoreface-connected sand ridges to sand extraction on an inner shelf

Abstract

Shoreface-connected sand ridges are rhythmic bedforms that occur on many storm-dominated inner shelves. The ridges span several kilometers, are a few meters high, and they evolve on a timescale of centuries. A process-based model is used to gain a fundamental insight into the response of these ridges to extraction of sand. Different scenarios of sand extraction (depth, location, and geometry of the extraction area; multiple sand extractions) are imposed. For each scenario, the response timescale as well as the characteristics of the new equilibrium state are determined. Results show that ridges partially restore after extraction, i.e., the disturbed bathymetry recovers on decadal timescales. However, in the end, the ridge original sand volume is not recovered. Initially, most sand that accomplishes the infill of the pit originates from the area upstream of the extraction, as well as from the areas surrounding the pit. The contribution of the latter strongly decreases in the subsequent time period. Depending on the location of the pit, additional sand sources contribute: First, if the pit is located close to the downstream trough, the pit gains sand by reduction of sand transport from the ridge to this trough. Second, if the pit is located close to the adjacent outer shelf, the ridge recovery is stronger due to an import of sand from that area. Furthermore, pits that are located close to the nearshore zone have a weak recovery, deeper pits have longer recovery timescales, wide and shallow pits recover most sand, while multiple sand pits slow down the recovery process.

Appendices

Appendix 1: Bottom evolution

The evolution of the bottom is governed by

$$(1-p)\frac{\partial z_{b}}{\partial t} + {\boldsymbol\nabla}\cdot {\bf q}_{\rm{tot}}=0\, ,$$

(24)

where p(∼0.4) is the bed porosity, z _b is the bed level, and q _tot is the sediment transport. In this study, the latter quantity is decomposed into three contributions:

$${\bf q}_{\rm{tot}} = {\bf q} + {\bf q}_{\rm{mbs}} + {\bf q}_{\rm{wa}}\, .$$

Here, q is defined in Eqs. (15) and (16), q _mbs is the sediment transport due to the mean bottom slope (directed offshore), and q _wa is the sediment transport induced by wave asymmetry (directed onshore). It is assumed that q _mbs and q _wa are alongshore uniform, so they depend only on x, t. Furthermore, z _b = −H(x)+h(x, y, t), where H(x) is the mean depth profile in the case of a naturally evolving system.

Averaging Eq. (24) over longshore coordinate y yields

$$\frac{\partial <h>}{\partial t} + \frac{\partial}{\partial x}\left(<{\bf q}> + {\bf q}_{\rm{mbs}} + {\bf q}_{\rm{wa}} \right) = 0\, .$$

Note that, in the case of a naturally evolving system, < h > = 0 by construction. This means that q _mbs + q _wa = −< q > . Substitution of this result into Eq. (24) thus yields the mass balance given in Eq. (14) of the main paper.

Appendix 2: Recovery time t _r versus slope β

Simulations (Fig. 14) show that recovery time t _r is inversely proportional to bottom slope β, i.e., t _r ∼ β ⁻¹. This relationship can be understood from Eq. (23), in which, for simplicity, only the dominant term (T ₁) is considered. The result is

$$\frac{\partial h}{\partial t}\sim\frac{K_{b}v}{D}\frac{\partial h}{\partial y}.$$

(25)

Upon decomposing velocity vector v and depth D in terms of their unperturbed components (which occur in the absence of bottom perturbations h and only depend on x) and perturbed components, i.e.

$$v\, =\, V_{0}(x)+v'(x, y, t), \!\!\!\!\!\quad u\, =\, u(x, y, t)\!\!\!\!\quad D\, =\, H(x)-h(x, y, t),$$

(26)

and using the fact that ratio h/H << 1, Eq. (25) is approximated as

$$\frac{\partial h}{\partial t}\sim\frac{K_{b}V_{0}}{H}\frac{\partial h}{\partial y}+\frac{K_{b}v'}{H}\frac{\partial h}{\partial y}.$$

(27)

From this, an equation governing the domain-averaged potential energy of the bottom perturbations is derived by multiplying the above equation with h and integrating over the domain (Garnier et al. 2006). The result is

$$\frac{\partial}{\partial t}\overline{\left(\frac{1}{2}h^{2}\right)}\sim\overline{\frac{K_{b}V_{0}}{H}\frac{\partial}{\partial y}\left(\frac{1}{2}h^{2}\right)}+\overline{\frac{K_{b}v'}{H}\frac{\partial}{\partial y}\left(\frac{1}{2}h^{2}\right)},$$

(28)

in which the overbar denotes domain averaging (i.e., $\frac {1}{LL_{y}}\int \int \cdot ~ dx dy$). The first term on the r.h.s is 0 because of periodic boundary conditions at the lateral boundaries (note that K _b = K _b (x)). An estimate of the recovery time t _r follows from

$$[t_{\rm{r}}]=\frac{\overline{\frac{1}{2}h^{2}}}{\frac{\partial}{\partial t}\overline{\frac{1}{2}h^{2}}}\sim\frac{\left[\overline{\frac{1}{2}h^{2}}\right]}{\left[\overline{\frac{K_{b}v'}{H}\frac{\partial}{\partial y}\frac{1}{2}h^{2}}\right]},$$

(29)

where scales are denoted by brackets [.]. A scale for the term in the denominator, $\left [\overline {\frac {K_{b}v'}{H}\frac {\partial }{\partial y}\frac {1}{2}h^{2}}\right ]$, is found using the mass balance Eq. (11) (assuming rigid-lid, i.e., ∇⋅(D v)≃0):

$$\overbrace{H\frac{\partial u}{\partial x}\, +\, H\frac{\partial v'}{\partial y}-V_{0}\frac{\partial h}{\partial y}}^{\text{dominant}}\, +\, \overbrace{u\frac{\partial H}{\partial x}-u\frac{\partial h}{\partial x}-h\frac{\partial u}{\partial x}\, -\, h\frac{\partial v'}{\partial y}-v'\frac{\partial h}{\partial y}}^{\text{order~ smaller}}=0.$$

(30)

Note that the first three terms on the l.h.s. dominate over the last five terms, because $\frac {\partial H}{\partial x}=\beta \sim 10^{-3} << 1$, h << H and (u, v′) << V ₀. Hence, term $[v'\frac {\partial h}{\partial y}]\sim [u\frac {\partial H}{\partial x}]\sim \beta [u]$. Since $[u]\sim \frac {[h]}{H_{0}}[V_{0}]$, it follows that $[v'\frac {\partial h}{\partial y}]\sim \beta \frac {[h]}{H_{0}}[V_{0}]$. Scale [t _r ] can thus be approximated to be

$$[t_{\rm{r}}]\sim\frac{[h]^{2}}{\beta \frac{[h]^{2}}{H_{0}}[V_{0}]}\sim\beta^{-1}.$$

(31)