Feedback loop structure of cliff dynamics
From the literature reviewed, the main processes and feedback loops that determine cliff erosion and shore platform evolution are summarized below. How each of these processes is represented using CLDs is first individually presented (see Figs. 5 and 6) and at the end of the section collapsed into an overall feedback loop structure (see Fig. 7.).
The main processes and feedback loops considered are:
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Lithology, mechanical strength, discontinuities. Backwearing and downwearing erosion rates are influenced by rock resistance, which depends on rock lithology, mechanical strength, and the presence of discontinuities (Naylor and Stephenson 2010; Sunamura 1994)
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Shear depletion loop. Wave energy dissipation on a shore platform is mostly controlled by the water depth (Thornton and Guza 1982). An increase in the water depth due to downwearing, reduces the energy dissipation rate and therefore the downwearing erosion rate. A change on the energy dissipation rate over the platform indirectly influences the energy reaching the cliff toe.
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Cliff toe energy depletion. Backwearing of the cliff toe precludes further erosion by widening the shore platform that dissipates energy more effectively.
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Feedback between wetting-drying cycles and platform downwear. Higher numbers of wetting and drying cycles on platform surfaces have been linked to increased downwearing rates (Stephenson and Kirk 2000). With other factors held constant, a vertically eroding platform increases its water depth, which reduces the number of wet-dry cycles, balancing back the erosion rate.
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Cover protection loop. Fine sediment eroded from the cliff and platform may be removed from the system as suspended sediment, whereas beach-grade sediment may be deposited at the cliff toe. Beach sediments might impede direct wave attack, reducing backwearing rates, and cover the shore platform reducing downwearing rates (Castedo et al. 2012; Trenhaile 2005; Walkden and Hall 2005).
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Abrasion enhancement loop. Beach material may also increase the abrasive capacity of the incoming waves, both at the cliff toe (Robinson 1977; Trenhaile 2005) increasing the backwearing rate, and also on the platform surface increasing the downwearing rate (Walkden and Hall 2005).
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Feedback loop between pore pressure release and backwearing erosion rate. The build-up and release of pore pressure has been identified (measured and modelled) as a balancing feedback of backwearing rate (Castedo et al. 2012; Fort et al. 2000). The associated zone of depressed pore pressure to cliff backwearing prevents the pore pressure build-up over time.
Figure 5 represents an initial subset of the processes that have been identified from existing literature. By definition, an increase of both backwearing and downwearing erosion rates decreases the cliff and shore platform mass content and is therefore represented as a negative influence link. To avoid cluttering the diagram, the influence of lithology, mechanical strength and discontinuities on the erosion rates is acknowledged as an influence link (dashed line), but their sign (positive or negative) is not shown in the diagram. This implies that, well-known processes such as fatigue due to temperature (frost/defrost cycles) are not explicitly represented in the diagram but implicitly included within this influence link. Platform downwearing due to weathering can be accomplished by a variety of mechanisms, but the general efficacy of platform-weathering processes is often modulated by tidal wetting and drying cycles (Trenhaile 2005). This is represented as a positive link between intertidal exposure time and weathering rate. Downwearing of the intertidal platform, in the absence of abrasive material and bioerosional organisms, has been attributed generally to weathering and removal of the fine-grained debris by waves (Porter et al. 2010). Most of the mechanical downwearing of the submerged platform is due to breaking waves, the downwearing erosion rate has been often conceptualized as a decaying function of water depth (Sunamura 1992; Trenhaile 2005). Since the exponential decay with depth is controlled by the wave energy dissipation rate, the influence of wave breaking is represented as a positive link between wave energy dissipation rate and downwearing. An increase in the energy reaching the cliff toe increases the backwearing rate (positive influence) by increasing undercutting. The influence of the debris at the toe of the cliff on the backwearing erosion rate is lately included within the overall feedback loop structure. The rainfall accumulated as groundwater increases the pore pressure eventually triggering the shear failure of the cliff (Castedo et al. 2012). Fort et al. (2000) found that the associated unloading resulted in a zone of the depressed pore pressures which temporarily prevent deep-seated instability of the over-steepened cliff. This is represented as negative link between backwearing and pore pressure resulting in a balanced loop between backwearing erosion rate and pore pressure. The backwearing erosion rate is balanced back by widening the platform. Platform width is understood as the horizontal distance between the cliff toe and the sea edge at which offshore waves start to be transformed by interaction with the bottom. Given that the wave energy at the gravity band frequency decreases shoreward from the platform sea edge (Sunamura 1992) an increase of platform width will have a negative influence in the energy reaching the cliff toe. The cliff toe energy depletion loop is further explained in the next section.
Role of shore platform geometry
The influence of shore platform geometry (i.e. gradient, elevation and width) has long been identified as an important control on the rate of cliff and platform erosion, but it has been only broadly conceptualized. For instance, platform gradient has been studied in association with tidal range and despite a great deal of attention, the link between platform morphology and tidal range and duration remains ambiguous (Stephenson 2000). Simple conceptualizations of wave propagation on simulation models proven useful are not applicable to typical geometries. For instance, Trenhaile (2010) used linear wave theory to propagate offshore waves and compute the bed shear stress over cohesive clay platforms. While for gently sloping shore platforms (e.g. cohesive, mixed clay-sand), linear theory might suffice, but it does not capture the rapid bottom orbital velocity changes observed on irregular geometries that occur on hard-rock coasts. The typical Type-B morphology of hard-rock coasts includes an abrupt seaward scarp and a near-horizontal to gently sloping (~1 deg) platform. This morphology is similar to that which occurs on coasts with coral reef platforms. Theoretical and experimental observations on such shores suggests that the so-called non-linearity wave parameter, F
co
, based on deep water wave height and a representative depth over the reef-platform geometry is a suitable parameter for classifying wave transformation on a number of platform geometries (Massel and Gourlay 2000; Swart and Loubser 1979). The non-linearity parameter increases with offshore wave height and period, and decreases with the representative water depth. The representative depth might be defined as the still water depth over the reef-edge. An increase of F
co
increases the wave energy dissipation and decreases the wave transmission (i.e. decreases wave energy onshore) (Fig. 6.).
While the use of the wave non-linearity parameter is relatively new within the rock coast morphodynamic literature (e.g. Ogawa 2012), we found this concept useful to represent the overall feedback structure. An increase of the offshore wave non-linearity parameter in any case decreases the wave energy reaching the cliff toe (i.e. due to an increase in reflection and dissipation) and, is represented as a negative link among F
co
and Energy at cliff toe. On the shore platform, an increase on the offshore non-linearity parameter increases the water depth relative to the wave base. The wave base is defined here as the water depth at which the wave bottom shear-stress might potentially induce bed erosion. The wave base increases with the wave period and wave height. The complex interaction between long waves and, short waves and platform geometry might lead to counter-intuitive energy transformation at the cliff toe which is yet not fully understood (Dickson et al. 2013; Ford et al. 2012, 2013; Payo and Muñoz-Perez 2013). The unknown influence of the cliff and shore platform geometry (elevation, gradient and width) is acknowledged by an influence link between the mass content and the offshore wave non-linearity parameter in Fig. 7.
The overall shore platform feedback structure including beach-platform interaction is shown in Fig.
7
. The bed material eroded from cliff backwearing and platform downwearing is deposited as beach sediment. Fine material eroded from the cliff and shore platform increases the suspended sediment content and is lost from the system. Beach and surf-zone sediment protects the cliff and shore platform through balancing feedback loops (cliff deposit protection and cover protection loops). Cliff-foot deposits can provide protection to the cliff or they can act as abrasives inducing a reinforced feedback loop (abrasion enhancement loop). The beach and surf-zone volume are further controlled (not shown in Fig. 7) by the alongshore sediment transport gradient and the balance between diffusive and anti-diffusive processes (Murray and Ashton 2013). The role of mean water level and offshore wave energy in the feedback structure is shown within the overall feedback structure. If the water depth relative to the wave base increases, the influence of the surface water waves on the platform surface decreases. This is represented as a positive link between the offshore wave non- linearity parameter and the relative water depth to the wave base. If mean sea water levels rise too quickly, an eroding platform can be abandoned if the wave base no longer reaches the platform. By reasoning on the influence pathway linking cliff and shore platform mass content with the relative water depth to wave base two balancing feedback loops are identified, termed here the weathering limited and the shear depletion loops. A decrease of mass content due to vertical erosion, increases the relative water depth that, for a given tidal range, decrease the intertidal exposure time (starting from mass content the pathways read as −−++ − = −). A decrease of mass content also reduces wave erosional potential, by reducing energy dissipation rate, balancing back the initial downwearing erosion rate.
Strength of cliff toe energy depletion loop
In the following the strength of the cliff toe energy depletion loop is estimated by reasoning on the current understanding of the causal pathway. Based on the feedback structure identified in the previous section and the methodology proposed, three main quantities need to be estimated to assess the strength of this balancing loop (Fig. 8.): (1) how the backwearing erosion rate (E) increases with the energy reaching the cliff toe; (2) how the platform width (W) increases with the backwearing erosion rate; (3) how the energy at the cliff toe decreases with the platform width. The way these estimates are obtained is described below. Ultimately, the feedback factor strength is then obtained from the product of these three quantities.
Influence of incident wave energy on backwearing erosion rate.
Hackney et al. (2013), proposed and validated with field data a simple relationship between soft cliff erosion rate and wave energy. Based on the premise that an ‘Accumulated Excess Energy’ (AEE) parameter can be used to represent the process of hydraulic erosion at the base of the cliff, the relationship between backwearing erosion rate and wave energy is given by;
$$ E={\displaystyle {\int}_{t=0}^t}\left[f\left(\left(\Omega (t)-{\Omega}_c\right)+{c}_2\right)\right]dt $$
(1)
Where E = the amount of erosion (m) during the time interval t, Ω = applied wave energy (J/m3), Ωc = threshold wave energy (J/m3) required to initiate erosion, and c
2
is a calibration coefficient. The average energy of a wave per unit surface area, Ω, is;
$$ \Omega =\frac{1}{8}\rho g{H}_s^2 $$
(2)
Where Ω is energy (J/m3), ρ is the density of sea water (kg/m3), g is the gravitational potential energy of the wave (m2/s) and H
s
is the significant wave height (m). The time interval is bounded to the underlying theoretical concept of Basal End Point Control (BEPC). The principal implication of BEPC, that the rate of retreat is governed by hydraulic activity at the toe, applies over time scales encompassing multiple cycles of slumping and toe erosion. Accordingly, it is important to ensure that the calibration of any model based on the principles of BEPC is conducted over a large enough temporal scale (i.e., multiple cycles of hydraulically controlled undercutting, mass wasting, and removal of the slumped debris) to ensure that hydraulic activity at the toe is the dominant factor controlling slope retreat. For a time scale of about 10 year, and for the different wave environments and geologies around the Isle of Wight and Suffolk coast in UK, Hackney et al. (2013) found that Eq. (1) is best fitted by;
$$ E={c}_1\left(\Omega (t)-{\Omega}_c\right)+{c}_2 $$
(3)
Where c
1
is a fitted constant of the order of 10−6 to 10−7 for the predominantly clay-cliff study sites. They further notice that the failure of Eq. (1) to produce significant relationships in the more resistant units may indicate that the ~10-year timeframe employed may be too short. The change of erosion rate with changes on the energy reaching the cliff toe can be then estimated as
$$ \frac{dE}{d\Omega (t)}={c}_1 $$
(4)
Influence of erosion rate on platform width.
Platform width is defined after Sunamura (1992) for the two main types of shore platforms, namely sloping platforms and near-horizontal platforms. Sloping platforms, which most notably occur in meso to macro-tidal areas, slope continuously (1° to 10° slope) from the cliff toe beneath sea level. Near horizontal platforms (<1° slope), mainly occur in micro-tidal and low meso-tidal environments and which are characterized by an abrupt scarp at their outer edge. In either case the platform width is defined as the horizontal distance between the sea-edge of the platform and the cliff toe as shown in Error! Reference source not found.. At any instant of time the platform width can be estimated as the sum of the initial platform width and the added platform due to backwearing erosion as: w = w(t = 0) + E(t). The change of platform width relative to erosion rate is obtained from this relationship as: dw/dE = + 1.
Influence of width platform on energy reaching the cliff toe.
The simple analytical solution developed by Gerritsen (2011) that relates the wave energy with the distance to the seaward limit of the shore platform is used here (Eq. 5). This solution is valid for platforms of about constant water depth and at the region where bottom friction can be neglected.
$$ H(x)=H(0) exp\left(-\frac{B}{A}\right) $$
(5)
Where H(0) is the reference wave height at a reef-edge, x is the distance from the reef-edge and the coefficients A and B are;
$$ A=\frac{1}{4}\rho g{C}_g\approx \frac{1}{4}\rho gC $$
(6)
$$ B=\frac{\xi \rho g}{4\sqrt[]{2}T} $$
(7)
Coefficient ξ is of an order of magnitude one and is given by
$$ \xi =\frac{\alpha {H}_{*}\sqrt[]{2+{H}_{*}}}{F_r\sqrt[]{1+{H}_{*}}} $$
(8)
in which
$$ {F}_r=\frac{C}{\sqrt[]{gh(x)}}\kern0.36em and\kern0.24em {H}_{*}=\frac{H(x)}{h(x)} $$
(9)
where C is the phase velocity that satisfies the dispersion relationship and h(x) is the water depth over the shore platform, α is the breaking coefficient that describes the deviation of actual breaking from the periodic bore form on the O(1).
The change of energy reaching the cliff toe with platform width is obtained as the difference between the energy at the seaward edge of the platform and the energy at the cliff toe for different platform widths. The cliff is assumed to retreat, while maintaining a constant platform slope. The energy reaching the cliff toe has been estimated by Eq. (5) for an eroding platform of 1°, 5° and 10° slope. Equation (5) is an implicit equation (i.e. H(x) is at both sides of the equation) that has been solved numerically using the MATLAB solve function. The water depth at the seaward edge, do, has been assumed equal to 4.5 m. Wave height at the seaward limit, Ho, has been assumed equal to the maximum wave height before breaking to ensure that waves were breaking at seaward platform edge and T = 10s. The breaking coefficient is assumed α = 0.8 and the decay of wave height has been estimated for a range of platform width values from 1 m to x = do/tan(slope). Equation (5) is an approximation that is valid where wave breaking dominates the bottom friction. From Massel and Gourlay (2000) this dominance can be assessed by the non-linearity index assuming that the representative platform water depth, h
r
, is the mean water depth from seaward edge to platform width. For the presented set-up, values of F
co
vary between 700 up and 2400, indicating that wave breaking dominates the bottom friction for all cases, and therefore supports the validity of the approximation.
Figure 9 shows the variation of wave energy reaching the cliff toe for different platform widths. The energy gradient has been obtained from solutions of Eq. (5), converting wave height into wave energy by Eq. (2), given ρ = 1024 kg/m3 and g = 9.81 m2/s. As expected, the maximum gradients of energy dissipation with platform width are obtained for the steeper slope, O(−500 J/m4), and the smaller for the gentle slope, O(−50 J/m4), being the 5° slope in between with O(−300 J/m4). For a given platform width of, for instance 25 m, the energy decrease is maximum for the slope of 10°, and two and three orders of magnitude bigger than for platforms of slope 5° and 1° respectively, which is consistent with results presented by Thornton and Guza (1982). To assess the sensitivity of these results to different wave periods, the above estimation was made for wave periods of 5 s and 30s. Similar orders of magnitude were found (not shown). It is noticed that the non-linearity index for the 5 s waves was within the region where breaking does not dominate bottom friction (F
co
< 450), meaning that the approximation from Eq. (5) overestimates the dissipation rates.
Figure 10 summaries the feedback factor strength of the cliff toe energy depletion loop derived from the above estimates. A range of g values from ~ −10−10 to ~ −10−4 is obtained as the product of all partial derivatives that determines the loop strength. The six order of magnitude difference between the minimum and maximum estimated values is mostly due to the geometry of the retreating platform. A nearly horizontal retreating platform (1° slope) presents the minimum absolute g value of −10−10, which corresponds with a total effect of about 1 (i.e. total effect is equal to the direct effect). A sloping platform (10° slope) presents the maximum absolute value of g of −10−4 which corresponds to a total effect of about 0.9999 (i.e. total effect is slightly smaller than the direct effect). The important of these differences for modelling cliff toe position over decades and longer is discussed in the following section.