# Effects of stochastic wave forcing on equilibrium shoreline modelling across the 21st century

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## Global Sensitivity Analysis

Throughout this work, we address the sensibility of long-term shoreline predictions to several sources of uncertainties. The uncertainties associated to model assumptions (epistemic) and unpredictability of climate forcing (intrinsic) propagate through the model resulting in uncertain model results. For a given set of n uncertain input variables (X1, …, Xn), uncertainties on each variable can be represented by a probability distribution covering the range of possible values associated to the variable (Figure 2.6). An ensemble of model results can be the distribution of modelled shoreline positions at a given time (Figure 2.6). While epistemic uncertainties can be reduced by acquiring new knowledge and/or removing simplifying assumptions, intrinsic uncertainties derive from the random nature of some physical processes and cannot be reduced. Regardless of their nature, the uncertainties affecting the input variables can have a different influence on the model outcome.
Figure 2.6 Schematic representation of the propagation of uncertainties on n input variables (X1, …, Xn) through a model, producing uncertain outputs (Y), e.g. a distribution of modelled shoreline positions (coloured lines) at a time t=t*. Uncertainties on each variable are represented through the respective probability density function (black curves) and their variance.
In variance-based approaches, the uncertainties associated with a variable X are represented by its variance (σ2), defined as: = 1 ( − ) (2.9).
where xi is the i-th sampled value of X, N is the number of samples, and µ is X’s median. We quantified the different contributions to the uncertainties of modelled shoreline change at Truc Vert beach by performing a variance-based GSA (Saltelli et al., 2008). The variance-based GSA explores the full range of realistic combinations of n uncertain model inputs (X1, …, Xn) with the aim to divide the uncertainties (i.e. variance) of probabilistic model results into several portions, each one attributed to an uncertain model input (or a set of inputs). The resulting contribution that an uncertain input Xi provides to the uncertainties in model predictions (Y) is given by the main effect (first-order sensitivity), and by interaction terms (higher-order sensitivity). The main effect measures the sensitivity of model results to variations of the Xi value alone, while the higher-order components measure the additional effects that simultaneous variations of Xi with other uncertain inputs (Xj) have on the results’ variance. The latter accounts for the fact these simultaneous variations may drive a larger contribution to the results variance than the cumulated main effects (individual variations) of Xi’s.
Figure 2.7 illustrates schematically the variance-based GSA framework for a set of uncertain input variables (X1, …, Xn), represented by the respective probability density functions, producing an uncertain model output with its associated variance (σ2). If an uncertain input is fixed to its true value (e.g. X1= X*1), the effects of the uncertainties on X1 and interactions with other uncertain variables on the results variance (conditional to X1= X*1) are removed (see red lines in Figure 2.7).

### Sea-Level-Driven Recession and Equilibrium Shoreline Models

The Bruun Rule (1962), introduced in Section 2.2.2, models the long-term (>decadal) shoreline adjustment to SLR. The model assumes that: (i) a theoretical equilibrium beach profile exists and is maintained over time; (ii) sediment volume is conserved cross-shore between the limits of the active beach profile (down to a so-called “depth of closure”); (iii) SLR always induces long-term shoreline retreat (as the beach profile migrates upward and landward with SLR in order to conserve sediment volume); and (iv) there is unlimited available sediment from the subaerial beach (Bruun, 1988, 1983).
The theoretical equilibrium profile shape is determined by the local sediment and wave climate characteristics (Dean, 1991). Therefore, the preservation of the equilibrium profile shape implies that the local wave climate characteristics remain roughly constant. According to the Bruun model, under the assumptions listed above, the long-term beach profile responds to SLR with a landward and upward translation (Figure 3.2), producing a total shoreline retreat (YBruun) given by: = tan( ) (3.1).
where tan(β) is the mean slope of the active beach profile between the offshore limit of sediment exchange (i.e. the depth of closure) and the stable emerged beach location (e.g. emerged berm, dune toe or dune crest). The depth of closure is determined by the local sediment size and the frequency of extreme waves over a given period of time, and thus depends on the local wave climate (Hallermeier, 1978). This last point highlights another strong link between the Bruun model and the wave climate. As the depth of closure depends on the time window observed and the respective distribution of extreme wave events (Nicholls, 1998), the assumption of a constant depth of closure implies a long-term stationarity of the wave climate.
The derivation of the original Bruun model was essentially based on an intuitive unification of physical beach characteristics and nearshore processes (Bruun, 1988). An implicit assumption of the Bruun model is that, in response to SLR, the sediment is redistributed over the profile by the long-term integrated effects of wave action (Bruun, 1988). This mechanism can be interpreted as the sea level setting a new non-equilibrium beach profile state, and the incident wave action mobilizing or relaxing the beach back toward equilibrium.

#### A physical interpretation of the Beach response to Sea-Level Rise

Consider an idealized equilibrium beach profile, where the shoreline (Y) is defined as the instantaneous intersection of the beach profile and mean sea level (Figure 3.3a). Here, the foreshore is defined as the intertidal portion of beach between the mean high water (MHW) level and the mean low water (MLW) level (Figure 3.3). With a stable mean sea level, the local wave climate produces seasonal fluctuations in the cross-shore beach profile around its theoretical equilibrium position. A rise in sea level floods the foreshore (Figure 3.3b), leading to a shoreline recession (Passive Flooding, YPF) proportional to the average foreshore slope, as follows: = tan( ) (3.2).
where tan(α) is the mean foreshore slope. The shoreline recession due to passive flooding is purely geometric: it is the simple result of the change of the reference shoreline position (e.g. MSL), and occurs in the absence of sediment transport (Anderson et al., 2018). However, with respect to the new MSL (MSL’ in Figure 3.3b), the beach slope across the profile is generally higher when the profile is concave (Figure 3.3b), as is the case for the majority of real beaches. A generally steeper (more accreted) profile is, on average, more balanced with low energy waves and less balanced with high energy waves (Wright and Short, 1984). Hence, the elevated mean sea-level position (MSL’) brings the “beach–wave climate” system into a disequilibrium state where the current wave conditions are larger than in the equilibrium state. Consequently, continuous wave action tends to reshape the beach profile toward its equilibrium shape (Figure 3.3c), driving a further landward displacement of the shoreline position (Wave Reshaping, YWR). Recall that the beach profile in equilibrium with MSL’ (Figure 3.3c) represents the mean position of the seasonal profile fluctuations. Therefore, depending on the energy provided by the incident wave conditions after a rise in sea level, the resulting, time-dependent effect of wave reshaping (here called dYWR) may not immediately reach YWR (Figure 3.3c), but would be gradually achieved with the cumulative wave action. In this mechanism, YPF occurs simultaneously with SLR, while the associated YWR develops on the time scale of several wave events.

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Disequilibrium and beach profile relaxation in equilibrium shoreline models

Equilibrium shoreline models (ESMs) assume the existence of an equilibrium beach profile and rely on the concept of disequilibrium between the current beach state and the current wave conditions (Wright and Short, 1984). A general ESM schematization is given by: = / (3.4).
where F is the incident wave thrust function, ΔD the disequilibrium state, and k+/- a model free response rate parameter that has different values for accretion (ΔD > 0) and erosion (ΔD < 0) events, and can assume different physical meanings based on the formulation of F and ΔD (e.g. equilibrium time scale, or wave power efficiency). The time-varying disequilibrium condition ΔD can be defined in reference to the current beach profile position (Banno et al., 2015; Falqués et al., 2007; Jara et al., 2015; Miller & Dean, 2004; Toimil et al., 2017; Vitousek et al., 2017b; Yates et al., 2009) or to the wave history (Davidson et al., 2013; Robinet et al., 2018; Splinter et al., 2014b). Recalling Section 2.2.1, typical expressions of disequilibrium conditions based on shoreline position (ΔD1) and wave history (ΔD2) are shown in Equations (3.5) and (3.7), respectively: = ( ) − (3.5) where E is the incident wave energy, Y is the current shoreline position and Eeq is the wave energy in equilibrium with Y and the beach profile, i.e. the wave energy that would cause no change to the current shoreline position Y, defined, for instance, by a linear relationship: = + (3.6) with a and b being two empirical parameters, as in Yates et al.(2009): = − (3.7).
where Hs is the incident significant wave height, Tp is the peak wave period and (Hs/ Tp)eq is an equilibrium wave condition defined as function of the past wave conditions (e.g. a weighted average of the past Hs/ Tp over a period Φ, calibrated for the site of application; see, e.g. Splinter et al. 2014b).
Depending on the approach, ΔD can thus be quantified in different ways, using either equilibrium wave energy or wave steepness, as in Equations (3.5) and (3.7), respectively. Each approach to evaluate ΔD (and F) is associated to a different physical interpretation to the k parameter (Vitousek et al., 2021). However, regardless of the way it is defined, the disequilibrium state ΔD at a given time expresses the direction of shoreline response as well as the efficiency of the incoming waves in shaping the profile toward its theoretical equilibrium state.

Sea-Level-Rise-Driven Disequilibrium within ESMs

In the context of SLR, the explicit relationship between the disequilibrium state of the beach (ΔD) and the wave-driven shoreline change provided by ESMs can be used to quantify the wave reshaping effect (dYWR) responding to the sea-level-induced disequilibrium. For instance, consider an ESM based on the current shoreline position (Equation 3.5), where the equilibrium wave energy (Eeq) is expressed by Equation 3.6. An increase in MSL (dMSL) induces passive flooding (dYPF), corresponding to a landward shift of the reference shoreline (Figure 3.3b), i.e. a smaller Y value. Recall that this shift represents only a translation of the reference system, with no morphologic change to the beach profile. The second consequence of SLR, observed in Figure 3.3b, is a general increase in the bottom slope relative to the local wave climate propagating over the new MSL. In morphologic terms, this steeper beach profile is no longer in equilibrium for the mean wave climate at the site of interest. In particular, the steeper beach profile is closer to the equilibrium profile for low energy wave conditions, and farther to the equilibrium profile for high energy wave conditions (Wright and Short, 1984). If the mean wave climate does not change, wave events are more likely to result in an overall (erosive) disequilibrium of the “beach–wave climate” system (Figure 3.3c). As the equilibrium condition is defined in terms of wave energy (E), the SLR-induced disequilibrium can be expressed as a decrease in equilibrium wave energy (dEeq,SLR <0), and can be injected in Equation 3.6: =++, (3.8).
This indicates that, for a given shoreline position (Y), less wave energy is required to erode the beach and restore the equilibrium. Or, alternatively, for a given wave condition, the mean erosive efficiency of waves is increased, which likely results from an increased bottom slope across the beach profile due to SLR and wave breaking tending to occur closer to the shore. Further insights on dEeq,SLR are provided in Section 3.3.2 and Appendix B2.

1. Introduction
1.1. Context
1.2. Objectives and approach
1.3. Thesis outline
2. Study site, material and method
2.1. Truc Vert beach, past and future environmental conditions
2.2. Shoreline change models
2.3. Method
3. Integrating SLR-driven recession into equilibrium shoreline models
3.1. Introduction
3.2. Sea-Level-Driven Recession and Equilibrium Shoreline Models
3.3. Integrating Sea-Level Rise in Equilibrium Shoreline Models
3.4. Model Application
3.5. Discussion
3.6. Conclusions
4. Impact of model free parameters and sea-level rise uncertainties on a 20-year shoreline hindcast
4.1. Introduction
4.2. Data, models and method
4.3. Input probability distributions
4.4. Results
4.5. Discussion and conclusions
4.6. Conclusions
5. Uncertainties on future shoreline projections to 2100
5.1. Introduction
5.2. Material and method
5.3. Input probability distributions for future projections
5.4. Results
5.5. Discussion
5.6. Conclusions
6. Effects of stochastic wave forcing on equilibrium shoreline modelling across the 21st century
6.1. Introduction
6.2. Material and Methods
6.3. Input Uncertainties
6.4. Results
6.5. Discussion
6.6. Conclusions
7. Conclusions and perspectives
7.1. Conclusions
7.2. Research perspectives
Publications and presentations
Research articles
Conference presentations
References
Appendix A Methodology
A1. Direct formulation for wave refraction and breaking
A2. Global Sensitivity Analysis
Appendix B Supporting information for Chapter 3
B1. Passive Flooding, Wave Reshaping and the Bruun Rule
B2. Integrating SLR-Driven Disequilibrium into ESMs with Feedback
Appendix C Supporting information for Chapter 5
C1. Wave projections correction
C2. Construction of the high-end sea level scenario
C3. Model free parameters probability distribution
C4. Ensemble waves test with ShoreFor
Appendix D Supporting information for Chapter 6
D1. Wave ensemble and hindcast wave data comparison
D2. Sea-Level Rise
D3. Past scenario simulation with the Yates model

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