In this section I outline the patterns in wave energy density (ξ) conditions during the study period and describe and explain the differences in ξ between sites from the offshore to the cliff toe.
5.4.2.1. Patterns in wave energy
A selected 12-day time series of the variation in ξ between offshore (ξoffshore) and
cliff toe (ξcliff) energies is displayed in Figure 5.12. This date range was chosen
due to the occurrence of a wide range of wave conditions within a short period. As wave energy is proportional to H2 (Holthuijsen, 2009), the variation in ξ is considerably higher than the variation in Hs. Values for ξduring storms (offshore
Hs = 5.0 m, ξoffshore = 250 kJ m-2) are two orders of magnitude higher than when
wave heights are lower (offshore Hs = 0.5 m, ξoffshore = 2.5 kJ m-2). There is a
clear reduction in ξ between the offshore measurements and cliff toe sites for the majority of the time series shown. Exceptions to this occurred on 29/04/17 and 07/05/17 (arrows on Figure 5.12), whereby ξcliff exceeded ξoffshore at site 5.
Site elevation is inversely proportional to inundation duration, hence ξ = 0 for sites 2 and 3 when the tide levels are low.
Figure 5.12: Selected wave energy density (ξ) time series between 28/04/2017 and 10/05/17 illustrating typical variation in site energy (ξcliff) compared with offshore energy (ξoffshore) (black line). Blue line shows water depth (dcliff) maxima at each high tide relative to ODN: neap tide occurred on 05/05/17. The maximum value of ξoffshore on 08/05/17 corresponded to Hs = 4.2 m; minimum on 29/04/17 was Hs = 0.7 m. Arrows show when ξcliff > ξoffshore.
5.4.2.2. Wave energy transfer to the cliff toe
Over the full study period, values of ξdifference (Equation 5-1) fit an exponential
distribution, described by:
𝑃(𝜉𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒) = 𝜆 𝑒−𝜆 𝜉𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 (5-9)
The rate parameter (λ) describes the curvature of the distribution. A higher value of λ shows a smaller number of values where ξdifference > 100. These data
range from ξdifference < 1%, indicating negligible energy density is observed at the
cliff toe relative to offshore conditions, to ξdifference > 500%, indicating
considerably greater energy density is observed at the cliff toe than is observed offshore. Different values of λ occur between sites (Figure 5.13A-E). Sites 1, 4 and 5 exhibit larger values of λ that can be seen in the increased gradient of the semi-log exponential PDF (Figure 5.13F), denoting an increased contribution of smaller ξdifference values.
Figure 5.13: A-E) Fitted probability density functions (p > 0.001) for ξdifference for all sites plotted on semi-log axes, alongside the fitted exponential distribution and the associated rate parameter (λ); F) Each site plotted together to highlight the variation in the λ value. Values above ξdifference = 100% are those where ξcliff >
ξoffshore.
5.4.2.3. Wave energy amplification
In the following analysis I consider what determines the occurrence and degree of both wave energy reduction and amplification of waves at the cliff toe relative to the offshore conditions.
The data in Figure 5.8 and Figure 5.13 indicate that some values of cliff toe Hs
cliff during the shoaling process but prior to breaking (Beetham and Kench, 2011). Waves that are smaller at the cliff toe than offshore are indicative of waves that have broken prior to reaching this cliff toe. In this situation, the break point is seaward of the cliff toe and wave height attenuation has occurred. In this scenario, the foreshore bathymetry acts dissipatively, leading to wave height reduction and energy density loss (Kline et al., 2014). Stephenson et al. (2018) observed a large amplification of cliff toe wave energy density over a 96- hour study at a macrotidal site in South Wales under moderate wave conditions (Hs = 1.00 - 1.43 m). This was due to the large tidal range (11 m), low cliff toe
elevation relative to the tidal range and the resulting high tide water depths across the foreshore and at the cliff toe (1.4 m). The difference between deep water and cliff toe wave energy density ξdifference was in excess of 300 %. This is
contrary to the cliff toe wave conditions observed on microtidal, horizontal shore platforms (Stephenson and Thornton, 2005; Ogawa et al., 2012), where waves break at the platform edge and continually dissipate energy as they propagate shorewards up to the cliff toe. Waves may also reform multiple times during propagation (Collins and Sitar, 2008) but this was not considered in my investigation, which is a simplification previously used in other studies of these phenomena (Beetham and Kench, 2011; Ogawa et al., 2011; Stephenson et al., 2018).
The data presented in this chapter demonstrate that the shore platform at Staithes can act both to amplify and dissipate wave energy density depending on incident wave height, water depth, and platform morphology, as summarised in Figure 5.10. The total ξdifference (ξtotal) describes the total offshore energy
transferred to the cliff toe as a percentage of offshore energy over the full study period during conditions where the cliff toe is inundated. The variation in ξtotal
with platform morphology can be seen in Figure 5.14. Values for ξtotal are seen
to increase with lower cliff toe elevations (Figure 5.14A) and shorter platforms (Figure 5.14B). The widest platform (w = 166 m) dissipates the most energy overall (ξtotal = 3%), compared with the shortest platform (w = 36 m, ξtotal = 23%).
The importance of tidal water depth fluctuation is also demonstrated: ξtotal is
consistently higher at high tide (64% at high tide, 23% overall for the shortest platform). Waves propagating across platforms are more likely to dissipate
energy when shallow depths and wide platforms force breaking, leading to wider surf zones prior to reaching the cliff toe (Marshall and Stephenson, 2011; Poate et al., 2016).
Figure 5.14: Variation in total offshore energy transferred to the cliff toe (ξtotal) by A) elevation (Etoe) and B) platform width (w).
The exponential distribution of ξdifference demonstrated in Figure 5.13 shows that
although cliff toe ξdensity can exceed that of offshore ξdensity, the majority of the
time wave transformation produces smaller energy densities at the cliff toe. During the study period, conditions where ξdifference > 100% and where dcliff > 0
occurred during 18% of measurements. Furthermore, these amplified wave energy conditions are considerably more likely to occur during relatively quiet wave energy conditions. This is illustrated in Figure 5.15; the probability of ξdifference > 100% occurring is higher with smaller offshore wave heights and zero
where Hs > 2.6 m. As such, the wave transformation process at the field site is
substantially more likely to amplify wave energy density during low energy conditions and attenuate energy density during high energy storm conditions. This does not, however, indicate that considerable wave energy cannot reach the cliff toe in storm conditions, as has been shown at other macrotidal shore platforms (Earlie et al., 2015).
Figure 5.15: Probability density function of overall offshore Hs (dashed line); where cliff toe energy density exceeds offshore energy density (orange); where cliff toe energy density is lower than offshore energy density (black).
Stephenson et al. (2018) measured values of ξdifference during incident wave
conditions of Hs = 1.00 - 1.43 m on a macrotidal platform. My results tentatively
imply that the wave conditions under which this would occur would be exceedingly rare. However, it is more likely that differences in platform morphology and water depths will alter the distributions of ξdifference. In
Stephenson et al.'s (2018) study, cliff toe water depths exceeded 8 m, allowing larger waves to shoal up to the cliff toe without breaking. This is associated with an increase in the elevation of the HAT in the power law parameter model (Figure 5.11), allowing larger waves to propagate further across the shore
platform. Therefore, the contribution of conditions where ξdifference > 100% will
increase; the λ value for the exponential distribution of ξdifference in that site
(Figure 5.13) will be correspondingly smaller. This is supported by the variation in offshore and cliff toe Hs due to cliff toe elevation seen in Figure 5.8 and
Figure 5.10. Sites with lower cliff toe elevations, and therefore larger water depths (such as those of Stephenson et al. (2018) and Sites 4 and 5), exhibit a
greater number of larger amplified waves than sites with higher elevations. This supports the notion that shorter, lower platforms experience more intense wave conditions on average for any given offshore sea state.
With regard to wave transformation, this study demonstrates that wave dissipation on macrotidal shore platforms is considerably less than that observed in micro- and meso-tidal platforms. In macrotidal environments, smaller waves are more likely to propagate and reach the cliff prior to breaking, sometimes larger than their offshore height. Whilst larger waves undergo depth- controlled breaking and surf zone dissipation, greater water depths mean that cliff toe waves will be larger than experienced in microtidal environments.