Wave, Tide and Topographical Controls on Headland Sand Bypassing

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61
Embayed beaches separated by irregular rocky headlands represent around 50% of the world's 62 shoreline and are important zones ecologically and commercially (Short & Masselink, 1999). Accurate 63 determination of sediment budgets is necessary for prediction of coastal change over long timescales 64 in these zones. It has been recognised that the traditional view of embayed beaches as closed littoral 65 cells is not accurate for many embayments, where sediment can enter and exit the system via 66 headland bypassing (Goodwin et  Headland bypassing is defined as the process of sand transport around headlands, which act as 69 obstructions to longshore sediment transport, forced by wave, tide and wind action (Evans, 1943;  Short, 1985), and are associated with high-energy conditions with major 89 storm events thought to be an important driving force of headland-attached rip bypassing (Short & 90 Masselink, 1999). Embayment length is important in determining the flushing of the surf zone via 91 headland rips with wider embayments allowing greater development of longshore drift in oblique 92 wave conditions, resulting in stronger flushing at the downwave headland (Castelle & Coco, 2013). 93

154
The North Coast of Cornwall is situated in the South West United Kingdom, on the Northwest 155 European Continental Shelf (Figure 1). Resonant effects contribute to large tidal amplitudes over the 156 whole Celtic shelf, with a mean spring tide range (MSTR) in the study area of ca. 5m in the Southwest 157 and increasing to >7m at Hartland Point (Uncles, 2010). Modelled regional scale bed shear stresses, 158 tidal residual currents and sand transport pathways indicate residual sand transport towards the 159 northeast along this coastline, progressively weakening as it moves up coast (Pingree & Griffiths, 1979;160 Holt et al., 2001;Uncles, 2010;King et al., 2019a). Strong tidal currents (around 1.5 ms -1 at springs) 161 drive a net residual current of up to 15 cms -1 towards the northeast immediately adjacent to large 162 coastal promontories. This residual is broken up by multiple headland-bound embayments, resulting 163 in areas of low residual tidal transport close to shore. In combined wave and tide conditions, sand 164 transport is wave dominated for median waves in these areas where tidal forcing is weakest and is 165 wave dominated across the whole North Coast under extreme waves (King et al., 2019a). 166 A 75-km stretch of this coast was selected for this study (Figure 1). This section of coastline is 167 comprised of embayed beaches separated by irregular rocky headlands (29 embayments were 168 selected for this study). Beaches in the study area are comprised of medium quartz sand (Prodger et 169 al., 2016). These embayments comprise a wide range of wave exposures, embayment lengths, degrees 170 of embaymentisation and headland morphologies. This coast is directly exposed to the Atlantic, 171 bringing waves with potential fetch lengths of 6000km (Collins, 1987). Winter storm H s at nearshore 172 wave buoys along the North Coast can exceed 6m (Scott et al., 2016b). Average H s based on a 10-year 173 hindcast of WAM is ~1.5m along this section of coast, with H s of ~2m further offshore (Bricheno et al., 174 2015;King et al., 2019a). The wave climate in the region has experienced an increase in extreme (99th 175 percentile) H s of up to 1% per annum between 1985 and 2008, and has also experienced an increase 176 in winter wave height and interannual variability (Young et al., 2011;Castelle et al., 2018). 177 The large tidal range, high degree of wave exposure and diversity of headland and embayment 178 morphologies make this a suitable site for an investigation into the impacts of different environmental 179 and morphological forcing conditions on potential headland bypassing. 180 The inset around Godrevy Point shows the computational grid as an example of the localised grid refinement around headlands. Headlands are numbered from southwest to northeast as indicated, and their names are included below the map. Other locations of reference are annotated. ADCP deployments (+) and wave buoy locations (Δ) are marked, alongside their name as referred to in the text. Open model boundaries are marked with a solid red line. A wave rose of the wave climate at the Wave Hub between 01-June-2015 and 31-May-2018 is inset bottom-right, showing principle wave directions. An example aerial image of headlands 9-13 is included for reference (bottom right). For the purpose of this study, upcoast is defined as towards the northeast (increasing headland number). The process-based numerical model Delft 3D was used to model the North Coast ( Figure 1). The FLOW 184 hydrodynamic module was 2-way coupled to a SWAN third-generation spectral wave model packaged 185 as Delft3D WAVE with an identical grid. Three-dimensional hydrodynamics are calculated using the 186 unsteady shallow-water equations, following the Boussinesq approximation with the vertical 187 momentum equation reduced to the hydrostatic pressure relation, assuming that vertical 188 accelerations are small relative to gravitational acceleration (Lesser et al., 2004). The contribution of 189 3D turbulent eddies is modelled using a k-ε turbulence model. SWAN, packaged as Delft3D-WAVE, is 190 a third-generation phase-averaged wave model based on fully spectral representation of the action 191 balance equation, accounting for wave-current interaction through radiation stress, refraction, wind 192 generation, whitecapping, nonlinear wave-wave interactions, bottom dissipation, and depth-induced 193 breaking (Booij et al., 1999). 194 The North Coast model was one-way nested within a regional fully coupled hydrodynamic, wave and 195 sand transport model validated and presented in King et al. (2019a). Grid resolution of the North Coast 196 model was ca. 50m in the vicinity of headlands, and the model was run in 3D hydrodynamic mode 197 with 10 sigma-levels in the vertical. The WAVE grid was extended two grid cells out from the FLOW 198 grid. Bathymetry was derived from merged high-resolution multibeam data from the UK Hydrographic 199

Methods
Office and lidar data Plymouth Coastal Observatory, corrected to Mean Sea Level 2000 datum using 200 the Vertical Offshore Reference Frame  and merged with coarser EMODnet 201 bathymetry offshore (EMODnet Bathymetry Consortium, 2016; Figure 1). Bathymetry at the 202 boundaries matched the bathymetry of the regional forcing model. High-resolution bathymetry was 203 assigned to the grid using spatial averaging, while lower resolution EMODnet bathymetry was assigned 204 to the grid using triangular interpolation. A uniform grain size of 330 m (Prodger et al., 2017) 205 throughout the domain was used, allowing cross-comparison of different embayments. 206 The hydrodynamic model has two water level boundaries and one velocity boundary to the south-207 west. This combination of forcing types provided the best agreement with observations during 208 calibration. Boundaries were situated far from the headlands of interest. Boundary conditions were 209 linearly interpolated from the regional model, which was itself one-way nested within the Atlantic step was 12 s. Wind fields were interpolated linearly from 0.25° resolution scatterometer blended 6-212 hourly mean wind fields retrieved from the Copernicus Marine Service (Bentamy & Fillon, 2012). 213 Atmospheric pressure was interpolated linearly to the model grid from the 0.5° resolution Climate 214 Forecast System version 2 model (Saha et al., 2014). 215 The wave model was forced with parametric boundary conditions (H s , T p , direction, directional 216 spreading) linearly interpolated from the regional model at 1km resolution at the open boundaries. 217 For calibration and validation the regional model in turn forced by the UK Met Office Wave Watch III 218 continental shelf model (King, 2019a;Saulter, 2017). The wave time-step was 10 minutes, with a 219 coupling interval between WAVE and FLOW of 1 hour, where wave forces are passed based on energy 220 dissipation rate radiation stresses, bed shear stresses, Stokes drift and bottom orbital velocity, and 221 receiving water levels and velocities. The wave model had a directional resolution of 5° (72 bins over 222 a full circle) and 24 frequency bins between 0.05 and 1 Hz. 223  This method assumes areas of sediment are vertically smoother than rock over a 100m window. Some 247 sediment features were highlighted as rock due to their large vertical expression (such as large 248 sandwaves west of St Ives). These were identifiable due to their linear, repeating pattern, and included 249 in the sediment polygons. Perranporth has a sand-gravel transition at ca. -26m ODN (Valiente et al., 250 2019a). This was identifiable in the data as a border with elevated maximum standard deviation and 251 was used to define to offshore sand polygon boundary. Similar borders elsewhere were also used for 252 this purpose. The purpose of this was not to determine the exact spatial extent of sediment across 253 this region, as this would require a more detailed observational campaign to determine sediment 254 physical characteristics and spatial extent. Rather, the method was used as a means of generating an 255 approximate sediment distribution to test the effect of a realistic pattern of sediment spatial coverage 256 on headland bypassing rates versus a uniform, homogeneous sand bed. As such, this method was 257 considered sufficient for the purpose of this study. 258 In general, the model has good or excellent skill for both depth-integrated and near-bed instantaneous 286 and residual (low-pass filtered) northward velocity components. The lowest performing residual 287 northward velocity skill is the near-bed velocity at A17, which has "reasonable" skill. Eastward velocity 288 components at both ADCP deployments were very small, which resulted in lower Brier Skill and R 2 289

Sediments
Polygons of spatial sand extent in embayments of interest, determined by eye from (c), also indicating areas of land.  Note. Brier skill scores are coded for excellent and good (bold), reasonable (italic) and poor (underlined) model skill. Eastward and Northward velocity components are denoted by "-E" and "-N" respectively. Near-bed currents are denoted by "(bed)". "ALL" indicates performance for combined data from A17 and A25.

metrics. A more informative metric at these sites is
* Circular correlation coefficient for directional data.

Simulated scenarios 320
The wave climate was characterised near the offshore boundary using wave buoy data from the Wave 321 Hub (Figure 1)  Wave-only, tide-only and coupled wave-tide scenarios were conducted. Wave-only scenarios were 333 conducted for two water levels corresponding approximately to spring high water (SHW) and spring 334 low water (SLW) to give maximum variability in tidal elevations tested. Tidal scenarios were conducted 335 over a spring-neap cycle and times where water levels were at Spring High or Spring Low were 336 extracted for analysis. Velocities at these times ranged from 0.02 -1 ms -1 . All scenarios were simulated 337 for a uniform homogeneous sand bed and for the spatially variable sand distribution demonstrated in 338 Wave-only scenarios were run for 72 hours, and sand transport was averaged over the final 24 hours. 351 In tidal scenarios, times of spring high and low water were defined as when the median water level 352 across each transect was > + 3m or < -3m relative to mean sea level (MSL2000 datum) respectively. 353 Tide range increases towards the northeast; therefore, the number of points satisfying this criteria 354 increased moving up-coast. Bypassing rates were averaged over all times where the water level was 355 within the SHW or SLW depth bin at each headland. shore sediment extent X sed and the area of sediment coverage adjacent to the headland A sed . This was 373 used to determine a ratio R sed defined as: 374 Where A DoT is the total area between the adjacent and apex transects, bounded by the headland face 375 and DoT. Depth off the headland toe Z toe was determined 50m offshore of the headland apex along 376 the apex transect. This was nondimensionalised across all headlands by dividing by 50m to give the 377 slope of the headland toe m t : 378 , surf zone width X surf , cross-shore headland length X head , beach length L b , headland toe depth Z toe , sediment area adjacent to headland A sed , total area between headland and maximum depth of transport A DoT , and sand bypassing rate Q b .  shore length X head , headland toe depth Z toe , beach length L b and sediment ratio R sed . Stacked bar graphs (a, c, e, g) show parameter values per headland for spring high water (SHW) and spring low water (SLW), and for the upcoast orientation (up) and downcoast orientation (down). Box plots (b, d, f, h) show summary statistics for each water level and headland orientation. The main body of the boxes span the 25 th and 75 th percentiles, the horizontal bar shows the median, the mean is shown (black dots), whiskers span up to 1.5 × inter quartile range, and outliers are shown (black crosses). The impact of tidal elevation on headland bypassing rates was independent of wave direction ( Figure  434 7a-c). The relative impact of tidal elevation changes was greatest during median wave conditions, 435 where in some cases bypassing was activated only at SLW. In other cases, bypassing direction changed 436 between SHW and SLW, mainly for median wave conditions. The impact of changing water levels 437 decreases as wave height increases (Figure 7d). For median waves, bypassing at SLW has a median 438 increase in magnitude of ca. 4 × relative to SHW, whereas this is reduced to ca. 2.5 × for large waves 439 and ca. 1.5 × for extreme waves. The mean increase in bypassing at SLW is influenced by several large outliers (beyond the axes scale) for median waves where bypassing increased from a very low level, 441 otherwise it is in close agreement with the median for large and extreme waves. 442

Headland bypassing
Bypassing rates were strongly dependent upon the cross-shore headland extent relative to surf zone 443  (Figure 8c). An exponential term was fitted of the form: 476 Where a and b were calibration parameters. The best fit was found for a = 3.5 and b = 0.7, shown in 477 Figure 8d. This improved the MAE of the parameter to a factor 3.5 and RMSE to a factor 5.2.
Bypassing directions were generally predicted correctly as a function of breaking wave direction 479 relative to shore normal. The percentage of scenarios where bypassing was predicted correctly is 480 shown in Figure 8e (grey bars) for each headland. Where there was no bypassing under any conditions, 481 no bars are shown. Coloured bars with negative percentages indicate the percentage of scenarios 482 where bypassing direction was wrongly predicted. The colours indicate the wave conditions where 483 bypassing direction was predicted wrongly. For over half of headlands that had at least one bypassing 484 direction wrongly predicted, the direction was wrong for median wave, low bypassing conditions or 485 for only one or two scenarios. Six headlands had bypassing direction wrongly predicted for over 50% 486 of cases. These are discussed in section 5. The impact of introducing spatially variable sediments was determined for each headland using the 490 ratio: 491 Where Q b_Sed represents bypassing for the non-uniform sediment distribution scenarios, and Q b_Uni 492 represents bypassing for the uniform sediment scenarios. Results are presented in Figure 9a. The main 493 impact of introducing a realistic sediment spatial distribution was that bypassing rates were generally 494 reduced, or bypassing ceased altogether (ratio = -1). There was only one headland (headland 6; Figure  495 9a) where bypassing direction was predicted to change (ratio < -1) between the uniform and spatially 496 variable sediment scenarios. This occurred for median waves and a low bypassing magnitude. 497 Eight headlands exhibited an activation of net bypassing in the case of non-uniformly distributed 498 sediment for at least one wave condition, and a further five exhibited an increase in net bypassing 499 rate, although this tended to be relatively small, never more than a factor 2 (ratio = 1). In these cases, 500 gross transport along the apex transect was greater for uniform sediments, however net bypassing in 501 was low or zero/ divergent. This was due to a relatively large magnitude divergent transport off the 502 headland toe in the uniform sediment scenario which opposed alongshore transport past the 503 headland further offshore, resulting in zero or low net bypassing for uniform sediment coverage. This 504 nearshore transport divergence was of a much lower magnitude when sediment was unavailable for 505 resuspension off the headland toe, and bypassing further offshore in the suspended load dominated 506 (example: headland 11 - Figure 9d, e). 507 Two conditions were determined that were indicative of where a sediment availability parameter 508 should be applied. Firstly, if X head < 1.5 X surf then bypassing was approximately equal to the uniform 509 sediment availability case and a sediment availability parameter need not be applied. Likewise, if 510 sediment is available off the headland toe (in this case tested at 100 m from the headland toe) then 511 bypassing can be approximated using the uniform sediment parameterisation and the sediment 512 parameter need not be applied. These conditions account for the headlands with zero or very small 513 relative change in Figure 9a. 514 incorrectly predicted (coloured bars). Colours represent the wave conditions where bypassing direction was wrongly predicted. No bars are shown where no bypassing occurred, and percentages were calculated relative to the number of cases where bypassing occurred.
For cases where these conditions indicate a change in bypassing rate due to sediment availability, a 515 number of parameters were tested for influence on bypassing rates, including: cross-shore extent of 516 sediment adjacent to the headland, sediment coverage ratio R sed , X head , Z toe , headland alongshore 517 length and headland perimeter length. No parameters indicated a clear correlation with changes in 518 bypassing rates predicted by the model. A uniform reduction of an order of magnitude performed best 519 when applied to Q b_Toe (Equation 6). 520 When applying the criteria discussed above with this parameter, the MAE for all headlands under the 521 spatially variable sediment scenarios was reduced from a factor of 5.5 to a factor of 4.6 (Figure 9b, c). 522 This indicates the parameters applied thus far are able to capture the order of magnitude of wave-523 forced instantaneous headland bypassing for different headland morphologies, at different tidal 524 elevations, and for spatially variable sediments with an overall R 2 of 0.66. It remains to test the 525 influence of tidal currents on bypassing rates. The scenarios described above were repeated with the inclusion of tidal currents. Tidal currents at the 529 times of SHW and SLW extracted for processing ranged between 0.02 and 1 ms -1 in magnitude off the 530 headland apexes, with greater magnitude off larger promontories. Whilst these were not the peak 531 ebb and flood currents, they represent a large range of velocities for the assessment of the impact of 532 tidal currents on instantaneous bypass rates. Bypass rates were averaged over all times of SHW or 533 SLW respectively. Example results are presented in Figure 10  at SHW is provided in supplementary Figure FS1. 536 Tidally-driven bypassing, in the absence of wave forcing, had a maximum magnitude of ca. 10 -3 m 3 s -1 537 across SHW and SLW in the case of uniform sediments (Figure 10a-b). The greatest bypassing 538 magnitude for uniform sediments was off Trevose Head (headland 23), the largest promontory in the 539 domain. Tide-driven bypassing directions sometimes opposed the wave-driven bypassing. In this case, 540 for median waves (Figure 10a), bypassing under combined wave-tide forcing tended to follow the tide-541 driven bypassing direction, indicating that median waves act to enhance sand transport in the tidal 542 direction. For extreme waves (Figure 10b), bypassing direction rarely changed between wave-only and 543 wave-tide scenarios (headland 15 at SHW only), and there was generally only a minor enhancement 544 of bypassing magnitudes relative to wave-only scenarios. In some cases (headlands 20 (SHW) and 26) 545 bypassing was switched off with the addition of tidal forcing. 546 Tide-driven bypassing was greatly reduced when non-uniform sediment distributions were included 547 (Figure 10c-d). Bypassing was switched off across most headlands, and only active for seven headlands 548 in total between SHW and SLW (1, 2, 10, 14, 21, 23 & 27). In these cases, bypassing was generally 549 downcoast (with the exception of 10 and 21 at SLW) and of very low magnitude. The greatest 550 magnitude was for headlands 1 and 2 at SHW, which indicated tidally driven sand transport out of St 551 Ives Bay to the west, in agreement with transport directions reported in King et al., (2019a). Regardless 552 of low tide-only bypassing rates, tidal currents were able to induce reversals in the median wave 553 bypassing directions (Figure 10c) indicating that wave-current interactions are important during 554 median waves, even when tide-only bypassing may be negligible. 555 availability. (d, e) Example headland (11 -Kelsey Head) where net bypassing was divergent for uniform sediments but upcoast for non-uniform sediments, with sand transport magnitude and vectors shown. Colours and vectors are log-scaled. The condition shown is extreme waves from 292.5° at SHW. Dashed white lines in (e) indicate the offshore limit of sand coverage. This is indicated in the relative change bar plots for uniform sediments (Figure 10e) and non-uniform 556 sediments (Figure 10f). Here, relative differences were averaged over the SHW and SLW scenarios and 557 all wave directions. The largest relative differences tended to be for median waves (blue bars). There 558 was also a widespread activation of bypassing under the wave-tide forcing when wave-only bypassing 559 was nil, particularly for median and large waves. 560 Figure 10: Comparison between tide-only, wave-only and wave-tide bypassing rates, for waves from the modal wave direction 281.25°. Instantaneous bypassing rates are presented for median and To quantify the relative impact of waves, tides and their non-linear interactions, bypassing rates were 561 used to determine their wave-tide dominance classification as per King et al., (2019a). This indicates 562 whether the dominant driver of sand transport is tidal forcing (T), wave forcing (W) or the non-linear 563 interactions between the waves and tides (N) using two ratios: 564 Where W represents wave-only bypassing rate, T represents tide-only bypassing rate and N represents 566 the contribution of non-linear wave-current interactions to bypassing, calculated as: 567 Where WT is the bypass rate under coupled wave-tide forcing. Results of the classification over all 568 scenarios are presented in Figure 11. Lower-case letters indicate a sub-dominant contribution from 569 the denoted forcing mechanism. There was no appreciable difference between wave directions, 570 therefore all directions were aggregated to calculate the percentage of data in each class for each 571 scenario wave scenario (columns) and waver level (rows). Median waves exhibit non-linear wave-tide 572 interaction dominance of bypassing rates under all scenarios for the majority of headlands. At SLW 573 around 10% of headlands shift from non-linear dominated to wave dominated under median waves, 574 reflecting greater wave impacts at low water. The relative influence of tides under these waves is 575 greatest at SHW, mainly manifested as a subdominant tidal contribution, denoted by a lower-case "t" 576 (e.g. Nt). This reduces to < 5 % of data at SLW. 577 Dominant forcing shifts towards wave-dominance as the wave exceedance increases (median → large 578 → extreme). For large and extreme waves, the majority of bypassing is wave-dominated in this 579 macrotidal environment at both SHW and SLW. For these waves and uniform sediments there is a 580 secondary, tide dominated mode of sand transport for ~ 18 % of data at SLW (Figure 11 e -f). This 581 occurs where wave-only bypassing was weak or negligible, for example at headland 23. This signal is 582 extreme waves for uniform sediments (a, b respectively) and non-uniform sediments (c, d respectively) for tide-only (black solid line), wave-only (coloured solid line) and wave-tide (coloured dashed line). Positive values represent upcoast bypassing, and downcoast bypassing for negative values. Values are for each headland. (e, f) Relative differences for uniform sediments (e) and nonuniform sediments (f) per headland. Values are an average over all water levels and wave directions. Bars are coloured for each wave condition. Symbols indicate wave conditions where bypassing was activated by wave-tide forcing Q b_WT but not by wave only forcing Q b_WO for at least one condition. The y-scale increases in log 2 increments. much reduced, or negligible, for non-uniform sediment distributions (Figure 11 g -l), reflecting the 583 much reduced tidally driven bypassing when sediment is not available off large headland 584 promontories. For extreme waves and non-uniform sediments (Figure 11i, l), wave-current 585 interactions have a greatest impact at SLW, shifting the class of bypassing from W to Wn for around 586 30 % of the data. 587 Figure 11: Wave-tide dominance classification as per King et al. (2019a). Classifications range from tide-dominate ("T" -red) through dominance of non-linear wave-tide interactions ("N" -green) to

589
This paper tested the influence of wave, tide and morphological controls on instantaneous headland 590 sand bypassing using a coupled wave-tide numerical model, and tested the performance of an existing 591 parameterisation when applied to realistic headland morphologies and sediment coverage, making 592 recommendations for additional terms to improve model performance. We discuss connectivity 593 between embayments via headland bypassing along this stretch of coast in the context of previous 594 work in this region and globally (Section 5.1). We then discuss the assumptions and limitations of the 595 proposed bypassing parameterisation (Section 5.2), before outlining practical considerations for the 596 application of a headland bypassing parameter with recommendations for further research (Section 597 5.3). 598 between spring and neap tide conditions. Results presented here suggest that, in macrotidal 604 environments, bypassing during energetic events (deep water H s ≥ 6 m) is wave-dominated; however, 605 during median wave events (deep water H s = 2 m) bypassing rates are dominated by non-linear wave-606 current interactions between waves and tidal velocities, with waves enhancing bypassing in the tidal 607 direction and activation of sand transport when tide-only bypassing is negligible. 608

Headland bypassing on embayed coastlines
Non-uniform sediment availability reduces tide-only bypassing when sand is not available adjacent to 609 the headland apex, where tidal currents are amplified (King et al., 2019a). Bypassing in these situations 610 was in the suspended load. Tides have a greatest impact for median waves: tidal elevations modulate 611 bypassing by a factor of 4 between SHW and SLW because of modulation of headland cross-shore 612 length, whilst the impact of currents is generally not more than a factor of 2 for non-uniform sand 613 coverage, matching the minor tidal control reported by Valiente et al. (2020). The primary control on 614 bypassing rates is the cross-shore length of the headland relative to surf zone width, and low bypassing 615 rates for X head / X surf > 3 matches McCarroll et al. (In Review). 616 wave dominated ("W" -blue), and mixed ("M" -purple). Lower-case letters denote a subdominant contribution from the denoted process. Data for all three wave directions were aggregated into median (50% exceedance, column 1), large (5% exceedance) and extreme (12h exceedance) wave conditions for simplicity. Classifications are shown for uniform (a -f) and non-uniform (g -l) sediment distribution. Water levels are denoted by SHW and SLW for spring high and low water respectfully. Reduced depth off the headland toe increases headland bypassing rates following the relationship in 617 Equation 6. McCarroll et al. (In Review) report an increase in bypassing magnitude of a factor 1.5 for 618 headlands with sub-aqueous ridges of around 1 to 3 m prominence, resulting from increased orbital 619 velocities at the bed off the headland. Equation 6 predicts this, as a decrease in depth off the headland 620 toe of ca. 2 m for depths between 3 and 10 m results in an increase in bypassing of a factor between 621 1.3 and 1.8. This acts as an additional term to the parameterisation (Equations 5 & 6). 622 The parameterisation of the form of Equations 1 and 5 had previously been shown to apply for an 623 isolated headland with uniform offshore bathymetry, sediments and wave-only forcing (McCarroll et  624 al., In Review). The alteration of the exponent between Equations 1 and 5 reduces the rate of decay 625 of the bypassing parameter as headlands extend beyond the surf zone (X heas / X surf > 1). This implies, 626 for realistic headland morphologies and bathymetric expressions, headland bypassing occurs for 627 greater relative headland cross-shore extents than predicted through idealised scenarios with a linear 628 shoreface gradient. 629 We show that with this minor adaptation, and the addition of terms for variable headland toe depth 630 Else if X head ≤ 1.5 X surf or there is sand at the headland toe: 635 Where Q 0 is the uninhibited longshore transport formulation of van Rijn (2014): 636