Sand waves are large scale features that move under the influence of the incoming wave climate. Inclusion of these features into the model may be significant in explaining the erosion behaviour observed along the Durban coastline. Given that the basic coastline model is based on a diffusive mechanism, a variation of the Pelnard-Considere equation was implemented to replicate the advective nature of sand waves, namely:
∂y
∂t + Sx,0 (Dc+Db)B
∂y
∂x = 1 Dc+Db
∂Q
∂x (4.23)
where Sx,0 is the longshore transport rate at the zero degree coastline orientation and B is the amplitude of the nourishment. The nourishment is advected by a velocity equal to that of the second term on the left hand side of equation 5.6. The value of
∂y/∂x is calculated using the central difference theorem.
For application in the model, a baseline coast was used as the reference plane for added nourishments (see sect. 4.4.4). For model stability and accuracy, a numerical process was devised to ensure that sand waves behave appropriately. The first step involved calculating the relative orientations of coastline segments without the nour- ishment added onto the baseline coast. The nourishment was thereafter added to the baseline coast. Given that coastline positions will only change for the region where the nourishment was added, it was possible to isolate baseline coast orientations within the region of the nourishment. These base coastline orientations were averaged to obtain essentially a tangent to the coastline where the nourishment is situated (see fig. 4.9).
The tangent is a straight line that is inclined at the previously calculated average coastline orientation. Use of a straight baseline is advantageous as ∂y/∂x values are simpler to calculate. Furthermore, corrections due to over-erosion is made simpler.
Considering the relative size of nourishments with regards to the total domain, it was assumed that curvature effects would not be significant.
Fig. 4.9 Visualisation of tangent method used for diffusing and advecting sand waves.
4.7 Nourishment Diffusion & Advection The process then assumed that the nourishment was placed along the tangent coastline, and was thereafter adjusted to account for diffusion effects as detailed in section 4.5.5. Alteration of the tangent baseline is prevented by using the tangent as a hard boundary limit (see sect. 4.7.1). The diffused nourishment is thereafter advected using equation 5.6. Using the incoming wave angle, Sx,0 is computed using for the tangent coastline orientation by subtracting the wave angle from the tangent normal (see fig. 4.10). B is the maximum value of the plume, which does not vary along the coastline. Finally, ∂y/∂x is thereafter computed. The tangent base is thereafter subtracted from the now diffused and advected coastline inclusive of the nourishment. This isolates the nourishment such that it may be added back onto the curved baseline. Figure 4.11 shows the advection and diffusion of an arbitrary nourishment along a straight coastline over a period given a fixed deepwater wave angle.
Fig. 4.10 Coastline orientations used for calculating Sx,0.
Fig. 4.11 Example of an advected and diffused sand wave.
4.7.1 Vertical Boundary Conditions
Given that this study investigates coastline evolution due to varying sediment supply schemes, the possibility of beach profiles being completely eroded must be accounted for in terms of the model. The inclusion of an erosion limit is important when considering availability of sediment for longshore transport and the resulting coastal evolution. Numerically, the computed longshore transport rate is independent of available sediment, hence a computed rate within a cell that has been exhausted of sediment could provide unrealistic results. This limit is introduced in the form of a vertical boundary condition where the erosion limit essentially represents a seawall.
Hanson and Kraus (1986) detail a procedure for implementing a seawall boundary condition into the capabilities of a one-line model. They base their method on the following principles:
• The shoreline in front of a seawall may not recede landward of the structure.
• Sand volume must be conserved.
• Direction of alongshore sand transport must be conserved in accordance with the natural direction of the potential local transport.
This erosion limit was introduced by using beach widths obtained from eThekwini Municipality’s beach survey program by averaging measured widths for 41 profiles.
During the initialisation process, the model interpolated beach widths from the entered data and subtracted them from the zero contour. This essentially produced an offset beach width contour that provided limits for erosion activity. Should sufficient erosion occur such that this limit is exceeded in any cell, the following process is implemented.
Individual grid cells breaching the limit are analysed to identify whether they are regular, minus or plus area cells. Figure B.1 depicts the differences between these classifications. This classification is based on the direction of longshore transport rates.
Following this classification, the necessary adjustment to the longshore transport rate is carried out using the appropriate formula. Minus cells (fig. 4.13a) use equation 4.24 to adjust transport rates whereas regular cells (fig. 4.13b) use equation 4.25 to adjust transport rates. Notably, plus area cells require no adjustment of transport rates as they represent a convergence of sediment.
Qi+1∗ =Qi+1yi−ysi
yi−yi′ (4.24)
Qi+1∗ =Qi− ysi−yi
2B (4.25)
4.7 Nourishment Diffusion & Advection Longshore transport rates with asterisks correspond to adjusted rates equations 4.24 and 4.25, with yi referring to the original coordinate, yi′ being the shifted coordinate using equation 4.20 and ysi referring to the hard boundary position. Corrections to longshore transport rates often propagate through the system, hence this correction is continued throughout the model domain until either the end or a plus area cell is reached (Hanson and Kraus, 1986). For this section of the model, the pseudocode provided by Hanson and Kraus (1986) was converted from FORTRAN to Python and implemented. See appendix D for more details.
(a) (b)
Fig. 4.12 Conceptual diagram showing cell classifications for minus and plus area (a) and regular (b) cells (Hanson and Kraus, 1986).
(a) (b)
Fig. 4.13 Conceptual diagram showing shoreline and transport corrections for minus (a) and regular (b) (Hanson and Kraus, 1986).