COVE

The conservation equation for beach sediment expressed in terms of local coordinates states that the change in position of the shoreline \(d\eta\), perpendicular to the local shoreline orientation \(s\) through time \(t\) is a function of the divergence of alongshore sediment flux \(Q_{ls}\):

\(\frac{d\eta}{dt} = f\left(\frac{dQ_{ls}}{ds}\right)\)

The nature of the function is dependent on the geometry of shoreline cells, which in COVE are not rectilinear, but rather triangular, trapezoidal or polygonal. The change of shoreline position for such cells is calculated by inverting quadratic and cubic equations for the volume of sediment in these cells (see Hurst et al., 2015).

Bulk alongshore sediment flux is driven by waves breaking on the shoreface. Typically in alongshore transport laws, flux depends on the height \(H_b\) and angle \(\alpha_b\) of breaking waves. For example, in the simplest case of fine/medium sand, COVE uses the CERC equation:

\(Q_{ls} = K_{ls} H_b^{5\over2} \sin 2\alpha_b\)

where \(K_{ls}\) is a transport coefficient. The transport coefficient \(K_{ls}\) may be modified to account for the size of beach material (\(D_{50}\)). Calibration of this coefficient can be made from estimates of bulk alongshore transport or by calibration against a historical record of coastal change (e.g. Barkwith et al. 2014a).

Cliffs are represented in the model as a separate line. The cliffline and coastline interact to determine how wide the beach is locally. Eroded cliff material is provided to the adjacent beach and causes the shoreface to advance. Cliff erosion is controlled by beach width since a wider beach provide energy dissipation and protection from approaching waves. Figure X shows a schematic graph of this relationship, as well as a conceptual diagram of the representation and relationship of the cliff and the beach.

The result is that we can run simlutaions at decadal timescales to explore the interactions between coastal erosion and alongshore sediment dynamics.

The COVE code is under continuous development. As we publish scientific papers that use the model, we will release the model code associated. The tar.gz release version and .zip release version is the version used by Hurst et al. (2015) to explore the sensitivity of crenulate-shaped bays to variation in wave climate. Once downloaded, extract the contents to an appropriate workspace.

The code can be compiled in a Linux environment from the command line, using one of the makefiles. These are contained in the 'driver_files' subdirectory. The driver files are C++ scripts that control the initiation, running and saving of a COVE model run. In this tutorial we will use the example for running a spiral bay as used in Hurst et al. (2015).

In a terminal, navigate to the 'driver_files' subdirectory:

Compile COVE for running a spiral bay by launching the makefile:

This will create an executable 'spiral_bay.out' which can be launched from the command line to run the model. First, let’s move the executable to the parent directory, and navigate to the same directory:

The file spiral_bay.out generated by compiling the code can be launched from the command line:

Running it in this way will result in it terminating with an error, which will tell you that the program requires a number of input arguments in order to run. In the spiral bay example, the offshore wave climate is represented with three Gaussian distributions, for wave period, height and direction. Each of these is described by a mean and standard deviation, and these are fed to the model as arguments. To run the model with mean wave period of 6 seconds, standard deviation 1 second, mean wave height 1 metre, standard deviation 0.1 metre, and mean wave direction 035^o and standard deviation 25^o:

The model should then run for a hundred years. This example evolves a crenulate-shaped bay from a straight initial coastline between two fixed headlands or sea walls. Sediment is transported out of the model domain by alongshore sediment flux and the shoreline gradually adjusts to the distribution of wave directions. The bay eventually reaches a state of equilibrium where the net alongshore flux is close to zero everywhere. The model is setup to run for 100 years, more than enough time for an equilibrium bay configuration to form.

While running the model will print the current model time to screen, it may also print some other messages, particularly including intersections in the coastline. The intersection analysis detects when the coastline intersects itself, such as when it erodes back behind the headland. Once this has happened the coastline is prevented from eroding any further.

We make plots of the resulting coastline evolution using the python matplotlib library. To use them you will need a python IDE such as Spyder. A series of plotting functions are included in the subdirectory 'plotting_functions'. To plot the results of your spiral bay model run, open the file 'plot_coastline_evolution_figure.py' in your favourite python IDE, and run. You should get the following figure:

Additionally, below will be a link to a video of a spiral bay evolving, which will be hosted on Vimeo once I have time to work out how to do it (MDH).