Overview
HāpuaMod is a pseudo 2D model consisting of three interlinked 1D models:
The fluvial model component simulates flow and bedload transport in the lower part of the river upstream of the lagoon, through the online part of the lagoon and through the outlet channel to the sea. The fluvial model is discretised into rectangular cross-sections, each of which is defined by a bed level and width. Additionally the 'offline' parts of the lagoon are included as additional area for water storage attached to the first and last cross-sections of the 'online' part of the lagoon. At each hydraulic time-step the model depth and velocity at each cross-section is calculated using the full 'unsteady' St-Venant equations of open channel flow. River inflow and sea/tide level provide the hydraulic boundary conditions for the model. Bed roughness is specified using a globally constant Manning's 'n' value. In order to maintain a stable solution the hydraulic time-step needs to short (seconds) but a longer morphological time-step is used for all other calculations to reduce model run-time.
Bedload transport is calculated at each model cross-section based on flow depth and velocity using the Bagnold (1980) streampower approach. The rate of deposition/erosion at each cross-section is calculated by the difference in transport rate between adjacent sections. Equilibrium transport is assumed at the upstream cross-section (i.e. no deposition/erosion). Bedload transported through the downstream cross-section is assumed to be deposited onto the shoreline either side of the downstream end of the outlet channel.
Deposition in a cross-section is always accounted for by raising the cross-section bed level. The way erosion is accounted for depends on the cross-section. Within the river upstream of the lagoon, erosion is assumed to reduce the bed level. In the lagoon and outlet channel erosion is either applied to the channel bed (i.e. lowering) or the banks (i.e. widening) depending on the channel width-depth ratio. An equilibrium width-depth ration is specified in the model configuration. If the channel is over-wide erosion is applied to the bed, and if the channel is over deep it is applied to the banks. During bank erosion an equal volume of erosion is assumed to occur on each bank.
Full details of the fluvial model are given here.
HāpuaMod includes a very simple 1-line shoreline model to simulate longshore transport and horizontal movement (advance/retreat) of the shoreline. At each morphological timestep the model calculates the angle of each shoreline segment between transects. Based on this angle the model refracts wave energy to the breakpoint and calculates longshore transport rate using the CERC formula (Coastal Engineering Research Center 1984, Komar 1971). The rate of deposition/erosion of sediment at each transect is calculated from the difference in longshore transport between the shore segments either side of it. The volumetric deposition/erosion each timestep is converted into an advance/retreat distance by dividing by the shoreface area, calculated as the distance between transects multiplied by the depth from the barrier crest to a fixed 'closure depth' specified in the input file (see the transect schematisation figure below).
Full details of the 1-line shoreline model are given here
The cross-shore morphology model is very simple and only considers overtopping/overwashing effects on barrier height and width. Evolution of the shoreface profile shape is completely neglected. The process for calculating morphological evolution due to overtopping/overwashing is:
- Runup height above still water level is calculated using the equation for gravel beaches derived by Poate et al 2016.
- Overwash potential is calculated as sea level plus runup minus barrier crest height (as in Matias et al. 2012).
- If overwash potential is positive (i.e. runup greater than barrier crest height) then cross-shore transport rate is calculated as a function of overwash potential.
- Cross-shore transport is partitioned between overtopping (sediment deposited onto top of barrier) and overwashing (sediment deposited on back of barrier). The partitioning depends on the current width and height of the barrier compared to user specified target height/width.
- Morphology is updated by taking sediment off the foreshore and moving it to the barrier top/back.
Full details of the cross-shore transport model are given here.
HāpuaMod converts the shoreline, outlet channel and lagoon geometry from real world Eastings and Northings into a model coordinate system where:
- the X-axis is a straight line fitted to the initial shoreline position;
- the origin is the point on this line closest to where the river enters the lagoon;
- the Y-axis is directed positive offshore.
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| Example map view of model initial conditions showing location and orientation of the model co-ordinate system. |
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| The same lagoon after conversion into the model coordinate system. |
HāpuaMod tracks the position of the shoreline, outlet channel and lagoon using a set of shore normal transects. The transects extend in the model Y-direction and are evenly spaced in the model X-direction. The HāpuaMod transect schematisation has similarities to the barrier retreat model of Lorenzo-Trueba and Ashton (2014). At each transect the model tracks the:
- Y-coordinate of the shoreline position.
- Y-coordinates of the seaward and lagoonward edges of the outlet channel (null if there is no outlet channel at a given transect).
- Y-coordinate of the seaward edge of the lagoon (i.e. the barrier backshore).
- Y-coordinate of the backshore cliff toe.
- Elevation of the barrier crest (either side of the outlet channel if present).
- Elevation of the outlet channel bed (if outlet channel present at transect).
- Elevation of the lagoon bed.
- Water level and velocity in the online parts of the lagoon and outlet channel The backshore cliff top elevation and closure depth for shoreline modelling are assumed constant in HāpuaMod.
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| Schematised transect through the hāpua barrier and lagoon (outlet channel banks/outlet bed level/inner barrier height, and lagoon edge/lagoon bed level are only present for transects intersected by outlet channel or lagoon respectively) |
Aspects of the outlet channel and river which do not fit neatly onto the transect system and are tracked separately are the:
- X-coordinate of the outlet channel centreline at its upstream and downstream ends (to enable continuous tracking of their position as they migrate).
- Width and bed elevation of the ends of the outlet channel.
- Bed elevation of each river cross-section upstream of the lagoon.
The fluvial hydrodynamics change rapidly with tidal changes in sea level and require a short timestep (seconds). Morphological changes in the model (e.g. changes in bed/barrier level, shoreline/channel position etc) happen much more slowly (minutes to hours). To reduce simulation time, whilst allowing the hydrodynamics to be solved with a sufficiently short timestep, HāpuaMod decouples the hydrodynamic and morphological timesteps, with morphological timesteps typically one to two orders of magnitude greater than hydrodynamic timesteps.
To further reduce simulation time HāpuaMod uses an adaptive morphological timestep to accelerate periods where little morphological change occurs. The morphological timestep is adjusted, within user specified limits, to maximise the timestep whilst keeping the rate of adjustment of elevation and position everywhere in the model below a user specified threshold.
Further explanation of how the adaptive timestep is calculated is given here.
Bagnold R.A. (1980) An empirical correlation of bedload transport rates in flumes and natural rivers. Proc R Soc Lond A Math Phys Sci 372(October):453–473. http://www.jstor.org/stable/2397042
Coastal Engineering Research Center (1984) Shore protection manual. US Army Corps of Engineers, Vicksburg, Mississippi. https://doi.org/10.5962/bhl.title.47830
Komar P.D. (1971) The Mechanics of Sand Transport on Beaches. J Geophys Res 76(3):713–721. https://doi.org/10.1029/JC076i003p00713
Lorenzo-Trueba J., Ashton A.D. (2014) Rollover, drowning, and discontinuous retreat: Distinct modes of barrier response to sea-level rise arising from a simple morphodynamic model. J Geophys Res Earth Surf 119. https://doi.org/10.1002/2013JF002941
Matias A., Williams J.J., Masselink G., Ferreira Ó. (2012) Overwash threshold for gravel barriers. Coast Eng 63:48–61. http://www.sciencedirect.com/science/article/pii/S0378383911001980
Poate T.G., McCall R.T., Masselink G. (2016) A new parameterisation for runup on gravel beaches. Coast Eng 117:176–190. http://doi.org/10.1016/j.coastaleng.2016.08.003