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Model instability and channel connectivity for 2D coastal marsh simulations

Abstract

Reduced freshwater inflow into a coastal marsh can result in environmental stress through episodic hypersalinity. Hydrodynamic models can be used to evaluate salinity-control strategies when freshwater supplies are constrained by climate or increasing urban demands. However, there remain significant scientific, engineering, and technical barriers to correctly modeling salinity transport in such systems. In particular, the numerical instability at the wetting/drying front caused by strong wind stress and steep surface gradient and the inappropriate representation of the complex channels at practical computational scales are unsolved problems. This study documents recent achievements in modeling the time–space evolution of shallow marsh salinity using the Fine Resolution Environmental Hydrodynamic model (Frehd) applied to the Nueces River Delta (Texas, USA). The 2D depth-integrated model is tested across a variety of bathymetric representations derived from high-resolution lidar data to evaluate the effects of grid refinement and a variety of bathymetry processing methods. Novel treatments are proposed and tested to suppress unrealistic velocities and scalar concentrations caused by rapid wetting/drying and strong wind stress. The model results are compared with the field data collected at 12 spatially-distributed locations across the marsh, yielding good model-data agreements for free surface elevation and reasonable agreements for salinity. Analyses of results indicate that the critical difficulty for capturing salinity transport is in correctly representing connectivity effects (both blocking and channel features) at fine scales on the coarse grid without overestimating fluxes. Modeled water surface elevations are relatively robust to poor representation of connectivity whereas the salinity distribution is strongly affected, particularly at key choke points. This study defines a set of future challenges in developing automated methods for evaluating and preserving geometric connectivity at practical model grid resolution.

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Notes

  1. 1.

    https://cbiweb.tamucc.edu/TCOON/.

  2. 2.

    http://www.nueces-ra.org/CP/CITY/rincon/.

  3. 3.

    https://www.mathworks.com/matlabcentral/fileexchange/29025-ordinary-kriging.

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Acknowledgements

This work has been supported by the Texas Water Development Board under interagency cooperation contracts 1400011719 and 1600011928. Field work was conducted by the TWDB with funding from the US Army Corps of Engineers.

Author information

Correspondence to Z. Li.

Appendices

Appendix A. Model spin up

The “spin-up” time for a model is the simulation time that it takes for the results to be independent of the initial conditions. Spin-up for a fresh/salt marsh simulation is inherently challenging. Unless the sampling period includes a complete flush of the system, we cannot start from a “clean slate” of zero salinity and expect to reach the actual salinity distribution by some date. In contrast, the hydraulic memory of the velocity and elevation fields is relatively short and can be approximated by the time it takes for an increase in tidal elevation to be seen throughout the domain, which allows uniform conditions to be readily applied as a starting guess. To examine the spin-up behavior of the NDHM, we conducted test simulations starting two weeks apart (from Nov. 15 and Dec. 1, 2012 respectively) to evaluate when the model results are independent of the initial starting date. The start date was chosen based on availability of field data and to ensure sufficient spin-up time prior to the pump operations in summer. Note that during the winter, the secular water elevations in the Nueces and Corpus Christi Bay systems are declining towards a semi-annual low in January [50], and it can be expected that spin-up times during other stages of the secular cycle might be different. For the spin-up simulations, the NDHM was started from quiescent water (\(u_j = 0, j \in \left\{ 1,2 \right\}\)) with a uniform free-surface elevation equal to the tidal boundary condition. The initial condition for salinity was computed using the ordinary kriging method,Footnote 3 in which the salinity field was interpolated based on measurements from the 12 stations at the beginning of the model start date.

We consider an adequate spin-up time as the interval when the residual (difference between 2 simulation results) is less than 2 psu for salinity and 0.002 m for free surface. Using this metric, the spin-up times as well as a final salinity residual after 60 days from Dec. 1, 2012 are listed in Table 5. As expected, the spin-up times for free surface elevation are significantly shorter than that for salinity. The poorest result was at Nueces4 in the upstream tidal flat known as West Lake (Fig. 1) that is poorly flushed during the secular low tidal period in winter. In contrast, the locations in the Rincon Bayou main channel and marsh areas close to the open boundary (e.g. Nueces1, 5, 6, 7, 9, 10 and 11) see more consistent flushing through the winter and hence the spin-up times are much shorter. As a conservative measure, all data analyses herein is based on model results after 60 days of spin-up time.

Table 5 Spin-up time for free surface elevation, salinity and 60-day residual for salinity at each TWDB station

Appendix B. The irrelevance of global drag calibration

Model calibration for a 2D shallow-water model is generally accomplished by adjusting either coefficients of a turbulence model (e.g. \(\nu\)) or drag (\(C_D\)) that control energy dissipation [34, 53]. This is typically conducted using global values: the calibration problem cannot be reasonably constrained if every model grid cell has an independent \(C_D\) and field data is available at limited locations. Our analyses (not shown) indicate the NDHM results are relatively insensitive to the choice of a global \(\nu\) or \(C_D\). The minor variability of results obtained in our calibration exercise does not allow rejection of the hypothesis that our a priori selected baseline values are acceptable. Similar conclusions have been reached by other researchers for simulations with complex bathymetries [9, 35].

Arguably, there are two principal reasons for the insensitivity of the model to calibration: (i) the numerical dissipation associated with our 1st-order upwind advection scheme [14, 15], and (ii) the “topographic” dissipation associated with the convoluted channels in a marsh. We have not seen this latter topic specifically addressed in the literature, but it follows from simple consideration of how momentum turns a corner with the hydrostatic approximation. In the real world, the pressure gradients at a channel bend serve to redirect momentum, i.e. the \(dp/dx_1\) required to slow momentum in the \(x_1\) direction increases the pressure p on the outer edge of the channel, and results in \(dp/dx_2\) that accelerates the flow in the \(x_2\) direction around the bend. Thus, streamwise momentum is not lost around a corner, but is smoothly transferred from \(x_1\) to \(x_2\) coordinate directions through the pressure gradients and nonlinear terms. Indeed, the 1D Saint-Venant equations for channel flow are essentially the mathematical embodiment of this idea [21]. However, when a narrow channel bend is represented by a single set of grid cells in a 2D hydrostatic model, only a small part of the momentum change in the \(x_1\) direction will be recovered in the hydrostatic pressure and returned to the \(x_2\) direction. The fundamental problem is that insufficient grid resolution at the channel scale creates an inability to have smooth transition of pressure gradients and nonlinear terms between coordinate directions. Thus, every bend in a narrow channel causes the flow to stop its streamwise acceleration in the \(x_1\) direction and restart the streamwise acceleration in the \(x_2\) direction. If the marsh system were strongly nonlinear with high channel velocities, then increased grid resolution would be necessary for a reasonable approximation of the fluxes. However, velocities in the marsh channels are slow and only weakly nonlinear, so relatively coarse grid resolution of the channels is an acceptable trade-off for computational efficiency. The main consequence is that the topographic dissipation from channel bends plays a major role in dissipation of energy, which makes \(C_D\), \(\nu\), and drag calibration nearly irrelevant.

It can be argued that a depth-dependent drag model (e.g. Chezy–Manning \(C_D = g{\hat{n}}^2 h^{-1/3}\), where \({\hat{n}}\) is Manning’s n) would be an appropriate baseline model. However, our calibration exercises showed that field data could not adequately discriminate between competing models. Thus, we appeal to Occam’s Razor and use the simplest possible drag model. This model is a baseline uniform \(C_D\) that is only depth-dependent in thin layers (as discussed in Sect. 2.5) where the depth dependency has a clear impact. Nevertheless, we do not consider this the final answer. Results with the automated channel bathymetry treatment (discussed below) indicate that some form of spatially-distributed drag might be useful, although it is not clear that simple depth-dependency such as Chezy–Manning is necessarily the solution. We speculate that a local drag coefficient might be linked to the approximations made in the channel connectivity algorithm and calibrated with some form of global coefficient. This idea remains an area of ongoing research.

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Li, Z., Hodges, B.R. Model instability and channel connectivity for 2D coastal marsh simulations. Environ Fluid Mech 19, 1309–1338 (2019). https://doi.org/10.1007/s10652-018-9623-7

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Keywords

  • Shallow coastal marsh
  • Numerical modeling
  • Bathymetric error
  • Salinity transport