Model Description
To analyse salt marsh behaviour under increasing rates of sea level rise, a partial differential equations (PDE) model is proposed based on Morris and others (2002) and Kirwan and Murray (2007). This model considers two spatial dimensions, that is, the elevation of the marsh platform (the marsh height) and the distance from the marsh platform to outlet along the water flow direction. The model describes salt marsh height development and encompasses three processes: sedimentation during inundation due to vegetation sediment capture, erosion due to water flow over the salt marsh surface, and sediment dispersal, which is implemented in the model as diffusion. Our model closely follows the model by Kirwan and Murray (2007), with the difference that only one spatial dimension is considered, that is water flow path, allowing for more detailed numerical analysis and yielding analytical tractability.
The model consists of two differential equations that describe the development over time of the salt marsh height (zSM) and the mean high tide (zHT), both in millimetres. The rate of change in mean high tide is assumed to be equal to the rate of sea level rise r in mm year−1, which in turn is assumed to be constant. The rate of change of the marsh platform depends on the depth of the marsh below the mean high tide, as well as the amount of biomass. This can be summarized in the following equations:
$$ \frac{{dz_{HT} }}{dt} = r $$
(1)
$$ \frac{{dz_{SM} }}{dt} = S - E + F $$
(2)
in which S is the sedimentation rate, E is the erosion rate, and F is a diffusion term, all with the unit mm year−1. The full model description is given in online Appendix 1.
To analytically study the model described above, the salt marsh height can be considered at the centre of the marsh platform. At this place, there will be no water flowing over the marsh (the centre of the marsh is analogous to the water divide) which allows us to set the erosion and diffusion terms to zero. With combining Eqs. (1) and (2), we can determine the critical depth Dc and the critical rate of sea level rise rc above which the marsh will go through a rate-induced critical transition and drown. The mathematical analysis can be found in online Appendix 2. Furthermore, analysing the model at the water divide allows for the analytical calculation of the perturbation growth rate of the system, which indicates the stability of the system given a specific rate of sea level rise r. The calculation of the perturbation growth rate can be found in online Appendix 3.
Despite the apparent simplicity of the model, it does show an important characteristic of salt marshes, that is, the marsh will become submerged when exposed to a rate of sea level rise too high for the system to cope with (Morris and others 2002). This is due to a maximum biomass productivity, which corresponds to a maximum sediment capture ability at an optimal rate of sea level rise. When the rate of sea level rise in the model exceeds this optimum, the productivity decreases again, which leads to a lower sediment capture ability of the marsh vegetation, which in turn leads to a slower increase in marsh height with less available sediment. Furthermore, this model forms a cliff edge between the marsh and the submerged area, when made spatially explicit.
Numerical Implementation
When the sedimentation, erosion and diffusion terms are all included, the full model represents a transect through a salt marsh. The model is discretized into one row of n = 600 cells which have a length Δx = 5 m, hereby modelling a water flow path of L = 3000 m. Within each cell, deposition and erosion of sediment are governed as described above; sediment diffusion occurs every time step by redistributing the sediment driven by gravitation, with Δt = 10 years. The rate of sea level rise r in the model is varied to study how the system will respond to rising sea level with a continuous rate. In the one-dimensional salt marsh model, a rate-induced critical transition will occur when sea level rise is higher than the rate of sediment deposition in equilibrium. The model parameterization used in this study is based on Kirwan and Murray (2007) and Kirwan and Guntenspergen (2010). An overview of the values used in the numerical simulations is given in Table 1.
Table 1 Parameter Values Used in This Study. Early Warning Signals
We tested several approaches to examine how proximity to a rate-induced critical transition could be signalled by changes in the salt marsh height. If the system state is far away from this rate-induced critical transition, the system will quickly return to its equilibrium state after a perturbation. However, as the rate of sea level rise approaches its critical value, the perturbation growth rate of the system approaches zero. As a consequence, the system will become increasingly slow in recovering from perturbations in the system (Wissel 1984; Scheffer and others 2009; Ritchie and Sieber 2016). This can be demonstrated by analytically calculating the return time from perturbations, the temporal autocorrelation and the temporal variance when noise is applied to the salt marsh height state variable (Siteur and others 2016). These early warning signals are related to the perturbation growth rate λ via the following formulae:
$$ t_{r} = - \frac{1}{\lambda } $$
(3a)
$$ ac1 = e^{{{\uplambda }\Delta t}} $$
(3b)
$$ var = \frac{{\sigma^{2} }}{{1 - ac1^{2} }} $$
(3c)
in which tr is the return time, ac1 is the lag-1 autocorrelation, var is the variance, Δt is the time interval with which noise is applied on the state variables, and σ is the standard deviation of the applied noise.
If the perturbation growth rate of a system cannot be determined analytically, the autocorrelation and the variance can still be calculated numerically, as is done for the numerical model simulations. We calculate the numerical autocorrelation AC1 with the following formula:
$$ AC1 = \frac{{\mathop \sum \nolimits_{i = 1}^{N} \left( {z_{SM} \left( t \right) - \overline{{z_{SM} }} } \right)\left( {z_{SM} \left( {t + \Delta t} \right) - \overline{{z_{SM} }} } \right)}}{{\sqrt {\mathop \sum \nolimits_{i = 1}^{N} \left( {z_{SM} \left( t \right) - \overline{{z_{SM} }} } \right)^{2} \mathop \sum \nolimits_{i = 1}^{N} \left( {z_{SM} \left( {t + \Delta t} \right) - \overline{{z_{SM} }} } \right)^{2} } }} $$
(4)
in which N is the number of time steps Δt during which noise is applied to the state variable and \(\stackrel{-}{{z}_{SM}}\) is the average salt marsh height over time. In the same manner, the numerical variance VAR is calculated using the formula:
$$ VAR = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \left| {z_{SM} \left( t \right) - \overline{{z_{SM} }} } \right|^{2} $$
(5)
Three additional early warning signals were identified which are specific for tidal marsh ecosystems (based on the results of Kirwan and Guntenspergen 2010), that is, the marsh vs the water channel depth with rising sea level, the creek cliff steepness, and the ratio of total marsh area to creek area. The depth D of the marsh and channel is calculated as the difference between the salt marsh height (zSM) and the mean high tide (zHT). The creek cliff steepness is calculated by taking the highest value of the slope of the salt marsh height zSM with respect to the distance x (that is, ΔzSM/Δx, analogous to the derivative of the marsh height zSM over the distance x). The creek area is calculated by calculating the percentage of the salt marsh which has a depth D lower than the critical depth Dc through the marsh transect; this is then compared to the total marsh area (the modelled transect through the marsh x is taken representative of marsh and creek area).