## Abstract

The paper summarizes the theoretical and practical needs for cellular automata (CA)-type models in coastal simulation, and describes early steps in the development of a CA-based model for estuarine sedimentation. It describes the key approaches and formulae used for tidal, wave and sediment processes in a prototype integrated cellular model for coastal simulation designed to simulate estuary sedimentary responses during the tidal cycle in the short-term and climate driven changes in sea-level in the long-term. Results of simple model testing for both one-dimensional and two-dimensional models, and a preliminary parameterization for the Blackwater Estuary, UK, are shown. These reveal a good degree of success in using a CA-type model for water and sediment transport as a function of water level and wave height, but tidal current vectors are not effectively simulated in the approach used. The research confirms that a CA-type model for the estuarine sediment system is feasible, with a real prospect for coupling to existing catchment and nearshore beach/cliff models to produce integrated coastal simulators of sediment response to climate, sea-level change and human actions.

## 1. Introduction and aims

A major challenge for environmental science is to develop dynamic models that can simulate future environmental responses to the combined effect of human activities and environmental change. Through the simulation of scenarios, dynamic models should be capable of providing the foundation for judging alternative policy options. For most types of environmental systems, however, the challenge is far from trivial. Complex environmental systems, such as those represented by coastal zones (Leont'yev 2003), are characterized by interacting networks of processes and conditions operating through feedback mechanisms that may give rise to nonlinear dynamical behaviour (e.g. Amsterdam Declaration on Global Change 2001). The success of nonlinear systems dynamics research over the past two decades has provided us with a range of mathematical tools and concepts to study complex environmental systems (e.g. Phillips 1998; Scheffer *et al*. 2001), but has also reinforced the need to develop simulation models that emulate or capture realistically the nonlinear nature of complex natural environments. Without the capacity of coastal simulation tools to model nonlinear responses there is the serious probability that coastal management policies will fail because they underestimate, ignore or are unable to account for the possibility of unpredictable events and dramatic shifts in trends. In human terms, these failings may translate to inadequate or poor investments, costly remediation, new environmental hazards and loss of life.

The need to develop new systems-based frameworks for coastal systems that address long-term interactions between social and biophysical systems is widely recognized. In the UK, the establishment in 2001 of a Regional Coastal Simulator as a Core Project of the UK's Tyndall Centre has been followed by a series of reports to the Office of Science and Technology (Watkinson 2003) and Department for Environment, Food and Rural Affairs (Conlan *et al*. 2002; Walkden 2005): all driven by the national awareness that sections of the UK coastline are at great threat from erosion and flooding as a result of sea-level change and global warming. A series of UK reports by DEFRA (2002*a*,*b*, 2005) point to the mismatch between the demand for simulation tools generated by the different management plans for the coastal zone, shoreline and estuary and the status of current predictive models. English Nature (2004), too, emphasizes the need ‘to complete the cultural shift from coastal defence to coastal management’, drawing attention to the demand for decision-support tools that can help to optimize Integrated Coastal Zone Management and implement the EU Water Framework Directive. In the US, the Centre for Sponsored Coastal Ocean Research through the Coastal Ocean Program is developing a three-stage approach to modelling the accumulated impacts of sea-level change and human activities on estuarine ecosystems and coastal regions. These programmes emphasize the need for improving ‘the capability to predict how coastal ecosystems will change under certain scenarios [so that] management can become proactive in facing the problems confronting the coastal ocean’. Therefore, a major global consideration in the design of decision-support tools is how to capture the evolution of coastlines and estuaries within complex socio-environmental systems, driven by external forcings, such as sea-level and climate change, over decadal and centennial time-scales.

This paper has two main aims. First, it seeks to make a robust case for the development of decision-support tools in coastal systems based on cellular automata (CA) as an alternative to conventional dynamic modelling approaches. Second, it describes preliminary steps in the development of a cellular model for simulating estuarine sediment dynamics with the focus on the Blackwater Estuary in southeast England.

## 2. The case for cellular automata

Despite increased computer capacity, conventional hydrodynamical modelling has not succeeded in predicting coastal morphological response beyond short time-scales represented by days and months, particularly in tidal basins (Stive & Wang 2003; UK DEFRA-funded EstProc programme). The fundamental nonlinear nature of coastal morphodynamics with complex mutual co-adjustment of topography and fluid dynamics involving sediment transport is well known (Wright & Thom 1977; Leont'yev 2003). Coastal evolution should be regarded as a cumulative process in a system where input data from the preceding (or initial) condition influences the nature and outcome of the next time-step. If we take our evolutionary view from nonlinear dynamical theory, it could be argued that we might be setting our predictive sights too high: evolving complex systems are essentially ‘unpredictable’. But, paradoxically, the inherent ‘unpredictability’ should not negate learning about the likely behaviour of the system under different future scenarios, not least for long-term decision-making where the generalities of system behaviour may be far more important than the details. Realistic simulation models that can capture the spatial and temporal dimensions and dynamics of nonlinear behaviour should be able to inform us about how external forcings may give rise to threshold-dependent change (the ‘big surprises’), the likely time-scales over which change may evolve or, indeed, which of the alternative actions under our control should be avoided or selected: essentially trading quantitative accuracy and precision for realism. A key feature of complex systems is the *emergence* of phenomena at a macroscale through the iterations of relatively simple interactive processes at a microscale. The nature, timing and time-scales of interactions between the microscale components and its changing external environment are all significant in determining the trajectory of a system, and any simulation model that purports to capture nonlinear behaviour has to emulate these interactions from the bottom upwards (e.g. Dearing in press). If there is one single reason why hydrodynamic models often fail to produce stable outputs over time-scales longer than days and months, it may be that the structure of the current generation of models does not explicitly allow for the evolution of emergent features; especially, in the topography across which they operate.

CA models (including a number of variants) appear to satisfy many bottom-up requirements because they simulate interactions between processes through fundamental rules (Wolfram 2002), and importantly allow the emergence of spatially defined macroscale phenomena. CAs were originally created as toy models to simulate the complexity of hypothetical systems, but have graduated to many applications in the social (e.g. Benenson & Torrens 2004) and physical (e.g. Favis-Mortlock 2004) sciences. At their basis lies a spatially explicit landscape defined as a series of contiguous cells. Each cell has a number of rules that determine how neighbouring cells will change. At each time-step, the state and conditions of each cell are updated to provide new states and conditions for the rules to operate on. Through continuous interaction, the rules generate emergent patterns and features, capturing along the way the feedbacks, time-lags and -leads that prove intractable to alternative methods. In biophysical systems, Tucker & Slingerland (1997) and Coulthard *et al*. (2002) have pioneered the application of mathematical CA modelling approaches in catchment hydrology. The basic cell in Coulthard's cellular automaton slope and river model (CAESAR) is a cube with edges ranging from 1 to 50 m, sub-divided to represent the land surface and sub-surface horizons. Each divided section of cube has mathematical algorithms to characterize hydrology, hydraulics and sediment transport processes. The interactions between cubes for any defined catchment are driven by regional rainfall, temperature and land use records (or their reconstructed equivalents), acting as inputs to the algorithms at each time-step. Environmental changes are expressed as sequential maps or as time-series of outputs from the whole catchment, and validation is achieved through comparing evolved system behaviour with long-term instrument or reconstructed time-series (e.g. Coulthard *et al*. 2002). The limited number of CA coastal models has resulted in the successful generation of emergent or self-organized phenomena, such as estuarine channel changes and shoal patterns (Hibma *et al*. 2003) and beach cusps (Coco *et al*. 2000, 2001). A mixed modelling approach that includes a CA component for tidal channels and peat formation is used in SimDelta, a spatially interactive but non-process based model of the geographical and geomorphological development of the Dutch Delta from the year 6500 BC to AD 2000 (SimDelta 2003). B. Straatman, A. Hagen, C. Power, G. Engelen & R. White (2001, unpublished report) describe three small-scale modelling experiments of morphological processes based on the interaction of sediment and water particles through CA, but currently there is no fully operational and validated process-based CA or CA-type model of estuarine sediment dynamics.

## 3. The development of a CA-type model for estuarine sediment dynamics

Here, a number of developmental stages in a project designed to produce a cellular model for coastal simulation (CEMCOS) are described. The intention was to initiate the development of a fully integrated cellular process model that could ultimately be used in all wave- and tide-dominated environments to simulate the future combined effects of climate, sea-level, land use and coastal protection measures on inland tidal, shore and nearshore environments. A key requirement was to develop a model that could capture the medium long-term (10^{1}–10^{2} years) nonlinear changes and responses inherent in such a complex dynamic system, such as coastline morphology and estuarine channels, as well as provide acceptable accuracy in its simulation of key variables, such as mudflat erosion rates and coastal flooding, over shorter time-scales (10^{−1}–10^{1} years). Ultimately, the test of simulation skill will be based on the model's ability to capture mapped and reconstructed system behaviour from the past (*palaeo-validation*). Model development and testing has been focused primarily on the Blackwater Estuary (figure 1; table 1) in southeast England (part of the Tyndall Centre's Regional Coastal Simulator), but also on the Dee Estuary and Liverpool Bay in northwest England where the modelling approach could take advantage of previous research.

### (a) CEMCOS model structure

The basic structure of CEMCOS relates sediment entrainment and deposition to hydrodynamic processes (tides and waves) and volumetric sediment transport vectors as a function of tidal elevation relative to bathymetry, in order to obtain the resulting ‘sedimentary outcome’ following each tidal cycle. The main priority is to achieve direct cell interactions (*strictly CA*) for sedimentary responses at fine spatio-temporal scales. Tidal and wave functions have been developed that can be applied to each cell to determine the current due to the tide and the significant wave height in each cell. The tide model calculates the current in the cell using specified tidal amplitude, determined from local tide-gauge data at Sheerness, Essex, and the wave model uses input wave heights to determine the significant wave height in each cell. At this stage in development of the CEMCOS model, neither the tide nor wave model is CA, as at each time-step, the new current and wave height in a given cell is dependent only on global inputs (tide-gauge input, boundary wave heights, wind speed and direction) and cell water depth. This is analogous to current geomorphical CA models that apply uniform climate forcings to all cells at each update (e.g. Coulthard *et al*. 2002). However, water depth and sediment properties interact with the tide and wave parameterization to determine bed and suspended sediment response to local shear stress as a function of tidal elevation and wave height. The rate and direction of both water and sediment transport are then determined by the differential bed elevation (estuary morphology) and the tidal current vector. In this way, the tide model component can capture direct and continuous feedback resulting from water depth changes associated with the erosion and deposition of sediment.

The cell-based transitional rules developed and used in CEMCOS address relationships between water depth, flow velocity, shear stress (tide- and wave-induced) and sediment entrainment, transport and deposition. Field studies indicate that suspended sediment profiles are highly variable (Stoddart *et al*. 1987), thus CEMCOS estimates the vertical suspended sediment distribution by subdividing each cell into five vertical sub-cells. It is assumed that 40% of the suspended sediment remains close to the bed, as a function of turbulence, increased frictional drag in proximity to the bed and the gravitational settling of particles, with the remainder distributed linearly through the water column and no suspended sediment in the uppermost vertical sub-cell. This cell structure was considered to best represent empirical data on the vertical distribution of suspended sediment (e.g. Severn Estuary (Kirby 1986)). In each cell, bed shear stress due to the tide and waves may be calculated from established relationships (e.g. Whitehouse *et al*. 2000). Erosion and deposition thresholds are determined by the interactions between flow velocity and properties of the sediment/bed; for example, critical shear stress for non-cohesive particles, with the potential addition of consolidation, bed roughness, the binding effects of algae and ‘bioarmouring’ effects due to vegetation (increased roughness length, wave attenuation, litter layer, root networks). At present, CEMCOS models the sedimentary outcome of tide-induced shear stresses and vectors for a single non-cohesive grain size (i.e. medium silt of 16 μm diameter). This grain size was selected from published literature as being representative of a ‘typical’ grain size present in the Blackwater Estuary (Emmerson *et al*. 1997). The following sections document the development of the tide, wave and sediment model components, with examples of tests, before demonstrating the functioning of this CA-type sediment model in the Blackwater Estuary.

### (b) Tide component

For estuaries that are long and narrow with tidal height *Z*(*x*, *t*)=*Z*^{∧} cos(*kx*−*ωt*), the gradient of the sea-surface can be reasonably approximated by a transversally constant axial slope of(3.1)where *Z*^{∧} is the tidal amplitude and *k* represents the wave number (2*π*/*cP*), *ω* is the tidal wave frequency (2*π*/*P*), *c* the celerity ((*gD*)^{1/2}) and *P* is the tidal wave period. This ‘synchronous’ solution assumes that the slope associated with tidal phase variation greatly exceeds that associated with axial variation in tidal elevation amplitude. This assumption is valid in most estuaries with depths less than about 15 m. For the main channel, setting the depth, *D*, as the average across the section, localized values of tidal current amplitude can be calculated from an algebraic solution to the following (Prandle 2003):(3.2)The equivalent equation for transverse flow is approximated by(3.3)with the algebraic solution , where *Ω* is the Coriolis parameter, *f* the bed friction coefficient (*ca* 0.0025) and *R*=(*U*^{2}+*V*^{2})^{1/2}. For more localized simulations of intertidal zones, from conservation of mass close to the high water mark, the transverse velocity must supply the rate of change of surface height, i.e.(3.4)where *S* is the slope of the inter-tidal section. To select between the applicability of these two formulations for *V*, equation (3.4) is used when it indicates an (absolute) value of *V* greater than the velocity *V* indicated in equation (3.3).

### (c) Wave component

A parameterized (cell-based) wave model has been developed using the Simulating WAves Nearshore (SWAN) model (Ris *et al*. 1999). To enable model development for a range of coastal settings from open coast to enclosed estuary, initial tests were carried out for the Liverpool Bay and River Dee Estuary area (Wolf 2003) for three different coastal zones.

*Scenario 1. Open coast*. Two approaches to modelling significant wave height were tested for open coasts (the Tucker and Kitaigorodskii methods). It was found that the output was similar in both cases and, as it is simpler to calculate, Tucker's scaling (Tucker 1994) was selected to model the open coast environment in the parameterized wave model. The relationships used by CEMCOS for the open coast are:(3.5)where *H*_{sb} and *k*_{pb} are the wave height and wave number at the boundary, respectively. In order to calculate the peak wave number, the following approximation has been used:(3.6)where(3.7)

*Scenario 2. Estuary*. The formulae (equations (3.8) and (3.9)) for fetch-limited growth in shallow water, taken from the Shore Protection Manual (USACE 1984), were tested for estuary conditions(3.8)(3.9)where, , , , , *f* is in m and *U*_{10} is the wind speed in m s^{−1} at 10 m above the sea-surface. Two tests were carried out for these relationships, and it was found for the Dee Estuary that results were similar if a realistic fetch or constant fetch was used. This simplifies the parameterization as fetch need not be calculated for different wind directions. However, more detailed modelling of the Blackwater Estuary (Wolf 2004), which is more enclosed and narrower than the Dee, gave better results with an effective fetch (see below).

*Scenario 3. Nearshore*. Due to the complex processes occurring in the nearshore zone, the open coast model was found to break down for water depths below 10 m. A formula (equation (3.10)) was developed for this area (as a correction to be applied after using Tucker's scaling) to model significant wave height in the nearshore:(3.10)A comparison of output from SWAN and the parameterized wave model shows reasonable agreement (see figure 2). However, despite the use of a correction (equation (3.10)), the largest discrepancy between the parameterized model and SWAN model is still for the nearshore zone. This is mainly the result of the complex refraction of the waves by large depth gradients occurring in that area, where the full detail of SWAN is required to reliably model the wave height. The parameterized model does not account for wave propagation, but it assumes a local balance of generation and dissipation.

Results obtained for the Blackwater Estuary (see figure 3) indicate that the parameterized model (using scenario 2) is able to predict wave height and period in reasonable agreement with SWAN. Here, it was found that an effective fetch assumption (equation (3.11)) gives better results than either a constant or realistic local fetch. Constant fetch assumes a fixed fetch value independent of real world conditions; realistic local fetch is the correct fetch calculated for the actual point location within the model, for each wind direction; while effective fetch makes allowance for the fact that in a narrow estuary or bay, the waves may grow preferentially in an off-wind direction if this has a longer fetch than the actual wind direction. Thus,(3.11)where *F*_{W} is the fetch in the wind direction and *F*_{W+45} and *F*_{W−45} are the fetch in the +/−45 degree directions respectively.

### (d) Sediment component

There are several components to the sediment model, including erosion, deposition and transport. Unless stated otherwise, the method has been adapted from Whitehouse *et al*. (2000). This section focuses on the method used to relate sediment response to tidal currents. Erosion by waves could be important, especially, at the mouth of the estuary, but this was not included in the prototype model.

*Erosion*. The first step in calculating erosion in each cell at each time-step is the calculation of the friction velocity that requires the iterative solution of equation (3.12),(3.12)where *u*_{*} is the friction or shear velocity in m s^{−1}.

It is noted that the method used in this approach to calculate the friction velocity is for a hydrodynamically smooth flow. The bed shear stress (*τ*_{o}) and erosion shear stress (*τ*_{e}) due to currents can then be calculated using established relationships (Whitehouse *et al*. 2000),(3.13)

(3.14)The constants *E*1 and *E*2 in equation (3.14) are site-specific dimensional coefficients and default values from Whitehouse *et al*. (2000) are currently used (*E*1=0.0012, *E*2=1.2). A value of 70 kg m^{−3} is used for the dry density of mud (*C*_{M}). The rate of erosion (d*m*/d*t*) is calculated using equation (3.15). The cell erosion constant (*m*_{e}) is currently constant in CEMCOS. However, it is anticipated that varying the value of this parameter will be used to model different environments, e.g. modifying the constant so that it is more difficult to erode sediment (when vegetation is present) or making it easier to erode sediment (for recently deposited beds where the sediment is less well consolidated),(3.15)The mass of sediment eroded in each cell during the time-step is then calculated (equation (3.16)) by(3.16)where *A*_{cell} is the cell size in m, which can have any user-defined value. Changes in cell elevation due to erosion are calculated as:(3.17)

*Deposition*. Deposition during each time-step for each cell is calculated as:(3.18)where SS_{bed} is the amount of suspended sediment near the bed per unit area, SS_{subcell1} is the amount of suspended sediment in a vertical sub-cell lying closest to the bed, *A*_{cell} is the area of the cell, d*m*_{d}/d*t* is the rate of deposition, *τ*_{o} is the bed shear stress due to the current, *τ*_{d} is the critical bed shear stress for deposition (default value of 0.08 N m^{−2}), *v* is the median settling velocity (default of 1 mm s^{−1}) and *M*_{dep} is the mass deposited per time-step, *t*. Change in cell elevation due to deposition is calculated using a relationship similar to equation (3.17).

*Sediment transport*. Sediment transport uses a CA approach, with a Moore neighbourhood, where each cell can interact with each of its eight neighbours, but only those neighbours. As a result, transport can occur in any direction, but only to nearest neighbour cells; there is no long distance transport during a single time-step. For each cell, the percent change in water depth from *t*−1 to *t* is determined. From the resulting volumetric changes as a function of cell water depth at each iteration, the corresponding amount of total suspended sediment is then transported out of the cell. For an outgoing tide, the sediment can be transported into any of the three neighbouring cells in the direction of the open coast. If all of the neighbouring cells are submerged, equal quantities of sediment are transported into each cell. However, if one or more of the possible destination cells are dry, CEMCOS transports the sediment into only those cells that are submerged. In the current model, the propagation of the tide is instantaneous and uni-directional across the area, but in future versions there will be a propagation direction to allow apportioning of sediment to neighbouring cells. With the present set-up, the prototype model is still able to transport sediment down narrowing or widening channels and around islands, etc. Sediment transport into an estuary works in a similar way, but with all the directions reversed over ebb and flood.

## 4. Prototype CEMCOS models

All three model components described above can be integrated into one-dimensional or two-dimensional dynamic models, according to a standard sequence (see figure 4). For each time-step, the program loops over all cells in the model area, calculating for each cell the tidal current, the wave height and then the associated sediment response. The following paragraphs illustrate the first steps undertaken in the development of experimental one-dimensional and two-dimensional models, and the application of a two-dimensional model driven by tidal currents only to the Blackwater Estuary.

*One-dimensional model*. A simple one-dimensional model was developed to explore model skill in handling the following:

erosion/deposition under variable currents;

erosion/deposition under waves;

sediment transport by variable currents;

distribution of sediment through the water column;

conservation of sediment;

change in bed elevation due to erosion and deposition;

variable cell size.

Figure 5 shows three time slices (0, 2 and 4 h) in a continuous sequence of tide-driven suspended sediment run at 15 min intervals. The upper sections show the amount of suspended sediment in each cell, the lower sections show the changing water depth driven by tide for a simple slope geometry. When run continuously, it is notable that even in a simple one-dimensional model, the relationship between suspended sediment and water depth is highly nonlinear. In multiple runs, the distribution of sediment obtained is strongly dependent on the time-step used: the sediment distribution is similar in each case, but shorter time-steps capture significantly more details (and minimize the restriction on exchanges confined to neighbouring cells over one time-step).

*Two-dimensional model*. The one-dimensional model was adapted to run within a two-dimensional grid of cells on which simulation could include realistic estuary morphologies. Components of the final two-dimensional model include:

detailed tidal model calculating the current using local tide-gauge data;

fully developed model of sediment response to currents, including:

erosion;

deposition;

sediment distribution through the water column;

CA sediment transport model for variable geometries, including channels, variable slope, islands for incoming, outgoing and full tidal cycles;

variable cell size;

changing cell water depth due to erosion and deposition.

Figure 6 shows two-dimensional model output for one of the more complex cases, with a channel of variable width, constant slope and an island. The run was initialized with a large concentration of sediment in one cell at the edge of the model area and active boundary conditions adding sediment to the system at each time-step. The distribution of suspended sediment is shown for different time-steps. Note that no sediment is transported into cells that are dry, while sediment is successfully transported along the non-uniform channel, and sediment is transported around the island. The example shown is for a tide entering the estuary; equivalent results are obtained for transport out of an estuary.

*Blackwater two-dimensional model*. Following simple generic testing, it is straightforward to convert the prototype two-dimensional model to run on the true morphology (estuary shape, intertidal slope, etc.) of the Blackwater Estuary, using digitized data from the 2003 edition Admiralty Chart (No. 3741) at a 500 m grid cell resolution. The Blackwater two-dimensional model differs from the two-dimensional prototype by permitting an increased number of cells and reading file input for the initial bathymetry. A first step was to establish realistic tidal movements through the estuary. Example tidal output for the Blackwater Estuary (see figure 7) shows high and low tide in the estuary for 1 January 1990. The time resolution was 1 h steps, and although the data are presented in plan view (500 m×500 m cells), they are fully three-dimensional in form. Note that at high tide, both Northey Island and Osea Island are resolved at this spatial scale. This highlights the importance and relevance of tests, ensuring that the sediment transport method can successful handle complex geometries.

## 5. Further developments

The current state of CEMCOS shows promise, but further key developments are required before it can be used to simulate future scenarios. First, an alternative tidal vector parameterization approach has been identified as an essential future development, where a look-up table is used, which lists either the near bed current or bed shear stress according to water depth and change in tide amplitude. Values in this table are determined from other model results (as with the SWAN-based wave parameterization) or estimated based on field/experimental data. We will also consider using local rules in relation to a propagating tidal wave to model tide-generated shear stress and direction. This is important in the modelling of tidal currents around islands and emergent shoals, and in channel creek networks during the flood tide. Once achieved, the ‘sediment component’ of CEMCOS may be upgraded to include combined shear stress due to tides and waves. Second, further development will include cohesive sediment and changing sediment consolidation, as well as the bioarmouring effects of algae on mudflats and vegetation on saltmarshes. Elevation within the tidal frame will determine threshold altitudes where bioarmouring is switched on, for example where saltmarsh environments emerge above mean high water, and the time elapsed since deposition of sediment will determine consolidation. Cohesion, consolidation and bioarmouring will be parameterized using a cell erosion constant with values selected or calculated from available literature. Third, validation of the model will be undertaken through comparison of the simulated outputs with the observed sedimentary response between the Admiralty Chart editions of 1847, 1918/9 and 2003. The skill of the modelled sedimentary outcome will be assessed in terms of trends and periods of vertical net accretion or loss. Simulated changes in elevation will be compared with observed changes in tidal datums (from the Sheerness/Southend tide-gauge), and mapped changes in the form and location of the main low water channel, saltmarsh, intertidal mudflats, sandbanks and beach profiles.

## 6. Discussion and conclusions

We have shown that key components of the estuarine sediment system can be modelled within a CA-type framework. Initial model development suggests that there seems to be much value and practicality in developing hybrid models, where high level or global rules from sub-models (e.g. wave, tide or climate) also apply to large numbers of cells. A strictly CA approach to estuary modelling may be feasible and would be advantageous where existing model structures are incompatible with contiguous cellular structures. The key is to retain low level and local rules for driving inter-cell exchange and response, interaction and feedback *for those processes of interest*, while maintaining full compatibility between the different model components. Indeed, this compatibility is a crucial element for developing an integrated coastal simulator encompassing river, estuary, open coast, shelf sea and policy-/decision-making environments. Certainly, even at an early stage of development, CEMCOS casts doubt on the criticisms levelled at bottom-up CA approaches. Certainly, the assertion of Neal *et al*. (2003) that ‘bottom-up models based on hydrodynamics and sediment properties become unstable within a small number of tidal cycles and fail to account for long-term trends in estuarine function’ neither holds for the early CA results presented in this paper nor the simulation of estuarine form by Hibma *et al*. (2003).

Although there have been few attempts to develop bottom-up CA or CA-type models for estuaries (EMPHASYS Consortium 2000), evaluation of cell-based simulation techniques in the context of grid computing is identified as a fundamental priority in estuarine research (DEFRA/EA 2002*b*). However, much previous modelling of coastal morphology has tended towards methods based on partial differential equations, where quantities change smoothly with space and time, and numerical solutions are obtained using finite-element or difference techniques. These techniques may have similarities to CA approaches, in that they produce optimum solutions through ‘discretization’, but these are not strictly discrete systems (Wolfram 2002) and therefore ‘the overall state of the system will not be properly prepared for the next step of cellular automaton evolution’ (p. 1130). Finite difference models require a large amount of computational effort to integrate long time-scales (although the solution to partial differential equations may be simplified using time- or depth-averaging), and unlike CA methods, which are strictly discrete, do not have a built-in evolutionary capacity to generate continuous feedback and the production of emergent phenomena (Wolfram 2002).

Additionally, CA types of model have the advantages of very rapid integration times and transferability due to interactions existing only between neighbouring cells and the low levels of site-specific data and parameterization that are required. This contrasts with the emerging Lagrangian solutions (encouraged by the advancement of computer speed and parallel computation), where individual grains are modelled as particles that move vertically under the action of gravity (fall velocity) and diffusion (random walk) (e.g. POL3DD, Black 2003).

Despite their advantages, there are concerns that fully operational CA models may provide realistic simulations without a full understanding of the physics (‘the right results for the wrong reasons’). The transitional rules for the CA component may be based on physical principles or derived empirically from observations, but must be relatively simple. A characteristic of the classical CA approach is that it gives a qualitative description, capturing some of the patterns of self-organization such as flow channels, without necessarily having a quantitative predictability. However, if we want to extend this into a more verifiable model, it needs to be quantifiable. The cellular model approach outlined here may be regarded as a hybrid model, as it combines elements of finite difference modelling with a rule-based approach, which allows calculation of the evolution of self-organization in the model system while including more complex physics where essential. Indeed, Leont'yev (2003) identifies the advantages of such hybrid models for long-term coastal evolution modelling by combining a traditional ‘integral’ approach operating with sediment balance with a ‘local’ approach, describing long-term evolution as a cumulative result of short-term processes proceeding against a background trajectory (e.g. Roelvink *et al*. 2001). In the present structure, the tidal component of CEMCOS is not truly CA, as the tides are calculated ‘offline’ assuming an equilibrium morphology to be updated at selected intervals. Hence, the priority for future development is a tidal model based on local cell-based rules to enable the effects of emergent mudflats and shoals to be incorporated. Even so, the effects of such emergence are presently incorporated in the volumetric transport of water and sediment relative to bathymetry at the cellular level.

As an example of the advantages in the implementation of simple models over state-of-the-art more physically realistic models, we note some similarities between the approach presented here and the so-called second-generation wave models used by the UK Meteorological Office (http://www.metoffice.com/research/ncof/wave/), in which the shape of the wave spectrum is assumed. This allows much faster computation than the state-of-the-art physics in third-generation models, e.g. WAM (Komen *et al*. 1994), which allow the full evolution of the spectrum from first principles. Analogously, the value of a model like CEMCOS should be judged according to the accuracy and precision with which it simulates the spatio-temporal complexity of the overall system. In the present context, this might initially involve decadal-long driving of the model with instrumented sea-level, tide and wave data from the mid-twentieth century onwards, in order to simulate the mapped or reconstructed patterns in estuarine channel shape and distribution of tidal mudflats up to the present: a potentially powerful tool for dealing with contemporary management issues.

A key objective of the Tyndall Centre's Integrated Regional Coastal Simulator programme is to provide a coastal management decision-making tool with a view to considering future coastal change with reference to Intergovernmental Panel on Climate Change scenarios. Some developments have been made in constructing decision-support tools for coastal environments. Costanza & Ruth (1998) and Reyes *et al*. (2000) describe the use of the generic STELLA computing language to develop the coastal ecosystem landscape spatial simulation model of the Louisiana coastal wetlands. Set up with a spatial scale of 1 km^{2}, the model simulates the changing nature of the Louisiana coast over 50–100 year time-scales as a function of management alternatives and climate variations. Embedding decision-making in an integrated cellular coastal model can be viewed on at least two different levels: where decision-making is represented by transitional rules, or where decision-making is through setting scenarios for the model to simulate. The former would mean linking existing biophysical models to agent-based models, and this would be feasible for catchment land use in systems where the catchment flux to the sea is a major driver of the estuarine and nearshore environment. However, for many coastlines, the link between society and environmental change is less interactive and more responsive, with concerns surrounding the optimal combinations of defence and managed retreat. Thus, links between a biophysical cellular simulator and user communities are more likely to develop towards setting scenarios in terms of Shoreline Management Plans (as well as Coastal Habitat Management Plans and Estuary Management Plans), with accompanying computer visualization of landscape changes. In this sense, the use of cellular-based modelling frameworks with their expression in Geographical Information Systems and the potential to include users in model development (e.g. Costanza & Ruth 1998) may be seen as a major advantage. Ultimately, outreach to the non-specialist stakeholder and local coastal community should be considered through the production of a web-based resource, in which the modelling environment is clearly explained and illustrated, and the user is able to interrogate a series of scenario-based outcomes.

Currently, our understanding of land–ocean fluxes is based on analysis of the coherence in outputs from different models (cf. Shennan *et al*. 2003). There is, however, the immediate prospect of improving upon this methodology through direct coupling of existing cellular models across the land–ocean interface. A first stage might be through simple output–input linkage, where cellular fluvial models (e.g. CAESAR) drive inputs at synchronized time-steps into the CEMCOS estuary, where the main drivers are downscaled regional climate inputs from Global Circulation Models over the next few decades (e.g. Hulme *et al*. 2002). More long-term development would work towards the full integration of cell-based transitional rules, where one set of rules deals with all cells above the water line (catchment slopes, floodplain, saltmarsh, mudflats, sandbanks) and another set models processes in submerged cells (channels, sea-bed, inundated surfaces). There is much to be gained from an integrated catchment-coastal model, particularly over longer time periods where there are macroscale signals to be resolved, such as changes in land cover and sea- and base-levels.

## Acknowledgments

This work was supported by the Tyndall Centre for Climate Change Research, Project RT3.41. We thank Pedro Osuna, Roger Proctor, Alex Souza, Andrew Watkinson and Jon Williams for useful discussions on different aspects of the research. We also acknowledge useful discussions between members of a Tyndall Centre coastal community workshop held in Liverpool, June 2004.

## Footnotes

↵† Present address: Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0225, USA.

One contribution of 20 to a Theme Issue ‘Sea level science’.

- © 2006 The Royal Society