Modelling coastal hydrodynamics

Hydrodynamic modelling has greatly advanced the understanding of processes that shape and condition the coastal zone. Results of hydrodynamic modelling largely contribute to explanations and results on topics dealt with in Coastal Wiki articles. Several articles have an appendix where basic features of the underlying models are explained in mathematical terms. This article aims to present an overview of major hydrodynamic modelling approaches and tools that are used in support of coastal management. Indications are given of strengths and limitations of different approaches and possible fields of application. The article focuses on models that simulate hydrodynamic processes, leaving aside empirical or purely data-based model approaches (see Data analysis techniques for the coastal zone). The combination of process models with observations through data assimilation is briefly discussed. The elementary introductions in the appendix offer a glimpse of the mathematics that drive numerical process models currently used in practice.

Introduction

Hydrodynamic models are used to understand, simulate and predict water motion in coastal waters, estuaries and the nearshore zone. They represent the physical processes through which external and internal forces, together with the constraints imposed by coastal geometry and seabed topography, generate and modify tides, storm surges, waves and currents. Hydrodynamic modelling also provides essential information for studies of sediment transport, morphological evolution, coastal flooding, water quality and ecosystem functioning.

Three complementary modelling approaches are used:

• Physical modelling: experiments in laboratory facilities that reproduce the relevant hydrodynamic processes, generally at reduced scale.
• Numerical modelling: computer simulations based on mathematical equations describing the physical processes that govern water motion.
• Composite modelling: the coordinated use of physical and numerical models, exploiting the strengths of each approach.

Field observations are indispensable for understanding hydrodynamic processes and for evaluating model performance. The appropriate modelling approach depends on the processes involved, the spatial and temporal scales, the required accuracy and the available resources.

Coastal hydrodynamic processes

Coastal hydrodynamics results from the action of external and internal forces on seawater, subject to constraints imposed by water depth, coastal geometry and seabed topography. The main driving forces include gravitational forces associated with astronomical tides and water-level gradients, wind stress, atmospheric pressure gradients and buoyancy forces associated with density differences. Wind stress also generates surface gravity waves.

The resulting water motions include tides and tidal currents, storm surges, wind-driven currents, surface gravity waves and density-driven circulation. Their characteristics are determined by physical processes such as wave propagation, shoaling, refraction, diffraction and breaking, frictional dissipation, turbulent mixing and the interaction between waves and currents.

Different water motions and the processes governing them are often coupled. For example, tidal and surge-related water-level variations change the water depth and thereby influence wave propagation and breaking. Currents modify wave propagation through refraction and Doppler shifting, while waves influence currents through momentum transfer and changes in bottom stress. Tide–surge interaction modifies coastal water-level variations, while density stratification can suppress vertical turbulent mixing and modify the current structure.

Hydrodynamic models represent these processes through mathematical equations, using the driving forces, initial conditions and boundary conditions as inputs to calculate the resulting water motion. Different modelling approaches resolve different processes and spatial and temporal scales. For example, regional tidal models generally do not resolve individual surface waves, whereas detailed wave models may neglect large-scale circulation or represent it through prescribed boundary conditions.

The selection of a model therefore begins with identifying the driving forces, hydrodynamic processes and resulting water motions that are essential for the problem under investigation.

Physical modelling

Physical models reproduce hydrodynamic processes in laboratory facilities such as wave flumes, wave basins and current facilities. They are particularly valuable when the relevant processes are strongly nonlinear, involve complex interactions with structures or are insufficiently understood for reliable numerical representation.

Physical experiments offer several advantages:

• Processes can be studied under controlled and repeatable conditions.
• Detailed measurements can be obtained in regions that are difficult to observe in nature.
• The influence of individual parameters can be investigated systematically.
• Experimental results provide data for developing and validating numerical models.

Similarity requirements and scale effects

Physical models generally operate at reduced geometric scale. Their validity depends on reproducing the dominant physical forces in appropriate proportions.

For flows dominated by gravity and inertia, Froude similarity is commonly applied. However, simultaneous similarity of gravity, viscous and surface-tension forces is generally impossible in reduced-scale models. Consequently, Reynolds and Weber similarity (referring to viscosity and capillarity, respectively) cannot normally be satisfied together with Froude similarity.

These limitations can be important for wave breaking, air entrainment, wave overtopping, flow through porous structures and sediment transport.

The selection of model scale therefore depends on the processes being investigated. Large-scale facilities can reduce scale effects, but do not automatically eliminate them. Further details can be found in the article Scaling Issues in Hydraulic Modelling.

A distinction should be made between:

• Scale effects, arising because the relative importance of physical forces differs between model and prototype.
• Model effects, arising from differences in geometry, boundary conditions, wave generation, measurement procedures or other aspects of the experimental setup.

For example, comparisons of laboratory and prototype measurements of wave overtopping at rubble-mound breakwaters have shown that differences in armor configuration, toe geometry and incident wave characteristics can affect the results, in addition to genuine scale effects.

Physical modelling remains especially useful for investigating wave loads, overtopping, structure stability and other processes involving complex interactions between waves, turbulence and structures.

Numerical modelling

Numerical hydrodynamic models calculate water motion from equations expressing conservation of mass and momentum (Appendix A1). The general mathematical foundation is provided by the Navier–Stokes equations, which describe the motion of fluids under the influence of pressure gradients, gravity, friction and other forces. Together with appropriate boundary and initial conditions, these equations determine the evolution of the flow.

Direct numerical simulation of all turbulent motions (DNS) is computationally impractical for most coastal applications. Turbulence is therefore generally represented by averaging or filtering the equations. The Reynolds-averaged Navier–Stokes (RANS) equations describe the flow averaged over turbulent fluctuations, with the influence of unresolved turbulence represented by Reynolds stresses. Large-eddy simulation (LES) resolves the larger turbulent motions and parameterizes the smaller ones . Both approaches can retain the oscillations of individual surface waves.

For practical coastal modelling, further simplifications are generally required. These depend on the spatial and temporal scales and on the physical processes that must be represented. Examples are the hydrostatic approximation, depth integration, linearization and averaging over the phases of individual waves. Different combinations of these approximations lead to different model formulations.

The main families of coastal hydrodynamic models are discussed below. Their characteristics are summarized in Appendix A10. Representative equations and their underlying assumptions are presented in the Appendices A1-A9.

Models of tides, surges and coastal circulation

Tides, storm surges and large-scale coastal currents generally have horizontal length scales much greater than the water depth. Under these conditions, vertical accelerations are usually small compared with gravitational acceleration. The vertical momentum equation can then be simplified by assuming a hydrostatic pressure distribution.

With this hydrostatic approximation, pressure at a given depth is determined by the weight of the overlying water, modified where relevant by density variations (Appendix A3). The resulting horizontal pressure gradients, together with wind stress, Coriolis acceleration, friction and other forces, determine the horizontal water motion.

For many applications, the vertical variation of horizontal velocity can also be eliminated by integrating the equations over the water depth. This yields the depth-averaged shallow-water equations, which calculate water levels and depth-averaged currents (Appendix A4).

These equations can represent the propagation of tides and storm surges, including nonlinear effects associated with changing water depth, advection and bottom friction. They are widely used for investigating tidal motion in shelf seas, tide–surge interaction, tidal propagation in estuaries and tidal rivers, and coastal flooding. They also provide the basis for many tsunami propagation and inundation models.

The shallow-water equations neglect frequency dispersion arising from non-hydrostatic pressure variations. Consequently, they cannot describe all details of rapidly varying free-surface flows. A tidal bore, for example, can be represented as a propagating hydraulic jump, but its internal structure, including undulations and breaking, requires additional modelling assumptions or a more detailed formulation. The same distinction applies to tsunamis: hydrostatic models often describe their propagation and inundation adequately, while dispersive models are needed when non-hydrostatic effects significantly influence wave evolution.

Depth-averaged nonlinear shallow-water models have also been applied to wave run-up and overtopping, using hydraulic-jump approximations for breaking waves. Their computational efficiency is an advantage, but their simplified representation of vertical flow structure and wave breaking can limit accuracy for complex wave–structure interactions.

Three-dimensional hydrostatic models retain the vertical variation of horizontal velocity, salinity and temperature. They are particularly relevant for estuarine circulation, density stratification, vertical mixing and wind-driven circulation. Their mathematical foundation includes the hydrostatic pressure approximation, but not depth averaging of the horizontal momentum equations.

Vertical turbulent momentum and mass transport must be represented by turbulence closure models (Appendix A2). Bottom friction and the effects of unresolved bedforms and small-scale topographic features also require parameterization (see Bed roughness and friction factors in estuaries and Bedforms and roughness). These representations can strongly influence calculated currents and water levels, especially in shallow coastal waters.

Applications of two- and three-dimensional hydrostatic models are discussed in several Coastal Wiki articles, for example Geostrophic flow, Ekman transport, Tidal motion in shelf seas, River tides and Estuarine circulation.

Numerical models commonly use computational meshes with spatially varying resolution. Unstructured meshes allow local refinement in regions with complex geometry or strong gradients, while retaining coarser resolution elsewhere. This reduces, but does not eliminate, the need for nested models.

Phase-averaged spectral wave models

Wind-generated sea states consist of many wave components with different frequencies, directions, amplitudes and phases. A statistical description of such wave fields is provided by the directional wave spectrum, which specifies how wave variance or energy is distributed over frequency and direction. See Statistical description of wave parameters.

Spectral wave models calculate the evolution of this distribution without resolving the oscillations of individual waves. Their governing equation expresses conservation of wave action, accounting for propagation through geographical and spectral space and for processes that generate, redistribute or dissipate wave energy.

The principal source and sink processes include wind input, nonlinear interactions between wave components, whitecapping, bottom friction and depth-induced breaking. Wave propagation is influenced by water depth and currents, producing effects such as shoaling, refraction and frequency shifting.

The main advantage of phase averaging is that the rapid oscillations of individual waves need not be calculated. Spectral models can therefore simulate wave conditions over large oceanic and coastal domains and over long periods at substantially lower computational cost than phase-resolving models.

The main limitation is that information about the relative phases of wave components is lost. Spectral models cannot directly reproduce individual wave crests, their interference patterns or instantaneous wave impacts. Processes that depend strongly on wave phase, such as detailed diffraction and reflection around structures, generally require phase-resolving models.

Spectral wave models are widely used to determine offshore and nearshore wave conditions, including the wave forcing for coastal sediment transport, beach erosion and coastal protection studies. When coupled to circulation models, they provide wave-induced momentum forcing for calculating wave setup and wave-driven currents (see Radiation stress). Their results can also be used as input to empirical methods or more detailed models for wave run-up, wave overtopping, swash and local scour.

The wave-action balance equation and its principal terms are explained in Appendix A7. For wave propagation in shallow water, see also Shallow-water wave theory and Wave transformation.

Phase-resolving wave models

Phase-resolving models calculate the evolution of the water-surface elevation and the associated flow, retaining information about the phase of individual waves. They can therefore represent interference, reflection, diffraction and nonlinear interactions between wave components.

Several mathematical formulations are used, depending on the importance of nonlinearity, frequency dispersion, vertical accelerations and three-dimensional flow structure.

Some models (e.g. XBEACH) resolve infragravity waves while using a spectral representation of short waves.

Mild-slope models

The classical mild-slope equation is derived from linear wave theory, assuming that water depth varies gradually over distances comparable with the wavelength (Appendix A6). It combines wave refraction, caused by variations in wave propagation speed, with diffraction, caused by horizontal variations in wave amplitude and phase.

Mild-slope models are particularly suitable for calculating wave propagation, diffraction in coastal areas and around harbor structures where linear wave theory is applicable. The elliptic model version simulates also reflection.

The classical formulation does not represent strong nonlinear wave deformation or the detailed processes of wave breaking. Extended formulations relax some of these restrictions.

Boussinesq-type models

The nonlinear shallow-water equations neglect frequency dispersion because they assume a hydrostatic pressure distribution. Boussinesq-type equations retain additional terms representing the effects of non-hydrostatic pressure on wave propagation (Appendix A5).

In their classical form, these equations are derived for relatively long waves with weak nonlinearity and weak frequency dispersion. They describe the combined influence of nonlinear wave deformation and dispersive propagation.

The dispersive terms allow wave components of different frequencies to propagate at different speeds. They are therefore essential for describing phenomena such as the development of undular bores and the dispersive evolution of long waves.

Extended Boussinesq formulations provide improved dispersion characteristics over a wider range of relative water depths. Fully nonlinear formulations relax the weak-nonlinearity assumption. Boussinesq-type models thus constitute a family of models with different ranges of applicability.

They are used for nonlinear wave transformation, infragravity waves, wave interaction with coastal structures, wave run-up and dispersive tsunami propagation. They can also describe the development of undular tidal bores (see Tidal bore dynamics). Strongly breaking waves generally require additional parameterizations of breaking and energy dissipation.

Non-hydrostatic models

Non-hydrostatic models retain vertical accelerations by calculating departures from the hydrostatic pressure distribution. This allows them to represent frequency dispersion and nonlinear wave propagation without relying on the same long-wave expansions as conventional Boussinesq formulations.

The accuracy with which dispersive effects are represented depends on the vertical discretization and numerical formulation. Models with several vertical layers can represent a wider range of vertical flow structures than models with only one layer.

Non-hydrostatic models are used for nonlinear wave propagation, surf-zone processes, infragravity waves and rapidly varying flows. Their computational requirements are generally greater than those of hydrostatic shallow-water models.

Wave breaking requires special consideration. Some formulations represent breaking through hydraulic-jump approximations or numerical dissipation. Others resolve more details of the breaking process, but do not necessarily reproduce overturning, air entrainment and turbulent energy dissipation explicitly.

Detailed computational fluid dynamics models

Computational fluid dynamics (CFD) is the general term for numerical simulation of fluid motion. In coastal engineering, the term is often used more specifically for detailed numerical models that resolve three-dimensional flow structures and complex interactions between waves, currents and structures.

These models generally solve the Navier–Stokes equations, using RANS or LES formulations to represent turbulence. They retain vertical accelerations and can represent flow separation, vortices and strongly nonlinear free-surface motions (Appendix A8).

Two principal descriptions of fluid motion are used:

• Eulerian description: flow properties are calculated at locations in space, generally using a computational mesh. The free surface may be represented by methods such as Volume of Fluid (VOF), which calculates the fraction of each computational cell occupied by water.
• Lagrangian description: the motion of fluid particles is followed through time. Particle-based methods such as Smoothed Particle Hydrodynamics (SPH) use this description and are particularly suitable for strongly deforming free surfaces.

Both descriptions are based on the same physical conservation principles. Their differences concern the numerical representation of the flow and the treatment of moving boundaries and free surfaces.

Detailed CFD models can simulate wave breaking, wave impact, overtopping, flow through or around coastal structures and local flow processes associated with scour. They are particularly valuable where vertical accelerations, turbulence and complex free-surface deformation are important. Applications include numerical investigations of wave overtopping at rubble-mound breakwaters and simulations of spilling and plunging breakers using large-eddy simulation.

Their principal limitations are high computational requirements and uncertainties associated with turbulence closure, air entrainment, porous-flow resistance and numerical resolution. Consequently, detailed CFD applications generally concern smaller domains and shorter simulation periods than regional circulation or spectral wave models.

Numerical wave tanks can reproduce prototype-scale conditions without the geometric scale effects of reduced-scale laboratory experiments. However, they introduce numerical and modelling uncertainties that must be evaluated against experimental or field observations.

Flow through permeable coastal structures, such as rubble-mound breakwaters and revetments, requires special treatment. Resolving the flow around individual stones is computationally demanding and generally impractical for engineering-scale simulations. An alternative is to represent the porous material as a continuum, using spatially averaged flow equations with additional resistance terms, combined with a VOF representation of the free surface.

This approach is implemented in the Volume-Averaged Reynolds-Averaged Navier–Stokes (VARANS) equations. It combines Reynolds averaging, which represents the effects of unresolved turbulent fluctuations, with volume averaging over a region containing both water and solid material. The resulting equations include resistance forces associated with flow through the pores, generally represented by linear and nonlinear drag terms. VARANS models can therefore simulate wave propagation, breaking, reflection, transmission and overtopping in combination with flow through permeable structures. Their accuracy depends on the representation of porosity, hydraulic resistance and turbulence within the porous material.

Composite modelling

Physical and numerical models have complementary strengths. Physical models can reproduce complex local processes, while numerical models can simulate large domains, investigate many alternative conditions and provide spatially extensive information.

Composite modelling combines these approaches within a coordinated investigation.

For example, a numerical model may reproduce the regional wave and current field and provide boundary conditions for a physical experiment investigating wave interaction with a coastal structure. Experimental measurements may subsequently be used to improve the representation of the structure in the numerical model.

Another application is the combination of detailed local experiments with regional numerical simulations to assess the consequences of coastal structures for wave transformation, sediment transport and morphological development.

Composite modelling is particularly valuable when the required processes cannot be represented with sufficient reliability by either approach alone.

However, combining models does not automatically improve accuracy. The exchange of information between models requires compatible variables, appropriate spatial and temporal scales, and an assessment of the uncertainties introduced by the coupling.

Selecting a modelling approach

Different modelling approaches represent different aspects of coastal hydrodynamics. Table 1 summarizes their principal strengths, applications and limitations. Several approaches may be appropriate for the same coastal problem, depending on the processes to be investigated, the required spatial and temporal resolution and the quantities to be predicted. Different approaches may also be combined within a single investigation.

Table 1. Principal strengths, applications and limitations of coastal hydrodynamic modelling approaches

Modelling approach	Principal strengths and applications	Main limitations and considerations
Physical modelling	Detailed investigation of wave breaking, wave loads, overtopping, structure stability and complex local flow processes. Controlled experiments provide data for understanding processes and validating numerical models.	Scale effects, laboratory boundary conditions and limitations of experimental facilities.
2D depth-averaged hydrodynamic models	Tides, storm surges, coastal flooding, tidal propagation in rivers and estuaries, depth-averaged circulation and long-wave propagation.	Vertical flow structure is not resolved. Hydrostatic approximation; friction and unresolved processes require parameterization.
3D hydrostatic circulation models	Estuarine and coastal circulation, density stratification, vertical mixing, wind-driven circulation and transport of dissolved or suspended substances.	Vertical accelerations are neglected. Turbulent mixing and bottom friction require parameterization; computational demand is greater than for 2D models.
Phase-averaged spectral wave models	Wave generation and transformation over large domains and long periods; wave climates, nearshore wave conditions and wave forcing for circulation and sediment-transport models.	Individual waves and their interference are not resolved. Detailed diffraction, reflection, run-up and wave impacts require other approaches.
Mild-slope wave models	Wave refraction, diffraction, interference and reflection, particularly around harbor entrances and coastal structures.	Classical formulation assumes linear waves and gradually varying bathymetry; breaking and strong nonlinearities are not represented explicitly.
Boussinesq-type wave models	Nonlinear and dispersive wave propagation, infragravity waves, undular bores, wave transformation, run-up and dispersive tsunami processes.	Range of validity depends on the formulation. Breaking and associated dissipation generally require additional treatment.
Non-hydrostatic wave models	Dispersive and nonlinear wave propagation, surf-zone processes, infragravity waves and rapidly varying flows.	Accuracy depends on vertical resolution and numerical formulation; overturning and air entrainment may not be resolved.
Detailed CFD models	Local three-dimensional flows, turbulence, wave breaking, wave–structure interaction, wave impacts, overtopping and local scour processes. VARANS equations combine Reynolds averaging with volume averaging to represent turbulent flow through porous structures, using parameterized resistance forces.	High computational cost; sensitivity to turbulence modelling, numerical resolution and treatment of air–water interfaces. Parameterization of porous-media resistance.
Composite modelling	Combination of regional numerical simulations with detailed physical experiments or local numerical models, exploiting complementary strengths.	Requires consistent boundary conditions, compatible scales and assessment of uncertainties introduced by coupling.

The choice of modelling approach should be based on the processes that must be represented and the required reliability of the results. A complex model is not necessarily preferable to a simpler one. For example, a depth-averaged model may adequately predict tidal water levels, whereas a three-dimensional model is needed to investigate the associated vertical circulation. Similarly, spectral wave models can provide the offshore wave conditions required as boundary forcing for detailed wave–structure interaction models.

Model calibration, validation and uncertainty

Hydrodynamic models contain uncertainties arising from input data, boundary conditions, model parameters, numerical approximations and the representation of physical processes. Calibration involves adjusting uncertain parameters to improve agreement with observations, whereas validation assesses model performance using independent observations not employed for calibration.

Model performance is commonly quantified by comparing simulated and observed values at corresponding locations and times. A widely used measure is the root-mean-square error (RMSE), based on the least-squares criterion:

[math]RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(M_i-O_i)^2},[/math]

where [math]M_i[/math] and [math]O_i[/math] are modelled and observed values and [math]N[/math] is the number of comparisons. The Brier skill score compares model performance with a reference prediction:

[math]BSS=1-\dfrac{\sum_i(M_i-O_i)^2}{\sum_i(R_i-O_i)^2},[/math]

where [math]R_i[/math] is the reference prediction. A score of 1 indicates perfect agreement, 0 indicates performance equal to the reference, and negative values indicate poorer performance. This formulation is a squared-error skill score for continuous variables.

These measures are sensitive to temporal phase errors. A small phase shift between simulated and observed tidal or wave oscillations may produce a large error even when amplitudes are reproduced accurately. For periodic motions, amplitude and phase should therefore also be assessed separately, for example by harmonic or spectral analysis, or by cross-correlation. Graphical comparison of observed and simulated time series can reveal systematic discrepancies that are not apparent from a single performance score. The importance of phase errors depends on the application.

Sensitivity analysis investigates how uncertain parameters and inputs influence model results. Parameters can be varied individually, but Monte Carlo methods, involving repeated simulations with parameter values sampled from prescribed ranges or distributions, can also reveal parameter interactions and identify influential parameters. The spread of model outcomes provides an indication of prediction uncertainty, subject to the assumed parameter distributions and the sources of uncertainty included in the analysis.

Model performance should be assessed for the quantities and conditions relevant to the intended application. Agreement with observed water levels, for example, does not guarantee accurate currents or discharges, and calibration under ordinary conditions does not establish reliability during extreme events. Model complexity is no substitute for adequate calibration, validation and uncertainty assessment.

Data assimilation

Calibration adjusts model parameters once, after which the model runs without further use of observations. Data assimilation goes a step further: observations are combined with the model continuously, while the simulation proceeds. The aim is to obtain the best possible estimate of the actual state of the water body (water levels, currents, waves) by using both sources of information: the model, which fills the gaps between observation points and observation times, and the observations, which correct the model where it deviates from reality.

The basic idea is a weighted average. At a given moment, the model provides a prediction and an observation provides a measured value. Both contain errors. The best estimate lies between the two, closer to the one that is believed to be the more accurate. A simple example is given in Appendix A9. In a real model the same principle is applied to many thousands of variables at once. The correction at an observation point is then spread to neighbouring locations and to other variables, according to how model errors are expected to be related in space and time. For example, an observed water level that is higher than predicted may lead to a correction of water levels along a whole stretch of coast and of the associated currents.

Two main families of methods are used:

• Sequential methods correct the model each time new observations become available, after which the model continues from the corrected state. The Kalman filter is the classical example. In large nonlinear models, the required error statistics are often estimated from an ensemble of model runs with perturbed forcing or parameters, the ensemble Kalman filter. A simpler sequential method is nudging, in which an extra term in the model equations gradually pulls the model towards the observations.
• Variational methods search for the model state (and possibly model parameters or boundary conditions) that gives the best overall fit to all observations within a time window, while respecting the model equations. In four-dimensional variational assimilation (4D-Var) this is done over a period of time rather than at a single moment. The same approach can be used to estimate uncertain parameters such as bottom friction coefficients, which makes it a systematic form of calibration.

In coastal hydrodynamics, data assimilation is used for:

• Storm-surge and water-level forecasting. Kalman filtering of tide-gauge observations in shelf-sea tide and surge models was developed early in the Netherlands. The correction mainly improves forecasts for the first hours, because the influence of the initial state fades as the model is increasingly driven by the forecast wind and the tide.
• Global and regional tide models. Tide atlases such as TPXO and FES combine hydrodynamic tide models with satellite altimetry and tide-gauge observations.
• Wave forecasting. Wave-buoy and satellite measurements of wave height are assimilated in operational wave models to correct the initial wave field.
• Coastal and shelf-sea circulation. Regional ocean models assimilate observations of sea level, temperature, salinity and surface currents (for example from HF radar).

Data assimilation does not remove the need for a good model. The method relies on estimates of the model and observation errors, which are themselves uncertain. If these are poorly specified, assimilation can degrade rather than improve the results. Corrections that compensate for systematic model errors may also mask deficiencies in the model physics. Assimilation is therefore most effective when combined with careful calibration and validation, as described above. A general overview of methods is given by Carrassi et al. (2018).

A more detailed discussion of Data assimilation is given in the article Reduction of uncertainties through Data Model Integration (DMI).

Available numerical modelling tools

A wide range of numerical modelling tools is available, including open-source and commercial software. Table 2 provides representative examples organized according to their principal modelling approach.

The table is intended as an orientation guide rather than a ranking. Model performance depends strongly on the application, implementation, resolution, parameterization and available validation data.

Table 2. Principal model types, representative software and literature references

Model type	Representative software and references	Principal applications	Computational demand	Availability
2D/3D coastal circulation	Delft3D FM; TELEMAC; SCHISM; COAWST	Tides, surges, currents, estuarine circulation	Moderate to high	Open source
Coupled surge and wave modelling	ADCIRC–SWAN; other coupled coastal modelling systems	Coastal flooding, tide–surge–wave interaction	Moderate to high	Open source or license-dependent
Spectral wave modelling	SWAN; WAVEWATCH III	Offshore and coastal wave climates, wave transformation	Low to moderate	Open source
Boussinesq-type wave modelling	FUNWAVE-TVD; Celeris	Nonlinear waves, diffraction, run-up, inundation	Moderate to high	Open source
Non-hydrostatic wave modelling	SWASH	Surf-zone processes, infragravity waves, wave–structure interaction	Moderate to high	Open source
Nearshore hydro-morphodynamic modelling	XBeach	Storm erosion, infragravity waves, overwash, inundation and nearshore dynamics	Moderate to high	Open source
Mild-slope equation	ARTEMIS (TELEMAC)	Diffraction and reflection in coastal areas and around harbor structures	Low to moderate	Open source
CFD: grid-based	OpenFOAM-based wave solvers, RANS/VARANS and VOF	Detailed wave impact, overtopping, local flow and scour	High to very high	Open source (several other commercial codes also exist)
CFD: particle-based	DualSPHysics	Violent free-surface flows, wave impact, floating bodies	High to very high	Open source
Commercial coastal modelling suites	MIKE	Hydrodynamics 2D, 3D, spectral waves, circulation, flooding and coupled coastal processes	Varies	Commercial

Computational demand is qualitative and refers to typical applications rather than intrinsic software performance. References describe representative model formulations or software versions; subsequent versions may incorporate additional capabilities.

Conclusions

Physical, numerical and composite modelling provide complementary means of investigating coastal hydrodynamic processes.

Physical modelling is especially valuable for complex local processes and interactions with structures. Numerical modelling permits simulations over a wide range of spatial and temporal scales, with the appropriate mathematical formulation determined by the processes to be resolved. Composite modelling can combine their strengths when a single approach is insufficient.

The reliability of model predictions depends not only on the mathematical formulation and numerical methods, but also on the quality of the input data, the representation of unresolved processes and the adequacy of calibration and validation.

Model selection should therefore be guided primarily by the physical problem and the required reliability of the results, rather than by model sophistication or software availability alone.

Appendix: Mathematical foundations of coastal hydrodynamic models

The equations below illustrate the mathematical principles underlying the model formulations discussed in this article. They are representative simplified equations, not complete descriptions of operational models.

The purpose is to identify the physical meaning of the principal terms and to show how different approximations lead to different model capabilities. For clarity, most equations are written in one horizontal dimension. Additional terms may be required to represent two- or three-dimensional flow, rotation, density variations and turbulence.

A1. Conservation of mass and momentum

The fundamental equations of fluid motion express conservation of mass and momentum.

For incompressible flow in two dimensions, conservation of mass is expressed by

[math] \dfrac{\partial u}{\partial x} + \dfrac{\partial w}{\partial z} =0, \qquad (A1) [/math]

where [math]u[/math] and [math]w[/math] are the horizontal and vertical velocity components, respectively.

A simplified horizontal momentum equation is

[math] \underbrace{\dfrac{\partial u}{\partial t}}_{(1)} + \underbrace{\dfrac{\partial u^2}{\partial x} +\dfrac{\partial u \, w}{\partial z}}_{(2)} = \underbrace{-\dfrac{1}{\rho} \dfrac{\partial p}{\partial x}}_{(3)} + \underbrace{\nu \left( \dfrac{\partial^2u}{\partial x^2} + \dfrac{\partial^2u}{\partial z^2} \right)}_{(4)}. \qquad (A2) [/math]

Here [math]p[/math] is pressure, [math]\rho[/math] is water density and [math]\nu[/math] is kinematic viscosity. The water density has been assumed uniform and constant in these equations.

The numbered terms represent:

(1) Local acceleration: change of velocity with time at a fixed location.

(2) Advective acceleration: change of velocity caused by transport through a spatially varying flow field.

(3) Pressure-gradient force: acceleration caused by spatial differences in pressure.

(4) Viscous momentum diffusion: redistribution of momentum by molecular viscosity.

Equation (A2) omits Coriolis acceleration, horizontal body forces and the third spatial dimension. Gravity acts vertically and enters the horizontal momentum balance through the pressure distribution.

The vertical momentum equation contains the same types of acceleration and viscous terms, together with gravitational acceleration.

A2. Reynolds averaging and turbulence closure

In turbulent flow, velocity fluctuates over a wide range of spatial and temporal scales. Reynolds averaging separates the velocity into a mean component and a turbulent fluctuation:

[math] u=\overline{u}+u', \qquad w=\overline{w}+w'. \qquad (A3) [/math]

Substitution into the Navier–Stokes equations and averaging produces additional terms representing the transport of momentum by turbulent fluctuations.

For example, the vertical turbulent transport of horizontal momentum is represented by the Reynolds stress

[math] \tau_{xz}^{R} = -\rho\,\overline{u'w'} \, , \qquad (A4) [/math]

where the overline [math]\overline{..}[/math] denotes the average over the temporal and spatial scales of turbulent fluctuations. This stress must be related to the resolved flow through a turbulence closure model. A commonly used approximation is the eddy-viscosity formulation

[math] \tau_{xz}^{R} = \rho\nu_t \dfrac{\partial\overline{u}}{\partial z}, \qquad (A5) [/math]

where [math]\nu_t[/math] is the turbulent eddy viscosity.

The eddy viscosity is not a fluid property like molecular viscosity. It depends on the turbulent flow and must be estimated from a closure model. An often used simple turbulence closure model is the Prandtl mixing-length model [math]\nu_t = l_m^2 |\dfrac{\partial \overline{u}}{\partial z}|[/math], where the mixing length scale [math]l_m[/math] is related to the scale of the most effective eddies. A more elaborate model (among others) involves separate equations for the kinetic energy [math]k[/math] and the dissipation rate [math]\epsilon[/math] from which an estimate of [math]\overline{u'w'}[/math] can be derived.

Reynolds averaging does not necessarily eliminate surface-wave oscillations. When the averaging is intended to remove turbulent fluctuations while retaining the wave motion, RANS models can calculate individual waves. Additional averaging over wave phases leads to different model formulations.

A3. Hydrostatic approximation

The vertical momentum equation describes the balance between vertical acceleration, pressure forces, gravity and viscous or turbulent stresses.

When vertical accelerations and vertical stress gradients are sufficiently small, this equation reduces to

[math] \underbrace{\dfrac{\partial p}{\partial z}}_{(1)} = \underbrace{-\rho g}_{(2)}. \qquad (A6) [/math]

The numbered terms represent:

(1) Vertical pressure gradient.

(2) Gravitational force per unit volume.

For constant density, integration from the free surface gives

[math] p=p_a+\rho g(\eta-z), \qquad (A7) [/math]

where [math]p_a[/math] is atmospheric pressure, [math]\eta[/math] is the water-surface elevation and [math]z[/math] is the vertical coordinate measured upward from a fixed datum.

The horizontal pressure-gradient force then contains a contribution proportional to the water-surface slope:

[math] -\dfrac{1}{\rho}\dfrac{\partial p}{\partial x} = -\dfrac{1}{\rho}\dfrac{\partial p_a}{\partial x} -g\dfrac{\partial\eta}{\partial x}. \qquad (A8) [/math]

Equation (A8) assumes constant density. In stratified water, horizontal density gradients also contribute to the pressure-gradient force.

The hydrostatic approximation is fundamental to many tidal, storm-surge and coastal circulation models.

A4. Depth-averaged shallow-water equations

The shallow-water equations are obtained by integrating the conservation equations over the water depth, using the hydrostatic approximation.

Derivation of the 1D momentum and mass balance equations. The expressions (A9) and (A10) are obtained by letting [math]\Delta t \rightarrow 0 \, , \; \Delta x \rightarrow 0[/math], such that [math]h_2=h_1+(\partial \eta / \partial dx) dx \, , \; U_2=U_1+(\partial U / \partial dx) dx [/math].

For one-dimensional flow, conservation of water volume gives

[math] \underbrace{\dfrac{\partial\eta}{\partial t}}_{(1)} + \underbrace{\dfrac{\partial(hU)}{\partial x}}_{(2)} =0, \qquad (A9) [/math]

where [math]h[/math] is the instantaneous water depth and [math]U[/math] is the depth-averaged horizontal velocity.

The numbered terms represent:

(1) Local change of water level.

(2) Divergence of the depth-integrated water discharge per unit width.

A simplified depth-averaged momentum equation is

[math] \underbrace{\dfrac{\partial U}{\partial t}}_{(1)} + \underbrace{U\dfrac{\partial U}{\partial x}}_{(2)} = \underbrace{-g\dfrac{\partial\eta}{\partial x}}_{(3)} + \underbrace{\dfrac{\tau_s-\tau_b}{\rho h}}_{(4)}. \qquad (A10) [/math]

Here [math]\tau_s[/math] and [math]\tau_b[/math] are the surface and bottom shear stresses in the horizontal momentum balance.

The numbered terms represent:

(1) Local acceleration.

(2) Advective acceleration.

(3) Acceleration caused by the water-surface slope.

(4) Momentum input by surface stress and momentum loss through bottom friction.

Equation (A10) assumes that the depth-averaged advective momentum flux can be approximated by [math]hU^2[/math]. It neglects atmospheric pressure gradients, Coriolis acceleration, horizontal turbulent stresses and additional contributions arising from vertical variations of horizontal velocity.

The nonlinear shallow-water equations can represent finite-amplitude long waves and hydraulic jumps, but do not reproduce frequency dispersion caused by non-hydrostatic pressure variations.

For small-amplitude waves over a horizontal bottom of constant depth [math]h_0[/math], neglecting friction and nonlinear terms gives

[math] \dfrac{\partial\eta}{\partial t} + h_0\dfrac{\partial U}{\partial x}=0, \qquad (A11) [/math]

[math] \dfrac{\partial U}{\partial t} + g\dfrac{\partial\eta}{\partial x}=0. \qquad (A12) [/math]

Combining these equations yields

[math] \dfrac{\partial^2\eta}{\partial t^2} = gh_0\dfrac{\partial^2\eta}{\partial x^2} \, , \qquad (A13) [/math]

which has solutions of the form [math]\eta(x,t)=f_1(x-Ct)+f_2(x+Ct)[/math]. Here [math]C[/math] is the wave propagation speed given by

[math] C=\sqrt{gh_0}. \qquad (A14) [/math]

In this approximation, the wave propagation speed is independent of wavelength. A more general introduction to the theory of surface gravity waves in the nearshore, is presented in Shallow-water wave theory.

A5. Dispersive wave equations

The hydrostatic shallow-water equations neglect the influence of vertical acceleration on the pressure distribution. For waves that are not sufficiently long relative to water depth, this approximation causes errors in wave propagation speed.

Boussinesq-type equations retain additional non-hydrostatic terms that account for weak frequency dispersion. Extended Boussinesq formulations provide improved dispersion characteristics over a wider range of relative water depths.

A representative one-dimensional weakly nonlinear Boussinesq formulation over a horizontal bottom is

[math] \dfrac{\partial\eta}{\partial t} + \dfrac{\partial[(h_0+\eta)U]}{\partial x} =0, \qquad (A15) [/math]

[math] \underbrace{\dfrac{\partial U}{\partial t}}_{(1)} + \underbrace{U\dfrac{\partial U}{\partial x}}_{(2)} + \underbrace{g\dfrac{\partial\eta}{\partial x}}_{(3)} - \underbrace{\dfrac{h_0^2}{3} \dfrac{\partial^3U}{\partial x^2\partial t}}_{(4)} =0. \qquad (A16) [/math]

The numbered terms represent:

(1) Local acceleration.

(2) Nonlinear advective acceleration.

(3) Hydrostatic pressure-gradient acceleration.

(4) Dispersive correction associated with non-hydrostatic pressure.

The above Boussinesq equations can be derived from the mass and momentum conservation equations by expanding the solution in powers of the vertical coordinate [math]z[/math] and retaining terms to prescribed orders in the small parameters [math](h/L)^2[/math] (long-wave approximation) and [math]a/h[/math] (weak nonlinearity), where [math]L[/math] is the wavelength and [math]a[/math] the wave amplitude.

The dispersive term makes wave propagation speed dependent on wavelength. The classical formulation is valid within its long-wave and weak-nonlinearity assumptions; extended and fully nonlinear formulations use different equations and have wider ranges of applicability.

Equation (A16) is an illustrative leading-order formulation. Its nonlinear and dispersive terms are retained consistently only within the accuracy of the underlying asymptotic approximation.

Non-hydrostatic models take a different approach. Instead of introducing dispersive corrections through a long-wave expansion, they calculate the non-hydrostatic pressure contribution as part of the flow solution.

The total pressure is decomposed as

[math] p=p_h+p_{nh}, \qquad (A17) [/math]

where [math]p_h[/math] is hydrostatic pressure and [math]p_{nh}[/math] is the non-hydrostatic contribution.

The vertical momentum equation is then used to determine the pressure correction required to represent vertical accelerations. The resulting dispersion characteristics depend on the vertical resolution and numerical formulation.

A6. Mild-slope equation

For linear waves propagating over gradually varying water depth, the combined effects of refraction and diffraction can be described by the mild-slope equation. The mild-slope equation is derived from linear potential-flow theory by integrating over the water depth, assuming that the vertical structure of the flow locally corresponds to that of a progressive wave over a flat bottom. Where diffraction is negligible, it reduces to conservation of wave energy flux along wave rays.

A representative form for a monochromatic wave is

[math] \underbrace{ \dfrac{\partial}{\partial x} \left(CC_g\dfrac{\partial\hat{\eta}}{\partial x}\right) }_{(1)} + \underbrace{ \dfrac{\partial}{\partial y} \left(CC_g\dfrac{\partial\hat{\eta}}{\partial y}\right) }_{(2)} + \underbrace{k^2CC_g\hat{\eta}}_{(3)} =0. \qquad (A18) [/math]

Here [math]\hat{\eta}[/math] is the complex wave-elevation amplitude, [math]C[/math] is phase velocity, [math]C_g[/math] is group velocity and [math]k[/math] is wave number.

The numbered terms represent:

(1) Wave propagation and amplitude variation in the x-direction.

(2) Wave propagation and amplitude variation in the y-direction.

(3) Local wave propagation determined by the dispersion relation.

The spatial derivatives jointly represent refraction and diffraction. The equation is based on linear wave theory and the assumption that water depth varies gradually.

A7. Spectral wave-action equation

Spectral wave models represent wave energy distributed over frequencies and propagation directions. They do not calculate the instantaneous elevation of individual waves.

The governing equation expresses conservation of wave action. Wave action is defined as

[math] N=\dfrac{E(f)}{f}, \qquad (A19) [/math]

where [math]E(f)[/math] is the wave energy density spectrum and [math]f[/math] is intrinsic wave frequency, see Statistical description of wave parameters.

For illustration, a simplified one-dimensional wave-action balance equation is

[math] \underbrace{\dfrac{\partial N}{\partial t}}_{(1)} + \underbrace{\dfrac{\partial(c_g+U) N}{\partial x}}_{(2)} = \underbrace{\dfrac{F}{f}}_{(3)}. \qquad (A20) [/math]

The numbered terms represent:

(1) Local change of wave action.

(2) Transport of wave action by wave propagation relative to the water and advection by the current.

(3) Wave-action sources and sinks associated with energy generation, redistribution and dissipation.

The complete directional spectral equation also includes propagation in the second horizontal direction and transport through frequency and directional space, representing processes such as refraction and frequency shifting.

The source term commonly includes wind input, nonlinear wave interactions, whitecapping, bottom friction and depth-induced breaking.

The advantage of the wave-action formulation is that it describes the evolution of the wave spectrum without resolving individual wave oscillations. The loss of wave-phase information limits its applicability to processes requiring detailed wave interference or instantaneous wave motion.

A8. Detailed CFD formulations

Detailed CFD models solve the fluid-motion equations with sufficient spatial and temporal resolution to represent local three-dimensional flow structures and strongly deforming free surfaces.

Volume-of-Fluid (VOF) models consider the dynamic interaction of different flow constituents. Wave breaking, for example, involves mixing of water and air through the interface. The spatial distribution of water and air is described by a transport equation of the form

[math] \underbrace{\dfrac{\partial\alpha}{\partial t}}_{(1)} + \underbrace{\dfrac{\partial(\alpha u)}{\partial x} +\dfrac{\partial(\alpha w)}{\partial z}}_{(2)} =0, \qquad (A21) [/math]

where [math]\alpha[/math] represents the local volume fraction of water.

The numbered terms represent:

(1) Local change of water volume fraction.

(2) Transport of water volume fraction by the resolved flow.

The free surface is represented by the transition between regions occupied by water and air. Practical VOF implementations may include additional interface-compression and numerical stabilization terms.

In Lagrangian particle methods, the fluid is represented by moving particles. Their positions [math]\mathbf{r}_i(t)[/math] and velocities [math]\mathbf{v}_i(t)[/math] evolve according to

[math]\dfrac{d \mathbf{r}_i}{dt} = \mathbf{v}_i \, , \qquad (A22) [/math]

where the subscript [math]i[/math] identifies a fluid particle and bold characters indicate vectors.

The particle accelerations are calculated from discretized momentum equations, with pressure computed from density through an equation of state. The Eulerian and Lagrangian formulations describe the same physical conservation principles but differ in their numerical representation of the flow and moving boundaries.

In the SPH formalism, fluid particles (positions, velocities, density) are smoothed by a user specified Kernel function [math]W(\mathbf{r}, l_s)[/math] in space [math]V[/math] that defines a kernel with compact support, normalized such that [math]\int_V W(\mathbf{r},l_s) dV =1[/math]. The parameter [math]l_s[/math] defines the smoothing length of the Kernel. Each function [math]f(\mathbf{r})[/math] is expressed in SPH as

[math]f(\mathbf{r}) = \int_V f(\mathbf{r}') W(\mathbf{r}-\mathbf{r}',l_s) dV = \int_V \dfrac{f(\mathbf{r}')}{\rho(\mathbf{r}')} W(\mathbf{r}-\mathbf{r}',l_s) \rho(\mathbf{r}') dV \approx \sum_i \dfrac{f(\mathbf{r}_i)}{\rho(\mathbf{r}_i)} W(\mathbf{r}-\mathbf{r}_i,l_s) \rho(\mathbf{r_i}) \Delta V_i = \sum_i \dfrac{m_i}{\rho_i} f(\mathbf{r}_i) W(\mathbf{r}-\mathbf{r}_i,l_s) \, , [/math]

where the sum over [math]i[/math] comprises all particles in the domain delimited by the Kernel function, each with mass [math]m_i=\rho_i \Delta V_i[/math]. The equations of motion are of the form

[math]\dfrac{d \mathbf{v}_i}{dt} = - \sum_j m_j \left( \dfrac{p_i}{\rho_i^2} + \dfrac{p_j}{\rho_j^2} + \Pi_{i,j} \right) \dfrac{\partial W_{i,j}}{\partial \mathbf{r}_i} + \mathbf{g} \quad ,[/math] where [math]p_i = p(\mathbf{r}_i,t)[/math] is the pressure, [math]\Pi_{i,j}[/math] is a viscosity term, [math]\mathbf{g}[/math] is the gravitational acceleration vector and [math]W_{i,j} = W(\mathbf{r}_i - \mathbf{r}_j,l_s)[/math].

Flow through porous structures: VARANS

In porous media, the flow is constrained by the solid material. Instead of resolving the velocity around individual stones, the governing equations can be averaged over a representative volume containing both water and solid material. This introduces the porosity [math]n[/math], defined as the fraction of the volume occupied by water.

For incompressible flow through a homogeneous porous medium, conservation of mass can be expressed as

[math]\nabla \cdot (n\mathbf{U})=0,[/math]

where [math]\mathbf{U}[/math] is the intrinsic volume-averaged velocity, averaged over the water-filled part of the representative volume.

The volume-averaged momentum equations contain additional terms representing resistance exerted by the solid material. This resistance term in the momentum equation is often approximated by a linear and a quadratic contribution:

[math]\mathbf{F}_r = - a \mathbf{U}-b |\mathbf{U}|\mathbf{U},[/math]

where [math]a[/math] and [math]b[/math] are resistance coefficients whose values depend on the porosity, characteristic stone dimensions and flow conditions. The coefficients and the precise form of the resistance terms depend on the adopted formulation.

In VARANS models, the volume-averaged equations are combined with Reynolds averaging to represent turbulent momentum transport. Additional closure assumptions are required for turbulence and resistance within the porous material. The resulting equations allow wave-induced flow through rubble-mound structures to be calculated without resolving the individual pores.

A9. Data assimilation: a simple example

The following explanations are based on the textbook by Kalnay (2003 ). Consider a single quantity, for example the water level at a tide gauge at a given moment. The model predicts the value [math]x_f[/math] (the forecast), with an expected error variance [math]\sigma_f^2[/math]. The gauge measures the value [math]y[/math], with error variance [math]\sigma_o^2[/math]. Assuming that the two errors are random, unbiased and independent, the estimate with the smallest expected error variance is

[math] x_a = x_f + K\,(y - x_f) , \qquad K=\dfrac{\sigma_f^2}{\sigma_f^2+\sigma_o^2} . \qquad (A23) [/math]

The corrected value [math]x_a[/math] is called the analysis. The weight [math]K[/math] (the gain) lies between 0 and 1:

• If the observation is much more accurate than the model ([math]\sigma_o \ll \sigma_f[/math]), then [math]K \approx 1[/math] and the analysis is close to the observation.
• If the model is much more accurate than the observation ([math]\sigma_f \ll \sigma_o[/math]), then [math]K \approx 0[/math] and the observation is largely ignored.

The error variance of the analysis is

[math] \sigma_a^2 = (1-K)\,\sigma_f^2 = \dfrac{\sigma_f^2\,\sigma_o^2}{\sigma_f^2+\sigma_o^2} , \qquad (A24) [/math]

which is smaller than both [math]\sigma_f^2[/math] and [math]\sigma_o^2[/math]. Under the stated assumptions, combining two imperfect sources of information thus gives a better estimate than either source alone.

In a numerical model, [math]x[/math] is a vector containing the values of all model variables at all grid points, and the observations [math]\mathbf{y}[/math] are available at only a limited number of locations. Equation (A23) then becomes (see Reduction of uncertainties through Data Model Integration (DMI))

[math] \mathbf{x}_a = \mathbf{x}_f + \mathbf{K}\,(\mathbf{y}-\mathbf{H}\mathbf{x}_f) , \qquad \mathbf{K}=\mathbf{P}\mathbf{H}^T\left(\mathbf{H}\mathbf{P}\mathbf{H}^T+\mathbf{R}\right)^{-1} , \qquad (A25) [/math]

where the matrix [math]\mathbf{H}[/math] selects (or interpolates) the model values at the observation locations, [math]\mathbf{P}[/math] is the covariance matrix of the model errors and [math]\mathbf{R}[/math] is the covariance matrix of the observation errors. The off-diagonal elements of [math]\mathbf{P}[/math] describe how errors at different locations and in different variables are related. Through these elements, a correction at an observation point is spread to the surrounding grid points and to unobserved variables such as currents.

In the Kalman filter, [math]\mathbf{P}[/math] is updated in time with the model equations; in the ensemble Kalman filter it is estimated from the spread of an ensemble of model runs. Variational methods obtain the same analysis by minimizing a cost function that measures the weighted misfit to both the forecast and the observations,

[math] J(\mathbf{x}) = \tfrac{1}{2}(\mathbf{x}-\mathbf{x}_f)^T\mathbf{P}^{-1}(\mathbf{x}-\mathbf{x}_f) + \tfrac{1}{2}(\mathbf{y}-\mathbf{H}\mathbf{x})^T\mathbf{R}^{-1}(\mathbf{y}-\mathbf{H}\mathbf{x}) . \qquad (A26) [/math]

For linear models and Gaussian errors, the minimum of [math]J[/math] coincides with the analysis of Eq. (A25). In 4D-Var, the misfit to observations is summed over a time window, and the model equations link the model state at different times.

A10. Relation between model formulations

The different model formulations can be understood through the assumptions used to simplify the governing equations.

• Navier–Stokes equations: general conservation equations for fluid motion.
• RANS equations: average over turbulent fluctuations while retaining the resolved mean or wave motion; unresolved turbulence requires closure.
• Hydrostatic models: neglect vertical accelerations in the vertical momentum balance.
• Depth-averaged shallow-water models: combine the hydrostatic approximation with integration over water depth.
• Boussinesq-type models: retain selected non-hydrostatic corrections to represent frequency dispersion.
• Non-hydrostatic models: calculate departures from hydrostatic pressure through the vertical momentum balance.
• Mild-slope models: use linear wave theory and slowly varying bathymetry to represent refraction and diffraction.
• Spectral wave models: describe the statistical evolution of wave energy or action without resolving individual wave phases.
• Detailed CFD models: resolve local flow structures and complex free-surface motions, with turbulence represented according to the selected numerical approach.

These formulations are not necessarily mutually exclusive. For example, a phase-resolving numerical wave tank may solve RANS equations with a VOF free-surface representation. Similarly, a three-dimensional coastal circulation model may use hydrostatic equations with a turbulence closure based on Reynolds averaging.

The appropriate model formulation is determined by the processes that must be represented explicitly, the processes that can be parameterized and the spatial and temporal scales of the intended application.

Related articles

Reduction of uncertainties through Data Model Integration (DMI)

Scaling Issues in Hydraulic Modelling

Bedforms and roughness

Bed roughness and friction factors in estuaries

Geostrophic flow

Shelf sea exchange with the ocean

Ekman transport

Ocean and shelf tides

Coriolis and tidal motion in shelf seas

Estuary and river tides

Ocean circulation

Shoreface profile

Breaker index

Wave set-up

Wave overtopping

Tidal bore dynamics

Tsunami

Shallow-water wave theory

Statistical description of wave parameters

Data analysis techniques for the coastal zone

Further reading

Fringer, O.B., Dawson, C.N., He, R., Ralston, D.K. and Zhang, Y.J. 2019. The future of coastal and estuarine modeling: findings from a workshop. Ocean Modelling 143: 101458

The main authors of this article are Job Dronkers, Prinos, Panayotis, De Rijcke, Maarten, Steendam, Gosse Jan, Zanuttigh, Barbara, Lopez, Lara and Andersen, Thomas Lykke Please note that others may also have edited the contents of this article. For other articles by this author see Category:Articles by Job Dronkers For other articles by this author see Category:Articles by Prinos, Panayotis For other articles by this author see Category:Articles by De Rijcke, Maarten For other articles by this author see Category:Articles by Steendam, Gosse Jan For other articles by this author see Category:Articles by Zanuttigh, Barbara For other articles by this author see Category:Articles by Lopez, Lara For other articles by this author see Category:Articles by Andersen, Thomas Lykke