Modelling algae transport in coastal areas with a shallow water equation model including wave effects

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Abstract

The transport of algae in the marine environment is becoming a major issue for integrated sustainable coastal management and can be nowadays investigated through numerical modelling. The Shallow Water Equations are a suitable tool for this purpose. In the framework of the present study, they are used to perform predictions of algae transport in the vicinity of a harbour, combined to a 3rd generation spectral wave model to take account of near-shore currents induced by wave breaking in the surf zone. The simulated paths of algae have been proved to be very sensitive to the effects of those wave-induced currents and exhibit a behaviour in acceptable agreement with on site observations. In particular, the effects of waves on currents is required to qualitatively predict the large recirculation pattern observed in the vicinity of one of the dykes of the harbour. This recirculation is suspected to be one of the major mechanisms of algae dynamics in this area of the coast. Stochastic (random-walk) diffusion was also tested to account for local dispersion of algae due to large turbulent eddies.

Introduction

During summer, algal blooms can be a major problem for the sustainability of beaches, harbours and on-shore industry factories. The numerical modelling of their growth and transport through scalar advection–diffusion is nowadays currently performed for predictions over large-scale areas (i.e. typically 10–100 km, see e.g. Donaghay and Osborn, 1997) and large time scale (several months), but it does not solve the question of the motion of algae at a smaller length scale (i.e. 10 m to 1 km) and time scale (i.e. 1–24 h) in the vicinity of local coastal structure such as harbours. For doing so, looking at individual motion is a better practice, which can be carried out from the study of algae paths in a shallow water model.

Among the phenomena which affect the motion of algae in the marine environment, one should mention tides and waves, but also diffusion under turbulent fluctuations. In our zone of interest, a harbour located on French coasts of the British Channel, it was observed that algae tend to accumulate in preferred areas, typically near the west dyke of the harbour (see Fig. 1), moving according to large recirculation patterns. It also appeared that they form long and persistent “filaments” instead of making two-dimensional “clouds” as one could have expected. This behaviour was also observed in the case of oil spills, and will be explained below from the numerical experience.

The purpose of this study is to prove the ability of numerical modelling to predict the motion of algae at small length and time scales, using the Shallow Water Equations (SWE) combined with a spectral wave model. It will be shown that the effect of waves is necessary to correctly assess this issue. Random-walk motion of individual algae will be then tested to account for turbulent dispersion.

Section snippets

The shallow water equation model Telemac-2D

Developed at EDF R&D (Electricité de France, R&D branch) since 1987, the Telemac-2D code solves the two-dimensional Shallow Water Equation (SWE), or depth-integrated Navier–Stokes equations in the following (conservative) form:ht+·(hU)=0hUt+·(hUU)=ghη+·(hD+hR)+γhez×U+F+Swhere h denotes the water depth, U = (Ux,Uy) the two-dimensional depth-averaged velocity vector, g the gravity acceleration, η the free surface elevation, γ = 2 Ωsin λ the Coriolis parameter (Ω being the Earth's

The spectral wave model Tomawac

Besides the Telemac-2D code, EDF R&D also develops (among others) the finite-element code Tomawac, for modelling sea states. It is a 3rd generation model solving the equation governing the spectral wave action density N(x,k) (see e.g. Komen et al., 1994):Nt+x,k·(NV)=i=16Qiwhere V=(x˙,k˙) is a four-dimensional vector, x being the spatial position vector and k the wave vector (both having two components), and ∇x,k a four-dimensional gradient operator. The components of V represent the

Stochastic diffusion

In all simulations achieved here, algae are modelled as water particles, which is reasonable considering that algae have a density close to that of water. However, the previous model would be appropriate to model algae bloom transport in the absence of diffusion. In reality, turbulence affects the motion of individual algae through eddies, making their path random-like. Due to the large size and inertia of algae, not all the eddies will have a significant effect on their motion, but the largest

Application to algae motion

The models presented above are used here to simulate combined waves and currents in the area presented on Fig. 2, the extension of which is about 20 km longshore and 10 km offshore. Two cases are considered to model algae motion: a purely hydrodynamic simulation and a second considering stochastic diffusion. The purely hydrodynamic simulation was performed in two stages: a tidal hydrodynamic case (without wave effect) has been first carried out with Telemac-2D. Free surface elevation as well as

Conclusions

A model of combined waves and currents have been built and used to predict the behaviour of algal blooms near a harbour. It provides predictions in a satisfactory agreement with observations. In particular, it was shown that an accumulation area exists near the west dyke of the harbour, which cannot be properly modelled without taking account of wave-induced currents due to radiation stresses. A random-walk Langevin-type model was also tested to account for turbulent mixing due to large eddies.

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