Copepod diffusion within multifractal phytoplankton fields

Abstract

Oceanic turbulence has been considered for a while as one of the main sources of the heterogeneity of the phytoplankton field over a wide range of scales. However, it is only recently that the intermittency of turbulence has been taken into account, although it is rather indispensable in the explanation of the observed patchiness of the plankton field. In order to improve the understanding and characterization of the diffusion of copepods within an heterogeneous phytoplankton field, we developed a model based on particle diffusion in a multifractal field. After discussing this model with some detail, we used it and the corresponding numerical simulations in order to investigate the fundamental question: does the strong heterogeneity of the phytoplankton generate a anomalous diffusion of the copepods? Although a positive answer is obtained in a rather straightforward manner for a one-dimensional multifractal field of phytoplankton concentration, the answer is rather more involved for a greater (topological) dimension of the field, contrary to some previous claims.

Introduction

Understanding the effect of the oceanic turbulence on secondary production, i.e. the transfer of organic matter from phytoplankton to zooplankton, is a challenging objective. In particular, one would like to know how much the development of turbulence increases the predator–prey contact rate, favorising the zooplankton. The existence of intimate relationships between physical and biological processes has been previously considered (Denman and Powell, 1984; Legendre and Demers, 1984; Mackas et al., 1985), due to the observed coupling between the distribution of phytoplankton populations and the structure of their physical environment over a wide range of spatial and temporal scales (Haury et al., 1978; Steele, 1985). However, in order to demonstrate it and understand it better, one needs to simulate the involved biological and physical processes on a similar range of scale. Unfortunately, this cannot be done by direct numerical simulations due at least to the rather limited memory size of our computers. The classical approach (Malchow and Shigesada, 1994) corresponds to truncating the original set of governing equations in a narrow band of scales (e.g. with a scale ratio of the order which is at best of one hundred on supercomputers), which is turn requires ad hoc parametrizations (the so called sub-grid modelling). Instead, we consider a multifractal modelling of the fields, i.e. a stochastic modelling physically based on the scale symmetries of the original equations. These symmetries are immediately lost as soon as the corresponding equations are truncated. Space and/or time multifractal modelling has been originally developed for hydrodynamic turbulence, clouds and rain fields. More precisely, these techniques will be used in order to simulate the phytoplankton field advected by oceanic turbulence, as discussed in Section 2. The copepod diffusion —diffusion being understood in the sense of displacement of inert or living particles within a media (Okubo, 1980)— will be controlled by a local diffusivity depending on the local concentration of the phytoplankton (see Section 3). Corresponding to the notion of zooplankton grazing, the general rule to be followed is that this diffusivity should decrease with phytoplankton concentration, since copepods linger where the food is most abundant, and move away from where the food is scarce. Therefore, we have to study the diffusion of particles in a multifractal media. This diffusion is expected to behave quite differently from the classical and so called `normal' diffusion in homogeneous media, which corresponds to Brownian particle walks (e.g. Gouyet, 1992). Indeed, this anomalous diffusion should reflect at a given level the intermittency of the medium, i.e. particle walks should have bursts of speedy diffusion which do not exist in the case of normal diffusion. We will discuss how to characterize this anomalous diffusion and we will show that one cannot rely on the classical method of estimating the scaling of the average distance travelled by particles as a function of time.