Analysis and modeling of scale-invariance in plankton abundance

The power spectrum, S, of horizontal transects of plankton abundance are often observed to have a power-law dependence on wavenumber, k, with exponent close to −2: S(k) ∝ k over a wide range of scales. I present power spectral analyses of aircraft lidar measurements of phytoplankton abundance from scales of 1 to 100 km. A power spectrum S(k) ∝ k is obtained. As a model for this observation, I consider a stochastic growth equation where the rate of change of plankton abundance is determined by turbulent mixing, modeled as a diffusion process in two dimensions, and exponential growth with a stochastically variable net growth rate representing a fluctuating environment. The model predicts a lognormal distribution of abundance and a power spectrum of horizontal transects S(k) ∝ k, close to the observed spectrum. The model equation predicts that the power spectrum of variations in abundance in time at a point in space is S(f) ∝ f (where f is the frequency). Time series analysis of local variations of phytoplankton and zooplankton yield a power-law power spectrum with exponents −1.3 and −1.2, respectively from time scales of one hour to one year. These values are roughly consistent with the model prediction of −1.5. The distribution of abundances is nearly lognormal as predicted. The model may be more generally applicable than for the spatial distribution of plankton. I relate the model predictions to observations of spatial patchiness in vegetation.