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: $S(k)\propto k^{-2}$ 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 $S(k)\propto k^{-2}$ 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 $S(k)\propto k^{-1.8}$, 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 $S(f)\propto f^{-1.5}$ (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.