Analysis and modeling of scale-invariance in plankton abundance
Abstract
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
−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)∝
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)∝
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)∝
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.