Critical Times for the Critical Depth Theory

Abstract

Critical Depth Hypothesis is arguably one of the longest standing biophysical theories in oceanography and is the earliest mathematically formulated theory aimed at explaining the phenomenon of phytoplankton blooms. It introduces a depth horizon, termed the critical depth, at which the integrated primary production from the surface to that depth equals the integrated loss terms within the same layer. In mixed layers deeper than the critical depth, losses dominate photosynthesis and vice versa. A related horizon in case of week mixing is the compensation depth, where the rate of photosynthesis matches the loss rate. In this paper, the effect of phytoplankton light attenuation on the critical depth is examined, showing that it creates a bio-optical feedback in the model. A new differential equation, derived for the time evolution of the compensation depth reveals that the light intensities at both the compensation depth and the critical depth are constants of motion. Exact solutions for average and total mixed layer biomass at steady state are derived, and their stability properties are analyzed. An existence of a bio-optical bifurcation is shown, in which the mixed layer depth acts as the bifurcation parameter and the critical depth is identified as the bifurcation point. Transients between steady states are also explored, and it is shown that the relation between the initial condition and the final steady state is paramount in determining whether a shallowing or deepening of the mixed layer will lead to a rise or a decline in biomass over time.