Sverdrup meets Lambert: analytical solution for Sverdrup’s critical depth

Abstract

Abstract In his 1953 paper, Harald Sverdrup argued that the development of a spring bloom in the ocean depends on the juxtaposition of two depth horizons: the mixed-layer depth and the critical depth. Mixed-layer depth shallower than the critical depth favours phytoplankton growth in the layer and vice versa. However, mathematically, Sverdrup left the problem unsolved in the form of a transcendental equation. In spite of the high number of citations that this paper has garnered, the solution to this equation has not been found, until now. In this work, we present an analytical solution for the critical depth, as originally defined by Sverdrup. The paper opens with the definition of the critical depth and the description of the Lambert W function. The analytical solution for critical depth follows. Sverdrup’s original model is extended to include the effect of light attenuation by phytoplankton and the analytical solution for steady-state biomass in the mixed layer is derived. The expression for mixed-layer production at steady state is also presented. Two novel variants of the critical depth are defined: the optically uncoupled critical depth and the optically coupled critical depth. It is demonstrated that at steady state the optically coupled critical depth equals the mixed-layer depth and that the irradiance at the base of the mixed layer equals the irradiance at the optically uncoupled critical depth. Competitive exclusion is demonstrated to hold and the optically uncoupled critical depth is linked to the critical light intensity in multi-species competition. Finally, a conservation principle for the critical depth is found.