Carrying capacity for finfish aquaculture. Part I-Near-field semi-analytic solutions

The ecosystem carrying capacity for aquaculture cage farming in South Australia is based on guidelines that the maximum feed rates (and farmed fish biomass) be determined such that the concentration c of a given dissolved nutrient does not exceed a prescribed value (say cP). The problem then is one of relating the nutrient flux F, due to feeding, to the tracer concentration c. To this end the evolution of concentration is modelled using the depth-averaged advection-diffusion equation for a constant source flux F over a finite area cage (or lease) and for both constant and time dependent (tidal) velocities. The divergence theorem is applied to this equation to obtain a new scale estimate of the relation between the flux F and the maximum concentration cmax of a nutrient in the cage region: cmax ≈ F·T*, where T* is a time scale of cage “flushing” that involves both advection and diffusion. The maximum allowed nutrient flux F (and carrying capacity of fish biomass) can then be simply estimated from: F ≈ cP/T*. New semi-analytic solutions of the advection-diffusion equation for a finite (cage) source are then derived to explore the physics of concentration evolution for constant and tidally varying currents, and to show that the estimate cmax ≈ F·T* is surprisingly robust and generally within 40% of the exact values for a wide set of advective/diffusive parameters. The results generally should find application in other finite source flux problems in the coastal oceans including desalination plants and waste water outfalls.