Dual oil concentration estimates in ecologically sensitive zones

Propagation of the oil spilling from a damaged oil tanker is considered in a limited sea area. Direct and adjoint estimates of the average oil concentration in special zones are derived by using solutions of the 2-D main and adjoint oil transport problems, respectively. The dual estimates complement each other nicely in studying the oil spill consequences. While the direct estimates are preferable to get a comprehensive idea of the oil spill impact on the whole area, the adjoint ones are specially useful and economical when the accident site-dependence or/and the oil spill rate-dependence of the oil concentration is/are studied only in a few ecologically sensitive zones. Indeed, each adjoint estimate explicitly relates the average oil concentration in a zone to the oil spill rate using the adjoint solution values at the accident site. Being independent of the two parameters (the accident site and oil spill rate), the adjoint solution can be found for each zone regardless of a concrete accident and used repeatedly for various possible values of these parameters. Several examples explain how to decide between two estimates.Thanks to special boundary conditions, the main and adjoint problems are both well-posed according to Hadamard (1923). The dual estimates can be generalized to the three dimensions. The balanced, absolutely stable and compatibie main and adjoint 3-D numerical algorithms by Skiba (1993) can easily be adapted to the problem discussed here.