Environmental Modeling & Assessment

, Volume 22, Issue 4, pp 379–395

# Application of Adjoint Approach to Oil Spill Problems

• Yuri N. Skiba
• David Parra-Guevara
Article

## Abstract

Adjoint method is applied to various oil spill problems. A three-dimensional model for describing the dispersion of a quasi-passive substance (a pollutant or a nutrient) and its adjoint model are considered in a limited sea region. Direct and adjoint estimates are used to get dual (equivalent) estimates of the mean concentration of the substance in important zones of the region. The role of dual estimates is illustrated with a few examples. They include such oil spill problems as the search of the most dangerous point of the oil tanker route, the oil dispersion with a climatic velocity, and the dependence of the oil concentration estimates on the oil spill rate. One more example is the application of optimal bioremediation strategy for cleaning a few zones polluted by oil. In this case, instead of oil, the model describes the dispersion of a nutrient released to marine environment. Balanced, unconditionally stable second-order finite-difference schemes based on the splitting method for the solution of the dispersion model and its adjoint are suggested. The main and adjoint difference schemes are compatible in the sense that at every fractional step of the splitting algorithm, the one-dimensional split operators of both schemes satisfy a discrete form of Lagrange identity. In the special unforced and non-dissipative case, each scheme has two conservation laws. Every split one-dimensional problem is solved by Thomas’ factorization method.

## Keywords

Pollution transport Direct and adjoint estimates Oil spill problems Bioremediation

## Notes

### Acknowledgements

This work was supported by the grants 14539 and 25170 of National System of Researches (CONACyT, México).

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© Springer International Publishing Switzerland 2016

## Authors and Affiliations

• Yuri N. Skiba
• 1
• David Parra-Guevara
• 1
1. 1.Centro de Ciencias de la AtmósferaUniversidad Nacional Autónoma de MéxicoMéxico, D. F.México