The direct and inverse problem for two-dimensional turbulent diffusion

Summary

This paper examines the problem of the diffusion of a pollutant in the atmosphere considering the effect of the wind and turbulence. The following equation is considered:

$$u\left( z \right)\frac{{\partial c\left( {z,x} \right)}}{{\partial x}} = \frac{\partial }{{\partial z}}\left( {K\left( z \right)\frac{{\partial c\left( {z,x} \right)}}{{\partial z}}} \right)\left( {z,x} \right) \in \Omega ,$$

whereΩ={(z, x)|z>0, x>0}, with the initial condition

$$c\left( {z,0} \right) = \delta \left( {z - H_s } \right)z \geqslant 0,$$

and the boundary condition

$$\frac{{\partial c\left( {0,x} \right)}}{{\partial z}} = 0,x \geqslant 0,$$

whereσ(z−H _s) is Dirac's σ distribution concentrated at the pointz=H _s>0, and whereu(z) andK(z) are real valued piecewise constant functions with a finite number of jumps. For this problem an integral representation ofc(z, x) is obtained; using this integral representation we consider an inverse problem. That is from the knowledge ofc(0, x) forx≥0 we will recover the product ofu(z) K(z) forz>0. Finally the behaviour ofc(0, x), is studied analytically and numerically in some particular cases.