Asymptotic Theory of Classical Impurity Transport in Inhomogeneous Media. Fermat's Principle

An asymptotic theory of impurity transport based on advection-diffusion in media with large-scale inhomogeneities is developed. The expression for the concentration is reduced to one-dimensional integrals along the characteristic line called the trajectory of the concentration signal. The trajectory itself is determined from the variational principle-an analog of Fermat's principle in geometrical optics-which leads to a first-order ordinary differential equation for the unit tangent vector to the trajectory. The asymptotic theory is applicable at distances from the impurity source much larger than the size of the main distribution region of the impurity.