An analytic solution of the time-dependent, one-dimensional diffusion equation in the atmospheric boundary layer

Abstract The diffusion equation with a variable K -coefficient describes the dispersion of passive contaminants by turbulence. Only a few analytic solutions of this equation are known. These are based on the assumption that the K -coefficient can be expressed in the form of a power law. Such an assumption is not realistic in the atmosphere, where the K -coefficient must approach zero both at the surface and at the top of the boundary layer. We present here an analytic solution of the time-dependent, one-dimensional diffusion equation in which the K -coefficient is given by a more realistic profile expressed by the equation K = cu ∗ z (1 − z / h ), where c is a constant, u ∗ is the friction velocity, h is the boundary-layer height and z is the vertical coordinate. The solution contains a series of Legendre polynomials. The difference between this solution and the analytic solution obtained with a power-law profile for K is investigated.