Contaminant contraction in two-dimensional oscillatory flows

If the vertically-mixing time is comparable with the period of oscillatory current, the contaminant contraction may occur, then the coefficient of shear dispersion will be negative (singularity). According to the two-dimensional delay-diffusion equation derived by Liu in another paper, The equations for X(t), Y(t), central displacement as well as dispersion tensor were derived. Generally speaking, δDij/δτ is positive when τ is small; but when τ is large, the memory functions may be negative. The analytical expressions for Dij and X,Y were also given.