Contaminant contraction in two-dimensional oscillatory flows

Abstract

If the vertically-mixing time is comparable with that of period oscillatory current, the contaminant contraction may occur. The coefficient of shear dispersion is negative (singularity). According to the two-dimensional delay-diffusion equation derived by the author^[7]:

$$\begin{gathered} \frac{{\partial \bar C}}{{\partial t}} + \bar u\frac{{\partial \bar C}}{{\partial x}} + \bar v\frac{{\partial \bar C}}{{\partial y}} = \bar K_{zz} \frac{{\partial ^2 \bar C}}{{\partial x^2 }} + \bar K_{yy} \frac{{\partial ^2 \bar C}}{{\partial y^2 }} + \int_0^\infty {\left[ {\frac{\partial }{{\partial \tau }}} \right.} D_{zz} \frac{{\partial ^2 }}{{\partial x^2 }} \hfill \\ \left. { + \frac{\partial }{{\partial \tau }}\left( {D_{zy} + D_{yz} } \right)\frac{{\partial ^2 }}{{\partial x\partial y}} + \frac{\partial }{{\partial \tau }}D_{yy} \frac{{\partial ^2 }}{{\partial y^2 }}} \right]\bar C\left( {x - X,y - Y,t - \tau } \right)d\tau \hfill \\ \end{gathered} $$

where$\bar u\left( t \right),\bar v\left( t \right)$ are vertically-averaged velocities, the equations for X(t), Y(t), central displacement, dispersion tensor, had been derived. ∂D_ij/∂τ is positive when τ is small. If the τ is large, the memory functions may be negative. Also the expressions for D_ij and X, Y had been obtained.