  Many lamellar systems exhibit strongly anisotropic diffusion. When the
diffusion across the lamellae is slow, an alternative mechanism for transverse
transport becomes important. A tracer particle can propagate in the direction
normal to the lamellae, never leaving a particular layer, by going around a
screw dislocation. Given the density of positive and negative screw
dislocations, we calculate the statistical properties of the transverse
transport. When either positive or negative dislocations are in excess, the
tracer moves ballistically normally to the layers with the mean square of the
displacement growing like the square of time T^2. When the average dislocation
charge is zero, the mean square of the normal displacement grows like T log T
for large times. To obtain this result, the trajectory of the tracer must be
smoothed over distances of order of the dislocation core size.
