  The evolution of a passive scalar field is considered for a slowly varying
stratified medium, which is convected in an incompressible sheared flow with
many overlapping static flux islands. Within the quasilinear/random phase
approximation, a multiple scale expansion is made. Due to the rapid spatial
variation of the temperature, the "ensemble" averaged/ slowly varying part of
the solution is not described by the arithmetic average of the oscillatory
evolution equation. The standard Markovian and continuum approximations are
shown to be invalid. For times of order $N$, where there are $O(N^2)$ excited
modes, most of the time dependent perturbation phase mixes away and the fluid
reaches a new saturated state with small time oscillations about the
temperature. This saturated state has smaller resonance layers, (corresponding
to magnetic islands) than those that occur in the isolated resonant
perturbation case. Thus the quasilinear response to the resonant interactions
reduces the effective size of the perturbations. The temperature gradient of
the saturated state vanishes at all the excited resonance surfaces but has a
nonzero average. Thus either the quasi-linear approximation ceases to be valid
on long time scales, or the fluid remains essentially in this modified
equilibrium and does not evolve diffusively. Thus collisionless, driftless fast
particles will not be lost rapidly in equilibria with many small islands.
