We construct a simple model for diurnal mixing in the planetary boundary layer $($PBL$)$ by dividing the PBL vertically into two superimposed domains: $(1)$ the mixed layer and $(2)$ the remnant PBL $($see figure below$).$ These two domains are separated by an inversion, and a second inversion caps the remnant PBL$.$ We assume that the domains are individually well mixed and that there is no vertical exchanore arrose the inversions

Provide a brief justification for this model, and for the diurnal variation in the sizes of the two domains. Why is there a mixed layer at night? $($Hint: buoyancy is not the only source of vertical turbulent mixing$).$

Consider an inert pollutant $x$ emitted from the surface with a constant emission flux beginning at $t=0($midnight$).$ Plot the change in the concentration of $x$ from $t=0$ to $t=24$ hours in domains $(1)$ and $(2),$ starting from zero concentrations at $t=0$ in both domains.