We show that the flux of mass crossing in one direction (the gross flux) through any specified surface S that divides anadvective-diffusive flow in a closed domain is infinite. That is, the flux, ${cal F}$ (τ), through S of the fluid mass that spent at least time τ on one side of S diverges like τ-1/2 as τ → 0, in the continuum limit. The gross flux is completely dominated by fluid elements residing infinitesimally short times on one side of S before re-crossing to the other side. This general result puts into context the widely varying estimates of gross mass flux across the midlatitude tropopause. Such estimates are dominated by the smallest resolved scales, leading us to conclude that gross mass flux is not a useful diagnostic of stratosphere-troposphere exchange. The function ${cal F}$ (τ), however, provides important information on transport across the tropopause.