We quantify the flow stratification in the Earth's mid-mantle (600--1500 km) in terms of a stratification index for the vertical mass flux, Sf(z)=1-f(z)/fref(z), in which the reference value fref(z) approximates the local flux at depth z expected for unstratified convection (Sf=0). Although this flux stratification index cannot be directly constrained by observations, we show from a series of two-dimensional convection simulations that its value can be related to a thermal stratification index ST(z) defined in terms of the radial correlation length of the temperature-perturbation field ΔT(z,&OHgr;). ST is a good proxy for Sf at low stratifications (Sf<0.2), where it rises with stratification strength much more rapidly than Sf. Assuming that the shear-speed variations Δβ(z,&OHgr;) imaged by seismic tomography are primarily due to convective temperature fluctuations, we can approximate ST by Sβ, the analogous index for the radial correlation length of Δβ, and thereby construct bounds on Sf. We discuss several key issues regarding the implementation of this strategy, including finite resolution of the seismic data, biases due to the parameterization of the tomographic models, and the bias and variance due to noise. From the comparison of the numerical simulations with recent tomographic structures, we conclude that it is unlikely that convection in the Earth's mantle has Sf≳0.15. We consider the possibility that this estimate is biased because mantle convection is intermittent and therefore that the present-day tomographic snapshot may differ from its time average. Although this possibility cannot be dismissed completely, we argue that values of Sf≳0.2 can be discounted under a weak version of the Uniformitarian Principle. The bound obtained here from global tomography is consistent with local seismological evidence for slab flux into the lower mantle; however, the total material flux has to be significantly greater (by a factor of 2--3) than that due to slabs alone. A stratification index, Sf≲0.2, is sufficient to exclude many stratified convection models still under active consideration, including most forms of chemical layering between the upper and lower mantle, as well as the moreextreme versions of avalanching convection governed by a strong endothermic phase change.¿ 1997 American Geophysical Union