One-dimensional checkerboards and blending heights

We analyze the checkerboard problem of many alternating surfaces with different properties, on scales up to (say) 3,000 m. Power-law representations of the vertical profiles of mean wind speed and eddy diffusivity lead to solutions in terms of Kelvin and trigonometric functions.These solutions are used to determine blending heights (_*), where deviations from the mean of concentration, or of vertical flux density, fall to some small fraction, , of their value at the surface. Values of _*are important for regional and larger-scale meteorological models. In smaller scale micrometeorological studies, they may serve also as the top levels of surface boundary layers.An important result for both theoretical and experimental contexts is that deviations of flux persist with elevation much more strongly than those of concentration, so that, in general, _* should be based on flux rather than concentration. Representative values of _*, for = 0.05, are of order 5 and 30 m for surface pattern wavelengths of 10² and 10³ m, respectively. Values of _* are robust to changes in adopted power-law indices, and are independent of wind speed. Surface roughness has a mild but calculable effect.