We construct an idealized one-dimensional model of the rectifier effect and solve it analytically. The model equations and solutions depend on a single ``rectifier" parameter. This parameter represents the degree of day-night difference in the eddy diffusivity of the atmosphere.
Using this model, it is relatively straightforward to perform 1D inverse calculations that account for the diurnal rectifier effect. However, the inverse equations reveal that this particular inversion problem has some potential pitfalls. Specifically, a surface measurement alone is insufficient to infer the depth of vertical transport. Rather, measurements at two or more altitudes are required, from a tower for instance. This is because surface CO2 time series are fundamentally ambiguous in this 1D setting: the same time series may arise from strong surface flux and transport, or weak flux and transport. Furthermore, when CO2 is measured at two and only two altitudes, there may remain further ambiguities. If the measurement altitude aloft is too low, and the CO2 mixes deep into the atmosphere, the inverse estimate may be imprecise. If the altitude of measurement is approximately the same as the depth over which CO2 mixes, then the inverse estimate may be non-unique.
Supplementary URL: http://www.uwm.edu/~vlarson