The artifact is clearly demonstrated by the fact that the equation always leads to increases in iWUE. Simulated (random) δ13C values, used here to replace tree-ring measurements, are obviously independent from physiological responses and yet lead to increasing iWUE over time. This is explained by the fact that Ca is used as an independent variable in the calculation, artificially producing positive responses as shown in Figure 1.
Indeed according to Fick's law A = g (Ca − Ci) with CO2 and H2O related by a constant factor that links gas exchange of carbon and water vapor. However, in the scenario proposed in the comment above, Ci would have to follow exactly the same increase in Ca to produce a constant Ca − Ci, which is not realistic given that gas exchange and carboxylation continue to occur. The Ci/Ca ratio, on the other hand, reflects the balance between A and g, so the assumption is that the linear relationship between Ci/Ca and 13C discrimination (eq. 2) can be used to calculate iWUE (A/g).
Based on the recent literature, we consider our results in the context of expected physiological responses (such as increases in A or declines g), which should cause changes in 13C discrimination. We found, however, that increases in iWUE would occur in any analysis independently of such physiological responses and respective changes in discrimination. The final outcome of our model reflects (based on the existing equations) what would be observed if Ci increased proportionally with Ca. This is a conservative scenario compared with, for example, constant Ci which would represent much stronger increases in iWUE.
For these reasons, we maintain that previously reported trends in iWUE do not necessarily reflect coherent physiological responses to rising CO2. This conclusion is further supported by the lack of consistent growth responses across biomes (see references cited).