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There is as yet no theoretical framework to guide the search for emergentconstraints. As a result, there are significant risks that indiscriminatedata-mining of the multidimensional outputs from GCMs could lead to spuriouscorrelations and less than robust constraints on future changes. To mitigateagainst this risk, Cox et al (hereafter CHW18) proposed a theory-motivatedemergent constraint, using the one-box Hasselmann model to identify a linearrelationship between ECS and a metric of global temperature variabilityinvolving both temperature standard deviation and autocorrelation ($\Psi$). Anumber of doubts have been raised about this approach, some concerning thetheory and the application of the one-box model to understand relationships incomplex GCMs which are known to have more than the single characteristictimescale. We illustrate theory driven testing of emergent constraints usingthis as an example, namely we demonstrate that the linear $\Psi$-ECSproportionality is not an artifact of the one-box model and rigorously featuresto a good approximation in more realistic, yet still analytically solubleconceptual models, namely the two-box and diffusion models. Each of theconceptual models predict different power spectra with only the diffusionmodel's pink spectrum being compatible with observations and the complex CMIP5GCMs. We also show that the theoretically predicted $\Psi$-ECS relationshipexists in the \texttt{piControl} as well as \texttt{historical} CMIP5experiments and that the differing gradients of the proportionality areinversely related to the effective forcing in that experiment.

Theoretical foundations of emergent constraints: relationships between climate sensitivity and global temperature variability in conceptual models