So the BEST papers are assuming that the earth has at each instant in time a "mean temperature" $\theta(t)$ which is affected by a purely spatial additive term $C(x)$ (loosely "climate") and a spatially and temporally varying term $W(x,t)$ (loosely "weather") to give the measure temperature $T(x,t)$:

$T(x,t) = \theta(t) + C(x) + W(x,t)$

with some additional cross-site constraints on the C and W terms. (They do say in passing in the paper that some effects which would conventionally be regarded as climate are included in their weather term). Is that reasonable in terms of physics? (Clearly there is an average global temperature at any time, the question is whether the temperatures all over the globe are accurately modelled by additive modifications which are unchanging.)