The proportionality of temperature to forcing in transient climate change

As the radiative forcing from greenhouse gases increases with time, the surface layer of the ocean (top 50 m or so) quasi-equilibrates with this forcing on a relatively short time scale of 5 years or so. The deep ocean, on the other hand, takes centuries to equilibrate. This separation of timescales implies that on intermediate, multi-decadal timescales, the surface temperature anomaly is simply proportional to the forcing, where the proportionality constant is a well-studied quantity known as the "transient climate response", or TCR. Here we review the physics of this simple model and demonstrate that knowledge of Earth's TCR, along with a projection of future forcing, is sufficient to predict when the Earth will cross a given temperature target, such as the internationally agreed upon 1.5 or 2 C targets.