Emergent constraints for the climate system as effective parameters of bulk differential equations

Planning for the impacts of climate change requires accurate projections by Earth system models (ESMs). ESMs, as developed by many research centres, estimate changes to weather and climate as atmospheric greenhouse gases (GHGs) rise, and they inform the influential Intergovernmental Panel on Climate Change (IPCC) reports. ESMs are advancing the understanding of key climate system attributes. However, there remain substantial inter-ESM differences in their estimates of future meteorological change, even for a common GHG trajectory, and such differences make adaptation planning difficult. Until recently, the primary approach to reducing projection uncertainty has been to place an emphasis on simulations that best describe the contemporary climate. Yet a model that performs well for present-day atmospheric GHG levels may not necessarily be accurate for higher GHG levels and vice versa. A relatively new approach of emergent constraints (ECs) is gaining much attention as a technique to remove uncertainty between climate models. This method involves searching for an inter-ESM link between a quantity that we can also measure now and a second quantity of major importance for describing future climate. Combining the contemporary measurement with this relationship refines the future projection. Identified ECs exist for thermal, hydrological and geochemical cycles of the climate system. As ECs grow in influence on climate policy, the method is under intense scrutiny, creating a requirement to understand them better. We hypothesise that as many Earth system components vary in both space and time, their behaviours often satisfy large-scale differential equations (DEs). Such DEs are valid at coarser scales than the equations coded in ESMs which capture finer high-resolution grid-box-scale effects. We suggest that many ECs link to such effective hidden DEs implicit in ESMs and that aggregate small-scale features. An EC may exist because its two quantities depend similarly on an ESM-specific internal bulk parameter in such a DE, with measurements constraining and revealing its (implicit) value. Alternatively, well-established process understanding coded at the ESM grid box scale, when aggregated, may generate a bulk parameter with a common “emergent” value across all ESMs. This single emerging parameter may link uncertainties in a contemporary climate driver to those of a climate-related property of interest. In these circumstances, the EC combined with a measurement of the driver that is uncertain constrains the estimate of the climate-related quantity. We offer simple illustrative examples of these concepts with generic DEs but with their solutions placed in a conceptual EC framework.