59models. Below, results are shown with and without consideration of these three
terms. No lag is imposed for these three terms since the indices used to describe
these processes vary slowly with respect to time.
The coefficients (C 1 to C 6 ) that multiply the various model terms, as well as the
constant term C 0 and the variable γ, are found using multiple linear regression, which
provides numerical values for each of these parameters such that the Cost Function
(Eq. 2.1) has the smallest possible value. The term γ in Eq. 2.2 is the dimensionless
climate sensitivity parameter. If the net response of changes in humidity, lapse rate,
clouds, and surface albedo that occur in response to anthropogenic ΔRF of climate is
positive, as is most often the case, then the value of γ is positive.
The estimate of QOCEAN is based on finding the value of the final model output
parameter κ, the ocean heat uptake efficiency coefficient with units of W m−^2 °C−^1
(Raper et al. 2002 ) that best fits a time series of ocean heat content (OHC), where:
QOCEANiiii
P= GHGRFAerosol RF LUCRF+
k --++-g
lDDD
1
() 72 72 72
(2.3)
The subscripts i − 72 in Eq. 2.3 represent a 6 year (or 72 month) lag between the
anthropogenic ΔRF perturbation and the export of heat to the upper ocean. The
numerical estimate of this lag is based on the simulations described by Schwartz
( 2012 ); the projections of global warming found using the EM-GC framework are
insensitive to any reasonable choice for the this lag. Since the model is based on
matching perturbations in RF of climate to variations in temperature, the flow of
heat from the atmosphere to the ocean is modeled as a perturbation to the mean state
induced by anthropogenic RF of climate (i.e., QOCEAN in Eq. 2.2 depends only on
“delta” terms that represent human influence on climate). Finally, the net effect of
human activity on ΔT is the sum of GHG warming, aerosol cooling, very slight
cooling due to land use change, and ocean heat export:
D
lTGHUMAN gDHG RF AerosolRFLDDUC RF
Pii=+ ++ii-QOCEANi1
[( 1 )( )]
(2.4)
Equations 2.1–2.4 constitute our Empirical Model of Global Climate. Of the
model inputs, the aerosol ΔRF term is the most uncertain. As shown below, there is
a strong relation between the value of the climate sensitivity parameter γ and the
magnitude of aerosol ΔRF. This dependency is well known in the climate commu-
nity, as discussed for example by Kiehl ( 2007 ). Also, there is a wide variation in the
value of κ, depending on which dataset is used to specify OHC.
Figures 2.4 and 2.5 provide a graphical illustration of how the model works. The
simulations in these figures use estimates for GHG and aerosol ΔRF from RCP 4.5,
tied to the best estimate for aerosol ΔRF in year 2011 (AerRF 2011 ) of −0.9 W m−^2
from IPCC ( 2013 ), and a time series for OHC in the upper 700 m of the global
oceans that is an average of six published studies. In the interest of keeping the
attention of those reading this far, we describe a few simulations prior to delving
into further details about the model parameters.
2.2 Empirical Model of Global Climate