In the following, we test the validity of a one-box climate model as an
emulator for atmosphere–ocean general circulation models (AOGCMs). The one-box
climate model is currently employed in the integrated assessment models FUND,
MIND, and PAGE, widely used in policy making. Our findings are twofold.
Firstly, when directly prescribing AOGCMs' respective equilibrium climate
sensitivities (ECSs) and transient climate responses (TCRs) to the one-box
model, global mean temperature (GMT) projections are generically too high by
0.5 K at peak temperature for peak-and-decline forcing scenarios, resulting
in a maximum global warming of approximately 2 K. Accordingly, corresponding
integrated assessment studies might tend to overestimate mitigation needs and
costs. We semi-analytically explain this discrepancy as resulting from the
information loss resulting from the reduction of complexity. Secondly, the
one-box model offers a good emulator of these AOGCMs (accurate to within 0.1 K
for Representative Concentration Pathways, RCPs, namely RCP2.6, RCP4.5, and
RCP6.0), provided the AOGCM's ECS and TCR values are universally mapped onto
effective one-box counterparts and a certain time horizon (on the order of the
time to peak radiative forcing) is not exceeded. Results that are based on
the one-box model and have already been published are still just as informative
as intended by their respective authors; however, they should be
reinterpreted as being influenced by a larger climate response to forcing
than intended.
Introduction
Climate–economy integrated assessment models (IAMs) are used to derive
welfare-optimal climate policy scenarios (Kunreuther et al., 2014) or
constrained welfare-optimal scenarios that comply with a prescribed policy
target (Clarke et al., 2014). Most of them employ relatively simple climate
modules emulating sophisticated climate models, atmosphere–ocean general
circulation models (AOGCMs). These climate modules (hereafter “simple
climate models” – SCMs) offer computational efficiency and hence allow
researchers to examine a broader set of scenarios in orders of magnitude
less time. For IAMs based on a decision-analytic framework involving
intertemporal welfare optimization, SCMs are in fact indispensable, as these
IAMs' numerical solvers may need to access the climate module anywhere from
10 000 to 100 000 times before numerical convergence is flagged.
The need to qualify the degree of accuracy with which SCMs mimic AOGCMs or
properly represent ensembles of AOGCMs is increasingly being recognized
(Calel and Stainforth, 2017; van Vuuren et al., 2011a), as this aspect
might have immediate monetary consequences in connection with derived policy
scenarios (Calel and Stainforth, 2017). In previous work, van Vuuren et
al. (2011a) found that IAMs tend to underestimate the effects of greenhouse gas emissions.
Due to the centennial-scale quasi-linear properties of AOGCMs' global mean
temperature (GMT) dynamics, SCMs have proven capable of emulating AOGCMs'
behavior regarding GMT change, with deviations being a function of spread of
forcing, SCM complexity (Meinshausen et al., 2011a), and quality of SCM
calibration. The climate component of the Model for the Assessment of Greenhouse Gas Induced Climate Change (MAGICC; Meinshausen et al., 2011a)
represents the most complex SCM currently in use. In some sense one could
even call MAGICC an Earth system model of intermediate complexity. It has
demonstrated its capacity to emulate all AOGCMs' GMT even more precisely
than the standard deviation of interannual GMT variability (Meinshausen et
al., 2011a), with a fixed set of parameters, utilized for the whole range of
Representative Concentration Pathways (RCPs) (see van Vuuren et al., 2011b).
This represents the current gold standard of AOGCM emulation using SCMs.
The most extreme opposite end of the scale of complexity within the model
category of SCMs is provided by the one-box model as introduced by
Petschel-Held et al. (1999) (hereafter “PH99”), converting a radiative
forcing time series into a GMT time series. The current role of this model
as assessed in the literature is as follows: by fitting PH99 to GMT time
series, it can be used as a diagnostic instrument, as Andrews and Allen (2008)
have done. However, its main application is as an emulator of AOGCMs.
In conjunction with the most parsimonious carbon cycle model (described in
Petschel-Held et al., 1999 as well), PH99 has been used to derive
“admissible” greenhouse gas emission scenarios in view of prescribed GMT
targets (Bruckner et al., 2003; Kriegler and Bruckner, 2004). Furthermore,
the following climate–economic IAMs are currently utilizing PH99: FUND
(Anthoff and Tol, 2014), MIND (Edenhofer et al., 2005), and PAGE (Hope,
2006) – the last of which was used in the “Stern Review” for the
UK government (Stern, 2007). While MIND has since been succeeded by the IAM
REMIND (Luderer et al., 2011) when it comes to spatial resolution or
representing the energy sector by dozens of technologies, it currently
serves as a state-of-the-art IAM for decision-making under uncertainty (Held
et al., 2009; Lorenz et al., 2012; Neubersch et al., 2014; Roth et al.,
2015) or joint mitigation–solar radiation management analyses (Roshan et
al., 2019; Stankoweit et al., 2015).
Kriegler and Bruckner (2004) validated PH99 in conjunction with a simple
carbon cycle model. When diagnosing the effect of the IS92a emissions
scenario (Kattenberg et al., 1996) on GMT, they demonstrated deviations of
less than 0.2 K for the 21st century (see their Fig. 5). Recently, Calel
and Stainforth (2017) highlighted the potential future role of PH99 and
hence further validation of its behavior is warranted.
In this article, we ask by what calibration procedure is PH99's
temperature response to radiative forcing able to correctly map globally
averaged radiative forcing anomalies onto GMT anomalies? In this article,
“correctly” refers to an accuracy on the order of magnitude of the standard
deviation of natural variability, i.e., ∼0.1 K. Furthermore,
in the context of this article we would judge a deviation of 0.5 K as
inacceptable because a proclaimed goal of the 2015 Paris Agreement (UNFCCC,
2016) is “… holding the increase in the global average
temperature to well below 2 ∘C above preindustrial levels and
pursuing efforts to limit the temperature increase to 1.5 ∘C above
preindustrial levels …” In the policy domain, a difference of 0.5 K matters.
We believe that further validation of PH99 is necessary and possible, at a
higher level of consistency than has been performed previously. Firstly, the
respective GMT time series as checked in Kriegler and Bruckner (2004) is
convexly increasing. However in the context of scenario generation in
keeping with the well-below 2 K target (UNFCCC, 2016), validation along GMT
stabilization or even peaking scenarios is crucial, as these scenarios
display a qualitatively different shape from IS92a. Secondly, in Kattenberg
et al. (1996) the forcing was reconstructed by the additional assumption
that non-CO2 greenhouse gas forcing approximately balances aerosol cooling.
Here we employ recently diagnosed forcings for 14 CMIP5 AOGCMs by Forster et
al. (2013). As a main finding we diagnose that in the context of 2 K
stabilization scenarios, it would be necessary to implement a smaller equilibrium climate
sensitivity (ECS)
value in PH99 than the diagnosed ECS value of the very AOGCM which PH99 is
supposed to emulate. Hence previous work based on PH99 (see Hope, 2006;
Anthoff and Tol, 2014, and all the MIND-based work on decision-making under
ECS uncertainty – see citations above) requires a reinterpretation.
Needless to say, we are not claiming that the previously published IAM-based
work mentioned above is “worthless”. Rather, we argue that the parameters
and probability density distributions need to be interpreted as transformed
ones, essentially because a response has been sampled which is higher than
that of the corresponding AOGCM. To resolve this, we propose calibrating
PH99 by mapping AOGCMs' ECS and TCR to respective effective values, which
are suitable for a centennial time horizon, before using them in PH99.
In this way, PH99 could complement the use of increasingly complex climate
modules, ranging from DICE's two-box model (Nordhaus, 2013) to the complex
upwelling–diffusion climate module used in MAGICC (Meinshausen et al.,
2011a). The potential benefits of doing so are twofold: firstly, the most
parsimonious SCM, PH99, ensures maximum comprehensibility. Secondly, in the
context of numerically solving decision-making under climate response
uncertainty (Kunreuther et al., 2014), having to simultaneously deal with
dozens, hundreds, or even thousands of alternate climate “states of the
world” (the economist's term for the uncertain system property) poses a
significant challenge for numerical solvers and memory. In this regard, PH99
appears particularly attractive. Keeping the state space as slim as possible
proves particularly relevant for decision-making under uncertainty with
endogenous learning. For that reason, Traeger (2014) utilizes a one-box rather
than a two-box model, however with an exogenously given time series somewhat
mimicking the existence of a deep ocean layer.
Finally, our article represents a warning: if PH99 is to be used in the
future, it should be done in a re-scaled manner, adjusted to the time
horizon under investigation.
This article is organized as follows. Section 2 introduces the data-based
part of our analysis. We call for a three-step procedure, including (i) a
conventional, though not naïve, calibration of PH99 with regard to
climate sensitivity and transient climate response (i.e., the GMT change in
response to a 1 % yr-1 increase in the CO2 concentration until
doubling compared to the preindustrial value); (ii) an AOGCM-specific
calibration; and (iii) the validation of (ii). In Sect. 3 we first
demonstrate that (i) would lead to emulation errors of up to 0.5 K for
scenarios approximately compatible with the 2 K target. We then show that
this emulation error can be reduced to 0.1 K when choosing AOGCM-specific
calibrations of PH99. This calibration is subsequently validated by
independent scenarios. Note that, in Sect. 3, we focus on only the RCP2.6
scenario for calibration, use RCP4.5 and RCP8.5 for validation, and leave
further analyses, which show that PH99 can be generally calibrated to and
validated by a variety of scenarios, to Appendix B. In Sect. 4 we present a
scheme of how to calibrate PH99 for a given ECS, thereby avoiding
AOGCM-specific calibrations. This results in a larger emulation error than
achieved in Sect. 3 but one that would nevertheless suffice for most
applications. In Sect. 5 we explain the observed discrepancy between PH99
and AOGCMs as reported for step one of Sect. 2 by pursuing a
semi-analytical, physically based approach. In Sect. 6 we discuss the
implications of our findings for the integrated assessment community, while
Sect. 7 presents our conclusions and outlines further research needs.
Before we proceed, a brief note on the role of AOGCM data in our article is
in order. We compare PH99 to AOGCM data because we utilize AOGCMs here as
the entities closest to “reality” available on the “model market”. We do
not, however, claim that IAM modelers were using them or should be using
them. AOGCM data are used to demonstrate how ECS and TCR data can skew the
calibration of PH99 and how it should be corrected. The same correction
should in principle be used for ECS data inferred from any source,
e.g.,
abstract distributions such as those presented in Bindoff et al. (2013).
Mirroring PH99 in AOGCM data, however, is currently the most direct way to
infer the quality of a (not) recalibrated PH99.
Method
This section introduces the analytic structure of PH99, relates it to ECS
and TCR, and then describes a three-step scheme for PH99–AOGCM intercomparison.
PH99 projects the atmospheric GMT anomaly compared to its preindustrial
level. Petschel-Held et al. (1999) specified the model for a CO2-only
forcing scenario and accordingly PH99 reads
dTdt=μln(c)-αT.
Here T denotes the GMT anomaly, c is the CO2 concentration in units of
its preindustrial level, and α and μ are constant tuning parameters.
From Eq. (1) we can readily read the ECS, the equilibrium temperature
anomaly in response to a doubling of the CO2 concentration compared to
its preindustrial value:
ECS=μαln(2),
also in line with Petschel-Held et al. (1999) and Kriegler and Bruckner (2004).
In Appendix A we briefly derive the TCR (GMT) from a stylized
experiment after the CO2 concentration has been exponentially increased
with the rate γ (of 1 % yr-1) until the concentration has doubled
for this model:
TCR=μγα2-1+2-αγ+αγln(2)=γECSαln(2)-1+2-αγ+αγln(2).
In the following we propose a three-step validation approach to clarify PH99's
range of applicability.
Step one
We first check whether simply calibrating PH99 from AOGCM-specific ECS and
TCR data would deliver good emulations (i.e., accurate to within 0.1 K) for
scenarios compatible with the
2 K target. After a technical derivation, we summarize
this method of mapping AOGCMs' ECS and TCR onto PH99's two parameters.
Some difficulty arises due to the fact that AOGCMs have not been run for
2 K-target-compatible scenarios for CO2-only forcing but solely for a
plethora of simultaneous forcings that would add up to a total forcing.
Hence we generalize Eq. (1) to its total-forcing counterpart (see
Eqs. 4–7) to be driven by total forcing time series as reconstructed in
Forster et al. (2013). Accordingly, we utilize scenarios generated by
14 AOGCMs (see Table 1) from CMIP5. From Forster et al. (2013), we also take
the ECS and TCR for these 14 models to derive model-specific α
and μ, utilizing Eqs. (2) and (3).
PH99 parameters (α and μ) and feedback response times
(1/α) utilizing data (ECS and TCR) from AOGCMs.
In order to generalize Eq. (1), we recall its derivation from an energy
balance approach, as summarized in Kriegler and Bruckner (2004), allowing
for a physical interpretation of the model. We start by introducing the
general energy balance equation, expressing the change in oceanic heat
content as the difference of ingoing (F) and outgoing (λT)
radiative flux while h denotes the constant effective oceanic heat capacity
(see also Geoffroy et al., 2013, Eqs. 1–4).
hdTdt=F(t)-λT(t)F also represents the total radiative forcing as applied in Forster et al. (2013).
However the equation could still not be integrated as h and
λ are yet to be determined. In order to solve the posed problem
(CO2-only versus total forcing), we note that h and λ represent
universal parameters of PH99 in the sense that their numerical values would
not depend on the mix of substances (i.e., CO2, other greenhouse gases,
aerosols) causing the total radiative forcing. Therefore, h and
λ can be determined by considering the CO2-only case and,
hence, by tracing them back to the already determined α and μ.
For the CO2-only case, Eq. (4) reads
hdTdt=-λT(t)+Q2lnc(t)ln2.Q2 denotes the additional forcing from the doubling of the CO2
concentration compared to its preindustrial value and is listed for all of
the AOGCMs (see Forster et al., 2013, Table 1).
If we then divide by h, we obtain
dTdt=-λhT(t)+Q2hlnc(t)ln2.
A comparison with Eq. (1) readily reveals
α=λhandμ=Q2hln2.
These equations would allow for the determination of h=Q2/(μln2) and
λ=αh. Utilizing these equations and Eq. (4), we generate
PH99's temperature response to the total radiative forcing as specified in
Forster et al. (2013).
The derivation displayed so far can be summarized in terms of the following
recipe to generate PH99's parameters on the basis of AOGCMs' ECS and TCR:
set PH99's ECS and TCR equal to the selected AOGCM's ECS and TCR;
numerically invert Eq. (3), right-hand-side expression, to find α
(no analytic expression possible);
invert Eq. (2) to find μ;
derive h and λ from Eq. (7), and then utilize Eq. (4), divided by h.
Finally, to avoid differences occurring over the historical period (pre-2006
for the RCPs), we need to initialize PH99 with each AOGCM's 2006 temperature
anomaly with respect to the preindustrial value. To do this, for each AOGCM
we calculate the mean temperature over the period 1881–1910 and set this as
the preindustrial value. We then calculate the mean temperature over the
period 1991–2020 and use this as an indicator for the 2006 temperature
level. The difference between these two values is fixed as the initial
temperature anomaly for PH99.
Each temperature trajectory should be compared to the temperature data from
the corresponding AOGCM. As for GMT-target-constrained economic
optimizations (Clarke et al., 2014; Edenhofer et al., 2005), the maximum GMT
(rather than the whole time series) is of special importance. Hence we use
the difference between the respective 2071–2100 GMT time averages of PH99
and the AOGCM as an error metric. If the deviations are tolerable (accurate
to within 0.1 K), the climate module is validated; if they are intolerable,
we proceed with steps two and three.
Step two
For each AOGCM, α and μ are tuned such that the difference between
PH99 and the AOGCM GMT anomaly for the RCP2.6 scenario in the period 2006–2100
is minimized using a least-squares approach. For further
diagnostics we then determine the new “effective” ECS and TCR from Eqs. (2)
and (3). As in step one, the deviations in 2071–2100 means of GMT
between PH99 and the respective AOGCM are determined as an accuracy check.
Step three
Lastly, we validate the PH99 model versions generated in step two. For this
purpose, independent temperature and forcing paths must be run as a
nontrivial test to check whether the trained climate module can accurately
project other temperature data trajectories. To do so, the values
for α and μ determined in step two are implemented in PH99, the latter
then being driven by the total climate forcing of the RCP4.5 and RCP8.5
scenarios. Similar to steps one and two, the deviations in 2071–2100 means
of GMT between PH99 and the respective AOGCM are determined as an accuracy check.
One might be interested in seeing if the calibrated module is capable of
mimicking other scenarios such as RCP6.0 or if PH99 was calibrated to
RCP4.5 or others. Stating that, in general, the procedure outlined above
brings about similar results, for the sake of brevity of the main text, we
present the respective results in Appendix B.
Results
Table 1 shows the calculated α and μ together with the feedback
response time 1/α in step one. For all of the indicators we also
compute the mean values and standard deviations of the samples. The mean
value of the ECS for GCM data is 3.35 K, with a minimum and maximum of
2.11 and 4.67 K, respectively. The mean value of the timescales is roughly 35 years.
Figure 1 represents the projected PH99 temperature
evolution for the scenario RCP2.6 of each GCM in 2006–2100, using the data
from Table 1 and RCP2.6's forcings. PH99 clearly
overestimates the temperature anomaly for all GCMs, especially over the last
30 years. The absolute values of the deviations of mean temperature over the
last 30 years (hereafter MTD) from the AOGCM data are shown in
Fig. 2. The MTD ranges from 0.22 K for MRI-CGCM3
to approximately 0.79 K for HadGEM2-ES. On average, the deviations are
ca. 0.45 K. This is clearly a large error, in both units of annual GMT standard
deviation as well as the climate policy dimension. Accordingly, we must
proceed with step two.
Comparison of temperature paths (K) projected by PH99 (black curve),
calibrated by an AOGCM's ECS and TCR, to the corresponding AOGCM's temperature
paths (red curve). Deviations on the order of 0.5 K for 2100 are observed.
In step two, for each of the GCMs, we tune α and μ such that the GMT
deviations for the whole period 2006–2100 are minimized in a least-squares
manner as represented in Figs. 3 and 4. From the thereby adjusted α
and μ we derive the ECS and TCR, which are presented in
Table 2. MTDs for the various AOGCMs are shown in Fig. 2.
Modulus of deviations of GMT (K) mean values of PH99 over the
period 2071–2100 from corresponding AOGCM means. The red bars show the
deviations for RCP2.6 when α and μ are from Table 1 and not fitted.
The cyan bars show the deviations in RCP2.6 when α and μ are fitted
to the AOGCM's RCP2.6 data. The light blue bars show the deviations for RCP4.5
when α and μ are kept at their RCP2.6-fitted values (validation).
The dark blue bars show the deviations for RCP8.5 when α and μ are
kept at their RCP2.6-fitted values (validation).
The results tell us three main things. Firstly, the average of the absolute
values of deviations is significantly reduced when α and μ are tuned.
Indeed, the MTD average drops to below 0.02 K. Secondly, while the average
ECS decreases by 0.9 K (from 3.35 to 2.46 K), the average TCR increases by
0.14 K (from 1.90 to 2.04 K). Thirdly, the mean value of feedback response
times decreases significantly, from roughly 35 years to less than 12 years.
For validation we move on to step three. We utilize the RCP4.5 temperature
and forcing data as provided by Forster et al. (2013). In Figs. 3 and 4 the
respective GMT trajectories for any AOGCM are contrasted with the
PH99-generated ones, where α and μ are fixed to their values as
determined in step two. The MTDs are shown in Fig. 2. The results confirm
that the climate module is sufficiently well trained
in the second step that it can suitably mimic the actual temperatures
(accurate to within 0.1 K) for RCP4.5 and RCP8.5. As shown, the average MTD
is approximately 0.05 K for RCP4.5 and about 0.14 K for RCP8.5. For RCP4.5,
the deviations for three of the GCMs, namely CCSM4, CNRM-CM5, and NorESM1-M,
are even better than those diagnosed for RCP2.6 in step two. See Appendix B
for further analyses.
Comparison of temperature evolutions (K) projected by the climate
module PH99 (solid and dotted black curves) to the actual AOGCM's temperature
(solid and dotted red curves). α and μ have been tuned to fit the
PH99 temperature path (solid black curve) to the respective AOGCM's RCP2.6
temperature path (solid red curve). Using the fitted α and μ, and
taking the forcing reconstructed for RCP4.5 into account, PH99 also reproduces
the projected RCP4.5 (dotted black curve). The dotted red curve shows the actual
RCP4.5 temperatures.
Comparison of temperature evolutions (K) projected by the climate
module PH99 (black solid curves) in the RCP8.5 scenario to the actual AOGCM's
temperature (red solid curves) in the RCP8.5 scenario. α and μ are taken
from the second step, in which PH99 is calibrated to the RCP2.6 scenario.
A mapping of ECS onto their PH99-specific counterparts α and μ
Finally, we attempt to abstract from fitting PH99 to individual AOGCMs and
provide an approximate way to calibrate PH99 within the cloud of AOGCMs
simply by knowing the ECS. Then PH99 could be utilized for any ECS in
analyses in which the ECS is uncertain.
An existing mapping for PH99
Before diving into our suggestions, we examine one of the existing options
(a reader solely interested in our improved method of utilizing PH99 can
move straight on to Sect. 4.2). We inspect the curve suggested by Lorenz
et al. (2012), which correlates α and μ to ECS. Using a sample from
Frame et al. (2005) and assuming a strict relationship between 1/μ and ECS,
Lorenz et al. (2012) suggest the following approximation:
1μ≈1μ‾-10exp(-0.5ECS),
where μ‾ is the mean value of μ in the sample (see
Fig. 7 in Lorenz et al., 2012; all quantities measured in the units utilized
in Kriegler and Bruckner, 2004). Knowing μ, Eq. (2) is used to
determine α. In turn, Eqs. (2) and (8) have been repeatedly used
in studies employing MIND and concerning uncertainties and ECS (Neubersch et
al., 2014; Roshan et al., 2019; Roth et al., 2015).
Modulus of mean temperature deviations (K) over the period 2071–2100 (MTD)
for PH99 from AOGCMs when α, μ, ECS, and TCR from Table 2 are related
to ECS and TCR in Table 1. Using linear (yellow bars), quadratic (light green
bars), and cubic functions (dark green bars), α and μ are related to
ECS when the outlier is put out for the linear case. Using linear fits, ECS and TCR
are related to ECS (blue bars). Using linear fits, ECS and TCR are related to
ECS and TCR, respectively (light blue bars). The dark blue bars show the
deviations for RCP2.6 when α and μ are from Table 1 and not fitted
(the same as Fig. 2). The orange bars indicate MTD using Lorenz's curve.
We employ Eqs. (2) and (8) for all ECSs from
Table 1 and show the MTDs for the RCP2.6 scenario
in Fig. 5. Note that TCR can readily be
calculated using Eq. (3). Clearly, on average, employing Lorenz's curve does
not result in a better situation than step one. However, this might not
necessarily be a case of comparing like with like. At the time of Frame et
al. (2005), the two-dimensional uncertainty information was obtained by
reconstructing the 20th century's warming signal from fingerprinting by
means of a single AOGCM and then using these observational data as a
constraint. It is well known that observational constraints may lead to
different distributions than ensembles of AOGCMs do (Andrews and Allen,
2008). Nevertheless we include this piece of information here for the sake
of completeness.
PH99 parameters (α and μ), climate sensitivities (ECS and
TCR), and feedback response times (1/α) after fitting PH99 GMT time
series to AOGCM RCP2.6 GMT time series.
Given the inferred estimates in Table 2, one can
directly relate α and μ to the ECS. To do so, we generate polynomial
fits (of orders of 2 and 3) of α and μ against all AOGCMs' ECSs.
Predicting a two-dimensional manifold from ECS alone implicitly exploits the
fact that AOGCMs' TCRs can be predicted well using ECSs (see
e.g., Meinshausen et al., 2009) in a statistical sense. Another option would be to
derive α and μ analytically (like in the first step) when the inferred
ECS and TCR are correlated to the ECS and TCR of AOGCMs.
Figure 6 relates α and μ (from Table 2) to the ECS (from
Table 1), using linear, quadratic, and cubic
polynomial approximations. For the case of a linear approximation, we put
the model GISS_E2_R out as an outlier. Figure 5 indicates that on average all
approximations mimic the actual temperature paths better than a non-fitted
one. The cubic estimation projects significantly smaller deviations compared
to the quadratic approximation and slightly smaller deviations compared to
the linear approximation. The maximum MTD in the cubic approximation is
0.3 K for IPSL-CM5A-LR, which is roughly a third of the maximum in the quadratic
approximation that is revealed for CSIRO-Mk3-6-0.
Quadratic (a, b), cubic (c, d), and linear (e, f)
relationships of μ(a, c, e) and α(b, d, f) in
Table 2 to ECS in Table 1. Notice that in the linear case the model GISS_E2_R,
as an outlier, is out.
We also consider alternative ways to map ECS and TCR from the 14 utilized
AOGCMs onto PH99-intrinsic properties, going beyond the scheme displayed in
Fig. 6. As one option, shown in Fig. 7, we linearly regress the ECS and TCR
values inferred from step two against their original AOGCM counterparts
and obtain
ECSPH99≈aECSAOGCM+b,
with a=0.5846, b=0.5095 K, and R2=0.8158, as long as
ECSPH99< ECSAOGCM and
TCRPH99≈cTCRAOGCM+d,
with c=0.9763, d=0.1829 K, and R2=0.667.
Inferred effective TCR (K) vs. AOGCMs' TCR (K) (a), inferred
effective ECS (K) vs. AOGCMs' ECS (K) (b), and inferred effective
TCR (K)
vs. AOGCMs' ECS (K) (c). While the TCRs differ by less than 0.2 K,
the ECSs differ by up to 2 K. This opens the door for a discussion as to
whether PH99 should be calibrated using scenario-class-adjusted effectively
lower ECS values.
The other option consists in using Eq. (9) along with a linearly regressed
TCRPH99 over ECSAOGCM, that is
TCRPH99≈mECSAOGCM+n,
with m=0.4582, n=0.5044 K, and R2=0.7876.
The respective MTDs are shown in Fig. 5. Although
both approximations mimic the actual temperature paths better than a
non-fitted one, regressing both the inferred effective ECS and TCR solely
against AOGCMs' ECS (hereafter ETE) clearly offers the best overall approximation.
Using the ETE has four major advantages over all other options dealt with
here, especially for the IAM community. Firstly, its approximation is better
than all options but the cubic fit. Secondly the ETE still has an advantage
over the cubic fit because one can easily use a broader range of climate
sensitivities, for example, from 1 to 9 K, which may not be accurately
determined by the cubic fit. Even though the cubic fit may yield a better
approximation, in our analysis it is only better by 0.03 K at the expense of
a nonintuitive shape that might result in even worse deviations for out-of-sample data. Thirdly, prior knowledge regarding the TCR is no longer a
decisive factor. Note that prior knowledge regarding the TCR can make
approximations better. However, as we tested, for example, in the case of
linearly regressing both the inferred effective ECS and TCR against both
AOGCMs' ECS and TCR, the R squares for Eqs. (9) and (11) only improve by
6 % and 7 % respectively, and the MTD is no better than the ETE.
Finally, in the case of ETE, we do not need to re-evaluate our sample and
possibly drop any model as an outlier. Given the explorations already carried
out
and their performance, we leave explorations beyond the linear approximation
for future research.
An analytic interpretation of the AOGCM–PH99 intercomparison
In the following, we explain why PH99 systematically overestimates maximum
GMT for peaking scenarios when fitted for exponentially growing scenarios.
As an AOGCM is analytically not accessible, we investigate an intermediate
step of model replacement by moving from a one-box to a two-box SCM (as utilized
in DICE; Nordhaus, 2013). In fact we qualitatively trace back the effects
reported so far to the information loss incurred by replacing a two-box SCM
with a one-box SCM like PH99. We then also investigate the quality of
alternative fitting schemes based on our semi-analytic analysis, which
complements our previously mentioned AOGCM-based validation.
Following Geoffroy et al. (2013) we introduce a two-box SCM as a more
universal emulator of AOGCMs' mapping from radiative forcing onto temperature.
CdT2Bdt=F-λ2BT2B-δT2B-T0C0dT0dt=δT2B-T0T2B denotes the two-box analogue of the one-box temperature T in Eq. (1).
The upper and the lower equations represent the upper and the lower ocean, respectively.
Total radiative forcing (anthropogenic plus natural) for RCPs – supporting
the original names of the four pathways, as there is a close match among
peaking, stabilization, and 2100 levels for RCP2.6 (also called RCP3-PD), RCP4.5
and RCP6, and RCP8.5, respectively (taken from Meinshausen et al., 2011b).
In order to contrast PH99 with this two-box model, we search for analytic
approximations of generic shapes of the forcing F(t) and examine the long-term
projections under various RCPs as depicted in Meinshausen et al. (2011b) – an
excerpt is included in Fig. 8 for the reader's
convenience. Particularly in view of the peaking, mitigation-oriented lowest
forcing scenario, we approximate forcing paths in three phases: zero
forcing, linear increase, and linear decrease, under a continuity assumption.
F(t)=0fort<0k1tfor0≤t≤t1k2t-t1+k1t1fort>t1
We approximately identify t1 with the year 2035 and t=0 with 100 years
earlier, i.e., we assume a ramp-up time t1 for the forcing of roughly
100 years. Furthermore, k2<0 and |k2/k1|=:ε≪1. From
Fig. 8 we approximate a generic value of
ε=0.2. For 0≤t≤t1 we
draw on Geoffroy et al. (2013 – see their Eq. 14):
T2B0≤t≤t1=k1λ2Bt-τfaf1-e-tτf-τsas1-e-tτs.
This represents two linear modes of amplitudes af and as (with a sum
equal to 1), delayed by the characteristic timescales of a fast and a slow
mode, τf and τs, respectively, and continuously
matched to the initial condition “0” by an exponential. In Geoffroy et
al. (2013) the two-box model is fitted to 16 AOGCMs. After having reviewed their
results, we can make the following two simplifying assumptions: (i) both
amplitudes af and as approximately equal 1/2 (see their
Fig. 3a – amplitudes range from 0.35 to 0.65) and (ii) τf≈0 (values
range from 1 to 5.5 years; see their Table 4; for centennial effects,
this mode would nearly match the equilibrium response). Furthermore we can
see that τs ranges from 100 to 300 years for 15 out of
16 AOGCMs. Hence the two-box model is characterized by a marked timescale
separation between the two linear modes. With the aid of these two
approximations, the last equation can be simplified to
T2B0≤t≤t1≈k1λ2Bt-τ21-e-tτwithτ:=τs.
We then extend the analytic range of that formula, given the two
approximations above, for t>t1 (for a derivation; see Appendix C):
T2Bt>t1≈k1λ2B(-εt+1+εt1+τ2ε+e-tτ-(1+ε)e-t-t1τ.
The analogous expressions for the one-box model read
T0≤t≤t1=k1λt-θ1-e-tθ,θ:=1α,λfromEq.(7),
and
Tt>t1=k1λ(-ε(t-θ)+(1+ε)t1+θe-tθ-(1+ε)e-t-t1θ.
Explaining the PH99–AOGCM discrepancy for equal ECS and TCR values
We are now prepared to mimic step one in Sect. 2: we calibrate the one-box
model such that it is characterized by the same ECS and TCR as the two-box
model. As λ=Q2/ECS2B, equal ECS values for both models
deliver λ=λ2B.
Determining the second degree of freedom of PH99 (e.g., as expressed by θ)
from some transient property proves more intricate. We choose
TtTCR=T2BtTCR,
where we introduce tTCR as the moment in time when T needs to be
evaluated in order to determine TCR. In Appendix A we note, by definition,
that tTCR=(ln2)/γ≈70 years for a growth rate
γ=1 % yr-1 of the carbon dioxide concentration; hence
0<tTCR<t1. Therefore, when exploiting Eq. (20), Eqs. (16)
and (18) (rather than Eqs. 17 and 19) apply and result in the expression
hθtTCR=12hτtTCR,
with h denoting the auxiliary function (see Fig. 9)
h(x):=1-e-1xx,
where
limx→0h(x)=0,limx→∞h(x)=1,h(x)≈xforx≪1.
From this, we can already get a first impression of the scale of θ,
prior to numerical inversion: as τ is generically markedly larger
than tTCR, the right-hand side of the defining equation above
approximates 1/2. Further, if we boldly assume a slight timescale separation
between θ and tTCR, the former being smaller than the latter,
then the linear approximation of h would apply and θ≈tTCR/2≈35 years.
For a centered value of τ=250 years, this
approximation is confirmed in a direct numerical treatment of Eq. (21).
The auxiliary function h(x), which links the slow timescale of the
two-box model and the timescale of the one-box model.
Hence from the twin timescale separation of “the one-box model mode”,
“defining timescale for TCR”, and the “slow mode of the two-box model” we
have explained why TCR-oriented fitting exercises of the one-box model would
generically result in timescales of roughly 30 to 40 years (see
e.g., Anthoff and Tol, 2014; Kriegler and Bruckner, 2004). The factor 1/2
between the one-box model's timescale and the TCR-defining timescale goes back to the observation of Geoffroy et al. (2013) that the fast and
the slow modes both enter the superposition result with approximately equal
weights of 1/2. The slow mode is then too slow to be of much
relevance for TCR – a phenomenon not revealed by the one-box model.
We are now equipped to compare the two models' temperature projections and
apply the three-phase forcing as defined above for ε=0.2.
a1/λ is chosen such that peak temperatures enter the 2 K regime
for illustrative purposes. We exploit the coincidence that tTCR just
happens to approximately correspond to our starting year 2006 for PH99
(because 2035-100+70=2005). Hence the formulas for the one-box model do
not need to be adapted for an explicit initial condition for this purpose.
Figure 10 shows that by construction, both
temperature responses match at tTCR≈70 years, although the
one-box model's maximum exceeds the maximum by 0.5 K. This phenomenon can be
explained as follows. As the one-box model responds with a finite timescale,
its derivative must be continuous in response to a continuous forcing. Hence
the leading term is quadratic when the forcing starts. In contrast, the
two-box model contains a virtually degenerate timescale (the fast one); hence
its leading term is linear. If the two curves are to nevertheless match
at tTCR, the one-box model's derivative at tTCR must transcend the two-box
model's derivative. This, together with the right-bending kink in the two-box
model's response at t1, leads to a larger maximum in the one-box model. In
summary, on timescales much smaller than the slow mode, the slow mode,
compared to the fast mode, cannot develop yet; hence the fast mode will
dominate the slow mode. As such, fitting a one-mode model in a convex regime
is likely to yield poor predictions of a temperature maximum for mitigation-based forcings.
One-box vs. two-box model in response to kink-linear forcing as a stylized
interpretation of mitigation-oriented forcing paths and for equal levels of ECS
and TCR in both models. Kink-linear curve: two-box model; smooth curve: one-box
model. The temperature development of the one-box model overshoots the maximum
of the two-box model by roughly 50 %.
This explains the discrepancies found in our PH99–AOGCM comparison when
directly transferring AOGCMs' ECS and TCR onto PH99.
Figure 10 further suggests that if PH99 were used
to predict correct maxima and emulate AOGCMs in this time regime, it would
need to be used with a markedly smaller timescale. However, a simple
reduction in timescale would lead to a new inter-model discrepancy before
the kink; hence the overall amplitude of PH99's response would need to be
reduced as well. The latter scales with the ECS. Thus the ECS must be
reduced by a certain factor towards a new “effective ECS”, which could also
be called a “transient climate sensitivity”.
Testing the validity of a recalibrated PH99 for a two-box model
In Sect. 5.1 we derived an analytic explanation for why a naïve
transfer of an AOGCM's ECS and TCR to PH99 results in a maximum GMT, which is
too large when driven by a mitigation forcing scenario. However we show in
Sects. 3 and 4 that PH99 in fact is a good emulator of an AOGCM within 0.1 K
if it were either directly fitted to that AOGCM or if the AOGCM's ECS and
TCR were transformed into effective quantities for PH99. Hereby “good
emulator” expresses the fact that the same parameter set can be utilized for
any RCP (2.6, 4.5, 6.0, 8.5). From a practical point of view, we could stop
our analysis here and suggest that this type of validation might be
sufficient to generate trust in PH99 as an emulator for any forcing scenario.
However for further validation, in this subsection we would like to exploit
the fact that for a two-box–one-box intercomparison we can validate PH99 for
an order of magnitude larger set of forcing scenarios. We systematically
test the previously suggested adjustment formulas Eqs. (9) to (11) for a
range of t1 and ε values, hence varying mitigation
scenarios, given the alternative ECS and slow mode's timescale τ for the
two-box model. We find numerically that θ is on the order of 10 years,
and the ECS needs to be reduced by 1/4 to 1/3. We test for the centered ECS
values of 3 and 4 K and a slow mode's timescale, ranging from 100 to
300 years (see Geoffroy et al., 2013).
Comparing GMT (K) maxima of the two-box model and the one-box model, the
latter being adjusted to the former by prescribing the linearly transformed ECS
and TCR according to the scheme ETE. Abscissa is ε and ordinate is changed
peaking year t1, transformed to years however, for the two-box ECS of 3 and
4 K, and τ=100, 200, 300 years. The relative error (max. GMT
difference normalized by the max. GMT of the two-box model) is markedly smaller
than for the case of prior adjustment.
In principle, for any forcing scenario characterized by varying t1
and ε, we would need to compare GMT as calculated by
Eqs. (18) and (19) vs. Eqs. (16) and (17). However all of these equations derive GMT for the
boundary condition of zero temperature at t=0. Conversely, our
validation scheme as utilized in Sects. 3 and 4 fix PH99 to the AOGCM at the
year 2006. The latter point in time we denote by t0(≈tTCR).
Having transformed ECS and TCR according to Eqs. (9)–(11), we
cannot expect that T(t0)=T2B(t0) any longer. Therefore we have
to force the solution of PH99 to match the solution of the two-box model at t0
and call the thereby initialized solution of PH99 “Tinit”:
Tinitt0=T2Bt0.
We generate Tinit(t) from T(t) (see Eqs. 18 and 19) by adding a suitably
scaled solution of the homogenous counterpart of Eq. (4):
Tinitt≥t0=T(t)+T2Bt0-Tt0e-t-t0θ.
Figure 11 shows the relative deviations of the GMT
maxima of the one-box and the two-box model for the extrapolation scheme ETE
(Eqs. 9 and 11). In a certain regime, the extrapolation delivers
sufficiently accurate results, however, not everywhere. When utilizing the
mapping scheme represented by Eqs. (9) and (10), the results look similar.
The overall impression is that the mapping removes the bias. However, it
does not deliver a universal correction as found for the direct
intercomparison between PH99 and AOGCMs. Hence we cannot exclude the
possibility that AOGCMs are easier to emulate as they contain many more timescales than the two-box model and their effects might in part cancel.
While we observe a qualitative gain, Fig. 11
reveals there is still room for improvement. Accordingly, we further
transform the ECS to request perfect matching for t1=100 years,
ε=0.2; the results can be seen in
Fig. 12. The fit is much further improved such
that a major fraction of (t1, ε) values would lead to a
relative error of <5 %, and another large fraction would lead to a relative
error of <10 %. As the standard deviation of annual GMT is
between 0.1 and 0.2 ∘C and a typical application
might be a cost-effectiveness analysis of the 2 ∘C target, such
errors might still seem tolerable. However we observe structural problems
for very small values of ε, the latter implying very late
assumption of a maximum. In this case, the slow mode becomes more relevant,
and hence the quality of the calibration deteriorates. We find that the
calibration is valid for a time horizon on the order of t1 to 2 t1,
i.e., on the order of the time to peak forcing.
Similar to the previous figure (relative max. GMT error with
abscissa of ε and ordinate of t1 in years), however for a further adjusted
ECS of the one-box model, such that perfect matching is achieved for t1=100 years,
ε=0.2, and a one-box timescale of 12 years. For most of the parameter
settings, the relative error is below 10 %.
Discussion
The previous section offers a key mechanism to explain why, for given ECS
and TCR, GMT responses generated by PH99 in response to peak-and-decline
forcing scenarios are biased towards higher temperatures. How does this
relate to the observation that PH99 tends to underestimate the effect of
greenhouse gas emissions (van Vuuren et al., 2011a) as mentioned in our
introduction? In fact, van Vuuren et al. (2011a) describe a different
forcing experiment: a step function (see their Fig. 3). Here FUND, based on
PH99, displays a GMT lower than that of MAGICC-4 by more than 0.8 K at
certain times during the most transient phase, although both models share
the same ECS. This can be explained by the lack of timescales faster than
35 years (the latter characterizing PH99 in standard calibrations) within
PH99. Whether PH99 over- or underestimates GMT is hence a strong function of
the functional shape of forcing. Our article highlights the effects of
naïvely calibrating PH99 when assessing mitigation scenarios.
Additional mechanisms are also possible. Firstly, the statistical errors in
determining AOGCMs' ECS, TCR, and Q2 may lead, mediated through the
nonlinear mapping to PH99's parameters, to an overall bias in PH99's GMT.
Furthermore, diagnosing the total radiative forcing active in an AOGCM is a
complex undertaking (see, e.g., Meinshausen et al., 2011a, for a discussion).
A bias to the high end here would also result in inaccurately large GMT
responses by PH99.
However, in the context of this article, we contend that the information
loss when moving from a two-box to a one-box model is the key source of the
observed discrepancy – we find Fig. 10
compelling in this regard. Complying with the latter interpretation raises a
key question: can PH99 be seen as a “physical model” and if so, what are the
implications for users? It is readily apparent that a one-box model cannot
mimic a two-box model, characterized by a marked timescale separation for all
forcings at all times. However it is equally clear that the simplest
temperature equation is in fact the one that treats the ocean as a single
box. It would still explain warming with forcing in a quasi-linear manner,
though with some delay. If we are willing to accept that the calibration of
PH99 is time horizon specific, then PH99 still holds some semi-physical
meaning. If, however, this is seen as unacceptable, then we would have to
recognize that PH99 is more an efficient emulator than a physical model. In
this context we would like to recall that virtually every model has a
limited range of validity – and as such, PH99 is no different from most other models.
When investigating the one-box and two-box models' differences, our research also
suggests that within the class of peak-and-decline scenarios PH99 provides a
good emulation (accurate to within 0.2 K for a generic AOGCM setting such as
ECS = 4 K, a peaking of forcing between 2020 and 2100, and a ratio of slopes
of pre- and post-peaking forcing of 0.1 to 0.4). For the AOGCM–PH99
intercomparison, PH99 performs even better: for RCP2.6, RCP4.5, RCP6.0
(∼0.1 K) and approximately 0.2 K for RCP8.5.
What are the ramifications of our findings for previous publications based
on PH99? Those authors who claimed to have worked with PH99 in conjunction
with ECS = 3 K have effectively worked with a more complex model in
conjunction with ECS ≈ 4 K for the centennial time horizon.
Much of the work performed based on MIND in conjunction with PH99 and the
lognormal distribution for ECS by Wigley and Raper (2001) has essentially
been based on a lognormal distribution shifted to larger ECS values. The
5 %, 50 %, and 95 % quantiles of the lognormal distribution by Wigley
and Raper (2001) are 1.2, 2.6, and 5.8 K, respectively. When
interpreting these values as PH99 values, as they have in fact been utilized
in PH99 for the MIND model since Lorenz et al. (2012), in the sense of a
rough estimate one could ask what the corresponding effective ECS
values of a more complex model according to our
Fig. 7 were. The respective values are 1.2, 3.6, and
9.0 K. From Fig. 13, which reflects IPCC
AR5's synopsis of current knowledge regarding ECS (Bindoff et al., 2013), we
can see that these are still in line with the range spanned by instrumental
studies. Hence the results obtained by PH99 in conjunction with the
distribution by Wigley and Raper (2001) are not erroneous but simply need
to be reinterpreted as rather high-end representatives within the
collection of ranges as seen in IPCC AR5.
Probability density distributions of ECS according to IPCC AR5 WG-I
(Bindoff et al., 2013, Fig. 10.20).
For future applications we can conclude that PH99 must be applied and
interpreted with greater care – utilizing transformed values for ECS and
TCR – than in the past, if it is not to be replaced by at least a two-box
model as suggested by Geoffroy et al. (2013) and implemented in DICE
(Nordhaus, 2013). One-box models like PH99 can be crucial for modeling
decision-making under uncertainty and anticipated future learning. As an
illustration, execution of the MIND model currently demands between hours
and days for 20 different values of climate sensitivity in conjunction with
one learning step (Elnaz Roshan, personal communication, 2018). The execution time needed will
grow exponentially with the number of learning steps and at least linearly
with the number of state variables influenced by uncertainty. For endogenous
learning in a recursive design, computation time scales factorially with the
numerical resolution per state variable. The change from a one-box to a two-box
model might hence imply an order of magnitude larger execution time
(Christian Traeger, personal communication, 2018, in conjunction with Traeger, 2014). So a one-box model
will remain an attractive alternative in numerical applications addressing
decision-making under anticipated future learning. Users who would like to
go that road might, however, also consider the augmented one-box model by
Traeger (2014) as an alternative to PH99, employing an additional exogenous
forcing of that single box to somewhat emulate two boxes.
Summary and conclusion
We utilize recent data on total radiative forcing (Forster et al., 2013)
from 14 state-of-the-art CMIP5 atmosphere ocean general circulation models (AOGCMs)
in order to test the validity of the one-box climate module by
Petschel-Held et al. (1999, “PH99”) for scenarios approximately compatible with the
2∘ target. PH99 is currently utilized within the integrated
assessment models FUND, MIND, and PAGE.
We find that when prescribing the equilibrium climate sensitivity (ECS) and
transient climate response (TCR) of these AOGCMs to the emulator PH99,
global mean temperature (GMT) is generically projected 0.5 K higher by PH99
than by the corresponding AOGCM. In contrast, by directly fitting PH99 to
the RCP2.6 time series and validating with the RCP4.5 and RCP6.0 series, we
find that PH99 can emulate AOGCMs to a degree of accuracy better than 0.1 K.
Even for RCP8.5 the error is on the same order of magnitude, although
somewhat larger (up to 0.2 K).
We numerically demonstrate that PH99 can be used to excellently emulate
AOGCMs (accurate to within 0.1 K on average) within centennial-scale
integrated assessment of the 2 K target, provided its ECS and TCR are
reinterpreted as effective values and mapped from original ECS and TCR
values. We suggest such a mapping.
Furthermore we explain the observed discrepancies and the need to reduce
PH99's ECS compared to the AOGCM's ECS as being due to the information loss
produced by approximating a two-box-based energy balance model with a
one-box-based model. The key point is that PH99 has a fundamentally different
response shape to an AOGCM and hence ECS alone does not allow one to easily
move between the two. The transformation we propose adjusts PH99's ECS,
sacrificing agreement in the long-term response in order to gain agreement
in the centennial response (which is useful given it is more often than not
the timescale of interest).
In fact the slow mode of the two-box model is so slow that in a
climate-policy-relevant context it can unfold only up to a relatively small
extent; hence for practical purposes the two-box model's ECS cannot fully
develop. Accordingly, adjusting the ECS to lower values also proves to be
compatible with reducing PH99's response time. When comparing PH99 and
AOGCMs, the match is even better – a phenomenon for which the explanation
is beyond the scope of this article.
Hence older work based on PH99, executed within FUND, MIND, and PAGE, may
need to be reinterpreted in the sense that a response had been sampled
that is higher than that of the corresponding AOGCM. This effect, in turn,
proves equivalent to utilizing higher ECS values in the more complex model.
Even when having dealt with distributions of ECS as for the MIND model, ECS
values reinterpreted in that sense are still within the range outlined by
IPCC AR5 (see Fig. 13). Accordingly, we see this
reinterpretation as a mere numerical fix. In terms of the underlying
physics, we stress that using ECS alone to characterize climate response on
a timescale of a few hundred years is fundamentally flawed, given that ECS takes
on the order of 1000 years to emerge.
For future work, we propose the following steps: (i) by comparison with more
sophisticated, multi-box climate modules it should be tested again whether
the effect of a transient climate sensitivity (and TCR) alone could explain
our observed PH99–AOGCM discrepancy; (ii) future discussions with the AOGCM
community should illuminate to what extent the further explanations we
suggested might also apply, thereby potentially reducing the need to correct
for PH99; (iii) an AOGCM- and scenario class-independent yet centennial-timescale-specific two-dimensional mapping from ECS and TCR on to ECS and TCR and
designed for PH99 should be derived in conjunction with two-dimensional
distributions inferred from observations as performed in Frame et al. (2005). The
IAM community could then be offered both options for emulation: the one
presented here, trained by AOGCMs, and one based on observational data and
mediated by more complex SCMs.
In summary, PH99 could continue to be used as the most parsimonious emulator
of AOGCMs and is especially efficient for decision-making under climate
response uncertainty. However its calibration proves to be much more
involved than previously assumed. Future users should carefully consider
whether they actually want to use PH99 or whether they prefer a less parsimonious solution.
Data availability
For data sets please contact the corresponding author for
Forster et al. (2013).
An analytic expression of TCR in PH99
We rearrange Eq. (1) as
T˙=μln(c)-αT.
TCR is defined as the temperature change in response to a 1 % yr-1 increase
in CO2 concentration, starting from preindustrial conditions. Hence the
concentration, expressed in units of the preindustrial concentration, reads
c=exp(γt),
with γ denoting the above rate of change. As Eq. (A1) represents a
linear ordinary differential equation with constant coefficients, and the
initial temperature anomaly is to vanish, its solution reads
T=μγexp(-αt)∫texp(αt)dt=exp(-αt)μγ(1+exp(αt)⋅(-1+αt))α2.
Temperature should be evaluated at t2 when the concentration is doubled.
t2 is determined by c(t2)=2⇒t2=ln2/γ. From this
and Eq. (A3) we conclude Eq. (3). (In fact
we find the same result using an expression provided in Andrews and Allen (2008)
when we plug in our expression for t2 into theirs, which is phrased in terms of ECS.)
Further analysis on calibration and validation
As further validation of the trained PH99 calibrated to RCP2.6,
Fig. B1 shows the respective GMT trajectories of
AOGCMs for the RCP6.0 scenario contrasted with its respective PH99-generated
ones for which α and μ are fixed to their value as determined in
step two. MTDs are shown in the third columns of
Table B1. The missing models are due to either lack
of temperature trajectories for AOGCM or lack of total forcing. Notice
that first, second, and fourth columns are exactly the numbers related to Fig. 2. The results confirm that the climate
module is so well trained in the second step that it can appropriately mimic
the actual temperatures (accurate to within 0.1 K) for RCP6.0. As shown, the
average value of MTD is about 0.06 K for RCP6.0.
Column 5 thereafter in Table B1 shows MTDs in the
situations when PH99 is calibrated to the other RCP scenarios and is
validated against the others.
The comparison of temperature evolutions projected by the climate
module PH99 (black solid curves) in the RCP6.0 scenario to the actual AOGCM's
temperature (red solid curves) in the RCP6.0 scenario. α and μ are taken
from the second step, in which PH99 is calibrated to the RCP2.6 scenario.
Modulus of mean temperature deviations over the period 2071–2100 (MTD)
for PH99 from the corresponding AOGCM. In the first four columns, PH99 is calibrated
to RCP 2.6. In the second four columns, PH99 is calibrated to RCP4.5.
Calibrated to RCP2.6 Calibrated to RCP4.5 MTDMTDMTDMTDMTDMTDMTDMTDRCP2.6RCP4.5RCP6.0RCP8.5RCP2.6RCP4.5RCP6.0RCP8.5bcc_csm1_1_m0.0290.0400.2360.0180.0070.154bcc_csm1_10.0090.0660.0520.0640.0210.059CanESM20.0010.0210.0430.0390.0030.018CCSM40.0330.0030.0690.1320.0240.0050.0640.128CNRM_CM50.0140.0010.2010.0050.0120.273CSIRO_Mk3_6_00.0360.1150.0400.0630.0170.0150.1680.278GISS_E2_R0.0080.1140.0940.1440.0640.0030.0270.015HadGEM2_ES0.0180.1030.0360.1310.0570.0200.0970.211IPSL_CM5A_LR0.0200.0430.0500.2010.1210.0130.0170.033MIROC50.0150.0440.0290.0890.0320.0090.0340.106MIROC_ESM0.0280.1040.0790.2380.1400.0120.0440.241MPI_ESM_LR0.0170.0470.1190.1080.0150.060MRI_CGCM30.0150.0610.0830.2080.0010.0070.0040.061NorESM1_M0.0110.0000.0260.0430.0240.0000.0060.082Multimodel mean0.0180.0540.0560.1360.0350.0050.0390.078Standard deviation0.0100.0410.0250.0710.0440.0060.0530.093Calibrated to RCP6.0 Calibrated to RCP8.5 MTDMTDMTDMTDMTDMTDMTDMTDRCP2.6RCP4.5RCP6.0RCP8.5RCP2.6RCP4.5RCP6.0RCP8.5bcc_csm1_1_m0.2870.2570.027bcc_csm1_10.0910.0080.039CanESM20.0080.0250.010CCSM40.0380.0670.0180.0860.0590.0040.0100.004CNRM_CM50.1170.1510.005CSIRO_Mk3_6_00.1610.1990.0190.0620.1190.0190.0340.015GISS_E2_R0.0410.0370.0190.0460.0450.0230.0110.001HadGEM2_ES0.1460.2330.0210.0630.1460.2520.0730.017IPSL_CM5A_LR0.0160.0770.0010.0950.0520.0780.0300.002MIROC50.0670.0790.0110.0320.0250.0060.0190.019MIROC_ESM0.1870.0700.0050.1980.3090.2350.1400.007MPI_ESM_LR0.0110.0820.012MRI_CGCM30.0920.0680.0030.0420.0080.0140.0550.027NorESM1_M0.0680.0210.0160.1360.0700.0550.0540.013Multimodel mean0.0910.0950.0070.0840.0960.0860.0290.014Standard deviation0.0600.0720.0080.0530.0960.0960.0410.011Derivation of Eqs. (16)–(18)
We start by rewriting Eq. (15) in a way that it is most consequently
decomposed into the contributions from the two modes i∈{f, s}
(for “slow” and “fast” modes, respectively).
T2B0≤t≤t1=k1λ2B∑iait-τi+τie-tτi
One could derive Eq. (17) from an intuitive perspective by noticing that for
any of the modes i, its contribution to the temperature response would
consist of an equilibrium response, delayed by τi, and a summand of
exponential decay that would ensure continuity with respect to the initial
condition. This very principle can be followed again for the time horizon
beyond t1.
However, for those readers who would like to see a more formal derivation,
we provide the following ansatz: for t>t1, we decompose T2B
into three contributions, according to the superposition principle
for linear differential equations.
T1 is induced by a forcing k2(t-t1) with T1(t1)=0.
This contribution can be treated analogously to T2B(0<t<t1) when
noticing the replacements k1→k2, t→t-t1. From Eq. (C1) we inferT1t≥t1=k2λ2B∑iait-τi+τie-t-t1τi.
T2 is induced by a constant forcing k1t1 with
T2(t1)=0. This problem has also been solved by Geoffroy et al. (2013)
in terms of their Eq. (9), which we rewrite in our notation:T2t≥t1=k1t1λ2B∑iai1-e-t-t1τi.
T3 is the decaying initial condition at t=t1. For reasons of
continuity, this initial condition is identical to the terminal condition
according to Eq. (C1). Hence,T3t≥t1=k1λ2B∑iait1-τi+τie-t1τie-t-t1τi.
When we add these three components, we receive
T2Bt≥t1=1λ2B(∑iai(k1t1+k2t-t1-τi+e-tτik1τi-et1τik1-k2τi.
Allowing for the limit τf→0 and noticing that
k2=-εk1, we verify Eq. (17) by a summand-by-summand comparison.
Allowing for τf=τs=θ (i.e., simulating a one-box
setting by a two-box approach), we obtain Eq. (18) from Eq. (C1) and Eq. (19) from Eq. (C5).
Author contributions
MMK performed the statistical analysis. HH provided the
analytic analysis. MMK suggested and developed the alternative scheme. Both
participated in the writing of the article.
Competing interests
The authors declare that they have no conflicts of interest.
Acknowledgements
The authors would like to thank Jochem Marotzke for drawing their attention to
the Forster et al. (2013) article, discussing these results on total forcing
and providing the relevant data. In addition, the authors would like to
thank Chao Li for supporting the data handling process and making the authors
aware of Geoffroy et al. (2013), who discuss negligible AOGCM drift. The
authors are also grateful to Elnaz Roshan for her help with the visualizations
and providing quantiles of the distribution of Wigley and Raper (2001) on ECS.
We thank Matt Fentem for proofreading the second version of our paper
from a native speaker's perspective as well as Benjamin Blanz and
Manuel Wifling for further proofreading. All remaining errors are ours. Mohammad M. Khabbazan was supported by
the Cluster of Excellence “Integrated Climate System Analysis and
Prediction” (CliSAP, DFG-EXC177). Finally, the authors would like to thank the
three anonymous referees for their valuable criticism and constructive suggestions.
Edited by: Fubao Sun
Reviewed by: three anonymous referees
References
Andrews, D. G. and Allen, M. R.: Diagnosis of climate models in terms of
transient climate response and feedback response time, Atmos. Sci. Lett.,
9, 7–12, 2008.Anthoff, D. and Tol, R. S. J.: The Climate Framework for Uncertainty, Negotiation
and Distribution (FUND): Technical description, Version 3.6, available at:
http://www.fund-model.org (last access: 30 November 2016), 2014.
Bindoff, N. L., Stott, P. A., AchutaRao, K. M., Allen, M. R., Gillett, N.,
Gutzler, D., Hansingo, K., Hegerl, G., Hu, Y., Jain, S., Mokhov, I. I., Overland, J.,
Perlwitz, J., Sebbari, R., and Zhang, X.: Detection and Attribution of Climate
Change: from Global to Regional, in: Climate Change 2013: The Physical Science
Basis, Contribution of Working Group I to the Fifth Assessment Report of the
Intergovernmental Panel on Climate Change, edited by: Stocker, T. F., Qin, D.,
Plattner, G.-K., Tignor, M., Allen, S. K., Boschung, J., Nauels, A., Xia, Y.,
Bex, V., and Midgley, P. M., Cambridge University Press, Cambridge, UK and
New York, NY, USA, 2013.
Bruckner, T., Petschel-Held, G., Leimbach, M., and Toth, F. L.: Methodological
aspects of the tolerable windows approach, Climatic Change, 56, 73–89, 2003.
Calel, R. and Stainforth, D. A.: On the Physics of three Integrated Assessment
Models, B. Am. Meteorol. Soc., 98, 1199–1216, 2017.
Clarke, L., Jiang, K., Akimoto, K., Babiker, M., Blanford, G., Fisher-Vanden,
K., Hourcade, J.-C., Krey, V., Kriegler, E., Löschel, A., McCollum, D.,
Paltsev, S., Rose, S., Shukla, P. R., Tavoni, M., van der Zwaan, B. C. C., and
van Vuuren, D. P.: Assessing Transformation Pathways, in: Climate Change 2014:
Mitigation of Climate Change, Contribution of Working Group III to the Fifth
Assessment Report of the Intergovernmental Panel on Climate Change, edited by:
Edenhofer, O., Pichs-Madruga, R., Sokona, Y., Farahani, E., Kadner, S., Seyboth,
K., Adler, A., Baum, I., Brunner, S., Eickemeier, P., Kriemann, B., Savolainen,
J., Schlömer, S., von Stechow, C., Zwickel, T., and Minx, J. C., Cambridge
University Press, Cambridge, UK and New York, NY, USA, 2014.
Edenhofer, O., Bauer, N., and Kriegler, E.: The impact of technological change
on climate protection and welfare: Insights from the model MIND, Ecol. Econ.,
54, 277–292, 2005.Forster, P. M., Andrews, T., Good, P., Gregory, J. M., Jackson, L. S., and
Zelinka, M.: Evaluating adjusted forcing and model spread for historical and
future scenarios in the CMIP5 generation of climate models, J. Geophys.
Res.-Atmos., 118, 1139–1150, 10.1002/jgrd.50174, 2013.Frame, D. J., Booth, B. B. B., Kettleborough, J. A., Stainforth, D. A., Gregory,
J. M., Collins, M., and Allen, M. R.: Constraining climate forecasts: The role
of prior assumptions, Geophys. Res. Lett., 32, L09702, 10.1029/2004GL022241, 2005.Geoffroy, O., Saint-Martin, D., Olivié, D. J. L., Voldoire, A., Bellon, G.,
and Tytéca, S.: Transient Climate Response in a Two-Layer Energy-Balance
Model. Part I: Analytical Solution and Parameter Calibration Using CMIP5 AOGCM
Experiments, J. Climate, 26, 1841–1857, 10.1175/JCLI-D-12-00195.1, 2013.
Held, H., Kriegler, E., Lessmann, K., and Edenhofer, O.: Efficient climate
policies under technology and climate uncertainty, Energy Econ., 31, S50–S61, 2009.Hope, C.: The Marginal Impact of CO2 from PAGE2002: An Integrated
Assessment Model Incorporating the IPCC's Five Reasons for Concern, Integrat.
Assess. J., 6, 19–56, 2006.
Kattenberg, A., Giorgi, F., Grassl, H., Meehl, G. A., Mitchell, J. F., Stouffer,
R. J., Tokioka, T., Weaver, A. J., and Wigley, T. M.: Climate models – projections
of future climate, in: Climate Change 1995: The Science of Climate Change,
Contribution of Working Group I to the Second Assessment Report of the
Intergovernmental Panel on Climate Change, Cambridge University Press, Cambridge,
New York, and Melbourne, 285–357, 1996.
Kriegler, E. and Bruckner, T.: Sensitivity analysis of emissions corridors for
the 21st century, Climatic Change, 66, 345–387, 2004.
Kunreuther, H., Gupta, S., Bosetti, V., Cooke, R., Dutt, V., Ha-Duong, M., Held,
H., Llanes-Regueiro, J., Patt, A., Shittu, E., and Weber, E.: Integrated Risk
and Uncertainty Assessment of Climate Change Response Policies, in: Climate
Change 2014: Mitigation of Climate Change, Contribution of Working Group III
to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change,
edited by: Edenhofer, O., Pichs-Madruga, R., Sokona, Y., Farahani, E., Kadner,
S., Seyboth, K., Adler, A., Baum, I., Brunner, S., Eickemeier, P., Kriemann, B.,
Savolainen, J., Schlömer, S., von Stechow, C., Zwickel, T., and Minx, J. C.,
Cambridge University Press, Cambridge, UK and New York, NY, USA, 2014.Lorenz, A., Schmidt, M. G. W., Kriegler, E., and Held, H.: Anticipating Climate
Threshold Damages, Environ. Model Assess., 17, 163–175, 10.1007/s10666-011-9282-2, 2012.Luderer, G., Leimbach, M., Bauer, N., and Kriegler, E.: Description of the
ReMIND-R model, Potsdam Institute for Climate Impact Research, available at:
https://www.pik-potsdam.de/research/sustainable-solutions/models/remind/REMIND_Description.pdf
(last access: 30 November 2018), 2011.Meinshausen, M., Meinshausen, N., Hare, W., Raper, S. C. B., Frieler, K., Knutti,
R., Frame, D. J., and Allen, M. R.: Greenhouse-gas emission targets for limiting
global warming to 2 ∘C, Nature, 458, 1158–1162, 2009.Meinshausen, M., Raper, S. C. B., and Wigley, T. M. L.: Emulating coupled
atmosphere–ocean and carbon cycle models with a simpler model, MAGICC6 – Part 1:
Model description and calibration, Atmos. Chem. Phys., 11, 1417–1456,
10.5194/acp-11-1417-2011, 2011a.Meinshausen, M., Smith, S. J., Calvin, K., Daniel, J. S., Kainuma, M. L. T.,
Lamarque, J.-F., Matsumoto, K., Montzka, S. A., Raper, S. C. B., Riahi, K.,
Thomson, A., Velders, G. J. M., and van Vuuren, D. P. P.: The RCP greenhouse gas
concentrations and their extensions from 1765 to 2300, Climatic Change, 109,
213–214, 10.1007/s10584-011-0156-z, 2011b.
Neubersch, D., Held, H., and Otto, A.: Operationalizing climate targets under
learning: An application of cost-risk analysis, Climatic Change, 126, 305–318, 2014.
Nordhaus, W. D.: The climate casino: Risk, uncertainty, and economics for a
warming world, Yale University Press, New Haven, USA and London, UK, 2013.Petschel-Held, G., Schellnhuber, H.-J., Bruckner, T., Toth, F. L., and Hasselmann,
K.: The tolerable windows approach: Theoretical and methodological foundations,
Climatic Change, 41, 303–331, 1999.
Roshan, E., Khabbazan, M. M., and Held, H.: Cost-Risk Trade-off of Mitigation
and Solar Geoengineering – Considering Regional Disparities under Probabilistic
Climate Sensitivity, Environ. Resour. Econ., 72, 263–279, 2019.
Roth, R., Neubersch, D., and Held, H.: Evaluating Delayed Climate Policy by
Cost-Risk Analysis, EAERE, Helsinki, 24–27 June 2015.
Stankoweit, M., Schmidt, H., Roshan, E., Pieper, P., and Held, H.: Integrated
mitigation and solar radiation management scenarios under combined climate
guardrails, in: EGU General Assembly Conference Abstracts, 12–17 April 2015,
Vienna, Austria, 2015.
Stern, N.: The Stern Review – The Economics of Climate Change, Cambridge, UK, 2007.Traeger, C.: A 4-Stated DICE: Quantitatively Addressing Uncertainty Effects in
Climate Change, Environ. Resour. Econ., 59, 1–37, 10.1007/s10640-014-9776-x, 2014.
UNFCCC: United Nations Framework Convention on Climate Change. Adoption of
the Paris Agreement, in: Conference of the Parties on its twenty-first session,
30 November–11 December 2015, Paris, France, 21932, 2016.van Vuuren, D. P., Lowe, J., Stehfest, E., Gohar, L., Hof, A. F., Hope, C.,
Warren, R., Meinshausen, M., and Plattner, G.-K.: How well do integrated
assessment models simulate climate change?, Climatic Change, 104, 255–285,
10.1007/s10584-009-9764-2, 2011a.van Vuuren, D. P., Edmonds, J. A., Kainuma, M., Riahi, K., and Weyant, J.: A
special issue on the RCPs, Climatic Change, 109, 1–4, 10.1007/s10584-011-0157-y, 2011b.Wigley, T. M. and Raper, S. C.: Interpretation of high projections for
global-mean warming, Science, 293, 451–454, 10.1126/science.1061604, 2001.