ESDEarth System DynamicsESDEarth Syst. Dynam.2190-4987Copernicus PublicationsGöttingen, Germany10.5194/esd-8-617-2017Flexible parameter-sparse global temperature time profiles that stabilise at
1.5 and 2.0 ∘CHuntingfordChrischg@ceh.ac.ukYangHuiHarperAnnahttps://orcid.org/0000-0001-7294-6039CoxPeter M.https://orcid.org/0000-0002-0679-2219GedneyNicolahttps://orcid.org/0000-0002-2165-5239BurkeEleanor J.https://orcid.org/0000-0002-2158-141XLoweJason A.HaymanGarryhttps://orcid.org/0000-0003-3825-4156CollinsWilliam J.https://orcid.org/0000-0002-7419-0850SmithStephen M.Comyn-PlattEdwardhttps://orcid.org/0000-0001-7821-4998Centre for Ecology and Hydrology, Benson Lane, Wallingford, Oxfordshire, OX10 8BB, UKDepartment of Ecology, School of Urban and Environmental Sciences, Peking University, Beijing, 100871, P.R. ChinaCollege of Engineering and Environmental Science, Laver Building, University of Exeter, North Park Road, Exeter, EX4 4QF, UKMet Office Hadley Centre, Joint Centre for Hydrometeorological Research, Maclean Building, Wallingford, OX10 8BB, UKMet Office, FitzRoy Road, Exeter, Devon, EX1 3PB, UKDepartment of Meteorology, University of Reading, Earley Gate, P.O. Box 243, Reading, RG6 6BB, UKCommittee on Climate Change, 7 Holbein Place, London, SW1W 8NR, UKChris Huntingford (chg@ceh.ac.uk)14July20178361762624February20172March201729May2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://esd.copernicus.org/articles/8/617/2017/esd-8-617-2017.htmlThe full text article is available as a PDF file from https://esd.copernicus.org/articles/8/617/2017/esd-8-617-2017.pdf
The meeting of the United Nations Framework Convention on Climate Change
(UNFCCC) in December 2015 committed parties at the convention to hold the
rise in global average temperature to well below 2.0 ∘C above
pre-industrial levels. It also committed the parties to pursue efforts to
limit warming to 1.5 ∘C. This leads to two key questions. First,
what extent of emissions reduction will achieve either target? Second, what
is the benefit of the reduced climate impacts from keeping warming at or below
1.5 ∘C? To provide answers, climate model simulations need to follow
trajectories consistent with these global temperature limits. It is useful to
operate models in an inverse mode to make model-specific estimates of
greenhouse gas (GHG) concentration pathways consistent with the prescribed
temperature profiles. Further inversion derives related emissions pathways
for these concentrations. For this to happen, and to enable climate research
centres to compare GHG concentrations and emissions estimates, common
temperature trajectory scenarios are required. Here we define algebraic
curves that asymptote to a stabilised limit, while also matching the
magnitude and gradient of recent warming levels. The curves are deliberately
parameter-sparse, needing the prescription of just two parameters plus the final
temperature. Yet despite this simplicity, they can allow for temperature
overshoot and for generational changes, for which more effort to decelerate
warming change needs to be made by future generations. The curves capture
temperature profiles from the existing Representative Concentration Pathway
(RCP2.6) scenario projections by a range of different Earth system models
(ESMs), which have warming amounts towards the lower levels of those that
society is discussing.
Introduction
The conventional approach to understand climate change for
possible futures is to force Earth system models (ESMs) with either
emissions scenarios e.g. or prescribed future
atmospheric greenhouse gas (GHG) concentrations
e.g.. However, recent UNFCCC meetings have
placed a focus on prescribed temperature thresholds. This has mainly focused
on how to avoid crossing 2.0 ∘C of global warming since
pre-industrial times. Furthermore, the December 2015 Paris Conference of the
Parties (COP21) meeting suggested an additional aspiration of remaining below
a 1.5∘C warming threshold. To achieve the latter could in particular
involve major changes in energy demand or production
and extensive reliance on artificial carbon
removal such as biofuels combined with carbon
capture and storage. Equilibrium temperatures associated with even current
GHG concentrations may already correspond to warming levels near
1.5 ∘C . Therefore, given the likely
difficulty of fulfilling the 1.5 ∘C target, there is a focus on
understanding what is to be gained climatically from achieving that lower
threshold and the impacts of any temporary overshoot beforehand. There is a
related need to calculate the amount of flexibility between different
mixtures of greenhouse gas emissions that will achieve the same eventual
stabilisation levels. Forward modelling by prescription of emissions or GHG
concentrations cannot answer these questions directly, as there is no
guarantee that a particular simulation will asymptote precisely to an
increase of 1.5 or 2.0 ∘C. Instead, climate modelling needs to
develop inversion methods that follow predefined future warming profiles.
Existing ESM projections e.g. from the CMIP5
database; can be scaled to these, for instance by pattern
scaling e.g. . Here we move towards that by
presenting families of temperature profiles that eventually stabilise. The
use of common future warming trajectories may lead to easier discussion and
comparison between projects designed to assess a range of implications of
either the 1.5 or 2.0 ∘C target.
Temperature profiles that asymptote to prescribed temperature limitsOne-parameter profiles
Derived are profiles of global warming above pre-industrial levels, ΔT(t) (∘C), dependent on time t (yr) and with t=0 as year 2015.
Three boundary conditions are satisfied, with two related to present-day
warming. One is an estimate of warming between pre-industrial times and the
year 2015, ΔT0 (∘C). The second is an estimate of the
current rate of global warming, β=dΔT/dt|t=0 (∘Cyr-1). The values of these two parameters are
derived from the HadCRUT4 dataset . We use the
median from the 100 HadCRUT4 decadally-smoothed realisations of global
temperature rise estimates (see Data Availability below; HadCRUT4 smoothing
is with a 21-point binomial filter applied to annual values). Values in that
dataset normalise against the period 1961–1990; we renormalise to the period
1850–1900 as a proxy for pre-industrial times, giving ΔT0=0.89∘C. For further discussion of this value, see
. The recent gradient in warming is from
regression fitting of the last 21 years (1995–2015 inclusive), giving β=0.0128∘Cyr-1. We note, though, that when using
HadCRUT4 as our observationally based starting point, it is necessary to be
aware of its non-global spatial extent. Additionally, it is compiled from a
mix of air and sea surface temperatures, as described in
. The third boundary condition is the final
prescribed warming level ΔTLim (∘C), i.e.
1.5 or 2.0 ∘C. This is an eventual stabilisation level that our
profiles ΔT approach asymptotically. The specification of the
temperature thresholds in the COP21 statements could have other
interpretations, including eventual stabilisation at even lower warming
levels or long-term temperature fluctuations, but which remain below
prescribed limits. We do however allow the possibility of a near-term
temporary overshoot of ΔTLim, as described below.
We search for a parameter-sparse family of curves and consider a path that
moves away from a linear temperature rise (via parameter γ) and
towards a stabilisation level. Characterising different curves with an
adaptation parameter μ (yr-1) leads to
ΔT=ΔT0+γt-1-e-μtγt-ΔTLim-ΔT0.
A larger (positive) value for μ represents greater societal capability to
adjust the temperature pathway towards a stable temperature state. The value
of 1/μ (yr) is an approximate e-folding time in moving from a non-zero
positive gradient (in time) of global warming and towards levelling off at
ΔTLim. Taking the time derivative of Eq. ()
(Appendix, Eq. ) and matching to the historical record at
year t=0 gives
γ=β-μΔTLim-ΔT0.
Hence, γ is not the current rate of warming, i.e. γ≠β.
From Eq. () and for 0<μ<2β/(ΔTLim-ΔT0), this gives d2ΔT/dt2|t=0<0.0, corresponding to no acceleration of the
warming rate in the immediate future. Solutions require μ>0 for
convergence.
Profiles for different μ values and for ΔTLim values of 2.0 or
1.5 ∘C are presented in Fig. . For the three values
selected, varying behaviours occur. The lower value of μ=0.0074yr-1 is sufficiently small that stabilisation can only be
achieved after overshoot. The middle value of μ=0.03yr-1
achieves stabilisation without overshoot. The value of μ=0.05yr-1 also achieves stabilisation without overshoot; it
corresponds to the strongest ability by society to adjust temperature. For
this μ value, there is significant initial acceleration, particularly for
ΔTLim=2.0∘C.
The effect of changing μ in the single-parameter temperature
profiles, designed to asymptote to either 2.0 ∘C (left panel) or
1.5 ∘C (right panel). Values of μ as given in the
legend.
The effect of changing μ0 and μ1 in the two-parameter
temperature profiles, designed to asymptote to either 2.0 ∘C (left
panel) or 1.5 ∘C (right panel). Values of μ0 and μ1 as
given in the legend.
Two-parameter profiles
Whilst aiming to create profiles that are simple and mathematically
tractable, allowing just one parameter may be overly restrictive. For
example, society might be much more able to reduce emissions (corresponding
to high μ values) further in the future, but may be less able to in the
near future. To capture differences in generational approaches to fossil fuel
usage, one additional degree of freedom is introduced, setting μ(t) as a
function of time:
μ(t)=μ0+μ1t.
Matching the first derivative (Appendix, Eq. ) at year
t=0 gives
γ=β-μ0ΔTLim-ΔT0.
Profiles for different μ0 (yr-1) and μ1
(yr-2) values are presented in Fig. . Curves can
approach the warming target rapidly, then quickly asymptote to it through an
increasingly large value in time of μ (e.g. red curve, 2.0 ∘C
target). Similarly, increasing μ values offer the opportunity to have
overshoot occurrences followed by rapid convergence to the desired warming
level (e.g yellow curve, 1.5 ∘C target).
The dependence of the time to stabilisation and any overshoot
magnitude (where present, white space otherwise) on the parameters μ0
and μ1 in the temperature profiles, with ΔTLim=2.0∘C. The scale of the colour bar is nonlinear. The grey region
in
the bottom left corner of the right-hand panel is where temperatures become
higher than the target of 2.0 ∘C and increase throughout the
500 years; thus, peak warming is not attained in that
time.
The dependence of the time to stabilisation and any overshoot
magnitude (where present, white space otherwise) on the parameters μ0
and μ1 in the temperature profiles, with ΔTLim=1.5∘C. The scale of the colour bar is nonlinear. The grey region
in the bottom left corner of the right-hand panel is where temperatures become
higher than the target of 1.5 ∘C and increase throughout the
500 years; thus, peak warming is not attained in that
time.
The left-hand panel in Fig. presents the time from
the year 2015 to achieve stabilisation, defined as within 0.01 ∘C of
the target temperature threshold of 2.0 ∘C. The right-hand panel
shows the maximum additional overshoot temperature should ΔTLim be crossed. Figure shows the same for
ΔTLim=1.5∘C. These look-up charts enable the
selection of a balance between general action on moving away from a
business-as-usual approach to emissions (via parameter μ0) and leaving
more change to future generations (via parameter μ1). Lower μ0
and μ1 values take longer to reach stabilisation levels although they
risk temporary overshoot of the temperature target. The grey shading in the
right-hand panels of Figs.
and is where overshoot happens, and the temperature rises throughout the 500-year period – hence peak warming occurs after
that time. Overshoot is considered present if any year has a temperature
of more than 0.01 ∘C above the target level. By definition, solutions of
μ0<0.0 and μ1=0 never converge.
One potential evolution of global temperature could be a rapid rise to
2.0 ∘C of global warming, followed by strong efforts to quickly reduce
and stabilise at 1.5 ∘C. To achieve this on a single-century timescale, with the curve structure of
Eqs. () and () and ΔTLim=1.5∘C, μ0 must be slightly negative, combined with
high values of μ1. This influences the selection of the ranges of
μ0 and μ1 in Fig. .
Fit of Eq. () (oranges curves) for the years after
2015 and for three representative ESM simulations (red curves) that
correspond to the RCP2.6 scenario of atmospheric gas changes. The blue curve
is the linear fit to the ESM for the period 1995–2015. Annotated in each panel
is the modelling centre and the ESM name. The fit to all the RCP2.6
simulations is given in Fig. .
Fitting to existing ESM simulations
Equations (), () and ()
generate a range of future temperature pathways towards prescribed warming
limits. For these, the related changes in atmospheric gas concentrations and
emissions can be determined. However, many ESMs have been operated in forward
mode, forced with scenarios of atmospheric GHG concentrations that
correspond to heavy mitigation of fossil fuel burning. The RCP2.6 scenario
gives ESM-based estimates of the stabilisation
of global warming around 2.0 ∘C warming since pre-industrial times.
We fit our model to these ESM projections of the RCP2.6 scenario. Parameters
β and ΔT0 are tuned to their projections of temperature for
the years 1995–2015, whilst ΔTLim, μ0 and
μ1 are fitted to the years 2016–2100. Figure
shows this curve calibration against three representative ESM RCP2.6
projections, expanded to the full set of 25 ESMs in Fig. .
Over the years from 2016 to 2100 and for each individual ESM, the RMSE of the differences between the fit and the ESM simulation
is calculated. The mean of these RMSE values is 0.11 ∘C. This value
is similar to the SD of measurement and model estimates
of interannual variability in global temperature after detrending e.g.
Table 1b of . This confirms that our curves can reproduce
the RCP2.6 high-mitigation ESM projections. Otherwise, any systematic
differences would cause the RMSE deviations to be higher than those of the
interannual variability only; the latter is not represented in our profiles.
We additionally fit our curves to pathways in which the emissions are
generated using integrated assessment models (IAMs) and the related global
temperature profiles are created using a simple climate model. This has been
done for warming profiles from the IPCC scenario database
(https://tntcat.iiasa.ac.at/AR5DB/) and for the marker scenarios of the
more recent shared socioeconomic pathways (SSPs) database
(https://tntcat.iiasa.ac.at/SspDb). We demonstrate that the functional
forms used here can also represent these IAM-based scenarios to a good level
of accuracy (see the Supplement).
Accounting for uncertainty in warming rates
The relatively low rate of warming increase since the year 1998 has been the
subject of debate and is sometimes referred to as the “warming hiatus”.
The possibility of such a warming hiatus occurring has been assessed in detail e.g.
. If a natural decadal-timescale fluctuation has
temporarily suppressed the background warming trend, then our HadCRUT-based
warming rate β could be too small. The MAGICC climate impacts model,
parameterised against a range of ESMs, typically projects the recent warming
as around β=0.025∘Cyr-1. As a sensitivity
study, we reproduce Figs. and
using that higher warming rate as Figs.
and , respectively.
Applications
Our profiles enable a common framework for the discussion of warming
trajectories that stabilise to predefined temperature limits. Regional
climate change corresponding to these global temperatures can be estimated
from interpolation of ESM projections e.g. by pattern scaling;
. Such scaling techniques can be linked to impact
models e.g. . In the comprehensive review
of methods for identifying regional differences associated with alternative
global warming targets, note pattern scaling as
a key technique. The accuracy of this interpolation system has been recently
reviewed in detail by and with
enhancements proposed by . In the other
approaches of , the central issue of how
to interpret existing simulations, which even for identical forcings, project
a range of different future final warming levels, remains.
Emissions profiles can be calculated to fulfil the ESM-dependent radiative
forcings associated with any prescribed global temperature stabilisation
profile. These can include different mixtures of individual GHG
emissions, whilst accounting for any perturbed land–atmosphere and
ocean–atmosphere gas exchanges. The sum of the radiation changes for altered
individual atmospheric greenhouse gas combinations must equal the
ESM-dependent radiative forcing. Although our analytical forms are generic
and can be calculated for any prescribed final stabilised temperature ΔTLim, the emphasis here is placed on the 1.5 or 2.0 ∘C
targets. This is due to their strong current discussion in policy circles
regarding clean energy e.g..
Understanding the significance between stabilising global warming at either
1.5 or 2.0 ∘C is a complex and multi-dimensional problem. There are
implications for regional climate changes and impacts and for ”allowable”
emissions, including the range of potential mixes between emitted greenhouse
gases. These factors will also depend on the time evolution of global warming
towards such warming thresholds. Each of these issues requires study,
ideally in a way that enables findings to be compared in a common framework.
The application of these curves is to work towards such a framework by
offering a set of possible future warming pathways for utility in research
initiatives and that can be readily defined through a limited set of
parameters.
Conclusions
Presented in this work are parameter-sparse algebraic curves
that match contemporary levels and the rate of change of global mean
temperature and that asymptote to prescribed warming thresholds. These
represent a smooth transition from current rates of warming through to
stabilised temperature levels. They can include an initial overshoot of
temperatures above any desired final warming level. Their relative simplicity
makes them transparent and open to discussion. If common temperature
scenarios are adopted by a range of studies (by selection of μ0,
μ1 and ΔTLim values), this may allow easier
comparison of either the impacts of or emissions to achieve 1.5 or
2.0 ∘C warming stabilisation. At this stage, we do not associate any
particular parameter combinations (or ranges) with their feasibility of
fulfilment by society.
The curves have five parameters, with three of these constrained by the
current warming level ΔT0, the current rate of warming change
β and the final stabilised state ΔTLim. The remaining
two parameters μ0 and μ1, offering two degrees of freedom, give
flexibility to the pathway shape before asymptoting to the temperature
ΔTLim. Our curves allow for the possibility of temporary
overshoot. This enables the characterisation of the illustrative scenarios
proposed in Fig. 4, and their
metric of dangerous anthropogenic interference (DAI) defined as the
integrated time and magnitude spent overshooting a safe upper limit. Where an
impact study is for a period ahead that is much less than the time to
stabilisation, then these curves allow for the possibility of gradually rising
or declining temperatures through any analysis period.
Some very specific pathways may require further versatility. For instance,
defining a pathway asymptoting to 1.5 ∘C and allowing warming
overshoot to 2.0 ∘C constrains one degree of freedom. If the
difference between the speed of approaching 2.0 ∘C is specified as
either much quicker or much slower than the time from that peak to
1.5 ∘C, then two more degrees of freedom are required, giving three
in total. To satisfy situations such as this, further curve forms could, for
instance, include specification of μ as a quadratic function of time.
The Python scripts leading to any of the diagrams are
available on request to Chris Huntingford (chg@ceh.ac.uk).
The global warming amount to the present day, along with
the estimates of its gradient, comes from the HadCRUT dataset. In particular,
the global annual anomalies are used from the median of the 100-member
ensemble. These values are column 2 (column 1 is date) of
http://www.metoffice.gov.uk/hadobs/hadcrut4/data/current/time_series/HadCRUT.4.5.0.0.annual_ns_avg_smooth.txt.
First and second derivatives
Here we present the first and second derivatives for the one- and
two-parameter profiles.
One-parameter profiles
The first derivative of Eq. () satisfies
dΔTdt=γ-1-e-μt[γ]-[-e-μt⋅(-μ)]⋅[γt-(ΔTLim-ΔT0)],
which, at t=0, gives
dΔTdt|t=0=β=γ+μ(ΔTLim-ΔT0).
The second derivative of Eq. () is found by differentiating
Eq. () with respect to t, giving
d2ΔTdt2=-(-e-μt⋅-μ)γ--e-μt⋅-μγ--e-μt⋅(-μ)⋅(-μ)⋅γt-(ΔTLim-ΔT0)=-2μγe-μt+μ2e-μtγt-(ΔTLim-ΔT0),
which, at time t=0, gives
d2ΔTdt2|t=0=-2μγ-μ2(ΔTLim-ΔT0).
Substitution of condition () into Eq. () gives
d2ΔTdt2|t=0=-2μβ+μ2(ΔTLim-ΔT0).
Two-parameter profiles
The first derivative of Eq. () with time-dependent μ as
given in Eq. () satisfies
dΔTdt=γ-1-e-[μ0+μ1t]tγ--e-[μ0+μ1t]t⋅(-μ0-2μ1t)γt-(ΔTLim-ΔT0),
which, at t=0, gives
dΔTdt|t=0=β=γ+μ0(ΔTLim-ΔT0).
The second derivative is found by differentiating Eq. ()
with respect to t, giving
d2ΔTdt2=--e-μ0+μ1tt⋅-μ0-2μ1tγ--e-μ0+μ1tt⋅-μ0-2μ1tγ--e-μ0+μ1tt⋅-μ0-2μ1t⋅-μ0-2μ1t-e-μ0+μ1tt⋅-2μ1⋅γt-ΔTLim-ΔT0=-2μ0+2μ1γ+-μ0-2μ1t2-2μ1γt-ΔTLim-ΔT0⋅e-μ0+μ1tt.
At time t=0, this gives
d2ΔTdt2|t=0=-2μ0γ-[μ02-2μ1](ΔTLim-ΔT0).
Additional figures
Figure repeats Fig. , but showing
the fit of curves and related parameters (ΔTLim, μ0,
μ1, β and ΔT0) for 25 ESM simulations of the RCP2.6
scenario. For these future fits, there is some interplay between parameter
values that can achieve a good fit. The values fitted were constrained such
that in all cases, 0.0≤ΔTLim≤4.0∘C,
-0.02 ≤μ0≤0.08yr-1 and 0.0≤μ1≤0.0006yr-2. A visual scan suggests a generally good fit for all
ESMs except the GFDL_CM3 model.
Figure shows the dependence of time to
convergence and any overshoot amount on μ0 and μ1, whilst
converging to 2.0 ∘C of global warming. The recent rate of warming
is set to β=0.025∘Cyr-1.
Figure is also for this higher β value,
converging to 1.5 ∘C of global warming.
Identical to Fig. except showing fitted curves
for a larger set of 25 ESMs. Annotated in each panel is the modelling centre,
ESM name and values of ΔTLim (∘C), μ0
(yr-1), μ1 (yr-2), ΔT0
(∘C) and β
(∘Cyr-1).
Identical to Fig. but with β=0.025∘Cyr-1.
Identical to Fig. but with β=0.025∘Cyr-1.
The Supplement related to this article is available online at https://doi.org/10.5194/esd-8-617-2017-supplement.
CH created the mathematical profiles and designed the paper.
All authors helped discuss the expected requirements of the curves for
research into differences between achieving the 1.5 and 2.0 ∘C
targets. All authors made suggestions as to diagram format and aided in
writing the paper.
The authors declare that they have no conflict of
interest.
Acknowledgements
Chris Huntingford acknowledges the NERC national capability fund. All authors
(except SMS) received support from the UK Natural Environmental Research
Council program “Understanding the Pathways to and Impacts of a
1.5 ∘C Rise in Global Temperature”, through specific projects
NE/P014909/1, NE/P014941/1 and NE/P015050/1. We acknowledge the World Climate
Research Programme's Working Group on Coupled Modelling, which is responsible
for CMIP, and we thank the climate modelling groups for producing and making
available their model output.
Edited by: Axel Kleidon
Reviewed by: two anonymous referees
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