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Journal topic
**Climate of the Past**
An interactive open-access journal of the European Geosciences Union

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- Abstract
- Introduction
- The energy balance climate model
- Applications of the energy balance model
- Conclusions and future work
- Code availability
- Appendix A: Empirical calibration of EBM parameters
- Appendix B: Equilibrium climate sensitivity
- Author contributions
- Competing interests
- Acknowledgements
- Review statement
- References

**Research article**
21 Mar 2019

**Research article** | 21 Mar 2019

An energy balance model for paleoclimate transitions

- Department of Mathematics and Statistics, University of Guelph, 50 Stone Road West, Guelph, ON, N1G 2W1, Canada

- Department of Mathematics and Statistics, University of Guelph, 50 Stone Road West, Guelph, ON, N1G 2W1, Canada

**Correspondence**: William F. Langford (wlangfor@uoguelph.ca)

**Correspondence**: William F. Langford (wlangfor@uoguelph.ca)

Abstract

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A new energy balance model (EBM) is presented and is used to study paleoclimate
transitions. While most previous EBMs only dealt with the globally averaged climate, this
new EBM has three variants: Arctic, Antarctic and tropical climates. The EBM incorporates
the greenhouse warming effects of both carbon dioxide and water vapour, and also includes
ice–albedo feedback and evapotranspiration. The main conclusion to be inferred from this
EBM is that the climate system may possess multiple equilibrium states, both warm and
frozen, which coexist mathematically. While the actual climate can exist in only one of
these states at any given time, the EBM suggests that climate can undergo transitions
between the states via mathematical saddle-node bifurcations. This paper proposes that
such bifurcations have actually occurred in Paleoclimate transitions. The EBM is applied
to the study of the *Pliocene paradox*, the *glaciation of Antarctica* and
the so-called *warm, equable climate problem* of both the mid-Cretaceous Period
and the Eocene Epoch. In all cases, the EBM is in qualitative agreement with the
geological record.

How to cite

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How to cite.

Dortmans, B., Langford, W. F., and Willms, A. R.: An energy balance model for paleoclimate transitions, Clim. Past, 15, 493–520, https://doi.org/10.5194/cp-15-493-2019, 2019.

1 Introduction

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For approximately 75 % of the last 540 million years of the paleoclimate
history of the Earth, the climate of both polar regions was mild and free
of permanent ice-caps (Cronin, 2010; Crowley, 2000; Hubert et al., 2000).
Today, both the North and South poles are ice-capped; however, there is
overwhelming evidence that these polar ice-caps are melting.
The Arctic is warming faster than any other region on Earth.
The formation of the present-day Arctic and Antarctic ice-caps occurred
abruptly, at widely separated times in the geological history of the Earth.
This paper explores some of the underlying mechanisms and forcing
factors that have caused climate transitions in the past, with a focus on
transitions that were *abrupt*.
The understanding gained here will be applied in a subsequent paper
to the important problem of determining whether anthropogenic
climate change may lead to an abrupt transition in the future.

We present a new two-layer energy balance model (EBM) for the
climate of the Earth.
General knowledge of climate and of climate change has been
advanced by many studies employing simple EBMs
(Budyko, 1968; Kaper and Engler, 2013; McGehee and Lehman, 2012; North et al., 1981; Payne et al., 2015; Sagan and Mullen, 1972; Sellers, 1969; Stap et al., 2017).
In general, these EBMs facilitate exploration of the relationship between
specific climate forcing mechanisms and the resulting climate changes.
The EBM presented here includes a more accurate representation of the role of
greenhouse gases in climate change than has been the case for previous EBMs.
The model is based on fundamental principles of atmospheric physics,
such as the Beer–Lambert law, the Stefan–Boltzmann law, the
Clausius–Clapeyron equation and the ideal-gas equation.
In particular, the modelling of water vapour acting as a greenhouse gas in the
atmosphere, presented in Sect. 2.3.3, is more physically accurate
than in previous EBMs and it shows that *water vapour feedback*
is important in climate change.
Also, *ice–albedo feedback* plays a central role in this EBM.
The nonlinearity of this EBM leads to *bistability*
(existence of multiple stable equilibrium states), to
*hysteresis* (the climate state realized in the
model depends on the past history) and to *bifurcations*
(abrupt transitions from one state to another).
Forcing factors that are included explicitly in the model include
insolation, CO_{2} concentration, relative humidity,
evapotranspiration, ocean heat transport and atmospheric heat transport.
Other factors that may affect climate change, such as geography,
precipitation, vegetation and continental drift, are not included
explicitly in the EBM but are present only implicitly in so far as
they affect those included factors.

During the Pliocene Epoch, 2.6–5.3 Ma, the climate of the Arctic
region of Earth changed abruptly from ice-free to ice-capped.
The major climate forcing factors (solar constant, orbital parameters,
CO_{2} concentration and locations of the continents)
were all very similar to today (Fedorov et al., 2006, 2010; Haywood et al., 2009; Lawrence et al., 2009; De Schepper et al., 2015).
Therefore, it is difficult to explain why the early Pliocene climate
was so different from that of today.
That problem has been called the *Pliocene paradox*
(Cronin, 2010; Fedorov et al., 2006, 2010).
Currently, there is great interest in mid-Pliocene climate as a natural analogue
of the future warmer climate expected for Earth, due to
anthropogenic forcing.
In recent years, significant progress has been achieved in understanding
Pliocene climate, e.g. by
Haywood et al. (2009); Salzmann et al. (2009); Ballantyne et al. (2010); Steph et al. (2010); von der Heydt and Dijkstra (2011); Zhang and Yan (2012); Zhang et al. (2013); Sun et al. (2013); De Schepper et al. (2013); Tan et al. (2017); Chandan and Peltier (2017, 2018) and Fletcher et al. (2018); see also references therein.
This paper proposes that bistability and bifurcation may have played
a fundamental role in determining the Pliocene climate.

The waning of the warm ice-free paleoclimate at the South Pole,
leading eventually to abrupt glaciation of Antarctica
at the Eocene–Oligocene transition (EOT)
about 34 Ma, is believed to have been caused primarily by two major
geological changes (although other factors played a role).
One is the movement of the continent of Antarctica from the Southern Pacific
Ocean to its present position over the South Pole, followed by the development
of the Antarctic Circumpolar Current (ACC), thus drastically reducing
ocean heat transport to the South Pole (Cronin, 2010; Scher et al., 2015).
The second is the gradual draw down in CO_{2}
concentration worldwide (DeConto et al., 2008; Goldner et al., 2014; Pagani et al., 2011).
The EBM of this paper includes both CO_{2} concentration
and ocean heat transport as explicit forcing factors.
The results of this paper suggest that the abrupt glaciation of Antarctica
at the EOT was the result of a *bifurcation* that occurred as both
of these factors changed incrementally; and furthermore that both are
required to explain the abrupt onset of Antarctic glaciation.

In the mid-Cretaceous Period (100 Ma) the climate of the entire Earth was much more
*equable* than it is today. This means that, compared to today, the
pole-to-Equator temperature gradient was much smaller and also the summer/winter
variation in temperature at mid- to high-latitudes was much less (Barron, 1983). The
differences in forcing factors between the Cretaceous Period and modern times appeared to
be insufficient to explain this difference in the climates. Eric Barron called this
*the warm, equable Cretaceous climate problem* and he explored this problem in a
series of pioneering papers (Barron, 1983; Barron et al., 1981, 1995; Sloan and Barron, 1992). In a similar vein,
Huber and Caballero (2011) and Sloan and Barron (1990, 1992) observed that the early Eocene (56–48 Ma)
encompasses the warmest climates of the past 65 million years, yet climate modelling
studies had difficulty explaining such warm and equable temperatures. Therefore, this
situation was called the *early Eocene warm, equable climate problem*. The EBM of
the present paper suggests that the mathematical mechanism of *bistability*
provides plausible answers to both the mid-Cretaceous and the early Eocene equable
climate problems. In fact, from the perspective of this simple model, these two are
mathematically the same problem. Therefore, we call them collectively the *warm, equable Paleoclimate problem*. Recent progress in paleoclimate science has succeeded in
narrowing the gap between proxy and general circulation model (GCM) estimates for the
Cretaceous climate (Donnnadieu et al., 2006; Ladant and Donnadieu, 2016; O'Brien et al., 2017) and for the Eocene climate
(Baatsen et al., 2018; Hutchinson et al., 2018; Lunt et al., 2016, 2017).

The principal contribution of this paper is a simple climate EBM, based on fundamental
physical laws, that exhibits *bistability*, *hysteresis* and
*bifurcations*. We propose that these three phenomena have occurred in the
paleoclimate record of the Earth and they help to explain certain paleoclimate
transitions and puzzles as outlined above. A key property of this EBM is that its
underlying physical principles are highly *nonlinear*. As is well known, nonlinear
equations can have multiple solutions, unlike linear equations, which can have only one
unique solution (if well-posed). In our EBM, the same set of equations can have two or
more coexisting stable solutions (bistability); for example, an ice-capped solution (like
today's climate) and an ice-free solution (like the Cretaceous climate), even with the
same values of the forcing parameters. The determination of which solution is actually
realized by the planet at a given time is dependent on past history (hysteresis). Changes
in forcing parameters may drive the system abruptly from one stable state to another, at
so-called “tipping points”. In this paper, these tipping points are investigated
mathematically, and are shown to be *bifurcation points*, which are investigated
using mathematical bifurcation theory. Bifurcation theory tells us that the existence of
bifurcation points is preserved (but the numerical values may change) under small
deformations of the model equations. Thus, even though this conceptual model may not give
us precise quantitative information about climate changes, qualitatively there is good
reason to believe that the *existence* of the bifurcation points in the model will
be preserved in similar more refined models and in the real world.

Previous climate models exhibiting multiple equilibrium states
have indicated that bifurcations
may cause abrupt climate transitions (Budyko, 1968; Sellers, 1969; North et al., 1981; Paillard, 1998; Alley et al., 2003; Rial et al., 2004; Lindsay and Zhang, 2005; Ferreira et al., 2010; Thorndike, 2012; Payne et al., 2015; Stap et al., 2017). None of these authors employ the recently developed
mathematical theory of bifurcations to the extent used in this paper. Abrupt transitions
in Quaternary glaciations have been studied by Ganopolski and Rahmstorf (2001); Paillard (2001a, b); Calov and Ganopolsky (2005) and
Robinson et al. (2012). These glacial cycles are strongly influenced by orbital forcings, which
are not within the scope of this paper. Scientists have long considered that abrupt
transitions (or bifurcations) in the Atlantic Meridional Overturning Circulation (AMOC)
were possible, and that this could contribute to abrupt climate change
(Stommel, 1961; Rahmstorf, 1995; Ganopolski and Rahmstorf, 2001; Lenton et al., 2012). This type of change in the AMOC is sometimes called a
*Stommel bifurcation*. However, that phenomenon is also outside the
scope of
the EBM of this paper, which does not include ocean geography or meridional dependence.
(An extension of this EBM, to a partial differential equation (PDE) spherical shell
model, is planned.)

With regard to the EOT,
a variety of indicators, including the analysis of fossil plant stomata
(Steinthorsdottir et al., 2016), imply that decreasing atmospheric *p*CO_{2}
in the Eocene preceded the large shift in oxygen isotope records
that characterize the EOT.
It was hypothesized in Steinthorsdottir et al. (2016) that at the EOT, a certain threshold
of *p*CO_{2} was crossed, resulting in an abrupt change
of climate mode.
Possible mechanisms for the threshold that they hypothesized
are studied in Sect. 3.2.

Even for changes as recent as the mid-Holocene, there is a debate, for example, over whether the abrupt desertification of the Sahara is due to a bifurcation, as suggested in Claussen et al. (1999) using an Earth model of intermediate complexity (EMIC), or is a transient response of the AMOC to a sudden termination of freshwater discharge to the North Atlantic, as proposed in Liu et al. (2009), using a coupled atmosphere–ocean GCM (AOGCM). Similarly, for the glaciation of Greenland at the Pliocene–Pleistocene transition, recent work in Tan et al. (2018), using a coupled GCM–ice-sheet model, shows good agreement with proxy records without the need for bifurcations. The model of this paper does not adapt to these two situations, which are localized and away from the poles where the axis of symmetry restricts the dynamics and facilitates the analysis presented here.

Further investigation of climate changes, using a range of climate models from EBM and EMIC to GCM and AOGCM, is warranted to clarify the underlying mechanisms of abrupt climate change. Very sophisticated GCMs, which include many 3-D processes, are only able to run a few climate trajectories; while EBMs and EMICs may explore more possibilities and investigate climate transitions (tipping points) with major simplifications and with less effort. More rigorous mathematical analysis is possible on small models, which may then suggest lines of inquiry on large models. The Earth climate system is extremely complex. For best results, a hierarchy of climate models is necessary.

The energy balance model presented here is conceptual and qualitative.
It contains many simplifying assumptions and is not
intended to give a detailed description of the climate of the Earth
with quantitative precision.
It complements but does not replace more detailed GCMs.
Geographically, this model is as simple as possible.
It follows in a long tradition of *slab models* of the atmosphere.
Previous slab models represented the atmosphere of the Earth as a
globally averaged uniform slab at a single temperature *T*.
The temperature *T* is determined in those models by a global energy
balance equation of the form *energy in* = *energy out*.
Such models are unable to differentiate between different climates
at different latitudes; for example, if the polar climate is changing
more rapidly than the tropical climate.
In this new model, the forcing parameters of the slab atmosphere are chosen to
represent each one of three particular latitudes: Arctic, Antarctic or tropics.
Each of these regions is represented by its own slab model, with its own forcing
parameters and its own surface temperature *T*_{S}.
In addition, each region has its own variable *I*_{A} representing the intensity
of the radiation re-emitted by the atmosphere.
The two independent variables *I*_{A} and *T*_{S} are
determined in each model by two energy balance equations, expressing energy
balance in the atmosphere and energy balance at the surface, respectively.
In this way, the different climate responses of these three regions to their
respective forcings can be explored.

The role played by greenhouse gases in climate change is a particular
focus of this model.
Greenhouse gases trap heat emitted by the surface and are major contributors
to global warming.
The very different roles of the two principal greenhouse gases in the
atmosphere, carbon dioxide and water vapour, are
analyzed here in Sects. 2.3.2 and 2.3.3, respectively.
The greenhouse warming effect of CO_{2} increases with the density
of the atmosphere but is independent of temperature, while the greenhouse
warming of H_{2}O increases with temperature but is
independent of the density (or partial pressure) of the other gases present.
The greenhouse warming of methane (CH_{4}) acts in a similar
fashion to that of CO_{2}; therefore, CH_{4} can be
incorporated into the CO_{2} concentration.
As an increase in CO_{2} concentration causes
climate warming, this warming causes an increase in evaporation of
H_{2}O into the atmosphere, which further increases the climate
warming beyond that due to CO_{2} alone
(this is true both in the model and in the real atmosphere).
This effect is known as *water vapour feedback*.
The energy balance model presented here is the first EBM to incorporate
these important roles of the greenhouse gases in such detail.

The paper concludes with two appendices.
In Appendix Appendix A, model parameters that are difficult or impossible to
determine for paleoclimates are calibrated using the abundant satellite
and surface data available for today's climate.
In addition, justification is given for parameter values chosen for the model.
In Appendix Appendix B the paleoclimate model of this paper is adapted to
modern-day conditions and its equilibrium climate sensitivity (ECS) is
determined.
Here, ECS is the change in global mean temperature produced by
a doubling of CO_{2} in the model, starting from the
pre-industrial value of 270 ppm.
For this EBM, the ECS is determined to be Δ*T*=3.3 ^{∘}C,
which is at the high end of the range accepted by the IPCC (IPCC, 2013).

2 The energy balance climate model

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In this EBM, the atmosphere and surface
are each assumed to be in energy balance.
Short-wave radiant energy from the sun is partly reflected by the atmosphere back into space,
a small portion is absorbed directly by the atmosphere and the remainder passes through to the surface.
The surface reflects some of this short-wave energy (which is assumed to escape to space)
and absorbs the rest, re-emitting long-wave radiant energy of intensity *I*_{S} upward into the atmosphere.
The atmosphere is modelled as a slab, with greenhouse gases,
that absorbs a fraction *η* of the radiant energy *I*_{S}
from the surface.
The atmosphere re-emits radiant energy of total intensity *I*_{A}.
Of this radiation *I*_{A}, a fraction *β* is directed downward to
the surface, and the remaining fraction (1−*β*) goes upward
and escapes to space.

This model is based on the uniform slab EBM used in Payne et al. (2015), modified as shown in Fig. 1. In our case, the “slab” is a uniform column of air of unit cross section, extending vertically above the surface to the tropopause, and located either at a pole or at the Equator. The symbols in Fig. 1 are defined in Table 1.

This section presents the mathematical derivation of the EBM. Readers interested only in the climate applications of the model may skip this section and go directly to Sect. 3. A preliminary version of this EBM was presented in a conference proceedings paper (Dortmans et al., 2018). The present model incorporates several important improvements over that model. The differences between that model and the one presented here are indicated where appropriate in the text. Furthermore, the previous paper (Dortmans et al., 2018) considered only the application of the model to the Arctic climate and the Pliocene paradox; it did not study Antarctic or tropical climate or the Cretaceous warm, equable climate problem, as does the present paper.

The model consists of two energy balance equations, one for the
atmosphere and one for the surface.
In Fig. 1, the so-called *forcings* are shown as
arrows, pointing in the direction of energy transfer.
From Fig. 1, the energy balance equations
for the atmosphere and surface are given, respectively, by

$$\begin{array}{}\text{(1)}& {\displaystyle}\mathrm{0}& {\displaystyle}={F}_{\mathrm{A}}+{F}_{\mathrm{C}}+{\mathit{\xi}}_{\mathrm{A}}Q+\mathit{\eta}{I}_{\mathrm{S}}-{I}_{\mathrm{A}},\phantom{\rule{1em}{0ex}}\mathrm{and}\text{(2)}& {\displaystyle}\mathrm{0}& {\displaystyle}={F}_{\mathrm{O}}-{F}_{\mathrm{C}}+(\mathrm{1}-\mathit{\alpha}){F}_{\mathrm{S}}-{I}_{\mathrm{S}}+\mathit{\beta}{I}_{\mathrm{A}}.\end{array}$$

Symbols and parameter values for the model are defined in Table 1.
Appendix A4 provides derivations and justification for the values
of the empirical parameters.
The forcings *F*_{O} and *F*_{A} represent ocean and atmosphere heat
transport, respectively, and are specified as constants for each region of interest.
Heat transport by conduction/convection from the
surface to the atmosphere is denoted *F*_{C}.
This quantity will be largely dependent on surface temperature, *T*_{S}.
As described in Appendix Appendix A we have modelled it as a hyperbola that is
mostly flat for temperatures below freezing, and grows roughly linearly for temperatures above
freezing so that

$$\begin{array}{}\text{(3)}& {F}_{\mathrm{C}}={A}_{\mathrm{1}}\left({T}_{\mathrm{S}}-{T}_{\mathrm{R}}\right)+\sqrt{{A}_{\mathrm{1}}^{\mathrm{2}}({T}_{\mathrm{S}}-{T}_{\mathrm{R}}{)}^{\mathrm{2}}+{A}_{\mathrm{2}}^{\mathrm{2}}},\end{array}$$

where *A*_{1} and *A*_{2} are constants.
Since the model is concerned with temperatures around the freezing point of
water, we set this as a reference temperature, *T*_{R}=273.15 K.

The annually averaged intensity of solar radiation striking a surface parallel to
the Earth's surface but at the top of the atmosphere is *Q*.
The value of *Q* at either Pole is $Q=\mathrm{173.2}\phantom{\rule{0.125em}{0ex}}{\mathrm{Wm}}^{-\mathrm{2}}$ and at the
Equator is 418.8 Wm^{−2} (McGehee and Lehman, 2012; Kaper and Engler, 2013).
A fraction
*ξ*_{R} of this short-wave radiation is reflected by the atmosphere back into space
and a further fraction
*ξ*_{A} is directly absorbed by the atmosphere;
the remainder penetrates to the surface.
See Appendix A1 for the derivation of values for *ξ*_{R} and *ξ*_{A}.
The solar radiation striking the surface of the Earth is

$$\begin{array}{}\text{(4)}& {F}_{\mathrm{S}}=(\mathrm{1}-{\mathit{\xi}}_{\mathrm{R}}-{\mathit{\xi}}_{\mathrm{A}})Q.\end{array}$$

The surface *albedo* is the fraction, *α*, of this solar radiation
that reflects off the surface back into space.
Thus the solar forcing absorbed by the surface is (1−*α*)*F*_{S},
and the solar radiation reflected back to space is *α**F*_{S}.
Typical values of the surface albedo *α* are 0.6–0.9 for snow,
0.4–0.7 for ice, 0.2 for crop land and 0.1 or less for open ocean.
In this paper we introduce a smoothly varying albedo
given by the hyperbolic tangent function:

$$\begin{array}{}\text{(5)}& \mathit{\alpha}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left([{\mathit{\alpha}}_{\mathrm{W}}+{\mathit{\alpha}}_{\mathrm{C}}]+[{\mathit{\alpha}}_{\mathrm{W}}-{\mathit{\alpha}}_{\mathrm{C}}]\mathrm{tanh}\left({\displaystyle \frac{{T}_{\mathrm{S}}-{T}_{\mathrm{R}}}{\mathrm{\Omega}}}\right)\right),\end{array}$$

where *α*_{C} and *α*_{W} are the albedo values for cold and warm temperatures, respectively,
and the parameter Ω determines the steepness of the transition between
*α*_{C} and *α*_{W}.
See Appendix A2 for a full explanation of Eq. (3).

The emission of long-wave radiation, *I*, from a body is
governed by the *Stefan–Boltzmann law*, *I*=*ϵ**σ**T*^{4}, where
*ϵ* is the emissivity,
*σ* is the Stefan–Boltzmann constant, and *T* is temperature.
The surface of the Earth acts as a black-body radiator; so for the radiation emitted from
the surface *ϵ*=1, thus

$$\begin{array}{}\text{(6)}& {I}_{\mathrm{S}}=\mathit{\sigma}{T}_{\mathrm{S}}^{\mathrm{4}}.\end{array}$$

Previous authors have postulated an idealized uniform atmospheric temperature
*T*_{A} for the slab model, so that the intensity of radiation emitted by
the atmosphere, *I*_{A}, is

$$\begin{array}{}\text{(7)}& {I}_{\mathrm{A}}=\mathit{\u03f5}\mathit{\sigma}{T}_{\mathrm{A}}^{\mathrm{4}}.\end{array}$$

The emissivity is *ϵ*=0.9 since the atmosphere is an imperfect black-body radiator.
A uniform temperature for the atmosphere
does not exist in the real world, where *T*_{A} varies strongly
with height, unlike *T*_{S}, which has a single value.
Previous two-layer EBMs have used (*T*_{S},*T*_{A}) as the two independent
variables in the two energy balance Eqs. (1) and (1).
Here instead, we use (*T*_{S},*I*_{A}) as the two independent variables,
and then we formally let *T*_{A} be *defined* by Eq. (5).
The fraction of *I*_{A} that reaches the surface is *β*=0.63; see
Appendix A6.

The parameter *η* represents the fraction of the infrared radiation *I*_{S}
from the surface that is absorbed by the atmosphere and is called
*absorptivity*.
The major constituents of the atmosphere are nitrogen and oxygen
and these gases do not absorb any infrared radiation.
The gases that do contribute to the absorptivity *η* are called
*greenhouse gases*.
Chief among these are carbon dioxide and water vapour.
The contribution of these two greenhouse gases to *η* are analyzed
in Sects. 2.3.2 and 2.3.3, respectively.
Although both contribute to warming of the climate,
the underlying physical mechanisms of the two are very different.
In general, the contribution to *η* from water vapour is a function of
temperature. Another major contributor to absorption is the liquid and solid water in clouds. We
model this portion of the absorption as constant, since we do not include any data on cloud cover
variation. However, we experimented with making this portion vary with surface temperature, and
the results were not qualitatively different than those presented here.

We rescale temperature by the reference temperature *T*_{R}=273.15 K and define
new nondimensional temperatures and new parameters

$$\begin{array}{ll}{\displaystyle}{\mathit{\tau}}_{\mathrm{S}}& {\displaystyle}={\displaystyle \frac{{T}_{\mathrm{S}}}{{T}_{\mathrm{R}}}},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{i}_{A}={\displaystyle \frac{{I}_{\mathrm{A}}}{\mathit{\sigma}{T}_{\mathrm{R}}^{\mathrm{4}}}},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}q={\displaystyle \frac{Q}{\mathit{\sigma}{T}_{\mathrm{R}}^{\mathrm{4}}}},{f}_{O}={\displaystyle \frac{{F}_{\mathrm{O}}}{\mathit{\sigma}{T}_{\mathrm{R}}^{\mathrm{4}}}},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{f}_{A}={\displaystyle \frac{{F}_{\mathrm{A}}}{\mathit{\sigma}{T}_{\mathrm{R}}^{\mathrm{4}}}},\\ \text{(8)}& {\displaystyle}& {\displaystyle}{f}_{C}={\displaystyle \frac{{F}_{\mathrm{C}}}{\mathit{\sigma}{T}_{\mathrm{R}}^{\mathrm{4}}}},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{a}_{\mathrm{1}}={\displaystyle \frac{{A}_{\mathrm{1}}}{\mathit{\sigma}{T}_{\mathrm{R}}^{\mathrm{3}}}},{a}_{\mathrm{2}}={\displaystyle \frac{{A}_{\mathrm{2}}}{\mathit{\sigma}{T}_{\mathrm{R}}^{\mathrm{4}}}},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathit{\omega}={\displaystyle \frac{\mathrm{\Omega}}{{T}_{\mathrm{R}}}}.\end{array}$$

After normalization, the freezing temperature of water is represented by
*τ*=1 and the atmosphere and surface energy balance
Eqs. (1)–(4)
simplify to

$$\begin{array}{}\text{(9)}& {\displaystyle}{i}_{A}& {\displaystyle}={f}_{A}+{f}_{C}\left({\mathit{\tau}}_{\mathrm{S}}\right)+{\mathit{\xi}}_{\mathrm{A}}q+\mathit{\eta}\left({\mathit{\tau}}_{\mathrm{S}}\right){\mathit{\tau}}_{\mathrm{S}}^{\mathrm{4}},\text{(10)}& {\displaystyle}{i}_{A}& {\displaystyle}={\displaystyle \frac{\mathrm{1}}{\mathit{\beta}}}\left[{f}_{C}\left({\mathit{\tau}}_{\mathrm{S}}\right)-{f}_{O}-(\mathrm{1}-\mathit{\alpha}({\mathit{\tau}}_{\mathrm{S}}\left)\right)(\mathrm{1}-{\mathit{\xi}}_{\mathrm{R}}-{\mathit{\xi}}_{\mathrm{A}})q+{\mathit{\tau}}_{\mathrm{S}}^{\mathrm{4}}\right],\end{array}$$

where, from Appendix Appendix A,

$$\begin{array}{}\text{(11)}& {\displaystyle}{f}_{C}\left({\mathit{\tau}}_{\mathrm{S}}\right)& {\displaystyle}={a}_{\mathrm{1}}\left({\mathit{\tau}}_{\mathrm{S}}-\mathrm{1}\right)+\sqrt{{a}_{\mathrm{1}}^{\mathrm{2}}({\mathit{\tau}}_{\mathrm{S}}-\mathrm{1}{)}^{\mathrm{2}}+{a}_{\mathrm{2}}^{\mathrm{2}}},\text{(12)}& {\displaystyle}\mathit{\alpha}\left({\mathit{\tau}}_{\mathrm{S}}\right)& {\displaystyle}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left([{\mathit{\alpha}}_{\mathrm{W}}+{\mathit{\alpha}}_{\mathrm{C}}]+[{\mathit{\alpha}}_{\mathrm{W}}-{\mathit{\alpha}}_{\mathrm{C}}]\mathrm{tanh}\left({\displaystyle \frac{{\mathit{\tau}}_{\mathrm{S}}-\mathrm{1}}{\mathit{\omega}}}\right)\right).\end{array}$$

The range of surface temperatures *T*_{S} observed on Earth is
restricted to an interval around the freezing point
of water, 273.15 K, and
therefore the nondimensional temperature *τ* lies
in an interval around *τ*=1.
In this paper, we assume $\mathrm{0.8}\le \mathit{\tau}\le \mathrm{1.2}$, which
corresponds approximately to a range in more familiar degrees Celsius of
$-\mathrm{54}\phantom{\rule{0.125em}{0ex}}{}^{\circ}\mathrm{C}\le T\le +\mathrm{54}$ ^{∘}C.
Another reason for the upper limit on temperature is that the
Clausius–Clapeyron law used in Sect. 2.3.3 fails to apply at
temperatures above the boiling point of water.

The goal of this section is to define the absorptivity parameter *η* in the EBM
Eq. (6) (or Eq. 1) in such a way
that the atmosphere in the uniform slab model will absorb the
same fraction *η* of the long-wave radiation *I*_{S} from the surface as does the
real nonuniform atmosphere of the Earth.
This absorption is due primarily to water vapour, carbon dioxide and clouds.
Previous energy balance models have assigned a constant value to *η*,
often determined by climate data.
In the present EBM, *η* is not constant but is a function of other more
fundamental physical quantities, such as $\mathit{\mu},\mathit{\delta},{k}_{\mathrm{C}},{k}_{\mathrm{W}}$ and *T*.
This function is determined by classical physical laws.
In this way, the present EBM adjusts automatically to changes in these
physical quantities, and represents a major advance over previous EBMs.

The Beer–Lambert law states that when a beam of radiation (or light) enters a
sample of absorbing material, the absorption of radiation at any
point *z* is proportional to the intensity of the radiation *I*(*z*)
and also to the concentration or density of the absorber *ρ*(*z*).
This bilinearity fails to hold at very high intensity of radiation or high density
of absorber, neither of which is the case in the Earth's atmosphere.
Whether this law is applied to the uniform slab model or to the nonuniform
real atmosphere, it yields the same differential equation

$$\begin{array}{}\text{(13)}& {\displaystyle \frac{\mathrm{d}I}{\mathrm{d}z}}=-k\mathit{\rho}\left(z\right)I\left(z\right),\end{array}$$

where *k* (m^{2} kg^{−1}) is the absorption coefficient of the material,
*ρ* (kg m^{−3}) is the density of the absorbing substance such as
CO_{2} and *z* (*m*) is distance along the path.
The differential Eq. (6) may be integrated from *z*=0 (the surface)
to *z*=*Z* (the tropopause) to give

$$\begin{array}{}\text{(14)}& {\displaystyle \frac{{I}_{\mathrm{T}}}{{I}_{\mathrm{S}}}}={e}^{-{\int}_{\mathrm{0}}^{Z}k\mathit{\rho}\left(z\right)\phantom{\rule{0.125em}{0ex}}\mathrm{d}z}\equiv {e}^{-\mathit{\lambda}},\phantom{\rule{1em}{0ex}}\mathrm{where}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathit{\lambda}\equiv \underset{\mathrm{0}}{\overset{Z}{\int}}k\mathit{\rho}\left(z\right)\phantom{\rule{0.125em}{0ex}}\mathrm{d}z.\end{array}$$

Here *I*_{T}≡*I*(*Z*) is the intensity of radiation escaping to space at the tropopause
*z*=*Z*, and *λ* is the so-called *optical depth* of the material.
Note that *λ* is dimensionless.
The absorptivity parameter *η* in Eqs. (1) and (6)
represents the fraction of the outgoing radiation *I*_{S} from the surface
that is absorbed by the atmosphere
(not to be confused with the absorption coefficient *k*).
It follows from the Beer–Lambert law that *η* is completely
determined by the corresponding optical depth parameter
*λ*; that is

$$\begin{array}{}\text{(15)}& \mathit{\eta}={\displaystyle \frac{{I}_{\mathrm{S}}-{I}_{\mathrm{T}}}{{I}_{\mathrm{S}}}}=\mathrm{1}-{e}^{-\mathit{\lambda}}.\end{array}$$

For a mixture of *n* attenuating materials, with densities *ρ*_{i}, absorption
coefficients *k*_{i} and corresponding optical depths *λ*_{i}, the
Beer–Lambert law extends to

$$\begin{array}{}\text{(16)}& {\displaystyle \frac{{I}_{\mathrm{T}}}{{I}_{\mathrm{S}}}}={e}^{-{\mathrm{\Sigma}}_{i=\mathrm{1}}^{n}{\int}_{\mathrm{0}}^{Z}{k}_{i}{\mathit{\rho}}_{i}\left(z\right)\phantom{\rule{0.125em}{0ex}}\mathrm{d}z}\equiv {e}^{-{\mathrm{\Sigma}}_{i=\mathrm{1}}^{n}{\mathit{\lambda}}_{i}},\end{array}$$

so that

$$\begin{array}{}\text{(17)}& {\displaystyle}\mathit{\eta}=\mathrm{1}-{e}^{-{\mathrm{\Sigma}}_{i=\mathrm{1}}^{n}{\mathit{\lambda}}_{i}}=\mathrm{1}-\prod _{i=\mathrm{1}}^{n}{e}^{-{\mathit{\lambda}}_{i}}=\mathrm{1}-\prod _{i=\mathrm{1}}^{n}(\mathrm{1}-{\mathit{\eta}}_{i}),\end{array}$$

where ${\mathit{\eta}}_{i}=\mathrm{1}-{e}^{-{\mathit{\lambda}}_{i}}$.
Equation (10) is the key to solving the problem posed in the first
sentence of this subsection.
For the *i*th absorbing material in the slab model, we set its optical
depth *λ*_{i} to be equal to the value of the optical depth
that this gas has in the Earth's atmosphere as given by
Eq. (7), and then combine them using Eq. (9).
This calculation is presented for the case of CO_{2}
in Sect. 2.3.2 and for water vapour in Sect. (2.3.3).
For the third absorbing material in our model, clouds, we assume a constant value *η*_{Cl}.

The two principal greenhouse gases are carbon dioxide (CO_{2}) and water vapour
(H_{2}O). Because they act in different ways, we determine the absorptivities
*η*_{C} and *η*_{W}, and optical depths *λ*_{C} and *λ*_{W} of
CO_{2} and H_{2}O separately, and then combine their effects, along with the
absorption due to clouds, *η*_{Cl}, using the Beer–Lambert law for mixtures,
Eq. (10). Methane acts similarly to CO_{2} and can be included in the
optical depth for CO_{2}. Other greenhouse gases have only minor influence and are
ignored in this paper.

Although it is well-known that gases like CO_{2} and
H_{2}O absorb infrared radiation *I*_{S}
only at specific wavelengths (spectral lines), in this paper
the *grey gas approximation* is used; that is, the
absorption coefficient *k*_{C} or *k*_{W} is given as a single number
averaged over the infrared spectrum (Pierrehumbert, 2010).
The thesis by Dortmans (2017) presents a survey of values in the literature for
the absorption coefficients *k*_{C} and *k*_{W} of CO_{2}
and H_{2}O, respectively, in the grey gas approximation.
The values used in this paper are
given in Table 1; and are derived as described in
Appendix A5.

The concentration of CO_{2} in the atmosphere is usually
expressed as a ratio, in molar parts per million (ppm) of CO_{2}
to dry air, and written as *μ*.
There is convincing evidence that *μ* has varied greatly in the
geological history of the Earth, and has decreased slowly over the past
100 million years; however, today *μ* is increasing due to human activity.
The value before the industrial revolution was *μ*=270 ppm, but today
*μ* is slightly above 400 ppm.

Although traditionally *μ* is measured as a ratio of molar concentrations, in practice
both the density *ρ* and the absorption coefficient *k* are expressed in mass units of
kilograms. Therefore, before proceeding, *μ* must be
converted from a molar ratio to a ratio of masses in units of kilograms. The mass of one
mole of CO_{2} is approximately $m{m}_{C}=\mathrm{44}\times {\mathrm{10}}^{-\mathrm{3}}$ kg mol^{−1}. The dry
atmosphere is a mixture of 78 *%* nitrogen, 21 *%* oxygen and 0.9 *%* argon,
with molar masses of 28, 32 and 40 g mol^{−1}, respectively. Neglecting other trace
gases in the atmosphere, a weighted average gives the molar mass of the dry atmosphere as
mm${}_{\mathrm{A}}=\mathrm{29}\times {\mathrm{10}}^{-\mathrm{3}}$ kg mol^{−1}. Therefore, the CO_{2}
concentration *μ* measured in molar ppm is converted to mass concentration in kg ppm
by multiplication by the ratio mm${}_{C}/{\mathrm{mm}}_{\mathrm{A}}\approx \mathrm{1.52}$. If
*ρ*_{A}(*z*) is the density of the atmosphere at altitude *z* in
kg m^{−3}, then the mass density of CO_{2} at the same altitude, with
molar concentration *μ* ppm, is

$$\begin{array}{}\text{(18)}& {\mathit{\rho}}_{C}\left(z\right)=\mathrm{1.52}\phantom{\rule{0.125em}{0ex}}{\displaystyle \frac{\mathit{\mu}}{{\mathrm{10}}^{\mathrm{6}}}}\phantom{\rule{0.125em}{0ex}}{\mathit{\rho}}_{\mathrm{A}}\left(z\right)\phantom{\rule{1em}{0ex}}\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{3}}.\end{array}$$

It is known that CO_{2} disperses rapidly throughout the
Earth's atmosphere, so that its concentration *μ* may be assumed
independent of location and altitude (IPCC, 2013).
As the density of the atmosphere decreases with altitude,
the density of CO_{2} decreases at exactly the same rate,
according to Eq. (11).
Substituting Eq. (11) into Eq. (7) determines
the optical depth *λ*_{C} of CO_{2}

$$\begin{array}{}\text{(19)}& {\mathit{\lambda}}_{C}=\mathrm{1.52}\phantom{\rule{0.125em}{0ex}}{\displaystyle \frac{\mathit{\mu}}{{\mathrm{10}}^{\mathrm{6}}}}\phantom{\rule{0.125em}{0ex}}{k}_{\mathrm{C}}\underset{\mathrm{0}}{\overset{Z}{\int}}{\mathit{\rho}}_{\mathrm{A}}\left(z\right)\phantom{\rule{0.125em}{0ex}}\mathrm{d}z.\end{array}$$

Now consider a vertical column of air, of unit cross section, from
surface to tropopause.
The integral in Eq. (12) is precisely the total mass
of this column.
The atmospheric pressure at the surface is
*P*_{A}, and this is
the total weight of the column. Therefore
${\int}_{\mathrm{0}}^{Z}\mathit{\rho}\left(A\right)\phantom{\rule{0.125em}{0ex}}\mathrm{d}z={P}_{\mathrm{A}}/g$, where
*g* is acceleration due to gravity.
Therefore, the optical depth *λ*_{C} of CO_{2}
in the actual atmosphere in Eq. (12) is

$$\begin{array}{}\text{(20)}& {\mathit{\lambda}}_{C}=\mathit{\mu}{G}_{C},\phantom{\rule{2em}{0ex}}\text{where}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{G}_{C}\equiv \mathrm{1.52}\times {\mathrm{10}}^{-\mathrm{6}}\phantom{\rule{0.125em}{0ex}}{k}_{\mathrm{C}}{\displaystyle \frac{{P}_{\mathrm{A}}}{g}}.\end{array}$$

With *λ*_{C} so determined, it follows from Eq. (8) that

$$\begin{array}{}\text{(21)}& {\mathit{\eta}}_{C}=\mathrm{1}-{e}^{-{\mathit{\lambda}}_{C}}=\mathrm{1}-\mathrm{exp}(-\mathit{\mu}{G}_{C}).\end{array}$$

As listed in Table 1 and derived in Appendix A5,
the calibrated value for *k*_{C} is 0.07424, and therefore the value for
the greenhouse gas parameter for CO_{2} is ${G}_{C}=\mathrm{1.166}\times {\mathrm{10}}^{-\mathrm{3}}$.

The *dry atmosphere EBM* is obtained by assuming there is no water vapour and no clouds in the
atmosphere. Hence *η*=*η*_{C} and $\mathit{\delta}=\mathrm{0}.$ We assume also that ${\mathit{\xi}}_{\mathrm{R}}={\mathit{\xi}}_{\mathrm{A}}=\mathrm{0}$.
Making these substitutions in the EBM
Eqs. (6)–(6)
yields

$$\begin{array}{}\text{(22)}& {\displaystyle}{i}_{A}& {\displaystyle}={f}_{A}+{f}_{C}\left({\mathit{\tau}}_{\mathrm{S}}\right)+[\mathrm{1}-\mathrm{exp}(-\mathit{\mu}{G}_{C}\left)\right]\cdot {\mathit{\tau}}_{\mathrm{S}}^{\mathrm{4}},\text{(23)}& {\displaystyle}{i}_{A}& {\displaystyle}={\displaystyle \frac{\mathrm{1}}{\mathit{\beta}}}\left({f}_{C}\left({\mathit{\tau}}_{\mathrm{S}}\right)-{f}_{O}-[\mathrm{1}-\mathit{\alpha}({\mathit{\tau}}_{\mathrm{S}}\left)\right]q+{\mathit{\tau}}_{\mathrm{S}}^{\mathrm{4}}\right).\end{array}$$

Due to the nonlinearity of the ice–albedo function in 3,
there can be in fact up to three points of intersection as shown in
Fig. 2a.
If the CO_{2} level *μ*
decreases sufficiently, the warm state (*τ*_{S}>1) disappears, while as
CO_{2} increases, the frozen state (*τ*_{S}<1) may disappear.
The CO_{2} level *μ* is quite elevated in Fig. 2, but
the effects of water vapour as a greenhouse gas and of clouds have been ignored.

Figure 2b introduces a *bifurcation diagram*
(Kuznetsov, 2004),
in which the two EBM Eqs. (15)–(15)
have been solved for the surface temperature *τ*_{S}, which is then plotted
as a function of the parameter *μ*.
Figure 2b shows the three distinct solutions
*τ*_{S} of the EBM: a warm solution (*τ*_{S}>1), a frozen solution
(*τ*_{S}<1), and a third solution that crosses through *τ*_{S}=1
and connects the other two solutions in *saddle-node bifurcations*
(Kuznetsov, 2004).
Stability analysis (Dortmans, 2017) shows that the warm and frozen solutions
are *stable* (in a dynamical systems sense), while the third solution
(denoted by a dashed line) is unstable.

In this section, we determine the absorptivity of water vapour
as a function of temperature, *η*_{W}(*τ*_{S}),
using fundamental physical laws including the Clausius–Clapeyron equation,
the ideal-gas law and the Beer–Lambert law (Pierrehumbert, 2010).
We also assume the idealized
lapse rate of the International Standard Atmosphere (ISA) as defined by ICAO (ICAO, 1993).
Unlike CO_{2}, the concentration of water vapour
(H_{2}O) in the atmosphere varies widely with location and altitude.
This is because the partial pressure of H_{2}O varies
strongly with the local temperature.
In fact, it is bounded by a
maximum saturated value, which itself is a nonlinear function of temperature,
${P}_{\mathrm{W}}^{\mathrm{sat}}\left(T\right)$.
The actual partial pressure of water vapour is then a fraction, *δ*, of this
saturated value,

$$\begin{array}{}\text{(24)}& {P}_{\mathrm{W}}\left(T\right)=\mathit{\delta}\phantom{\rule{0.125em}{0ex}}{P}_{\mathrm{W}}^{\mathrm{sat}}\left(T\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{0}\le \mathit{\delta}\le \mathrm{1},\end{array}$$

where *δ* is called *relative humidity*. While ${P}_{\mathrm{W}}^{\mathrm{sat}}\left(T\right)$
varies greatly with *T* in the atmosphere, the relative humidity *δ* is
comparatively constant. If the actual *P*_{W}(*T*) exceeds the saturated value
${P}_{\mathrm{W}}^{\mathrm{sat}}\left(T\right)$ (i.e. *δ*>1), the excess water vapour condenses out
of the atmosphere and falls as rain or snow.

The saturated partial pressure at temperature *T* is determined by the
*Clausius–Clapeyron equation* (Pierrehumbert, 2010),

$$\begin{array}{}\text{(25)}& {P}_{\mathrm{W}}^{\mathrm{sat}}\left(T\right)={P}_{\mathrm{W}}^{\mathrm{sat}}\left({T}_{\mathrm{R}}\right)\phantom{\rule{0.125em}{0ex}}\mathrm{exp}\left({\displaystyle \frac{{L}_{\mathrm{v}}}{{R}_{\mathrm{W}}}}\left[{\displaystyle \frac{\mathrm{1}}{{T}_{\mathrm{R}}}}-{\displaystyle \frac{\mathrm{1}}{T}}\right]\right),\end{array}$$

where *T*_{R} is the reference temperature, here chosen to be the freezing point
of water (273.15 K); *L*_{v} is the latent heat of vaporization of water and
*R*_{W} is the ideal-gas constant for water, see Table 1. The
actual partial pressure of water vapour at relative humidity *δ* and temperature *T*
is then given by combining Eqs. (15) and (16).

We may use the *ideal-gas law* in the form
*P*_{W}=*ρ*_{W}*R*_{W}*T* to convert the partial pressure *P*_{W} of
water vapour at temperature *T* to mass density *ρ*_{W} of
water vapour at that temperature.
Substituting into Eqs. (15) and (16) gives

$$\begin{array}{}\text{(26)}& {\mathit{\rho}}_{\mathrm{W}}\left(T\right)=\mathit{\delta}\phantom{\rule{0.125em}{0ex}}{\displaystyle \frac{{P}_{\mathrm{W}}^{\mathrm{sat}}\left({T}_{\mathrm{R}}\right)}{{R}_{\mathrm{W}}T}}\phantom{\rule{0.125em}{0ex}}\mathrm{exp}\left({\displaystyle \frac{{L}_{\mathrm{v}}}{{R}_{\mathrm{W}}}}\left[{\displaystyle \frac{\mathrm{1}}{{T}_{\mathrm{R}}}}-{\displaystyle \frac{\mathrm{1}}{T}}\right]\right).\end{array}$$

Transforming to the dimensionless temperature $\mathit{\tau}=T/{T}_{\mathrm{R}}$ as in Sect. 2.1.1, this becomes

$$\begin{array}{}\text{(27)}& {\mathit{\rho}}_{\mathrm{W}}\left(\mathit{\tau}\right)=\mathit{\delta}\left({\displaystyle \frac{{P}_{\mathrm{W}}^{\mathrm{sat}}\left({T}_{\mathrm{R}}\right)}{{R}_{\mathrm{W}}{T}_{\mathrm{R}}}}\right){\displaystyle \frac{\mathrm{1}}{\mathit{\tau}}}\mathrm{exp}\left({\displaystyle \frac{{L}_{\mathrm{v}}}{{R}_{\mathrm{W}}{T}_{\mathrm{R}}}}\left[{\displaystyle \frac{\mathit{\tau}-\mathrm{1}}{\mathit{\tau}}}\right]\right).\end{array}$$

The Beer–Lambert law in Sect. 2.2 implies that
the absorptivity of a greenhouse gas *η*_{i} is completely determined
by its optical depth *λ*_{i}.
For water vapour, from Eqs. (8) and (7),

$$\begin{array}{}\text{(28)}& {\mathit{\eta}}_{\mathrm{W}}=\mathrm{1}-{e}^{-{\mathit{\lambda}}_{W}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{where}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{\mathit{\lambda}}_{W}\equiv \underset{\mathrm{0}}{\overset{Z}{\int}}{k}_{\mathrm{W}}{\mathit{\rho}}_{\mathrm{W}}\left(z\right)\phantom{\rule{0.125em}{0ex}}\mathrm{d}z.\end{array}$$

Here *k*_{W} is the absorption coefficient of water vapour; see
Appendix A5. In order to evaluate the integral in Eq. (19), we
need to know how *ρ*_{W} varies with height *z*. We have shown that
*ρ*_{W} is a function of temperature, given by Eqs. (17)
or (18). Therefore, we need an expression for the variation in
temperature *T* with height *z*. Under normal conditions, the temperature
*T* decreases with height in the troposphere. This rate of decrease is called the
*lapse rate* Γ, and is defined as

$$\begin{array}{}\text{(29)}& \mathrm{\Gamma}\equiv -{\displaystyle \frac{\mathrm{d}T}{\mathrm{d}z}}.\end{array}$$

Normally, Γ is positive and is close to constant in value from the surface to the tropopause. The International Civil Aviation Organization has defined, for reference purposes, the ISA, in which Γ is assigned the constant value $\mathrm{\Gamma}=\mathrm{6.49}\times {\mathrm{10}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}\mathrm{K}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{1}}$ (ICAO, 1993). Using this assumption, the variation in temperature with height is given as

$$\begin{array}{}\text{(30)}& T\left(z\right)={T}_{\mathrm{S}}-\mathrm{\Gamma}z\phantom{\rule{1em}{0ex}}\mathrm{or}\phantom{\rule{1em}{0ex}}\mathit{\tau}\left(z\right)={\mathit{\tau}}_{\mathrm{S}}-\mathit{\gamma}z,\end{array}$$

where the normalized lapse rate is
$\mathit{\gamma}=\mathrm{\Gamma}/{T}_{\mathrm{R}}=\mathrm{2.38}\times {\mathrm{10}}^{-\mathrm{5}}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{1}}$.
The tropopause height *Z* ranges from 8 to 11 km at the poles and
16 to 18 km at the Equator, based on satellite measurements (Kishore et al., 2006).
Therefore, both *T*(*Z*) and *τ*(*Z*) are positive.
For this paper we will take the height to be *Z*=9 km at the poles
and *Z*=17 km at the Equator.

Equation (21) may be used to change the variable of integration
in Eq. (19) for the optical depth of water vapour,
from *z* to *τ*.
The result is

$$\begin{array}{}\text{(31)}& {\mathit{\lambda}}_{W}={k}_{\mathrm{W}}\underset{\mathrm{0}}{\overset{Z}{\int}}{\mathit{\rho}}_{\mathrm{W}}\left(\mathit{\tau}\right(z\left)\right)\phantom{\rule{0.125em}{0ex}}\mathrm{d}z={\displaystyle \frac{{k}_{\mathrm{W}}}{\mathit{\gamma}}}\underset{{\mathit{\tau}}_{\mathrm{S}}-\mathit{\gamma}Z}{\overset{{\mathit{\tau}}_{\mathrm{S}}}{\int}}{\mathit{\rho}}_{\mathrm{W}}\left(\mathit{\tau}\right)\phantom{\rule{0.125em}{0ex}}\mathrm{d}\mathit{\tau}.\end{array}$$

Now substitute Eq. (18) into the integral Eq. (22) and simplify to

$$\begin{array}{}\text{(32)}& {\mathit{\lambda}}_{W}\left({\mathit{\tau}}_{\mathrm{S}}\right)\phantom{\rule{0.33em}{0ex}}\phantom{\rule{0.33em}{0ex}}=\phantom{\rule{0.33em}{0ex}}\phantom{\rule{0.33em}{0ex}}\mathit{\delta}\phantom{\rule{0.33em}{0ex}}{G}_{\mathrm{W}\mathrm{2}}\underset{{\mathit{\tau}}_{\mathrm{S}}-\mathit{\gamma}Z}{\overset{{\mathit{\tau}}_{\mathrm{S}}}{\int}}{\displaystyle \frac{\mathrm{1}}{\mathit{\tau}}}\mathrm{exp}\left({G}_{\mathrm{W}\mathrm{1}}\left[{\displaystyle \frac{\mathit{\tau}-\mathrm{1}}{\mathit{\tau}}}\right]\right)\phantom{\rule{0.125em}{0ex}}\mathrm{d}\mathit{\tau},\end{array}$$

where the greenhouse gas parameters *G*_{W1} and *G*_{W2}
for water vapour are defined as

$$\begin{array}{}\text{(33)}& {G}_{\mathrm{W}\mathrm{1}}\equiv {\displaystyle \frac{{L}_{\mathrm{v}}}{{R}_{\mathrm{W}}{T}_{\mathrm{R}}}}\phantom{\rule{1em}{0ex}}\mathrm{and}\phantom{\rule{1em}{0ex}}{G}_{\mathrm{W}\mathrm{2}}\equiv {\displaystyle \frac{{k}_{\mathrm{W}}{P}_{\mathrm{W}}^{\mathrm{sat}}\left({T}_{\mathrm{R}}\right)}{{R}_{\mathrm{W}}{T}_{\mathrm{R}}\mathit{\gamma}}}.\end{array}$$

Finally, using the Beer–Lambert law, the absorptivity *η*_{W}
of water vapour in Eq. (19) is
determined by its optical depth, and is now a function of
the surface temperature,

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathit{\eta}}_{\mathrm{W}}\left({\mathit{\tau}}_{\mathrm{S}}\right)=\mathrm{1}-\mathrm{exp}\left[-{\mathit{\lambda}}_{W}\left({\mathit{\tau}}_{\mathrm{S}}\right)\right]\\ \text{(34)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}=\mathrm{1}-\mathrm{exp}\left[-\mathit{\delta}\phantom{\rule{0.33em}{0ex}}{G}_{\mathrm{W}\mathrm{2}}\underset{{\mathit{\tau}}_{\mathrm{S}}-\mathit{\gamma}Z}{\overset{{\mathit{\tau}}_{\mathrm{S}}}{\int}}{\displaystyle \frac{\mathrm{1}}{\mathit{\tau}}}\mathrm{exp}\left({G}_{\mathrm{W}\mathrm{1}}\left[{\displaystyle \frac{\mathit{\tau}-\mathrm{1}}{\mathit{\tau}}}\right]\right)\mathrm{d}\mathit{\tau}\right].\end{array}$$

The definite integral in this expression is easily evaluated numerically in the
process of solving the atmosphere and surface EBM equations.
As given in Table 1 and derived in Appendix A5, the calibrated
value of *k*_{W} is 0.05905 m^{2} kg^{−1} and the greenhouse gas parameters
for H_{2}O are

$${G}_{\mathrm{W}\mathrm{1}}=\mathrm{17.89}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{G}_{\mathrm{W}\mathrm{2}}=\mathrm{12.05}.$$

Figure 3a shows that the function *η*_{W}(*τ*_{S}) in
Eq. (25)
increases rapidly from near 0 to near 1 as *τ*_{S} increases past 1, and it
steepens towards a step function as *δ* increases.
This implies that if the surface temperature *τ*_{S} in the surface
energy balance Eq. (6) increases past *τ*_{S}=1,
then the absorptivity of water vapour *η*_{W}(*τ*_{S}), acting as a greenhouse
gas in Eq. (6),
increases rapidly, thus amplifying the heating of the atmosphere by the
radiation ${I}_{\mathrm{S}}=\mathit{\sigma}{T}_{\mathrm{S}}^{\mathrm{4}}$ from the surface.
Energy balance requires a corresponding increase in the radiation *I*_{A}
transmitted from the atmosphere back to the surface, further increasing
the surface temperature *τ*_{S}.
This is a positive feedback loop called *water vapour feedback*.
This positive feedback is manifested in Fig. 3b
in the additional increase in *T*_{S} on the warm branch,
beyond that due to the ice–albedo bifurcation.
Compare this with Fig. 2, where no water vapour is present.
This rapid nonlinear change is due to the relatively large size of the
greenhouse constant ${G}_{\mathrm{W}\mathrm{1}}=\frac{{L}_{\mathrm{v}}}{{R}_{\mathrm{W}}{T}_{\mathrm{R}}}=\mathrm{17.89}$
in the exponent of the Clausius–Clapeyron Eq. (18),
as it reappears in Eq. (25).

The combined effect of two greenhouse gases
is determined by the Beer–Lambert law as shown in Sect. 2.2.
If *η*_{C} is the absorptivity of CO_{2} as in Eq. (14)
and *η*_{W} is the absorptivity of H_{2}O as in Eq. (19), then the combined absorptivity *η* of these two is
obtained by adding the two corresponding optical depths
*λ*_{C} and *λ*_{W}. The overall absorptivity of the atmosphere including these two
greenhouse gases and the (assumed) constant effect of clouds is, by Eq. (10),

$$\begin{array}{ll}{\displaystyle}\mathit{\eta}\left({\mathit{\tau}}_{\mathrm{S}}\right)& {\displaystyle}=\mathrm{1}-(\mathrm{1}-{\mathit{\eta}}_{C})(\mathrm{1}-{\mathit{\eta}}_{\mathrm{W}})(\mathrm{1}-{\mathit{\eta}}_{\mathrm{Cl}})\\ {\displaystyle}& {\displaystyle}=\mathrm{1}-\mathrm{exp}\left[-{\mathit{\lambda}}_{C}-{\mathit{\lambda}}_{W}\left({\mathit{\tau}}_{\mathrm{S}}\right)\right](\mathrm{1}-{\mathit{\eta}}_{\mathrm{Cl}})\\ {\displaystyle}& {\displaystyle}=\mathrm{1}-\mathrm{exp}[-\mathit{\mu}\cdot {G}_{C}-\mathit{\delta}\phantom{\rule{0.33em}{0ex}}{G}_{\mathrm{W}\mathrm{2}}\\ \text{(35)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}\underset{{\mathit{\tau}}_{\mathrm{S}}-\mathit{\gamma}Z}{\overset{{\mathit{\tau}}_{\mathrm{S}}}{\int}}{\displaystyle \frac{\mathrm{1}}{\mathit{\tau}}}\mathrm{exp}\left({G}_{\mathrm{W}\mathrm{1}}\left[{\displaystyle \frac{\mathit{\tau}-\mathrm{1}}{\mathit{\tau}}}\right]\right)\mathrm{d}\mathit{\tau}\left]\right(\mathrm{1}-{\mathit{\eta}}_{\mathrm{Cl}}).\end{array}$$

The full nondimensional two-layer EBM is therefore specified by Eqs. (6)–(6) and (25), and the parameter values in Table 1.

The above analysis shows that there are two highly nonlinear
positive feedback mechanisms in this EBM.
Both serve to amplify an increase (or decrease) in surface temperature
*T*_{S} near the freezing point as follows.
Consider the case of rising temperature.
The first feedback is *ice–albedo feedback*, due to a change in
albedo in the surface equilibrium equation.
As illustrated in Fig. A1, if the surface temperature
increases slowly through the freezing point, causing a drop in albedo,
there is a large increase in solar energy absorbed, leading to an
abrupt increase in temperature.
The second is *water vapour feedback* in the atmosphere
equilibrium equation.
As shown in Fig. 3,
if the surface temperature continues to increase above freezing,
then the absorptivity of water vapour increases dramatically,
strengthening the greenhouse effect for water vapour and further
increasing the temperature.
Both of these mechanisms act independently of the concentration of
CO_{2} itself in the atmosphere.
However, if the concentration of CO_{2} goes up,
causing a rise in temperature, then each of these two positive feedback
mechanisms can *amplify* the increase in temperature that
would occur due to CO_{2} alone.

3 Applications of the energy balance model

Back to toptop
The goal of this section is an exploration of the underlying causes of abrupt climate changes that have occurred on Earth in the past 100 million years, using, as a tool, the EBM developed in Sect. 2. The most dramatic climate changes occurred in the two polar regions of the Earth. The climate of the tropical region of the Earth has changed relatively little in the past 100 million years. In this section, the EBMs for the two polar regions are applied to the Arctic Pliocene paradox, the glaciation of Antarctica, the warm, equable Cretaceous problem and the warm, equable Eocene problem. For comparison, a control EBM with parameter values set to those of the tropics shows no abrupt climate changes in the tropical climate for the past 100 million years.

The Arctic region of the Earth's surface has had ice cover year-round for only the past
few million years. For at least 100 million years prior to about 3 Ma, the Arctic had no
permanent ice cover, although there could have been seasonal snow in the winter.
Recently, investigators have found plant and animal remains, in particular on the
farthest northern islands of the Canadian Arctic Archipelago, which demonstrate that
there was a wet temperate rainforest there for millions of years, similar to that now
present on the Pacific Northwest coast of North America
(Basinger et al., 1994; Greenwood et al., 2010; Struzik, 2015; West et al., 2015; Wolfe et al., 2017). The relative humidity has been estimated at
67 % (Jahren and Sternberg, 2003), and this value has been chosen for *δ* in the Arctic EBM of
this section. The change from an ice-free to an ice-covered condition in the Arctic
occurred abruptly, during the late Pliocene and early Pleistocene. This is sometimes
called the Pliocene–Pleistocene transition (PPT 3.0–2.5 Ma; Willeit et al., 2015; Tan et al., 2018). It
has been a longstanding challenge to explain this dramatic change in the climate;
however,
significant progress has been made recently for the case of
the Greenland ice sheet (GIS).
The authors of Willeit et al. (2015) have solved the “inverse problem” for the GIS by finding a
schematic *p*CO_{2} concentration scenario from an ensemble of transient
simulations using an EMIC that gives the best fit to data from 3.2 to 2.4 Ma, taking
into account the obliquity cycle. Meanwhile, Tan et al. (2018) have used an
AOGCM asynchronously coupled with sophisticated ice sheet models to
reproduce the waxing and waning of the GIS across the PPT, obtaining good qualitative
agreement with ice rafted debris data reconstructions.

Currently, there is great interest in the mid-Pliocene climate, because it is the most
recent paleoclimate that resembles the future warmer climate now predicted for the Earth.
Significant progress in understanding the Pliocene climate has been achieved in recent
years. The Pliocene Research, Interpretation and Synoptic Mapping (PRISM) project of the
US Geological Survey has contributed to this goal (Dowsett et al., 2011, 2013, 2016), as has the
Pliocene Model Intercomparison Project (PlioMIP;
Haywood et al., 2011, 2016; Zhang et al., 2013).
Advances in the extraction and interpretation of proxy data
have given a clearer picture of the warm Pliocene climate
(Haywood et al., 2009; Salzmann et al., 2009; Ballantyne et al., 2010; Steph et al., 2010; Seki et al., 2010; Bartoli et al., 2011; De Schepper et al., 2013; Knies et al., 2014; O'Brien et al., 2014; Brierley et al., 2015; Fletcher et al., 2018).
At the same time, computer models have achieved closer
agreement with proxy data, see for example
Haywood et al. (2009); Dowsett et al. (2011); Zhang and Yan (2012); Sun et al. (2013); Willeit et al. (2015); Tan et al. (2017, 2018); Chandan and Peltier (2017, 2018),
and other references therein.
Using a fully coupled atmosphere–ocean GCM, Lunt et al. (2008) considered the following forcing
factors contributing to late Pliocene glaciation: decreasing carbon dioxide concentration, closure
of the Panama seaway, end of a permanent El Niño state, tectonic uplift and changing orbital
parameters. They concluded that falling CO_{2} levels were primarily responsible for the formation
of the Greenland ice sheet in the late Pliocene.
Recently, Tan et al. (2018) have strengthened this conclusion.

During the Pliocene Epoch, important forcing factors that
determine climate were very similar to those of today.
The Earth orbital parameters, the CO_{2} concentration,
solar radiation intensity, position of the continents, ocean currents and
atmospheric circulation all had values close to the values they have today.
Yet, in the early/mid-Pliocene, 3.5–5 million years ago, the Arctic climate
was much milder than that of today.
Arctic surface temperatures were 8–19 ^{∘}C
warmer than today and global sea levels were 15–20 m
higher than today, and yet CO_{2} levels are estimated to
have been 340–400 ppm,
about the same as 20th century values (Ballantyne et al., 2010; Csank et al., 2011; Tedford and Harington, 2003).
As mentioned in the Introduction section, the problem of explaining how such
dramatically different climates could exist with such similar forcing
parameter values has been called the *Pliocene paradox*
(Cronin, 2010; Fedorov et al., 2006, 2010; Zhang and Yan, 2012).

Another interesting fact concerning polar glaciation is that
although both poles have transitioned abruptly from ice-free to ice-covered conditions,
they did so at very different geological times.
The climate forcing conditions of Earth are highly symmetric between the
two hemispheres, and for most of the past 200 million years (or more)
the climates of the two poles have been similar.
However, there was an anomalous interval of about 30 million years, from
the EOT, 34 Ma, to the early Pliocene, 4 Ma,
when the Antarctic was largely ice covered but the Arctic was
largely land ice free.
Because CO_{2} disperses rapidly in the Atmosphere,
its concentration *μ* must be the same everywhere at any given time.
Therefore we seek a forcing factor other than *μ* to account
for this 30-million-year period of broken symmetry.
One obvious difference is geography.
Since the Eocene, the South Pole has been land-locked in Antarctica,
while the North Pole has been in the Arctic Ocean.
Therefore, our two EBMs for the North Pole and South Pole have very
different values for ocean heat transport *F*_{O}.
We will show that this difference is sufficient to account for the gap of
30 million years between the Antarctic and Arctic glaciations.

Our Pliocene Arctic EBM brackets the Pliocene Epoch between the
mid-Eocene (50 Ma) and pre-industrial modern times, and it models
the effects of the slow decrease in both CO_{2} and
ocean heat transport *F*_{O} in the Arctic over this long time interval.
In this Arctic model, abrupt glaciation of the Arctic is inevitable,
due to the existence of a bifurcation point.

During the Eocene (56–34 Ma), temperatures were much higher than today, especially in
the Arctic (Greenwood et al., 2010; Wolfe et al., 2017; Huber and Caballero, 2011); also, CO_{2} concentration *μ* was higher than today. Estimates of Eocene
CO_{2} concentration *μ* vary from 1000–1500 ppm (Pagani et al., 2005, 2006) to 490 ppm
(Wolfe et al., 2017). For this EBM, we set mid-Eocene CO_{2} at *μ*=1000 (Pagani et al., 2005).
Both temperature and CO_{2} concentration have decreased steadily but not
monotonically, with many fluctuations from their Eocene values to pre-industrial modern
values. The overall decrease in CO_{2} concentration observed since the Eocene may
be attributed to decreased volcanic activity, increased absorption and sequestration by
vegetation and the oceans, continental erosion, and other sinks.

The changes in ocean heat transport to the Arctic are more complicated
and derive from many factors only summarized here
(De Schepper et al., 2015; Haug et al., 2004; Knies et al., 2014; Lunt et al., 2008; Zhang et al., 2013).
There was a slow drop in global sea level, in large part due to the gradual
accumulation of vast amounts of water in the form of ice and snow on Antarctica.
It has been estimated that the total amount of ice today in the Antarctic is
equivalent to a change in sea level of about 58 m (Fretwell et al., 2013; IPCC, 2013).
This drop in sea level likely reduced the flow of warm tropical ocean
water into the Arctic.
Against this background, several other factors came into play
due to changing geography.
In the Eocene, the North Atlantic Ocean was not always connected to
the Arctic, but the Turgai Sea existed between Europe and Asia
and connected the warm Indian Ocean to the Arctic,
until about 29 Ma (Briggs, 1987).
By the Oligocene, the Turgai Sea had closed and the North Atlantic had
opened between Greenland and Norway, forming a deep-water connection
to the Arctic Ocean.
The Bering Strait opened and closed.
During the Pliocene the formation of the Isthmus of Panama about 3.5 Ma
cut off a warm equatorial current that had existed between the Atlantic and
Pacific, at least since Cretaceous times.
On the Atlantic side of the isthmus, the sea water became warmer, and
became more saline due to evaporation.
The Gulf Stream carried this warm salty water to western Europe.
One might expect the Gulf Stream to transport more heat into the Arctic.
However, some believe that the Gulf Stream actually contributed
to glaciation in the Arctic (Haug et al., 2004; Bartoli et al., 2005) as follows.
Evaporation from the Gulf Stream waters contributed to rainfall across
northern Europe and Siberia, increasing the flow of fresh water in rivers
emptying into the Arctic Ocean.
This reduced the salinity, and hence the density, of the Arctic Ocean waters.
The Gulf Stream waters in the North Atlantic now cooler, and denser due to
high salinity, were forced downward by the less dense Arctic waters,
which began to flow into the North Atlantic on the surface.
The denser Gulf Stream waters returned southward as a deep ocean current,
without having conveyed much heat to the Arctic.
Meanwhile the low salinity Arctic surface water, with a higher freezing
temperature, began to freeze,
resulting in higher albedo and accelerating Arctic glaciation.
In large measure, these changing geographical factors partially
cancelled each other in their contributions to ocean heat transport to the Arctic.
In the EBM, we summarize all of the above heat transport mechanisms
by specifying a slow overall decrease in ocean heat transport to the
Arctic, represented by the single forcing parameter *F*_{O}.

Figure 4a shows graphs of the surface and atmosphere
equilibrium equations for the Arctic Pliocene EBM, for varying values of
*μ*.
The figure shows only one surface equilibrium curve (red), with
${F}_{\mathrm{O}}=\mathrm{50}\phantom{\rule{0.125em}{0ex}}\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}$, because the change in *F*_{O}
is relatively small.
The blue atmosphere equilibrium curves represent values of
CO_{2} concentration *μ* falling from 1100 to 200 ppm,
from the top to bottom blue curves.
It is clear that there may exist up to three
points of intersection of a given atmosphere equilibrium curve (blue) with
the surface equilibrium curve (red); namely, a warm equilibrium state
*τ*_{s}>1, a frozen equilibrium state *τ*_{S}<1 and a third
(intermediate) solution, which is always unstable (when it exists).
As *μ* decreases and the warm equilibrium state *τ*_{s}>1
approaches the local minimum on the red S-shaped curve, the unstable intermediate equilibrium state
moves down the middle branch of the red S-shaped curve.
When they meet, these two equilibria coalesce, then disappear,
via a saddle-node bifurcation.
Beyond this saddle-node bifurcation, only one equilibrium state remains,
which is the stable frozen state on the left in the Figure.
Dynamical systems theory tells us that following this bifurcation,
the system will transition rapidly to that frozen equilibrium state.
The paleoclimate record shows that CO_{2} concentration
was trending downward for millions of years before and during the Pliocene.
Therefore, Fig. 4a predicts that an abrupt drop
in temperature to a frozen state would be inevitable if this trend
continued far enough.

In order to explore this downward trend further, we bracket the Pliocene
Epoch between the mid-Eocene Epoch (50 Ma) and the pre-industrial
modern era (300 years ago), and define a surrogate time variable *ν* by

$$\begin{array}{}\text{(36)}& t=\mathrm{50}(\mathrm{1}-\mathit{\nu})\phantom{\rule{1em}{0ex}}\mathrm{Ma}.\end{array}$$

As reviewed above, it is believed that ocean heat transport *F*_{O}
decreased modestly over this time period, mainly due to the drop in
global sea level, while the CO_{2} concentration *μ*
decreased more significantly.
Therefore, we express both *μ* and *F*_{O} as decreasing
functions of *bifurcation parameter* *ν*

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}\mathit{\mu}=\mathrm{1000}-\mathrm{730}\cdot \mathit{\nu}\phantom{\rule{1em}{0ex}}\mathrm{ppm}\\ \text{(37)}& {\displaystyle}& {\displaystyle}{F}_{\mathrm{O}}=\mathrm{60}-\mathrm{10}\cdot \mathit{\nu}\phantom{\rule{1em}{0ex}}{\mathrm{Wm}}^{-\mathrm{2}}.\end{array}$$

Here, *ν*=0 corresponds to estimated mid-Eocene values of *μ* and
*F*_{O} (Pagani et al., 2005; Wolfe et al., 2017; Barron et al., 1981), while *ν*=1 corresponds to modern
pre-industrial values (IPCC, 2013). Equation (26) defines *μ* and
*F*_{O} as linear functions of *ν*. In the real world, neither *μ* nor
*F*_{O} decreased linearly. This is not an obstacle for our bifurcation analysis.
What is important is that, somewhere between *ν*=0 and *ν*=1, a bifurcation point is
crossed.

Figure 4b is a bifurcation diagram,
which shows the dependence of surface temperature *τ*_{S}
on this bifurcation parameter *ν*.
Note that for *ν*=0 (mid-Eocene values, 50 Ma) only the warm equilibrium state exists.
At about *ν*=0.116 (44 Ma) both warm and
frozen equilibrium states exist.
However, as *ν* increases toward *ν*=1, the warm equilibrium
state disappears in a saddle-node (or fold) bifurcation, leaving only the
frozen equilibrium state to the right of the saddle-node bifurcation.
This bifurcation occurs at approximately *ν*=0.91, which
corresponds to ${F}_{\mathrm{O}}=\mathrm{50.9}\phantom{\rule{0.125em}{0ex}}{\mathrm{Wm}}^{-\mathrm{2}}$ and *μ*=336,
which is in good agreement with the determination (Seki et al., 2010)
that the warm Pliocene *p*CO_{2} was between 330 and
400 ppm, similar to today.
The temperatures of the warm state and frozen state, at the bifurcation
value of *ν*=0.91 in the EBM, are +3.9 and
−26 ^{∘}C, respectively.
The surrogate time of the bifurcation point, *ν*=0.91, corresponds
to a geological time of *t*=4.5 Ma, from Eq. (25),
which is close to
the time of glaciation of the Arctic in the geological record.

The EBM plotted in Fig. 4 provides
a plausible explanation for the Pliocene paradox.
For millions of years up to the mid-Pliocene, while the Arctic
temperature remained above freezing on the warm solution
branch in Fig. 4b, the climate
change was incremental.
Then the slowly acting physical forcings of decreasing
CO_{2} concentration and decreasing ocean heat
transport, *F*_{O},
were amplified by the mechanisms of ice–albedo feedback
and water vapour feedback, both of which act very strongly
when the temperature crosses the freezing point of water.
The EBM suggests that when the freezing temperature was
approached, the Arctic climate changed abruptly via a
saddle-node bifurcation as in Fig. 4b,
from a warm state to a frozen state.

Another explanation has been proposed for the Pliocene paradox. There is convincing evidence that at the beginning of the Pliocene, there was a permanent El Niño condition in the tropical Pacific Ocean (Fedorov et al., 2006, 2010; Steph et al., 2010; Cronin, 2010; von der Heydt and Dijkstra, 2011). However, some have disputed this finding (Watanabe et al., 2011). It has been suggested that a permanent El Niño condition could explain the warm early Pliocene, and that the onset of the El Niño–Southern Oscillation (ENSO) was the cause of sudden cooling of the Arctic during the Pliocene. Today, it is known that ENSO can influence weather patterns as far away as the Arctic. However, the present authors propose that bistability and bifurcation provide a more satisfactory explanation for the Pliocene paradox, and suggest that the concurrent change in ENSO may have been a consequence, not the cause, of the changing Pliocene Arctic climate (work in progress).

Antarctica in the mid-Cretaceous Period was ice free, and it remained so for the
remainder of the Cretaceous Period and into the Paleocene
and early Eocene.
However, recent investigations in paleoclimate science have shown
that there was an abrupt drop in temperature and an onset of
glaciation in Antarctica at the EOT
about 34 Ma
(Katz et al., 2008; Lear et al., 2008; Miller et al., 2008; Scher et al., 2011, 2015; Ladant et al., 2014; Ruddiman, 2014).
In the mid-Cretaceous, the continent of Antarctica was in the South
Pacific Ocean and the South Pole was located in open ocean,
warmed by South Pacific Ocean currents (Cronin, 2010).
Then, the continent of Antarctica moved poleward and began to encroach
upon the South Pole towards the end of the Cretaceous, fully covering
the South Pole before the end of the Eocene (Briggs, 1987).
The diminishing marine influence on the South Pole coincided with the onset of
cooling of Antarctica about 45 Ma (mid-Eocene).
The opening of both the Drake Passage (between South America and Antarctica)
and of the Tasmanian Gateway (between Antarctica and Australia)
near the end of the Eocene (34 Ma) led to the development of the ACC, which further isolated the South Pole from warm
ocean heat transport and accelerated the cooling and glaciation of Antarctica
(Cronin, 2010).
Therefore, from the early Eocene to the Oligocene, the ocean heat transport
to the South Polar region decreased significantly, from near the Cretaceous value,
about 100 Wm^{−2} (Barron et al., 1981), to a much lower value,
estimated here to be 30 Wm^{−2}.
The fact that the glaciation of the Antarctic took place 30 million years before the glaciation of the Arctic is very likely due primarily to the
much larger decrease in ocean heat transport that took place at the South Pole.

Because the atmosphere is well mixed, at any given time the
CO_{2} level in the Antarctic is the same as elsewhere.
For this EBM, we estimate the early Eocene global
CO_{2} level as *μ*=1100 ppm, (Pagani et al., 2005, 2006; Cronin, 2010);
decreasing to approximately modern levels by the end of the Oligocene,
23 Ma, that is to *μ*=400 ppm.

In the recent literature, there has been a discussion of whether the primary cause of the
onset of Antarctic glaciation is the slow decline in CO_{2} concentration or the
decrease in poleward ocean heat transport due to the opening of ocean gateways and the
development of the ACC (DeConto et al., 2008; Goldner et al., 2014; Pagani et al., 2011; Scher et al., 2015). In fact, Ladant et al. (2014) state
“The reasons for this greenhouse–icehouse transition have long been debated, mainly
between the 'tectonic-oceanic' hypothesis and the CO_{2} hypothesis”. In the work
by Scher et al. (2015), it is observed that the onset of the ACC coincided with major changes
in global ocean circulation, which probably contributed to the drawdown in CO_{2}
concentration in the atmosphere. Since both CO_{2} concentration and ocean heat
transport *F*_{O}
are explicit parameters in the EBM presented here, they can be varied
independently in the model to investigate which one played a primary role.

Figure 5 shows the solutions of the EBM equations
for Antarctic parameter values, including *δ*=0.67,
${F}_{\mathrm{A}}=\mathrm{45}\phantom{\rule{0.125em}{0ex}}{\mathrm{Wm}}^{-\mathrm{2}}$, $Q=\mathrm{173.2}\phantom{\rule{0.125em}{0ex}}{\mathrm{Wm}}^{-\mathrm{2}}$,
*α*_{W}=0.15, and *Z*=9 km.
Points of intersection of one red line (surface equation) and one
blue line (atmosphere equation) are the equilibrium solutions of the system.
As noted earlier, there can be up to three equilibrium solutions.
In Fig. 5a the ocean heat transport *F*_{O}
is held fixed, while CO_{2} concentration *μ* is
decreased from the top to bottom blue lines.
In Fig. 5b the reverse is true; the
CO_{2} concentration *μ* is held fixed while
the ocean heat transport *F*_{O} is decreased from bottom to top
red lines.
For a sufficiently small value of *F*_{O},
the warm solution disappears in Fig. 5b.
This occurs when the warm solution meets the unstable
intermediate solution in a saddle-node bifurcation.
Beyond this bifurcation, only the frozen solution (*τ*_{S}<1)
exists. In contrast, reducing *μ* in Fig. 5a
does not affect the existence
of the warm equilibrium, only the existence of the cold equilibrium.

Figure 6 displays bifurcation diagrams
in which the surface temperature *τ*_{S}, which has been determined
as a solution of the EBM, is plotted while parameters in the EBM are
varied smoothly, from the early Eocene values, 55 million years ago, to
the late Oligocene values, 23 million years ago.
These dates are chosen to bracket the time of the glaciation of Antarctica.
First we introduce a *bifurcation parameter* *ν*, which acts as a
surrogate time variable.
The bifurcation parameter *ν* is related to geological time by

$$\begin{array}{}\text{(38)}& t=\mathrm{55}-\mathrm{32}\cdot \mathit{\nu}\phantom{\rule{0.125em}{0ex}}\mathrm{Ma}.\end{array}$$

Thus, *ν*=0 corresponds to the early Eocene, 55 Ma, and *ν*=1
corresponds to the late Oligocene, 23 Ma.
During this time, it is known that both CO_{2} concentration
*μ* and ocean heat transport *F*_{O} were decreasing.
To study the effects of the simultaneous reduction of *μ* and *F*_{O},
we express both as simple linear functions of the bifurcation parameter
*ν*, as follows:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}\mathit{\mu}=\mathrm{1100}-\mathrm{700}\cdot \mathit{\nu},\\ \text{(39)}& {\displaystyle}& {\displaystyle}{F}_{\mathrm{O}}=\mathrm{100}-\mathrm{70}\cdot \mathit{\nu},\phantom{\rule{2em}{0ex}}\mathrm{0}\le \mathit{\nu}\le \mathrm{1}.\end{array}$$

Here, *ν*=0 corresponds to the high early Eocene values
of *μ*=1100 ppm and ${F}_{\mathrm{O}}=\mathrm{100}\phantom{\rule{0.125em}{0ex}}{\mathrm{Wm}}^{-\mathrm{2}}$, while *ν*=1 corresponds
to the low Oligocene–Miocene boundary values of *μ*=400 ppm
and ${F}_{\mathrm{O}}=\mathrm{30}\phantom{\rule{0.125em}{0ex}}{\mathrm{Wm}}^{-\mathrm{2}}$.
Thus, in Eq. (27), as *ν* increases from 0 to 1,
the climate forcing factors *μ* and *F*_{O} decrease linearly from
their Eocene values to late Oligocene values.
Strictly speaking, these forcings did not change linearly in time;
however, the important fact is that overall they were decreasing,
For our bifurcation results the decrease does not need to be linear
in the EBM.
The atmospheric heat transport parameter *F*_{A} is held constant
at ${F}_{\mathrm{A}}=\mathrm{45}\phantom{\rule{0.125em}{0ex}}{\mathrm{Wm}}^{-\mathrm{2}}$. The solar radiation is $Q=\mathrm{173.2}\phantom{\rule{0.125em}{0ex}}{\mathrm{Wm}}^{-\mathrm{2}}$,
the warm albedo value is *α*_{W}=0.15 and the tropopause height is *Z*=9 km.
Other parameters are as in Table 1.

Figure 6a shows a saddle-node bifurcation
at *ν*=0.779, corresponding to forcing parameter values
*μ*=555 ppm and ${F}_{\mathrm{O}}=\mathrm{45.5}\phantom{\rule{0.125em}{0ex}}\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}$ in the model,
and to geological time about 30 Ma, assuming the linear time
relation in Eq. (26).
This corresponds closely to best estimates of the timing of
glaciation in the geological record of about 34 Ma at the EOT.
This quantitative agreement should not be taken very seriously, as it is largely
due to assumptions made in the modelling.
However, the *existence* of the bifurcation, implying an abrupt
glaciation of Antarctica, can be taken seriously, as this is a robust
(structurally stable) property of the model.
The warm state and frozen state temperatures coexisting at the bifurcation
point are +3.5 and −22.1 ^{∘}C.
As *ν* increases past the bifurcation point *ν*=0.779,
the warm climate state ceases to exist, and the climate system
transitions (“falls”) rapidly from the saddle-node point to the frozen
state.
Thus the EBM reproduces the abrupt transition to the glaciation of
Antarctica, which is seen in the geological record at the EOT.

Figure 6b explores the relative importance of
decreasing CO_{2} concentration *μ* and decreasing
ocean heat transport *F*_{O} in the glaciation of Antarctica, a subject
that has been much debated in the literature
(DeConto et al., 2008; Goldner et al., 2014; Ladant et al., 2014; Miller et al., 2008; Pagani et al., 2011; Scher et al., 2011, 2015).
The green curve represents a scenario in which *μ* decreases
as in Eq. (27) but *F*_{O} is held fixed at its
Eocene value, and the magenta curve represents a case in which *μ*
is held fixed at its Eocene value while *F*_{O} decreases according to
Eq. (27).
In neither case does a glaciation event occur.
The analysis of this paper implies that significant decreases in *both*
CO_{2} concentration *μ* *and* ocean heat transport *F*_{O}
are required to achieve a saddle-node bifurcation, reproducing the observed transition
to a frozen Antarctic state at the EOT.

While the glaciation of Antarctica is an accepted fact in paleoclimate science
(Miller et al., 2008; Lear et al., 2008; Pagani et al., 2011; Scher et al., 2011; Goldner et al., 2014; Scher et al., 2015), the suddenness of the climate change that
occurred in Antarctica near the Eocene–Oligocene transition (34 Ma), a time when the
forcing parameters were changing slowly, has been difficult to explain. The bifurcation
analysis presented here gives a simple but plausible explanation for the abruptness of
this event. Furthermore, this EBM supports the hypothesis that both falling CO_{2}
concentration *μ* and decreasing ocean heat transport *F*_{O} (due to gateway
openings and development of the ACC) are essential to an explanation of the sudden
glaciation of Antarctica at the EOT.

In the tropics, many of the values of the forcing parameters are different from their values in the Arctic and Antarctic, see Table 2. The geological record shows little change in the tropical climate over the past 100 million years, other than minor cooling. Even when the Arctic climate changed dramatically in the late Pliocene, the tropical climate changed very little.

Figure 7a shows solutions of the EBM equations for tropical parameter values. Note that at no value of the forcings used here does the warm equilibrium point approach a saddle-node bifurcation point. Thus, our EBM is in agreement with the geological record.

A bifurcation diagram is constructed for the tropics EBM,
spanning mid-Cretaceous to modern pre-industrial times,
similar to those for the Antarctic and Arctic glaciations.
Here *ν*=0 corresponds to mid-Cretaceous values and
*ν*=1 corresponds to modern pre-industrial values.
We let *μ* and *F*_{O} decrease linearly with *ν*.

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}\mathit{\mu}=\mathrm{1130}-\mathrm{860}\cdot \mathit{\nu},\\ \text{(40)}& {\displaystyle}& {\displaystyle}{F}_{\mathrm{O}}=-\mathrm{55}+\mathrm{29}\cdot \mathit{\nu},\phantom{\rule{2em}{0ex}}\mathrm{0}\le \mathit{\nu}\le \mathrm{1}.\end{array}$$

The mid-Cretaceous CO_{2} concentration
*μ*=1130 ppm is as determined by Fletcher et al. (2008),
see Table 3.
The ocean heat transport ${F}_{\mathrm{O}}=-\mathrm{55}\phantom{\rule{0.125em}{0ex}}{\mathrm{Wm}}^{-\mathrm{2}}$, from the tropics in the
mid-Cretaceous, is from Barron et al. (1981).
Astrophysicists have determined that solar luminosity is slowly increasing
with time (Sagan and Mullen, 1972).
For the mid-Cretaceous, insolation was approximately 1 % less than
it is today (Barron, 1983).
This difference is considered too small to be significant in our model.
The bifurcation diagram for the tropics EBM is
shown in Fig. 7b.
Note that no bifurcations occur for parameter values relevant
to the tropics.
This is in agreement with paleoclimate records that show little
change in tropical climate, even when polar climates changed
dramatically.

One hundred million years ago, in the mid-Cretaceous Period,
the climate of Earth was much more *equable* than today.
“More equable” means that the pole-to-Equator temperature gradient
was much smaller, and also the seasonal summer/winter temperature
variations were much smaller.
The climate in the tropics was only slightly warmer than today, but the
climate at both poles was much warmer than today.
An abundance of plant and animal life thrived under these conditions,
from the Equator to both poles, including of course dinosaurs.
The question of how this globally ice-free climate could have been
maintained was explored in pioneering work of Barron and coworkers
(Barron, 1983; Barron et al., 1981, 1995; Sloan and Barron, 1992).
Barron called this the *warm, equable Cretaceous climate problem*.
Early GCMs of the 1980s adjusted to
mid-Cretaceous forcing parameter values, had difficulty giving good
agreement with climate proxies.
In order to obtain polar temperatures in agreement with mid-Cretaceous
values, these simulations typically assumed increased CO_{2}
levels more than 4 times modern levels; but then the tropical temperatures
predicted by the models were too high.
Today, the issue of reproducing an equable Equator-to-pole
temperature gradient remains a challenge for modellers.

There have been many investigations of the correlation between
early climate models and proxy data for the Cretaceous climate
(Pagani et al., 2005; Bice et al., 2006; Donnnadieu et al., 2006; Cronin, 2010; Ruddiman, 2014).
Recent studies have succeeded in narrowing the gap
(Craggs et al., 2012; Bowman et al., 2014; O'Brien et al., 2017; Lunt et al., 2016; Ladant and Donnadieu, 2016).
Based on ocean drilling samples, Bice et al. (2006) estimate
Cretaceous CO_{2} concentrations between 600
and 2400 ppmv, and tropical Atlantic upper ocean temperatures
between 33 and 42 ^{∘}C.
Based on fossil samples, Fletcher et al. (2008) estimate
mid-Cretaceous atmospheric CO_{2}
concentrations of 1130 ppmv and we choose their estimate for
use in this EBM, see Table 3.

Paleoclimate data from the mid-Cretaceous show little difference in
climate between the two warm poles at that time (Barron, 1983; Barron et al., 1981).
The major difference in forcings between the two poles at that time
was that ocean heat transport *F*_{O} was much higher in the
Antarctic than the Arctic, as shown in
Table 3.
This is due to the location of the South Pole in open
ocean during the Cretaceous, as the continent of
Antarctica had not yet drifted to its present position over
the South Pole.
Estimates for atmospheric transport are also different (Barron et al., 1981).
Therefore, we model the mid-Cretaceous climate at the two poles
using Eqs. (6)–(6), with the values from
Table 3.
The EBM equilibrium curves are shown in Fig. 8,
where it is clear that a warm (*τ*_{S}>0)
and a frozen (*τ*_{S}<0)
equilibrium state exists at both poles for the given parameter
values.
The Antarctic warm equilibrium state is warmer than in the Arctic because
of the higher ocean heat transport *F*_{O} to the Antarctic.
From Sect. 3.3, the tropical region of Earth also
had a warm equilibrium state under mid-Cretaceous conditions.
Therefore, the EBM of this paper implies the existence of a
pole-to-pole warm, equable Cretaceous climate,
as is seen in the
geological record.

Figure 8 also implies that a frozen equilibrium state
is mathematically possible at each pole during the Cretaceous.
In that case, the tropics could remain in the warm state, thus giving
the mathematical possibility of a Cretaceous climate that is
*not equable*, but has warm tropics and ice-covered poles,
like today's climate.
This may help explain why some computer simulations, originally designed
for today's climate conditions, found a mathematically existing Cretaceous
climate that resembled today's climate more than the warm, equable
climate that physically existed on Earth in the Cretaceous.

The climate of the Eocene Epoch (≈55 million years ago)
was the warmest of the past 65 million years, but a little
cooler than the Cretaceous (Sloan and Barron, 1992; Pagani et al., 2005; Cronin, 2010; Huber and Caballero, 2011; Hutchinson et al., 2018).
Both poles were ice-free and the pole-to-Equator temperature
gradient was much smaller than today.
As for the mid-Cretaceous, early computer simulations, based on
modern climate conditions, had difficulty in
reproducing the early Eocene warm, equable climate
(Sloan and Barron, 1990, 1992; Sloan and Rea, 1995; Jahren and Sternberg, 2003; Huber and Caballero, 2011).
This discrepancy has been called the *early Eocene warm, equable climate problem*.
Here we combine these two “problems” under one name as the
*warm, equable Paleoclimate problem*.

The main difference in forcings between early Eocene conditions and
those of the mid-Cretaceous is that the global CO_{2}
concentration *μ* may have been a little lower and the ocean heat
transport *F*_{O} to the Antarctic was less.
Referring to Fig. 8,
this means that the blue and green atmosphere equilibrium curves move downward slightly,
and the magenta Antarctic surface equilibrium curve moves upward slightly.
The orange Arctic surface equilibrium curve does not change
significantly.
With these small changes, all of the equilibrium climate states
(intersection points) persist in Fig. 8.
The figure for the Eocene EBM (not shown here) is topologically
the same as Fig. 8 for the mid-Cretaceous EBM,

Thus, the EBM predicts that a warm ice-free equilibrium state exists (mathematically) at both poles in the mid-Cretaceous and early Eocene. From Sect. 3.3, the equatorial region also has a warm equilibrium state for Eocene conditions. Therefore, this EBM study supports the existence of a pole-to-pole warm, equable climate in both the mid-Cretaceous and the early Eocene. Additionally, in the EBM there is the coexistence of a nonequable climate, with cold poles, under exactly the same forcing conditions, which suggests the following plausible solution to the “warm, equable paleoclimate problem” in both the mid-Cretaceous (Barron, 1983; Barron et al., 1981, 1995; Sloan and Barron, 1990) and the early Eocene (Sloan and Barron, 1990, 1992; Sloan and Rea, 1995; Jahren and Sternberg, 2003; Huber and Caballero, 2011). While the Earth's climate existed in the warm, equable climate state of the EBM, computer simulations of Barron and others may have correctly computed the coexisting nonequable solution.

4 Conclusions and future work

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This paper presents a new energy balance model (EBM) for the climate of Earth, one that elucidates the distinctive roles of carbon dioxide and water vapour as greenhouse gases, and also the role of ice–albedo feedback, in climate change. Nonlinearity of the EBM leads to multiple solutions of the mathematical equations and to bifurcations that represent transitions between coexisting stable equilibrium states. This EBM sheds new light on several important problems of paleoclimate science; namely, the Pliocene paradox, the abrupt glaciation of Antarctica, and the warm, equable mid-Cretaceous and early Eocene climate problems. Predictions of the EBM are in qualitative agreement with the paleoclimate record.

There has been a range of values of paleoclimate forcing
parameters in the literature, in particular for mid-Cretaceous
and Eocene CO_{2} concentrations and
temperatures.
The specific choices made here, while informed by proxies,
are somewhat arbitrary.
However, this fact does not affect the validity of our main conclusions.
The *coexistence of multiple solutions* and the
*bifurcations* demonstrated in the EBM are robust phenomena.
That is, the existence of these multiple solutions and bifurcations
will persist over a range of values of the forcing parameters.
The main conclusion of this paper that sudden and significant changes
in climate have occurred, even while forcing parameters were
changing very gradually, follows from the existence of mathematical
bifurcations in the EBM, not from particular choices of the
forcing parameters.

As the paleoclimate record becomes clearer, there is growing evidence
for small but rapid fluctuations in some parameter values, including
CO_{2} concentrations, over the geological time period studied here
(Cronin, 2010; Pagani et al., 1999, 2005).
The model of this paper assumes a smooth decline in CO_{2} concentration.
However, this does not invalidate our main conclusions.
The theory of “stochastic bifurcation” (Namachchivaya, 1990; Arnold et al., 1996) tells us that if
stochastic noise is added parametrically to a deterministic bifurcation problem,
then typically the location of the bifurcation (in terms of the bifurcation parameter)
may change; but the *existence* of the bifurcation is preserved.

Further work on this EBM is in progress.
Having demonstrated the applicability of the EBM to known paleoclimate
transitions, this EBM is now being applied to anthropogenic climate
change with the goal of predicting the future climate effects of
continued increases in CO_{2} concentration.

The most significant question is whether a bifurcation is to be
expected in our future.
This bifurcation would lead to climate warming even stronger
than that currently underway, leading possibly to a
*warm, equable climate*, at least in the Northern Hemisphere,
similar to the climate of Earth before the Pliocene.

The equilibrium climate sensitivity (ECS) of the EBM, adapted to present-day satellite data, has been calculated and is presented in Appendix Appendix B of this paper.

This scalar EBM will be generalized to a two-point boundary value problem (BVP) in the
altitude variable *z*, using the Schwarzschild equations to replace the ICAO
International Standard Atmosphere (ISA) approximation for lapse rate. That change will
more accurately model the behaviour of the greenhouse gases in the atmosphere. Next, that
one-dimensional EBM BVP will be incorporated into a generalization of the spherical shell
PDE model of Lewis and Langford (2008) and Langford and Lewis (2009). We anticipate that this 3-D zonally symmetric
Navier–Stokes Boussinesq model will confirm the fundamental predictions relating to
bistability and bifurcation of the present simple EBM. Also, it will enable the study of
a third positive feedback mechanism (in addition to the two studied in this paper);
namely, Hadley cell convection feedback, which influences atmospheric heat transport
*F*_{A}.

Code availability

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Code availability.

Appendix A: Empirical calibration of EBM parameters

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Some of the parameters used in the EBM have standard values, available in textbooks and reported in Table 1. Others, however, are determined empirically in this appendix. Modern values of these parameters may be determined from today's abundant satellite and land-based data. It is assumed that the values determined here are also valid for paleoclimates.

The primary source for these empirical calibrations are the data presented by Wild et al. (2013, Fig. 4) and Trenberth et al. (2009, Fig. 3). The data from these authors are globally averaged values. We are applying our model to specific regions (the Arctic, Antarctic or tropics) and therefore adjust some of these values as discussed below. We also employ the model with globally averaged values to give an equilibrium climate sensitivity (ECS) calculation in Appendix Appendix B.

The value of the annually averaged solar radiation at either pole is
$Q=\mathrm{173.2}\phantom{\rule{0.125em}{0ex}}{\mathrm{Wm}}^{-\mathrm{2}}$ and at the Equator is 418.8 Wm^{−2} (McGehee and Lehman, 2012; Kaper and Engler, 2013).
Values from Fig. 4 of Wild et al. (2013) indicate the globally averaged solar radiation
is $Q=\mathrm{340}\phantom{\rule{0.125em}{0ex}}\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}$. Of this, 79 W m^{−2} is
directly absorbed by the
atmosphere, and 76 W m^{−2} is reflected by the atmosphere
back into space
(both primarily due to clouds). Hence we define the two solar radiation fractions

$$\begin{array}{}\text{(A1)}& {\mathit{\xi}}_{\mathrm{A}}={\displaystyle \frac{\mathrm{79}}{\mathrm{340}}}=\mathrm{0.2324}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{\mathit{\xi}}_{\mathrm{R}}={\displaystyle \frac{\mathrm{76}}{\mathrm{340}}}=\mathrm{0.2235}.\end{array}$$

Since we incorporate no information on varying cloud cover in the model, and since there is very
little such information for paleoclimates,
we assume these values are constant around the globe.
Clearly cloud cover is correlated with surface temperature. We also tried making *ξ*_{A} and
*ξ*_{R} vary with *T*_{S} in a manner similar to how we model heat conduction/convection, *F*_{C} below,
but found no qualitative differences in the results. For this reason, and to keep the model simple,
we have kept these values of *ξ*_{A} and *ξ*_{R} as global constants in our model.

The Earth's albedo, *α*, varies considerably depending on the surface features.
Typical values are 0.6–0.9 for snow,
0.4–0.7 for ice, 0.2 for crop land and 0.1 or less for open ocean.
In previous papers, including Dortmans et al. (2018), the polar albedo *α*
is assumed to jump between two discrete values, a cold albedo
*α*_{C} for the ice/snow-covered surface, when below the
freezing temperature, and a
warm albedo *α*_{W} corresponding to land or open ocean
above the freezing temperature; that is,

$$\begin{array}{}\text{(A2)}& \mathit{\alpha}=\left\{\begin{array}{ll}{\mathit{\alpha}}_{\mathrm{C}}& \text{if}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{T}_{\mathrm{S}}\le {T}_{\mathrm{R}},\\ {\mathit{\alpha}}_{\mathrm{W}}& \text{if}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{T}_{\mathrm{S}}>{T}_{\mathrm{R}}.\end{array}\right.\end{array}$$

This discontinuous albedo function is conceptually simple but it is not an accurate representation of what would actually happen if the polar region cooled from ice-free to ice-covered conditions. Recall that this model represents the annually averaged climate. As the polar region cools, there will be a transition period in which warm ice-free summers get shorter and cold ice-covered winters get longer. The annually averaged albedo, therefore, would not jump abruptly from low to high constant values as in Eq. (A2); it would transition more smoothly between the summer and winter extreme values. Therefore, in this paper we introduce a more realistic (and smooth) sigmoidal albedo given by the hyperbolic tangent function, Eq. (3), and reproduced here:

$$\begin{array}{}\text{(A3)}& \mathit{\alpha}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left([{\mathit{\alpha}}_{\mathrm{W}}+{\mathit{\alpha}}_{\mathrm{C}}]+[{\mathit{\alpha}}_{\mathrm{W}}-{\mathit{\alpha}}_{\mathrm{C}}]\mathrm{tanh}\left({\displaystyle \frac{{T}_{\mathrm{S}}-{T}_{\mathrm{R}}}{\mathrm{\Omega}}}\right)\right).\end{array}$$

The parameter Ω determines the steepness of the transition between
*α*_{C} and *α*_{W}.
We have chosen $\mathit{\omega}\equiv \mathrm{\Omega}/{T}_{\mathrm{R}}=\mathrm{0.01}$.
With this choice, the albedo changes by 80 % (that is, it goes from
0.1 to 0.9 of the difference between *α*_{C} and *α*_{W})
over a span of 6 C in annual mean temperature, which is realistic;
see Fig. A1.

Figure A1 shows the nondimensional energy balance Eq. (6) (in blue) and Eq. (6) (in red) for a range of values of the parameter $\mathit{\omega}=\mathrm{\Omega}/{T}_{\mathrm{R}}$ in Eq. (A3). The blue (atmosphere) curve may have up to three intersections with a red (surface) curve, implying the existence of multiple equilibrium states.

Both Wild et al. (2013) and Trenberth et al. (2009) indicate that the globally
averaged amount of infrared radiation
being emitted by the surface of the Earth is 398 W m^{−2}.
Since the EBM assumes the Earth is a black-body radiator, the
Stefan–Boltzmann law dictates that the corresponding temperature is

$$\begin{array}{}\text{(A4)}& {T}_{\mathrm{S}}^{\mathrm{avg}}={\left({\displaystyle \frac{\mathrm{398}}{\mathit{\sigma}}}\right)}^{\mathrm{1}/\mathrm{4}}=\mathrm{289.45}\phantom{\rule{0.33em}{0ex}}\mathrm{K}.\end{array}$$

This temperature is equivalent to 16.3 ^{∘}C, which is a little higher than the accepted average
surface temperature of about 14 ^{∘}C. This is not surprising since the former is obtained by
essentially averaging *T*^{4} while the latter is obtained from averaging actual temperatures.
We use the above value for ${T}_{\mathrm{S}}^{\mathrm{avg}}$ in the calibrations described below, since it is
consistent with the EBM.

*Evapotranspiration* (ET) is the transport of water from
the surface to the atmosphere in the form of water vapour.
It combines the effects of evaporation from the surface and
transpiration by plants.
Globally, the largest contributor is evaporation from the
surface of the oceans and the main determining factor there
is the ocean surface temperature.
The ET process also transports heat from the surface
to the atmosphere, in the form of latent heat.
Recently Wang et al. (2010) have shown that global ET
has been increasing over the past several decades.

The forcing term *F*_{C} in the EBM represents transport of heat away from the
surface to the atmosphere by conduction plus convection plus change of state of water.
The most important of these is the upward transport of latent heat. Surface water
evaporates, taking heat from the surface. As warm moist air rises and cools, the water
vapour condenses, releasing its latent heat into the surrounding atmosphere. According to
both Wild et al. (2013) and Trenberth et al. (2009), the magnitude of this forcing is ${F}_{\mathrm{C}}=\mathrm{104}\phantom{\rule{0.125em}{0ex}}\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}$. However, this is a globally averaged value and, due to strong
dependence on temperature, is not likely valid at temperatures other than the globally
averaged surface temperature ${T}_{\mathrm{S}}^{\mathrm{avg}}$. In the past several decades, ET
has increased by 0.6 W m^{−2} per decade (Wang et al., 2010). Extrapolating back
100 years, this would be 6 W m^{−2}. In the past century the average global
surface temperature has risen about 1 ^{∘}C. Thus it is reasonable to assume the
dependence of *F*_{C} on the surface temperature *T*_{S} should have a
slope near ${T}_{\mathrm{S}}={T}_{\mathrm{S}}^{\mathrm{avg}}$ of about 6. If the dependence of
*F*_{C} on *T*_{S} was simply linear, then, given that
${F}_{\mathrm{C}}\left({T}_{\mathrm{S}}^{\mathrm{avg}}\right)=\mathrm{104}$, this slope would predict that
*F*_{C} was negative for ${T}_{\mathrm{S}}<{T}_{\mathrm{S}}^{\mathrm{avg}}-\mathrm{17.3}\approx -\mathrm{3}$ ^{∘}C. Since negative values for *F*_{C} are unreasonable, and since
there likely is still some heat transport even near freezing, instead of a linear
function, we have modelled the dependence of *F*_{C} on temperature as a
hyperbolic function that is nearly zero for temperatures below freezing, and increases
roughly linearly for temperatures above freezing. Specifically, we use the function

$$({T}_{\mathrm{S}}-{T}_{\mathrm{R}})={\displaystyle \frac{\mathrm{1}}{\mathrm{2}{A}_{\mathrm{1}}}}\left({F}_{\mathrm{C}}-{\displaystyle \frac{{A}_{\mathrm{2}}^{\mathrm{2}}}{{F}_{\mathrm{C}}}}\right),$$

or equivalently,

$$\begin{array}{}\text{(A5)}& {F}_{\mathrm{C}}\left({T}_{\mathrm{S}}\right)={A}_{\mathrm{1}}({T}_{\mathrm{S}}-{T}_{\mathrm{R}})+\sqrt{{A}_{\mathrm{1}}^{\mathrm{2}}({T}_{\mathrm{S}}-{T}_{\mathrm{R}}{)}^{\mathrm{2}}+{A}_{\mathrm{2}}^{\mathrm{2}}},\end{array}$$

where *A*_{1} and *A*_{2} are constants.
In terms of the nondimensional variables and parameters Eq. (6),
this equation becomes

$$\begin{array}{}\text{(A6)}& {f}_{C}\left({\mathit{\tau}}_{\mathrm{S}}\right)={a}_{\mathrm{1}}({\mathit{\tau}}_{\mathrm{S}}-\mathrm{1})+\sqrt{{a}_{\mathrm{1}}^{\mathrm{2}}({\mathit{\tau}}_{\mathrm{S}}-\mathrm{1}{)}^{\mathrm{2}}+{a}_{\mathrm{2}}^{\mathrm{2}}}.\end{array}$$

The constants *A*_{1} and *A*_{2} are chosen so that the forcing at the global average surface
temperature ${T}_{\mathrm{S}}^{\mathrm{avg}}$ is 104, and so that the forcing at the freezing point is 20 % of this value, that is, ${F}_{\mathrm{C}}\left({T}_{\mathrm{R}}\right)=\mathrm{0.2}{F}_{\mathrm{C}}\left({T}_{\mathrm{S}}^{\mathrm{avg}}\right)$.
Thus

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{A}_{\mathrm{2}}=\mathrm{0.2}\cdot \mathrm{104}=\mathrm{20.8},\\ \text{(A7)}& {\displaystyle}& {\displaystyle}\text{and}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{A}_{\mathrm{1}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}({T}_{\mathrm{S}}^{\mathrm{avg}}-{T}_{\mathrm{R}})}}\left(\mathrm{104}-{\displaystyle \frac{{A}_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{104}}}\right)=\mathrm{3.063},\end{array}$$

or in terms of the nondimensional parameters,

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{a}_{\mathrm{1}}={\displaystyle \frac{{A}_{\mathrm{1}}}{\mathit{\sigma}{T}_{\mathrm{R}}^{\mathrm{3}}}}=\mathrm{2.650},\\ \text{(A8)}& {\displaystyle}& {\displaystyle}\text{and}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{a}_{\mathrm{2}}={\displaystyle \frac{{A}_{\mathrm{2}}}{\mathit{\sigma}{T}_{\mathrm{R}}^{\mathrm{4}}}}=\mathrm{6.590}\times {\mathrm{10}}^{-\mathrm{2}}.\end{array}$$

Figure A2 shows a plot of Eq. (A5) for these parameter values. With these choices of the constants, the slope at ${T}_{\mathrm{S}}^{\mathrm{avg}}$ is 5.9, agreeing well with the above argument that it should be about 6.

Two of the empirical parameters in the EBM are *k*_{W}
and *k*_{C}, defined in Sect. 2.2 as the grey gas
absorption coefficients for water vapour and carbon dioxide, respectively.
More precisely, the absorption coefficient *k* of a gas is its
proportionality coefficient in the Beer–Lambert law, as defined in
Eq. (6) of Sect. 2.2.
In the *grey gas approximation*, it is assumed that the absorption
coefficient *k* is constant for all frequencies of incident radiation,
and depends only on the total intensity of that radiation.
In reality though, *k* is not an absolute physical constant
because gases absorb radiation at discrete spectral lines.
The total amount of energy absorbed depends on the distribution of energy
in the radiation at each of these spectral wavelengths.
Values of *k*_{W} and *k*_{C} for the particular spectrum of
incident radiation experienced by the atmosphere may be obtained empirically
from modern era satellite and land-based measurements.
We assume that *k*_{W} and *k*_{C} for the atmosphere do not change
with time, even over the geological timescales considered here.

Atmospheric absorption, *η*, of infrared radiation emitted by the Earth is, in this EBM, attributed to
the two primary greenhouse gases: water vapour and carbon dioxide, and to the liquid and solid
water making up clouds.
According to Trenberth et al. (2009), at present as a global average, the Earth emits 398 W m^{−2}
of which 40.1 W m^{−2} passes through the atmosphere to space.
Thus approximately 90 % of the radiation is absorbed in the atmosphere, that is *η*=0.9.
The present-day
absorption factor *η* can be attributed as follows (Schmidt et al., 2010):
water vapour 50 %, clouds 25 % and CO_{2} (plus other gases) 25 %.
Furthermore, they determined that these ratios remain unchanged, even
after a doubling of CO_{2}.
Therefore, setting $x={\mathit{\eta}}_{C}={\mathit{\eta}}_{\mathrm{Cl}}$ and *η*_{W}=2*x* in
Eq. (10) we get

$$\mathrm{0.9}=\mathrm{1}-(\mathrm{1}-x)(\mathrm{1}-\mathrm{2}x)(\mathrm{1}-x)=\mathrm{2}{x}^{\mathrm{3}}-\mathrm{5}{x}^{\mathrm{2}}+\mathrm{4}x.$$

This cubic has one real root, $\widehat{x}$,

$$\begin{array}{}\text{(A9)}& \widehat{x}\approx \mathrm{0.3729}.\end{array}$$

In this model, since we have no data on cloud cover for paleoclimates, we make the assumption that absorption due to clouds is a constant globally and temporally, and set

$$\begin{array}{}\text{(A10)}& {\mathit{\eta}}_{\mathrm{Cl}}=\widehat{x}.\end{array}$$

We also tried varying *η*_{Cl} with temperature using a hyperbolic function like that used for
*F*_{C} in Eq. (A5); however, we found no qualitative change in the results and therefore, for
simplicity, have left *η*_{Cl} as a constant.
To calibrate the absorption coefficient *k*_{C} for the greenhouse gas carbon dioxide, we set
${\mathit{\eta}}_{C}=\widehat{x}$ in
Eq. (14) yielding

$$\begin{array}{ll}{\displaystyle}\widehat{x}& {\displaystyle}=\mathrm{1}-{e}^{-\mathit{\mu}\cdot \mathrm{1.52}\times {\mathrm{10}}^{-\mathrm{6}}{k}_{\mathrm{C}}{P}_{\mathrm{A}}/g}\\ {\displaystyle}\u27f9& {\displaystyle}{k}_{\mathrm{C}}={\displaystyle \frac{g\mathrm{ln}(\mathrm{1}-\widehat{x})}{\mathit{\mu}(\mathrm{1.52}\times {\mathrm{10}}^{-\mathrm{6}}){P}_{\mathrm{A}}}}.\end{array}$$

Using the present-day value of *μ*=400 ppm, the value of $\widehat{x}$ from Eq. (A7),
the standard atmospheric pressure value of ${P}_{\mathrm{A}}=\mathrm{101.3}\times {\mathrm{10}}^{\mathrm{3}}\phantom{\rule{0.33em}{0ex}}\mathrm{Pa}$,
and the acceleration due to gravity $g=\mathrm{9.8}\phantom{\rule{0.33em}{0ex}}\mathrm{m}\phantom{\rule{0.33em}{0ex}}{\mathrm{s}}^{-\mathrm{2}}$ we get

$$\begin{array}{ll}{\displaystyle}{k}_{\mathrm{C}}& {\displaystyle}={\displaystyle \frac{\left(\mathrm{9.8}\right)\mathrm{ln}(\mathrm{1}-\widehat{x})}{\left(\mathrm{400}\right)(\mathrm{1.52}\times {\mathrm{10}}^{-\mathrm{6}})(\mathrm{101.3}\times {\mathrm{10}}^{\mathrm{3}})}}\\ \text{(A11)}& {\displaystyle}& {\displaystyle}=\mathrm{0.07424}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}.\end{array}$$

With this value for *k*_{C}, the coefficient *G*_{C}, defined in Eq. (13) has the value

$$\begin{array}{}\text{(A12)}& {G}_{C}=\mathrm{1.52}\times {\mathrm{10}}^{-\mathrm{6}}{k}_{\mathrm{C}}{\displaystyle \frac{{P}_{\mathrm{A}}}{g}}=\mathrm{1.166}\times {\mathrm{10}}^{-\mathrm{3}}.\end{array}$$

To calibrate *k*_{W} we proceed as follows.
The latent heat of vaporization of water is ${L}_{\mathrm{v}}=\mathrm{2.2558}\times {\mathrm{10}}^{\mathrm{6}}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{2}}$ and the
ideal-gas constant specific to water vapour is ${R}_{\mathrm{W}}=\mathrm{461.5}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$.
Thus from Eq. (24) we have

$$\begin{array}{}\text{(A13)}& {G}_{\mathrm{W}\mathrm{1}}={\displaystyle \frac{{L}_{\mathrm{v}}}{{R}_{\mathrm{W}}{T}_{\mathrm{R}}}}=\mathrm{17.89}.\end{array}$$

Dai (2006) indicates that an average value for relative humidity, *δ*, from 1976 to 2004 for the region
of the Earth between 60^{∘} S and 75^{∘} N is 0.74. Over the same latitudes, Dai reports
that averages over water and over land are 0.79 and 0.65, respectively, except that over deserts the
humidity drops to 30 %–50 %. The polar regions are typically drier; however, there is a significant
amount of water north of 75^{∘} N and south of 60^{∘} S, hence
we have chosen *δ*^{avg}=0.74 as the average global relative
humidity for purposes of calibrating *k*_{W}.
In Eq. (25) we set
${\mathit{\eta}}_{\mathrm{W}}=\mathrm{2}\widehat{x}$ with $\widehat{x}$ given by Eq. (A7), *τ*_{S} to the present normalized global average temperature
${T}_{\mathrm{S}}^{\mathrm{avg}}/{T}_{\mathrm{R}}$, *δ* to the present average global relative humidity
*δ*^{avg}, *Z* to an average global tropopause height of *Z*^{avg}=14000 m,
and the normalized lapse rate to $\mathit{\gamma}=\mathrm{\Gamma}/{T}_{\mathrm{R}}=\mathrm{2.38}\times {\mathrm{10}}^{-\mathrm{5}}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{1}}$.
Using these values and inverting Eq. (25) to isolate *k*_{W} we get

$$\begin{array}{ll}{\displaystyle}{k}_{\mathrm{W}}& {\displaystyle}=-\mathrm{ln}(\mathrm{1}-\mathrm{2}\widehat{x})\mathit{\gamma}{\left[\mathit{\delta}{\displaystyle \frac{{P}_{\mathrm{W}}^{\mathrm{sat}}\left({T}_{\mathrm{R}}\right)}{{R}_{\mathrm{W}}{T}_{\mathrm{R}}}}\underset{{T}_{\mathrm{S}}^{\mathrm{avg}}-\mathit{\gamma}{Z}^{\mathrm{avg}}}{\overset{{T}_{\mathrm{S}}^{\mathrm{avg}}}{\int}}{\displaystyle \frac{\mathrm{1}}{\mathit{\tau}}}\mathrm{exp}\left({G}_{\mathrm{W}\mathrm{1}}\left[{\displaystyle \frac{\mathit{\tau}-\mathrm{1}}{\mathit{\tau}}}\right]\right)\phantom{\rule{0.125em}{0ex}}\mathrm{d}\mathit{\tau}\right]}^{-\mathrm{1}}\\ \text{(A14)}& {\displaystyle}& {\displaystyle}=\mathrm{0.05905}\phantom{\rule{0.33em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.33em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}.\end{array}$$

With this value for *k*_{W}, the second greenhouse gas parameter for water vapour is

$$\begin{array}{}\text{(A15)}& {G}_{\mathrm{W}\mathrm{2}}\equiv {\displaystyle \frac{{k}_{\mathrm{W}}{P}_{\mathrm{W}}^{\mathrm{sat}}\left({T}_{\mathrm{R}}\right)}{\mathit{\gamma}{R}_{\mathrm{W}}{T}_{\mathrm{R}}}}=\mathrm{12.05}.\end{array}$$

In previous slab models, including Dortmans et al. (2018) and Payne et al. (2015), it was
assumed that the upward and downward radiation intensities
from the atmosphere are equal; that is, $\mathit{\beta}=\frac{\mathrm{1}}{\mathrm{2}}$
in Eq. (1).
This would be the case for an actual uniform slab; however, the real
atmosphere is not uniform in temperature nor density, and
in fact both the temperature and density of the atmosphere are much
higher in value near the surface than near the tropopause.
The net effect of this nonuniformity is that, of the total radiation intensity *I*_{A}
emitted by the atmosphere, almost two-thirds goes back to the surface and
only a little more than one-third escapes to space, according to satellite
and surface data (Trenberth et al., 2009; Wild et al., 2013).
From Trenberth et al. (2009), the atmosphere and clouds emit
$\mathrm{169.9}+\mathrm{29.9}\phantom{\rule{0.125em}{0ex}}\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}$ upward to space
and emit 340.3 W m^{−2} downward to the Earth.
Therefore, in order to have the model in this paper represent the atmosphere
more realistically, we set

$$\begin{array}{}\text{(A16)}& \mathit{\beta}={\displaystyle \frac{\mathrm{340.3}}{\mathrm{340.3}+\mathrm{169.9}+\mathrm{29.9}}}\approx \mathrm{0.63}.\end{array}$$

instead of 0.50.

Appendix B: Equilibrium climate sensitivity

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Equilibrium climate sensitivity (ECS) is a useful measure of the
sensitivity of a given climate model to an increase in
CO_{2} concentration *μ*.
It is usually defined as the change Δ*T* in the global mean
temperature $\stackrel{\mathrm{\u203e}}{T}$, resulting from a doubling of *μ*,
starting from the accepted pre-industrial value *μ*=270 ppm
(IPCC, 2013; Forster, 2016; Knutti et al., 2017; Proistosescu and Huybers, 2017).
The ECS provides a first-order estimate of the amount of present-day
global warming predicted by a given model as *μ* increases.
Because ECS is a single number, it facilitates comparisons
between models.
It has been used extensively in the assessment reports of the IPCC
(IPCC, 2013), where Δ*T* values in the range 1.5 to
4.5 ^{∘}C
have been documented.
Some have cited this wide range of ECS values as a sign of weakness
of the IPCC methodology.
Recently these results have been reconciled (Proistosescu and Huybers, 2017).
They showed that a linear statistical analysis of historical data gives low
estimates in the 1.5 to 3 ^{∘}C range, while nonlinear models give
higher estimates.

In terms of our EBM, to apply it to a global average we make the
following settings.
The global mean insolation at the top of the atmosphere
is $Q=\mathrm{340}\phantom{\rule{0.125em}{0ex}}{\mathrm{Wm}}^{-\mathrm{2}}$ (Wild et al., 2013; Trenberth et al., 2009).
They indicate that
161 W m^{−2} of sunlight is absorbed by the Earth while
24 W m^{−2} is reflected back on average. Thus we take

$${\mathit{\alpha}}_{\mathrm{W}}={\displaystyle \frac{\mathrm{24}}{\mathrm{161}+\mathrm{24}}}=\mathrm{0.13}.$$

For the tropopause height *Z* we take an average value of *Z*^{avg}=14 km.
For the global model, the ocean and atmospheric transport terms, *F*_{O} and *F*_{A}, will be zero.

The ECS value we calculate will be the difference between the warm equilibrium temperatures
when *μ*=270 and *μ*=540.
For these calculations we took the relative humidity to be $\mathit{\delta}={\mathit{\delta}}^{\mathrm{avg}}=\mathrm{0.74}$, in
accord with the value obtained in Dai (2006).
The graphs of the surface and atmosphere equilibrium equations are
shown in Fig. B1.
The equilibrium solutions of this global mean version of the EBM yield
*T*_{S}(270)=14.3, *T*_{S}(540)=17.6 and
Δ*T*=3.3 ^{∘}C. This is in excellent agreement with
accepted values (IPCC, 2013).

Author contributions

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Author contributions.

WFL contributed the original hypothesis on which the investigation is based, guided the work and wrote much of the original paper. BD carried out the computations required for this research and wrote an MSc thesis on his work, which was the starting point for this paper. ARW co-supervised BD's work, significantly extended his calculations, contributed many insights and guidance for this paper, and wrote much of the major revision of this paper.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

The authors acknowledge financial support of this work by the Natural Sciences and Engineering Research Council of Canada, and thank two anonymous referees for many valuable suggestions that greatly improved the quality of the paper. The authors thank Kolja Kypke for his contributions to the final revisions of the manuscript. William F. Langford gratefully acknowledges very helpful discussions with James F. Basinger and David R. Greenwood in the early stages of this work.

Review statement

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Review statement.

This paper was edited by Qiuzhen Yin and reviewed by two anonymous referees.

References

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Short summary

In geology and in paleoclimate science, most changes are caused by well-understood forces acting slowly over long periods of time. However, in highly nonlinear physical systems, mathematical bifurcation theory predicts that small changes in forcing can cause major changes in the system in a short period of time. This paper explores some sudden changes in the paleoclimate history of the Earth, where it appears that bifurcation theory gives a more satisfying explanation than uniformitarianism.

In geology and in paleoclimate science, most changes are caused by well-understood forces acting...

Climate of the Past

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