# Emergent constraints on climate sensitivities

Mark S. Williamson  College of Engineering, Mathematics and Physical Sciences,
University of Exeter, UK  Global Systems Institute, University of Exeter, UK
Chad W. Thackeray  Department of Atmospheric and Oceanic Sciences, University
of California, Los Angeles, CA, USA  Peter M. Cox  College of Engineering,
Mathematics and Physical Sciences, University of Exeter, UK  Alex Hall
Department of Atmospheric and Oceanic Sciences, University of California, Los
Angeles, CA, USA  Chris Huntingford  Centre for Ecology and Hydrology,
Wallingford, UK  Femke J. M. M. Nijsse  College of Engineering, Mathematics
and Physical Sciences, University of Exeter, UK

###### Abstract

Despite major advances in climate science over the last 30 years, persistent
uncertainties in projections of future climate change remain. Climate
projections are produced with increasingly complex models which attempt to
represent key processes in the Earth system, including atmospheric and oceanic
circulations, convection, clouds, snow, sea-ice, vegetation and interactions
with the carbon cycle. Uncertainties in the representation of these processes
feed through into a range of projections from the many state-of-the-art
climate models now being developed and used worldwide. For example, despite
major improvements in climate models, the range of equilibrium global warming
due to doubling carbon dioxide still spans a range of more than three. Here we
review a promising way to make use of the ensemble of climate models to reduce
the uncertainties in the sensitivities of the real climate system. The
emergent constraint approach uses the model ensemble to identify a
relationship between an uncertain aspect of the future climate and an
observable variation or trend in the contemporary climate. This review
summarises previous published work on emergent constraints, and discusses the
huge promise and potential dangers of the approach. Most importantly, it
argues that emergent constraints should be based on well-founded physical
principles such as the fluctuation-dissipation theorem. It is hoped that this
review will stimulate physicists to contribute to the rapidly-developing field
of emergent constraints on climate projections, bringing to it much needed
rigour and physical insights.

###### Contents

  1. I Introduction
  2. II How relationships in model ensembles might ‘emerge’
    1. II.1 Commonly used ESM ensembles: Multi-model (MME) and perturbed physics (PPE) ensembles
    2. II.2 Null hypothesis: Emergent relationships occur by chance
    3. II.3 Low dimensional relationships ‘emerge’ from high dimensional ESMs
    4. II.4 Range in response due to the same physical process having a wide range across ESMs
    5. II.5 What is needed for an EC?
      1. II.5.1 Observable ($X$) range and uncertainty
      2. II.5.2 Response ($Y$) range and uncertainty
      3. II.5.3 Relationship between $X$ and $Y$
      4. II.5.4 Large ensemble size $n$
  3. III Underlying theory for emergent constraints based-on temporal variability
    1. III.1 Relationships between variability of fluxes and the sensitivity of stores
    2. III.2 Theoretical emergent relationships for idealised time-varying forcing
      1. III.2.1 Sinusoidal forcing
      2. III.2.2 Linearly-increasing forcing
      3. III.2.3 White-noise forcing
  4. IV How might emergent constraints go wrong and how to guard against it?
    1. IV.1 Risks of purely using data mining
    2. IV.2 The risk of $p$-hacking and overconfidence
    3. IV.3 Missing process in all current models, measurement errors and model compensating errors
    4. IV.4 System passes through a tipping point
    5. IV.5 Problems with common code across many models and implications for ‘out-of-sample’ testing
    6. IV.6 What to do when different ECs are found for the same quantity, but differ in value, or differ between ensembles?
    7. IV.7 ECs may cause future CMIP-type climate model ensembles to have much less spread in projections
    8. IV.8 Inability to verify an EC
    9. IV.9 Lack of perturbed physics experiments with ESMs
  5. V Emergent constraints found in the Earth system
    1. V.1 Equilibrium Climate Sensitivity
    2. V.2 Cloud Feedbacks
    3. V.3 Carbon Cycle
    4. V.4 High-latitude Processes
    5. V.5 Hydrologic Cycle
  6. VI Statistical underpinnings
    1. VI.1 Uncertainty in observations
    2. VI.2 Uncertainty from internal variability
    3. VI.3 Uncertainty in the functional form of the relationship
    4. VI.4 Uncertainty from imperfect models
    5. VI.5 Combining sources of uncertainty in an EC
    6. VI.6 Combining multiple constraints
  7. VII Outlook
    1. VII.1 Key gaps in ECs to date
    2. VII.2 Targeted model development
    3. VII.3 Use of conceptual models as the basis of emergent relationships and understanding of more complex ESMs
    4. VII.4 Multidimensional ECs and nonlinear emergent relationships
    5. VII.5 Continued improvement of climate projections and impacts-led requirements
    6. VII.6 Better understanding of the effects of parametric and structural uncertainty on ECs
    7. VII.7 Machine Learning
    8. VII.8 Building connections to other fields
  8. VIII Conclusion

##  I Introduction

Numerical methods have become a standard technique to simulate complex
systems. The equations governing components of such systems may be well-known.
But their solutions cannot be solved analytically, creating a need for
numerical approaches. Because of the discretization of time and space inherent
in numerical techniques, modelling complex systems must involve ways to
include effects of unresolved processes. Often there is no ‘first principles’
approach to do this. Typically, the effects of unresolved processes are
included by resorting to quasi-empirical relationships between them and
explicitly-resolved variables, otherwise known as ‘parameterization’. There
are usually multiple defensible ways to parameterize unresolved processes.
Thus, independently developed models of the same complex system might
incorporate different parameterization choices. The more models that are
independently developed, the greater the diversity of approaches for modelling
the same natural system.

A classic example of this model diversity is the use of numerical models of
the atmosphere to predict hurricane development. Initial conditions are
imposed on a model at some time, and it is integrated forward in time to
produce simulations of critical hurricane features such as intensity and
track. This approach might be replicated for multiple models incorporating
different parameterization choices, producing an ‘ensemble’ of hurricane
forecasts. The spread in the forecasts is a measure of the uncertainty in the
future hurricane behaviour, given the range of plausible approaches to
atmospheric modelling. Over time, with enough hurricanes and associated
predictions, the various models can be evaluated for their prediction skill.
Certain parameterization choices may emerge as producing systematically better
predictions. The models can then be rebuilt or re-calibrated with the better
choices. Over time the ensemble will become more skilful, with less spread. In
fact, this is approximately the process that has resulted in dramatic
improvements not only in hurricane prediction, but in weather forecasting
generally over the past seven decades.

Earth’s climate is another example of a complex system whose governing
equations can only be solved through numerical methods. (In fact, the
dynamical equations for the atmospheric component of a climate model are
typically almost identical to those in the hurricane models referred to
above.) As expected, there are a variety of plausible approaches to
parameterization in the components of Earth system models (ESMs)111State-of-
the-art climate models are also commonly called general circulation models
(GCMs). This was particularly true in the past when they consisted of just an
atmosphere and sometimes an ocean. As time has progressed and more processes
have been included the term ESM has become more common (previously ESM may
have referred to models of reduced complexity featuring a carbon cycle). In
this review we use the term ESM to mean a full complexity dynamical state-of-
the-art climate model although this could be used interchangeably with GCM..
Thus, modelling groups throughout the world have built a few dozen ESMs with
different approaches to parameterization. Because of these differences, these
models produce different future climate states, even when the same scenario of
radiative forcing is imposed (associated, for example, with an increase in
greenhouse gases).

A classic climate change experiment is to double CO2 concentrations in the
atmospheric component of an ESM, and measure the surface warming that occurs
after the simulation has equilibrated Manabe and Wetherald (1975), an
important number in climate science referred to as equilibrium climate
sensitivity (ECS). Reaching a true equilibrium with a full complexity ESM
requires long, computationally expensive simulations of thousands of years so
ECS is usually estimated from shorter duration CO2 doubling experiments. When
these experiments are done with contemporary ESMs, the spread in values across
the ensemble of ESMs is large, between 1.5 and 6 degrees Celsius Forster  _et
al._ (2019), although true equilibrium values are higher (median 17% higher,
Rugenstein  _et al._ (2020)). The international climate science community has
organised itself to generate scenarios of greenhouse gas (GHG) emissions that
result in more realistic future radiative forcing than the CO2 doubling
experiment. These correspond to scenarios of controls (or lack thereof) on
future GHG emissions Moss  _et al._ (2010); Riahi  _et al._ (2017). The
scenarios are designed so that the same radiative forcing time series is
imposed on each ESM. This allows for inter-comparison of ESM response at some
future specified time, say the end of the 21st century. The ESM responses vary
significantly across the ensemble, an indicator of deep uncertainty in Earths’
climate future, simply due to the variety of plausible ways to handle
parameterization within ESM components.

If each ESM is treated as being an equally plausible analogue of the real
climate system, known as ‘model democracy’, an undesirably large factor of
three difference in future response to man-made GHG response currently
results. Apart from the subject of this review, there are several techniques
used to reduce this uncertainty and much research effort has been dedicated to
this important task. One approach is to give ESMs that simulate the real world
‘better’ (according to some chosen metric) more weight in the future
projection. There are also methods that do not use ESMs at all, by
constraining future response from past observations and those that use a
combination of all of these. Many more details can be found in e.g. Eyring
_et al._ (2019); Sherwood  _et al._ (2020) and chapters 9 and 11-14 in IPCC
(2013).

The approach taken to improve hurricane forecasting models – to evaluate their
performance and adjust parameterizations over the course of multiple
prediction cycles – is unfortunately impractical in the case of projecting
future climate with ESMs. With future climate, we have only one realisation of
the real system’s trajectory, and our mandate is to predict it as best we can
now. To compound the problem, the climate system involves many more components
than the atmosphere alone, including the ocean, the land surface, glaciers and
ice sheets, and the marine and terrestrial biospheres. The engineering
required to model each component and allow them to interact in a simulation is
impressive. But the components can interact with one another in information
flows are so complex that it is not easy to predict how behavior in one
component might affect behavior in another. For example, parameterization
reformulation or adjustment in one component can significantly affect the
simulated state of another component for reasons that are not always clear
(e.g. Donner  _et al._ (2011)).

To circumvent the impossibility of directly evaluating ESMs for their ability
to simulate a future climate state which has not yet been observed, a new
technique has emerged over the past decade and a half, known as the _emergent
constraint_ (EC) approach (Figure 1 shows a schematic). The basic idea is to
identify an element ($X$) of the observable climate that varies significantly
across the ESM ensemble, and which exhibits a statistically-significant
relationship with variations in some important variable ($Y$) describing the
ESM’s future simulated state. If we call this relationship $f$, then
$Y=f(X)+\varepsilon$, where $\varepsilon$ is a relatively small departure from
$f$. Since $X$ is an element of the observable climate, it is a quantity that
can be measured. The relationship $f$ may then place a useful constraint on
$Y$, provided the measurement uncertainty in $X$ is small compared to the
range of simulated values. This constraint is ‘emergent’ because the _emergent
relationship_ $f$ cannot be diagnosed from a single ESM. It only becomes
apparent when the full ensemble is analysed. If the relationship $f$ arises
from model physics or dynamics common to the ESMs, then reducing the spread in
$X$ through reformulation or adjustment of parameterizations ought to result
in spread reduction in $Y$. Corresponding reduction in the spread in other
future climate variables affected by $Y$ should also occur. If the process is
repeated for enough variables $X$ and $Y$, then we can imagine that overall
simulated spread in multiple aspects of future climate would gradually be
reduced. We also note that ECs can be used directly to ascertain the most
likely values for a particular $Y$.

Figure 1: Schematic showing the most common procedure used to derive emergent
constraints on Earth system sensitivities. An ensemble of ESMs (each red dot
is an individual ESM) running the same experiment (the pdf on the right-hand
$y$-axis represents the spread in the ensemble) is used to identify an
emergent relationship (black dashed line with grey uncertainty range), between
an uncertain Earth system sensitivity $Y$ ($y$-axis) and an observed trend or
variation $X$ ($x$-axis). An observation of the trend or variation (blue pdf
on the $x$-axis) can then be combined with the model-based emergent
relationship to derive an emergent constraint on the Earth system sensitivity
(green pdf on the left-hand $y$-axis).

To further illustrate the concept, the subject of the first published EC well
established across multiple generations of ESMs, is used as an example Hall
and Qu (2006). In this case, the future climate variable of interest ($Y$) is
the snow albedo feedback (SAF), a climate mechanism characterized by the
retreat of highly reflective snow cover under climate warming and the
associated reduction in surface albedo, which amplifies warming and promotes
further snow melt Hall (2004); Bony  _et al._ (2006); Thackeray and Fletcher
(2016). However, modern ESMs disagree on the strength of this feedback,
exhibiting a nearly threefold spread Qu and Hall (2014); Thackeray  _et al._
(2018). SAF also occurs each spring in the current climate when Northern
Hemisphere snow cover recedes from its winter peak to its summer minimum,
enhancing seasonal warming in the process ($X$). A comparison between
simulated SAF under climate change ($Y$) and in the seasonal cycle context
($X$), uncovers a strong linear relationship in three different generations of
ESM ensembles (Figure 2). The strength of this relationship tells us that
seasonal SAF is likely highly predictive of climate change SAF, while the
consistency across model generations illustrates robustness to out-of-sample
testing. Furthermore, we can quantify the seasonal SAF using satellite-derived
observations, thus allowing for model bias to be properly assessed. For
example, an ESM that underestimates observed SAF in the seasonal context is
likely to underestimate SAF in future climate. Producing a more accurate SAF
in ESMs is more than just an academic exercise as variability in SAF can
account for a significant portion of uncertainty in 21st century projections
of warming across Northern Hemisphere extratropical land Qu and Hall (2014).
Thus, by reducing variability in SAF we can expect similar reductions in
projections of regional warming. Figure 3 highlights the first application of
this and several other ECs from the literature (pertaining to the carbon cycle
and climate sensitivity), all of which will be further discussed in Section V.

Figure 2: Emergent relationship between springtime snow albedo feedback (SAF)
across Northern Hemisphere land under climate change ($Y$) and an observable
snow albedo feedback associated with the current climate’s seasonal cycle
($X$). An observational estimate derived from satellite data is shown as a
vertical bar. Each point represents an individual climate model from the three
most recent generations (CMIP3, CMIP5, and CMIP6). Methodology for calculating
SAF is adapted from Qu and Hall (2014) (further details in Thackeray  _et al._
(submitted)) Figure 3: Emergent constraints on Earth System sensitivities
based on some key examples published in the literature: (a) snow-albedo
feedback, from snow seasonal cycle Hall and Qu (2006); (b) sensitivity of
tropical land carbon to global warming, from interannual variability in CO2
Cox  _et al._ (2013); (c) atmospheric CO2 concentration at 2060, from
atmospheric CO2 concentration at 2010 Hoffman  _et al._ (2014); (d) CO2
fertilization of plant photosynthesis, from changes in the seasonal cycle of
CO2 Wenzel  _et al._ (2016); (e) sensitivity of tropical ocean primary
production to warming, from interannual variability Kwiatkowski  _et al._
(2017); (f) global ocean carbon sink in the 2090s, from the current day carbon
sink in the Southern Ocean Kessler and Tjiputra (2016); (g) Equilibrium
climate sensitivity, from interannual variability of temperature Cox  _et al._
(2018a); (h) Transient climate response, from increase in global mean
temperature Nijsse  _et al._ (2020). In each case the emergent constraint was
reconstructed from data available in the literature or provided directly by
the authors. The model ensemble used in each original study is shown in the
brackets after the panel title.

In this review article, we synthesise what is now understood about ECs. We
discuss how relationships $f$ might emerge in ESM ensembles in section II, as
well as the theory that underlies ECs in section III. In section IV we note
pitfalls associated with ECs and how to guard against them. In section V we
survey the ECs that have been found in the climate system, and describe the
statistics that can be used to apply them rigorously in section VI. Finally,
we discuss how techniques similar to ECs have been or might be used in other
fields, and consider the outlook for development of the technique going
forward in section VII before concluding in section VIII.

##  II How relationships in model ensembles might ‘emerge’

ECs are possible because of ‘emergent relationships’ appearing in an ensemble
of ESMs. An emergent relationship, $Y=f(X)+\varepsilon$, between an element of
the observable climate $X$, something that can be measured now in the real
world, and the unknown future response $Y$ one wants to know, can then be used
to place a constraint on the real world value of $Y$ via observations of $X$
to give an EC. For example, in Cox  _et al._ (2018a) the observable was the
variability in global annual mean surface air temperature during roughly the
last one hundred years and the unknown future response was equilibrium climate
sensitivity (ECS, defined as the magnitude of the Earth’s warming to doubled
CO2 levels), one of the oldest and most important numbers in climate science.
Despite decades of research the latest ESMs still vary widely in their
predictions of this number by a factor of three Forster  _et al._ (2019);
Zelinka  _et al._ (2020).

How might a relationship, $f$, between observable and response ‘emerge’ in an
ensemble of ESMs? In the Cox  _et al._ (2018a) EC on ECS, an analytically
soluble energy balance model, much reduced in complexity compared to the ESMs
in the ensemble, predicted a functional form between $X$ and $Y$ for the
emergent relationship (although this provoked a lively debate, see Brown  _et
al._ (2018); Po-Chedley  _et al._ (2018); Rypdal  _et al._ (2018); Cox  _et
al._ (2018b); Williamson  _et al._ (2019)). This mechanism from high to low
dimensional relationship emergence and others are given in the following
subsections. First we discuss the differences in the types of climate model
ensembles commonly used to find emergent relationships.

###  II.1 Commonly used ESM ensembles: Multi-model (MME) and perturbed physics
(PPE) ensembles

Model simulations are not perfect reproductions of the system they are
designed to emulate. Ensembles of models are used to get a handle on (i)
initial condition uncertainty, (ii) parametric uncertainty and (iii)
structural uncertainty.

Uncertainty in the initial state is particularly important for numerical
weather prediction. As the equations governing the weather are chaotic,
forecast solutions with very small differences in their initial states,
equally likely to be the ‘real’ initial state, can diverge strongly after just
a few days. This sensitivity to initial conditions means running the weather
model just once, even if the model is a perfect reproduction of the real
world, may produce a very different forecast from the one actually
experienced. By running the same weather model many times with different, but
equally plausible initial states chosen to sample the most unstable and
divergent regions of the model phase space, the likelihood of experiencing a
particular weather forecast can be constructed. Ensemble forecasting is now a
standard tool in numerical weather prediction Epstein (1969); Leith (1974);
Toth and Kalnay (1993); Molteni  _et al._ (1996). Initial value ensembles in
climate applications are usually used to assess natural climate variability
Kay  _et al._ (2015); Maher  _et al._ (2019).

Climatic variables are (generally speaking) the long term statistics of
weather and their prediction is less impacted by initial state uncertainty
relative to parametric and structural uncertainties Hawkins and Sutton (2009,
2011). Parametric uncertainty arises from uncertainty in the values of
constants (‘parameters’) in quasi-empirical relations (‘parameterizations’)
used to model the effects of unresolved, subgrid scale processes. Parametric
uncertainty can be sampled by running the same model with different values of
these parameters. ESM model ensembles that do this are known as perturbed
physics or perturbed parameter ensembles (PPEs, Murphy  _et al._ (2004)).
Examples of ESM PPEs are the climateprediction.net experiment Stainforth  _et
al._ (2005) and the Met Office’s Quantifying Uncertainties in Model
Projections (QUMP) ensemble Collins  _et al._ (2011) that both used the HadCM3
ESM.

Structural uncertainty arises from uncertainty in the functional form of
equations. Although a lot of the model equations in ESMs are well known, there
are multiple, equally defensible parameterization schemes for unresolved
processes (e.g. convective precipitation and cloud radiative properties) and
so different equations can appear in different ESMs. ESM ensembles that sample
structural uncertainty are called multi-model ensembles (MMEs) and include the
coupled model intercomparison project (CMIP) ensembles Meehl  _et al._
(2007a); Taylor  _et al._ (2011); Eyring  _et al._ (2016) used to inform the
Intergovernmental Panel on Climate Change (IPCC) reports Cubasch  _et al._
(2001); Meehl  _et al._ (2007b); Collins  _et al._ (2013) and used extensively
in much of the EC literature.

The CMIP MME ensembles consist of a number of different ESMs developed by
different international groups thus sampling structural uncertainty. Some CMIP
groups also submit multiple initial value runs and occasionally multiple
physics parameter runs performed with the same model, sampling initial
condition and parametric uncertainty to some degree. Numerical weather
prediction ensemble forecasting has also incorporated more parametric and
structural uncertainty into their forecasting as time has gone on Palmer
(2019).

Most ECs have derived from MME ensembles to date although there are some
examples from PPEs e.g. Knutti  _et al._ (2006). ECs derived from MMEs are, in
our opinion, the most believable. Each model in a MME could be thought of as
that particular group’s best guess of the Earth system. PPEs on the other
hand, generally vary only a handful of parameters and thus sample just a few
dimensions of the model’s phase space. Due to the few varied dimensions,
relationships hard coded in the fixed equations between model variables become
easy to find (based on the authors experience with the QUMP PPE Lambert  _et
al._ (2013)). These ECs need to be backed up with physical arguments and
ideally tested in MMEs. Collins  _et al._ (2011) found that long wave cloud
feedback was highly correlated to climate sensitivity in the QUMP HadCM3 PPE
ensemble, however this correlation was absent in (smaller sample size) MMEs.
Yokohata  _et al._ (2010) also found that strong correlations in a PPE with
HadSM3 were not always present in the equivalent MIROC3.2 PPE. Further,
Yokohata  _et al._ (2010) found the mechanism for variations in climate
sensitivity was different.

Values of parameters chosen to correctly give the proper balance between
opposing processes in MMEs could be skewed to one of the processes by
simultaneously varying them in a PPE thus reducing the realism of each
ensemble member solution. However, PPEs can be useful for testing ECs (e.g.
Kamae  _et al._ (2016); Wagman and Jackson (2018) and section IV.9) and
quantifying parametric uncertainty (section VII.6).

###  II.2 Null hypothesis: Emergent relationships occur by chance

A starting null hypothesis is that emergent relationships occur by chance and
are not indicative of a deeper mechanistic relationship Hall  _et al._ (2019).
Data mining an ESM ensemble for high correlations between pairs of variables
may fall into this category (see also section IV.1). If an ESM ensemble is
reasonably small and the number of variables output by each ESM is high, the
expected number of variable pairs with high correlation is high purely by
chance Caldwell  _et al._ (2014).

More precisely, an ESM labelled by the index $i$ in an ensemble of $n$ ESMs
$i=\\{1,2,\ldots,n\\}$ calculates a large number $m$ of different variables
$X^{l}$, each variable labelled with index $l$ and $l=\\{1,2,\ldots,m\\}$. In
this data mining scenario, a pair of variables $(X^{l}_{i},X^{k}_{i})$ is
chosen for the prospective emergent constraint, one for the observable $X^{l}$
and the other, the response $X^{k}$ one wishes to know in the real world
(previously written $Y$). In an indiscriminate data mining approach, every
possible pair of variables are used to create $\tfrac{1}{2}m(m-1)$ data sets,
each labelled $\mathcal{S}^{kl}$. Each dataset has $n$ elements,
$\mathcal{S}^{kl}=\\{(X^{l}_{1},X^{k}_{1}),(X^{l}_{2},X^{k}_{2}),\ldots,(X^{l}_{n},X^{k}_{n})\\}$
with each element corresponding to one of the $n$ ESMs. If the pair of
variables making up $\mathcal{S}^{kl}$ is a good candidate for an emergent
relationship and we assume this relationship is linear, then the correlation
$r^{kl}$ in $\mathcal{S}^{kl}$ should be high. Correlation here is defined as
$r^{kl}=\text{cov}(X^{l},X^{k})/\sigma_{X^{l}}\sigma_{X^{k}}$ and $\sigma_{X}$
is the standard deviation of $X$.

The number of ESMs in a model ensemble, $n$, is typically small being around
10 to 40 in the state-of-the-art CMIP ensembles although numbers increase with
each successive generation Meehl  _et al._ (2007a); Taylor  _et al._ (2011);
Eyring  _et al._ (2016). If we calculate the correlation $r^{kl}$ within every
one of the $\tfrac{1}{2}m(m-1)$ possible datasets, some will have high
correlations purely by chance. The likelihood increases as the number of
models in the ensemble $n$ gets smaller giving less data points in each
$\mathcal{S}^{kl}$. While the fraction of datasets with high correlations
$|r|>|r_{high}|$, does not increase as the number of variables $m$ gets
larger, the total number of datasets with correlation above $|r_{high}|$ will
increase simply because there are more possible variable pairs to correlate.
Thus one must be careful that any prospective emergent constraint makes
physical sense, particularly in current, small ESM ensembles to avoid this
pitfall.

To illustrate this point in a worst case type scenario where every ESM output
variable is uncorrelated to any other output variable or any of the other
ESMs, we can calculate the correlations expected between all possible pairs of
variables as a function of ensemble size $n$. Figure 4 shows the fraction of
all possible datasets with a particular correlation for a ‘toy’ ESM ensemble
of $n$ models where each of the toy ESMs is a modelled as a random number
generator. This is a rather extreme and unrealistic example, each ESM
producing $m$ random output variables, the $l$th one labelled $X^{l}_{i}$,
drawn from a normal distribution with unit standard deviation,
$X^{l}_{i}\in\mathcal{N}(0,1)$. Even though there are no mechanistic
relationships in this example by construction, a fraction of the datasets have
strong correlations, particularly in smaller ensemble sizes (small $n$). This
is purely by chance.

Figure 4: High correlations between pairs of variables across small ESM
ensembles are expected by chance. To illustrate this point, the fraction of
all possible pairs of variables across a ‘toy’ ESM ensemble of size $n$
plotted against correlation $r$. Each ESM is not a real ESM; it is represented
by a random variable drawn from a normal distribution of unit standard
deviation and each of the $n$ ESMs produces $m=100$ output variables giving
4950 different possible variable pairs to correlate with each other. This
example is designed to be a worst case scenario yet still shows calculated
correlation even between uncorrelated objects can be large in small enough
ensembles. In the left panel, histograms of fraction of datasets with
correlation falling in a particular interval are plotted. In the right hand
panel, the total fraction of datasets with magnitude of correlation greater
than $|r|$ is plotted. Ensemble size is varied for toy ESM ensemble sizes of
$n=5$, $n=30$ and $n=100$,

###  II.3 Low dimensional relationships ‘emerge’ from high dimensional ESMs

In the last example even though all $n\times m$ ‘toy’ ESM variables in the
ensemble were uncorrelated from each other by construction, high correlations
were still possible. This is more likely in small ensembles. In reality _there
are_ mechanistic relationships between $X$ and $Y$ in any particular ESM. Real
ESMs are not random number generators, the output of a ESM is the numerical
solution of a large set of coupled, nonlinear equations codifying
relationships between variables representing the ocean and atmosphere dynamics
and thermodynamics and biogeochemical feedbacks222Many of these equations are
well known from other fields of physics such as the fluid dynamics Navier-
Stokes equations. See for example Ghil and Lucarini (2020) and references
within for the equations that commonly feature in ESMs.. However the exact
relationship between any pair of $X$ and $Y$ is not solely a function of just
these two variables, it is generally also a function of many others that
prescribe the exact state of the ESM at every point in space and time on the
ESM grid. The minimal number of (non-unique) variables, $d$, required to
describe the exact state of the ESM at time $t$ can be thought of as a point
in $d$ dimensional phase space with coordinates $(x_{1},x_{2},\ldots,x_{d})$
and in the case of a ESM, the dimension of this space is extremely high333To
give an idea of a lower bound on the dimension of the phase space required to
describe a typical CMIP6 era ESM, the atmosphere is typically cut into a total
number of $\sim 200\times 200\times 100=4\times 10^{6}$ spatial cubes. A
single atmospheric prognostic variable will need at least this many phase
space dimensions at each point in time to specify it exactly. A lower estimate
for the number of prognostic variables required just to specify the physical
state of the atmosphere is $\mathcal{O}(10)$. This already gives $d\sim
10^{7}$ even before including the ocean, land and biogeochemisty.. As the ESM
state changes with time, a trajectory is drawn between these phase space
coordinates connecting the past and present states. Variables describing the
precise state of the ESM such as temperature or precipitation, the $X$ and the
$Y$, may be functions of all the $d$ phase space coordinates as well as time
$(d+1)$ i.e. $X=f_{X}(t,x_{1},x_{2},\ldots,x_{d})$ and
$Y=f_{Y}(t,x_{1},x_{2},\ldots,x_{d})$.

Although the mechanistic relationship between $X$ and $Y$ is generally a
function of all $d+1$ coordinates, climatic variables of interest are usually,
although not exclusively, long term temporal and/or spatial averages that can
be thought of as (time invariant) attractors of the phase space reducing the
information relative to a precise configuration at any one time. In many
circumstances the mechanistic relation between $X$ and $Y$ can be approximated
to a good degree by a much smaller dimensional subset of the full phase space
reducing the effective dimension.

The success of statistical mechanics and science in general444Low dimensional
models can often mimic the responses of more complex ones to a good
approximation and this is essentially why science is successful i.e. we can
comprehend and predict the high dimension real world using (relatively) simple
models. relies on this effective dimension reduction, that certain properties
of systems with many degrees of freedom can be well approximated by fewer
state parameters. For example, a container of gas is, as far as we know at
present, most completely described by the equations of quantum mechanics. In
principle, one could solve these equations for each of the $N$ gas molecules,
at each position and time in the container. The dimension of this space
increases exponentially with $N$, requiring a huge number of degrees of
freedom to specify the state exactly even for just a handful of molecules.
This could be useful if the question one wanted to answer required detailed
knowledge of the full quantum state. However, if we ‘only’ want to know the
configuration of positions and momenta of the gas molecules, one can use an
approximation of the quantum theory. Treating the gas as a bunch of identical
spheres each with a mass and a different position and momentum, the state can
be described by a point in 6$N$ dimensional phase space, a reduction in the
phase space dimension yet still a massive number of dimensions for a realistic
gas.

Generally detailed knowledge of a precise system state is not desired or
required. Bulk properties describing the mean overall state are a useful way
of reducing information and making sense of complex systems. Going back to the
container of gas, kinetic theory provides a way of mapping the many degrees of
freedom from each of the molecules to the few bulk parameters of the ideal gas
law, under reasonable assumptions (see for example Schekochihin (2015)). That
is, the $6N$ dimensions effectively reduces to the three encoded by the
relationship $pV=nRT$. The bulk variable pressure $p$ felt as a force per unit
area on the container wall and is related to how often and how vigorously on
average molecules with mean square velocity $\langle v^{2}\rangle$ hit the
sides of the container ($p=\frac{n}{3V}\langle v^{2}\rangle$, $n$ is the
number of moles in the container of volume $V$), whereas the temperature $T$
is a measure of the most probable kinetic energy of a molecule in the
container. $R$ is the ideal gas constant. This approach works well when $N$ is
large as one finds more of the $6N$ states correspond to the ideal gas law.

Effective dimension reduction in ESMs can also happen when we want answers to
questions about bulk mean states of the climate. For instance, relations
between bulk variables such as the annual global mean surface air temperature
and the net incoming radiation are well modelled by the few degrees of freedom
in a simple energy balance model (see for example Hasselmann (1976); Wigley
and Raper (1990); Gregory (2000)) even though the exact state of a ESM is
given by all prognostic variables at every point and multiple times on its
spatial grid. If one wanted to ask detailed questions about the state of a ESM
such as ‘How many days in March does it rain more than $x$ mm in Madagascar?’
then the full ESM theory is best placed, however if the question was ‘How much
does the global mean temperature increase from a spatially uniform doubling of
atmospheric CO2’, a lot of that extra detail seems unnecessary. The analogy we
attempt to present here is that ESMs are analogous to the solutions to the
laws of quantum mechanics and the effective, reduced dimension relationship
for a particular ESM $i$, $Y=f_{i}(X)+\epsilon$ is analogous to the ideal gas
law. The analogy is far from perfect, ESMs are of course not as simple as
containers of large numbers of identical molecules and there is, as yet, no
such elegant route between the two solutions provided by the statistical
mechanical or kinetic theory recipes. The best candidates for ESM ideal gas
laws to serve as theoretical bases for emergent relationships are simplified,
analytically soluble models of the climate or its subsystems such as the
energy balance models previously mentioned Hasselmann (1976); Wigley and Raper
(1990); Gregory (2000). These act more as testable, plausible hypotheses of
the bulk behaviour of ESMs rather than derivations from the basic laws that
constitute them.

So far we have discussed how a physically plausible, mechanistic, low
dimensional relationship $f_{i}$ between $X$ and $Y$ may appear from a
particular ESM $i$, $Y=f_{i}(X)+\varepsilon$. We have not yet discussed how
low dimensional relationships, the _emergent relationship_ , could appear
across an ensemble of ESMs. Although each ESM is different, they should be
equally plausible models of the $(X,Y)$ relationship in the real world and it
should not be a surprise that ESM solutions across a model ensemble should be
also be mechanistically related. This is because many of equations are based
on well established and tested physics (such as the Navier-Stokes equations)
and are common amongst ESMs. Indeed, if the all the equations were well known
and could be numerically integrated exactly, the ESMs should produce identical
outputs (and the problem of climate modelling would be solved and emergent
relationships would disappear).

However, some equations can not be exactly numerically integrated due to
limitations on temporal and spatial resolution. One then has to parameterize
subgrid scale processes and this may be a source of difference between ESM
solutions, each model giving a different $(X,Y)$ pair (see section II.1).
Although they are our best attempts at understanding the climate, some ESM
responses diverge on important questions. Such a scenario is where the
emergent constraint approach may be helpful. To have a chance of finding an
emergent relationship, the set of $n$ $(X_{i},Y_{i})$ data points should be
different (i.e. $(Y_{1},X_{1})\neq(Y_{2},X_{2})$) and span a wide enough range
for the emergent relationship to become apparent. The model relationship
$f_{i}$ also needs to be shared amongst the model ensemble ($f_{i}(X)\approx
f_{j}(X)\forall i,j=\\{1,2,\ldots,n\\}$). Provided such a relationship exists,
the emergent relationship $f(X)$ can be determined. If this model relation is
also shared by the real world, the real world’s response can also be
determined (along with its uncertainty) to give an emergent constraint (see
section II.5).

###  II.4 Range in response due to the same physical process having a wide
range across ESMs

A related mechanism for a low dimensional relationship to emerge between $X$
and $Y$ occurs when the magnitude of the same physical process, correlated to
the observable $X$, differs appreciatively across the ESM ensemble and the
magnitude of this process heavily determines the size of the response $Y$.
This is also a case of effective dimension reduction - the range in response
is dominated by the dimensions of phase space of just that physical process.

An example of this happening was provided by Caldwell  _et al._ (2018) who
evaluated ECs on equilibrium climate sensitivity (ECS) in the CMIP3 and CMIP5
ESM ensembles. That is, the targeted unknown response, $Y$, amongst these ECs
was the same, ECS, although different observables $X$ were found to be well
correlated to it. Examples of a few of the $X$ used in the different ECs were
the strength of resolved-scale mixing between the boundary layer and lower
troposphere in the tropical east Pacific and Atlantic Sherwood  _et al._
(2014), error in the distribution of cloud-top pressure and optical thickness
for regions between 60oN/S Klein  _et al._ (2013), fraction of tropical clouds
with tops below 850 mb whose tops are also below 950 mb Brient  _et al._
(2016) and variability in global mean air temperature Cox  _et al._ (2018a).
Caldwell  _et al._ (2018) showed all of these observables were also highly
correlated to the short wave (SW) cloud feedback, a strong and uncertain
feedback on the resulting global temperature. The SW cloud feedback in turn
was shown to be highly correlated to intermodel variations in ECS within the
CMIP5 ensemble. To paraphrase Caldwell  _et al._ (2018), ‘…intermodel
variations in cloud feedback were so big that they left a strong imprint on
intermodel variations in ECS. This means that fields that are strongly
correlated with ECS are probably correlated with the SW cloud feedback (and
vice versa).’

###  II.5 What is needed for an EC?

In addition to having an emergent relationship in a model ensemble, an EC also
ideally requires a few other things to be useful.

####  II.5.1 Observable ($X$) range and uncertainty

The range of $X$ in the ESM ensemble should be large relative to the
uncertainty in the value of the real world observable. Uncertainty in each of
the model $X$ should also be small relative to the range in model $X$. Ideally
the real world value of the observable $X_{\text{obs}}$ should also fall well
within the range of values in the ESM ensemble to avoid extrapolation issues.

####  II.5.2 Response ($Y$) range and uncertainty

One would like a large range in the values of the ESM ensemble responses
relative to the uncertainty in each of the individual ESM responses. This
reduces the uncertainty in the real world value of $Y$ derived from the EC.

####  II.5.3 Relationship between $X$ and $Y$

A particular ESM labelled $i$ should have a physically plausible relationship
$f_{i}$ between the observable and response $Y=f_{i}(X)+\varepsilon$. It
should be based on a simplified theory that predicts a functional response and
has clear, testable assumptions that can be falsified independently. In
addition, all models in the ensemble should share roughly the same relation
between observable and response i.e. $f_{i}=f_{j}$. The stronger this
relationship is, the more useful the emergent constraint due to the potential
uncertainty reduction in $Y$. _And_ this relationship should also be shared by
the real world.

####  II.5.4 Large ensemble size $n$

Ideally you would have a large number of independent ESMs in your ensemble
(larger sample, lower error in estimates, less chance for high correlations by
chance). See section IV.5 for further discussion of ESM ensemble independence.

##  III Underlying theory for emergent constraints based-on temporal
variability

Many published EC studies relate the longer-term sensitivity of interest to
aspects of the mean climate simulation in models - see Table I (Volodin, 2008;
Kidston and Gerber, 2010; Massonnet  _et al._ , 2012; Tian, 2015; Brient and
Schneider, 2016; Simpson and Polvani, 2016; Lin  _et al._ , 2017; Lipat  _et
al._ , 2017; Siler  _et al._ , 2018; Selten  _et al._ , 2020). However, a
growing number of papers are also now relating Earth System sensitivities to
observable temporal variations, such as trends (Boe  _et al._ , 2009; Jiménez-
de-la Cuesta and Mauritsen, 2019; Nijsse  _et al._ , 2020; Tokarska  _et al._
, 2020), interannual variability (Clement  _et al._ , 2009; O’Gorman, 2012;
Cox  _et al._ , 2013; Wenzel  _et al._ , 2014; Qu  _et al._ , 2015;
Kwiatkowski  _et al._ , 2017; Cox  _et al._ , 2018a) and seasonal cycles (Hall
and Qu, 2006; Knutti  _et al._ , 2006; Qu and Hall, 2014; Zhai  _et al._ ,
2015; Wenzel  _et al._ , 2016; Thackeray and Hall, 2019). This section
discusses the theoretical basis for emergent constraints based on such
temporal variations.

In the case of changes in relatively fast variables, such as seasonal snow-
cover Qu and Hall (2007), or marine phytoplankton concentration Kwiatkowski
_et al._ (2017), there may be a fairly straightforward near one-to-one
relationship between the short-term variability and the longer-term
sensitivity, because the fast variable will be in a quasi-equilibrium state
even with short-term climate variations. For slower variables (such as the
forest carbon storage), short-term variations are more likely to measure
fluxes (or equivalently the rate of change of the store). In this case,
finding a constraint on future changes in the store requires multiplying the
flux sensitivity to short-term variations by a characteristic timescale for
each model. In some cases, the characteristic timescale may be similar across
the model spectrum, leading to a simple emergent relationship between the
short-term flux sensitivity and the long-term sensitivity of the store Cox
_et al._ (2013); Wenzel  _et al._ (2014). In general though, converting a flux
sensitivity to a store sensitivity requires an independent estimate of the
characteristic timescale of the store, which itself requires a theoretical
basis Williamson  _et al._ (2019).

The Fluctuation-Dissipation Theorem (FDT) provides one such theoretical
framework Kubo (1966). The FDT relates the sensitivity of a system to a small
external forcing to the response of the same system to its own internally-
generated fluctuations. FDT is therefore a potential theoretical basis for
ECs, as it links the natural variability of a system to its sensitivity. The
size of the forcing is important because the theorem only strictly applies to
near-equilibrium linear systems. Nevertheless, FDT-based approaches have had a
huge impact in statistical physics, including Einstein’s work on Brownian
motion Einstein (1905), and the understanding of Johnson-Nyquist noise in
electrical circuits Johnson (1928); Nyquist (1928) .

The first proposal to apply FDT to the climate system came almost 40 years ago
Leith (1975), but there has been a recent resurgence of interest in this area
fuelled by methodological advances and detailed comparison of FDT-derived
estimates of climate change to climate model simulations Majda  _et al._
(2010). In principle, it may be possible to use FDT to get good estimates of
the response of the real climate system to small forcing (e.g. that due to
doubling CO2) purely from accurate long-term climate observations that reveal
the full spectrum of natural fluctuations of the climate Bell (1980); Schwartz
(2007). Unfortunately though, the length of the detailed climate records
required to achieve this are typically much longer than those available (Kirk-
Davidoff, 2009), and include the contemporary period when the mean climate is
changing.

In the EC technique model projections are instead used to define emergent
relationships between observable variations and future climate (Hall  _et al._
, 2019), and then specific observations provide a selection principle to
constrain the range of future climate projections from these model-defined
relationships (Eyring  _et al._ , 2019). The EC technique has yielded many
proposed constraints on aspects of the future climate (Hall  _et al._ , 2019)
and the carbon cycle (Cox, 2019). However, this power also comes with a danger
- that blind data-mining of multidimensional model outputs could lead to
spurious and misleading constraints (Caldwell  _et al._ , 2014). Protection
from this risk can come from tests of robustness across different model
ensembles, and from basing the search for ECs on firm theoretical and
mathematical foundations. Theory-led ECs can also be considered as hypotheses
that can be tested against the ensemble of complex ESMs. The remainder of this
section aims to describe the reasons why relationships between Earth System
variability and sensitivity are ubiquitous, and to provides some examples of
the emergent relationships to be expected under different types of time-
varying forcing (such as seasonal cycles and longer-term trends).

###  III.1 Relationships between variability of fluxes and the sensitivity of
stores

We can write the time evolution of a dynamical system variable $V$ in the
general form:

$\frac{dV}{dt}=F(V,z_{i})$ (1)

where $F$ is the net flux into the system, and $z_{i}$ are the environmental
variables that affect this net flux. Equation 1 represents a local
conservation law as it implies that the variable $V$ can only change if the
net flux is non-zero. When the environmental variables are in a steady state,
$z_{i}=z_{i}(0)$, the variable $V$ has an equilibrium state $V=V(0)$, defined
by:

$F(V(0),z_{i}(0))=0$ (2)

In climate science we are typically interested in how a climate system
variable, such as global mean temperature or land carbon storage in the
tropics, varies with environmental factors, such as the atmospheric carbon
dioxide concentration or the climate in the tropics. Where the associated
perturbations $\Delta z_{i}$ are small compared to $z_{i}(0)$, equation 1 can
be linearly expanded about the equilibrium state:

$\frac{d\Delta V}{dt}=\frac{\partial F}{\partial V}\,\Delta V+\frac{\partial
F}{\partial z_{i}}\,\Delta z_{i}$ (3)

where $\Delta V$ is the resulting perturbation to the state variable, and the
partial derivatives are calculated around the initial equilibrium
$\left\\{V(0),z_{i}(0)\right\\}$. The initial equilibrium is stable if
$\frac{\partial F}{\partial V}<0$, and we can then define an effective
timescale to perturbations of $V$ as $\tau=(-\tfrac{\partial F}{\partial
V})^{-1}$. Equation 3 can therefore be written as:

$\frac{d\Delta V}{dt}+\frac{\Delta V}{\tau}=\frac{\partial F}{\partial
z_{i}}\,\Delta z_{i}$ (4)

The righthand-side of equation 4 can be viewed as the ‘external’ forcing
factors that produce changes in $V$ (e.g. the radiative forcing due to
increasing atmospheric CO2). For a permanent time-invariant change in the
environmental variable $\Delta z_{i}$, $V$ will be changed by an amount
$\Delta V_{eq}$ where:

$\Delta V_{eq}=\tau\,\frac{\partial F}{\partial z_{i}}\,\Delta z_{i}$ (5)

Defining the sensitivity of $V$ to the environmental variable $z_{i}$ as
$\xi_{i}=\partial V_{eq}/\partial z_{i}$, we can rewrite this equation as:

$\xi_{i}=\tau\,\mu_{i}$ (6)

where $\mu_{i}=\partial F/\partial z_{i}$. Equation 6 implies that the
sensitivity of $V$ to a step change in $z_{i}$ will be proportional to the
sensitivity of the net flux $F$, with a constant of proportionality which is
the characteristic lifetime of perturbations to $V$. In climate change
research, the equilibrium sensitivities ($\xi_{i}$) are often the things we
would most like to constrain (e.g. Equilibrium Climate Sensitivity, ECS).

###  III.2 Theoretical emergent relationships for idealised time-varying
forcing

For emergent constraints we therefore need to find relationships between
sensitivities $\xi_{i}$ and observable variations in the climate system. In
this subsection we derive candidate emergent relationships for different time-
variations in the environmental variables $z_{i}$. First we rewrite equation 4
in the form:

$\frac{d\Delta V}{dt}+\frac{\Delta V}{\tau}=\mu_{i}\,\Delta z_{i}$ (7)

As we limit ourselves here to this linear model with a single timescale
$\tau$, the emergent relationships presented below are intended to be
illustrative, or as simple hypotheses to be tested against the outputs from
complex models.

####  III.2.1 Sinusoidal forcing

As an idealised representation of the response of our system to diurnal and
seasonal forcing, we first consider sinusoidal environmental variations of
angular frequency $\omega$ and amplitude $a_{z}$, for which $\Delta
z_{i}=a_{z}\,e^{i\omega t}$. The solution to equation 7 under these
circumstances (after initial transients have died-down) is:

$\Delta
V=\mu_{i}\,a_{z}\,\frac{\tau}{\sqrt{1+\omega^{2}\tau^{2}}}\,e^{i(\omega
t-\phi)}$ (8)

where $\phi=\arctan{(\omega\tau)}$. Therefore $\Delta V$ also has a sinusoidal
variation but with a phase-lag relative to the forcing of $\phi$ which
asymptotes to $\pi/2$ as $\omega\tau\rightarrow\infty$. Substituting $\Delta
z_{i}=A_{z}\,e^{i\omega t}$. The amplitude of the sinusoidal variation in
$\Delta V$, $a_{v}$, is proportional to $a_{v}$ but also depends on the
frequency of the sinusoidal forcing:

$\frac{a_{v}}{a_{z}}=\mu_{i}\,\frac{\tau}{\sqrt{1+\omega^{2}\tau^{2}}}$ (9)

Using equation 6 we can therefore write the sensitivity $\xi_{i}$ in terms of
the ratio of the sinusoidal amplitudes:

$\xi_{i}=\frac{a_{v}}{a_{z}}\,\sqrt{1+\omega^{2}\tau^{2}}$ (10)

In the limit of slowly-varying forcing compared to the system timescale
($\omega\tau\rightarrow 0$) this formula reduces to :

$\xi_{i}\rightarrow\frac{a_{v}}{a_{z}}\mbox{ \hskip 28.45274pt ; for \hskip
14.22636pt}\omega\tau\rightarrow 0$ (11)

This is the limit of very fast variables that are in a quasi-equilibrium with
the sinusoidal forcing, such as seasonal snow-cover (Qu and Hall, 2007) or
marine primary production (Kwiatkowski  _et al._ , 2017). However, in the
opposite high-frequency limit ($\omega\tau\gg 1$) the emergent relationship
also depends linearly on the frequency of the forcing:

$\displaystyle\xi_{i}\rightarrow\frac{a_{v}}{a_{z}}\,\omega\tau\mbox{ \hskip
28.45274pt ; for \hskip 14.22636pt}\omega\tau\gg 1$ (12)

####  III.2.2 Linearly-increasing forcing

Contemporary climate change is largely being driven by an approximately
exponential rate of increase in atmospheric carbon dioxide above the pre-
industrial level, which yields a radiative forcing that is approximately
linear in time. This has motivated a long-running series of idealised climate
model experiments which prescribe an exponential increase in carbon dioxide,
including the _1% per year runs_ which are used to define the concept of the
_Transient Climate Response (TCR)_ . In addition, many other Earth System
changes can be approximated by linear trends. In this sub-section we therefore
consider emergent relationships under a linear increase in the environmental
variable $z_{i}$:

$\Delta z_{i}=\gamma\,t$ (13)

where $\gamma$ is the linear rate of increase $z_{i}$ which begins at time
$t=0$. Under this idealised forcing the solution to equation 7 is

$\Delta V(t)=\tau\mu_{i}\,\gamma\,\left\\{t-\tau(1-e^{-t/\tau})\right\\}$ (14)

From equation 5 we note that the sensitivity $\xi_{i}=\tau\mu_{i}$, so
equation 14 can be rewritten as:

$\xi_{i}=\frac{\Delta
V(t)}{\gamma}\,\frac{1}{\left\\{t-\tau(1-e^{-t/\tau})\right\\}}$ (15)

The exponential decay-term here is a transient response to the sudden switch-
on of the linear trend at $t=0$. Once this term has died-out the transient
solution lags the quasi-equilibrium solution by $\tau$ years such that:

$\xi_{i}=\frac{\Delta V(t)}{\gamma\,(t-\tau)}$ (16)

This equation represents a potential emergent relationship between the
sensitivity $\xi_{i}$ and a transient change $\Delta V(t)$.

####  III.2.3 White-noise forcing

Emergent relationships have also been proposed between interannual variability
and sensitivities (Schwartz, 2007; Cox  _et al._ , 2013; Nijsse  _et al._ ,
2019), assuming that the environmental variable $z_{i}$ is approximately
gaussian white-noise. Under those circumstances equation 7 becomes the widely-
used Ornstein-Uhlenbeck equation of statistical physics Uhlenbeck and Ornstein
(1930). Standard solutions relating the variance of $V$ to the variance of
$z_{i}$ can be derived by integrating the sinusoidal solution given by
equation 8 over all frequencies $\omega$. For interannual variability this
yields the following relationship for the ratio of standard deviations:

$\frac{\sigma_{v}}{\sigma_{z}}=\mu_{i}\,\sqrt{\frac{\tau}{2}}=\xi_{i}\sqrt{\frac{1}{2\tau}}$
(17)

and an even simpler relationship for the lag-1 autocorrelation of $Y$:

$\alpha_{v1}=e^{-1/\tau}$ (18)

These equations can be combined to yield an equation for the sensitivity
$\xi_{i}$ without needing to know the timescale $\tau$ (Cox  _et al._ , 2018a)
:

$\xi_{i}=\frac{\sigma_{v}}{\sigma_{z}}\,\sqrt{\frac{2}{-\ln{(\alpha_{v1})}}}$
(19)

Similar formulae can be derived for more sophisticated representations, such
as two-box and semi-infinite heat diffusion models of ocean heat uptake
(Williamson  _et al._ , 2019).

##  IV How might emergent constraints go wrong and how to guard against it?

Uncertainty in future projections makes adaptation planning difficult, and so
there is tremendous pressure on climate researchers to provide much more
refined predictions of expected large-scale environmental change as
atmospheric greenhouse gases (GHGs) rise. However, in the absence of a full
knowledge of all climate processes and their parameterizations, such
deficiencies will continue to cause substantial ESM differences. For this
reason, the method of emergent constraints has attracted substantial
attention. Such interest is because ECs offer a method to potentially ‘short
cut’ current deficiencies in process understanding, by providing better
estimates of change. Or, as a minimum, ECs offer more reliable estimates of
bulk aggregated parameters of the climate system of interest to policy. We
believe ECs do provide a route to reduce uncertainty, and consider it is
appropriate for the technique to substantially underpin societal decisions
both regarding adaptation planning and parallel mitigation programs to reduce
GHG emissions. However this brings tremendous responsibility, and so the
method must be reliable and robust. It is therefore prudent to discuss the
multiple circumstances in which the method may fail, thus altering to where
caution is needed. We now describe such situations.

###  IV.1 Risks of purely using data mining

One approach with ECs is to simply ‘data mine’ within climate model ensembles,
such as CMIP5 and CMIP6, searching for two quantities in each model that when
analyzed form a statistically significant inter-ESM regression. In these
circumstances, the researcher is not led by process understanding, a
hypothesis to be tested, or indeed intuition (see section II.2 for an extreme
example). However, there is a logic that in some circumstances, this approach
might be valid. In the climate system, there may be links between different
parts of the Earth system, and including between a contemporary measurable
quantity and an attribute change in a future climate state, that are not
immediately obvious as being connected.

One set of examples could be similar to the on-going discovery of
teleconnections in the climate system, where strong correlations are noted
between variations in two parts that are a substantial geographical distance
apart Nigam and Baxter (2015). Besides spatial distances, discovered
teleconnections can contain lags, although unlike ECs that are designed to
project decades ahead, the delays are months. These delays are often used to
estimate expected meteorological conditions in the months ahead. For example,
the status of El Niño, can strongly weight the probabilities of particular
weather features for land regions and across the tropics and for the season
ahead. Although El Niño is an obvious system fluctuation to investigate for
its impacts elsewhere, over the last couple of decades, researchers have been
investigating other candidates e.g. Feldstein (2000).

Given the sometimes diverse form of teleconnections, their discovery can
require a data mining approach, as comparison between two different parts of
the climate system, additionally offset in space and time, are unlikely to be
found through intuition-led inspection of ESMs. If novel ECs exist, between
diverse parts of the Earth system, then such mining methods may be needed to
aid their discovery. If ECs are discovered by statistical methods, then
subsequent process analysis may reveal the connection to have a strong
physical basis. Hall  _et al._ (2019) refers to this as ‘EC confirmation’,
although this may take many years to achieve, and especially if additional ESM
diagnostics are needed to confirm the size of any transport equation terms
between different spatial locations.

However, whilst such an approach is expected to reveal new ECs, caution is
needed. The concern is that statistically-significant regressions are likely
to be found in a small number of instances, simply by chance, and the
resultant EC is therefore not a robust indicator of future change (i.e.
section II.2). That is, and presented loosely, if statistical mining finds a
large number of new ECs, all with non-zero regressions forming them and at a
90% confidence level, then roughly one-in-ten will be invalid. This highlights
the need to undertake process-based confirmation of ECs found through
statistical means.

###  IV.2 The risk of $p$-hacking and overconfidence

The term ‘$p$-hacking’ was first introduced by Nuzzo (2014), and alerts to the
risk of self-selecting only scientific findings that are statistically
significant. In the context here, that would be ECs having their regressions
at small $p$ values, suggestive of a low probability of an $X-Y$ relation
occurring merely by chance. The concept has strong similarities to the dangers
noted in data mining, where scanning across a range of potential ECs could
lead investigators to only concentrate on a few with low $p$-values, with the
attendant risk they occur by chance. However, $p$-hacking goes further and
cautions over other decisions that researchers may make, in a rush to find a
significant result. Behaviours could include a deliberate selection of one-
sided tests and the stopping of sampling upon finding a low $p$-value. It
could also involve adjusting the parameters, relationships between data, or
time frequencies considered until lower $p$-values are discovered. In the case
of the climate system, and as an example, the assessment of attributes of the
global hydrological cycle offers many time scales of interest. Precipitation
statistics with potential to form different $X$ and $Y$ quantities of ECs,
range from short, intense rainfall events through to seasons or even decades.
To account for overconfidence Bretherton and Caldwell (2020) suggests that ECs
be corrected by scaling up the unexplained variance by a user-defined factor
across the board.

###  IV.3 Missing process in all current models, measurement errors and model
compensating errors

ECs most frequently link a fluctuating quantity that is measurable for the
contemporary period, to either a future change (e.g. extent of polar ice sheet
cover), or an invariant system attribute that describes change (e.g. climate
sensitivity). A potential concern for the validity of an EC is if there is a
missing process in every model that affects the regression forming the basis
for the EC. If, for instance, that error affected the $X$-axis fluctuating
quantity, but the measured quantity was accurate, then this would introduce a
bias in to the projection of the future $Y$-axis quantity.

Such a missing process could itself impact ESM performance at all modelled
levels of atmospheric GHG concentrations. One potential example is that many
ESMs underestimate interception loss, by vegetation, of rainfall e.g. Yang
_et al._ (2018); Lian  _et al._ (2018). Interception loss is the return of
water from rainfall to the atmosphere, and that has not passed through soils
but instead is temporarily held on leaves and branches instead. If an emergent
constraint on, for instance, a feature of the atmospheric part of the
hydrological cycle that depended on the overall land-atmosphere exchange of
water, then the EC could have an overly strong dependence on plant evaporation
(i.e. to compensate for the low modelled values of interception loss). If,
then, the true $X$ value was taken from data of actual plant evaporation, this
would introduce a bias in the estimate of the $Y$ quantity. A straightforward
corollary to this is if an EC depends on an $X$ quantity that is modelled
well, but measurements of it contain a bias, then this too would lead to
inaccurate estimates of the value of the $Y$ quantity.

Alternatively, other process omissions could become more important as GHG
levels rise. Until recently, an example of a component frequently not modelled
in ESMs is the terrestrial nitrogen cycle Thomas  _et al._ (2015). Nitrogen
limitation could become an increasingly important factor as CO2 concentrations
rise, by suppressing levels of vegetation fertilisation from the increased CO2
in the atmosphere. The lack of nitrogen cycle inclusion in the land components
of ESMs could therefore affect ECs linking current variations of the carbon
cycle to future carbon stores, leading to potential overestimates of the
capability of terrestrial ecosystems to offset future anthropogenic CO2
emissions. That said, in Wenzel  _et al._ (2014), two ESMs that did have the
nitrogen cycle included were close to an EC regression line in an ensemble of
carbon cycle only ESMs. That analysis linked contemporary fluctuations in
temperature and atmospheric CO2 concentrations to the long-term stability of
terrestrial carbon stores under global warming.

A corollary to single model errors is the case where two contemporary
processes are defined, but together they allow a model to perform well on some
metrics as a consequence of compensating errors. There is quite substantial
anecdotal evidence that many ESMs have these issues, as model developers often
discover that improvement of one component, by removing the compensation,
actually results in less accurate contemporary climate simulations. Poor
performance is only then resolved when the process representation of the
second feature is also upgraded. Such compensations present a risk for ECs.
Specifically, if the $X$-axis quantity of an EC relates to just one of the two
process that compensate, then that will introduce a bias. Returning to the
interception example, if we consider the deficiency recognised but accounted
for by parameterizing overly strong plant evaporation amounts, any EC
dependent on plant evaporation ($X$) would introduce a bias in prediction of
$Y$ if actual measurements of $X$ are used.

Ultimately, climate science requires a set of ESMs that make similar
projections and are highly accurate. Hence the on-going attempt to include all
relevant processes in ESMs, and to remove compensating errors, remains an
appropriate activity to advance such predictive tools. However, a co-benefit
is that such ESM enhancements will likely raise the accuracy of any ECs that
are used to constrain understanding where uncertainty remains.

###  IV.4 System passes through a tipping point

It is arguable that moderate human adjustment to the climate system is
relatively linear, and so the expected meteorological changes increase
proportionally with changes to the combined radiative forcing of atmospheric
GHG concentrations. If this same linearity extends to higher frequency system
responses such as interannual variability, then observations of contemporary
fluctuations may give a strong indication of the system’s response to a more
permanent forcing from increased GHGs. One example is Cox  _et al._ (2013),
where the change to tropical atmospheric CO2 concentration in response to
annual temperature variation may project any expected loss of rainforest
carbon stores under sustained global warming. Hence, there is an assumption of
linearity present in each individual ESM, or at least a relationship that is
monotonic in increasing temperature. When plotting the simultaneous variations
in temperature and CO2 ($X$) against response of future carbon stores in
response to warming ($Y$), then a further linear inter-ESM relationship is
found, and this is the EC itself.

A legitimate question, therefore, is what happens should the planetary system
move beyond linear responses, and further, start to pass through tipping
points Lenton  _et al._ (2008)? Many examples of such potential nonlinear
behaviours are conjectured, where a relatively small increase in radiative
forcing could cause major system changes. Potential climate tipping points
such as the collapse of the Atlantic meridional circulation Stocker and Wright
(1991), Amazon dieback Cox  _et al._ (2000), Greenland ice sheet melt Toniazzo
_et al._ (2004) and others Drijfhout  _et al._ (2015) are presently thought to
be low probability, high risk events although there is evidence subsystems of
the climate have ‘tipped’ many times in the past Bond  _et al._ (1992). Most
of these ‘tips’ have resulted in regional rather than global climate changes
and have been emulated through simpler equation systems, amenable to
bifurcation analysis to characterise the nonlinearity (e.g. Dijkstra (2013)).

Once past a large-scale tipping point, the Earth System might have radically
different responses and feedbacks, and so behave very differently. This is
likely to impact on the EC approach is two ways. First, for each individual
ESM, for sufficiently changed climate, the contemporary fluctuating quantity
may no longer give accurate information in that model on future store size.
Second, if there are system tipping points, but an EC is only validated for
moderate changes to the climate system (so, for instance, the $Y$-axis
quantity is only derived from ESMs operated under lower emission scenarios),
then using it to project to more severe climate alteration might be
inappropriate.

A more abstract but relevant question is whether information from pre-tipping
point conditions has any validity at constraining climatic conditions beyond
one? Do tipping points invalidate ‘information exchange’ between different GHG
levels when passing through one? Two ESMs may have very similar responses up
to a tipping point, and thereby appear as points near each other on a standard
EC $X-Y$ plot, but marked different responses beyond a tipping point, thus
weakening an EC-based inter-ESM regression. Also in the abstract is a view
that ECs are capitalising on the ‘hard-wiring’ implicit in ESMs, as based on
model parameterizations constrained from contemporary measurement datasets.
Beyond any major tipping point represents a world that has not yet been
witnessed and so not measured for equation parameterization purposes.

We hope that climate researchers can be encouraged to investigate further
whether the planet passing through a tipping point invalidates ECs, and
possibly also the period just beforehand. This could be in the form of a
conceptual model, or by analysis of ESMs in the CMIP5 ensemble. ESM
simulations with a tipping point could be split in to a future period pre-
tipping point, and one post-tipping point for a high emissions scenario,
testing the EC on both modelled time frames. One obstacle to undertaking such
analysis is that although ESMs do exhibit tipping points, they vary markedly
between models regarding their location, component of the Earth system
affected, and level of global warming Drijfhout  _et al._ (2015). Notable is
that the strong regional features of tipping points implies that any ECs that
may be affected by their presence also have to be localised in application.
Possibly of more concern is that if tipping points affect the predictive
capability of ECs, there will be little inter-ESM consensus on when and how
this might occur. This lack of understanding is due to major model differences
in predictions of tipping point occurrence, or even existence Drijfhout  _et
al._ (2015), and so an ‘EC-type’ modulation factor to the original EC and
common across models is unlikely to be discovered.

###  IV.5 Problems with common code across many models and implications for
‘out-of-sample’ testing

ECs rely on the statistics of regressions, which in turn assume an
independence of data points. However, individual ESMs may not be completely
independent. For parts of the climate system known to respond to well-
established and well-understood physical processes, commonality in models is
to be expected. The concern is for the model parts that are suspected as
characterising actual processes less well, and where ECs seek to constrain
this uncertainty. The lack of independence may take the form of individual
research centres offering multiple model versions but at different
resolutions. Alternatively, some components of ESMs are shared between
research centres, or are coded in similar ways Knutti  _et al._ (2013). The
lack of independence could appear in the range of effective bulk parameters
sampled, such as equilibrium climate sensitivity. Another possibility is there
may be a common component existing between models, and that an EC seeks to
reveal, but the ESM differences to reveal this are in the fluctuating
forcings. These fluctuating quantities could have relatively low sampling due
to similarities between how ESMs calculate their values.

As part of developing comprehensive process understanding of discovered ECs,
Hall  _et al._ (2019) encourage ‘out-of-sample’ testing. Such testing is where
the EC relationship is checked to be valid in additional ESMs that are not
part of the original set used to initially find the EC. New simulations from
individual modelling centres can be analysed as they become available. A more
comprehensive test is to search within new ensembles, and so for instance an
EC found in the CMIP5 climate model ensemble can be checked for its presence
in the newly-released CMIP6 set of ESM simulations. However this too may not
be a completely independent assessment, if modelling centres retain
substantial amounts of model code and parameterizations between ensemble
contributions. We note that the out-of-sampling testing performed by Schlund
_et al._ (2020), that assesses if the ECs for climate sensitivity, ECS, found
in the CMIP5 ensemble remain valid for the CMIP6 models. As commented
elsewhere, they find that for CMIP6, the CMIP5-based ECs have less predictive
capability, and also give generally higher ECS values. Schlund  _et al._
(2020) note that the majority of emergent constraints for ECS are related to
some extent to cloud feedbacks, and this is a major on-going area of climate
research to create numerical cloud schemes with strong predictive skill. As
many cloud schemes are currently under development, then their newness implies
that this is an example of little carrying of common code from CMIP5 to CMIP6.
From that perspective, CMIP6 models have independent features from CMIP5,
validating using this more recent ensemble for out-of-sample testing. However,
if it is the new cloud schemes that cause the CMIP6-derived ECs to have less
predictive capability, then this suggests that previously estimated EC-based
uncertainties on ECS from the CMIP5 models are overly narrow. This form of
error differs from that of Section IV.3 that considers how a missing process
in all models may create a systematic bias in an EC projection. Here, instead,
the suspected missing processes associated with cloud dynamics and feedbacks
are such that there likely remains substantial uncertainty in how to model
them accurately. Alternative cloud schemes may weaken ECs by causing a larger
spread around their regression lines, expanding the bounds on predictions of
quantity $Y$ for quantity $X$ and for which contemporary data also exists.

###  IV.6 What to do when different ECs are found for the same quantity, but
differ in value, or differ between ensembles?

In some instances, multiple and different contemporary measurements have been
suggested, via ECs, to be able to predict the same quantity. This is
particularly the situation where a broad range of climate attributes have been
used to evaluate equilibrium climate sensitivity (ECS). However, and notably,
cloud feedback Sherwood  _et al._ (2014); Brient and Schneider (2016); Brient
_et al._ (2016); Zhai  _et al._ (2015) or top-of-atmosphere (TOA) radiation
flux Brown and Caldeira (2017) based emergent constraints tend to project
higher ECS values than surface temperature based constraints Cox  _et al._
(2018a). Investigations need to continue to ascertain why these ECs disagree.
Some disagreements may be due to spurious correlations, affecting Y values.
However, Caldwell  _et al._ (2018) show that these emergent relationships are
highly correlated. With high correlation, but different ECS estimates, this
implies that differences may instead be due to measurement biases in some X
value observations. Alternatively, the real world may not be sharing the same
responses as the models, which could suggest a persistent error across ESMs
for at least one of the ECs.

As ECs come under increasing scrutiny due to their growing widespread use,
more needs to be understood about how they operate and any limitations. In
that context, we suggest understanding the differences between projections of
the same quantity will be a highly rewarding research path. However in order
to generate a single EC-based projection, Bretherton and Caldwell (2020)
present statistical methods to merge multiple ECs of the same quantity in to a
single range of uncertainty for the quantity being estimated. That analysis
focuses on combining the multiple ECs used by other researchers to estimate
ECS.

As noted in Section IV.5, new ensembles provide an ‘out-of-sample’ test for
existing ECs. The question then is how to use that new value, and especially
if it is substantially different to the value for an earlier ensemble. One
possibility is to use the methods of Bretherton and Caldwell (2020) to merge
the findings from two ensembles. A further possibility is to introduce a
weighting to the Bretherton and Caldwell (2020) approach, but with a high
influence for the more recent ensemble, corresponding to the hope that newer
ESMs are better models.

###  IV.7 ECs may cause future CMIP-type climate model ensembles to have much
less spread in projections

In the most general terms, a reduction of spread of climate models is to be
highly welcomed, and especially if their convergence is on to projections that
are accurate estimates of future change. Such convergence is good if what is
learnt from ECs applied to previous ensembles is accurate, and this has caused
climate modellers to make their new simulations achieve the constrained $Y$
values (see section VII.2). Convergence may also occur simply because
previously uncertain parts of the Earth system have become better understood.
However, the rapid development of ECs could cause a reduction in model spread
that is potentially unwelcome, precluding further progress with the
methodology.

As ECs are discovered, and estimates of future change are refined, this may
cause climate modelling groups to, either consciously or unconsciously,
parameterize new ESMs so as to estimate changes that fall within those EC-
based bounds. This convergence of projections could be because many modelling
groups might not wish to estimate climatological changes that fall outside a
consensus view based on previous EC-based estimates. However, this could have
two detrimental side effects, and potentially cause future EC calculations to
be less accurate. First, re-noting that it is almost a paradox of ECs, as
designed to constrain estimates of change, that they can only work well with
large inter-ESM differences. It is a substantial spread of ESM estimates that
enable ‘strong’ regressions. Hence any clustering of projections will reduce
the capability to re-test ECs for new ensembles of ESMs, and certainly make
them less reliable. Second, it risks that if new processes are believed to be
needed in the majority of new models, then other balancing process could be
tuned, incorrectly, to balance these changes an in order to fit the earlier
EC. For example, introducing the terrestrial nitrogen cycle more routinely
into ESMs may suppress projections of future land carbon stores. But to remain
in alignment with earlier EC estimates and as using ESMs mainly without
nitrogen suppression, then instead in new ESMs this could encourage
parameterization of an offsetting and incorrect overly strong CO2
fertilisation effect for vegetation.

Despite the aim of ECs being to reduce the effects of ESM uncertainty, we
suggest that climate science remains alert to these two concerns, where the
existence of an EC may overly reduce the model spread in any new ensemble
(although this is not something that we have encountered to date). Such a
reduction in a new ensemble, where model developers may wish to replicate the
earlier EC, risks, in particular, the propagation of errors that might be
prevalent in the earlier ensemble.

###  IV.8 Inability to verify an EC

In Section IV.1 we heed the dangers of ECs based on data mining, noting that
Hall  _et al._ (2019) encourage that ECs discovered statistically should first
move to having potential, i.e. contain an element of intuition as their
confirmation. Detailed process understanding, maybe even as far as analytical
assessment of related key climate-based differential equations, ideally then
move an EC from having ‘potential’ to being ‘verified’. At the same time, we
suggest a potential danger whereby it is not possible to follow this cascade
of increased certainly in validity of an EC. For example, there may be
insufficient saved diagnostics from ESMs to build and study offline the
process interactions that need understanding to achieve EC verification. The
risk here is that important and informative potential ECs are dismissed as
their understanding cannot be developed. Although this may initially appear
opposite to the suggestions in Section IV.1, we advise such ECs are not cast
aside, thereby with the danger of losing valuable insights into the climate
system. Instead such potential ECs from data mining can generate requests to
climate research centres to provide related additional ESM outputs, to allow
additional testing.

Any additional ESM diagnostics may also reveal the extent to which a potential
EC is dependent on one major physical process, or is caused by an amalgamation
of multiple interacting effects. If more than one major process is present,
then this may make it more difficult to transition from ‘plausible’ to
‘verified’.

###  IV.9 Lack of perturbed physics experiments with ESMs

The operation of ESMs is hugely computationally expensive and often there is
only capacity in research centres to perform simulations for standardised
scenarios of changes in future atmospheric greenhouse gas concentrations (e.g.
the Representative Concentration Pathways, RCPs; Meinshausen  _et al._
(2011)). Unfortunately, this precludes performing simulations with varied
parameters, and especially for those where there is substantial uncertainty
surrounding their true value (or effective value, if grid-dependent). We
suggest that this lack of model investigation restricts further testing of
ECs, and in a way that may either preclude gaining extra confidence in their
reliability or alternatively fails to alert to one that may not be as robust
as initially believed. We suggest that PPEs may require interpretation in two
ways, and that are converse to each other. If a parameter value is changed,
and if it relates to a part of the Earth system believed to be independent of
the underlying processes associated with a particular emergent constraint,
then for that ESM the $Y$ value should not change in the EC. Such invariance
both confirms EC robustness and helps illustrates that the EC aligns to the
parts of the model expected from intuition or process understanding. The
alternative situation is where a parameter believed to relate to an EC is
perturbed, and potentially adjusted to the outer bounds of its expected value
or beyond. The expectation then may be that for that particular ESM, the point
moves to the bounds of the EC regression. Such an anomaly confirms that the EC
is affected by the part of the climate system the parameter pertains to, and
may indicate that outlier parameter values are not present in the other ESMs.

If spare computational cycles become available, we suggest perturbed physics
experiments with ESMs will be beneficial to the development and testing of
ECs.

##  V Emergent constraints found in the Earth system

Numerous ECs have been identified across physical and biogeochemical
components of the Earth System (a handful of which have been discussed thus
far), with a substantial increase in their number over the past decade. In
order to capture the breadth of its application in the geosciences, we discuss
previously documented ECs based on their broad geophysical classification
here. A list of EC studies is provided in Table LABEL:table:ECs. Note that
this list is not exhaustive and may be easily superseded. Moreover, some of
the references in Table LABEL:table:ECs offer more than one potential current
climate quantity to explain $Y$, in which case we list only one example. For
example, Sherwood  _et al._ (2014) provide three metrics that describe various
aspects of lower-tropospheric mixing to explain equilibrium climate
sensitivity. Rather than list all three metrics, we only discuss the one that
appears most robust Caldwell  _et al._ (2018); Schlund  _et al._ (2020). An
important criterion for establishing credibility of an EC is to show
robustness to the choice of ensemble. Therefore, Table LABEL:table:ECs also
documents the model ensemble(s) for which each EC appears to have value (i.e.,
a statistically significant emergent relationship is present). In several
cases this is only the ensemble from which the EC was derived as many of the
more recent examples have not undergone out-of-sample verification at the time
of this paper. We expect several upcoming publications to assess the validity
of previously published ECs with the newer CMIP6 ensemble (e.g. Schlund  _et
al._ (2020); Pendergrass (2020)).

A vast majority of ECs pertain to one of the following general topics: climate
sensitivity, cloud feedbacks, cryospheric feedbacks and change, carbon cycle
feedbacks, and the hydrologic cycle, but in theory the methodology can be used
for any number of applications. There are also additional examples relating to
topics such as radiative forcing Bowman  _et al._ (2013), regional air and sea
surface temperature change Lin  _et al._ (2017); Selten  _et al._ (2020);
Sgubin  _et al._ (2017), frequency of temperature extremes Donat  _et al._
(2018), and atmospheric circulation changes Kidston and Gerber (2010); Simpson
and Polvani (2016). For the purpose of this review, we will focus primarily on
the main applications to date. Along with the differing applications, ECs can
also be sorted by the type of constraint (Figure 5). One way to think of this
is to group ECs by the time-scale of information that defines their $X$ (e.g.,
from multi-decadal to extreme events). For example, a rather simple class of
ECs relates climatological biases or multi-decadal trends in some quantity
($X$) to the future change in $Y$. Similarly, a collection of ECs use seasonal
or interannual variations in $X$ to constrain future changes in $Y$. In many
of these instances, variability in some quantity to seasonal or interannual
temperature variability is related to sensitivities of that same quantity to
future warming. Constraints using climatological biases or short-term
sensitivities are also common for attempting to constrain invariant properties
such as equilibrium climate sensitivity. Applications to future changes in
higher-order climate statistics (e.g., extreme events) are more rare. This
highlights a relatively unexplored area of EC research.

Figure 5: Schematic showing the different types of emergent constraints
proposed in the literature to date. Each number corresponds to a study listed
in Table 1. ECs are sorted by the timescale of their current climate quantity
and the type of future constrained quantity. ECs are also sorted by whether
the future quantity relates to an impact or a physical or geochemical
quantity.

###  V.1 Equilibrium Climate Sensitivity

By far the most popular application of the EC approach has been in relation to
equilibrium climate sensitivity (ECS) (Table LABEL:table:ECs). ECS is
seemingly an ideal candidate for this technique because of its importance to
predicting future warming rates and its consistently large spread across
various GCMs, which has been a persistent feature of all model generations.
(The likely range of ECS according to the IPCC (1.5 to 4.5K) has been largely
unchanged for decades). A series of contemporary quantities ($X$) have been
identified as being closely linked to ECS, many of which relate to simulated
cloud feedbacks or properties. In general, these cloud-based constraints tend
to suggest relatively high ECS values (3.5-4.1K) Knutti  _et al._ (2017);
Brient (2020). Relevant examples include the height of tropical low clouds
Brient  _et al._ (2016), the sensitivity of the reflection by subtropical low
clouds to sea-surface temperature Brient and Schneider (2016), seasonal
sensitivity of low cloud between 20-40°N and 20-40°S to sea-surface
temperature Zhai  _et al._ (2015), variability in relative humidity and cloud
extent Fasullo and Trenberth (2012), vertically resolved zonally-average
relative humidity and clouds between 45S-40N Su  _et al._ (2014), and the
climatological difference between tropical and Southern Hemisphere mid-
latitude cloud fraction Volodin (2008).

Another suite of studies relate ECS to intermodel differences in the
characteristics of historical radiative fluxes, such as climatological TOA
energy balance in the Southern Hemisphere Trenberth and Fasullo (2010),
features of TOA radiation fluxes (Huber  _et al._ , 2011; Tett  _et al._ ,
2013; Brown and Caldeira, 2017), and cloud-sky radiative flux sensitivity to
temperature Lutsko and Takahashi (2018). Alternatively, studies have also used
temperature characteristics defined over a variety of time-scales to constrain
ECS. This approach tends to suggest relatively weaker ECS values than the
others Knutti  _et al._ (2017). Examples of $X$ include the seasonal cycle of
temperature Knutti  _et al._ (2006), 20th century warming Annan and Hargreaves
(2006), and statistics of interannual variability Cox  _et al._ (2018a).
Similar to this, there is a collection of studies that have evaluated
temperature changes over much longer timescales (e.g., global-mean cooling
during the Last Glacial Maximum: 19-23 ka before present) as derived from
paleoclimate data Hargreaves  _et al._ (2012); Schmidt  _et al._ (2014). These
paleoclimate-based constraints suggest ECS of 2.3 and 3.1 K, respectively.
However, there is a large amount of uncertainty surrounding the model
simulations of the LGM, the proxies used to constrain the intermodel
relationship, and the validity of using data from these time periods as
proxies for anthropogenic warming Harrison  _et al._ (2015); Brient (2020). A
variety of other metrics have also been proposed as being strongly related to
ECS with large variability in their constrained predictands. The highest value
(4.5K) is derived from a constraint using vertical mixing between the boundary
layer and lower troposphere over tropical oceans Sherwood  _et al._ (2014).
Tian (2015) use climatological precipitation in the Intertropical Convergence
Zone (ITCZ) region to suggest a slightly weaker ECS (4.1K), while the weakest
non-paleo estimate uses the climatological latitude of SH Hadley Cell edge in
Dec-Feb to suggest an ECS of only 2.5K Lipat  _et al._ (2017).

As discussed in Section IV.6, having such a large number of ECs for the same
quantity can complicate their interpretation when the constraints suggest
differing $Y$ values. In this instance, rigorous testing of various proposed
ECs is necessary to better understanding their validity. Along these lines,
ECs on ECS have been heavily scrutinized over the past few years with
assessments tied to their plausibility and robustness across both multimodel
ensembles Grise  _et al._ (2015); Caldwell  _et al._ (2018); Schlund  _et al._
(2020) and, to a lesser extent, perturbed physics ensembles Kamae  _et al._
(2016); Zhao  _et al._ (2016); Wagman and Jackson (2018). Caldwell  _et al._
(2018) provide the most robust assessment to date on the credibility of
various constraints on ECS. In doing so, they find that several of the
aforementioned constraints lack a physically plausible mechanism connecting
the $X$ quantity with ECS (Volodin, 2008; Su  _et al._ , 2014; Tian, 2015;
Siler  _et al._ , 2018). This limits how much faith can be put into the EC
itself as the statistical relationship could be fortuitous. Furthermore, many
of the constraints are found to be closely related Caldwell  _et al._ (2018).
For example, one strongly correlated group relates aspects of present-day
Southern Hemisphere cloud cover with ECS, suggesting they may all be capturing
a single SH mechanism (Lipat  _et al._ , 2017; Trenberth and Fasullo, 2010;
Volodin, 2008; Siler  _et al._ , 2018). Additional testing of EC robustness
comes in the form of out-of-sample verification (i.e., evaluating the emergent
relationship in a different model ensemble to the one from which the EC was
originally derived). Although there are some known limitations to this
approach related to the existence of common model code across generations
(Section IV.5), it is still seen as a valuable exercise for evaluating ECs.
For example, Grise  _et al._ (2015) show that the Trenberth and Fasullo (2010)
EC derived from CMIP3 is not valuable in CMIP5 because the emergent
relationship only exists in a subset of models with unrealistic cloud
properties in the Southern Hemisphere subtropics. Caldwell  _et al._ (2018)
extend this type of analysis to show that Fasullo and Trenberth (2012) is also
non-existent in CMIP5. On the other hand, they find the Volodin (2008) EC to
be present in both CMIP3 and CMIP5, despite lacking a physically plausible
explanation. Schlund  _et al._ (2020) recently assessed a number of the
constraints on ECS documented here using the CMIP6 ensemble. They find that
most of these constraints appear less skilful at predicting ECS in CMIP6,
largely tied to differing representation of cloud processes. Three of the ECs
discussed here Brient and Schneider (2016); Sherwood  _et al._ (2014); Volodin
(2008) still exhibit statistically significant skill, while two others no
longer have value Lipat  _et al._ (2017); Su  _et al._ (2014). Lastly, two ECs
are somewhat in between, what the authors define as “indeterminate”, including
Cox  _et al._ (2018a); Tian (2015) Schlund  _et al._ (2020). Further testing
of these ECs has utilised perturbed physics ensembles to better sample the
parametric uncertainty space associated with various emergent relationships
(section II.1). Of particular interest to these studies is the EC proposed by
Sherwood  _et al._ (2014). Results from three different PPEs show that the
physical mechanism proposed by this study may not be valid, while the emergent
relationship between lower tropospheric mixing and ECS only exists when
certain convection schemes are used Kamae  _et al._ (2016); Zhao  _et al._
(2016); Wagman and Jackson (2018). Similar doubts have also been reported for
the Fasullo and Trenberth (2012) EC Wagman and Jackson (2018).

This collection of analyses implies that the most robust constraints on ECS
(determined by successful out-of-sample testing and plausible physical
mechanisms) are tied to the present-day response of subtropical low clouds to
SST variability Brient and Schneider (2016); Zhai  _et al._ (2015). However,
one factor that may limit confidence in these constraints is the short
observational record of low cloud characteristics Brient (2020). ECs from
Sherwood  _et al._ (2014); Volodin (2008) are also statistically significant
in three ensembles (Table LABEL:table:ECs), but they face credibility issues
tied to their physical mechanism (or lack thereof) Caldwell  _et al._ (2018);
Wagman and Jackson (2018). Despite these signs of robustness, the much weaker
correlation values associated with these ECs in CMIP6 Schlund  _et al._ (2020)
supports the belief that it is unlikely that a single predictor will be able
to physically explain a large amount of ECS variability Caldwell  _et al._
(2018); Hall  _et al._ (2019). Rather, several constraints on various feedback
components will likely be needed to make a difference for such complex
processes. In this fashion, a recent comprehensive assessment of ECS gave
little weight to ECs on ECS when attempting to determine its likely value
instead favouring observations of actual temperature change and climate
feedbacks Sherwood  _et al._ (2020). This is also believed to be true for the
global cloud feedback (described in more detail below) Klein and Hall (2015)
given that the radiative effect of clouds in response to warming is expected
to vary by cloud type and regime Gettelman and Sherwood (2016).

Similar research efforts have recently suggested that historical decadal
warming trends are strongly predictive of the transient climate response (TCR;
defined as the amount of warming that occurs at the time of atmospheric carbon
dioxide doubling, having increased by 1% each year) Jiménez-de-la Cuesta and
Mauritsen (2019); Tokarska  _et al._ (2020); Nijsse  _et al._ (2020). TCR is
of interest because it more closely resembles the way carbon dioxide
concentrations have changed in the past. These constraints suggest that the
best estimate of TCR is 1.67K in CMIP5 Jiménez-de-la Cuesta and Mauritsen
(2019) and between 1.60K and 1.68K in CMIP6 Tokarska  _et al._ (2020); Nijsse
_et al._ (2020). The difference between the latter two studies largely stems
from the choice of historical time period evaluated Nijsse  _et al._ (2020).
All three of these estimates are lower than the raw ensemble statistics
(median of 1.95K in CMIP6). Similar constraints on future warming have also
been developed for regional temperature changes. Both Lin  _et al._ (2017) and
Selten  _et al._ (2020) find that present-day summer temperatures across
central USA and Europe, respectively, are strongly tied to future continental
summer warming. The authors hypothesise that the same physical mechanisms
operating in the current climate (tied to precipitation biases and soil
hydrology) also govern the future climate response. Incorporating observations
allows for the future spread in European summer warming to be reduced by
nearly 50% Selten  _et al._ (2020).

###  V.2 Cloud Feedbacks

One of the primary contributors to intermodel spread in climate sensitivity is
the cloud feedback, which encompasses changes in how clouds modulate the
radiation budget in response to warming. (For a recent synthesis of cloud
feedbacks see Section 3.3 of Sherwood  _et al._ (2020)). The global mean cloud
feedback is the most uncertain component of the total climate feedback Bony
_et al._ (2006); Ceppi  _et al._ (2017); Sherwood  _et al._ (2020), varying
from -0.36 to 1.17 W m-2 K-1 in the latest generation of climate models
Zelinka  _et al._ (2020). Strengthening of ensemble mean cloud feedback
strength in CMIP6 is also thought to be largely responsible for a recent ECS
increase in many models Zelinka  _et al._ (2020). Thus, many of the
aforementioned constraints for ECS include cloud effects either directly or
indirectly Qu  _et al._ (2018); Brient and Schneider (2016); Brient  _et al._
(2016); Fasullo and Trenberth (2012); Su  _et al._ (2014); Volodin (2008);
Zhai  _et al._ (2015). This also means that there is great interest in
constraining cloud feedbacks as well Brient and Bony (2013); Qu  _et al._
(2014); Gordon and Klein (2014); Zhou  _et al._ (2015); Siler  _et al._
(2018). For example, Siler  _et al._ (2018) propose a constraint on future
global mean cloud feedback using the climatological latitudinal gradient in
cloud reflectivity (a quantity that can be derived from satellite
observations). In general, this EC suggests that ESMs with lower cloud albedo
in warm-SST regions and higher cloud albedo in cool-SST regions will exhibit
greater cloud feedback (and to a lesser extent ECS). It suggests a global
cloud feedback in the upper range of ESM estimates (0.58 $\scriptstyle\pm$0.31
W m-2 K-1). However, this constraint lacks a testable physical explanation
Caldwell  _et al._ (2018). Another study that seeks to constrain the global
mean cloud feedback is Zhou  _et al._ (2015). This paper highlights strong
similarities between the global mean cloud feedback and a present-day analogue
cloud feedback derived from interannual climate variability Zhou  _et al._
(2015). This follows the assumption of several ECs that processes operating on
seasonal to interannual timescales are also likely to be occurring on longer
timescales. However, the usefulness of this EC is limited because the
observational uncertainty in the present-day quantity is large relative to
inter-model spread.

As previously mentioned, it is seen as unlikely that a single current climate
quantity can account fully explain variability in the global cloud feedback
Klein and Hall (2015). Thus, a number of studies have focused on isolating the
individual components of the cloud feedback (high-cloud altitude, tropical
marine low-cloud, tropical anvil cloud-area, land cloud amount, mid-latitude
marine low cloud amount, and high-latitude low-cloud optical depth). For
example, Po-Chedley  _et al._ (2019) narrow in on the changes to upper
troposphere clouds and relative humidity across the tropics. They show that
future changes in upper troposphere cloud fraction under climate warming are
strongly tied to the vertical gradient in climatological mean cloud fraction
and relative humidity. Elsewhere, much of this research concerns tropical low
clouds as ESMs struggle to represent them, they account for nearly half of the
variance in the global mean cloud feedback (Zelinka  _et al._ , 2016; Klein
_et al._ , 2017). Moreover, intermodel spread in the low cloud feedback is
closely tied to variability in ECS Sherwood  _et al._ (2020). An early EC
along these lines showed that the future low cloud feedback sign ($Y$) is
related to current low cloud sensitivity as a result of natural variability
($X$) in CMIP3 Clement  _et al._ (2009). Unlike most other ECs, this example
simply suggests the sign of a feedback rather than specific values. Thus, it
is not always mentioned in reviews of this nature Klein and Hall (2015);
Brient (2020). More traditional ECs relating to this topic include the link
between low cloud optical depth changes with warming and the optical depth
response to temperature anomalies associated with natural variability Gordon
and Klein (2014). This feedback is of limited importance globally, but makes
an important contribution at high latitudes Klein and Hall (2015). The Gordon
and Klein (2014) EC also has a simple thermodynamic explanation, which makes
it a promising example. Low cloud amount changes under future warming play a
much larger role in the global cloud feedback so this has naturally been an
area of focus Qu  _et al._ (2014, 2015); Brient and Schneider (2016). Qu  _et
al._ (2014) identified an EC on this low cloud response through the low cloud
response to temperature and stability anomalies derived from interannual
variability. Although weaker in CMIP5 than CMIP3 Klein and Hall (2015), this
EC has value in both ensembles and features a strong physical basis explaining
the relationship Qu  _et al._ (2014, 2015). This type of process based
evidence can then be used in combination to assemble a best estimate of the
more complex global mean cloud feedback. Figure 7 of Sherwood  _et al._ (2020)
illustrates this concept to get a global cloud feedback of 0.45
$\scriptstyle\pm$0.33 W m-2 K-1. Going forward, it is imperative that we
continue to improve our process-based understanding of individual cloud
components, where potential physical mechanisms are more likely to be
uncovered.

###  V.3 Carbon Cycle

Elsewhere, the EC technique has been used extensively to constrain elements of
climate-carbon cycle feedbacks (Table I #28-36) (Wenzel  _et al._ , 2014,
2016; Cox  _et al._ , 2013; Hoffman  _et al._ , 2014; Mystakidis  _et al._ ,
2017). Climate-carbon cycle feedbacks are characterised by changes in
terrestrial and ocean carbon storage in response to climate change, which
modify the atmospheric CO2 concentration, thus enhancing warming. Intermodel
differences in feedback strengths lead to highly variable projections of
future atmospheric carbon dioxide (more than 300 ppmv of intermodel spread at
the end of the 21st century for a high-emissions scenario Cox (2019)). A
rather simple relationship was uncovered between the mid-century atmospheric
CO2 concentration in ESMs and the simulated present-day concentration, but it
was shown to weaken as the influence of varying climate-carbon feedbacks
becomes more apparent later in the 21st century Hoffman  _et al._ (2014). This
EC suggests a much narrower likely range for atmospheric CO2 concentrations at
the end of the century (947 $\scriptstyle\pm$35 ppm) than the full ensemble
(980 $\scriptstyle\pm$161 ppm).

Much of the model uncertainty in future atmospheric CO2 has been shown to stem
from differences in the land carbon sink Cox  _et al._ (2013); Wenzel  _et
al._ (2014). Therefore, significant effort has gone into constraining this
feature of the carbon cycle. A prominent example uses the historical
sensitivity of annual atmospheric carbon dioxide growth-rate to temperature
variability (which is strongly impacted by tropical land carbon storage
fluctuations) to constrain the future loss of terrestrial tropical carbon Cox
_et al._ (2013); Wenzel  _et al._ (2014). As with prior examples, the CO2
growth rate sensitivity is an observable quantity, thus allowing for the EC to
suggest future tropical land carbon stability. This EC has been proven robust
across both C4MIP and CMIP5 ensembles Wenzel  _et al._ (2014)555It should be
noted that Wang  _et al._ (2014) also attempt to evaluate this EC in CMIP5,
but their use of RCP8.5 to diagnose the sensitivity of tropical land carbon
loss to tropical warming is not comparable to the work of Cox  _et al._
(2013); Wenzel  _et al._ (2014). Given that there is land-use change in
RCP8.5, the authors conflate changes in tropical land carbon due to climate
change, with changes in land carbon due to deforestation (for which there is
no physical reason to expect a correlation with X)., but has yet to undergo
testing pertaining to CMIP6. This is an important EC because there was a
fourfold simulated spread in tropical land carbon reductions per degree of
warming in C4MIP (29-133 GtC K-1) Jones  _et al._ (2016), with the higher-end
models suggesting potentially catastrophic dieback of the Amazon rainforest
Cox  _et al._ (2000, 2004). However, the Cox  _et al._ (2013) EC suggests
tropical land carbon reductions slightly weaker than the ensemble mean
(53$\scriptstyle\pm$17 GtC K-1).

Another carbon cycle EC uses the change in seasonal atmospheric CO2 amplitude
to constrain CO2 fertilisation of photosynthesis on the extratropics Wenzel
_et al._ (2016). On a more regional basis, Winkler  _et al._ (2019a) highlight
a strong relationship between future increases in terrestrial Arctic gross
primary productivity and historical increases in leaf area index (greening).
These two ECs both suggest that most models are underestimating future changes
in gross primary productivity across the high latitudes Wenzel  _et al._
(2016); Winkler  _et al._ (2019a). While these ECs have yet to undergo out-of-
sample verification, Winkler  _et al._ (2019b) provides detailed evaluation of
the factors contributing to uncertainty in the Winkler  _et al._ (2019a)
constraint. They point to choices relating to the temporal period of the
predictor variable, choice of observational dataset, and the rate of CO2
forcing as particularly large sources of uncertainty Winkler  _et al._
(2019b). Lastly, additional research identified constraints on terrestrial
carbon cycle feedbacks using interannual variability in evapotranspiration,
net biome productivity, and gross primary productivity. This EC suggests a 40%
reduction in the climate-carbon feedback and a 30% reduction in the
concentration-carbon feedback Mystakidis  _et al._ (2017).

When it comes to understanding how the ocean carbon cycle will respond to
climate change, one of the main challenges is to reduce uncertainty in
estimates of tropical ocean primary productivity. Satellite observations of
the interannual variability in ocean productivity resulting from ENSO-driven
SST anomalies have been used to constrain highly uncertain projected changes
in tropical marine primary productivity with warming Kwiatkowski  _et al._
(2017). There is a strong inverse relationship between NPP and SST anomalies
in the observational record, but the number of ENSO events in the satellite
record is limited. This EC suggests a substantial reduction in the long-term
tropical NPP sensitivity (-3.4% K-1 to -2.4 % K-1) from the unconstrained
ensemble average (-4.0 $\scriptstyle\pm$2.2 % K-1). Another recently published
EC by Kessler and Tjiputra (2016) relates future global ocean carbon uptake to
the contemporary carbon uptake by the Southern Ocean. In this example, models
with anomalously low uptake in the current climate project low global uptake
over the course of the 21st century. Uncertainty in simulated ocean uptake
also translates to uncertainty in projections of future ocean acidification,
the impacts of which are expected to be greatest in the Arctic ecosystem
Terhaar  _et al._ (2020). Recent work uses CMIP5 models to identify an EC on
future Arctic ocean acidification using the simulated density of Arctic ocean
surface waters in the current climate Terhaar  _et al._ (2020). Observations
of sea surface density when coupled with this emergent relationship imply that
future Arctic Ocean acidification will be greater than previously expected.

###  V.4 High-latitude Processes

The Earthﬂs high-latitude regions are rapidly warming and there is a great
deal of uncertainty in how various components of the cryosphere (Earthﬂs
frozen surfaces) will evolve in the future Mudryk  _et al._ (2020); Notz  _et
al._ (2020); Meredith  _et al._ (2019). Thus, ECs are an intriguing option for
reducing intermodel spread here. One of the earliest documented examples of an
emergent constraint is for the snow albedo feedback Hall and Qu (2006), which
was introduced in Section I. This is a leading example of the EC technique
because of the simplicity behind its physical mechanism and its robustness
across several generations of models (Figure 2). An interesting point on the
SAF EC is that despite substantial time since its initial publication, we have
yet to see a meaningful reduction in its current climate quantity ($X$).

Following on from its application to SAF, recent work has identified that the
emergent constraint approach can also be applied to a similar process over
Arctic sea ice Thackeray and Hall (2019). The sea ice albedo feedback (SIAF)
is characterized by the enhancement of future warming through the reduction of
surface albedo as a result of decreasing sea ice Holland and Bitz (2003); Hall
(2004). An analogue of this process operates in the seasonal cycle during the
transition from maximum to minimum ice extent ($X$). These two processes are
found to be closely linked across the CMIP5 ensemble, forming the basis of an
EC. However, given the projected rapid loss of sea ice in the 21st century,
this constraint exhibits a regime-dependence whereby the relationship between
seasonal cycle and climate change SIAF begins to break down in the latter half
of the 21st century, when most models exhibit an ice-free Arctic Thackeray and
Hall (2019). This is similar to the concept of tipping points discussed in
Section IV.4.

In addition to the aforementioned radiative feedbacks, there is a large amount
of uncertainty when it comes to the fate of a number of high-latitude Earth
system components, notably sea ice and permafrost Stroeve  _et al._ (2012);
Slater and Lawrence (2013). Given this uncertainty and the potential
widespread implications associated with these changes, ECs can provide
valuable information here. Notably, a number of studies have proposed
constraints on projections of future Arctic sea ice (Boe  _et al._ , 2009;
Massonnet  _et al._ , 2012; Liu  _et al._ , 2013). Boe  _et al._ (2009) used
the observed historical trend in September Arctic sea ice extent over the
satellite era to constrain the time in the 21st century when the Arctic is
likely to become ice-free during summer. Similar research by Massonnet  _et
al._ (2012) used a series of historical ice characteristics and trends in an
attempt to constrain the time period when the Arctic is likely to become
seasonally ice-free. Their results suggest ice-free conditions for September
to begin somewhere between 2041-2068 under a high-emissions scenario, a
significant reduction from the full CMIP5 spread that spans nearly 100 years.
The presence of constraints between historical ice properties and the future
change in both CMIP3 and CMIP5 is a promising sign, but we are not aware of
CMIP6 testing for this EC thus far. Similar historical sea-ice metrics have
also been used to inform future changes in high-latitude temperature
variability Borodina  _et al._ (2017a). Permafrost underlies a significant
portion of the Northern Hemisphere high-latitudes and is highly sensitive to
warming Lawrence  _et al._ (2012), with its degradation expected to have
detrimental climate and developmental effects Teufel and Sushama (2019).
However, modern ESMs struggle to agree upon future changes in permafrost area
Slater and Lawrence (2013). Chadburn  _et al._ (2017) use a relationship
between mean annual air temperature and permafrost area during the historical
period ($X$) to constrain projections of future permafrost thaw ($Y$). This
constraint suggests that roughly 20% more permafrost will be lost per degree
of warming than previously expected Chadburn  _et al._ (2017).

###  V.5 Hydrologic Cycle

Future changes to precipitation are of great interest to the climate science
community because of wide-ranging impacts on natural and human systems.
However, precipitation is highly variable in space and time, thus making it
difficult to both observe and predict. These factors contribute to large
intermodel differences in future projections of precipitation. Reducing this
uncertainty is therefore vital for those making policy and infrastructure
decisions based on model projections. ECs have been applied to multiple
aspects of the hydrologic cycle across various scales (DeAngelis  _et al._ ,
2015; O’Gorman, 2012; Li  _et al._ , 2017; Rowell, 2019; Watanabe  _et al._ ,
2018). First, we consider the globally-averaged changes to the hydrologic
cycle. In response to rising temperatures and increasing atmospheric water
vapour, global-mean precipitation is projected to increase. This is because
the atmosphere radiatively cools to space and this radiative cooling must be
balanced by latent heat release from precipitation, thus setting the
radiative-convective balance Pendergrass and Hartmann (2014); DeAngelis  _et
al._ (2015). Under a high-emission scenario, models project a future increase
of between 2 to 10 % in global-mean precipitation Kharin  _et al._ (2013).
This spread can be partially explained by differing rates of future warming
Fläschner  _et al._ (2016), although a threefold spread still exists when the
precipitation change is normalised by warming Kharin  _et al._ (2013). In an
attempt to minimise this spread, DeAngelis  _et al._ (2015) used the
sensitivity of clear-sky shortwave radiation absorption to changes in column
water vapour to constrain the global mean precipitation increase with warming
(termed hydrologic sensitivity). This constraint suggests a 35 % reduction in
the ensemble spread and a 40 % reduction in the ensemble mean hydrologic
sensitivity. This work was followed by Watanabe  _et al._ (2018), who use the
climatological global mean surface longwave cloud radiative effect (CRE) to
constrain the future surface longwave CRE feedback. In combination with the
most-likely sensitivity of clear-sky shortwave radiative absorption to water
vapour DeAngelis  _et al._ (2015), this EC suggests a constrained estimate of
1.8 % K-1 for global hydrologic sensitivity, down from an ensemble mean of 2.6
% K-1 Watanabe  _et al._ (2018). However, a recently published assessment of
these constraints in CMIP6 reveals that although the DeAngelis  _et al._
(2015) EC is still present, its strength is weakened Pendergrass (2020).
Moreover, the Watanabe  _et al._ (2018) EC on HS is found to be non-existent
in the new ensemble. This highlights the importance of applying out-of-sample
testing to the remainder of the ECs on hydrologic cycle components discussed
below.

Another key component of the global hydrologic cycle is terrestrial
evapotranspiration (ET), which is made up of evaporation and biological
transpiration (T). The ratio of T to ET is used to estimate land water fluxes,
while because similar processes influence T and land-atmosphere carbon
exchanges, better estimates of T/ET help to reduce uncertainty in global
carbon cycle projections Lian  _et al._ (2018). Climate models exhibit large
variability in the global strength of T/ET, spanning a factor of 3. Lian  _et
al._ (2018) found a strong relationship ($r^{2}=0.93$) between simulated local
T/ET (averaged over grid-cells corresponding to in situ measurements) and
global mean T/ET. This EC reveals that the constrained global T/ET ratio is
significantly higher than the CMIP5 ensemble mean.

Alternatively, several studies have attempted to use this technique to
constrain the change in aspects of the hydrologic cycle over specific regions.
Regional constraints have the potential for more policy and impacts relevance
than globally averaged metrics. As a first attempt, O’Gorman (2012) proposed
an EC on the change in future daily tropical precipitation extremes using the
observed sensitivity of tropical precipitation extremes to temperature
variability. They found a strong relationship between the sensitivity of very
heavy precipitation (99.9th percentile) to both seasonal and future
temperature changes. The constrained estimate based on observations (6-14%
K-1) was substantially narrower than the intermodel spread across the CMIP3
ensemble (2-23% K-1). Similarly, Borodina  _et al._ (2017b) found that the
scaling of annual maximum one day precipitation (Rx1day) with warming global
land temperatures can be used to constrain the future intensification of heavy
rainfall across extratropical regions with sufficient data records. This EC
suggests that the CMIP5 models are likely underestimating the future change in
extreme rainfall where climatological rainfall intensity is high.

Another example links biases in simulated historical climate and future Indian
Summer Monsoon (ISM) rainfall Li  _et al._ (2017). In this case, the authors
find that models overestimating historical precipitation across the tropical
western Pacific also exhibit larger increases in Monsoon rains in the future
Li  _et al._ (2017). Using observations of western Pacific rainfall reduces
the projected increase in ISM rainfall by nearly 50%. CMIP5 models also differ
greatly in their projections of future total East African Long Rains (March-
April-May mean spanning a region from northwest Tanzania to southwest
Ethiopia) – from a 20% decrease to a 120% increase Rowell and Chadwick (2018).
Rowell (2019) use an EC between present day interannual SST-low level cloud
sensitivity and future SST change over the Indian Ocean to assess the
credibility of projections of East African precipitation. They find that one
outlier model, which projects a doubling of seasonal precipitation is likely
unreliable because of unrealistic SST/low cloud processes. This reduction in
intermodel spread is another example of the value that an EC approach can
provide. Along these lines, Lehner  _et al._ (2019) recently suggested that
the sensitivity of historical runoff to temperature and precipitation change
across three watersheds in the Western US is closely tied to projections of
future runoff across a series of ESMs. It is expected that regional
applications of ECs to highly uncertain quantities like precipitation change
will be a key area of future research.

Table 1: Collection of existing ECs. Note that some of these ECs involve correlations that are lower than those portrayed in Figure 1, with correspondingly less potential for uncertainty reduction. The ensemble(s) for which the EC appears to have value is also listed (note that many have only been tested on the ensemble that they were developed on). |  Future constrained quantity (Y) |  Current climate quantity (X) |  Ensemble |  Reference
---|---|---|---|---
1 |  Equilibrium Climate Sensitivity |  Seasonal cycle of temperature |  CMIP1 PPE |  Knutti  _et al._ (2006); Covey  _et al._ (2000)
2 |  Equilibrium Climate Sensitivity |  Difference in cloud fraction between the tropics and SH midlatitudes |  CMIP3 CMIP5 CMIP6 |  Volodin (2008)
3 |  Equilibrium Climate Sensitivity |  TOA radiation balance in the SH |  CMIP3 |  Trenberth and Fasullo (2010)
4 |  Equilibrium Climate Sensitivity |  Features of TOA radiation fluxes |  CMIP3 |  Huber  _et al._ (2011); Tett  _et al._ (2013)
5 |  Equilibrium Climate Sensitivity |  Variability in climatological May-August relative humidity and cloud extent |  CMIP3 |  Fasullo and Trenberth (2012)
6 |  Equilibrium Climate Sensitivity |  Last Glacial Maximum Cooling |  CMIP3/ PMIP2 CMIP5/ PMIP3 |  Hargreaves  _et al._ (2012); Schmidt  _et al._ (2014)
7 |  Equilibrium Climate Sensitivity |  Vertical mixing strength between the boundary layer and lower troposphere over tropical oceans |  CMIP3 CMIP5 CMIP6 |  Sherwood  _et al._ (2014)
8 |  Equilibrium Climate Sensitivity |  Vertically resolved relative humidity and clouds between 45°S and 40°N |  CMIP5 |  Su  _et al._ (2014)
9 |  Equilibrium Climate Sensitivity |  Precipitation in the double-ITCZ region |  CMIP3 CMIP5 |  Tian (2015)
10 |  Equilibrium Climate Sensitivity |  Seasonal sensitivity of low cloud to SSTs (20-40°latitude) |  CMIP3 CMIP5 CMIP6 |  Zhai  _et al._ (2015)
11 |  Equilibrium Climate Sensitivity |  Height of tropical low clouds |  CMIP5 |  Brient  _et al._ (2016)
12 |  Equilibrium Climate Sensitivity |  Sensitivity of subtropical low cloud albedo to SSTs |  CMIP5 CMIP6 |  Brient and Schneider (2016)
13 |  Equilibrium Climate Sensitivity |  Climatological latitude of the SH Hadley Cell edge in winter |  CMIP5 |  Lipat  _et al._ (2017)
14 |  Equilibrium Climate Sensitivity |  Statistics of interannual temperature variability |  CMIP5 |  Cox  _et al._ (2018a)
15 |  Equilibrium Climate Sensitivity |  Cloudy-sky radiative flux sensitivity to temperature |  CMIP5 |  Lutsko and Takahashi (2018)
16 |  Future global warming (mid-century or end of century) |  Recent warming trend in CMIP models (1981-2017) |  CMIP6 |  Tokarska  _et al._ (2020)
17 |  Transient climate response and Equilibrium Climate Sensitivity |  Recent warming in CMIP models |  CMIP5 CMIP6 |  Jiménez-de-la Cuesta and Mauritsen (2019); Nijsse  _et al._ (2020)
18 |  Low cloud feedback sign |  Low cloud sensitivity to Pacific variability |  CMIP3 |  Clement  _et al._ (2009)
19 |  Global mean cloud feedback and ECS |  Climatological latitudinal gradient in the reflectivity of clouds |  CMIP5 |  Siler  _et al._ (2018)
20 |  Low cloud optical depth change per degree climate warming |  Low cloud optical depth response to temperature anomalies |  CMIP3 CMIP5 |  Gordon and Klein (2014)
21 |  Subtropical low cloud cover change under climate warming |  Subtropical low cloud cover response to inter-annual temperature and stability anomalies |  CMIP3 CMIP5 |  Qu  _et al._ (2014, 2015)
22 |  Global mean cloud feedback under climate change |  Global mean cloud feedback derived from interannual variability in current climate |  CMIP5 |  Zhou  _et al._ (2015)
23 |  Change in upper troposphere relative humidity and cloud fraction |  Vertical gradient in climatological mean relative humidity and cloud fraction |  CMIP5 |  Po-Chedley  _et al._ (2019)
24 |  Snow albedo feedback |  Seasonal snow albedo feedback (spring) |  CMIP3 CMIP5 CMIP6 |  Hall and Qu (2006); Qu and Hall (2014)
25 |  Sea ice albedo feedback |  Seasonal sea ice albedo feedback (summer) |  CMIP5 |  Thackeray and Hall (2019)
26 |  Timing of Arctic ice-free summer |  Historical September sea ice trend |  CMIP3 |  Boe  _et al._ (2009)
27 |  Timing of Arctic ice-free summer |  Historical sea ice characteristics |  CMIP5 |  Massonnet  _et al._ (2012)
28 |  Arctic thermal feedbacks |  Different aspects of Northern Latitude regional temperatures |  CMIP3 CMIP5 |  Bracegirdle and Stephenson (2013)
29 |  Sensitivity of tropical land carbon storage to warming |  Short-term sensitivity of CO2 to interannual temperature variability |  C4MIP CMIP5 |  Cox  _et al._ (2013); Wenzel  _et al._ (2014)
30 |  CO2 fertilisation of photosynthesis on the extratropics |  Seasonal fluctuations in CO2 concentrations |  CMIP5 |  Wenzel  _et al._ (2016)
31 |  Global land carbon feedback strength |  Interannual variations in evapotranspiration, net biome productivity, and gross primary productivity |  CMIP5 |  Mystakidis  _et al._ (2017)
32 |  Change in tropical primary production to temperature anomalies |  Sensitivity of tropical primary production to interannual SST anomalies |  CMIP5 |  Kwiatkowski  _et al._ (2017)
33 |  Future permafrost thaw |  Relationship between mean annual air temperature and permafrost area |  CMIP5 |  Chadburn  _et al._ (2017)
34 |  Future CO2 concentration |  Simulated CO2 concentration by 2010 |  CMIP5 |  Hoffman  _et al._ (2014)
35 |  Change in Arctic Gross Primary Productivity |  Sensitivity of annual maximum leaf area index to increasing CO2 |  CMIP5 |  Winkler  _et al._ (2019a)
36 |  Global Ocean Carbon Uptake |  Present-day Southern Ocean carbon uptake |  CMIP5 |  Kessler and Tjiputra (2016)
37 |  Future Arctic Ocean acidification |  Present-day Arctic sea surface density |  CMIP5 |  Terhaar  _et al._ (2020)
38 |  Intensification of heavy rainfall across certain extratropical regions |  Scaling of annual maximum daily precipitation with global land temperatures |  CMIP5 |  Borodina  _et al._ (2017b)