One of the approaches to constrain uncertainty in climate models is the identification of emergent constraints. These are physically explainable empirical relationships between a particular simulated characteristic of the current climate and future climate change from an ensemble of climate models, which can be exploited using current observations. In this paper, we develop a theory to understand the appearance of such emergent constraints. Based on this theory, we also propose a classification for emergent constraints, and applications are shown for several idealized climate models.

Improving the accuracy of climate projections is one of the most important
challenges in climate modelling. The uncertainty can be reduced by the
development of more and more sophisticated global climate models, capturing
more processes and scales. However, the societal importance of climate
projections calls for a faster pace of improvement and alternative approaches
that aim to better determine the accuracy of existing models. One of the
proposed methods to accomplish this has been the use of so-called emergent
constraints, where current observations are used to constrain future
projections

In multi-model ensembles of complex climate models, an apparent linear
relation can be found between short-term and long-term changing variables.
More credibility is attached to models that match the observed variability or
trend well over the recent period. In this way, current observations provide
a constraint to long-term trends. The observed variable is called the
predictor, while the variable that is to be constrained is called the
predictand

A prominent example is the emergent constraint found in

However, a more general dynamical picture on how emergent constraints occur
in multi-model ensembles or even in a parameter ensemble of a single model is
still lacking. Under which circumstances are these constraints expected to
arise? Some emergent constraints may be spurious and could arise because of
shared errors in a particular multi-model ensemble

Here, we investigate how and under what conditions emergent constraints
appear and what can be learned about the physics of the climate system. We
will use linear response theory (LRT) to address the problem of
forcing-response relations on different timescales

The paper is organized as follows. To obtain an understanding of emergent constraints, we start by formulating a mathematical framework in terms of susceptibilities by making use of LRT (Sect. 2). This results in explicit expressions for the appearance of emergent constraints in terms of susceptibility functions. In Sect. 3, a classification scheme for emergent constraints is proposed. Then, in Sect. 4, applications are presented for conceptual climate models, such as Ornstein–Uhlenbeck processes in one and two dimensions, an energy balance model and the PlaSim model. The results are summarized and discussed in Sect. 5.

In this section, explicit expressions are given for response functions of the
state of a dynamical system which depends on a single parameter and which is
subjected to a non-stationary forcing. Such response functions are used in
the following section to classify the different emergent constraints.
Rigorous results for linear response properties of large class of general
stochastic systems was obtained by

We illustrate the approach using the general one-dimensional forced
stochastic differential equation (SDE):

The probability density function of the unforced (

In the Appendix, it is shown that when we take the identity operator

The amplitude

Although a wide set of different emergent constraints has been found, no attempts have been made to classify them so far using dynamical criteria. Here, a classification is proposed based on the time characteristics of the predictor and on the relationship between the predictor and the predictand. Using this classification, assessment of their applicability becomes easier. Furthermore, a classification is a prerequisite for a dynamical description of emergent constraints.

The emergent constraint on snow-albedo feedback

Firstly, an emergent constraint can be either direct or indirect. In the direct case, the predictor and predictand are the same observable, while in the indirect case they are not. In the latter case, the predictor variable and predictand variable have to be closely linked, for instance, via a physical process. We make a further distinction between static and dynamic emergent constraints. In a dynamic emergent constraint, a response to a known, or sometimes even unknown, forcing in the (present-day) predictor is linked to the response of the (future) predictand under the same (or a similar) forcing. For example, the forcing can be the annual cycle of solar radiation but can also be caused by ENSO or historical climate change. In a static emergent constraint, a relationship between the time-independent quantity of the unforced system in the present-day (predictor) is linked to the response in a quantity under climate change.

Application of our classification of emergent constraints to a
selection of examples found in literature. DD is a direct dynamical
constraint, DS is a direct static constraint, and IS is an indirect static
constraint, while ID denotes indirect dynamical emergent constraints.
Abbreviations are as follows: RH: relative humidity, ITCZ: Intertropical
Convergence Zone, TOA: top of atmosphere, SH: Southern Hemisphere,
ECS: equilibrium climate sensitivity, LLC: low-level cloud, SAF: snow-albedo
feedback, SAT: surface air temperature; GMST: global mean surface temperature. The emergent constraint found by

As an illustration, we apply our classification to examples of emergent
constraints found in the literature in Table

Based on the response function theory in Sect. 2, we further elaborate on the classification and also discuss conditions for each type of constraint for a dynamical system with varying parameters (which defines the ensemble of models).

For a direct dynamical emergent constraint, in the standard case of a linear
relationship, the relation has the following form: Predictand

One variable (

Physically, we expect that the same mechanism is responsible for the
response at a short and long timescale to obtain this type of emergent
constraint. The system should have response times smaller than the timescale
of the forcing or equivalently: the generator should have eigenvalues

Mathematically, the ratio in Eq. (

In the case of indirect dynamical emergent constraints, a relationship
between a predictor

Static direct constraints link the mean of an observable (predictor) to a
change in the system under a specific forcing (predictand). Note that the
susceptibility only contains information about the response to such forcing.
Even in the limit of

For static emergent constraints, the linear relationship between the
predictand and the predictor is not expected to pass through the origin,
since the predictor will in general be non-zero. Therefore, an additional
term

As an illustration of the theory from Sect.

From the previous sections, it appears that the computation of the eigensolutions of the generator of the dynamical system are central to determine whether an emergent constraint will appear or not. In this section, we will provide examples using idealized climate models.

The eigenvalues and eigenfunctions of the generator were numerically
determined using the fact that the eigenvalues of the Fokker–Planck
operator

Eigenvalue spectrum for

First, the one-dimensional Ornstein–Uhlenbeck process is considered with SDE

In the two-dimensional Ornstein–Uhlenbeck case, the same forcing

In Fig.

The results in Fig.

In this section, a specific emergent constraint is examined in more detail,
namely the one pertaining to the SAF first described
by

Constants for the energy balance model.

With constant albedo, the energy balance model reads

Before examining the snow-albedo feedback, note that, for some variables,
notably the climate sensitivity, a simple energy balance model (EBM) can react differently to
forcing from solar insolation or greenhouse gases. This can be determined
from, with

Sensitivity to greenhouse forcing decreases when albedo decreases, while
sensitivity to solar insolation (seasonal sensitivity) increases for an
increasing albedo, using typical values for

To mimic the physical mechanism behind the emergent constraint, the albedo is
taken to be temperature dependent; i.e. for low (high) temperatures, the
albedo is high (low). A logistic function is used to model this effect:

In the first case, the insolation forcing is given by

As mentioned above, application of Eq. (

Instead, the SAF can be described by two observables: SAF is determined by
taking the ratio of the susceptibilities of albedo to temperature. Therefore,
we use the modified Eq. (

In Fig.

One can extend the energy balance model by representing the response of snow
and ice explicitly as a relaxation towards the logistic reference albedo
function

Extending the model with an explicit albedo function does not change the
dynamics of the system significantly, nor the eigenvalues and eigenvectors.
Figure

To bridge the gap between parameter ensembles in simple dynamical systems and
Earth system models, the SAF emergent constraint is further examined in
PlaSim. PlaSim is a numerical model of intermediate complexity, developed at
the University of Hamburg to provide a fairly realistic present climate which
can still be simulated on a personal computer

In this climate model, snow albedo is a function of surface
temperature

In Fig.

Same as Fig.

In this paper, we have presented a dynamical framework behind the occurrence of emergent constraints in parameter-dependent stochastic dynamical systems. In these systems, emergent constraints are related to ratios of response functions which can be determined using linear response theory. It was shown that for a large class of systems, these ratios could be expressed in terms of eigenvalues and projections on eigenvectors of the generator of the system.

A classification of emergent constraints was given and several types could be distinguished depending on whether similar (direct) or different (indirect) observables are considered and whether a response in present-day climate (dynamical) or the time-independent part of present-day climate (static) is linked to a response of the future climate system. For a linear dynamical emergent constraint, the ratio of susceptibilities at the two frequencies under consideration should be a positive constant over the ensemble. When the response is computed with respect to an internal variable (in contrast to an external forcing), a condition is posed on the susceptibilities of the two observables in the system. Static constraints are encountered when a linear relationship is found between the expectation value of the observable and the susceptibility at the frequency of the forcing.

Examples were given using several idealized climate models. In particular, the
emergent constraints involving the snow-albedo feedback was considered in
detail. We found that linear dynamical emergent relationships can occur when
the timescale of the system, indicated by the eigenvalues, changes with the
parameter and is smaller than the forcing timescales. This is of particular
interest because differences in response size between climate models is often
determined by feedback strength in climate systems. Larger feedbacks give
rise to larger timescales

Modelling emergent constraints with conceptual models is justified when different Earth system models (ESMs) are closely related and structural differences can be parameterized. This can, for instance, be tested using an intermediate complexity model with full parameterization of the process under consideration.

The classification of emergent constraints provided gives a hint to which
kind of emergent constraints one can look out for in an ensemble of
high-dimensional global climate models (GCMs). To find an emergent constraint
for climate sensitivity by data mining in a CMIP5 ensemble proved fruitless

In a high-dimensional dynamical system, eigenfunctions and eigenvalues can be
accessed with the help of transfer operators, associated with the propagation
of probability densities associated with the Fokker–Planck operator. The
eigenfunctions that lie on the invariant measure are then computed by making
use of the ergodic properties of the climate system. To overcome the burden
of high dimensionality, a reduced transfer operator can be computed from a
very long simulation, from which the eigenfunctions on the attractor are
approximated

In conclusion, while the current theoretical framework provides an understanding on how emergent constraints may arise in low-dimensional stochastic dynamical systems, its application to output from GCMs, in particular in finding novel and useful emergent constraints, is a challenging issue for future work.

All the code used in this paper is available upon request.

For

For reversible processes, these eigenvalues are real, positive and discrete
under the inner product

Repeating the derivation with a general observable

FJMMN and HAD designed the study, FJMMN performed the numerical calculations and drafted the manuscript. HAD supervised the project. Both authors discussed the results and contributed to the final manuscript.

The authors declare that they have no conflict of interest.

We thank Frank Lunkeit for his help with the PlaSim simulations and Alexis Tantet is thanked for useful discussions. The authors also thank Valerio Lucarini and an anonymous reviewer whose suggestions helped improve and clarify the manuscript. Both authors acknowledge support by the Netherlands Earth System Science Centre (NESSC), financially supported by the Ministry of Education, Culture and Science (OCW), grant no. 024.002.001. Edited by: Andrey Gritsun Reviewed by: Valerio Lucarini and one anonymous referee