We examine what can be learnt about climate sensitivity from variability in the surface air temperature record over the instrumental period, from around 1880 to the present. While many previous studies have used trends in observational time series to constrain equilibrium climate sensitivity, it has also been argued that temporal variability may also be a powerful constraint. We explore this question in the context of a simple widely used energy balance model of the climate system. We consider two recently proposed summary measures of variability and also show how the full information content can be optimally used in this idealised scenario. We find that the constraint provided by variability is inherently skewed, and its power is inversely related to the sensitivity itself, discriminating most strongly between low sensitivity values and weakening substantially for higher values. It is only when the sensitivity is very low that the variability can provide a tight constraint. Our investigations take the form of “perfect model” experiments, in which we make the optimistic assumption that the model is structurally perfect and all uncertainties (including the true parameter values and nature of internal variability noise) are correctly characterised. Therefore the results might be interpreted as a best-case scenario for what we can learn from variability, rather than a realistic estimate of this. In these experiments, we find that for a moderate sensitivity of 2.5

For many years, researchers have analysed the warming of the climate system as observed in the modern instrumental temperature record (spanning the mid-19th to early-21st century), in order to understand the response of the climate system to external forcing. For the most part, the focus has been on the long-term energy balance as constrained by the warming trend in atmospheric and oceanic temperatures

More recently however,

In this paper, we explore the question of to what extent temporal variability in the globally and annually averaged temperature record can be used to constrain equilibrium climate sensitivity. We consider both the internal variability in the climate system itself and also the total variability including deviation from a linear trend due to the forced response. Our investigations are performed in the paradigm of a simple idealised modelling framework, using a two-layer energy balance model which has been widely used to simulate the climate system and which generalises and improves on the performance of the zero-dimensional model. As part of our investigations, we examine the relationship between the

In the next section, we present the two-layer energy balance model and briefly outline the experimental methods used in this paper. We first focus on internal variability, that is to say, the temporal variability arising entirely from internal dynamics of the climate system in the absence of forcing. We evaluate the power of the

The basic underpinning of previous work is energy balance modelling of the climate system, from which it is anticipated that interannual variability may be informative regarding the equilibrium sensitivity. While previous research was based on analysis of the simplest possible zero-dimensional single-layer planetary energy balance, there is evidence that the behaviour of the climate system over the historical period is poorly modelled by such a system

This is a two-layer globally averaged energy balance model which simulates the mixed (

The values of the various adjustable parameters are listed in Table

Adjustable parameters and default values.

Our investigations are performed within the paradigm of Bayesian estimation. In general, the Bayesian approach provides us with a way to estimate a set of unknown parameters

Here

Formally, the value of the observations is fully summarised by the likelihood function

While this study primarily focusses on the behaviour of the simple energy balance model, we also use and present some data from external sources. In order to perform simulations of the historical period, we force our climate model with annual time series for the major forcing factors based on

For comparison with our simple model results, we also present some results calculated from historical simulations performed by climate models in the CMIP5

We now present some investigations into the relationship between

Figure

When we consider the two-layer model using the standard parameter value of

We now directly consider the question of how useful an observed value

Posterior estimates of sensitivity inferred by using observations of

Our experiments take the form of a perfect model scenario, where the model is assumed to be a perfect representation of the system under consideration, with no structural imperfections. Our uncertainties here are due solely to unknown parameter values and internal variability noise. In these experiments, we assume that

Strictly, when considering the strength of the constraint obtained from the variability, we should focus on the likelihood

The pdf's plotted in the top panel of Fig.

Although the results in Fig.

This approach requires us to calculate the likelihood for the full set of observations,

It is worth emphasising that this calculation represents an absolute best-case scenario for using the time series of temperature anomalies as a constraint. There can be no diagnostic or statistical summary of the observations that provides more information than the full set of observations themselves contain. Thus, we cannot hope to obtain a better constraint by some alternative analysis of the temperature time series.

Posterior estimates for the climate sensitivity from Bayesian estimation using the full time series of annual mean surface temperatures. Main plot: results from 150-year unforced simulations as discussed in Sect.

Figure

Thus, there appears to be the potential for internal variability, as represented by the full temperature time series, to provide a slightly better constraint than that obtained by a summary statistic alone, but the improvement is marginal, and even our optimal calculation, which uses the exact likelihood of the full time series, cannot accurately diagnose equilibrium sensitivity except when the true value is very low. These results again show a skew similar to that obtained when

While the theoretical underpinning of

In this section, we perform a series of analyses based on historical forced simulations, in order to investigate more fully the potential for such forced effects to improve the constraint. We force the climate model with annual time series for the major forcing factors based on

Simulations of instrumental period with two-layer model. Thick lines are forced response excluding internal variability; thin lines are five replicates of each parameter set including internal variability. Blue lines:

Figure

When we hold other parameters at default levels, best agreement between model and data (defined here simply by RMS difference between the two time series) is achieved for a rather low sensitivity of 1.78

In order to assess what we can learn about sensitivity from the variability in historical temperature observations, we first consider the utility of

Grey dots and error bars indicate results obtained when only

Blue and red crosses also shown in Fig. 5 show results obtained from the CMIP5 and CMIP6 ensembles. The CMIP models appear to generate slightly lower values of

Figure

When we use the observational value of

Finally, we repeat the approach of Sect.

The calculation is similar to that of Sect.

The results of multiple replicates are shown in Fig.

We have explored the potential for using interannual temperature variability in estimating equilibrium sensitivity. While – as

Forced variability, such as that occurring during the instrumental period, does provide additional information in our experiments, and therefore we could in theory hope to calculate a narrower posterior range, with a typical width of around 4

Data and code to reproduce all figures and analysis are included in the Supplement.

The supplement related to this article is available online at:

JDA and JCH undertook the research and wrote the article. TS and BS contributed to the conception of the study and provided critical feedback to the research and article.

The authors declare that they have no conflict of interest.

We would like to thank the editor and two reviewers for their helpful comments.

This research was supported by the MPI-Hamburg. Thorsten Mauritsen was also supported by the European Research Council (ERC) (grant no. 770765) under the European Commission Horizon 2020 research and innovation programme (grant no. 820829).

This paper was edited by Steven Smith and reviewed by two anonymous referees.