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A low-order stochastically forced two-layer global energy balance model (EBM) admitting an analytical solution is developed for studying natural inter-annual, decadal and multi-decadal climate variability, and ultimately to better understand forced climate change. The EBM comprises upper and lower oceanic layers with a diffusive coupling, a radiative damping term including feedbacks and stochastic atmospheric forcing. The EBM is used to analyse the influence of radiative forcing, feedbacks and climate system inertia on the global mean surface temperature variance (climate variability) and to understand why Coupled Model Intercomparison Project, Phase 5 (CMIP5) models exhibit such a wide range in the level of variability in globally averaged surface air temperature. We examine the influence of the model parameters on the climate variability on different timescales. To this end, we derive the Fokker–Planck equation for the EBM and then obtain the analytical expression that quantifies the sensitivity coefficients for all model parameters. For all timescales, the most influential factors are as follows: (1) the magnitude of the stochastic forcing, (2) the feedback mechanisms, (3) the upper layer depth, (4) the diffusion parameter and (5) the lower ocean depth. Results from the EBM imply that the range of stochastic forcing in the CMIP5 climate models is around twice as important as the strength of radiative feedback or upper layer depth in causing the model-to-model spread in the magnitude of globally averaged climate model variability.

Global temperatures are characterised by large global-scale ‘natural variability’ on timescales from months to decades (Power et al.,

It has long been known that climate change projections under a range of projected greenhouse gas increases vary by a factor of around three in coupled general circulation models (CGCMs) (e.g. Bony et al.,

Part of the challenge is the complexity of processes underlying natural variability, including features such as El Niño–Southern Oscillation (ENSO) and the Interdecadal Pacific Oscillation (Power et al.,

A major advantage of the use of simplified, for example, energy balance models (EBMs) is that the hugely complex ‘multi-dimensionality’ of CGCM response can potentially be reduced to a few simple parameters, allowing straightforward exploration and understanding of model sensitivities. Under climate change scenarios, simple EBMs have been extensively used and it has been found that they can describe much of the multi-decadal temperature evolution under very different idealised external forcing, such as instantaneously quadrupled CO_{2}, or 1% compounded increases (Geoffroy et al.,

A key point of interest in the present study is to explore the relative roles of radiative climate feedbacks versus oceanic heat uptake in setting the level of climate variability and the magnitude of the annual to multi-decadal response to external forcing. There is now considerable evidence supporting a significant role for radiative feedbacks in influencing inter-annual and decadal variability (Minschwaner and Dessler,

With this ultimate goal in mind, and following the discussion above, we develop a simple two-layer EBM of the climate system, and then we examine the ‘natural variability’ evident in the EBM that arises in response to stochastic radiative forcing. We will obtain analytical solutions for the second-moment statistics of the model (i.e. the global mean surface temperature variance) under stochastic forcing. We will then explore the sensitivity of the EBM variability on different timescales to the parameter settings. This will help us to understand the relative importance of the different physical processes included in the EBM, such as radiative feedbacks, mixed layer depth and deep oceanic mixing. We will then use the EBM to try and better understand the reasons why current CGCMs exhibit such a wide range in globally averaged surface air temperature (GSAT) and variability.

In summary, we emphasise that we are not seeking a fully detailed representation of variability in the real world or in GCMs, but rather seeking to understand the simplest possible model that can represent some of the key features of global-scale variability (and response to external forcing) and understand the characteristics of that model.

Section

To represent the earth’s climate system, we formulate a two-layer EBM that consists of a pair of coupled linear subsystems (Gregory and Mitchell, ^{−2}K^{−1}), ^{−2}K^{−1}), while ^{−2}).

A key motivation of using such a model is that similar two-layer models have been considered and analysed in a number of papers considering the response to forced climate change. For example, Geoffroy et al. (_{2} experiment) and one with the 1pctCO_{2} scenario (atmospheric CO_{2} increasing at 1% per year). Despite the fact that the two-layer model is one of the simplest tools to mimic climate dynamics under external radiative forcing, it was able to simulate the evolution of average global surface temperature over time in response to both abrupt and time-dependent forcing reasonably accurately (Geoffroy et al.,

Indeed, the degree of complexity of the EBM is chosen in the present study very deliberately. The advantages, quite apart from the close links to EBMs used in climate change, include the simplicity, and the evidence that stochastic forcing dominates global decadal variability (as discussed in the Introduction). Further refinements could of course be added, for example as discussed by Held et al. (

In accordance with the theory of sensitivity in dynamical systems (e.g. Eslami, ^{−2}K^{−1}). Meanwhile, following the Intergovernmental Panel on Climate Change (IPCC, _{2} concentration. To avoid confusion, herein we will refer to this as the ‘ECS’. To understand the role of climate feedbacks, it is useful to introduce a ‘reference climate sensitivity parameter’, ^{−3} K^{−4} the Stephan–Boltzmann constant,

We assume that the deterministic radiative forcing _{2}) atmospheric concentrations. For the classic 1% compounding CO_{2} experiment, we assume _{2} concentration), which gives (Geoffroy et al., ^{−1} growth in CO_{2} concentration until

To describe the natural variability, we include an additive stochastic radiative forcing

Since our objective is to study climate variability on annual, decadal and multi-decadal (30 years) timescales, the normally distributed stochastic forcing is assumed to be uncorrelated on monthly timescales. Estimates of stochastic forcing from the CMIP5 CGCMs were calculated by: (1) first detrending pre-industrial (PI) experiment annual mean temperatures and top of atmosphere (TOA) and surface radiation; (2) removing the mean seasonal cycles; and (3) removing TOA and surface radiation fluctuations correlated with global mean temperature. This latter removal was carried out on the assumption that radiative perturbations on monthly timescales contain a component (particularly in the long wave) which is in response to the temperature perturbations and is therefore not ‘forcing’ the system. This component was removed by linearly regressing radiation change against temperature, then subtracting the temperature-dependent fraction. The fraction removed was typically very small, at around 5% of the total standard deviation of net radiation on average across the CGCMs.

The remaining variation is taken to represent the stochastic forcing in the models and is found to be dominated by shortwave variations (not shown). As the result, we have obtained the value of 0.60 Wm^{−2} at the TOA and ∼0.46 W m^{−2} at the surface for the monthly timescale ^{−2} on monthly (annual) timescales (Trenberth et al., ^{2} for PI inter-annual, decadal and multi-decadal (30 years) variances of the global mean surface temperature, respectively. These values will be used in the analysis of two-layer model discussed in this article.

The two-layer model has two state time-dependent variables

Note that we would expect parameter values to determine the ‘timescale’ of the temperature response to forcing. Specifically, there are two timescales set by the relaxation times _{D}

The question arises on possible variations of these parameters on multi-decadal timescales. In a stationary climate, there is no

The values of

The feedback factor,

Equations

The associated Fokker–Planck or forward Kolmogorov equation that describes the time evolution of the probability density function

Multiplying both sides of Equation _{D}

The partial derivative by

The second-moment equation for the variable

For the steady-state probability distribution, the Equations

It follows from Equation _{2}. Therefore, the parameter

Note that the study of the variance of the model is confined to that in the PI climate (i.e. without external forcing). A full analysis of the evolution of variability under climate change (e.g. using the approach in Majda and Gershgorin (

Model parameters, denoted by the vector

Sensitivity coefficients for different model parameters, however, can naturally be expected to differ in magnitude, simply because of differences in their units. For assessing, and ranking, the relative influence of parameter uncertainties on the climate variability, we use the relative sensitivity coefficients defined as

The relative sensitivity, then, describes how temperature variance is affected by fractional changes in a particular variable – for example, how a 10% change in mixed layer depth changes the variance, which can then be compared, say with a 10% change in the ocean deep layer.

To evaluate the two-layer model and in particular to rank the impact of model parameters on the surface temperature variance

The OVAT approach requires maximum and minimum values, and base values for all model parameters. Geoffroy et al. (_{2} quadrupling). For reference, we display the three values for each parameter used by Geoffroy et al. (

Maximum, minimum and multi-model mean values of radiative feedback parameter _{D}_{s}

^{−2}K

^{−1})

^{−2}K

^{−1})

^{−2}K

^{−1})

_{D}

^{−2}K

^{−1})

^{−2})

Maximum, minimum and multi-model mean values of model parameters used in calculations.

^{−2}K

^{−1})

^{−2}K

^{−1})

_{D}

^{−2}K

^{−1})

^{−2})

Shown values for the parameter

Figures

Surface temperature variance as a function of (a) upper layer effective heat capacity _{D}

Surface temperature variance as a function of parameter

To explore how the model reproduces climate variability from annual to multi-decadal timescales under the influence of stochastic radiative forcing, we calculated the surface temperature variance as a function of the upper layer heat capacity. This was chosen, as we would expect this to set the timescale of response, over short time periods at least to vary sensitively with this variable (Gregory, ^{2}_{D} ≫ γC

Figure

Figure

The heat exchange parameter, γ, also noticeably affects the climate variability, as shown in Fig.

A less significant effect on the climate variability of various climate timescales is provided by the parameter _{D}_{D}_{D}_{D}

Figure

We now compare the EBM variability with variability in the CGCMs. To do this, we first set the variance of the stochastic forcing to the mean of the CGCM values. Verifying the calculated temperature variance against the PI inter-annual, decadal and multi-decadal variances (see Section _{D}_{D}

Calculated surface temperature variance

^{-2}K

^{-1})

_{D}

_{D}

Calculated sensitivity coefficients with respect to the model parameters are illustrated in Figs. _{D}

Sensitivity of surface temperature variance to (a) upper layer effective heat capacity _{D}

Sensitivity of the surface temperature variance to parameter

Each sensitivity coefficient, as shown in Figs. ^{−2} K^{−1}, which is equal to 5% change with respect to the base value, then for inter-annual timescales we obtain ^{2}, while for decadal – ^{2} since the sensitivity coefficient ^{2} because of the growth of the sensitivity coefficient from

Sensitivity coefficients allow us to estimate the influence of model parameter uncertainty on the uncertainty of calculated surface temperature variance, but the absolute effects depend on the scale of the parameter being varied. To assess the relative effects of parametric uncertainty, we shall assume that the parameter variations are 5% of their base values, that is _{D}

Changes in the absolute value of surface temperature variance

^{−2}K

^{−1})

^{−2}K

^{−1})

^{−2}K

^{−1})

^{−2})

^{−4}

^{−4}

^{−4}

^{−6}

^{−4}

Along with the absolute sensitivity coefficients analysed above, we also calculated relative (normalised) sensitivity coefficients

Relative sensitivity of surface temperature variance to (a) upper layer effective heat capacity _{D}

Relative sensitivity coefficients calculated for annual, decadal and multi-decadal surface temperature variability.

_{D}

_{S}

^{−2}K

^{−1}

^{−2}K

^{−1}

^{−2}K

^{−1}

^{−2}K

^{−1}

The results discussed above are for the base values of model parameters. However, a key question is how the range of model parameters found from fitting GCM results (Geoffroy et al., ^{2}) followed by the upper layer heat capacity ^{2} and 0.0053 K^{2} respectively). The impact of the diffusion parameter range is around half the value again. Finally, the range in the deep ocean heat capacity is the least significant, at 0.0002 K^{2}.

Surface temperature variances

^{−2}K

^{−1})

^{−2}K

^{−1})

^{−2}K

^{−1})

_{D}

^{−2}K

^{−1})

_{s}

^{−2})

The values for _{D}

Global mean surface temperature variances simulated by CMIP5 CGCMs deviate from each other substantially, on inter-annual, decadal and multi-decadal timescales (e.g. Colman and Power,

A key aim of the study was to understand the relative importance of parameters uncertainty, and in particular, uncertainty in the radiative feedback, to the range of variability implied by the EBM. We found that for all but extremely long timescales, the most important parameters were in descending order: the magnitude of the stochastic forcing, the radiative feedback parameter, the depth of the upper ocean layer, the diffusion parameter and the depth of the deep ocean layer. However, the relative importance of these parameters varied strongly across timescales, as the timescale lengthened fractional changes in the feedback parameter and depth of the deep ocean became more important, and the depth of the upper ocean and diffusion parameter less so.

Finally, we compared the range of temperature variance that is implied by the range in model parameters obtained from the study of Geoffroy et al. (

Of course, there are important caveats to our conclusions. The most important is of course that the EBM is highly simplified and while we can tune the EBM to replicate CGCM variability, the EBM might be simulating the right level of variability for the wrong reasons. Another is that despite the similarity in overall magnitude of radiative feedbacks under climate change and variability (Colman and Hanson,

We thank Scott Power, Meelis Zidikheri and two anonymous reviewers for helpful comments. We acknowledge the World Climate Research Programme’s Working Group on Coupled Modelling, which is responsible for CMIP, and we thank the climate modelling groups for producing and making available their model output. For CMIP the US Department of Energy’s Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals.

No potential conflict of interest was reported by the authors.

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_{2}concentration: some dynamical scenarios