Frequency distributions of transient regional climate change from perturbed physics ensembles of general circulation model simulations

Abstract

Large ensembles of coupled atmosphere–ocean general circulation model (AOGCM) simulations are required to explore modelling uncertainty and make probabilistic predictions of future transient climate change at regional scales. These are not yet computationally feasible so we have developed a technique to emulate the response of such an ensemble by scaling equilibrium patterns of climate change derived from much cheaper “slab” model ensembles in which the atmospheric component of an AOGCM is coupled to a mixed-layer ocean. Climate feedback parameters are diagnosed for each member of a slab model ensemble and used to drive an energy balance model (EBM) to predict the time-dependent response of global surface temperature expected for different combinations of uncertain AOGCM parameters affecting atmospheric, land and sea-ice processes. The EBM projections are then used to scale normalised patterns of change derived for each slab member, and hence emulate the response of the relevant atmospheric model version when coupled to a dynamic ocean, in response to a 1% per annum increase in CO₂. The emulated responses are validated by comparison with predictions from a 17 member ensemble of AOGCM simulations, constructed from variants of HadCM3 using the same parameter combinations as 17 members of the slab model ensemble. Cross-validation permits estimation of the spatial and temporal dependence of emulation error, and also allows estimation of a correction field to correct discrepancies between the scaled equilibrium patterns and the transient response, reducing the emulation error. Emulated transient responses and their associated errors are obtained from the slab ensemble for 129 pseudo-HadCM3 versions containing multiple atmospheric parameter perturbations. These are combined to produce regional frequency distributions for the transient response of annual surface temperature change and boreal winter precipitation change. The technique can be extended to any surface climate variable demonstrating a scaleable, approximately linear response to forcing.

Appendix

This section provides an estimate of the variance in the frequency distribution of the transient regional response emulated from the slab ensemble. For a given region and time, the emulated response P _j in Eq. 10 is assumed to be equivalent to:

$$P_{j} = \Delta T_{j} {\left({s_{j} + \varepsilon _{j} } \right)}.$$

(14)

where s _j is the signal in the combined slab and correction field pattern for member j, and ɛ_j is random internal variability in the normalised slab pattern uncorrelated with s _j . Internal variability in the slab pattern arises since we have used 20 year means to estimate the response. The variance of the emulated frequency distribution is:

$$V = \frac{1}{{N - 1}}{\sum\limits_{j = 1}^N {{\left[ {{\left( {P_{j} + \delta _{j} } \right)} - \overline{{{\left( {P_{j} + \delta _{j} } \right)}}} } \right]}^{2} } }$$

(15)

where to each emulated response P _j we add random noise δ _j , sampled from a normal distribution with some yet to be determined variance δ². In the limit (N−1)/N →1, and with $\overline{{\delta _{j} }} = 0$ this becomes

$$V = \overline{{P^{2} }} - \overline{P} ^{2} + \overline{{\delta ^{2} }}. $$

(16)

Equation 14 for P _j is used in Eq. 16 to give

$$V = \overline{{\delta ^{2} }} + \overline{{s^{2} \Delta T^{2} }} - \overline{{s\Delta T}} ^{2} + \overline{{\varepsilon ^{2} \Delta T^{2} }} + 2\overline{{s\varepsilon \Delta T^{2} }} - \overline{{\varepsilon \Delta T}} ^{2} - 2\overline{{\varepsilon \Delta T}} {\text{ }}\overline{{s\Delta T}}.$$

(17)

Random internal variability in the slab pattern is assumed to be uncorrelated with EBM projections for ΔT, so the last three terms in Eq. 17 equal zero. Defining the variance V_s in scaled prediction of the true slab pattern signal:

$$V_{\rm s} = \frac{1}{N}{\sum\limits_{j = 1}^N {{\left[ {s_{j} \Delta T_{j} - \overline{{s_{j} \Delta T_{j} }} } \right]}^{2} } } = \overline{{s^{2} \Delta T^{2} }} - \overline{{s\Delta T}} ^{2} $$

(18)

the variance in the emulated frequency distribution becomes

$$V = V_{\rm s} + \overline{{\varepsilon ^{2} \Delta T^{2} }} + \overline{{\delta ^{2} }}.$$

(19)

The second term in Eq. 19, with internal variability scaled by the EBM global temperature response ΔT ², represents variance due to our uncertain knowledge of the true slab signal. The remaining variance $\overline{{\delta ^{2} }}$ will include a contribution V _em arising from uncertainty in the emulation, and variance $\overline{{\eta ^{2} }}$ due to random internal variability in the transient response being emulated:

$$\overline{{\delta ^{2} }} = V_{{\rm em}} + \overline{{\eta ^{2} }}.$$

(20)

The variance δ² can be estimated from the cross validation ensemble. Similarly to Eq. 14, we assume the AOGCM simulated response M _j for a given region and time is equal to a model signal m _j , plus random internal variability η _j uncorrelated with m _j :

$$M_{j} = m_{j} + \eta _{j}. $$

(21)

Substituting e _j =P _j −M _j into the definition for σ² in Eq. 8, expanding, and setting the means of uncorrelated terms such as $\overline{{s\eta }}$ to zero, the variance in cross validation error is given by

$$\begin{aligned}\sigma ^{2} &= \frac{1}{{M - 1}}{\sum\limits_{j = 1}^M {{\left[ {{\left( {\Delta T_{j} s_{j} - m_{j} } \right)} - \overline{{{\left( {\Delta T_{j} s_{j} - m_{j} } \right)}}} } \right]}^{2} } }\\ &\quad + \frac{1}{{M - 1}}{\sum\limits_{j = 1}^M {\eta ^{2}_{j} } } + \frac{1}{{M - 1}}{\sum\limits_{j = 1}^M {\varepsilon ^{2}_{j} \Delta T^{2}_{j} } }.\end{aligned}$$

(22)

The first term in Eq. 22 is the variance of the emulation error for the true signal, which we identify as V _em, i.e.:

$$V_{{\rm em}} = \frac{1}{{M - 1}}{\sum\limits_{j = 1}^M {{\left[ {{\left( {\Delta T_{j} s_{j} - m_{j} } \right)} - \overline{{{\left( {\Delta T_{j} s_{j} - m_{j} } \right)}}} } \right]}^{2} } }.$$

(23)

V _em includes error due to non-linearities in the simulated AOGCM signal m _j , and error due to discrepancy between m _j and the slab signal scaled by global temperature from the EBM. The second term in Eq.  22 represents uncertainty in the validation due to random internal variability in the AOGCM transient simulations. The third term represents emulation uncertainty due to scaling of random internal variability in the slab patterns by the temperature response. A quadratic dependence on ΔT is indeed demonstrated in Fig. 9, where the variance in emulation error σ² for Northern Europe surface temperature is shown (blue curve).

Equation 22 can be rewritten:

$$\sigma ^{2} = V_{{\rm em}} + \overline{{\eta ^{2} }} + \overline{{\varepsilon ^{2} \Delta T^{2} }}$$

(24)

and combining this with Eq. 20, we arrive at an expression that relates δ² to the variance of cross validation error:

$$\overline{{\delta ^{2} }} = \sigma ^{2} - \overline{{\varepsilon ^{2} }} \overline{{\Delta T^{2} }} $$

(25)

assuming ɛ _j to be uncorrelated with ΔT _j . The required variance in random uncertainty associated with each emulation P _j is therefore equal to the variance in cross validation error, minus the variance associated with scaling of random internal variability in the slab patterns by the global temperature response. An estimate for $\overline{{\varepsilon ^{2} }}$ is obtained from thirty 20 year means from a 600 year control simulation with the HadSM3 standard slab model configuration.