Emulating AOGCM results using simple climate models

Abstract

Three simple climate models (SCMs) are calibrated using simulations from atmosphere ocean general circulation models (AOGCMs). In addition to using two conventional SCMs, results from a third simpler model developed specifically for this study are obtained. An easy to implement and comprehensive iterative procedure is applied that optimises the SCM emulation of global-mean surface temperature and total ocean heat content, and, if available in the SCM, of surface temperature over land, over the ocean and in both hemispheres, and of the global-mean ocean temperature profile. The method gives best-fit estimates as well as uncertainty intervals for the different SCM parameters. For the calibration, AOGCM simulations with two different types of forcing scenarios are used: pulse forcing simulations performed with 2 AOGCMs and gradually changing forcing simulations from 15 AOGCMs obtained within the framework of the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. The method is found to work well. For all possible combinations of SCMs and AOGCMs the emulation of AOGCM results could be improved. The obtained SCM parameters depend both on the AOGCM data and the type of forcing scenario. SCMs with a poor representation of the atmosphere thermal inertia are better able to emulate AOGCM results from gradually changing forcing than from pulse forcing simulations. Correct simultaneous emulation of both atmospheric temperatures and the ocean temperature profile by the SCMs strongly depends on the representation of the temperature gradient between the atmosphere and the mixed layer. Introducing climate sensitivities that are dependent on the forcing mechanism in the SCMs allows the emulation of AOGCM responses to carbon dioxide and solar insolation forcings equally well. Also, some SCM parameters are found to be very insensitive to the fitting, and the reduction of their uncertainty through the fitting procedure is only marginal, while other parameters change considerably. The very simple SCM is found to reproduce the AOGCM results as well as the other two comparably more sophisticated SCMs.

Appendices

Appendix 1 : The SCM_TLSE and analytical solutions for special forcings

SCM_TLSE has two prognostic variables: the change in global-mean surface air temperature (T _a), and the change in global-mean volumetric ocean temperature (T _o). The system’s response to an external radiative forcing F is given by

$$ \left\{ \begin{array}{lll} \tau_{\text{a}} {\frac{{\hbox{d}} T_{\text{a}}} {\hbox{d} t}} + T_{\text{a}} =f_{\text{a}} F(t) + \alpha T_{\text{o}} \\ \tau_{\text{o}} {\frac{{\hbox{d}} T_{\text{o}}} {\hbox{d} t}} + T_{\text{o}} =f_{\text{o}} F(t) . \end{array} \right. $$

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See Section 2.1 for a description of the different parameters.

For the purposes of solving the system, it is instructive to rewrite it as

$$ {\frac{\hbox{d}} {\hbox{d}t}} \left( \begin{array}{c} T_{\text{a}} \\ T_{\text{o}} \end{array} \right) = \left( \begin{array}{cc} -{\frac{1} {\tau_{\text{a}}}} & {\frac{\alpha} {\tau_{\text{a}}}}\\ 0 & -{\frac{1} {\tau_{\text{o}}}} \\ \end{array} \right) \left( \begin{array}{c} T_{\text{a}} \\ T_{\text{o}} \end{array} \right) + F(t) \left( \begin{array}{c} {\frac{f_{\text{a}}} {\tau_{\text{a}}}} \\ {\frac{f_{\text{o}}} {\tau_{\text{o}}}} \end{array} \right) . $$

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In the following we will give analytical solutions of this system for three special cases of forcings: an exponential decaying forcing, a step forcing and a δ-forcing.

1.1 Exponential forcing

Assume an exponentially decaying forcing, given by

$$ F(t)= \left\{ \begin{array}{lll} 0 &\hbox{if} & t < 0 \\ \exp{\frac{-t} {\tau_{\text{p}}}}& \hbox{if} & t > 0. \end{array} \right. $$

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We assume that the relaxation time τ_p of the forcing is different from the atmosphere and ocean response times τ_a and τ_o. For the special cases where τ_p = τ_a or τ_p = τ_o, an analytical solution can easily be derived, too, but these cases will not be treated here. These solutions can also be found by carefully taking the limits of the general solution given below.

If the initial conditions are

$$ \begin{aligned} T_{\text{a}}(0-)&=0\\ T_{\text{o}}(0-)&=0 , \end{aligned} $$

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then the response to an exponentially decaying forcing for t > 0 is given by

$$ \begin{aligned} T_{\text{a}}(t)=&{\frac{f_{\text{a}}} {1-{\frac{\tau_{\text{a}}} {\tau_{\text{p}}}}}} \left(\exp{\frac{-t} {\tau_{\text{p}}}}- \exp{\frac{-t} {\tau_{\text{a}}}} \right)\\ &+ {\frac{f_{\text{o}}\alpha} {1-{\frac{\tau_{\text{o}}} {\tau_{\text{p}}}}}} \left[{\frac{1} {1-{\frac{\tau_{\text{a}}} {\tau_{\text{p}}}}}} \left(\exp{{\frac{-t} {\tau_{\text{p}}}}} - \exp{{\frac{-t} {\tau_{\text{a}}}}} \right) -{\frac{1} {1-{\frac{\tau_{\text{a}}} {\tau_{\text{o}}}}}} \left(\exp{{\frac{-t} {\tau_{\text{o}}}}}- \exp{{\frac{-t} {\tau_{\text{a}}}}} \right) \right] \\ T_{\text{o}}(t)=&{\frac{f_{\text{o}}} {1-{\frac{\tau_{\text{o}}} {\tau_{\text{p}}}}}} \left(\exp{\frac{-t} {\tau_{\text{p}}}}- \exp{\frac{-t} {\tau_{\text{o}}}} \right) . \end{aligned} $$

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Note that the response of the atmosphere depends on both, τ_a and τ_o (and τ_p), whereas the response of the ocean depends only on τ_o (and τ_p). In general τ_a≪ τ_o, implying that there is a slow component in the response of the atmosphere. In cases where one only has an atmospheric time series, it is often hard to estimate τ_o. However, with the above formulation of a model, τ_o can be derived from the ocean time series, making it more reliable.

In the following we will derive some characteristics of the response to an exponentially decaying pulse. If we assume that α = 0, both equations reduce to

$$ T(t)={\frac{f} {1-{\frac{\tau} {\tau_{\text{p}}}}}} \left(\exp{{\frac{-t} {\tau_{\text{p}}}}}- \exp{{\frac{-t} {\tau}}} \right), $$

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where τ is assumed to be either τ_a of τ_o.

For t being small [t ≤ min(τ, τ_p)], this solution can be approximated by,

$$ T(t)\simeq{\frac{f t} {\tau}}, $$

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while for t being large [t ≥ max(τ, τ_p)], the solution can be approximated by

$$ T(t)\simeq \left\{ \begin{array}{lll} {\frac{f} {1-{\frac{\tau} {\tau_{\text{p}}}}}} \exp{{\frac{-t} {\tau_{\text{p}}}}} & \hbox{if} & \tau < \tau_{\text{p}} \\ {\frac{f} {{\frac{\tau} {\tau_{\text{p}}}}-1}} \exp{{\frac{-t} {\tau}}} & \hbox{if} & \tau > \tau_{\text{p}} . \end{array} \right. $$

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This implies that for t being large [t ≥ max(τ, τ_p)] the response will decay at a rate determined by the maximum of τ and τ_p.

Note that the solution in Eq. 22 is maximal at time

$$ t_{\text{m}}={\frac{\ln \tau - \ln \tau_{\text{p}}} {{\frac{1} {\tau_{\text{p}}}}-{\frac{1}{\tau}}}} . $$

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One can verify from the above expression that t _m increases with τ_p.

1.2 Step forcing

This is just a special case of an exponential forcing with τ_p → ∞. A step forcing can be expressed by

$$ F(t)= \left\{ \begin{array}{lll} 0 &\hbox{for} & t < 0 \\ 1 &\hbox{for} &t > 0 . \end{array} \right. $$

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The general solution can easily be derived from Eq. 21 by taking the limit τ_p → ∞, giving for t >  0

$$ \begin{aligned} T_{\text{a}}(t)&=f_{\text{a}} \left(1- \exp{{\frac{-t} {\tau_{\text{a}}}}} \right) + f_{\text{o}} \alpha \left[ \left(1 - \exp{{\frac{-t} {\tau_{\text{a}}}}} \right) -{\frac{1} {1-{\frac{\tau_{\text{a}}} {\tau_{\text{o}}}}}} \left(\exp{{\frac{-t} {\tau_{\text{o}}}}}- \exp{{\frac{-t} {\tau_{\text{a}}}}} \right) \right] \\ T_{\text{o}}(t)&=f_{\text{o}} \left(1- \exp{{\frac{-t} {\tau_{\text{o}}}}} \right) . \end{aligned} $$

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It is interesting to determine the response in the limit t → ∞. The solution follows immediately by noting that at t → ∞, the time derivatives in Eq. 17 are 0, such that the system can be written as

$$ \begin{aligned} T_{\text{a}}(\infty)&= f_{\text{a}} + \alpha T_{\text{o}}(\infty) \\ T_{\text{o}}(\infty)&= f_{\text{o}} , \end{aligned} $$

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which gives

$$ \begin{aligned} T_{\text{a}}(\infty)&= f_{\text{a}} + \alpha f_{\text{o}} \\ T_{\text{o}}(\infty)&= f_{\text{o}} . \end{aligned} $$

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This means that the climate sensitivity of this system is

$$ \lambda=f_{\text{a}} + \alpha f_{\text{o}} . $$

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1.2.1 δ-forcing

Although a δ-forcing is hard to impose in an AOGCM, deriving its theoretical solution is instructive, as it can serve as a building block for the solution to a general forcing.

Assume a forcing F(t) such that

$$ F(t)= 0 \quad \hbox{for} \quad t \neq 0 $$

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and

$$ \int\limits_{-\infty}^\infty F(t) \hbox{d}t = 1 . $$

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Assuming that the initial conditions are

$$ \begin{aligned} T_{\text{a}}(0-)&=0\\ T_{\text{o}}(0-)&=0 , \end{aligned} $$

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then the response to this δ-forcing for t >  0 is given by

$$ \begin{aligned} T_{\text{a}}(t)&={\frac{f_{\text{a}}} {\tau_{\text{a}}}} \exp{{\frac{-t} {\tau_{\text{a}}}}} +{\frac{f_{\text{o}} \alpha} {\tau_{\text{o}} \left(1-{\frac{\tau_{\text{a}}} {\tau_{\text{o}}}} \right)}} \left(\exp{{\frac{-t} {\tau_{\text{o}}}}}- \exp{{\frac{-t} {\tau_{\text{a}}}}} \right)\\ T_{\text{o}}(t)&={\frac{f_{\text{o}}} {\tau_{\text{o}}}} \exp{{\frac{-t} {\tau_{\text{o}}}}} . \end{aligned} $$

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Again, the response of the ocean only depends on its own (i.e., slow) response time, while the response of the atmosphere depends on the response times of both, atmosphere and ocean.

The expression for the atmosphere, which consists of two exponentials, agrees with the responses suggested in Hasselman et al. (1993) and Sausen and Schumann (2000). The main difference to the approach presented here is that we do not only obtain these equations as solutions, but also suggest the differential equations by which the system can be described.

Appendix 2: General response using exponential response function

Pulse scenario experiments performed with an AOGCM can be used to deduce the AOGCM response to an arbitrary forcing.

If we assume that the model’s response to a radiative forcing is linear, and if χ^e(t) is the response of a specific variable χ to an exponentially from unity decaying pulse radiative forcing (as described in Eq. 19), then the response to an arbitrary forcing F(t) can be written as

$$ \Uppsi(t)=\left(\chi^e \otimes \left({\frac{1} {\tau_f}} + {\frac{\hbox{d}} {\hbox{d} t}} \right) F \right)(t) = \int\limits_{-\infty}^t \chi^e(t-t')\left({\frac{F(t')} {\tau_f}}+{\frac{\hbox{d} F} {\hbox{d}t}}(t')\right){\hbox{d}}t' . $$

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As the response χ^e(t) is often only known for discrete times, the above integral has to be calculated numerically. If t _i (i =  − ∞, ..., ∞) is a set of equidistant points in time (time interval Δt) then

$$ \Uppsi(t_i)= \Updelta t \sum_{j=-\infty}^{j=i} \chi^e(t_i-t_j) \left({\frac{F(t_j)} {\tau_f}}+{\frac{\hbox{d} F} {\hbox{d}t}}(t_j) \right) . $$

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Appendix 3: Results from emulating CMIP3 AOGCM responses

See Figs. 14, 15, 16. Tables 9, 10, 11, 12, 13, 14, 15

Table 9 Overview of the climate sensitivity parameters for the different CMIP3 models

Table 10 A priori and a posteriori estimates of SCM_READ’s parameters for emulating the data of fifteen different CMIP3 models

Table 11 As Table 10 but for SCM_OSLO

Table 12 As Table 10 but for SCM_TLSE

Table 13 Mean a posteriori correlation matrix for the parameters of SCM_READ when using CMIP3 AOGCM data

Table 14 As Table 13 but for SCM_OSLO

Table 15 As Table 13 but for SCM_TLSE

Fig. 14

Tsurf, OHC, Tland, Tocean, and ocean temperature at 4 different depths for the scenarios Cg2 and Cg4 as simulated by CGCM3.1(T47) (grey symbols) and using the standard parameters (dotted lines) or the fitted parameters (solid lines) for SCM_READ (blue), SCM_OSLO (red) and SCM_TLSE (green)

Fig. 15

As Fig. 14 using CNRM-CM3 data

Fig. 16

Tsurf (left) and OHC (right) for the scenarios Cg2 and Cg4 as simulated by the different CMIP3 models (grey symbols) and using the standard parameters (dotted lines) or the fitted parameters (solid lines) for SCM_READ (blue), SCM_OSLO (red) and SCM_TLSE (green). For the results of CGCM3.1(T47) and CNRM-CM3, see Figs. 14 and 15