LONG RUN SURFACE TEMPERATURE DYNAMICS OF AN A-

OGCM: THE HADCM3 4×CO2 FORCING EXPERIMENT

REVISITED

Sile Li and Andrew Jarvis

5 Lancaster Environment Centre, Lancaster University, Lancaster UK.

Correspondence: Dr. Andrew Jarvis, Lancaster Environment Centre, Lancaster

University, Lancaster, UK. LA1 4YQ.

email: A.Jarvis@lancaster.ac.uk

Tel: +44(0)1524 593280

10 Fax: +44(0)1524 593985

1

Abstract

The global mean surface temperature response of HadCM3 to a 1000 year 4×CO2

forcing is analysed using a transfer function methodology. We identify a third order

transfer function as being an appropriate characterisation of the dynamic relationship

between the radiative forcing input and global mean surface temperature output of this

A-OGCM model. From this transfer function the equilibrium climate sensitivity is

estimated as 4.62 (3.92 – 11.88) K which is significantly higher than previously

estimated for HadCM3. The response is also characterised by time constants of 4.5

(3.2 – 6.4), 140 (78 – 191) and 1476 (564 – 11737) years. The fact that the longest

time constant element is significantly longer than the 1000 year simulation run makes

estimation of this element of the response problematic, highlighting the need for a

significantly longer model runs to express A-OGCM behaviour fully. The transfer

function is interpreted in relation to a three box global energy balance model. It was

found that this interpretation gave rise to three fractions of ocean heat capacity with

effective depths of 63.0 (46.7 – 85.4), 1291.7 (787.3 – 2955.3) and 2358.0 (661.3 –

17283.8) meters of seawater, associated with three discrete time constants of 4.6 (3.2

– 6.5), 107.7 (68.9 – 144.3) and 585.4 (196.2 – 1243.1) years. Given this accounts for

approximately 94 percent of the ocean heat capacity in HadCM3, it appears HadCM3

could be significantly more well mixed than previously thought when viewed on the

millennial timescale.

5

10

15

20

Keywords

Transfer Function; Global Energy Balance; Radiative Forcing; CO2

2

1.0 Introduction

The release of greenhouse gases by humans impose disturbances on the global climate

system, initiating dynamics which play out over a broad spectrum of timescales from

months to millennia (e.g. Watts et al. 1994; Stouffer 2004). Understanding these

dynamics is central to informing the climate change debate and, as a result, a

significant investment has been made in the development of Atmosphere-Ocean

General Circulation Models (A-OGCMs) which attempt to simulate the detail of the

climate system response to anthropogenic forcing (Harvey et al. 1997; McGuffie and

Henderson-Sellers 2001). However, given the complexity of such models, it is often

necessary to summarise their broad dynamic behaviour using measures which are

more amenable to communicating the relevant dynamic information. Invariably this is

achieved using simple proxy models of the A-OGCM calibrated to reproduce aspects

of the A-OGCM behaviour (e.g. Hasselmann et al. 1997; Huntingford and Cox 2000;

Raper et al. 2001; Zickfeld et al. 2004; Meinhaussen et al. 2008) and this has been the

preferred method of obtaining estimates of the equilibrium climate sensitivity of A-

OGCMs in the IPCC-AR4 (Randall et al. 2007).

5

10

15

20

25

30

Simple model proxies of A-OGCMs appear to fall into one of two classes. Several

studies have tuned global energy balance models (GEBMs) to match the response of

A-OGCMs and hence infer globally aggregated climate system properties such as

effective climate sensitivity, equilibrium climate sensitivity and ocean heat uptake

rates (e.g. Raper et al. 2001; Grieser and Schönwiese 2001; Eickhout et al. 2004;

Wigley et al. 2005; Meinhaussen et al. 2008). Alternatively, some have elected to use

entirely black-box response functions calibrations to summarise the output of global

models for the carbon cycle (e.g. Joos et al. 1996); cloud cover, precipitation or sea

level (e.g. Hooss et al. 2001; Enting and Trudinger 2002) and, more specifically, the

global mean surface temperature (GMST) response of A-OGCMs (Hasselman et al.

1997; Hooss et al. 2001; Lowe 2003).

The advantage of using GEBMs to capture the dynamic behaviour of A-OGCMs is

that, in addition to providing model diagnostics which are interpretable (Raper et al.

2001), the structure of the GEBM provides a constraint which helps facilitate

calibration from partially equilibrated A-OGCM runs. This is important because near-

equilibrated fully coupled A-OGCM runs are scarce due to run-time costs. However,

3

when one has near-equilibrated A-OGCM run data then response functions should

also be considered for evaluating A-OGCM dynamics because this affords an

opportunity to apply a more objective data-led methodology for inferring the dynamic

behaviour of the A-OGCM. Using response functions for more than an empirical

description of a system is not widely accepted in the climate literature (Enting 2007).

However, in other disciplines it has been known for some time that, extreme non-

linearity aside, complex dynamic systems can often express dominant modes of

dynamic behaviour in their response to perturbations and that this can be exploited to

derive reduced order interpretations of those systems (Moore, 1981; Godfrey, 1982;

Young et al., 1996; Dowell, 1996; Young, 1999; Tang et al., 2001; Garnier et al.,

2003; Young and Garnier, 2006). Furthermore, because response function parameter

estimation is significantly more straightforward than calibrating GEBMs,

interpretation of the response function parameterisation, when possible, should be less

prone to the effects of bias in the parameter estimates and uncertainty measures for

these parameters should be easier to obtain.

5

10

15

20

25

30

In this paper we analyse the GMST dynamics of a 1000 year, 4×CO2 run of HadCM3

using a response function methodology. These data were originally characterised by

Lowe (2003) using the sum of exponential (SE) response function framework for the

UNFCCC Brazil proposal. However, rather than simply use the response function as

an empirical device to mimic the behaviour of HadCM3, the aim will be to interrogate

the reduced order model we identify and, in particular, explore its properties in

relation to its GEBM counterpart. To facilitate this we exploit a transfer function

framework which, we argue, is amenable to interpreting the dynamic behaviour being

expressed in these data.

2.0 Response function identification and estimation

Figure 1 shows the HadCM3 4×CO2 data in question which are annual average

perturbations in aggregate surface land-ocean temperatures relative to a zero forcing

control run. The forcing data ramp up to 2×3.74 W m-2 over a 70 year period and are

held constant thereafter, where 3.74 W m-2 is the estimated ‘standard’ radiative

forcing associated with a doubling in atmospheric CO2 concentration in HadCM3 (see

Gregory et al. 2004). Taking forcing as the input u(t) and the associated HadCM3

GMST perturbations as the output y(t) the aim here is to find the transfer function H

4

which translates input to output. This is then assumed to represent the dominant

dynamic elements of HadCM3 GMST in response to this forcing. For this we exploit

the following linear, continuous time transfer function structure,

...( )( )( ) ...

−−

−−

+ + + += =

+ + + +

m mm

n nn n

b s b s b s bx sH su s s a s a s a

11 2 1

11 1

m

5

10

15

20

(1)

where u(s) and x(s) are the Laplace transforms of the forcing input u(t) (W m-2) and

GMST perturbation signal output x(t) (K); the a’s and b’s are the transfer function

coefficients and s is the Laplace operator. In the zero initial conditions case

considered in this paper s is simply the derivative operator, sm,n = dm,n/dtm,n, and hence

m and n denote the dynamic order of transfer function. For those who are not familiar

with transfer functions, the relationship between Eq. (1) and sum of exponential

response functions is given in Appendix A along with the relationship of both to their

ordinary differential equation (ODE) parent structures. The interested reader is also

pointed to general texts such as Nise (2004) for a useful introduction to transfer

functions and there use in linear systems analysis.

Lowe (2003) concluded that the data in Figure 1 were adequately captured by the sum

of two first order exponential terms i.e. m = n = 2. Similarly, Hooss et al. (2001) and

Grieser and Schönwiese (2001) both evaluated the GMST dynamics of ECHAM3 to a

2×CO2 forcing and concluded they were also second order (m = n = 2) a , whilst

Hasselmann et al. (1997) concluded that, for the same forcing, ECHAM3 was third

order (m = n = 3), although on close inspection one of the exponential elements they

included had such a short time constant as to be in effect instantaneous and hence m =

n ≈ 2. Therefore, we started by fitting an m = n = 2 transfer function to the data in

Figure 1 i.e.

ˆ( ) ( )b s bx ss a s a

u s+=

+ +1 2

21 2

(2)

a In Grieser and Schönwiese (2001) the cascade model they use has three layers, but the top

atmospheric layer has no inertia, hence making their system second order m = n = 2.

5

To estimate H(s) we used a least squares gradient search for the a’s and b’s. For this

x̂ (t) is simulated using the lsim tool in Matlab with a first order hold on u(t) over the

annual sample interval. The response error residuals e(t) = y(t) – x̂ (t) were found to

be significantly autocorrelated (partial correlation at lag 1 year = 0.4556 ± 0.0302).

Therefore, to avoid any bias this would introduce to the estimates of the a’s and b’s

we modelled the noise as AR(1) and used this model to whiten the residuals being

minimised. These whitened residuals were neither autocorrelated or cross correlated

with either u(t) or y(t) and passed a standard Lillefors test for normality at a 95

percent significance level. The values of the a’s and b’s in Eq. (2) are provided in

Table 1a.

5

10

15

20

25

We rejected the m = n = 2 transfer function structure for two related reasons. From

Figure 2 we see that, despite passing a test for normality, the model residuals have a

significant non-zero trend. More specifically, this trend is particularly pronounced

after 500 years reflecting the inability of this structure to account for the dynamics in

the tail of the HadCM3 response. In affect, the m = n = 2 response function predicts

the data are approaching equilibrium after 1000 years. Although this appears possible

in Figure 1, it is unlikely to be the case. In Figure 3 we have redrawn the GMST data

compressing the time axis using a log scale to remove the exponential character of the

tail of the response. From Figure 3 it is clear that the probability that HadCM3 is

approaching equilibrium after 1000 years is low and, therefore, either the response is

non-stationary, or higher order. We assume the later, although it must be appreciated

that non-stationarity in the HadCM3 response is a distinct possibility, especially if

drift in the control run has not been removed completely.

Because the m = n = 2 case accounts for more than 99% of the variance in y(t) there is

clearly not much signal remaining on which to constrain any additional dynamics and

we found that the m = n = 3 transfer function was the highest order response we could

estimate from these data i.e.

ˆ( ) ( )b s b s bx ss a s a s a

+ +=

+ + +1 2 3

3 21 2 3

u s2

(3)

6

The estimated parameter values for Eq. (3) are given in Table 1b. From Figures 2 and

3 we see that Eq. (3) appears to capture the HadCM3 response better than the m = n =

2 case, particularly its tail.

Because the HadCM3 response is still some distance from equilibrium after 1000

years these estimates of the a’s and b’s are rather uncertain, particularly a3 (see Table

1b), highlighting the need to run A-OGCMs for several thousand years in order to

express these dynamics fully (Stouffer and Manabe 1999; Raper et al. 2002). To carry

this uncertainty forward in our analysis we drew 104 parameter combinations from the

estimated covariance structure given in Table 1b and then culled all unstable

5

10

15

20

a

parameter combinations given these were deemed to be physically untenable (Roe and

Baker 2007). The following discussions are based on the remaining 7×103 stable

parameter combinations.

3.0 Response function diagnosis

For a stable response, at equilibrium all derivatives are zero and so sm,n = 0. Therefore,

the equilibrium gain G of Eq. (3) is given by b3/a3 = 1.2352 (1.0470 - 3.1756)b K (W

m-2). For a 2×CO2 forcing of 3.74 W m-2 this gives an equilibrium climate sensitivity

S = 3.74G K = 4.62 (3.92 – 11.88) K. This is significantly higher than the estimate of

3.79 (3.74 – 3.85) K obtained from the m = n = 2 transfer function and the 3.3 K

offered for HadCM3 in Randal et al. (2007). However, it is indistinguishable from the

4.1 ± 0.1 K estimated by Gregory et al. (2004) for this same HadCM3 run using a

technique based on linear regression of GMST on the net radiative flux at the

tropopause, although we note that their estimate falls in the lower 40th percentile of

our estimate, the probability density of which is shown in Figure 4. Note how the

mode and best estimate of S differ due to the asymmetry arising from the lower limit

a A rational continuous time transfer function is stable only if none of its poles lies in the right hand

portion of the complex s plane. That is to say, for a stable response (i.e., an impulse response that

decays to zero) each pole must be less than zero. For a conservative response (i.e., an impulse response

that stabilises at a non-zero value) one or more poles may be equal to zero and if any pole is greater

than zero then the impulse response is unstable and will grow exponentially. b The range given in the brackets is the 95 percent confidence interval derived from the 7x103

ensemble.

7

constraint placed on S by the data (for a stable response, the climate sensitivity cannot

be below the 3.6 K warming observed after 1000 years). The upper limit is much less

constrained by the data due to the HadCM3 response being un-equilibrated, hence the

long tail in the distribution for S. Given the transfer function Eq. (3) does not include

any explicit feedback process this uncertainty is not due to feedbacks (Roe and Baker

2007) but rather the lack of equilibration in the data.

5

10

15

20

25

30

Because m = n, Eq. (3) can be expressed in an equivalent sum of first order

(exponential) form through partial fraction decomposition of this transfer function.

This gives three first order elements arranged in parallel, each with associated time

constants and gains. The partial fraction decomposition parameter values are

presented in Table 2 which reveals that, when expressed in this way, the system is

stiff i.e. it is characterised by a broad range of time constants; 4.5 (3.2 6.4), 140 (78.23

191.0) and 1476 (564.1 11737) years. Again, note the longest time constant element

of 1476 years is highly uncertain highlighting the inadequacy of the 1000 year data

series in fully constraining the estimate of this dynamic element. From Figure 3 we

can see that a run of more than 5000 years would be needed for this. The longest time

constant is largely determined by a3 in Eq. (3), hence the uncertainty in this parameter

estimate in particular. However, not all this uncertainty is translated into the estimates

of the equilibrium G because of the strong covariance between a3 and b3 (see Table

1b), indicating that the data, although not ideal, usefully constrain the ratio b3/a3.

Using the partial fraction decomposition, G and hence S can be partitioned across the

three time constant timescales thus; 43 percent for the 4.5 year response; 18 percent

for the 140 year response and 39 percent for the 1476 year response, which is useful

for gauging the relative importance of these timescales following a disturbance in

radiative forcing, anthopogenic or otherwise.

4.0 Response function interpretation

As touched on in section 1.0, there is a possibility that Eq. (3) is not just a black box

description of the HadCM3 response, but instead may capture some of the emergent

energy balance characteristics of this A-OGCM. We can explore this by considering

further decompositions of the aggregate third order system Eq. (3) into constituent

first order components. In particular, the one decomposition that appears to make

8

sense to explore is the layer-cascade structure because this readily maps onto a three

box GEBM.

Consider the following three box GEBM ode system (e.g. Dickinson 1981),

( ) ( ) ( ) { ( ) ( )}dx tc u t x t k x t xdt

λ= − − −11 1 1 1 t2 (4a)

( ) { ( ) ( )} { ( ) ( )}dx tc k x t x t k x t xdt

= − − −22 1 1 2 2 2 t35 (4b)

c3

dx3(t)dt

= k2{x2(t) − x3(t)} (4c)

where xi(t) (K) are the aggregate temperatures of each box; ki (Wm-2K-1) the effective

heat exchange coefficients between boxes; ci (Wm-2K-1) the effective heat capacities

of each box and λ = G-1 (W m-2 K-1). Assuming Eq. (4a) describes GMST dynamics

i.e. x1(t) = x(t), by taking the Laplace transform of Eq. (4) assuming zero initial

conditions this ODE system can be rearranged into Eq. (3) where,

10

b1 =1c1

(5a)

c1c2c3

b2 =k2c3+ k2c2 + k1c3 (5b)

b3 =k2k1

c1c2c3

(5c)

c1c2c3

a1 =k2c3c1+ k1c3c1 + k2c2c1 + c2c3k1 + c2c3λ (5d) 15

a2 =k2c3λ+ k1c2k2 + k1c3λ + k2c3k1 + k1k2c1 + c2k2λ

c1c2c3

(5e)

9

a3 =k2k1λc1c2c3

(5f)

The physical parameters ki, ci and λ can be retrieved by solving the five simultaneous

equations Eq. (5a – f) using the transfer function parameters in Table 1b. These are

given in Table 3.

5 Assuming the thermal inertia of each of the three boxes is dominated by the thermal

properties of sea water then,

=Δ

p w ii

c dc

ρ (6)

where cp and ρw are the specific heat capacity and density of sea water (3989.8 J Kg-1

K-1 and 1025.98 Kg m-3), Δ is the number of seconds in each annual time sample

(31104000 year-1)a and di is the effective depth in metres of each oceanic box. This

yields effective depths of each box of 63.0 (46.7 – 85.4) m, 1291.7 (787.3 – 2955.3) m

and 2358.0 (661.3 – 17283.8) m respectively.

10

15

20

An effective depth of approximately 63 m is in accord with values generally

employed to describe an aggregate land-atmosphere-well mixed surface ocean

compartment (e.g. Hoffert et al. 1980; Schneider and Thompson 1981; Harvey and

Schneider 1985; Wigley and Raper 1987). Because GMST is comprised of the

composite effects of the atmosphere, land and well mixed surface ocean thermal

inertia, the heat capacity of this aggregate box will be distorted by surface ocean-land-

atmosphere feedbacks (Dickinson 1981). This distortion can be accounted for by

comparing the surface ocean and atmosphere-land-surface ocean energy balances

which gives (see Appendix B),

1 1 o as

s o as

f kdd f k

λβ

+= (7)

a The HadCM3 systematically uses artificial calendar that consists of 360 days in 12 months of 30 days

each. The 360-day calendar is used in long climate simulations for internal organizational convenience.

10

where ds (m) is the well mixed surface ocean depth, f0 is the fraction of earth surface

that is sea, kas (Wm-2K-1) is the atmosphere-sea exchange coefficient and β is the

proportionality between GMST and the surface ocean mixed layer temperature, which

is 1.475 for this experimenta. If kas >> λ, because the rate of temperature-dependent

energy exchange of sensible and latent fluxes at the ocean atmosphere interface is

large (e.g. Dickinson 1981; Wigley and Schlesinger 1985; Harvey 2000), then ds ≈

βd1 = 92.9 (68.9 – 126.0) m which is consistent with previous studies that variously

place the global average well mixed surface ocean depth in the range 50 – 150 m (e.g.

Hoffert et al. 1980; Wigley and Raper 1987; Watterson 2000; Grieser and Schönwiese

2001).

5

10

15

20

25

The three oceanic boxes collectively provide a total effective heat capacity equivalent

to 92.9 + 1291.7 + 2358.0 = 3742.6 (1626.7 – 20282.2) m of seawater, associated with

three time constants of 4.6 (3.2 – 6.5), 107.7 (68.9 – 144.3) and 585.4 (196.2 – 1243.1)

years respectively. It transpires that the estimate of the total effective heat capacity is

very sensitive to the estimate of b1, hence we must be slightly cautious in any

inference we may draw from these results. However, if the global average ocean depth

is of the order of 4000 m (Kester 2006), an estimate of 3743 m would represent

approximately 94 percent of the of the heat capacity in HadCM3. This result presents

somewhat of a paradox because it infers that the poorly mixed ocean HadCM3 is

designed to model maps relatively well to three well mixed compartments configured

as a layer cascade. It would be tempting to offer ‘surface’, ‘intermediate’ and ‘deep’

as labels for these three boxes, and indeed, from the previous paragraph we might be

relatively comfortable with ‘surface’ as one of the labels. However, it would be naive

to label the two remaining fractions ‘intermediate’ and ‘deep’, not least because of the

simplistic nature of the interconnections specified in the layer cascade in relation to

the circulation patterns described in HadCM3.

An alternative way of viewing the aggregation implied in Eq. (4b and c) would be to

consider this not in space (i.e. layers) but in time. Ocean circulation processes that

a β is estimated by regressing GMST on the grid aggregation of the surface ocean mixed layer

temperature (Parker et al. 1995).

11

have similar response timescales but not necessarily similar locations or dimensions

could aggregate to form distinct distributions of timescale effects. It can be shown that

providing these distributions are symmetric, then they can always be represented by

their first moment (Li 2009), these would be 108 and 585 years for HadCM3.

Upwelling-diffusion as a process does not generate symmetric timescale distributions

(see Kirchner et al. 2001). However, circulation, which implies some form of return

flow or feedback, does (Li 2009). Given there is significant quantities of circulation in

the ocean component of general circulation models like HadCM3, one could envisage

this could be sufficient to account for the aggregation we have observed. The only

other condition implied by the layer cascade is that the longer 585 year timescale

circulation effects interact with the intermediate 108 year timescale circulation effects

and not the well mixed surface ocean. Because of this somewhat unusual

interpretation of the behaviour of the ocean in HadCM3 the exchange coefficients ki

become difficult quantities to interpret in this framework.

5

10

15

20

25

30

5.0 Discussion and conclusion

Clearly 1000 years is not enough time to fully express the spectrum of dynamic

behaviour of HadCM3 and stabilization runs of the order of 5000 years are needed for

this. The consequence of using the partially equilibrated data in a response function

framework is that the response function estimates we have derived of the A-OGCM

dynamics are rather uncertain. However, this uncertainty can be quantified and it

appears that it is not so large as to render the results meaningless.

The transfer function framework we have exploited is linear. It is not uncommon that

complex models exhibit locally linear behaviour for small perturbations about a set

point. Whether a four-fold increase in atmospheric CO2 burden can be considered

small is a matter of debate, but it still remains somewhat surprising that the GMST

response of HadCM3 to this excitation is so well captured by such a simple model as

Eq. (3), and suggests that the important nonlinearities in this A-OGCM are either not

heavily excited by this disturbance or tend to cancel out. As mentioned in the

introduction, this is not a unique result given there are a number of examples where

linear response functions have successfully captured A-OGCM dynamics.

The estimate of 3.3 K offered for S for HadCM3 in Randal et al. (2007) appears

incompatible with the data analysed here which exceed this sensitivity even before 12

1000 years. If our estimate for S of 4.62 (3.92 – 11.88) K is indeed more accurate then

obviously this has important consequences for assessing the risks attached to

anthropogenic forcing of climate, although the millennial timescale associated with

full equilibration of GMST to forcing is somewhat beyond the scope our current

political machinery. 5

10

15

20

25

30

We appreciate that the well-mixed paradigm used in this paper for the interpretation

of the ocean energy balance is at variance with the contemporary view of this system.

Such a view was prevalent before the early 80’s when well mixed (albeit two fraction)

GEBMs were the norm (Schneider and Thompson 1981; Gilliland 1982). It is

interesting to chart the evolution of ocean heat modelling subsequently. Seminal

papers such as Harvey and Schneider (1985) steered the paradigm away from well

mixed 0d models toward the 1d poorly mixed up-welling diffusion models, and

subsequent increases in computing resources allowed for more ‘realistic’ 2d and later

3d fluid dynamic models to be developed (e.g. Henderson-Sellers and McGuffie 1987;

Claussen et al. 2002). The evolution toward fluid dynamic modelling of the ocean has

inevitably involved the incorporation of descriptions of circulation. We argue that it is

this circulation, and the local feedbacks this implies, that leads to the emergence of

the simpler box-type behaviour we have observed through the transfer function

framework. Rather than viewing such interpretations as being in violation of our

current understanding of how the poorly mixed ocean should behave, we would

suggest that this offers an opportunity to develop new model diagnostics which

characterise the emergent properties of the ocean heat transport system. Such

diagnostics could prove valuable in refining A-OGCMs, particularly in relation to

global scale observations. Further research on the long run dynamics of A-OGCMs is

needed to resolve this.

Acknowledgement

We would like to express our gratitude to Tim Johns in the Hadley Centre who

provided both HadCM3 simulation data and useful discussion.

References

Claussen M et al (2002) Earth system models of intermediate complexity: closing the

gap in the spectrum of climate system models. Climate Dynamics 18: 579-586.

13

Cubasch U et al (2001) Projections of future climate change. Climate Change 2001:

The Scientific Basis. Contribution of Working Group I to the Third Assessment

Report of the Intergovernmental Panel on Climate Change, Houghton JT et al.

(Eds) Cambridge University Press, 526–582.

5

10

15

20

25

Dickinson RE (1981) Convergence Rate and Stability of Ocean-Atmosphere Coupling

Schemes with a Zero-Dimensional Climate Model. Journal of the Atmospheric

Sciences 38: 2112-2120.

Eickhout B, den Elzen MGJ, Kreileman GJJ (2004) The Atmosphere-Ocean System

of IMAGE 2.2 A global model approach for atmospheric concentrations, and

climate and sea level projections. RIVM report 481508017/2004.

Enting IG, Trudinger CM (2002) Modelling earth system change: I. Validating

parameterisations for attribution calculations. CSIRO Atmospheric Research

Technical Paper 56.

Enting IG (2007) Laplace transform analysis of the carbon cycle. Environmental

Modelling & Software 22: 1488-1497.

Gilliland RL (1982) Solar, volcanic, and CO2 forcing of recent climatic changes.

Climatic Change 4: 111-131.

Gregory JM, Ingram WJ, Palmer MA, Jones GS, Stott PA, Thorpe RB, Lowe JA,

Johns TC, Williams KD (2004) A new method for diagnosing radiative forcing

and climate sensitivity. Geophysical Research Letters 31: L03205,

doi:10.1029/2003GLO18747.

Grieser J, Schönwiese C-D (2001) Process, forcing, and signal analysis of global

mean temperature variations by means of a three-box energy balance model.

Climatic Change 48: 617-646.

Harvey LDD, Schneider SH (1985) Transient climate response to external forcing on

100 – 104 year time scales part 1: Experiments with globally averaged, coupled

atmosphere and ocean energy balance models. Journal of Geophysical Research

90: 2191 – 2205

14

Harvey LDD, Gregory J, Hoffert M, Jain A, Lal M, Leemans R, Raper S, Wigley T,

de Wolde J, (1997) An introduction to simple climate models used in the IPCC

Second Assessment Report. IPCC Technical Paper 2.

Harvey LDD (2000) Global Warming: The Hard Science. Prentice Hall, Edinburgh.

pp. 233. 5

10

15

20

25

Hasselmann K, Hasselmann S, Giering R, Ocana V, Storch HV (1997) Sensitivity

study of optimal CO2 emission paths using a simplied structural integrated

assessment model (SIAM). Climatic Change 37: 345-386.

Hoffert MI, Calligari AJ, Hseich C-T (1980) The role of deep sea heat storage in the

secular response to climate forcing. Journal of Geophysical Research 85: 6667–

6679.

Henderson-Sellers A, McGuffie K (1987) A climate modeling primer. John Wiley,

Chichester, pp. 55.

Hooss G (2001) Aggregate models of climate change: development and applications.

PhD thesis. Max-Planck-Institut.

Hooss G, Voss R, Hasselman K, Maier-Reimer E, Joos F (2001) A nonlinear impulse

response model of the coupled carbon cycle-climate system (NICCS). Climate

Dynamics 18: 189-202.

Huntingford C, Cox PM (2000) An analogue model to derive additional climate

change scenarios from existing GCM simulations. Climate Dynamics 16: 575-

586.

Jarvis AJ, Young PC, Taylor CJ, Davies WJ (1999) An analysis of the dynamic

response of stomatal conductance to a reduction in humidity over leaves of

Cedrella odorata. Plant Cell and Environment 22: 913-924.

Joshi MM, Gregory JM, Webb MJ, Sexton DMH, Johns TC (2008) Mechanism for

the land/sea warming contrast exhibited by simulations of climate change.

Climate Dynamics 30: 455-465.

15

Joos F, Bruno M, Fink R, Siegenthaler U, Stocker TF, LeQuere C (1996) An efficient

and accurate representation of complex oceanic and biospheric models of

anthropogenic carbon uptake. Tellus 48B: 397-417.

Kester, DR. (2006) The World Oceans, World Book Encyclopedia Volume 14.

Kirchner JW, Feng X and Neal C (2001). Catchment scale advection dispersion as a

mechanism for fractal scaling in stream tracer concentrations. Journal of

Hydrology, 254, 82-101.

5

10

ml

Li S (2009). Simplicity from complexity: An Earth systems perspective. PhD thesis.

Lancaster University, UK.

Li S, Jarvis AJ and Leedal D (2008) Are response function representations of the

global carbon cycle ever interpretable? Tellus B: DOI: 10.1111/j.1600-

0889.2008.00401.x.

Lowe J (2003) Parameters for tuning a simple climate model. (accessed

from http://unfccc.int/resource/brazil/climate.ht ).

15

20

25

McGuffie K, Henderson-Seller A (2001) Forty years of numerical climate modeling.

International Journal of Climatology 21: 1067-1109.

Murphy JM (1995) Transient response of the Hadley Centre coupled ocean-

atmosphere model to increasing carbon dioxide: Part III. Analysis of global-mean

response using simple models. Journal of Climate 8: 496-514.

Nise NS (2004). Control Systems Engineering. John Wiley, New York.

Parker DE, Jones PD, Bevan A, Folland CK (1995) Interdecadal changes of surface

temperature since the late 19th century. Journal of Geophysical Research 99:

14373-14399.

Price L, Young PC, Berckmans D, Janssens K, Taylor CJ (1999) Data-Based

Mechanistic Modelling (DBM) and Control of Mass and Energy transfer in

agricultural buildings. Annual Reviews in Control 23: 71-82.

16

Randall DA et al. (2007) Climate Models and Their Evaluation. In Climate Change

2007: The Physical Basis of Climate Change. Contribution of Working Group I

to the Fourth Assessment Report of the IPCC, edited by Solomon S et al.

Cambridge and New York: Cambridge University Press, pp. 589-662.

5

10

15

20

25

Raper SCB, Gregory JM, Osborn TJ (2001) Use of an upwelling-diffusion energy

balance climate model to simulate and diagnose A/OGCM results. Climate

Dynamics 17: 601-613.

Raper SCB, Gregory JM, Stouffer RJ (2002) The role of climate sensitivity and ocean

heat uptake on AOGCM transient temperature response. Journal of Climate 15:

124-130.

Roe GH, Barker MB (2007) Why is climate sensitivity so unpredictable? Science 318:

629-632.

Schneider SH, Mastrandrea MD (2005) “Probabilistic assessment of ‘dangerous’

climate change and emissions pathways.” Proceedings of the National Academy

of Sciences 102: 15728-15735.

Schneider SH, Thompson SL (1981) Atmospheric CO2 and climate – Importance of

the transient response. Journal of Geophysical Research 86: 3135-3147.

Sellers PJ, Dickinson RE, Randall DA, Betts AK, Hall FG, Hall FG, Berry JA,

Collatz GJ, Denning AS, Mooney HA, Nobre CA, Sato N, Field CB, Henderson-

Sellers A, (1997) Modeling the exchanges of energy, water, and carbon between

continents and the atmosphere. Sciene 275: 502-509.

Stouffer RJ, Manabe S (1999) Response of a coupled ocean-atmosphere model to

increasing atmospheric carbon dioxide: sensitivity to the rate of increase. Journal

of Climate 12: 2224-2237.

Stouffer RJ (2004) Time scales of climate response. Journal of Climate 17: 209-217.

Watterson IG (2000) Interpretation of simulated global warming using a simple model.

Journal of Climate 13: 202-214.

17

18

5

10

15

Watts RG, Morantine MC, Achutarao K. (1994) Timescales in energy balance climate

models. 1. The limiting case solutions. Journal of Geophysical Research 99:

3631-3641.

Wigley TML, Schlesinger ME (1985) Analytical solution for the effect of increasing

CO2 on global mean temperature. Nature 315: 649-652.

Wigley TML, Raper SCB (1987) Thermal expansion of sea water associated with

global warming. Nature 330: 127-131.

Wigley TML, Ammann CM, Santer BD, Raper SCB (2005) Effect of climate

sensitivity on the response to volcanic forcing. Journal of Geophysical Research

110 (D9): Art. No. D09107.

Young PC, Parkinson S, Lees MJ (1996) Simplicity out of complexity: Occam's razor

revisited. Journal of Applied Statistics 23: 165-210.

Young PC, Garnier H (2006) Identification and estimation of continuous-time, data-

based mechanistic models for environmental systems. Environmental Model &

Software 21: 1055-1072.

Zickfeld K, Slawig T, Rahmstorf S (2004) A low-order model for the response of the

Atlantic thermohaline circulation to climate change. Ocean Dynamics 54: 8-26.

Appendix A. The Equivalence between transfer function, sum of

exponential and ordinary differential equation representations of linear

dynamic systems.

In Eq. (1) we presented the generic linear, continuous time transfer function structure

with zero initial conditions i.e., 5

...( )( )( ) ...

−−

−

−+ + + += =

+ + + +m

n nn n

b s b s b s bx sH su s s a s a s a

1 2 11

1 1

m mm

1

n

(A1)

If we take sm,n = dm,n/dtm,n in this zero initial condition case then Eq. (A1) can be re-

arranged to give a generic linear, continuous time ordinary differential equation of the

form,

d x(t)

dtn + a1

d x(t)dtn−1

n−1

+ ...+ anx(t) = b0

d u(t)dtm

m

+ ...+ bmu(t) (A2) 10

For the m = n case, Eq. (A1) can also be re-expressed in a partial fraction expansion

form,

( ) n

n

rr rH ss p s p s p

= + + +− − −

1 2

1 2

… (A3)

where pi {i = 1…n} are the poles of H(s). Providing pi < 0 and real, then -pi-1 are the

time constants (or e-folding times) Ti of each first order element in Eq. (A3). The

equilibrium gains (or amplitudes) Gi of each first order element are simply -ri/pi.

15

n

20

Taking the inverse Laplace transform L-1 of Eq. (A3) for this m = n case gives the sum

of exponentials response function,

(A4) L−1{H (s)}= riepit

i=1∑

19

Appendix B. The relationship between surface ocean mixed layer heat

capacity (cs) and coupled atmosphere-surface ocean mixed layer heat

capacity (c1).

Assuming the thermal inertia of the atmosphere is approximately zero and ignoring

any deep ocean feedbacks, the atmosphere and surface ocean mixed layer energy

balances can be written in the following TF forms,

5

( ) { ( ) ( )}ao as

o as sx s u s f kf kλ

= ++

x s1 (B1a)

{ } ( ) ( ) ( ) ( )s s o as a s sd sc s x s f k x s x s k x s⋅ ⋅ = − −

10

(B1b)

where xa is the atmospheric temperature response. Inserting (B1a) into (B1b), one

obtains

xs(s) = 1cs

fokas

λ + fokas

⋅ s +fokas(λ + ksd ) + λ ⋅ ksd

fokas

u(s) (B2)

Likewise, expressing the coupled atmosphere-surface ocean mixed layer energy

balance in its TF form, again ignoring any deep ocean feedback (c.f. Eq. (5a)) gives,

11 1

( ) ( )( )1x s

c s k λ=

+ +u s

15

(B3)

Assuming xa xs (Hoffert et al. 1980; Parker et al. 1994; Eickhout et al. 2004; Joshi et

al. 2008) then x1 = βxs, which holds for the HadCM3 experiment considered here.

Then, equating βc1 in Eq. (B3) with cs/f0kas/(λ+f0kas) in Eq. (B2) gives,

∝

20

c1

cs

=d1

ds

=1βλ + fokas

fokas

(B4)

21

Table 1. The parameter values for Eq. (2) (a) and Eq. (3) (b) derived from least-squares fitting to the data in Figure 1. The figures in parentheses are estimated standard deviations. A discrete time AR(1) noise model was used to account for serial correlation in the model residuals y(t) – x(t) with the AR(1) parameter included within the optimisation scheme (4.3865×10-1 (7.4473×10-4)). Also shown below is the associated covariance matrix of the parameter estimates, the roots of the diagonal of which are the estimated standard deviations of each parameter.

5

a.

b1 = 7.6153×10-2

(3.7595×10-5) a1 = 1.2989×10-1

(1.2450×10-4) b2 = 3.1667×10-4

(1.2776×10-9) a2 = 3.1229×10-4

(1.3428×10-9) b1 b2 a1 a2

3.7595× 10-5 2.0063× 10-7 6.8159 × 10-5 2.0193× 10-7

2.0063× 10-7 1.2776 × 10-9 3.7894 × 10-7 1.3084 × 10-9

6.8159 × 10-5 3.7894 × 10-7 1.2450 × 10-4 3.8269 × 10-7

2.0193× 10-7 1.3084 × 10-9 3.8269 × 10-7 1.3428 × 10-9

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

10

b.

b1 = 1.2063×10-1 (2.0574×10-2) a1 = 2.2952×10-1 (4.5118×10-2) b2 = 1.3446×10-3 (6.6392×10-4) a2 = 1.7346×10-3 (1.0350×10-3) b3 = 1.3220×10-6 (2.1022×10-6) a3 = 1.0703×10-6 (2.0608×10-6) b1 b2 b3 a1 a2 a3

-4 -5 -8 -4 -5 -8

-5 -7 -9 -5 -7 -9

-8 -9 -12 -8 -9 -1

4.2327 10 1.0277 10 2.2308 10 9.1774 10 1.4969 10 2.0837 101.0277 10 4.4079 10 1.3106 10 2.5026 10 6.8503 10 1.2677 102.2308 10 1.3106 10 4.4191 10 5.8683 10 2.0942 10 4.3292 10

× × × × × ×

× × × × × ×

× × × × × × 2

-4 -5 -8 -3 -5 -8

-5 -7 -9 -5 -6 -9

-8 -9 -12 -8 -9 -1

9.1774 10 2.5026 10 5.6883 10 2.0356 10 3.7002 10 5.5325 101.4969 10 6.8503 10 2.0942 10 3.7002 10 1.0711 10 2.0313 102.0837 10 1.2677 10 4.3292 10 5.5325 10 2.0313 10 4.2467 10

× × × × × ×

× × × × × ×

× × × × × × 2

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

22

Table 2. SE response function R(t) inferred from inverse Laplace transform of H(s) derived from the partial fraction decomposition of Eq. (2). L-1 denotes the inverse of Laplace transform. The figures in parentheses denote the 95 percent parameter confidence interval generated by means of 7×103 stable model random draws from the covariance matrix structure in Table 1. ri are residues, pi are poles and Ti are time constants, i.e. 1/-pi.

5

L−1{H (s)}= L−1{

Gi

Tis +1i=1

n

∑ }= R(t) = riepit

i=1

n

∑

G1 = 0.1188 (0.0861 0.1615) T1 = 4.510 (3.191 6.436) G2 = 0.0015 (0.0010 0.0020) T2 = 140.3 (78.23 191.0) G3 = 0.0003 (0.0002 0.0007) T3 = 1476 (564.1 11737)

23

Table 3. The layer cascade and GEBM parameter values derived analytically from Eq. (3) and the associated parameter values in Table 1. The figures in parentheses denote 95 percent uncertainty range again derived from 7×103 ‘stable’ parameter sets drawn from the covariance matrix in Table 1b.

Model Representation Parameter values

Layer cascade

G1 = 0.55240 (0.52480 0.57876)

G2 = 0.63434 (0.53960 0.68545)

G3 = 0.57710 (0.44178 0.87288)

T1 = 4.5793 (3.2339 6.5448)

T2 = 107.71 (68.853 144.29)

T3 = 537.05 (196.16 1243.1)

GEBM

c1 = 8.2898 (6.1088 11.399)

c2 = 170.02 (103.62 399.91)

c3 = 310.35 (82.907 2351.7)

λ = 0.8096 (0.3149 - 0.9551)

k1 = 1.0007 (0.81196 1.4924)

k2 = 0.57787 (0.32509 1.9105)

5

24

Figure 1. The 4×CO2 radiative forcing u(t) (dashed) and HadCM3 global mean temperature perturbation y(t) (•). Also shown is the calibrated output x(t) of the response functions Eq. (2) (grey solid) and Eq. (3) (black solid) fitted to these data. 5

25

Figure 2. Standard error bounds of the model residuals from Figure 1 for both Eq. (2) (grey) and Eq. (3) (black solid).

5

26

Figure 3. GMST data and response functions redrawn from Figure 1.

27

Figure 4. The estimated probability density distribution for the HadCM3 equilibrium climate sensitivity S associated with the transfer function Eq (3). The estimates were derived from 7×103 ‘stable’ random draws of a3 and b3 from the covariance matrix in Table 1b. 5

28