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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: [email protected]
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.
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Keywords
Transfer Function; Global Energy Balance; Radiative Forcing; CO2
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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).
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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.
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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
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(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.
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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.
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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)
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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
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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.
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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.
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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
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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,
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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)
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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.
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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.
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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).
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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).
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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.
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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
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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.
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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