Improved regional temperature predictions
from a minimalist model
Karen
and Peter
McKinnon1
Huybers1
Department of Earth and Planetary Science, Harvard University
1
The general circulation models (GCMs) used to simulate future climate are under-constrained,
1, 2as evidenced by the fact that factor-of-two changes in rates of ocean heat uptake, magnitudes of radiative forcing anomalies3, 4, and the strength of various feedbacks5–7 can
be
tr
a
d
e
d
of
f
a
g
ai
n
st
o
n
e
a
n
ot
h
e
r
to
yi
el
d
r
e
s
ul
ts
th
at
al
l
a
g
r
e
e
.
T
hi
s
in
determinacy, although inevitable given finite observations and incomplete theory,
prompts
the question of whether retaining many of the inherently under-constrained
processes represented within GCMs is useful for predicting future climate. Here we present a
simple
energy balance model (EBM) that is tuned only to the seasonal cycle and driven
by the
10, 11same
radiative perturbations as each of 18 GCMs, and show that it
reproduces the
spatial and temporal record of surface temperature between 1850 and 2009 better
than
any of the GCMs or their ensemble average. A major discrepancy appears in the
Arctic,
where the GCMs give more warming than observed or produced by the EBM. On
the
basis of a better fit with the historical observations, and noting that both the GCMs
and
EBM predict warming patterns that are essentially linear extrapolation of their
historical
anomaly patterns, we suggest that the EBM’s predictions of surface temperature
are more
credible.
1
Model complexity can retard predictive capacity when parameters are
under-constrained.
As an example, consider the case of extrapolating a third-order polynomial from noisy
observations. Greater predictive accuracy can often be obtained by fitting and extrapolating a secondorder polynomial to the data, even though the realizations come from a higher-order process,
because inclusion of the poorly constrained third-order term leads to larger prediction error
(i.e. the bias-variance tradeoff, Fig. S1). In the case of climate models, the predictions associated with more complete models will tend to be more variable because of under-constrained
parameters, whereas those associated with simple models will be biased because of missing
processes. Although the scope of the climate problem admits for construction of arbitrarily
complex models, optimal prediction requires some trade-off between completeness and parsimony.
To explore a more parsimonious approach, we posit a simple yet spatially resolved energy balance model (EBM), constrain the small number of free model parameters by fitting
to the observed seasonal cycle of surface temperature, and then use the constrained EBM to
predict inter-annual surface temperature variability. This approach is unique among applications of simple models for prediction of climate8, because the spatial dependence is
12–14
wholly
constrained by fitting to the seasonal cycle. The seasonal cycle has been shown to offer some
constraint upon GCM parameters15, 16, but no simple relationship has been deduced between
17how
well GCMs fit observations and their climate sensitivity. This situation may arise be-
cause of the large number of under-determined processes represented within GCMs, making
it interesting to explore constraining parameters using the seasonal cycle in a relatively simple
model.
2
Temperatures in each
5◦
◦x
5
grid box are simulated by four equations representing
a
single atmospheric layer that radiatively exchanges energy with land and surface ocean endmembers, and where the latter communicates with a deep ocean box (see methods). Only
two
spatially adjustable parameters are used for fitting the EBM to the seasonal cycle: a heat flux
convergence term, F c( x, y ) , and a mixing ratio between land and surface-ocean
temperatures,
m ( x, y ) , that represents the degree of continentality18. This fit captures the majority of the
structure of the climatological seasonal cycle (1961–1990) of Earth’s surface
temperature—the
spatial average Pearson’s correlation is =0. 82. The seasonal cycle over Northern
r2
Hemisphere continents is almost perfectly reproduced, whereas the model fairs more poorly along
the sea-ice edge in the Southern Ocean and atop the East Antarctic Plateau, and completely
fails in the deep tropics where the relative influence of the seasonal cycle of insolation upon
temperature is small compared to the influence of tropical dynamics.
One test of the EBM’s predictive skill is to compare the inferred values of the heat flux
convergence, F c, against satellite-derived observations19. Summing F c across longitude and
then taking the cumulative sum between the poles gives a combined ocean-atmosphere
heat
flux that agrees within the error of the satellite
(Fig. 1a). Furthermore,
observations20
whereas
the satellite observations only conserve global energy to within 6 W/m2, the model energy
fluxes balance to within 1 W/m2, without having been explicitly constrained to do so. The
model’s pattern of heat flux convergence also appears physically reasonable, with the largest
losses of heat occurring in eastern oceanic upwelling zones and the largest gains in the
vicinity
of the northern North Atlantic (Fig. S3). An obvious error again occurs over the East Antarctic
Plateau, where heat loss is inferred, resulting from the use of a globally constant albedo and
an
3
inability to generate atmospheric inversions.
The seasonal fit also affords estimates of continentality and the basic sensitivity to radiative perturbations. The most continental conditions are associated with subtropical arid
regions
and the western interiors of major continents in Northern mid-latitudes (Fig. 1b), and these regions will respond most rapidly to radiative perturbations. The basic sensitivity to radiative perturbations is diagnosed from the model by specifying a perturbation in downward atmospheric
radiative forcing and allowing the ocean and land components to re-equilibrate (Fig. 1c).
Radiative sensitivity averages 0.31
21) globally, consistent with that found in GCMs, and
K/(W/m2
is
larger in cold regions because the magnitude of the negative Planck feedback is smaller22.
That
is, if the radiation absorbed by the surface is perturbed by d F , the change in temperature
needed
to reestablish equilibrium, d T , is greater in cold regions, d F = d T 4 s T3, atleast insomuch
as
blackbody radiation holds and where T is in Kelvin.
To apply the EBM toward predicting interannual temperature changes requires the additional prescription of radiative forcing anomalies, feedbacks upon temperature changes, and
23ocean
heat uptake, all of which are uncertain. To allow for direct comparison against more
complex models, the EBM is run multiple times using the diagnosed radiative forcing and net
feedback from each of 18 GCMs10, 11. In addition, heat diffusivity between the surface and
deep
ocean is estimated by minimizing the mean square error between GCM and EBM global average temperature anomalies. (Fig. S4). Note that the radiative forcing, feedback, and linear
diffusivity are each applied as global constants so that the EBM’s spatial patterns of
temperature
variability are derived solely from terms obtained by fitting to the seasonal cycle.
4
The GCM and EBM estimates of global temperature anomalies are similar, having an
average across each model version of 0.82. However, the EBMs produces a better fit
r2
with
the observed global average temperatures =0. 74) than do the GCMs
=0.
( r2
(average r 2
63) ,
even though the EBM has not been tuned to interannual temperature observations. The fit is
also better when assessed using a mean-square-error approach. Note that the EBM and
GCMs
share some conspicuous misfits with the observations, particularly near 1950, which are likely
24related
to errors in the observational record.
Model skill can be further assessed with respect to spatial variability, and the EBM consistently outperforms its GCM counterparts in reproducing the observed space-time variability
of surface temperature (Fig. 2). In fact, the worst performing EBM still out-performs the best
fitting GCM. The discrepancy in performance generally exists over the entire instrumental period, but becomes strongest in the last 30 years when observations are more complete and
the
anthropogenic warming signal is strongest. The discrepancy also holds whether the fit is assessed using mean square error or Pearson’s correlation, the global average is first removed
for
each year, or temperature variability is smoothed using 5, 11, or 21 year tapered windows or
not smoothed. Finally, although taking the ensemble average across all 18 GCMs improves
the
fit with the observations, the ensemble EBM average still gives the better fit (Fig. 2).
The superior EBM results even after heavy temporal smoothing and ensemble
averaging
indicates that the discrepancy in performance does not arise merely from the greater internal
25variability
generated by the GCMs. Evidently, the EBM better reproduces the observations
through suppressing all but a few relatively well-understood energy balance processes, a
situation similar to the polynomial fitting example given earlier. There is a parallel in this behavior
5
in that averaging across all the GCMs is expected to suppress processes that vary between
models because they are uncertain, and the better performance of the ensemble than any
individual
GCM again indicates that this suppression improves model performance. The construction
and
running of multiple GCMs is an enormously complex undertaking, and we note that initial
omission of uncertain processes appears advantageous in the present circumstance and is
certainly
much more efficient.
Much of the misfit between the GCMs and observations arises because of discrepancies
in
Arctic warming. For purposes of specificity, we discuss differences in temperature over the
last
decade (1999-2009) relative to a base period (1900-1980) and between latitudes above
65◦N and
those between North and South (Fig. 3), though other choices give broadly similar
40◦
results.
Instrumental observations show 0 . 31◦C more Arctic warming than at low latitudes, whereas
the
GCMs average almost three times this value at 0 . 90 ± 0 . 40◦C. The EBMs are more
consistent
with 0 . 23 ± 0 . 06◦C of additional Arctic warming. To make the comparisons consistent,
model
differences are computed masking all model output not corresponding to observational data,
but if the full spatial field is considered, the excess Arctic warming in the GCMs and EBMs is
1.07◦C and 0.34◦C, respectively.
Existing hypotheses for Arctic amplification include decreased surface albedo, increased
heat transport, and increased polar cloudiness26, but in the EBM it results solely from greater
continentality (Fig. 1b) and a greater sensitivity to radiative forcing (Fig. 1c). The relative im-
portance of these two processes depends on timescale. The instantaneous response to
radiative
perturbations only depends on effective heat capacity, which is inversely related to
continentality, whereas under fully equilibrated conditions the magnitude of warming depends on
radiative
6
sensitivity and the strength of feedbacks. These same processes also dictate that greatest
warming will occur over land, because of the lower effective heat capacity, and during winter and
spring, because of the weaker Planck feedback, which is consistent with the observed pattern
27of
warming. Obviously, the GCMs include the same basic physics that gives rise to Arctic
amplification in the EBM but apparently also include processes that tend to over-amplify that
warming.
The EBM gives more accurate simulation of historical patterns of warming, as judged by
comparison against instrumental observations, and this has important implications for prediction, particularly because past temperature anomalies simulated by both the EBM and GCM
ensembles are almost perfectly linearly related to future anomaly patterns. For instance, the
GCM ensemble average warming pattern between 2040-2050 is very nearly the 1999-2009
pattern multiplied by 2.12 (the two patterns have an =0. 96), and a similar multiplicative
r2
rela-
the EBM ensem
ti
o
n
s
h
i
p
o
f
2
.
1
5
h
o
l
d
s
f
o
r
worse
=
at capturing regional trends in surface temperature, all o
EBM
0
.
9
predictions
are credible.
9
)
.
Through the end of this century, the EBM predicts that the
T
h
northern continental interiors, whereas the GCM ensemble pre
i
s
most (Fig. 3b, c). Although both patterns could be incorrect, the
e
7
s
s
e
n
t
i
a
ll
y
li
n
e
a
r
e
x
t
r
a
p
o
l
a
t
i
o
n
of temperature anomalies extends over the
entire century and indicates that skill in simulating historical warming provides an upper bound
on the skill in predicting future temperature
changes, where actual skill could be lower
insomuch as actual anomalies diverge from
linearity.
The linearity of the ensemble GCM predictions
also suggests that linearized models will be no
a better historical representation of surface temperature, particularly in the Arctic. The degree
to which the Arctic and/or Northern continental interiors warm in the coming decades has profound societal and environmental implications, and it will be important to better discriminate
between these two scenarios through continued observational analysis and model
development.
Finally, to make a more optimal prediction of future global average temperature
changes,
the ocean diffusivity parameter, κ , is re-tuned to the global average temperature observations
(Fig. S8). Though it has little effect on the spatial pattern of warming, re-tuning κ provides
for close agreement between the EBM and the observations in the global average, whereas
the
GCMs average 0.1◦C warmer during the last decade. By 2050, the discrepancy between the
EBM and GCM ensembles grows to 0.2◦C, and although this is much smaller than the intermodel difference, it nonetheless suggests that the GCMs have a slight warming bias (Fig.
3a).
Although we have presented several demonstrations of how the EBM better represents
surface temperature anomalies, this model is perhaps best thought of as a filter for the
physics
in the more complete GCMs that only passes some simple processes depending on radiative
balance. Ultimately, if we are to confidently predict future climate, our models require a
reasonably
complete depiction of the relevant climate processes, a parsimony that permits for
constraining
parameters, and demonstrable skill in reproducing observational data. With regard to the last
point, the EBM presented here could be used in a manner like persistence in weather forecasting: predictions using more complex models would be judged skillful insomuch as they
outperform this minimalist model.
8
M
et
h
o
ds
S
u
m
m
ar
y
T
h
e
E
n
er
gy
B
al
a
nc
e
M
o
d
el
(E
B
M
)
co
ns
ist
s
of
fo
ur
e
q
u
ati
o
ns
,
4os
+
sT
a
l 4l - 4
dT adt =
sw +
dT ldt =
sT
4
a
sT -s
T
1
dT osdt
=
sw +
T
C l , (2)
s a - s T4 4
os - κ
os - T
1
C
4
a , (1)
Fc + s
l ), ocean surface (T
T (1 - a )
F (1 - a ) F
4 1C
od
os , (3)
T
dT dt = κ
Tos - Tod +
1C , (4)
od
d
od
respectively representing the temperature of the atmosphere ( T a ),
os
land ( T
),
and deep ocean (T ). The seasonal cycle of surface temperature at each grid box, ( x, y )
od
, is
then simulated by driving the model with the seasonal cycle of incoming solar radiation 2
8
( y ) ; specifying a heat flux convergence, F c ( x, y ) ;
and
a
defining
a linear mixture, m ( x, y ) , between land and surface ocean temperature, T =
p
mT
p l+
r
o
p
ri
at
e
fo
r
a
gi
v
e
n
la
tit
u
d
e,
F
s
w
(1 - m ) T os. The values of m and are determined by finding the combination that
Fc
minimizes
2
the mean square residual between T and the climatological seasonal 9 for each grid
box.
cycle
All other parameters are held globally constant.
E
qs
.
24
ar
e
lin
e
ar
iz
e
d
to
re
pr
es
e
nt
int
er
a
n
n
u
al
ch
a
n
g
es
in
te
m
p
er
at
ur
e,
T
os (1 fg )
o
d
λ ( x, y ) - κ
os T od
T
d
T
o
s
dt
= =
dT l gF g
dt
,
(6
)
d
T
o
d
dt
=
κg
T
FT
os - T
1C l (1 - fg )λ
( x, y )
, (5)
1
C os
os
,
1
C
o
d
.
(7
)
Temperature at each grid box are again represented as a linear
mixture, T
usin
g
the
valu
es
of m
dete
rmin
ed
from
the
sea
son
al
cycl
e.
Also
take
n
from
the
sea
son
al
cycl
e
9
= mTl +(1- m ) T
model is the basic sensitivity to radiative perturbations, λ ( x, y ) , as described in the text. The
other three terms that require specifying are taken directly from each GCM, g : the net
feedback
upon changes in
10, f g , anomalies in radiative forcing10, F g , following the
temperature
A1B
; and the magnitude of the ocean heat diffusivity, κg, which is determined
e
m minimizing the residual between global average EBM and GCM temperatures between
by
is
1850
si
o 2009.
and
n
s
c
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Supplemental Information is linked to the online version of the paper at www.nature.com/nature.
Acknowledgments Piers Forster provided the forcing histories for the CMIP3 models.
Thomas Laepple, Alexander Stine, Martin Tingley, and Carl Wunsch provided useful comments. Funding was provided by the Packard Foundation.
Author Information The author declares no competing financial interests. Correspondence and requests for materials should be addressed to phuybers@fas.harvard.edu.
13
Figure 1: Three inferences from fitting the EBM to the seasonal cycle. (a) The zonally integrated heat flux inferred from the model (red) agrees with that derived from satellite
radiation
19budget
(black) to within the satellite standard error
(gray
observations
estimates20
shading).
(b) The distribution of continentality, expressed as a land/ocean mixing ratio, m , that varies
from zero to one. Note the west-to-east penetration of marine conditions into continents asso30ciated
with mid-latitude storm tracks. (c) The temperature sensitivity to radiative forcing in
K/(W/m2), which excludes any feedbacks and is larger in colder regions.
Figure 2: A measure of the GCM and EBM skill in reproducing historical observations. Correspondence between model temperature variability and all available observations between
18502009 is measured using Pearson’s correlation coefficient. Correlations shown here are after
smoothing variability at each grid point using an eleven year running average. The dashed
black
line indicates equal correlation, and in every case the EBM yields a higher correlation than its
counter-part (crosses), even after averaging together all the EBM and GCM results (circle).
14
Figure 3: Changes in zonal average temperature from the observations (black), GCMs
(green),
and versions of the EBM (red). Changes are calculated for 1999-2009 relative to a base
period
of 1900-1980. Observational uncertainty estimates (black dashed lines) are estimated assuming that average temperatures in the last decade and base period have an uncertainty of 1◦C
( 1 s ) and that errors in each interval and each grid box are uncorrelated. All model output not
corresponding to observational data is omitted prior to averaging.
Figure 4: Historical and predicted changes in surface temperature. (a) Observed global temperature anomalies (black), and those averaged across the ensemble of EBM (red) and GCM
(green) results, where the dashed lines indicate the interval containing 90% of the respective
model results. (b) Temperature anomalies for 2040-2050 averaged across the ensemble of 18
GCMs, and (c) the same for the EBM. The EBM predicts less warming than the GCMs, and
that the warming will be more localized to the interiors of high latitude continents. Anomalies
are with respect to a 1900-1980 reference interval.
15
M
et
h
o
ds
S
e
as
o
n
al
m
o
d
el:
T
h
e
E
n
er
gy
B
al
a
nc
e
M
o
d
el
(E
B
M
)
co
ns
ist
s
of
fo
ur
e
q
u
ati
o
ns
,
4os
+
sT
a
l 4l - 4
dT adt =
sw +
sT 1C
l , (9)
4
sT
a
dT ldt =
dT osdt
=
s
T
sw +
T
s a - s T4
4
os - κ
os - T
1
C
4
a , (8)
Fc + s
l ), ocean surface (T
T (1 - a )
F (1 - a ) F
4 1C
od
os , (10)
T
dT dt = κ
Tos - Tod +
1C , (11)
od
d
od
respectively representing the temperature of the atmosphere ( T a ),
os
land ( T
),
and deep ocean (T ). The seasonal cycle of surface temperature at each grid box, ( x, y )
od
, is
then simulated by driving the model with the seasonal cycle of incoming solar radiation 2
8
( y ) ; specifying a heat flux convergence, F c ( x, y ) ;
and
a
defining
a linear mixture, m ( x, y ) , between land and surface ocean temperature, T =
p
mT
p l+
r
o
p
ri
at
e
fo
r
a
gi
v
e
n
la
tit
u
d
e,
F
s
w
(1 - m ) T os. The values of m and are determined by finding the combination that
Fc
minimizes
the mean square residual between T and the climatological seasonal 2 for each grid
9
cycle
box.
Al
l
ot
h
er
p
ar
a
m
et
er
s
ar
e
h
el
d
gl
o
b
al
ly
c
o
n
st
a
nt
,
a
s
fo
ll
o
w
s:
al
b
e
d
o,
a
=
0.
3
;
at
m
o
s
p
h
er
ic
emissivity,
C land
=0. 8 ; and heat capacities,
1
0
= C atm =2m × 4 . 2 × 10 6J/(K m), C =
os
0m × 4 . 2 × 106J/(K m), and C od
performed
=
at daily resolution
3
9 a seasonal equilibrium is reached, after which results are averaged to monthly
until
0
resolution.
0
m
×
earized about their mean annual temperatures, allowing for more rapid simulation and for
4
ap16
.
κ , is set to zero for the purposes of
simulating the seasonal cycle,
making arbitrary the offset
between surface and deep ocean
temperature, d . Model runs are
Interannual model: To represent interannual changes in temperature, Eqs. 2-4 are lin-
2
×
1
0
6
J
/
(
K
m
)
.
O
c
e
a
n
h
e
a
t
d
i
f
f
u
s
i
v
i
t
y
,
pending the net feedback terms estimated
elsewhere,
dT
T os (1 - fg )
od λ ( x, y ) - κ
os
dt
= = FT
os - T
1C l (1 - fg )λ
gF
g
( x, y )
dT l
os T T od
C
os
, (12)
1
dt
, (13) dT od dt= κg T1 Cod . (14)
Temperature at each grid box are again represented as a linear
mixture, T
= mT l +(1- m )
T os ,
using the values of m determined from the seasonal
cycle. Also taken from the seasonal cycle
model is the basic sensitivity to radiative
perturbations, λ ( x, y ) , which is deduced by
specifying
a perturbation in downward atmospheric radiative
forcing and allowing the ocean and land components to re-equilibrate. The other three terms that
require specifying are taken directly from
each GCM, g . Specifically, the net feedback upon
changes in temperature, f g , and anomalies
in radiative forcing inclusive of the effects of aerosols, F g , are those
diagnosed by
m
a
g
ni
tu
d
e
of
th
e
o
c
e
a
n
h
e
at
di
ff
u
10;
and
the
sivity, κg
,
y minimizing the residual be
global
is
average EBM and GCM temperatures between 1850 and 200
possible
d
to
et
e
r
m
in
e
d
b
spatially modify f g according to the values of λ ( x, y ) but here a single global average is
applied. Radiative forcing follows from the changes in
greenhouse gas inventories specified by the
A1B emission scenario11. All 20 GCM results for
which forcing and feedbacks were diagnosed
are included, except for the MIROC3.2 medium
resolution model, whose temperatures greatly
differ from the historical observations, and
FGOALSg1.0, whose diagnosed radiative forcing
is difficult to distinguish from noise. Errors in diagnosing GCM radiative forcing
pected to degrade EBM results, which further
distinguishes the EBM’s superior representation
of surface temperature variability.
17
1
0
are
ex-
Gridding and normalization: Each GCM result is placed onto the same × 5◦ grid
5◦
used
for the CRU observations by first interpolating to × 1◦ resolution and then averaging.
1◦
For
purposes of interpolation, longitude is wrapped and persistance is specified at latitudes
poleward
of the highest reported grid points. Each model and observational temperature time-series is
normalized to be zero mean between 1900-1980 and averages are area-weighted. This
longer
29and
earlier interval than the standard 1960-1990 interval used by the Climate Research Unit
gives a more stable estimate and decreases the magnitude of negative temperature
anomalies
that otherwise generally appear prior to 1960. Nevertheless, the EBM similarly out-performs
the GCMs if the 1960-1990 standardization interval is used. All model results are masked to
only have entries corresponding to observations in the interval between 1850-2009, making
the
normalization equivalent for model and observational results. Annual averages are computed
from the CRU observations only when data from all 12 months are present.
18
(a)
4 2 0 2 4 6
50 0 50 6latitude
(b) W 120
oW
6
0
0
(
c
)
60N
30oo
30N oS
0
o
60S
o
o
oW
6
0
oW
180
180
0.8
0.6
l
o
n
g
i
t
u
d
e
0.4
0.2
0
0.45
60N
o
o oE
o
E
o
120
0.4
30oS
0 30N
o
o
60S
Fig. 1
180oW
60o
120oW
W
0
o
60longitudeoE
180oW
120oE
at transport (petta Watts)
0.35
0.3
uared cross correlation
temperature anomaly (°C)
2.5
(a)
2
1.5
1
0.
5
1950 2000 2050 0year
(b)
o
60N
o
30N
W
120
oW
(
c
)
oW
o
o oE
180 oW
6
0
4
6
0
0
3
0o
o
30S
o
60S
o
2
180
1
3
o
60N
o
30N
o
5
E
120
2.5
2
0o
30S
W
120
W
60 W
0
o
60
60S
180o
E
120
o
E
o
180 W
o
o
o
o
° C)
1.5
1
perature anomaly (
Fig. 4