Energy balance climate modeling: Comparison of
radiative and dynamic feedback mechanisms
By TZVI GAL-CHEN,’ Advanced Study Program and STEPHEN H. SCHNEIDER,
Climate Project, National Center f o r Atmospheric Research,2 Boulder, Colorado 80303, U S A
(Manuscript received January 27; in final form July 1, 1975)
ABSTRACT
The time dependent energy balance climate model of Schneider and Gal-Chen (1973)
is extended to consider the relative importance of radiative and dynamic parameterizations on the sensitivity of the model’s equilibrium climate t o perturbations in solar
input. The albedo-temperature feedback parameterization of Sellers ( 1969) is used t o
test the sensitivity of the model’s global temperature and equator-to-poletemperature
gradient to solar input changes with six different dynamical parameterizations. The
nonlinear eddy flux parameterization used by Stone (1973) appears to give the best
results, but the assumption that global average static stability remains constant
during climatic changes (on earth) is not supported by our experiments; but also cannot
be ruled out as a possibility. The thermodynamic processes of ice-albedo-temperature
feedback and moist adiabatic convective adjustment are found to dominate the effects
of large-scale eddies in controlling the behavior of the globally-averagedlapse rate
during climatic changes. This conclusion is drawn after comparisons of our results
with those computed by Wetherald and Manabe with a three-dimensional general
circulation model. Our findings suggest that thermodynamic processes must be central
elements of any climatic theory, and that interpretation of the results of complex
general circulation models can be made easier by drawing on the experience gained
with simpler models like the energy balance varieties used here.
1. Introduction
Considerable interest has been expressed
recently in climate modeling as a primary tool
in the development of a quantitative theory of
climate (GARP, 1975). Climate models can be
classified in a hierarchy generally based on
geometric degrees of freedom ranging from
globally-averaged vertical column models and
zonally-averaged energy balance types t o
highly detailed three-dimensional simulations
of the general atmospheric circulation (GCMs)
interacting with oceans and sea ice (e.g., see
Chapt. 6 of SMIC, 1971). I n an extensive recent
survey of climate modeling Schneider &
Dickinson (1974) have argued t h a t the primary
role of the simpler models is in “making tentative estimates of the sensitivity of long-term
1 Present affiliation: Department of Physics, University of Toronto, Toronto, Ontario, Canada.
The National Center for Atmospheric Research
is sponsored by the National Science Foundation.
conditions of t,he atmosphere-land-ocean -cryosphere system t o changes in various known
thermodynamical and transport processes with
a view identifying those of greatest importance’’. This step is clearly essential t o t h e
design and interpretation of more complicated
and more realistic interactive models. It is the
purpose of this paper to illustrate the important
insights t h a t can be gained from numerical
experimentation with a relatively simple model,
and t o show how experience with this model
can be extremely helpful in pulling together t h e
results of several numerical experiment,s with
more complex models of different investigators.
In particular, we have extended our earlier
work (Schneider & Gal-Chen, 1973) t o show
the relative importance of various radiative
and dynamic feedback mechanisms on the
sensitivity of t h e model’s surface temperature
distribution t o perturbations in energy input.
We experiment with different parametric
representations of ice-albedo-temperature feedTellus XXVIII (1976), 2
109
ENERGY BALANCE CLIMATE MODELING
back, eddy heat transport, and transport of
latent heat of vaporization. Despite the wellknown simplicity of these energy balance
models, we feel that these kinds of experiments
have, nonetheless, helped us to combine the
application of baroclinic theory to climate
modeling (used by Stone, 1973) with ice-albedotemperature feedback (of Sellers, 1969) in such
a way that the effect of changes in solar input
on the equator-to-pole temperature gradient
and lapse rate may be more easily understood.
We then use this experience to help us interpret
some results of the general circulation model
(GCM) of the Geophysical Fluid Dynamics
Laboratory (GFDL) (in Princeton, New Jersey).
It is our hope that the kind of approach we
report here might encourage other workers to
examine and compare the results of different
models of varying complexity for the purpose
of helping to build a quantitative theory of
climate. To that end, the computer code used
here has been expanded and complied into a
portable and easily used package by Clifford
Mass, Dept. of Atmospheric Science, University
of Washington, Seattle, Washington, USA.
This package can be made available t o other
researchers upon their request t o S. H. Schneider, Climate Project, National Center for
Atmospheric Research, Boulder, Colorado, USA.
2. Modeling assumptions
The basic modeling assumptions are similar
to that of Schneider and Gal-Chen (1973). A
time dependent version of the zonally-averaged,
vertically integrated energy equation is
the albedo, Fm is the outgoing infrared radiation flux to space:
FIR= c(4)aT4[1 - m tanh (19Te x 10-ls)]
(2)
c(4) is a consistency factor, designed t o make
the present climate an exact steady state solution of the finite difference analogue of eq. (1)
when no perturbations (external or internal)
are present. The role of c(4) has been discussed
thoroughly by Schneider & Gal-Chen (1973). m
is an opacity factor ( = 0.5), F,, FA,Fq are zonal
heat fluxes due to ocean currents, atmospheric
motion, and the transport of latent heat, respectively. 0 is the colatitude [ = (n/2) 41, y = d,
and a is the earth’s radius. The parameterization of albedo is given by
-
(3)
with the restriction: 0.25 < a < 0.85, regardless
of To.
To is the ground temperature and is detmmined from sea level temperature based on
zonally averaged topography (e.g., Sellers,
1969). a(+) are empirical coefficients designed
to fit eq. (3)t o present observed albedos. Thus,
b ( 4 ) are uniquely determined for given a, C,,
TFand To.
Eq. (3) is a parameterized mimic of the
observed fact that a decrease in the surface
temperature in mid or high latitudes would
usually be accompanied by an increase of
snowfall or sea ice, and therefore an increase
of surface albedo. The converse would apply
for an increase of surface temperature. Eq. (3)
represents a “positive feedback mechanism”
and a potential source of instability. Here,
with one noted exception, the Sellers (1969)
values for TF 283.16 (the feedback temperature) and CT == 0.009 (the feedback rate parameter) are used. It must be borne in mind
that (by definition)
i=
R is the thermal inertia of the ocean. Its value
is chosen based on 25 m mixed layer in the
ocean and linearly varying profile down to
zero degrees Celsius at 125 m. The value of R
could also be interpreted as a scaling factor for
the time scale t. At any rate the value of R
docs not change the final steady state of the
model. t is the time, T, sea level temperature,
Qw(4) is the yearly averaged, zonally averaged
value of solar energy input a t latitude 4, a is
Tellus XXVIII (1976), 2
is the crucial parameter that governs the
intensity of the ice feedback. Sellers (1969)
choice of C, = 0.009 was based on a comparison
of albedos at similar latitudes in the northern
and southern hemispheres, despite the climate
differences of these two hemispheres. A po-
110
T. GAL-CHEN AND 5. H. SCHNEIDER
tentially better way to determine C , and T,,
might be to use satellite radiation and surface
temperature observations a t the same latitude,
and then to try to correlate observed albedo
changes to temperature changes. It is also
possible that satellite observations (which
include cloudiness effects as well as surface
albedo) would suggest a functional relationship
between albedo and surface temperature different from the linear one (with discontinuous
first derivative) implied by (3). Leith (1974)
has suggested another approach-that
one
might eliminate T , from both a(T,) and F,( T,)
and derive an empirical expression for a as a
function of FIR.Such an analysis could be
attempted with use of satellite observations.
The sensitivity of the results to changes in C,
am discussed later.
Six dynamical parameterizations have been
constructed. The first one labeled ( S )is
q ( T )is the water vapor mixing ratio, 8 is some
vertically integrated mean mcridional velocity
and is pararnetcrized as function of the local
and global temperature gradients (see Sellers,
1969). The K’s are “austauch” or eddy diffusion
coefficients. I n order to derive eq. (4 d ) , it is
necessary to assume that release of latent heat
occurs when the relative humidity exceeds
certain empirical value. This value of critical
relative humidity is absorbed among many
other constants in K , in order that Fa agrees
with observations for itself and for present
values of T ( 4 ) . When these assumptions are
made and one uses the Clausiiis-Clapeyron rq.,
one can thus arrive at eq. (4d).
The sccond formulation labeled (ST’) is
similar to that of ( 8 )except 0-0. The third
formulation labeled ( S T )is like ( S )bat the K’s
are nonlinear as suggested by Stone (1973)
and usrd by Sellers (1973); namely,
Kim I AT1
(5)
where in our case K , is either K,, K,, or K Oand
AT is the temperature difference between zones.
Green (1970) uses a similar form for the K , ,
but in his formulation I AT I denotes the global
equator-to-pole temperature gradient, not the
local temperature gradient.
The fourth formulation labeled (ST V ) is
similar to ( S T )but a = 0.
The fifth formulation labeled ( S T C ) is
similar to ( S T )but
instead of (4d); that is, the nonlinear ClausiusClapeyron term p ( T ) / T 2
is removed as a factor
in the K , term.
The sixth formulation labeled ( S T C V ) is
0.
similar to ( S T C )but
The values of the K’s are computed based
on the observed fluxes F,, FA, Fa and their
numerical values depend on the particular
dynamical parameterization.
One of our aims in considering these six
dynamical parameterizations is to find out
what effect, if any, different dynamical parameterizations have in determining the “climate”, i.e., an equilibrium value of T,(4) in
the model. This can be compared to previous
calculations by Budyko (1969), Sellers (1969)
and Schneider & Gal-Chen (1973), in which it
was found that the sensitivity of the model’s
climates to perturbation in energy input are
very dependent on the albedo parameterization
(e.g., see Fig. 6 of Sellers or Table 3 of Schneider
& Gal-Chen). I n addition, results obtained by
Gordon & Davies (1974) seem to indicate that
the sensitivity of Budyko’s model is quite
dependent on whether one uses Budyko’s
original infrared parameterization, or the one
proposed by Smagorinsky (1963) in his twolevel GCM.
I n all the parametcrizations used here the
distinction between K Oand K , is only formal
since all thc empirical constants (see Sellers,
1969) can be absorbed into one effective eddy
coefficient Keff= K O K,. I n formulation (STC)
and ( S T C l.), the sum Keff= K O+ K , + K , can
be cornbind into one effective eddy diffusion
coefficient since tho nonlinear factor q( T ) / T 2
is dropped [althotlgh the numerical value of
+
Trllus X-YVIII (1976), 2
ENERGY BALANCE CLIMATE MODELING
Kerfis different for ( S T C )and ( S T C V )because
the latter has t7 01.
I n formulation ( S ) , (ST),and ( S T C ) the
same value of t7 is used for meridional advection
of sensible and latent heat fluxes. This is
physically questionable as pointed out by
Sellers (1969),Robinson (1971),and Schneider
& Gal-Chen (1973). Sellers has justified the
use of the .I) parameterization as a means of
eliminating negative diffusion coefficients ( K , )
of latent heat in the tropics, where P, and
aT/ay can be of opposite sign (eq. 4d). It must
be borne in mind, however, that there is nothing
conceptually and physically wrong with individual negative K4’s as long as
-
111
of the eddy diffusion coefficients, and the
rationale for this approach t o climate modeling.
Those interested in further details (or additional
references) should consult this reference and
Schneider & Dickinson (1974).[For an analytic
treatment of the Budyko model, see North
(1975).]
3. Discussion of results
I n the following sections the response of
some parameters in the models t o changes in
solar constant is discussed. I n some cases the
response is qualitatively independent of the
particular dynamical parameterization, in
other cases strong dependence in the particular
q (T)
dynemical parameterization is found. Therefore
K , t KA + K ,
model validation studies are very important.
T‘ ’
formulations ( S V ) (, S T V ) ,( S ) (, S T ) (7) Unfortunately, the mere fact that the models
Kerf = ’
reproduce the observed zonally averaged surface temperature distribution cannot be taken
K , t KA + K,,
as an indication of the validity of the models
, formulation ( S T C V ) ,(STC)
simply because the models are based on, in fact
deliberately tuned to, empirical data. Ideally
is positive. For negative Kerf,the whole problem one would like to verify the models against
becomes mathematically ill posed (e.g., Richt- differing climatic regimes, with simultaneous
myer & Morton, 1967, p. 4 and p. 69). In knowledge of the values of QsC(d),but obviously
formulations (S),(ST),( S T C ) , and ( S T C V ) , this is not yet possible. Despite this limitation
the initial positiveness for Kerfwill ensure the we have singled out the dynamical parameteriwell posedness of the problem for a later time. zations ( S T C V ) as being the “best” in some
This is not necessarily the case for models senses. We have then used this particular
( S V ) and ( S T V ) in which initially negative formulation to help us interpret some results
K,’s can be amplified by a factor q ( T ) / T BI.n of the three-dimensional GFDL GCM. We have
fact, our numerical experiments have shown also used this model to show some circumthat for an increase in the solar constant stances in which the assumption that static
>lo%, models ( S V ) and ( S T V ) become ill stability remains constant during climatic
posed. When ice feedback has been omitted, a changes may be invalid.
“blow up” occurs in these two models for an
increase > 15 %. Thus in order to be able t o
3.1. Surface temperature albedo coupling:
experiment with greater solar constant perturI m p l i c a t i o n s for climate stability
bations, Kerf is set to zero whenever eq. (7)
Fig. 1 summarizes typical results of the
indicates negative values. A 10% increase in
the solar constant would seem unrealistically energy balance model reported by Schneider
BE Gal-Chen (1973). The dynamical paralarge, nevertheless energy balance is a universal
principle and as such is applicable to other meterization which is used is that of Sellers
planets (e.g., Stone, 1972; Sagan et al., 1973) (8).Curve a in Fig. 1 is the asymptotic equilibrium climate for the present solar constant,
with appreciably different solar constant.
The previous paper by Schneider & Gal-Chen with a mean sea level temperature of 287.43 K.
(1973)discusses the polar boundary conditions I n the previous paper (Schneider & Gal-Chen,
(see also the recent paper by Hantel, 1974), 1973), slightly different numerical values were
the crucially important convergence criteria, obtained due to the lower precision of the
initialization (or “consistency”) procedure, IBM 360/95 computer used then compared to
method of determining the numerical values the CDC 7600 which is used now. The initial
~
Tellus
XXVIII (1976), 2
112
T. GAL-CHEN AND S. H. SCHNEIDER
310
-5
I
300
-
290
-
I
I
I
l
l
I
I
I
-
270 -
-
280
-
u
IY
3
5
260
-
b
a
b
C
a
250
-
240
-
a
!
!
5
-
-
C
Curve
W
Global Mean Temp. (OK)
207.43
290.25
282.05
175.55
+ I a/.
-I
a/.
I70
i
1
d
90"N 70
QeC(1
< -1.6%
1
'801
alone
Solar I n p u t
No Change
2001
190
-
m
50
30
j
10 0 10
30
50
70 90'
LATITUDE
Fig. 1. Zonal distribution of asymptotic steady
state sea level temperature computed as a function
of solar input for the dynamical parameterization
(S),as discussed in the text.
temperature distribution used to produce the
equilibrium state had a mean temperature of
287.30 K in both papers. The closeness in values
between the initial and final steady state is a
result of the model parameterization being
defined in terms of the initial state. With an
infinite arithmetic precision the initial and
final states should coincide. (The present
accuracy is quite sufficient, however, for our
purposes.) Curves b and c result from a 1%
increase and decrease, respectively, in the
value of the solar constant. We see that 1%
increase in solar input causes an increase in
mean global temperature by 2.82 K, which is
1.4 K increase
significantly larger than
which would be obtained if there were no
positive feedback between the albedo and
temperature (i.e., C,=O in eq. (3)). On the
other hand, a 1% decrease in solar constant
shown by curve c yields a mean global temperature decrease of 5.38 K, a significant
amplification of the decrease in temperature.
For a decrease in solar constant of more than
-
1.6%, we obtain a planetary temperature of
175.55 K, which corresponds to an entirely
ice-covered earth, with a uniform albedo of
0.85. The completely ice-covered solution
appears to be a possible steady state solution
of the most general energy balance equations.
Indeed, it has been obtained by detailed atmospheric general circulation models (GCM) which
satisfy the surface energy balance requirements
(e.g., Manabe, private communication). To
see that this is a consistent result, one notes
that upon global averaging, the global tem-
-4T))= p m ( T )
Substituting tc =0.85, one gets a global temperature 175 K.
The comparative results for the other dynamical parameterizations are listed in Tables
1 and 2. The control temperature (Table 1) is
the final steady state which is obtained for
present day solar constant. The global temperature is the final steady state for a 1%
reduction in solar constant. I n comparing the
K + K ( A T ) parameterizations [(S) and ( S V ) ]
to those using eq. ( 5 ) [ ( S T ) (, S T V ) ,( S T C ) ,and
( S T C V ) ] ,one notes that the latter cases reduce
the severity of the temperature decrease in
polar and midlatitudes by up to a factor of two
(as Stone, 1973 had noted should be the case).
The temperature decrease in the tropical
latitudes, however, is comparable for both
linear and nonlinear eddy viscosity formulation
(which is not surprising considering that A T
is small in the tropics). The net result is that the
change in equator-to-poletemperature gradient
with solar constant in the K + K ( A T ) parameterizations is greater than that of Stone
( K = K ( A T ) ) .This is so because in the nonlinear eddy flux parameterization the nonconstant eddy diffusion coefficients are increased when the temperature gradient is
increased; therefore the inherent negative feedback (smoothing of gradients) associated with a
diffusion process is increased.
Table 2 shows the factor by which the solar
constant must be multiplied so as to just be
able to produce the ice-covered earth model
instability for the six dynamical parameterizations. The ( S T C V ) case is less sensitive to
negative perturbations in Qsc than the other
Tellus XXVIII (1976), 2
113
ENERGY BALANCE CLIMATE MODELING
Table 1. Surface temperature decreases for a one percent decrease in sokr input for zonal and global
horizontal averaging and for six different dynamical pararneterizatiom (see text)
80
70
60
50
40
30
20
10
0
- 10
- 20
- 30
- 40
- 50
.- 60
- 70
- 80
- 7.70
- 7.71
- 6.89
- 6.87
- 7.86
- 7.32
- 6.65
- 5.90
- 4.53
- 4.08
- 3.88
- 3.87
- 3.95
- 5.10
- 8.08
- 8.38
- 8.06
- 3.57
- 5.38
- 4.68
- 3.64
- 3.21
- 3.34
- 3.08
282.05
282.82
283.84
284.25
284.21
284.46
287.43
287.50
287.48
287.46
287.55
287.54
- 3.14
cases, but all parameterizations exhibit the
unstable behavior characteristic of these energy
balance models with ice feedback (i.e., eq. (3)).
Due to spherical shape of the earth a decrease
in the solar constant means that the absolute
reduction of QSc(+)in the tropics is greater than
that of the poles. Therefore one might expect
that the temperature decrease in the tropics
will be greater than that of the poles for a
decrease in solar input above the atmosphere.
A glimpse a t Table 1 reveals, however, that
this does not happen in our calculations. The
ice feedback of the albedo formulation (eq. 3)
Table 2. Value o f factor to which solar constant
must be multiplied so as to just be able to produce
ice-covered earth instability for six different
dynamical parameterizations
S
SV
ST
STV
STC
STCV
0.986
0.984
0.980
0.980
0.979
0.979
Tellus XXVIII (1976), 2
8 - 762892
- 3.54
- 3.53
-3.10
'- 3.18
-3.19
-3.18
- 3.34
- 3.70
-4.14
- 4.47
- 4.68
-. 4.86
- 5.79
- 6.97
Global temperature for
1 % decrease
in Qec
Control
temperature
- 3.91
-4.12
- 3.73
- 3.66
- 3.71
- 3.48
- 3.57
- 3.98
- 4.81
- 5.76
- 6.51
- 6.79
- 7.03
- 6.04
- 4.58
difference
- 3.87
- 4.63
- 4.58
- 4.29
- 3.86
- 3.29
-4.14
-4.12
- 4.01
- 3.87
- 3.48
- 3.19
- 3.05
- 3.02
- 2.98
- 2.92
- 2.98
- 3.03
- 3.40
- 3.72
- 3.95
- 4.03
- 4.05
- 4.08
Global
average
- 4.61
- 3.84
- 3.65
- 3.45
- 3.07
- 2.96
- 2.93
- 3.01
- 3.01
- 2.98
- 3.09
- 3.25
- 3.42
- 3.48
- 3.54
- 3.47
- 3.42
- 3.20
- 3.05
- 2.93
- 2.86
- 2.82
- 2.84
- 2.95
- 2.97
- 3.19
- 3.32
- 3.41
- 3.40
- 3.40
results in a greater decrease in Qec(+) (1 - a)
(the short wave radiation flux which is absorbed
by the zonal column) in the polar latitudes than
in the tropics. On the other hand, if ice feedback is excluded from the model, we get the
expected result of temperature decrease in the
tropics which is greater than that of the poles.
Our numerical experiments show that the
temperature increase in the polar and midlatitudes with ice feedback is also greater than that
of the tropics for an increase of solar constant.
Again the tropical change is relatively larger if
ice feedback is excluded. Important implications of these differences are discussed later.
Some paleoclimatic data (e.g., Budyko, 1972)
reveal that the previous ice age in the Northern
Hemisphere was more severe than that of the
Southern Hemisphere. This difference might
result from a slower expansion of glaciers in the
more oceanic hemisphere. Inspection of Table 1
reveals, however, that this feature reported by
Budyko is not reproduced in our energy balance
model. The failure could be due to the inade-
114
T. GAL-CHEN AND 9. H. SCHNEIDER
quate representation of the oceans in this
model; or to the fact that in our calculations
the solar constant was decreased uniformly at
comparable latitudes in each hemisphere in
contrast t o the asymmetric decrease which
would have resulted from actual earth orbital
variation. Clearly, we are not trying here to
explain ice ages with these energy balance
models, but rather to use them as tools to
investigate some fundamental sensitivities in
the climate system.
Perhaps the most significant information in
Table 1 is the global mean surface temperature
decrease ATs which, according to the various
models, results from a 1% decrease AQBcin the
solar constant. This decrease is a useful measure
of the sensitivity of a climatic model to any
perturbation in an energy term, and has many
uses (see, for example, Schneider & Dennett,
1975 for an application of climate model sensitivities to the problem of thermal pollution).
This number in Table 1 lies in the range 3.08 <
A T , <5.38. From these values we infer that
the radiation balance is the primary factor
which determines the global surface temperature sensitivity to changes in energy input but
that dynamical effects can modulate this
global sensitivity by nearly a factor of 2 (e.g.,
compare parameterizations ( 8 ) and ( S T C V ) ) .
Furthermore previous numerical experiments
by Schneider & Gal-Chen (1973) and Sellers
(1969) have given other examples of much
greater sensitivity to radiation balance parameterizations than to dynamical ones. For
instance, Sellers found that with C, = 0.005
(eq. 3) the completely ice-covered solution
occurs for solar constant decrease > 10% as
compared to a decrease > 2 % of Q,, with
C, =0.009 (his Table 2). Schneider & Gal-Chen
found that with Faegre’s (1972)albedo formulation, which essentially corresponds to TF= 300
(eq. 3), the present “climate” is unstable with
respect to small negative perturbations in
initial or external conditions.
It is perhaps worth noting that the smallest
value ATs = 3.08 (i.e., Stone’s parameterization
( S T C V ) ) is closest to the one calculated by
Wetherald & Manabe (1975) using a threedimensional GCM that computes the dynamics
of large-scale eddies in explicit detail. Their
model calculates rather than specifies sea surface temperature and accounts for snow-albedotemperature feedback, but fixes cloud cover.
It seems likely that a careful tuning of C, or TF
(eq. 3) would give results which could be in
almost exact agreement with that of Manabe.
It is our opinion, however, that such a tuning
exercise is of virtually no intellectual content
in light of other inherent major assumptions
here (e.g., fixed clouds). Tuning to observations,
on the other hand, can be a worthwhile exerciesprovided the “tuned” parameters remain within
the bounds of observational uncertainties.
3.2. Response of the equator-to-pole gradient to
perturbations i n solar input with various
dynamical parameterizations
Dynamical processes (e.g., baroclinic eddies,
direct circulation, dry and moist convection)
tend to combine so as to destroy the differential
heating that created them; but they are not
able to completely wipe it out. Thus, under the
restriction of no ice feedback, a + a ( T ) , an
increase/decrease in the differential heating
should mean an increase/decrease in the horizontal temperature difference and the vigor of
the circulation systems that accompany it. The
implication for the earth atmosphere system
is that, for unaltered radiative properties (e.g.,
no ice and cloud feedback which could make tc
a function of T ) , the equator-to-pole gradient
should be a monotonically increasing function
of the solar constant.
Figs. 2 and 3 display the globally averaged
equator-to-pole gradient as a function of the
solar constant for the six dynamical parameterizations, and with ice feedback omitted
(i.e., the albedos are prescribed functions of
latitude-such that C , = 0 in eq. (3)). It can
be seen that models (S)and ( S T ) are suspect
immediately, since they show a strange maximum around the present climate. Recall that
these two models distinguish between sensible
and latent heat transport in a manner such
that latent heat transport is parameterized in
terms of surface relative humidity (which is
lumped into the eddy diffusion coefficient) and
a nonlinear factor q ( T ) / T a derived
,
from the
Clausius-Clapeyron relation. More importantly,
in these models some mean meridional velocity
and “advective” transport is included. The
omission of such mean meridional motion (e.g.,
in models ( S V ) and (STT’)) eliminates this
peculiar baroclinicity maximum at) current
values of Q,, that we find on Fig. 2 for models
(S)and ( S T ) ,although a strange drop occurs
Tellus XXVIII (1976), 2
115
ENERGY BALANCE CLIMATE MODELING
‘S
0’23’
0:s
019
l.AO
1.10
1.10 O:I
SOLAR INPUT FACTOR [ I + ( A Q / Q ) ]
require, as a consequence of the thermal wind
relation, a n equatorial jet stream with a wind
shear which would be an order of magnitude
greater than the present midlatituda jet stream
(see e.g., discussion by Dickinson (1971a, a)).
Perhaps as the model suggests, a 4 % increase
in the earth solar constant may cause such a
strong tropical jet, but we are, rather, skeptical
of this model formulation.
Fig. 3 reveals that when latent heat transport
is not parameterized in terms of ClausiusClapeyron relation, the equator-to-polegradient
is a smooth monotonic function of the solar
constant. It must be reemphasized that in this
case there is no real distinction made between
latent and sensible heat transport. The less
peculiar performance of models (STC) and
( S T C V ) is perhaps not surprising if one remembers that in these semiempirical models
everything is parameterized in terms of surface
temperature and ita derivatives. Thus, this
parameterization may incorrectly describe a
particular energy transport term, but it may
parameterize correctly the net effect of all the
( 1 = present d u e of solar input 1
Pig.2. Equator-to-polesurface temperature gradient
[K/100 km] (globally-averaged)as a function of solar
input for four different dynamical parameterizations
(seetext). Note that none of these yields amonotonically varying function, which would be expectedfrom
baroclinic theory considerations. Ice feedback is
excluded.
0
t
.
3
3
0.32
for solar constant increases which lie in the range
1.15 Q,, $Qsc < 1.20 Q8,
This can be understood when we recall that
whenever Kerf (eq. 7) becomes negative it is
set to zero; this option has been invoked here
(for fixed albedos) when the solar constant
increase was greater than about 15 %.
On a first glance ons might tentatively conclude that for solar constant increases less than
15%, models ( S V ) and ( S T V ) do give the
expected monotonic relationship between solar
constant and equator-to-pole gradient. However, when we checked the zonal dependence
of the surface temperature we found that for
solar constant increases > 4 % , a local zonal
temperature gradient as large as 7 K/1 000 km
developed in the southern tropics. This gradient
led to a great asymmetry between the temperature profiles in the Northern and Southern
Hemispheres. Such a “tropical front” would
Tellus XXVIII (1976), 2
0 23
0.8
0.9
1.00
1.10
1.20
1.30
SOLAR INPUT FACTOR [I+ ( A O / O ) ]
II = present value of solar input 1
Pig. 3. Same as Fig. 2 but for two dynamioal parameterizations that do not include Clausius-Clapeyron factor in the moisture diffusion term.
116
T. GAL-CHEN AND S. H. SCHNEIDER
transports together. An interesting analogue
to this phenomenon has been discussed by
Manabe & Terpstra (1974). They discuss the
relative importance of stationary and transient
eddies in transporting heat in the GFDL GCM
model atmosphere. According to their results
the contribution of stationary eddies in the
model with mountains is very important whereas
that of transient eddies becomes dominant in
the mountainless model. However, the total
eddy heat transport is affected relatively little
by the presence of mountains. This result
suggests that, while transient eddies might
compensate for stationary eddies in transporting
heat, transport by the stationary eddy component may not be parameterized simply in
terms of zonally-averaged temperature gradients
alono. Since stationary eddies are responsible
for much of the non-zonal aspects of climatic
statistics, the parameterization (eq. 5 ) is
probably not sufficient! for models with a
longitudinal coordinate. Nevertheless, the
combined zonally-averaged heat transport may
well be adequately parameterized in terms of
zonal temperature gradients by using baroclinic theories of the sort discussed by Green
(1970), Saltzrnan & Vernekar (1972) or Stone
(1973). Returning to our results (Figs. 2 and 3),
it seems unlikely that such a complex phenomenon as latent heat transport, which involves
among other things moist convection in the
tropics, can be parameterized simply in terms
of the local relative humidity, ClausiusClapeyron relation, and the local temperature
gradient. Yet, we concede that it is the differential heating which fundamentally drives the
atmospheric motions and therefore one may
speculate that the total heat transport may be
parameterized as some function of the temperature gradient alone, though the extent to
which this approach will prove accurate remains
to be fully demonstrated-through comparisons
of parameterized transport calculations with
observations and the statistics generated by
more explicit dynamical models (e.g., GCMs).
3.3. Does static stability remain constant during
climatic changes?
One of the most common assumptions in
climate modeling is that static stability remains
constant during climatic changes. For instance,
Rasool & Schneider (1971) have invoked this
assumption in order to estimate the global
temperature changes that may result from an
increase in the amount of carbon dioxide or
aerosols, but no estimate was presented as to
the sensitivity of their results to this assumption. Cess (1975) has examined a few aspects
of this question, yet any possible feedback
between the lapse rate and the surface temperature remains an important unsolved problem in climate theory. The results of our studies
suggest that the assumption of constant static
stability during global climate changes is
difficult to justify, particularly when ice feedback is included. The results of the Geophysical
Fluid Dynamics Laboratory’s (GFDL) GCM
with surface temperature computation and
snow-albedo-temperature foedback [Wetherald
& Manabe (1975), reprinted in Schneider &
Dickinson (1974) and Smagorinsky (1974)l
suggests that the lines of thinking which have
led to this assumption may be incomplete. At
any rate, some assumptions about static stability
is a necessary closure assumption in the simple
(and useful) one-dimensional (vertical coordinate) radiation balance models.
Nevertheless, in order to put the reliability
of results derived from these models in perspective, one must perform studies to determine the
dependence of the model’s sensitivity to changes
in internal parameters (e.g., static stability).
The argument for the near constancy of
static stability under climate changes goes as
follows. An increase of the solar constant will
(as mentioned before because of the spherical
shape of the earth) lead to an increase of the
globally-averaged equator-to-pole temperature
gradient; therefore, increased baroclinic activity
and more horizontal sensible eddy heat flux
(V’T’). Stone, using a theory of baroclinic
eddies, reasoned that an increase in tho horizontal flux will be associated with an increase
in vertical eddy heat flux (w’!””). This latter
condition will tend to make tho lapse rate less
steep. However, the increase in radiation alone
will tend to make the lapse rate more steep
because the increase in the ground temperature
leads to a shorter radiative relaxation time
which makes the radiation processes more
efficient a t destabilizing the atmosphere (i.e.,
increasing the lapse rate). According to Stone
(1973), these two competing effects almost
cancel each other and leave the static stability
relatively unchanged over a wide range of
Tellus XXVIII (1976), 2
117
ENERQY BALANCE CLIMATE MODELING
z
k-
w
0
a
8
0 301
I
I
I
I
I
I
I
0.29-
W
lK
k3
zr
0.28-
W
z
4-W
0.27
-
W
0
_J
9-
e
0
IT
s
0.26 0.25 -
:Without I c e Feedback
w
0
024'
0:o
0:o
oio
Id0
I
/o
1o:
140
SOLAR I N P U T FACTOR [I + (AWQI]
II :present value of solar constant 1
Fig. 4. Equator-to-polesurface temperature gradient
[K/100km] as a function of solar input for (S!Z'CV)
parameterization with and without ice-albedo-temperature feedback as given by eq. (3)with G, = 0.009.
values of Q,,. Exactly the reverse arguments
can be applied to the case of a decrease in
solar constant, and both cases are evident on
our Fig. 4. Suppose, for the sake of argument,
that the equator-to-pole temperature gradient
were incremed (rather than decreased as suggested by the above arguments) as a result of a
decrease in the solar constant; then baroclinic
eddies and radiation would work on the lapse
rate in the same direction and a decrease in the
solar constant would work toward a less steep
lapse rate (i.e., increased static stability).
[Again, an increase in the solar constant would
work toward decreased static stability for this
case.] Fig. 4 shows the response of the equatorto-pole gradient to changes in solar constant
with and without ice feedback. The particular
dynamical parameterization used here is
( S T C V ) , i.e., Stone's dynamical parameterization. It is clear from this figure that without
ice feedback the arguments presented by
Stone (1973) suggest cancelling influences that
would tend toward a constancy of the lapse
rate (since the equator-to-pole gradient increases with Qac thereby opposing the radiative
effects on static stability). However, when icetemperature-albedo feedback (eq. 3) is included, the same essential arguments presented
above will lead t o the conclusion that static
stability cannot remain constant during climatic
changes. [The curve stops a t solar input factor
-0.98 of the ice feedback case because of the
Tellus XXVIII (1976), 2
instability that leads to the ice-covered earth
solution discussed earlier (Table 2).]
It is interesting to compare the results just
described to the results obtained from a numerical model that contains both radiative and
baroclinic eddy influences on the lapse rate,
and the change in moist adiabatic lapse rate
associated with changing surface temperature,
namely: the GFDL GCM. [We have obtained
these GFDL GCM results through the courtesy
of Wetherald & Manabe (1975). They have also
been published and discussed by Smagorinsky
(1974) and Schneider k Dickinson (1974) from
which additional references and model details
can be obtained.] I n this mod01 snow feedback
is taken into account explicitly, but through
the hydrological cycle and a snow prediction
equation. Therefore, this model does not add an
empirical relation between changes in albedo
and changes in surface temperature as we have
done in eq. (3).
Fig. 5 shows the difference between the
standard control run and a run with a 2 %
increase in Qac, for this GFDL model. It is
clear from the figure that an increase of 2y6 in
the solar constant has caused an increase in
surface temperature everywhere, the largest
increase occurring near the pole. Wetherald
& Manabe (1975) also found that the converse
was true for a decrease in solar constant. The
+2%-STANDARD
r----
,336
.500
,664
,811
,926
80.
70.
60'
500 40. 30.
LATITUDE
20.
10.'
0.
Fig. 5. Zonally-averaged temperature difference
(K) between a standard case and a case where
solar input has been increased by two percent for
the GFDL GCM. Surface temperature, moist convection, eddy heat fluxes and snow-albedo-tem-
perature processes are all explicitly computed in
this model (Wetheraid & Manabe, 1976; reprinted
by Smagorinsky, 1974 and Schneider & Dickinson,
1974).
118
T. GAL-CHEN AND 9. H. SCHNEIDER
strong warming response in the polar surface
layers seen on Fig. 5 is caused by the relatively
large static stability characteristic of this
region and by the effect of snow-albedo-temperature feedback. Thus the results of the more
complex GCM model (Fig. 5 ) and simple energy
balance models with ice feedback (Fig. 4) both
indicate that the equator-to-pole temperature
gradient does not behave as might be expected
from pure baroclinic theory and radiation
alone (Fig. 3). This suggests that vertical eddy
heat fluxes, driven by the reversed change in
equator-to-poletemperature gradient, could act
to enhance rather than to dampen changes in the
lapse rate originating from changes in the radiation input. Most surprisingly, however, the
GFDL GCM result in Fig. 5 shows an increase in
hemispheric average tropospheric stability, i.e.,
a less steep lapse rate, resulting from an increase in solar input; whereas the chain of
arguments suggested by adding ice feedback
to Stone’s theory would suggest a destabilization, i.e., a steeper lapse rate. To understand
this apparent contradiction one must note that
the energy balance approach can account for
radiative heating or cooling and (through
baroclinic theory) for vertical eddy fluxes, but
not for lapse rate changes associated with moist
convection. The static stability of the GFDL
GCM, however, appears to be controlled primarily by moist convection, particularly in the
tropics-and much less so in midlatitudes where
the baroclinic influences described by Stone
would be expected to have a stronger impact.
It must also be mentioned that this particular
GCM neglects (as does our energy balance
model) the changes in cloudiness which may
result from a change in the climatic state
arising from the changes in solar constant.
Changes in cloudiness will in turn change the
radiation balance. The latter could possibly be
a t least as important a feedback mechanism as
the snow and ice albedo coupling. However,
even the direction of this effect is difficult to
determine (Schneider, 1972), so all these results
are, of course, tentative. Nevertheless, use of
these models suggests that the near constancy of
the static stability during climatic changes cannot
be msumed (but also cannot be ruled out as a
possibility), and that thermodynamic influences
(such as moist adiabatic lapse rates or ice
feedback) must be considered a8 central elements
in any climatic theory.
Despite the standard reservations, intercomparison of these climatic models helped to
develop physical insights. Thus, this discussion
illustrates the need to check inferences from
simple models against more elaborate numerical
simulations, which in turn can be more easily
interpreted by a mechanistic model such as
ours or Stone’s. A climate modeling methodology that stresses intercomparison of models
of differing complexity applied to the same
problem can sharpen intuition and reduce
misinterpretation of results (GARP, 1975; and
Schneider & Dickinson, 1974).
I n our energy balance models ( S T C V ,S T C )
the decrease in the equator-to-pole gradient
associated with an increase in the solar constant
is a result of the particular functional relationship we used between albedo and temperature.
Thus, if C, =0.009 (eq. 3) ice feedback dominates, but, if C, = 0 baroclinic eddies dominato
and the equator-to-pole gradient becomes a
monotonically increasing function of the solar
constant as suggested by Stone. The natural
question which arises then is what is the crossover point value of C, a t which a change in
solar constant would have no influence on the
gradient? I n general wo have found by numerical experimentation that this point is a function
of both the magnitude of the solar constant
perturbation, and C,. Nevertheless for small
solar constant perturbations [0.98 < ( 1 + A&,,/
Q,,) < 1.021 the value C , = 0.004 corresponds
roughly to the crossover point. Not surprisingly,
however, with such a value of C, the degree to
which the ice feedback enhances the response
of the surface temperature to changes in Q,,
is significantly reduced in comparison to the
“standard” case of C, = 0.009. This is evident
from Table 3 which displays the average
equator-to-pole gradient and the global temperature as functions of various solar constants
and C, for the case ( S T C V ) .For instance, 1 %
reduction in the solar constant has reduced the
planetary temperature by 1.35 K for C, =
0, 1.72 K for C, =0.004 and 3.08 K for C, =
0.009. The different values of the control runs
(i.e., 1 +AQ,,/Q,, = 1.00) are simply due to small
imbalances in the initial data, due to rounding
errors. Recall that the initial temperature
which produced these control runs was 287.30 K
and we see that the imbalances tend to be somewhat amplified in the presence of strong ice
feedback.
Tellus XXVIII (1976), 2
119
ENERGY BALANCE CLIMATE MODELING
Table 3. Equator-to-pole temperature gradient (globally-averaged)and global average surface tempemture for three different values of ice feedback parameter C, and for present value, one percent increoae,
and one percent decreme i n sohr constant [for ( S T C V ) parameterization]
CT
1.oo
0.009
aT
--
aY
0.99
= 0.2791
“K
- 3
100 km
T = 287.54 K
0.004
--
aT
aY
0.2796
a- T
=
aY
aY
“K
aT
100 km
aY
-
- = 0.2795
“K
T = 287.42 K
aT
- =
aY
(1) The radiation balance, in particular the
functional relationship between albedo and
temperature, seems to dominate the sensitivity
of the “climate” to perturbation in energy
inputs in these energy balance models. The
dynamical parameterizations are able to modulate sensitivity by roughly factors of 2, but
thermodynamic parameterizations control the
order of magnitude response of these zonal
climatic models. I n addition, our numerical
experiments indicate that once the dynamics
are parameterized in terms of the thermodynamics the “simplest” parameterization,
i.e., the one which does not distinguish between
various (i.e. ocean, atmosphere or latent heat)
transport terms, is apparently the most reasonable choice (at least insofar as the reproduction of a monotonically varying, smooth graph
of equator-to-pole temperature gradient versus
solar input is concerned). With respect to this
conclusion one obvious extension of our energy
balance model seems to be a multilayer treatment of the radiation. I n such a treatment a
distinction can be made between surface albedo,
which causes the ice feedback effect, and
atmospheric albedo. Increasing the solar zenith
aY
”K
aT
-=
100 km
aY
-
0.2785
T = 286.07 K
4. Conclusions and suggestions for
further work
aT
- =
“K
0.2763 --.__
100 km
T = 289.76 K
“K
0.2795 100 km
T = 289.04 K
T = 285.76 K
0.2798 __
100 km
Tellus XXVIII (1976), 2
“K
0.2834 ___
100 km
T = 284.46 K
T = 287.48 K
0.000
aT
~
1.01
“K
aT
- =
100 km
aY
~
0.2810
“K
100 km
T = 288.74 K
angle reduces the effect of changes in the surface
albedo on the earth-atmosphere system albedo
(Coakley & Schneider, 1974) and should be
accounted for since ice-albedo-temperature
feedback is most active in high latitudesprecisely the regions of high solar zenith angle.
Ideally one would also like to improve the
infrared treatment as well but here one encounters the difficulty of having to specify the
lapse rate (see discussion belov:).
(2) The reduction in the global average surface
temperature as a result of 1 % decrease in the
solar constant was found to be in the range
- 5.35 6 AT, < - 3.08, where the smaller change
is for Stone’s dynamical parameterization
which is based on baroclinic theory, and the
larger values are computed for Sellers (1969)
K + K ( T)parameterization. The smaller number
is also closer to that obtained with the GFDL
GCM. (These all include ice-albedo feedback.)
(3) The assumption that static stability
remains constant during climatic changes
cannot be simply justified when both dynamical
and thermodynamical processes are considered
simultaneously. However, it is not possible to
evaluate even the direction of static stability
change with surface temperature change unless
an assessment is made on the relative importance of all the factors which determine the
120
T. QAL-CHEN AND S. H. SCHNEIDER
static stability. Such factors include baroclinic
eddies, radiative relaxation time, ice feedback,
moist convection, a n d cloudiness.
Acknowledgements
We wish to acknowledge m a n y valuable
discussions with Robert E. Dickinson, in-
eluding his suggestion to perform the computations displayed i n Table 3. W e also appreciate
his critical comments and suggestions for improvement of t h e original manuscript as well as
those of John S. A. Green, Cecil E. Leith and
Peter H. Stone. W e are indebted t o R . T.
Wetherald and S. Manabe for sending us prepublication copies of their GCM results (our
Fig. 5 ) .
REFERENCES
Budyko, M. I. 1969. The effect of solar radiation
variations on the climate of the earth. Tellus 21,
611-619.
Budyko, M. I. 1972. The future climate. Trans.
Amer. Geophys. Union 53, 868-874.
Cess, R. D. 1975. Global climate change: An investigation of atmospheric feedback mechanisms.
Tellus 27, 193-198.
Coakley, J . A., Jr & Schneider, S. H. 1974. I n fluence of solar zenith angle on the albedotemperature fredback (abstract). Eos Trans.
AGU 55, 266.
Dickinson, R. E. 1971a. Analytic model for zonal
winds in the tropics. 1. Details of the model and
simulation of gross features of the zonal mean
troposphere. Mon. Wea. Rev. 99, 501-510.
Dickinson, R. E. 19716. Analytic model for zonal
winds in the tropics. 2. Variation of the tropospheric mean structure with season and differences between hemispheres. Mon. Wea. Rev. 99,
511-523.
Faegre, A. 1972. An intransitive model of the
earth-atmosphere-ocean system. J . A p p l . Meteorol. 11, 4-6.
GARP 1975. Report of the GARP Study Conference on The Physical Basis of Climate and
Climate Modelling. World Meteorological Organization, Geneva GARP Series no. 16.
Gordon, H. B. & Davies, D. R. 1974. The effect of
changes in solar radiation on climate. Quart J .
Roy. Meteorol. SOC.100, 123-126.
Green, J . S. A. 1970. Transfer properties of the
large-scale eddies and the general circulation of
the atmosphere. Quart. J . Roy. Meteorol. SOC.96,
157-185.
Hantel, M. 1974. Polar boundary conditions in
zonally averaged global climate models. J . A p p l .
Meteorol. 13, 752-759.
Leith, C. E. 1974. Unpublished note.
Manabe, S. & Terpstra, T. B. 1974. The effects of
mountains on the general circulation of the
atmosphere as identified by numerical experiments. J . Atmoa. Sci. 31, 3-42.
North, G. 1975. Theory of energy balance climate
models. J . Atmos. Sci. 32, 2033-2043.
Rasool, S. I. & Schneider, S. H. 1971. Atmospheric
carbon dioxide and aerosols: Effects of large
increases on global climate. Science 173 ,138-141.
Richtmyer, R. D. & Morton, K. W. 1967. Diflerence
methods for initial-value problems. Wiley, New
York.
Robinson, G. D. 1971. Review of climate models.
I n Man’s impact o n the climate (ed. W. H. Matthews, W. W. Kellogg and G. D. Robinson),
pp. 205-215. MIT Press, Cambridge, Mass.
Sagan, C . , Toon, 0. B. & Gierasch, P. J. 1973.
Climate change on Mars. Science 181, 1045-1049.
Saltzman, B. & Vernekar, A. D. 1972. Global
equilibrium solution for the zonally averaged
macroclimate. J . Geophys. Res. 77, 3936-3945.
Schneider, S. H. 1972. Cloudiness as a global climatic
feedback mechanism: The effects on the radiation
balance and surface temperature of variations
in cloudiness. J . Atmos. Sci. 29, 1413-1422.
Schneider, S. H. & Gal-Chen, T. 1973. Numerics1
experiments in climate stability. J . Geophye.
Res. 78, 6182-6194.
Schneider, S. H. & Dickinson, R. E. 1974. Climate
modeling. Rev. Geophys. Space Phys. 12, 447-493.
Schneider, S. H. & Dennett, R. D. 1975. Climatic
barriers to long term energy growth. Ambio 4,
65-74.
Sellers, W. D. 1969. A global climatic model based
on the energy balance of the earth-atmosphere
system. J . A p p l . Meteorol. 8, 392-400.
Sellers, W. D. 1973. A new global climatic model.
J . Appl. Meteorol. 12, 241-254.
Smagorinsky, J. 1963. General circulation experiments with the primitive equations: The basic
experiment. Mon. Wea. Rev. 99, 99-164.
Smagorinsky, J. 1974. Global atmospheric modeling
and the numerical simulation of climate. I n
Weather modification (ed. W. N. Hcss). John
Wiley, New York.
Stone, P. H. 1972. A simplified radiative-dynamical
model for the static stability of rotating atmoepheres. J . Atmos. Sci. 29, 405-418.
Stone, P. H. 1973. The effect of large-scale eddies
on climatic change. J . Atmos. Sci. 30, 621-529.
Study of Man’s Impact on Climate (SMIC) 1971.
Inadvertent climate modification. MIT Press,
Cambridge, Mass.
Wetherald, R. T. & Manabe, S. 1975. The effects
of changing the solar constant on the climate of
a general circulation model. J . Atmos Sci. 32,
2044-2059.
Tellus XXVIII (1976), 2
ENERGY BALANCE CLIMATE MODELING
KJIHMATBqECKOE MOAEJIBPOBAHHE EAJIAHCA 3HEPI'HH: CPABHEHHE
PAJ(HAqHOHHOI'0 H AHHAMHYECKOI'O MEXAHHBMOB O E P A T H O n CBFIBH
Tellus XXVIII (1976), 2
121