A Study of the Stability of the Global Surface Temperature
by
Tsuyoshi Sasaka
B.A., Kyoto University, 1980
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Basic Science
1993
Atf

This thesis for the Master of Basic Science
degree by
Tsuyoshi Sasaka
has been approved for the
Program of
the Master of Basic Science
by
Clyde Zaidins
Thomas Russell
Date

Sasaka, Tsuyoshi (M.B.S., Basic Science)
A Study of the Stability
of the Global Surface Temperature
Thesis directed by Professor Clyde Zaidins
ABSTRACT
This project addresses the basic question about
the factors determining the surface temperature of the
earth system and the stability of the global surface
temperature. This project used approaches, quantitative
(numerical) and qualitative.
The quantitative approach is computer modeling to
simulate the fundamentals of the system of the earth
temperature restriction on its surface. This was done
by the analysis and the construction of the radiation
energy budget on the surface of the simulated earth. In
this model, the emissivity of the incoming and outgoing
radiation, the heat capacity of the system, and the
horizontal heat conductivity determine central part of
the simulation. Seven versions of this model were
calculated.
The qualitative approach is the feedback analysis
of the assumed critical components interacting on the
global radiation energy budget, and hence the surface
temperature of the earth. Facing the global warming
trend of the present days, many 'factors are thought to
be acting either directly on the radiation emissivity
or indirectly on it through influences on each other.
Needless to say, the so called greenhouse gases,
vegetation, and clouds are among them and of major
concern. Ice and snow play a role on albedo
(reflectivity to the incoming solar radiation).
Anthropogenic impacts are now significant and a
1

complicating factor. In particular, water plays a most
important part about both emissivity and thermal
inertia in every phase (vapor, liquid, and solid). All
these factors are dependent on each other. This
qualitative analysis is done so that we can visualize a
whole picture of these interacting factors, and perform
analyses about the structure of the feedback loops.
2

MASTER OF BASIC SCIENCE THESIS
DATE
12 MAY 1993
TITLE
A STUDY OF THE STABILITY
OF THE GLOBAL SURFACE TEMPERATURE
NAME
TSUYOSHI SASAKA
MAJOR (OPTION)
BASIC SCIENCE (APPLIED SCIENCE)

TABLE OF CONTENTS
1. INTRODUCTION 1
2. A SIMPLE EARTH MODEL ABOUT TEMPERATURE RESTRICTION
A) METHOD(A COMPUTER MODEL) ..............................3
(1) First version 3
(2) Second to Sixth version .........................6
(3) Seventh version 6
(4) Error Estimation of COSÂ£ ............6
B) RESULTS AND DISCUSSION 9
(1) Second version ...............10
(2) Fourth version ...............10
(3) Seventh version ...............11
(4) Extension of the Interpretation of the Model....14
3. A FEEDBACK ANALYSIS
A) METHOD 15
(1) Directness and Indirectness ....................15
(2) Definition of a Relation 16
(3) Feedback Structure on GMST (a Hypothesis).......16
B) DISCUSSION 18
(1) Basic Assumptions ..............................18
(2) Loop Property ...............20
(3) Feedback on GMST and Stabilizing Factors ...24
4. CONCLUSION AND PERSPECTIVE 28
5. REFERENCE 29
6. APPENDIX(A) PROGRAM CLIM07.FOR
APPENDIX(B) CLIMDATA TABLE
APPENDIX(C) VIEWGRAPHS OF THE ERROR OF COS$
APPENDIX(D) A PAIR OF LOOPS(LENGTH 2)

1. INTRODUCTION
This project started from the basic question about the
factors restricting the surface temperature of the earth
system and the stability of the global surface temperature.
There are two possible approaches, quantitative (numerical)
and qualitative. I used both approaches.
The quantitative approach is computer modeling to
simulate the fundamentals of the system of the earth
temperature restriction on its surface. This was done by the
analysis and the construction of the radiation energy budget
on the surface of the simulated earth. In this model, three
variables (the emissivity of the incoming and outgoing
radiation, the heat capacity of the system cthermal
inertia>, and the horizontal heat conductivity) play the
central part of the simulation. The model started from a
fixed sphere of the earth radius at the earth's position
from the sun and its surface temperature was computed
recursively for some time period. Then developments have
been done to enable it to simulate the earth's rotation and
revolution and the several processes occurring on and over
the earth's surface. Now it has seven versions, each with
increasing complexity.
The qualitative approach is the feedback analysis of
the assumed-to-be critical components interacting on the
global radiation energy budget, hence the surface
temperature of the earth.
If we assume the system as a box, Â£ (emissivity, which
is 1-reflectivity ) gives major impacts on both the input
energy into the box and the output energy out of it. (X(heat
conductivity) relates to energy movement to or from adjacent
boxes. fl (heat capacity or thermal inertia) is the major
determinant of the time required for the box to reach an
equilibrium with respect to the temperature behavior
following an abrupt change of the net radiant energy budget
of the box. Â£ controling the output energy from inside the
1

box is also playing a role in determining the time interval
to reach an equilibrium.
Thus, g and Q are acting as principal factors
regarding to the heat gain and heat loss of the system.
Facing the global warming trend of the present days, many
factors are thought to be acting either directly on the
emissivity or indirectly on it through influences on each
other. Needless to say, the so called greenhouse gases,
vegetation, and clouds are among them and of major concern.
Ice and snow play a role on albedo (reflectivity to the
incoming solar radiation). Anthropogenic impacts are now
significant and a complicating factor. In particular, water
plays a most important part about both emissivity and
thermal inertia in every phase (vapor, liquid, and solid).
But all these factors are dependent on each other.
My effort in this qualitative analysis is done so that
we can visualize a whole, if it is a rough sketch, picture
of these interacting factors and hopefully extract some
important traits of their interdependency and collective
feedbacks to the emissivity, hence to the temperature.
Specifically, analyses about the structure of the feedback
loops and its property has been done.
Collective works in 'J.Hansen and T.Takahashi, "Climate
Processes and Climate Sensitivity", 1987' discussed relevant
important topics. Those include the studies by Hansen
(feedback mechanisms, a comprehensive study), Hall (albedo
and surface temperature relation), Stone (moist convection),
Dickinson (vegetation-albedo feedbacks), and Hartmann (ice-
albedo and vegetation-albedo feedbacks). These subjects
continue to be studied in Houghton 1991 (stability and
variability in a Coupled Ocean-Atmosphere Climate Model),
Xue 1991 (vegetation-albedo-moisture relations), and Cohen
1991 (snow cover-surface temperature feedback). Other
important, yet not clearly understood, processes are clouds'
effects on the climate feedback. The studies concerning this
topic include Cess 1987, Wetherald and Manabe 1988, Ardanuy
1992, Molnar 1992, and Dutton 1992.
The idea of studying a complicated feedback structure
by means of signed digraphs in discrete modeling method can
be seen in F.Roberts 1971. A signed digraph is an arc-
directed graph consisting of a set of vertices and a set of
arcs between vertices, with each arc signed either
positively or negatively. If we replace 'signed' with
'weighted' in the previous sentence, it defines a weighted
digraph. In such an application, the vertices of a digraph
are taken to be variables in feedback processes being
studied. He also studied in mid 70's the stability of a
weighted digraph relative to some simple pulse process,
using the eigenvalues of the adjacency matrix that
represents the set of weighted arcs of the digraph
concerned. He called this 'pulse stable' or 'pulse
unstable'. [F.Roberts,1975,1976] Though his method gives us
a convenient way to predict the stability of a digraph
representing a whole feedback structure, it is difficult for
2

us to tell the relation of each component as a vertex with
the total system and the other components. So, in this
paper, I developed an alternative way to analyze them in
Ch.3.
2. A SIMPLE EARTH MODEL ABOUT TEMPERATURE RESTRICTION
A) METHOD( A COMPUTER MODEL)
This is the computer-dependent project. The language
is Fortran. Seven versions have been developed.
(1) First version
The first version is the base model and the main
setting was done here. Suppose we have a sphere of the earth
radius (R). The sphere is divided on the surface into 22
zones. < Fig2.1 > and < Fig2.2 >. All physical quantities
are in the SI units.
^the center
of
the earth
< Fig2.1 > < Fig2.2 >
The sole radiant energy source is the solar insolation
onto the earth surface. The insolation is incident upon the
center of the 10th zone in < Fig2.1 > No rotation of the
sphere is introduced in this version. So, the sphere and
each zone stay fixed relative to the incoming solar rays.
Thus, there are three types of zones with respect to the
radiant incident angles; i.e., polar(l, 2, 21, 22),
midlatitudinal(3 through 8), and equatorial(9 through 14).
The longitudinal differences will disappear when the sphere
is rotated in the fourth version.
The boundary between polar and midlatitudinal zones is
set at 60 degree N or S. The boundary between midlatitudinal
and equatorial ones is set at 20 degree N or S. The
longitudinal boundaries for polar groups are 0 and 180
degrees, which correspond to the terminator of the day and
the night. Those for midlatitude and equatorial groups are
3

obtained by equally dividing 360 degree into 6 sections in
each group. In this way, the area of each zone in the same
group is equalized.
For each zone, the recursive calculations of the
surface mean temperature were done with the interval of At,
given the assigned initial temperature for each zone, which
was taken as 280K. The calculation is done by the
corresponding computer program, using the following
difference equation.
In < E 2.1 >, At is the time interval. aT[ is the
temperature change during at for the ith zone. S is the
solar constant at the distance of one astronomical unit from
the sun. (7 is the Stephan-Boltzmann constant. 6 is
emissivity( 0< 6 <1 ) (X is the conductivity between
neighboring zones. A zone is actually assumed as an
atmospheric column of 8000 m height. Q is defined as the
specific heat/m2 surface, accounting for the thermal inertia
of each zone.
< Table 2.1 >
< E 2.1
j
The recursion is given by the next equation.
c
constants S(solar constant)
at(time interval)
0( Stephan-Boltzmann)
R(Earth radius)
8640*104sec(a normal day)
1.36*103watts/m2
5.67*10"8w/rn2K4
6.37*106m
e (emissivity) 0.7, l-6 =ALBEDO
variables Q[(horiz.cond. Atm.) 2.6*105w/mK
g(spec.heat/m2 Atm.) l.05*107J/m2
4

T^, measured in Kelvin, is the surface temperature of
each zone on the sphere, which corresponding to the
effective radiating temperature at the mean radiating height
of a more realistic earth model. is the area of a zone.
Lij is the boundary length between ith and jth zone.
Cos is the cosign of the angle of the incident radiation
at the center of the ith zone. The center is the
intersection of the median longitude and latitude for each
zone. Thus, the heat gain of a zone by the solar insolation,
the heat loss of the zone by the long wave emission to the
space, and the heat fluxes between adjacent zones are given
by the following quantities respectively. (SAie-cosi,.)^t (heat gain)
< E 2,4 > ( (heat loss)
< E 2.5 > | (X 2 Llj (T; -Tj))'At (heat flux>
It is necessary to describe a little more detail of the
term that governs the heat flow between the zones, that is
< E2.5 >. This term is also given by:
OTijLij ( Ti Tj )
where Ti is the temperature of, the ith zone, Tj is the
temperature of one of the neighboring zones labeled by j,
Lij is the length of the boundary between the ith and the jth
zone, Of^j is a heat transfer coefficient discussed, and the
sum is taken over all zones that share a boundary with the
ith zone. The particular form of this expression implies
that the net heat flow is from the ith zone if the expressed
numerical value is positive. The heat flow model used here
is based on an assumption that all the heat transfer takes
place through a thin layer between zones. Although it is
based on a conductivity model, it is also a good
approximation to convection and radiation processes as well.
The assumption here is that the heat transfer is
proportional to the temperature difference between the
zones. Since a fixed width of the boundary is chosen and
this value is much smaller than the dimensions of the zone,
5

the usual temperature gradient term is replaced by the
temperature difference, and the linear dimension is absorbed
into the denominator of the coefficient. The numerical
value of Ckij depends upon the "conductivity" and the height
of the atmosphere in the numerator and the boundary width in
the denominator, and has the dimensions of
Watts/meter*Kelvin. The value of (X ^ depends upon the
boundary involved. The original modelJ used only a single
value of a for all boundaries.
(2) Second to Sixth version
In the second version, the boundaries between polar and
midlatitude(m.l.) zones and between m.l. and equatorial ones
are changed from 20 and 60 degree to 16 and 55 degree
respectively. The areas of all zones have been equalized in
this way.
In the third and fourth versions, the simulations
became possible for the daily rotation and the annual
revolution of the earth, with the recursive calculations of
the changing insolation angles. This makes us possible to
observe the simple seasonal variations.
In the fifth version, a had two distinctive weights
(HFPM, HFEM) HFPM is the coefficient of the heat flow
between polar and m.l. zones. HFEM is that between m.l. and
equatorial ones. Also, the distinction was made between the
emissivity of the incoming radiation ( ) and that of the
outgoing radiation ( Q0). In this way, it became possible to
count on different radiative forcings, such as cloud
radiative forcing on both the shortwave and longwave
radiation, the greenhouse effect by greenhouse gases, and
the influence on albedo by the ocean, ice, and snow. The
term "forcing" means the effect on the net radiation energy
budget of the earth system by a component of interest;
radiative forcings are given by Watts/m2 surface, and albedo
forcing is given by scalar.
In the sixth version, 6[ and 60 are classified into
three types, relative to different latitudinal zones,
respectively.
(3) Seventh version
The seventh version is the current one. The same thing
was done for Q value (heat capacity) here as 0- and 0O
in the sixth version.
Now, g and Q have nine degrees of freedom, instead
of two. So, tropical, temperate, and polar properties can be
captured, regarding the radiative forcing and the thermal
inertia.
(4) The Error Estimation of COS^
6

When the incoming energy term < E 2.3 > is calculated
using cos ^ , some skew is occurring since cos $ is
represented by the center of each zone. For example, while
the sun, relative to the earth, oscillates between 23.5 N
and 23.5 S the sun is right over the equator at the
equinox. See < Fig 2.3 >
< Fig 2.3 >
Then, the input energy of the equatorial zones is
calculated with the largest cos ( =1 ). So, it tends to be
overestimated. Then, the question is what is the true
average value of cos ^ for each zone( denote it as ) Cos is given by sin Q* sin (p where 9 is
the angle of the surface point with Z-axis, and (p is the
one with X-axis, assuming that the north pole is on the
positive Z-axis and the sun is right on the positive Y-axis
in the initial versions. Later the sphere is rotated while
the axes are fixed relative to the sun. See < Fig 2.4 >
7

The area weighted average of cos $> is given by the
integral of cos J *dA/A, where dA= sin 9 *d8*dip for a sphere
of radius of unit length. < Fig2.5 >
Thus,
< E 2.6 > =
Sin
d 9
siny Sin 5
2 5 Sin 0 Sin Â£
6-
Sin2Â£
2
4. sin2Â£. sin2@
,where 0 Â£ and (f> Â§ represent the boundaries, with
Â£ and Â£ arbitrarily small.
When we rotate or revolve the earth model relative to
the fixed sun, the new coordinates of the surfaces points
are obtained by a multiplication of three matrices. And
sin 9 's and sin (p 's for the new angles can be obtained
theoretically. However, sin Q might become very close to
zero in the motion of the model, which in turn might cause a
computational stop in the calculation of the surface
temperature if I use < cos ^ > for the incoming energy
term. So, I decided not to use it in the program. As a
compensation, the assessment was done for the error range of
cos ^ from < cos ^ >., by computing and observing the
following two quantities for the three latitude regions,
respectively.
< E 2.7 >
Difference = < cos | > sin 8 sin

note, sin 8 sinCp = cos Â£
and
< E 2.8 >
Relative Dif.
< cos ^ > sin 8 * sin ip
sin Q sin (p
8

Both quantities are computed for the ranges of the center of
three types of zones.
0.17 ft Â£ 9 Â£ 0.43 71 ), and equatorial ( 0.37 ft Â£ 9 Â£ 0.5ft ),
and 0 < cp < ft for all>
See the viewgraphs in Appendix(C).
<1> Dif.Pol. <2> R.D.Pol. <3> Dif.Mid. <4> R.D.Mid.
<5> Dif.Equ. <6> R.D.Equ.
For the graphs of the difference, the abscissa(X-axis )
is sin 9 *sin (p and the ordinate (Y-axis) is < cos ^ >.
For the graphs of R.D., the quantity is given as the
surface with respect to the plane, with $ (p extending
over the above ranges that are shown in twelve portions.
The polar region shows the worst case of the three. For
the large sin 9 *sin (p is overestimated up to
about 0.38, and for the small values the average is
underestimated up to about 0.2. The net effect seems to be a
slight overestimation. But, since this is only for the
center on the day side. When the center is on the night
side, some amount of underestimation should occur.
Therefore, is slightly overestimated in the summer
and vice versa in the winter for the polar zones.
For the temperate zones, it can be seen in Dif.graphs
that sin 0*sin

values.
For the equatorial zones, though very slight
overestimations are occurring, it is clear from R.D.graphs
that deviations are very small.
Thus, for both Mid. and Equ. zones we are making very
good estimations for the true average value.
(B) RESULTS AND DISCUSSION
I will focus on the results of the 2nd, 4th, and 7th
versions, since major adjustments can be summarized by them.
See < Table 2.2 >
< Table 2.2 >
version .features
2 No motion, Boundaries( EM, MP 16, 55 degrees)
4 Daily rotation, Annual revolution
7 rotation, revolution, (X (HFPM, HFEM) ,
latitudinal distinction in 0;, 0O and 3
note: g(emissivity), Of (horizontal heat cond.),
g(heat capacity-thermal inertia)
9

(1) Second version ( Appendix(B)-2A,2B,2C )
This is the version fixed in the position of < Fig 2.1>
Three tests cases are shown in Appendix(B).
First test (2A.DAT)
The values for are equal to those in ,
except at. At is taken as 1.0*105sec. The recursion of T's
computation is 100. About 116 days passed after the last
step. In the output data, the first step and the last step
are shown. The initial assigned temperature for each zone is
280K (the zones tabulated are, from the left, 1 through 13
(see). Also, the computed surface mean
temperature(SMT) at each step is presented. Reasonably, the
10th zone became the hottest. Interestingly, however, the
coldest zone was the 7 th, not the 13 th. This might be
because the system has not reached an equilibrium yet, or
simply an artifact of this discrete modeling scheme. The
average temperature didn't reach an equilibrium. It is still
decreasing quite rapidly after 116 days.
Second test (2B.DAT)
Only A t was changed from the first test. A t is now
5*105sec. So, the last step is the 579th day. SMT is still
decreasing at the end. The intermediate steps, as well as
the first and the last, are given for each zone. The two
hottest zones (4 th and 10th) seem to have reached
equilibria, since their last step values are the same as in
2A. On the other hand, the colder zones are still quickly
going down. Notice the extreme coldness in the 7th.
Furthermore, it may be seen that the neighboring zones of
the colder are falling in temperature due to the loss of
heat to their colder neighbors. Observe the 1st zone. It
once increased its temperature, then is declining. It may be
said that either the outgoing energy to space is too large
or the horizontal heat flux is too inefficient; or probably
both may be true.
Third test (2C.DAT)
The passed time is the same as in the second test (579
days) Considering the result of the 2nd, a greater heat
capacity was tested. It was increased from 1.05*107(J/m2K)
to 5.0*107 to reduce or delay the loss of heat. Though SMT
clearly slowed down its declining, it did not show an
equilibrium yet. The hottest regions were the same in their
values as before. The colder zones are still quite cold.
However, the extreme value of the cold (7th) disappeared.
The colder regions leveled off with each other. I infer from
this that the greater Q value enabled more heat energy to
move horizontally rather than simply to be emitted into the
space.
(2) Fourth version
( Appendix(B)-4NOTILT,4SUMMER,4AUTUMN,4WINTER )
10

In the fourth version, daily rotation and annual
revolution became possible. In the output data, the last two
values at the top of data means the axis tilt and its
initial angle with respect to the sun. a t is one normal
solar day. The recursion of the computation is 30. So, it
shows one month of phenomena. Now all 22 zones appear on the
data. The effect of the rotation is evident in SMT. SMT does
not drop rapidly from the beginning. It is obvious even in
"NOTILT" test. Notilt allows only daily rotation since the
annual revolution doesn't mean anything. All 22 zones were
greatly leveled off, compared with the second version. Why
can the mere daily rotation control the fall of SMT? It is,
I suppose, because the rotation prevented the hotter zones
from becoming extremely hot; hence it reduced the outgoing
energy quantity of those zones.
The seasonal features are also to be seen clearly. The
summer and the winter tests are exactly in reverse order
with each other, therefore they are basically equivalent.
Notice the slight difference of the last SMT's between
"Summer" and "Autumn", which is 279 vs 278. Also compare the
hottest and the coldest zones of the two; 296 around the
north pole in the summer of the northern hemisphere vs 289
of the equatorial zones in the autumn, and 238 around the
southpole in its winter vs 247 around the north pole one
month after the autumn equinox. It might be that a little
overestimation of is occurring in the summer pole,
and vice versa in the winter pole as noted in the assessment
of
(3) Seventh version
( Appendix(B)-7NORMAL,7ALPHA1,7ALPHA2,
7EI1,7E01,7E02,7B1,7T1,7T2,7T3 )
Emissivity was assigned the distinct terms regarding
the incoming and the outgoing radiation in the 5th version
( 6i, 60) Bj., Â£0, and fi (heat capacity) can take three
different values respectively corresponding to the three
latitude types (polar, temperate, and equatorial). Qf (heat
conductivity) can bear additional distinct weights between
different latitudes (HFPM, HFEM). HFPM is the weight between
the polar and the temperate. HFEM is that between the
temperate and the equatorial. In the output data, (HFPM,
HFEM) appears in the third raw, ( Q, ip, eim, Bie, eop, eom,
6oe) are in the fourth, and ( fl 3 mr Be) are the
fifth, with all in these order from the left.
7N0RMAL.DAT
This test data stands for the basis and the reference
to be compared with all other tests with the 7th version. HF
is taken as 1.0, meaning no weights for (X . Â£ 's are all
taken as 0.7, the average normal data. Q 's are all taken as
1.05*107, assuming the system model of the atmospheric
11

columns. Thus, the data should show the same results as
those of the 4th summer version, and so they are.
7ALPHA.DAT
First test HFPM is 0.1 and HFEM is 1.0.
Second test & is increased by the factor of 10 from
2*105 to 2*10, and HF's are as of 1st.
They were done to inquire into the sensitivity of the
model response to this magnitude of OC Thus, it matters
how much latitudinal variation is caused. The increase of
a in the second brought the midlatitudinal temperatures
closer to SMT than can be seen in the first, while it
created no apparent change in SMT. Compare the last step
values for each zone in "ALPHA1" with those in "NORMAL".
Though they show the changes of the order of 1 to 2 degrees
in several zones, the changes are not as large as those
between "ALPHA1" and "ALPHA2". It seems to indicate the
order of 105 of a is a little too small to allow enough heat
flux horizontally, consistent with the limiting assumption
of the system as the atmospheric columns.
7E.DAT
First test 6in's were taken as 0.69.(AlbedoO.31)
Second test eout's were taken as 0.69.
Third test g,out's were taken as 0.699. (* The format
of the output rounded them to .70, but
the input were .699 and the computation
was done so. To see that, please compare
the last step with that of "NORMAL".)
In the first test, more than half of the zones showed
the decrease of the temperature by one degree. SMT also
showed a quicker decrease than that of "NORMAL". This
magnitude is more than significant in terms of SMT. The
differences between the second and the third shows clearly
the temperature dependence on Â£0 regionally and globally.
But the small differences between "E02" and "NORMAL" may
mean more than those between "EOl" and "E02". Notice the
last step of the winter pole both in "E02" and "NORMAL".
Also notice that SMT's drop to 279 is one step later in
"E02" than in "NORMAL". It is encouraging that we have the
model sensitivity to eo change of the order of 102. When we
speculate on the real climate, the variations of in the
order of 10_1 and that of &0 in the order of 10~2 should be
reasonable since the cloud radiative forcing can be a few to
ten times greater than that by doubling carbon dioxide in
the atmosphere.[Ramanathan 1989]
7B.DAT
To check about the impact of the heat capacity variable
on the thermal inertia of the system, Â£ of the polar region
was raised by the factor of about 2, with all Â£ 's as
"NORMAL". This should enable us to make bridge to the large
12

thermal inertia which had been kept low in this modeling.
The results shows the control of the temperature rise
around the summer pole and the sustaining of the temperature
around the winter pole. SMT was given a strong impact, too.
It seems to be near an equilibrium.
7T.DAT
Based on the results so far, the variables' changes
were combined to see if the model approaches to the real
globe.
First test
Go were all taken as .695, with the green house trend
in mind. 8 polar was taken as 2*107. With this short time
period(one month), the difference of SMT from "Bl.DAT" did
not show up. The summer polar zones are still higher in
temperature than the temperate zones of the opposite
hemisphere.
Second test
To improve "Tl.DAT", (X was increased to 3.*105, with
an atmospheric-ocean column in mind. HF were weighted
heavily for two interlatitudinal fluxes in the opposite
direction with each other, inferring the water vapor
movement and its phase change. Speculating on polar ice and
snow and equatorial cloud, I assigned distinct values to
both Â£'s. As a result, the equatorial region was improved.
But the shortcoming in "Tl.DAT" remained in this data, too.
Third test
On the same line of conjecture, 's were changed
further. 0.4 of bipolar is the result of the consideration
of the surface albedo, since it may vary from 0.8 or more
for fresh snow to 0.4 or less over old, melting sea ice.
[Mitchell, pl23] The difference between both Â£ 's for
equatorial zones may be a little too large. But this is
assigned with the image of the large amount of water vapor,
not as clouds, over the tropical region.
The result shows a good shape of the temperature
distribution, comparable to the real globe. SMT is stable.
The shortcomings left are about the temperate zones on the
winter side, which is a little too cold. I suspect this is
due to the difficulty of modeling the impact created by the
interlatitudinal movement of water vapor in the atmosphere.
Because the water cycle change its phase with its large
latent heat exchange with the surroundings, it may be hard
to model that effect by the simple term of < E2.5 >.
As for SMT, it is too early to judge if it is behaving
close to the real system. If it is in the equilibrium at
28OK, it would be 7 or 8 degree lower than the observed
global mean temperature. But it takes an experimentation
over very long period, since a slight change could become
more than significant, considering the impact of a few
degree increase of SMT on the climatic system. Also, if we
13

run the model over a few year period, it might sustain the
temperature of the midlatitude in the winter using a better
estimation of the thermal inertia for which it takes to
understand the large and complicated impacts of the global
water cycle.
(4) Extension of the interpretation of the model
Suppose each zone of this model as a box. Then the heat
energy fluxes of the system is < Fig2.6 >.
The model in the original(first) version assumed the
box contained only the atmosphere. a and Q values were
originally obtained assuming 8000m of atmospheric columns.
This imposes rather strict constraint on Of and ft
e
< Fig2.6 >
outp*
< Fig2.7 >
14

In addition, the box doesn't have any internal
structure. The inside of the box is uniform. It puts
serious limitations on the flexibility of Â£ cx and $ ,
since there is not any basis to assume regional(latitudinal,
in particular) differences, beside the solar angle, between
the boxes as zones. Hence it limits the room for
adjustments. To avoid this difficulty I reinterpret the
box as Atmosphere-Ocean(land) model to assume an internal
structure and hence the difference with the other boxes than
itself. This reinterpretation should be allowed within the
system of the given energy fluxes, because it does not
change any fluxes qualitatively with outside the system.
Thus the basic structure of the model and the computation go
unchanged, while more room for the interpretation
of e, and g is brought in. < Fig2.8 >
I
< Fig2.8 >
3. A FEEDBACK ANALYSIS
A) METHOD
(1) Directness and Indirectness
The atmospheric increase of the greenhouse gases(G.G.)
like C02 and CH4 is a fact among most researchers who study
the greenhouse warming. But it is not yet certain how much
increase of the global mean temperature it will result in.
So, we should not confuse the certainty of the buildup of
these gases in the atmosphere with the uncertainty of its
precise results in the environment.
For example, the journal 'Science' reports this
confusion in a research news ( Science, Aug 1990, p481).
After pointing out the media coverage of a few extreme views
of the greenhouse skeptics, it refers to an IPCC report (the
International Panel on Climate Change) as saying 'there is
virtual unanimity among greenhouse experts that a warming is
on the way and that the consequence will be serious'. Still
considerable uncertainties remain in the prediction of the
temperature rise.
15

Suppose G.G. directly increase the surface
temperature(S.T.). < Fig 3.1 > Then, the average temperature
rise on account of the enhancement of these gases should be
well determined, provided that good data on their
concentrations exist. But it is not. In fact, G.G.
indirectly relate to S.T. They increase the temperature by
blocking the long wave emission from the surface into the
space.< Fig3.2 >
Q.Q. OLR T
-->-------->
M . T outgoing
---r------- Longwave.
Radiation
< Fig3.1 > < Fig3.2 >
Another path, however, may exist in G.G.-S.T. relation.
It is conceivable for C02, the major G.G., to cause the
terrestrial vegetation to increase.( Note:3 to 4 giga tons
of atmospheric carbon is currently disappearing into an(?)
unknown sink every year. ) If land
plants are consuming a some portion of added C02 in the
atmosphere, they are secretly increasing and possibly
controling the aridity of the land surface, or at least
delaying the buildup of C02 in the atmosphere. As a result,
it could reduce the size of the temperature increase.<
Fig3.3 > oLR>
< Fig3.3 > G.S.

Therefore it may be said that the indirect relation of
G.G. to the surface temperature is. a cause of uncertainty in
the prediction of S.T. increase. Because there may be a lot
of pathways which cause a conflicting effects on the
resulting temperature.
So, I will make the following definition before I try
to draw a hypothetical picture about the feedback mechanism
relating to the surface temperature.
(2) Definition of a Relation
a i
)
+
1) If there is an arc from a to b, then a directly
influences b.
2) + sign means that if a increases, then b also
increases.
- sign means that if a increases, then b decreases.
(3) The Feedback Structure ( a Hypothesis )
16

In the quantitative model discussion (Ch.2), three
variables(emissivity, heat capacity, and conductivity of the
heat flux) had the direct influences on S.T. Among them, the
importance of 6 and fl were noted. J. Mitchell of the
Meteorological Office in England also makes clear their
importance on the response of the earth system (Climate
Feedbacks). [Mitchell, Ch.5,6]
So, I claim a hypothetical picture of the climate
feedback on the global mean surface temperature (GMST) as
follows. < Fig3.4 >
T : A U (internal heat change),
AT (Temperature change in i zone)
&Li radiant energy input (emissivity, 1-albedo)
eQ: radiant energy output (emissivity)
Note: To avoid complication, AU and AT are absorbed in the
temperature component. The concept of emissivity is
translated into the incoming radiative energy( eL) and
the outgoing radiative energy ( CQ) to enable a
flexible interpretation.
G : greenhouse gases (H^O vapor, C02,CH4,N20,and CFC's)
V : terrestrial vegetation
N0X: nitric oxide(NO), nitrogen dioxide(N02)
S0X: sulfur dioxide(S02), sulfate(S04)
C : cloud
S : snow
I : ice
H : human activity

H
< Fig3.4 >
The arc definition is by section (2)
17

To simplify < Fig3.4 >, I separate it into two
components. < Fig3.5 > is mainly of the carbon cycle, though
it can contain a part of the water cycle and non-carbon
gases. < Fig3.6 > is mainly of the water cycle. + and -
signs are defined by section(2).
H
N
i + dJ -j- (2)
r
< Fig3.5 > < Fig3.6 >
Note that only 6i and 60 go into T in both < Fig3.5 > and
< Fig3.6 >. So, Q. L and e0 are the exclusive terms that
directly influence T. Also note that there is no incoming
arc into H, hence no loop including H exists. A question
mark means the sign is uncertain.
B) DISCUSSION
(1) Basic Assumptions
Since each arc (especially those of V, G and C) of the
above figures constitutes a hot field of study, it is not
the purpose of this paper to determine the sign of arcs. But
assumptions are necessary to analyze the structure, so I
will make the brief justifications for them.
About< Fio3.5 >
For arcs <1> and <2>, the signs should be clear. The
increase of ^ increases T,and the other way around foreo.
<3> is not easily determined at all. My assumption of the
sign is positive. Observing that more than half of
terrestrial biomass is tropical, the increase of temperature
may benefit plants in the colder regions.[Whittaker and
Likens, 1973, in Schlesinger 1991, pl21] C02 concentration
in the atmosphere oscillates seasonally, low in the summer
18

and high in the winter both at Mauna Loa of Hawaii and at
Barrow of Alaska, indicating more uptake of C02 by
vegetation in the summer.[Schlesinger, p52, and NOAA data in
Sievering(the Carbon Cycle)] However, counter arguments
exist. One example is that the total respiration of an
ecosystem increases exponentially with an ambient
temperature rise while GPP( gross primary production)
increases linearly, resulting in NEP(net ecosystem
productivity) reduction by the temperature rise. As to <4>,
the two principal greenhouse gases namely water vapor and
carbon dioxide, both may increase with a higher temperature.
Climate models generally predict a more humid world, in
which the movement of water in the hydrologic cycle is
enhanced through evaporation and precipitation.[Mitchell,
Ch4,5, and Schlesinger, p304] On the land, more C02 will be
released by the enhanced respiration and the decomposition
of organic matter. In the ocean, a warmer surface water
would dissolve less C02, since C02 is about twice as soluble
at 0 degreeC as at 20 degreeC. [Broecker 1974 in
Schlesinger, p274] Thus, the assumption is positive for <4>.
In <5>, a land surface covered with vegetation contains more
water content than a drier land that is a result of rapid
release of water into the atmosphere, and more C02 is taken
up by the photosynthesis of plants. Then, the sign should be
negative. With almost the identical reasons as <5> in the
reverse fashion, higher productivity of the terrestrial
vegetation may result from a higher moisture and higher
concentration of C02 in the atmosphere due to the direct
effect on the photosynthesis. <6> is thus positive. Arc<7>
is difficult to make an assumption. Albedo forcing under the
clear sky in the deserts is higher than that of the land
covered by vegetation. [Hartmann et al., pl292] On the other
hand, evapotranspiration of the latter directs the incoming
energy into the latent heat of vaporization of water instead
of the absorption by the surface. Though this assumption
might be controversial since this process includes the water
cycle, I tend toward the negative sign for <7>, thinking of
the daytime scorch of the desert. <8> should be obvious.
Greenhouse gases trap the long wave radiation emitted from
the surface. It's negative.
As for <9> and <10>, human activity is currently
causing deforestation and the loss of the living biomass by
burning them (primitive agriculture, fuel wood,
'developments', tourism, etc.), while increasing C02, CH4,
N^O, and CFC's by fossil fuel burning, motorization,
nitrogen fertilizer, and industrial processes. Hence, <9> is
negative and <10> is positive. Human activity is also
increasing NOx and SOx in the similar processes to those of
<10>. NOx and SOx, in turn, are causing serious damage to or
death of vegetation. Thus, <11>, <12> are positive, and
<13>, <14> are negative. S0X is very important component in
< Fig3.6 >, too. But the arcs<9>, <10>, <11>, and<12> depend
on human activity, not solely on the physical laws.
[Scientific American, 1989]
19

About< Fiq3.6 >
The cloud formation and their radiative properties
depend on many microphysical parameters. The
parameterization of them is one of the largest sources of
uncertainty in the determination of the climate sensitivity.
[Mitchell,pl24] But, more cloud cover of both high and low
altitudes is observed in the summer than in the winter,
globally.[Hartmann et.al., pl283 to 1284] Therefore, the
arc<15> is positive. More cloud cover in a warmer winter of
the higher latitudes may lead to heavier snow fall. So, <16>
is positive. A higher temperature, however, will directly
reduce the snow cover on the land surface. This relation
also applies to the area of ice cover in the colder regions.
Hence, <17> and <18> are negative. Cloud, snow, and ice
cover all increase the albedo on the surface because of its
whitish color. So, <19>, <20>, and <21> all should be
assigned the negative sign. Cloud cover reduces the emission
of the outgoing longwave radiation from the surface. This is
because a cloud absorbs the OLR emitted from the surface
while it emits less OLR from its colder top layer. Then <22>
is negative.
Human activity is not only increasing sulfur compounds
in the atmosphere as mentioned in < Fig 3.5 > but also have
clearly outcompeted the natural source as the sulfur
source.[Sievering 1993] In addition, J.Lovelock pointed out
in his 'Gaian hypothesis' that a higher temperature enables
more plankton biomass in the ocean which produce dimethyl
sulfide that could lead to a significant increase of sulfate
particles.[Lovelock 1972,1979] So, <12> and <23> are
positive. Sulfates as sulfate haze are a direct cause of the
negative radiative forcing of the incoming short waves. It
is also one of the most efficient cloud condensation nuclei.
Then, <24> is negative and <25> is positive.
(2) Loop Property
In < Fig3.4 >, there can be seen different loops and
quasi-loops. A real loop is a cycle which enables us to go
back to the starting point along the path directed by arcs.
For example, < Fig3.7 > shows real loops while < Fig3.8 >
does not.
A
< Fig3.7 > < Fig3.8 >
20