Chapter 4
Iteration
4.1
introduction
Iteration is the process of performing the same mathematical operation repeatedly, usually for the
purpose to refining the outcome of a particular calculation, until some criterion is met. In general,
iterative computations are required in the solution of non-linear equations and are also used in the
solution of large systems of simultaneous equations.
Where y is the exact solution to
dy
= a (x) y
dx
(4.1)
we compute approximate solutions y 1 , y 2 , . . . , y n ,. . . that converge toward y. Framing the solution
of a non-linear equation in iterative form in effect recasts it as a linear equation. Because iterative
solutions rely on an initial estimate of unknown quality and proceed until a convergence criterion
is met, they can proceed for an unpredictable number of calculations. Different initial estimates
may yield different results and a very poor initial estimate may yield no solution at all.
Many iterative techniques are available. Selection of an appropriate technique depends on the
nature of the equation(s) to be solved. We don’t have time in this class to discuss iterative
techniques in any detail. The goal of this lab assignment is to gain some experience in using
iteration in a computational model and to introduce the idea of a zero-dimensional model.
4.2
4.2.1
the basics
control or flow
A control flow statement in a computer program results in a choice (the control) about which of
two or more paths (the flow) should be followed. Implementation can be a simple repetition of a
63
64
CHAPTER 4. ITERATION
set of commands for a predetermined number of times, or a more sophisticated evaluation of the
progress of a calculation using relational operators and logic. The latter might be used to stop a
calculation when a certain threshold is met. The flow of control in numerical models can become
quite involved. As always, the time you spend writing implementation and pseudocode algorithms
will be saved when it comes to writing your program. The more complicated the flow becomes,
the more likely the programmer is to make a mistake.
The Matlab flow control commands include: if, else, elseif, end, for, while, switch.
A paired for and end causes a statement or group of statements to be repeated a fixed number
of times. A paired while and end repeats a group of statements an indefinite number of times,
according to a logical condition. A paired if and end evaluates a logical expression and executes
a group of statements when the expression is true. Optional elseif and else commands allow the
execution of alternate groups of statements:
while l o g i c a l e v a l u a t i o n
statements
end
A paired if and end evaluates a logical expression and executes a group of statements when the
expression is true. Optional elseif and else commands allow the execution of alternate groups of
statements.
if logical evaluation
statements
elseif relational evaluation
statements
else
statements
end
Flow control statements are accompanied by relational and logical operators that are used to
determine when some criterion has been met.
The six relational operators are:
equal
not equal
less than
greater than
less than or equal
greater than or equal
==
˜=
<
>
<=
>=
The common logical operators are:
and
or
not
&
|
˜
4.2. THE BASICS
4.2.2
65
an example
Suppose we have an exact equation for the dependent variable x:
x = a + b
(4.2)
where the coefficient a depends on x:
1
a = cx 2
(4.3)
and the coefficients b and c are constant.
The solution to equation (4.3) must be found iteratively. The simplest pathway forward is to use
repeated substitution:
1. Make an initial estimation a1
2. Use a to compute the corresponding value of x.
3. Use the new x to update the estimate of a.
4. Compare the old value a1 with the new value of a.
While a1 and a differ by more than a specified convergence criterion, replace a1 with the
updated value a and return to step 2.
Other, more sophisticated schemes are possible. The speed with which the iteration converges on
a solution to the equation depends in part on the quality of the initial estimate. A good strategy
is to use a likely value of the dependent variable to make your first estimate of the coefficient upon
which you are iterating.
The steps outlined above are easy to implement in Matlab using a while statement and a relational
evaluation.
% program t o s o l v e x = a+b where a = cx ˆ ( 1 / 2 )
b =2;
c =2;
% constant
% constant
a =2;
a1 =0;
a t o l =0.0001;
% i n i t i a l guess
% updated v a l u e
% tolerance for convergence t e s t
t a l l y =1;
% c o u n t e r t o k e e p t r a c k o f t h e number o f e s t i m a t e s
w h i l e ( a−a1 ) / a > a t o l
a1=a ;
x=a1 + b ;
a=c ∗ x ˆ ( 1 / 2 ) ;
t a l l y = t a l l y +1;
end
66
CHAPTER 4. ITERATION
It is important to remember that the solution you find depends on both the initial estimate and
the convergence criterion you set. Here are a few solutions to equation with different tolerances
and initial estimates of a. The above program, with an initial estimate of a = 2 and a tolerance of
0.0001 finds a solution x = 7.46 and a = 5.46 in 11 steps (including the initial estimate). With a
tolerance of 0.01, a solution of x = 7.39 and a = 5.44 is found in 6 steps.
4.3
a zero-dimensional planetary energy balance model
You may encounter the word simple associated with a computational model of an Earth system.
In this context, simple means low dimensional, low order, or low complexity. Low order means a
model with limited resolution and low complexity means that complicated mathematical expressions or detailed processes have been approximated with parameterized expressions that embrace
key elements in a manageable way. While these models may be “simple,” they are often quite
elegant and can lead to important insights into physical processes.
Zero-dimensional models are ones in which the spatial dimensions of a physical system have been
collapsed to one single point. These models are generated by approximating spatially-varying
properties with mean properties of a system, thereby avoiding the need to integrate derivatives of
dependent variables. The zero-dimensional approximation may at first blush seem overly simple
but in fact, these models are quite useful. They are easy to implement and allow us to study
broad-scale characteristics of a system. The American Institute of Physics has published a very
interesting history of simple climate modeling and its importance to climatology at http://www.
aip.org/history/climate/simple.htm
In this section, a zero-dimensional model of planetary energy balance is derived. The model
describes the balance among the radiative energy (solar) incident at the surface of a planet’s
atmosphere, its reflection, absorption, and re-radiation. The model can be used to study how
phenomena such as the greenhouse effect, ice-albedo feedback, and perturbations such as volcanic
eruptions effect global mean temperature. Zero-dimensional and one-dimensional (zonal) energy
balance models were very important to early computational studies of Earth’s climate and are still
widely employed today.
4.3.1
conservation of energy
The first law of thermodynamics is a statement of the conservation of energy. The change in the
energy of a system is equal to the amount of heat added to the system minus the work done by
the system on its surroundings.
For our problem, we need to consider heat flow in and out of the planet due to radiative transfers.
Our conservation of energy statement is:
dE
=
dt
Z
Q ds
s
(4.4)
4.3. A ZERO-DIMENSIONAL PLANETARY ENERGY BALANCE MODEL
67
where E represents the energy stored in the system, t represents time, Q represents the heat flow,
and s is the surface area of the planet. Making the zero-dimension simplification, we replace the
integration with the sum of heat flows into and out of the planet:
dE
= Qin − Qout
dt
(4.5)
where we have adopted a sign convention with positive flows into the planet and negative flows out
of the planet.
4.3.2
heat flow to the planet
The intensity of solar radiation arriving at the top of a planet’s atmosphere is proportional to the
inverse of the square of the distance between the planet and its sun. We will use the mean distance
in this derivation, but note that the model we construct could accommodate annual or longer time
scale variations in the sun-planet distance. The solar constant Se at the top of Earth’s atmosphere
is 1370 W m−2 . The intensity of the solar radiation at any other planet in our solar system may
be computed:
Sp = Se
re2
rp2
(4.6)
where the subscript e denotes Earth and the subscript p indicates another planet. One astronomical
unit (AU) is defined to be the mean distance between Earth and our sun.
Some portion of the incoming radiation is reflected directly back to space by atmospheric aerosols
and by the land surface. The albedo of a material is the fraction of incoming radiation that it
reflects. The net amount of solar radiation is then Se (1 − α) where α represents the albedo.
Albedos for some typical Earth surfaces are listed in Table 1.
A spherical planet appears as a disk over the long distance between Earth and the sun. The surface
area over which incoming heat is collected is the area of that disk. The total amount of heat flow
into the planet is thus:
Qin = π Re2 Se (1 − α)
(4.7)
where Re represents the mean radius of Earth.
4.3.3
heat flow out from the planet
Any object that is warmer than absolute zero (0 Kelvin) radiates energy in proportion to the fourth
power of its temperature. This relationship was discovered experimentally by Jozef Stefan in 1879
and derived theoretically by Ludwig Boltzmann in 1884. The Stefan-Boltzmann Law tells us that
the outgoing heat flow is:
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CHAPTER 4. ITERATION
surface type
clouds
ocean
soil
deciduous forest
desert
Antarctica
new snow
albedo
0.01 to 0.70
0.05
0.05 to 0.40
0.13
0.25
0.80
0.90
Table 4.1: Albedos of some typical earth surfaces
Qout = σ T 4
4 π Re2
(4.8)
where is the emissivity of the radiating object, σ is the Stefan-Boltzmann constant, T represents
the temperature of the planet’s surface, Re is Earth’s mean radius and the term in parentheses is
the surface area. The value of σ in mks units is 5.67 x 10−8 W m−2 K−4 . The emissivity of an
ideal blackbody is 1. Objects with emissivities of less than 1 are called greybody objects.
4.3.4
energy balance for a blackbody planet
The steady-state temperature of an object is a temperature such that the heat flow into the object
isequal to the heat flow out of the object. That is,
dE
= 0.
dt
(4.9)
In this case, the steady-state version of equation (4.4):
Qin = Q
(4.10)
out
πRe2 Se2 (1 − α) = σ T 4 4 π Re2
(4.11)
is easily solved for the mean surface temperature T
T =
Se (1 − α)
4σ
1
4
(4.12)
4.3. A ZERO-DIMENSIONAL PLANETARY ENERGY BALANCE MODEL
constant
σ
Se
αEarth
To
value
5.67 × 10−8
1
1370
0.31
273.15
69
units
W m−2 K−4
W m−2
degrees
Table 4.2: constants for planetary energy balance
4.3.5
radiative transfer in the atmosphere
Earth’s mean surface temperature according to equation (4.10) with the coefficients defined above
is about 254 K (-19 ◦ C). Earth’s measured mean surface temperature is about 288 K. The source
of the difference is Earth’s atmosphere.
The wavelength at which a material radiates energy depends on its temperature. Earth receives
shortwave radiation from our sun because the sun’s surface is very hot. Earth’s surface is relatively
cool so when it radiates energy back to space, it does so at longer wavelengths. Wien’s displacement
law shows the inverse relationship between the peak emission wavelength λmax and the temperature
of a blackbody object:
λmax =
0.002898
T
where the empirical constant has mks units of m K. (Review Plank’s law of blackbody radiation
for a more complete understanding of this relationship.)
How a material interacts with that radiation depends on its molecular structure. For the most
part, the gasses that make up Earth’s atmosphere do not absorb energy at short wavelengths so
the radiation passes through the atmosphere and is absorbed by surface materials. An exception is
ozone (O3 ), which absorbs ultraviolet radiation. Several of the gasses in Earth’s atmosphere (H2 O,
CO2 , N2 O, CH4 , O3 , CCl3 F, and CCl2 F2 ) are strong absorbers at the longer, outgoing energy
wavelengths. These greenhouse gasses absorb the outgoing longwave radiation and reradiate it,
warming the atmosphere. In order to correctly simulate planetary energy balance, our model must
account for this radiative transfer in the atmosphere.
A complete atmospheric radiative transfer model would integrate from the initial energy radiation
from the planet’s surface to the near-surface air, up to the top of the atmosphere, wavelength by
wavelength, molecule by molecule. Such models must include atmospheric attributes including
the different physical properties of the greenhouse gasses, changing atmospheric temperature and
pressure (and thus gas density) with altitude, the roles of various atmospheric particulates, and the
various properties of different cloud types. Such models are difficult to build and time-consuming
to run. They also play an important role in understanding global warming and climate change.
We will adopt a simple, linearized formulation to embody radiative transfer in the atmosphere.
This empirical expression will take the place of the blackbody term in equation (4.8):
70
CHAPTER 4. ITERATION
Qout =
4 π Re2 (A + BTC )
(4.13)
in which A is a constant, the coefficient B expresses the temperature-dependence of the radiative
transfer, and TC is now the temperature in degrees Celsius.
The derivation of (A + BTC ) makes use of a handy trick from Calculus, binomial expansion. The
quantity (1+x)n is approximately equal to (1+nx) if x is much smaller than 1. So for the radiative
part of equation (4.8) we can use the binomial expansion to re-write T ,
4
TK
= (To + TC )4
To4 = To4 1 +
'
To4
TC
To
4 TC
1 +
To
4
(4.14)
in which To is the value used to convert from Kelvin to degrees Celsius, 273.15. Equation (4.14) is
then used to rewrite the radiative transfer:
σ To4 1 +
4 TC
To
C
It is easy to see that the coefficients are A = σ To4 and B = 4 σ To3 for a blackbody planet.
The linearization is a reasonable simplification for the troposphere, where temperature and pressure
decrease linearly with altitude. Ozone chemistry in the stratosphere causes it to warm, relative to
the top of the troposphere. Fortunately, 80% of the mass of Earth’s atmosphere (and most of the
greenhouse gas content) is in the troposphere so we can neglect everything above the tropopause.
The formulation in equation (4.13) was first proposed by Mikhail Budyko (1920-2001), a climatologist at the Leningrad Geophysical Observatory. His work, from the 1940’s onward, moved
climatology from the qualitative to the quantitative. Budyko calculated values for the coefficients
A and B using on observations of outgoing longwave radiation and surface temperature at a few
hundred locations (Budyko, 1969). Since that time, additional observations have improved the
present-day estimates of A and B. It is important to recognize that the empirical coefficients
implicitly contain a dependence on the atmospheric greenhouse gas concentrations. Values for the
coefficients for various scenarios can also be derived from the output of radiative transfer models.
William Sellers, at the University of Arizona, built on Budyko’s recognition of the importance of
greenhouse gasses and albedo to global climate (Sellers, 1969). The simple, elegant class of zero(and one-) dimensional models are often called Budyko-Sellers models, recognizing the important
contributions of both scientists. Interestingly, Budyko, who worked at the Leningrad Geophysical
Observatory, was concerned about the possibility that albedo-temperature feedbacks (section 4.3.6)
4.3. A ZERO-DIMENSIONAL PLANETARY ENERGY BALANCE MODEL
71
Figure 4.1: Scatter plot of OLR versus surface temperature from 30◦ N to 90◦ N from the 10 year
data set. By Graves et. al.
72
CHAPTER 4. ITERATION
could lead the way to another ice age, and saw the warming of the planet, and consequent reduction
of Arctic sea ice, as a positive development for the Soviet Union. Sellers’ concern was the opposite.
His 1969 paper warned that “man’s increasing industrial activities may eventually lead to a global
climate much warmer than today.”
The energy balance equation can be re-written to include radiative heating in the atmosphere using
equations (4.7), (4.10), and (4.13):
π Re2 Se (1 − α) =
4 π Re2 (A + BTC )
(4.15)
and rearranged to solve for T :
T = To +
1
4 Se (1
− α) − A
(4.16)
B
in mks units of Kelvin.
4.3.6
4.3.6.1
adding a temperature-albedo feedback
temperature-dependent albedo
One of the early, important, uses of zero-dimensional energy balance models was to explore the
connection between global temperature and global albedo. The essence of this idea is that as the
planet cools, the fraction of its land surface covered by ice increases, and visa versa. There are a
number of ways to implement this notion, with varying levels of sophistication. We will consider a
simple rule in which albedo varies with temperature between two threshold values. Below the cold
threshold temperature Ti , albedo is fixed at a value αi that represents a snow and ice-covered land
surface and an ocean with widespread sea-ice cover. Above the warm threshold Th , the albedo is
fixed at a value αh slightly lower than the present-day albedo. Mathematically, we can write this
rule:
α =



αi
αi + (αh − αi ) TTh−−TTii


αh
if T < Ti
if Ti < T < Th
if T > Th
(4.17)
The inclusion of the albedo-temperature feedback creates a non-linearity in the energy balance
equation. We now have a coefficient that depends on the dependent variable. Equation (4.14)
must be solved iteratively. The feedback also gives rise to the possibility of multiple steady-state
solutions. The recognition of multiple steady states in zero- and one-dimensional energy balance
climate models led to the suggestion that Earth may have experienced extreme cold and extreme
warm conditions in the past. The geologic record suggests that “snowball Earth” conditions may
have existed in the Precambrian (700 Ma) and the Huronian (2000 Ma), although this notion is
controversial.
4.3. A ZERO-DIMENSIONAL PLANETARY ENERGY BALANCE MODEL
4.3.6.2
73
the iteration
In order to solve equation (4.16) for the global mean temperature T , we must iterate on the albedo
α until a match between T and α is found. A Matlab script to perform the iteration is included
below. An initial α is used to generate a first approximation of the steady-state T . That value
of T is then used to update the value of α. If the initial and second values of α agree within
some tolerance, then the computation is complete, the steady-state T has been found. If not, the
iteration continues. A while loop is used to control the iteration.
Note that ellipsis notation . . . is used to continue a math statement from one line to the next in
the Matlab script.
% z e r o −d i m e n s i o n a l e n e r g y b a l a n c e model
% p l a n e t w i t h an E a r t h − l i k e a t m o s p h e r e and t e m p e r a t u r e −d e p e n d e n t a l b e d o
clear
Se =1370;
% s o l a r c o n s t a n t W/mˆ 2 , E a r t h = 1370
A= 2 0 2 . 1 ;
B= 1 . 9 ;
% r a d i a t i v e heat l o s s c o e f f i c i e n t
% r a d i a t i v e heat l o s s c o e f f i c i e n t
%∗ c o n s t a n t s f o r t e m p e r a t u r e −d e p e n d e n t a l b e d o p a r a m e t e r i z a t i o n
a l b e d o i =0.6;
% a l b e d o o f i c e & snow s u r f a c e
albedo h =0.3;
% albedo of land surface
T i =263;
T h =283;
Tnot = 2 7 3 . 1 5 ;
% mean s u r f a c e t e m p e r a t u r e f o r s n o w b a l l e a r t h
% mean s u r f a c e t e m p e r a t u r e f o r s m a l l −i c e e a r t h
% c o n v e r t from K t o d e g r e e s C
albedo =0.4;
a l b e d o e s t =0;
a l b e d o t o l =0.05;
% i n i t i a l value for albedo
% a r r a y to t r a c k e s t i m a t e s of albedo
% tolerance for p r e c i s i o n in albedo estimate
t a l l y =1;
% t e m p e r a t u r e −d e p e n d e n t a l b e d o
w h i l e abs ( ( a l b e d o −a l b e d o e s t ( t a l l y ) ) / a l b e d o )> a l b e d o t o l
a l b e d o e s t ( t a l l y +1)= a l b e d o ;
T=Tnot + ( 0 . 2 5 ∗ Se ∗(1− a l b e d o )−A) /B ;
T e s t ( t a l l y )=T ;
a l b e d o =(T<T i ) ∗ a l b e d o i . . .
+ (T>T i \& T<T h ) ∗ ( a l b e d o i + ( a l b e d o h −a l b e d o i ) . . .
∗ ( ( T−T i ) / ( T h−T i ) ) ) + (T>T h ) ∗ a l b e d o h ;
t a l l y = t a l l y +1;
end
74
CHAPTER 4. ITERATION
a l b e d o e s t =[ a l b e d o e s t ( 2 : t a l l y ) a l b e d o ] ;
T e s t =[ T e s t T ] ;
figure (2)
clf
plot ( a l b e d o e s t , T est , ’ b . : ’ )
hold on
plot ( albedo , T, ’ ro ’ )
xlabel ( ’ albedo ’ )
y l a b e l ( ’ t e m p e r a t u r e (K) ’ )
legend ( ’ g u e s s e s ’ , ’ s t e a d y −s t a t e ’ )
t i t l e ( ’ i t e r a t i o n t o s t e a d y −s t a t e t e m p e r a t u r e w i t h t e m p e r a t u r e −d e p e n d e n t \
albedo ’ )
4.3.6.3
a few words about equilibrium solutions discovered by iteration
Our albedo-temperature feedback model has found a solution when the value of the albedo ceases to
change from update to update. In nonlinear models (models with feedbacks between the dependent
variable and coefficients) in is possible that more than one such equilibrium exists. It is also possible
to choose an initial estimate that will never lead to a stable solution. Simple models of the sort
we have developed here are especially likely to find multiple solutions. It is important to evaluate
the significance and the stability of these solutions in the post-model analysis phase of your work.
Formal mathematical stability analyses are possible but we can also examine the stability of our
solutions by following a more immediately accessible definition: a solution is stable if the system
returns to that solution after a small perturbation. You will have an opportunity to examine this
issue in exercise 3, below.
4.4
exercises
1. The Matlab code presented in section 4.3.4 is easily expanded to compute steady-state
blackbody temperatures for other planets in our solar system. The intensity of incoming
solar radiation arriving at any planet is computed by scaling Se according to the ratio of the
squares of the mean distances between the sun and Earth re and the sun and the planet rp ,
according to equation (4.8).
Variation in albedo from planet to planet must also be considered. A Matlab script that
computes the steady-state blackbody temperature for Earth and four other planets is written
below. The script may be downloaded at the class website. The script stores the parameters
needed to solve equation (4.12) for each planet in arrays called rp and albedo. Formatting
information and labels used to plot the results are stored in arrays called rps and rls .
(a) Please expand this script to include one additional planet. In your answer to this
exercise, report the name of the planet, the sun-planet distance in AU, and the value
you used for the planet’s mean albedo. Please also include a plot of the steady-state
4.4.
EXERCISES
75
blackbody temperatures for Mercury, Venus, Earth, Mars, Saturn, Neptune, and the
planet of your choosing.
(b) The mean annual temperature at Earth’s surface is about 287 K and the mean annual
temperature at the surface of Mars is about 210 K (from http://en.wikipedia.org).
Compare those temperatures with the temperatures computed by the steady-state blackbody model. What can you infer about the Martian atmosphere from this comparison?
% z e r o −d i m e n s i o n a l s t e a d y s t a t e e n e r g y b a l a n c e model
% f o r a blackbody planet
s i g m a =5.67 e −8;
e p s i l o n =1;
Se =1370;
% S t e f a n −Boltzmann c o n s t a n t , W/mˆ2/Kˆ4
% e m i s s i v i t y f o r blackbody object
% s o l a r c o n s t a n t W/mˆ 2 , E a r t h = 1370
r e =1;
% d i s t a n c e from sun t o E a r t h i n AU
% l i s t of planets
rp =[0.387 0.723 r e 1.524 9.529 3 0 . 0 8 7 ] ;
% mean a l b e d o o f p l a n e t s u r f a c e
albedo =[0.12 0.65 0.31 0.15 0.47 0 . 4 1 ] ;
%p l o t t i n g i n f o
r p s =[ ’ r o ’ ; ’ go ’ ; ’ co ’ ; ’ bo ’ ; ’mo ’ ; ’ r+ ’ ] ;
r l s =[ ’ Mercury ’ ; ’ Venus ’ ; ’ E a r t h ’ ; ’ Mars ’ ; ’ S a t u r n ’ ; ’ Neptune ’ ] ;
Tnot = 2 7 3 . 1 5 ;
% c o n v e r t between K and d e g r e e s C
% z e r o −D s t e a d y −s t a t e b l a c k b o d y t e m p e r a t u r e
T= ( 0 . 2 5 ∗ ( Se ∗ r e ˆ 2 . / r p . ˆ 2 ) . ∗ ( 1 − a l b e d o ) / e p s i l o n / s i g m a ) . ˆ ( 1 / 4 )
figure (1)
clf
a x i s ( [ 0 max( r p ) min (T)−5 max(T) + 5 ] )
hold on
f o r n =1: l e n g t h (T)
p l o t ( r p ( n ) , T( n ) , r p s ( n , : ) )
end
y l a b e l ( ’ t e m p e r a t u r e (K) ’ )
x l a b e l ( ’ d i s t a n c e from Sun (AU) ’ )
t i t l e ( ’ blackbody temperatures of s o l a r system o b j e c t s ’ )
legend ( r l s )
2. The energy balance equation for a planet with an atmosphere, (4.15) can be used to investigate the global temperature effects of changes in any of its coefficients. Easy implementation
for sensitivity studies is one of the many attractive features of zero-dimensional models. In
order to implement the improved model, we must add some parameters to our basic Matlab
code (in section 4.3.4) and modify the equation that it solves.
% z e r o −d i m e n s i o n a l e n e r g y b a l a n c e model
% p l a n e t w i t h an E a r t h − l i k e a t m o s p h e r e and t e m p e r a t u r e −d e p e n d e n t a l b e d o
Se =1370;
% s o l a r c o n s t a n t W/mˆ 2 , E a r t h = 1370
76
CHAPTER 4. ITERATION
albedo =0.31;
A=202;
B= 1 . 4 5 ;
Tnot = 2 7 3 . 1 5 ;
%
%
%
%
constant albedo
r a d i a t i v e heat
r a d i a t i v e heat
c o n v e r t from K
, Earth = 0.31
loss coefficient
loss coefficient
to degrees C
% z e r o −D s t e a d y −s t a t e t e m p e r a t u r e
T=Tnot + ( 0 . 2 5 ∗ Se ∗(1− a l b e d o )−A) /B ;
(a) Graves et al., (1993) suggest radiative heat loss coefficients of A = 202.1 W m−2 and B
= 1.9 W m−2 K−1 . North et al. (1981) estimated pre-industrial values for the coefficients
to be A = 203.34 W m−2 and B = 2.09 W m−2 K−1 . Use the zero-order energy balance
model to compute the mean global temperature change from pre-industrial to the 1970’s.
Use an albedo of 0.32. How does your model-derived temperature change compare to the
observed global mean warming over that time period? (Try looking up “global warming”
at http: // en. wikipedia. org .)
(b) Before we get too excited about the result of exercise 2a (or of any model result), we
need to examine how sensitive our model is to the selection of values for its various
coefficients. The numbers for A and B depend on combinations of observation and
complicated radiative transfer models that are beyond our grasp. We can, however, test
out values suggested by other climatologists. Repeat the calculation in exercise 2a using
Budyko’s (1969) original values of A = 202 W m−2 and B = 1.45 W m−2 K−1 . What
do Budyko’s parameters suggest about global warming?
(c) So we better not make any dramatic predictions about next year’s temperatures based
on the zero-dimensional model. That’s OK, that’s not what zero-dimension models are
built to do, they are built to study sensitivity of a system to variations in its parameters.
One final question, then. Using the Graves et al. (1993) values for A and B, compute
Earth’s mean temperature sensitivity to changes in the energy balance for a global
albedo of 0.32. Your result should have units of K per W m−2 . How does the sensitivity
change when you use the North et al. (1981) values for A and B?
3. In section 4.3.6, we changed the model so that it could be used to investigate temperaturealbedo feedbacks. A script at the course website called zeroD ebm aa ss.m implements that
model. Use the Graves et al. (1993) values for A and B.
(a) Run the model with tolerance of 0.05 for the iteration on the albedo and an initial albedo
estimate of 0.4 . What are the final, steady-state albedo and temperature?
(b) What happens when you change the tolerance to 0.01?
(c) An important conceptual outcome of the positive feedbacks in the temperature-dependent albedo model (cooling reinforces cooling and warming reinforces warming) is that
multiple steady states are possible. Run zeroD ebm aa ss.m for a range of initial albedos
and a tolerance of 0.01. How many steady-state albedo and temperature combinations
do you find and what are they?
You can answer this question by changing the initial estimate of the albedo manually
and re-running the model or you can modify the script to test many initial values
4.5.
REFERENCES
77
sequentially. To do the latter, you would need to define an array of the initial estimates:
alb init =[0.3:0.01:0.4];
and then construct a for loop around the while loop:
% t e m p e r a t u r e −d e p e n d e n t a l b e d o
f o r n =1: l e n g t h ( a l b i n i t )
a l b e d o= a l b i n i t ( n ) ;
a l b e d o e s t =0;
T e s t =0;
t a l l y =1;
w h i l e abs ( ( a l b e d o −a l b e d o e s t ( t a l l y ) ) / a l b e d o )> a l b e d o t o l
%i m p o r t a n t s t u f f h e r e
end
a l b e d o e s t =[ a l b e d o e s t ( 2 : t a l l y ) a l b e d o ] ;
T e s t =[ T e s t T ] ;
figure (2)
plot ( a l b e d o e s t , T est , ’ b . : ’ )
hold on
plot ( albedo , T, ’ ro ’ )
end
g r i d on
The figure plotting commands have been simplified in order to preserve the results of
each calculation.
(d) Are all the steady-state solutions you found in exercise 3c stable? Please explain your
reasoning.
4.5
References
Budyko, M.I., 1969, The effect of solar radiation variations on the climate of the Earth, Tellus 21,
611-619.
Graves, C.E., W.H. Lee and G.R. North, 1993, New parameterizations and sensitivities for simple
climate models”, J. Geophy. Res., 98 (D3), 5025-5036.
Sellers, W. D., 1969, A global climatic model based on the energy balance of the Earth-atmosphere
system, J. Applied Meteorology, 8, 392-400.
North,G.R., Cahalan, R.F. and Coakley, J.A., 1981. Energy balance climate models. Review of
Geophys. and Space Phys., 19, 91-121.
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CHAPTER 4. ITERATION