A Zero Dimensional Energy Balance Model for Earth Systems

1. A Zero-Dimensional
Energy Balance Model
For Earth Systems
Geneva Porter
San Diego State University
geneva.porter@gmail.com
14 April 2016
Abstract
Global warming and climate change has been an issue of growing concern in the 21st
century. With empirical evidence of melting ice caps, rising sea levels, and increasing average
temperatures around the globe, meteorologists are worried that our environment may be vulnerable
to a rejection of viable human habitation conditions. Scientists have developed a tool called an
Energy Balance Model (EBM) to predict how Earth’s diverse systems, along with human
contributions to the environment, may affect the rise in global temperature. These models are useful
and cost-efficient resources that can help policy-makers decide on implementing mitigating actions
to combat global warming. Initiatives like increasing the Earth’s albedo and radiation emissivity can
have a significant impact on the rising global temperature.

2. Introduction
We will construct a zero-dimensional Energy Balance Model for planet Earth. An Energy
Balance Model is an approximation of Earth’s global temperature average based on the relationship
between incoming energy, such as heat from the sun, and outgoing energy, such as radiation emitted
from the planet’s surface. To make a zero-dimensional EBM, we do not take into account variations
in the complexity of Earth systems, such as land or water surface area, latitude, longitude, or altitude,
among other considerations. Since earth is treated as a single point using a single average
temperature, we use no dimensions of variation, thus we call this model “zero-dimensional.” We can
use mathematical models to give values to the incoming and outgoing energies in order to predict
how the earth’s temperature changes given a few simple variations, such as cloud cover or ice cap
presence. We will use theoretical models formulated by the meteorological academic community1, 2, 3,
4, 5
as a basis for building our own model. Because these models are scrutinized and peer reviewed,
they are considered highly reliable as sources for EBM research. Through studying these sources, we
can give a reasonable perspective on Earth’s temperature in relation to the input and output of
energy on earth’s surface.
This research model is important because we must observe Earth’s global trends to ensure
future habitation is possible. While our results will lack precision because of the simplicity of the
zero-dimensional model, we can still gain valuable information on Earth’s energy systems, and
hopefully, some insight into the global warming phenomenon. There is already empirical data that
shows Earth’s temperature average has been on the rise for the last 100 years, and suggests that this
trend may continue. Below, in Fig.(1), is the linear regression model based on global land and ocean
surface temperature (SAT) data, gathered from James Hansen’s NASA research group on climate
studies6
. Hansen and his team collected data on temperature anomalies, or temperatures that were
above or below the standard global mean. The results are interpreted as follows.

3. Fig.(1) A linear regression model illustrating SAT trends over the last 165 years.
These data tell us an important story. If we average the change in temperature anomalies
over the globe and chart the change linearly, we can see that temperature averages have significantly
increased, most notably over the last 40 years. By highlighting these rates of change over increasingly
longer time intervals, the path to global warming seems set. However, to understand why this is, we
must better understand the energy balance in Earth’s systems.

4. Data and Methods
To better illustrate the zero-dimensional EBM for Earth, we will establish how energy
interacts with the planet. The relationship between incoming and outgoing energy, or radiation,
determines global temperature. The Earth absorbs shortwave radiation from the sun, denoted by S,
the “solar constant” representing radiation per square meter. S an approximate value of 1370
W/m2
.1
The energy from the sun hits the earth through a disk with an area of π×R2
, with R being
the radius of planet earth (approx. 64,000 meters). However, as it rotates through space, energy is
absorbed by Earth around its entire surface area, represented by the expression 4×π×R2
. See Fig.(2)
for an illustration of this. If we say that the earth receives S amount of energy over an area of π×R2
,
then the strength of that energy is mitigated by the area over which it is absorbed, 4×π×R2
. We can
represent this relationship by the expression: S×(π×R2
)/(4×π×R2
), or simplified, S/4. Inserting our
value of S = 1370, S/4 = 342.5.
Fig.(2) Taken from Principles of Mathematical Modeling, S. Shen (2016)6

5. To add precision to our model, we will include the element of reflectivity, denoted by α, a
measure of how much radiation Earth reflects back into space. We call this reflectivity “albedo.”
This radiation never reaches the surface, so we can scale down our value of S/4 by reducing it by a
certain percentage. Our model supposes that the albedo of Earth is .32,6
meaning 32% of incoming
radiation does not reach the surface. This value is an approximation, and fluctuates significantly
based on cloud cover and seasonal changes. Adjusting our expression, we can now measure Earth’s
incoming shortwave radiation by: (1- α)×S/4. Later, we will investigate how changing the value
associated with albedo will affect the global temperature.
Regarding outgoing radiation, or longwave radiation, we will use the “black body” model to
approximate these values. The black body model supposes that Earth absorbs all radiation that hits
the surface. However, according to the laws of thermodynamics, every body must emit radiation in
relation to its temperature to the 4th power. The current approximation of Earth’s average global
temperature is 287 degrees Kelvin,6
denoted by Ts. We assign a value to the emissivity constant ε,
representing the emissivity of longwave radiation from Earth. Since 70% of Earth is covered in
water, we can simplify our model by calculating for a completely water-covered earth. For this
model, ε is approximately .6,6
meaning 60% of earth’s outgoing radiation escapes the atmosphere.
The total amount of longwave radiation escaping Earth can be represented by something called the
Stefan-Boltzmann expression2, 4, 6
: ε×σ×Ts
4
, with σ as a value called the Stefan-Boltzmann constant,
defined as 5.670373*10-8
. The Stefan-Boltzmann constant was formulated by much research and
study on the emission of radiation from Earth, and has the units W×m-2
×K-4
. Later, we will
investigate how Earth’s emissivity can affect the global temperature.
In addition, the laws of thermodynamics state that the change in internal energy must be
balanced with the rate that a body is heated. C will represent Earth’s heat capacity, the the difference
between the absorbed shortwave radiation vs. the emitted longwave radiation. If Earth’s systems are

6. in an ideal balance, this difference is zero. Looking at the rate of change of Earth’s heat capacity
gives us insight into the temperature trends of the planet. We can represent this by ΔC = (1-α)×S/4
- ε×σ×Ts
4
.
If we receive more incoming radiation that outgoing radiation, then ΔC will be positive,
indicating that the temperature will rise globally. In the opposite case, where the amount of outgoing
radiation is greater than the incoming radiation, Earth will experience a trend of declining global
temperature. If our incoming radiation and outgoing radiation are equal, then the change in
temperature is zero, and we will have equilibrium. Therefore, equilibrium is achieved when ΔC = 0,
or (1-α)×S/4 = ε×σ×Ts
4
.
This simple model does not account for the change in albedo that surely occurs with a global
change in temperature. A planet in an ice age would have a much higher albedo than present-day
earth, due to its high snow cover. When considering albedo as a changing variable, one model1
gives
albedo the value of α = .5-.2×tanh((Ts - 265)/10). With this new element in our model creation, our
new balance equation is (1-.5-.2×tanh((Ts - 265)/10))×S/4 = ε×σ×Ts
4
. Before exploring a variable
albedo, let us fist examine this EBM without one.

7. -20°C
0°C
20°C
40°C
60°C
80°C
30% 40% 50% 60% 70% 80% 90%
Earth'sTemperature
Emissivity Percentage
Earth's Temperature
vs. Emissivity
Results
Our goal was to evaluate the temperature of Earth based on the equilibrium equation in
order to model an ideal energy balance. Solving for temperature, we get Ts = (1-α)×S/(4× ε×σ)]¼
.
Here, Ts is given in degrees Kelvin. To convert to Celsius, which is more relatable, all we must do is
subtract 273 from our equation, giving us:
Ts = (1-α)×S/(4× ε×σ)]¼
- 273.
Given our assumptions that α = .32, S = 1368, ε = .6, and σ = 5.670373×10-8
, we can model
earth’s current temperature:
Ts = {(1-.32)× 1368/(4× .6×5.670373×10-8
)]}¼
- 273 ≈ 14°C
14°C is reasonably close to earth’s average temperature. This projection assumes an ideal
balance. But that may not be the case. What if Earth’s albedo or emissivity changes? How will this
affect our temperature? First, let’s keep our balanced EBM and look at the changes in temperature
that result by varying albedo and emissivity. The table below, Tab.(1), and the corresponding graph,
Fig.(3), show us how varying our emissivity will change the global temperature average.
Tab.(1) Ts vs. ε (given .32 α)
ε % from .6 Ts [°C] % from 14°C S
.3 -50.00% 68.75 +378.16% -0.55%
.4 -33.33% 45.04 +213.22% -0.65%
.5 -16.67% 27.78 +93.21% -0.75%
.6 0.00% 14.38 0.00% 0.00%
.7 +16.67% 3.51 -75.56% -0.92%
.8 +33.33% -5.56 -138.70% -1.00%
.9 +50.00% -13.32 -192.66% -1.08%
Fig.(3) Ts as a function of ε

8. -20°C
-10°C
0°C
10°C
20°C
30°C
40°C
10% 20% 30% 40% 50% 60%
Earth'sTemperature
Albedo Percentage
Earth's Temperature
vs. Albedo
We can see that Earth’s temperature changes significantly between emissivity fluctuations of
10%. Furthermore, Tab.(2) and Fig.(4) below show the effect of changing albedo. Like emissivity,
an increase in albedo points towards a decline in Earth’s temperature.
Tab.(2) Ts vs. α (given .6 ε)
ε % from .32 Ts [°C] % from 14°C S
.1 -68.75% 35.24 +145.08% -1.05%
.2 -37.50% 26.30 +82.88% -1.01%
.3 -6.25% 16.47 +14.54% -0.96%
.32 0.00% 14.38 0.00% 0.00%
.4 +25.00% 5.53 -61.57% -0.90%
.5 +56.25% -6.88 -147.88% -0.85%
.6 +87.50% -21.32 -248.30% -0.78%
Fig.(4) Ts as a function of α
By varying the reflectivity and emissivity of Earth, our “black body” mass, we can model the
theoretical temperature that changes the result. But what if temperature affects albedo? Wouldn’t an
ice-covered earth have higher reflectivity than a warm one? If we treat temperature as a function of
albedo, then we see that their relationship is not as straightforward as the simple exponential model
we have just reviewed. As stated previously, our balanced EBM equation with a varying albedo will
be (1-.5-.2×tanh((Ts )/10))×S/4 = ε×σ×Ts
4
. In order to utilize this information, we must first start
with our albedo vs. temperature equation, α = .5-.2×tanh((Ts )/10). The plot of this function is
shown below, in Fig.(5).

9. Fig.(5) Albedo percentage as a function of Earth’s average global temperature
Here we can see that Earth’s albedo has a minimum and maximum within a broad range of
temperatures on earth. When Earth’s temperature is below about -25°C, like during an ice age,
albedo plateaus at .7. Likewise, upwards of 25°C, albedo holds a steady .3. However, between these
temperatures, Earth can move from ice-covered to our present climate relatively quickly, as this area
is unstable. We can find the most ideal and stable Earth temperatures by going one step further, and
graphically solve our incoming energy, (1-.5-.2×tanh((Ts )/10))×S/4 and outgoing energy,
ε×σ×(Ts+265)4
, in relation to temperature. These date are shown in Fig.(6).
20%
30%
40%
50%
60%
70%
80%
-100°C -75°C -50°C -25°C 0°C 25°C 50°C 75°C 100°C
AlbedoPercentage
Earth's Temperature
Var. Albedo vs. Earth's Temperature

10. Fig.(6) Incoming and outgoing energy as a function of Earth’s temperature
Where the plots for incoming energy and outgoing energy cross are our solutions to Earth’s
EBM. Earth’s temperature is stable around -39°C, -9°C, and 16°C. However, we can see that the
first and third solutions are much more stable than the second. Stability can be seen here by the
relative consistency of temperature below -39°C and above 16°C. At -9°C, a flux in temperature can
push the energy balance to a negative rate, resulting in an ice age, or a positive rate, resulting in a
climate similar to our own. 16°C is a stable solution that we humans would like, and is reasonably
close to our current global average of 14°C. However, we are currently at the point in this model
where our incoming energy is greater than our outgoing energy. Will global warming push us to a
more stable temperature balance, or the opposite?
0
50
100
150
200
250
300
-100°C -75°C -50°C -25°C 0°C 25°C 50°C 75°C 100°C
Energy[W/m2]
Earth Temperature
EBM Solution Using Varying Albedo
Incoming Energy
Outgoing Energy

11. Conclusion
EBMs have important information for Earth’s atmospheric viability. With the decline
emissivity caused by the increasing effect of global warming, it is necessary to look at the long-term
effects of Earth’s energy balance. Since this is a zero-dimensional model, it is considered the
simplest, and therefore, least precise. This model does not take into account varying temperatures
over different land masses, bodies of water, human contribution, and latitude, among the many
other factors that influence Earth’s atmospheric temperature.
Using this model, we can see that Earth’s temperature is greatly affected by disturbances in
albedo. In order to keep a habitable planet, we must also take care to monitor the greenhouse effect.
Although this model shows 16°C as our ideal balance state, it is not necessarily precise. One theory
on the progression of global warming is that as the ice caps melt and the water levels rise, the earth
will begin to cool rapidly, possibly sending us into an ice age more quickly than ever before in
Earth’s history. Whether this is a serious threat or not, there are great advantages to preventing the
buildup of greenhouse gasses that cause global warming. Increasing Earth’s albedo by using more
reflective building materials could be one way to significantly mitigate decreased emissivity due to
greenhouse gasses, and keep our earth in a more stable energy state. So do your part, and paint your
roof white!

12. References
1. Shen, S., (2016). Principles of Mathematical Modeling. San Diego State University
Department of Mathematics.
2. Silva et al., (2009). Development of a Zero-Dimensional Mesoscale Thermal Model for
Urban Climate. Journal of Applied Meteorology and Climatology, V48, 657-668.
3. North, G., (1975). Theory of Energy-Balance Climate Models. Journal of the Atmospheric
Sciences, 32, 2033–2043.
4. Mann, M., Gaudet, B. (2015). From Meteorology to Mitigation: Understanding Global
Warming. The Pennsylvania State University Open Educational Resources, METEO 469
Lessons.
5. Benzi, R., Parisi, G., Sutera, A., and Vulpiani, A. (1982). Stochastic resonance in climatic
change. Tellus, 34(1), 10-16.
6. Hansen, J., (2014). NASA GISS Surface Temperature Analysis. Carbon Dioxide Information
Analysis Center, Center for Climate Systems Research.