DifferentialEquations

6.1 Climate Change

6.1.1 The Zero-Dimensional Energy Balance Model

Here we introduce the simplest model, also known as the Zero-Dimensional En-

ergy Balance Model, where solely globally averaged quantities are used.

The theory originates from independent works of Budyko (1920–2001)1 and Sellers

(1955–2016).2

The heat equation of a body of any arbitrary shape is given by

dQ

dt = Pgain −Ploss.

The quantity of heat is Q = C · M · T where T is the temperature in Kelvin, M the

mass of the heat exchanging body, and C its mean specific heat capacity with di-

mension J/(kg K). Next we determine the power of heat gain Pgain and heat loss Ploss

of the earth. The total power absorbed by the earth equals the short wave radiation

received from the sun, hence

Pgain = π R2 ·S0 ·(1 −α ).

Here π R2 is the cross-sectional area of the earth, S0 = 1370 W/m2 is the total power

per square meter received from the sun in the form of radiation (Total Solar Irradi-

ance, TSI) and α ≈ 0.32 is the fraction that is reflected back into space, the so-called

planetary albedo.

1 Mikhail Ivanovich Budyko was a Russian climatologist, geophysicist and geographer. He is con-

sidered one of the leading climate researchers. Numerous models and predictions on global warm-

ing were incited by his research.

2 Piers John Sellers was a British-American climate researcher and astronaut. After his three space

shuttle flights from 2002 to 2010 he became a director of the Earth Science Division at Goddard

Space Flight Center of NASA.

167© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021

A. Fässler, Fast Track to Differential Equations,

https://doi.org/10.1007/978-3-030-83450-0_6

168 6 Climate Change and Epidemics

The total loss of power due to the emitted long wave radiation is given by the Stefan-

Boltzmann law3

Ploss = 4π R2 ·ε ·σ ·g·T4.

Here ε = 0.97 is the emissivity, σ = 5.67 · 10−8 W

m2K4 the Boltzmann constant, T

the temperature on the earth’s surface in Kelvin, and g the crucial radiation factor,

which can be influenced and models the greenhouse gas effects.4

About 34 of the earth’s surface is covered with water that has a specific heat capacity

of 4187 Jkg·K . Considering the fact that some materials of the earth’s crust have lower

values, the global mean specific heat capacity is estimated at C ≈ 3440 Jkg·K .

Hence the temperature varies with time t, in seconds, according to the equation

R2 ·Δ R·ρ ·C · dT

dt = Pgain −Ploss.

Here ρ is the average density of the layer Δ R of the earth’s surface that is involved

in the heat exchange and is estimated at ρ ≈ 1500 kg/m3.

Dividing by 4π R2 yields a nonlinear autonomous differential equation for the func-

tion T(t) that is also found in [15], Subsection 3.2.1:5

CE · dT

dt = S0

4 (1 −α )−ε ·σ ·g·T4.

The constant CE = Δ R·ρ ·C, measured in J/(m2 K), may be interpreted as the spe-

cific heat capacity per m2.

According to the findings of the climate sciences the prevailing power of radiationS0

4 (1 −α ) in the future will increase by the so-called radiative forcing

Δ F = 5.35 W

m2 ·ln

( c

c0

)

caused by feedbacks in the atmosphere due to greenhouse gases.

3 Josef Stefan (1835–1893) was an Austrian mathematician and physicist with Slovenian roots,

Ludwig Boltzmann (1844–1906) was an Austrian physicist.

4 The scaling factor g is a result of the fact that the emission temperature in the real atmosphere,

i.e. the temperature that is observed in the universe, does not correspond to the temperature on the

earth’s surface.

The reason is that a considerable portion of the infrared radiation emitted by the earth’s surface

is trapped in the atmosphere, as it is absorbed by greenhouse gases, mainly steam and carbon

dioxide CO2, in the atmosphere. These particles radiate the absorbed energy in all directions, also

back down towards the earth, thus heating up the earth’s surface and, via convection, the entire

atmosphere as well.

In the convective atmosphere the temperature falls with rising altitude (see Section 3.11.2),

mainly because the air expands adiabatically with decreasing pressure.

Due to infrared radiation back into the universe, which depends on the altitude, the earth sys-

tem can maintain its balance between incoming and outgoing radiation even in the presence of

greenhouse gases. With the current concentration of greenhouse gases, infrared radiation into the

universe takes place at an altitude of 5 km. In our simple model of energy balance we account for

the heat loss by infrared emission by using the radiation factor g < 1. It should be noted that its

exact value depends on many feedbacks of the earth system, for instance on the increase of the

amount of steam in the atmosphere that heats up the earth.

5 Reference provided by Prof. Dr. Stefan Br ̈onnimann, Climatology Group of the Institute of

Geography at the University of