Lecture 2: The Global Mean Energy Budget
Before considering a climate in equilibrium, we can write the heat budget for an average
meter squared column of the Earth’s climate system as:
dQ
 Qabs  T 4
dt
(1)
The left side of (1) is the rate change of heat content (Q) of the climate system, Qabs
represents the solar energy absorbed, and T4 taken from the Stephan-Boltzmann Law
represents the longwave emission by the planet out to space. T is the emission
temperature of the Earth.
We can write the heat content of the climate system as:
Q = mcwT ,
where m is the relevant mass of the climate system, cw is the relevant specific heat (if we
are considering the ocean to regulate the heat capacity of sea water cw~4000 J K-1 kg-1. If
we are considering just the atmosphere, it would be the specific heat of air cp = 1004 J K-1
kg-1).
Therefore, a globally averaged heat balance can be written as:
mcw
dT
= Qabs - s T 4
dt
(2)
Sometimes, the product mcw is called the heat capacity of the climate system.
For a climate system in equilibrium that is neither gaining nor losing heat (dT/dt=0), an
equilibrium state will hold that can be used to calculate the equilibrium emission
temperature (TE) of the planet:
Qabs = s TE4
where Qabs is the amount of solar radiation absorbed by the Earth.
We will show later that calculation of an equilibrium temperature for the Earth is indeed
justifiable. However, let’s first look at the global mean energy balance.
Remember that at the average radius of the Earth’s orbit around the Sun, the solar
constant is:
So = 1363 W m-2.
1
How much of the solar flux is intercepted by the Earth?
Before considering absorption of solar radiation specifically, we need to calculate the
amount of solar radiation that the Earth’s disk intercepts. We use the concept of a shadow
area to calculate the intercepted solar flux:
Figure 2.1 provides a graphical view of the concept of shadow area.
The shadow area of the Earth that intercepts solar radiation is given by:
Shadow Area=  a 2 ,
where a= Earth’s radius = 6.37x106 m
Thus, the total solar energy flux intercepted by the Earth is given by:
Qintercepted =So a2
This is the solar flux intercepted by the disk of the Earth, and not necessarily the amount
that is absorbed.
Why might not all of this solar energy be absorbed by the climate system?
The Earth is not a blackbody and does not absorb all solar radiation impinging on it. This
is because the Earth has:
1) clouds
2) aerosols
3) ice
4) snow
5) land surfaces
6) an absorption dependence on zenith angle.
All of these factors are responsible for reflecting some of the incident solar flux that
impinges on the top of the Earth’s atmosphere.
Reflection of solar radiation can easily be seen by looking at a visible satellite image
(Figure 2.2). Visible light is reflected off of various features in Earth’s atmosphere,
ocean, and land surface.
To account for this portion of solar radiation that is reflected by the Earth’s climate
system and not absorbed, we define a term called albedo.
albedo= = the fraction of incoming solar radiation that is reflected.
2
The fraction of solar radiation that is absorbed is simply given by one minus the fraction
that is reflected:
Fraction absorbed = (1-)
Figure 2.3 shows the albedo of the Earth as a function of season and the location on the
Earth:
Higher latitude regions and regions of clouds generally have the highest albedo. Higher
latitude regions have high albedos due both to 1) the presence of ice and snow and 2) the
high zenith angle of incoming solar radiation. Zenith angle is defined as the angle of the
Sun relative to the local vertical.
Notice also that deserts have high albedos due to the bright sand surface.
Albedos in general have a wide variety of values depending on the type of surface. This
is apparent in the following table of albedo versus surface type:
Table 1: Albedo as a function of surface type.
As can be seen, low latitude oceans have the lowest albedos, and ice surfaces have the
highest albedos. Even the albedo of the ocean surface varies, however. If it is windy and
we are at high latitude (high zenith angle), the albedo of the ocean surface can be as high
as 0.2.
For the Earth as a whole, the global mean albedo comes out to be about 30%.
Global mean albedo= p = 0.3.
Solar Radiation Absorbed by Climate System
Thus, to calculate the amount of solar radiation absorbed by the Earth, we can use the
amount of radiation intercepted by the Earth, and also account for the planetary albedo
p:
Earth’s Total absorbed solar energy per second= S o p a2 (1- a p )
For a basic understanding of the Earth’s global energy balance, it is instructive to average
the amount of absorbed radiation over the entire globe to get a globally-averaged solar
energy flux, which also provides the units for use in (2).
Surface Area of Earth= 4 a 2 ,
3
Thus, dividing by the surface area of the Earth, we can get the global mean flux of solar
radiation absorbed by the climate system in W m-2:
Absorbed Energy Flux per
m2
= Qabs 
S oa 2 (1   p )
4a 2
And simplifying:
Qabs 
So
(1   p )
4
(3)
Equilibrium Global Mean Energy Balance
As discussed earlier, for a climate in equilibrium, the amount of solar radiation absorbed
must be balanced by the emission of terrestrial radiation by the planet in the global mean:
Qabs=Qout ,
where Qout= Longwave radiation emitted by Earth.
Let calculate emitted terrestrial radiation using an equilibrium blackbody emission
temperature:
Qout  TE
4
Therefore for exactly as much energy coming in as going out,
TE 4 
So
(1   p )
4
Solving for TE, TE=255 K
The value of 255 K is the equilibrium emission temperature for Earth, and would also be
the equilibrium global mean surface temperature if Earth did not have an atmosphere (but
of course the albedo would then likely be different).
However, the Earth has an atmosphere with radiatively active trace gases, clouds, and
aerosols that absorb effectively in the infrared part of the spectrum, but are relatively
transparent to shortwave radiation from the Sun. The behavior for trace gases can be seen
by examining the absorption spectrum of trace gases as a function of wavelength as
shown in Figure 2.4, where the Sun and Earth emission spectra are also plotted as a
function of wavelength. Earth’s emission spectrum peaks in the infrared, the Sun’s in
much shorter wavelengths in the visible.
4
Given this distinction between the emission spectra of the Sun and Earth, we will often
use the following terminology to account for the different wavelengths of solar and
terrestrial radiation:
Solar radiation: “Shortwave”
Terrestrial radiation: “Longwave”
Before we use the concept of radiative equilibrium to describe Earth’s global mean
energy balance, how justified are we in using the concept of equilibrium?
To estimate this, we need to calculate a time constant that determines how quickly the
climate system responds to a radiative perturbation. Before we consider the climate
system as a whole, let’s consider only the mass of the atmosphere.
Derivation of a simple globally-averaged energy balance model:
As before, Let’s consider an atmosphere that is not in equilibrium such that:
dQ
 Qabs  T 4
dt
Assuming that all of this excess heating goes into increasing the temperature of the
atmosphere:
mc p
dT
 Qabs  T 4 ,
dt
where m is the mass of the atmosphere and cp=1004 J K-1 kg-1 is the specific heat at
constant pressure. cp represents the amount of energy necessary to raise one kilogram of
dry air one degree. As a hint on how to compute the atmospheric mass, the mass of the
atmosphere per meter squared can be derived by integrating in height the hydrostatic
balance relationship:
dp
= -r g
dz
where =density, z = height, g = gravitational acceleration, and p = pressure.
Consider perturbations about an equilibrium temperature TE to estimate the timescale on
which the atmosphere responds to radiative forcing. In this case, let’s consider a change
in solar forcing. Related calculations can be done for other forcings such as increased
greenhouse gas forcing.
Essentially, we want to ask the question:
How quickly do temperature perturbations grow?
5
To answer the question, we can linearize atmospheric temperature about an equilibrium
temperature (TE) and perturb the solar forcing slightly about its background value (So).
Hence, let:
T  TE  T  , S  S o  S 
So is the solar constant, and S’ is a perturbation solar forcing. S   S o . We can plug
these expression into the heat budget equation above, where Qabs is given by (3) above.
Using a series of steps including a Taylor series expansion, and solving a simple ODE
using an integrating factor, an atmospheric time constant can be derived:
t a » 31 days
Thus, the atmosphere responds relatively slowly (compared to diurnal forcing) to a
radiative perturbation, which helps to justify use of the global mean energy balance
models we use above and below. The diurnal cycle should not significantly affect the
assumption we make about absorbed solar radiation being approximately uniformly
distributed across the entire surface area of the globe. Nor is the global mean balance
disrupted by small transient forcings.
The climate system as a whole responds much more sluggishly than the atmosphere in
response to a radiative forcing. The atmosphere has a heat capacity of only about a 2.5
meter deep ocean.
The ocean and therefore the climate system respond much more sluggishly to an imposed
radiative forcing since the oceans have a much greater heat capacity than the atmosphere.
Approximately, the top 70 meters of the ocean regulate the interactions of the atmosphere
and ocean on timescales of a year. The time constant of such a 70 meter deep mixed layer
ocean is about:
 70m  2.3 years.
ocean
As we will highlight when we discuss monsoons, ocean temperature on large scales
varies relatively little over an annual cycle compared to land surface temperature, which
is predicted by the relatively long timescale of 2.3 years given above. Contributions of
the deeper ocean to Earth’s energy balance make adjustment of the climate system to
external forcing even longer than 2.3 years, in fact on the order of decades. We’ll get
back to this in our discussions of the ocean general circulation.
In reality, the method of determining the time constant of the climate system as described
above is inadequate, because it does not include feedback processes that affect the
response of the climate system to external forcing. With feedbacks, the time constant of
6
the climate system is longer than that derived above. We’ll discuss this further also once
we get later in the course. Essentially, the time constant that is derived for the climate
system as a whole helps to explain statements such that we have already “bought into” a
certain level of warming due to our current levels of greenhouse gases and associated
radiative forcing in the atmosphere.
Heuristic Energy Balance with an Atmosphere.
Above, we calculated the effective equilibrium emission temperature of the Earth’s
climate system as a whole, ignoring the distinctions between the atmosphere, surface, and
other parts of the climate system. We came up with an effective emission temperature of
the Earth as a whole of about 255 K. However, this clearly cannot be the surface
temperature of the Earth, otherwise life as we know it would not exist. Therefore, the
radiative emission temperature of the Earth as a whole must be different than the surface
temperature. Let’s in a relatively crude way help to explain this.
Here, we will construct a simple heuristic model for the global mean energy balance that
includes an atmosphere. We will use a 1-layer atmosphere that is assumed to be
isothermal. The simple model can be described as follows:
1) Shortwave radiation is not absorbed by the atmosphere and penetrates to the surface
(after accounting for the albedo).
2) We assume that the one-layer atmosphere absorbs/emits longwave radiation as a
blackbody.
If the atmosphere and surface are in equilibrium, we can derive the following balances:
Surface:
 TS 4   Ta 4 
So
(1   )
4
7
Atmosphere:
2 Ta   Ts
4
4
So
4
(1   )   Ta (note that this is the same as the balance we
4
used above to find Earth’s equilibrium emission temperature).
Top of Atmosphere:
Solving for Ta and Ts using the atmosphere and top-of-atmosphere balances, we obtain
for So=1363 W m-2 and =0.3:
TA=255 K, TS=303 K
This is a simple heuristic description of the “greenhouse effect”. While defined in
different ways, we define the greenhouse effect as the difference between upward
longwave radiation emitted by the surface and atmosphere:
“Greenhouse effect” = TS 4  Ta 4
The greenhouse effect due to absorption by atmospheric trace gases can be considered a
climate forcing.
We will discuss later in the course a term called the climate sensitivity parameter that
gives the change in surface temperature that results from a 1 W m-2 climate forcing.
Climate sensitivity parameter=
TS
 f
F
Ultimate determination of climate sensitivity includes not only the direct effects of the
forcing itself on temperature, but also any feedbacks associated with changes in
temperature, as will be discussed later.
Deficiencies in the Simple Heuristic Radiative Equilibrium Model Shown Above
Many deficiencies exist in this model, including the fact that the atmosphere can’t be
simulated by a single isothermal layer that behaves like a blackbody.
More broadly, the idea that the climate system is in simple radiative equilibrium is an
oversimplification.
Global mean surface temperatures in the model are not commensurate with observations.
Heuristic model: Ts~303 K
Observed global mean temperature: Ts~288 K
The heuristic model is not that bad in that the observed internal fluxes of energy between
the atmosphere and surface are also dominated by longwave radiation.
8
Figure 2.5 shows a recent estimation of the global mean energy budget from Trenberth et
al. 2009 (their Figure 1).
There are a few take-away points from this plot:
1) The heuristic model we derived above is not that bad in that the observed internal
fluxes of energy between the atmosphere and surface are also dominated by longwave
radiation. Note that there is a reasonable amount of uncertainty in the precise values of
the external longwave exchanges between the atmosphere and surface. Wild et al. (2013)
has upward surface longwave at 398 W m-2 and downward longwave absorbed at the
surface at 342 W m-2 (they also added error bars). Hence, these values are somewhat
different than the Trenberth number, but the qualitative picture is the same.
2) A significant portion of the solar flux is actually absorbed by the atmosphere (~78 W
m-2), as opposed to the heuristic model that assumes all solar radiation absorbed by the
climate system is absorbed at the surface. Most of the solar absorption by the atmosphere
occurs due to water vapor and clouds, and the near-infrared part of the electromagnetic
spectrum is key to this solar absorption.
3) Latent (evaporative) and sensible heat fluxes play an important role in fluxing heat
from the surface. Without these surface latent and sensible heat fluxes, the Earth’s
global mean surface temperature would be much hotter (~303 K) than the
approximately 288 K it is now. After input of these fluxes into the atmospheric boundary
layer, deeper atmospheric convection then plays a key role in redistributing this heat from
the boundary layer to upper portions of the troposphere.
The term “global warming” may be thus an incomplete description of the influence of
increasing greenhouse gases on Earth’s climate. A more proper term may be “global
heating”. Some of the anthropogenic radiative forcing due to increased greenhouse gases
may go into increased evaporative and sensible heat fluxes that may enhance the
hydrologic cycle (increase precipitation) rather than warming the surface.
Window-Gray Heuristic Model for Global Mean Energy Balance
The model is similar to the heuristic energy balance model described above, except that
in this case, the atmosphere is not assumed to absorb (or emit) as a blackbody. Let’s first
define some terms:
Window: The atmosphere is transparent to all incident shortwave radiation that is not
reflected.
Gray: The atmosphere absorbs only a fraction of the longwave radiation from below that
impinges on it. This fraction is called absorptivitya. For the “gray” approximation, the
absorptivity is uniform across all infrared wavelengths.
Kirchoff’s Law
9
In developing models such as this, a nice simplification can be made due to Kirchoff’s
Law.
Kirchoff’s Law states that for a given wavelength of electromagnetic radiation:
Absorptivity=Emissivity
Stated another way:
a   
 is the fraction of radiation emitted relative to the radiation that would be emitted by a
blackbody at the same temperature.
We can use Kirchoff’s Law in the window-gray model.
An isothermal atmospheric layer that has an emissivity can be modeled as follows,
where F is a longwave flux impinging on the bottom of the layer:
The following discussion may be very useful when considering your modeling project for
this semester, where you will essentially be developing a multi-level window-gray model.
First, let’s assume that the atmospheric layer is not in equilibrium. The heat balance for
the atmospheric layer is thus:
mc p
dTa
= Flux _ absorbed - Flux _ emitted
dt
The mass of the atmospheric layer is given by the pressure difference across the top and
bottom of the layer divided by gravity (g=9.8 m/s2):
10
m=
Dp
g
Thus, the equation for layer heat tendency is given as (remember a=):
c p Dp dTa
= e F - 2es Ta4
g dt
In equilibrium the balance for this atmospheric layer would be:
2es Ta4 = e F
You should start thinking about how you would develop a multi-layer model, with the
balance for each individual layer as given above for the time-dependent equation. For a
multi-level model, you would also need to take into account the longwave radiation
impinging on the top of individual layers being emitted from atmospheric layers above.
11