Lecture 3: Latitudinal Gradients in the
Top-of-Atmosphere Energy Budget
Up to now, we have only considered the global mean energy budget. However, we know
that the Earth is not heated by the Sun uniformly.
Meridional differences in insolation, the downward solar flux impinging on the top of the
atmosphere, exist as a function of latitude and season.
The solar constant that we derived earlier in the class (So=1363 W m-2) is the flux density
for a surface perpendicular to the nearly parallel rays of the Sun.
In reality, the Earth’s surface is generally tilted relative to the solar beam.
Figure 3.1 Demonstrates the angle dependence of solar radiation.
As before, s= zenith angle which gives the angle of the solar beam at a given location
relative to vertical.
In a similar manner to what we did for the entire planet, we can calculate the amount of
flux intercepted by the surface element A, given by the shadow area.
You will show in your homework that the solar flux impinging on the top of the Earth’s
atmosphere is given by:
2
Qsolar
d 
 S o   cos  s with d=d(t), where
d 
Qsolar is the solar flux per unit area impinging on the top of atmosphere, and a factor of
d 
 
d 
 
2
accounts for the elliptical nature of Earth’s orbit and differences in Earth-Sun distance
during an annual cycle.
The zenith angle s is a function of various geometric factors:
1) latitude 
2) season = declination angle
= 23.45 at summer solstice (June 21)
= -23.45 at winter solstice (Dec 21)
Note that  corresponds to the relative tilt toward the Sun at a given point in Earth’s orbit:
3) hour angle h = longitude of subsolar point relative to noon. h=0o at noon.
The cosine of zenith angle (cos s ) can be prescribed as follows:
cos s  sin  sin   cos  cos  cos h
Figure 3.2 gives a sense of the ugly geometry that goes into determining the zenith angle.
We can average Qsolar over an entire day to get the daily-average top-of-atmosphere
insolation Q day .
For a non-circular orbit, which is what the Earth actually experiences, the derivation of
Q day produces the following:
Qday
2
d 
 S o   ho sin  sin   cos  cos  sin ho 
 d 
1
The expression for Qday can be plotted to give the distribution of daily-average
insolation as a function of latitude and day shown in Figure 3.3.
Some notable features about the distribution in Figure 3.3 include:
1) A difference in maximum magnitude of insolation between the two poles,
consistent with the fact that the Earth is closer to the Sun in January than June.
2) The lack of insolation in winter, up to the Arctic and Antarctic circles.
3) Summertime insolation is greatest at the poles in the daily average (this does not
mean most solar radiation is absorbed there, as we will see below).
4) A relatively constant insolation in the tropics throughout the year.
Remember, Figure 3.3 shows the insolation, and not the solar radiation that is actually
absorbed.
Actual shortwave absorption depends not only on insolation, but also on the albedo,
which varies with cloudiness, surface type, zenith angle, etc.
Spatial Distribution of Solar Absorption
The following plots show the actual solar flux absorbed by the climate system on average
in W m-2.
Figure 3.4 shows the June, July, August average of absorbed solar radiation
Figure 3.5 shows an annual mean absorbed solar radiation
Of particular note in these figures:
1) The highest absorbed shortwave radiation is not at poles during summer,
different than insolation. Again, this is because actual shortwave absorption
depends not only on insolation, but also on the albedo, which varies with
cloudiness, surface type, zenith angle. The high latitudes have snow and ice, quite a
bit of cloudiness, and high zenith angles relative to other latitudes.
2) Cloudy regions of the Tropics are characterized by suppressed shortwave
absorption
3) Deserts, with sand surface and high albedo are regions of relatively lower solar
absorption.
4) Highest absorption tends to occur in cloud-free subtropical/tropical oceans.
Spatial Distribution of Outgoing Longwave Radiation at Top of Atmosphere
We can create similar plots for the top of atmosphere longwave flux emitted from Earth
(Outgoing Longwave Radiation [OLR]). Figure 3.6 shows the annual mean outgoing
longwave radiation (OLR)
Longwave emission from the planet depends primarily upon:
1) Surface and atmospheric temperatures (remember, longwave emission from an
object depends upon temperature [T4]).
2) The vertical distributions and thicknesses of clouds
3) Vertical distributions and concentrations of trace gas species.
Notable features from Figure 3.6 include:
1) Reduced OLR in regions of deep tropical clouds (e.g. equatorial west Pacific,
Indian Oceans, Africa, South America). These cloud tops are at extremely low
temperature and thus have reduced longwave emission to space.
2) Low cloud regions off the West Coasts of N. America and S. America have high
OLR. These clouds are low, and thus at high temperature. Their tops have strong
longwave emission.
3) Desert areas with hot surfaces and few clouds have very high longwave emission
(e.g. Sahara).
4) Colder midlatitude and polar regions have lower OLR.
Net Incoming Radiation at the Top of Atmosphere (TOA)
At the top of the atmosphere, the net incoming radiation per m2 is given by:
RTOA= Incoming Solar Flux - Outgoing Longwave Flux
RTOA is the net rate of energy gain (or loss) in a column consisting of atmosphere, ocean,
land, and/or ice per m2.
As we will see later, the surface energy balance is more complicated than this, since we
must also consider evaporative and sensible heat fluxes.
The net incoming radiation input into the Earth’s climate system is shown by the attached
Movie detailing the Net Incoming Radiation Seasonal Cycle from the ERBE satellite
mission.
This movie nicely shows the seasonal cycle in the TOA energy balance, with the equator
of peak heating moving back and forth across the actual equator during an annual cycle.
The midlatitudes have a very strong seasonal cycle, with TOA energy loss during winter,
and TOA energy gain during summer.
In the annual mean, the TOA energy balance looks like that shown in Figure 3.7.
Notable features of this figure include:
1) Net incoming radiation is positive in the Tropics and negative in the high
latitudes in the annual mean.
2) Other interesting features can be seen, including the TOA radiative loss from the
Sahara in the annual mean (this must be balanced by a net heat transport into the
atmosphere above the Sahara from other regions).
The annual mean TOA energy balance can be averaged around latitude belts to nicely
show the equator-to-pole gradient in heating of the climate system:
Figure 3.8 shows the zonal average OLR, absorbed solar, and net annual mean TOA
energy balance.
Some notable points from Figure 3.8 include:
1) In the annual mean, the Tropics absorb more solar than is emitted in longwave.
2) High latitudes emit more longwave than absorbed in solar.
**Thus, in the annual mean, there is a net top of atmosphere energy loss at the poles,
and a net top of atmosphere energy gain in the Tropics.**
Meridional (North-South) Heat Transport
For a climate system that is in equilibrium, the latitudinal heating gradients indicated in
Fig. 3.8 cannot be maintained. Otherwise, tropical temperatures would be continually
increasing, and high latitude temperature would be continually falling.
IMPORTANT POINT: The latitudinal heating gradients drive the general circulation of
the atmosphere and ocean, in which a net transport of heat occurs from the Tropics to the
high latitudes.
Atmosphere and oceanic heat transports are thus required to maintain energy balance.
We will discuss the atmosphere and ocean general circulations later in the course.
However, the consequences of these general circulations can be seen here.
More formally:
E AO
 RTOA  FAO ,
t
E AO
is the rate of energy gain by the climate system in a column, RTOA is the net
t
incoming radiation, and FAO is the divergence of heat due to ocean and atmosphere
circulations.
where
Averaging over a sufficiently large integral number of years, for a climate in equilibrium,
E AO
 0 at every location on the globe.
t
Since no energy is gained or lost in an annual average in equilibrium, in an annual
average sense,
RTOA  FAO
This states that if we know the net top of atmosphere incoming radiation in the annual
mean, we can determine the required heat transport by the ocean and atmosphere general
circulations to keep a balanced climate.
The required heat transport across a latitude band () can be calculated as follows
(Figure 3.9):
 2
F 
  RTOA a

2
cos  d d ,
0
2
where  is longitude, a is Earth’s radius. The integration goes from the South Pole to
latitude . A negative value of this integral implies a southward flux of heat. i.e. the polar
cap is losing energy in the annual mean and a transport from regions closer to the equator
is necessary to maintain energy balance. A positive value implies a northward flux of
heat. This integral should be negative (southward) until some point in the tropics, where
it switches to positive, indicating a northward heat transport.
Figure 3.10 shows the meridional transports of energy required to maintain the Earth’s
energy balance, along with the estimated contributions from the atmosphere and ocean.
This figure was calculated in the early 1970’s, and updated measures of the atmospheric
and ocean contributions to transport will be discussed below.
Traditional Ways of Partitioning the Atmosphere and Ocean Heat Transport
Determining heat transport based on the top-of-atmosphere flux tells you how much
energy must flux across latitude belts to maintain energy balance. However, it does not
say what the partitioning of this energy flux is between ocean and atmosphere.
How the partitioning of transport between ocean and atmosphere was traditionally done is
to take radiosonde (balloon) data or gridded atmosphere datasets and calculate the
atmospheric heat transport from equator-to-pole. The total transport is mandated by the
net top-of-atmosphere radiation (RTOA, shortwave in minus longwave out), usually
derived from satellite data. Thus, the ocean contribution was calculated as the residual
transport required to achieve the total required equator-to-pole transport. Figure 3.10
from Vonder Haar and Oort (1973) used this method (and employed radiosonde data to
estimate the atmospheric transport), and this early attempt indicated a nearly equal
contribution from the ocean and atmosphere to meridional heat transport.
The partitioning of the transport between ocean and atmosphere is only as good as the
atmospheric datasets used to calculate the atmosphere energy budget.
The early attempt by Vonderhaar and Oort was limited by the fact that it used radiosonde
data, which was mainly found in land regions away from the storm tracks. Much of the
equator-to-pole atmospheric heat transport occurs in the storm tracks (e.g. see Figure
3.11), which made these early estimates of atmospheric transport problematic.
Atmospheric heat transport estimates can be improved by using models to “fill-in”
atmospheric fields where actual observations from balloons, satellites, buoys, etc. are
sparse.
Note on atmospheric analysis datasets:
Observations of the atmosphere are not available everywhere on the globe. Where they
do exist, they are sometimes sparse. Thus, observations must be assimilated into
dynamical models to produce physically consistent representations of the “observed”
atmospheric state.
Gridded atmospheric datasets are generated by using available observations (surface land
stations, radiosondes, satellite, buoys, etc), along with a dynamical model of the
atmosphere to generate a near-physically consistent gridded global atmospheric state
(Figure 3.12). Many use these so-called “analysis” and “reanalysis” datasets as
observations. Some of the observational analyses extend back into the 1940s.
The Reanalyses are only as good as the observations used to constrain the model, and our
ability to parameterize processes that occur on scales smaller than those of the model grid
(Figure 3.13 Shows cumulus parameterization as an example). These models are thus
imperfect and have some systematic biases that produce errors in the analysis.
Climate modeling uses similar modeling techniques for the atmospheric component,
although the simulations are not constrained by observations (except possibly by
observed climate forcings).
As mentioned above, traditionally the atmosphere and ocean have been thought to
contribute about equal portions of the heat transport toward the poles. We will look now
at more recent estimates of heat transport to see if this is still thought to be the case.
Newer Attempts at Partioning the Ocean and Atmosphere Heat Transport
(Trenberth and Caron 2001)
Recent studies suggest that the atmosphere may take up a relatively larger portion of the
heat transport from equator to pole than the ocean. For example, let’s discuss the recent
estimates from one of your readings this week (Trenberth and Caron 2001, hereafter
TC01):
1) TC01 used Earth Radiation Budget Experiment satellite data from 1985-1989 to
calculation the TOA radiative energy balance. They first estimate the required heat
transport and atmospheric components from two global reanalysis datasets (NCEP and
ECMWF) (Figure 3.14). It can be seen that the atmospheric transport varies somewhat
depending on the atmospheric dataset that is used (due to model differences, differences
in how data was ingested into the models, etc).
2) Atmospheric heat transports are greatest in the winter hemisphere, where the equatorto-pole temperature gradient is strongest. It is no coincidence that this is also the period
of strongest midlatitude cyclone activity (i.e. our typical winter storms) (Figure 3.15).
We will talk about the mechanisms for heat transport later.
3) The oceanic heat transports were then estimated using three ways distinct from the
traditional way:
a) As the required transport to balance the surface heat exchange at the air-sea
interface, as calculated from reanalysis datasets (i.e. the net “top-of-ocean”
heat balance).
b) From coupled climate models.
c) From direct ocean measurements of the product of currents and temperatures
(often making some significant assumptions about the structure of ocean
currents given the sparse sampling), see Figure 3.16
The oceanic heat transport for the Atlantic and World Oceans using all three of these
methods is shown in Figure 3.17. While there are some significant differences between
these methods, they all seem to agree that oceanic heat transport is more modest than that
found in previous studies. In the plots, NCEP and ECMWF refer to ocean transport
calculated using method a), CSM and HADCM3 are climate models in method b), and
the error bars and symbols represent the direct ocean measurements c).
The best estimates of ocean and atmospheric heat transport according to TC01 are shown
in Figure 3.18 [using method a)], where the ocean heat transport peaks near 20N with a
maximum of 2 PW, and the atmospheric transport peaks near 40N and 40S with a
maximum near 5 PW. The atmospheric contribution to meridional heat transport is thus
significantly larger than the oceanic contribution and larger than thought from earlier
attempts at such partitioning (e.g. Figure 3.10)
We will get into the details of meridional heat transport in much more detail when we
discuss the general circulations of the ocean and atmosphere.