Climate Change/Science/Distribution of Insolation

../Sun-Earth System/ should make it clear that the geometry of the
Sun-Earth system plays a key role in how much sunlight reaches Earth
and where it arrives. This section briefly describes the pattern of
incoming solar radiation, or insolation, over the course of the
year. This pattern is the direct result of the geometrical factors
described above, and since the insolation is the energy source for the
entire climate system, where the energy enters the system is of
fundamental importance to the subsequent distribution of energy. That
is to say, where the sun shines has a direct impact on weather and
climate. 
In an idealized system, where the orbit is circular and the planet's
spin axis is perpendicular to the ecliptic (i.e. obliquity equal to
zero), the problem becomes nearly trivial because every day is
identical. However, the spherical shape of the planet requires some
consideration. A first guess at why the poles of this hypothetical
planet are cold compared to the equator might be because the pole is
farther from the sun than the equator (by a distance equal to the
radius of the planet); in fact, this is a common mistake people make
about the real Earth as well. To convince us that this can not be the
case, consider the change in incoming energy in say 6800 km versus the
Earth-sun distance: it is minuscule. However, if we allow the solar
constant to be constant at the equator and the poles, there is still
an important effect of geometry, namely the angle between the incoming
photons (which we can think of as parallel rays of light) and the
direction normal to the surface (which can be thought of as the local
vertical direction, looking straight up, the sun is not always
overhead). The insolation is reduced by the cosine of this angle,
which is known as Lambert's cosine law. This is not an
atmospheric effect, but simply an optical one, so consider this the
insolation at the top of the atmosphere instead of the surface. The
principle is simply that the curvature of the Earth means that the
same radiance (or photon flux or sunshine) gets spread over a larger
area as the angle between the photons and the normal to the surface
increases from zero to 90 degrees at the poles. So even though the sun
is just as bright everywhere in the world, the power per area incident
at the top of the atmosphere changes as the sun appears lower in the
sky. The result is a modification to the local insolation, I, 
where formula_2 is latitude. 
Thus far, we have neglected Earth's spin. Of course, only half the
planet faces the sun at any instant in time. Consider a snapshot in
time. Half the planet faces the sun, half is in darkness. From
everyday experience, we know that the sun appears at different
distances above the horizon over the course of the day. At dawn, the
sun comes over the eastern horizon, traveling in an arc across the
sky, reaching its maximum height at local noon, and then descending toward the
western horizon. Let us define the "zenith" as the point directly
overhead; the angle from this zenith point to the sun's current
position is called the "zenith angle". In the hypothetical planet
described above, where seasons do not exist, the sun only appears
directly overhead along the equator at noon; moving away from the
equator, the sun sinks lower and lower toward the equatorward
horizon. At the south pole, the sun is just at the northward horizon
at local noon, providing essentially no insolation. Define a new
angle, that between Earth's equator and the highest local position of
the sun (the position at local noon), and call this the declination
angle; it is essentially a measure of the height above the horizon
that the sun will reach each day, and is equal to the latitude at
which the sun is directly overhead at noon. In the hypothetical world
above, the conditions are perpetually equinox, so the declination is
zero because the sun is overhead directly on the equator every
day. Earth's obliquity is about formula_3 (0.409
radians), which combined with the Earth's revolution of 360 degrees
per 365 days provides an expression for the declination angle,
where N is the Julian day of the year. This expression is an
approximation, but will serve our purposes. 
With the latitude and declination, an estimate of insolation can be
made given one additional piece of information related to the geometry
of the system: the time of day. The time of day is needed since it
affects the overall angle between the local vertical and the suns
rays, as is clear by noting the difference between night, dawn/sunset,
and noon. To describe the time of day, an additional angle in
introduced, conveniently denoted the "hour angle,"
where, formula_6 is the rotation rate (i.e.,
formula_7) and LT is the local time. Note that at
local noon, H = 0. These calculations are normally done in radians,
but since most people more intuitively understand angles, we use them
here. 
Putting all these together, the actual solar zenith angle can be
described in terms of the above angles
This relation can be derived without explicitly expressing the
declination and hour angles, purely from the geometry. These angles
make sense physically, so are included here.
Returning to Lambert's cosine law, we can write a simple expression
for the insolation: formula_9. This describes the
flux of energy into the climate system at any particular moment, given
the location and time of day.