In Part Three we had a very brief look at the orbital factors that affect solar insolation.

Here we will look at these factors in more detail. We start with the current situation.

### Seasonal Distribution of Incoming Solar Radiation

The earth is tilted on its axis (relative to the plane of orbit) so that in July the north pole “faces” the sun, while in January the south pole “faces” the sun.

Here are the TOA graphs for average incident solar radiation at different latitudes by month:

*Figure 1*

And now the average values first by latitude for the year, then by month for northern hemisphere, southern hemisphere and the globe:

*Figure 2*

We can see that the southern hemisphere has a higher peak value – this is because the earth is closest to the sun (perihelion) on January 3rd, during the southern hemisphere summer.

This is also reflected in the global value which varies between 330 W/m² at aphelion (furthest away from the sun) to 352 W/m² at perihelion.

### Eccentricity

There is a good introduction to planetary orbits in Wikipedia. I was saved from the tedium of having to work out how to implement an elliptical orbit vs time by the Matlab code kindly supplied by Jonathan Levine. He also supplied the solution to the much more difficult problem of insolation vs latitude at any day in the Quaternary period, which we will look at later.

Here is the the TOA solar insolation by day of the year, as a function of the eccentricity of the orbit:

*Figure 3 – Updated*

The earth’s orbit currently has an eccentricity of 0.0167. This means that the maximum variation in solar radiation is 6.9%.

Perihelion is 147.1 million km, while aphelion is 152.1 million km. The amount of solar radiation we receive is “the inverse square law”, which means if you move twice as far away, the solar radiation reduces by a factor of four. So to calculate the difference between the min and max you simply calculate: (152.1/147.1)² = 1.069 or a change of 6.9%.

Over the past million or more years the earth’s orbit has changed its eccentricity, from a low close to zero, to a maximum of about 0.055. The period of each cycle is about 100,000 years.

Here is my calculation of change in total annual TOA solar radiation with eccentricity:

*Figure 4*

Looking at figure 1 of Imbrie & Imbrie (1980), just to get a rule of thumb, eccentricity changed from 0.05 to 0.02 over a 50,000 year period (about 220k years ago to 170k years ago). This means that the annual solar insolation dropped by 0.1% over 50,000 years or 3 mW/m² per century. (This value is an over-estimate because it is the peak value with sun overhead, if instead we take the summer months at high latitude the change becomes 0.8 mW/m² per century)

It’s a staggering drop, and no wonder the strong 100,000 year cycle in climate history matching the Milankovitch eccentricity cycles is such a difficult theory to put together.

### Obliquity & Precession

To understand those basics of these changes take a look at the Milankovitch article. Neither of these two effects, precession and obliquity, changes the **total** annual TOA incident solar radiation. They just change its **distribution**.

Here is the last 250,000 years of solar radiation on July 1st – for a few different latitudes:

*Figure 5 – Click for a larger image*

Notice that the equatorial insolation is of course lower than the mid-summer polar insolation.

Here is the same plot but for October 1st. Now the equatorial value is higher:

*Figure 6 - Click for a larger image*

Let’s take a look at the values for 65ºN, often implicated in ice age studies, but this time for the beginning of each month of the year (so the legend is now 1 = January 1st, 2 = Feb 1st, etc):

*Figure 7 - Click for a larger image*

And just for interest I marked one date for the last inter-glacial – the Eemian inter-glacial as it is known.

Come up with a theory:

- peak insolation at 65ºN
- fastest rate of change
- minimum insolation
- average of summer months
- average of winter half year
- average autumn 3 months

Then pick from the graph and let’s start cooking.. Having trouble? Pick a different latitude. Southern Hemisphere – no problem, also welcome.

As we will see, there are a lot of theories, all of which call themselves “Milankovitch” but each one is apparently incompatible with other similarly-named “Milankovitch” theories.

At least we have a tool, kindly supplied by Jonathan Levine, which allows us to compute any value. So if any readers have an output request, just ask.

One word of caution for budding theorists of ice ages (hopefully we have many already) from Kukla et al (2002):

..

The marine isotope record is commonly tuned to astronomic chronology, represented by June insolation at the top of the atmosphere at 60′ or 65′ north latitude. This was deemed justified because the frequency of the Pleistocene gross global climate states matches the frequency of orbital variations....The mechanism of the climate response to insolation remains unclear and the role of insolation in the high latitudes as opposed to that in the low latitudes is still debated..

..In either case, the link between global climates and orbital variations appears to be complicated and not directly controlled by June insolation at latitude 65′N. We strongly discourage dating local climate proxies by unsubstantiated links to astronomic variations..

[Emphasis added].

I’m a novice with the historical records and how they have been constructed, but I understand that SPECMAP is tuned to a Milankovitch theory, i.e., the dates of peak glacials and peak inter-glacials are set by astronomical values.

### Articles in the Series

Part One - An introduction

Part Two – Lorenz - one point of view from the exceptional E.N. Lorenz

Part Three – Hays, Imbrie & Shackleton - how everyone got onto the Milankovitch theory

Part Five – Obliquity & Precession Changes - and in a bit more detail

Part Six – “Hypotheses Abound” - lots of different theories that confusingly go by the same name

Part Seven – GCM I - early work with climate models to try and get “perennial snow cover” at high latitudes to start an ice age around 116,000 years ago

Part Seven and a Half – Mindmap - my mind map at that time, with many of the papers I have been reviewing and categorizing plus key extracts from those papers

Part Eight – GCM II - more recent work from the “noughties” – GCM results plus EMIC (earth models of intermediate complexity) again trying to produce perennial snow cover

Part Nine – GCM III - very recent work from 2012, a full GCM, with reduced spatial resolution and speeding up external forcings by a factors of 10, modeling the last 120 kyrs

Part Ten – GCM IV - very recent work from 2012, a high resolution GCM called CCSM4, producing glacial inception at 115 kyrs

Pop Quiz: End of An Ice Age - a chance for people to test their ideas about whether solar insolation is the factor that ended the last ice age

Eleven – End of the Last Ice age - latest data showing relationship between Southern Hemisphere temperatures, global temperatures and CO2

Twelve – GCM V – Ice Age Termination - very recent work from He et al 2013, using a high resolution GCM (CCSM3) to analyze the end of the last ice age and the complex link between Antarctic and Greenland

Thirteen – Terminator II - looking at the date of Termination II, the end of the penultimate ice age – and implications for the cause of Termination II

### References

Last Interglacial Climates, Kukla et al, *Quaternary Research* (2002)

Modeling the Climatic Response to Orbital Variations, John Imbrie & John Z. Imbrie, *Science* (1980)

## “Blah blah blah” vs Equations

Posted in Basic Science, Commentary on January 30, 2012 | 455 Comments »

It is not surprising that the people most confused about basic physics are the ones who can’t write down an equation for their idea.

The same people are the most passionate defenders of their beliefs and I have no doubts about their sincerity.

I’ll meander into what it is I want to explain..

I found an amazing resource recently –

iTunes Ushort foriTunes University. Now I confess that I have been a little confused about angular momentum. I always knew what it was, but in the small discussion that followed The Coriolis Effect and Geostrophic Motion I found myself wondering whether conservation of angular momentum was something independent of, or a consequence of, linear momentum or some aspect of Newton’s laws of motion.It seemed as if conservation of angular momentum was an orphan of Newton’s three laws of motion. How could that be? Perhaps this conservation is just another expression of these laws in a way that I hadn’t appreciated? (Knowledgeable readers please explain).

Just around this time I found iTunes U and searched for “mechanics” and found the amazing series of lectures from MIT by Prof. Walter Lewin. A series of videos. I recommend them to anyone interested in learning some basics about forces, motion and energy. Lewin has a gift, along with an engaging style. It’s nice to see chalk boards and overhead projectors because they are probably no more in use (? young people please advise).

These lectures are not just for iPhone and iTunes people – here is the weblink.

The gift of teaching science is not in accuracy – that’s a given – the gift is in showing the principle via experiment and matching it with a theoretical derivation, and “why this should be so” and thereby producing a conceptual idea in the student.

I haven’t got to

Lecture 20: Angular Momentumyet, I’m at about lecture 11. It’s basic stuff but so easy to forget (yes, quite a lot of it has been forgotten). Especially easy to forget how different principles link together and which principle is used to derive the next principle.For example, in deriving the work done on an object, Lewin integrates force over the distance traveled and comes up with the equation for kinetic energy.

While investigating the oscillation of a mass on a spring, the equation for its harmonic motion is derived.

Every principle has an equation that can be written down.

Over the last few days, as at many times over the past two years, people have arrived on this blog to explain how radiation from the atmosphere can’t affect the surface temperature because of

blah blah blah. Whereblah blah blahsounds like it might be some kind of physics but is never accompanied by an equation.Here’s the equation I find in textbooks.

Energy absorbed from the atmosphere by the surface, E

_{a}:E

_{a}= αR_{L↓}….[eqn 1]where α = absorptivity of the surface at these wavelengths, R

_{L↓}= downward radiation from the atmosphereAnd this energy absorbed, once absorbed, is indistinguishable from the energy absorbed from the sun. 1 W/m² absorbed from the atmosphere is identical to 1 W/m² absorbed from the sun.

That’s my equation. I have provided six textbooks to explain this idea in a slightly different way in Amazing Things we Find in Textbooks – The Real Second Law of Thermodynamics.

It’s also produced by Kramm & Dlugi, who think the greenhouse effect is some unproven idea:

Now the equation shown is a pretty simple equation. The equation reproduced in the graphic above from Kramm & Dlugi looks a little more daunting but is simply adding up a number of fluxes at the surface.

Here’s what it says:

Solar radiation absorbed + longwave radiation absorbed – thermal radiation emitted – latent heat emitted – sensible heat emitted + geothermal energy supplied = 0

Or another way of thinking about it is

energy in = energy out(written as “energy in – energy out = 0“)Now one thing is not amazing to me - of the tens (hundreds?) of concerned citizens commenting on the many articles on this subject who have tried to point out my “basic mistake” and tell me that the atmosphere can’t

blah blah blah,not a single one has produced an equation.The equation might look something like this:

E

_{a}= f(α,T_{atm}-T_{sur}).R_{L↓}….[eqn 2]where T

_{atm}= temperature of the atmosphere, T_{sur}= temperature of the surfaceWith the function f being defined like this:

f(α,T

_{atm}-T_{sur}) = α, when T_{atm}≥ T_{sur}andf(α,T

_{atm}-T_{sur}) = 0, when T_{atm}< T_{sur}In English, it says something like energy from the atmosphere absorbed by the surface = 0 when the temperature of the atmosphere is less than the temperature of the surface.

I’m filling in the blanks here. No one has written down such ridiculous unphysical nonsense because it would look like ridiculous unphysical nonsense. Or perhaps I’m being unkind. Another possibility is that no one has written down such ridiculous unphysical nonsense because the proponents have no idea what an equation is, or how one can be constructed.

## My Prediction

No one will produce an equation which shows how

noatmospheric energy can be absorbed by the surface. Or how atmospheric energy absorbedcannotaffect internal energy.This is because my next questions will be:

## My Challenge

Here’s my challenge to the many people concerned about the “dangerous nonsense” of the atmospheric radiation affecting surface temperature -

Supply an equation.

If you can’t, it is because you don’t understand the subject.

It won’t stop you talking, but everyone who is wondering and reads this article will be able to join the dots together.

## The Usual Caveat

If there were only two bodies – the warmer earth and the colder atmosphere (no sun available) – then of course the earth’s temperature would decrease towards that of the atmosphere and the atmosphere’s temperature would increase towards that of the earth until both were at the same temperature – somewhere between the two starting temperatures.

However, the sun does actually exist and the question is simply whether the presence of the (colder) atmosphere affects the surface temperature compared with if no atmosphere existed. It is The Three Body Problem.

## My Second Prediction

The people not supplying the equation, the passionate believers in

blah blah blah, will not explain why an equation is not necessary or not available. Instead, continue toblah blah blah.Read Full Post »