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I just do not "get" the "maths" of climate science.
Hello All,
Recently there have been quite a few interesting and involving discussions (well heated debates on various threads really) of
the basic principles of AGW and the present "state" of climate science, and it's "maths" and therefore the "physics" also.
Maxwell vs Planck
A brief glance at AGW science.

Frank Davis
December 28th, 2010

and, here, here, here. Plus too many other places to mention, except this one of course..

For myself there is a recurring theme to all these "discussions" which I just do not "get".
Namely the "maths" of climate science.
I will try to explain my "not understanding" of the "maths" by using a very simple "sum" that we all know the actual answer to.

The "sum".

Half a pint of water at 20 degrees Celsius,
a half pint of water at 40 degrees Celsius,
one pint of water at 30 degrees Celsius.

OK, there are a couple of "efficiency" assumptions in that "sum", but I hope most understand and accept it "as is".
It is, I happily admit, a gross oversimplification to help me illustrate what I want to convey.
The main point of this "sum" is to illustrate that what we see every day, all around us, in every form of heat / energy flows and,
that these are all relative heat / energy flows.
Specifically that heat only goes from hot to cold, never from cold to hot.
Cold can go to hot, and cool the hotter, which is really the same as saying hot can go to cold, and warm the cooler.
The two half pints in the "sum" above show this,
cool cools warmer, and warm warms cooler.
This is also the same as saying, cooler never warms hotter, and hotter never cools cooler, without outside intervention / energy input.

The "sum" shows what we all know from our every day lives and experiences, but
how does climate science tackle the above "sum"?
"Climate science" tackles the "sum" in two distinctly different ways,
"Planck" as described by Frank Davis is the accepted "consensus" view of present "climate science".
"Maxwellian" in Frank Davis's piece describes the "new" way that Claes Johnson has recently proposed.
I "get" niether of these versions of the "maths", as became even more apparent to me on this thread whilst
further discussing the issue/s with Climate Realist. Thank you Climate Realist.
So, I will explain a third version I will call "Relatively" here.

"Relatively" is, well, errr, my understanding of what the "maths" should be, that no one appears to agree with, yet
it is what I see, observe and, experience all around me, every day in the real world, as the above "sum" will be used to illustrate.

The "Planck" calculation of the "sum".
This is the present "AGW paradigm" of climate science view of how the "sum" should be calculated.
This view is most clearly depicted in the Kiehl and Trenberth type global energy budgets, one of which (from NASA ie Gavin Schmidt) is shown below.
The basis of these budgets is that the annual total figures (guesstimates generally) for each flow of energy
is expressed as a per second rate, rather than by any other method.
This method in itself produces some rather peculiar results, that I will try to illustrate using the "sum".
[Image: global_energy_budget_components.png]
Please note this version is a little awkward to use as all the figures have been converted to percentages of 340 W/m2.
(Solar input at 93,5 million miles is actually (about, but constantly varying) 1366W/m2, so
340W/m2 is for a (in constant flux to and fro) saucer shaped disc of the same surface area as planet earth,
but 374 million miles from the sun..)

For example the "absorbed by surface; 48%, means 48% of 340, which is a figure of 163.2 W/m2 absorbed by the surface.
As can also be seen from the plot, another 100% is also absorbed by the surface from "back radiation", which would be 340 W/m2.
The plot also depicts that, 7% is reflected by the surface, so it is niether absorbed or emitted by the surface, so
for the surface sum received and emitted W/m2, that part which is reflected by the surface can be ignored.
There is however 5%, 25%, and 117% emitted by the surface, which totals to 147% emitted by the surface.
The resulting 1% difference is described as 0.9 W/m2 being absorbed by the surface in another version of this type of plot, as shown below.
[Image: radiation_budget_kiehl_trenberth_20.jpg]
In this IPCC version (I believe) of this type of plot the figures in W/m2 are used, which is somewhat easier to understand.

In both of the above versions of the K&T type plots it can be seen that the W/m2 per second rates are simply added together.
This clearly shows how "Planck" calculates.

As some will no doubt be aware I have recently produced a free to all excel workbook in conjunction with a pdf
as available on this thread.
Within the excel workbook, on sheet 4, cells T26 and T27 there is a "calculator".
The answer for an input of P in W/m2 (which you put into cell T26) is in cells W26 (in Kelvin) and cell X26 (in Celsius) and
is for the equation (P in Wm2/5.6704)^0.25*100 = K.
The same applies to cells T27, W27, and X27 respectively on that sheet.

A similar calculator is also contained within sheet 1 of the workbook.
OK, so how does "Planck" calculate? We have a calculator, and examples, let us see.
For ease of calculation I shall use the IPCC example above.
What is the sum used for surface temperature depicted by the plot?
The sum appears to be,
161W/m2 solar radiation absorbed by surface + 333W/m2 back radiation absorbed by the surface.
This would equal 494W/m2, which if calculated using the calculator equals 32.362 degrees celsius surface temperature.
This is too hot, global mean temperature (which should be global near surface mean temperature really...)
is supposed to be about 16 degrees Celsius.
So, what have I missed in this sum so far ?
Alan Siddons stated sometime ago in a private communication to me that,

" The Kiehl-Trenberth budget that you cite assigns 161 W/m² to the surface,
which corresponds to a very frigid minus 42°C.
But back-radiation brings the surface up to 396 W/m², i.e., 16°C.

Obviously 161 + 333 is not 396, there is a 98 difference, how is this accounted for.
Alan Siddons clarified this later in another email, that,

" What you’re missing is that non-radiative heat losses (evaporative cooling and convective uplift) remove a total of 97 from the surface,
leaving it with 64.
NOW... that 64 gets added to the downwelling 333, bringing the surface to 397.
But wait, that’s one unit too many. Right,
1 unit creeps into the surface and is not radiated, as indicated by "net absorbed 0.9." Bizarre.

We are left in no doubt that K&T type plots, and therefore "Planck" "AGW climate science" does indeed add together radiative transfers.
This seems completely at odds with reality.

" Here’s one clue for clarifying Trenberth’s muddle: If an object is already
radiating 100 W/m², then radiating 101 W/m² at it will only raise its output by 1 W/m².
Radiative transfers do not add; transfer occurs according to the difference.
Trenberth’s 333 coming out of the sky, then, could only bring the surface to 333.
For all his weird efforts, he’s STILL left the surface too cold!

To put this in terms of my "sum" to be used here to illustrate, the figures go something like this,
(Please use excel workbook calculators to check)

half a pint of water at 20 degrees Celsius = a rate of emission of 418.74 W/m2.

half a pint of water at 40 degrees Celsius = a rate of emission of 545.25 W/m2.

So my "sum" would be according to "Planck",
418.74 W/m2 + 545.25 W/m2 = 963.99 W/m2 = 87.95 degrees Celsius for our pint of water.....

If we include the same percentage of non-radiative heat losses (evaporative cooling and convective uplift), 97 of 161, 60.25% losses, then,
418.74 becomes 252.29 W/m2, and, 545.25 becomes 328.51 W/m2.
So, my "sum" according to "Planck" in AGW climate science's "real world" is,

252.29 + 328.51 = 580.8 W/m2 = 44.98 degrees Celsius.

We know that is a wrong answer, and by quite some margin, or
margins depending on what "we" allow "Planck" to "correct" for.

Another way some people interpret the "Planck" AGW way to calculate is
to simply add the temperatures together, as the radiative transfers are,
in the approved "Planck" AGW way..
In this case, my "sum" would simply be,
20C + 40C = 60C.

I am sure people who know, will know how to correct my calculations according to "Planck" here, but that is not the point.
I know as well as anyone else emissivity, etc, should be taken into account,
but the point is the K&T type plots, and therefore AGW does not in these plots.
The K&T type plots clearly depict flows from and to gases, liquids and solids at all stages from and to gases, liquids and solids,
without emissivity, etc, being taken into account,
AND that radiative transfers are simply added together.
So, I have calculated my "sum" in the same way,
"Planck" and AGW it seems to me produces the wrong answer/s.
I simply do not "get" "Planck" and AGW "maths" in regards to climate science.

The "Maxwellian" calculation of the "sum".
Professor of applied mathematics Claes Johnson has his own blog.
Claes Johnson on Mathematics and Science
He has explained therein in great depth his works and reasoning in regards to
the problems with the "maths" of climate science at present.
I have only read some of his postings there and have to admit I do not really understand his work in depth.

I have also tried to grasp the basics of the problems Professor Johnson describes, and the solutions he suggests in this paper.
Computational Blackbody radiation.
Claes Johnson
September 16, 2010

For myself however, an easier to understand description of the scale of the problems physics and
therefore the "maths" are having to deal with is described in The Desperation of Planck.
" Modern physics and fluid mechanics was born out of a crisis of classical physics
at the turn into the 20th century caused by the seemingly unsolvable problems:

1. Second Law of Thermodynamics.
2. Cut-off of high-frequency spectrum of black-body radiation.
3. Michelson-Morley experiment showing observer-independent speed of light.
4. Nature of gravitation.
5. D'Alembert's paradox of non-zero drag in fluids with very small viscosity.

The problems were "solved" by introducing new non-classical basic laws:

1. Boltzmann: Molecular chaos.
2. Planck: Smallest quantum of energy.
3. Einstein: Special relativity based on Lorentz transformation of space-time coordinates.
4. Einstein: General relativity based on curved spacetime and equivalence of heavy and inertial mass.
5. Prandtl: Substantial effects from vanishingly thin boundary layers.

The nature of these basic laws is "modern" in the sense that they cannot be verified experimentally.

But how does this all relate to my "sum"? I am not sure, in all honesty.
If I have understood correctly then Claes Johnson's approach is to say that
all radiation received by an object at a lower "temperature" than the object is at can effectively be ignored.
So, the temperature difference is what is used to calculate the difference.
Excerpt from Computational Blackbody radiation.
" in order for energy to be stored as internal/heat energy,
it is required that the incoming radiation energy is bigger than the outgoing.
in effect, emitting IR cools an object by the rate of emission, and volume is included.
However, Computational Blackbody radiation. also states,
" It can also be viewed as a 2nd Law of Radiation stating that
radiative heat transfer is possible only from warmer to cooler.

So, in this "Maxwellian" version of my "sum", we get...
A difference of 20C shared between the two half pints, but
radiative transfer would only occur from the hotter to the cooler.

1/2 pint of water at 20C plus 10C + 1/2 pint of water at 40C minus 10C = 1 pint of water at 30 degrees Celsius.
The right answer.

However, I am not sure at all about the description of heat / energy ONLY going from the hotter to the cooler,
in that it does not allow at all for the cooler object cooling the hotter.
In fact, the descriptions as I understand them say this can not happen.
This is what I do not "get" with the "Maxwellian" maths,
it seems to be the right answer by the wrong route to me.
A sort of overall "net" way of calculating.

Before I started writing this part of the piece, I was somewhat concerned about
how to handle approaching the "Maxwellian" way of calculating, but
the excerpt from The Desperation of Planck above
gave me some encouragement because it appears that no one really knows.
" The nature of these basic laws is "modern" in the sense that they cannot be verified experimentally. "
So questioning the reasoning appears to be perfectly in order, not out of order.

To my mind, given my understanding to date of Professor Claes Johnson's approach
using Computational Blackbody radiation when all has been said and done is that it is a massive,
quantum leap forwards for climate science "maths" compared to the "Planck" approach.
Maybe I should not of used the word quantum there...

My "Relative" calculation of the "sum".
Unsurprisingly this is the "sum" I am least sure about, or rather how to calculate it.
That said, for a pint, and we English are supposed to like warm beer, so warm water ain't so bad really, I will dive in..

As I said at the start of this piece it does seem from every day life that,
hotter warms cooler, AND cooler cools hotter.
Apart from the problems within physics itself as explained above I am not aware of any reason
that is definitely, absolutely, proven that cooler can not, and does not cool hotter, as far as radiative transfer is concerned.
This seems to happen for every other heat flow we can observe, so
if it is true for radiative transfer that cooler can not cool hotter,
as both "Planck" and "Maxwellian" approaches seem to say,
then why has it not been shown to date?
Maybe such has been shown, and I am unaware of it.

The way I look at this calculation is from a "worm's eye" point of view.
If I were a small part of the surface of one of the objects, what would I experience.?
If I was on the hotter objects surface I would presumably experience the feeling of emitting at the objects temperature, or rather energy level.
But I would also be receiving some incoming energy from below and the sides, as I emitted and cooled, or rather lowered my energy level.
Every so often I might receive a lower energy "incoming" from a nearby cooler object.
What would this feel like, cold presumably, it would lower my energy level by
half of the difference between me and the "incoming" allowing for the mass differences between "us".

If I was on the cooler objects surface, I would presumably have a very similar experience, but
with far more incoming hotter, errr, "incomings", so I would warm overall.
But I would also be giving warmth to the sides and below, instead of
receiving from the sides and below as I was on the hotter objects surface.

So, the calculation for my "sum" would look something like,

1/2 pint of water at 20C - emission of energy + received energy from hotter object
1/2 pint of water at 40 C - received cooler energy from cooler object - emission of energy
I pint of water at 30 C.

In the real world radiative transfer between the objects is lost "in transit"
as it is either absorbed by, or deflected away by something else in between the objects.
Nasif Nahle's work in regards of the mean free path length of a photon should be remembered and included,
as also discussed at the Climate Realists blog.

Unfortunately there appears no way to me at present to make my "sum" balance using this approach because,
the emission of the cooler object (which further cools the cooler object), if it cools the hotter object as well, means the hotter object can not emit enough,
without cooling itself too much to raise the cooler objects temperature upto 30C.

For example,
20 - 1 + 11 = 30,
but that means,
40 - 1 - 11 = 28.

I think this might be a further problem for the "Maxwellian" approach as well,
as it possibly means, or could indicate that below cut off is ignored for "other" reasons.

This might be, and is probably nothing more than a "snapshot" or "timing" view point problem.
The objects untill 30C is attained, are constantly in a flux of one mostly cooling and the other mostly warming.
Whereas the answer of a pint of water at 30C is a "net" or overall view point answer.
In the end I have to say that I do not "get" my "Relative" approach either to the "maths" in regard of climate science.

The best conclusion I can arrive at in regard of climate science "maths" is..
Given the three approaches described above, the "Maxwellian" approach of Professor Claes Johnson is
by far and a way the most correct, and best approach for the overall or "net" answer,
however it seems to me that his approach does not necessarily describe how the objects get there correctly.

Attached Files
.pdf   Computational Blackbody Radiation - 16th Sep 2010 - Claes Johnson.pdf (Size: 214.55 KB / Downloads: 394)
.pdf   I just do not get the maths of climate science - pdf..pdf (Size: 271.23 KB / Downloads: 334)
.pdf   Mean Free Path length of a Photon - 12th July 2010 - Nasif Nahle.pdf (Size: 508.12 KB / Downloads: 1,590)
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