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In Part One we had a look at some introductory ideas. In this article we will look at one of the ground-breaking papers in chaos theory – Deterministic nonperiodic flow, Edward Lorenz (1963). It has been cited more than 13,500 times.

There might be some introductory books on non-linear dynamics and chaos that don’t include a discussion of this paper – or at least a mention – but they will be in a small minority.

Lorenz was thinking about convection in the atmosphere, or any fluid heated from below, and reduced the problem to just three simple equations. However, the equations were still non-linear and because of this they exhibit chaotic behavior.

Cencini et al describe Lorenz’s problem:

Consider a fluid, initially at rest, constrained by two infinite horizontal plates maintained at constant temperature and at a fixed distance from each other. Gravity acts on the system perpendicular to the plates. If the upper plate is maintained hotter than the lower one, the fluid remains at rest and in a state of conduction, i.e., a linear temperature gradient establishes between the two plates.

If the temperatures are inverted, gravity induced buoyancy forces tend to rise toward the top the hotter, and thus lighter fluid, that is at the bottom. This tendency is contrasted by viscous and dissipative forces of the fluid so that the conduction state may persist.

However, as the temperature differential exceeds a certain amount, the conduction state is replaced by a steady convection state: the fluid motion consists of steady counter-rotating vortices (rolls) which transport upwards the hot/light fluid in contact with the bottom plate and downwards the cold heavy fluid in contact with the upper one.

The steady convection state remains stable up to another critical temperature difference above which it becomes unsteady, very irregular and hardly predictable.

Willem Malkus and Lou Howard of MIT came up with an equivalent system – the simplest version is shown in this video:

Figure 1

Steven Strogatz (1994), an excellent introduction to dynamic and chaotic systems – explains and derives the equivalence between the classic Lorenz equations and this tilted waterwheel.

L63 (as I’ll call these equations) has three variables apart from time: intensity of convection (x), temperature difference between ascending and descending currents (y), deviation of temperature from a linear profile (z).

Here are some calculated results for L63  for the “classic” parameter values and three very slightly different initial conditions (blue, red, green in each plot) over 5,000 seconds, showing the start and end 50 seconds – click to expand:

Lorenz63-5ksecs-x-y-vs-time-499px

Figure 2 – click to expand – initial conditions x,y,z = 0, 1, 0;  0, 1.001, 0;  0, 1.002, 0

We can see that quite early on the two conditions diverge, and 5000 seconds later the system still exhibits similar “non-periodic” characteristics.

For interest let’s zoom in on just over 10 seconds of ‘x’ near the start and end:

Lorenz63-5ksecs-x-vs-time-zoom-499px

Figure 3

Going back to an important point from the first post, some chaotic systems will have predictable statistics even if the actual state at any future time is impossible to determine (due to uncertainty over the initial conditions).

So we’ll take a look at the statistics via a running average – click to expand:

Lorenz63-5ksecs-x-y-vs-time-average-499px

Figure 4 – click to expand

Two things stand out – first of all the running average over more than 100 “oscillations” still shows a large amount of variability. So at any one time, if we were to calculate the average from our current and historical experience we could easily end up calculating a value that was far from the “long term average”. Second – the “short term” average, if we can call it that, shows large variation at any given time between our slightly divergent initial conditions.

So we might believe – and be correct – that the long term statistics of slightly different initial conditions are identical, yet be fooled in practice.

Of course, surely it sorts itself out over a longer time scale?

I ran the same simulation (with just the first two starting conditions) for 25,000 seconds and then used a filter window of 1,000 seconds – click to expand:

Lorenz63-25ksecs-x-time-1000s-average-499px

 Figure 5 – click to expand

The total variability is less, but we have a similar problem – it’s just lower in magnitude. Again we see that the statistics of two slightly different initial conditions – if we were to view them by the running average at any one time –  are likely to be different even over this much longer time frame.

From this 25,000 second simulation:

  • take 10,000 random samples each of 25 second length and plot a histogram of the means of each sample (the sample means)
  • same again for 100 seconds
  • same again for 500 seconds
  • same again for 3,000 seconds

Repeat for the data from the other initial condition.

Here is the result:

Lorenz-25000s-histogram-of-means-2-conditions

Figure 6

To make it easier to see, here is the difference between the two sets of histograms, normalized by the maximum value in each set:

Lorenz-25000s-delta-histogram

Figure 7

This is a different way of viewing what we saw in figures 4 & 5.

The spread of sample means shrinks as we increase the time period but the difference between the two data sets doesn’t seem to disappear (note 2).

Attractors and Phase Space

The above plots show how variables change with time. There’s another way to view the evolution of system dynamics and that is by “phase space”. It’s a name for a different kind of plot.

So instead of plotting x vs time, y vs time and z vs time – let’s plot x vs y vs z – click to expand:

Lorenz63-first-50s-x-y-z-499px

Figure 8 – Click to expand – the colors blue, red & green represent the same initial conditions as in figure 2

Without some dynamic animation we can’t now tell how fast the system evolves. But we learn something else that turns out to be quite amazing. The system always end up on the same “phase space”. Perhaps that doesn’t seem amazing yet..

Figure 7 was with three initial conditions that are almost identical. Let’s look at three initial conditions that are very different: x,y,z = 0, 1, 0;   5, 5, 5;   20, 8, 1:

Lorenz63-first-50s-x-y-z-3-different-conditions-499px

Figure 9 - Click to expand

Here’s an example (similar to figure 7) from Strogatz – a set of 10,000 closely separated initial conditions and how they separate at 3, 6, 9 and 15 seconds. The two key points:

  1. the fast separation of initial conditions
  2. the long term position of any of the initial conditions is still on the “attractor”
From Strogatz 1994

From Strogatz 1994

Figure 10

A dynamic visualization on Youtube with 500,000 initial conditions:

Figure 11

There’s lot of theory around all of this as you might expect. But in brief, in a “dissipative system” the “phase volume” contracts exponentially to zero. Yet for the Lorenz system somehow it doesn’t quite manage that. Instead, there are an infinite number of 2-d surfaces. Or something. For the sake of a not overly complex discussion a wide range of initial conditions ends up on something very close to a 2-d surface.

This is known as a strange attractor. And the Lorenz strange attractor looks like a butterfly.

Conclusion

Lorenz 1963 reduced convective flow (e.g., heating an atmosphere from the bottom) to a simple set of equations. Obviously these equations are a massively over-simplified version of anything like the real atmosphere. Yet, even with this very simple set of equations we find chaotic behavior.

Chaotic behavior in this example means:

  • very small differences get amplified extremely quickly so that no matter how much you increase your knowledge of your starting conditions it doesn’t help much (note 3)
  • starting conditions within certain boundaries will always end up within “attractor” boundaries, even though there might be non-periodic oscillations around this attractor
  • the long term (infinite) statistics can be deterministic but over any “smaller” time period the statistics can be highly variable

References

Deterministic nonperiodic flow, EN Lorenz, Journal of the Atmospheric Sciences (1963)

Chaos: From Simple Models to Complex Systems, Cencini, Cecconi & Vulpiani, Series on Advances in Statistical Mechanics – Vol. 17 (2010)

Non Linear Dynamics and Chaos, Steven H. Strogatz, Perseus Books  (1994)

Notes

Note 1: The Lorenz equations:

dx/dt = σ (y-x)

dy/dt = rx – y – xz

dz/dt = xy – bz

where

x = intensity of convection

y = temperature difference between ascending and descending currents

z = devision of temperature from a linear profile

σ = Prandtl number, ratio of momentum diffusivity to thermal diffusivity

r = Rayleigh number

b = “another parameter”

And the “classic parameters” are σ=10, b = 8/3, r = 28

Note 2: Lorenz 1963 has over 13,000 citations so I haven’t been able to find out if this system of equations is transitive or intransitive. Running Matlab on a home Mac reaches some limitations and I maxed out at 25,000 second simulations mapped onto a 0.01 second time step.

However, I’m not trying to prove anything specifically about the Lorenz 1963 equations, more illustrating some important characteristics of chaotic systems

Note 3: Small differences in initial conditions grow exponentially, until we reach the limits of the attractor. So it’s easy to show the “benefit” of more accurate data on initial conditions.

If we increase our precision on initial conditions by 1,000,000 times the increase in prediction time is a massive 2½ times longer.

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There are many classes of systems but in the climate blogosphere world two ideas about climate seem to be repeated the most.

In camp A:

We can’t forecast the weather two weeks ahead so what chance have we got of forecasting climate 100 years from now.

And in camp B:

Weather is an initial value problem, whereas climate is a boundary value problem. On the timescale of decades, every planetary object has a mean temperature mainly given by the power of its star according to Stefan-Boltzmann’s law combined with the greenhouse effect. If the sources and sinks of CO2 were chaotic and could quickly release and sequester large fractions of gas perhaps the climate could be chaotic. Weather is chaotic, climate is not.

Of course, like any complex debate, simplified statements don’t really help. So this article kicks off with some introductory basics.

Many inhabitants of the climate blogosphere already know the answer to this subject and with much conviction. A reminder for new readers that on this blog opinions are not so interesting, although occasionally entertaining. So instead, try to explain what evidence is there for your opinion. And, as suggested in About this Blog:

And sometimes others put forward points of view or “facts” that are obviously wrong and easily refuted.  Pretend for a moment that they aren’t part of an evil empire of disinformation and think how best to explain the error in an inoffensive way.

Pendulums

The equation for a simple pendulum is “non-linear”, although there is a simplified version of the equation, often used in introductions, which is linear. However, the number of variables involved is only two:

  • angle
  • speed

and this isn’t enough to create a “chaotic” system.

If we have a double pendulum, one pendulum attached at the bottom of another pendulum, we do get a chaotic system. There are some nice visual simulations around, which St. Google might help interested readers find.

If we have a forced damped pendulum like this one:

Pendulum-forced

Figure 1 – the blue arrows indicate that the point O is being driven up and down by an external force

-we also get a chaotic system.

What am I talking about? What is linear & non-linear? What is a “chaotic system”?

Digression on Non-Linearity for Non-Technical People

Common experience teaches us about linearity. If I pick up an apple in the supermarket it weighs about 0.15 kg or 150 grams (also known in some countries as “about 5 ounces”). If I take 10 apples the collection weighs 1.5 kg. That’s pretty simple stuff. Most of our real world experience follows this linearity and so we expect it.

On the other hand, if I was near a very cold black surface held at 170K (-103ºC) and measured the radiation emitted it would be 47 W/m². Then we double the temperature of this surface to 340K (67ºC) what would I measure? 94 W/m²? Seems reasonable – double the absolute temperature and get double the radiation.. But it’s not correct.

The right answer is 758 W/m², which is 16x the amount. Surprising, but most actual physics, engineering and chemistry is like this. Double a quantity and you don’t get double the result.

It gets more confusing when we consider the interaction of other variables.

Let’s take riding a bike [updated thanks to Pekka]. Once you get above a certain speed most of the resistance comes from the wind so we will focus on that. Typically the wind resistance increases as the square of the speed. So if you double your speed you get four times the wind resistance. Work done = force x distance moved, so with no head wind power input has to go up as the cube of speed (note 4). This means you have to put in 8x the effort to get 2x the speed.

On Sunday you go for a ride and the wind speed is zero. You get to 25 km/hr (16 miles/hr) by putting a bit of effort in – let’s say you are producing 150W of power (I have no idea what the right amount is). You want your new speedo to register 50 km/hr – so you have to produce 1,200W.

On Monday you go for a ride and the wind speed is 20 km/hr into your face. Probably should have taken the day off.. Now with 150W you get to only 14 km/hr, it takes almost 500W to get to your basic 25 km/hr, and to get to 50 km/hr it takes almost 2,400W. No chance of getting to that speed!

On Tuesday you go for a ride and the wind speed is the same so you go in the opposite direction and take the train home. Now with only 6W you get to go 25 km/hr, to get to 50km/hr you only need to pump out 430W.

In mathematical terms it’s quite simple: F = k(v-w)², Force = (a constant, k) x (road speed – wind speed) squared. Power, P = Fv = kv(v-w)². But notice that the effect of the “other variable”, the wind speed, has really complicated things.

To double your speed on the first day you had to produce eight times the power. To double your speed the second day you had to produce almost five times the power. To double your speed the third day you had to produce just over 70 times the power. All with the same physics.

The real problem with nonlinearity isn’t the problem of keeping track of these kind of numbers. You get used to the fact that real science – real world relationships – has these kind of factors and you come to expect them. And you have an equation that makes calculating them easy. And you have computers to do the work.

No, the real problem with non-linearity (the real world) is that many of these equations link together and solving them is very difficult and often only possible using “numerical methods”.

It is also the reason why something like climate feedback is very difficult to measure. Imagine measuring the change in power required to double speed on the Monday. It’s almost 5x, so you might think the relationship is something like the square of speed. On Tuesday it’s about 70 times, so you would come up with a completely different relationship. In this simple case know that wind speed is a factor, we can measure it, and so we can “factor it out” when we do the calculation. But in a more complicated system, if you don’t know the “confounding variables”, or the relationships, what are you measuring? We will return to this question later.

When you start out doing maths, physics, engineering.. you do “linear equations”. These teach you how to use the tools of the trade. You solve equations. You rearrange relationships using equations and mathematical tricks, and these rearranged equations give you insight into how things work. It’s amazing. But then you move to “nonlinear” equations, aka the real world, which turns out to be mostly insoluble. So nonlinear isn’t something special, it’s normal. Linear is special. You don’t usually get it.

..End of digression

Back to Pendulums

Let’s take a closer look at a forced damped pendulum. Damped, in physics terms, just means there is something opposing the movement. We have friction from the air and so over time the pendulum slows down and stops. That’s pretty simple. And not chaotic. And not interesting.

So we need something to keep it moving. We drive the pivot point at the top up and down and now we have a forced damped pendulum. The equation that results (note 1) has the massive number of three variables – position, speed and now time to keep track of the driving up and down of the pivot point. Three variables seems to be the minimum to create a chaotic system (note 2).

As we increase the ratio of the forcing amplitude to the length of the pendulum (β in note 1) we can move through three distinct types of response:

  • simple response
  • a “chaotic start” followed by a deterministic oscillation
  • a chaotic system

This is typical of chaotic systems – certain parameter values or combinations of parameters can move the system between quite different states.

Here is a plot (note 3) of position vs time for the chaotic system, β=0.7, with two initial conditions, only different from each other by 0.1%:

Forced damped harmonic pendulum, b=0.7: Start angular speed 0.1; 0.1001

Forced damped harmonic pendulum, b=0.7: Start angular speed 0.1; 0.1001

Figure 1

It’s a little misleading to view the angle like this because it is in radians and so needs to be mapped between 0-2π (but then we get a discontinuity on a graph that doesn’t match the real world). We can map the graph onto a cylinder plot but it’s a mess of reds and blues.

Another way of looking at the data is via the statistics – so here is a histogram of the position (θ), mapped to 0-2π, and angular speed (dθ/dt) for the two starting conditions over the first 10,000 seconds:

Histograms for 10k seconds

Histograms for 10,000 seconds

Figure 2

We can see they are similar but not identical (note the different scales on the y-axis).

That might be due to the shortness of the run, so here are the results over 100,000 seconds:

Pendulum-0.7-100k seconds-2 conditions-hist

Histogram for 100,000 seconds

Figure 3

As we increase the timespan of the simulation the statistics of two slightly different initial conditions become more alike.

So if we want to know the state of a chaotic system at some point in the future, very small changes in the initial conditions will amplify over time, making the result unknowable – or no different from picking the state from a random time in the future. But if we look at the statistics of the results we might find that they are very predictable. This is typical of many (but not all) chaotic systems.

Orbits of the Planets

The orbits of the planets in the solar system are chaotic. In fact, even 3-body systems moving under gravitational attraction have chaotic behavior. So how did we land a man on the moon? This raises the interesting questions of timescales and amount of variation. Planetary movement – for our purposes – is extremely predictable over a few million years. But over 10s of millions of years we might have trouble predicting exactly the shape of the earth’s orbit – eccentricity, time of closest approach to the sun, obliquity.

However, it seems that even over a much longer time period the planets will still continue in their orbits – they won’t crash into the sun or escape the solar system. So here we see another important aspect of some chaotic systems – the “chaotic region” can be quite restricted. So chaos doesn’t mean unbounded.

According to Cencini, Cecconi & Vulpiani (2010):

Therefore, in principle, the Solar system can be chaotic, but not necessarily this implies events such as collisions or escaping planets..

However, there is evidence that the Solar system is “astronomically” stable, in the sense that the 8 largest planets seem to remain bound to the Sun in low eccentricity and low inclination orbits for time of the order of a billion years. In this respect, chaos mostly manifest in the irregular behavior of the eccentricity and inclination of the less massive planets, Mercury and Mars. Such variations are not large enough to provoke catastrophic events before extremely large time. For instance, recent numerical investigations show that for catastrophic events, such as “collisions” between Mercury and Venus or Mercury failure into the Sun, we should wait at least a billion years.

And bad luck, Pluto.

Deterministic, non-Chaotic, Systems with Uncertainty

Just to round out the picture a little, even if a system is not chaotic and is deterministic we might lack sufficient knowledge to be able to make useful predictions. If you take a look at figure 3 in Ensemble Forecasting you can see that with some uncertainty of the initial velocity and a key parameter the resulting velocity of an extremely simple system has quite a large uncertainty associated with it.

This case is quantitively different of course. By obtaining more accurate values of the starting conditions and the key parameters we can reduce our uncertainty. Small disturbances don’t grow over time to the point where our calculation of a future condition might as well just be selected from a randomly time in the future.

Transitive, Intransitive and “Almost Intransitive” Systems

Many chaotic systems have deterministic statistics. So we don’t know the future state beyond a certain time. But we do know that a particular position, or other “state” of the system, will be between a given range for x% of the time, taken over a “long enough” timescale. These are transitive systems.

Other chaotic systems can be intransitive. That is, for a very slight change in initial conditions we can have a different set of long term statistics. So the system has no “statistical” predictability. Lorenz 1968 gives a good example.

Lorenz introduces the concept of almost intransitive systems. This is where, strictly speaking, the statistics over infinite time are independent of the initial conditions, but the statistics over “long time periods” are dependent on the initial conditions. And so he also looks at the interesting case (Lorenz 1990) of moving between states of the system (seasons), where we can think of the precise starting conditions each time we move into a new season moving us into a different set of long term statistics. I find it hard to explain this clearly in one paragraph, but Lorenz’s papers are very readable.

Conclusion?

This is just a brief look at some of the basic ideas.

Other Articles in the Series

Part Two – Lorenz 1963

References

Chaos: From Simple Models to Complex Systems, Cencini, Cecconi & Vulpiani, Series on Advances in Statistical Mechanics – Vol. 17 (2010)

Climatic Determinism, Edward Lorenz (1968) – free paper

Can chaos and intransivity lead to interannual variation, Edward Lorenz, Tellus (1990) – free paper

Notes

Note 1 – The equation is easiest to “manage” after the original parameters are transformed so that tω->t. That is, the period of external driving, T0=2π under the transformed time base.

Then:

Pendulum-forced-equation

where θ = angle, γ’ = γ/ω, α = g/Lω², β =h0/L;

these parameters based on γ = viscous drag coefficient, ω = angular speed of driving, g = acceleration due to gravity = 9.8m/s², L = length of pendulum, h0=amplitude of driving of pivot point

Note 2 – This is true for continuous systems. Discrete systems can be chaotic with less parameters

Note 3 – The results were calculated numerically using Matlab’s ODE (ordinary differential equation) solver, ode45.

Note 4 – Force = k(v-w)2 where k is a constant, v=velocity, w=wind speed. Work done = Force x distance moved so Power, P = Force x velocity.

Therefore:

P = kv(v-w)2

If we know k, v & w we can find P. If we have P, k & w and want to find v it is a cubic equation that needs solving.

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In The “Greenhouse” Effect Explained in Simple Terms I list, and briefly explain, the main items that create the “greenhouse” effect. I also explain why more CO2 (and other GHGs) will, all other things remaining equal, increase the surface temperature. I recommend that article as the place to go for the straightforward explanation of the “greenhouse” effect. It also highlights that the radiative balance higher up in the troposphere is the most important component of the “greenhouse” effect.

However, someone recently commented on my first Kramm & Dlugi article and said I was “plainly wrong”. Kramm & Dlugi were in complete agreement with Gerlich and Tscheuschner because they both claim the “purported greenhouse effect simply doesn’t exist in the real world”.

If it’s just about flying a flag or wearing a football jersey then I couldn’t agree more. However, science does rely on tedious detail and “facts” rather than football jerseys. As I pointed out in New Theory Proves AGW Wrong! two contradictory theories don’t add up to two theories making the same case..

In the case of the first Kramm & Dlugi article I highlighted one point only. It wasn’t their main point. It wasn’t their minor point. They weren’t even making a point of it at all.

Many people believe the “greenhouse” effect violates the second law of thermodynamics, these are herein called “the illuminati”.

Kramm & Dlugi’s equation demonstrates that the illuminati are wrong. I thought this was worth pointing out.

The “illuminati” don’t understand entropy, can’t provide an equation for entropy, or even demonstrate the flaw in the simplest example of why the greenhouse effect is not in violation of the second law of thermodynamics. Therefore, it is necessary to highlight the (published) disagreement between celebrated champions of the illuminati – even if their demonstration of the disagreement was unintentional.

Let’s take a look.

Here is the one of the most popular G&T graphics in the blogosphere:

From Gerlich & Tscheuschner

From Gerlich & Tscheuschner

Figure 1

It’s difficult to know how to criticize an imaginary diagram. We could, for example, point out that it is imaginary. But that would be picky.

We could say that no one draws this diagram in atmospheric physics. That should be sufficient. But as so many of the illuminati have learnt their application of the second law of thermodynamics to the atmosphere from this fictitious diagram I feel the need to press forward a little.

Here is an extract from a widely-used undergraduate textbook on heat transfer, with a little annotation (red & blue):

From Incropera & DeWitt (2007)

From “Fundamentals of Heat and Mass Transfer” by Incropera & DeWitt (2007)

Figure 2

This is the actual textbook, before the Gerlich manoeuvre as I would like to describe it. We can see in the diagram and in the text that radiation travels both ways and there is a net transfer which is from the hotter to the colder. The term “net” is not really capable of being confused. It means one minus the other, “x-y”. Not “x”. (For extracts from six heat transfer textbooks and their equations read Amazing Things we Find in Textbooks – The Real Second Law of Thermodynamics).

Now let’s apply the Gerlich manoeuvre (compare fig. 2):

Fundamentals-of-heat-and-mass-transfer-post-G&T

Not from “Fundamentals of Heat and Mass Transfer”, or from any textbook ever

Figure 3

So hopefully that’s clear. Proof by parody. This is “now” a perpetual motion machine and so heat transfer textbooks are wrong. All of them. Somehow.

Just for comparison, we can review the globally annually averaged values of energy transfer in the atmosphere, including radiation, from Kiehl & Trenberth (I use the 1997 version because it is so familiar even though values were updated more recently):

From Kiehl & Trenberth (1997)

From Kiehl & Trenberth (1997)

Figure 4

It should be clear that the radiation from the hotter surface is higher than the radiation from the colder atmosphere. If anyone wants this explained, please ask.

I could apply the Gerlich manoeuvre to this diagram but they’ve already done that in their paper (as shown above in figure 1).

So lastly, we return to Kramm & Dlugi, and their “not even tiny point”, which nevertheless makes a useful point. They don’t provide a diagram, they provide an equation for energy balance at the surface – and I highlight each term in the equation to assist the less mathematically inclined:

Kramm-Dlugi-2011-eqn-highlight

 

Figure 5

The equation says, the sum of all fluxes – at one point on the surface = 0. This is an application of the famous first law of thermodynamics, that is, energy cannot be created or destroyed.

The red term – absorbed atmospheric radiation – is the radiation from the colder atmosphere absorbed by the hotter surface. This is also known as “DLR” or “downward longwave radiation, and as “back-radiation”.

Now, let’s assume that the atmospheric radiation increases in intensity over a small period. What happens?

The only way this equation can continue to be true is for one or more of the last 4 terms to increase.

  • The emitted surface radiation – can only increase if the surface temperature increases
  • The latent heat transfer – can only increase if there is an increase in wind speed or in the humidity differential between the surface and the atmosphere just above
  • The sensible heat transfer – can only increase if there is an increase in wind speed or in the temperature differential between the surface and the atmosphere just above
  • The heat transfer into the ground – can only increase if the surface temperature increases or the temperature below ground spontaneously cools

So, when atmospheric radiation increases the surface temperature must increase (or amazingly the humidity differential spontaneously increases to balance, but without a surface temperature change). According to G&T and the illuminati this surface temperature increase is impossible. According to Kramm & Dlugi, this is inevitable.

I would love it for Gerlich or Tscheuschner to show up and confirm (or deny?):

  • yes the atmosphere does emit thermal radiation
  • yes the surface of the earth does absorb atmospheric thermal radiation
  • yes this energy does not disappear (1st law of thermodynamics)
  • yes this energy must increase the temperature of the earth’s surface above what it would be if this radiation did not exist (1st law of thermodynamics)

Or even, which one of the above is wrong. That would be outstanding.

Of course, I know they won’t do that – even though I’m certain they believe all of the above points. (Likewise, Kramm & Dlugi won’t answer the question I have posed of them).

Well, we all know why

Hopefully, the illuminati can contact Kramm & Dlugi and explain to them where they went wrong. I have my doubts that any of the illuminati have grasped the first law of thermodynamics or the equation for temperature change and heat capacity, but who could say.

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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 U short for iTunes 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 Momentum yet, 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.

What caught my attention for the purposes of this article was how every principle had an equation.

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. Where blah blah blah sounds 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, Ea:

Ea = αRL↓ ….[eqn 1]

where α = absorptivity of the surface at these wavelengths, RL↓ = downward radiation from the atmosphere

And 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:

Ea = f(α,Tatm-Tsur).RL↓ ….[eqn 2]
where Tatm = temperature of the atmosphere, Tsur = temperature of the surface

With the function f being defined like this:

f(α,Tatm-Tsur) = α, when Tatm ≥ Tsur and

f(α,Tatm-Tsur) = 0, when Tatm < Tsur

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 no atmospheric energy can be absorbed by the surface. Or how atmospheric energy absorbed cannot affect internal energy.

This is because my next questions will be:

  1. Please supply a textbook or paper with this equation
  2. Please explain from fundamental physics how this can take place

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 to blah blah blah.

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The Rotational Effect

Climate scientists think that the rotation of the earth is responsible for a lot of the atmospheric and ocean effects that we see. In fact, most climate scientists think it is easy to prove. (Although not as simple as proving that radiatively-active gases affect the climate).

Now suppose the earth’s rotation speed was reducing by X% per year as a result of some important human activity (just suppose, for the sake of this mental exercise) and had been for 100 years or so.

Then atmospheric physics papers and textbooks would comment on the effect of the current speed of rotation of the planet – quantifying its effect by analyzing what climate would be like without rotation. This would be just as an introduction to the effect of rotation on climate. Let’s say that the mean annual equator-arctic temperature differential is currently 35°C (I haven’t checked the exact value) but without rotation it might be thought to be 45°C. So we will describe the rotational effect as being responsible for a 10°C arctic-equatorial temperature differential.

More specifically the rotational effect might be quantified as the number of petawatts of equatorial to polar heat transported vs the value calculated for a “no rotational” earth. But by way of introduction the temperature differential is an easier value to grasp than the change in petawatts.

Various researchers would attempt to calculate the much smaller changes likely to occur in the climate as a result of the rotational changes that might take place over the next 10-20 years. They would use GCMs and other models that would be exactly like the current ones.

And of course there would be many justifiable questions about how accurate the models are – like now.

And many from the general public, not understanding how to follow the equations of motion in rotational frames, or the thermal wind equation, or Ekman pumping, or baroclinic instability, or pretty much anything relating to atmospheric & ocean dynamics might start saying:

The rotational effect doesn’t exist

Many of these people would be skeptical about the small changes to climate that could result from an impercetible change in the rotation rate.

Many blogs would spring up with people using hand-waving arguments about the climatic effects of rotation being vastly overstated.

Other blogs would write that climate science makes massively simplistic assumptions in its calculations and uses the geostrophic balance as its complete formula for climate dynamics. Many other people unencumbered with any knowledge from climate science textbooks, or any desire to read one, would curiously label themselves as skeptics and happily repeat these “facts” without ever checking them.

People with some scientific qualifications, but without solid understanding of the complete field of oceanic or atmospheric dynamics, would write poor quality papers explaining how the rotational effect was much less than climate science calculated and produce some incomplete or incorrectly derived equations to demonstrate this.

These scientists and their new work would be lauded by many blogs as being free from the simplistic assumptions that has dogged climate science and yes, finally, accurate and high quality work has been done!

Other blogs would claim that climate science was ignoring the huge effects of absorption and emission of radiation on the climate.

Then some more serious scientists would come along and write lengthy papers to argue that the rotational effect as defined by climate science does not exist because the “no rotation” result is incorrectly defined, or is not possible to accurately calculate.

Papers of incalculable value.

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In Kramm & Dlugi On Illuminating the Confusion of the Unclear I pointed out that the authors of Scrutinizing the atmospheric greenhouse effect and its climatic impact are in agreement with climate science on the subject of “back radiation” from the atmosphere contributing to the surface temperature.

No surprise to people familiar with the basics of radiative heat transfer. However, Kramm & Dlugi are apparently “in support of” Gerlich & Tscheuschner, who famously proposed that radiation from the atmosphere affecting the temperature of the ground was a violation of the second law of thermodynamics. A perpetual motion machine or something. (Or they were having a big laugh). For more on the exciting adventures of Gerlich & Tscheuschner, read On the Miseducation of the Uninformed..

The first article on the Kramm & Dlugi paper was short, highlighting that one essential point.

Given the enthusiasm that new papers which “cast doubt” on the inappropriately-named “greenhouse” effect are lapped up by the blogosphere, I thought it was worth explaining a few things from their complete paper.

If I sum it up in simple terms, it is a paper which will annoy climate scientists and add confusion to scientifically less clear folk who wonder about the “greenhouse” effect.

And mostly, I have to say, without actually being wrong – or not technically wrong (note 1). This is its genius. Let’s see how they “dodge the bullet” of apparently slaying the “greenhouse” effect without actually contradicting anything of real significance in climate science.

Goody & Yung’s Big Mistake

Regular readers of this blog will know that I have a huge respect for Richard M. Goody, who wrote the seminal Atmospheric Radiation: Theoretical Basis in 1964. (The 2nd edition from 1989 is coauthored by Goody & Yung).

However, they have a mistake in a graph on p.4:

Kramm & Dlugi say:

..This figure also shows the atmospheric absorption spectrum for a solar beam reaching the ground level (b) and the same for a beam reaching the temperate tropopause (c) adopted from Goody and Yung [30]. Part (a) of Figure 5 completely differs from the original twin-peak diagram of Goody and Yung. We share the argument of Gerlich and Tscheuschner [2,4] that the original one is physically misleading..

I have the same argument about this one graph from Goody & Yung’s textbook. You can see my equivalent graph in 4th & 5th figures of The Sun and Max Planck Agree – Part Two.

There is nothing in the development of theory by Goody & Yung that depends on this graph. Kramm & Dlugi don’t demonstrate anything else in error from Goody & Yung. However, I’m sure that someone who wants to devote enough time to the subject will probably find another error in their book, or at least, an incautious statement that could imply that they have carelessly tossed away their knowledge of basic physics. This is left as an exercise for the interested reader..

To clarify the idea for readers – the energy emitted by the climate system to space is approximately equal to the energy absorbed from the sun by the climate system. This is not in dispute.

Kramm & Dlugi point out that one should be careful when attempting to plot equal areas on logarithmic graphs. Nice point.

Kepler & Milankovitch

Kramm & Dugli spend some time deriving the equations of planetary motion. These had been lost by climate science so it is good to see them recovered.

They also comment on Milankovitch’s theory in terms that are interesting:

Thus, on long-term scales of many thousands of years (expressed in kyr) we have to pay attention to Milankovitch’s [33] astronomical theory of climatic variations that ranks as the most important achievement in the theory of climate in the 20th century [10].

The theory definitely has a lot of mainstream support as being the explanation for the ice ages. However, as a comment to be developed one day when I understand enough to write about it, there isn’t one Milankovitch theory, there are many, and of necessity they contradict each other.

Interesting as well to suggest it as the most important achievement in the theory of climate last century – as the consequence of accepting Milankovitch’s theory is that climate is very sensitive to small peturbations in radiative changes in particular regions at particular times. In essence, the Milankovitch theory appears to rely on quite a high climate sensitivity.

Anyway, I’m not criticizing Kramm & Dugli or saying they are wrong. It’s just an interesting comment. And excellent that Kepler’s theories are no longer lost to the world of climate science.

Energy Conversion in the Atmosphere & at the Surface

The authors devote some time to this study (with no apparent differences to standard climate science) with the conclusion:

..Note that the local flux quantities like Q(θ, φ), H(θ, φ), G(θ, φ) and RL↑(θ, φ) are required to calculate global averages of these fluxes, but not global averages of respective values of temperature and humidity.

An important point.

They also confirm – as noted in Kramm & Dlugi On Illuminating the Confusion of the Unclear – that the energy balance at the surface is affected by the energy radiated by the atmosphere. Just helping out the many blog writers and blog commenters – be sure to strike Kramm & Dlugi off your list of advocates of the imaginary second law of thermodynamics.

The Gulags for Everyone? – Climatology Loses Its Rational Basis

The authors cite this extract from the WMO website about the “greenhouse” effect:

In the atmosphere, not all radiation emitted by the Earth surface reaches the outer space. Part of it is reflected back to the Earth surface by the atmosphere (greenhouse effect) leading to a global average temperature of about 14°C well above –19°C which would have been felt without this effect.

This website statement is incorrect as the radiation emitted by the Earth’s surface is absorbed and re-emitted by the atmosphere – not reflected. This is a very basic error.

Kramm & Dlugi say:

Note that the argument that “part of it is reflected back to the Earth surface by the atmosphere” is completely irrational from a physical point of view. Such an argument also indicates that the discipline of climatology has lost its rational basis. Thus, the explanation of the WMO is rejected..

[Emphasis added]

Well, we could argue that if one person writing a website for one body writes one thing that is not technically correct then that whole discipline has lost its rational basis. We could.

Seems uncharitable to me. Although I have to confess that on occasion I am a little bit uncharitable. I wrote that Gerlich & Tscheuschner had lost their marbles, or were having a big laugh, with their many ridiculous and unfounded statements. We all have our off days.

I think if we want to uphold high standards of defendable technical accuracy we would say that the person that wrote this website and the person that reviewed this website are not technically sound as far as the specifics of radiative physics go. I’m hard pressed to think it is justified to cast stones at say Prof. Richard M Goody for this particular travesty. Or Prof. R. Lindzen. Or Prof. V. Ramanathan. Or Prof. F.W. Taylor. Otherwise it might be a bit like Stalin with the Gulag. Everyone and their mother gets tarred with the sins of the fellow down the road and 30 million people wind up digging rocks out of the ground in a very cold place..

But let’s stay on topic. If indeed there is one.

The Main Point

Now that we have found a graph in Goody that is wrong, a website that has a mistake and have rediscovered Kepler’s equations of motion, we turn to the main course.

Kramm & Dlugi turn to perhaps their main point, about the surface temperature of the earth with and without radiatively-active gases.

As a clarification for newcomers, average temperature has many problems. Due to the non-linearity of radiative physics, if we calculate the average radiation from the average temperature we will get a different answer compared with calculating the radiation from the temperature at each location/time and then taking the average.

For more on this basic topic see under the subheading How to Average in Why Global Mean Surface Temperature Should be Relegated, Or Mostly Ignored

First citing Lacis et al:

The difference between the nominal global mean surface temperature (TS = 288 K) and the global mean effective temperature (TE = 255 K) is a common measure of the terrestrial greenhouse effect (GT = TS – TE = 33 K).

The authors develop some maths, of which this is just a sample:

Using Eq. 3.8 and ignoring G(θ,φ) will lead to:

<Ts> = 23/2Te/5 ≈ 144K (3.9)

for a non-rotating Earth in the absence of its atmosphere, if S = 1367 W/m² , α (Θ0, θ, φ) = αE = 0.30 and ε(θ, φ) = ε = 1 are assumed [2]

Ts = 153 K if αE = 0.12 and Ts = 155 K if αE = 0.07

It might surprise readers that these particular points are not something novel or in contradiction to the “greenhouse” effect. In fact, you can see similar points in two articles (at least) on this blog:

- In The Hoover Incident we had a look at what would happen to the climate if all the radiatively-active gases (= “greenhouse” gases) were removed from the atmosphere. Here is an extract:

..And depending on the ice sheet extent and whether any clouds still existed the value of outgoing radiation might be around 1.0 – 1.5 x 1017 W. This upper value would depend on the ice sheets not growing and all the clouds disappearing which seems impossible, but it’s just for illustration.

Remember that nothing in all this time can stop the emitted radiation from the surface making it to space. So the only changes in the energy balance can come from changes to the earth’s albedo (affecting absorbed solar radiation).

And given that when objects emit more energy than they absorb they cool down, the earth will certainly cool. The atmosphere cannot emit any radiation so any atmospheric changes will only change the distribution of energy around the climate system.

What would the temperature of the earth be? I have no idea..

Notice the heresy that without “greenhouse” gases we can’t say for sure what the surface temperature would be.. (It’s definitely going to be significantly lower though).

- In Atmospheric Radiation and the “Greenhouse” Effect – Part One:

..The average for 2009 [of outgoing longwave radiation] is 239 W/m². This average includes days, nights and weekends. The average can be converted to the total energy emitted from the climate system over a year like this:

Total energy radiated by the climate system into space in one year = 239 x number of seconds in a year x area of the earth in meters squared..

ETOA= 3.8 x 1024 J

The reason for calculating the total energy in 2009 is because many people have realized that there is a problem with average temperatures and imagine that this problem is carried over to average radiation. Not true. We can take average radiation and convert it into total energy with no problem..

[Emphasis added]

The point here is that the total emitted top of atmosphere radiation is much lower than the total surface emitted radiation. It can be calculated. In that article I haven’t actually attempted to do it accurately – it would require some work (spatial and temporal temperature across a year and the longwave emissivity of the surface around the globe) – it is a straightforward yet tedious calculation. (See note 2).

A note in passing that this difference between the top of atmosphere radiation and the surface radiation is also derided by the internet imaginary second law advocates as being a physical impossibility because it “creates energy”.

Now I am not in any way a “representative of climate science” despite the many claims to this effect, it’s just that the basics are.. the basics. And radiative transfer in the atmosphere is a technical yet simple subject which can be easily solved with the aid of some decent computing power. So I have no quarrel with anything of substance that I have so far read in textbooks or papers on radiative physics. Yet I appear to have stated similar points to Kramm & Dlugi.

Perhaps Kramm & Dlugi have not yet stated anything controversial on the inappropriately-named “greenhouse” effect.

They take issue with what I would call the “introduction to the greenhouse effect” where a simple comparison is drawn. This is where the “greenhouse” effect is highlighted as “effective temperature”.

It could more accurately be highlighted as “difference in average flux between surface and TOA” or “difference in total flux between surface and TOA”

Is it of consequence to anything in climate science if we agreed that the difference between the TOA radiation to space and the upward surface radiation is a better measure of the “greenhouse” effect?

Kramm & Dlugi comment on a paper by Ramanathan et al:

“At a surface temperature of 288 K the long-wave emission by the surface is about 390 W/m², whereas the outgoing long-wave radiation at the top of the atmosphere is only 236 W/m² (see Figure 2 [here presented as Figure 17]). Thus the intervening atmosphere causes a significant reduction in the long-wave emission to space. This reduction in the long-wave emission to space is referred to as the greenhouse effect”

As discussed before, applying the power law of Stefan and Boltzmann to a globally averaged temperature cannot be justified by physical and mathematical reasons.

Thus, the argument that at a surface temperature of 288 K the long-wave emission by the surface is about 390 W/m² is meaningless.

Just for interest here is how Ramanathan et al described their paper:

The two primary objectives of this review paper are (1) to describe the new scientific challenges posed by the trace gas climate problem and to summarize current strategies for meeting these challenges and (2) to make an assessment 0f the trace gas effects on troposphere-stratosphere temperature trends for the period covering the pre-industrial era to the present and for the next several decades. We will rely heavily on the numerous reports..

We could assume they don’t understand science basics, despite their many excellent papers demonstrating otherwise. Or we could assume that someone writing their 100th paper in the field of climate science doesn’t need to demonstrate that something called the “greenhouse” effect exists, or quantify it accurately in some specific way unless that is necessary for the specific purpose of the paper.

However, this is the genius of Kramm & Dlugi’s paper..

Dodging the Bullet

Casual readers of this paper (and people who rely on the statements of others about this paper) might think that they had demonstrated that the “greenhouse” effect doesn’t exist. They make a claim in their conclusion, of course, but they haven’t proven anything of the sort.

Instead they have written a paper explaining what everyone in climate science already knows.

So, to clarify matters, what is the emission of radiation from the top of atmosphere to space in one year?

ETOA= 3.8 x 1024 J

What is the emission of radiation from the surface in one year?

Esurface = ?

My questions to Kramm & Dlugi:

Is  Esurface significantly greater than ETOA ?

Obviously I believe Kramm & Dlugi will answer “Yes” to this question. This confirms the existence of the greenhouse effect, which they haven’t actually disputed except in their few words at the conclusion of their paper.

Hopefully, the authors will show up and confirm these important points.

Conclusion

The authors have shown us:

  • that a graph in the seminal Goody & Yung textbook is wrong
  • Kepler’s laws of planetary motion
  • that a website describes the “greenhouse” effect inaccurately
  • that without any “greenhouse” gases the effective albedo of the earth would be different
  • the average temperature of the earth’s surface can’t be used to calculate the average upward surface radiation

However, the important calculations of “radiative forcing” and various effects of increasing concentrations of radiatively-active gases are all done without using the “33K greenhouse effect”.

Without using the 33K “greenhouse” effect, we can derive all the equations of radiative transfer, solve them using the data for atmospheric temperature profiles, concentration of “greenhouse” gases, spectral line data from the HITRAN database and get:

  • the correct flux and spectral intensity at top of atmosphere
  • the correct flux and spectral intensity of downward radiation at the surface

We can also do this for changes in concentrations of various gases and find out the changes in top of atmosphere and downward surface flux. (Feedback and natural climate variations are the tricky part).

The discussions about average temperature are an amusing sideshow.

They are of no consequence for deriving the “greenhouse” effect or for determining the changes that might take place in the climate from increases or decreases in these gases.

Notes

Note 1: I didn’t check everything, so there could be mistakes. As the full article makes clear, not much need to check. I don’t endorse their last paragraph, as my conclusion – and article – makes clear.

Note 2: The calculation in that article for total annual global surface radiation doesn’t take into account surface emissivity. The value of ocean emissivity is incorrectly stated (see Emissivity of the Ocean). There are probably numerous other errors which I will fix one day if someone points them out.

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Many people are confused about science basics when it comes to the inappropriately-named “greenhouse” effect.

This can be easily demonstrated in many blogs around the internet where commenters, and even blog owners, embrace multiple theories that contradict each other but are somehow against the “greenhouse” effect.

Recently a new paper: Scrutinizing the atmospheric greenhouse effect and its climatic impact by Gerhard Kramm & Ralph Dlugi was published in the journal Natural Science.

Because of their favorable comments about Gerlich & Tscheuschner and the fact that they are sort of against something called the “greenhouse” effect I thought it might be useful for many readers to find out what was actually in the paper and what Kramm & Dlugi actually do believe about the “greenhouse” effect.

Much of the comments on blogs about the “greenhouse” effect are centered around the idea that this effect cannot be true because it would somehow violate the second law of thermodynamics. If there was a scientific idea in Gerlich & Tscheuschner, this was probably the main one. Or at least the most celebrated.

So it might surprise readers who haven’t opened up this paper that the authors are thoroughly 100% with mainstream climate science (and heat transfer basics) on this topic.

It didn’t surprise me because before reading this paper I read another paper by Kramm – A case study on wintertime inversions in Interior Alaska with WRF, Mölders & Kramm, Atmospheric Research (2010).

This 2010 paper is very interesting and evaluates models vs observations of the temperature inversions that take place in polar climates (where the temperature at the ground in wintertime is cooler than the atmosphere above). Nothing revolutionary (as with 99.99% of papers) and so of course the model used includes a radiation scheme from CAM3 (=Community Atmospheric Model) that is well used in standard climate science modeling.

Here is an important equation from Kramm & Dlugi’s recent paper for the energy balance at the earth’s surface.

Lots of blogs “against the greenhouse effect” don’t believe this equation:

Figure 1

The highlighted term is the downward radiation from the atmosphere multiplied by the absorptivity of the earth’s surface (its ability to absorb the radiation). This downward radiation (DLR) has also become known as “back radiation”.

In simple terms, the energy balance of Kramm & Dlugi adds up the absorbed portions of the solar radiation and atmospheric longwave radiation and equates them to the emitted longwave radiation plus the latent and sensible heat.

So the temperature of the surface is determined by solar radiation and “back radiation” and both are treated equally. It is also determined of course by the latent and sensible heat flux. (And see note 1).

As so many people on blogs around the internet believe this idea violates the second law of thermodynamics I thought it would be helpful to these readers to let them know to put Kramm & Dlugi 2011 on their “wrong about the 2nd law” list.

Of course, many people “against the greenhouse thing” also – or alternatively – believe that “back radiation” is negligible. Yet Kramm & Dlugi reproduce the standard diagram from Trenberth, Fasullo & Kiehl (2009) and don’t make any claim about “back radiation” being different in value from this paper.

“Back radiation” is real, measurable and affects the temperature of the surface – clearly Kramm & Dlugi are AGW wolves in sheeps’ clothing!

I look forward to the forthcoming rebuttal by Gerlich & Tscheuschner.

In the followup article, Kramm & Dlugi On Dodging the “Greenhouse” Bullet, I will attempt to point out the actual items of consequence from their paper.

Further reading –  Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part One and New Theory Proves AGW Wrong!

Note 1 – The surface energy balance isn’t what ultimately determines the surface temperature. The actual inappropriately-named “greenhouse” effect is determined by:

  • the effective emission height to space of outgoing longwave radiation which is determined by the opacity of the atmosphere (for example, due to increases in CO2 or water vapor)
  • the temperature difference between the surface and the effective emission height which is determined by the lapse rate

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A long time ago I started writing this article. I haven’t yet finished it.

I realized that trying to write it was difficult because the audience criticism was so diverse. Come to me you huddled masses.. This paper, so simple in concept, has become somehow the draw card for “everyone against AGW”. The reasons why are not clear, since the paper is nothing to do with that.

As I review the “critiques” around the blogosphere, I don’t find any consistent objection. That makes it very hard to write about.

So, the reason for posting a half-finished article is for readers to say what they don’t agree with and maybe – if there is a consistent message/question – I will finish the article, or maybe answer the questions here. If readers think that the ideas in the paper somehow violate the first or second law of thermodynamics, please see note 1 and comment in those referenced articles. Not here.

==== part written article ===

In 1997, J. T. Kiehl and Kevin Trenberth’s paper was published, Earth’s Annual Global Mean Energy Budget. (Referred to as KT97 for the rest of this article).

For some reason it has become a very unpopular paper, widely criticized, and apparently viewed as “the AGW paper”.

This is strange as it is a paper which says nothing about AGW, or even possible pre-feedback temperature changes from increases in the inappropriately-named “greenhouse” gases.

KT97 is a paper which attempts to quantify the global average numbers for energy fluxes at the surface and the top of atmosphere. And to quantify the uncertainty in these values.

Of course, many people criticizing the paper believe the values violates the first or second law of thermodynamics. I won’t comment in the main article on the basic thermodynamics laws – for this, check out the links in note 1.

In this article I will try and explain the paper a little.  There are many updates from various researchers to the data in KT97, including Trenberth & Kiehl themselves (Trenberth, Fasullo and Kiehl 2009), with later and more accurate figures.

We are looking at this earlier paper because it has somehow become such a focus of attention.

Most people have seen the energy budget diagram as it appears in the IPCC TAR report (2001), but here it is reproduced for reference:

From Kiehl & Trenberth (1997)

From Kiehl & Trenberth (1997)

History and Utility

Many people have suggested that the KT97 energy budget is some “new invention of climate science”. And at the other end of the spectrum at least one commenter I read was angered by the fact that KT97 had somehow claimed this idea for themselves when many earlier attempts had been made long before KT97.

The paper states:

There is a long history of attempts to construct a global annual mean surface–atmosphere energy budget for the earth. The first such budget was provided by Dines (1917).

Compared with “imagining stuff”, reading a paper is occasionally helpful. KT97 is simply updating the field with the latest data and more analysis.

What is an energy budget?

It is an attempt to identify the relative and absolute values of all of the heat transfer components in the system under consideration. In the case of the earth’s energy budget, the main areas of interest are the surface and the “top of atmosphere”.

Why is this useful?

Well, it won’t tell you the likely temperature in Phoenix next month, whether it will rain more next year, or whether the sea level will change in 100 years.. but it helps us understand the relative importance of the different heat transfer mechanisms in the climate, and the areas and magnitude of uncertainty.

For example, the % of reflected solar radiation is now known to be quite close to 30%. That equates to around 103 W/m² of solar radiation (see note 2) that is not absorbed by the climate system. Compared with the emission of radiation from the earth’s climate system into space – 239 W/m² – this is significant. So we might ask – how much does this reflected % change? How much has it changed in the past? See The Earth’s Energy Budget – Part Four – Albedo.

In a similar way, the measurements of absorbed solar radiation and emitted thermal radiation into space are of great interest – do they balance? Is the climate system warming or cooling? How much uncertainty do we have about these measurements.

The subject of the earth’s energy budget tries to address these kind of questions and therefore it is a very useful analysis.

However, it is just one tiny piece of the jigsaw puzzle called climate.

Uncertainty

It might surprise many people that KT97 also say:

Despite these important improvements in our understanding, a number of key terms in the energy budget remain uncertain, in particular, the net absorbed shortwave and longwave surface fluxes.

And in their conclusion:

The purpose of this paper is not so much to present definitive values, but to discuss how they were obtained and give some sense of the uncertainties and issues in determining the numbers.

It’s true. There are uncertainties and measurement difficulties. Amazing that they would actually say that. Probably didn’t think people would read the paper..

AGW – “Nil points”

What does this paper say about AGW?

Nothing.

What does it say about feedback from water vapor, ice melting and other mechanisms?

Nothing.

What does it say about the changes in surface temperature from doubling of CO2 prior to feedback?

Nothing.

Top of Atmosphere

Since satellites started measuring:

  • incoming solar (shortware) radiation
  • reflected solar radiation
  • outgoing terrestrial (longwave) radiation

- it has become much easier to understand – and put boundaries around – the top of atmosphere (TOA) energy budget.

The main challenge is the instrument uncertainty. So KT97 consider the satellite measurements. The most accurate results available (at that time) were from five years of ERBE data (1985-1989).

From those results, the outgoing longwave radiation (OLR) from ERBE averaged 235 W/m² while the absorbed solar radiation averaged 238 W/m². Some dull discussion of error estimates from earlier various papers follows. The main result being that the error estimates are in the order of 5W/m², so it isn’t possible to pin down the satellite results any closer than that.

KT97 concludes:

Based on these error estimates, we assume that the bulk of the bias in the ERBE imbalance is in the shortwave absorbed flux at the top of the atmosphere, since the retrieval of shortwave flux is more sensitive than the retrieval of longwave flux to the sampling and modeling of the diurnal cycle, surface and cloud inhomogeneities.

Therefore, we use the ERBE outgoing longwave flux of 235 W/m² to define the absorbed solar flux.

What are they saying? That – based on the measurements and error estimates – a useful working assumption is that the earth (over this time period) is in energy balance and so “pick the best number” to represent that. Reflected solar radiation is the hardest to measure accurately (because it can be reflected in any direction) so we assume that the OLR is the best value to work from.

If the absorbed solar radiation and the OLR had been, say, 25 W/m² apart then the error estimates couldn’t have bridged this gap. And the choices would have been:

  • the first law of thermodynamics was wrong (150 years of work proven wrong)
  • the earth was cooling (warming) – depending on the sign of the imbalance
  • a mystery source of heating/cooling hadn’t been detected
  • one or both of the satellites was plain wrong (or the error estimates had major mistakes)

So all the paper is explaining about the TOA results is that the measurement results don’t justify concluding that the earth is out of energy balance and therefore they pick the best number to represent the TOA fluxes. That’s it. This shouldn’t be very controversial.

And also note that during this time period the ocean heat content (OHC) didn’t record any significant increase, so an assumption of energy balance during this period is reasonable.

And, as with any review paper, KT97 also include the results from previous studies, explaining where they agree and where they differ and possible/probable reasons for the differences.

In their later update of their paper (2009) they use the results of a climate model for the TOA imbalance. This comes to 0.9 W/m². In the context of the uncertainties they discuss this is not so significant. It is simply a matter of whether the TOA fluxes balance or not. This is something that is fundamentally unknown over a given 5-year or decadal time period.

As an exercise for the interested student, if you review KT97 with the working assumption that the TOA fluxes are out of balance by 1W/m², what changes of note take place to the various values in the 1997 paper?

Surface Fluxes

This is the more challenging energy balance. At TOA we have satellites measuring the radiation quite comprehensively – and we have only radiation as the heat transfer mechanism for incoming and outgoing energy.

At the surface the measurement systems are less complete. Why is that?

Firstly, we have movement of heat from the surface via latent heat and sensible heat – as well as radiation.

Secondly, satellites can only measure only a small fraction of the upward emitted surface radiation and none of the downward radiation at the surface.

Surface Fluxes – Radiation

To calculate the surface radiation, upward and downward, we need to rely on theory, on models.

You mean made up stuff that no one has checked?

Well, that’s what you might think if you read a lot of blogs that have KT97 on their hit list. It’s easy to make claims.

In fact, if we want to know on a global annual average basis what the upward and downward longwave fluxes are, and if we want to know the solar (shortwave) fluxes that reach the surface (vs absorbed in the atmosphere), we need to rely on models. This is simply because we don’t have 1,000’s of high quality radiation-measuring stations.

Instead we do have a small network of high-quality monitoring stations for measuring downward radiation – the BSRN (baseline surface radiation network) was established by the World Climate Research Programme (WCRP) in the early 1990’s. See The Amazing Case of “Back Radiation”.

The important point is that, for the surface values of downward solar and downward longwave radiation we can check the results of theory against measurements in the places where measurements are available. This tells us whether models are accurate or not.

To calculate the values of surface fluxes with the resolution to calculate the global annual average we need to rely on models. For many people, their instinctive response is that obviously this is not accurate. Instinctive responses are not science, though.

Digression – Many Types of Models

There are many different types of models. For example, if we want to know the value of the DLR (downward longwave radiation) at the surface on Nov 1st, 2210 we need to be sure that some important parameters are well-known for this date. We would need to know the temperature of the atmosphere as a function of height through the atmosphere – and also the concentration of CO2, water vapor, methane – and so on. We would need to predict all of these values successfully for Nov 1st, 2210.

The burden of proof is quite high for this “prediction”.

However, if we want to know the average value of DLR for 2009 we need to have a record of these parameters at lots of locations and times and we can do a proven calculation for DLR at these locations and times.

An Analogy – It isn’t much different from calculating how long the water will take to boil on the stove – we need to know how much water, the initial temperature of the water, the atmospheric temperature and what level you turned the heat to. If we want to predict this value for the future we will need to know what these values will be in the future. But to calculate the past is easy – if we already have a record of these parameters.

See Theory and Experiment – Atmospheric Radiation for examples of verifying theory against experiment.

End of Digression

And if we want to know the upward fluxes we need to know the reflected portion.

Related Articles

Kiehl & Trenberth and the Atmospheric Window

The Earth’s Energy Budget – Part One – a few climate basics.

The Earth’s Energy Budget – Part Two –  the important concept of energy balance at top of atmosphere.

References

Earth’s Annual Global Mean Energy Budget, Kiehl & Trenberth, Bulletin of the American Meteorological Society (1997) – free paper

Earth’s Global Energy Budget, Trenberth, Fasullo & Kiehl, Bulletin of the American Meteorological Society (2009) – free paper

Notes

Note 1 – The First Law of Thermodynamics is about the conservation of energy. Many people believe that because the temperature is higher at the surface than the top of atmosphere this somehow violates this first law. Check out Do Trenberth and Kiehl understand the First Law of Thermodynamics? as well as the follow-on articles.

The Second Law of Thermodynamics is about entropy increasing, due to heat flowing from hotter to colder. Many have created an imaginary law which apparently stops energy from radiation from a colder body being absorbed by a hotter body. Check out these articles:

Amazing Things we Find in Textbooks – The Real Second Law of Thermodynamics

The Three Body Problem

Absorption of Radiation from Different Temperature Sources

The Amazing Case of “Back Radiation” – Part Three and Part One and Part Two

Note 2 – When comparing solar radiation with radiation emitted by the climate system there is a “comparison issue” that has to be taken into account. Solar radiation is “captured” by an area of πr² (the area of a disc) because the solar radiation comes from a point source a long way away. But terrestrial radiation is emitted over the whole surface of the earth, an area of 4πr². So if we are talking about W/m² either we need to multiply terrestrial radiation by a factor of 4 to equate the two, or divide solar radiation by a factor of 4 to equate the two. The latter is conventionally chosen.

More about this in The Earth’s Energy Budget – Part One

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In Part Two we looked at the claimed relationship ED=AA in Miskolczi’s 2007 paper.

  • Ed = downward atmospheric radiation absorbed by the surface
  • Aa = surface radiation absorbed in the atmosphere

I showed that they could not be exactly equal. Ferenc Miskolczi himself has just joined the discussion and confirmed:

I think I was the first who showed the AA≈ED relationship with reasonable quantitative accuracy.

That is, there is not a theoretical basis for equating AA=ED as an identity.

There is a world of difference between demonstrating a thermodynamic identity and an approximate experimental relationship. In the latter situation, it is customary to make some assessment of how close the values are and the determining factors in the relationship.

But in reviewing the 2007 paper again I noticed something very interesting:

From Miskolczi (2007)

Figure 1

Now the point I made in Part Two was that AA ≠ ED because the atmosphere is a little bit cooler than the surface – at the average height of emission of the atmosphere. So we would expect ED to be a a little less than AA.

Please review the full explanation in Part Two to understand this point.

Now take a look at the graph above. The straight line drawn on is the relationship ED=AA.

The black circles are for an assumption that the surface emissivity, εG = 1. (This is reasonably close to the actual emissivity of the surface, which varies with surface type. The oceans, for example, have an emissivity around 0.96).

In these calculated results you can see that Downwards Emittance, ED is a little less than AA. In fact, it looks to be about 5% less on average. (And note that is  ED = Absorbed Downwards Emittance)

Of course in practice, εG < 1. What happens then?

Well, in the graph above, with εG = 0.96 the points appear to lie very close to the line of ED=AA.

I think there is a calculation error in Miskolczi’s paper – and if this is true it is quite fundamental. Let me explain..

Here is the graphic for explaining Miskolczi’s terms:

From Miskolczi (2007)

Figure 2

When the surface is a blackbody (εG =1), SU = SG  – that is, the upwards radiation from the surface = the emitted radiation from the ground.

The terms and equations in his 2007 are derived with reference to the surface emitting as a blackbody.

When εG < 1, some care is needed in rewriting the equations. It looks like this care has not been taken and the open circles in his Fig 2 (my figure 1) closely matching the ED=AA line are an artifact of incorrectly rewriting the equations when ε< 1.

That’s how it looks anyway.

Here is my graphic for the terms needed for this problem:

Figure 3

As much as possible I have reused Miskolczi’s terms. Because the surface is not a blackbody, the downward radiation emitted by the atmosphere is not completely absorbed. So I created the term EDA for the emission of radiation by the atmosphere. Then some of this, Er, is reflected and added to SG to create the total upward surface radiation, SU.

Note as well that the relationship emissivity = absorptivity is only true for the same wavelengths. See note 4 in Part Two.

Some Maths

Now for some necessary maths – it is very simple. All we are doing is balancing energy to calculate the two terms we need. (Updated note – some of the equations are approximations – the real equation for emission of radiation is a complex term needing all of the data, code and a powerful computer – but the approximate result should indicate that there is an issue in the paper that needs addressing – see comment). 

And the objective is to get a formula for the ratio ED/AA – if ED=AA, this ratio = 1. And remember that in Figure 1, the relationship ED/AA=1 is shown as the straight line.

First, instead of having the term for atmospheric temperature, let’s replace it with:

TA = TS – ΔT      [1]

where ΔT represents the idea of a small change in temperature.

Second, the emitted atmospheric downward radiation comes from the Stefan-Boltzmann law:

EDA = εAσ(TS – ΔT)4      [2]

Third, downward atmospheric radiation absorbed by the surface:

ED = εGEDA      [3]

Fourth, the upward surface radiation is the emitted radiation plus the reflected atmospheric radiation. Emitted radiation is from the Stefan-Boltzmann law:

SU = εGσTS4 + (1-εG) EDA    [4]

Fifth, the absorbed surface radiation is the upward surface radiation multiplied by the absorptivity of the atmosphere (= emissivity at similar temperatures):

AA = εASU      [5]

So if we put [2] -> [3], we get:

ED = εGεAσ(TS – ΔT)4    [6]

And if we put [4] -> [5], we get:

AA = εGεAσTS4 + EDεA(1-εG)/εG   [7]

We are almost there. Remember that we wanted to find the ratio ED/AA. Unfortunately, the AA term includes ED and we can’t eliminate it (unless I missed something).

So let’s create the ratio and see what happens. This is equation 6 divided by equation 7 and we can eliminate εA that appears in each term:

ED/AA = [ εGσ(TS - ΔT)4 ] / [ εGσTS4 + ED(1-εG)/εG ]    [8]

And just to make it possibly a little clearer, we will divide top and bottom by εG and color code each part:

ED/AA = [ σ(TS - ΔT)4 ] / [ σTS4 + ED(1-εG)/εG2 ]      [8a]

And so the ratio = blackbody radiation at the atmospheric temperature divided by

( blackbody surface radiation plus a factor of downward atmospheric radiation that increases as εreduces )

We didn’t make a blackbody assumption, it is just that most of the emissivity terms canceled out.

What Does the Maths Mean?

Take a look at the green term – if εG = 1 this term is zero (1-1=0) and the equation simplifies down to:

ED/AA = (TS – ΔT)4  /  TS4

Which is very simple. If ΔT = 0 then ED/AA = 1.

Let’s plot ED vs AA for a few different values of ΔT and for TS = 288K:

Figure 4

Compare this with figure 1 (Miskolczi’s fig 2).

Note: I could have just cited the ratios of ED/AA, which – in this graph – are constant for each value of ΔT.

And we can easily see that as ΔT →0, ED/AA →1. This is “obvious” from the maths for people more comfortable with equations.

That’s the simplest stuff out of the way. Now we want to see what happens when εG < 1. This is the interesting part, and when you see the graph, please note that the axes are not the same as figure 4. In figure 4, the graph is of ED vs AA, but now we will plot the ratio of ED/AA as other factors change.

Take a look back at equation 8a. To calculate the ratio we need a value of Ed, which we don’t have. So I use some typical values from Miskolczi – and it’s clear that the value of Ed chosen doesn’t affect the conclusion.

Figure 5

You can see that when  εG = 1 the ratio is almost at 0.99. This is the slope of the top line (ΔT=1) in figure 4.

But as surface emissivity reduces, ED/AA reduces

This is clear from equation 8a – as εG  reduces below 1, the second term in the denominator of equation 8a increases from zero. As this increases, the ratio must reduce.

In Miskolczi’s graph, as εG changed from 1.0 → 0.96 the calculated ratio increased. I believe this is impossible.

Here is another version with a different value of ΔT:

Figure 6

Conclusion

Perhaps I made a mistake in the maths. It’s pretty simple – and out there in the open, so surely someone can quickly spot the mistake.

Of course I wouldn’t have published the article if I thought it had a mistake..

On conceptual grounds we can see that as the emissivity of the surface reduces, it absorbs less energy from the atmosphere and reflects more radiation back to the atmosphere.

This must reduce the value of ED and increase the value of AA. This reduces the ratio ED/AA.

In Miskolczi’s 2007 paper he shows that as emissivity is reduced from a blackbody to a more realistic value for the surface, the ratio goes in the other direction.

If my equations are correct then the equations of energy balance (for his paper) cannot have been correctly written for the case εG <1.

This one should be simple to clear up.

Update May 31st – Ken Gregory, a Miskolczi supporter appears to agree – and calculates ED/AA=0.94 for a real world surface emissivity.

Other articles in the series

The Mystery of Tau – Miskolczi – introduction to some of the issues around the calculation of optical thickness of the atmosphere, by Miskolczi, from his 2010 paper in E&E

Part Two – Kirchhoff – why Kirchhoff’s law is wrongly invoked, as the author himself later acknowledged, from his 2007 paper

Part Three – Kinetic Energy – why kinetic energy cannot be equated with flux (radiation in W/m²), and how equation 7 is invented out of thin air (with interesting author comment)

Part Five – Equation Soufflé – explaining why the “theory” in the 2007 paper is a complete dog’s breakfast

Part Six – Minor GHG’s – a less important aspect, but demonstrating the change in optical thickness due to the neglected gases N2O, CH4, CFC11 and CFC12.

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In Part One we looked at the calculation of total atmospheric optical thickness.

In Part Two we looked at the claim that the surface and atmosphere exchanged exactly equal amounts of energy by radiation. A thermodynamics revolution if it is true, as the atmosphere is slightly colder than the surface. This claim is not necessary to calculate optical thickness but is a foundation for Miskolczi’s theory about why optical thickness should be constant.

In this article we will look at another part of Miskolczi’s foundational theory from his 2007 paper, Greenhouse Effect in Semi-Transparent Planetary Atmospheres, Quarterly Journal of the Hungarian Meteorological Service.

For reference of the terms he uses, the diagram from the 2007 paper:

From Miskolczi (2007)

Figure 1

On pages 6-7, we find this claim:

Regarding the origin, EU is more closely related to the total internal kinetic energy of the atmosphere, which – according to the virial theorem – in hydrostatic equilibrium balances the total gravitational potential energy. To identify EU as the total internal kinetic energy of the atmosphere, the EU = SU / 2 equation must hold.

Many people have puzzled over the introduction of the virial theorem (note 1), which relates total kinetic energy of the atmosphere to total potential energy of the atmosphere. Generally, there is a relationship between potential energy and kinetic energy of an atmosphere so I don’t propose to question it, we will accept it as a given.

By the way, on the diagram SU = SG, i.e. SU = upwards radiation from the surface. And EU = upwards radiation from the atmosphere (cooling to space).

Kinetic Energy of a Gas

For people who don’t like seeing equations, skip to the statement in bold at the end of this section.

Here is the equation of an ideal gas:

pV = nkT (also written as pV = NRT)   [1]

where p = pressure, V = volume, n = number of molecules, k = 1.38 x 10-23 J/K = Boltzmann’s constant, T = temperature in K

This equation was worked out via experimental results a long time ago. Our atmosphere is a very close approximation to an ideal gas.

If we now take a thought experiment of some molecules “bouncing around” inside a container we can derive an equation for the pressure on a wall in terms of the velocities of the molecules:

pV = Nm<vx²>     [2]

where m = mass of a molecule, <vx²> = average of vx², where vx = velocity in the x direction

Combining [1] and [2] we get:

kT = m<vx²>, or

m<vx²>/2 = kT/2     [3]

The same considerations apply to the y and z direction, so

m<v²>/2 = 3KT/2      [4]

This equation tells us the temperature of a gas is equal to the average kinetic energy of molecules in that gas divided by a constant.

For beginners, the kinetic energy of a body is given by mv²/2 = mass x velocity squared divided by two.

So temperature of a gas is a direct measure of the kinetic energy.

The Kinetic Error

So where on earth does this identity come from?

..To identify EU as the total internal kinetic energy of the atmosphere..

EU is the upwards radiation from the atmosphere to space.

To calculate this value, you need to solve the radiative transfer equations, shown in Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations. These equations have no “analytic” solution but are readily solvable using numerical methods.

However, there is no doubt at all about this:

EU ≠ 3kTA/2   [5]

where TA = temperature of the atmosphere

that is, EU ≠ kinetic energy of the atmosphere

As an example of the form we might expect, if we had a very opaque atmosphere (in longwave), then EU = σTA4 (the Stefan-Boltzmann equation for thermal radiation). As the emissivity of the atmosphere reduces then the equation won’t stay exactly proportional to the 4th power of temperature. But it can never be linearly proportional to temperature.

A Mystery Equation

Many people have puzzled over the equations in Miskolczi’s 2007 paper.

On p6:

The direct consequences of the Kirchhoff law are the next two equations:
EU = F + K + P    (M5)
SU − (F0 + P0 ) = ED − EU   (M6)

Note that I have added a prefix to the equation numbers to identify they as Miskolczi’s. As previously commented, the P term (geothermal energy) is so small that it is not worth including. We will set it to zero and eliminate it, to make it a little easier to see the problems. Anyone wondering if this can be done – just set F’ = F0 + P0 and replace F0 with F’ in the following equations.

So:

EU = F + K    (M5a)
SU − F0 = ED − EU   (M6a)

Please review figure 1 for explanation of the terms.

If we accept the premise that AA = ED then these equations are correct (the premise is not correct, as shown in Part Two).

M5a is simple to see. Taking the incorrect premise that surface radiation absorbed in the atmosphere is completely re-emitted to the surface: therefore, the upward radiation from the atmosphere, EU must be supplied by the only other terms shown in the diagram – convective energy plus solar radiation absorbed by the atmosphere.

What about equation M6a? Physically, what is the downward energy emitted by the atmosphere minus the upward energy emitted by the atmosphere? What is the surface upward radiation minus the total solar radiation?

Well, doesn’t matter if we can’t figure out what these terms might mean. Instead we will just do some maths, using the fact that the surface energy must balance and the atmospheric energy must balance.

First let’s write down the atmospheric energy balance:

AA + K + F = EU + ED   [10]   –  I’m jumping the numbering to my equation 10 to avoid referencing confusion

This just says that Surface radiation absorbed in the atmosphere + convection from the surface to the atmosphere + absorbed solar radiation in the atmosphere = energy radiated by the atmosphere from the top and bottom.

Given the (incorrect) premise that AA = ED, we can rewrite equation 10:

K + F = EU    [10a]

We can see that this matches M5a, which is correct, as already stated.

So first, let’s write down the surface energy balance:

F0 – F + ED = SU + K    [11]

This just says that Solar radiation absorbed at the surface + downward atmospheric radiation = surface upward radiation + convection from the surface to the atmosphere.

Please review Figure 1 to confirm this equation.

Now let’s rewrite equation 11:

SU – F0 = ED – F – K    [11a]

and inserting eq 10a, we get:

SU – F0 = ED -EU    [11b]

Which agrees with M6a.

And as an aside only for people who have spent too long staring at these equations – re-arrange the terms in 11b:

Su – Ed = F0 – Eu; The left side is surface radiation – absorbed surface radiation in the atmosphere (accepting the flawed premise) = transmitted radiation. The right side is total absorbed solar radiation – upward emitted atmospheric radiation. As solar radiation is balanced by OLR, the right side is OLR – upward emitted atmospheric radiation = transmitted radiation.

Now, let’s see the mystery step :

In Eq. (6) SU − (F0 + P0 ) and ED − EU represent two flux terms of equal magnitude, propagating into opposite directions, while using the same F0 and P0 as energy sources. The first term heats the atmosphere and the second term maintains the surface energy balance. The principle of conservation of energy dictates that:
SU − (F0) + ED − EU = F0 = OLR   (M7)  

This equation M7 makes no sense. Note that again I have removed the tiny P0 term.

Let’s take [11b], already demonstrated (by accepting the premise) and add (ED -EU) to both sides:

SU – F0 + (ED – EU) = ED – EU+ (ED -EU) = 2(ED -EU)   [12]

So now the left side of eq 12 matches the left side of M7.

The M7 equation can only be correct if the right side of eq 12 matches the right side of M7:

2(ED -EU) = F0      [13] – to be confirmed or denied

In concept, this claim is that downward radiation from the atmosphere minus upward radiation from the atmosphere = half the total planetary absorbed solar radiation.

I can’t see where this has been demonstrated.

It is not apparent from energy balance considerations – we wrote down those two equations in [10] and [11].

We can say that energy into the climate system = energy out, therefore:

F0 = OLR = EU + ST    [14]   (atmospheric upward radiation plus transmitted radiation through the atmosphere)

Which doesn’t move us any closer to the demonstration we are looking for.

Perhaps someone from the large fan club can prove equation 7. So many people have embraced Miskolczi’s conclusion that there must be a lot of people who understand this step.

Conclusion

I’m confused about equation 7 of Miskolczi.

Running with the odds, I expect that no one will be able to prove it and instead I will be encouraged to take it on faith. However, I’m prepared to accept that someone might be able to prove that it is true (with the caveat about accepting the premise already discussed).

The more important point is equating the kinetic energy of the atmosphere with the upward atmospheric radiation.

It’s a revolutionary claim.

But as it comes with no evidence or derivation and would overturn lots of thermodynamics the obvious conclusion is that it is not true.

To demonstrate it is true takes more than a claim. Currently, it just looks like confusion on the part of the author.

Perhaps the author should write a whole paper devoted to explaining how the upwards atmospheric flux can be equated with the kinetic energy – along with dealing with the inevitable consequences for current thermodynamics.

Update 31st May: The author confirmed in the ensuing discussion that equation 7 was not developed from theoretical considerations.

Other Articles in the Series:

The Mystery of Tau – Miskolczi – introduction to some of the issues around the calculation of optical thickness of the atmosphere, by Miskolczi, from his 2010 paper in E&E

Part Two – Kirchhoff – why Kirchhoff’s law is wrongly invoked, as the author himself later acknowledged, from his 2007 paper

Part Four – a minor digression into another error that seems to have crept into the Aa=Ed relationship

Part Five – Equation Soufflé – explaining why the “theory” in the 2007 paper is a complete dog’s breakfast

Part Six – Minor GHG’s – a less important aspect, but demonstrating the change in optical thickness due to the neglected gases N2O, CH4, CFC11 and CFC12.

Further Reading:

New Theory Proves AGW Wrong! – a guide to the steady stream of new “disproofs” of the “greenhouse” effect or of AGW. And why you can usually only be a fan of – at most – one of these theories.

References

Greenhouse Effect in Semi-Transparent Planetary Atmospheres, Miskolczi, Quarterly Journal of the Hungarian Meteorological Service (2007)

Notes

Note 1 – A good paper on the virial theorem is on arXivThe Virial Theorem and Planetary Atmospheres, Victor Toth (2010)

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