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## Turbulence, Closure and Parameterization

In Latent heat and Parameterization I showed a formula for calculating latent heat transfer from the surface into the atmosphere, as well as the “real” formula. The parameterized version has horizontal wind speed x humidity difference (between the surface and some reference height in the atmosphere, typically 10m) x “a coefficient”.

One commenter asked:

Why do we expect that vertical transport of water vapor to vary linearly with horizontal wind speed? Is this standard turbulent mixing?

The simple answer is “almost yes”. But as someone famously said, make it simple, but not too simple.

Charting a course between too simple and too hard is a challenge with this subject. By contrast, radiative physics is a cakewalk. I’ll begin with some preamble and eventually get to the destination.

There’s a set of equations describing motion of fluids and what they do is conserve momentum in 3 directions (x,y,z) – these are the Navier-Stokes equations, and they conserve mass. Then there are also equations to conserve humidity and heat. There is an exact solution to the equations but there is a bit of a problem in practice. The Navier-Stokes equations in a rotating frame can be seen in The Coriolis Effect and Geostrophic Motion under “Some Maths”.

Simple linear equations with simple boundary conditions can be re-arranged and you get a nice formula for the answer. Then you can plot this against that and everyone can see how the relationships change with different material properties or boundary conditions. In real life equations are not linear and the boundary conditions are not simple. So there is no “analytical solution”, where we want to know say the velocity of the fluid in the east-west direction as a function of time and get a nice equation for the answer. Instead we have to use numerical methods.

Let’s take a simple problem – if you want to know heat flow through an odd-shaped metal plate that is heated in one corner and cooled by steady air flow on the rest of its surface you can use these numerical methods and usually get a very accurate answer.

Turbulence is a lot more difficult due to the range of scales involved. Here’s a nice image of turbulence:

Figure 1

There is a cascade of energy from the largest scales down to the point where viscosity “eats up” the kinetic energy. In the atmosphere this is the sub 1mm scale. So if you want to accurately numerically model atmospheric motion across a 100km scale you need a grid size probably 100,000,000 x 100,000,000 x 10,000,000 and solving sub-second for a few days. Well, that’s a lot of calculation. I’m not sure where turbulence modeling via “direct numerical simulation” has got to but I’m pretty sure that is still too hard and in a decade it will still be a long way off. The computing power isn’t there.

Anyway, for atmospheric modeling you don’t really want to know the velocity in the x,y,z direction (usually annotated as u,v,w) at trillions of points every second. Who is going to dig through that data? What you want is a statistical description of the key features.

So if we take the Navier-Stokes equation and average, what do we get? We get a problem.

For the mathematically inclined the following is obvious, but of course many readers aren’t, so here’s a simple example:

Let’s take 3 numbers: 1, 10, 100:   the average = (1+10+100)/3 = 37.

Now let’s look at the square of those numbers: 1, 100, 10000:  the average of the square of those numbers = (1+100+10000)/3 = 3367.

But if we take the average of our original numbers and square it, we get 37² = 1369. It’s strange but the average squared is not the same as the average of the squared numbers. That’s non-linearity for you.

In the Navier Stokes equations we have values like east velocity x upwards velocity, written as uw. The average of uw, written as $\overline{uw}$ is not equal to the average of u x the average of w, written as $\overline{u}.\overline{w}$. For the same reason we just looked at.

When we create the Reynolds averaged Navier-Stokes (RANS) equations we get lots of new terms like$\overline{uw}$. That is, we started with the original equations which gave us a complete solution – the same number of equations as unknowns. But when we average we end up with more unknowns than equations.

It’s like saying x + y = 1, what is x and y? No one can say. Perhaps 1 & 0. Perhaps 1000 & -999.

### Digression on RANS for Slightly Interested People

The Reynolds approach is to take a value like u,v,w (velocity in 3 directions) and decompose into a mean and a “rapidly varying” turbulent component.

So $u = \overline{u} + u'$, where $\overline{u}$ = mean value;  u’ = the varying component. So $\overline{u'} = 0$. Likewise for the other directions.

And $\overline{uw} = \overline{u} . \overline{w} + \overline{u'w'}$

So in the original equation where we have a term like $u . \frac{\partial u}{\partial x}$, it turns into  $(\overline{u} + u') . \frac{\partial (\overline{u} + u')}{\partial x}$, which, when averaged, becomes:

$\overline{u} . \frac{\partial \overline{u}}{\partial x} +\overline{u' . \frac{\partial u'}{\partial x}}$

So 2 unknowns instead of 1. The first term is the averaged flow, the second term is the turbulent flow. (Well, it’s an advection term for the change in velocity following the flow)

When we look at the conservation of energy equation we end up with terms for the movement of heat upwards due to average flow (almost zero) and terms for the movement of heat upwards due to turbulent flow (often significant). That is, a term like $\overline{\theta'w'}$ which is “the mean of potential temperature variations x upwards eddy velocity”.

Or, in plainer English, how heat gets moved up by turbulence.

..End of Digression

### Closure and the Invention of New Ideas

“Closure” is a maths term. To “close the equations” when we have more unknowns that equations means we have to invent a new idea. Some geniuses like Reynolds, Prandtl and Kolmogoroff did come up with some smart new ideas.

Often the smart ideas are around “dimensionless terms” or “scaling terms”. The first time you encounter these ideas they seem odd or just plain crazy. But like everything, over time strange ideas start to seem normal.

The Reynolds number is probably the simplest to get used to. The Reynolds number seeks to relate fluid flows to other similar fluid flows. You can have fluid flow through a massive pipe that is identical in the way turbulence forms to that in a tiny pipe – so long as the viscosity and density change accordingly.

The Reynolds number, Re = density x length scale x mean velocity of the fluid / viscosity

And regardless of the actual physical size of the system and the actual velocity, turbulence forms for flow over a flat plate when the Reynolds number is about 500,000. By the way, for the atmosphere and ocean this is true most of the time.

Kolmogoroff came up with an idea in 1941 about the turbulent energy cascade using dimensional analysis and came to the conclusion that the energy of eddies increases with their size to the power 2/3 (in the “inertial subrange”). This is usually written vs frequency where it becomes a -5/3 power. Here’s a relatively recent experimental verification of this power law.

From Durbin & Reif 2010

Figure 2

In less genius like manner, people measure stuff and use these measured values to “close the equations” for “similar” circumstances. Unfortunately, the measurements are only valid in a small range around the experiments and with turbulence it is hard to predict where the cutoff is.

A nice simple example, to which I hope to return because it is critical in modeling climate, is vertical eddy diffusivity in the ocean. By way of introduction to this, let’s look at heat transfer by conduction.

If only all heat transfer was as simple as conduction. That’s why it’s always first on the list in heat transfer courses..

If have a plate of thickness d, and we hold one side at temperature T1 and the other side at temperature T2, the heat conduction per unit area:

$H_z = \frac{k(T_2-T_1)}{d}$

where k is a material property called conductivity. We can measure this property and it’s always the same. It might vary with temperature but otherwise if you take a plate of the same material and have widely different temperature differences, widely different thicknesses – the heat conduction always follows the same equation.

Now using these ideas, we can take the actual equation for vertical heat flux via turbulence:

$H_z =\rho c_p\overline{w'\theta'}$

where w = vertical velocity, θ = potential temperature

And relate that to the heat conduction equation and come up with (aka ‘invent’):

$H_z = \rho c_p K . \frac{\partial \theta}{\partial z}$

Now we have an equation we can actually use because we can measure how potential temperature changes with depth. The equation has a new “constant”, K. But this one is not really a constant, it’s not really a material property – it’s a property of the turbulent fluid in question. Many people have measured the “implied eddy diffusivity” and come up with a range of values which tells us how heat gets transferred down into the depths of the ocean.

Well, maybe it does. Maybe it doesn’t tell us very much that is useful. Let’s come back to that topic and that “constant” another day.

### The Main Dish – Vertical Heat Transfer via Horizontal Wind

Back to the original question. If you imagine a sheet of paper as big as your desk then that pretty much gives you an idea of the height of the troposphere (lower atmosphere where convection is prominent).

It’s as thin as a sheet of desk size paper in comparison to the dimensions of the earth. So any large scale motion is horizontal, not vertical. Mean vertical velocities – which doesn’t include turbulence via strong localized convection – are very low. Mean horizontal velocities can be the order of 5 -10 m/s near the surface of the earth. Mean vertical velocities are the order of cm/s.

Let’s look at flow over the surface under “neutral conditions”. This means that there is little buoyancy production due to strong surface heating. In this case the energy for turbulence close to the surface comes from the kinetic energy of the mean wind flow – which is horizontal.

There is a surface drag which gets transmitted up through the boundary layer until there is “free flow” at some height. By using dimensional analysis, we can figure out what this velocity profile looks like in the absence of strong convection. It’s logarithmic:

Figure 3 – for typical ocean surface

Lots of measurements confirm this logarithmic profile.

We can then calculate the surface drag – or how momentum is transferred from the atmosphere to the ocean – using the simple formula derived and we come up with a simple expression:

$\tau_0 = \rho C_D U_r^2$

Where Ur is the velocity at some reference height (usually 10m), and CD is a constant calculated from the ratio of the reference height to the roughness height and the von Karman constant.

Using similar arguments we can come up with heat transfer from the surface. The principles are very similar. What we are actually modeling in the surface drag case is the turbulent vertical flux of horizontal momentum $\rho \overline{u'w'}$ with a simple formula that just has mean horizontal velocity. We have “closed the equations” by some dimensional analysis.

Adding the Richardson number for non-neutral conditions we end up with a temperature difference along with a reference velocity to model the turbulent vertical flux of sensible heat $\rho c_p . \overline{w'\theta'}$. Similar arguments give latent heat flux $L\rho . \overline{w'q'}$ in a simple form.

### Now with a bit more maths..

At the surface the horizontal velocity must be zero. The vertical flux of horizontal momentum creates a drag on the boundary layer wind. The vertical gradient of the mean wind, U, can only depend on height z, density ρ and surface drag.

So the “characteristic wind speed” for dimensional analysis is called the friction velocity, u*, and $u* = \sqrt\frac{\tau_0}{\rho}$

This strange number has the units of velocity: m/s  – ask if you want this explained.

So dimensional analysis suggests that $\frac{z}{u*} . \frac{\partial U}{\partial z}$ should be a constant – “scaled wind shear”. The inverse of that constant is known as the Von Karman constant, k = 0.4.

So a simple re-arrangement and integration gives:

$U(z) = \frac{u*}{k} . ln(\frac{z}{z_0})$

where z0 is a constant from the integration, which is roughness height – a physical property of the surface where the mean wind reaches zero.

The “real form” of the friction velocity is:

$u*^2 = \frac{\tau_0}{\rho} = (\overline{u'w'}^2 + \overline{v'w'}^2)^\frac{1}{2}$,  where these eddy values are at the surface

we can pick a horizontal direction along the line of the mean wind (rotate coordinates) and come up with:

$u*^2 = \overline{u'w'}$

If we consider a simple constant gradient argument:

$\tau = - \rho . \overline{u'w'} = \rho K \frac{\partial \overline{u}}{\partial z}$

where the first expression is the “real” equation and the second is the “invented” equation, or “our attempt to close the equation” from dimensional analysis.

Of course, this is showing how momentum is transferred, but the approach is pretty similar, just slightly more involved, for sensible and latent heat.

### Conclusion

Turbulence is a hard problem. The atmosphere and ocean are turbulent so calculating anything is difficult. Until a new paradigm in computing comes along, the real equations can’t be numerically solved from the small scales needed where viscous dissipation damps out the kinetic energy of the turbulence up to the large scale of the whole earth, or even of a synoptic scale event. However, numerical analysis has been used a lot to test out ideas that are hard to test in laboratory experiments. And can give a lot of insight into parts of the problems.

In the meantime, experiments, dimensional analysis and intuition have provided a lot of very useful tools for modeling real climate problems.

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## The Atmosphere Cools to Space by CO2 and Water Vapor, so More GHGs, More Cooling!

The atmosphere cools to space by radiation. Well, without getting into all the details, the surface cools to space as well by radiation but not much radiation is emitted by the surface that escapes directly to space (note 1). Most surface radiation is absorbed by the atmosphere. And of course the surface mostly cools by convection into the troposphere (lower atmosphere).

If there were no radiatively-active gases (aka “GHG”s) in the atmosphere then the atmosphere couldn’t cool to space at all.

Technically, the emissivity of the atmosphere would be zero. Emission is determined by the local temperature of the atmosphere and its emissivity. Wavelength by wavelength emissivity is equal to absorptivity, another technical term, which says what proportion of radiation is absorbed by the atmosphere. If the atmosphere can’t emit, it can’t absorb (note 2).

So as you increase the GHGs in the atmosphere you increase its ability to cool to space. A lot of people realize this at some point during their climate science journey and finally realize how they have been duped by climate science all along! It’s irrefutable – more GHGs more cooling to space, more GHGs mean less global warming!

Ok, it’s true. Now the game’s up, I’ll pack up Science of Doom into a crate and start writing about something else. Maybe cognitive dissonance..

Bye everyone!

Halfway through boxing everything up I realized there was a little complication to the simplicity of that paragraph. The atmosphere with more GHGs has a higher emissivity, but also a higher absorptivity.

Let’s draw a little diagram. Here are two “layers” (see note 3) of the atmosphere in two different cases. On the left 400 ppmv CO2, on the right 500ppmv CO2 (and relative humidity of water vapor was set at 50%, surface temperature at 288K):

Figure 1

It’s clear that the two layers are both emitting more radiation with more CO2.More cooling to space.

For interest, the “total emissivity” of the top layer is 0.190 in the first case and 0.197 in the second case. The layer below has 0.389 and 0.395.

Let’s take a look at all of the numbers and see what is going on. This diagram is a little busier:

Figure 2

The key point is that the OLR (outgoing longwave radiation) is lower in the case with more CO2. Yet each layer is emitting more radiation. How can this be?

Take a look at the radiation entering the top layer on the left = 265.1, and add to that the emitted radiation = 23.0 – the total is 288.1. Now subtract the radiation leaving through the top boundary = 257.0 and we get the radiation absorbed in the layer. This is 31.1 W/m².

Compare that with the same calculation with more CO2 – the absorption is 32.2 W/m².

This is the case all the way up through the atmosphere – each layer emits more because its emissivity has increased, but it also absorbs more because its absorptivity has increased by the same amount.

So more cooling to space, but unfortunately more absorption of the radiation below – two competing terms.

### So why don’t they cancel out?

Emission of radiation is a result of local temperature and emissivity.

Absorption of radiation is the result of the incident radiation and absorptivity. Incident upwards radiation started lower in the atmosphere where it is hotter. So absorption changes always outweigh emission changes (note 4).

### Conceptual Problems?

If it’s still not making sense then think about what happens as you reduce the GHGs in the atmosphere. The atmosphere emits less but absorbs even less of the radiation from below. So the outgoing longwave radiation increases. More surface radiation is making it to the top of atmosphere without being absorbed. So there is less cooling to space from the atmosphere, but more cooling to space from the surface and the atmosphere.

If you add lagging to a pipe, the temperature of the pipe increases (assuming of course it is “internally” heated with hot water). And yet, the pipe cools to the surrounding room via the lagging! Does that mean more lagging, more cooling? No, it’s just the transfer mechanism for getting the heat out.

That was just an analogy. Analogies don’t prove anything. If well chosen, they can be useful in illustrating problems. End of analogy disclaimer.

If you want to understand more about how radiation travels through the atmosphere and how GHG changes affect this journey, take a look at the series Visualizing Atmospheric Radiation.

### Notes

Note 1: For more on the details see

Note 2: A very basic point – absolutely essential for understanding anything at all about climate science – is that the absorptivity of the atmosphere can be (and is) totally different from its emissivity when you are considering different wavelengths. The atmosphere is quite transparent to solar radiation, but quite opaque to terrestrial radiation – because they are at different wavelengths. 99% of solar radiation is at wavelengths less than 4 μm, and 99% of terrestrial radiation is at wavelengths greater than 4 μm. That’s because the sun’s surface is around 6000K while the earth’s surface is around 290K. So the atmosphere has low absorptivity of solar radiation (<4 μm) but high emissivity of terrestrial radiation.

Note 3: Any numerical calculation has to create some kind of grid. This is a very course grid, with 10 layers of roughly equal pressure in the atmosphere from the surface to 200mbar. The grid assumes there is just one temperature for each layer. Of course the temperature is decreasing as you go up. We could divide the atmosphere into 30 layers instead. We would get more accurate results. We would find the same effect.

Note 4: The equations for radiative transfer are found in Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations. The equations prove this effect.

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## Natural Variability and Chaos – Two – Lorenz 1963

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:

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:

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:

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:

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:

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:

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:

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:

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

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

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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## The “Greenhouse” Effect Explained in Simple Terms

Over the last few years I’ve written lots of articles relating to the inappropriately-named “greenhouse” effect and covered some topics in great depth. I’ve also seen lots of comments and questions which has helped me understand common confusion and misunderstandings.

This article, with huge apologies to regular long-suffering readers, covers familiar ground in simple terms. It’s a reference article. I’ve referenced other articles and series as places to go to understand a particular topic in more detail.

One of the challenges of writing a short simple explanation is it opens you up to the criticism of having omitted important technical details that you left out in order to keep it short. Remember this is the simple version..

### Preamble

First of all, the “greenhouse” effect is not AGW. In maths, physics, engineering and other hard sciences, one block is built upon another block. AGW is built upon the “greenhouse” effect. If AGW is wrong, it doesn’t invalidate the greenhouse effect. If the greenhouse effect is wrong, it does invalidate AGW.

The greenhouse effect is built on very basic physics, proven for 100 years or so, that is not in any dispute in scientific circles. Fantasy climate blogs of course do dispute it.

Second, common experience of linearity in everyday life cause many people to question how a tiny proportion of “radiatively-active” molecules can have such a profound effect. Common experience is not a useful guide. Non-linearity is the norm in real science. Since the enlightenment at least, scientists have measured things rather than just assumed consequences based on everyday experience.

### The Elements of the “Greenhouse” Effect

Atmospheric Absorption

1. The “radiatively-active” gases in the atmosphere:

• water vapor
• CO2
• CH4
• N2O
• O3
• and others

absorb radiation from the surface and transfer this energy via collision to the local atmosphere. Oxygen and nitrogen absorb such a tiny amount of terrestrial radiation that even though they constitute an overwhelming proportion of the atmosphere their radiative influence is insignificant (note 1).

How do we know all this? It’s basic spectroscopy, as detailed in exciting journals like the Journal of Quantitative Spectroscopy and Radiative Transfer over many decades. Shine radiation of a specific wavelength through a gas and measure the absorption. Simple stuff and irrefutable.

Atmospheric Emission

2. The “radiatively-active” gases in the atmosphere also emit radiation. Gases that absorb at a wavelength also emit at that wavelength. Gases that don’t absorb at that wavelength don’t emit at that wavelength. This is a consequence of Kirchhoff’s law.

The intensity of emission of radiation from a local portion of the atmosphere is set by the atmospheric emissivity and the temperature.

Convection

3. The transfer of heat within the troposphere is mostly by convection. The sun heats the surface of the earth through the (mostly) transparent atmosphere (note 2). The temperature profile, known as the “lapse rate”, is around 6K/km in the tropics. The lapse rate is principally determined by non-radiative factors – as a parcel of air ascends it expands into the lower pressure and cools during that expansion (note 3).

The important point is that the atmosphere is cooler the higher you go (within the troposphere).

Energy Balance

4. The overall energy in the climate system is determined by the absorbed solar radiation and the emitted radiation from the climate system. The absorbed solar radiation – globally annually averaged – is approximately 240 W/m² (note 4). Unsurprisingly, the emitted radiation from the climate system is also (globally annually averaged) approximately 240 W/m². Any change in this and the climate is cooling or warming.

Emission to Space

5. Most of the emission of radiation to space by the climate system is from the atmosphere, not from the surface of the earth. This is a key element of the “greenhouse” effect. The intensity of emission depends on the local atmosphere. So the temperature of the atmosphere from which the emission originates determines the amount of radiation.

If the place of emission of radiation – on average – moves upward for some reason then the intensity decreases. Why? Because it is cooler the higher up you go in the troposphere. Likewise, if the place of emission – on average – moves downward for some reason, then the intensity increases (note 5).

More GHGs

6. If we add more radiatively-active gases (like water vapor and CO2) then the atmosphere becomes more “opaque” to terrestrial radiation and the consequence is the emission to space from the atmosphere moves higher up (on average). Higher up is colder. See note 6.

So this reduces the intensity of emission of radiation, which reduces the outgoing radiation, which therefore adds energy into the climate system. And so the climate system warms (see note 7).

That’s it!

It’s as simple as that. The end.

### A Few Common Questions

CO2 is Already Saturated

There are almost 315,000 individual absorption lines for CO2 recorded in the HITRAN database. Some absorption lines are stronger than others. At the strongest point of absorption – 14.98 μm (667.5 cm-1), 95% of radiation is absorbed in only 1m of the atmosphere (at standard temperature and pressure at the surface). That’s pretty impressive.

By contrast, from 570 – 600 cm-1 (16.7 – 17.5 μm) and 730 – 770 cm-1 (13.0 – 13.7 μm) the CO2 absorption through the atmosphere is nowhere near “saturated”. It’s more like 30% absorbed through a 1km path.

You can see the complexity of these results in many graphs in Atmospheric Radiation and the “Greenhouse” Effect – Part Nine – calculations of CO2 transmittance vs wavelength in the atmosphere using the 300,000 absorption lines from the HITRAN database, and see also Part Eight – interesting actual absorption values of CO2 in the atmosphere from Grant Petty’s book

The complete result combining absorption and emission is calculated in Visualizing Atmospheric Radiation – Part Seven – CO2 increases – changes to TOA in flux and spectrum as CO2 concentration is increased

CO2 Can’t Absorb Anything of Note Because it is Only .04% of the Atmosphere

See the point above. Many spectroscopy professionals have measured the absorptivity of CO2. It has a huge variability in absorption, but the most impressive is that 95% of 14.98 μm radiation is absorbed in just 1m. How can that happen? Are spectroscopy professionals charlatans? You need evidence, not incredulity. Science involves measuring things and this has definitely been done. See the HITRAN database.

Water Vapor Overwhelms CO2

This is an interesting point, although not correct when we consider energy balance for the climate. See Visualizing Atmospheric Radiation – Part Four – Water Vapor – results of surface (downward) radiation and upward radiation at TOA as water vapor is changed.

The key point behind all the detail is that the top of atmosphere radiation change (as CO2 changes) is the important one. The surface change (forcing) from increasing CO2 is not important, is definitely much weaker and is often insignificant. Surface radiation changes from CO2 will, in many cases, be overwhelmed by water vapor.

Water vapor does not overwhelm CO2 high up in the atmosphere because there is very little water vapor there – and the radiative effect of water vapor is dramatically impacted by its concentration, due to the “water vapor continuum”.

The Calculation of the “Greenhouse” Effect is based on “Average Surface Temperature” and there is No Such Thing

Simplified calculations of the “greenhouse” effect use some averages to make some points. They help to create a conceptual model.

Real calculations, using the equations of radiative transfer, don’t use an “average” surface temperature and don’t rely on a 33K “greenhouse” effect. Would the temperature decrease 33K if all of the GHGs were removed from the atmosphere? Almost certainly not. Because of feedbacks. We don’t know the effect of all of the feedbacks. But would the climate be colder? Definitely.

See The Rotational Effect – why the rotation of the earth has absolutely no effect on climate, or so a parody article explains..

The Second Law of Thermodynamics Prohibits the Greenhouse Effect, or so some Physicists Demonstrated..

See The Three Body Problem – a simple example with three bodies to demonstrate how a “with atmosphere” earth vs a “without atmosphere earth” will generate different equilibrium temperatures. Please review the entropy calculations and explain (you will be the first) where they are wrong or perhaps, or perhaps explain why entropy doesn’t matter (and revolutionize the field).

See Gerlich & Tscheuschner for the switch and bait routine by this operatic duo.

And see Kramm & Dlugi On Dodging the “Greenhouse” Bullet – Kramm & Dlugi demonstrate that the “greenhouse” effect doesn’t exist by writing a few words in a conclusion but carefully dodging the actual main point throughout their entire paper. However, they do recover Kepler’s laws and point out a few errors in a few websites. And note that one of the authors kindly showed up to comment on this article but never answered the important question asked of him. Probably just too busy.. Kramm & Dlugi also helpfully (unintentionally) explain that G&T were wrong, see Kramm & Dlugi On Illuminating the Confusion of the Unclear – Kramm & Dlugi step up as skeptics of the “greenhouse” effect, fans of Gerlich & Tscheuschner and yet clarify that colder atmospheric radiation is absorbed by the warmer earth..

And for more on that exciting subject, see Confusion over the Basics under the sub-heading The Second Law of Thermodynamics.

Feedbacks overwhelm the Greenhouse Effect

This is a totally different question. The “greenhouse” effect is the “greenhouse” effect. If the effect of more CO2 is totally countered by some feedback then that will be wonderful. But that is actually nothing to do with the “greenhouse” effect. It would be a consequence of increasing temperature.

As noted in the preamble, it is important to separate out the different building blocks in understanding climate.

Miskolczi proved that the Greenhouse Effect has no Effect

Miskolczi claimed that the greenhouse effect was true. He also claimed that more CO2 was balanced out by a corresponding decrease in water vapor. See the Miskolczi series for a tedious refutation of his paper that was based on imaginary laws of thermodynamics and questionable experimental evidence.

Once again, it is important to be able to separate out two ideas. Is the greenhouse effect false? Or is the greenhouse effect true but wiped out by a feedback?

If you don’t care, so long as you get the right result you will be in ‘good’ company (well, you will join an extremely large company of people). But this blog is about science. Not wishful thinking. Don’t mix the two up..

Convection “Short-Circuits” the Greenhouse Effect

Let’s assume that regardless of the amount of energy arriving at the earth’s surface, that the lapse rate stays constant and so the more heat arriving, the more heat leaves. That is, the temperature profile stays constant. (It’s a questionable assumption that also impacts the AGW question).

It doesn’t change the fact that with more GHGs, the radiation to space will be from a higher altitude. A higher altitude will be colder. Less radiation to space and so the climate warms – even with this “short-circuit”.

In a climate without convection, the surface temperature will start off higher, and the GHG effect from doubling CO2 will be higher. See Radiative Atmospheres with no Convection.

In summary, this isn’t an argument against the greenhouse effect, this is possibly an argument about feedbacks. The issue about feedbacks is a critical question in AGW, not a critical question for the “greenhouse” effect. Who can say whether the lapse rate will be constant in a warmer world?

### Notes

Note 1 – An important exception is O2 absorbing solar radiation high up above the troposphere (lower atmosphere). But O2 does not absorb significant amounts of terrestrial radiation.

Note 2 – 99% of solar radiation has a wavelength <4μm. In these wavelengths, actually about 1/3 of solar radiation is absorbed in the atmosphere. By contrast, most of the terrestrial radiation, with a wavelength >4μm, is absorbed in the atmosphere.

Note 3 – see:

Density, Stability and Motion in Fluids – some basics about instability
Potential Temperature – explaining “potential temperature” and why the “potential temperature” increases with altitude
Temperature Profile in the Atmosphere – The Lapse Rate – lots more about the temperature profile in the atmosphere

Note 4 – see Earth’s Energy Budget – a series on the basics of the energy budget

Note 5 – the “place of emission” is a useful conceptual tool but in reality the emission of radiation takes place from everywhere between the surface and the stratosphere. See Visualizing Atmospheric Radiation – Part Three – Average Height of Emission – the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions.

Also, take a look at the complete series: Visualizing Atmospheric Radiation.

Note 6 – the balance between emission and absorption are found in the equations of radiative transfer. These are derived from fundamental physics – see Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations – the equations of radiative transfer including the plane parallel assumption and it’s nothing to do with blackbodies. The fundamental physics is not just proven in the lab, spectral measurements at top of atmosphere and the surface match the calculated values using the radiative transfer equations – see Theory and Experiment – Atmospheric Radiation – real values of total flux and spectra compared with the theory.

Also, take a look at the complete series: Atmospheric Radiation and the “Greenhouse” Effect

Note 7 – this calculation is under the assumption of “all other things being equal”. Of course, in the real climate system, all other things are not equal. However, to understand an effect “pre-feedback” we need to separate it from the responses to the system.

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## Radiative Atmospheres with no Convection

If we open an introductory atmospheric physics textbook, we find that the temperature profile in the troposphere (lower atmosphere) is mostly explained by convection. (See for example, Things Climate Science has Totally Missed? – Convection)

We also find that the temperature profile in the stratosphere is mostly determined by radiation. And that the overall energy balance of the climate system is determined by radiation.

Many textbooks introduce the subject of convection in this way:

• what would the temperature profile be like if there was no convection, only radiation for heat transfer
• why is the temperature profile actually different
• how does pressure reduce with height
• what happens to air when it rises and expands in the lower pressure environment
• derivation of the “adiabatic lapse rate”, which in layman’s terms is the temperature change when we have relatively rapid movements of air
• how the real world temperature profile (lapse rate) compares with the calculated adiabatic lapse rate and why

We looked at the last four points in some detail in a few articles:

Density, Stability and Motion in Fluids – some basics about instability
Potential Temperature – explaining “potential temperature” and why the “potential temperature” increases with altitude
Temperature Profile in the Atmosphere – The Lapse Rate – lots more about the temperature profile in the atmosphere

In this article we will look at the first point.

All of the atmospheric physics textbooks I have seen use a very simple model for explaining the temperature profile in a fictitious “radiation only” environment. The simple model is great for giving insight into how radiation travels.

Physics textbooks, good ones anyway, try and use the simplest models to explain a phenomenon.

The simple model, in brief, is the “semi-gray approximation”. This says the atmosphere is completely transparent to solar radiation, but opaque to terrestrial radiation. Its main simplification is having a constant absorption with wavelength. This makes the problem nice and simple analytically – which means we can rewrite the starting equations and plot a nice graph of the result.

However, atmospheric absorption is the total opposite of constant. Here is an example of the absorption vs wavelength of a minor “greenhouse” gas:

From Vardavas & Taylor (2007)

Figure 1

So from time to time I’ve wondered what the “no convection” atmosphere would look like with real GHG absorption lines. I also thought it would be especially interesting to see the effect of doubling CO2 in this fictitious environment.

This article is for curiosity value only, and for helping people understand radiative transfer a little better.

We will use the Matlab program seen in the series Visualizing Atmospheric Radiation. This does a line by line calculation of radiative transfer for all of the GHGs, pulling the absorption data out of the HITRAN database.

I updated the program in a few subtle ways. Mainly the different treatment of the stratosphere – the place where convection stops – was removed. Because, in this fictitious world there is no convection in the lower atmosphere either.

Here is a simulation based on 380 ppm CO2, 1775 ppb CH4, 319 ppb N2O and 50% relative humidity all through the atmosphere. Top of atmosphere was 100 mbar and the atmosphere was divided into 40 layers of equal pressure. Absorbed solar radiation was set to 240 W/m² with no solar absorption in the atmosphere. That is (unlike in the real world), the atmosphere has been made totally transparent to solar radiation.

The starting point was a surface temperature of 288K (15ºC) and a lapse rate of 6.5K/km – with no special treatment of the stratosphere. The final surface temperature was 326K (53ºC), an increase of 38ºC:

Figure 2

The ocean depth was only 5m. This just helps get to a new equilibrium faster. If we change the heat capacity of a system like this the end result is the same, the only difference is the time taken.

Water vapor was set at a relative humidity of 50%. For these first results I didn’t get the simulation to update the absolute humidity as the temperature changed. So the starting temperature was used to calculate absolute humidity and that mixing ratio was kept constant:

Figure 3

The lapse rate, or temperature drop per km of altitude:

Figure 4

The flux down and flux up vs altitude:

Figure 5

The top of atmosphere upward flux is 240 W/m² (actually at the 500 day point it was 239.5 W/m²) – the same as the absorbed solar radiation (note 1). The simulation doesn’t “force” the TOA flux to be this value. Instead, any imbalance in flux in each layer causes a temperature change, moving the surface and each part of the atmosphere into a new equilibrium.

A bit more technically for interested readers.. For a given layer we sum:

• upward flux at the bottom of a layer minus upward flux at the top of a layer
• downward flux at the top of a layer minus downward flux at the bottom of a layer

This sum equates to the “heating rate” of the layer. We then use the heat capacity and time to work out the temperature change. Then the next iteration of the simulation redoes the calculation.

And even more technically:

• the upwards flux at the top of a layer = the upwards flux at the bottom of the layer x transmissivity of the layer plus the emission of that layer
• the downwards flux at the bottom of a layer = the downwards flux at the top of the layer x transmissivity of the layer plus the emission of that layer

End of “more technically”..

Anyway, the main result is the surface is much hotter and the temperature drop per km of altitude is much greater than the real atmosphere. This is because it is “harder” for heat to travel through the atmosphere when radiation is the only mechanism. As the atmosphere thins out, which means less GHGs, radiation becomes progressively more effective at transferring heat. This is why the lapse rate is lower higher up in the atmosphere.

Now let’s have a look at what happens when we double CO2 from its current value (380ppm -> 760 ppm):

Figure 6 – with CO2 doubled instantaneously from 380ppm at 500 days

The final surface temperature is 329.4, increased from 326.2K. This is an increase (no feedback of 3.2K).

The “pseudo-radiative forcing” = 18.9 W/m² (which doesn’t include any change to solar absorption). This radiative forcing is the immediate change in the TOA forcing. (It isn’t directly comparable to the IPCC standard definition which is at the tropopause and after the stratosphere has come back into equilibrium – none of these have much meaning in a world without convection).

Let’s also look at the “standard case” of an increase from pre-industrial CO2 of 280 ppm to a doubling of 560 ppm. I ran this one for longer – 1000 days before doubling CO2 and 2000 days in total- because the starting point was less in balance. At the start, the TOA flux (outgoing longwave radiation) = 248 W/m². This means the climate was cooling quite a bit with the starting point we gave it.

At 180 ppm CO2, 1775 ppb CH4, 319 ppb N2O and 50% relative humidity (set at the starting point of 288K and 6.5K/km lapse rate), the surface temperature after 1,000 days = 323.9 K. At this point the TOA flux was 240.0 W/m². So overall the climate has cooled from its initial starting point but the surface is hotter.

This might seem surprising at first sight – the climate cools but the surface heats up? It’s simply that the “radiation-only” atmosphere has made it much harder for heat to get out. So the temperature drop per km of height is now much greater than it is in a convection atmosphere. Remember that we started with a temperature profile of 6.5K/km – a typical convection atmosphere.

After CO2 doubles to 560 ppm (and all other factors stay the same, including absolute humidity), the immediate effect is the TOA flux drops to 221 W/m² (once again a radiative forcing of about 19 W/m²). This is because the atmosphere is now even more “resistant” to the escape of heat by radiation. The atmosphere is more opaque and so the average emission of radiation of space moves to a higher and colder part of the atmosphere. Colder parts of the atmosphere emit less radiation than warmer parts of the atmosphere.

After the climate moves back into balance – a TOA flux of 240 W/m² – the surface temperature = 327.0 K – an increase (pre-feedback) of 3.1 K.

Compare this with the standard IPCC “with convection” no-feedback forcing of 3.7 W/m² and a “no feedback” temperature rise of about 1.2 K.

Figure 7 – with CO2 doubled instantaneously from 280ppm at 1000 days

Then I introduced a more realistic model with solar absorption by water vapor in the atmosphere (changed parameter ‘solaratm’ in the Matlab program from ‘false’ to ‘true’). Unfortunately this part of the radiative transfer program is not done by radiative transfer, only by a very crude parameterization, just to get roughly the right amount of heating by solar radiation in roughly the right parts of the atmosphere.

The equilibrium surface temperature at 280 ppm CO2 was now “only” 302.7 K (almost 30ºC). Doubling CO2 to 560 ppm created a radiative forcing of 11 W/m², and a final surface temperature of 305.5K – that is, an increase of 2.8K.

Why is the surface temperature lower? Because in the “no solar absorption in the atmosphere” model, all of the solar radiation is absorbed by the ground and has to “fight its way out” from the surface up. Once you absorb solar radiation higher up than the surface, it’s easier for this heat to get out.

### Conclusion

One of the common themes of fantasy climate blogs is that the results of radiative physics are invalidated by convection, which “short-circuits” radiation in the troposphere. No one in climate science is confused about the fact that convection dominates heat transfer in the lower atmosphere.

We can see in this set of calculations that when we have a radiation-only atmosphere the surface temperature is a lot higher than any current climate – at least when we consider a “one-dimensional” climate.

Of course, the whole world would be different and there are many questions about the amount of water vapor and the effect of circulation (or lack of it) on moving heat around the surface of the planet via the atmosphere and the ocean.

When we double CO2 from its pre-industrial value the radiative forcing is much greater in a “radiation-only atmosphere” than in a “radiative-convective atmosphere”, with the pre-feedback temperature rise 3ºC vs 1ºC.

So it is definitely true that convection short-circuits radiation in the troposphere. But the whole climate system can only gain and lose energy by radiation and this radiation balance still has to be calculated. That’s what current climate models do.

It’s often stated as a kind of major simplification (a “teaching model”) that with increases in GHGs the “average height of emission” moves up, and therefore the emission is from a colder part of the atmosphere. This idea is explained in more detail and less simplifications in Visualizing Atmospheric Radiation – Part Three – Average Height of Emission – the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions.

A legitimate criticism of current atmospheric physics is that convection is poorly understood in contrast to subjects like radiation. This is true. And everyone knows it. But it’s not true to say that convection is ignored. And it’s not true to say that because “convection short-circuits radiation” in the troposphere that somehow more GHGs will have no effect.

On the other hand I don’t want to suggest that because more GHGs in the atmosphere mean that there is a “pre-feedback” temperature rise of about 1K, that somehow the problem is all nicely solved. On the contrary, climate is very complicated. Radiation is very simple by comparison.

All the standard radiative-convective calculation says is: “all other things being equal, an doubling of CO2 from pre-industrial levels, would lead to a 1K increase in surface temperature”

All other things are not equal. But the complication is not that somehow atmospheric physics has just missed out convection. Hilarious. Of course, I realize most people learn their criticisms of climate science from people who have never read a textbook on the subject. Surprisingly, this doesn’t lead to quality criticism..

On more complexity  – I was also interested to see what happens if we readjust absolute humidity due to the significant temperature changes, i.e. we keep relative humidity constant. This led to some surprising results, so I will post them in a followup article.

### Notes

Note 1 – The boundary conditions are important if you want to understand radiative heat transfer in the atmosphere.

First of all, the downward longwave radiation at TOA (top of atmosphere) = 0. Why? Because there is no “longwave”, i.e., terrestrial radiation, from outside the climate system. So at the top of the atmosphere the downward flux = 0. As we move down through the atmosphere the flux gradually increases. This is because the atmosphere emits radiation. We can divide up the atmosphere into fictitious “layers”. This is how all numerical (finite element analysis) programs actually work. Each layer emits and each layer also absorbs. The balance depends on the temperature of the source radiation vs the temperature of the layer of the atmosphere we are considering.

At the bottom of the atmosphere, i.e., at the surface, the upwards longwave radiation is the surface emission. This emission is given by the Stefan-Boltzmann equation with an emissivity of 1.0 if we consider the surface as a blackbody which is a reasonable approximation for most surface types – for more on this, see Visualizing Atmospheric Radiation – Part Thirteen – Surface Emissivity – what happens when the earth’s surface is not a black body – useful to understand seeing as it isn’t..

At TOA, the upwards emission needs to equal the absorbed solar radiation, otherwise the climate system has an imbalance – either cooling or warming.

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## On Uses of A 4 x 2: Arrhenius, The Last 15 years of Temperature History and Other Parodies

As a friend of mine in Florida says:

You can’t kill stupid, but you can dull it with a 4×2

Some ideas are so comically stupid that I thought there was no point writing about them. And yet, one after another, people who can type are putting forward these ideas on this blog.. At first I wondered if I was the object of a practical joke. Some kind of parody. Perhaps the joke is on me. But, just in case I was wrong about the practical joke..

If you pick up a textbook on heat transfer that includes a treatment of radiative heat transfer you find no mention of Arrhenius.

If you pick up a textbook on atmospheric physics none of the equations come from Arrhenius.

Yet there is a steady stream of entertaining “papers” which describe “where Arrhenius went wrong”, “Arrhenius and his debates with Fourier”. Who cares?

Likewise, if you study equations of motion in a rotating frame there is no discussion of where Newton went wrong, or where he got it right, or debates he got right or wrong with contemporaries. Who knows? Who cares?

History is fascinating. But if you want to study physics you can study it pretty well without reading about obscure debates between people who were in the formulation stages of the field.

Here are the building blocks of atmospheric radiation:

• The emission of radiation – described by Nobel prize winner Max Planck’s equation and modified by the material property called emissivity (this is wavelength dependent)
• The absorption of radiation by a surface – described by the material property called absorptivity (this is wavelength dependent and equal at the same wavelength and direction to emissivity)
• The Beer-Lambert law of absorption of radiation by a gas
• The spectral absorption characteristics of gases – currently contained in the HITRAN database – and based on work carried out over many decades and written up in journals like Journal of Quantitative Spectroscopy and Radiative Transfer
• The theory of radiative transfer – the Schwarzschild equation – which was well documented by Nobel prize winner Subrahmanyan Chandrasekhar in his 1952 book Radiative Transfer (and by many physicists since)

The steady stream of stupidity will undoubtedly continue, but if you are interested in learning about science then you can rule out blogs that promote papers which earnestly explain “where Arrhenius went wrong”.

Hit them with a 4 by 2.

Or, ask the writer where Subrahmanyan Chandrasekhar went wrong in his 1952 work Radiative Transfer. Ask the writer where Richard M. Goody went wrong. He wrote the seminal Atmospheric Radiation: Theoretical Basis in 1964.

They won’t even know these books exist and will have never read them. These books contain equations that are thoroughly proven over the last 100 years. There is no debate about them in the world of physics. In the world of fantasy blogs, maybe.

There is also a steady stream of people who believe an idea yet more amazing. Somehow basic atmospheric physics is proven wrong because of the last 15 years of temperature history.

The idea seems to be:

More CO2 is believed to have some radiative effect in the climate because of the last 100 years of temperature history, climate scientists saw some link and tried to explain it using CO2, but now there has been no significant temperature increase for the last x years this obviously demonstrates the original idea was false..

If you think this, please go and find a piece of 4×2 and ask a friend to hit you across the forehead with it. Repeat. I can’t account for this level of stupidity but I have seen that it exists.

An alternative idea, that I will put forward, one that has evidence, is that scientists discovered that they can reliably predict:

• emission of radiation from a surface
• emission of radiation from a gas
• absorption of radiation by a surface
• absorption of radiation by a gas
• how to add up, subtract, divide and multiply, raise numbers to the power of, and other ninja mathematics

The question I have for the people with these comical ideas:

Do you think that decades of spectroscopy professionals have just failed to measure absorption? Their experiments were some kind of farce? No one noticed they made up all the results?

Do you think Max Planck was wrong?

It is possible that climate is slightly complicated and temperature history relies upon more than one variable?

Did someone teach you that the absorption and emission of radiation was only “developed” by someone analyzing temperature vs CO2 since 1970 and not a single scientist thought to do any other measurements? Why did you believe them?

Bring out the 4×2.

Note – this article is a placeholder so I don’t have to bother typing out a subset of these points for the next entertaining commenter..

Update July 10th with the story of Fred the Charlatan

Let’s take the analogy of a small boat crossing the Atlantic.

Analogies don’t prove anything, they are for illustration. For proof, please review Theory and Experiment – Atmospheric Radiation.

We’ve done a few crossings and it’s taken 45 days, 42 days and 46 days (I have no idea what the right time is, I’m not a nautical person).

We measure the engine output – the torque of the propellors. We want to get across quicker. So Fred the engine guy makes a few adjustments and we remeasure the torque at 5% higher. We also do Fred’s standardized test, which is to zip across a local sheltered bay with no currents, no waves and no wind – the time taken for Fred’s standarized test is 4% faster. Nice.

So we all set out on our journey across the Atlantic. Winds, rain, waves, ocean currents. We have our books to read, Belgian beer and red wine and the time flies. Oh no, when we get to our final destination, it’s actually taken 47 days.

Clearly Fred is some kind of charlatan! No need to check his measurements or review the time across the bay. We didn’t make it across the Atlantic in less time and clearly the ONLY variable involved in that expedition was the output of the propellor.

Well, there’s no point trying to use more powerful engines to get across the Atlantic (or any ocean) faster. Torque has no relationship to speed. Case closed.

Analogy over.

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## Ghosts of Climates Past – Nineteen – Ice Sheet Models I

In Part Seven – GCM I  through Part Ten – GCM IV we looked at GCM simulations of ice ages.

These were mostly attempts at “glacial inception”, that is, starting an ice age. But we also saw a simulation of the last 120 kyrs which attempted to model a complete ice age cycle including the last termination. As we saw, there were lots of limitations..

One condition for glacial inception, “perennial snow cover at high latitudes”, could be produced with a high-resolution coupled atmosphere-ocean GCM (AOGCM), but that model did suffer from the problem of having a cold bias at high latitudes.

The (reasonably accurate) simulation of a whole cycle including inception and termination came by virtue of having the internal feedbacks (ice sheet size & height and CO2 concentration) prescribed.

Just to be clear to new readers, these comments shouldn’t indicate that I’ve uncovered some secret that climate scientists are trying to hide, these points are all out in the open and usually highlighted by the authors of the papers.

In Part Nine – GCM III, one commenter highlighted a 2013 paper by Ayako Abe-Ouchi and co-workers, where the journal in question, Nature, had quite a marketing pitch on the paper. I made brief comment on it in a later article in response to another question, including that I had emailed the lead author asking a question about the modeling work (how was a 120 kyr cycle actually simulated?).

Most recently, in Eighteen – “Probably Nonlinearity” of Unknown Origin, another commented highlighted it, which rekindled my enthusiasm, and I went back and read the paper again. It turns out that my understanding of the paper had been wrong. It wasn’t really a GCM paper at all. It was an ice sheet paper.

There is a whole field of papers on ice sheet models deserving attention.

### GCM review

Let’s review GCMs first of all to help us understand where ice sheet models fit in the hierarchy of climate simulations.

GCMs consist of a number of different modules coupled together. The first GCMs were mostly “atmospheric GCMs” = AGCMs, and either they had a “swamp ocean” = a mixed layer of fixed depth, or had prescribed ocean boundary conditions set from an ocean model or from an ocean reconstruction.

Less commonly, unless you worked just with oceans, there were ocean GCMs with prescribed atmospheric boundary conditions (prescribed heat and momentum flux from the atmosphere).

Then coupled atmosphere-ocean GCMs came along = AOGCMs. It was a while before these two parts matched up to the point where there was no “flux drift”, that is, no disappearing heat flux from one part of the model.

Why so difficult to get these two models working together? One important reason comes down to the time-scales involved, which result from the difference in heat capacity and momentum of the two parts of the climate system. The heat capacity and momentum of the ocean is much much higher than that of the atmosphere.

And when we add ice sheets models – ISMs – we have yet another time scale to consider.

• the atmosphere changes in days, weeks and months
• the ocean changes in years, decades and centuries
• the ice sheets changes in centuries, millennia and tens of millenia

This creates a problem for climate scientists who want to apply the fundamental equations of heat, mass & momentum conservation along with parameterizations for “stuff not well understood” and “stuff quite-well-understood but whose parameters are sub-grid”. To run a high resolution AOGCM for a 1,000 years simulation might consume 1 year of supercomputer time and the ice sheet has barely moved during that period.

### Ice Sheet Models

Scientists who study ice sheets have a whole bunch of different questions. They want to understand how the ice sheets developed.

What makes them grow, shrink, move, slide, melt.. What parameters are important? What parameters are well understood? What research questions are most deserving of attention? And:

Does our understanding of ice sheet dynamics allow us to model the last glacial cycle?

To answer that question we need a model for ice sheet dynamics, and to that we need to apply some boundary conditions from some other “less interesting” models, like GCMs. As a result, there are a few approaches to setting the boundary conditions so we can do our interesting work of modeling ice sheets.

Before we look at that, let’s look at the dynamics of ice sheets themselves.

### Ice Sheet Dynamics

First, in the theme of the last paper, Eighteen – “Probably Nonlinearity” of Unknown Origin, here is Marshall & Clark 2002:

The origin of the dominant 100-kyr ice-volume cycle in the absence of substantive radiation forcing remains one of the most vexing questions in climate dynamics

We can add that to the 34 papers reviewed in that previous article. This paper by Marshall & Clark is definitely a good quick read for people who want to understand ice sheets a little more.

Ice doesn’t conduct a lot of heat – it is a very good insulator. So the important things with ice sheets happen at the top and the bottom.

At the top, ice melts, and the water refreezes, runs off or evaporates. In combination, the loss is called ablation. Then we have precipitation that adds to the ice sheet. So the net effect determines what happens at the top of the ice sheet.

At the bottom, when the ice sheet is very thin, heat can be conducted through from the atmosphere to the base and make it melt – if the atmosphere is warm enough. As the ice sheet gets thicker, very little heat is conducted through. However, there are two important sources of heat for surface heating which results in “basal sliding”. One source is geothermal energy. This is around 0.1 W/m² which is very small unless we are dealing with an insulating material (like ice) and lots of time (like ice sheets). The other source is the shear stress in the ice sheet which can create a lot of heat via the mechanics of deformation.

Once the ice sheet is able to start sliding, the dynamics create a completely different result compared to an ice sheet “cold-pinned” to the rock underneath.

Some comments from Marshall and Clark:

Ice sheet deglaciation involves an amount of energy larger than that provided directly from high-latitude radiation forcing associated with orbital variations. Internal glaciologic, isostatic, and climatic feedbacks are thus essential to explain the deglaciation.

..Moreover, our results suggest that thermal enabling of basal flow does not occur in response to surface warming, which may explain why the timing of the Termination II occurred earlier than predicted by orbital forcing [Gallup et al., 2002].

Results suggest that basal temperature evolution plays an important role in setting the stage for glacial termination. To confirm this hypothesis, model studies need improved basal process physics to incorporate the glaciological mechanisms associated with ice sheet instability (surging, streaming flow).

..Our simulations suggest that a substantial fraction (60% to 80%) of the ice sheet was frozen to the bed for the first 75 kyr of the glacial cycle, thus strongly limiting basal flow. Subsequent doubling of the area of warm-based ice in response to ice sheet thickening and expansion and to the reduction in downward advection of cold ice may have enabled broad increases in geologically- and hydrologically-mediated fast ice flow during the last deglaciation.

Increased dynamical activity of the ice sheet would lead to net thinning of the ice sheet interior and the transport of large amounts of ice into regions of intense ablation both south of the ice sheet and at the marine margins (via calving). This has the potential to provide a strong positive feedback on deglaciation.

The timescale of basal temperature evolution is of the same order as the 100-kyr glacial cycle, suggesting that the establishment of warm-based ice over a large enough area of the ice sheet bed may have influenced the timing of deglaciation. Our results thus reinforce the notion that at a mature point in their life cycle, 100-kyr ice sheets become independent of orbital forcing and affect their own demise through internal feedbacks.

[Emphasis added]

In this article we will focus on a 2007 paper by Ayako Abe-Ouchi, T Segawa & Fuyuki Saito. This paper is essentially the same modeling approach used in Abe-Ouchi’s 2013 Nature paper.

### The Ice Model

The ice sheet model has a time step of 2 years, with 1° grid from 30°N to the north pole, 1° longitude and 20 vertical levels.

Equations for the ice sheet include sliding velocity, ice sheet deformation, the heat transfer through the lithosphere, the bedrock elevation and the accumulation rate on the ice sheet.

Note, there is a reference that some of the model is based on work described in Sensitivity of Greenland ice sheet simulation to the numerical procedure employed for ice sheet dynamics, F Saito & A Abe-Ouchi, Ann. Glaciol., (2005) – but I don’t have access to this journal. (If anyone does, please email the paper to me at scienceofdoom – you know what goes here – gmail.com).

How did they calculate the accumulation on the ice sheet? There is an equation:

Acc=Aref×(1+dP)Ts

Ts is the surface temperature, dP is a measure of aridity and Aref is a reference value for accumulation. This is a highly parameterized method of calculating how much thicker or thinner the ice sheet is growing. The authors reference Marshall et al 2002 for this equation, and that paper is very instructive in how poorly understood ice sheet dynamics actually are.

Here is one part of the relevant section in Marshall et al 2002:

..For completeness here, note that we have also experimented with spatial precipitation patterns that are based on present-day distributions.

Under this treatment, local precipitation rates diminish exponentially with local atmospheric cooling, reflecting the increased aridity that can be expected under glacial conditions (Tarasov and Peltier, 1999).

Paleo-precipitation under this parameterization has the form:

P(λ,θ,t) = Pobs(λ,θ)(1+dp)ΔT(λ,θ,t) x exp[βp.max[hs(λ,θ,t)-ht,0]]       (18)

The parameter dP in this equation represents the percentage of drying per 1C; Tarasov and Peltier (1999) choose a value of 3% per °C; dp = 0:03.

[Emphasis added, color added to highlight the relevant part of the equation]

So dp is a parameter that attempts to account for increasing aridity in colder glacial conditions, and in their 2002 paper Marshall et al describe it as 1 of 4 “free parameters” that are investigated to see what effect they have on ice sheet development around the LGM.

Abe-Ouchi and co-authors took a slightly different approach that certainly seems like an improvement over Marshall et al 2002:

So their value of aridity is just a linear function of ice sheet area – from zero to a fixed value, rather than a fixed value no matter the ice sheet size.

How is Ts calculated? That comes, in a way, from the atmospheric GCM, but probably not in a way that readers might expect. So let’s have a look at the GCM then come back to this calculation of Ts.

### Atmospheric GCM Simulations

There were three groups of atmospheric GCM simulations, with parameters selected to try and tease out which factors have the most impact.

Group One: high resolution GCM – 1.1º latitude and longitude and 20 atmospheric vertical levels with fixed sea surface temperature. So there is no ocean model, the ocean temperature are prescribed. Within this group, four experiments:

• A control experiment – modern day values
• LGM (last glacial maximum) conditions for CO2 (note 1) and orbital parameters with
• no ice
• LGM ice extent but zero thickness
• LGM ice extent and LGM thickness

So the idea is to compare results with and without the actual ice sheet so see how much impact orbital and CO2 values have vs the effect of the ice sheet itself – and then for the ice sheet to see whether the albedo or the elevation has the most impact. Why the elevation? Well, if an ice sheet is 1km thick then the surface temperature will be something like 6ºC colder. (Exactly how much colder is an interesting question because we don’t know what the lapse rate actually was). There will also be an effect on atmospheric circulation – you’ve stuck a “mountain range” in the path of wind so this changes the circulation.

Each of the four simulations was run for 11 or 13 years and the last 10 years’ results used:

From Abe-Ouchi et al 2007

Figure 1

It’s clear from this simulation that the full result (left graphic) is mostly caused by the ice sheet (right graphic) rather than CO2, orbital parameters and the SSTs (middle graphic). And the next figure in the paper shows the breakdown between the albedo effect and the height of the ice sheet:

From Abe-Ouchi et al 2007

Figure 2 – same color legend as figure 1

Now a lapse rate of 5K/km was used. What happens if the lapse rate of 9K/km was used instead? There were no simulations done with different lapse rates.

..Other lapse rates could be used which vary depending on the altitude or location, while a lapse rate larger than 7 K/km or smaller than 4 K/km is inconsistent with the overall feature. This is consistent with the finding of Krinner and Genthon (1999), who suggest a lapse rate of 5.5 K/km, but is in contrast with other studies which have conventionally used lapse rates of 8 K/km or 6.5 K/km to drive the ice sheet models..

Group Two – medium resolution GCM 2.8º latitude and longitude and 11 atmospheric vertical levels, with a “slab ocean” – this means the ocean is treated as one temperature through the depth of some fixed layer, like 50m. So it is allowing the ocean to be there as a heat sink/source responding to climate, but no heat transfer through to a deeper ocean.

There were five simulations in this group, one control (modern day everything) and four with CO2 & orbital parameters at the LGM:

• no ice sheet
• LGM ice extent, but flat
• 12 kyrs ago ice extent, but flat
• 12 kyrs ago ice extent and height

So this group takes a slightly more detailed look at ice sheet impact. Not surprisingly the simulation results give intermediate values for the ice sheet extent at 12 kyrs ago.

Group Three – medium resolution GCM as in group two, and ice sheets either at present day or LGM, with nine simulations covering different orbital values, different CO2 values of present day, 280 or 200 ppm.

There was also some discussion of the impact of different climate models. I found this fascinating because the difference between CCSM and the other models appears to be as great as the difference in figure 2 (above) which identifies the albedo effect as more significant than the lapse rate effect:

From Abe-Ouchi et al 2007

Figure 3

And this naturally has me wondering about how much significance to put on the GCM simulation results shown in the paper. The authors also comment:

Based on these GCM results we conclude there remains considerable uncertainty over the actual size of the albedo effect.

Given there is also uncertainty over the lapse rate that actually occurred, it seems there is considerable uncertainty over everything.

Now let’s return to the ice sheet model, because so far we haven’t seen any output from the ice sheet model.

### GCM Inputs into the Ice Sheet Model

The equation which calculates the change in accumulation on the ice sheet used a fairly arbitrary parameter dp, with (1+dp) raised to the power of Ts.

The ice sheet model has a 2 year time step. The GCM results don’t provide Ts across the surface grid every 2 years, they are snapshots for certain conditions. The ice sheet model uses this calculation for Ts:

Ts = Tref + ΔTice + ΔTco2 + ΔTinsol + ΔTnonlinear

Tref is the reference temperature which is present day climatology. The other ΔT (change in temperature) values are basically a linear interpolation from two values of the GCM simulations. Here is the ΔTCo2 value:

So think of it like this – we have found Ts at one value of CO2 higher and one value of CO2 lower from some snapshot GCM simulations. We plot a graph with Co2 on the x-axis and Ts on the y-axis with just two points on the graph from these two experiments and we draw a straight line between the two points.

To calculate Ts at say 50 kyrs ago we look up the CO2 value at 50 kyrs from ice core data, and read the value of TCO2 from the straight line on the graph.

Likewise for the other parameters. Here is ΔTinsol:

So the method is extremely basic. Of course the model needs something..

Now, given that we have inputs for accumulation on the ice sheet, the ice sheet model can run. Here are the results. The third graph (3) is the sea level from proxy results so is our best estimate of reality, with (4) providing model outputs for different parameters of d0 (“desertification” or aridity) and lapse rate, and (5) providing outputs for different parameters of albedo and lapse rate:

From Abe-Ouchi et al 2007

Figure 4

There are three main points of interest.

Firstly, small changes in the parameters cause huge changes in the final results. The idea of aridity over ice sheets as just linear function of ice sheet size is very questionable itself. The idea of a constant lapse rate is extremely questionable. Together, using values that appear realistic, we can model much less ice sheet growth (sea level drop) or many times greater ice sheet growth than actually occurred.

Secondly, notice that the time of maximum ice sheet (lowest sea level) for realistic results show sea level starting to rise around 12 kyrs, rather than the actual 18 kyrs. This might be due to the impact of orbital factors which were at quite a low level (i.e., high latitude summer insolation was at quite a low level) when the last ice age finished, but have quite an impact in the model. Of course, we have covered this “problem” in a few previous articles in this series. In the context of this model it might be that the impact of the southern hemisphere leading the globe out of the last ice age is completely missing.

Thirdly – while this might be clear to some people, but for many new to this kind of model it won’t be obvious – the inputs for the model are some limits of the actual history. The model doesn’t simulate the actual start and end of the last ice age “by itself”. We feed into the GCM model a few CO2 values. We feed into the GCM model a few ice sheet extent and heights that (as best as can be reconstructed) actually occurred. The GCM gives us some temperature values for these snapshot conditions.

In the case of this ice sheet model, every 2 years (each time step of the ice sheet model) we “look up” the actual value of ice sheet extent and atmospheric CO2 and we linearly interpolate the GCM output temperatures for the current year. And then we crudely parameterize these values into some accumulation rate on the ice sheet.

### Conclusion

This is our first foray into ice sheet models. It should be clear that the results are interesting but we are at a very early stage in modeling ice sheets.

The problems are:

• the computational load required to run a GCM coupled with an ice sheet model over 120 kyrs is much too high, so it can’t be done
• the resulting tradeoff uses a few GCM snapshot values to feed linearly interpolated temperatures into a parameterized accumulation equation
• the effect of lapse rate on the results is extremely large and the actual value for lapse rate over ice sheets is very unlikely to be a constant and is also not known
• our understanding of ice sheet fundamental equations are still at an early stage, as readers can see by reviewing the first two papers below, especially the second one

### Articles in this Series

Part One – An introduction

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

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

Part Four – Understanding Orbits, Seasons and Stuff – how the wobbles and movements of the earth’s orbit affect incoming solar radiation

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

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

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

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

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

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

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

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

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

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

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

Fourteen – Concepts & HD Data – getting a conceptual feel for the impacts of obliquity and precession, and some ice age datasets in high resolution

Fifteen – Roe vs Huybers – reviewing In Defence of Milankovitch, by Gerard Roe

Sixteen – Roe vs Huybers II – remapping a deep ocean core dataset and updating the previous article

Seventeen – Proxies under Water I – explaining the isotopic proxies and what they actually measure

Eighteen – “Probably Nonlinearity” of Unknown Origin – what is believed and what is put forward as evidence for the theory that ice age terminations were caused by orbital changes

### References

Basal temperature evolution of North American ice sheets and implications for the 100-kyr cycle, SJ Marshall & PU Clark, GRL (2002) – free paper

North American Ice Sheet reconstructions at the Last Glacial Maximum, SJ Marshall, TS James, GKC Clarke, Quaternary Science Reviews (2002) – free paper

Climatic Conditions for modelling the Northern Hemisphere ice sheets throughout the ice age cycle, A Abe-Ouchi, T Segawa, and F Saito, Climate of the Past (2007) – free paper

Insolation-driven 100,000-year glacial cycles and hysteresis of ice-sheet volume, Ayako Abe-Ouchi, Fuyuki Saito, Kenji Kawamura, Maureen E. Raymo, Jun’ichi Okuno, Kunio Takahashi & Heinz Blatter, Nature (2013) – paywall paper

### Notes

Note 1 – the value of CO2 used in these simulations was 200 ppm, while CO2 at the LGM was actually 180 ppm. Apparently this value of 200 ppm was used in a major inter-comparison project (the PMIP), but I don’t know the reason why. PMIP = Paleoclimate Modelling Intercomparison Project, Joussaume and Taylor, 1995.

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