From Marshall & Plumb (2008)

Figure 1 – Click for a larger image

We see that the equator is warmer than the poles and the surface is warmer than the upper troposphere (“troposphere” = lower atmosphere). No surprises.

Here is “potential temperature”, whatever that is..

From Marshall & Plumb (2008)

Figure 2 – Click for a larger image

We see that – whatever “potential temperature” is – the equator is warmer than the poles, but this version of temperature increases with height.

Why does temperature decrease with height? What is potential temperature? And why does it increase with height?

The Lapse Rate

Atmospheric pressure decreases with height. This is because as you go higher up there is less air above you, and therefore less downward force due to the weight of this air.

Because pressure decreases – and because air is a compressible fluid - air that rises expands (and air that sinks contracts).

Air that expands does “work” against its surroundings and because of the first law of thermodynamics (conservation of energy) this work needs to be paid for. So internal energy is consumed in expanding the parcel of air outwards against the atmosphere. And a reduction in internal energy means a reduction in temperature.

• Air that rises expands
• Expanding air cools

A little bit more technically.. adiabatic expansion is what we are talking about. An adiabatic process is one where no heat is exchanged with the surroundings. This is a reasonable approximation for typical rising air. It is reasonable because conduction is an extremely slow process (= negligible) in the atmosphere and radiative heat transfer is quite slow.

So if heat can’t be exchanged between a “parcel of air” and its surroundings it is relatively simple to calculate how the temperature changes. An example which contains way too much detail (because it is debunking a “debunking”) at Paradigm Shifts in Convection and Water Vapor?

The essence of the calculation is to equate internal energy changes with work done on the environment.

Textbooks usually start off with the simplest version, the dry adiabatic lapse rate, or DALR. (The “lapse rate” is the change in temperature with height of a parcel of air).

The DALR is for air without any water vapor. Now water vapor is very influential in our climate. The reason for neglecting it and starting off with this simplification is:

• the calculation is easy and everyone (almost) can understand it
• it represents one extreme of the atmosphere (polar climates and upper troposphere)

The result from this simplification:

Change in temperature with height = -g/c_p ≈ -10 °C/km, where g = acceleration due to gravity = 9.8 m/s² and c_p = heat capacity of air at constant pressure ≈ 1 J/kg.K

So for every km we displace air upwards it cools by about 10°C – so long as we displace it reasonably quickly. Well, this is true if it is dry.

A note on conventions – dry parcels of air moved upwards cool by 10°C per km, but the lapse rate is usually written as a positive number. So a cooling of 10 °C/km =  -10 °C/km, but by convention, equals a “lapse rate” of +10 °C/km. This makes it very confusing when people say things like “the environmental lapse rate must be less than the adiabatic lapse rate“. Are we talking about the number with the minus sign in front? Or not?

It’s not easy to think about negative numbers being less than other negative numbers when the “less than” test is applied after they have been made into positive numbers. Not for me anyway. I have to write it down each time.

The Saturated Lapse Rate

If a parcel of air contains water vapor and it cools sufficiently then the water vapor condenses. This releases latent heat.

As a result, moist rising air cools slower than dry rising air

So the saturated adiabatic lapse rate is “less than” the dry adiabatic lapse rate.

E.g. the change in temperature with height of a dry parcel of air ≈ -10 °C/km, while the change in temperature with height of a moist parcel of air in the tropics near the surface ≈ -4 °C/km.

Conventionally we say that the saturated adiabatic lapse rate is less than the dry adiabatic lapse rate. Because we write them as positive numbers.

Now note the caveats around the value for the moist parcel of air rising. I said “..in the tropics near the surface..”, but for the DALR there are no caveats. That’s because once we consider moisture we have to consider how much water vapor and the amount varies hugely depending on temperature (and also on other factors – see Clouds and Water Vapor – Part Three).

The maths is somewhat harder for the saturated adiabatic lapse rate but it’s not conceptually more difficult, there is just an addition of energy (from condensing water vapor) to offset the work done.

Potential Temperature

Potential temperature is usually written with the Greek letter θ.

θ = T.(p₀/p)^k

where T = (real) temperature, p = pressure, p₀ = reference pressure (usually at 1000 mbar) and k = R/c_p = 2/7 for our atmosphere (more on this in a later article)

With a bit of tedious maths we can prove that θ stays constant under adiabatic conditions (for dry air).

Let’s look at what that means.

Suppose the surface (1000 mbar) temperature = 288 K (15°C) so also θ = 288K.

Now the air is moved (adiabatically) to 800 mbar, so T = 270 K. That’s what you expect – temperature falls with height. And no change to potential temperature, so θ = 288 K.

Now we move the air to 600 mbar, and T = 249 K. More reduction of temperature. And still θ = 288 K.

So is this a useful parameter – move the air (adiabatically) and the potential temperature stays the same?

The parameter is mathematically sound, but whether it is useful remains to be seen. As an artificial construct no doubt many people will be shaking their heads..

Stability and Potential Temperature Profile

In Density, Stability and Motion in Fluids we saw that for a fluid to be stable, lighter fluid must be above heavier fluid. No surprise to anyone.

And we saw that in mechanical terms equilibrium is different from stability.

An unstable equilibrium can exist, but a slight displacement will turn the instability into motion. Whereas with a stable equilibrium a slight displacement (or a large displacement) will result in a restoring force back to its original position. For the simplest case – an incompressible fluid – this means that the temperature must increase with height.

If you watched the accompanying video of a tank of water being heated from below you would have seen that the instability caused turbulent motion until finally the tank was well-mixed.

We left the more complex case of compressible fluids (like air) until today. What we will find is that with a compressible fluid potential temperature is effectively the same as “real” temperature for an incompressible fluid.

So if potential temperature increases with height the fluid is stable, but if potential temperature decreases with height the fluid is unstable.

Let’s look at two examples:

Figure 3

On the left hand side we see an example where potential temperature decreases with height. At the surface, θ = 288 K but at 800 mbar, θ = 275 K. A parcel of air displaced adiabatically from the surface to 800 mbar will keep its potential temperature of 288 K. Now we convert that to real temperatures. The environmental temperature at 800 mbar is 258 K, but the parcel of air cools to only 270 K. This means the displaced parcel is warmer than the surroundings, so it is less dense – and therefore it keeps rising.

This case is unstable – clearly any air that starts rising or falling (perhaps due to atmospheric winds, pressure differentials, etc) will keep rising or falling.

On the right hand side we see potential temperature decreasing with height. The parcel of air displaced from the surface to 800 mbar reaches the same temperature as on the left – 270 K. But here the environmental temperature is 281 K. So the parcel of air is cooler than the surrounding air, so it is more dense – and so it falls.

This case is stable – any air that starts rising or falling experiences a restoring force.

So the potential temperature profile with height tells us whether the atmosphere is stable, neutral or unstable. If potential temperature increases with height the atmosphere is stable, and if potential temperature decreases with height the atmosphere is unstable.

This is exactly the same as comparing the actual temperature change with the lapse rate.

Both answer the same question about atmospheric stability.

Moist Potential Temperature

The previous section slightly over-simplified things because potential temperature is with reference to dry air and yet moisture changes the way in which temperature decreases with height.

So here is the real deal – moist potential temperature. This is also known as equivalent potential temperature:

From Marshall & Plumb (2008)

Figure 4 – Click for a larger image

Here we see the “real potential temperature” and notice that especially in the tropics moist potential temperature is almost constant with height – up to the tropopause at 200 mbar. This is due to convection creating a well-mixed atmosphere. In the polar regions we see that the atmosphere is still quite stratified, which is due to the lack of convective mixing.

Conclusion

Potential temperature is very useful. It is a method of comparing the temperature of air at two different heights.

And if potential temperature is constant or increasing with height then the atmosphere is stable.

The atmosphere is mostly stable for dry air. If you refer back to figure 2 you see that (dry) potential temperature is quite stratified which means any displaced air experiences a restoring force. So it is moisture in the air that is the enabler for most of the convection that takes place. Figure 4 shows us that the atmosphere is “finely” balanced as far as moist convection is concerned.

(Remember of course that these graphs are annual mean values. It doesn’t mean that dry convection does not occur).

Potential temperature is also a useful metric because the change of potential temperature with height can be used to calculate the strength of the restoring force on displaced air. The result is the buoyancy frequency and the period of internal gravity waves.

So in advance of that article, here are some basics on stability and density in fluids. Science, unlike for example most of history and politics, is one subject built on another. If we fail to grasp fundamental concepts clearly then the more difficult subjects, the next steps, will always be a mystery.

Therefore, if you reach the end of this article and don’t feel that the subject has been clearly explained, ask away.

Stability

This can be an involved subject but we only want to consider a very simple aspect of stability. Take a look at the two cases below. In both cases the ball is not moving. It is in equilibrium:

Figure 1

Everyone can appreciate the difference between the two from common experience.

The first case is stable – push the ball a little to the left or right and it moves back to the center, overshoots, comes back, overshoots and so on.. and eventually ends up stationary at its starting point. It oscillates. Real world friction dissipates the energy injected into the system from the initial disturbance and causes the ball to finally return to rest.

The second case is unstable – push the ball a little to the left or right and it accelerates away and never returns to its starting point.

The reason for stating the apparent obvious is that both are in equilibrium. It is only with some kind of disturbance that the unstable situation “rearranges the energy of the system”. Without some kind of disturbance nothing happens. Of course, depending on the exact setup the disturbance needed might only be tiny.

Another way to think about stability is potential energy vs kinetic energy.

For newcomers to physics or mechanics, a non-technical description is:

• potential energy is the energy stored in a system
• kinetic energy is the energy from movement

Often with simple motion we can think about potential energy being converted into kinetic energy and vice-versa. Take a bouncy ball released from height so that it drops to the floor. It starts with zero kinetic energy and by the time the ball has hit the floor the potential energy (height) has been turned into kinetic energy (motion). Then the ball bounces up in the air, stops at a point and returns to the floor.

Energy is dissipated along the way by the friction due to air, and the energy lost in the impact so eventually the ball finishes at rest on the floor.

When a mass is moved higher in a gravitational field the potential energy is increased. When a mass is moved lower in a gravitational field the potential energy is reduced.

When potential energy is reduced, the change can be released as kinetic energy (like a falling ball). If however you increase the height of the mass then the increased potential energy can only consume kinetic energy (or it has to be supplied from elsewhere).

Take a look back at the two scenarios in figure 1. In the first scenario a disturbance increases potential energy so we have to do work to move it up the side of the container. Therefore, it is stable.

In the second scenario a disturbance reduces potential energy so we don’t need to do any work to create kinetic energy, or motion.  (Only the small work needed to disturb the ball from its position). Therefore, it is unstable.

Hopefully, this is so clear that readers wonder why I am still explaining it..

Density in an Incompressible Fluid

Let’s look at the simple case of an incompressible fluid like water where density depends on temperature but not on pressure.

And let’s look at why it is that lighter fluids actually rise:

From Marshall & Plumb (2008)

Figure 2

The description in the figure 2 caption is the best (i.e., simplest) explanation of why lighter fluids rise.

Displacement of a Parcel in An Incompressible Fluid 

So consider a fluid parcel being displaced upwards. For now, it doesn’t matter why or how. Just that it is moved. If the movement is done quickly enough it will not lose or gain heat during its journey (because heat exchange can only take place by diffusion, which is a very slow process in most liquids).

We call this movement without exchange of heat (with the surroundings) an adiabatic process.

And because the fluid is incompressible it will do no work on its surroundings, so its internal energy will be conserved. As a result its temperature will stay the same and so its density will also stay the same.

If the density of the surrounding fluid increases with height then this parcel will accelerate upwards, because the parcel has a lower density than its environment and so (as in figure 2) the differential pressure will “push it” upwards.

If the density of the surrounding fluid reduces with height then the parcel will be slowed down and experience a restoring force back to where it came from.

Now usually density is related to temperature. In the case of water, as temperature is increased density is decreased.

So if the temperature of the surrounding fluid increases with height, the density decreases with height and displaced parcels of fluid have a restoring force back to their origin. So in this case (this normal case) the fluid is stable.

If the temperature of the surrounding fluid decreases with height, the density increases and displaced parcels accelerate in the direction in which they started moving. So in this case (this strange case) the fluid is unstable. Clearly the instability will result in fluid movement and therefore ultimately in stability.

So it’s all about the difference between the change in temperature of a displaced parcel vs the change in temperature with height of the environment. And the displacement can take place for many different reasons. Don’t think about the reason for the initial displacement – as with figure 1 just think about whether a displacement will result in a restoring force back to the starting point, or in an acceleration away from the starting point.

Here is a very educational example of convection as a result of heating a liquid from below. This example comes from the webpages accompanying the Marshall & Plumb (2008) textbook – I recommend the video link within that page (but read the text and diagrams first):

Figure 3 – Click for movie

And in the next article we will consider what happens with a compressible fluid like air. In that case when a parcel of air moves upwards, and expands because of lower pressure, its temperature drops. That makes the stability question slightly more complicated, but the same principles apply.

Further reading – new article: Potential Temperature

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, E_a:

E_a = αR_L↓ ….[eqn 1]

where α = absorptivity of the surface at these wavelengths, R_L↓ = 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:

E_a = f(α,T_atm-T_sur).R_L↓ ….[eqn 2]
where T_atm = temperature of the atmosphere, T_sur = temperature of the surface

With the function f being defined like this:

f(α,T_atm-T_sur) = α, when T_atm ≥ T_sur and

f(α,T_atm-T_sur) = 0, when T_atm < T_sur

In English, it says something like energy from the atmosphere absorbed by the surface = 0 when the temperature of the atmosphere is less than the temperature of the surface.

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

My Prediction

No one will produce an equation which shows how 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.

If you roll a ball along a flat frictionless surface it keeps going in the same direction. This is because objects that have no forces on them continue in the same direction at the same speed. (The combination of direction and speed is known as velocity, which is a vector. A vector consists of a magnitude (e.g. speed) and a direction).

Well, that statement was not strictly true – because it wasn’t specific enough.

If you get onto a merry go round and launch your same ball in one direction you observe it move away in a curved arc. But someone above the merry go round, perhaps someone who had climbed up a pole and was looking down, would observe the ball moving in a straight line.

It’s all about frames of reference.

Now we live on planet that is rotating so we have to consider the “merry go round” effect.

There are two approaches for a mathematical basis (and we will keep the maths separated):

• consider everything from an inertial frame – as if all motion was viewed from space (note 1)
• consider everything from the surface of the planet

If we considered everything from space then the problem would actually be more difficult. On the plus side thrown balls would go in a straight line (as normal). On the minus side the boundaries of the oceans, mountains and everything else important would be constantly on the move and we would need mathematical trickery beyond most people’s comprehension.

So everyone goes for option b – consider motion from the surface of the planet. This means the frame of reference is constantly on the move.

Coriolis

The excellent Atmosphere, Ocean and Climate Dynamics by Marshall & Plumb (2008) comes with a number of accompanying web pages most of which have some videos.

See GFDLab V: Inertial Circles – visualizing the Coriolis force for some detail and the video link, or click on the image below for the video link:

Figure 1 – Click for the video

• the left hand video is the inertial frame of reference – stationary camera
• the right hand video is the rotational frame of reference – the camera is moving with the turntable

This is the best video I have found for making clear what happens in a rotating frame.

With some relatively simple maths, the equations of motion in an inertial frame get transformed into a rotating frame of reference.

Two new terms get introduced:

• the Coriolis acceleration = “stuff appears to veer off to the side as far as I can tell” effect
• centrifugal acceleration = ”things get thrown outwards like on a merry-go-round that goes very fast” effect

The centrifugal acceleration is not so significant, just a slight modifier of magnitude and direction to the very strong gravitational effect. But the Coriolis effect is very significant.

Now the Coriolis effect is easy to demonstrate on a rotating table, but we live on a rotating sphere and so there are some complexities that require the use of vector maths to calculate.

Mathematically it is easy to show that the Coriolis effect is modified by a factor relating to latitude. Specifically the effect is multiplied by the sine of the latitude, which means that at the equator the Coriolis effect is zero (sin 0° = 0), and at 30° it is half the maximum (sin 30°=0.5) and at the poles it has the full effect (sin 90° = 1.0).

I found it difficult to come up with a conceptual model which helps readers see why this is so. Readers who have had to think about the effect of resolving forces and rotations into orthogonal directions might be able to provide a conceptual picture – so please add comment if you think so. (Note 2).

Some Maths

The Coriolis effect has to be seen in the light of the other terms in the equation of motion.

The intimidating version, for those not used to the equations of motion for fluids in a Lagrangian formulation (note 3):

Du/Dt + 1/ρ.∇p + ∇φ + fz x u = F_r …..[1]

where bold characters are vectors, z is the unit vector in the upward direction, u = velocity vector (u,v,w), φ = gravitational potential modified by the centrifugal force, ρ = density, p = pressure and f = Coriolis parameter.

And in not-quite-plain English, the change in velocity with time (following a moving parcel of fluid) plus pressure force plus gravitional force plus the coriolis force equals the frictional force (note that the terms are effectively for unit mass).

The Coriolis parameter:

f = 2Ω sinφ …..[2]

where Ω = the rotational speed of the earth (in radians/sec) = 2 π / (24*3600) = 7.3 x 10^-5 /s

And the simpler version in each local x,y,x direction with some simplifications applied (like the hydrostatic equilibrium approximation):

Du/Dt + 1/ρ . ∂p/∂x –  f.v = F_x ….(local x-direction) …[3a]

Dv/Dt + 1/ρ . ∂p/∂y + f.u = F_y ….(local y-direction) …[3b]

                  1/ρ . ∂p/∂z  + g = 0 ….(local z-direction) …[3c]

Geostrophic Balance and the Magnitude of the Coriolis Effect

Analysis of fluid flows is often carried out via non-dimensional ratios.

The Rossby number is the ratio of acceleration terms to the Coriolis force, and in the atmosphere at mid-latitudes is typically 0.1.

Another way of saying this is that the acceleration terms in equation 3 are a lot smaller than the Coriolis term. And in the free atmosphere (away from the boundary layer with the earth’s surface) the friction terms are negligible. This simplifies equation 3:

u_g = – 1/fρ . ∂p/∂y ….[4a]

v_g =   1/fρ . ∂p/∂x ….[4b]

With u_g, v_g defining the solution – geostrophic balance – to these simplified equations. This tells us that the E-W wind speed is proportional to the pressure change in the N-S direction, and the N-S wind speed is proportional to the pressure change in the E-W direction.

From Marshall & Plumb (2008)

Figure 2 – Colored text added

What might be surprising is the instead of the wind flowing from high to low pressure, it flows at right angles – along the lines of constant pressure.

So of course we have to ask whether these simplifications are justified..

Here is a sample of the 500 mbar wind and geopotential height:

From Marshall & Plumb (2008)

Figure 3

We can see that the wind at 5oo mbar (about 5km high) is quite close to geostrophic balance.

By contrast, if we look at surface winds:

From Marshall & Plumb (2008)

Figure 4

Here we see that the wind is flowing more across the pressure field from high to low pressure – this is because of the effect of friction at the surface. The friction term in equation 3 cannot be ignored when we want to calculate the motion near boundary layers.

Conclusion

This is just an interesting part of climate science. The large scale atmospheric and oceanic motion is fascinating and also necessary for understanding the science of climate.

Notes

Note 1: Even watching the planet from space is not an inertial frame of reference as the earth is rotating around the sun, and the sun is rotating around the center of the galaxy, etc, etc.. To avoid this article being a 100 page unfathomable treatise on rederiving the equations of motion, there are necessarily many simplifications, offered without caveat or explanation.

Note 2: The components of the Coriolis force on the surface of a sphere are calculated from Ω x u (where the “x” is the vector cross product, not “times”).

Ω x u = (0,  Ωcosφ,  Ωcosφ) x (u,  v,  w)

            = (Ωcosφ.w - Ωsinφ.v,   Ωsinφ.u,  -Ωcosφ.u)

w is the vertical component of wind and is generally very small compared with horizontal components. So when at the equator (φ=0°), then:

Ω x u = (Ωcosφ.w,   0,  -Ωcosφ.u)

the u-direction (W-E) is very small because w is very small, and the w-direction (vertical) is not important because it competes with the much larger gravity term

Note 3: The term D/Dt has a specific meaning that might be new to many people. This is the Lagrangian differential, which is the change in the property of a fluid following that element of fluid. Rather than the change in property of a fluid at a fixed point in space.

D/Dt ≡ ∂/∂t + u∂/∂x + v∂/∂y + w∂/∂z, where u = (u,v,w) is the velocity vector

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.

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 R_L↑(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:

<T_s> = 2^3/2T_e/5 ≈ 144K (3.9)

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

T_s = 153 K if α_E = 0.12 and T_s = 155 K if α_E = 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..

E_TOA= 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?

E_TOA= 3.8 x 10²⁴ J

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

E_surface = ?

My questions to Kramm & Dlugi:

Is  E_surface significantly greater than E_TOA ?

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.

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

In Part One I created a Matlab model which reproduced the same problems as Spencer & Braswell (2008) had found. This model had one layer  (an “ocean slab” model) to represent the MLD with a “noise” flux into the deeper ocean (and a radiative noise flux at top of atmosphere). Murphy & Forster claimed that longer time periods require an MLD of increased depth to “model” the extra heat flow into the deeper ocean over time:

Because heat slowly penetrates deeper into the ocean, an appropriate depth for heat capacity depends on the length of the period over which Eq. (1) is being applied (Watterson 2000; Held et al. 2010). For 80-yr global climate model runs, Gregory (2000) derived an optimum mixed layer depth of 150 m. Watterson (2000) found an initial global heat capacity equivalent to a mixed layer of 200 m and larger values for longer simulations.

This seems like it might make sense – if we wanted to keep a “zero dimensional model”. But it’s questionable whether the model retains any value with this “fudge”. So because heat actually moves from the mixed layer into the deeper ocean (rather than the mixed layer increasing in depth) I instead enhanced the model to create a heat flux from the MLD through a number of ocean layers with a parameter called the vertical eddy diffusivity to determine this heat flux.

So the model is now a 1D model with a parameterized approach to ocean convection.

Eddy Diffusivity

The concept here is the analogy of conductivity but when convection is instead the primary mover of heat.

Heat flow by conduction is governed by a material property called conductivity and by the temperature difference. Changes in temperature are governed by heat flow and by the heat capacity. The result is this equation for reference and interest – so don’t worry if you don’t understand it:

∂T / ∂t = α . ∂²T / ∂z²  - the 1-d version (see note 1)

where T = temperature, t = time, α = thermal diffusivity and z = depth

What it says in almost plain English is that the change in temperature with respect to time is equal to the thermal diffusivity times the change in gradient of temperature with depth. Don’t worry if that’s not clear (and there is a explanation of the simple steps required to calculate this in note 1).

Now the thermal diffusivity, α:

α = k/c_pρ, where k = conductivity, c_p = heat capacity and ρ = density

So, an important bit to understand..

• if the conductivity is high and the heat capacity is low then temperature can change quickly
• if the conductivity is high and the heat capacity is high then it slows down temperature change, and
• if the conductivity is low and the heat capacity is high then temperature takes much longer to change

Many researchers have attempted to measure an average value for eddy diffusivity in the ocean (and in lakes). The concept here, an explained in Part Two, is that turbulent motions of the ocean move heat much more effectively than conduction. The value can’t be calculated from first principles because that would mean solving the problem of turbulence, which is one of the toughest problems in physics. Instead it has to be estimated from measurements.

There is an inherent problem with eddy diffusivity for vertical heat transfer that we will come back to shortly.

There is also a minor problem of notation that is “solved” here by changing the notation. Usually conductivity is written as “k”. However, most papers on eddy diffusivity write diffusivity as “k”, sometimes “K”, sometimes “κ” (Greek ‘kappa’) – creating potential confusion so I revert back to “α”. And to make it clear that it is the convective value rather than the conductive value, I use α_eddy. And for the equivalent parameter to conductivity, k_eddy.

k_eddy = α_eddyc_pρ

because c_p= 4200 J/K.kg and ρ ≈ 1000 kg/m³:

k_eddy =4.2 x 10⁶  α_eddy – it’s useful to be able to see what the diffusivity means in terms of an equivalent “conductivity” type parameter

Measurements of Eddy Diffusivity

Oeschger et al (1975):

α is an apparent global eddy diffusion coefficient which helps to reproduce an average transport phenomenon consisting of a series of distinct and overlapping mechanisms.

Oeschger and his co-workers studied the problem via the diffusion into the ocean of ¹⁴C from nuclear weapons testing.

The range they calculated for α_eddy = 1 x 10^-4 – 1.8 x 10^-4 m²/s.

This equates to k_eddy = 420 – 760 W/m.K, and by comparison, the conductivity of still water, k = 0.6 W/m.K – making convection around 1,000 times more effective at moving heat vertically through the ocean.

Broecker et al (1980) took a similar approach to estimating this value and commented:

We do not mean to imply that the process of vertical eddy mixing actually occurs within the body of the main oceanic thermocline. Indeed, the values we require are an order of magnitude greater than those permitted by conventional oceanographic wisdom (see Garrett, 1979, for summary).

The vertical eddy coefficients used here should rather be thought of as parameters that take into account all the processes that transfer tracers across density horizons. In addition to vertical mixing by eddies, these include mixing induced by sediment friction at the ocean margins and mixing along the surface in the regions where density horizons outcrop.

Their calculation, like Oeschger’s, used a simple model with the observed values plugged in to estimate the parameter:

Anyone familiar with the water mass structure and ventilation dynamics of the ocean will quickly realize that the box-diffusion model is by no means a realistic representation. No simple modification to the model will substantially improve the situation.

To do so we must take a giant step in complexity to a new generation of models that attempt to account for the actual geometry of ventilation of the sea. We are as yet not in a position to do this in a serious way. At least a decade will pass before a realistic ocean model can be developed.

The values they calculated for eddy diffusivity were broken up into different regions:

• α_eddy(equatorial) = 3.5 x 10^-5 m²/s
• α_eddy(temperate) = 2.0 x 10^-4 m²/s
• α_eddy(polar) = 3.0 x 10^-4 m²/s

We will use these values from Broecker to see what happens to the measurement problems of climate sensitivity when used in my simple model.

These two papers were cited by Hansen et al in their 1985 paper with the values for vertical eddy diffusivity used to develop the value of the “effective mixed depth” of the ocean.

In reviewing these papers and searching for more recent work in the field, I tapped into a rich vein of research that will be the subject of another day.

First, Ledwell et al (1998) who measured eddy diffusivity via SF₆ that they injected into the ocean:

The diapycnal eddy diffusivity K estimated for the first 6 months was 0.12 ± 0.02 x10^-4 m²/s, while for the subsequent 24 months it was 0.17 ± 0.02 x10^-4 m²/s.

[Note: units changed from cm²/s into m²/s for consistency]

It is worth reading their comment on this aspect of ocean dynamics. (Note that isopycnal = contact density surfaces and diapycnal = across isopycnal):

The circulation of the ocean is severely constrained by density stratification. A water parcel cannot move from one surface of constant potential density to another without changing its salinity or its potential temperature. There are virtually no sources of heat outside the sunlit zone and away from the bottom where heat diffuses from the lithosphere, except for the interesting hydrothermal vents in special regions. The sources of salinity changes are similarly confined to the boundaries of the ocean. If water in the interior is to change potential density at all, it must be by mixing across density surfaces (diapycnal mixing) or by stirring and mixing of water of different potential temperature and salinity along isopycnal surfaces (isopycnal mixing).

Most inferences of dispersion parameters have been made from observations of the large-scale fields or from measurements of dissipation rates at very small scales. Unambiguously direct measurements of the mixing have been rare. Because of the stratification of the ocean, isopycnal mixing involves very different processes than diapycnal mixing, extending to much greater length scales. A direct approach to the study of both isopycnal and diapycnal mixing is to release a tracer and measure its subsequent dispersal. Such an experiment, lasting 30 months and involving more than 10⁵ km² of ocean, is the subject of this paper.

From Jayne (2009):

For example, the Community Climate Simulation Model (CCSM) ocean component model uses a form similar to Eq. (1), but with an upper-ocean value of 0.1 x 10^-4 m²/s and a deep-ocean value of 1.0 x 10^-4 m²/s, with the transition depth at 1000 m.

However, there is no observational evidence to suggest that the mixing in the ocean is horizontally uniform, and indeed there is significant evidence that it is heterogeneous with spatial variations of several orders of magnitude in its intensity (Polzin et al. 1997; Ganachaud 2003).

More on eddy diffusivity measurements in another article – the parameter has a significant impact on modeling of the ocean in GCMs and there is a lot of current research into this subject.

Eddy Diffusivity and Buoyancy Gradient

Sarmiento et al (1976) measured isotopes near the ocean floor:

Two naturally occurring isotopes can be applied to the determination of the rate of vertical turbulent mixing in the deep sea: ²²²Rn (half-life 3.824 days) and ²²⁸Ra (half-life 5.75 years). In this paper we discuss the results from fourteen ²²²Rn and two ²²⁸Ra profiles obtained as part of the GEOSECS program.

From these results we conclude that the most important factor influencing the vertical eddy diffusivity is the buoyancy gradient [(g/p)(∂ρ_pot/∂z)]. The vertical diffusivity shows an inverse proportionality to the buoyancy gradient.

Their paper is very much about the measurements and calculations of the deeper ocean, but is relevant for anywhere in the ocean, and helps explain why the different values for different regions were obtained by Broecker that we saw earlier. (Prof. Wallace S. Broecker was a co-author on this paper as well, and has authored/co-authored 100′s of papers on the ocean).

What is the buoyancy gradient and why does it matter?

Cold fluids sink and hot fluids rise. This is because cold substances contract and so are more dense. So in general, in the ocean, the colder water is below and the warmer water above. Probably everyone knows this.

The buoyancy gradient is a measure of how strong this effect is. The change in density with depth determines how resistant the ocean is to being overturned. If the ocean was totally stable no heat would ever penetrate below the mixed layer. But it does. And if the ocean was totally stable then the measurements of ¹⁴C from nuclear testing would be zero below the mixed layer.

But it is not surprising that the more stable the ocean is due to the buoyancy gradient the less heat diffuses down by turbulent motion.

And this is why the estimates by Broecker shown earlier have a much lower value of diffusivity for the tropics than for the poles. In general the poles are where deep convection takes place – lots of cold water sinks, mixing the ocean – and the tropics are where much weaker upwelling takes place – because the ocean surface is strongly heated. This is part of the large scale motion of the ocean, known as the thermohaline circulation. More on this another day.

Now water is largely incompressible which means that the density gradient is only determined by temperature and salinity. This creates the problem that eddy diffusivity is a value which is not only parameterized, but also dependent on the vertical temperature difference in the ocean.

Heat flow also depends on temperature difference, but with the opposite relationship. This is not something to untangle today. Today we will just see what happens to our simple model when we use the best estimates of vertical eddy diffusivity.

Modeling, Non-Linearity and Climate Sensitivity Measurement Problems

Murphy & Forster agreed in part with Spencer & Braswell about the variation in radiative noise from CERES measurements. I quote at length, because the Murphy & Forster paper is not freely available:

For the parameter N, SB08 use a random daily shortwave flux scaled so that the standard deviation of monthly averages of outgoing radiation (N – λT) is 1.3 W/m².

They base this on the standard deviation of CERES shortwave data between March 2000 and December 2005 for the oceans between 20 °Nand 20 °S.

We have analyzed the same dataset and find that, after the seasonal cycle and slow changes in forcing are removed, the standard deviation of monthly means of the shortwave radiation is 1.24 W/m², close to the 1.3 W/m² specified by SB08. However, longwave (infrared) radiation changes the energy budget just as effectively from the earth as shortwave radiation (reflected sunlight). Cloud systems that might induce random fluctuations in reflected sunlight also change outgoing longwave radiation. In addition, the feedback parameter λ is due to both longwave and shortwave radiation.

Modeled total outgoing radiation should therefore be compared with the observed sum of longwave and shortwave outgoing radiation, not just the shortwave component. The standard deviation of the sum of longwave and shortwave radiation in the same CERES dataset is 0.94 W/m². Even this is an upper limit, since imperfect spatial sampling and instrument noise contribute to the standard deviation.

[Note I change their α (climate feedback) to λ for consistency with previous articles].

And they continue:

We therefore use 0.94 W/m² as an upper limit to the standard deviation of outgoing radiation over the tropical oceans. For comparison, the standard deviation of the global CERES outgoing radiation is about 0.55 W/m².

All of these points seem valid (however, I am still in the process of examining CERES data, and can’t comment on their actual values of standard deviation. Apart from the minor challenge of extracting the data from the netCDF format there is a lot to examine. A lot of data and a lot of issues surrounding data quality).

However, it raised an interesting idea about non-linearity. Readers who remember on Part One will know that as radiative noise increases and ocean MLD decreases the measurement problem gets worse. And as the radiative noise decreases and ocean MLD increases the measurement problem goes away.

If we average global radiative noise and global MLD, plug these values into a zero-dimensional model and get minimal measurement problem what does this mean?

Due to non-linearity, it tells us nothing.

Averaging the inputs, applying them to a global model (i.e., a zero-dimensional model) and calculating λ_est (from the regression) gets very different results from applying the inputs separately to each region, averaging the results and calculating λ_est

I tested this with a simple model – created two regions, one 10% of the surface area, the other 90%. In the larger region the MLD was 200m and the radiative noise was zero; and in the smaller region the MLD was 20m and the (standard deviation of) radiative noise was varied from 0 – 2. The temperature and radiative flux were converted into an area weighted time series and the regression produced large deviations from the real value of λ.

A similar run on a global model with an MLD of 180m and radiative noise of 0-0.2 shows an accurate assessment of λ.

This is to be expected of course.

So with this in mind I tested the new 1D model with different values of ocean depth eddy diffusivity,  radiative noise, and an AR(1) model for the radiative noise. I used values for the tropical region as this is clearly the area most likely to upset the measurement – shallow MLD, higher radiative noise and weaker eddy diffusivity.

As best as I could determine from de Boyer Montegut’s paper, the average MLD for the 20°N – 20°S region is approximately 30m.

Here are the results using Oeschger’s value of eddy diffusivity for the tropics and the tropical value of radiative noise from MF2010 – varying ocean depth around 30m and the value of the AR(1) model for radiative noise:

Figure 1

For reference, as it’s hard to read off the graph, the value at 30m and φ=0.5 is λ_est = 2.3.

Using the current CCSM value of eddy diffusivity for the upper ocean:

Figure 2

For reference,  the value at 30m and φ=0.5 is λ_est = 0.2. (Compared with the real value of 3.0)

Note that these values are only for one region, not for the whole globe.

Another important point is that I have used the radiative noise value as the standard deviation of daily radiative noise. I have started to dig into CERES data to see whether such a value can be calculated, and also what typical value of autoregressive parameter should be used (and what kind of ARMA model), but this might take some time.

Yet smaller values of eddy diffusivity are possible for smaller regions, according to Jochum (2009). This would likely cause the problems of estimating climate sensitivity to become worse.

Simple Models

Murphy & Forster comment:

Although highly simplified, a single box model of the earth has some pedagogic value. One must remember that the heat capacity c and feedback parameter λ are not really constants, since heat penetrates more deeply into the ocean on long time scales and there are fast and slow climate feedbacks (Knutti et al. 2008).

It is tempting to add a few more boxes to account for land, ocean, different latitudes, and so forth. Adding more boxes to an energy balancemodel can be problematic because one must ensure that the boxes are connected in a physically consistent way. A good option is to instead consider a global climate model that has many boxes connected in a physically consistent manner.

The point being that no one believes a slab model of the ocean to be a model that gives really useful results. Spencer & Braswell likewise don’t believe that the slab model is in any way an accurate model of the climate.

They used such a model just to demonstrate a possible problem. Murphy & Forster’s criticism doesn’t seem to have solved the problem of “can we measure climate sensitivity?“

Or at least, it appears easy to show that slightly different enhancements of the simple model demonstrate continued problems in measuring climate sensitivity – due to the impact of radiative noise in the climate system.

Conclusion

I have produced a simple model and apparently demonstrated continued climate sensitivity measurement problems. This is in contrast to Murphy & Forster who took a different approach and found that the problem went away. However, my model has a more realistic approach to moving heat from the mixed layer into the ocean depths than theirs.

My model does have the drawback that the massive army of Science of Doom model testers and quality control champions are all away on their Xmas break. So the model might be incorrectly coded.

It’s also likely that someone else can come along and take a slightly enhanced version of this model and make the problem vanish.

I have used values for MLD and eddy diffusivity that seem to represent real-world values but I have no idea as to the correct values for standard deviation and auto-correlation of daily radiative noise (or appropriate ARMA model). These values have a big impact on the climate sensitivity measurement problem for reasons explained in Part One.

A useful approach to determining the effect of radiative noise on climate sensitivity measurement might be to use a coupled atmosphere-ocean GCM with a known climate sensitivity and an innovative way of removing radiative noise. These kind of experiments are done all the time to isolate one effect or one parameter.

Perhaps someone has already done this specific test?

I see other potential problems in measuring climate sensitivity. Here is one obvious problem – as the temperature of the mixed layer increases with continued increases in radiative forcing the buoyancy gradient increases and the eddy diffusivity reduces. We can calculate radiative forcing due to “greenhouse” gases quite accurately and therefore remove it from the regression analysis (see Spencer & Braswell 2008 for more on this). But we can’t calculate the change in eddy diffusivity and heat loss to the deeper ocean. This adds another “correlated” term that seems impossible to disentangle from the climate sensitivity calculation.

An alternative way of looking at this is that climate sensitivity might not be a constant – as already noted in Part One.

References

Potential Biases in Feedback Diagnosis from Observational Data: A Simple Model Demonstration, Spencer & Braswell, Journal of Climate (2008) – FREE

On the accuracy of deriving climate feedback parameters from correlations between surface temperature and outgoing radiation, Murphy & Forster, Journal of Climate (2010)

A box diffusion model to study the carbon dioxide exchange in nature, Oeschger et al, Tellus (1975)

Modeling the carbon system, Broecker et al, Radiocarbon (1980) – FREE

Climate response times: dependence on climate sensitivity and ocean mixing, Hansen et al, Science (1985)

The study of mixing in the ocean: A brief history, MC Gregg, Oceanography (1991) – FREE

Spatial Variability of Turbulent Mixing in the Abyssal Ocean, Polzin et al, Science (1997) – FREE

The Impact of Abyssal Mixing Parameterizations in an Ocean General Circulation Model, Steven R. Jayne, Journal of Physical Oceanography (2009)

The relationship between vertical eddy diffusion and buoyancy gradient in the deep sea, Sarmiento et al, Earth & Planetary Science Letters (1976)

Mixing of a tracer in the pycnocline, Ledwell et al, JGR (1998)

Impact of latitudinal variations in vertical diffusivity on climate simulations, Jochum, JGR (2009) – FREE

Mixed layer depth over the global ocean: An examination of profile data and a profile-based climatology, de Boyer Montegut et al, JGR (2004)

Notes

Note 1: The 1D version is really:

∂T / ∂t = ∂/∂z (α.∂T/∂z)

due to the fact that α can be a function of z (and definitely is in the case of the ocean).

Although this looks tricky – and it is tricky to find analytical solutions – solving the 1D version numerically is very straightforward and anyone can do it.

In plain English is looks something like:

- Heat flow into cell X = temperature difference between cell X and cell X-1

- Heat flow out of cell X = temperature difference between cell X and cell X+1

- Change in temperature = (Heat flow into cell X – Heat flow out of cell X) x time / heat capacity

Note 2: I am in the process of examining CERES data. Apart from the challenge of extracting the data from the netCDF format there is a lot to examine. A lot of data and a lot of issues surrounding data quality.

Using a simple model Spencer & Braswell (2008) had demonstrated that even when the value of “climate sensitivity” is constant and known, measurement of it can be obscured for a number of reasons.

The simple model was a “slab model” of the ocean with a top of atmosphere imbalance in radiation.

Murphy & Forster (2010) criticized Spencer & Braswell for a few reasons including the value chosen for the depth of this ocean mixed layer. As the mixed layer depth increases the climate sensitivity measurement problems are greatly reduced.

First, we will consider the mixed layer in the context of that simple model. Then we will consider what it means in real life.

The Simple Model of Climate Sensitivity

The simple model used by Spencer & Braswell has a “mixed ocean layer” of depth 50m.

Figure 1

In the model the mixed layer is where all of the imbalance in top of atmosphere radiation gets absorbed.

The idea in the simple model is that the energy absorbed from the top of atmosphere gets mixed into the top layer of the ocean very quickly. In reality, as we will see, there isn’t such a thing as one layer but it is a handy approximation.

Murphy & Forster commented:

For the heat capacity parameter c, SB08 use the heat capacity of a 50-m ocean mixed layer. This is too shallow to be realistic.

Because heat slowly penetrates deeper into the ocean, an appropriate depth for heat capacity depends on the length of the period over which Eq. (1) is being applied (Watterson 2000; Held et al. 2010).

For 80-yr global climate model runs, Gregory (2000) derived an optimum mixed layer depth of 150 m. Watterson (2000) found an initial global heat capacity equivalent to a mixed layer of 200 m and larger values for longer simulations.

Held et al. (2010) found an initial time constant τ = c/α of about four yr in the Geophysical Fluid Dynamics Laboratory global climate model. Schwartz (2007) used historical data to estimate a globally averaged mixed layer depth of 150 m, or 106 m if the earth were only ocean.

The idea is an attempt to keep the simplicity of one mixed layer for the model, but increase the depth of this mixed layer for longer time periods.

There is always a point where models – simplified versions of the real world – start to break down. This might be the case here.

The initial model was of a mixed layer of ocean, all at the same temperature because the layer is well-mixed – and with some random movement of heat between this mixed layer and the ocean depths. In a more realistic scenario, more heat flows into the deeper ocean as the length of time increases.

What Murphy & Forster are proposing is to keep the simple model and “account” for the ever increasing heat flow into the deeper ocean by using a depth of the mixed layer that is dependent on the time period.

If we do this perhaps the model will work, perhaps it won’t. By “work” we mean provide results that tell us something useful about the real world.

So I thought I would introduce some more realism (complexity) into the model and see what happened. This involves a bit of a journey.

Real Life Ocean Mixed Layer

Water is a very bad conductor of heat – as are plastic and other insulators. Good conductors of heat include metals.

However, in the ocean and the atmosphere conduction is not the primary heat transfer mechanism. It isn’t even significant. Instead, in the ocean it is convection - the bulk movement of fluids – that moves heat. Think of it like this – if you move a “parcel” of water, the heat in that parcel moves with it.

Let’s take a look at the temperature profile at the top of the ocean. Here the first graph shows temperature:

Soloviev & Lukas (1997)

Figure 2

Note that the successive plots are not at higher and higher temperatures – they are just artificially separated to make the results easier to see. During the afternoon the sun heats the top of the ocean. As a result we get a temperature gradient where the surface is hotter than a few meters down. At night and early morning the temperature gradient disappears. (No temperature gradient means that the water is all at the same temperature)

Why is this?

Once the sun sets the ocean surface cools rapidly via radiation and convection to the atmosphere. The result is colder water, which is heavier. Heavier water sinks, so the ocean gets mixed. This same effect takes place on a larger scale for seasonal changes in temperature.

And the top of the ocean is also well mixed due to being stirred by the wind.

A comment from de Boyer Montegut and his coauthors (2004):

A striking and nearly universal feature of the open ocean is the surface mixed layer within which salinity, temperature, and density are almost vertically uniform. This oceanic mixed layer is the manifestation of the vigorous turbulent mixing processes which are active in the upper ocean.

Here is a summary graphic from the excellent Marshall & Plumb:

From Marshall & Plumb (2008)

Figure 3

There’s more on this subject in Does Back-Radiation “Heat” the Ocean? – Part Three.

How Deep is the Ocean Mixed Layer?

This is not a simple question. Partly it is a measurement problem, and partly there isn’t a sharp demarcation between the ocean mixed layer and the deeper ocean. Various researchers have made an effort to map it out.

Here is a global overview, again from Marshall & Plumb:

Figure 4

You can see that the deeper mixed layers occur in the higher latitudes.

Comment from de Boyer Montegut:

The main temporal variabilities of the MLD [mixed layer depth] are directly linked to the many processes occurring in the mixed layer (surface forcing, lateral advection, internal waves, etc), ranging from diurnal [Brainerd and Gregg, 1995] to interannual variability, including seasonal and intraseasonal variability [e.g., Kara et al., 2003a; McCreary et al., 2001]. The spatial variability of the MLD is also very large.

The MLD can be less than 20 m in the summer hemisphere, while reaching more than 500 m in the winter hemisphere in subpolar latitudes [Monterey and Levitus, 1997].

Here is a more complete map by month. Readers probably have many questions about methodology and I recommend reading the free paper:

From de Boyer Montegut et al (2004)

Figure 5 – Click for a larger image

Seeing this map definitely had me wondering about the challenge of measuring climate sensitivity. Spencer & Braswell had used 50m MLD to identify some climate sensitivity measurement problems. Murphy & Forster had reproduced their results with a much deeper MLD to demonstrate that the problems went away.

But what happens if instead we retest the basic model using the actual MLD which varies significantly by month and by latitude?

So instead of “one slab of ocean” at MLD = choose your value, we break up the globe into regions, have different values in each region each month and see what happens to climate sensitivity problems.

By the way, I also attempted to calculate the global annual (area weighted) average of MLD from the maps above, by eye. I also emailed the author of the paper to get some measurement details but no response.

My estimate of the data in this paper was a global annual area weighted average of 62 meters.

Trying Simple Models with Varying MLD

I updated the Matlab program from Measuring Climate Sensitivity – Part One. The globe is now broken up into 30º latitude bands, with the potential for a different value of mixed layer depth for each month of the year.

I created a number of different profiles:

Depth Type 0 – constant with month and latitude, as in the original article

Type 1 – using the values from de Boyer’s paper, as best as can be estimated from looking at the monthly maps.

Type 2 – no change each month, with scaling of 60ºN-90ºN = 100x the value for 0ºN – 30ºN, and 30ºN – 60ºN = 10x the value for 0ºN – 30ºN – similarly for the southern hemisphere.

Type 3 – alternating each month between Type 2 and its inverse, i.e., scaling of 0ºN – 30ºN = 100x the value for 60ºN-90ºN and 30ºN – 60ºN = 10x the value for 60ºN-90ºN.

Type 4 – no variation by latitude, but  month 1 = 1000x month 4, month 2 = 100x month 4, month 3 = 10x month 4, repeating 3 times  per year.

In each case the global annual (area weighted) average = 62m.

Essentially types 2-4 are aimed at creating extreme situations.

Here are some results (review the original article for some of the notation), recalling that the actual climate sensitivity, λ = 3.0:

Figure 6

Figure 7 – as figure 6 without 30-day averaging

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

What’s the message from these results?

In essence, type 0 (the original) and type 1 (using actual MLDs vs latitude and month from de Boyer’s paper) are quite similar – but not exactly the same.

However, if we start varying the MLD by latitude and month in a more extreme way the results come out very differently – even though the global average MLD is the same in each case.

This demonstrates that the temporal and area variation of MLD can have a significant effect and modeling the ocean as one slab – for the purposes of this enterprise – may be risky.

Non-Linearity

We haven’t considered the effect of non-linearity in these simple models. That is, what about interactions between different regions and months. If we created a yet more complex model where heat flowed between regions dependent on the relative depths of the mixed layers what would we find?

Losing the Plot?

Now, in case anyone has lost the plot by this stage – and it’s possible that I have – don’t get confused into thinking that we are evaluating GCM’s and gosh aren’t they simplistic.. No, GCM’s have very sophisticated modeling.

What we have been doing is tracing a path that started with a paper by Spencer & Braswell. This paper used a very simple model to show that with some random daily fluctuations in top of atmosphere radiative flux, perhaps due to clouds, the measurement of climate sensitivity doesn’t match the actual climate sensitivity.

We can do this in a model – prescribe a value and then test whether we can measure it. This is where this simple model came in. It isn’t a GCM.

However, Murphy & Forster came along and said if you use a deeper mixed ocean layer (which they claim is justified) then the measurement of climate sensitivity does more or less match the actual climate sensitivity (they also had comment on the values chosen for radiative flux anomalies, a subject for another day).

What struck me was that the test model needs some significant improvement to be able to assess whether or not climate sensitivity can be measured. And this is with the caveat – if climate sensitivity is a constant.

The Next Phase – More Realistic Ocean Model

As Murphy & Forster have pointed out, the longer the time period, the more heat is “injected” into the deeper ocean from the mixed layer.

So a better model would capture this better than just creating a deeper mixed layer for a longer time. Modeling true global ocean convection is an impossible task.

As a recap, conducted heat flow:

q” = k.ΔT/d

where q” = heat flow per unit area, k = conductivity, ΔT = temperature difference, and d = depth of layer

Take a look at Heat Transfer Basics – Part Zero for more on these basics.

For water, k = 0.6 W/m².K. So, as an example, if we have a 10ºC temperature difference across 1 km depth of water, q” = 0.006 W/m². This is tiny. Heat flow via conduction is insignificant. Convection is what moves heat in the ocean.

Many researchers have measured and estimated vertical heat flow in the ocean to come up with a value for vertical eddy diffusivity. This allows us to make some rough estimates of vertical heat flow via convection.

In the next version of the Matlab program (“in press”) the ocean is modeled with different eddy diffusivities below the mixed ocean layer to see what happens to the measurement of climate sensitivity. So far, the model comes up with wildly varying results when the eddy diffusivity is low, i.e., heat cannot easily move into the ocean depths. And it comes up with normal results when the eddy diffusivity is high, i.e., heat moves relatively quickly into the ocean depths.

Due to shortness of time, this problem has not yet been resolved. More in due course.

This article is already long enough, so the next part will cover the estimated values for eddy diffusivity because it’s an interesting subject

Conclusion

Regular readers of this blog understand that navigating to any kind of conclusion takes some time on my part. And that’s when the subject is well understood. I’m finding that the signposts on the journey to measuring climate sensitivity are confusing and hard to read.

And that said, this article hasn’t shed any more light on the measurement of climate sensitivity. Instead, we have reviewed more ways in which measurements of it might be wrong. But not conclusively.

Next up we will take a detour into eddy diffusivity, hoping in the meantime that the Matlab model problems can be resolved. Finally a more accurate model incorporating eddy diffusivity to model vertical heat flow in the ocean will show us whether or not climate sensitivity can be accurately measured.

Possibly.

References

Potential Biases in Feedback Diagnosis from Observational Data: A Simple Model Demonstration, Spencer & Braswell, Journal of Climate (2008)

On the accuracy of deriving climate feedback parameters from correlations between surface temperature and outgoing radiation, Murphy & Forster, Journal of Climate (2010)

Observation of large diurnal warming events in the near-surface layer of the western equatorial Pacific warm pool, Soloviev & Lukas, Deep Sea Research Part I: Oceanographic Research Papers (1997)

Atmosphere, Ocean and Climate Dynamics: An Introductory Text, Marshall & Plumb, Elsevier Academic Press (2008)

Mixed layer depth over the global ocean: An examination of profile data and a profile-based climatology, de Boyer Montegut et al, JGR (2004)

In the context of Science of Doom all my time has been diverted into work-related activities and I’m not sure when this will ease up.

Unless someone hands me this machine, and for a price well below market worth, I am not sure when my next post will take place.

I have lots of ideas, but like to do research and gain understanding before writing articles.

Normal service will eventually be resumed.