The Science of Doom

This post is intended to help readers better understand how changes in temperature and water vapor at different locations affect the radiation balance of the planet, primarily outgoing longwave radiation (OLR).

A lot of questions on this blog come about because people have trouble visualizing the process of radiative transfer. This is not surprising – it’s not an intuitive subject.

Basic energy balance for the planet is covered in The Earth’s Energy Budget – Part One and Part Two. If some change to the climate causes more energy to be radiated into space then the climate system will cool. Likewise, if some change causes less energy to be radiated into space then the climate system will warm (assuming constant absorbed solar radiation). We use the term OLR (outgoing longwave radiation) for the radiation from the climate system into space.

Heating the Atmosphere and the Surface

Here is a calculation of how changing temperature in the atmosphere affects OLR at different latitudes and different pressures (1000mbar at the bottom of the graph is the surface):

Figure 1 – Italic text is added

Understanding the Terminology

The top two graphs show the same effect under two different conditions. The top one is “All Sky” which is all conditions. And the second one is “Clear Sky”, i.e., the subset of conditions when clouds are not present. From left to right we have latitude and from bottom to top we have pressure. 1000mbar is the surface pressure = zero altitude, and 200mbar is the pressure at the top of the troposphere, and is around 14km (it varies depending on the latitude).

The units (the values are shown as colors) are in W/m².K.100hPa. (See Note 1). Already some readers are lost?

The important part is W/m² – this is flux (radiation in less technical language) or watts per square meter. We can call it power per unit area.

W/m².K is watts per square meter per 1K temperature change. So this asks – how much does the power per unit area increase (or decrease) for each 1K of temperature change?

If we asked, how much does the OLR increase for 1K increase in the whole atmosphere we would use the units of W/m².K. But when we want to look at how different layers in the atmosphere affect the OLR we are considering just once “slice”. So we have to have watts per meter squared per Kelvin per slice. In this case we are considering 100mbar (=100hPa) slices so this is how we get W/(m².K.100hPa).

Understanding the Results

So now the basics are out of the way, what do the graphs show us?

At the simplest level if the whole atmosphere heats up by 1K, the graphs show us the relative contribution of different latitudes and altitudes to OLR.

Let’s suppose we increase the temperature of the atmosphere at the equator between 1000 and 900 mbar by 1°C (=1K). This means we have taken a “layer” of the atmosphere and somehow just increased its temperature. What is the effect on the OLR? All bodies, including gases, emit according to their temperature and their emissivity.

Increase the temperature and the radiation (flux) increases. In our graph, at that location, for 1°C the flux increases by about 0.3 W/(m².100hPa).

Of course this means the climate cools which should be totally unsurprising. Increase the temperature of the atmosphere and it radiates more energy away into space. This is negative feedback. You can see from the graph that no matter where you heat the atmosphere it increases the flux into space – cooling the climate back down.

The bottom graph shows the result of heating the earth’s surface for clear sky and for all sky conditions. Note the difference. Under clear skies the increased flux emitted by the surface more easily escapes to space. When clouds are present the increased surface radiation is absorbed by clouds and the clouds emit at the cloud top temperature. (The cloud top temperature is high up in the atmosphere, is cooler than the surface, and so the emission to space is reduced by the presence of clouds).

At this point let’s make it clear what the graphs are not showing. They are not showing the ultimate result of heating the surface or a slice of the atmosphere after the whole climate has come into a new “equilibrium”. They are simply showing what happens directly to radiation balance as a result of a change in temperature of a “portion” of the climate.

If you’ve understood why the all sky/clear sky results in the surface graph are different then the difference between the first and second graph might be clear. The first graph is under all sky conditions (including clouds) and so the cloud tops are the region where a 1K increase has the greatest effect on OLR. Lower down in the atmosphere an increase in flux (due to hotter conditions) can be masked by clouds.

In contrast, under clear sky conditions changes in the lower atmosphere have a similar effect to changes in the upper atmosphere.

The authors say:

Under total-sky conditions the longwave fluxes are most sensitive to temperatures at the level of cloud tops that are exposed to space. This results in an obvious maximum just beneath the tropopause, where convectively detrained cirrus anvils are common, and along the top of the cloud topped boundary layer. By masking the surface, clouds also diminish the surface contribution to K^T

Adding Water Vapor to the Atmosphere

More water vapor in the atmosphere generally reduces the outgoing longwave radiation which has a heating effect on the atmosphere. (The opposite of higher atmospheric temperatures).

The reason for this is with more water vapor the atmosphere becomes more opaque to longwave radiation. So, for example, with more water vapor in the upper atmosphere, radiation from the surface or the lower atmosphere is absorbed by the water vapor higher up.

Another way of looking at the problem is to say that the more opaque the atmosphere the higher up the effective radiation to space. And higher altitudes have colder temperatures. This is the essence of the inappropriately-named “greenhouse” effect. For more on this see The Earth’s Energy Budget – Part Three.

Figure 2 – Italic text is added

Any calculation / visualization of the climate effect of increased water vapor has a choice – do you show the effect from absolute or relative changes in water vapor? As water vapor concentration reduces by more than 1000 times as you go up through the atmosphere showing relative change is generally preferred over showing absolute change.

The authors of this paper have chosen relative changes and calculated the change in OLR if temperature changes by 1K and relative humidity stays constant.

Some of the readers might be tempted to jump in here thinking that some unproven claim is being used as a premise for a climate calculation. But this is not so. It is simply a convenient way of illustrating OLR changes with water vapor changes.

Reviewing the graphs, we can see that under clear skies the deep tropics have the dominant water vapor response. This is not surprising as the tropics have so much more water vapor than the rest of the globe. See Clouds and Water Vapor – Part Two for discussion on this.

Under all sky conditions the effects of clouds are seen. The subtropics become more important than the tropics because the subtropics are mostly cloud-free. In the deep tropics the clouds “mask” the effects of lower levels in the atmosphere.

The authors comment:

By masking underlying water vapor perturbations, clouds reduce the sensitivity of OLR to water vapor changes and increase the relative importance of upper-tropospheric moistening to the total feedback.

Water vapor also absorbs solar radiation. If there was no water vapor wouldn’t the surface absorb all the solar radiation anyway? Does it make a difference? The surface doesn’t absorb all the solar (shortwave) radiation, and especially over snow/ice covered areas the proportion of reflected solar radiation is high. Therefore, solar absorption by water vapor (as water vapor increases) has a relatively larger impact at the poles.

Figure 3 – Italic text is added

And Out of Interest..

This graph below wasn’t the intent of the article, but is in the Soden et al paper and is quite interesting.

This article is aimed at showing how the net radiative climate balance changes when atmospheric temperature and water vapor changes under clear and cloudy skies.

But once a climate model is used to compute changes in temperature we can see what different climate models show for the difference between the feedbacks:

This graph shows the results of various climate models.

As an example, Lapse rate feedback is the feedback from changes in the atmospheric temperature profile.

Final results from climate models are much more complex than determining how changes in water vapor or atmospheric temperature affect the emission of thermal radiation into space.

Conclusion

This article is aimed at increasing understanding of how changes in temperature and water vapor change the net radiation balance of the climate before any feedback.

The whole paper is well worth reading.

Further reading

        Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Ten

References

Quantifying Climate Feedbacks Using Radiative Kernels, Soden et al, Journal of Climate (2008) – Free Link

Notes

Note 1 – The units in figure 1 are Wm^-2K^-1. To non-mathematicians this notation can be difficult to understand. Most people know what m² means – it means meters squared. m^-2 means per meter squared. So we can write Wm^-2 or W/m². Both mean the same thing.

The mathematical convention of Wm^-2K^-1 is better because it is more precise (when I write W/m²K does it mean watts per meter squared per Kelvin, or Watts per meter squared times Kelvin?)

But I write the slightly less mathematically precise way to provide better readability for the non-mathematicians.

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Here is the annual mean temperature as a function of pressure (=height) and latitude:

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..

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:

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.

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The coriolis effect isn’t the easiest thing to get your head around, but it is an essential element in understanding the large scale motions of the atmosphere and the oceans.

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.

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:

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:

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

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In a discussion a little while ago on What’s the Palaver? – Kiehl and Trenberth 1997, one of our commenters asked about the surface forcing and how it could possibly lead to anything like the IPCC-projected temperature change for doubling of CO2.

Following a request for clarification, he added:

..We first look at the RHS. We believe that the atmosphere will also increase in temperature by roughly the same amount, so there will be no change in the conductive term. The increase in the Radiative term is roughly 5.5W/m².

The increase in the evaporative term is much more difficult, but is believed to be in the range 2-7%/DegC. So the increase in the evaporative term is 1.5 to 5.5W/m², for a total change on the RHS of 7 to 11 W/m².

Since balance is an assumption, the LHS changed by the same amount. The surface sensitivity is therefore 0.095 to 0.15 DegC/W/m².

Note that this is the sensitivity to changes in Surface Forcing, whatever the source. It is NOT the response to Radiative Forcing – there is no response of the surface to Radiative Forcing, it can only respond to Sunlight and Back-Radiation.

[See the whole comment and exchange for the complete picture].

These are good questions and no doubt many people have similar ones. The definition of radiative forcing (see CO2 – An Insignificant Trace Gas? Part Seven – The Boring Numbers) is at the tropopause, which is the top of the troposphere (around 12km above the surface).

Why is it at the tropopause and not at the surface? The great Ramanathan explains (in his 1998 review paper):

..Manabe & Wetherald’s [1967] paper, which convincingly demonstrated that the CO2-induced surface warming is not solely determined by the energy balance at the surface but by the energy balance of the coupled surface-troposphere-stratosphere system.

The underlying concept of the Manabe-Wetherald model is that the surface and the troposphere are so strongly coupled by convective heat and moisture transport that the relevant forcing governing surface warming is the net radiative perturbation at the tropopause, simply known as radiative forcing.

In essence, the reason we consider the value at the tropopause is that it is the best value to tell us what will happen at the surface. It is now an idea established for over 40 years, although for some it might sound bizarre. So we will try and make sense of it here.

Here is a schematic originating in Ramanathan’s 1981 paper, but extracted here from his 1998 review paper:

Figure 1

The first thing to pay attention to is the right hand side – 1. CO2 direct surface heating – which is shown as 1.2 W/m².

The surface forcing from a doubling of CO2 is around 1 W/m² compared with around 4 W/m² at the tropopause. The surface forcing is a lot less than at the top of atmosphere!

Before too much joy sets in, let’s consider what these concepts represent. They are essentially idealized quantities, derived from considering the instantaneous change in concentrations of CO2.

As CO2 shows a steady increase year on year, the idea of doubling overnight is clearly not in accord with reality. However, it is a useful comparison point and helps to get many ideas straight. If instead we said, “CO2 increasing by 1% per year”, we would need to define a time period for this 1% annual increase, plus how long after the end before a new balance was restored. It wouldn’t make solving the problem any easier – and it would make the results harder to understand – by contrast GCM’s do consider a steadily rising CO2 level according to whatever scenario they are considering.

So, with the idea of an instantaneous doubling, if the surface increase in radiative forcing is less than the tropopause increase in radiative forcing, this must mean that the balance of energy is absorbed within the atmosphere. And also, we have to consider what happens as a result of the surface energy imbalance.

The numbers I use here are Ramanathan’s numbers from his 1981 paper. Later, and more accurate, numbers have been calculated but don’t affect the main points of this analysis. The reason for reviewing his analysis is because some (but not all) of the inherent responses of the climate system are explicitly calculated – making it easier to understand than the output of  GCM.

Immediate Response

The immediate result of this doubling of CO2 is a reduced emission of radiation (OLR = outgoing longwave radiation) from the climate system into space. See the Atmospheric Radiation and the “Greenhouse” Effect series for detailed explanations of why.

At the tropopause the OLR reduces by 3.1 W/m², and downward emission from the stratosphere into the troposphere increases by 1.2 W/m².

This results in a net forcing at the tropopause of 4.3 W/m². Most of the radiation from the atmosphere to the surface (as a result of more CO2) is absorbed by water vapor. So at the surface the DLR (downward long radiation) increases by only 1.2 W/m² – this is the (immediate) surface forcing. Here is a simple graphical explanation of why the OLR decreases and the DLR increases:

Figure 2 – Click for a larger image

Response After a Few Months

The stratosphere cools and reaches a new radiative equilibrium. This reduces the downward emission from the stratosphere by a small amount. The new value of radiative forcing at the tropopause = 4.2 W/m².

Response After Many Decades

The surface-troposphere warms until a new equilibrium is reached – the radiative forcing at the tropopause has returned to zero.

The Surface

So let’s now consider the surface. Take a look at Figure 1 again. The values/ranges we will consider are calculated by a model. This doesn’t mean they are correct. It means that applying well-understood processes in a simplistic way gives us a “first order” result. The reason for assessing this kind of approach is because our mental models are usually less accurate than a calculated result which draws on well-understood physics.

As Ramanathan says in his 1998 paper:

As a caveat, the system we considered up to this point to elucidate the principles of warming is a highly simplified linear system. Its use is primarily educational and cannot be used to predict actual changes.

Process 1 is as already described – the surface forcing increases by just over 1 W/m². But the balance of 3 W/m² goes into heating the troposphere.

Process 2 – The warming of the troposphere results in increases downward radiation to the surface (because the hotter the body, the higher the radiation emitted). The calculated value is an additional 2.3 W/m², so the surface imbalance is now 3.5 W/m² and the surface temperature must increase in response. Upwards surface radiation and/or sensible and latent heat will increase to balance.

Process 3 – The surface emission of radiation increases at around 5.5 W/m² for every 1°C of surface temperature increase. But this is almost balanced by increased downward radiation from the atmosphere (“back radiation”). The net effect is only about 10% of the change in upward radiation. So latent heat and sensible heat increase to restore the energy balance, but this also heats the troposphere.

Process 4 – The tropospheric humidity increases. This increases the emissivity of the atmosphere near the surface, which increases the back radiation.

So essentially some cycles are reinforcing each other (=positive feedback). The question is about the value of the new equilibrium point.

Figure 3

In Ramanathan’s 1981 paper he gives some basic calculations before turning to GCM results. The basic calculations are quite interesting because one of the purposes of the paper was to explain why some model results of the day produced very small equilibrium temperature changes.

Sadly for some readers, a little maths is necessary to reproduce the result. It is simple maths because it is based on simple concepts – as already presented. As much as possible I follow the equation numbers and notations from Ramanathan’s 1981 paper.

Calculations

Energy balance at an “average” surface:

Upward flux = Downward flux

→  LH + SH + F↑ = F↓ + S + ΔR  ….[2]

where LH = latent heat, SH = sensible heat, F↑ = surface emitted upward radiation, F↓ = surface downward radiation from the atmosphere, S = solar radiation absorbed, ΔR = instantaneous change in energy absorbed at the surface due to an increase in CO2

And see note 1. We have simple formulas for the left hand side.

F↑ = σT⁴….[3a]

Latent heat and sensible heat flux have “bulk aerodynamic formulas” (note 2):

LH = ρLC_DV (q*_M – q_S)   ….[3b]

SH = ρc_pC_DV (T_M – T_S)   ….[3c]

Where ρ = density of air = 1.3 kg/m, L = latent heat of water vapor = 2.5 x 10⁶, C_D = empirically determined coefficient ≈ 1.3 x10^-3,  V = average wind speed at some reference height above the surface ≈ 5 m/s, q*_M = specific humidity at saturation at the surface temperature of the ocean,  q_S = specific humidity at the reference height,  T_M = temperature of the ocean at the surface,  T_S = temperature of the air at the reference height (typically 10m).

To give an idea of typical values, for every 1°C difference between the surface and the air at the reference height, SH = 8.5 W/m²K, and with a relative humidity of 80% at the reference height (and 100% at the ocean surface), LH = 55 W/m²K.

Now we consider changes.

T_M‘ is the change in the surface temperature of the ocean as the result of the increased CO2, and similar notation for other changes in values. Missing out a few steps that you can read in the paper:

T_M‘ =                                    ΔR(0) + ΔF↓(2) + ΔF↓(3)                               ….[13]

      [ ∂LH/∂T_M + ∂SH/∂T_M + 4σT_M³] + [  ∂LH/∂T_S +  ∂SH/∂T_S ].T_S‘/T_M‘

This probably seems a little daunting to a lot of readers.. so let’s explain it:

• The parameter on the top line in black, ΔR(0) is the surface radiative forcing from the increase in CO2
• The red terms are the changes in downward radiation as a result of process 2 and 3 described above
• The blue terms are the changes in upward flux due to only the ocean surface temperature changing
• The green terms are the changes in upward flux due to only the atmospheric temperature near the surface changing
• The blue term ≈ 30 W/m²K @ 15°C; the green term ≈ -8.5 W/m²K @ 15°C (note 3)

And the smaller the total under the line, the higher the increase in temperature. And there are two competing terms:

• As the surface temperature of the ocean increases the heat transfer from the ocean to the atmosphere increases
• As the atmospheric temperature (just above the ocean surface) increases the heat transfer from the ocean to the atmosphere decreases

As an interesting comparison, Ramanathan reviewed the methods and results of Newell & Dopplick (1979) who found a changed surface temperature, Tm’ = 0.04 °C as a result of CO2 doubling. Effectively, very little change in surface temperature as a result of doubling of CO2.

Ramanathan states that the calculations of Newell & Dopplick had ignored the red terms and the green terms. Ignoring the red terms means that the heating of the atmosphere is ignored. Ignoring the green terms means that the effect of the ocean surface heating is inflated – if the ocean surface heats and the atmosphere just above somehow stayed the same then the heat transferred would be higher than if the atmospheric temperature also increased as a result. (Because heat transfer depends on temperature difference).

I expect that many people doing their own estimates will be working from similar assumptions.

Later Work

Here is a graphic from Andrews et al (2009), reference and free link below, which shows the simplified idea:

Figure 4

The paper itself is well worth reading and perhaps will be the subject of another article at a later date.

Conclusion

I haven’t demonstrated that the surface will warm by 3°C for a doubling of CO2. But I hope I have demonstrated the complexity of the processes involved and why a simplistic calculation of how the surface responds immediately to the surface forcing is not the complete answer. It is nowhere near the complete answer.

The surface temperature change as a result of doubling of CO2 is, of course, a massively important question to answer. GCM’s are necessarily involved despite their limitations.

Re-iterating what Ramanathan said in his 1998 paper in case anyone thinks I am making a case for a 3°C surface temperature increase:

As a caveat, the system we considered up to this point to elucidate the principles of warming is a highly simplified linear system. Its use is primarily educational and cannot be used to predict actual changes.

References

Trace Gas Greenhouse Effect and Global Warming, V. Ramanathan, Ambio (1998)

The role of ocean-atmosphere interactions in the CO2 climate problem, V Ramanathan, Journal of Atmospheric Sciences (1981)

Thermal equilibrium of the atmosphere with a given distribution of atmospheric humidity, Manabe & Wetherald, Journal of Atmospheric Sciences (1967)

A Surface Energy Perspective on Climate Change, Andrews, Forster & Gregory, Journal of Climate (2009)

Notes

Note 1: The equation ignores the transfer of heat into the ocean depths

Note 2: The “bulk aerodynamic formulas” – as they have become known – are more usable versions of the fundamental equations of heat and water vapor flux. Upward sensible heat flux, SH = ρc_p<wT>, where w = vertical velocity, T = temperature, so <wT> is the time average of the product of vertical velocity and temperature. However, turbulent motions are so rapid, changing on such short time intervals that measurement of these values is usually impossible (or requires intensive measurement with specialist equipment in one location). We can write,

w = <w> + w’, where <w> = mean vertical velocity and w’ = deviation of vertical velocity from the mean, likewise T = <T> + T’.

So:

<wT> = <w><T> + <w’ T’> or, Total = Mean + Eddy

Near the surface the mean vertical motion is very small compared with the turbulent vertical velocity and so the turbulent component, <w’ T’>, dominates. Therefore,

SH = c_p ρ <w’ T’>

LH = L ρ <w’ T’>

where  c_p = specific heat capacity of air, ρ = density of air, L = latent heat of water vapor

By various thermodynamic arguments, and especially by lots of empirical measurements, an estimate of heat transfer can be made via the bulk aerodynamic formulas shown above, which use the average horizontal wind speed at the surface in conjunction with the coefficients of heat transfer, which are related to the friction term for the wind at the ocean surface.

Note 3: The calculation of each of the partial derivative terms is not shown in the paper, these are my calculations. I believe that ∂LH/∂T_S = 0, most of the time – this is because if the atmosphere at the reference height is not saturated then an increase in the atmospheric temperature, T_S, does not change the moisture flux, and therefore, does not change the latent heat. I might be wrong about this, and clearly some of the time this assumption I have made is not valid.

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During a discussion following one of the six articles on Ferenc Miskolczi someone pointed to an article in E&E (Energy & Environment). I took a look and had a few questions.

The article is question is The Thermodynamic Relationship Between Surface Temperature And Water Vapor Concentration In The Troposphere, by William C. Gilbert from 2010. I’ll call this WG2010. I encourage everyone to read the whole paper for themselves.

Actually this E&E edition is a potential collector’s item because they announce it as: Special Issue – Paradigms in Climate Research.

The author comments in the abstract:

 The key to the physics discussed in this paper is the understanding of the relationship between water vapor condensation and the resulting PV work energy distribution under the influence of a gravitational field.

Which sort of implies that no one studying atmospheric physics has considered the influence of gravitational fields, or at least the author has something new to offer which hasn’t previously been understood.

Physics

Note that I have added a WG prefix to the equation numbers from the paper, for ease of referencing:

First let’s start with the basic process equation for the first law of thermodynamics
(Note that all units of measure for energy in this discussion assume intensive properties, i.e., per unit mass):

dU = dQ – PdV ….[WG1]

where dU is the change in total internal energy of the system, dQ is the change in thermal energy of the system and PdV is work done to or by the system on the surroundings.

This is (almost) fine. The author later mixes up Q and U. dQ is the heat added to the system. dU is change in internal energy which includes the thermal energy.

But equation (1) applies to a system that is not influenced by external fields. Since the atmosphere is under the influence of a gravitational field the first law equation must be modified to account for the potential energy portion of internal energy that is due to position:

dU = dQ + gdz – PdV ….[WG2]

where g is the acceleration of gravity (9.8 m/s²) and z is the mass particle vertical elevation relative to the earth’s surface.

[Emphasis added. Also I changed “h” into “z” in the quotes from the paper to make the equations easier to follow later].

This equation is incorrect, which will be demonstrated later.

The thermal energy component of the system (dQ) can be broken down into two distinct parts: 1) the molecular thermal energy due to its kinetic/rotational/ vibrational internal energies (CvdT) and 2) the intermolecular thermal energy resulting from the phase change (condensation/evaporation) of water vapor (Ldq). Thus the first law can be rewritten as:

dU = CvdT + Ldq + gdz – PdV ….[WG3]

where Cv is the specific heat capacity at constant volume, L is the latent heat of condensation/evaporation of water (2257 J/g) and q is the mass of water vapor available to undergo the phase change.

Ouch. dQ is heat added to the system, and it is dU which is the internal energy which should be broken down into changes in thermal energy (temperature) and changes in latent heat. This is demonstrated later.

Later, the author states:

This ratio of thermal energy released versus PV work energy created is the crux of the physics behind the troposphere humidity trend profile versus surface temperature. But what is it that controls this energy ratio? It turns out that the same factor that controls the pressure profile in the troposphere also controls the tropospheric temperature profile and the PV/thermal energy ratio profile. That factor is gravity. If you take equation (3) and modify it to remove the latent heat term, and assume for an adiabatic, ideal gas system CpT = CvT + PV, you can easily derive what is known in the various meteorological texts as the “dry adiabatic lapse rate”:

dT/dz = –g/Cp = 9.8 K/km ….[WG5]

[Emphasis added]

Unfortunately, with his starting equations you can’t derive this result.

What I am talking about?

The Equations Required to Derive the Lapse Rate

Most textbooks on atmospheric physics include some derivation of the lapse rate. We consider a parcel of air of one mole. (Some terms are defined slightly differently to WG2010 – note 1).

There are 5 basic equations:

The hydrostatic equilibrium equation:

dp/dz = -ρg ….[1]

where p = pressure, z = height, ρ = density and g = acceleration due to gravity (=9.8 m/s²)

The ideal gas law:

pV = RT ….[2]

where V = volume, R = the gas constant, T = temperature in K, and this form of the equation is for 1 mole of gas

The equation for density:

ρ = M/V ….[3]

where M = mass of one mole

The First Law of Thermodynamics:

dU = dQ + dW ….[4]

where dU = change in internal energy, dQ = heat added to the system, dW = work added to the system

..rewritten for dry atmospheres as:

dQ = C_vdT + pdV ….[4a]

where C_v = heat capacity at constant volume (for one mole), dV = change in volume

And the (less well-known) equation which links heat capacity at constant volume with heat capacity at constant pressure (derived from statistical thermodynamics and experimentally verifiable):

C_p = C_v + R ….[5]

where C_p = heat capacity (for one mole) at constant pressure

With an adiabatic process no heat is transferred between the parcel and its surroundings. This is a reasonable assumption with typical atmospheric movements. As a result, we set dQ = 0 in equation 4 & 4a.

Using these 5 equations we can solve to find the dry adiabatic lapse rate (DALR):

dT/dz = -g/c_p ….[6]

where dT/dz = the change in temperature with height (the lapse rate), g = acceleration due to gravity, and c_p = specific heat capacity (per unit mass) at constant pressure

dT/dz ≈ -9.8 K/km

Knowing that many readers are not comfortable with maths I show the derivation in The Maths Section at the end.

And also for those not so familiar with maths & calculus, the “d” in front of a term means “change in”. So, for example, “dT/dz” reads as: “the change in temperature as z changes”.

Fundamental “New Paradigm” Problems

There are two basic problems with his fundamental equations:

• he confuses internal energy and heat added to get a sign error
• he adds a term for gravitational potential energy when it is already implicitly included via the pressure change with height

A sign error might seem unimportant but given the claims later in the paper (with no explanation of how these claims were calculated) it is quite possible that the wrong equation was used to make these calculations.

These problems will now be explained.

Under the New Paradigm – Sign Error

Because William Gilbert mixes up internal energy and heat added, the result is a sign error. Consult a standard thermodynamics textbook and the first law of thermodynamics will be represented something like this:

dU = dQ + dW

Which in words means:

The change in internal energy equals the heat added plus the work done on the system.

And if we talk about dW as the work done by the system then the sign in front of dW will change. So, if we rewrite the above equation:

dU = dQ – pdV

By the time we get to [WG3] we have two problems.

Here is [WG3] for reference:

dU = CvdT + Ldq + gdz – PdV ….[WG3]

The first problem is that for adiabatic process, no heat is added to (or removed from) the system. So dQ = 0. The author says dU = 0 and makes dQ = change in internal energy (=CvdT + Ldq).

Here is the demonstration of the problem using his equation..

If we have no phase change then Ldq = 0. The gdz term is a mistake – for later consideration – but if we consider an example with no change in height in the atmosphere, we would have (using his equation):

CvdT – PdV = 0 ….[WG3a]

So if the parcel of air expands, doing work on its environment, what happens to temperature?

dV is positive because the volume is increasing. So to keep the equation valid, dT must be positive, which means the temperature must increase.

This means that as the parcel of air does work on its environment, using up energy, its temperature increases – adding energy. A violation of the first law of thermodynamics.

Hopefully, everyone can see that this is not correct. But it is the consequence of the incorrectly stated equation. In any case, I will use both the flawed and the fixed version to demonstrate the second problem.

Under the New Paradigm – Gravity x 2

This problem won’t appear so obvious, which is probably why William Gilbert makes the mistake himself.

In the list of 5 equations, I wrote:

dQ = C_vdT + pdV ….[4a]

This is for dry atmospheres, to keep it simple (no Ldq term for water vapor condensing). If you check the Maths Section at the end, you can see that using [4a] we get the result that everyone agrees with for the lapse rate.

I didn’t write:

dQ = C_vdT + Mgdz + pdV ….[should this instead be 4a?]

[Note that my equations consider 1 mole of the atmosphere rather than 1 kg which is why “M” appears in front of the gdz term].

So how come I ignored the effect of gravity in the atmosphere yet got the correct answer? Perhaps the derivation is wrong?

The effect of gravity already shows itself via the increase in pressure as we get closer to the surface of the earth.

Atmospheric physics has not been ignoring the effect of gravity and making elementary mistakes. Now for the proof.

If you consult the Maths Section, near the end we have reached the following equation and not yet inserted the equation for the first law of thermodynamics:

pdV – Mgdz = (C_p-C_v)dT ….[10]

Using [10] and “my version” of the first law I successfully derive dT/dz = -g/cp (the right result). Now we will try using William Gilbert’s equation [WG3], with Ldq = 0, to derive the dry adiabatic lapse rate.

0 = CvdT + gdz – PdV ….[WG3b]

and rewriting for one mole instead of 1 kg (and using my terms, see note 1):

pdV = C_vdT + Mgdz ….[WG3c]

Inserting WG3c into [10]:

C_vdT + Mgdz – Mgdz = (C_p-C_v)dT ….[11]

which becomes:

C_v = (C_p-C_v) ↠   C_p = C_v/2 ….[11a]

A New Paradigm indeed!

Now let’s fix up the sign error in WG3 and see what result we get:

0 = CvdT + gdz + PdV ….[WG3d]

and again rewriting for one mole instead of 1 kg (and again using my terms, see note 1):

pdV = -C_vdT – Mgdz ….[WG3e]

Inserting WG3e into [10]:

-C_vdT – Mgdz – Mgdz = (C_p-C_v)dT ….[12]

which becomes:

-C_vdT – 2Mgdz = C_pdT – C_vdT ….[12a]

and canceling the -C_vdT term from each side:

-2Mgdz = C_pdT ….[12b]

So:

dT/dz = -2Mg/C_p, and because specific heat capacity, c_p = C_p/M

dT/dz = -2g/c_p ….[12c]

The result of “correctly including gravity” is that the dry adiabatic lapse rate ≈ -19.6 K/km. 

Note the factor of 2. This is because we are now including gravity twice. The pressure in the atmosphere reduces as we go up – this is because of gravity. When a parcel of air expands due to its change in height, it does work on its surroundings and therefore reduces in temperature  – adiabatic expansion. Gravity is already taken into account with the hydrostatic equation.

The Physics of Hand-Waving

The author says:

As we shall see, PV work energy is very important to the understanding of this thermodynamic behavior of the atmosphere, and the thermodynamic role of water vapor condensation plays an important part in this overall energy balance. But this is unfortunately often overlooked or ignored in the more recent climate science literature. The atmosphere is a very dynamic system and cannot be adequately analyzed using static, steady state mental models that primarily focus only on thermal energy.

Emphasis added. This is an unproven assertion because it comes with no references.

In the next stage of the “physics” section, the author doesn’t bother with any equations, making it difficult to understand exactly what he is claiming.

Keeping this gravitational steady state equilibrium in mind, let’s look again at what happens when latent heat is released (condensation) during air parcel ascension.

Latent heat release immediately increases the parcel temperature. But that also results in rapid PV expansion which then results in a drop in parcel temperature. Buoyancy results and the parcel ascends and is driven by the descending pressure profile created by gravity.

The rate of ascension, and the parcel temperature, is a function of the quantity of latent heat released and the PV work needed to overcome the gravitational field to reach a dynamic equilibrium. The more latent heat that is released, the more rapid the expansion / ascension. And the more rapid the ascension, the more rapid is the adiabatic cooling of the parcel. Thus the PV/thermal energy ratio should be a function of the amount of latent heat available for phase conversion at any given altitude. The corresponding physics shows the system will try to force the convecting parcel to approach the dry adiabatic or “gravitational” lapse rate as internal latent heat is released.

For the water vapor remaining uncondensed in the parcel, saturation and subsequent condensation will occur at a more rapid rate if more latent heat is released. In fact if the cooling rate is sufficiently large, super saturation can occur, which can then cause very sudden condensation in greater quantity. Thus the thermal/PV energy ratio is critical in determining the rate of condensation occurring. The higher this ratio, the more complete is the condensation in the parcel, and the lower the specific humidity will be at higher elevations.

I tried (unsuccessfully) to write down some equations to reflect the above paragraphs. The correct approach for the author would be:

• A. Here is what atmospheric physics states now (with references)
• B. Here are the flaws/omissions due to theoretical consideration i), ii), etc
• C. Here is the new derivation (with clear statement of physics principles upon which the new equations are based)

One point I think the author is claiming is that the speed of ascent is a critical factor. Yet the equation for the moist adiabatic lapse rate doesn’t allow for a function of time in the equation.

The (standard) equation has the form (note 2):

dT/dz = g/c_p {[1+Lq*/RT]/[1+βLq*/c_p]} ….[13]

where q* is the saturation specific humidity and is a function of p & T (i.e. not a constant), and β = 0.067/°C. (See, for example: Atmosphere, Ocean & Climate Dynamics by Marshall & Plumb, 2008)

And this means that if the ascent is – for example – twice as fast, the amount of water vapor condensed at any given height will still be the same. It will happen in half the time, but why will this change any of the thermodynamics of the process?

It might, but it’s not clearly stated, so who can determine the “new physics”?

I can see that something else is claimed to do with the ratio CvdT /pV but I don’t know what it is, or what is behind the claim.

Writing the equations down is important so that other people can evaluate the claim.

And the final “result” of the hand waving is what appears to be the crux of the paper – more humidity at the surface will cause so much “faster” condensation of the moisture that the parcel of air will be drier higher up in the atmosphere. (Where “faster” could mean dT/dt, or could mean dT/dz).

Assuming I understood the claim of the paper correctly it has not been proven from any theoretical considerations. (And I’m not sure I have understood the claim correctly).

Empirical Observations

The heading is actually “Empirical Observations to Verify the Physics”. A more accurate title is “Empirical Observations”.

The author provides 3 radiosonde profiles from Miami. Here is one example:

Figure 1 – “Thermal adiabat” in the legend = “moist adiabat”

With reference to the 3 profiles, a higher surface humidity apparently leads to complete condensation at a lower altitude.

This is, of course, interesting. This would mean a higher humidity at the surface leads to a drier upper troposphere.

But it’s just 3 profiles. From one location on two different days. Does this prove something or should a few more profiles be used?

A few statements that need backing up:

The lower troposphere lapse rate decreases (slower rate of cooling) with increasing system surface humidity levels, as expected. But the differences in lapse rate are far less than expected based on the relative release of latent heat occurring in the three systems.

What equation determines “than expected”? What result was calculated vs measured? What implications result?

The amount of PV work that occurs during ascension increases markedly as the system surface humidity levels increase, especially at lower altitudes..

How was this calculated? What specifically is the claim? The equation 4a, under adiabatic conditions, with the additional of latent heat reads like this:

C_vdT + Ldq + pdV = 0 ….[4a]

Was this equation solved from measured variables of pressure, temperature & specific humidity?

Latent heat release is effectively complete at 7.5 km for the highest surface humidity system (20.06 g/kg) but continues up to 11 km for the lower surface humidity systems (18.17 and 17.07 g/kg). The higher humidity system has seen complete condensation at a lower altitude, and a significantly higher temperature (−17 ºC) than the lower humidity systems (∼ −40 ºC) despite the much greater quantity of latent heat released.

How was this determined?

If it’s true, perhaps the highest humidity surface condition ascended into a colder air front and therefore lost all its water vapor due to the lower temperature?

Why is this (obvious) possibility not commented on or examined??

Textbook Stuff and Why Relative Humidity doesn’t Increase with Height

The radiosonde profiles in the paper are not necessarily following one “parcel” of air.

Consider a parcel of air near saturation at the surface. It rises, cools and soon reaches saturation. So condensation takes place, the release of latent heat causes the air to be more buoyant and so it keeps rising. As it rises water vapor is continually condensing and the air (of this parcel) will be at 100% relative humidity.

Yet relative humidity doesn’t increase with height, it reduces:

Figure 2

Standard textbook stuff on typical temperature profiles vs dry and moist adiabatic profiles:

Figure 3

And explaining why the atmosphere under convection doesn’t always follow a moist adiabat:

Figure 4 

The atmosphere has descending dry air as well as rising moist air. Mixing of air takes place, which is why relative humidity reduces with height.

Conclusion

The “theory section” of the paper is not a theory section. It has a few equations which are incorrect, followed by some hand-waving arguments that might be interesting if they were turned into equations that could be examined.

It is elementary to prove the errors in the few equations stated in the paper. If we use the author’s equations we derive a final result which contradicts known fundamental thermodynamics.

The empirical results consist of 3 radiosonde profiles with many claims that can’t be tested because the method by which these claims were calculated is not explained.

If it turned out that – all other conditions remaining the same – higher specific humidity at the surface translated into a drier upper troposphere, this would be really interesting stuff.

But 3 radiosonde profiles in support of this claim is not sufficient evidence.

The Maths Section – Real Derivation of Dry Adiabatic Lapse Rate

There are a few ways to get to the final result – this is just one approach. Refer to the original 5 equations under the heading: The Equations for the Lapse Rate.

From [2], pV = RT, differentiate both sides with respect to T:

↠ d(pV)/dT = d(RT)/dT

The left hand side can be expanded as: V.dp/dT + p.dV/dT, and the right hand side = R (as dT/dT=1).

↠ Vdp + pdV = RdT  ….[7]

Insert [5], C_p = C_v + R, into [7]:

Vdp + pdV = (C_p-C_v)dT ….[8]

From [1] & [3]:

Vdp = -Mgdz ….[9]

Insert [9] into [8]:

pdV – Mgdz = (C_p-C_v)dT ….[10]

From 4a, under adiabatic conditions, dQ = 0, so C_vdT + pdV = 0, and substituting into [10]”

-C_vdT – Mgdz = C_pdT – C_vdT

and adding C_vdT to both sides:

-Mgdz = C_pdT, or dT/dz = -Mg/C_p ….[11]

and specific heat capacity, c_p = C_p/M, so:

dT/dz = g /c_p ….[11a]

The correct result, stated as equation [6] earlier.

Notes

Note 1: Definitions in equations. WG2010 has:

• P = pressure, while this article has p = pressure (lower case instead of upper case0
•  C_v = heat capacity for 1 kg, this article has C_v = heat capacity for one mole, and c_v = heat capacity for 1 kg.

Note 2: The moist adiabatic lapse rate is calculated using the same approach but with an extra term, Ldq, in equation 4a, which accounts for the latent heat released as water vapor condenses.

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