Basic Atmospheric Thermodynamics as 2nd Law

The debate on the temperature distribution in the atmosphere is going around in never-ending circulation just like the air in the atmosphere. Let us here recall the basic statements of my chapter Climate Thermodynamics in a famous book, which is condensed as the 2nd law of thermodynamics expressed in the following form with the dot signifying time differentiation:

• $\dot K+\dot P = W-D$
• $\dot E = -W + D$,  

where $K$ is kinetic energy, $P$ potential energy, $W$ work, $E$ heat energy and $D\ge 0$ is turbulent dissipation with $W > 0$ under expansion and $W < 0$ under compression. The sign of $D$ sets the direction of time with always transfer of energy from $K+P$ to $E$.

There are two basic temperature distributions with linear decrease with height as lapse rate (assuming zero heat conductivity): 

• Isothermal atmosphere with zero lapse rate: $D$ maximal with $W=D$.
• Maximal (dry adiabatic) lapse rate $=9.8\, C/km$ with $D=0$ minimal.

The observed lapse rate (of about 6.5 C/km) is somewhere between maximal and minimal. We note:

1. Lapse rate may increase by slow laminar vertical circulation with ascending air cooling and descending air warming with $D=0$.
2. Lapse rate may decrease by turbulent dissipation $D>0$ heating upper layers.
3. A (partially) transparent atmosphere (like on Earth) heated from below may naturally develop a positive lapse rate by 1. 
4. An opaque atmosphere (like on Venus) heated from above may become isothermal by heat conduction and may then develop a positive lapse rate by 1.  

The lapse rate is basic to planetary climate since it determines the surface temperature from the temperature at the top of the troposphere, and its dependence on the radiative properties of the atmosphere is a key question in global climate science.

Compare with the previous post Lapse Rate by Gravitation: Loschmidt or Boltzmann/Maxwell?

Lapse Rate by Gravitation: Loschmidt or Boltzmann/Maxwell?

Will an atmosphere under the action of gravity assume a linear temperature profile with slope equal to the dry adiabatic lapse rate? Loschmidt said yes, while Boltzmann and Maxwell claimed that the atmosphere would be isothermal. Graeff (2007) has made experiments supporting Loschmidt and so it is natural to seek a theoretical explanation. 

Consider a horizontal closed insulated tube filled with still air at uniform temperature. Let the tube be turned into an upright position. Alternatively, we may consider a vertical tube with gravitation gradually being turned on from zero, or a horizontal tube being rotated horizontally starting from rest. During increasing gravitational force the air will be compressed and knowing that compression of air causes heating, we expect to see a temperature increase. How big will it be? Well, the 2nd Law of Thermodynamics states that under adiabatic and isentropic transformation (no external heat source and no turbulent dissipation):

• c_vdT + pdV =0   

where c_v is heat capacity of air under constant volume, dT is change of temperature T, p is pressure and dV is change of volume V. Recalling the differentiated form of the gas law pV = RT with R a gas constant

• pdV + Vdp = RdT

and the equilibrium equation in still air with x a vertical coordinate 

• dp = -g rho dx or Vdp = - gdx

where g is the gravity constant, rho = 1/V is density, we find

• (c_v + R) dT = -gdx or c_p dT/dx = - g, 

where c_p = c_v + R is heat capacity under constant pressure.

We thus find still air solutions with a dry adiabatic lapse rate dT/dx= - g/c_p = - g with c_p = 1 for air, as a consequence of compression by gravitation, using

1. work by compression stored as heat energy
2. pressure balancing gravity (hydrostatic balance).

A corresponding family of stationary still air solutions is given by (assuming x = 0 corresponds to the bottom of the tube):

• p(x) ~ (1 - gx)^(a+1)
• rho(x) ~ (1 - gx)^a
• T(x) ~ (1 - gx)

with a >0 a constant. In the absence of heat conduction such solutions may remain as stationary still air solutions. We thus find support of Loschmidt's conjecture of still air solutions with the dry adiabatic lapse rate, in the absence of heat conduction. In the presence of (small) heat conduction, it appears that a (small) external source will be needed to maintain the lapse rate. Of course, in planetary atmospheres external heat forcing from insolation is present.

Returning the tube to a horizontal position would in the present set up without turbulent dissipation, restore the isothermal case. Turning the tube upside down from the vertical position would then establish a reverse lapse rate passing through the horizontal isothermal case.

Further, it seems that without heat source, the isothermal case of Boltzmann/Maxwell will take over under the action of heat conduction, with p(x) ~ exp( - cx) and rho(x) ~ exp ( - cx) with c>0 a constant.  

For the Euler/Navier-Stokes equations for a compressible gas subject to gravitation, see Computational Thermodynamics and the chapter Climate Thermodynamics in Slaying the Sky Dragon.

Faint Young Sun Paradox Resolved

The faint young Sun paradox describes the apparent contradiction between observations of liquid water early in the Earth's history and the astrophysical expectation that the Sun's output would be only 70% as intense during that epoch as it is during the modern epoch. The issue was raised by astronomers Carl Sagan and George Mullen in 1972.

The analysis of the lapse rate in earlier posts suggests the following resolution of the paradox:

A reduction of 30% of the insolation could mean a reduction from 180 to 125 W/m2 absorbed by the Earth surface, and a reduction of 140 to 100 W/m2 to be radiated from the tropopause, assuming 40 W/m2 directly radiated through the atmospheric window in both cases. 

This would require a drop in the temperature of the tropopause from - 50 C to - 68 C (from 223 K to 205 K by Stefan-Boltzmann with 223 =(140/5.66)^0.25 x 100). The Earth surface temperature could then remain at + 15 C if the lapse rate increased from the present 6.5 C/km to 8.3 C/km (with the tropopause at 10 km altitude). 

The maximal lapse rate is 10 C/km and could be attained in an atmosphere without thermodynamics, in an atmosphere at rest without motion of the air, with heat transfer by conduction and radiation but no thermodynamics of convection and evaporation/condensation. 

An effect of thermodynamics in the present atmosphere can thus be viewed as a reduction of the lapse rate from 10 to 6.5 C/km with the difference increasing with the vigor of the thermodynamics. With a less vigorous atmosphere the lapse rate could thus increase from 6.5 to 8.3 C/km and thus sustain the same surface temperature with only 70% of the input from the Sun of today. 

In the extreme case of an atmosphere without motion with a lapse rate of 10 C/km,  a 50% Sun would thus be enough to sustain comfortable organic life at + 15 C, thus very early in the history of the solar system.  Organic life is supposed to be 4 billion years old, apparently ignited by a young 50% Sun. 

The argument supports the idea of the thermodynamics of the atmosphere as an air conditioner acting to reduce the lapse rate and thus cool the Earth surface as the inside, with the the tropopause as the outside with a temperature set by the input via Stefan-Boltzmann.

PS Several unsuccessful attempts to resolve the paradox have been presented recently:

• Faint Young Sun Paradox remains (Nature June 2011)
• Why early Earth was no snowball (Potsdam Institute Dec 2012)
• Hypothesis for Faint Young Sun Paradox (Purdue Univ May 2012) 
• Faint Young Sun Paradox not resolved (NASA May 2012)
• Paradox: Is there even life on Earth (Smithsonian Astrophysical Observatory Sept 2012)
• Faint Young Sun Problem (Geophysical Research Letters 2012).

Solutions attempts include:

• Early Earth Atmosphere: Right mix of greenhouse gases (Nathan Sheldon) 
• Geology: Much more geothermal energy 
• Biology: Life developed on a cold planet (John Priscu)
• Fundamental Physics: e.g., gravitational constant has varied 
• Astrophysical Solutions: Young Sun was not faint. 

Lapse Rate by Thermodynamics

In the previous post I asked Roy Spencer about the correct description of the GreenHouse Gas Effect GHGE, which he claims is misunderstood by many people, but got no answer.

It seems that Spencer, WUWT, Lubos and of course all climate alarmists, take the existence of a GHGE for granted, maybe because it has an acronym, while the nature of this effect remains elusive, which makes serious scientific discussion difficult or even meaningless.

Spencer gives the hint that GHGE "is an infrared effect" which connects to the common (mis)conception that the GHGE is an effect of radiation, more precisely, that the lapse rate, or temperature drop with atmospheric altitude, is determined by radiation.

This connects to the question of the lapse rate of a thermally isolated column of air subject to gravitation, to which Loschmidt answered the dry adiabatic lapse rate of 10 C/km, while Maxwell and Boltzmann claimed zero lapse rate or isothermal air, without ever reaching any conclusion. 

The observed lapse rate of the atmosphere is 6.5 C/km which keeps the Earth surface at a comfortable + 15 C, while the temperature at the tropopause at 10 Km is chilling - 50 C. 

The surface temperature is thus determined by the temperature at the tropopause and the lapse rate. What determines then the lapse rate? Thermodynamics or radiation? I argue that it is thermodynamics, while people assuming that the GHGE is real seem to argue that it is radiation. 

In a previous post I gave an argument in favor of thermodynamics, which I here return to. Consider thus the column air discussed by Loschmidt and Maxwell/Boltzmann: 

If the column is isolated without heat supply and the air is still, then heat conduction will equalize the temperature into zero lapse rate in accordance with Maxwell/Boltzmann. However, if there is a (small) heat supply Q at the bottom of the column, then by conduction in still air a linear any lapse rate - dT/dx will be established by the balance equation  

• - alpha x dT/dx = Q 

where alpha is a (small) coefficient of heat conduction. With alpha sufficiently small, this will make -dT/dx bigger than the dry adiabatic lapse rate of 10 C/km, and convective overturning will be initiated (to eventually give an observed lapse rate of 6.5 C/m).

If the coefficient of heat conduction in still air is small, which is the case, then a small heat source at the bottom will prior to convective overturning establish a lapse rate equal to the dry adiabatic lapse in accordance with Loschmidt, a lapse rate thus determined by thermodynamics and not radiation.

To sum up: Loschmidt was more correct in the setting of the Earth atmosphere, since the Earth surface is heated, and the lapse rate is thus determined by thermodynamics and not by radiation. A GHGE based on a lapse rate determined by radiation, is just an acronym and no a real effect.

With the lapse rate determined by thermodynamics and not radiation, the role of GHG is limited to an effect on the temperature at the tropopause, which may not easily be changed by small perturbations in the radiative properties of the atmosphere, thus suggesting small climate sensitivity as discussed in previous posts.

What Is Roy Spencer's Definition of The Greenhouse Effect?

The top Google definition of the "greenhouse effect" is:

• The trapping of the sun’s warmth in a planet’s lower atmosphere due to the greater transparency of the atmosphere to visible radiation from the sun than to infrared radiation emitted from the planet’s surface.

Roy Spencer ironizes over this definition:

• Actually, the greenhouse effect would still operate even if the atmosphere absorbed just as much solar as it does infrared. When even Google gets the definition so wrong, how can mere mortals be expected to understand it?
• The so-called greenhouse effect, which is an infrared effect, is admittedly not as intuitively obvious to us as solar heating. Every layer of the atmosphere becomes both a “source” as well as a “sink” of IR energy, which is a complication not faced with understanding solar heating, with the sun as the source.
• To understand the greenhouse effect’s impact on surface temperature and the atmospheric temperature profile, there are some basic concepts which I continue to see misunderstandings about. If we can’t agree on these basics, then there really is no reason to continue the discussion because we are speaking different languages, with no way to translate between them.

Spencer then lists 6 basic concepts related to the greenhouse effect which "people most commonly misunderstand". But Spencer does not reveal his own definition of the greenhouse effect, which makes the discussion meaningless.

So Roy, what is then the correct description of the greenhouse effect, which you suggest is a real effect? I have sent the question to Roy and await response.

PS Roy does not give any response at all, not even to any of the many comments on his blog. Roy throws out a bunch of propositions to the crowd and then runs to hide behind a wall of silence. Is this what scientific debate is supposed to be? 

Negative Climate Sensitivity: Global Cooling 2

Here is a remark connecting to the previous posts Negative Climate Sensitivity: Global Cooling 1 and  Leaked Climate Sensitivity of 0.3 C.

Climate sensitivity as the effect on the Earth surface temperature of doubled atmospheric CO2,  is by IPCC estimated to an alarming + 3 C. The idea is that CO2 by acting like a "greenhouse gas" blocks radiation from the Earth and thus causes warming. This is a very primitive idea and as such it may well be wrong.

Let us see what basic thermodynamics says:

1. The Earth surface is heated by incoming energy from the Sun, and the Earth-atmosphere system radiates an equal amount of energy from an outer boundary or top of the atmosphere TOA at the tropopause at a pressure of 0.2 - 0.3 bar.
2. The surface temperature is determined by the lapse rate from the temperature at TOA.  
3. In an atmosphere without thermodynamics of advection (still air) energy would be transported from the Earth surface to TOA by a combined process of conduction and radiation, which would require a linear temperature profile with constant lapse rate equal to the dry adiabatic lapse rate of 10 C/km as the maximal rate of a stable atmosphere without advective overturning (thus establishing the Loschmidt gravito-thermal effect).
4. The observed lapse rate in the real atmosphere with thermodynamics of advection is 6.5 C/km.
5. Increasing thermodynamics would thus tend to decrease the lapse rate and with a temperature of TOA unchanged, would thus cause global cooling.
6. The logic is that more vigorous thermodynamics would transport energy more efficiently from the Earth surface and thus cause cooling.

Doubled CO2 could increase the temperature of TOA, by decreasing the direct radiation to outer space from the Earth surface, but would also demand a more vigorous thermodynamics reducing the lapse rate.

The rationale is that conduction/radiation passively operates on a temperature gradient/lapse rate maintained by exterior forcing, while thermodynamics/advection actively works to decrease the gradient/lapse rate. To see the active part dominate the passive would not be surprising.

Climate sensitivity would thus come out by subtracting effects of radiation and thermodynamics, and not as suggested by IPCC by adding these effects. This may explain why the  +3 C by IPCC is not what is observed, and that what is observed is close to 0 or even negative.   

Negative Climate Sensitivity: Global Cooling 1

The preceding posts lead to the conclusion that the Earth (and Venus) including atmosphere up to a pressure of 0.2 - 0.3 bar have a TOA temperature at the tropopause equal to the bolometric temperature determined by the distance to the Sun, as a minimal temperature.

The surface temperature would then be determined by a lapse rate observed to be 6.5 C/km resulting from atmospheric thermodynamics driven by radiative forcing of the Earth surface,  to be compared with the dry adiabatic lapse rate of g/c_p = 9.8 C/km with g gravitational acceleration and c_p the heat capacity of air at constant pressure.

The thermodynamics in the atmosphere would thus have the effect of reducing the dry adiabatic lapse representing a possible state without radiative forcing and thermodynamics, and thus an effect of reducing the surface temperature.

Doubling the atmospheric CO2 is by IPCC estimated to correspond to a radiative forcing of 2- 4 W/m2, to be added to the 180 - 40 = 140 W/m2 effectively absorbed by the Earth surface with 180 incoming and 40 directly outgoing through the atmospheric window.  The effect on the surface temperature would then be determined by the lapse rate with the bolometric temperature of TOA at the tropopause  unchanged because the distance to the Sun is unchanged.

The effect of additional effective radiative forcing of the Earth surface would be more active thermodynamics which would tend to further reduce the lapse rate and thus the Earth surface temperature.

Climate sensitivity as the increase of the Earth surface temperature upon doubling of CO2, would thus be negative: More CO2 would tend to be cooling rather than warming, but the effect would probably be so small that it could not be observed. Climate sensitivity would thus seem to be non-positive and the risk of global warming would (very likely) be small (with a most likely value of 0).

(This insight is now quickly eating its way into the minds of both people and politicians and global warming hysteria is already history).

Compare with the climate sensitivity of + 3 C by IPCC, which is obtained by a combination (i) radiative forcing increasing the bolometric temperature, as if the Earth was moved closer to the Sun and (ii) positive thermodynamics feedback, as if thermodynamics could slow down by additional forcing.

The IPCC view is presented by its Swedish representative Lennart Bengtsson with the following key argument:

• .... the Earth energy balance can temporarily be changed by reduced radiation to outer space by increased concentration of greenhouse gases. 

We see here the idea of heating (less outgoing radiation) with necessarily a warming effect by greenhouse gases, which is the key of global warming propaganda. But LB eliminates the warming effect by stating that it is only temporary, and thus seems to say that the effect in the end is zero.

PS Notice that in the IPCC and LB greenhouse gas argument, the TOA would be put at 5 km at a bolometric temperature of - 18 C corresponding to 240 W/m2 outgoing radiation, and would then be shifted upwards to cooler levels under increased concentration of greenhouse gases and then eventually cause surface warming.  But there is no TOA other than the tropopause (as concerns thermodynamics), and shifting an artificial TOA up or down would lack physical meaning.

The Earth-Atmosphere System as Blackbody

Data suggests that the Earth including the troposphere can be viewed as a blackbody heated by 140 W/m2 absorbed by the Earth surface and emitting 140 W/m2 at a bolometric temperature of about -55 C  (according to Stefan-Boltzmann's Law) attained at the tropopause at 0.1 - 0.3 bar as its "outer boundary".

The 140 W/m2 absorbed by the Earth surface comes from 240 W/m2 absorbed by the Earth-atmosphere with an albedo of 0.3 out of a total of 340 W/m2 incoming from the Sun, minus 60 W/m2 absorbed and re-emitted by the atmosphere minus 40 W/m2 directly radiated from the Earth surface to outer space through the "atmospheric window".

Including the troposphere and the stratosphere in the Earth-atmosphere system makes the stratopause  at 0.001 bar the outer boundary with an observed temperature of 0 C with corresponding blackbody radiation of 320 W/m2, which is close to the total of 340 W/m2.

The internal thermodynamics/raditation of the Earth-atmosphere system is very complex and difficult to accurately model, but it thus appears that it is possible to view the system as a single blackbody in two ways:

1. Earth + troposphere 
2. Earth + troposphere + stratopshere 

with an observed temperature of the "outer boundary" in reasonable correspondence in both cases with the bolometric temperature as the blackbody temperature of the same irradiance.

Viewing the Earth + troposphere as a blackbody with outer boundary or top of the atmosphere TOA temperature determined as bolometric temperature, makes the surface temperature depend on the lapse rate, and with the lapse rate mainly determined by thermodynamics, the surface temperature would be insensitive to small changes in the internal radiative composition of the atmosphere. In other words, there would be no detectable greenhouse effect.

For Venus the observed temperature at 0.25 bar is -20 C which is again equal to the bolometric temperature.

We thus find evidence that for both the Earth and Venus including atmospheres up to about 0.3 bar, the TOA temperature is the bolometric temperature determined by distance to the Sun, and the surface temperature is determined by a lapse rates mainly set by thermodynamics.