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[sylas]

So, how high up in the atmosphere does the lapse rate apply? Obviously, it doesn't apply all the way to the moon since that would imply impossibly low temperatures. Instead, it only applies to the elevation at which there is no longer any significant water vapor and that elevation in turn is dictated by the level of CO2, CH4 and NOx in the atmosphere and the amount of heat being transferred.

Actually, the lapse rate applies up to the tropopause; and that is determined not by the presence or absence of water vapour, but by whether there is net heating or cooling from the effects of radiation transfers. Water vapour actually works to reduce the lapse rate, because the moist adiabat is significantly weaker than the dry adiabat.

A dry atmosphere has a much stronger lapse rate. But whether dry or moist, the lapse rate applies up until the atmosphere is back in a radiative equilibrium, and this transition marks tropopause, the end of vertical convection, and the start of the stratosphere (stratified) in which the "lapse rate" is governed by completely different principles, of radiative equilibrium rather than of adiabatic convection and radiative-convective equilibrium.

The theory of lapse rate and tropopause height is explained in Principles of Planetary Climate, in section 4.8 "Tropopause height for real gas atmospheres" (page 255). The theory is general, and applies for all kinds of atmospheres and planets. Other texts on atmospheric physics should explain the same ideas; I am consistently referring to this text (PoPC) primarily because it is online and easily accessible as a common reference point for discussion. There is a progression of material; dipping into a section will give useful conclusions; and for deeper understanding it's well worth working through the previous chapters.

In brief, if works like this. In the absence of any convection or conduction of heat, where the only energy flux is from radiation, we expect a sharp discontinuity in temperature at the surface. The so called natural "skin temperature" of the atmosphere is 2^-0.25 = 0.84 times the surface temperature. (See PoPC section 3.6 "Optically thin atmospheres: The skin temperature" page 141.) This is of course unstable; and there is a flow of heat up into the atmosphere from the surface by convection, and a natural temperature gradient is formed based on adiabatic energy transfers, up until the skin temperature is reached. In this region, the atmosphere is in "radiative-convective equilibrium", with net cooling of the atmosphere from radiation balanced exactly by net heating from convection. See also figure 3.14 from PoPC.

Above the tropopause the atmosphere is in a pure radiative equilibrium. I have also given some more discussion, with some extracts from PoPC, in other posts. See, for example [post=2311418]msg #154[/post] of thread " Need Help: Can You Model CO2 as a Greenhouse Gas (Or is This Just Wishful Thinking?)", and surrounding discussion.

Cheers -- sylas

 

[Xnn]

Thanks sylas;

At the tropopause, the atmosphere is dry and water vapor is no longer a significant constituent. However, the elevation of the tropopause is not constant. It varies about the Earth and is also rising as the levels of greenhouse gases rise. Hence, global warming.

Don't know if it's elevation could be deried from fundamental constant and physical properties or not.


The World Metrological Organization has the following definition for tropopause:

The lowest level at which the lapse rate decreases to 2 °C/km or less, provided that the average lapse rate between this level and all higher levels within 2 km does not exceed 2 °C/km.

 

[sylas]

Thanks sylas;

At the tropopause, the atmosphere is dry and water vapor is no longer a significant constituent. However, the elevation of the tropopause is not constant. It varies about the Earth and is also rising as the levels of greenhouse gases rise. Hence, global warming.

Yes; but the reduction in moisture is not the reason for the tropopause. You would still have a tropopause even if the atmosphere was completely dry all the way through.

The elevation of the tropospause is indeed not constant. It is highest at the equator... and the major reason for this is that the equator is more humid, and has a weaker lapse rate. Hence you go up much higher to get to skin temperature and radiative equilibrium. (PoPC page 258.)

The reason that global warming is leading to a rise in the tropopause is that the surface temperature increases, but the lapse rate remains about the same or perhaps a bit weaker. Hence there is a greater distance from the surface to the tropopause and radiative equilibrium.

Note also that a lapse rate is essential for greenhouse warming to work (PoPC Fig 3.6, section 3.3); but that it is not sufficient. An optically thin atmosphere provides negligible warming, but it still has an adiabatic lapse rate and a tropopause. (PoPC, section 3.6).

Don't know if it's elevation could be deried from fundamental constant and physical properties or not.

You can get a good quantified estimate. I cited previously Principles of Planetary Climate section 4.8. The calculations are based on well defined physical principles, but they are an approximation.

I recommend very highly the exercise of reading though the first four chapters of Principles of Planetary Climate; or indeed any other similarly detailed undergraduate level text on atmospheric physics and energy balance. I found it hard work, but it is an excellent way to develop a deeper understanding of how the temperature structure of the atmosphere works.

The World Metrological Organization has the following definition for tropopause:

This is a very Earth-centric definition. If you go through Principles of Planetary Climate a general physical theory is developed which can be applied in general to a whole range of different planets and different conditions. (See footnote 1 at the bottom of page 75, PoPC)

The major benefit of this is understanding better why there is a tropopause in the atmosphere, and why its altitude tends to increase with with greenhouse driven warming, and why the tropical tropopause is so much higher. When you can actually calculate an estimate for the height of the tropospause for given conditions, the level of understanding is greatly enhanced. I'd have to refer to notes to carry through such a calculation, at present.

Cheers -- sylas

 

[chriscolose]

I haven't worked through the details, but I think Ray Pierrehumbert loosely defines the tropopause as the height of convection.

 

[sylas]

I haven't worked through the details, but I think Ray Pierrehumbert loosely defines the tropopause as the height of convection.

I agree. See page 258 (my emphasis):

How does our computed tropopause height compare to the Earth’s actual tropopause? We have defined the tropopause as the height reached by convection, and in comparing this with atmospheric soundings one needs to recall that even above the convective region, the temperature continues to decrease with height, because temperature goes down with height even in pure infrared radiative equilibrium; in the Earth’s atmosphere, the temperature eventually begins to increase with height because of the effects of atmospheric solar absorption. Hence, the temperature minimum seen in Earth soundings, which is sometimes loosely called the tropopause, is always somewhat above the convectively defined tropopause (see Problem ?? ).[/color]​

I don't think this should be seen as a contrast to the formal definition established by the WMO; but as a generalization that is applicable to many planets and many atmospheres.

Note that the WMO definition is more sophisticated than the naïve simple definition of a temperature minimum. Like Pierrehumbert's dynamic definition, the WMO definition has temperature continuing to decrease with altitude above the tropopause, with the actual minimum value somewhat above the tropopause itself.

The number used in the WMO definition (lapse rate of 2˚C/km) is not arbitrary, but deliberately chosen, I presume in order to give a match with other ways of defining the tropopause. I've had a hunt around, and there seem to be three basic ways that are used to define the tropopause. There is temperature (the thermal tropopause, as defined by the WMO), and "potential vorticity" (the dynamical tropopause) and gradients of trace gases, especially ozone (chemical tropopause, or chemopause). All of these are ways of locating a well defined tropopause with available measurements, whereas the definition used by Pierrehumbert seems to be more about the physical nature of the beast. The text "Principles of Planetary Climate" deliberately omits details of fluid mechanics, apart from a brief introduction in chapter 9. Many of the subtleties of the tropopause seem to show up in mid latitudes at the "subtropical jet", and this level of complexity is not considered in PoPC.

A useful discussion of the alternative definitions is:

• Pan, L.L., et. al. (2004) Definitions and sharpness of the extratropical tropopause: A trace gas perspective, in J. Geophys. Res., 109, D23103, doi:10.1029/2004JD004982

Cheers -- sylas