An Alternative Derivation of the Static Dry Adiabatic
Temperature Lapse Rate
BSc, meteorologist, Stockholm University, Civil engineer, electronics, Royal Institute of Technology, Stockholm, PhD, institution of Paleogeophysics & geodynamics, Stockholm University
The ”static” dry adiabatic temperature lapse rate is derived for a hypothetical energetically isolated model atmosphere lacking advection and convection. The method of derivation is to investigate the energetic situation for two small equal atmospheric air masses at different altitudes in a vertical column of air. The difference of total energy between these masses is calculated. The ideal gas law is assumed to be valid. This derivation is just one version of several others but might be easier to understand for laymen. The adequate theory needed should have been learnt at high school in natural sciences.
The “dynamic” dry adiabatic temperature lapse rate can be and has been derived in different ways and it turns out to be dT/dz = –g/Cp* where g is the gravity constant and Cp* is the heat capacity of air at constant pressure. By “dynamic” it is meant that the derivation is based on following an energetically isolated “air parcel” of constant mass when moving vertically which is the reason for the epithet “dynamic”. One method to derive the result is to use the equation of state P = ρRT, take its total derivative and do a lot of calculus ( ref 1).
It has been shown by Jelbring 2003 (ref 2) that the an energetically closed planetary atmosphere under the impact of gravity which is allowed to come to rest for a long time (no winds and no temperature inversions) has to develop a “static” dry adiabatic temperature lapse rate that is equal to the “dynamic” one mentioned above. That derivation rests on the use of first principle physics (1:st and 2:nd law of thermodynamics).
This derivation considers two air parcels of equal and suitable mass (a billion molecules) which have to carry an equal amount of total energy regardless of their altitude if an adiabatic condition is assumed. It seems that the proof delivered in reference 2 has been hard to understand both by professional scientists and by laymen. Hence, the major reason for writing this article using an alternative method for the derivation. This is to make it easier for anybody to grasp the physics behind the observed temperature (energy) structure that can be observed in our atmosphere and in an even more illuminating way in the Venusian atmosphere.
It is conceptually important to accept that the “dynamic” and “static” dry adiabatic temperature lapse rates are approximately identical. The kinetic (vertical) energy in the first one is very small compared to other energies involved and can be omitted. The existence of the “static” adiabatic temperature lapse rate directly implies that there has to be a substantial “Greenhouse Effect” (GE) on any real planetary atmosphere as long as there exists agents in the atmosphere that are able to emit IR radiation to space from altitudes above the surface of the planet. Observational evidence shows this to be the case in all known planetary atmospheres. These agents can be any solid or liquid matter suspended in the atmosphere (dust, clouds and salt particles) and also so called “greenhouse gases”.
The energy states of two air parcels in a vertical air column with identical masses m1 = m0 and m2 = m0 at two locations (L1 and L2) are investigated. The altitudes of L1 and L2 is z1 and z2 where z2 > z1.m1 and m2 are carrying the total energies E1 and E2. ∆E = E1 – E2 is investigated and described with the help of mathematical formulae. The adiabatic relationship is found by setting ∆E = 0.
The following assumptions are made:
- The atmosphere is energetically insulated.
- There are no winds in the enclosed atmosphere.
- The atmosphere consists of a mixture of ideal gases and the Ideal Gas Law is valid.
- Gravity (g) is considered constant at L1 and L2.
- Cp* = Cv* + R* The star indicates that the dimension is (Joule/(kg K)).
Cp* = 1004, R* = 287 and Cv* = 717 Joule/(kg K) in air.
The static dry adiabatic temperature lapse rate dT/dz = – g/Cp* will develop in any model planetary atmosphere that is insulated from energy input and output at the surface and through a concentric spherical shell that surrounds the troposphere.
The following equations describe the energy situation for m1 and m2 both equal to m0.
E2 = E1 + m0 g ∆z + m0 Cv* ∆T + (P1 V1 – P2 V2) ( 1)
∆E = m0 g ∆z + m0 Cv* ∆T + (P1 V1 – P2 V2) (2)
The first term at the right hand side is gravitational potential energy difference. The second term is the increase (actually decrease since ∆T is negative) in molecular kinetic energy and the third term is the change in work done on the atmosphere by m1 and m2 at the two static locations under consideration.
It follows from P1 V1 = m0 R* T1 and P2 V2 = m0 R* T2 that
(P1 V1 – P2 V2) = m0 R* (T1-T2) where ∆T = T1-T2. Enter this into equation (2) and we get
∆E = m0 g ∆z + m0 Cv* ∆T + m0 R* ∆T (3)
R* = Cp* – Cv* gives
∆E = m0 g ∆z + m0 Cv* ∆T + m0 Cp* ∆T – m0 Cv * ∆T or
∆E = m0 g ∆z + m0 Cp* ∆T (4)
The definition of an adiabatic energy situation is given by ∆E = 0 which leads to
g ∆z + Cp* ∆T = 0 or (5)
∆T/ ∆z = -g/Cp*
When ∆z goes to zero, ∆T/ ∆z goes to dT/dt. Therefore we get:
dT/dz = -g/Cp*; dT/dz = – 9.81/1004; dT/dz = -0.00978 K/m or 9.78 K/km
It is obvious that the temperature profile at a specific location depends on latitude (solar insolation), orography, water vapour, tides and whether the surface happens to be land or ocean. In other words, the temperature profile at a specific location depends on several physical factors but the most important one is very probably the tendency for Earth’s (and the atmosphere of any planet) to seek an energetic equilibrium. This is what the existence of the static dry adiabatic temperature lapse rate shows and that is why it is important. On earth the dry adiabatic temperature lapse rate is best verified by observations during afternoons after a sunny day or in Antarctica where very strong katabatic winds prevail. In these situations dT/dz is close to -9.8 K/km. On the other hand the US 1976 Standard Atmosphere has a temperature lapse rate of -6.5 K/km under 10000 m altitude and is isothermal at the tropopause. Hence, the static dry adiabatic temperature lapse rate is directly confirmed on earth during specific physical conditions that don´t always prevail. But it does evolve close to Earth’s surface every sunny day over land.
The absolute best observational evidence of the impact of “energy dissipation” according to the second law of thermodynamics is found in the Venusian atmosphere where the lapse rate is close to the theoretical adiabatic one from the surface to about 40 km altitude. The reason is simply its thickness which facilitates an approximate even total energy distribution per mass unit in its atmosphere, from the surface to its upper troposphere.
It has been known for a 100 years that the surface of Earth is warmer than it should have been if radiating as a “black body” into space. This is the reason why a “Greenhouse Effect” (GE) has been suggested. The accepted value of GE is 33 K which is used by NASA. The value can be challenged. There is no doubt that most of this effect can be traced to the physical process of energy/mass equalization in the atmosphere of earth. It follows from this insight that other physical processes affecting GE must have a relatively small impact. It is possible today to quantitatively examine how much of the GE is created by a number of physical processes but there is little initiative to do so since the cause of GE creation has been almost solely attributed to the impact of “greenhouse gases”. IPPC should be more than ashamed to translate a complex scientific climate problem into a reductionistic one variable political statement relating to the unproven danger of carbon dioxide increase in our atmosphere.
- Holton J.R., An Introduction to dynamic Meteorology, second edition, Academic Press, 1979, pp 47-49
- Jelbring H. “The Greenhouse Effect as a Function of Atmospheric Mass.”, Energy & Environment, Vol 14, 2&3, 2003, 351-356