Abstract
This chapter provides the fundamentals of the thermodynamics of irreversible processes as they are used in meteorology. The discussion includes the laws of thermodynamics, kinetic gas theory, atmospheric stability, and thermodynamics charts. The idea of an air parcel and various thermodynamic systems as they are used in meteorology and climate modeling are introduced as well. Both adiabatic and diabatic processes are discussed. Thermodynamic potentials are introduced for explanation of thermodynamical, dynamical and chemical concepts throughout the book. The chapter covers dry and moist air, phase transition processes and genesis of air masses. In addition, various meteorological concepts important for atmospheric modeling, weather forecasting and analysis of meteorological measurements like the potential temperature, hydrostatic approximation, convective inhibition, convective available potential energy are presented.
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Notes
- 1.
e.g. cacao in a cup.
- 2.
In atmospheric chemistry, the focus is on transformation and reactions. It is required to characterize a fluid by the mass of its constituents and the amount (in mol). The molar mass or molecular/atomic weight (units kg kmol−1) allows the conversion from mass to the amount of the constituent. Physical tasks focus on forces and energy and usually prefer mass.
- 3.
Real gases cannot be approximated as ideal gases at high pressures and low temperatures because then intermolecular forces determine the properties of the gas.
- 4.
A mole is a unit of mass numerically equal to the molecular weight of a substance.
- 5.
In this book, \( \vartheta \) indicates that a temperature value is to be used in degree Celsius.
- 6.
Lorenzo Romano Amedeo Carlo Avogadro di Quaregna e di Cerreto, Count of Quaregna and Cerreto, Italian savant, 1776–1856.
- 7.
Note that in German literature, the Avogardo number is often called Loschmidt number after Johann Josef Loschmidt (Austrian chemist, physicist, 1821–1895). He was the first to mention the value of the Avogadro number when he estimated the average diameter of the molecules in air by a method similar to calculating the number of particles in a given volume of gas in 1865.
- 8.
Joseph Louis Gay-Lussac, also Louis Joseph Gay-Lussac, French chemist and physicist, 1778–1850.
- 9.
Jacques Alexandre César Charles, French inventor, scientist, mathematician, and balloonist, 1746–1823.
- 10.
John Dalton, English chemist, meteorologist and physicist, 1766–1844.
- 11.
Actio = reactio, Latin for action = reaction.
- 12.
The acceleration of gravity is great as compared to nearly all accelerations occurring in the atmosphere. If the Earth’s interior were uniform, acceleration of gravity would just depend on geographical latitude, \( \varphi \), and the distance from the surface, z. The variation caused by the inhomogeneity of the Earth’s interior is neglected in meteorological applications using \( g(\varphi,z) = 9.80665(1 - 0.0026373\cos \varphi + 0.0000059\cos ^{2}\varphi )(1 - 3.14 \cdot 10^{-7}z)\,\mathrm{m\,s^{-2}} \). This equation leads to an acceleration at sea level of 9. 83257, 9. 80665, 9. 78084 m s−2 at the poles, 45∘ latitude, and the equator, respectively.
- 13.
See Chap. 6 for a derivation from the vertical equation of motion.
- 14.
On the 500 hPa level map, contours of geopotential height are typically spaced at 60 m.
- 15.
In mountainous regions, pressure coordinates are disadvantageous as they intersect with the terrain.
- 16.
In an isothermal atmosphere, adiabatic sound-wave speed c s is independent of height.
- 17.
The thickness is proportional to the virtual temperature (Sect. 2.7.6) in the lower and mid-troposphere.
- 18.
The assumption of constant density is a far better approximation for the ocean than the atmosphere because of the sharp upper boundary and because the density of seawater is independent of pressure.
- 19.
Water is nearly incompressible.
- 20.
This procedure is how to measure fever, by establishing an equilibrium between the body and the thermometer.
- 21.
Work is force times path, i.e. \( \mathbf{F} \cdot d\mathbf{r} = F\cos \theta dr \) with θ being the angle between the vectors F and r.
- 22.
Chemical potentials differ among species. When, for instance, 1 kg water is added to a system, doing so changes the system’s energy by a smaller value than adding hydrogen despite both substances consist of the same atoms in the same number. However, the chemical potential of hydrogen exceeds that of water.
- 23.
Josiah Willard Gibbs, American mathematical-engineer, theoretical physicist, and chemist, 1839–1903. He is famous for his 1876 publication of On the Equilibrium of Heterogeneous Substances that graphically analyzes multi-phase chemical systems.
- 24.
All other components of the first law of thermodynamics are assumed to be zero in this example.
- 25.
The term ‘renewable energy’ is technically incorrect from a thermodynamic point of view. Actually, kinetic energy, in case of wind energy, or solar energy, in the cases of bio-fuels or solar panels, are used and taken from the system.
- 26.
\( c_{p} = 1,005.45 + 0.033T \) for − 100 ∘C ≤ T ≤ 60 ∘C.
- 27.
Simon Denis Poisson, French mathematician and physicist, 1781–1840.
- 28.
The first law of thermodynamics would permit that the space shuttle rises by cooling of its heat shield.
- 29.
Axioms cannot be proven.
- 30.
Many expressions of the second law of thermodynamics exist:
-
The Clausius statement: There is no \( \underline{\mathit{self - acting}} \) cyclic process or device that only removes heat from one reservoir and discharges an equal amount of heat to a reservoir at higher temperature.
-
The Kelvin-Planck statement: There is no cyclic process or device that only removes heat from a single reservoir and performs an equal amount of work.
-
Planck statement: Processes wherein friction occurs are irreversible.
-
A perpetuum mobile of second kind is impossible.
-
No cycle that operates between any two temperatures can have a higher efficiency than a Carnot cycle operating between the same two temperatures.
-
Each thermodynamic system has a state variable, the entropy \( S =\delta Q/T \) that can be calculated for reversible processes.
-
- 31.
Nicolas Léonard Sadi Carnot, French scientist, physicist and military engineer, 1796–1832.
- 32.
Irreversible processes lead to a permanent change in the environment in some way that even when the system returns to its exactly initial state. For instance, when a balloon bursts, its gases mix with the ambient air. The system is destroyed. The sudden expansion causes a sound wave carrying energy that finally dissipates by being converted to heat.
- 33.
Reversible processes have the best energy transfer for which they serve often as idealized reference processes in meteorology.
- 34.
Rudolf Julius Emmanuel Clausius, German physicist, 1822–1888.
- 35.
Hermann von Helmholtz, German physiologist and physicist, 1821–1894.
- 36.
Adrien-Marie Legendre, French mathematician, 1752–1833.
- 37.
When in an equation like \( du = Tds - pdv \) one of the independent variables is to be replaced by a dependent one (e.g. s or T), the product of the independent variables (here Ts) has to be subtracted from the dependent variable (here u). Doing so provides a new quantity that only depends on the two independent variables T and v.
- 38.
The relationship between the thermodynamic potentials and the state variables can be easily kept in mind by the Guggenheim Footnote 39 scheme where the four energies are framed by their natural state variables:
$$ \displaystyle{\begin{array}{ccc} \mathrm{S} & \mathrm{U} & \mathrm{V}\\ \mathrm{H} & & \mathrm{F} \\ \mathrm{p} & \mathrm{G} & \mathrm{T}\\ \end{array} } $$ - 39.
Edward Armand Guggenheim, English thermodynamicist and chemist, 1901–1970. He is famous for his 1933 publication of the Modern Thermodynamics by the Methods of Willard Gibbs.
- 40.
Benoît Paul Émile Clapeyron, French engineer and physicist, one of the founders of thermodynamics, 1799–1864.
- 41.
According to the World Meteorological Organization (WMO), climate is the average weather over 30 years. See also Chap. 7.
- 42.
Although in the SI-system, its unit is Pa, meteorologists traditionally use hPa.
- 43.
Under tropospheric conditions, absolute humidity (in g m−3) is approximately 0.8 times the water-vapor pressure in hPa.
- 44.
At 1, 000 hPa, the specific humidity (in g kg−1) is about 0.6 times the water-vapor pressure in hPa.
- 45.
Specific humidity is often approximated by \( q \approx 0.622\frac{e} {p} \) as p ≫ 0. 378e in the troposphere.
- 46.
The mixing ratio is often approximated as \( r \approx 0.622\frac{e} {p} \) as p ≫ e in the troposphere.
- 47.
This fact explains why more homeruns occur during humid and warm weather conditions. The ball experiences less resistance than under cold, dry conditions.
- 48.
Note that this approximation is not as good as for dry air because at conditions close to saturation, attractive forces between molecules become significant (Chap. 3).
- 49.
On the Celsius scale, the zero-point is defined so it corresponds to the freezing point of air-saturated water at 1, 013. 25 hPa.
- 50.
According to Gibbs, the number of independent variables, N for a heterogeneous system with C different, non-reactive components in P phases is \( N = C + 2 - P \).
- 51.
For the heat consumed during sublimation, vaporization and fusion the sign changes. The latent heats like the specific heats insignificantly depend on temperature \( L_{v} = 2.501 \cdot 10^{6}(\frac{T_{0}} {T} )^{b} \) with \( b = 0.167 + 3.67 \cdot 10^{-4}T \) for 233 K < T < 313 K; \( L_{m} = 0.3337 \cdot 10^{6} + 2,030.6\,T - 10.47\,T^{2} \) for − 50 ∘C < T < 0 ∘C. Typically, atmospheric scientists neglect this temperature dependency.
- 52.
James Prescott Joule, English physicist and brewer, 1818–1889; Joule formulated several physical laws. His second law states that the internal energy of an ideal gas is independent of its volume and pressure, and depends only on its temperature.
- 53.
The transformation of ice crystals to rounded grains of about 2 mm in diameter is referred to as snow metamorphism. It partly is due to water-vapor transfer from sharply curved edges to smoother surfaces of the ice crystals because of the dependency of water vapor pressure on curvature. Also, temperature gradients within the snow may result in water-vapor transfer and re-deposition within the snowpack.
- 54.
Heinrich Gustave Magnus, German physicist and chemist, 1802–1870.
- 55.
Sir John Murray, British oceanographer, 1841–1914.
- 56.
Many organic materials show expansion and shrinking that are functions of humidity.
- 57.
Warm water from a bath adds moisture to the bathroom and brings it to condensation at the mirrors and surfaces of the room, and fog develops.
- 58.
Many early Global Circulation and numerical weather prediction models used the pseudo-adiabatic assumption, i.e. that once condensed the water falls out immediately. This assumption led to incorrect moisture and temperature profiles as condensate might have undergone further phase-transition processes (e.g. freezing, evaporating below cloud base) thereby consuming heat or releasing latent heat after leaving the cloud.
- 59.
The equations of the moist atmosphere listed in this table can also be written for dq s instead of dr s because \( r \approx 0.622e/p \approx q \).
- 60.
Various formulas have been derived for equivalent potential temperature. A comparison and evaluation can be found in Davies-Jones (2009). The accuracy of the formulas can be examined by finding the imitation first law of thermodynamics. The accuracy of the various formulas ranges between 0. 015 and 0. 4 K for typical atmospheric conditions. The errors result from the temperature dependency of c p and L that are typically neglected in meteorology.
- 61.
e x ≈ 1 + x when x is very small.
- 62.
As a rule of thumb pressure and water-vapor pressure relatively decrease about 1.2 % per 100 m; saturation water vapor pressure relatively decreases by about 7 % per 100 m; relative humidity relatively increases by about 6 % per 100 m, e.g. for a relative humidity of 50 % this means an absolute increase dRH∕dz of 3 % per 100 m.
- 63.
The vertical development of clouds is related to the degree of instability. Less developed cumulus clouds indicate weak, strong developed cumulonimbus clouds strong instability (Chap. 3 for a cloud classification).
- 64.
Think, for instance, of a balloon ascending into the sky.
- 65.
When temperature is about 20 ∘C at a place in flat, homogeneous terrain, chances are high that temperature is not lower/higher 10 km away.
- 66.
Sir David Brunt, English meteorologist, 1886–1965.
- 67.
Vilho Väisälä, Finnish meteorologist, 1899–1969.
- 68.
Evapotranspiration is an artificial word used in meteorology and hydrology to describe the combined effects of evaporation of water and sublimation of ice/snow from various surface and the transpiration of water by plants.
- 69.
Material, concepts, ideas and problems of the following books and articles inspired this chapter. These sources are recommended for further reading.
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Mölders, N., Kramm, G. (2014). Thermodynamics. In: Lectures in Meteorology. Springer Atmospheric Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-02144-7_2
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