General Concepts of Atmospheric Thermodynamic Atmospheric Thermodynamic Theory The theory of thermodynamics is one of the important cornerstones of classical physics. It has applications not only in physics, chemistry, and the Earth sciences, but in subjects as diverse as biology and economics. Thermodynamics plays an important role in our quantitative understanding of atmospheric phenomena ranging from the smallest cloud microphysical processes to the general circulation of the atmosphere. The purpose of this course is to introduce some fundamental ideas and relationships in thermodynamics and to apply them to a number of simple, but important, atmospheric situations. 1.1 definitions: 1. Dry air is a theoretical sample of air that has no water vapor 2. Dry air is the air has a low relative humidity (less than 40%) Note: in winter season some people uses the humidifier as shown in figure 1 Fig1: Weather humidifier 1
1.2 Poisson Equation This equation gives the relation between (T and P) at any two different pressure levels. Where: Example 1: Find the temperature at 850 mb if the temperature at level of 950 mb is 265 K Solution Example 2: At what pressure level the temperature records 233 K if its values are 265 K at pressure level of 900 mb. Solution 2
1.3 Potential Temperature: Is the temperature of a parcel if it brought adiabatically to pressure 1000 mb Example 3 : An air parcel of temperature about 255 K at pressure level 700 mb, find the potential temperature of this parcel. Solution 3
1.4 Entropy (ϕ): It is a measure of the total energy of thermodynamic system including the internal energy U. 1.4.1 The relation between Entropy (ϕ) and potential temperature ( ) The change in Entropy (ϕ) is given by : But Supposing the air as Ideal gas For one mole (n=1) We compensate equation 2 in 1 We have ( We compensate equation 4 in 3 4
From other site: 5
From equation (І): By compensate in equation (ІІ) The Relationship between Entropy and Potential Temperature 6
1.4 Thermodynamic Diagrams 1.4.1 Stuve diagram Stuve diagram is a thermodynamic diagram depends on following equation: Which proved that the potential temperature ( ) in certain value, the pressure and temperature ( ) related by a linear relation, then the thermodynamic process presents as straight lines on thermodynamic Chart which have the coordinates ( ). This type of charts is suitable to describe atmospheric thermodynamic process and the line which will be constant on it called adiabat line. P P 1 P 2 Θ 1 Θ 1 Θ 1 T 1 T 2 T Fig2: Stuve diagram 1.4.2 Emagram diagram Emagram diagram is a thermodynamic diagram depends work, in meteorology the air describe by tow parameters pressure and temperature as shown in following figure with P and T coordinates according to the set of equations: Then we can write above equation as the following: 7
We notice that the value of the term is exact differential and will neglect and the work in this process will be as the following: And this result indicated to that the chart with T and lnp have a feature of real thermodynamic char Fig3: Emagram diagram 1.4.3 Tephigram (T-ϕ gram) diagram From entropy definition equation it is clear that the added heat on a thermodynamic process will be as shown: From that will the chart with coordinates T, or T and (T-ϕ gram chart) check the proportionality of Area with energy requirement that is required for the real thermodynamic chart. 8
Fig 4: T-ϕ gram diagram 1.5 Dry Adiabatic Lapse rate If a dry air sample is exposed to an adiabatic change in pressure, this process is described by the following equation: On this basis, the dry air parcel expands and rises and the following equation can be applied on it: The Pressure in an uncompressed air parcel will be immediately equal to surrounding pressure: Where ( ) represents density of the environment and given by the following equation: 9
It can be from the last three equations above to obtain the following equation: Where : =9.8 o C/km 1.5.1 Stability in Dry Atmosphere If an air parcel displaced vertically and continue to rising that is an indicator that the atmosphere is unstable. But if the air parcel returned to its original location this will indicate that the atmosphere is stable. Imagine an air parcel of temperature ( ) is in equilibrium with the environment (T), so that The initial temperature is equal, there will be no acceleration of air parcel of the air parcel displaced, will chance according to adiabatic lapse rate so that: At latitude Z, the environment temperature is: Where: is environmental lapse rate. The acceleration at altitude (z) is: 10
To assess the stability, the environmental lapse rate ( ) must be compared with the dry adiabatic lapse rate ( ) and this leads to the following stability criteria: Neutral air Unstable air Stable air Example1. A dry air parcel has a temperature of 20 o C. the environmental lapse rate is 5 o C/km. the air parcel forced to rise over a mountain that is 3 km high. a) What is the temperature of the air parcel at the top of the mountain? b) What is the temperature of the environment at the top of the mountain? c) What is the acceleration of the air parcel at the top of the mountain? d) Is the atmosphere is stable, unstable or neutral? Solution: a. Let b. c. 11
m/sec 2 d. The atmosphere is Stable where: ( ) Example2. A dry air parcel has a temperature of 42 o C. the environmental lapse rate is 12 o C/km. the air parcel forced to rise over a mountain that is 3 km high. a. What is the temperature of the air parcel at the top of the mountain? b. What is the temperature of the environment at the top of the mountain? c. What is the acceleration of the air parcel at the top of the mountain? d. Is the atmosphere is stable, unstable or neutral? Solution: a. Let b. 12
c. m/sec 2 d. The atmosphere is Unstable where: ( ) Example3. A dry air parcel has a temperature of 42 o C. the environmental lapse rate is 10 o C/km. the air parcel forced to rise over a mountain that is 3 km high. a. What is the temperature of the air parcel at the top of the mountain? b. What is the temperature of the environment at the top of the mountain? c. What is the acceleration of the air parcel at the top of the mountain? d. Is the atmosphere is stable, unstable or neutral? Solution: Let a. b. 13
c. e. The atmosphere is Unstable where: ( ) 14