SO345: Atmospheric Thermodynamics

1. SO345: Atmospheric Thermodynamics CHAPTER 19: SPECIAL ATMOSPHERES

2. SPECIAL ATMOSPHERES Now that we have added the hydrostatic equation to our (seemingly unending) list of important thermodynamic and meteorological formulas, we will look at some atmospheres of special interest.

3. LAPSE RATE Before jumping into these different types of atmospheres, let us first define an important meteorological term, the temperature lapse rate, or simply Alapse rate@. Lapse rate is the rate at which temperature >decreases= in the vertical. Our convention will be to use the symbol, gamma, to denote lapse rate. Capital gamma (G) will represent the decrease in temperature of an air parcel, while small gamma (?) will represent the atmosphere=s (or the environment=s) decrease in temperature.

4. LAPSE RATE The equations for lapse rate are therefore: (for an air parcel) G = -dT/dz, and (Eq 19.1) (for the environment) ? = -dT/dz. (Eq 19.2)

5. LAPSE RATE Pay very close attention to the negative sign, as this can sometimes be confusing, especially in the heat of battle (like during the middle of an exam). Since an air parcel=s temperature decreases upward during an adiabatic ascent, and the average tendency is for the environmental temperature (in the troposphere) to also decrease upward: positive lapse rate --------------> temperature decreases going up negative lapse rate --------------> temperature increases going up

6. MOIST ADIABATIC LAPSE RATE For the case of moist air (remember dry air + water vapor), technically the lapse rate is: Gmoist = -dT*/dz, or ?moist = -dT*/dz, (Eq 19.3) and this virtual temperature lapse rate is the most completely correct to use.

7. MOIST ADIABATIC LAPSE RATE However, the atmosphere and air parcels may be treated as essentially dry for most of our applications without sacrificing too much accuracy (recall the example in chapter 14 which differentiated between the adiabatic ascent of a dry air parcel and the adiabatic ascent of an unsaturated moist air parcel). The simplicity of using only T (versus T*) would make our calculations and use of thermodynamic diagrams much easier. (And you thought all we ever want to do is make things harder for the student).

8. SPECIAL ATMOSPHERES The special atmospheres we will be discussing are the: 1) homogeneous atmosphere, 2) isothermal atmosphere, 3) arbitrary constant lapse rate atmosphere, 4) dry adiabatic atmosphere, 5) U.S. Standard atmosphere.

9. SPECIAL ATMOSPHERES For each of these atmospheres, we will be looking at: - the temperature profile and corresponding equation, - the pressure profile and corresponding equation, - the atmosphere=s upper limit (if it exists), and - how realistic or unrealistic is this atmosphere. (The derivations of temperature and pressure profiles for these atmospheres can be found in Appendix N).

10. HOMOGENEOUS ATMOSPHERE (CONSTANT DENSITY) The homogeneous atmosphere assumes a constant air density (or constant ?) throughout. This very simplistic atmospheric model results in linear profiles for both pressure and temperature (Figure 19.1). This is the only atmosphere that can have linear relationships for both.

11. Homogeneous Atmosphere Fig. 19.1 Temperature and pressure profiles of the homogeneous atmosphere.

12. HOMOGENEOUS ATMOSPHERE The homogeneous atmosphere=s upper boundary is where either the temperature or pressure curve intersects at zero. Its upper limit is therefore defined by: � H = RT0 (Eq 19.4) g This height of the homogeneous atmosphere, H, is also known as Ascale height@. Assuming a surface temperature of 0EC (273EK), H.8 kilometers. Also if we solve for ? (or -dt/dz) we get a lapse rate, ? . 34EC/km. Does this seem like a realistic atmospheric model?

13. ISOTHERMAL ATMOSPHERE (CONSTANT TEMPERATURE) Since the temperature profile for this type atmosphere is a straight line pointing upward, the temperature would never reach zero, so this is an atmosphere of infinite vertical extent. Accordingly, pressure also does not reach zero, with an exponential decrease with height. The profiles are shown in Figure 19.2.

14. Isothermal Atmosphere Fig. 19.2Temperature and pressure profiles for the isothermal atmosphere. The scale height, H, here, is the height where pressure becomes 1/e of the value at the surface. Does this seem like a realistic atmospheric model?

15. ARBITRARY CONSTANT LAPSE RATE (?) ATMOSPHERE With T varying linearly with height, we actually have 3 cases for this type atmosphere: � 1) zero lapse rate 2) positive lapse rate 3) negative lapse rate

16. ZERO LAPSE RATE Notice that case 1, the zero lapse rate atmosphere, is the same as the isothermal case (T is constant, hence ? is zero), so we do not need to readdress that one. � .

17. POSITIVE LAPSE RATE � For case 2 (positive ?), temperature >decreases= linearly with height, so there is a finite upper limit to this atmosphere (once again, where temperature reaches 0EK). Pressure will also decrease with height and reach zero, but non-linearly (except for the specific case of ?=34EC/km -- the homogeneous atmosphere). See Figure 19.3.

18. Constant Lapse Rate (? =0) Atmosphere Fig. 19.3Temperature and pressure profiles of a positive lapse rate atmosphere.

19. NEGATIVE LAPSE RATE Case 3 (negative ?) is a less common case. This temperature inversion atmosphere has temperature linearly increasing with height (Figure 19.4). This is obviously another infinite atmosphere as temperature is actually moving further away from zero with increased height. Pressure must still decrease with height, but will do so non-linearly and not reach zero. Can you see why pressure can not increase with height? Do you expect these cases to be realistic?

20. Constant Lapse Rate (? <0) Atmosphere  Fig. 19.4 Temperature and pressure profiles of a negative lapse rate atmosphere 

21. DRY ADIABATIC ATMOSPHERE (CONSTANT POTENTIAL TEMPERATURE) Solving for the dry adiabatic lapse rate (Gd) from Poisson=s Equation, we get the expression: � ?d = Gd = g/cp. (Eq 19.5) With both g and cp being treated as constants, Gd also becomes a constant value. Substituting the values of g and cp in the formula, we get Gd=9.8EC/km, or rounded off to: � Gd . 10EC/km

22. DRY ADIABATIC ATMOSPHERE (CONSTANT POTENTIAL TEMPERATURE) Notice that we used both a capital and small gamma for this case. The small gamma (?) was used as a representation of an atmospheric lapse rate. However a dry adiabatic case is originally derived from the adiabatic ascent of a dry Aair parcel@ (to get Poisson=s Equation), hence the corresponding use of capital gamma (G).

23. DRY ADIABATIC ATMOSPHERE (CONSTANT POTENTIAL TEMPERATURE) As you see, this case is a specific subset of the Aconstant +? atmosphere@. The temperature and pressure profiles are therefore the same with the specific value of 10EC/km substituted for ?. Later you will see how this case becomes a major reference in distinguishing between stable and unstable atmospheres. Does this seem to be a fairly realistic atmosphere?

24. U.S. STANDARD ATMOSPHERE You may have had some misgivings as to the realism of some of the previous atmospheres. This final atmospheric model is meant to represent average normal conditions over the United States at 40EN latitude. Its basic features are: � 1) surface pressure = 1 standard atm = 1013.25 mb 2) surface temperature = 15.0EC = 285.0EK 3) acceleration of gravity = 9.80665 m/s2 (constant) 4) air is dry (assumed ideal gas) 5) from sea level to 10.77 km: ?=6.50EC/km ----------------------> troposphere 6) from 10.77 km to 32.00 km: ------> stratosphere ?=0 (T=-55EC = 218EK)

25. Fig. 19.5 The U.S. Standard Atmosphere.  Figure 19.5 shows the temperature and pressure profiles of this atmosphere. Note that it is a combination of two types of atmospheres which we have already discussed. Is this a good realistic atmosphere model? U.S. STANDARD ATMOSPHERE