Please show your work on all problems. Check that your results are reasonable.

1. Using the NCAR web site, calculate the density of air (in kg m^-3) at the Foothills and MESA Labs for the most recent data you can find.

2. Using the hypsometric equation, reduce each of the pressure measurements from 1 above to equivalent sea level pressure
 

3. A hot air balloon is lifted by the buoyancy force, which can be calculated by

where r _balloon is the density of the air in the balloon, r _env is the density of the displaced environmental air, V is the volume of the balloon, and g is the acceleration of gravity.

(a) For a typical hot air balloon with a volume of 2200 m³, what is the maximum total mass (note, this includes the material of the balloon) that can be lifted off the ground at the Foothills Lab on the morning of January 31, 2003 if the average air temperature inside the balloon is maintained at 100 ^oC?
 

(b) How much more does the pilot of the balloon need to heat the air in the balloon to keep this balloon afloat with this maximum payload at the altitude of the Mesa Lab (assuming that the temperature of ambient air at Mesa is the same as at Foothills)?

(c) Why do hot air balloonists like to fly on cold mornings?
 

4. (a) Compute the scale height for an isothermal atmosphere with a temperature of 250 K.

(b) Compare the pressure values in this isothermal atmosphere with the US standard atmosphere for altitudes of 5 km, 11 km, and 25 km.
 

5. (a) Use the hypsometric equation (with regular temperature) to calculate the heights in meters for the following Denver radiosonde profile. The surface elevation is 1625 m.

 Level     Pressure     Temperature     Water Vapor
                mb                     C                     g/kg

SFC         841                 18.8                     4.82
                700                    3.8                    4.65
                500                -13.9                    1.32
                400                -24.5                    0.07
                300                -40.7                    0.22

(b) What is the 700-500~mb thickness (in m)?

(c) Use virtual temperature instead of temperature for calculating the 700 mb height. Is there much difference?

(You may check your answers using the PAOS Weather Center web site; this sounding was for 0Z September 15, 1999.)

6. (a) Use the PAOS Weather Center sounding plotting web site (archives section, general sounding archive) to find recent examples of a well mixed boundary layer and a very stable boundary layer. Show the two corresponding Stuve diagrams.

(b) Support and explain your selection by quoting a few relevant potential temperatures (THETA) from the text outputs.

(c) Using the text output of the sounding, show that the lapse rate in the well mixed boundary layer (excluding the surface layer) is close to the dry adiabatic lapse rate. Explain why.

(d) Why does a shallow nocturnal inversion not imply, in general, that pollution will be trapped near the surface during the day?

7. Commercial aircraft normally cruise near 200 mb where the temperature is typically -60 ^oC.

(a) Calculate the potential temperature of the environmental air.

(b) Calculate the temperature of air if compressed adiabatically to the cabin pressure of 800 mb.

(c) How much heat per mass of air (J/kg) must be then be added/removed to maintain a cabin temperature of 25 ^oC? During a cross country flight perhaps 2000 kg of air are exchanged in the cabin. Express the total heat added (removed) during a flight in terms of the equivalent mass of water frozen (or ice melted).
 

(5710 questions)

8. Repeat problem 4, only using a surface temperature of 288 K and a lapse rate of -6.5 degrees km^-1.

9. A typical tropical sounding has the tropopause at 17 km with a pressure of 94 mb and a temperature of 195 K. Calculate the temperature of the parcel compressed adiabatically from the tropopause to sea level using both Poisson's equation and the dry adiabatic lapse rate. Note: the two methods give different results.

(a) Use hydrostatic balance and the ideal gas law to show that Poisson's equation and the dry adiabatic lapse rate will give the same parcel temperature only if the environmental lapse rate is equal to the dry adiabatic lapse rate (neutrally stable). Hint: find an expression for the height of a given pressure level in an atmosphere with a constant lapse rate.
 

(b) Carefully examine the derivation of the dry adiabatic lapse rate, and determine which assumption is to blame for getting the wrong parcel temperature.