Physical Climatology Programming Exercise 2

A moist air parcel at a temperature of 20°C and specific humidity of 10 g kg-1 is lifted adiabatically from the upwind base of a mountain, where the pressure is 1000 mbar, to the top, at 3000 m above the base, and is then brought down to the base on the other side of the mountain. Using appropriate equations given in class, calculate the following parameters:

(a) At the original base,

(1). pressure in mb (or mbar)
(2). specific humidity in kg kg-1
(3). mixing ratio in g kg-1
(4). relative difference between specific humidity and mixing ratio
(5). temperature in °F
(6). temperature in K
(7). gas constant, in Jkg-1K-1
(8). air density, in kg m-3
(9). vapor pressure, in mb
(10). the partial pressure for the dry air, in mb
(11). the saturated vapor pressure, in mb
(12). the vapor pressure deficit (= the saturated vapor pressure - vapor pressure), in mb 
(13). relative humidity, in %
(14). the saturated specific humidity, in g kg-1,  using both the exact and approximate forms of the equations in the class notes. Compare their values and explain why they are different
(15). dew point temperature [hint: inverting the formula for esat]
(16). latent heat of vaporization of water
(17). rate of change of saturated specific humidity with temperature
(18). dry and moist adiabatic lapse rate
(19). potential temperature, in °C

(b) The air parcel would become saturated at some level above the base. Find out

(20). the height at this level, in m; and the following parameters at this level
(21). the pressure, in mb
(22). temperature, dew point temperature, and virtual temperature, in °C
(23). potential and virtual potential temperature, in °C
(24). specific humidity, in g kg-1
(25). rate of change of saturated specific humidity with temperature
(26). relative humidity, in %
(27). the vapor pressure, in mb
(28). the saturated adiabatic lapse rate

[Hint: starting from the base level upwards, compute the temperature using the dry adiabatic lapse rate, and then compute the pressure using the formula discussed in class. Because specific humidity is constant before reaching the condensation level, all other parameters listed in (1)-(19) can be derived from these three basic parameters (i.e., P, T and q). The condensation level is where relative humidity (RH) is equal to 100%, or temperature is equal to dew point temperature. Initially, you may want to start with an interval of (say) 100 m, then examine how RH varies with height. To obtain a more accurate value of the condensation height, after some level, you may want to use a small interval of (say) 10m or 1 m. The level at which RH is equal to 100% is the condensation height.]

(c) The temperature of the parcel at the top of the mountain.

[Hint: using the moist adiabatic lapse rate. To ensure a greater accuracy, using a smaller height interval of (say) 10 m or 1 m. You will also need to calculate saturated vapor pressure, saturated specific humidity, the latent heat at the air temperature, and the rate of change of saturated specific humidity with temperature before re-calculating the new moist adiabatic lapse rate for use in the next height increment.

Assuming you are successful (it is worth a try since I will give marks for just trying), tabulate the parameters from (20) to (28) for different heights. Refer to LCL_MTOP.pdf for a sample output.]

(d) The temperature, pressure, vapor pressure, relative humidity, specific humidity, and dew point temperature, of the parcel at the downwind base, assuming that the parcel remained unsaturated during its entire descent.

[Hint: now using the dry adiabatic lapse rate.]

(e) Use the pseudoadiabatic chart (i.e. the Stüve diagram discussed in class) to solve the above problem by locating the starting point, the lifting condensation level, the mountain top, and the ending point. For each location, you should be able to determine temperature, pressure, height, mixing ratio, saturated mixing ratio, dew point temperature, relative humidity (approximately mixing ratio divided by saturated mixing ratio), and the rainout amount. Compare these values with those from what is calculated from your code and discuss. For example, if the numbers are different, why? What are the potential sources?  Do you expect this to be the real precipitation that would occur in nature? Why? Attach the chart as part of your completed homework.