Meteorology 3410 Exercise 5: Height Computations for Upper-air Soundings Due: October 6

Numerical weather prediction requires extensive information on the distribution of pressure, temperature, and water vapor in the upper atmosphere. An upper-air sounding is taken by releasing a balloon carrying a radiosonde. This is an inexpensive instrument which measures pressure, temperature, and relative humidity and transmits radio signals to a ground station. Such soundings are taken at least twice a day at several hundred locations around the world.

To make these soundings useful, one must determine the heights at which the measurements are made. To do this one uses the hydrostatic equation,

−\frac{dp}{dz} = ρg.

Here p is the pressure, z is height, ρ is the density, and g is the acceleration of gravity. This ordinary differential equation states that the rate of pressure decrease with height (−dp/dz) is equal to the density multiplied by the acceleration of gravity. We will use the hydrostatic equation in the form

dz = −\frac{dp}{ρg}.

We will assume that the air contains no water vapor. Then we can substitute for ρ from the equation of state for dry air (p = ρRT) to get

dz = −\frac{RT}{g} \frac{dp}{p},

where R = 287 J K^{−1} kg^{−1} is the specific gas constant for dry air. We will assume that g is a constant equal to 9.80665 m s^{−2} (it actually varies somewhat from place to place) and integrate this differential equation between two heights z_{1} and z_{2} where the pressures are p_{1} and p_{2} to obtain

z_{2} − z_{1} = −\frac{R}{g} ∫_{p_{1}}^{p_{2}} T \frac{dp}{p}.

One can always find a mean temperature T̄ so that

∫_{p_{1}}^{p_{2}} T \frac{dp}{p} = T̄ ∫_{p_{1}}^{p_{2}} \frac{dp}{p} = T̄(ln p_{2} − ln p_{1}) = T̄ ln \frac{p_{2}}{p_{1}} .

Then

z_{2} − z_{1} = −\frac{RT̄}{g} ln \frac{p_{2}}{p_{1}} .  (1)

We will assume that T varies linearly with ln p within each layer. Then

T̄ = \frac{1}{2}(T_{1} + T_{2}),  (2)

where T_{1} and T_{2} are the temperatures at the pressures p_{1} and p_{2}, respectively.

1. The Fortran (height.f) program in the class directory will be used as a template for the ”science” part of this exercise. Copy the program over to your directory and examine it. Now write a C program to do the same thing: compute the height from pressure and temperature data. Using the file defined in the program, run your C program and compare it to the output generated by the Fortran program. Make two plots using xmgr: (1) pressure (x-axis) versus height (y-axis) and (2) temperature versus height. Discuss very briefly the change of pressure and temperature with height in this instance.

2. Accurate radiosonde measurements of the pressure are not required to obtain accurate heights of pressure surfaces. It is only essential that the surface elevation and pressure be measured accurately. This makes it possible to manufacture radiosondes relatively cheaply, for a crude aneroid barometer costs far less than a precise one. If it were not for this fact, our present extensive network of sounding stations might be too expensive to maintain.

In this problem, we will explore the sensitivity of the height versus pressure calculations to errors in radiosonde pressure measurements. We will specify the pressure measurement error p′ to be zero at the surface (where the pressure is p_{s}) and to increase with height above the surface:

p′(p) = e(p_{s} − p).

In the C program, replace p with p + p′(p) when calculating z. Use the modified program to calculate z for the data provided previously with four values of e: 0.0, 0.01, 0.03, and 0.05. Pass e into the program as a command line variable and run the program 4 times for each possible value of e. (You may use other values too, if you wish.) These correspond to errors of 0, 1, 3, and 5 mb per 100 mb of pressure change. Plot all four pressure (x-axis) versus height (y-axis) curves on a single graph using xmgr. What accuracy (i.e., maximum value of e) is required in the radiosonde pressure measurements to determine the height of the 500 mb pressure surface to within 1 percent (about 50 meters)?