Assignment 4
Contents
Assignment 4
Due Friday October 13 2023 (at start of class, or submitted electronically via Brightspace before class)
Point values are indicated for each question.
Total points: 40
Corrected 10/10/2023
Question 1
4 points
(Based on Problem 4.3 of the text)
Show the mathematical steps leading from equation (4.17):
to equation (4.18):
where
HINT: Express the equation (4.17) in terms of a small non-dimensional variable \(x = \frac{z}{r_E}\) and use a power series approximation for \(g(x)\) about the reference point \(x=0\). See Appendix C.3 for a refresher on how that works.
Question 2
5 points
(Based on Problem 4.5 of the text)
a) (2 points) Show that for the constant density atmosphere, the top of the atmosphere must occur at a height \(H\) above the surface, where
where \(T_0\) is the surface temperature.
b) (2 points) What is the temperature at the top of this atmosphere, i.e. \(T(H)\)?
c) (1 point) Based on your results above, comment on whether you think a truly constant density atmosphere could actually exist.
Question 3
14 points
(Problem 4.6 of the text)
a) (3 points) Use the equation for \(p(z)\) for a constant density atmosphere to calculate the approximate height of the 1000, 850, and 500 hPa levels, assuming that the density thoughout the troposphere is the same as that at the surface (computed from \(T_0 = 15\)ºC and \(p_0 = 1013.2\) hPa).
b) (3 points) Repeat your calculation using the somewhat more accurate expression for \(p(z)\) in an isothermal atmosphere, assuming that the temperature of the entire atmosphere equals \(T_0\) as given above.
c) (3 points) Finally, repeat your calculation using the most accurate expression for \(p(z)\), which in this case is the one for a constant lapse rate atmosphere.
d) (2 points) For each of the pressure levels given, calculate the errors (approximate minus true value) in meters that results from using the simpler constant density and isothermal approximations. Use a table having the following format to present your results: rows correspond to pressure levels; first three columns correspond to three model atmospheres; final two columns give errors (approximate value minus true) for the constant density and isothermal atmospheres. Record all values to a precision of 1 meter.
e) (3 points) For the three model atmosphere (parts a, b, and c): From your 1000 and 850 hPa heights, obtain the layer thicknesses for this pair of pressure levels. Then, invert the hypsometric equation to find the layer mean virtual temperatures \(\overline{T_v}\).
Question 4
4 points
(Based on Problem 4.7 of the text)
Assuming a constant lapse rate of 6.5 K km\(^{-1}\) and a surface pressure near 1000 hPa, what value of the 1000-500 hPa thickness should correspond to surface temperatures near freezing?
Question 5
4 points
(Problem 4.8 of the text)
On a certain day, the atmosphere has a temperature \(T_0\) of 20ºC and a pressure \(p_0\) of 1000 hPa at the surface (\(z_0 = 0\)). The lapse rate is \(\Gamma_0 = 6\) K km\(^{-1}\) from the surface to 3 km altitude; \(\Gamma_1 = 3\) K km\(^{-1}\) from 3 km to 6 km altitude. Find the pressure \(p\) at an altitude of 5 km.
HINT: apply the piecewise linear temperature profile as in section 4.2.4. Start from the surface and work upward.
Question 6
9 points
(Problem 4.9 of the text)
The hypsometric equation (4.30) gives the thickness \(\Delta z\) of a layer bounded by two pressure levels \(p_1\) and \(p_2\) in terms of the layer mean virtual temperature \(\overline{T_v}\) (or just the layer mean temperature \(\overline{T}\) if the air is assumed to be dry).
a) (4 points) For the special case of a constant lapse rate layer with bottom and top temperatures \(T_1\) and \(T_2\), respectively, find an exact expression for the layer mean temperature \(\overline{T}\) required by the hypsometric equation.
b) (2 points) If \(T_1 = 280\) K and \(T_2 = 276\) K, find \(\overline{T}\) to the nearest 0.1 K. Compare your result with a mean temperature derived from the much simpler arithmetic average of \(T_1\) and \(T_2\) (add them together and divide by two)
c) (2 points) Repeat (b), but for \(T_2 = 240\) K.
d) (1 points) Based on your results for (b) and ©, speculate on when it is probably safe to use the simple arithmetic average of \(T_1\) and \(T_2\) as an approximation for the layer mean temperature.