# 1.12: Homework Exercises

- Page ID
- 9741

# 1.12.1. Broaden Knowledge & Comprehension

These questions allow you to solve problems using current data, such as satellite images, weather maps, and weather observations that you can download through the internet. With current data, exercises can be much more exciting, timely, and relevant. Such questions are more vague than the others, because we can’t guarantee that you will find a particular weather phenomenon on any given day.

Many of these questions are worded to encourage you to acquire the weather information for locations near where you live. However, the instructor might suggest a different location if a better example of a weather event is happening elsewhere. Even if the instructor does not suggest alternative locations, you should feel free to search the country, the continent, or the globe for examples of weather that are best suited for the exercise.

Web URL (uniform resource locator) addresses are very transient. Web sites come and go. Even a persisting site might change its web address. For this reason, the web-enhanced questions do not usually give the URL web site for any particular exercise. Instead, you are expected to become proficient with internet search engines. Nonetheless, there still might be occasions where the data does not exist anywhere on the web. The instructor should be aware of such eventualities, and be tolerant of students who cannot complete the web exercise.

In many cases, you will want to print the weather map or satellite image to turn in with your homework. Instructors should be tolerant of students who have access to only black and white printers. If you have black and white printouts, use a colored pencil or pen to highlight the particular feature or isopleths of interest, if it is otherwise difficult to discern among all the other black lines on the printout.

You should always list the URL web address and the date you used it from which you acquired the data or images. This is just like citing books or journals from the library. At the end of each web exercise, include a “References” section listing the web addresses used, and any of your own annotations.

A SCIENTIFIC PERSPECTIVE • Give Credit

Part of the ethic of being a good scientist or engineer is to give proper credit to the sources of ideas and data, and to avoid plagiarism. Do this by citing the author and the title of their book, journal paper, or electronic content. Include the international standard book number (isbn), digital object identifier (doi), or other identifying info.

B1. Download a map of sea-level pressure, drawn as isobars, for your area. Become familiar with the units and symbols used on weather maps.

B2. Download from the web a map of near-surface air temperature, drawn is isotherms, for your area. Also, download a surface skin temperature map valid at the same time, and compare the temperatures.

B3. Download from the web a map of wind speeds at a height near the 200 or 300 mb (= 20 or 30 kPa) jet stream level . This wind map should have isotachs drawn on it. If you can find a map that also has wind direction or streamlines in addition to the isotachs, that is even better.

B4. Download from the web a map of humidities (e.g., relative humidities, or any other type of humidity), preferably drawn is isohumes. These are often found at low altitudes, such as for pressures of 850 or 700 mb (85 or 70 kPa).

B5. Search the web for info on the standard atmosphere. This could be in the form of tables, equations, or descriptive text. Compare this with the standard atmosphere in this textbook, to determine if the standard atmosphere has been revised.

B6. Search the web for the air-pollution regulation authority in your country (such as the EPA in the USA), and find the regulated concentrations of the most common air pollutants (CO, SO_{2}, O_{3}, NO_{2}, volatile organic compounds VOCs, and particulates). Compare with the results in Table 1-2, to see if the regulations have been updated in the USA, or if they are different for your country.

B7. Search the web for surface weather station observations for your area. This could either be a surface weather map with plotted station symbols, or a text table. Use the reported temperature and pressure to calculate the density.

B8. Search the web for updated information on the acceleration due to gravity, and how it varies with location on Earth.

B9. Search the web for weather maps showing thickness between two pressure surfaces. One of the most common is the 1000 - 500 mb thickness chart (i.e., the 100 - 50 kPa thickness chart). Comment on how thickness varies with temperature (the most obvious example is the general thickness decrease further away from the equator).

B10. Access from the web an upper-air sounding (e.g., Stuve, Skew-T, Tephigram, etc.) that plots temperature vs. height or pressure for a location near you. We will learn details about these charts later, but for now look at only temperature vs. height. If the sounding goes high enough (up to 100 mb or 10 kPa or so) , can you identify the troposphere, tropopause, and stratosphere.

B11. Often weather maps have isopleths of temperature (isotherm), pressure (isobar), height (contour), humidity (isohume), potential temperature (adiabat or isentrope), or wind speed (isotach). Search the web for weather maps showing other isopleths. (Hint, look for isopleth maps of precipitation, visibility, snow depth, cloudiness, etc.)

# 1.12.2. Apply

These are essentially “plug & chug” exercises. They are designed to ensure that you are comfortable with the equations, units, and physics by getting hands-on experience using them. None of the problems require calculus.

While most of the numerical problems can be solved using a hand calculator, many students find it easier to compose all of their homework answers on a computer spreadsheet. It is easier to correct mistakes using a spreadsheet, and plotting graphs of the answer is trivial.

Some exercises are flagged with the symbol (§), which means you __should__ use a Spreadsheet or other more advanced tool such as R, Matlab, Mathematica, or Maple. These exercises have tedious repeated calculations to graph a curve or trend. To do them by hand calculator would be painful. If you don’t know how to use a spreadsheet (or other more advanced program), now is a good time to learn.

Most modern spreadsheets also allow you to add objects called text boxes, note boxes or word boxes, to allow you to include word-wrapped paragraphs of text, which are handy for the “Problem” and the “Exposition” parts of the answer.

A spreadsheet example is given in Figure 1.13. Normally, to make your printout look neater, you might use the page setup or print option to turn off printing of the row numbers, column letters, and grid lines. Also, the borders around the text boxes can be eliminated, and color could be used if you have access to a color printer. Format all graphs to be clear and attractive, with axes labeled and with units, and with tic marks having pleasing increments.

A1. Find the wind direction (degrees) and speed (m s^{–1}), given the (U, V) components:

a. (-5, 0) knots c. (-1, 15) mi h e. (8, 0) knots g. (-2, -10) mi h |
b. (8, -2) m s d. (6, 6) m s f. (5, 20) m s h. (3, -3) m s |

A2. Find the U and V wind components (m s^{–1}), given wind direction and speed:

a. west at 10 knots c. (-1, 15) mi h e. (8, 0) knots g. (-2, -10) mi h |
b. north at 5 m s d. (6, 6) m s f. (5, 20) m s h. (3, -3) m s |

A2. Find the U and V wind components (m s^{–1}), given wind direction and speed:

a. west at 10 knots c. 225° at 8 mi h e. east at 7 knots g. 110° at 8 mi h |
b. north at 5 m s d. 300° at 15 knots f. south at 10 m s h. 20° at 15 knots |

A3. Convert the following UTC times to local times in your own time zone:

a. 0000 c. 0610 e. 1245 g. 1800 |
b. 0330 d. 0920 f. 1515 h. 2150 |

A4. (i). Suppose that a typical airline window is circular with radius 15 cm, and a typical cargo door is square of side 2 m. If the interior of the aircraft is pressured at 80 kPa, and the ambient outside pressure is given below in kPa, then what are the magnitudes of forces pushing outward on the window and door?

(ii). Your weight in pounds is the force you exert on things you stand on. How many people of your same weight standing on a window or door are needed to equal the forces calculated in part a. Assume the window and door are horizontal, and are near the Earth’s surface.

a. 30 c. 20 e. 10 g. 0 |
b. 25 d. 15 f. 5 h. 40 |

A5. Find the pressure in kPa at the following heights above sea level, assuming an average T = 250K:

a. -100 m (below sea level) c. 11 km e. 30,000 ft g. 2 km |
b. 1 km d. 25 km f. 5 km h. 15,000 ft |

A6. Use the definition of pressure as a force per unit area, and consider a column of air that is above a horizontal area of 1 square meter. What is the mass of air in that column:

- above the Earth’s surface.
- above a height where the pressure is 50 kPa?
- between pressure levels of 70 and 50 kPa?
- above a height where the pressure is 85 kPa?
- between pressure levels 100 and 20 kPa?
- above height where the pressure is 30 kPa?
- between pressure levels 100 and 50 kPa?
- above a height where the pressure is 10 kPa?

A7. Find the virtual temperature (°C) for air of:

a. | b. | c. | d. | e. | f | g. | |

T (°C) | 20 | 10 | 30 | 40 | 50 | 0 | –10 |

r (g/kg) | 10 | 5 | 0 | 40 | 60 | 2 | 1 |

A8. Given the planetary data in Table 1-7

(i). What are the escape velocities from a planet for each of their main atmospheric components? (For simplicity, use the planet radius instead of the “critical” radius at the base of the exosphere.).

(ii). What are the most likely velocities of those molecules at the surface, given the average surface temperatures given in that table? Comparing these answers to part (i), which of the constituents (if any) are most likely to escape?

a. Mercury | b. Venus | c. Mars | d. Jupiter |

e. Saturn | f. Uranus | g. Neptune | h. Pluto |

Table 1-7. Planetary data | ||||

Planet | Radius (km) | T_{sfc} (°C) (avg.) |
Mass relative to Earth | Main gases in atmos. |
---|---|---|---|---|

Mercury | 2440 | 180 | 0.055 | H_{2}, He |

Venus | 6052 | 480 | 0.814 | CO_{2}, N_{2} |

Earth | 6378 | 8 | 1.0 | N_{2}, O_{2} |

Mars | 3393 | –60 | 0.107 | CO_{2}, N_{2} |

Jupiter | 71400 | –150 | 317.7 | H_{2}, He |

Saturn | 60330 | –185 | 95.2 | H_{2}, He |

Uranus | 25560 | –214 | 14.5 | H_{2}, He |

Neptune | 24764 | –225 | 17.1 | H_{2}, He |

Pluto* | 1153 | –236 | 0.0022 | CH_{4}, N_{2}, CO |

* Demoted to a “dwarf planet” in 2006.

A9. Convert the following temperatures:

a. 15°C = ?K | b. 50°F = ?°C | c. 70°F = ?K | d. 15°C = ?°F |

e. 303 K = ?°C | f. 250K = ?°F | g. 2000°C = ?K | h. –40°F = ?°C |

A10.

- What is the density (kg·m
^{–3}) of air, given P = 80 kPa and T = 0 °C ? - What is the temperature (°C) of air, given P = 90 kPa and ρ = 1.0 kg·m
^{–3}? - What is the pressure (kPa) of air, given T = 90°F and ρ = 1.2 kg·m
^{–3}? - Give 2 combinations of pressure and density that have a temperature of 30°C.
- Give 2 combinations of pressure and density that have a temperature of 0°C.
- Give 2 combinations of pressure and density that have a temperature of –20°C.
- How could you determine air density if you did not have a density meter?
- What is the density (kg·m
^{–3}) of air, given P = 50 kPa and T = –30 °C ? - What is the temperature (°C) of air, given P = 50 kPa and ρ = 0.5 kg·m
^{–3}? - What is the pressure (kPa) of air, given T = –25°C and ρ = 1.2 kg·m
^{–3}?

A11. At a location in the atmosphere where the air density is 1 kg m^{–3}, find the change of pressure (kPa) you would feel if your altitude increases by ___ km.

a. 2 | b. 5 | c. 7 | d. 9 | e. 11 | f. 13 |

g. 16 | h. –0.1 | i. –0.2 | j. –0.3 | k. –0.4 | l. –0.5 |

A12. At a location in the atmosphere where the average virtual temperature is 5°C, find the height difference (i.e., the thickness in km) between the following two pressure levels (kPa):

a. 100, 90 | b. 90, 80 | c. 80, 70 | d. 70, 60 |

e. 60, 50 | f. 50, 40 | g. 40, 30 | h. 30, 20 |

i. 20, 10 | j. 100, 80 | k. 100, 70 | l. 100, 60 |

m. 100, 50 | n. 50, 30 |

A13. Name the isopleths that would be drawn on a weather map to indicate regions of equal

a. pressure | b. temperature |

c. cloudiness | d. precipitation accumulation |

e. humidity | f. wind speed |

g. dew point | h. pressure tendency |

A14. What is the geometric height and geopotential, given the geopotential height?

a. 10 m | b. 100 m | c. 1 km | d. 11 km |

What is the geopotential height and geopotential, given the geometric height?

e. 500 m | f. 2 km | g. 5 km | h. 20 km |

A15. What is the standard atmospheric temperature, pressure, and density at each of the following geopotential heights?

a. 1.5 km | b. 12 km | c. 50 m | d. 8 km |

e. 200 m | f. 5 km | g. 40 km | h. 25 km |

A16. What are the geometric heights (assuming a standard atmosphere) at the top and bottom of the:

a. troposphere | b. stratosphere |

c. mesosphere | d. thermosphere |

A17. Is the inverse of an average of numbers equal to the average of the inverses of those number? (Hint, work out the values for just two numbers: 2 and 4.) This question helps explain where the hypsometric equation given in this chapter is only approximate.

A18(§). Using the standard atmosphere equations, re-create the numbers in Table 1-5 for 0 ≤ H ≤ 51 km.

# 1.12.3. Evaluate & Analyze

These questions require more thought, and are extensions of material in the chapter. They might require you to combine two or more equations or concepts from different parts of the chapter, or from other chapters. You might need to critically evaluate an approach. Some questions require a numerical answer — others are “short-answer” essays.

They often require you to make assumptions, because insufficient data is given to solve the problem. Whenever you make assumptions, justify them first. A sample solution to such an exercise is shown below.

**Sample Application – Evaluate & Analyze (E) **

What are the limitations of eq. (1.9a), if any? How can those limitations be eliminated?

**Find the Answer**

Eq. (1.9a) for P vs. z relies on an average temperature over the whole depth of the atmosphere. Thus, eq. (1.9a) is accurate only when the actual temperature is constant with height.

As we learned later in the chapter, a typical or “standard” atmosphere temperature is NOT constant with height. In the troposphere, for example, temperature decreases with height. On any given day, the real temperature profile is likely to be even more complicated. Thus, eq. (1.9a) is inaccurate.

A better answer could be found from the hypsometric equation (1.26b):

\(P_{2}=P_{1} \cdot \exp \left(-\frac{z_{2}-z_{1}}{a \cdot \bar{T}_{v}}\right) \quad\) with \(a=29.3 \mathrm{m} \mathrm{K}^{-1}\)

By iterating up from the ground over small increments Δz = z_{2} – z_{1}, one can use any arbitrary temperature profile. Namely, starting from the ground, set z_{1} = 0 and P_{1} = 101.325 kPa. Set z_{2} = 0.1 km, and use the average virtual temperature value in the hypsometric equation for that 0.1 km thick layer from z = 0 to 0.1 km. Solve for P_{2}. Then repeat the process for the layer between z = 0.1 and 0.2 km, using the new T_{v} for that layer.

Because eq. (1.9a) came from eq. (1.26), we find other limitations.

1) Eq. (1.9a) is for __ dry air__, because it uses temperature rather than virtual temperature.

2) The constant “a” in eq. (1.9a) equals = (1/29.3) K m^{–1}. Hence, on a different planet with different gravity and different gas constant, “a” would be different. Thus, eq. (1.9a) is limited to __ Earth__.

Nonetheless, eq. (1.9a) is a reasonable first-order approximation to the variation of pressure with altitude, as can be seen by using standard-atmosphere P values from Table 1-5, and plotting them vs. z. The result (which was shown in the Sample Application after Table 1-5) is indeed close to an exponential decrease with altitude.

E1. What are the limitations of the “standard atmosphere”?

E2. For any physical variable that decreases exponentially with distance or time, the e-folding scale is defined as the distance or time where the physical variable is reduced to 1/e of its starting value. For the atmosphere the e-folding height for pressure decrease is known as the scale height. Given eq. (1.9a), what is the algebraic and numerical value for atmospheric scale height (km)?

E3(§). Invent some arbitrary data, such as 5 data points of wind speed M vs. pressure P. Although P is the independent variable, use a spreadsheet to plot it on the vertical axis (i.e., switch axes on your graph so that pressure can be used as a surrogate measure of height), change that axis to a logarithmic scale, and then reverse the scale so that the largest value is at the bottom, corresponding to the greatest pressure at the bottom of the atmosphere.

Now add to this existing graph a second curve of different data of M vs. P. Learn how to make both curves appear properly on this graph because you will use this skill repeatedly to solve problems in future chapters.

E4. Does hydrostatic equilibrium (eq. 1.25) always apply to the atmosphere? If not, when and why not?

E5. a. Plug eqs. (1.1) and (1.2a) into (1.3), and use trig to show that U = U. b. Similar, but for V = V.

E6.What percentage of the atmosphere is above a height (km) of:

a. 2 | b. 5 | c. 11 | d. 32 |

e. 1 | f. 18 | g. 47 | h. 8 |

E7. What is the mass of air inside an airplane with a cabin size of 5 x 5 x 30 m, if the cabin is pressurized to a cabin altitude of sea level? What mass of outside air is displaced by that cabin, if the aircraft is flying at an altitude of 3 km? The difference in those two masses is the load of air that must be carried by the aircraft. How many people cannot be carried because of this excess air that is carried in the cabin?

E8.Given air of initial temperature 20°C and density of 1.0 kg m^{–3}.

- What is its initial pressure?
- If the temperature increases to 30°C in an isobaric process, what is the new density?
- If the temperature increases to 30°C in an isobaric process, what is the new pressure?
- For an isothermal process, if the pressure changes to 20 kPa, what is the new density?
- For an isothermal process, if the pressure changes to 20 kPa, what is the new T?
- In a large, sealed, glass bottle that is full of air, if you increase the temperature, what if anything would be conserved (P, T, or ρ)?
- In a sealed, inflated latex balloon, if you lower it in the atmosphere, what thermodynamic quantities if any, would be conserved?
- In a mylar (non stretching) balloon, suppose that it is inflated to equal the surrounding atmospheric pressure. If you added more air to it, how would the state change?

E9(§). Starting from sea-level pressure at z = 0, use the hypsometric equation to find and plot P vs. z in the troposphere, using the appropriate standard-atmosphere temperature. Step in small increments to higher altitudes (lower pressures) within the troposphere, within each increment. How is your answer affected by the size of the increment? Also solve it using a constant temperature equal to the average surface value. Plot both results on a semi-log graph, and discuss meaning of the difference.

E10. Use the ideal gas law and eq. (1.9) to derive the equation for the change of density with altitude, assuming constant temperature.

E11. What is the standard atmospheric temperature, pressure, and density at each of the following geopotential heights (km)?

a. 75 | b. 65 | c. 55 | d. 45 | e. 35 |

f. 25 | g. 15 | h. 5 | i. –0.5 |

E12. The ideal gas law and hypsometric equation are for compressible gases. For liquids (which are incompressible, to first order), density is not a function of pressure. Compare the vertical profile of pressure in a liquid of constant temperature with the profile of a gas of constant temperature.

E13. At standard sea-level pressure and temperature, how does the average molecular speed compare to the speed of sound? Also, does the speed of sound change with altitude? Why?

E14. For a standard atmosphere below H = 11 km:

- Derive an equation for pressure as a function of H.
- Derive an equation for density as a function of H.

E15. Use the hypsometric equation to derive an equation for the scale height for pressure, H_{p}.

E16. Plot & discuss graphs of temperature vs. height to compare the 4 different standard atmospheres.

# 1.12.4. Synthesize

These are “what if” questions. They are often hypothetical — on the verge of being science fiction. By thinking about “what if” questions you can gain insight about the physics of the atmosphere, because often you cannot apply existing paradigms.

“What if” questions are often asked by scientists, engineers, and policy makers. For example, “What if the amount of carbon dioxide in the atmosphere doubled, then how would world climate change?”

For many of these questions, there is not a single right answer. Different students could devise different answers that could be equally insightful, and if they are supported with reasonable arguments, should be worth full credit. Often one answer will have other implications about the physics, and will trigger a train of related ideas and arguments.

A Sample Application of a synthesis question is presented in the next page. This solution might not be the only correct solution, if it is correct at all.

S1. What if the meteorological angle convention is identical to that shown in Figure 1.2, except for wind directions which are given by where they blow towards rather than where they blow from. Create a new set of conversion equations (1.1 - 1.4) for this convention, and test them with directions and speeds from all compass quadrants.

S2. Find a translation of Aristotle’s Meteorologica in your library. Discuss one of his erroneous statements, and how the error might have been avoided if he had following the Scientific Method as later proposed by Descartes.

S3. As discussed in a Sample Application, the glass on the front face of CRT and old TV picture tubes is thick in order to withstand the pressure difference across it. Why is the glass not so thick on the other parts of the picture tube, such as the narrow neck near the back of the TV?

S4. Eqs. (1.9a) and (1.13a) show how pressure and density decrease nearly exponentially with height.

- How high is the top of the atmosphere?
- Search the library or the web for the effective altitude for the top of the atmosphere as experienced by space vehicles re-entering the atmosphere.

S5. What is “ideal” about the ideal gas law? Are there equations of state that are not ideal?

S6. What if temperature as defined by eq. (1.5) was not dependent on the molecular weight of the gas. Speculate on how the composition of the Earth’s atmosphere might have evolved differently since it was first formed.

S7. When you use a hand pump to inflate a bicycle or car tire, the pump usually gets hot near the outflow hose. Why? Since pressure in the ideal gas law is proportional to the absolute virtual temperature (P=ρ·ℜ_{d}·T_{v}), why should the tire-pump temperature warmer than ambient?

S8. In the definition of virtual temperature, why do water vapor and liquid water have opposite signs?

S9. How should equation (1.22) for virtual temperature be modified to also include the effects of airplanes and birds flying in the sky?

S10. Meteorologists often convert actual station pressures to the equivalent “sea-level pressure” by taking into account the altitude of the weather station. The hypsometric equation can be applied to this job, assuming that the average virtual temperature is known. What virtual temperature should be used below ground to do this? What are the limitations of the result?

S11. Starting with our Earth and atmosphere as at present, what if gravity were to become zero. What would happen to the atmosphere? Why?

S12. Suppose that gravitational attraction between two objects becomes greater, not smaller, as the distance between the two objects becomes greater.

- Would the relationship between geometric altitude and geopotential altitude change? If so, what is the new relationship?
- How would the vertical pressure gradient in the atmosphere be different, if at all?
- Would the orbit of the Earth around the sun be affected? How?

**Sample Application – Synthesize**

What if liquid water (raindrops) in the atmosphere caused the virtual temperature to increase [rather than decrease as currently shown by the negative sign in front of r_{L} in eq. (1.22)]. What would be different about the weather?

**Find the Answer**

More and larger raindrops would cause warmer virtual temperature. This warmer air would act more buoyant (because warm air rises). This would cause updrafts in rain clouds that might be fast enough to prevent heavy rain from reaching the ground.

But where would all this rain go? Would it accumulate at the top of thunderstorms, at the top of the troposphere? If droplets kept accumulating, they might act sufficiently warm to rise into the stratosphere. Perhaps layers of liquid water would form in the stratosphere, and would block out the sunlight from reaching the surface.

If less rain reached the ground, then there would be more droughts. Eventually all the oceans would evaporate, and life on Earth as we know it would die.

But perhaps there would be life forms (insects, birds, fish, people) in this ocean layer aloft. The reason: if liquid water increases virtual temperature, then perhaps other heavy objects (such as automobiles and people) would do the same.

In fact, this begs the question as to why liquid water would be associated with warmer virtual temperature in the first place. We know that liquid water is heavier than air, and that heavy things should sink. One way that heavy things like rain drops would not sink is if gravity worked backwards.

If gravity worked backwards, then all things would be repelled from Earth into space. This textbook would be pushed into space, as would your instructor. So you would have never been assigned this exercise in the first place.

Life is full of paradoxes. Just be careful to not get a sign wrong in any of your equations — who knows what might happen as a result.