Skip to main content
Geosciences LibreTexts

7.0: Homework Exercises

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    7.10.1. Broaden Knowledge & Comprehension

    B1. Search the web for any journal articles, conference papers, or other technical reports that have pictures of droplet or ice-crystal growth or fall processes (such as photos taken in vertical wind tunnels).

    B2. Can you find any satellite photos on the web showing haze? If so, which satellites and which channels on those satellites show haze the best? Do you think that satellites could be used to monitor air pollution in urban areas?

    B3. Search the web for microphotographs of ice crystals and snow flakes with different habits. What habits in those photos were not given in the idealized Fig. 7.12?

    B4. Find on the web a clear microphotograph of a dendrite snow flake. Print it out, and determine the fractal dimension of the snow flake. (Hint, see the Clouds chapter for a discussion of fractals.)

    B5. Search various government air-pollution web sites (such as the U.S. Environmental Protection Agency: ) for sizes of aerosol pollutants. Based on typical concentrations (or on concentrations specified in the air-quality standards) of these pollutants, determine the number density, and compare with Fig. 7.5.

    B6. Search the web for information about surface tension, and how it relates to Gibbs free energy.

    B7. Search the web for info on snow & snowflakes.

    B8. Search the web for IDF precipitation curves for your area, or for a location assigned by the teacher.

    B9. Search the web for climate statistics of actual annual precipitation last year worldwide (or for your country or region), and compare with Fig. 7.25.

    B10. Search the web for photos and diagrams of precipitation measurement instruments, and discuss their operation principles.

    7.10.2. Apply

    A1. Using Fig. 7.1, how many of the following droplets are needed to fill a large rain drop.

    1. small cloud droplet
    2. typical cloud droplet
    3. large cloud droplet
    4. drizzle droplet
    5. small rain droplet
    6. typical rain droplet

    A2. Find the supersaturation fraction and supersaturation percentage, given relative humidities (%) of:

    a. 100.1 b. 100.2 c. 100.4 d. 100.5 e. 100.7
    f. 101 g. 101.2 h. 101.8 i. 102 j. 102.5
    k. 103.3 l. 104.0 m. 105 n. 107 o. 110

    A3. For air at T = –12°C, find the supersaturation fraction, given a vapor pressure (kPa) of:

    a. 0.25 b. 0.26 c. 0.28 d. 0.3 e. 0.31
    f. 0.38 g. 0.5 h. 0.7 i. 0.9 j. 1.1
    k. 1.2 l. 1.4 m. 1.6 n. 1.8 o. 1.9

    Hint. Get saturation vapor pressure from the Water Vapor chapter.

    A4. For air at P = 80 kPa and T = –6°C, find the supersaturation percentage, given a mixing ratio (g kg–1) of:

    a. 6 b. 5.8 c. 5.6 d. 5.4 e. 5.2 f. 5.1 g. 5
    h. 4.9 i. 4.7 j. 4.5 k. 4.3 l. 4.1 m. 3.8 n. 3.6

    Hint. Get saturation mixing ratio from a thermo diagram (at the end of the Atmospheric Stability chapter).

    A5. For the previous problem, assume the given mixing ratios represent total water mixing ratio. Find the excess water mixing ratio (g kg–1).

    A6. For an air parcel with excess water mixing ratio of 10 g kg–1 at a geopotential height of 5 km above mean sea level, find the average radius (µm) of the hydrometeor assuming growth by condensation only, given a hydrometeor number density (# m–3) of:

    a. 1x108 b. 2x108 c. 3x108 d. 4x108 e. 5x108
    f. 6x108 g. 7x108 h. 8x108 i. 9x108 j. 1x109
    k. 2x109 l. 3x109 m. 4x109 n. 5x109 o. 8x109

    A7. If c = 5x106 µm3·m–3, use the Junge distribution to estimate the number density of CCN (# m–3) within a ∆R = 0.2 µm range centered at R (µm) of:

    a. 0.2 b. 0.3 c. 0.4 d. 0.5 e. 0.6 f. 0.8 g. 1.0
    h. 2 i. 3 j. 4 k. 5 l. 6 m. 8 n. 10

    A8. For pure water at temperature –20°C, use Kelvin’s equation to find the equilibrium RH% in air over a spherical droplet of radius (µm):

    a. 0.005 b. 0.006 c. 0.008 d. 0.01 e. 0.02
    f. 0.03 g. 0.04 h. 0.05 i. 0.06 j. 0.08
    k. 0.09 l. 0.1 m. 0.2 n. 0.3 o. 0.5

    A9 (§). Produce Köhler curves such as in Fig. 7.7b, but only for salt of the following masses (g) at 0°C:

    a. 10–18 b. 10–17 c. 10–16 d. 10–15 e. 10–14
    f. 10–13 g. 10–12 h. 10–11 i. 5x10–18 j. 5x10–17
    k. 5x10–16 l. 5x10–15 m. 5x10–14 n. 5x10–13

    A10(§). Produce Köhler curves for a solute mass of 10–16 g of salt for the following temperatures (°C):

    a. –35 b. –30 c. –25 d. –20 e. –15
    f. –10 g. –5 h. 2 i. 5 j. 10
    k. 15 l. 20 m. 25 n. 30

    A11. Find the critical radii (µm) and supersaturations at a temperature of –10°C, for

    1. 5x10–17 g of hydrogen peroxide
    2. 5x10–17 g of sulfuric acid
    3. 5x10–17 g of nitric acid
    4. 5x10–17 g of ammonium sulfate
    5. 5x10–16 g of hydrogen peroxide
    6. 5x10–16 g of sulfuric acid
    7. 5x10–16 g of nitric acid
    8. 5x10–16 g of ammonium sulfate
    9. 5x10–15 g of hydrogen peroxide
    10. 5x10–15 g of sulfuric acid
    11. 5x10–15 g of nitric acid
    12. 5x10–15 g of ammonium sulfate
    13. 5x10–14 g of hydrogen peroxide
    14. 5x10–14 g of sulfuric acid
    15. 5x10–14 g of nitric acid
    16. 5x10–14 g of ammonium sulfate

    A12. For the nuclei of the previous exercise, find the equilibrium haze droplet radius (µm) for the following relative humidities (%):

    (i) 70 (ii) 72 (iii) 74 (iv) 76 (v) 78
    (vi) 80 (vii) 82 (viii) 84 (ix) 86 (x) 88
    (xi) 90 (xii) 92 (xiii) 94 (xiv) 96 (xv) 98

    A13. How many CCN will be activated in maritime air at supersaturations (%) of

    a. 0.2 b. 0.3 c. 0.4 d. 0.5 e. 0.6 f. 0.8 g. 1.0
    h. 2 i. 3 j. 4 k. 5 l. 6 m. 8 n. 10

    A14. Find the average separation distances (µm) between cloud droplets for the previous problem.

    A15. What temperature is needed to immersionfreeze half the droplets of radius (µm)

    a. 10 b. 20 c. 30 d. 40 e. 50 f. 60 g. 70
    h. 80 i. 90 j. 125 k. 150 l. 200 m. 200 n. 300

    A16. Estimate the number density of active ice nuclei for the following combinations of temperature and supersaturation [T(°C), Sice(%)].

    a. –5, 3 b. –5, 5 c. –10, 5 d. –10, 10 e. –15, 10
    f. –10, 12 g. –15, 12 h. –20, 13 i. –15, 18 j. –20, 20
    k. –25, 20 l. –20, 23 m. –25, 23 n. –23, 25

    A17. For a supersaturation gradient of 1% per 2 µm, find the kinematic moisture flux due to diffusion.

    Given T = –20°C, and P = 80 kPa. Use D (m2·s–1) of:

    a. 1x10–6 b. 2x10–6 c. 3x10–6 d. 4x10–6
    e. 5x10–6 f. 6x10–6 g. 7x10–6 h. 8x10–6
    i. 9x10–6 j. 1x10–5 k. 3x10–5 l. 4x10–5
    m. 5x10–5 n. 6x10–5 o. 7x10–5 p. 8x10–5

    A18 (§). Compute and plot supersaturation (%) vs. distance (µm) away from drops of the following radii , given a background supersaturation of 0.5%

    1. 0.15 µm containing 10–16 g of salt
    2. 0.15 µm containing 10–16 g of ammon.sulfate
    3. 0.15 µm containing 10–16 g of sulfuric acid
    4. 0.15 µm containing 10–16 g of nitric acid
    5. 0.15 µm containing 10–16 g of hydrogen perox.
    6. 0.3 µm containing 10–16 g of salt
    7. 0.3 µm containing 10–16 g of ammon.sulfate
    8. 0.3 µm containing 10–16 g of sulfuric acid
    9. 0.3 µm containing 10–16 g of nitric acid
    10. 0.3 µm containing 10–16 g of hydrogen perox.
    11. 0.5 µm containing 10–16 g of salt
    12. 0.5 µm containing 10–16 g of ammon.sulfate
    13. 0.5 µm containing 10–16 g of sulfuric acid
    14. 0.5 µm containing 10–16 g of nitric acid
    15. 0.5 µm containing 10–16 g of hydrogen perox.
    16. 2 µm containing 10–16 g of salt
    17. 1 µm containing 10–16 g of ammon.sulfate
    18. 1 µm containing 10–16 g of sulfuric acid
    19. 1 µm containing 10–16 g of nitric acid
    20. 1 µm containing 10–16 g of hydrogen perox.

    A19. Find the diffusivity (m2·s–1) for water vapor, given [ P(kPa) , T(°C) ] of:

    a. 80, 0 b. 80, –5 c. 80, –10 d. 80, –20
    e. 70, 0 f. 70, –5 g. 70, –10 h. 70, –20
    i. 60, 0 j. 60, –5 k. 60, –10 l. 60, –20
    m. 50, 0 n. 50, –5 o. 50, –10 p. 50, –20

    A20 (§). For the previous exercise, plot droplet radius (µm) vs. time (minutes) for diffusive growth.

    A21. What phase (I - XIV) of ice is expected at the following locations in a standard atmosphere?

    a. Earth’s surface b. mid-troposphere
    c. tropopause d. mid stratosphere
    e. stratopause f. mid-mesosphere
    g. mesopause

    A22. What phase (I - XIV) of ice is expected for the following conditions of [ P(kPa) , T(°C) ]?

    a. 1, –250 b. 1, –150 c. 1, –50 d. 1, 50
    e. 103, –250 f. 103, –150 g. 103, –50 h. 103, 50
    i. 5x105, –250 j. 5x105, –150 k. 5x105, –30 l. 5x105, 50
    m. 107, –250 n. 107, –50 o. 107, 50

    A23. What crystal habit could be expected for the following combinations of [ ρve (g m–3) , T(°C) ]:

    a. 0.22, –25 b. 0.22, –20 c. 0.22, –13
    d. 0.22, –8 e. 0.22, –5 f. 0.22, –2
    g. 0.12, –25 h. 0.12, –20 i. 0.12, –13
    j. 0.12, –8 k. 0.12, –5 l. 0.12, –2
    m. 0.08, –25 n. 0.08, –20 o. 0.08, –13
    p. 0.08, –8 q. 0.08, –5 r. 0.08, –2

    A24. Suppose the following ice crystals were to increase mass at the same rate. Find the rate of increase with time of the requested dimension.

    1. effective radius of column growing in 3-D
    2. diameter of plate growing in 2-D
    3. length of needle growing in 1-D

    A25 (§). Given D = 2x10–5 m2·s–1 and ρv = 0.003 kg·m–3, plot ice-crystal mass (g) vs. time (minutes) for 3-D growth such as a hexagonal column. Use the following supersaturation fraction:

    a. 0.001 b. 0.002 c. 0.003 d. 0.004 e. 0.005
    f. 0.006 g. 0.007 h. 0.008 i. 0.009 j. 0.01
    k. 0.012 l. 0.014 m. 0.016 n. 0.018 o. 0.020

    A26 (§). Same as the previous problem, but for 2-D growth of a thin flat plate of thickness 10 µm.

    A27 (§). Use the Clausius-Clapeyon equation from the Water Vapor chapter to calculate the saturation vapor pressure over liquid water and ice for –50 ≤ T ≤ 0°C, and use that data to calculate and plot the difference. Namely, reproduce Fig. 7.16 with your own calculations.

    A28. Find the terminal velocity of cloud droplets of radius (µm):

    a. 0.2 b. 0.4 c. 0.6 d. 0.8 e. 1.0 f. 2 g. 3 h. 4
    i. 5 j. 7 k. 10 l. 15 m. 20 n. 30 o. 40 p. 50

    A29. Find the terminal velocity of rain drops of radius (µm):

    a. 100 b. 150 c. 200 d. 300 e. 400 f. 500 g. 600
    h. 700 i. 800 j. 900 k. 1000 l. 1200 m. 1500 n. 2000

    A30. Calculate the terminal velocity of hailstones of radius (cm):

    a. 0.25 b. 0.5 c. 0.75 d. 1 e. 1.25 f. 1.5 g. 1.75 h. 2
    i. 2.5 j. 3 k. 3.5 l. 4 m. 4.5 n. 5 o. 5.5 p. 6

    A31. What type of “meteor” is:

    a. a rainbow b. lightning c. corona d. dust e. a cloud f. a halo g. sand
    h. rain i. smoke j. snow k. fog l. haze m. dew n. frost

    A32. For a Marshall-Palmer rain-drop size distribution, if the rainfall rate is

    (i) 10 mm h–1, or (ii) 20 mm h–1,

    how many droplets are expected of radius (µm) greater than:

    a. 100 b. 200 c. 300 d. 400 e. 500
    f. 700 g. 1000 h. 1200 i. 1500 j. 2000

    A33. What is the rain intensity classification and the weather map symbol for rainfall rates (mm h–1) of:

    a. 0.02 b. 0.05 c. 0.1 d. 0.2 e. 0.5
    f. 1.0 g. 2 h. 3 i. 4 j. 5
    k. 6 l. 7 m. 8 n. 9 o. 10

    A34. For precipitation in the form of

    (i) drizzle, or (ii) snow,

    what is the precipitation intensity classification and weather map symbol for visibility (km) of:

    a. 0.1 b. 0.2 c. 0.3 d. 0.4 e. 0.5 f. 0.6 g. 0.7
    h. 0.8 i. 0.9 j. 1.0 k. 1.2 l. 1.5 m. 2 n. 5

    A35. Find the rainfall intensity (mm h–1) associated with the following return period (RP in years) and duration (D in hours) values. RP, D =

    a. 2, 0.2 b. 2, 0.5 c. 2, 2 d. 2, 6 e. 2, 12 f. 10, 0.2 g. 10, 0.5
    h. 10 2 i. 10, 6 j. 10, 12 k. 100, 0.2 l. 100, 0.5 m. 100 2 n. 100, 6

    A36. For a liquid-water equivalent precipitation value of 5 cm, find the snow depth if the snow density (kg m–3) is:

    a. 50 b. 75 c. 100 d. 150 e. 200 f. 250
    g. 300 h. 350 i. 400 j. 450 k. 500 l. 550
    m. 600 n. 650 o. 700 p. 800 q. 900

    A37. Find the mean annual precipitation for the following locations, given their longitudes, latitudes:

    a. 120°W, 50°N b. 120°W, 25°N c. 120°W, 10°N d. 120°W, 10°S e. 120°W, 30°S
    f. 60°W, 0°N g. 60°W, 20°N h. 60°W, 40°N i. 0°W, 50°N j. 0°W, 25°N
    k. 0°W, 5°N l. 0°W, 20°S m. 120°E, 25°S n. 120°E, 0°N o. 120°E, 30°N

    A38. What are the values of zonally averaged evaporation and precipitation rates at latitude:

    a. 70°N b. 60°N c. 50°C d. 40°N e. 30°N
    f. 20°N g. 10°N h. 0° i. 10°S j. 20°S
    k. 30°S m. 40°S n. 50°S o. 60°S p. 70°S

    7.10.3. Evaluate & Analyze

    E1. If saturation is the maximum amount of water vapor that can be held by air at equilibrium, how is supersaturation possible?

    E2. Fig. 7.2 shows how excess-water mixing ratio can increase as a cloudy air parcel rises. Can excess-water mixing ratio increase with time in a cloudy or foggy air parcel that doesn’t rise? Explain.

    E3. What can cause the supersaturation S in a cloud to be less than the available supersaturation SA?

    E4. Fig. 7.3 applies to cumulus clouds surrounded by clear air. Would the curves be different for a uniform stratus layer? Why?

    E5. An air parcel contains CCN that allow 109 m–3 hydrometeors to form. If the air parcel starts at P = 100 kPa with T = 20°C and Td = 14°C, find the average radius of cloud droplets due to condensation only, after the air parcel rises to P (kPa) of:

    a. 85 b. 82 c. 80 d. 78 e. 75 f. 72 g. 70
    h. 67 i. 63 j. 60 k. 58 l. 54 m. 50 n. 45

    Hint. Use a thermo diagram to estimate rE.

    E6. Derive eq. (7.8), stating and justifying all assumptions. (Hint: consider the volume of a spherical drop.)

    E7. Rewrite eq. (7.8) for hydrometeors that form as cubes (instead of spheres as was used in eq. 7.8). Instead of solving for radius R, solve for the width s of a side of a cube.

    E8. In Fig. 7.5, consider the solid curve. The number density (# cm–3) of CCN between any two radii R1 and R2 is equal to the average value of n/∆R within that size interval times ∆R (=R2 – R1). This works best when R1 and R2 are relatively close to each other. For larger differences between R1 and R2, just sum over a number of smaller intervals. Find the number density of CCN for droplets of the following ranges of radii (µm):

    a. 0.02 < R < 0.03 b. 0.03 < R < 0.04
    c. 0.04 < R < 0.05 d. 0.05 < R < 0.06
    e. 0.06 < R < 0.08 f. 0.08 < R < 0.1
    g. 0.2 < R < 0.3 h. 0.3 < R < 0.4
    i. 0.4 < R < 0.5 j. 0.5 < R < 0.6
    k. 0.6 < R < 0.8 l. 0.8 < R < 1
    m. 0.1 < R < 0.2 n. 1 < R <2
    o. 0.1 < R < 1 p. 0.1 < R < 10
    q. 0.02 < R < 0.1 r. 0.02 < R < 10

    E9. a. Find a relationship between the number density of CCN particles n, and the corresponding mass concentration c (µg·m–3), using the dashed line in Fig. 7.5 and assuming that the molecular weight is Ms.

    b. Use Table 7-1 to determine the molecular weight of sulfuric acid (H2SO4) and nitric acid (HNO3), which are two contributors to acid rain. Assuming tiny droplets of these acids are the particles of interest for Fig. 7.5, find the corresponding values or equations for mass concentration of these air pollutants.


    1. From the solid curve in Fig. 7.5, find the number density of CCN at R = 1 µm, and use this result to determine the value and units of parameter c in the CCN number density eq. (7.9)
    2. Plot the curve resulting from this calibrated eq. (7.9) on a linear graph, and on a log-log graph.
    3. Repeat (a) and (b), but for R = 0.1 µm.
    4. Repeat (a) and (b), but for R = 0.01 µm.
    5. Why does eq. (7.9) appear as a straight line in Fig. 7.5? What is the slope of the straight line in Fig. 7.5? [Hint: for log-log graphs, the slope is the number of decades along the vertical axis (ordinate) spanned by the line, divided by the number of decades along the horizontal axis (abscissa) spanned by the line. Decade means a factor of ten; namely, the interval between major tic marks on a logarithmic axis.]
    6. For number densities less than 1 cm–3, what does it mean to have less than one particle (but greater than zero particles)? What would be a better way to quantify such a number density?
    7. As discussed in the Atmospheric Optics chapter, air molecules range in diameter between roughly 0.0001 and 0.001 mm. If the dashed line in Fig. 7.5 were extended to that small of size, would the number density indicated by the Junge distribution agree with the actual number density of air molecules? If they are different, discuss why.

    E11. Show how the Köhler equation reduces to the Kelvin equation for CCNs that don’t dissolve.

    E12 (§). Surface tension σ for a cloud droplet of pure water squeezes the droplet, causing the pressure inside the droplet to increase. The resulting pressure difference ∆P between inside and outside the droplet can be found from the Young-Laplace equation (∆P = σ·dA/dV, where A is surface area and V is volume of the droplet). For a spherical droplet of radius R this yields:

    ∆P = 2·σ / R

    Plot this pressure difference (kPa) vs. droplet radius over the range 0.005 ≤ R ≤ 1 µm.

    E13. a. Using the equation from the previous exercise, combine it with Kelvin’s equation to show how the equilibrium relative humidity depends on the pressure excess inside the droplet.

    b. If greater pressure inside the drop drives a greater evaporation rate, describe why RH greater than 100% is needed to reach an equilibrium where condensation from the air balances evaporation from the droplet.

    E14. Using the full Köhler equation, discuss how supersaturation varies with:

    1. temperature.
    2. molecular weight of the nucleus chemical.
    3. mass of solute in the incipient droplet.

    E15. Consider a Köhler curve such as plotted in the INFO box on Droplet Growth. But let the RH = 100.5% for air. Starting with an aerosol with characteristics of point A on that figure, (a) discuss the evolution of drop size, and (b) explain why haze particles are not possible for that situation.

    E16. Considering Fig. 7.7, which type of CCN chemical would allow easier formation of cloud droplets: (a) a CCN with larger critical radius but lower peak supersaturation in the Köhler curve; or (b) a CCN with smaller critical radius but higher supersaturation? Explain.

    E17. What is so special about the critical radius, that droplets larger than this radius continue to grow, while smaller droplets remain at a constant radius?

    E18. The Kelvin curve (i.e., the Köhler curve for pure water) has no critical radius. Hence, there is no barrier droplets must get across before they can grow from haze to cloud droplets. Yet cloud droplets are easier to create with heterogeneous nucleation on solute CCN than with homogeneous nucleation in clean air. Explain this apparent paradox.

    E19. In air parcels that are rising toward their LCL, aerosol swelling increases and visibility decreases as the parcels get closer to their LCL. If haze particles are at their equilibrium radius by definition, why could they be growing in the rising air parcel? Explain.

    E20. How high above the LCL must air be lifted to cause sufficient supersaturation to activate CCN for liquid-droplet nucleation?

    E21. Suppose a droplet contained all the substances listed in Table 7-2. What is the warmest temperature (°C) that the droplet will freeze due to:

    a. contact freezing b. condensation freezing
    c. deposition freezing d. immersion freezing

    E22. Discuss the differences in nucleation between liquid water droplets and ice crystals, assuming the air temperature is below freezing.

    E23. Discuss the differences in abundance of cloud vs. ice nuclei, and how this difference varies with atmospheric conditions.

    E24. Can some chemicals serve as both water and ice nuclei? For these chemicals, describe how cloud particles would form and grow in a rising air parcel.

    E25. If droplets grow by diffusion to a final average radius given by eq. (7.8), why do we even care about the diffusion rate?

    E26. In the INFO box on Cubic Ice, a phase diagram was presented with many different phases of ice. The different phases have different natural crystal shapes, as summarized in the table below. In the right column of this table, draw a sketch of each of the shapes listed. (Hint, look in a geometry book or on the internet for sketches of geometric shapes.)

    Table 7-7. Crystal shapes for ice phases.
    Shape Phases Sketch
    hexagonal Ih
    cubic Ic, VII, X
    tetragonal III, VI, VIII, IX, XII
    rhombic II, IV
    orthorhombic XI
    monoclinic V

    E27. The droplet growth-rate equation (7.24) considers only the situation of constant background supersaturation. However, as the droplet grows, water vapor would be lost from the air causing supersaturation to decrease with time. Modify eq. (7.24) to include such an effect, assuming that the temperature of the air containing the droplets remains constant. Does the droplet still grow with the square root of time?

    E28. If ice particles grew as spheres, which growth rate equation would best describe it? Why?

    E29. Verify that the units on the right sides of eqs. (7.26) and (7.27) match the units of the left side.

    E30. Manipulate the mass growth rate eq. (7.26) to show that the effective radius of the ice particle grows as the square root of time. Show your work.

    E31. Considering the different mass growth rates of different ice-crystal shapes, which shape would grow fastest in length of its longest axis?

    E32. If the atmosphere were to contain absolutely no CCN, discuss how clouds and rain would form, if at all.

    E33. In Fig. 7.15 the thick black line follows the state of a rising air parcel that is cooling with time. Why does that curve show es decreasing with time?

    E34. If both supercooled liquid droplets and ice crystals were present in sinking cloudy air that is warming adiabatically, describe the evolutions of both types of hydrometeors relative to each other.

    E35. On one graph similar to Fig. 7.18, plot 3 curves for terminal velocity. One for cloud droplets (Stokes Law), another for rain drops, and a third for hail. Compare and discuss.

    E36. Can ice crystals still accrete smaller liquid water droplets at temperatures greater than freezing? Discuss.

    E37. Large ice particles can accumulate smaller supercooled liquid water drops via the aggregation process known as accretion or riming. Can large supercooled liquid water drops accumulate smaller ice crystals? Discuss.

    E38. Is it possible for snow to reach the ground when the atmospheric-boundary-layer temperature is warmer than freezing? Discuss.

    E39. Compare CCN size spectra with raindrop size spectra, and discuss.

    E40. For warm-cloud precipitation, suppose that all 5 formation factors are working simultaneously. Discuss how the precipitation drop size distribution will evolve with time, and how it can create precipitation.

    E41. Use the info in Fig. 7.24, or use the associated equations, to calculate and plot the total depth of rain (mm) vs. duration for any one return period.

    E42. Is it possible to have blizzard conditions even with zero precipitation rate? Discuss.

    E43. Precipitation falling out of a column of atmosphere implies that there was net latent heating in that column. Use the annual mean precipitation of Fig. 7.25 to discuss regions of the world having the greatest latent heating of the atmosphere.

    E44. According to Fig. 7.26, some latitudes have an imbalance between evaporation and precipitation. How can that be maintained?

    E45. Suppose that a disdrometer gives you information on the size of each hydrometeor that falls, and how many of each size hydrometeor falls per hour. Derive an equation to relate this information to total rainfall rate.

    7.10.4. Synthesize

    S1. What if the saturation vapor pressure over supercooled water and ice were equal.

    1. Discuss the formation of clouds and precipitation.
    2. Contrast with those processes in the real atmosphere.
    3. Discuss how the weather and climate might change, if at all.

    S2. In Fig. 7.15, suppose that the saturation vapor pressure over ice were greater, not less, than that over water. How would the WBF process change, if at all? How would precipitation and clouds change, if at all?

    S3. What if you were hired to seed warm (T > 0°C) clouds (i.e., to add nuclei), in order to create or enhance precipitation. Which would work better: (a) seeding with 105 salt particles cm–3, each with identical of radius 0.1 µm; or (b) seeding with 1 salt particle cm–3, each with identical radius of 0.5 µm; or (c) seeding with a range of salt particles sizes? Discuss, and justify.

    S4. What if you were hired to seed cold (T < 0°C) clouds (i.e., to add nuclei), in order to create or enhance precipitation. Would seeding with water or ice nuclei lead to the most precipitation forming most rapidly? Explain and justify.

    S5. Is it possible to seed clouds (i.e., add nuclei) in such a way as to reduce or prevent precipitation? Discuss the physics behind such weather modification.

    S6. What if all particles in the atmosphere were hydrophobic (i.e., repelled water). How would the weather and climate be different, if at all?

    S7. What if the concentration of cloud nuclei that could become activated were only one-millionth of what currently exists in the atmosphere. How would the weather and climate change, if at all?

    S8. Eq. (7.29) indicates that smaller droplets and aerosol particles fall slower. Does Stoke’s law apply to particles as small as air molecules? What other factors do air molecules experience that would affect their motion, in addition to gravity?

    S9. What if Stoke’s law indicated that smaller particles fall faster than larger particles. Discuss the nature of clouds for this situation, and how Earth’s weather and climate might be different.

    S10. What if rain droplet size distributions were such that there were more large drops than small drops. Discuss how this could possibly happen, and describe the resulting weather and climate.

    S11. Suppose that large rain drops did not break up as they fell. That is, suppose they experienced no drag, and there was no upper limit to rain drop size. How might plant and animal life on Earth have evolved differently? Why?

    S12. What if cloud and rain drops of all sizes fell at exactly the same terminal velocity. Discuss how the weather and climate might be different.

    S13. What if condensation and deposition absorbed latent heat (i.e., caused cooling) instead of releasing latent heat. How would clouds, precipitation, weather and climate be different, if at all.

    S14. Weather modification is as much a social issue as a scientific/technical issue. Consider a situation of cloud seeding (adding nuclei) to enhance precipitation over arid farm land in county X. If you wanted to make the most amount of money, would you prefer to be the:

    1. meteorologist organizing the operation,
    2. farmer employing the meteorologist,
    3. company insuring the farmer’s crop,
    4. company insuring the meteorologist, or
    5. lawyer in county Y downwind of county X, suing the meteorologist, farmer, and insurance companies?

    Justify your preference.

    S15. Suppose that you discovered how to control the weather via a new form of cloud seeding (adding nuclei). Should you ...

    1. keep your results secret and never publish or utilize them, thereby remaining impoverished and unknown?
    2. publish your results in a scientific journal, thereby achieving great distinction?
    3. patent your technique and license it to various companies, thereby achieving great fortune?
    4. form your own company to create tailored weather, and market weather to the highest bidders, thereby becoming a respected business leader?
    5. modify the weather in a way that you feel is best for the people on this planet, thereby achieving great power?
    6. allow a government agency to hold hearings to decide who gets what weather, thereby achieving great fairness and inefficiency?
    7. give your discovery to the military in your favorite country, thereby expressing great patriotism? (Note: the military will probably take it anyway, regardless of whether you give it willingly.)

      Discuss and justify your position. (Hint: See the “A SCIENTIFIC PERSPECTIVE” box at the end of this chapter before you answer this question.)


    The scenario of exercise S15 is not as far-fetched as it might appear. Before World War II, American physicists received relatively little research funding. During the war, the U.S. Army offered a tremendous amount of grant money and facilities to physicists and engineers willing to help develop the atomic bomb as part of the Manhattan Project.

    While the work they did was scientifically stimulating and patriotic, many of these physicists had second thoughts after the bomb was used to kill thousands of people at the end of the war. These concerned scientists formed the “Federation of Atomic Scientists”, which was later renamed the “Federation of American Scientists” (FAS).

    The FAS worked to discourage the use of nuclear weapons, and later addressed other environmental and climate-change issues. While their activities were certainly worthy, one has to wonder why they did not consider the consequences before building the bomb.

    As scientists and engineers, it is wise for us to think about the moral and ethical consequences before starting each research project.

    This page titled 7.0: Homework Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Roland Stull via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.