4.7: Homework Exercises

4.8.1. Broaden Knowledge & Comprehension

B1. Use the internet to acquire the current humidity at a weather station near you. What type of humidity variable is it?

B2. Use the internet to acquire a current weather map of humidity contours (isohumes) for your region. Print this map and label regions of humid and dry air.

B3. At the time this chapter was written, a web page was available from the National Weather Service in El Paso, Texas, that could convert between different weather variables (search on “El Paso weather calculator”). How do the formulas for humidity on this web page compare to the ones in this chapter?

B4. Use your internet search engine to find additional “weather calculators” that can convert between different units, other than the calculator mentioned in the previous problem. Which one do you like the best? Why?

B5. Use the internet to acquire the company names and model numbers of at least two different instruments for each of 3 different methods for sensing humidity, as was discussed on the previous page.

B6. Two apparent-temperature indices (humidex and heat index) describing heat stress or summer discomfort were presented in the Thermodynamics chapter. Use the internet to acquire journal articles or other information about any two additional indices from the following list:

• apparent temperature,
• discomfort index,
• effective temperature,
• humisery,
• humiture,
• index of thermal stress.
• livestock weather safety index,
• summer simmer index,
• temperature-humidity index (THI),
• wet-bulb globe temperature,

B7. Use the internet to acquire a weather map for your region showing isohumes (either at the surface, or at 85 or 70 kPa). For one Eulerian location chosen by your instructor, use the winds and horizontal humidity gradient to calculate the horizontal moisture advection. State if this advection would cause the air to become drier or more humid. Also, what other factors in the Eulerian water balance equation could counteract the advection (by adding or removing moisture to the Eulerian volume)?

B8. A meteogram is a graph of a weather variable (such as humidity or temperature) along the vertical axis as a function of time along the horizontal axis. Use the internet to either acquire such a humidity meteogram for a weather station near your location, or create your own meteogram from a sequence of humidity observations reported at different times from a weather station.

B9. Use the internet to search on “upper air sounding”, where a sounding is a plot of weather variables vs. height or pressure. Some of these web sites allow you to pick the upper-air sounding station of interest (such as one close to you), and to pick the type of sounding plot.

There are several different types of thermo diagram frameworks for plotting soundings, and so far we discussed only one (called an Emagram). We will learn about the other thermo diagrams in the next chapter -- for example, the Stuve diagram looks very similar to an Emagram. Find a web site that allows you to view and plot an emagram or Stuve for a location near you.

4.8.2. Apply

A1. Compare the saturation vapor pressures (with respect to liquid water) calculated with the Clausius-Clapeyron equation and with Tetens’ formula, for T (°C):

a. 45	b. 40	c. 35	d. 30	e. 25	f. 20	g. 15
h. 10	i. 5	j. 0	k. –5	l. –10	m. –15	n. –20

A2. Calculate the saturation vapor pressures with respect to both liquid water and flat ice, for T (°C) =

a. –3	b. –6	c. –9	d. –12	e. –15	f. –18
g. –21	h. –24	i. –27	j. –30	k. –35	l. –40

A3. Find the boiling temperature (°C) of pure water at altitudes (km) of:

a. 0.2	b. 0.4	c. 0.6	d. 0.8	e. 1.2	f. 1.4
g. 1.6	h. 1.8	i. 2.2	j. 2.4	k. 2.6	l. 2.8

A4. Calculate the values of e_s (kPa), r (g kg^–1), q (g kg^–1), ρ_v (g m^–3), RH (%), T_d (°C), LCL (km), T_w (°C), r_s (g kg^–1), q_s (g kg^–1), and ρ_vs (g m^–3), given the following atmospheric state:

	P (kPa)	T (°C)	e (kPa)
a.	100	45	5
b.	90	35	2
c.	80	25	1
d.	70	15	0.8
e.	80	5	0.5
f.	90	5	0.2
g.	80	15	1
h.	100	25	2.5

A5(§). Some of the columns in Table 4-1 depend on ambient pressure, while others do not. Create a new version of Table 4-1 for an ambient pressure P (kPa) of

a. 95	b. 90	c. 85	d. 80	e. 75	f. 70	g. 65
h. 60	i. 55	j. 50	k. 45	l. 40	m. 35	n. 30

A6. Given the following initial state for air outside your home. If your ventilation/heating system brings this air into your home and heats it to 22°C, what is the relative humidity (%) in your home? Assume that all the air initially in your home is replaced by this heated outside air, and that your heating system does not add or remove water.

T(°C), RH(%)	T(°C), RH(%)
a. 15, 80	g. 5, 90
b. 15, 70	h. 5, 80
c. 15, 60	i. 5, 70
d. 10, 80	j. –5, 90
e. 10, 70	k. –5, 80
f. 10, 60	m. –5, 70

A7. Given the temperatures and relative humidities of the previous exercise, what is the mixing ratio value (g kg^–1) for this “outside” air?

A8. For air with temperature and dew-point values given below in °C, find the LCL value (km).

T, Td	T, Td
a. 15, 12	g. 5, 4
b. 15, 10	h. 5, 0
c. 15, 8	i. 5, –5
d. 10, 8	j. 20, 15
e. 10, 5	k. 20, 10
f. 10, 2	m. 20, 5

A9. Given the following dry and wet-bulb temperatures, use equations (not graphs or diagrams) to calculate the dew-point temperature, mixing ratio, and relative humidity. Assume P = 95 kPa.

T, Tw	T, Tw
a. 15, 14	g. 5, 4
b. 15, 12	h. 5, 3
c. 15, 10	i. 5, 2
d. 10, 9	j. 20, 18
e. 10, 7	k. 20, 15
f. 10, 5	m. 20, 13

A10. Same as the previous exercise, but you may use the psychrometric graphs (Figs. 4.4 or 4.5).

A11. Given the following temperatures and dewpoint temperatures, use the equations of Normand’s Rule to calculate the wet-bulb temperature.

T, Td	T, Td
a. 15, 12	g. 5, 4
b. 15, 10	h. 5, 0
c. 15, 8	i. 5, –5
d. 10, 8	j. 20, 15
e. 10, 5	k. 20, 10
f. 10, 2	m. 20, 5

A12. Same as the previous exercise, but you may use a thermo diagram to apply Normand’s rule.

A13. For air at sea level, find the total-water mixing ratio for a situation where:

a. T = 3°C,	rL = 3 g kg^–1
b. r = 6 g kg^–1,	rL = 2 g kg^–1
c. T = 0°C,	rL = 4 g kg^–1
d. T = 12°C,	rL = 3 g kg^–1
e. r = 8 g kg^–1,	rL = 2 g kg^–1
f. r = 4 g kg^–1,	rL = 1 g kg^–1
g. T = 7°C,	rL = 4 g kg^–1
h. T = 20°C,	rL = 5 g kg^–1

A14. Given air with temperature (°C) and total water mixing ratio (g kg^–1) as given below. Find the amount of liquid water suspended in the air: r_L (g kg^–1). Assume sea-level.

T, rT	T, rT
a. 15, 16	g. 25, 30
b. 15, 14	h. 25, 26
c. 15, 12	i. 25, 24
d. 10, 12	j. 20, 20
e. 10, 11	k. 20, 18
f. 10, 10	m. 20, 16

A15. If the mixing ratio is given below in (g kg^–1), then use equations (not figures or graphs) find the dew point (°C), assuming air at P = 80 kPa.

a. 28	b. 25	c. 20	d. 15	e. 10	f. 5	g. 3
h. 2	i. 1	j. 0.5	k. 0.3	l. 0.2	m. 0.1	n. 0.05

A16. Same as the previous exercise, but you may use Figure 4.7.

A17. Given an air parcel starting at 100 kPa with dewpoint (°C) given below, use Figure 4.7 to find the parcel’s final dewpoint (°C) if it rises to a height where P = 60 kPa.

a. 40	b. 35	c. 30	d. 25	e. 20	f. 15	g. 10
h. 5	i. 0	j. –5	k. –10	l. –15	m. –20	n. –25

A18. What is the value of relative humidity (%) for air with the following state:

T (°C), Td (°C), P (kPa)	T (°C), Td (°C), P (kPa)
a. 15, 10, 90	g. 25, 20, 100
b. 15, 10, 90	h. 25, 15, 100
c. 15, 5, 80	i. 25, 10, 90
d. 10, 0, 80	j. 20, 15, 90
e. 10, 0, 70	k. 20, 10, 80
f. 10, –5, 70	m. 20, 5, 70

A19. Calculate the saturated adiabatic lapse rate (°C km^–1) at the temperatures and pressures given below. Use the equations, not the thermo diagram.

T (°C), P (kPa)	T (°C), P (kPa)
a. 15, 90	g. 25, 100
b. 15, 90	h. 25, 100
c. 15, 80	i. 25, 90
d. 10, 80	j. 20, 90
e. 10, 70	k. 20, 80
f. 10, 70	m. 20, 70

A20. Same as the previous exercise, but use equations to find the saturated adiabatic lapse rate as a change of temperature with pressure (°C kPa^–1).

A21. For air parcels with initial state as given below, use the thermo diagram (Figure 4.8) to find the final air-parcel temperature after it is lifted to an altitude where P = 50 kPa. Assume the air parcels are saturated at all times.

T (°C), P (kPa)	T (°C), P (kPa)
a. 15, 80	g. 25, 90
b. 15, 70	h. 25, 90
c. 15, 60	i. 25, 80
d. 10, 70	j. 20, 80
e. 10, 60	k. 20, 70
f. 10, 60	m. 20, 60

A22. For air parcels with initial state as given in the previous exercise, use the thermo diagram (Figure 4.8) to find the final air-parcel temperature after it is lowered to an altitude where P = 100 kPa. Assume the air parcels are saturated at all times.

A23. Using Figure 4.8 and other figures for dry adiabats in the Thermodynamics chapter, determine the values of the liquid-water potential temperature and equivalent potential temperature for the initial air parcel of exercise A21.

A24. Same as the previous exercise, except use equations instead of the thermo diagram.

A25. Given an air parcel that starts at a height where P = 100 kPa with T = 25°C and r = 12 g kg^–1 (i.e., it is initially unsaturated). After rising to its final height, it has an r_L (g kg^–1) value listed below. Assuming no precipitation falls out, find the final value for r (g kg^–1) for this now-saturated air parcel.

a. 0.5	b. 1	c. 1.5	d. 2	e. 2.5	f. 3	g. 3.5
h. 4	i. 4.5	j. 5	k. 5.5	l. 6	m. 6.5	n. 7

A26. Imagine a horizontally uniform wind given below, which is blowing in to the west side of a fixed cubic domain and blowing out of the east side. The cube is 200 km on each side. The total water mixing ratio for (incoming, outgoing) air is r_T = (12, 8) g kg^–1. What is the rate of change of r_T inside the volume due to this advection?

a. 2	b. 4	c. 5	d. 7	e. 10	f. 12	g. 15
h. 18	i. 20	j. 21	k. 23	l. 25	m. 27	n. 30

A27. Imagine a fixed cube of air 200 km on each side. Precipitation is falling at rate 4 mm h^–1 into the top of this volume, and is falling out of the bottom of the volume at the rate (mm h^–1) given below. What is the rate of change of r_T inside the volume due to this precipitation gradient?

a. 6	b. 5.5	c. 5	d. 4.5	e. 4.3	f. 4.2	g. 4.1
h. 4	i. 3.9	j. 3.8	k. 3.7	l. 3.5	m. 3	n. 2.5

A28. Given below the value of latent heat flux (W·m^–2) at the surface. Find the kinematic value of latent flux (K·m s^–1), the vertical flux of water vapor (kgwater·m^–2·s^–1), the vertical flux of water vapor in kinematic form (kgwater kg_air^–1)·(m s^–1), and the evaporation rate (mm d^–1).

a. 100	b. 150	c. 200	d. 250	e. 300	f. 350
g. 80	h. 75	i. 70	j. 60	k. 50	l. 40	m. 25

A29. For windy, overcast conditions, estimate the kinematic latent heat flux at the surface, assuming C_H = 5x10^–3.

	M (m s^–1)	rsfc (g kg^–1)	rair (g kg^–1)
a.	2	25	10
b.	2	20	10
c.	2	15	10
d.	5	25	10
e.	5	20	10
f.	5	15	10
g.	12	25	10
h.	12	20	10
i.	12	15	10

A30. For sunny, free-convective conditions, estimate the kinematic latent heat flux at the surface.

	wB (m s^–1)	rsfc (g kg^–1)	rML (g kg^–1)
a.	5	25	10
b.	5	20	10
c.	5	15	10
d.	12	25	10
e.	12	20	10
f.	12	15	10
g.	25	25	10
h.	25	20	10
i.	25	15	10

A31. Suppose the atmospheric mixed layer (ML) is as sketched in Figure 4.12, having |∆_sr_T| = 8 g kg^–1, |∆_sθ| = 5°C, ∆_zir_T (g kg^–1) as given below, and |∆_ziθ| = 3°C. The mean winds are calm for this daytime convective boundary layer, for which b_H·w_B = 0.02 m s^–1. What is the rate of change of total water mixing ratio in the ML due to the turbulence?

a. –10	b. –8	c. –5	d. –3	e. –2	f. –1	g. 0
h. +1	i. 2	j. 4	k. 6	l. 7	m. 9	n. 12

4.8.3. Evaluate & Analyze

E1. When liquid water evaporates into a portion of the atmosphere containing dry air initially, water vapor molecules are added into the volume as quantified by the increased vapor pressure e. Does this mean that the total pressure increases because it has more total molecules? If not, then discuss and justify alternative outcomes.

E2(§).

1. Make your own calculations on a spreadsheet to re-plot Figure 4.2 (both the bottom and top graphs).
2. What factor(s) in the equations causes the vapor pressure curves to differ for the different phases of water?

E3. Suppose that rain drops are warmer than the air they are falling through. Which temperature should be used in the Clausius-Clapeyron equation (T_air or T_water)? Why? (For this exercise, neglect the curvature of the rain drops.)

E4. Notice the column labels in Table 4-1. In that table, why are the same numbers valid for (T, r_s) and for (T_d, r)?

E5(§). a. Plot a curve of boiling temperature vs. height over the depth of the troposphere. b. The purpose of pressure cookers is to cook boiled foods faster. Plot boiling temperature vs. pressure for pressures between 1 and 2 times P_{sea level}.

E6(§). Plot r_s vs. T and q_s vs. T on the same graph. Is it reasonable to state that they are nearly equal? What parts of their defining equations allow for this characteristic? For what situations are the differences between the mixing ratio and specific humidity curves significant?

E7(§). On a graph of T vs. T_d, plot curves for different values of RH.

E8. Why is it important to keep thermometers dry (i.e., placing them in a ventilated enclosure such as a Stevenson screen) when measuring outside air temperature?

E9(§). For any unsaturated air-parcel, assume you know its initial state (P, T, r). Recall that the LCL is the height (or pressure) where the following two lines cross: the dry adiabat (starting from the known P, T) and the isohume (starting from the known P, r). Given the complexity of the equations for the dry adiabat and isohume, it is surprising that there is such a simple equation (4.16a or b) for the LCL.

Confirm that eq. (4.16) is reasonable by starting with a variety of initial air-parcel states on a thermo diagram, lifting each one to its LCL, and then comparing this LCL with the value calculated from the equation for each initial parcel state. Comment on the quality of eq. (4.16).

E10. Some of the humidity variables [vapor pressure, mixing ratio, specific humidity, absolute humidity, relative humidity, dew-point temperature, LCL, and wet-bulb temperature] have maximum or minimum limits, based on their respective definitions. For each variable, list its limits (if any).

E11. A swamp cooler is a common name for an airconditioning system that lowers the air temperature to the wet-bulb temperature by evaporating liquid water into the air. But this comes with the side-effect of increasing the humidity of the air. Consider the humidex as given in the Thermodynamics chapter, which states that humid air can feel as uncomfortable as hotter dry air. Suppose each cell in Table 3-5 represents a different initial air state. For which subset of cells in that table would a swamp cooler take that initial air state and change it to make the air feel cooler.

E12. We know that T_w = T for saturated air. For the opposite extreme of totally dry air (r = 0), find an equation for T_w as a function of (P, T). For a few sample initial conditions, does your equation give the same results you would find using a thermo diagram?

E13(§). Create a table of T_d as a function of (T, T_w). Check that your results are consistent with Figs. 4.4 and 4.5.

E14(§). Do eqs. (4.20) and (4.21) give the same results as using Normand’s Rule graphically on a thermo diagram (Figure 4.6)? Confirm for a few different initial air-parcel states.

E15. Derive eqs. (4.23) through (4.27) from eqs. (4.4, 4.7, and 4.10).

E16. Rain falls out of the bottom of a moving cloudy air parcel. (a) If no rain falls into the top of that air parcel, then what does eq. (4.35) tell you? (b) If rain falls out of the bottom at the same rate that it falls into the top, then how does this affect eq. (4.35).

E17(§). Create a thermo diagram using a spreadsheet to calculate isohumes (for r = 1, 3, 7, 10, 30 g kg^–1) and dry adiabats (for θ = –30, –10, 10, 30 °C), all plotted on the same graph vs. P on an inverted log scale similar to Figs. 3.3 and 4.7.

E18. Start with Tetens’ formula to derive equation (4.36). Do the same, but starting with the ClausiusClapeyron equation. Compare the results.

E19. It is valuable to test equations at extreme values, to help understand limitations. For example, for the saturated adiabatic lapse rate (eq. 4.37b), what is the form of that equation for T = 0 K, and for T approaching infinity?

E20(§). Create your own thermo diagram similar to Figure 4.8 using a spreadsheet program, except calculate and plot the following saturated adiabats (θ_w = –30, –10, 10, 30°C).

E21(§). Create a full thermo diagram spanning the domain (–60 ≤ T ≤ 40°C) and (100 ≤ P ≤ 10 kPa). This spreadsheet graph should be linear in T and logarithmic in P (with axes reversed so that the highest pressure is at the bottom of the diagram). Plot

• isobars (drawn as thin green solid lines) for P(kPa) = 100, 90, 80, 70, 60, 50, 40, 30, 20
• isotherms (drawn as thin green solid lines) for T(°C) = 40, 20, 0, –20, –40 °C.
• dry (θ, solid thick orange lines) and moist adiabats (θ_w, dashed thick orange lines) for the same starting temperatures as for T.
• isohumes (thin dotted orange lines) for r (g kg^–1) = 50, 20, 10, 5, 2, 1, 0.5, 0.2

E22. T and θ are both in Kelvins in eqs. (4.40 and 4.41). Does this mean that these two temperatures cancel each other? If not, then what is the significance of θ/T in those equations?

E23. Consider an air parcel rising adiabatically (i.e., no mixing and no heat transfer with its surroundings). Initially, the parcel is unsaturated and rises dry adiabatically. But after it reaches its LCL, it continues its rise moist adiabatically. Is θ_w or θL conserved (i.e., constant) below the LCL? Is it constant above the LCL? Are those two constants the same? Why or why not. Hint, consider the following: In order to conserve total water (r_T = r + r_L), r must decrease if r_L increases.

E24. For each of the saturated adiabats in Figure 4.8, calculate the corresponding value of θ_e (the equivalent potential temperature). Why is it always true that θ_e ≥ θ_L?

E25. a. By inspection of the horizontal advection terms in eq. (4.44), write the corresponding term for vertical advection. b. Which term of that equation could account for evaporation from a lake surface, if the Eulerian cube of air was touching the lake?

E26. Based on the full (un-simplified) Eulerian heat budget equation from the Thermodynamics chapter, create by inspection a full water-balance equation similar to eq. (4.44) but without the simplifications.

E27. Why does the condensation-caused latent heating term in the Eulerian heat balance equation (see the Thermodynamics chapter) have a different form (or purpose) than the precipitation term in the water balance eq. (4.44)?

E28(§). Plot curves kinematic latent flux vs. evaporation rate for different altitudes.

E29. The flux of heat and water due to entrainment at the top of the mixed layer can be written as F_{H zi} = w_e·∆_ziθ and F_{water zi} = w_e·∆_zir_T, respectively. Also, for free convection (sunny, calm) conditions in the mixed layer, a good approximation is F_{H zi} = 0.02·F_H. If the entrainment velocity is w_e, then show how the info above can be used to create eq. (4.54).

E30. Derive eq. (4.56) from (4.55).

E31. Which humidity sensors would be best suited for measuring the rapid fluctuations of humidity in the turbulent boundary layer? Why?

4.8.4. Synthesize

S1. Describe how the formation and evolution of clouds would differ if colder air could hold more water at saturation than warmer air. During a typical daytime summer day, when and at what altitudes would you expect clouds to form?

S2. Describe how isohumes on a thermo diagram would look if saturation mixing ratio depended only on temperature.

S3. Consider the spectral-absorption hygrometers described earlier (e.g., optical hygrometers, Lymanalpha hygrometer, krypton hygrometer, infrared hygrometers). What principle(s) or law(s) from the Solar and IR Radiation chapter describe the fundamental way that these instruments are able to measure humidity?

S4. Describe the shape (slope and/or curvature) of moist adiabats in a thermo diagram if water-vapor condensation released more latent heat than the cooling associated with adiabatic expansion of the air parcel. Describe any associated changes in climate and weather.

S5. What if the evaporation rate of water from the surface was constant, and did not depend on surface humidity, air humidity, wind speed, or solar heating (i.e., convection). Describe any associated changes in climate and weather.

S6.

1. Describe the depth of liquid water in rain gauges at the ground if all the water vapor in the troposphere magically condensed and precipitation out. Assume a standard atmosphere temperature profile, but with a relative humidity of 100% initially (before rainout), and with no liquid or solid water initially suspended in the atmosphere.
2. Oceans currently cover 70.8% of the Earth’s surface. If all the water from part (a) flowed into the oceans, describe the magnitude of ocean-depth increase.
3. Do a similar exercise to (a) and (b), but for a saturated standard-atmosphere stratosphere (ignoring the troposphere).

S7. The form of Clausius-Clapeyron equation presented near the beginning of this chapter included both T_o and T as arguments of the exponential function.

1. Use algebra to separate T_o and T into separate exponential functions. Once you have done that, your equation should look like: e_s = C·exp[–(L/ ℜ_v)·(1/T)], where C contains the other exponential. Write the expression for C.
2. The Boltzmann constant is k_B = 1.3806x10^–23 J·K^–1·molecule^–1 . This can be used to rewrite the water-vapor gas constant as ℜ_v ≈ k_B/m_v, where m_v represents the mass of an individual water-molecule. Substitute this expression for ℜ_v into e_s = C·exp[–(L/ ℜ_v)·(1/T)], leaving C as is in this eq.
3. If you didn’t make any mistakes in this alternative form for the Clausius-Clapeyron equation, and after you group all terms in the numerator and all terms in the denominator of the argument then your equation should contain a new form for the ratio in the argument of the exponential function. One can interpret the numerator of the argument as the potential energy gained when you pull apart the bond that holds a water molecule to neighboring molecules in a liquid, so as to allow that one molecule to move freely as water vapor. The denominator can be interpreted as the kinetic energy of a molecule as indicated by its temperature.

   With that in mind, the denominator is energy available, and numerator is energy needed, for one molecule of water to evaporate. Describe why this ratio is appropriate for understanding saturation vapor pressure as an equilibrium.

S8. Devise an equation to estimate surface water flux that works for a sunny windy day, which reduces to eqs. (4.51 & 4.52) in the limits of zero convection and zero mean wind, respectively.