# 18.8: Homework Exercises

- Page ID
- 10708

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# 18.8.1. Broaden Knowledge & Comprehension

B1. Access the upper-air soundings every 6 or 12 h for a rawinsonde station near you (or other site specified by your instructor). For heights every 200 m (or every 2 kPa) from the surface, plot how the temperature varies with time over several days. The result should look like Fig. 18.4, but with more lines. Which heights or pressure levels appear to be above the ABL?

B2. Same as the previous question, but first convert the temperatures to potential temperatures at those selected heights. This should look even more like Fig. 18.4.

B3. Access temperature profiles from the web for the rawinsonde station closest to you (or for some other sounding station specified by your instructor). Convert the resulting temperatures to potential temperatures, and plot the resulting θ vs z. Can you identify the top of the ABL? Consider the time of day when the sounding was made, to help you anticipate what type of ABL exists (e.g., mixed layer, stable boundary layer, neutral layer.)

B4. Access a weather map with surface wind observations. Find a situation or a location where there is a low pressure center. Draw a hypothetical circle around the center of the low, and find the average inflow velocity component across this circle. Using volume conservation, and assuming a 1 km thick ABL, what vertical velocity to you anticipate over the low?

B5. Same as previous question, but for a high-pressure center.

B6. Access a number of rawinsonde soundings for stations more-or-less along a straight line that crosses through a cold front. Identify the ABL top both ahead of and behind the front.

B7. Access the current sunrise and sunset times at your location. Estimate a curve such as in Fig. 18.10.

B8. Find a rawinsonde station in the center of a clear high pressure region, and access the soundings every 6 h if possible. If they are not available over N. America, try Europe. Use all of the temperature, humidity, and wind information to determine the evolution of the ABL structure, and create a sketch similar to Fig. 18.8, but for your real conditions. Consider seasonal affects, such as in Figs. 18.13 & 18.14.

B9. Access a rawinsonde sounding for nighttime, and find the best-fit parameters (height, strength) for an exponential potential temperature profile that best fits the sounding. If the exponential shape is a poor fit, explain why.

B10. Get an early morning sounding for a rawinsonde site near you. Calculate the anticipated accumulated heating, based on the eqs. in the Solar & IR Radiation & Heat chapters, and predict how the mixed layer will warm & grow in depth as the day progresses.

B11. Compare two sequential rawinsonde soundings for a site, and estimate the entrainment velocity by comparing the heights of the ABL. What assumptions did you make in order to do this calculation?

B12. Access the wind information from a rawinsonde site near you. Compare the wind speed profiles between day and night, and comment on the change of wind speed in the ABL.

B13. Access the “surface” wind observations from a weather station near you. Record and plot the wind speed vs. time, using wind data as frequent as are available, getting a total of roughly 24 to 100 observations. Use these data to calculate the mean wind, variance, and standard deviation.

B14. Same as the previous exercise, but for T.

B15. Same as the previous exercise, except collect both wind and temperature data, and find the covariance and correlation coefficient.

B16. Search the web for **eddy-correlation sensors**, **scintillometers**, **sonic anemometers**, or other instruments used for measuring vertical turbulent fluxes, and describe how they work.

B17. Search the web for eddy viscosity, eddy diffusivity, or K-theory values for K in the atmosphere.

# 18.8.2. Apply

A1(§). Calculate and plot the increase of cumulative kinematic heat (cooling) during the night, for a case with kinematic heat flux (K·m s^{–1}) of:

a. –0.02 | b. –0.05 | c. – 0.01 | d. –0.04 | e. –0.03 |

f. – 0.06 | g. – 0.07 | h. –0.10 | i. –0.09 | j. –0.08 |

A2 (§). Calculate and plot the increase of cumulative kinematic heat during the day, for a case with daytime duration of 12 hours, and maximum kinematic heat flux (K·m s^{–1}) of:

a. 0.2 | b. 0.5 | c. 0.1 | d. 0.4 | e. 0.3 |

f. 0.6 | g. 0.7 | h. 1.0 | i. 0.9 | j. 0.8 |

A3(§). For a constant kinematic heat flux of –0.02 K·m s^{–1} during a 12-hour night, plot the depth and strength of the stable ABL vs. time. Assume a flat prairie, with a residual layer wind speed (m s^{–1}) of:

a. 2 | b. 5 | c. 8 | d. 10 | e. 12 |

f. 15 | g. 18 | h. 20 | i. 22 | j. 25 |

A4(§). For the previous problem, plot the vertical temperature profile at 1-h intervals.

A5. Find the entrained kinematic heat flux at the top of the mixed layer, given a surface kinematic heat flux (K·m s^{–1}) of

a. 0.2 | b. 0.5 | c. 0.1 | d. 0.4 | e. 0.3 |

f. 0.6 | g. 0.7 | g. 0.7 | i. 0.9 | j. 0.8 |

A6. Find the entrainment velocity for a surface heat flux of 0.2 K·m s^{–1}, and a capping inversion strength of (°C):

a. 0.1 | b. 0.2 | c. 0.3 | d. 0.5 | e. 0.7 |

f. 1.0 | g. 1.2 | h. 1.5 | i. 2.0 | j. 2.5 |

A7. For the previous problem, calculate the increase in mixed-layer depth during a 6 h interval, assuming subsidence of – 0.02 m s^{–1}.

A8. Calculate the surface stress at sea level for a friction velocity (m s^{–1}) 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 |

A9. Find the friction velocity over a corn crop during wind speeds (m s^{–1}) of:

a. 2 | b. 3 | c. 4 | d. 5 | e. 6 | f. 7 | g. 8 |

h. 9 | i. 10 | j. 12 | k. 15 | m. 18 | n. 20 | o. 25 |

A12(§). Given M_{1} = 5 m s^{–1} at z_{1} = 10 m, plot wind speed vs. height, for z_{o} (m) =

a. 0.001 | b. 0.002 | c. 0.005 | d. 0.007 | e. 0.01 |

f. 0.02 | g. 0.05 | h. 0.07 | i. 0.1 | j. 0.2 |

k. 0.5 | m. 0.7 | n. 1.0 | o. 2.0 | p. 2.5 |

A13. Find the drag coefficients for problem A12.

A14(§). For a neutral surface layer, plot wind speed against height on linear and on semi-log graphs for friction velocity of 0.5 m s^{–1} and aerodynamic roughness length (m) of:

a. 0.001 | b. 0.002 | c. 0.005 | d. 0.007 | e. 0.01 |

f. 0.02 | g. 0.05 | h. 0.07 | i. 0.1 | j. 0.2 |

k. 0.5 | m. 0.7 | n. 1.0 | o. 2.0 | p. 2.5 |

A15. An anemometer on a 10 m mast measures a wind speed of 8 m s^{–1} in a region of scattered hedgerows. Find the wind speed at height (m):

a. 0.5 | b. 2 | c. 5 | d. 30 |

e. 1.0 | f. 4 | g. 15 | h. 20 |

A16. The wind speed is 2 m s^{–1} at 1 m above ground, and is 5 m s^{–1} at 10 m above ground. Find the roughness length. State all assumptions.

A17. Over a low crop with wind speed of 5 m s–1 at height 10 m, find the wind speed (m s^{–1}) at the height (m) given below, assuming overcast conditions:

a. 0.5 | b. 1.0 | c. 2 | d. 5 | e. 15 | f. 20 |

g. 25 | h. 30 | i. 35 | j. 40 | k. 50 | m. 75 |

A18. Same as previous problem, but during a clear night when friction velocity is 0.1 m s^{–1} and surface kinematic heat flux is –0.01 K·m s^{–1}. Assume |g|/T_{v} = 0.0333 m·s^{–2}·K^{–1}.

A19(§). Plot the vertical profile of wind speed in a stable boundary layer for roughness length of 0.2 m, friction velocity of 0.3 m s^{–1}, and surface kinematic heat flux (K·m s^{–1}) of:

a. –0.02 | b. –0.05 | c. – 0.01 | d. –0.04 | e. –0.03 |

f. – 0.06 | g. – 0.07 | h. –0.10 | i. –0.09 | j. –0.08 |

Assume |g|/T_{v} = 0.0333 m·s^{–2}·K^{–1}.

A20. For a 1 km thick mixed layer with |g|/T_{v} = 0.0333 m·s^{–2}·K^{–1}, find the Deardorff velocity (m s^{–1}) for surface kinematic heat fluxes (K·m s^{–1}) of:

a. 0.2 | b. 0.5 | c. 0.1 | d. 0.4 | e. 0.3 |

f. 0.6 | g. 0.7 | h. 1.0 | i. 0.9 | j. 0.8 |

A21§). For the previous problem, plot the wind speed profile, given u_{*} = 0.4 m s^{–1}, and M_{BL} = 5 m s^{–1}.

A22(§). For the following time series of temperatures (°C): 22, 25, 21, 30, 29, 14, 16, 24, 24, 20

- Find the mean and turbulent parts
- Find the variance
- Find the standard deviation

A23(§). Using the data given in the long Sample Application in the “Turbulent Fluxes and Covariances” section of this Chapter:

- Find the mean & variance for mixing ratio r.
- Find the mean and variance for U.
- Find the covariance between r and W.
- Find the covariance between U and W.
- Find the covariance between r and U.
- Find the correlation coefficient between r & W.
- Find the correlation coef. between u & W.
- Find the correlation coefficient between r & U.

A24(§). Plot the standard deviations of U, V, and W with height, and determine if and where the flow is nearly isotropic, given u_{*} = 0.5 m s^{–1}, w_{B} = 40 m s^{–1}, h = 600 m, and z_{i} = 2 km, for air that is statically:

- stable
- neutral
- unstable

Hint: Use only those “given” values that apply to your stability.

A25(§). Same as exercise A24, but plot TKE vs. z.

A26(§). Plot wind standard deviation vs. height, and determine if and where the flow is nearly isotropic, for all three wind components, for

- stable air with h = 300 m, u
_{*}= 0.1 m s^{–1} - neutral air with h = 1 km, u
_{* }= 0.2 m s^{–1} - unstable air with zi = 2 km, w
_{B}= 0.3 m s^{–1}

A27(§). Plot the turbulence kinetic energy per unit mass vs. height for the previous problem.

A28. Given a wind speed of 20 m s^{–1}, surface kinematic heat flux of –0.1 K·m s^{–1}, TKE of 0.4 m^{2} s^{–2}, and |g|/T_{v} = 0.0333 m·s^{–2}·K^{–1}, find

- shear production rate of TKE
- buoyant production/consumption of TKE
- dissipation rate of TKE
- total tendency of TKE (neglecting advection and transport)
- flux Richardson number
- the static stability classification?
- the Pasquill-Gifford turbulence type?
- flow classification

A29. Given the following initial sounding

z (m) | θ (°C) | M (m s^{–1}) |

700 | 21 | 10 |

500 | 20 | 10 |

300 | 18 | 8 |

100 | 13 | 5 |

Compute the

- value of eddy diffusivity at each height.
- turbulent heat flux in each layer, using K-theory
- the new values of θ after 30 minutes of mixing.
- turbulent kinematic momentum flux \(\ \overline{w^{\prime} u^{\prime}}\) in each layer, using K-theory. (Hint, generalize the concepts of heat flux. Also, assume M = U, and V = 0 everywhere.)
- the new values of M after 30 minutes of mixing.

# 18.8.3. Evaluate & Analyze

E1. Can the ABL fill the whole troposphere? Could it extend far into the stratosphere? Explain.

E2. Estimate the static stability near the ground now at your location?

E3. If the standard atmosphere was statically neutral in the troposphere, would there be a boundary layer? If so, what would be its characteristics.

E4. It is nighttime in mid winter with snow on the ground. This air blows over an unfrozen lake. Over the lake, what is the static stability of the air, and what type of ABL exists there.

E5. It is daytime in summer during a solar eclipse. How will the ABL evolve during the eclipse?

E6. Given Fig. 18.3, if turbulence were to become more vigorous and cause the ABL depth to increase, what would happen to the strength of the capping inversion? How would that result affect further growth of the ABL?

E7. The ocean often has a turbulent boundary layer within the top tens of meters of water. During a moderately windy, clear, 24 h period, when do you think stable and unstable (convective) ocean boundary layers would occur?

E8. Fig. 18.4 shows the free-atmosphere line touching the peaks of the ABL curve. Is it possible during other weather conditions for the FA curve to cross through the middle of the ABL curve, or to touch the minimum points of the ABL curve? Discuss.

E9. It was stated that subsidence cannot penetrate the capping inversion, but pushes the inversion down in regions of high pressure. Why can subsidence not penetrate into the ABL?

E10. Fig. 18.7a shows a cold front advancing. What if it were retreating (i.e., if it were a warm front). How would the ABL be affected differently, if at all?

E11. Draw a sketch similar to Fig. 18.8, except indicating the static stabilities in each domain.

E12. Copy Fig. 18.9, and then trace the nighttime curves onto the corresponding daytime curves. Comment on regions where the curves are the same, and on other regions where they are different. Explain this behavior.

E13. Use the solar and IR radiation equations from the Solar & IR Radiation chapter to plot a curve of daytime radiative heat flux vs. time. Although the resulting curve looks similar to half of a sine wave (as in Fig. 18.10), they are theoretically different. How would the actual curve change further from a sine curve at higher or lower latitudes? How good is the assumption that the heat flux curve is a half sine wave?

E14. Similar to the previous question, but how good is the assumption that nighttime heat flux is constant with time?

E15. In the explanation surrounding Fig. 18.10, the accumulated heating and cooling were referenced to start times when the heat flux became positive or negative, respectively. We did not use sunrise and sunset as the start times because those times did not correspond to when the surface heat flux changes sign. Use all of the relevant previous equations in the chapter (and in the Solar & IR Radiation chapter) to determine the time difference today at your town, between:

- sunrise and the time when surface heat flux first becomes positive.
- sunset and the time when surface heat flux first becomes negative.

E16. At nighttime it is possible to have well-mixed layers if wind speed is sufficiently vigorous. Assuming a well-mixed nocturnal boundary layer over a surface that is getting colder with time, describe the evolution (depth and strength) of this stable boundary layer. Also, for the same amount of cooling, how would its depth compare with the depth of the exponentially-shaped profile?

E17. Derive an equation for the strength of an exponentially-shaped nocturnal inversion vs. time, assuming constant heat flux during the night.

E18. Use eq. (18.3) and the definition of potential temperature to replot Fig. 18.15 in terms of actual temperature T vs. z. Discuss the difference in heights between the relative maximum of T and relative maximum of θ.

E19. For a linear early morning sounding (θ increases linearly with z), analytically compare the growth rates of the mixed layer calculated using the thermodynamic and the flux-ratio methods. For some idealized situations, is it possible to express one in terms of the other? Were any assumptions needed?

E20. Given an early morning sounding as plotted in Fig. 18.16a. If the daytime heat flux were constant with time, sketch a curve of mixed-layer depth vs. time, assuming the thermodynamic method. Comment on the different stages of mixed layer growth rate.

E21. Use the early-morning (6 AM) sounding given below with surface temperature 5°C. Find the mixed-layer potential temperature and depth at 11 AM, when the cumulative heating is 800 K·m. Assume the early-morning potential temperature profile is:

- ∆θ/∆z = 2 K km–1 = constant
- ∆θ(z) = (8°C)·exp(–z/150m)

Hint: Use the encroachment method.

E22. If the heating rate is proportional to the vertical heat-flux divergence, use Fig. 18.17b to determine the heating rate at each z in the mixed layer. How would the mixed-layer T profile change with time?

E23. Assume that equations similar to eq. (18.7) applies to moisture fluxes as well as to heat fluxes, but that eq. (18.9) defines the entrainment velocity based only on heat. Combine those two equations to give the kinematic flux of moisture at the top of the mixed layer as a function of potential temperature and mixing ratio jumps across the mixed-layer top, and in terms of surface heat flux.

E24. If the ABL is a region that feels drag near the ground, why can the winds at night accelerate, as shown in Fig. 18.18?

E25. Use eqs. (18.11) and (18.14a) to show how eq. (18.12) is derived from the log wind profile.

E26. Given the moderate value for u_{*} that was written after eq. (18.10), what value of stress (Pa) does that correspond to? How does this stress compare to sea-level pressure?

E27. Derive eq. (18.14b) from (18.14a).

E28. Given the wind-speed profile of the Sample Application in the radix-layer subsection, compare the bottom portion of this profile to a log wind profile.

Namely, find a best fit log-wind profile for the same data. Comment on the differences.

E29. Using eq. 18.21, find the average value (or simplify the notation ) of c·T, where c is a constant.

E30. Abbreviation “rms” means “root mean square”. Explain why “rms” can be used to describe eq. (18.23).

E31. For fixed w_{B} and z_{i} , plot eqs. (18.26) with z, and identify those regions with isotropic turbulence.

E32. Using eqs. (18.24 – 18.26), derive expressions for the TKE vs. z for statically stable, neutral, and unstable boundary layers.

E33. Derive an expression for the shear production term vs. z in the neutral surface layer, assuming a log wind profile.

E34. Knowing how turbulent heat flux varies with z (Fig. 18.17b) in a convective mixed layer, comment on the TKE buoyancy term vs. z for that situation.

E35. Given some initial amount of TKE, and assuming no production, advection, or transport, how would TKE change with time, if at all? Plot a curve of TKE vs. time.

E36. Use K-theory to relate the flux Richardson number to the gradient Richardson number (R_{i} = [(|g|/T_{v})·∆θ·∆z] / ∆M^{2} , see the Stability chapter).

E37. If shear production of TKE increases with time, how must the buoyant term change with time in order to maintain a constant Pasquill-Gifford stability category of E?

E38. Create a figure similar to Fig. 18.25, but for the log wind profile. Comment on the variation of momentum flux with z in the neutral surface layer.

E39. A negative value of eddy-correlation momentum flux near the ground implies that the momentum of the wind is being lost to the ground. Is it possible to have a positive eddy-correlation momentum flux near the surface? If so, under what conditions, and what would it mean physically?

# 18.8.4. Synthesize

S1. At the end of one night, assume that the stable ABL profile of potential temperature has an exponential shape, with strength 10°C and e-folding depth of 300 m. Using this as the early-morning sounding, compute and plot how the potential temperature and depth of the mixed layer evolve during the morning. Assume that D = 12 hr, and F_{Hmax} = 0.1 K·m s^{–1}. (Hint: In the spirit of the “Seek Solutions” box in a previous chapter, feel free to use graphical methods to solve this without calculus. However, if you want to try it with calculus, be very careful to determine which are your dependent and independent variables.)

S2. Suppose that there was no turbulence at night. Assume, the radiatively-cooled surface cools only the bottom 1 m of atmosphere by conduction. Given typical heat fluxes at night, what would be the resulting air temperature by morning? Also describe how the daytime mixed layer would evolve from this starting point?

S3. On a planet that does not have a solid core, but has a gaseous atmosphere that increases in density as the planet center is approached, would there be a boundary layer? If so, how would it behave?

S4. What if you were hired by an orchard owner to tell her how deep a layer of air needs to be mixed by electric fans to prevent frost formation. Create a suite of answers based on different scenarios of initial conditions, so that she can consult these results on any given day to determine the speed to set the fans (which we will assume can be used to control the depth of mixing). State and justify all assumptions.

S5. What if the Earth’s surface was a perfect conductor of heat, and had essentially infinite heat capacity (as if the Earth were a solid sphere of aluminum, but had the same albedo as Earth). Given the same solar and IR radiative forcings on the Earth’s surface, describe how ABL structure and evolution would differ from Fig. 18.8, if at all?

S6. What if all the air in the troposphere were saturated & cloudy. How would the ABL be different?

S7. Suppose that for some reason, the actual 2 km thick ABL in our troposphere was warm enough to have zero capping inversion on one particular day, but otherwise looked like Fig. 18.3. Comment on the evolution of the ABL from this initial state. Also, how might this initial state have occurred?

S8. If the ABL were always as thick as the whole troposphere, how would that affect the magnitude of our diurnal cycle of temperature?

S9. It was stated in this chapter that entrainment happens one way across the capping inversion, from the non-turbulent air into the turbulent air. Suppose that the capping inversion was still there, but that both the ABL below the inversion and the layer of air above the inversion were turbulent. Comment on entrainment and the growth of the mixed layer.

S10. Suppose that TKE was not dissipated by viscosity. How would that change our weather and climate, if at all?

S11. Suppose that wind shear destroyed TKE rather than generated it. How would that change our weather and climate, if at all?

S12. Suppose there were never a temperature inversion capping the ABL. How would that change our weather and climate, if at all?

S13. Suppose that the winds felt no frictional drag at the surface of the Earth. How would that change our weather and climate, if at all?

S14. Verify that K-theory satisfies the rules of parameterization.

S15. Positive values of K imply down-gradient transport of heat (i.e., heat flows from hot to cold, a process sometimes called the Zeroth Law of Thermodynamics). What is the physical interpretation of negative values of K?