17.13: Homework Exercises
- Page ID
- 10601
17.13.1. Broaden Knowledge & Comprehension
B1. Search the web for wind-rose graphs for a location (weather station or airport) near you, or other location specified by your instructor.
B2. Search the web for wind-speed distributions for a weather station near you. Relate this distribution to extreme or record-breaking winds.
Fourteenth century philosopher William of Occam suggested that “the simplest scientific explanation is the best”. This tenant is known as Occam’s Razor, because with it you can cut away the bad theories and complex equations from the good.
But why should the simplest or most elegant be the best? There is no law of nature that says it must be so. It is just one of the philosophies of science, as is the scientific method of Descartes. Ultimately, like any philosophy or religion, it is a matter of faith.
I suggest an alternative tenant: “a scientific relationship should not be more complex than needed.” This is motivated by the human body — an amazingly complex system of hydraulic, pneumatic, electrical, mechanical, chemical, and other physical processes that works exceptionally well. In spite of its complexity, the human body is not more complex than needed (as determined by evolution).
Although this alternative tenant is only subtly different from Occam’s Razor, it admits that sometimes complex mathematical solutions to physical problems are valid. This tenant is used by a data-analysis method called computational evolution (or gene-expression programming). This approach creates a population of different algorithms that compete to best fit the data, where the best algorithms are allowed to persist with mutation into the next generation while the less-fit algorithms are culled via computational natural selection.
B3. Find on the web climatological maps giving locations of persistent, moderate winds. These are favored locations for wind turbine farms. Also search the web for locations of existing turbine farms.
B4. Search the web for a weather map showing vertical velocities over your country or region. Sometimes, these vertical velocities are given as omega (ω) rather than w, where ω is the change of pressure with time experienced by a vertically moving parcel, and is defined in the Extratropical Cyclone chapter. What is the range of vertical velocities on this particular day, in m s–1?
B5. Search the web for the highest resolution (hopefully 0.5 km or better resolution) visible satellite imagery for your area. Which parts of the country have rising thermals, based on the presence of cumulus clouds at the top of the thermals?
B6. Search the web for lidar (laser radar) images of thermals in the boundary layer.
B7. Search the web for the highest resolution (hopefully 0.5 km or better resolution) visible satellite imagery for your area. Also search for an upper-air sounding (i.e., thermo diagram) for your area. Does the depth of the mixed layer from the thermo diagram agree with the diameter of thermals (clouds) visible in the satellite image? Comment.
B8. Access IR high resolution satellite images over cloud-free regions of the Rocky Mountains (or Cascades, Sierra-Nevada, Appalachians, or other significant mountains) for late night or early morning during synoptic conditions of high pressure and light winds. Identify those regions of cold air in valleys, as might have resulted from katabatic winds. Sometimes such regions can be identified by the fog that forms in them.
B9. Search the web for weather station observations at the mouth of a valley. Plotted meteograms of wind speed and direction are best to find. See if you can find evidence of mountain/valley circulations in these station observations, under weak synoptic forcing.
B10. Same as the previous problem, but to detect a sea breeze for a coastal weather station.
B11. Search the web for satellite observations of the sea breeze, evident as changes in cloudiness parallel to the coastline.
B12. Search the web for information on how tsunami on the ocean surface travel at the shallow-water wave speed as defined in this chapter, even when the waves are over the deepest parts of the ocean. Explain.
B13. Search the web for images of hydraulic jump in the atmosphere. If you can’t find any, then find images of hydraulic jump in water instead.
B14. Access images from digital elevation data, and find examples of short and long gaps through mountain ranges for locations other than Western Canada.
B15. Search the web for news stories about dangerous winds along the coast, but limit this search to only coastally-trapped jets. If sufficient information is given in the news story, relate the coastal jet to the synoptic weather conditions.
B16. Access visible high-resolution satellite photos of mountain wave clouds downwind of a major mountain range. Measure the wavelength from these images, and compare with the wind speed accessed from upper air soundings in the wave region. Use those data to estimate the Brunt-Väisälä frequency.
B17. Access from the web photographs taken from ground level of lenticular clouds. Also, search for iridescent clouds on the web, to find if any of these are lenticular clouds.
B18. Access from the web pilot reports of turbulence, chop, or mountain waves in regions downwind of mountains. Do this over several days, and show how these reports vary with wind speed and static stability.
B19. Access high-resolution visible satellite images from the web during clear skies, that show the smoke plume from a major source (such as Gary, Indiana, or Sudbury, Ontario, or a volcano, or a forest fire). Assume that this image shows a streakline. Also access the current winds from a weather map corresponding roughly to the altitude of the smoke plume, from which you can infer the streamlines. Compare the streamlines and streaklines, and speculate on how the flow has changed over time, if at all. Also, draw on your printed satellite photo the path lines for various air parcels within the smoke plume.
B20. From the web, access weather maps that show streamlines. These are frequently given for weather maps of the jet stream near the tropopause (at 20 to 30 kPa). Also access from the web weather maps that plot the actual upper air winds from rawinsonde observations, valid at the same time and altitude as the streamline map. Compare the instantaneous winds with the streamlines.
B21. From the web, access a sequence of weather maps of streamlines for the same area. Locate a point on the map where the streamline direction has changed significantly during the sequence of maps. Assume that smoke is emitted continuously from that point. On the last map of the sequence, plot the streakline that you would expect to see. (Hint, from the first streamline map, draw a path line for an air parcel that travels until the time of the next streamline map. Then, using the new map, continue finding the path of that first parcel, as well as emit a new second parcel that you track. Continue the process until the tracks of all the parcels end at the time of the last streamline map. The locus of those parcels is a rough indication of the streakline.)
B22. Access from the web information for aircraft pilots on how the pitot tube works, and/or its calibration characteristics for a particular model of aircraft.
B23. Access from the web figures that show the amount of destruction for different intensities of tornado winds. Prepare a table giving the dynamic pressures and forces on the side of a typical house for each of those different wind categories.
B24. Access from the web news stories of damage to buildings or other structures caused by Boras, mountain waves, or downslope windstorms.
B25. Access from the web data or images that indicate typical height of various mature crops (other than the ones already given at the end of the Numerical exercises).
B26. Access from the web the near-surface air temperature at sunrise in or just downwind of a large city, and compare with the rural temperature.
B27. Use info from the web to estimate the urban canopy H/W aspect ratio for the city center nearest to you. Then use Fig. 17.42 to estimate ∆TUHI_max.
17.13.2. Apply
A1.(§) Plot the probability of wind speeds using a Weibull distribution with a resolution of 0.5 m s–1, and Mo = 8 m s–1, for α =
| a. 1.0 | b. 1.5 | c. 2.0 | d. 2.5 | e. 3.0 | f. 3.5 | g. 4.0 |
| h. 4.5 | i. 5.0 | j. 5.5 | k. 6.0 | m. 7 | n. 8 | o. 9 |
A2. A wind turbine of blade radius 25 m runs at 35% efficiency. At sea level, find the theoretical power (kW) for winds (m s–1) of:
| a. 1 | b. 2 | c. 3 | d. 4 | e. 5 | f. 6 | g. 7 | h. 8 |
| i. 9 | j. 10 | k. 11 | m. 12 | n. 13 | o. 14 | p. 15 |
A3. Find the equilibrium updraft speed (m s–1) of a thermal in a 2 km boundary layer with environmental temperature 15°C. The thermal temperature (°C) is:
| a. 16 | b. 16.5 | c. 17 | d. 17.5 | e. 18 | f. 18.5 |
| g. 19 | h. 19.5 | i. 20 | j. 20.5 | k. 21 | m. 21.5 |
A4. Anabatic flow has a temperature excess of 4°C. Find the buoyant along-slope pressure gradient force per unit mass for a slope of angle (°):
| a. 10 | b. 15 | c. 20 | d. 25 | e. 30 | f. 35 | g. 40 |
| h. 45 | i. 50 | j. 55 | k. 60 | m. 65 | n. 70 | o. 75 |
A5(§). Plot katabatic wind speed (m s–1) vs. downslope distance (m) if the environment is 20°C and the cold katabatic air is 15°C. The slope angle (°) is:
| a. 10 | b. 15 | c. 20 | d. 25 | e. 30 | f. 35 | g. 40 |
| h. 45 | i. 50 | j. 55 | k. 60 | m. 65 | n. 70 | o. 75 |
A6. Find the equilibrium downslope speed (m s–1) for the previous problem, if the katabatic air is 5 m thick and the drag coefficient is 0.002.
A7. Find the depth (m) of the thermal internal boundary layer 2 km downwind of the coastline, for an environment with wind speed 8 m s–1 and γ = 4 K km–1. The surface kinematic heat flux (K·m s–1) is
| a. 0.04 | b. 0.06 | c. 0.08 | d. 0.1 | e. 0.12 |
| f. 0.14 | g. 0.16 | h. 0.18 | i. 0.2 | j. 0.22 |
A8. Assume Tv = 20°C. Find the speed (m s–1) of advance of the sea-breeze front, for a flow depth of 700 m and a temperature excess ∆θ (K) of:
| a. 1.0 | b. 1.5 | c. 2.0 | d. 2.5 | e. 3.0 | f. 3.5 | g. 4.0 |
| h. 4.5 | i. 5.0 | j. 5.5 | k. 6.0 | m. 7 | n. 8 | o. 9 |
A9. For the previous problem, find the sea-breeze wind speed (m s–1) at the coast.
A10. For a sea-breeze frontal speed of 5 m s–1, find the expected maximum distance (km) of advance of the sea-breeze front for a latitude (°) of
| a. 10 | b. 15 | c. 20 | d. 25 | e. 80 | f. 35 | g. 40 |
| h. 45 | i. 50 | j. 55 | k. 60 | m. 65 | n. 70 | o. 75 |
A11. What is the shallow-water wave phase speed (m s–1) for a water depth (m) of:
| a. 2 | b. 4 | c. 6 | d. 8 | e. 10 | f. 15 | g. 20 |
| h. 25 | i. 30 | j. 40 | k. 50 | m. 75 | n. 100 | o. 200 |
A12. Assume |g|/Tv = 0.0333 m·s–2·K–1. For a cold layer of air of depth 50 m under warmer air, find the surface (interfacial) wave phase speed (m s–1) for a virtual potential temperature difference (K) of:
| a. 1.0 | b. 1.5 | c. 2.0 | d. 2.5 | e. 3.0 | f. 3.5 | g. 4.0 |
| h. 4.5 | i. 5.0 | j. 5.5 | k. 6.0 | m. 7 | n. 8 | o. 9 |
A13. For the previous problem, find the value of the Froude number Fr1. Also, classify this flow as subcritical, critical, or supercritical. Given M = 15 m s–1.
A14. Assume |g|/Tv = 0.0333 m·s–2·K–1. Find the internal wave horizontal group speed (m s–1) for a stably stratified air layer of depth 400 m, given ∆θv/∆z (K km–1) of:
| a. 1.0 | b. 1.5 | c. 2.0 | d. 2.5 | e. 3.0 | f. 3.5 | g. 4.0 |
| h. 4.5 | i. 5.0 | j. 5.5 | k. 6.0 | m. 7 | n. 8 | o. 9 |
A15. For the previous problem, find the value of the Froude number Fr2. Also, classify this flow as subcritical, critical, or supercritical.
A16. Winds of 10 m s–1 are flowing in a valley of 10 km width. Further downstream, the valley narrows to the width (km) given below. Find the wind speed (m s–1) in the constriction, assuming constant depth flow.
| a. 1.0 | b. 1.5 | c. 2.0 | d. 2.5 | e. 3.0 | f. 3.5 | g. 4.0 |
| h. 4.5 | i. 5.0 | j. 5.5 | k. 6.0 | m. 7 | n. 8 | o. 9 |
A17. Assume |g|/Tv = 0.0333 m·s–2·K–1. For a twolayer atmospheric system flowing through a short gap, find the maximum expected gap wind speed (m s–1). Flow depth is 300 m, and the virtual potential temperature difference (K) is:
| a. 1.0 | b. 1.5 | c. 2.0 | d. 2.5 | e. 3.0 | f. 3.5 | g. 4.0 |
| h. 4.5 | i. 5.0 | j. 5.5 | k. 6.0 | m. 7 | n. 8 | o. 9 |
A18. Find the long-gap geostrophic wind (m s–1) at latitude 50°, given |g|/Tv = 0.0333 m·s–2·K–1 and ∆θv = 3°C, and assuming that the slope of the top coldair surface is given by the height change (km) below across a valley 10 km wide.
| a. 0.3 | b. 0.4 | c. 0.5 | d. 0.6 | e. 0.7 |
| f. 0.8 | g. 0.9 | h. 1.0 | i. 1.1 | j. 1.2 |
| k. 2.4 | m. 2.6 | n. 2.8 | o. 3 |
A19. Find the external Rossby radius of deformation (km) for a coastally trapped jet that rides against a mountain range of 2500 m altitude at latitude (°) given below, for air that is colder than its surroundings by 10°C. Assume |g|/Tv = 0.0333 m·s–2·K–1.
| a. 80 | b. 85 | c. 20 | d. 25 | e. 30 | f. 35 | g. 40 |
| h. 45 | i. 50 | j. 55 | k. 60 | m. 65 | n. 70 | o. 75 |
A20. Assume |g|/Tv = 0.0333 m·s–2·K–1. Find the natural wavelength of air, given
a. M = 2 m s–1, ∆T/∆z = 5 °C km–1
b. M = 20 m s–1, ∆T/∆z = –8 °C km–1
c. M = 5 m s–1, ∆T/∆z = –2 °C km–1
d. M = 20 m s–1, ∆T/∆z = 5 °C km–1
e. M = 5 m s–1, ∆T/∆z = –8 °C km–1
f. M = 2 m s–1, ∆T/∆z = –2 °C km–1
g. M = 5 m s–1, ∆T/∆z = 5 °C km–1
h. M = 2 m s–1, ∆T/∆z = –8 °C km–1
A21. For a mountain of width 25 km, find the Froude number Fr3 for the previous problem. Draw a sketch of the type of mountain waves that are likely for this Froude number.
A22. For the previous problem, find the angle of the wave crests, and the wave-drag force per unit mass. Assume H = 1000 m and hw = 11 km.
A23(§). Plot the wavy path of air as it flows past a mountain, given an initial vertical displacement of 300 m, a wavelength of 12.5 km, and a damping factor of
| a. 1.0 | b. 1.5 | c. 2.0 | d. 2.5 | e. 3.0 | f. 3.5 | g. 4.0 |
| h. 4.5 | i. 5.0 | j. 5.5 | k. 6.0 | m. 7 | n. 8 | o. 9 |
A24(§) Given a temperature dew-point spread of 1.5°C at the initial (before-lifting) height of air in the previous problem, identify which wave crests contain lenticular clouds.
A25. Cold air flow speed 12 m s–1 changes to 3 m s–1 after a hydraulic jump. Assume |g|/Tv = 0.0333 m·s–2·K–1. How high can the hydraulic jump rise if the exit velocity (ms–1) is
| a. 1.0 | b. 1.5 | c. 2.0 | d. 2.5 | e. 3.0 | f. 3.5 | g. 4.0 |
| h. 4.5 | i. 5.0 | j. 5.5 | k. 6.0 | m. 7 | n. 8 | o. 9 |
A26. Assuming standard sea-level density and streamlines that are horizontal, find the pressure change given the following velocity (m s–1) change:
| a. 1.0 | b. 1.5 | c. 2.0 | d. 2.5 | e. 3.0 | f. 3.5 | g. 4.0 |
| h. 4.5 | i. 5.0 | j. 5.5 | k. 6.0 | m. 7 | n. 8 | o. 9 |
A27. Wind at constant altitude decelerates from 12 m s–1 to the speed (ms–1) given below, while passing through a wind turbine. What opposing net pressure difference (Pa) would have caused the same deceleration in laminar flow?
| a. 1.0 | b. 1.5 | c. 2.0 | d. 2.5 | e. 3.0 | f. 3.5 | g. 4.0 |
| h. 4.5 | i. 5.0 | j. 5.5 | k. 6.0 | m. 7 | n. 8 | o. 9 |
A28. Air with pressure 100 kPa is initially at rest. It is accelerated isothermally over a flat 0°C snow surface as it is sucked toward a household ventilation system. What is the final air pressure (kPa) just before entering the fan if the final speed (m s–1) through the fan is:
| a. 1 | b. 2 | c. 3 | d. 4 | e. 5 | f. 6 | g. 7 | h. 8 |
| i. 9 | j. 10 | k. 11 | m. 12 | n. 13 | o. 14 | p. 15 |
A29. A short distance behind the jet engine of an aircraft flying in level flight, the exhaust temperature is 500°C. After the jet exhaust decelerates to zero, what is the final exhaust air temperature (°C), neglecting conduction and mixing, assuming the initial jet-blast speed (m s–1) is:
| a. 100 | b. 125 | c. 150 | d. 175 | e. 200 | f. 210 | g. 220 |
| h. 230 | i. 240 | j. 250 | k. 260 | m. 270 | n. 280 | o. 290 |
A30. An 85 kW electric wind machine with a 3 m radius fan blade is used in an orchard to mix air so as to reduce frost damage on fruit. The fan horizontally accelerates the air from calm to the speed (m s–1) given below. Find the temperature change (°C) across the fan, neglecting mixing with the environmental air.
| a. 6 | b. 6.5 | c. 7 | d. 7.5 | e. 8 | f. 8.5 | g. 9 |
| h. 9.5 | i. 10 | j. 10.5 | k. 11 | m. 12 | n. 13 | o. 14 |
A31. Tornadic air of temperature 25°C blows with speed (m s–1) given below, except that it stagnates upon hitting a barn. Find the final stagnation temperature (°C) and pressure change (kPa).
| a. 100 | b. 125 | c. 150 | d. 175 | e. 200 | f. 210 | g. 220 |
| h. 230 | i. 240 | j. 250 | k. 260 | m. 270 | n. 280 | o. 290 |
A32. Find the speed of sound (ms–1) and Mach number for Mair = 100 m s–1, given air of temperature (°C):
| a. –50 | b. –45 | c. –40 | d. –35 | e. –30 |
| f. –25 | g. –20 | h. –15 | i. –10 | j. –5 |
| k. 0 | m. 5 | n. 10 | o. 15 | p. 20 |
A33. Water flowing through a pipe with speed 2 ms–1 and pressure 100kPa accelerates to the speed (ms–1) given below when it flows through a constriction. What is the fluid pressure (kPa) in the constriction? Neglect drag against the pipe walls.
| a. 6 | b. 6.5 | c. 7 | d. 7.5 | e. 8 | f. 8.5 | g. 9 |
| h. 9.5 | i. 10 | j. 10.5 | k. 11 | m. 12 | n. 13 | o. 14 |
A34. For the bora Sample Application, redo the calculation assuming that the initial inversion height (km) is:
| a. 1.1 | b. 1.15 | c. 1.25 | d. 1.3 | e. 1.35 |
| f. 1.4 | g. 1.45 | h. 1.5 | i. 1.55 | j. 1.6 |
| k. 1.65 | m. 1.7 | n. 1.75 | o. 1.8 | p. 1.85 |
A35. Use a thermodynamic diagram. Air of initial temperature 10°C and dew point 0°C starts at a height where the pressure (kPa) is given below. This air rises to height 70 kPa as it flows over a mountain, during which all liquid and solid water precipitate out. Air descends on the lee side of the mountain to an altitude of 95 kPa. What is the temperature, dew point, and relative humidity of the air at its final altitude? How much precipitation occurred on the mountain? [Hint: use a thermo diagram.]
| a. 104 | b. 102 | c. 100 | d. 98 | e. 96 | f. 94 | g. 92 |
| h. 90 | i. 88 | j. 86 | k. 84 | m. 82 | n. 80 |
A36. Plot wind speed vs. height, for heights between 0.25 hc and 5 hc, where hc is average plant canopy height. Given:
| Plant | hc(m) | u*(m s–1) | attenuation coef. | |
| a. | Wheat | 1.0 | 0.5 | 2.6 |
| b. | Wheat | 1.0 | 0.75 | 2.6 |
| c. | Soybean | 1.0 | 0.5 | 3.5 |
| d. | Soybean | 1.0 | 0.75 | 3.5 |
| e. | Oats | 1.5 | 0.5 | 2.8 |
| f. | Oats | 1.5 | 0.75 | 2.8 |
| g. | Corn | 2.0 | 0.5 | 2.7 |
| h. | Corn | 2.0 | 0.75 | 2.7 |
| i. | Corn | 2.5 | 0.5 | 2.2 |
| j. | Corn | 2.5 | 0.75 | 2.2 |
| k. | Sunflower | 2.75 | 0.5 | 1.3 |
| m. | Sunflower | 2.75 | 0.75 | 1.3 |
| n. | Pine | 3.0 | 0.5 | 1.1 |
| o. | Pine | 3.0 | 0.25 | 1.1 |
| p. | Orchard | 4.0 | 0.5 | 0.4 |
| q. | Orchard | 4.0 | 0.25 | 0.4 |
| r. | Forest | 20. | 0.5 | 1.7 |
| s. | Forest | 20. | 0.25 | 1.7 |
A37. Estimate the max urban heat island temperature excess compared to the surrounding rural countryside, for a city with urban-canyon aspect ratio (H/W) of:
| a. 0.5 | b. 0.75 | c. 1.0 | d. 1.25 | e. 1.5 | f. 1.75 |
| g. 2.0 | h. 2.25 | i. 2.5 | j. 2.75 | k. 3.0 | m. 3.25 |
17.13.3. Evaluate & Analyze
E1. For a Weibull distribution, what is the value of the probability in any one bin as the bin size becomes infinitesimally small? Why?
E2(§). Create a computer spreadsheet with location and spread parameters in separate cells. Create and plot a Weibull frequency distribution for winds by referencing those parameters. Then try changing the parameters to see if you can get the Weibull distribution to look like other well-known distributions, such as Gaussian (symmetric, bell shaped), exponential, or others.
E3. Why was an asymmetric distribution such as the Weibull distribution chosen to represent winds?
E4. What assumptions were used in the derivation of Betz’ Law, and which of those assumptions could be improved?
E5. To double the amount of electrical power produced by a wind turbine, wind speed must increase by what percentage, or turbine radius increase by what percentage?
E6. For the Weibull distribution as plotted in Fig. 17.1, find the total wind power associated with it.
E7. In Fig. 17.5, what determines the shape of the wind-power output curve between the cut-in and rated points?
E8. List and explain commonalities among the equations that describe the various thermally-driven local flows.
E9. If thermals with average updraft velocity of W = 5 m s–1 occupy 40% of the horizontal area in the boundary layer, find the average downdraft velocity.
E10. What factors might affect rise rate of the thermal, in addition to the ones already given in this chapter?
E11. Anabatic and lenticular clouds were described in this chapter. Compare these clouds and their formation mechanisms. Is it possible for both clouds to occur simultaneously over the same mountain?
E12. Is the equation describing the anabatic pressure gradient force valid or reasonable in the limits of 0° slope, or 90° slope. Explain.
E13. Explain in terms of Bernoulli’s equation the horizontal pressure gradient force acting on anabatic winds.
E14. What factors control the shape of the katabatic wind profile, as plotted in Fig. 17.9?
E15. The Sample Application for katabatic wind shows the curves from eqs. (17.8) and (17.9) as crossing. Given the factors that appear in those equations, is a situation possible where the curves never cross? Describe.
E16. Suppose a mountain valley exits right at a coastline. For synoptically weak conditions (near zero geostrophic wind), describe how would the mountain/valley circulation and sea-breeze circulation interact. Illustrate with drawings.
E17. The thermal internal boundary layer can form both during weak- and strong-wind synoptic conditions. Why?
E18. For stronger land-sea temperature contrasts, which aspects of the sea-breeze would change, and which would be relatively unchanged? Why?
E19. At 30° latitude, can the sea-breeze front advance an infinite distance from the shore? Why?
E20. In the Southern Hemisphere, draw a sketch of the sea-breeze-vs.-time hodograph, and explain it.
E21. For what situations would open-channel hydraulics NOT be a good approximation to atmospheric local flows? Explain.
E22. Interfacial (surface) wave speed was shown to depend on average depth of the cold layer of air. Is this equation valid for any depth? Why?
E23. In deriving eq. (17.17) for internal waves, we focused on only the fastest wavelengths. Justify.
E24. In what ways is the Froude number for incompressible flows similar to the Mach number for compressible flows?
E25. If supercritical flows tend to “break down” toward subcritical, then why do supercritical flows exist at all in the atmosphere?
E26. Is it possible to have supercritical flow in the atmosphere that does NOT create an hydraulic jump when it changes to subcritical? Explain?
E27. Contrast the nature of gap winds through short and long gaps. Also, what would you do if the gap length were in between short and long?
E28. For gap winds through a long gap, why are they less likely to form in summer than winter?
E29. Can coastally trapped jets form on the east coast of continents in the N. Hemisphere? If so, explain how the process would work.
E30. It is known from measurements of the ionosphere that the vertical amplitude of mountain waves increases with altitude. Explain this using Bernoulli’s equation.
E31. What happens to the natural wavelength of air for statically unstable conditions?
E32. Why are lenticular clouds called standing lenticular?
E33. Compare and contrast the 3 versions of the Froude number. Do they actually describe the same physical processes? Why?
E34. Is there any max limit to the angle a of mountain wave crests (see Fig. 17.31)? Comment.
E35. If during the course of a day, the wind speed is constant but the wind direction gradually changes direction by a full 360°, draw a graph of the resulting streamline, streakline, and path line at the end of the period. Assume continuous emissions from a point source during the whole period.
E36. Identify the terms of Bernoulli’s equation that form the hydrostatic approximation. According to Bernoulli’s equation, what must happen or not happen in order for hydrostatic balance to be valid?
E37. Describe how the terms in Bernoulli’s equation vary along a mountain-wave streamline as sketched in Fig. 17.29.
E38. If a cold air parcel is given an upward push in a warmer environment of uniform potential temperature, describe how the terms in Bernoulli’s equation vary with parcel height.
E39. For compressible flow, show if (and how) the Bernoulli equations for isothermal and adiabatic processes reduce to the basic incompressible Bernoulli equation under conditions of constant density.
E40. In the Sample Application for the pressure variation across a wind turbine, hypothesize why the actual pressure change has the variation that was plotted.
E41. In Fig. 17.34, would it be reasonable to move the static pressure port to the top center of the darkly shaded block, given no change to the streamlines drawn? Comment on potential problems with a static port at that location.
E42. Design a thermometer mount on a fast aircraft that would not be susceptible to dynamic warming. Explain why your design would work.
E43. In Fig. 17.34, speculate on how the streamlines would look if the approaching flow was supersonic. Draw your streamlines, and justify them.
E44. Comment on the differences and similarities of the two mechanisms shown in this Chapter for creating Foehn winds.
E45. For Bora winds, if the upwind cold air was over an elevated plateau, and the downwind lowland was significantly lower than the plateau, how would Bora winds be different, if at all? Why?
E46. If the air in Fig. 17.38 went over a mountain but there was no precipitation, would there be a Foehn wind?
E47. Relate the amount of warming of a Foehn wind to the average upstream wind speed and the precipitation rate in mm h–1.
E48. How sensitive is the solution for wind speed above a plant canopy? [Hint: see the Sample Application in the canopy flow section.] Namely, if you have a small error in estimating displacement distance d, are the resulting errors in friction velocity u* and roughness length zo relatively small or large?
17.13.4. Synthesize
S1. Suppose that in year 2100 everyone is required by law to have their own wind turbine. Since wind turbines take power from the wind, the wind becomes slower. What effect would this have on the weather and climate, if any?
S2. If fair-weather thermals routinely rose as high as the tropopause without forming clouds, comment on changes to the weather and climate, if any.
S3. Suppose that katabatic winds were frictionless. Namely, no turbulence, no friction against the ground, and no friction against other layers of air. Speculate on the shape of the vertical wind profile of the katabatic winds, and justify your arguments.
S4. If a valley has two exists, how would the mountain and valley winds behave?
S5. Suppose that katabatic winds flow into a bowlshaped depression instead of a valley. Describe how the airflow would evolve during the night.
S6. If warm air was not less dense than cold, could sea breezes form? Explain.
S7. Why does the cycling in a sea-breeze hodograph not necessarily agree with the timing of the pendulum day?
S8. What local circulations would disappear if air density did not vary with temperature? Justify.
S9. Can a Froude number be defined based on deepwater waves rather than shallow-water waves? If so, write an equation for the resulting Froude number, and suggest applications for it in the atmosphere.
S10. What if waves could carry no information and no energy. How would the critical nature of the flow change, if at all?
S11. If the Earth did not rotate, compare the flow through short and long gaps through mountains.
S12. If no mountains existing along coasts, could there ever be strong winds parallel to the coast?
S13. If mountain-wave drag causes the winds to be slower, does that same drag force cause the Earth to spin faster? Comment.
S14. Suppose that mountain-wave drag worked oppositely, and caused winds to accelerate aloft. How would the weather & climate be different, if at all?
S15. Is it possible for a moving air parcel to not be traveling along a streamline? Comment.
S16. Suppose that Bernoulli’s equation says that pressure decreases as velocity decreases along a streamline of constant height. How would the weather and climate be different, if at all? Start by commenting how Boras would be different, if at all.
S17. Suppose you are a 2 m tall person in a town with average building height of 8 m. How would the winds that you feel be different (if at all) than the winds felt by a 0.2 m tall cat in a young corn field of average height 0.8 m?
S18. If human population continued to grow until all land areas were urban, would there be an urban heat island? Justify, and relate to weather changes.