# 15.6: Homework Exercises

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# 15.6.1. Broaden Knowledge & Comprehension

B1. Search the web for (and print the best examples of) photographs of:

2. landspouts
3. waterspouts
2. gust fronts and arc clouds
3. hailstones, and hail storms
4. lightning
1. CG
2. IC (including spider lightning, also known as lightning crawlers)
3. a bolt from the blue
5. damage caused by intense tornadoes

(Hint: search on “storm stock images photographs”.) Discuss the features of your resulting photos with respect to information you learned in this chapter.

B2. Search the web for (and print the best examples of) radar reflectivity images of

1. hook echoes
2. gust fronts

Discuss the features of your resulting image(s) with respect to information you learned in this chapter.

B3. Search the web for (and print the best example of) real-time maps of lightning locations, as found from a lightning detection network or from satellite. Print a sequence of 3 images at about 30 minute intervals, and discuss how you can diagnose thunderstorm movement and evolution from the change in the location of lightning-strike clusters.

B4. Search the web for discussion of the health effects of being struck by lightning, and write or print a concise summary. Include information about how to resuscitate people who were struck.

B5. Search the web for, and summarize or print, recommendations for safety with respect to:

1. lightning
3. straight-line winds and derechos
4. hail
5. flash floods
6. thunderstorms (in general)

B6. Search the web (and print the best examples of) maps that show the frequency of occurrence of:

2. lightning strike frequency & lightning deaths
3. hail
4. derechos

B7. Search the web for, and print the best example of, a photographic guide on how to determine Fujita or Enhanced Fujita tornado intensity from damage surveys.

B8. Search the web for (and print the best examples of) information about how different building or construction methods respond to tornadoes of different intensities.

B9. Search the web for, and print and discuss five tips for successful and safe tornado chasing.

B10. Use the internet to find sites for tornado chasers. The National Severe Storms Lab (NSSL) web site might have related info.

B11. During any 1 year, what is the probability that a tornado will hit any particular house? Try to find the answer on the internet.

B12. Search the web for private companies that provide storm-chasing tours/adventures/safaris for paying clients. List 5 or more, including their web addresses, physical location, and what they specialize in.

B13. Blue jets, red sprites, and elves are electrical discharges that can be seen as very brief glows in the mesosphere. They are often found at 30 to 90 km altitude over thunderstorms. Summarize and print info from the internet about these discharges, and included some images.

B14. Search the web for, and print one best example of, tornadoes associated with hurricanes.

B15. Search the web for a complete list of tornado outbreaks and/or tornado outbreak sequences. Print 5 additional outbreaks that were large and/or important, but which weren’t already included in this chapter in the list of outbreaks. Focus on morerecent outbreaks.

B16. With regard to tornadoes, search the web for info to help you discuss the relative safety or dangers of:

1. being in a storm shelter
2. being in an above-ground tornado safe room
3. being in a mobile home or trailer
4. hiding under bridges and overpasses
5. standing above ground near tornadoes

B17. Search the web for examples of downburst, gust front, or wind-shear sensors and warning systems at airports, and summarize and print your findings.

B18. Search the web for maps of tornado alley; namely, locations with frequent tornadoes.

# 15.6.2. Apply

A1. If a thunderstorm cell rains for 0.5 h at the precipitation rate (mm h–1) below, calculate both the net latent heat released into the atmosphere, and the average warming rate within the troposphere.

 a. 50 b. 75 c. 100 d. 125 e. 150 f. 175 g. 200 h. 225 i. 250 j. 275 k. 300 l. 325 m. 350 n. 375 o. 400

A2. Indicate the TORRO hail size code, and descriptive name, for hail of diameter (cm):

 a. 0.6 b. 0.9 c. 1.2 d. 1.5 e. 1.7 f. 2.1 g. 2.7 h. 3.2 i. 3.7 j. 4.5 k. 5.5 l. 6.5 m. 7.5 n. 8.0 o. 9.5

A3. Graphically estimate the terminal fall velocity of hail of diameter (cm):

 a. 0.6 b. 0.9 c. 1.2 d. 1.5 e. 1.7 f. 2.1 g. 2.7 h. 3.2 i. 3.7 j. 4.5 k. 5.5 l. 6.5 m. 7.5 n. 8.0 o. 9.5

A4. A supercooled cloud droplet of radius 40 µm hits a large hailstone. Using the temperature (°C) of the droplet given below, is the drop cold enough to freeze instantly (i.e., is its temperature deficit sufficient to compensate the latent heat of fusion released)? Based on your calculations, state whether the freezing of this droplet would contribute to a layer of clear or white (porous) ice on the hailstone.

 a. –40 b. –37 c. –35 d. –32 e. –30 f. –27 g. –25 h. –22 i. –20 j. –17 k. –15 l. –13 m. –10 n. –7 o. –5

A5. Given the sounding in exercise A3 of the previous chapter, calculate the portion of SB CAPE between altitudes where the environmental temperature is –10 and –30°C. Also, indicate if rapid hail growth is likely.

A6. Given the table below of environmental conditions, calculate the value of Significant Hail Parameter (SHIP), and state whether this environment favors the formation of hailstone > 5 cm diameter (assuming a thunderstorm indeed forms).

Exercise a b c d
MUCAPE (J kg–1) 2000 2500 3000 3500
rMUP (g kg–1) 10 12 14 16
γ70-50kPa (°C km–1) 2 4 6 8
T50kPa (°C) –20 –15 –10 –5
TSM0-6km (m s–1) 20 30 40 50

A7. Forthe downburst acceleration equation, assume that the environmental air has temperature 2°C and mixing ratio 3 g kg–1 at pressure 85 kPa. A cloudy air parcel at that same height has the same temperature, and is saturated with water vapor and carries liquid water at the mixing ratio (g kg–1) listed below. Assume no ice crystals.

(1) Find portion of vertical acceleration due to the combination of temperature and water vapor effects.

(2) Find the portion of vertical acceleration due to the liquid water loading only.

(3) By what amount would the virtual potential temperature of an air parcel change if all the liquid water evaporates and cools the air?

(4) If all of the liquid water were to evaporate and cool the air parcel, find the new vertical acceleration.

The liquid water mixing ratios (g kg–1) are:

 a. 20 b. 18 c. 16 d. 14 e. 12 f. 10 g. 9 h. 8 i. 7 j. 6 k. 5 l. 4 m. 3 n. 2

A8. Given the sounding from exercise A3 of the previous chapter, assume a descending air parcel in a downburst follows a moist adiabat all the way down to the ground. If the descending parcel starts at the pressure (kPa) indicated below, and assuming its initial temperature is the same as the environment there, plot both the sounding and the descending parcel on a thermo diagram, and calculate the value of downdraft CAPE.

 a. 80 b. 79 c. 78 d. 77 e. 76 f. 75 g. 74 h. 73 i. 72 j. 71 k. 70 l. 69 m. 68 n. 67

A9. Find the downdraft speed if the DCAPE (J kg–1) for a downburst air parcel is:

 a. –200 b. –400 c. –600 d. –800 e. –1000 f. –1200 g. –1400 h. –1600 i. –1800 j. –2000 k. –2200 l. –2400 m. –2600 n. –2800 o. –3000

A10. If a downburst has the same potential temperature as the environment, and starts with vertical velocity (m s–1, negative for descending air) given below, use Bernoulli’s equation to estimate the maximum pressure perturbation at the ground under the downburst.

 a. –2 b. –4 c. –6 d. –8 e. –10 f. –12 g. –14 h. –16 i. –18 j. –20 k. –22 l. –24 m. –26 n. –28 o. –30

A11. Same as the previous exercise, but in addition to the initial downdraft velocity, the descending air parcel is colder than the environment by the following product of virtual potential temperature depression and initial altitude (°C·km):

 (1) –0.5 (2) –1 (3) –1.5 (4) –2 (5) –2.5 (6) –3 (7) –3.5 (8) –4 (9) –4.5 (10) –5 (11) –5.5 (12) –6 (13) –6.5 (14) –7 (15) –7.5 (16) –8

A12. Find the acceleration (m s–2) of outflow winds from under a downburst, assuming a maximum mesohigh pressure (kPa) perturbation at the surface as given below, and a radius of the mesohigh of 3 km.

 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. 1.1 l. 1.2 m. 1.3 n. 1.4 o. 1.5

A13. How fast will a gust front advance, and what will be its depth, at distance 6 km from the center of a downburst. Assume the downburst has radius 0.5 km and speed 9 m s–1 , and that the environmental around the downburst is 28°C. The magnitude of the temperature deficit (°C) is:

 a. 1 b. 1.5 c. 2 d. 2.5 e. 3 f. 3.5 g. 4 h. 4.5 i. 5 j. 5.5 k. 6 l. 6.5 m. 7 n. 7.5 o. 8 p. 8.5 q. 9 r. 9.5

A14(§). Draw a graph of gust front depth and advancement speed vs. distance from the downburst center, using data from the previous exercise.

A15. Given a lightning discharge current (kA) below and a voltage difference between the beginning to end of the lightning channel of 1010 V, find (1) the resistance of the ionized lightning channel and (2) the amount of charge (C) transferred between the cloud and the ground during the 20 µs lifetime of the lightning stroke.

 a. 2 b. 4 c. 6 d. 8 e. 10 f. 15 g. 20 h. 40 i. 60 j. 80 k. 100 l. 150 m. 200 n. 400

A16. To create lightning in (1) dry air, and (2) cloudy air, what voltage difference is required, given a lightning stroke length (km) of:

 a. 0.2 b. 0.4 c. 0.6 d. 0.8 e. 1 f. 1.2 g. 1.4 h. 1.6 i. 1.8 j. 2.0 k. 2.5 l. 3 m. 4 n. 5

A17. For an electrical potential across the atmosphere of 1.3x105 V km–1, find the current density if the resistivity (Ω·m) is:

 a. 5x1013 1x1013 c. 5x1012 d. 1x1012 e. 5x1011 f. 1x1011 g. 5x1010 h. 1x1010 i. 5x109 j. 1x109 k. 5x108 l. 1x108 m. 5x107 n. 1x107 o. 5x106

A18. What is the value of peak current in a lightning stroke, as estimated using a lightning detection network, given the following measurements of electrical field E and distance D from the ground station.

 –E (V m–1) D (km) a. 1 10 b. 1 50 c. 2 10 d. 2 100 e. 3 20 f. 3 80 g. 4 50 h. 4 100 i. 5 50 j. 5 200 k. 6 75 l. 6 300

A19. For a power line struck by lightning, what is the probability that the lightning-generated current (kA) is greater than:

 a. 2 b. 4 c. 6 d. 8 e. 10 f. 15 g. 20 h. 40 i. 60 j. 80 k. 100 l. 150 m. 200 n. 400

A20. When lightning strikes an electrical power line it causes a surge that rapidly reaches its peak but then slowly decreases. How many seconds after the lightning strike will the surge have diminished to the fraction of the peak surge given here:

 a. 0.1 b. 0.15 c. 0.2 d. 0.25 e. 0.3 f. 0.35 g. 0.4 h. 0.45 i. 0.5 j. 0.55 k. 0.6 l. 0.65 m. 0.7 n. 0.75 o. 0.8

A21(§). If lightning heats the air to the temperature (K) given below, then plot (on a log-log graph) the speed (Mach number), pressure (as ratio relative to background pressure), and radius of the shock front vs. time given ambient background pressure of 100 kPa and temperature 20°C.

 a. 16,000 b. 17,000 c. 18,000 d. 19,000 e. 20,000 f. 21,000 g. 22,000 h. 23,000 i. 24,000 j. 25,000 k. 26,000 l. 27,000 m. 28,000 n. 29,000 o. 30,000

A22. What is the speed of sound in calm air of temperature (°C):

 a. –20 b. –18 c. –16 d. –14 e. –12 f. –10 g. –8 h. –6 i. –4 j. –2 k. 0 l. 2 m. 4 n. 6 o. 8 p. 10 q. 12 r. 14 s. 16 t. 18

A23(§). Create a graph with three curves for the time interval between the “flash” of lightning and the “bang” of thunder vs. distance from the lightning. One curve should be zero wind, and the other two are for tail and head winds of magnitude (m s–1) given below. Given Tenvironment = 295 K.

 a. 2 b. 4 c. 6 d. 8 e. 10 f. 12 g. 14 h. 16 i. 18 j. 20 k. 22 l. 24 m. 26 n. 28

A24(§). For a lightning stroke 2 km above ground in a calm adiabatic environment of average temperature 300 K, plot the thunder ray paths leaving downward from the lightning stroke, given that they arrive at the ground at the following elevation angle (°).

 a. 5 b. 6 c. 7 d. 8 e. 9 f. 10 g. 11 h. 12 i. 13 j. 14 k. 15 l. 16 m. 17 n. 18 o. 19 p. 20 q. 21 r. 22 s. 23 t. 24 u. 25

A25. What is the minimum inaudibility distance for hearing thunder from a sound source 7 km high in an environment of T = 20°C with no wind. Given a lapse rate (°C km–1) of:

 a. 9.8 b. 9 c. 8.5 d. 8 e. 7.5 f. 7 g. 6.5 h. 6 i. 5.5 j. 5 k. 4.5 l. 4 m. 3.5 n. 3 o. 2.5 p. 2 q. 1.5 r. 1 s. 0.5 t. 0 u. –1

A26. How low below ambient 100 kPa pressure must the core pressure of a tornado be, in order to support max tangential winds (m s–1) of:

 a. 20 b. 30 c. 40 d. 50 e. 60 f. 70 g. 80 h. 90 i. 100 j. 110 k. 120 l. 130 m. 140 n. 150

A27(§). For a Rankine Combined Vortex model of a tornado, plot the pressure (kPa) and tangential wind speed (m s–1) vs. radial distance (m) out to 125 m, for a tornado of core radius 25 m and core pressure deficit (kPa) 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. 1.1 l. 1.2 m. 1.3 n. 1.4 o. 1.5

A28. If the max tangential wind speed in a tornado is 100 m s–1, and the tornado translates at the speed (m s–1) given below, then what is the max wind speed (m s–1), and where is it relative to the center of the tornado and its track?

 a. 2 b. 4 c. 6 d. 8 e. 10 f. 12 g. 14 h. 16 i. 18 j. 20 k. 22 l. 24 m. 26 n. 28

A29. What are the Enhanced Fujita and TORRO intensity indices for a tornado of max wind speed (m s–1) of

 a. 20 b. 30 c. 40 d. 50 e. 60 f. 70 g. 80 h. 90 i. 100 j. 110 k. 120 l. 130 m. 140 n. 150

A30 Find the pressure (kPa) at the edge of the tornado condensation funnel, given an ambient near-surface pressure and temperature of 100 kPa and 35°C, and a dew point (°C) of:

 a. 30 b. 29 c. 28 d. 27 e. 26 f. 25 g. 24 h. 23 i. 22 j. 21 k. 20 l. 19 m. 18 n. 17

A31. For the winds of exercise A18 (a, b, c, or d) in the previous chapter, first find the storm movement for a

(1) normal supercell

(2) right-moving supercell

(3) left-moving supercell

Then graphically find and plot on a hodograph the storm-relative wind vectors.

A32. Same as previous exercise, except determine the (Us, Vs) components of storm motion, and then list the (Uj’, Vj’) components of storm-relative winds.

A33. A mesocyclone at 38°N is in an environment where the vertical stretching (∆W/∆z) is (20 m s–1) / (2 km). Find the rate of vorticity spin-up due to stretching only, given an initial relative vorticity (s–1) of

 a. 0.0002 b. 0.0004 c. 0.0006 d. 0.0008 e. 0.0010 f. 0.0012 g. 0.0014 h. 0.0016 i. 0.0018 j. 0.0020 k. 0.0022 l. 0.0024 m. 0.0026 n. 0.0028 o. 0.0030

A34. Given the hodograph of storm-relative winds in Fig. 15.40b. Assume that vertical velocity increases with height according to W = a·z, where a = (5 m s–1)/km. Considering only the tilting terms, find the vorticity spin-up based on the wind-vectors for the following pairs of heights (km):

 a. 0,1 b. 1,2 c. 2,3 d. 3,4 e. 4,5 f. 5,6 g. 0,2 h. 1,3 i. 2,4 j. 3,5 k. 4,6 l. 1,4 m. 2,5 n. 3,6 o. 1,5 p. 2,6 q. 1,6

A35. Same as the previous exercise but for the storm-relative winds in the hodograph of the Sample Application in the “Storm-relative Winds” subsection of the tornado section.

A36. Given the hodograph of winds in Fig. 15.40a. Assume W = 0 everywhere. Calculate the helicity H based on the wind-vectors for the following pairs of heights (km):

 a. 0,1 b. 1,2 c. 2,3 d. 3,4 e. 4,5 f. 5,6 g. 0,2 h. 1,3 i. 2,4 j. 3,5 k. 4,6 l. 1,4 m. 2,5 n. 3,6 o. 1,5 p. 2,6 q. 1,6

A37. Same as the previous exercise, but use the storm-relative winds from Fig. 15.40b to get the storm-relative helicity H’. (Hint, don’t sum over all heights for this exercise.)

A38. Given the hodograph of winds in Fig. 15.40a. Assume that vertical velocity increases with height according to W = a·z, where a = (5 m s–1)/km. Calculate the vertical contribution to helicity (eq. 15.54) based on the wind-vectors for the following pairs of heights (km):

 a. 0,1 b. 1,2 c. 2,3 d. 3,4 e. 4,5 f. 5,6 g. 0,2 h. 1,3 i. 2,4 j. 3,5 k. 4,6 l. 1,4 m. 2,5 n. 3,6 o. 1,5 p. 2,6 q. 1,6

A39. Use the storm-relative winds in the hodograph of the Sample Application in the “Storm-relative Winds” subsection of the tornado section. Calculate the total storm-relative helicity (SRH) graphically for the following height ranges (km):

 a. 0,1 b. 0,2 c. 0,3 d. 0,4 e. 0,5 f. 0,6 g. 1,2 h. 1,3 i. 1,4 j. 1,5 k. 1,6 l. 2,3 m. 2,4 n. 2,5 o. 2,6 p. 3,5 q. 3,6

A40. Same as the previous exercise, but find the answer using the equations (i.e., NOT graphically).

A41. Estimate the intensity of the supercell and tornado (if any), given a 0-1 km storm-relative helicity (m2 s–2) of:

 a. 20 b. 40 c. 60 d. 80 e. 100 f. 120 g. 140 h. 160 i. 180 j. 200 k. 220 l. 240 m. 260 n. 280 o. 300 p. 320 q. 340 r. 360 s. 380 t. 400

A42. Given a storm-relative helicity of 220 , find the energy-helicity index if the CAPE (J kg–1) is:

 a. 200 b. 400 c. 600 d. 800 e. 1000 f. 1200 g. 1400 h. 1600 i. 1800 j. 2000 k. 2200 l. 2400 m. 2600 n. 2800 o. 3000

A43. Estimate the likely supercell intensity and tornado intensity (if any), given an energy-helicity index value of:

 a. 0.2 b. 0.4 c. 0.6 d. 0.8 e. 1.0 f. 1.2 g. 1.4 h. 1.6 i. 1.8 j. 2.0 k. 2.2 l. 2.4 m. 2.6 n. 2.8 o. 3.0 p. 3.2 q. 3.4 r. 3.6 s. 3.8 t. 4.0

A44. If the tangential winds around a mesocyclone updraft are 20 m s–1, find the swirl ratio of the average updraft velocity (m s–1) is:

 a. 2 b. 4 c. 6 d. 8 e. 10 f. 12 g. 14 h. 16 i. 18 j. 20 k. 22 l. 24 m. 26 n. 28 o. 30 p. 32 q. 34 r. 36 s. 38 t. 40

A45. Given a mesocyclone with a tangential velocity of 20 m s–1 around the updraft region of radius 1000 m in a boundary layer 1 km thick. Find the swirl ratio and discuss tornado characteristics, given a radial velocity (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 l. 12 m. 13 n. 14 o. 15 p. 16 q. 17 r. 18 s. 19 t. 20

# 15.6.3. Evaluate & Analyze

E1. Why cannot hook echoes be used reliably to indicate the presence of a tornado?

E2. Cases of exceptionally heavy rain were discussed in the “Precipitation and Hail” section of this chapter and in the Precipitation chapter section on Rainfall Rates. However, most of those large rainfall rates occurred over exceptionally short durations (usually much less than an hour). Explain why longer-duration extreme-rainfall rates are unlikely.

E3. Use the info in Fig. 15.4 and the relationship between max likely updraft speed and CAPE, to plot a new graph of max possible hailstone diameter vs. total CAPE.

E4. In Fig. 15.7, what is the advantage to ignoring a portion of CAPE when estimating the likelihood of large hail? Explain.

E5. Explain why the various factors in the SHIP equation (15.4) are useful for predicting hail?

E6. Figures 15.5 and 15.10 show top and end views of the same thunderstorm, as might be seen with weather radar. Draw a side view (as viewed from the southeast by a weather radar) of the same thunderstorm. These 3 views give a blueprint (mechanical drawing) of a supercell.

E7. Cloud seeding (to change hail or rainfall) is a difficult social and legal issue. The reason is that even if you did reduce hail over your location by cloud seeding, an associated outcome might be increased hail or reduced rainfall further downwind. So solving one problem might create other problems. Discuss this issue in light of what you know about sensitive dependence of the atmosphere to initial conditions (the “butterfly effect”), and about the factors that link together the weather in different locations.

E8. a. Confirm that each term in eq. (15.5) has the same units.

b. Discuss how terms A and C differ, and what they each mean physically.

c. In term A, why is the numerator a function of ∆P’ rather than ∆P?

E9. a. If there were no drag of rain drops against air, could there still be downbursts of air?

b. What is the maximum vertical velocity of large falling rain drops relative to the ground, knowing that air can be dragged along with the drops as a downburst? (Hint: air drag depends on the velocity of the drops relative to the air, not relative to the ground.)

c. Will that maximum fall velocity relative to the ground be reached at the ground, or at some height well above ground? Why?

E10. A raindrop falling through unsaturated air will cool to a certain temperature because some of the drop evaporates. State the name of this temperature.

E11. Suppose that an altocumulus (mid-tropospheric) cloud exists within an environment having a linear, conditionally unstable, temperature profile with height. Rain-laden air descends from this little cloud, warming at the moist adiabatic rate as it descends. Because this warming rate is less than the conditionally unstable lapse rate of the environment, the temperature perturbation of the air relative to the environment becomes colder as it descends.

But at some point, all the rain has evaporated. Descent below this altitude continues because the air parcel is still colder than the environment. However, during this portion of descent, the air parcel warms dry adiabatically, and eventually reaches an altitude where its temperature equals that of the environment. At this point, its descent stops. Thus, there is a region of strong downburst that does NOT reach the ground. Namely, it can be a hazard to aircraft even if it is not detected by surface-based wind-shear sensors.

Draw this process on a thermo diagram, and show how the depth of the downburst depends on the amount of liquid water available to evaporate.

E12. Demonstrate that eq. (15.10) equates kinetic energy with potential energy. Also, what assumptions are implicit in this relationship?

E13. Eqs. (15.12) and (15.13) show how vertical velocities (wd) are tied to horizontal velocities (M) via pressure perturbations P’. Such coupling is generically called a circulation, and is the dynamic process that helps to maintain the continuity of air (namely, the uniform distribution of air molecules in space). Discuss how horizontal outflow winds are related to DCAPE.

E14. Draw a graph of gust-front advancement speed and thickness vs. range R from the downburst center. Do what-if experiments regarding how those curves change with

a. outflow air virtual temperature?

b. downburst speed?

E15. Fig. 15.21 shows large accumulations of electrical charge in thunderstorm clouds. Why don’t the positive and negative charge areas continually discharge against each other to prevent significant charge accumulation, instead of building up such large accumulations as can cause lightning?

E16. At the end of the INFO box about “Electricity in a Channel” is given an estimate of the energy dissipated by a lightning stroke. Compare this energy to:

a. The total latent heat available to the thunderstorm, given a typical inflow of moisture.

b. The total latent heat actually liberated based on the amount of rain falling out of a storm.

c. The kinetic energy associated with updrafts and downbursts and straight-line winds.

d. The CAPE.

E17. Look at both INFO boxes on electricity. Relate:

a. voltage to electrical field strength

b. resistance to resistivity

c. current to current density

d. power to current density & electrical potential.

E18. If the electrical charging process in thunderstorms depends on the presence of ice, then why is lightning most frequently observed in the tropics?

E19. a. Lightning of exactly 12 kA occurs with what probability?

b. Lightning current in the range of 8 to 12 kA occurs with what probability?

E20. How does the shape of the lightning surge curve change with changes of parameters $$\ \tau_1$$ and $$\ \tau_2$$?

E21. Show why eqs. (15.22) and (15.32) are equivalent ways to express the speed of sound, assuming no wind.

E22. Do you suspect that nuclear explosions behave more like chemical explosions or like lightning, regarding the resulting shock waves, pressure, and density? Why?

E23. The equations for shock wave propagation from lightning assumed an isopycnal processes. Critique this assumption.

E24. In Earth’s atmosphere, describe the conditions needed for the speed of sound to be zero relative to a coordinate system fixed to the ground. How likely are these conditions?

E25. What might control the max distance from lightning that you could hear thunder, if refraction was not an issue?

E26. Show how the expression of Snell’s law in an environment with gradually changing temperature (eq. 15.36) is equivalent to, or reduces to, Snell’s law across an interface (eq. 15.35).

E27. Show that eq. (15.39) for Snell’s Law reduces to eq. (15.34) in the limit of zero wind.

E28. Use Bernoulli’s equation from the Regional Winds chapter to derive the relationship between tornadic core pressure deficit and tangential wind speed. State all of your assumptions. What are the limitations of the result?

E29(§). Suppose that the actual tangential velocity in a tornado is described by a Rankine combined vortex (RCV). Doppler radars, however, cannot measure radial velocities at any point, but instead observe velocities averaged across the radar beam width. So the Doppler radar sees a smoothed version of the Rankine combined vortex. It is this smoothed tangential velocity shape that is called a tornado vortex signature (TVS), and for which the Doppler-radar computers are programmed to recognize. This exercise is to create the tangential velocity curve similar to Fig. 15.34, but for a TVS.

Let ∆D be the diameter of radar beam at some range from the radar, and Ro be the core radius of tornado. The actual values of ∆D and Ro are not important: instead consider the dimensionless ratio ∆D/Ro. Compute the TVS velocity at any distance R from the center of the tornado as the average of all the RCV velocities between radii of [(R/ Ro)–(∆D/2Ro)] and [(R/ Ro )+(∆D/2Ro)], and repeat this calculation for many values of R/Ro to get a curve. This process is called a running average. Create this curve for ∆D/Ro 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. 0.9 k. 0.8 l. 0.7 m. 0.6 n. 0.5

Hint: Either do this by analytically integrating the RCV across the radar beam width, or by brute-force averaging of RCV values computed using a spreadsheet.

E30(§). For tornadoes, an alternative approximation for tangential velocities Mtan as a function of radius R is given by the Burgers-Rott Vortex (BRV) equation:

\ \begin{align} \frac{M_{\tan }}{M_{\tan \max }}=1.398 \cdot\left(\frac{R_{o}}{R}\right) \cdot\left[1-e^{-\left(1.12 \cdot R / R_{o}\right)^{2}}\right]\tag{15.64}\end{align}

where Ro is the core radius.

Plot this curve, and on the same graph re-plot the Rankine combined vortex (RCV) curve (similar to Fig. 15.33). Discuss what physical processes in the tornado might be included in the BRV that are not in the RCV, to explain the differences between the two curves.

E31. For the Rankine combined vortex (RCV), both the tangential wind speed and the pressure deficit are forced to match at the boundary between the tornado core and the outer region. Does the pressure gradient also match at that point? If not, discuss any limitations that you might suggest on the RCV.

E32. Suppose a suction vortex with max tangential speed Ms tan is moving around a parent tornado of tangential speed Mp tan, and the parent tornado is translating at speed Mtr . Determine how the max speed varies with position along the resulting cycloidal damage path.

E33. The TORRO scale is related to the Beaufort wind scale (B) by:

\ \begin{align} \mathbf{B}=2 \cdot(\mathbf{T}+4)\tag{15.63}\end{align}

The Beaufort scale is discussed in detail in the Hurricane chapter, and is used to classify ocean storms and sea state. Create a graph of Beaufort scale vs. TORRO scale. Why cannot the Beaufort scale be used to classify tornadoes?

E34. Volcanic eruptions can create blasts of gas that knock down trees (as at Mt. St. Helens, WA, USA). The air burst from astronomical meteors speeding through the atmosphere can also knock down trees (as in Tunguska, Siberia). Explain how you could use the TORRO scale to classify these winds.

E35. Suppose that tangential winds around a tornado involve a balance between pressure-gradient force, centrifugal force, and Coriolis force. Show that anticyclonically rotating tornadoes would have faster tangential velocity than cyclonic tornadoes, given the same pressure gradient. Also, for anticyclonic tornadoes, are their any tangential velocity ranges that are excluded from the solution of the equations (i.e., are not physically possible)? Assume N. Hem.

E36. Does the outside edge of a tornado condensation funnel have to coincide with the location of fastest winds? If not, then is it possible for the debris cloud (formed in the region of strongest winds) radius to differ from the condensation funnel radius? Discuss.

E37. Given a fixed temperature and dew point, eq. (15.48) gives us the pressure at the outside edge of the condensation funnel.

a. Is it physically possible (knowing the governing equations) for the pressure deficit at the tornado axis to be higher than the pressure deficit at the visible condensation funnel? Why?

b. If the environmental temperature and dew point don’t change, can we infer that the central pressure deficit of a large-radius tornado is lower than that for a small-radius tornado? Why or why not?

E38. Gustnadoes and dust devils often look very similar, but are formed by completely different mechanisms. Compare and contrast the processes that create and enhance the vorticity in these vortices.

E39. From Fig. 15.40a, you see that the winds at 1 km above ground are coming from the southeast. Yet, if you were riding with the storm, the storm relative winds that you would feel at 1 km altitude would be from the northeast, as shown for the same data in Fig. 15.40b. Explain how this is possible; namely, explain how the storm has boundary layer inflow entering it from the northeast even though the actual wind direction is from the southeast.

E40. Consider Fig. 15.42.

a. What are the conceptual (theoretical) differences between streamwise vorticity, and the vorticity around a local-vertical axis as is usually studied in meteorology?

b. If it is the streamwise (horizontal axis) vorticity that is tilted to give vorticity about a vertical axis, why don’t we see horizontal-axis tornadoes forming along with the usual vertical tornadoes in thunderstorms? Explain.

E41. Compare eq. (15.51) with the full vorticity-tendency equation from the Extratropical Cyclone chapter, and discuss the differences. Are there any terms in the full vorticity equation that you feel should not have been left out of eq. (15.51)? Justify your arguments.

E42. Show that eq. (15.57) is equivalent to eq. (15.56). (Hint: use the average-wind definition given immediately after eq. (15.55).)

E43. Show mathematically that the area swept out by the storm-relative winds on a hodograph (such as the shaded area in Fig. 15.44) is indeed exactly half the storm-relative helicity. (Hint: Create a simple hodograph with a small number of wind vectors in easy-to-use directions, for which you can easily calculate the shaded areas between wind vectors. Then use inductive reasoning and generalize your approach to arbitrary wind vectors.)

E44. What are the advantages and disadvantages of eSRH and EHI relative to SRH?

E45. Suppose the swirl ratio is 1 for a tornado of radius 300 m in a boundary layer 1 km deep. Find the radial velocity and core pressure deficit for each tornado intensity of the

 a. Enhanced Fujita scale. b. TORRO scale.

E46. One physical interpretation of the denominator in the swirl ratio (eq. 15.60) is that it indicates the volume of inflow air (per crosswind distance) that reaches the tornado from outside. Provide a similar interpretation for the swirl-ratio numerator.

E47. The cycloidal damage sketched in Fig. 15.51 shows a pattern in between that of a true cycloid and a circle. Look up in another reference what the true cycloid shape is, and discuss what type of tornado behavior would cause damage paths of this shape.

# 15.6.4. Synthesize

S1. Since straight-line outflow winds exist surrounding downbursts that hit the surface, would you expect similar hazardous outflow winds where the updraft hits the tropopause? Justify your arguments.

S2. Suppose that precipitation loading in an air parcel caused the virtual temperature to increase, not decrease. How would thunderstorms differ, if at all?

S3. If hailstones were lighter than air, discuss how thunderstorms would differ, if at all.

S4. If all hailstones immediately split into two when they reach a diameter of 2 cm, describe how hail storms would differ, if at all.

S5. Are downbursts equally hazardous to both light-weight, small private aircraft and heavy fast commercial jets? Justify your arguments?

S6. Suppose that precipitation did not cause a downward drag on the air, and that evaporation of precipitation did not cool the air. Nonetheless, assume that thunderstorms have a heavy precipitation region. How would thunderstorms differ, if at all?

S7. Suppose that downbursts did not cause a pressure perturbation increase when they hit the ground. How would thunderstorm hazards differ, if at all?

S8. Suppose that downbursts sucked air out of thunderstorms. How would thunderstorms differ, if at all?

S9. Do you think that lightning could be productively utilized? If so, describe how.

S10. Suppose air was a much better conductor of electricity. How would thunderstorms differ, if at all?

S11. Suppose that once a lightning strike happened, the resulting plasma path that was created through air persists as a conducting path for 30 minutes. How would thunderstorms differ, if at all.

S12. Suppose that the intensity of shock waves from thunder did not diminish with increasing distance. How would thunderstorm hazards differ, if at all?

S13. Refraction of sound can make noisy objects sound quieter, and can amplify faint sounds by focusing them. For the latter case, consider what happens to sound waves traveling different paths as they all reach the same focus point. Describe what would happen there.

S14. The air outside of the core of a Rankine-combined-vortex (RCV) model of a tornado is moving around the tornado axis. Yet the flow is said to be irrotational in this region. Namely, at any point outside the core, the flow has no vorticity. Why is that? Hint: consider aspects of a flow that can contribute to relative vorticity (see the General Circulation chapter), and compare to characteristics of the RCV.

S15. What is the max tangential speed that a tornado could possibly have? What natural forces in the atmosphere could create such winds?

S16. Do anticyclonically rotating tornadoes have higher or lower core pressure than the surrounding environment? Explain the dynamics.

S17. Consider Fig. 15.36. If the horizontal pressure gradient near the bottom of tornadoes was weaker than that near the top, how would tornadoes be different, if at all?

S18. If a rapidly collapsing thunderstorm (nicknamed a bursticane, which creates violent downbursts and near-hurricane-force straight-line winds at the ground) has a rapidly sinking top, could it create a tornado above it due to stretching of the air above the collapsing thunderstorm? Justify your arguments.

S19. Positive helicity forms not only with updrafts and positive vorticity, but also with downdrafts and negative vorticity. Could the latter condition of positive helicity create mesocyclones and tornadoes? Explain.

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