# 10.12: Homework Exercises

Some of these questions are inspired by exercises in Stull, 2000: Meteorology for Sci. & Engr. 2nd Ed., Brooks/Cole, 528 pp.

# 10.12.1. Broaden Knowledge & Comprehension

For all the exercises in this section, collect information off the internet. Don’t forget to cite the web sites you use.

B1.

1. Find a weather map showing today’s sea-level pressure isobars near your location. Calculate pressure-gradient force (N) based on your latitude and the isobar spacing (km/kPa).
2. Repeat this for a few days, and plot the pressure gradient vs. time.

B2. Get 50 kPa height contour maps (i.e., 500 hPa heights) over any portion of the Northern Hemisphere. In 2 locations at different latitudes having straight isobars, compute the geostrophic wind speed. In 2 locations of curved isobars, compute the gradient wind speed. How do these theoretical winds compare with wind observations near the same locations?

B3. Similar exercise B2, but for 2 locations in the Southern Hemisphere.

B4.

1. Using your results from exercise B2 or B3, plot the geostrophic wind speed vs. latitude and pressure gradient on a copy of Fig. 10.10. Discuss the agreement or disagreement of your results vs. the lines plotted in that figure.
2. Using your results from exercise B4 or B5, show that gradient winds are indeed faster than geostrophic around high-pressure centers, and slower around low-pressure centers.

B5. Discuss surprising insights regarding Isaac Newton’s discoveries on forces and motion.

B6. Get a map of sea-level pressure, including isobar lines, for a location or date where there are strong low and high-pressure centers adjacent to each other. On a printed copy of this map, use a straight edge to draw a line connecting the low and high centers, and extend the line further beyond each center. Arbitrarily define the high center as location x = 0. Then, along your straight line, add distance tic marks appropriate for the map scale you are using. For isobars crossing your line, create a table that lists each pressure P and its distance x from the high. Then plot P vs. x and discuss how it compares with Fig. 10.14. Discuss the shape of your curve in the low- and high-pressure regions.

B7. Which animations best illustrate Coriolis Force?

B8.

1. Get a map of sea-level pressure isobars that also shows observed wind directions. Discuss why the observed winds have a direction that crosses the isobars, and calculate a typical crossing angle.
2. For regions where those isobars curve around cyclones or anticyclones, confirm that winds spiral into lows and out of highs.
3. For air spiraling in toward a cyclone, estimate the average inflow radial velocity component, and calculate WBL based on incompressible continuity.

B9. For a typhoon or hurricane, get a current or past weather map showing height-contours for any one isobaric level corresponding to an altitude about 1/3 the altitude of the storm (i.e., a map for any pressure level between 85 to 60 kPa). At the eye-wall location, use the height-gradient to calculate the cyclostrophic wind speed. Compare this with the observed hurricane winds at that same approximate location, and discuss any differences.

B10. Get a 500 hPa (= 50 kPa) geopotential height contour map that is near or over the equator. Compute the theoretical geostrophic wind speed based on the height gradients at 2 locations on that map where there are also observed upper-air wind speeds. Explain why these theoretical wind speeds disagree with observed winds.

# 10.12.2. Apply

A1. Plot the wind symbol for winds with the following directions and speeds:

 a. N at 5 kt b. NE at 35 kt c. E at 65 kt d. SE at 12 kt e. S at 48 kt f. SW at 105 kt g. W at 27 kt h. NW at 50 kt i. N at 125 kt

A2. How fast does an 80 kg person accelerate when pulled with the force given below in Newtons?

 a. 1 b. 2 c. 5 d. 10 e. 20 f. 50 g. 100 h. 200 i. 500 j. 1000 k. 2000

A3. Suppose the following force per mass is applied on an object. Find its speed 2 minutes after starting from rest.

 a. 5 N kg–1 b. 10 m·s–2 c. 15 N kg–1 d. 20 m·s–2 e. 25 N kg–1 f. 30 m·s–2 g. 35 N kg–1 h. 40 m·s–2 i. 45 N kg–1 j. 50 m·s–2

A4. Find the advective “force” per unit mass given the following wind components (m s–1) and horizontal distances (km):

1. U=10, ∆U=5, ∆x=3
2. U=6, ∆U=–10, ∆x=5
3. U=–8, ∆V=20, ∆x=10
4. U=–4, ∆V=10, ∆x=–2
5. V=3, ∆U=10, ∆y=10
6. V=–5, ∆U=10, ∆y=4
7. V=7, ∆V=–2, ∆y=–50
8. V=–9, ∆V=–10, ∆y=–6

A5. Town A is 500 km west of town B. The pressure at town A is given below, and the pressure at town B is 100.1 kPa. Calculate the pressure-gradient force/ mass in between these two towns.

 a. 98.6 b. 98.8 c. 99.0 d. 99.2 e. 99.4 f. 99.6 g. 99.8 h. 100.0 i. 100.2 j. 100.4 k. 100.6 l. 100.8 m. 101.0 n. 101.2 o. 101.4

A6. Suppose that U = 8 m s–1 and V = –3 m s–1, and latitude = 45° Calculate centrifugal-force components around a:

1. 500 km radius low in the N. hemisphere
2. 900 km radius high in the N. hemisphere
3. 400 km radius low in the S. hemisphere
4. 500 km radius high in the S. hemisphere

A7. What is the value of fc (Coriolis parameter) at:

1. Shanghai
2. Istanbul
3. Karachi
4. Mumbai
5. Moscow
6. Beijing
7. São Paulo
8. Tianjin
9. Guangzhou
10. Delhi
11. Seoul
12. Shenzhen
13. Jakarta
14. Tokyo
15. Mexico City
16. Kinshasa
17. Bangalore
18. New York City
19. Tehran
20. (a city specified by your instructor)

A8. What is the magnitude and direction of Coriolis force/mass in Los Angeles, USA, given:

 U (m s–1 ) V (m s–1) a. 5 0 b. 5 5 c. 5 –5 d. 0 5 e. 0 –5 f. –5 0 g. –5 –5 h. –5 5

A9. Same wind components as exercise A8, but find the magnitude and direction of turbulent drag force/ mass in a statically neutral atmospheric boundary layer over an extensive forested region.

A10. Same wind components as exercise A8, but find the magnitude and direction of turbulent drag force/ mass in a statically unstable atmospheric boundary layer with a 50 m/s buoyant velocity scale.

A11. Draw a northwest wind of 5 m s–1 in the S. Hemisphere on a graph, and show the directions of forces acting on it. Assume it is in the boundary layer.

 a. pressure gradient b. Coriolis c. centrifugal d. drag

A12. Given the pressure gradient magnitude (kPa/1000 km) below, find geostrophic wind speed for a location having fc = 1.1x10–4 s–1 and ρ = 0.8 kg m–3.

 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

A13. Suppose the height gradient on an isobaric surface is given below in units of (m km–1). Calculate the geostrophic wind at 55°N latitude.

 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 m. 1.2 n. 1.3 o. 1.4 p. 1.5

A14. At the radius (km) given below from a lowpressure center, find the gradient wind speed given a geostrophic wind of 8 m s–1 and given fc = 1.1x10–4 s–1.

 a. 500 b. 600 c. 700 d. 800 e. 900 f. 1000 g. 1200 h. 1500 i. 2000 j. 2500

A15. Suppose the geostrophic winds are Ug = –3 m s–1 with Vg = 8 m s–1 for a statically-neutral boundary layer of depth zi = 1500 m, where fc = 1.1x10–4 s–1. For drag coefficients given below, what is the atmos. boundary-layer wind speed, and at what angle does this wind cross the geostrophic wind vector?

 a. 0.002 b. 0.004 c. 0.006 d. 0.008 e. 0.010 f. 0.012 g. 0.014 h. 0.016 i. 0.018 j. 0.019

A16. For a statically unstable atmos. boundary layer with other characteristics similar to those in exercise A15, what is the atmos. boundary-layer wind speed, at what angle does this wind cross the geostrophic wind vector, given wB (m s–1) below?

 a. 75 b. 100 c. 50 d. 200 e. 150 f. 225 g. 125 h. 250 i. 175 j. 275

A17(§). Review the Sample Application in the “Atmospheric Boundary Layer Gradient Wind” section. Re-do that calculation for MABLG with a different parameter as given below:

 a. zi = 1 km b. CD = 0.003 c. G = 8 m s–1 d. fc = 1.2x10–4 s–1 e. R = 2000 km f. G = 15 m s–1 g. zi = 1.5 km h. CD = 0.005 i. R = 1500 km j. fc = 1.5x10–4 s–1

Hint: Assume all other parameters are unchanged.

A18. Find the cyclostrophic wind at radius (m) given below, for a radial pressure gradient = 0.5 kPa m–1:

 a. 10 b. 12 c. 14 d. 16 e. 18 f. 20 g. 22 h. 24 i. 26 j. 28 k. 30

A19. For an inertial wind, find the radius of curvature (km) and the time period (h) needed to complete one circuit, given fc = 10–4 s–1 and an initial wind speed (m s–1) of:

 a. 1 b. 2 c. 3 d. 4 e. 6 f. 7 g. 8 h. 9 i. 10 j. 11 k. 12 m. 13 n. 14 o. 15

A20. Find the antitriptic wind for the conditions of exercise A15.

A21. Below is given an average inward radial wind component (m s–1) in the atm. boundary layer at radius 300 km from the center of a cyclone. What is the average updraft speed out of the atm. boundarylayer top, for a boundary layer that is 1.2 km thick?

 a. 2 b. 1.5 c. 1.2 d. 1.0 e. –0.5 f. –1 g. –2.5 h. 3 i. 0.8 j. 0.2

A22. Above an atmospheric boundary layer, assume the tropospheric temperature profile is ∆T/∆z = 0. For a midlatitude cyclone, estimate the atm. boundary-layer thickness given a near-surface geostrophic wind speed (m s–1) of:

 a. 5 b. 10 c. 15 d. 20 e. 25 f. 30 g. 35 h. 40 i. 3 j. 8 k. 2 l. 1

A23(§). For atm. boundary-layer pumping, plot a graph of updraft velocity vs. geostrophic wind speed assuming an atm. boundary layer of depth 0.8 km, a drag coefficient 0.005 . Do this only for wind speeds within the valid range for the atmos. boundary-layer pumping eq. Given a standard atmospheric lapse rate at 30° latitude with radius of curvature (km) of:

 a. 750 b. 1500 c. 2500 d. 3500 e. 4500 f. 900 g. 1200 h. 2000 i. 3750 j. 5000

A24. At 55°N, suppose the troposphere is 10 km thick, and has a 10 m s–1 geostrophic wind speed. Find the internal Rossby deformation radius for an atmospheric boundary layer of thickness (km):

 a. 0.2 b. 0.4 c. 0.6 d. 0.8 e. 1.0 f. 1.2 g. 1.5 h. 1.75 i. 2.0 j. 2.5

A25. Given ∆U/∆x = ∆V/∆x = (5 m s–1) / (500 km), find the divergence, vorticity, and total deformation for (∆U/∆y , ∆V/∆y) in units of (m s–1)/(500 km) as given below:

 a. (–5, –5) b. (–5, 0) c. (0, –5) d. (0, 0) e. (0, 5) f. (5, 0) g. (5, 5) h. (–5, 5) i. (5, –5)

# 10.12.3. Evaluate & Analyze

E1. Discuss the relationship between eqs. (1.24) and (10.1).

E2. Suppose that the initial winds are unknown. Can a forecast still be made using eqs. (10.6)? Explain your reasoning.

E3. Considering eq. (10.7), suppose there are no forces acting. Based on eq. (10.5), what can you anticipate about the wind speed.

E4. We know that winds can advect temperature and humidity, but how does it work when winds advect winds? Hint, consider eqs. (10.8).

E5. For an Eulerian system, advection describes the influence of air that is blown into a fixed volume. If that is true, then explain why the advection terms in eq. (10.8) is a function of the wind gradient (e.g., ∆U/∆x) instead of just the upwind value?

E6. Isobar packing refers to how close the isobars are, when plotted on a weather map such as Fig. 10.5. Explain why such packing is proportional to the pressure gradient.

E7. Pressure gradient has a direction. It points toward low pressure for the Northern Hemisphere. For the Southern Hemisphere, does it point toward high pressure? Why?

E8. To help you interpret Fig. 10.5, consider each horizontal component of the pressure gradient. For an arbitrary direction of isobars, use eqs. (10.9) to demonstrate that the vector sum of the components of pressure-gradient do indeed point away from high pressure, and that the net direction is perpendicular to the direction of the isobars.

E9. For centrifugal force, combine eqs. (10.13) to show that the net force points outward, perpendicular to the direction of the curved flow. Also show that the magnitude of that net vector is a function of tangential velocity squared.

E10. Why does fc = 0 at the equator for an air parcel that is stationary with respect to the Earth’s surface, even though that air parcel has a large tangential velocity associated with the rotation of the Earth?

E11. Verify that the net Coriolis force is perpendicular to the wind direction (and to its right in the N. Hemisphere), given the individual components described by eqs. (10.17).

E12. For the subset of eqs. (10.1 - 10.17) defined by your instructor, rewrite them for flow in the Southern Hemisphere.

E13. Verify that the net drag force opposes the wind by utilizing the drag components of eqs. (10.19). Also, confirm that drag-force magnitude for statically neutral conditions is a function of wind-speed squared.

E14. How does the magnitude of the turbulenttransport velocity vary with static stability, such as between statically unstable (convective) and statically neutral (windy) situations?

E15. Show how the geostrophic wind components can be combined to relate geostrophic wind speed to pressure-gradient magnitude, and to relate geostrophic wind direction to pressure-gradient direction.

E16. How would eqs. (10.26) for geostrophic wind be different in the Southern Hemisphere?

E17. Using eqs. (10.26) as a starting point, show your derivation for eqs. (10.29).

E18. Why are actual winds finite near the equator even though the geostrophic wind is infinite there? (Hint, consider Fig. 10.10).

E19. Plug eq. (10.33) back into eqs. (10.31) to confirm that the solution is valid.

E20. Plug eqs. (10.34) back into eq. (10.33) to confirm that the solution is valid.

E21. Given the pressure variation shown in Fig. 10.14. Create a mean-sea-level pressure weather map with isobars around high- and low-pressure centers such that the isobar packing matches the pressure gradient in that figure.

E22. Fig. 10.14 suggests that any pressure gradient is theoretically possible adjacent to a low-pressure center, from which we can further infer that any wind speed is theoretically possible. For the real atmosphere, what might limit the pressure gradient and the wind speed around a low-pressure center?

E23. Given the geopotential heights in Fig. 10.3, calculate the theoretical values for gradient and/or geostrophic wind at a few locations. How do the actual winds compare with these theoretical values?

E24. Eq. 10.39 is an “implicit” solution. Why do we say it is “implicit”?

E25. Determine the accuracy of explicit eqs. (10.41) by comparing their approximate solutions for ABL wind against the more exact iterative solutions to the implicit form in eq. (10.39).

E26. No explicit solution exists for the neutral atmospheric boundary layer winds, but one exists for the statically unstable ABL? Why is that?

E27. Plug eqs. (10.42) into eqs (10.38) or (10.39) to confirm that the solution is valid.

E28(§).

1. Create your own spreadsheet that gives the same answer for ABLG winds as in the Sample Application in the ABLG-wind section.
2. Do “what if” experiments with your spreadsheet to show that the full equation can give the gradient wind, geostrophic wind, and boundary-layer wind for conditions that are valid for those situations.
3. Compare the results from (b) against the respective analytical solutions (which you must compute yourself).
4. Photocopy Fig. 10.13, and enhance the copy by drawing additional vectors for the atmospheric boundary-layer wind and the ABLG wind. Make these vectors be the appropriate length and direction relative to the geostrophic and gradient winds that are already plotted.

E30. Plug the cyclostrophic-wind equation into eq. (10.45) to confirm that the solution is valid for its special case.

E31. Find an equation for cyclostrophic wind based on heights on an isobaric surface. [Hint: Consider eqs. (10.26) and (10.29).]

E32. What aspects of the Approach to Geostrophy INFO Box are relevant to the inertial wind? Discuss.

E33.

1. Do your own derivation for eq. (10.66) based on geometry and mass continuity (total inflow = total outflow).
2. Drag normally slows winds. Then why does the updraft velocity increase in eq. (10.66) as drag coefficient increases?
3. Factor b varies negatively with increasing drag coefficient in eq. (10.66). Based on this, would you change your argument for part (b) above?

E34. Look at each term within eq. (10.69) to justify the physical interpretations presented after that equation.

E35. Consider eq. (10.70). For the internal Rossby deformation radius, discuss its physical interpretation in light of eq. (10.71).

E36. What type of wind would be possible if the only forces were turbulent-drag and Coriolis. Discuss

E37. Derive equations for Ekman pumping around anticyclones. Physically interpret your resulting equations.

E38. Rewrite the total deformation as a function of divergence and vorticity. Discuss.

# 10.12.4. Synthesize

S1. For zonal (east-west) winds, there is also a vertical component of Coriolis force. Using your own diagrams similar to those in the INFO box on Coriolis Force, show why it can form. Estimate its magnitude, and compare the magnitude of this force to other typical forces in the vertical. Show why a vertical component of Coriolis force does not exist for meridional (north-south) winds.

S2. On Planet Cockeyed, turbulent drag acts at right angles to the wind direction. Would there be anything different about winds near lows and highs on Cockeyed compared to Earth?

S3. The time duration of many weather phenomena are related to their spatial scales, as shown by eq. (10.53) and Fig. 10.24. Why do most weather phenomena lie near the same diagonal line on a log-log plot? Why are there not additional phenomena that fill out the relatively empty upper and lower triangles in the figure? Can the distribution of time and space scales in Fig. 10.24 be used to some advantage?

S4. What if atmospheric boundary-layer drag were constant (i.e., not a function of wind speed). Describe the resulting climate and weather.

S5. Suppose Coriolis force didn’t exist. Describe the resulting climate and weather.

S6. Incompressibility seems like an extreme simplification, yet it works fairly well? Why? Consider what happens in the atmosphere in response to small changes in density.

S7. The real Earth has locations where Coriolis force is zero. Where are those locations, and what does the wind do there?

S8. Suppose that wind speed M = c·F/m, where c = a constant, m = mass, and F is force. Describe the resulting climate and weather.

S9. What if Earth’s axis of rotation was pointing directly to the sun. Describe the resulting climate and weather.

S10. What if there was no limit to the strength of pressure gradients in highs. Describe the resulting climate, winds and weather.

S11. What if both the ground and the tropopause were rigid surfaces against which winds experience turbulent drag. Describe the resulting climate and weather.

S12. If the Earth rotated half as fast as it currently does, describe the resulting climate and weather.

S13. If the Earth had no rotation about its axis, describe the resulting climate and weather.

S14. Consider the Coriolis-force INFO box. Create an equation for Coriolis-force magnitude for winds that move:

 a. westward b. southward

S15. What if a cyclostrophic-like wind also felt drag near the ground? This describes conditions at the bottom of tornadoes. Write the equations of motion for this situation, and solve them for the tangential and radial wind components. Check that your results are reasonable compared with the pure cyclostrophic winds. How would the resulting winds affect the total circulation in a tornado? As discoverer of these winds, name them after yourself.

S16. What if F = c·a, where c = a constant not equal to mass, a = acceleration, and F is force. Describe the resulting dynamics of objects such as air parcels.

S17. What if pressure-gradient force acted parallel to isobars. Would there be anything different about our climate, winds, and weather maps?

S18. For a free-slip Earth surface (no drag), describe the resulting climate and weather.

S19. Anders Persson discussed issues related to Coriolis force and how we understand it (see Weather, 2000.) Based on your interpretation of his paper, can Coriolis force alter kinetic energy and momentum of air parcels, even though it is only an apparent force? Hint, consider whether Newton’s laws would be violated if your view these motions and forces from a fixed (non-rotating) framework.

S20. If the Earth was a flat disk spinning about the same axis as our real Earth, describe the resulting climate and weather.

S21. Wind shear often creates turbulence, and turbulence mixes air, thereby reducing wind shear. Considering the shear at the ABL top in Fig. 10.7, why can it exist without mixing itself out?

S22. Suppose there was not centrifugal or centripetal force for winds blowing around lows or highs. Describe the resulting climate, winds and weather.

S23. Suppose advection of the wind by the wind were impossible. Describe the resulting climate and weather.