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11.15: Homework Exercises

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  • 11.15.1. Broaden Knowledge & Comprehension

    B1. From hemispheric weather maps of winds near the tropopause (which you can access via the internet), identify locations of major global-circulation features including the jet stream, monsoon circulations, tropical cyclones and the ITCZ. 

    B2. Same as the previous exercise, except using water-vapor or infrared image loops from geostationary satellites to locate the features. 

    B3. From the web, find a rawinsonde sounding at a location in the trade-wind region, and confirm the wind reversal between low and high altitudes.

    B4. Use a visible, whole-disk image from a geostationary satellite to view and quantify the cloud-cover fraction as a function of latitude. Speculate on how insolation at the Earth’s surface is affected.

    B5. Download a series of rawinsonde soundings for different latitudes between the equator and a pole. Find the tropopause from each sounding, and then plot the variation of tropopause height vs. latitude.

    B6. Download a map of sea-surface temperature (SST), and discuss how SST varies with latitude.

    B7. Most satellite images in the infrared show greys or colors that are related to brightness temperature (see the legend in whole-disk IR images that you acquired from the internet). Use these temperatures as a function of latitude to estimate the corresponding meridional variation of IR-radiation out. Hint, consider the Stefan-Boltzmann law.

    B8. Download satellite-derived images that show the climatological average incoming and outgoing radiation at the top of the atmosphere. How does it relate to the idealized descriptions in this chapter?

    B9. Download satellite-derived or buoy & ship-derived ocean currents for the global oceans, and discuss how they transport heat meridionally, and why the oceanic transport of heat is relatively small at mid to high latitudes in the N. Hemisphere.

    B10. Use a satellite image to locate a strong portion of the ITCZ over a rawinsonde site, and then download the rawinsonde data. Plot (compute if needed) the variation of pressure with altitude, and discuss how it does or doesn’t deviate from hydrostatic.

    B11. Search the web for sites where you can plot “reanalysis data”, such as the NCEP/NCAR reanalysis or any of the ECMWF reanalyses. Pick a month during late summer from some past year in this database, and plot the surface pressure map. Explain how this “real” result relates to a combination of the “idealized” planetary and monsoonal circulations.

    B12. Same as B11, but for monthly average vertical cross sections that can be looped as movies. Display fields such as zonal wind, meridional wind, and vertical velocity, and see how they vary over a year.

    B13. Capture a current map showing 85 kPa temperatures, and assume that those temperatures are surrogates for the actual average virtual temperature between 100 and 70 kPa. Compute the thermal wind magnitude and direction for a location assigned by your teacher, and see if this theoretical relationship successfully explains the wind shear between 100 and 70 kPa. Justify your reasoning.

    B14 Capture a current map showing the thickness between 100 and 50 kPa, and estimate the thermal wind direction and magnitude across that layer.

    B15. Use rawinsonde soundings from stations that cross the jet stream. Create your own contour plots of the jet-stream cross section for (a) heights of key isobaric surfaces; (b) potential temperature; and (c) wind magnitude. Compare your plots with idealized sketches presented in this chapter.

    B16. What are the vertical and horizontal dimensions of the jet stream, based on weather maps you acquire from the internet.

    B17. Acquire a 50 kPa vorticity chart, and determine if the plotted vorticity is isentropic, absolute, relative, or potential. Where are positive-vorticity maxima relative to fronts and foul weather?

    B18. Calculate the values for the four types of vorticity at a location identified by your instructor, based on data for winds and temperatures. Namely, acquire the raw data used for vorticity calculations; do not use vorticity maps captured from the web. 

    B19. For the 20 kPa geopotential heights, use the wavy pattern of height contours and their relative packing to identify ridges and troughs in the jet stream. Between two troughs, or between two ridges, estimate the wavelength of the Rossby wave. Use that measured length as if it were the dominant wavelength to estimate the phase speed for baroclinic and barotropic waves. 

    B20. Confirm that the theoretical relationship between horizontal winds, temperatures, vertical velocities, and heights for baroclinic waves is consistent with the corresponding weather maps you acquire from the internet. Explain any discrepancies.

    B21. Confirm the three-band nature of the global circulation using IR satellite image movie loops. In the tropics, compare the motion of low (warm) and high (cold) clouds, and relate this motion to the trade winds and Hadley circulation. In mid-latitudes, find the regions of meandering jet stream with its corresponding high and low-pressure centers. In polar regions, relate cloud motions to the polar cell.

    B22. Are the ocean-surface current directions consistent with near-surface wind directions as observed in maps or animations acquired from the internet, given the dynamics describe for the Ekman spiral?

    11.15.2. Apply

    A1(§). For the “toy” model, make a graph of zonally-averaged temperature (°C) vs. latitude for the altitude (km) above ground level (AGL) given here:

    a. 0.5 b. 1 c. 1.5 d. 2 e. 2.5 f. 3 g. 3.5
    h. 4 i. 4.5 j. 5 k. 5.5 l. 6 m. 6.6 n. 7
    o. 8 p. 9 q. 10 r. 11 s. 12 t. 13 u. 14

    A2(§). For the “toy” model, make a graph of zonallyaveraged ∆T/∆y (°C km–1) vs. latitude for the altitude (km AGL) given here:

    a. 0.5 b. 1  c. 1.5 d. 2 e. 2.5 f. 3 g. 3.5
    h. 4 i. 4.5 j. 5 k. 5.5 l. 6 m. 6.6 n. 7
    o. 8 p. 9 q. 10 r. 11 s. 12 t. 13 u. 14

    A3. Estimate the annual average insolation (W m–2) at the following latitude:

    a. 90° b. 85° c. 80° d. 75° e. 70° f. 65° g. 60°
    h. 55° i. 50° j. 45° k. 40° l. 35° m. 30° n. 25°
    o. 20° p. 15° q. 10° r. 5° s. equator    

    A4. Estimate the annual average amount of incoming solar radiation (W m–2) that is absorbed in the Earth-ocean-atmosphere system at latitude:

    a. 90° b. 85° c. 80° d. 75° e. 70° f. 65° g. 60°
    h. 55° i. 50° j. 45° k. 40° l. 35° m. 30° n. 25°
    o. 20° p. 15° q. 10° r. 5° s. equator    

    A5. Using the idealized temperature near the middle of the troposphere (at z = 5.5 km), estimate the outgoing infrared radiation (W m–2) from the atmosphere at the following latitude:

    a. 90° b. 85° c. 80° d. 75° e. 70° f. 65° g. 60°
    h. 55° i. 50° j. 45° k. 40° l. 35° m. 30° n. 25°
    o. 20° p. 15° q. 10° r. 5° s. equator    

    A6. Using the results from the previous two exercises, find the net radiation magnitude (W m–2) that is input to the atmosphere at latitude:

    a. 90° b. 85° c. 80° d. 75° e. 70° f. 65° g. 60°
    h. 55° i. 50° j. 45° k. 40° l. 35° m. 30° n. 25°
    o. 20° p. 15° q. 10° r. 5° s. equator    

    A7. Using the results from the previous exercise, find the latitude-compensated net radiation magnitude (W m–2; i.e., the differential heating) at latitude:

    a. 90° b. 85° c. 80° d. 75° e. 70° f. 65° g. 60°
    h. 55° i. 50° j. 45° k. 40° l. 35° m. 30° n. 25°
    o. 20° p. 15° q. 10° r. 5° s. equator    

    A8. Assuming a standard atmosphere, find the internal Rossby deformation radius (km) at latitude:

    a. 90° b. 85° c. 80° d. 75° e. 70° f. 65° g. 60°
    h. 55° i. 50° j. 45° k. 40° l. 35° m. 30° n. 25°
    o. 20° p. 15° q. 10° r. 5° s. equator    

    A9. Given the following virtual temperatures at your location (20°C) and at another location, find the change of geostrophic wind with height [(m s–1)/ km]. Relative to your location, the other locations are:

    ∆x (km) ∆y(km) Tv(°C) | ∆x (km) ∆y(km) Tv(°C)
    a. 0 100 15 | k. 100 0 15
    b. 0 100 16 | l. 100 0 16
    c. 0 100 17 | m. 100 0 17
    d. 0 100 18 | n. 100 0 18
    e. 0 100 19 | o. 100 0 19
    f. 0 100 21 | p. 100 0 21
    g. 0 100 22 | q. 100 0 22
    h. 0 100 23 | r. 100 0 23
    i. 0 100 24 | s. 100 0 24
    j. 0 100 25 | t. 100 0 25

    A10. Find the thermal wind (m s–1) components, given a 100 to 50 kPa thickness change of 0.1 km across the following distances:

    ∆x(km) =  a. 200 b. 250 c. 300 d. 350 e. 400 f. 450 g. 550 h. 600 i. 650
    ∆y(km) =  j. 200 k. 250 l. 300 m. 350 n. 400 o. 450 p. 550 q. 600 r. 650

    A11. Find the magnitude of the thermal wind (m s–1) for the following thickness gradients:

    ∆TH(km) / ∆x(km) & ∆TH(km) / ∆y(km)
    a. –0.2 / 600 and – 0.1 / 400
    b. –0.2 / 400 and – 0.1 / 400
    c. –0.2 / 600 and + 0.1 / 400
    d. –0.2 / 400 and + 0.1 / 400
    e. –0.2 / 600 and – 0.1 / 400
    f. –0.2 / 400  and – 0.1 / 400
    g. –0.2 / 600 and + 0.1 / 400
    h. –0.2 / 400 and + 0.1 / 400

    A12. For the toy model temperature distribution, find the wind speed (m s–1) of the jet stream at the following heights (km) for latitude 30°:

    a. 0.5 b. 1 c. 1.5 d. 2 e. 2.5 f. 3 g. 3.5
    h. 4 i. 4.5 j. 5 k. 5.5 l. 6 m. 6.6 n. 7
    o. 8 p. 9 q. 10 r. 11 s. 12 t. 13 u. 14

    A13. If an air parcel from the starting latitude 5° has zero initial velocity relative to the Earth, then find its U component of velocity (m s–1) relative to the Earth when it reaches the following latitude, assuming conservation of angular momentum.

    a. 0° b. 2° c. 4° d. 6° e. 8° f. 10° g. 12°
    h. 14° i. 16° j. 18° k. 20° l. 22° m. 24° n. 26°

    A14. Find the relative vorticity (s–1) for the change of (U , V) wind speed (m s–1), across distances of ∆x = 300 km and ∆y = 600 km respectively given below.

    a. 50, 50 b. 50, 20 c. 50, 0  d. 50, –20 e. 50, –50
    f. 20, 50 g. 20, 20 h. 20, 0 i. 20, –20 j. 20, –50
    k. 0, 50 l. 0, 20 m. 0, 0 n. 0, –20 o. 0, –50
    p. –20, 50 q. –20, 20 r. –20, 0 s. –20, –20 t. –20, –50
    u. –50, 50 v. –50, 20 x. –50, 0 y. –50, –20 z. –50, –50

    A15. Given below a radial shear (∆M/∆R) in [(m s–1)/ km] and tangential wind speed M (m s–1) around radius R (km), find relative vorticity (s–1):

    a. 0.1, 30, 300 b. 0.1, 20, 300 c. 0.1, 10, 300 d. 0.1, 0, 300
    e. 0, 30, 300 f. 0, 20, 300 g. 0, 10, 300 h. –0.1, 30, 300
    i. –0.1, 20, 300 j. –0.1, 10, 300 k. –0.1, 0, 300  

    A16. If the air rotates as a solid body of radius 500 km, find the relative vorticity (s–1) for tangential speeds (m s–1) of:

    a. 10 b. 20 c. 30 d. 40 e. 50 f. 60 g. 70
    h. 80 i. 90 j. 100 k. 120 l. 140 m. 150  

    A17. If the relative vorticity is 5x10–5 s–1, find the absolute vorticity at the following latitude:

    a. 90° b. 85° c. 80° d. 75° e. 70° f. 65° g. 60°
    h. 55° i. 50° j. 45° k. 40° l. 35° m. 30° n. 25°
    o. 20° p. 15° q. 10° r. 5° s. equator    

    A18. If absolute vorticity is 5x10–5 s–1, find the potential vorticity (m–1·s–1) for a layer of thickness (km) of:

    a. 0.5 b. 1 c. 1.5 d. 2 e. 2.5 f. 3 g. 3.5
    h. 4 i. 4.5 j. 5 k. 5.5 l. 6 m. 6.6 n. 7
    o. 8 p. 9 q. 10 r. 11 s. 12 t. 13 u. 14

    A19. The potential vorticity is 1x10–8 m–1·s–1 for a 10 km thick layer of air at latitude 48°N. What is the change of relative vorticity (s–1) if the thickness (km) of the rotating air changes to:

    a. 9.5 b. 9 c. 8.5 d. 8 e. 7.5 f. 7 g. 6.5
    h. 10.5 j. 11 k. 11.5 l. 12 m. 12.5 n. 13  

    A20. If the absolute vorticity is 3x10–5 s–1 at 12 km altitude, find the isentropic potential vorticity (PVU) for a potential temperature change of ___ °C across a height increase of 1 km.

    a. 1 b. 2 c. 3 d. 4 e. 5 f. 6 g. 6.5
    h. 7 i. 8 j. 9 k. 10 l. 11 m. 12 n. 13

    A21. Find the horizontal circulation associated with average relative vorticity 5x10–5 s–1 over area (km2):

    a. 500 b. 1000 c. 2000 d. 5000 e. 10,000
    f. 20,000 g. 50,000 h. 100,000 i. 200,000  

    A22. For the latitude given below, what is the value of the beta parameter (m–1 s–1):

    a. 90° b. 85° c. 80° d. 75° e. 70° f. 65° g. 60°
    h. 55° i. 50° j. 45° k. 40° l. 35° m. 30° n. 25°
    o. 20° p. 15° q. 10° r. 5° s. equator    

    A23. Suppose the average wind speed is 60 m s–1 from the west at the tropopause. For a barotropic Rossby wave at 50° latitude, find both the intrinsic phase speed (m s–1) and the phase speed (m s–1) relative to the ground for wavelength (km) of:

    a. 1000 b. 1500 c. 2000 d. 2500 e. 3000
    f. 3500 g. 4000 h. 4500 i. 5000  j. 5500
    k. 6000 l. 6500 m. 7000 n. 7500 o. 8000

    A24. Plot the barotropic wave (y’ vs x’) from the previous exercise, assuming amplitude 2000 km.

    A25. Same as exercise A23, but for a baroclinic Rossby wave in an atmosphere where air temperature decreases with height at 4°C km–1.

    A26(§). Plot the baroclinic wave (y’ vs x’) from the previous exercise, assuming amplitude 2000 km and a height (km):

    (i) 2 (ii) 4 (iii) 6 (iv) 8 (v) 10

    A27. What is the fastest growing wavelength (km) for a baroclinic wave in a standard atmosphere at latitude:

    a. 90° b. 85° c. 80° d. 75° e. 70° f. 65° g. 60°
    h. 55° i. 50° j. 45° k. 40° l. 35° m. 30° n. 25°
    o. 20° p. 15° q. 10° r. 5° s. equator    

    A28. For the baroclinic Rossby wave of exercise A25 with amplitude 2000 km, find the wave amplitudes of the:

    1. vertical-displacement perturbation
    2. potential-temperature perturbation
    3. pressure perturbation
    4. U-wind perturbation
    5. V-wind perturbation
    6. W-wind perturbation

    A29(§). For a vertical slice through the atmosphere, plot baroclinic Rossby-wave perturbation amount for conditions assigned in exercise A28.

    A30. Find the latitude-weighted a·u’ momentum value (m s–1) for air that reaches destination latitude 50° from source latitude:

    a. 80° b. 75° c. 70° d. 65° e. 60° f. 55° g. 45° h. 40° i. 35°

    A31. Suppose the ____ cell upward and downward speeds are ___ and ___ mm s–1, respectively, and the north-south wind speeds are 3 m s–1 at the top and bottom of the cell. The cell is about __ km high by ___ km wide, and is centered at about ___ latitude. Temperature in the atmosphere decreases from about 15°C near the surface to –57°C at 11 km altitude. Find the vertical circulation.

    cell  Wup (mm s–1) Wdown (mm s–1) ∆z (km) ∆y (km) ϕ (°)
    a. Hadley 6 –4 17 3900 10
    b. Hadley 4 –4 15 3500 10
    c. Hadley 3 –3 15 3500 5
    d. Ferrel 3 –3 12 3000 45
    e. Ferrel 2 –2 11 3000 45
    f. Ferrel 2 –2 10 3000 50
    g. polar 1 –1 9 2500 75
    h. polar 1 –1 8 2500 75
    i. polar 0.5 –0.5 7 2500 80

    A32. Find the friction velocity at the water surface if the friction velocity (m s–1) in the air (at sea level for a standard atmosphere) is:

    a. 0.05 b. 0.1 c. 0.15 d. 0.2 e. 0.25
    f. 0.3 g. 0.35 h. 0.4 i. 0.45 j. 0.5
    k. 0.55 l. 0.6 m. 0.65 n. 0.7 o. 0.75

    A33. Find the Ekman-spiral depth scale at latitude 50°N for eddy viscosity (m2 s–1) of:

    a. 0.0002 b. 0.0004 c. 0.0006 d. 0.0008 e. 0.001
    f. 0.0012 g. 0.0014 h. 0.0016 i. 0.0018 j. 0.002
    k. 0.0025 l. 0.003 m. 0.0035 n. 0.004 o. 0.005

    A34(§). Create a graph of Ekman-spiral wind components (U, V) components for depths from the surface down to where the velocities are near zero, for near-surface wind speed of 8 m s–1 at 40°N latitude. 

    11.15.3. Evaluate & Analyze

    E1. During months when the major Hadley cell exists, trade winds cross the equator. If there are no forces at the equator, explain why this is possible.

    E2. In regions of surface high pressure, descending air in the troposphere is associated with dry (nonrainy) weather. These high-pressure belts are where deserts form. In addition to the belts at ±30° latitude, semi-permanent surface highs also exist at the poles. Are polar regions deserts? Explain.

    E3. The subtropical jet stream for Earth is located at about 30° latitude. Due to Coriolis force, this is the poleward limit of outflow air from the top of the ITCZ. If the Earth were to spin faster, numerical experiments suggest that the poleward limit (and thus the jet location) would be closer to the equator. Based on the spins of the other planets (get this info from the web or a textbook) compared to Earth, at what latitudes would you expect the subtropical jets to be on Jupiter? Do your predictions agree with photos of Jupiter?

    E4. Horizontal divergence of air near the surface tends to reduce or eliminate horizontal temperature gradients. Horizontal convergence does the opposite. Fronts (as you will learn in the next chapter) are regions of strong local temperature gradients. Based on the general circulation of Earth, at what latitudes would you expect fronts to frequently exist, and at what other latitudes would you expect them to rarely exist? Explain.

    E5. In the global circulation, what main features can cause mixing of air between the Northern and Southern Hemispheres? Based on typical velocities and cross sectional areas of these flows, over what length of time would be needed for the portion 1/e of all the air in the N. Hemisphere to be replaced by air that arrived from the S. Hemisphere?

    E6. In Fig. 11.4, the average declination of the sun was listed as 14.9° to 15° for the 4-month periods listed in those figures. Confirm that those are the correct averages, based on the equations from the Solar & Infrared Radiation chapter for solar declination angle vs. day of the year.

    E7. Thunderstorms are small-diameter (15 km) columns of cloudy air from near the ground to the tropopause. They are steered by the environmental winds at an altitude of roughly 1/4 to 1/3 the troposphere depth. With that information, in what direction would you expect thunderstorms to move as a function of latitude (do this for every 10° latitude)?

    E8. The average meridional wind at each pole is zero. Why? Also, does your answer apply to instantaneous winds such as on a weather map? Why?

    E9. Can you detect monsoonal (monthly or seasonal average) pressure centers on a normal (instantaneous) weather map analysis or forecast? Explain. 

    E10. Figs. 11.3a & 11.5a showed idealized surface wind & pressure patterns. Combine these and draw a sketch of the resulting idealized global circulation including both planetary and monsoon effects.

    E11. Eqs. (11.1-11.3) represent an idealized (“toy model”) meridional variation of zonally averaged temperature. Critically analyze this model and discuss. Is it reasonable at the ends (boundaries) of the curve; are the units correct; is it physically justifiable; does it satisfy any budget constraints (e.g., conservation of heat, if appropriate), etc. What aspects of it are too simplified, and what aspects are OK?

    E12. (a) Eq. (11.4) has the 3rd power of the sine times the 2nd power of the cosine. If you could arbitrarily change these powers, what values would lead to reasonable temperature gradients (∆T/∆y) at the surface and which would not (Hint: use a spreadsheet and experiment with different powers)? 

    (b) Of the various powers that could be reasonable, which powers would you recommend as fitting the available data the best? (Hint: consider not only the temperature gradient, but the associated meridional temperature profile and the associated jet stream.) Also, speculate on why I chose the powers that I did for this toy model.

    E13. Concerning differential heating, Fig. 11.9 shows the annual average insolation vs. latitude. Instead, compute the average insolation over the two-month period of June and July, and plot vs. latitude. Use the resulting graph to explain why the jet stream and weather patterns are very weak in the summer hemisphere, and strong in the winter hemisphere.

    E14. At mid- and high-latitudes, Fig. 11.9 shows that each hemisphere has one full cycle of insolation annually (i.e., there is one maximum and one minimum each year).

    But look at Fig. 11.9 near the equator. 

    1. Based on the data in this graph (or even better, based on the eqs. from the Solar & Infrared Radiation chapter), plot insolation vs. relative Julian day for the equator.
    2. How many insolation cycles are there each year at the equator?
    3. At the equator, speculate on when would be the hottest and coldest “seasons”.
    4. Within what range of latitudes near the equator is this behavior observed?

    E15. Just before idealized eq. (11.6), I mentioned my surprise that E2 was approximately constant with latitude. I had estimated E2 by subtracting my toy-model values for Einsol from the actual observed values of Ein. Speculate about what physical processes could cause E2 to be constant with latitude all the way from the equator to the poles.

    E16. How sensitive is the toy model for Eout (i.e., eq. 11.7) to the choice of average emission altitude zm? Recall that zm, when used as the altitude z in eqs. (11.1-11.3), affects Tm. Hint: for your sensitivity analysis, use a spreadsheet to experiment with different zm and see how the resulting plots of Eout vs. latitude change. (See the “A SCIENTIFIC PERSPECTIVE” box about model sensitivity.)

    E17(§). Solve the equations to reproduce the curves in figure:

    a. 11.10 b. 11.11 c. 11.12 d. 11.13

    E18. We recognize the global circulation as a response of the atmosphere to the instability caused by differential heating, as suggested by LeChatelier’s Principle. But the circulation does not totally undo the instability; namely, the tropics remain slightly warmer than the poles. Comment on why this remaining, unremoved instability is required to exist, for the global circulation to work.

    E19. In Fig. 11.12, what would happen if the surplus area exceeded the deficit area? How would the global circulation change, and what would be the end result for Fig. 11.12?

    E20. Check to see if the data in Fig. 11.12 does give zero net radiation when averaged from pole to pole.

    E21. The observation data that was used in Fig. 11.14 was based on satellite-measured radiation and differential heating to get the total needed heat transport, and on estimates of heat transport by the oceans. The published “observations” for net atmospheric heat transport were, in fact, estimated as the difference (i.e., residual) between the total and the ocean curves. What could be some errors in this atmosphere curve? (Hint: see the A SCIENTIFIC PERSPECTIVE box about Residuals.) 

    E22. Use the total heat-transport curve from Fig. 11.60. At what latitude is the max transport? For that latitude, convert the total meridional heat-flux value to horsepower.

    E23. For Fig. 11.15, explain why it is p’ vs. z that drive vertical winds, and not Pcolumn vs. z.


    1. Redraw Figs. 11.16 for downdraft situations.
    2. Figs. 11.16 both show updraft situations, but they have opposite pressure couplets. As you already found from part (a) both pressure couplets can be associated with downdrafts. What external information (in addition to the pressure-couplet sign) do you always need to decide whether a pressure couplet causes an updraft or a downdraft? Why?


    1. For the thermal circulation of Fig. 11.17(iv), what needs to happen for this circulation to be maintained? Namely, what prevents it from dying out?
    2. For what real-atmosphere situations can thermal circulations be maintained for several days?


    1. Study Fig. 11.18 closely, and explain why the wind vectors to/from the low- and high-pressure centers at the equator differ from the winds near pressure centers at mid-latitudes. 
    2. Redraw Fig. 11.5a, but with continents and oceans at the equator. Discuss what monsoonal pressures and winds might occur during winter and summer, and why.


    1. Redraw Fig. 11.19, but for the case of geostrophic wind decreasing from its initial equilibrium value. Discuss the resulting evolution of wind and pressure fields during this geostrophic adjustment.
    2. Redraw Fig. 11.19, but for flow around a lowpressure center (i.e., look at gradient winds instead of geostrophic winds). Discuss how the wind and pressure fields adjust when the geostrophic wind is increased above its initial equilibrium value. 

    E28. How would the vertical potential temperature gradient need to vary with latitude for the “internal Rossby radius of deformation” to be invariant? Assume constant troposphere depth.

    E29. In the Regional Winds chapter, gap winds and coastally-trapped jets are explained. Discuss how these flows relate to geostrophic adjustment.

    E30. At the top of hurricanes (see the Tropical Cyclones chapter), so much air is being continuously pumped to the top of the troposphere that a highpressure center is formed over the hurricane core there. This high is so intense and localized that it violates the conditions for gradient winds; namely, the pressure gradient around this high is too steep (see the Forces & Winds chapter). 

    Discuss the winds and pressure at the top of a hurricane, using what you know about geostrophic adjustment. Namely, what happens to the winds and air mass if the wind field is not in geostrophic or gradient balance with the pressure field?

    E31. In the thermal-wind relationship (eqs. 11.13), which factors on the right side are constant or vary by only a small amount compared to their magnitude, and which factors vary more (and are thus more important in the equations)?

    E32. In Fig. 11.20, how would it change if the bottom isobaric surface were tilted; namely, if there were already a horizontal pressure gradient at the bottom?

    E33. Draw a sketch similar to Fig. 11.20 for the thermalwind relationship for the Southern Hemisphere.

    E34. In maps such as Fig. 11.21, explain why thickness is related to average temperature. 

    E35. Redraw Fig. 11.22 for the case cold air in the west and warm air in the east. Assume no change to the bottom isobaric surface.

    E36. Copy Fig. 11.24. 

    1. On your copy, draw the G1 and G2 vectors, and the MTH vector at point B. Confirm that the thermal wind relationship is qualitatively satisfied via vector addition. Discuss why point B is an example of veering or backing.
    2. Same as (a) but calculate the actual magnitude of each vector at point B based on the spacing between isobars, thickness contours, or height contours. Again, confirm that the thermal wind relationship is satisfied. (1° latitude = 111 km)

    E37. Using a spreadsheet, start with an air parcel at rest at the tropopause over the equator. Assume a realistic pressure gradient between the equator and 30° latitude. Use dynamics to solve for acceleration of the parcel over a short time step, and then iterate over many time steps to find parcel speed and position. How does the path of this parcel compare to the idealized paths drawn in Fig. 11.26d? Discuss.

    E38. In the thunderstorms at the ITCZ, copious amounts of water vapor condense and release latent heat. Discuss how this condensation affects the average lapse rate in the tropics, the distribution of heat, and the strength of the equatorial high-pressure belt at the tropopause.

    E39. Summarize in a list or an outline all the general-circulation factors that make the mid-latitude weather different from tropical weather.

    E40. Explain the surface pressure patterns in Figs. 11.31 in terms of a combination of idealized monsoon and planetary circulations. 

    E41. Figs. 11.31 show mid-summer and mid-winter conditions in each hemisphere. Speculate on what the circulation would look like in April or October.

    E42. Compare Figs. 11.32 with the idealized planetary and monsoon circulations, and discuss similarities and differences.

    E43. Based on Figs. 11.32, which hemisphere would you expect to have strong subtropical jets in both summer and winter, and which would not. What factors might be responsible for this difference?

    E44. For the Indian monsoon sketched in Fig. 11.33, where are the updraft and downdraft portions of the major Hadley cell for that month? Also, what is the relationship between the trade winds at that time, and the Indian monsoon winds?

    E45. What are the dominant characteristics you see in Fig. 11.34, regarding jet streams in the Earth’s atmosphere? Where don’t jet streams go?

    E46. In Figs. 11.35, indicate if the jet-stream winds would be coming out of the page or into the page, for the: 

    1. N. Hemisphere,
    2. S. Hemisphere. 

    E47. Although Figs. 11.36 are for different months than Figs. 11.32, they are close enough in months to still both describe summer and winter flows.

    1. Do the near-tropopause winds in Figs. 11.36 agree with the pressure gradients (or height gradients) in Figs. 11.32?
    2. Why are there easterly winds at the tropopause over/near the equator, even though there is negligible pressure gradient there?

    E48. Describe the mechanism that drives the polar jet, and explain how it differs from the mechanism that drives the subtropical jet.

    E49. In Fig. 11.37b, we see a very strong pressure gradient in the vertical (indicated by the different isobars), but only small pressure gradients in the horizontal (indicated by the slope of any one isobar). Yet the strongest average winds are horizontal, not vertical. Why?

    E50. Why does the jet stream wind speed decrease with increasing height above the tropopause?

    E51. a. Knowing the temperature field given by the toy model earlier in this chapter, show the steps needed to create eq. (11.17) by utilizing eqs. (11.2, 11.4 and 11.13). b. For what situations might this jetwind-speed equation not be valid? c. Explain what each term in eq. (11.17) represents physically.

    E52. Why does an air parcel at rest (i.e., calm winds) near the equator possess large angular momentum?

    What about for air parcels that move from the east at typical trade wind speeds?

    E53. At the equator, air at the bottom of the troposphere has a smaller radius of curvature about the Earth’s axis than at the top of the troposphere. How significant is this difference? Can we neglect it?

    E54. Suppose that air at 30° latitude has no eastwest velocity relative to the Earth’s surface. If that air moves equatorward while preserving its angular momentum, which direction would it move relative to the Earth’s surface? Why? Does it agree with real winds in the general circulation? Elaborate.

    E55. Picture a circular hot tub of 2 m diameter with a drain in the middle. Water is initially 1.2 m deep, and you made rotate one revolution each 10 s. Next, you pull the plug, allowing the water depth to stretch to 2.4 m as it flows down the drain. Calculate the new angular velocity of the water, neglecting frictional drag. Show your steps.

    E56. In eq. (11.20), why is there a negative sign on the last term? Hint: How does the rotation direction implied by the last term without a negative sign compare to the rotation direction of the first term?

    E57. In the Thunderstorm chapters, you will learn that the winds in a portion of the tornado can be irrotational. This is surprising, because the winds are traveling so quickly around a very tight vortex. Explain what wind field is needed to gave irrotational winds (i.e., no relative vorticity) in air that is rotating around the tornado. Hint: Into the wall of a tornado, imagine dropping a neutrally-buoyant small paddle wheel the size of a flower. As this flower is translated around the perimeter of the tornado funnel, what must the local wind shear be at the flower to cause it to not spin relative to the ground? Redraw Fig. 11.43 to show what you propose.

    E58. Eq. (11.25) gives names for the different terms that can contribute toward vorticity. For simplicity, assume ∆z is constant (i.e., assume no stretching). On a copy of Fig. 11.44, write these names at appropriate locations to identify the dominant factors affecting the vorticity max and min centers.

    E59. If you were standing at the equator, you would be rotating with the Earth about its axis. However, you would have zero vorticity about your vertical axis. Explain how that is possible.

    E60. Eq. (11.26) looks like it has the absolute vorticity in the numerator, yet that is an equation for a form of potential vorticity. What other aspects of that equation make it like a potential vorticity?

    E61. Compare the expression of horizontal circulation C with that for vertical circulation CC.

    E62. Relate Kelvin’s circulation theorem to the conservation of potential vorticity. Hint: Consider a constant Volume = A·∆z .

    E63. The jet stream sketched in Fig. 11.49 separates cold polar air near the pole from warmer air near the equator. What prevents the cold air from extending further away from the poles toward the equator?

    E64. If the Coriolis force didn’t vary with latitude, could there be Rossby waves? Discuss.

    E65. Are baroclinic or barotropic Rossby waves faster relative to Earth’s surface at midlatitudes? Why?

    E66. Compare how many Rossby waves would exist around the Earth under barotropic vs. baroclinic conditions. Assume an isothermal troposphere at 50°N. 

    E67. Once a Rossby wave is triggered, what mechanisms do you think could cause it to diminish (i.e., to reduce the waviness, and leave straight zonal flow).

    E68. In Fig. 11.50 at point (4) in the jet stream, why doesn’t the air just continue turning clockwise around toward points (2) and (3), instead of starting to turn the other way? 

    E69. Pretend you are a newspaper reporter writing for a general audience. Write a short article describing how baroclinic Rossby waves work, and why they differ from barotropic waves. 

    E70. What conditions are needed so that Rossby waves have zero phase speed relative to the ground? Can such conditions occur in the real atmosphere?

    E71. Will Rossby waves move faster or slower with respect to the Earth’s surface if the tropospheric static stability increases? Why?

    E72. For a baroclinic wave that is meandering north and south, consider the northern-most point as the wave crest. Plot the variation of this crest longitude vs. altitude (i.e., x vs. z). Hint: consider eq. (11.40).

    E73. Use tropopause-level Rossby-wave troughaxes and ridge-axes as landmarks. Relative to those landmarks, where east or west is: (a) vertical velocity the greatest; (b) potential-temperature deviation the greatest; and (c) vertical displacement the greatest?

    E74. In Figs. 11.51 and 11.53 in the jet stream, there is just as much air going northward as there is air going southward across any latitude line, as required by mass conservation. If there is no net mass transport, how can there be heat or momentum transport?

    E75. For the Southern Hemisphere: (a) would a direct circulation cell have positive or negative CC? (b) for each term of eq. (11.51), what are their signs?

    E76. Compare definitions of circulation from this chapter with the previous chapter, and speculate on the relevance of the static stability and Earth’s rotation in one or both of those definitions. 

    E77. Consider a cyclonic air circulation over an ocean in your hemisphere. Knowing the relationship between ocean currents and surface winds, would you anticipate that the near-surface wind-driven ocean currents are diverging away from the center of the cyclone, or converging toward the center? Explain, and use drawings. Note: Due to mass conservations, horizontally diverging ocean surface waters cause upwelling of nutrient-rich water toward the surface, which can support ocean plants and animals, while downwelling does the opposite.

    11.15.4. Synthesize

    S1. Describe the equilibrium general circulation for a non-rotating Earth.

    S2. Circulations are said to spin-down as they lose energy. Describe general-circulation spin-down if Earth suddenly stopped spinning on its axis.

    S3. Describe the equilibrium general circulation for an Earth that spins three times faster than now.

    S4. Describe the spin-up (increasing energy) as the general circulation evolves on an initially non-rotating Earth that suddenly started spinning. 

    S5. Describe the equilibrium general circulation on an Earth with no differential radiative heating.

    S6. Describe the equilibrium general circulation on an Earth with cold equator and hot poles.

    S7. Suppose that the sun caused radiative cooling of Earth, while IR radiation from space caused warming of Earth. How would the weather and climate be different, if at all?

    S8. Describe the equilibrium general circulation for an Earth with polar ice caps that extend to 30° latitude.

    S9. About 250 million years ago, all of the continents had moved together to become one big continent called Pangaea, before further plate tectonic movement caused the continents to drift apart. Pangaea spanned roughly 120° of longitude (1/3 of Earth’s circumference) and extended roughly from pole to pole. Also, at that time, the Earth was spinning faster, with the solar day being only about 23 of our present-day hours long. Assuming no other changes to insolation, etc, how would the global circulation have differed compared to the current circulation?

    S10. If the Earth was dry and no clouds could form, how would the global circulation differ, if at all? Would the tropopause height be different? Why?

    S11. Describe the equilibrium general circulation for an Earth with tropopause that is 5 km high.

    S12. Describe the equilibrium general circulation for an Earth where potential vorticity isn’t conserved.

    S13. Describe the equilibrium general circulation for an Earth having a zonal wind speed halfway between the phase speeds of short and long barotropic Rossby waves.

    S14. Describe the equilibrium general circulation for an Earth having long barotropic Rossby waves that had slower intrinsic phase speed than short waves.

    S15. Describe the nature of baroclinic Rossby waves for an Earth with statically unstable troposphere.

    S16. Describe the equilibrium general circulation for an Earth where Rossby waves had no north-south net transport of heat, momentum, or moisture. 

    S17. Describe the equilibrium general circulation for an Earth where no heat was transported meridionally by ocean currents.

    S18. Describe the equilibrium ocean currents for an Earth with no drag between atmosphere and ocean. 

    S19. Suppose there was an isolated small continent that was hot relative to the surrounding cooler ocean. Sketch a vertical cross section in the atmosphere across that continent, and use thickness concepts to draw the isobaric surfaces. Next, draw a planview map of heights of one of the mid-troposphere isobaric surfaces, and use thermal-wind effects to sketch wind vectors on this same map. Discuss how this approach does or doesn’t explain some aspects of monsoon circulations.

    S20. If the Rossby wave of Fig. 11.50 was displaced so that it is centered on the equator (i.e., point (1) starts at the equator), would it still oscillate as shown in that figure, or would the trough of the wave (which is now in the S. Hem.) behave differently? Discuss.

    S21. If the Earth were shaped like a cylinder with its axis of rotation aligned with the axis of the real Earth, could Rossby waves exist? How would the global circulation be different, if at all?

    S22. In the subtropics, low altitude winds are from the east, but high altitude winds are from the west. In mid-latitudes, winds at all altitudes are from the west. Why are the winds in these latitude bands different?

    S23. What if the Earth did not rotate? How would the Ekman spiral in the ocean be different, if at all?