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12.11: Homework Exercises

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  • 12.11.1. Broaden Knowledge & Comprehension

    B1. Monitor the weather maps on the web every day for a week or more during N. Hemisphere winter, and make a sketch of each low center with the fronts extending (similar to Fig. 12.1). Discuss the variety of arrangements of warm, cold, and occluded fronts that you have observed during that week.

    B2. Same as B1, but for the S. Hemisphere during S. Hemisphere winter.

    B3. On a surface weather map, identify high-pressure centers, and identify ridges. Also, on a 50 kPa (500 hPa) chart, do the same. Look at the ratio of ridges to high centers for each chart, and identify which chart (surface or 50 kPa) has the largest relative number of high centers, and which has the largest relative number of ridges.

    B4. From weather maps showing vertical velocity w, find typical values of that vertical velocity near the center of highs. (If ω = ∆P/∆t is used instead of w, it is acceptable to leave the units in mb s–1). Compare this vertical velocity to the radial velocity of air diverging away from the high center in the boundary layer.

    B5. Use a sequence of surface weather maps every 6 or 12 h for a week, where each map spanning a large portion of the globe. Find one or more locations that exhibits the following mechanism for formation of a surface high pressure:

    1. global circulation
    2. monsoon
    3. Rossby wave 
    4. thunderstorm
    5. topographic

    B6. Do a web search to find surface weather maps that also include airmass abbreviations on them. Print one of these maps, and suggest how the labeled airmass moved to its present location from its genesis region (where you will need to make a reasonable guess as to where its genesis region was).

    B7. Use weather maps on the web to find a location where air has moved over a region and then becomes stationary. Monitor the development of a new airmass (or equivalently, modification of the old airmass) with time at this location. Look at temperature and humidity.

    B8. For a weather situation of relatively zonal flow from west to east over western N. America, access upper air soundings for weather stations in a line more-or-less along the wind direction. Show how the sounding evolves as the air flows over each major mountain range. How does this relate to airmass modification?

    B9. Access upper air soundings for a line of RAOB stations across a front. Use this data to draw vertical cross-sections of:

    1. pressure or height
    2. temperature
    3. humidity 
    4. potential temperature

    B10. Same as previous exercise, but identify the frontal inversions (or frontal stable layers) aloft in the soundings.

    B11. Access a sequence of weather maps that cover a large spatial area, with temporal coverage every 6 or 12 h for a week. Find an example of a front that you can follow from beginning (frontogenesis) through maturity to the end (frontolysis). Discuss how the front moves as it evolves, and what the time scale for its evolution is. 

    B12. Print from the web a weather map that shows both the analyzed fronts and the station plot data. On this map, draw your own analysis of the data to show where the frontal zones are. Discuss how the weather characteristics (wind, pressure, temperature, weather) across these real fronts compare with the idealized sketches of Figs. 12.11 and 12.12.

    B13. Access a sequence of surface weather maps from the web that show the movement of fronts. Do you see any fronts that are labeled as stationary fronts, but which are moving? Do you see any fronts labeled as warm or cold fronts, but which are not moving? Are there any fronts that move backwards compared to the symbology labeling the front (i.e., are the fronts moving opposite to the direction that the triangles or semicircles point)? [Hint: often fronts are designated by how they move relative to the low center. If a cold front, for example, is advancing cyclonically around a low, but the low is moving toward the west (i.e., backwards in mid-latitudes), then relative to people on the ground, the cold front is retreating.]

    B14. Access weather map data that shows a strong cold front, and compare the winds on both sides of that front to the winds that you would expect using geostrophic-adjustment arguments. Discuss.

    B15. For a wintertime situation, access a N. (or S.) Hemisphere surface weather map from the web, or access a series of weather maps from different agencies around the world in order to get information for the whole Hemisphere. Draw the location of the polar front around the globe. 

    B16. Access weather maps from the web that show a strengthening cold or warm front. Use the other weather data on this map to suggest if the strengthening is due mostly to kinematic, thermodynamic, or dynamic effects.

    B17. Access a sequence of weather maps that shows the formation and evolution of an occluded front. Determine if it is a warm or cold occlusion (you might need to analyze the weather map data by hand to help you determine this).

    B18. Use the web to access a 3-D sketch of a TROWAL. Use this and other web sites to determine other characteristics of TROWALs. 

    B19. Download upper-air soundings from a station under or near the jet stream. Use the data to see if there is a tropopause fold. [Hint: assuming that you don’t have measurements of radioactivity or isentropic potential vorticity, use mixing ratio or potential temperature as a tracer of stratospheric air.]

    B20. Search the web for surface weather maps that indicate a dryline in the S.W. USA. If one exists, then search the web for upper-air soundings just east and just west of the dryline. Plot the resulting soundings of potential temperature and humidity, and discuss how it relates to the idealized sketch of a dryline in this textbook.

    12.11.2. Apply

    A1. Identify typical characteristics of the following airmass:

    a. cAA b. cP c. cT d. cM e. cE
    f. mA g. mP h. mT i. mM j. mE
    k. cEw l. cAk m. cPw n. mPk o. mTw

    A2. List all the locations in the world where the following airmasses typically form.

    a. cAA b. cA c. cP d. mP e. mT f. cT g. mE

    A3.(§) Produce a graph of warm airmass depth zi and airmass potential temperature θML as a function of time, similar to Fig. 12.5. Use all the same conditions as in that figure (γ = 3.3 K km–1, β = 10–6 s–1, initial ∆θs = 10°C; see the Sample Application on the subsequent page) for genesis of a warm airmass, except with the following changes:

    a. initial ∆θs = 5°C b. initial ∆θs = 15°C
    c. initial ∆θs = 20°C d. initial ∆θs = 15°C
    e. β = 0 s–1 f. β = 10–7 s–1
    g. β = 5 x 10–7 s–1 h. β = 5 x 10–6 s–1
    i. β = 10–5 s–1 j. γ = 5 K km–1
    k. γ = 4 K km–1 l. γ = 3 K km–1
    m. γ = 2 K km–1 n. γ = 1 K km–1

    A4. Same as the previous exercise, but find the time scale \(\ \tau\)  that estimates when the peak airmass depth occurs for warm airmass genesis.

    A5. For the katabatic winds of Antarctica, find the downslope buoyancy force per unit mass for this location:

    Location ∆z/∆x ∆θ (K) Te (K)
    a. Interior  0.001 40 233
    b. Interior 0.002 35 238
    c. Interior 0.003 30 243
    d. Intermediate 0.004 27 245
    e. Intermediate 0.005 25 248
    f. Intermediate 0.006 24 249
    g. Intermediate 0.007 23 250
    h. Coast  0.008 22 251
    i. Coast 0.009 21 252
    j. Coast 0.010 20 253
    k. Coast 0.011 19 254

    A6.(§) An Arctic airmass of initial temperature –30°C is modified as it moves at speed 10 m s–1 over smooth warmer surface of temperature 0°C. Assume constant airmass thickness. Find and plot the airmass (mixed-layer) temperature vs. downwind distance for an airmass thickness (m) of:

    a. 100 b. 200 c. 300 d. 400 e. 500
    f. 600 g. 700 h. 800 i. 900 j. 1000
    k. 1200 l. 1400 m. 1500    

    A7. Find the external Rossby radius of deformation at 60° latitude for a cold airmass of thickness 500 m and ∆θ (°C) of:

    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

    Assume a background potential temperature of 300K.

    A8.(§) Find and plot the airmass depth and geostrophic wind as a function of distance from the front for the cases of the previous exercise. Assume a background potential temperature of 300 K.

    A9. Suppose that ∆θ/∆x = 0.02°C km–1, ∆θ/∆y = 0.01°C km–1, and ∆θ/∆z = 5°C km–1. Also suppose that ∆U/∆x = – 0.03 (m/s) km–1, ∆V/∆x = 0.05 (m/s) km–1, and ∆W/∆x = 0.02 (cm/s) km–1. Find the kinematic frontogenesis contributions from:

    a. confluence b. shear c. tilting d. and find the strengthening rate.

    A10. A thunderstorm on the warm side of a 300 km wide front rains at the following rate (mm h–1). Find the thermodynamic contribution to frontogenesis.

    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. 2.5 l. 3.0 m. 3.5 n. 4.0

    A11. Plot dryline movement with time, given the following conditions. Surface heat flux is constant with time at kinematic rate 0.2 K·m s–1. The vertical gradient of potential temperature in the initial sounding is γ. Terrain slope is s = ∆z/∆x.

    \(\gamma\) (K km–1) s \(\gamma\) (K km–1) s
    a. 8 1/500 b. 8 1/400
    c. 8 1/300 d. 8 1/200
    e. 12 1/500 f. 12 1/400
    g. 12 1/300 h. 12 1/200
    i. 15 1/500 j. 15 1/400

    12.11.3. Evaluate & Analyze

    E1. Would you expect there to be a physically- or dynamically-based upper limit on the number of warm fronts and cold fronts that can extend from a low-pressure center? Why?

    E2. Sketch a low with two fronts similar to Fig. 12.1, but for the Southern Hemisphere.

    E3. What are the similarities and differences between a ridge and a high-pressure center? Also, on a weather map, what characteristics would you look for to determine if a weather feature is a ridge or a trough?

    E4. In high-pressure centers, is the boundary-layer air compressed to greater density as subsidence pushes down on the top of the mixed layer? Explain.

    E5. In Fig. 12.3b, use the hypsometric relationship to explain why the ridge shifts or tilts westward with increasing altitude. Hint: consider where warm and cold air is relative to the surface high.

    E6. How would Fig. 12.3b be different, if at all, in Southern Hemisphere mid-latitudes. Hint: Consider the direction that air rotates around a surface high, and use that information to describe which side of the high (east or west) advects in warm air and which advects in colder air.

    E7. Consider the global-circulation, monsoon, and topographic mechanisms that can form high-pressure regions. Use information from the chapters on General Circulation, Extratropical Cyclones, and Regional Winds to identify 3 or more regions in the world where 2 or more of those mechanisms are superimposed to help create high-pressure regions. Describe the mechanisms at each of those 3 locations, and suggest how high-pressure formation might vary during the course of a year.

    E8. Out of all the different attributes of airmasses, why do you think that the airmass abbreviations listed in Table 12-1 focus on only the temperature (i.e., AA, A, P, T, M, E) and humidity (c, m)? In other words, what would make the relative temperature and humidity of airmasses be so important to weather forecasters? Hint: consider what you learned about heat and humidity in earlier chapters.

    E9. Fig. 12.4 shows the genesis regions for many different airmasses. At first glance, this map looks cluttered. But look more closely for patterns and describe how you can anticipate where in the world you might expect the genesis regions to be for a particular airmass type.

    E10. In this chapter, I described two very different mechanisms by which warm and cold airmasses are created. Why would you expect them to form differently? Hint: consider the concepts, not the detailed equations.

    E11. Fig. 12.5 shows how initially-cold air is transformed into a warm airmass after it becomes parked over a warm surface such as a tropical ocean. It is not surprising that the airmass temperature θML asymptotically approaches the temperature of the underlying surface. Also not surprising: depth zi of the changed air initially increases with time.

    But more surprising is the decrease in depth of the changed air at longer times. Conceptually, why does this happen? Why would you expect it to happen for most warm airmass genesis?

    E12. Using the concept of airmass conservation, formulate a relationship between subsidence velocity at the top of the boundary layer (i.e., the speed that the capping inversion is pushed down in the absence of entrainment) to the radial velocity of air within the boundary layer. (Hint: use cylindrical coordinates, and look at the geometry.)

    E13. For the toy model in the INFO Box on Warm Airmass Genesis, ∆Q is the amount of heat (in kinematic units) put into the mixed layer (ML) from the surface during one time step. By repeating over many time steps, we gradually put more and more heat into the ML. This heat is uniformly mixed in the vertical throughout the depth zi of the ML, causing the airmass to warm. 

    Yet at any time, such as 10 days into the forecast, the amount of additional heat contained in the ML (which equals its depth times its temperature increase since starting, see Fig. 12.5) is less than the accumulated heat put into the ML from the surface (which is the sum of all the ∆Q from time zero up to 10 days). 

    That discrepancy is not an error. It describes something physical that is happening. What is the physical process that explains this discrepancy, and how does it work?

    E14. In the Sample Application on warm airmass genesis, I iterate with very small time step at first, but later take larger and larger time steps. Why would I need to take small time steps initially, and why can I increase the time step later in the simulation? Also, if I wanted to take small time steps for the whole duration, would that be good or bad? Why?

    E15. Suppose that cold air drains katabatically from the center of Antarctica to the edges. Sketch the streamlines (lines that are parallel to the flow direction) that you would anticipate for this air, considering buoyancy, Coriolis force, and turbulent drag.

    E16.(§) Assume the katabatic winds of Antarctica result from a balance between the downslope buoyancy force, Coriolis force (assume 70°S latitude), and turbulent drag force at the surface (assume neutral boundary layer with CD = 0.002). The surface is smooth and the depth of the katabatic layer is 100 m. Neglect entrainment drag at the top of the katabatic flow. The slope, potential temperature difference, and ambient temperature vary according to the table from exercise N5. Find the katabatic wind speed at the locations in the table from exercise N5.

    E17. Where else in the world (Fig. 12.8) would you anticipate orographic airmass modification processes similar to those shown in Fig. 12.9? Explain.

    E18. For airmass modification (i.e., while an airmass is blowing over a different surface), why do we describe the heat flux from the surface using a bulk transfer relationship (CH ·M·∆θ), while for warm airmass genesis of stationary air we used a buoyancyvelocity approach (b ·wb·∆θ)?

    E19. The Eulerian heat budget from the Heat chapter shows how air temperature change ∆T is related to the heat input over time interval ∆t. If this air is also moving at speed M, then during that same time interval, it travels a distance ∆x = M·∆t. Explain how you can use this information to get eq. (12.3), and why the wind speed M doesn’t appear in that eq.

    E20. Look at Fig. 12.9 and the associated Sample Application. In the Sample Application, the air passes twice through the thermodynamic state given by point 1 on that diagram: once when going from point 0 to points 2, and the second time when going from points 2 to point 3. 

    Similarly, air passes twice through the thermodynamic state at point 3 on the thermo diagram, and twice through point 5.

    But air can achieve the same thermodynamic state twice only if certain physical (thermodynamic) conditions are met. What are those conditions, and how might they NOT be met in the real case of air traversing over these mountain ranges?

    E21. Background: Recall that a frontal zone separates warmer and cooler airmasses. The warm airmass side of this zone is where the front is drawn on a weather map. This is true for both cold and warm fronts.

    Issue: AFTER passage of the cold front is when significant temperature decreases are observed. BEFORE passage of a warm front is when significant warming is observed.

    Question: Why does this difference exist (i.e., AFTER vs. BEFORE) for the passage of these two fronts?

    E22. Overlay Fig. 12.11 with Fig. 12.12 by aligning the low-pressure centers. Do not rotate the images, but let the warm and cold fronts extend in different directions from the low center. Combine the information from these two figures to create a new, larger figure showing both fronts at the same time, extending from the same low. On separate copies of this merged figure, draw:

    a. isotherms b. isobars c. winds d. weather

    E23. Suppose you saw from your barometer at home that the pressure was falling. So you suspect that a front is approaching. What other clues can you use (by standing outside and looking at the clouds and weather; NOT by looking at the TV, computer, or other electronics or weather instruments) to determine if the approaching front is a warm or cold front. Discuss, along with possible pitfalls in this method.

    E24. Draw new figures (a) - (d) similar to (a) - (d) in Figs. 12.11 and 12.12, but for an occluded front.

    E25. Use the columns of temperature data in Fig. 12.13a, and plot each column as a separate sounding on a thermo diagram (See the Atmos. Stability chapter for blank thermo diagrams that you can photocopy for this exercise. Use a skew-T diagram, unless your instructor tells you otherwise). Describe how the frontal zone shows up in the soundings.

    E26. Draw isentropes that you might expect in a vertical cross-section through a warm front.

    E27. Other than drylines, is it possible to have fronts with no temperature change across them? How would such fronts be classified? How would they behave?

    E28. What clouds and weather would you expect with a cold anafront?

    E29. Why should the Rossby radius of deformation depend on the depth of the cold airmass in a geostrophic adjustment process?

    E30. For geostrophic adjustment, the initial outflow of cold air turns, due to Coriolis force, until it is parallel to the front. Why doesn’t it continue turning and point back into the cold air?

    E31. For geostrophic adjustment, what is the nature of the final winds, if the starting point is a shallow cylinder of cold air 2 km thick and 500 km radius?

    E32. Starting with Fig. 12.18, suppose that ABOVE the bottom contoured surface the temperature is horizontally uniform. Redraw that diagram but with the top two contoured surfaces sloped appropriately for the new temperature state.

    E33. By inspection, write a kinematic frontogenesis equation (similar to eq. 12.10), but for an east-west aligned front. 

    E34. Figs. 12.22 - 12.24 presume that temperature gradients already exist, which can be strengthened by kinematic frontogenesis. Is that presumption valid for Earth’s atmosphere? Justify.

    E35. For the fronts analyzed in the Sample Application in the section on Surface Fronts – Horizontal Structure, estimate and compare magnitudes of any kinematic, thermodynamic, and dynamic processes that might exist across those fronts, based on the plotted weather data (which includes info on winds, precipitation, etc.). Discuss the relative importance of the various mechanisms for frontogenesis.

    E36. Speculate on which is more important for dynamically generating ageostrophic cross-frontal circulations: the initial magnitude of the geostrophic wind or the change of geostrophic wind.

    E37. What happens in an occluded front where the two cold air masses (the one advancing behind the cold front, and the one retreating ahead of the warm front) have equal virtual temperature?

    E38. Draw isentropes that you expect in a vertical cross-section through a cold front occlusion (where a cold front catches up to a warm front), and discuss the change of static stability across this front.

    E39. What types of weather would be expected with an upper-tropospheric front that does not have an associated surface front? [Hint: track movement of air parcels as they ride isentropic surfaces.]

    E40. Draw a sketch similar to Fig. 12.35 showing the transverse circulations, but for a vertical cross-section in the Southern Hemisphere.

    E41. Would you expect drylines to be possible in parts of the world other than the S.W. USA? If so, where? Justify your arguments.

    12.11.4. Synthesize

    S1. Fig. 12.2 shows a ridge that is typical of mid-latitudes in the N. Hemisphere. Sketch a ridge for:

    1. Southern Hemisphere mid-latitudes
    2. N. Hemisphere tropics
    3. S. Hemisphere tropics

    Hint: Consider the global pressure patterns described in the General Circulation chapter.

    S2. What if there was no inversion at the top of the boundary layer? Redraw Fig. 12.3a for this situation. 

    S3. What if the Earth had no oceans? What mechanisms could create high-pressure centers and/or high-pressure belts?

    S4. For your location, rank the importance of the different mechanisms that could create highs, and justify your ranking.

    S5. Airmasses are abbreviated mostly by the relative temperature and humidity associated with their formation locations. Table 12-1 also describes two other attributes: returning airmasses, and airmasses that are warmer or colder than the underlying surface. What one additional attribute would you wish airmasses could be identified with, to help you to predict the weather at your location? Explain.

    S6. Fig. 12.4 shows many possible genesis regions for airmasses. But some of these regions would not likely exist during certain seasons, because of the absence of high-pressure centers. For the hemisphere (northern or southern) where you live, identify how the various genesis regions appear or disappear with the seasons. Also, indicate the names for any of the high-pressure centers. For those that exist year round, indicate how they shift location with the seasons. (Hint: review the global circulation info in the General Circulation chapter.)

    S7. What if airmasses remained stationary over the genesis regions forever? Could there be fronts and weather? Explain.

    S8. Would it be possible for an airmass to become so thick that it fills the whole troposphere? If so, explain the conditions needed for this to occur.

    S9. What if boundary-layer processes were so slow that airmasses took 5 times longer to form compared to present airmass formation of about 3 to 5 days? How would the weather and global circulation be different, if at all?

    S10. Cold-air drainage from Antarctica is so strong and persistent that it affects the global circulation. Discuss how this affect is captured (or not) in the global maps in the General Circulation chapter.

    S11. What if Antarctica was flat, similar to the Arctic? How would airmasses and Earth’s climate be different, if at all?

    S12. Consider Fig. 12.8. How would airmass formation and weather be different, if at all, given:

    1. Suppose the Rocky Mountains disappeared, and a new dominant mountain range (named after you), appeared east-west across the middle of N. America. 
    2. Suppose that the Alps and Pyrenees disappeared and were replaced by a new dominant north-south mountain range (named after you) going through Europe from Copenhagen to Rome.
    3. Suppose no major mountain ranges existed. 

    S13. Suppose a cross-section of the terrain looked like Fig. 12.9, except that the Great Basin region were below sea level (and not flooded with water). Describe how airmass modification by the terrain would be different, if at all. 

    S14. What if precipitation did not occur as air flows over mountain ranges? Thus, mountains could not modify air masses by this process. How would weather and climate be different from now, if at all?

    S15. A really bad assumption was made in eq. (12.3); namely, zi = constant as the airmass is modified. A better assumption would be to allow zi to change with time (i.e., with distance as it blows over the new surface). The bad assumption was made because it allowed us to describe the physics with a simple eq. (12.3), which could be solved analytically to get eq. (12.4). This is typical of many physical problems, where it is impossible to find an exact analytical solution to the full equations describing true physics (as best we know it).

    So here is a philosophical question. Is it better to approximate the physics to allow an exact analytical solution of the simplified problem? Or is it better to try to get an approximate (iterated or graphical solution) to the exact, more-complicated physics? Weigh the pros and cons of each approach, and discuss.

    S16. For a (a) cold front, or a (b) warm front, create station plot data for a weather station 100 km ahead of the front, and for another weather station 100 km behind the front. Hint: use the sketches in Figs. 12.11 and 12.12 to help decide what to plot. 

    S17. Suppose the width of frontal zones were infinitesimally small, but their lengths remained unchanged. How would weather be different, if at all?

    S18. What if turbulence were always so intense that frontal zones were usually 1000 km in width? How would weather be different, if at all?

    S19. Does the geostrophic adjustment process affect the propagation distance of cold fronts in the real atmosphere, for cold fronts that are embedded in the cyclonic circulation around a low-pressure center? If this indeed happens, how could you detect it?

    S20. The boxes at the top of Figs. 12.22 - 12.24 show the associated sign of key terms for kinematic frontogenesis. If those terms have opposite signs, draw sketches similar to those figures, but showing frontolysis associated with: 

    a. confluence b. shear c. tilting

    S21. Suppose that condensation of water vapor caused air to cool. How would the thermodynamic mechanism for frontogenesis work for that situation, if at all? How would weather be near fronts, if at all?

    S22. Suppose that ageostrophic motions were to experience extremely large drag, and thus would tend to dissipate quickly. How would frontogenesis in Earth’s atmosphere be different, if at all?

    S23. After cold frontal passage, cold air is moving over ground that is still warm (due to its thermal inertia). Describe how static stability varies with height within the cold airmass 200 km behind the cold front. Draw thermo diagram sketches to illustrate your arguments.

    S24. Suppose a major volcanic eruption injected a thick layer of ash and sulfate aerosols at altitude 20 km. Would the altitude of the tropopause adjust to become equal to this height? Discuss.

    S25. Suppose the slope of the ground in a dryline situation was not as sketched in Fig. 12.37. Instead, suppose the ground was bowl-shaped, with high plateaus on both the east and west ends. Would drylines exist? If so, what would be their characteristics, and how would they evolve?