chapter 10 Copyright © 2011, 2015 by Roland Stull. Meteorology for Scientists and Engineers, 3rd Ed. Dynamics contents We use winds to power our wind turbines, push our sailboats, cool our Winds and Weather Maps 290 houses, and dry our laundry on the Height Contours on Isobaric Surfaces 290 10 clothesline. But winds can also be destructive — in Plotting Winds 291 hurricanes, thunderstorms, or mountain downslope Newton’s Second Law of Motion 292 windstorms. We design our bridges and skyscrap- Lagrangian Momentum Budget 292 ers to withstand wind gusts. Airplane flights are Eulerian Momentum Budget 293 planned to compensate for headwinds and cross- Horizontal Forces 294 winds. Advection 294 Winds are driven by forces acting on air. But Pressure-Gradient Force 295 these forces can be altered by heat and moisture car- Centrifugal Force 296 ried by the air, resulting in a complex interplay we Coriolis Force 297 call weather. The relationship between forces and Turbulent-Drag Force 300 Summary of Forces 301 winds is called atmospheric dynamics. Newto- nian physics describes atmospheric dynamics well. Equations of Horizontal Motion 301 Pressure, drag, and advection are atmospheric Horizontal Winds 302 forces that act in the horizontal. Other forces, called Geostrophic Wind 302 apparent forces, are caused by the Earth’s rotation Gradient Wind 304 (Coriolis force) and by turning of the wind around a Boundary-Layer Wind 307 Boundary-Layer Gradient (BLG) Wind 309 curve (centrifugal and centripetal forces). Cyclostrophic Wind 311 These different forces are present in different Inertial Wind 312 amounts at different places and times, causing large Antitriptic Wind 312 variability in the winds. For example, Fig. 10.1 shows Summary of Horizontal Winds 313 changing wind speed and direction around a low- Horizontal Motion 314 pressure center. In this chapter we explore forces, Equations of Motion — Revisited 314 winds, and the dynamics that link them. Scales of Horizontal Motion 315 Vertical Forces and Motion 315 Mass Conservation 317 Continuity Equation 317 Incompressible Continuity Equation 318 Boundary-Layer Pumping 319 Kinematics 320 Measuring Winds 321 Summary 322 Threads 322 Exercises 322 Numerical Problems 322 Understanding & Critical Evaluation 324 Web-Enhanced Questions 326 Synthesis Questions 327 Z 1L1B “Meteorology for Scientists and Engineers, 3rd Edi- Y tion” by Roland Stull is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. To view a copy of the license, visit Figure 10.1 http://creativecommons.org/licenses/by-nc-sa/4.0/ . This work is Sketch of sea-level pressure (thin lines are isobars) & the result- available at http://www.eos.ubc.ca/books/Practical_Meteorology/ . ing near-surface winds (arrows). “L” is low pressure center. 289 290 chapter 10 DyNAMICS WinDs anD Weather maps B height contours on isobaric surfaces 1 Pressure-gradient force is the most important force because it is the only one that can drive winds L1B in the horizontal. Other horizontal forces can alter [ LN an existing wind, but cannot create a wind from calm air. All the forces, including pressure-gradient force, are explained in the next sections. However, to understand the pressure gradient, we must first understand pressure and its atmospheric variation. Y We can create weather maps showing values of the pressures measured at different horizontal locations all at the same altitude, such as at mean- sea-level (MSL). Such a map is called a constant- height map. However, one of the peculiarities of meteorology is that we can also create maps on oth- C [ er surfaces, such as on a surface connecting points of equal pressure. This is called an isobaric map. L1B Both types of maps are used extensively in meteo- rology, so you should learn how they are related. Z In Cartesian coordinates (x, y, z), height z is geo- metric distance above some reference level, such - as the ground or sea level. Sometimes we use geopotential height H in place of z, giving a coor- LN dinate set of (x, y, H) (see Chapter 1). However, an Y alternative coordinate system can use pressure P as the vertical coordinate, because pressure decreases monotonically with increasing height. Pressure coordinates consist of (x, y, P). A monotonic variable is one that changes only D [LNTVSGBDF E 1L1BTVSGBDF in one direction. For example, it increases or is constant, but never decreases. Or it decreases or is 1L1B [LN constant, but never increases. Pressure in the atmo- Z Z sphere always decreases with increasing height. A surface connecting points of equal pressure is an isobaric surface. In low-pressure regions, this surface is closer to the ground than in high-pressure regions (Fig. 10.2b). Thus, this surface curves up and down through the atmosphere. Although isobaric Y Y surfaces are not flat, we draw them as flat weather maps on the computer screen or paper (Fig. 10.2d). Low pressures on a constant-height map cor- respond to low heights on a constant-pressure Figure 10.2 surface. High pressures on a constant height map (a) Vertical slice through atmosphere, showing pressure values correspond to high heights on a constant pressure (kPa). Thick dashed line is the 70 kPa isobar. Thin straight line surface. Similarly, regions on a constant-height map is the 3 km height contour. The location of lowest pressure on that have tight packing (close spacing) of isobars the height contour corresponds to the location of lowest height correspond to regions on isobaric maps that have of the isobar. (b) 3-D sketch of the same 70 kPa isobaric sur- tight packing of height contours, both of which are face (shaded), and 3 km height surface (white). (c) Pressures regions of strong pressure gradients that can drive intersected by the 3 km constant height surface. (d) Heights crossed by the 70 kPa surface. The low-pressure center (L) in (c) strong winds. This one-to-one correspondence of matches the low-height center in (d). both types of maps (Figs. 10.2c & d) makes it easier for you to use them interchangeably. R. STULL • METEoRoLoGy FoR SCIENTISTS AND ENGINEERS 291 It is impossible for two different isobaric surfaces FOCUs • Why use isobaric maps? to cross each other. Also, these surfaces never fold back on themselves, because pressure decreases Constant pressure charts are used for five reasons. monotonically with height. However, they can in- First, the old radiosonde (consisting of weather sen- tersect the ground, such as frequently happens in sors hanging from a free helium balloon that rises mountainous regions. into the upper troposphere and lower stratosphere) Isobaric charts will be used extensively in the re- measured pressure instead of altitude, so it was easier mainder of this book when describing upper-air fea- to plot their measurements of temperature, humid- tures (mostly for historical reasons; see Focus box). ity and wind on an isobaric surface. For this reason, Fig. 10.3 is a sample weather map showing height upper-air charts (i.e., showing weather above the contours of the 30 kPa isobaric surface. ground) traditionally have been drawn on isobaric maps. Second, aircraft altimeters are really pressure plotting Winds gauges. Aircraft assigned by air-traffic control to a Symbols on weather maps are like musical notes specific “altitude” above 18,000 feet MSL will actually in a score — they are a shorthand notation that con- fly along an isobaric surface. Many weather observa- cisely expresses information. For winds, the symbol tions and forecasts are motivated by aviation needs. is an arrow with feathers (or barbs and pennants). Third, pressure is a measure of mass in the air, so The tip of the arrow is plotted over the observation every point on an isobaric map has the same number (weather-station) location, and the arrow shaft is of air molecules above it. aligned so that the arrow points toward where the Fourth, an advantage of using equations of motion wind is going. The number and size of the feath- in pressure coordinates is that you do not need to con- ers indicates the wind speed (Table 10-1, copied from sider density, which is not routinely observed. Table 9-9). Fig. 10.3 illustrates wind barbs. Fifth, some weather forecast models use pressure coordinate systems in the vertical. However, more and more routine upper-air sound- ings around the world are being made with modern Table 10-1. Interpretation of wind barbs. GPS (Global Positioning System, satellite Symbol Wind Speed Description triangulation method) sondes that can measure geo- calm two concentric circles metric height as well as pressure. Also, some of the modern weather forecast models do not use pressure 1 - 2 speed units shaft with no barbs as the vertical coordinate. In the future, we might see 5 speed units a half barb (half line) the large government weather data centers starting to produce upper-air weather maps on constant height 10 speed units each full barb (full line) surfaces. 50 speed units each pennant (triangle) • The total speed is the sum of all barbs and pennants. For example, indicates a wind from the west at 65$ speed 75 units. Arrow tip is at the observation location. - +VMZ • CAUTION: Different organizations use different 1L1B LN speed units, such as knots, m/s, miles/h, km/h, etc. [LN Look for a legend to explain the units. When in doubt, / assume knots — the WMO standard. For unit conver- sion, a good approximation is 1 m/s ≈ 2 knots. Solved Example Draw wind barb symbol for winds from the: (a) northwest at 115 knots; (b) northeast at 30 knots. / ) Solution 8 8 8 (a) 115 knots = 2 pennants + 1 full barb + 1 half barb. (b) 30 knots = 3 full barbs Figure 10.3 B C Weather map for a 30 kPa constant pressure surface over central Check: Consistent with Table 10-1.