1. 2/9/10MET 61 topic 02 1 MET 61 Topic 2 Atmospheric Dynamics

2. 2/9/10MET 61 topic 02 2 Goal(s)… Main goal = investigate motions = winds –Wind = air in motion 1.What causes air motions? –What causes any object to move? –A force! –“F = ma” –All we need to do is indentify forces in the atmosphere → ideas about why air moves

3. 2/9/10MET 61 topic 02 3 2.What air motions result? –Range from “simple” to highly complex –e.g., geostropic wind is a simple wind that we can derive –Sometimes a good approximation to real (observed) winds 3.How can we characterize air motions? –What properties of air motions are useful to know?

4. 2/9/10MET 61 topic 02 4 Kinematics of large-scale flows VIP note: we will focus on flows with 1)horizontal scales O(1000 km +) e.g., not a tornado 2)vertical scales O(10 km) E.g., not a Cu cloud 3)time scales O(one day +) e.g., not a tornado “on the order of”

5. 2/9/10MET 61 topic 02 5 Properties are listed in Table 7.1 Need to understand: Physics what the property is what information this conveys Math how to express / compute the property

6. 2/9/10MET 61 topic 02 6 Shear Physically A change in wind speed or direction in space Example: Winds across a front

7. 2/9/10MET 61 topic 02 7 Example: Associated with a thunderstorm Microburst example  A microburst is a very localized column of sinking air, producing damaging divergent and straight-line winds at the surface that are similar to but distinguishable from tornadoes which generally have convergent damage.straight-line windstornadoes

8. 2/9/10MET 61 topic 02 8 http://en.wikipedia.org/wiki/Eastern_Air_Lines_Flight_66

9. 2/9/10MET 61 topic 02 9 Math We invent and use natural coordinates Flow-following coordinates Fig. 7.1 s = direction along the flow n = direction  to flow (and to the left) Lower cases!

10. 2/9/10MET 61 topic 02 10 Math Thus: Here, V = wind speed (scalar) Wind direction is tracked via the coordinate system!

11. 2/9/10MET 61 topic 02 11 Curvature Physically A change in wind direction as one travels downstream Sign convention: Curvature is called positive if flow direction (vector) is turning anticlockwise in the NH. aka…cyclonic curvature (NH) anticyclonic curvature (SH) !!!

12. 2/9/10MET 61 topic 02 12 Math where  is an angle which defines the flow direction (relative to something…Fig. 7.1)

13. 2/9/10MET 61 topic 02 13 Diffluence / confluence Physically Relates to some measure of “spreading out” of flow direction  the flow (and vice versa) Example:

14. 2/9/10MET 61 topic 02 14 Math

15. 2/9/10MET 61 topic 02 15 Stretching Physically Relates to wind speed changes downstream Example:

16. 2/9/10MET 61 topic 02 16 Math

17. 2/9/10MET 61 topic 02 17 Streamlines Streamlines are lines drawn such that at any point, the actual (horizontal) wind is parallel to the streamline. Indicate flow direction. Can be made to indicate flow speed via spacing (closer spacing  higher wind speed)/ Only valid at an instant in time!

18. 2/9/10MET 61 topic 02 18 http://weather.unisys.com/surface/sfc_con_stream.html http://www.weatheronline.co.uk/cgi- bin/expertcharts?LANG=en&MENU=0000000000&CONT=samk&MOD ELL=gfs&MODELLTYP=1&BASE=- &VAR=w010&HH=48&ZOOM=0&ARCHIV=0

19. 2/9/10MET 61 topic 02 19 Vorticity Physically A measure of spin of an individual fluid parcel NOT (necessarily) rotation in the entire fluid! The sum of a shear effect and a curvature effect

20. 2/9/10MET 61 topic 02 20 Math Where u = east-west wind (u > 0 for eastward flow) v = north-south wind (v > 0 for northward flow) Typical values:  10 -5 s -1.

21. 2/9/10MET 61 topic 02 21 Fig. 7.2a – shear  vorticity imagine a “paddlewheel” in the flow spin indicates vorticity Fig. 7.2b – shear + curvature  vorticity Fig. 7.2c – zero shear, zero curvature  zero vorticity

22. 2/9/10MET 61 topic 02 22 Vorticity distributions and forecasting? Earliest forecasts were for future vorticity distributions!! http://www.aip.org/history/sloan/gcm/prehistory.html J.G. Charney et al integrated (solved) an equation for the evolution of vorticity  a primitive forecast (c. 1950, using ENIAC) ENIAC… http://en.wikipedia.org/wiki/ENIAC http://en.wikipedia.org/wiki/ENIAC and… http://en.wikipedia.org/wiki/Numerical_weather_prediction http://en.wikipedia.org/wiki/Numerical_weather_prediction

23. 2/9/10MET 61 topic 02 23 Vorticity distributions and “weather”? http://www.met.sjsu.edu/weather/gfsp.html High positive vorticity associated with troughs and regions of active “weather” High negative vorticity associated with ridges and regions of calm “weather”

24. 2/9/10MET 61 topic 02 24 Divergence Physically A measure of the tendency of flow to spread out (diverge) from a location or converge towards a location negative divergence = convergence The sum of diffluence and stretching effects. Example: Fig. 7.2c

25. 2/9/10MET 61 topic 02 25 Math Typical values:  10 -6 s -1.

26. 2/9/10MET 61 topic 02 26 Divergence values – positive or negative – difficult to discern from looking at wind obs Examples: http://weather.uwyo.edu/models/fcst/ukmet.html

27. 2/9/10MET 61 topic 02 27 Imagine putting an elastic sheet into a divergent flow. The flow will stretch & deform the sheet. The area of the sheet will increase.

28. 2/9/10MET 61 topic 02 28 It can be shown that divergence (  ) is related to area change by: Divergence (  > 0)  dA/dt > 0  area increases Convergence (  < 0)  dA/dt < 0  area decreases

29. 2/9/10MET 61 topic 02 29 Flow deformation Consider again this “stretchy area” idea… Fig. 7.3 shows how the area could be deformed by a complex flow involving all the elements in Table 7.1

30. 2/9/10MET 61 topic 02 30 Flow deformation can lead to sharpened gradients… Fig. 7.4a … north-south temperature gradient is sharpened by the flow  Frontal zone …zone in which frontal disturbances could develop (“frontogenesis”)

31. 2/9/10MET 61 topic 02 31 Streamlines and trajectories Streamlines are only valid at an instant in time! As time evolves, a parcel’s motion is defined by a trajectory. See example on the board…

32. 2/9/10MET 61 topic 02 32 Horizontal Equation of Motion As discussed earlier, we can: develop an equation of motion using Newton’s 2 nd Law of Motion… And use it to understand and forecast motions

33. 2/9/10MET 61 topic 02 33 Newton’s 2 nd Law of Motion where “Fi” refers to the various forces at work.

34. 2/9/10MET 61 topic 02 34 Step 1 = identify the forces physically –e.g., Newton’s 2 nd Law of Motion –e.g., Law of Gravitation mathematically –F = m.a –g = see below!

35. 2/9/10MET 61 topic 02 35 Step 2 = slot expressions into eqn. (*) above Step 3 = solve the equation Solution is wind vector at any location and future time Vector wind → u, v components

36. 2/9/10MET 61 topic 02 36 Real Forces These are very basic forces: a)gravity (same as gravitation??? No!) b)friction c)(air) pressure gradient force

37. 2/9/10MET 61 topic 02 37 Pressure gradient force We met the hydrostatic equation in §3.2 Equates the (downward directed) force of gravity with the (upward directed) pressure gradient force

38. 2/9/10MET 61 topic 02 38 The horizontal pressure gradient forces (PGF) can be expressed similarly: In vector form:

39. 2/9/10MET 61 topic 02 39 In MET 121A,B we will show that the PGF can be written as: when “z” is the vertical coordinate (p = pressure) when “p” is the vertical coordinate (z = altitude) when “p” is the vertical coordinate (Z = geopotential height, Z  z) when “p” is the vertical coordinate (  = geopotential)

40. 2/9/10MET 61 topic 02 40 Gravitational force Given by Newton’s Law of Gravitation: Gravitational force between two objects  product of two masses (e.g., earth and moon)  1 / (distance separating the masses) 2 What we experience is pure gravitation modified by earth’s rotation … see below!

41. 2/9/10MET 61 topic 02 41 Frictional force Maximizes @ surface where wind speeds MUST be zero! Insignificant above the boundary layer (typically about 1 km)

42. 2/9/10MET 61 topic 02 42 The value of the force is (see MET 121A,B etc.): Here,  is the shear stress = the stress associated with vertical wind shear Thus, stress  vertical wind shear

43. 2/9/10MET 61 topic 02 43 Apparent Forces These arise due to Earth’s rotation: a)Coriolis b)A centrifugal force which modifies pure gravitation → gravity (which we experience)

44. 2/9/10MET 61 topic 02 44 Coriolis force Causes a deflection of moving air Acts in a direction  direction of motion (i.e., left/right, up/down)

45. 2/9/10MET 61 topic 02 45 Value: With

46. 2/9/10MET 61 topic 02 46 f = 0 @ equator no Coliolis deflection @ equator hurricanes do not form @ equator f has max value @ poles  =7.292 x 10 -5 s -1 = 2  / “day” f > 0 in northern hemisphere (NH) f < 0 in SH

47. 2/9/10MET 61 topic 02 47 Centrifugal force Due to planet’s rotation, a parcel is subject to gravity and a centrifugal force Fig. 7.6 The net effect is what we call GRAVITY.

48. 2/9/10MET 61 topic 02 48 Equation of Motion Substituting all these expressions into (*) above gives:

49. 2/9/10MET 61 topic 02 49 In component form: NEXT…seek solutions! Too complicated  Instead – make simplifying assumptions

50. 2/9/10MET 61 topic 02 50 Looks like an ODE Is actually a PDE – much harder to solve One more thing…

51. 2/9/10MET 61 topic 02 51 We can show (METR 121) that Time derivative at a fixed location (e.g., as measured by instruments) = Eulerian derivative Spatial derivative at a fixed time = (negative of) advection Time derivative moving with the flow = Lagrangian derivative

52. 2/9/10MET 61 topic 02 52 Example with u wind (u) in our Equation of Motion on the LHS…

53. 2/9/10MET 61 topic 02 53 2 nd example with temperature (T) … Rate of change of T at a fixed location Rate of change of T following the motion advection

54. 2/9/10MET 61 topic 02 54 Large-scale winds Examine: http://www.met.sjsu.edu/weather/models/gfsone-00/all500relh.html We see: Flow roughly west → east with low heights on left (NH) With meanders Winds  contours “everywhere” Winds stronger when contours closely- spaced

55. 2/9/10MET 61 topic 02 55 What do the contours indicate? The height of the 500 hPa surface above MSL Compare to earth topo maps… http://www.digital-topo-maps.com/ Compare to 500 hPa maps… http://www.met.sjsu.edu/weather/models/gfsp/avort500f00.gif

56. 2/9/10MET 61 topic 02 56 Large-scale winds Examine: http://www.met.sjsu.edu/weather/models/gfsone-00/all500relh.html We see: Flow roughly west → east with low heights on left (NH) (why?) With meanders (why?) Winds  contours “everywhere” (why?) Winds stronger when contours closely- spaced (why?)

57. 2/9/10MET 61 topic 02 57 The geostrophic wind 1.Suppose we look at typical values of winds etc. for larger-scale motions in mid-latitudes. 2.And compute typical values of terms in (7.13) 3.And neglect smaller terms in (7.13) 4.And analyze the solution to modified-(7.13)

58. 2/9/10MET 61 topic 02 58 Wind speed  10 m/s And wind variations take place over a day  10 5 seconds Thus,  dV/dt   10 -4 m/s 2 But, f  10 -4, so  f V   10 -3 m/s 2 Now look back at (7.13) and neglect friction

59. 2/9/10MET 61 topic 02 59 10 -4 m/s 2 10 -3 m/s 2 We conclude that the pressure gradient term MUST also be of size  10 -3 ms -2

60. 2/9/10MET 61 topic 02 60 So an approximate equation is: In this case, we can actually solve for the wind components – u g and v g ! This wind is called the geostrophic wind.

61. 2/9/10MET 61 topic 02 61 The geostrophic wind.

62. 2/9/10MET 61 topic 02 62 Properties… The geostrophic wind blows parallel to isobars (height lines) with lower pressures (heights) on the left in the NH (right/SH). Closely spaced isobars (height lines)  stronger pressure gradient  stronger winds

63. 2/9/10MET 61 topic 02 63 Comparing with observations away from the surface (friction)  actual winds  geostrophic winds in speed & direction (to within about 10%) Thus we often “replace” actual winds with geostrophic winds

64. 2/9/10MET 61 topic 02 64 Adding friction… Observations http://www.met.sjsu.edu/weather/models/gfsone-12/allsfcdwpf.html Results same as with geostrophic wind BUT: Cross-isobar flow towards low pressure Cross-isobar flow away from high pressure

65. 2/9/10MET 61 topic 02 65 “F” acts to oppose flow 3-way force balance (“F”, “CF”, “PGF”) Fig. 7.10 Net results of adding friction: Speeds? –Reduced (as expected!)

66. 2/9/10MET 61 topic 02 66 Direction? –Friction induces flow towards lower pressure –This gives cross-isobar flow –Towards low pressure –Away from high pressure Cross-isobar flow angle can be as much as  45  Depends on surface roughness (parameter)

67. 2/9/10MET 61 topic 02 67 The gradient wind Similar to the geostrophic wind BUT with curvature effects included Flow around a LOW implies an extra force of the form: “cyclostrophic” component

68. 2/9/10MET 61 topic 02 68 Since flows are often curved (isobars and height lines are typically NOT straight), the gradient wind is a better approximation to the real wind. However – the math in this case is more complicated. {See 121A for derivation. See Eq. (7.17) for solution}

69. 2/9/10MET 61 topic 02 69 Some relationships we can derive: For cyclonic flow (NH), Gradient wind speed < Geostrophic wind speed We say the wind speed is subgeostrophic. Calculated from isobar spacing Calculated from isobar spacing and V 2 /R term

70. 2/9/10MET 61 topic 02 70 For anti cyclonic flow (NH), Gradient wind speed > Geostrophic wind speed Here, the wind speed is supergeostrophic. Calculated from isobar spacing Calculated from isobar spacing and V 2 /R term

71. 2/9/10MET 61 topic 02 71 The thermal wind Observations show that when there is a north-south temperature gradient There is also a vertical wind shear. Ditto with east-west temp. gradient WHY???

72. 2/9/10MET 61 topic 02 72 1 st - observations Best example = zonal average winds and temperatures (Fig. 1.11) = east-west average Strongest temperature gradient is in mid-latitudes Strongest vertical wind shear @ same location!

73. 2/9/10MET 61 topic 02 73 2 nd – “math” a)We take the geostrophic wind expressions (7.15b) b)We take the “vertical” derivatives c)We get a relationship between:  u g /  p and  T/  x and between  v g /  p and  T/  y (derive in 121A)

74. 2/9/10MET 61 topic 02 74 These allow us to write (7.19b): where Z is geopotential height (Z  z) and “2” and “1” refer to two different pressure levels.

75. 2/9/10MET 61 topic 02 75 Can use to find u @ one level, given u @ second level AND if the north-south height gradient is known. Can use to find the north- south height gradient if winds are known at two levels. Again…

76. 2/9/10MET 61 topic 02 76 3 rd – “physics” pressure  as z  - and more rapidly in a colder column of air Example: Consider the 1000-500 hPa layer Imagine this layer from the (cold) pole to the (warm) equator 1000 hPa 500 hPa Conclusion: slope =  Z/  y   T/  y Warm, “deep” layer Cold, shallow layer

77. 2/9/10MET 61 topic 02 77 Example Ex. 7.3 on p. 284 Given: zonally-averaged temperature gradient Given: geostrophic wind @ surface  zero Compute: geostrophic wind @ 200 hPa

78. 2/9/10MET 61 topic 02 78 First, expand Eq. 7.20 → components Only in this case