1. A&OS C110/C227: Review of thermodynamics and dynamics III Robert Fovell UCLA Atmospheric and Oceanic Sciences rfovell@ucla.edu 1

2. Notes Everything in this presentation should be familiar Please feel free to ask questions, and remember to refer to slide numbers if/when possible If you have Facebook, please look for the group “UCLA_Synoptic”. You need my permission to join. (There are two “Robert Fovell” pages on FB. One is NOT me, even though my picture is being used.) 2

3. Hydrostatic and hypsometric equations 3

4. Hydrostatic equation Represents balance (stalemate) between vertical pressure gradient force and gravity resulting (formally) in no vertical acceleration and (practically) in no vertical motion 4 g = 9.81 m s -2 at sea-level

5. Hypsometric equation Derived from the hydrostatic equation, the hypsometric (Greek, to measure height) or thickness equation is where T v is the layer mean virtual temperature of the p 0 -p 1 layer, p 0 > p 1, and ∆Z is the thickness of that layer Based on the idea that between two pressure levels, there is a fixed amount of mass and that mass occupies a greater depth as it becomes warmer and/or more moist ∆Z is really measured in “geopotential meters”, a gravity- adjusted height, but its units evaluate as meters 5

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7. 7 How temperature influences 1000-500 mb thickness

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9. 9 Temperature differences make pressure differences, which drive winds Note the 500 mb surface tilts downward towards the colder air. A pressure difference at constant height is directly relatable to a height difference on a constant pressure (isobaric) surface

10. Equations of motion 10

11. Forces: real and imaginary Real forces: PGF, gravity, friction Imaginary forces: Coriolis, centrifugal, curvature Vectors will be expressed with bold type or overlying arrows Centrifugal force owing to Earth rotation is  2 R, where  is the Earth’s angular velocity (2p radians per day) and R is the distance vector to the Earth’s axis of rotation Apparent gravity g = true gravity force (g*) + centrifugal force  2 R (see next slide) 11

12. Apparent gravity Picture a perfect sphere, in quarter section Note that true gravity g* is directed towards Earth’s center Note centrifugal force  2 R is directed perpendicular to rotation axis As a consequence, apparent gravity g does not point towards center of a perfect sphere Is it true that objects on the rotating Earth, influenced by apparent gravity, do not fall straight down? 12

13. Inertial and rotating frames 13 3D vector velocity relative to the rotating Earth 3D absolute velocity in the inertial reference frame Total derivative with respect to time in rotating Earth frame Absolute total derivative with respect to the in inertial frame

14. 14 Newton’s laws determine this What we need to make forecasts -1 x Coriolis -1 x centrifugal Due to Earth’s rotation Start with relationship for vector A between absolute motion and Earth-relative motion the “tool" We will soon see that…

15. Position vector Let r be the position vector of an object, extending from Earth center Object motion represents (1) that due to Earth rotation (2) that due to motion relative to Earth Use the “tool” with r as A 15

16. 16 Use the “tool” on U a : After utilizing (*) above and skipping steps, we find the expected equation (*)

17. 17 PGF true gravity friction Plug the above into (*), solve for dU/dt, and combine true gravity and centrifugal into apparent gravity g:

18. 18 Pressure gradient force Coriolis force

19. 19 Earth rotation gave rise to the Coriolis terms Earth’s sphericity give rise to the curvature terms Acceleration relative to flat Earth + acceleration due to curvature Flat Earth Curvature terms r = a + z a = Earth radius z = height above surface

20. Scale analysis of the equations of motion 20

21. Synoptic-scale scale analysis 21 Termvaluedescription  10 -4 s -1 Earth angular velocity a10000 kmEarth radius  45˚Nmidlatitudes U, V10 m/sHorizontal velocity W1 cm/sVertical velocity L10 6 mHorizontal length scale H10 4 mVertical length scale  1 kg m -3 Sea-level air density ∆p10 mb = 1000 PaHorizontal pressure difference ∆t100000 sTime scale = L/U

22. Horizontal equations’ 3 leading terms on synoptic scale 22 10 -4 s -1 10 -3 s -1 10 -3 s -1 You can make a horizontal time scale with L/U, so dU/dt = U/(L/U) = U 2 /L = (10) 2 (m/s) 2 /(10000 m) =10 -4 s -1 Break into x- and y-components, as below:

23. Horizontal equations’ 2 leading terms = geostrophic balance 23 where Geostrophic = Greek, Earth turns Note Coriolis is always pared with u, v; it is proportional to wind speed

24. Horizontal equations’ 3 leading terms, rewritten 24 Deviations from geostrophic balance cause accelerations Accelerations serve to bring atmosphere back towards geostrophy Note accelerations are an order of magnitude smaller than winds (see two slides back) CAUTION: only applies if our synoptic scale analysis is valid

25. Vertical equation of motion (part) 25 10 -7 s -1 10 s -1 10 s -1 On the synoptic scale, atmosphere is very nearly in hydrostatic balance For synoptic scale motions, we cannot use departures from hydrostatic balance to compute vertical accelerations or vertical velocity

26. How large-scale flow becomes geostrophic 26

27. 27 Start with an air parcel subjected to a PGF The parcel starts to move towards L pressure

28. 28 If the time scale of the motion is sufficiently large, Coriolis accelerations become important Coriolis always acts to the right following the motion, in the Northern Hemisphere

29. 29 This combination of forces causes the parcel to start deflecting away from low pressure

30. 30 Coriolis again acts to the right. Notice the change in the angle between the two force vectors

31. 31 As a consequence, the air parcel is turning even more away from L pressure

32. 32 You can see this process will continue until the parcel is traveling parallel to the isobars

33. 33 At that point, PGF and Coriolis are in opposition, and directional change ceases

34. 34 This adjustment process has brought the wind into geostrophic balance Typically, we see the already adjusted wind, but this process can be detected in diurnally-driven circulations, such as the sea-breeze

35. Isobar spacing represents PGF 35

36. Isobar spacing represents PGF 36

37. Gradient, cyclostrophic and inertial winds, and friction 37

38. Natural coordinates A coordinate system that “goes with the flow”, parallel to isobars Unit vector l points in direction of flow Unit vector n is perpendicular to the flow, positive to the left Wind speed V ≥ 0 R = radius of curvature R > 0 CCW R < 0 CW R = ∞ when isobars straight 38

39. Flow equation 39 Centripetal Coriolis PGF A B C Since the geostrophic wind can be defined as: when R = ∞ (straight-like flow) V = V g

40. Combinations 40 Centripetal Coriolis PGF A B C Combinations B, C = geostrophic wind (“Earth turns”) A, B, C = gradient wind A, C = cyclostrophic wind (“to turn in a circle”) A, B = inertial flow Alternate version:

41. Gradient vs. geostrophic wind 41 Cyclonic flow: R > 0 so gradient wind is subgeostrophic for same PGF Anticyclonic flow: R < 0 so gradient wind is supergeostrophic for same PGF

42. A closer look… Consider cyclonic (CCW) flow By itself, inertia takes a parcel on a straight- line path, at constant speed However, this path would cross isobars, towards higher pressure 42

43. 43 Start with geostrophic balance, straight-line flow parallel to isobars PGF is pointing to the left of the wind, towards L pressure. Coriolis force (CF) points to the right, following the motion, and opposite to PGF

44. 44 Now make the isobars curve Inertia is carrying our air parcel towards the curving isobars As it approaches, what happens to the PGF and Coriolis forces, and why?

45. 45 PGF always points directly towards lower pressure Note a component of the PGF is now pointing against the parcel’s motion… which slows it down The Coriolis force is reduced, as it is proportional to wind speed

46. 46 PGF gains the “upper hand” over Coriolis As a consequence, the parcel eases into a CCW turn, with Coriolis again to the right of the motion AND opposing the PGF To accomplish this CCW turn, the parcel must slow down

47. 47 Centripetal Coriolis PGF A B C In this point of view, the centripetal force is the force imbalance that develops between PGF and CF owing to isobar curvature Centripetal force does not exist until something has a spinning/curving motion. This explanation offers why the curving motion emerged

48. Cyclostrophic flow 48 For small-scale flow, Coriolis is negligible and curving flow represents PGF and centripetal forces Solving for V yields two roots, but V ≥ 0

49. Cyclostrophic: by itself, spin makes low pressure 49 L L V ≥ 0 always, n positive to left of flow Radicand must be positive Thus, R and dp/dn must have opposite signs

50. How friction affects the large-scale wind 50

51. 51 Start with geostrophic balance How would friction alter this balance? Friction slows the wind, thereby reducing the Coriolis force However, friction has no immediate effect on the PGF

52. 52 As a consequence, the wind slows and turns towards lower pressure Note the Coriolis force is acting to the right of the wind, and not opposite the PGF Now there is a wind component from H to L Because of surface friction, air can flow out of surface anticyclones, and into surface cyclones

53. 53 Gradient wind flow around synoptic-scale L and H Isobar spacing is the same, so the CCW flow around the L is somewhat slower than CW flow around H

54. 54 Flow around synoptic-scale L and H after inclusion of surface friction

55. Friction acts… …near the surface, generally in lowest 1 km or so, but varies with space and time On hot days, the boundary layer grows vertically, mixing more air downward to the surface In very stable, nocturnal boundary layers, winds closer to the surface are far less affected by friction When plotting isobaric charts, keep in mind how close a given pressure level may be from the ground! 55

56. Question for thought 56

57. Picture 1 57 For equal isobar spacing, flow around cyclones is slower than flow around anticyclones

58. Picture 2 58 In reality, PGFs and flows around cyclones are typically much stronger than around anticyclones. Why?

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