1. AOSS 401, Fall 2007 Lecture 28 November 30, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu 734-936-0502

2. Class News November 30, 2007 Homework 7 (Posted Thursday) –Due Next Friday Important Dates: –December 10: Final Exam –December 7: Go over homework Review session –December 5: Hurricanes Joint with AOSS 451

3. Weather National Weather Service –http://www.nws.noaa.gov/http://www.nws.noaa.gov/ –Model forecasts: http://www.hpc.ncep.noaa.gov/basicwx/day0- 7loop.html http://www.hpc.ncep.noaa.gov/basicwx/day0- 7loop.html Weather Underground –http://www.wunderground.com/cgi- bin/findweather/getForecast?query=ann+arborhttp://www.wunderground.com/cgi- bin/findweather/getForecast?query=ann+arbor –Model forecasts: http://www.wunderground.com/modelmaps/maps.asp ?model=NAM&domain=US http://www.wunderground.com/modelmaps/maps.asp ?model=NAM&domain=US

4. Material from Chapter 11 Tropics –Tropics versus middle latitudes –Features of the tropical circulation Tropical scale analysis Tropical waves –Kelvin waves –Equatorial Rossby Waves –Mixed Rossby-gravity Waves

5. Some remembering

6. What are the differences between the tropics and the middle latitudes on Earth? Tropics: –The area of the tropics – say + and – 30 degrees latitude is half the area of the Earth. –Might say the tropics is + and – 20 degrees of latitude, and subtropics are between 20 and 30 degrees of latitude. The importance of rotation, the Coriolis parameter. What else is different?

7. Differences between the tropics and middle latitudes The contrast between summer and winter is not as large as at middle and high latitudes. –There is lot of solar heating. There is a lot of water! –What is the “physical” difference between water and land? –Sea surface temperature is important to dynamics. What happens to water when it is warm?

8. Tropics and middle latitudes In middle latitudes the waves grow from the energy available in the baroclinic atmosphere. –horizontal temperature gradients scale is large latent heat release is on scales small compared to baroclinic energy convergence. In the tropics the horizontal temperature gradients are small.

9. What does importance of latent heat release mean. Diabatic processes are more important in the tropics. Hence, vertical velocity is more strongly related to diabatic heating than to temperature advection. –What about divergence? The scale of the forcing of motions is small –Related to the phase change of water.

10. Let’s get these ideas through scaling the equation.

11. Equations of motion in pressure coordinates (using Holton’s notation)

12. Equations of motion in log pressure coordinates (using Holton’s notation)

13. Introduce another vertical coordinate.

14. Scale factors for “large-scale” tropics MISTAKE IN LAST LECTURE. Corrected on ctools

15. Rossby number: Mid-latitudes Compare relative vorticity to planetary vorticity In mid-latitudes planetary vorticity is larger than relative vorticity.

16. Rossby number: Tropics Compare relative vorticity to planetary vorticity In tropics planetary vorticity is comparable to relative vorticity.

17. Coriolis force Can we say that the advection of planetary vorticity is less important? –Advection of planetary vorticity is comparable to advection of relative vorticity

18. Scaling: momentum equation

19. Geostrophic balance is not dominant. –How many km from the equator is geostrophic term no longer small? –What about  ? If the pressure gradient is balanced in the momentum equation, then...

20. This means something! For a similar scale disturbances in the tropics and middle latitudes the geopotential perturbation is a smaller by an order of magnitude in the tropics. What does this mean –for the scales of motion? –for the important physical terms?

21. Use the hydrostatic equation to say something about temperature The temperature variability in tropical systems of scale H, are very small.

22. Thermodynamic equation

23. Diabatic scale: Radiative

24. Go back to the scaling of the momentum equation Vertical advection is very small.

25. I need to pull all of this together. For a similar scale disturbances in the tropics and middle latitudes the geopotential perturbation is a smaller by an order of magnitude in the tropics. The temperature variability in tropical systems of scale H, are very small. Vertical advection is very small.

26. Remember vorticity and divergence. Remember that we earlier said that the flow could be defined as the sum of the rotational flow and the irrotational flow. –rotational  vorticity –irrotational  divergence

27. What is the scale of divergence and vorticity? So --- the divergence relative to the vorticity is even smaller than in middle latitudes. The flow is also quasi-nondivergent.

28. Pulling it all together For a similar scale disturbances in the tropics and middle latitudes the geopotential perturbation is a smaller by an order of magnitude in the tropics. The temperature variability in tropical systems of scale H, are very small. Vertical advection is very small. The flow is quasi-nondivergent.

29. Momentum equation: approximately

30. Make a vorticity equation: A VORTICITY EQUATION: Absolute vorticity conserved The temperature variability in tropical systems of scale H, are very small. The flow is quasi-nondivergent.Vertical advection is very small. That looks like a good study question: Derivation. Page 389-390, text.

31. Thinking about the tropics These disturbances are nearly barotropic. There is no mechanism for these disturbances to convert potential energy to kinetic energy. Yet, we know there are lots of disturbances in the tropics. What does this mean?

32. New Professor Trick Answer will be written in class.

33. Story time? Okay the last time I tried this, it didn’t really work, but it means that we are making something of a change of subject. Story topics: –Bowling? –What is your teddy bear named? –How about Britney’s new album?

34. Let’s think about waves some more We assume that dependent variables like u and v can be represented by an average and deviation from the average.

35. Let’s think about waves some more Some fundamental ideas

36. Linear perturbation theory Assume: variable is equal to a mean state plus a perturbation With these assumptions non-linear terms (like the one below) become linear: These terms are zero if the mean is independent of x. Terms with products of the perturbations are very small and will be ignored

37. Let’s think about waves some more Some fundamental ideas. –Waves have some sort of restoring force Buoyancy waves: gravity Rossby waves: The gradient of planetary vorticity –Think about the conservation of potential vorticity –Waves tend to grow and decay at the expense of the “energy” in the mean state. –Waves tend to respond to out of balance situations. –Waves tend to move things towards equilibrium –Waves propagate So they can communicate things happening in one part of the fluid to far away places.

38. Once long ago: Lecture 17

39. A simple version of potential vorticity Assume constant density and temperature. We can only do this for a SHALLOW layer of fluid.

40. A simple version of potential vorticity Integrate with height,z 1  z 2 over a layer of depth H.

41. A simple version of potential vorticity Integrate with height,z 1  z 2 over a layer of depth H. Why can we do this?

42. Most general form of the shallow water equations with variable bottom topography –h: depth of the fluid –h topo : height of bottom topography Shallow water equations h h topo

43. Shallow water equations The shallow water equations are a good framework for exploring waves. –In the tropics, for example, we have just seen that we have an approximately barotropic atmosphere  and the shallow water system is barotropic. –Could view that the atmosphere is a set of shallow water layers, one on top of another. This is a result of the hydrostatic balance.

44. Momentum equation: approximately

45. Going to consider equatorial waves Waves –Kelvin waves (trapped waves): coastal Kelvin waves (in the ocean!) equatorial Kelvin waves –Equatorial Rossby (ER) and Mixed Rossby- Gravity (MGR) waves

46. Kelvin waves are trapped gravity waves A trapped wave is one that decays exponentially in some direction Kelvin waves need a boundary to exist Observed in the ocean and the atmosphere –Coastal Kelvin waves –Equatorial Kelvin waves in the ocean and atmosphere Kelvin waves

47. Amplitudes decay away from the boundary (coastline) Coastal Kelvin waves

48. Amplitudes decay away from the ‘’boundary’’ (equator) Equatorial Kelvin waves

49. Connection between coastal and equatorial Kelvin waves: Coastal Kelvin waves can turn the corner and circulate counterclockwise in northern hemisphere around a closed basin Coastal & Equatorial Kelvin waves Important for El Nino

50. For coastal Kelvin waves assume: –Flat bottom topography –constant Coriolis parameter f 0 –Coast parallel to the y-axis –zonal velocity (normal to the coast): u=0 (everywhere) Coastal Kelvin waves: Derivation (1) u=0 coast

51. Linearize shallow water equations about a state at rest with mean height H Compute Coastal Kelvin waves: Derivation (2)

52. Yields wave equation Phase speed c is the shallow water gravity speed in a non-rotating fluid, non-dispersive General solution consists of two waves, traveling in the positive and negative y direction For solutions steadily translating at phase speeds ±c Coastal Kelvin waves: Derivation (3)

53. Continuity equation implies Utilize geostrophic balance (eq. 1) in the x-direction (perpendicular to the coast) Integrate with respect to x. Solutions with arbitrary dimensionless wave functions F and G (dependent on y only) and maximum velocity c (at x=0): Coastal Kelvin waves: Derivation (4)

54. Rossby radius of deformation Of the two independent solutions, the second increases exponentially with distance to the shore: declared physically unfit Therefore, most general (physical) solution with arbitrary (dimensionless) wave function F in y: Coastal Kelvin waves: Derivation (5)

55. Exponential decay away from coast: trapped The Rossby radius of deformation is a measure of the trapping distance In the longshore direction, the Kelvin wave travels without distortion at the speed of surface gravity waves In NH (f > 0): wave travels with coast on its right In SH (f < 0): wave travels with coast on its left Decay of Kelvin wave amplitude away from the coast manifested in the English Channel: –North Atlantic Tide enters Channel from the west, tide assumes the character of a Kelvin wave –Kelvin wave leans against the coast on its right (France), explains higher tides in France Coastal Kelvin waves: Facts

56. English Channel: Atlantic tides enter from the west Higher tide amplitudes (m) in France due to Kelvin waves Coastal Kelvin waves 6 5 4 4 5 6 3 2

57. Equatorial waves are important class of eastward and westward propagating disturbances Present in atmosphere and ocean Trapped about the equator (they decay away from the equator) Types of waves: –Equatorial Kelvin waves –Equatorial Rossby (ER), Mixed Rossby-Gravity (MWR), inertia-gravity waves Atmospheric equatorial waves excited by diabatic heating by organized tropical convection Oceanic equatorial waves excited by wind stresses Waves communicate effects of convective storms over large longitudinal distances Equatorial waves

58. For equatorial Kelvin waves assume: –Flat bottom topography –Coriolis parameter at the equator is approximated by equatorial  -plane (with f 0 =0, ) –Meridional velocity vanishes: v = 0 (everywhere) –Shallow water equations become: Equatorial Kelvin waves: Derivation (1) v=0 equator

59. Linearize shallow water equations about a state at rest with mean height H: Compute Yields Equatorial Kelvin waves: Derivation (2) c = phase speed

60. Seek wave solutions of the form (allow amplitudes to vary in y): Yields the system Rearrange (1): Plug (4) into (2): Equatorial Kelvin waves: Derivation (3)

61. (5) can be integrated immediately: amplitude function (with u 0 : amplitude of the perturbation at the equator) Solutions decaying away from the equator exist only for c > 0 with Therefore: Atmospheric Kelvin waves always propagate eastward Their zonal velocity and geopotential vary in latitude as Gaussian functions centered at the equator e-folding decay width is Equatorial Kelvin waves: Derivation (4) e.g. for c=30 m/s, Y k =1600 km

62. Amplitude of the height perturbations (use eq. 3): The physical solutions are Equatorial Kelvin waves: Derivation (5)

63. Equatorial Kelvin waves: Velocity and height perturbations L H Equator Kelvin wave animation: http://www.ems.psu.edu/%7Ebannon/mpegs/eqkel.mpg http://www.ems.psu.edu/%7Ebannon/mpegs/eqkel.mpg

64. Wow, that’s a lot! Professor Rood, Professor Rood, is there just one more wave we can learn about before the weekend?

65. The human waves follow the cheerleaders follow the cheerleaders No joke! Real research on stadium waves (La Ola) –http://www.sciencenews.org/articles/20020914/mathtrek.asphttp://www.sciencenews.org/articles/20020914/mathtrek.asp –http://angel.elte.hu/wave/index.cgi?m=modelshttp://angel.elte.hu/wave/index.cgi?m=models –Read the article in Nature http://angel.elte.hu/wave/download/article/MexWave.pdf http://angel.elte.hu/wave/download/article/MexWave.pdf