AOSS 401, Fall 2007 Lecture 28 November 30 , 2007

1. AOSS 401, Fall 2007Lecture 28November 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/ • Model forecasts: http://www.hpc.ncep.noaa.gov/basicwx/day0-7loop.html • Weather Underground • http://www.wunderground.com/cgi-bin/findweather/getForecast?query=ann+arbor • Model forecasts: 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-latitudesCompare relative vorticity to planetary vorticity In mid-latitudes planetary vorticity is larger than relative vorticity.

16. Rossby number: TropicsCompare 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. Scaling: momentum equation • Geostrophic balance is not dominant. • How many km from the equator is geostrophic term no longer small? • What about b? • 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 scalingof 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: That looks like a good study question: Derivation. Page 389-390, text. 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.

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? 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) becomelinear: 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,z1 z2 over a layer of depth H.

41. A simple version of potential vorticity Integrate with height,z1 z2 over a layer of depth H. Why can we do this?

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

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 • 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

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

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

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

50. Coastal Kelvin waves: Derivation (1) • For coastal Kelvin waves assume: • Flat bottom topography • constant Coriolis parameter f0 • Coast parallel to the y-axis • zonal velocity (normal to the coast): u=0 (everywhere) u=0 coast