Quasi-geostrophic Analysis of Atmospheric Dynamics

1. AOSS 401Geophysical Fluid Dynamics:Atmospheric DynamicsPrepared: 20131121Quasi-geostrophic / Analysis / Weather Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Cell: 301-526-8572

2. Class News • Ctools site (AOSS 401 001 F13) • Second Examination on December 10, 2013 • Homework • Homework due November 26, 2013

3. Weather • National Weather Service • Model forecasts: • Weather Underground • Model forecasts: • NCAR Research Applications Program

4. So when you are through • You will know how to do scale analysis • You will know how to derive the vorticity equation • You will know the wave equation and how to seek “wave-like” solution • Something more about weather and climate

5. Some fundamental notions you will learn. • The importance of the conservation equation • Atmospheric motions organize in distinct spatial and temporal scales • Most of the dynamic disturbances of the atmosphere can be classified as either: • Waves • Vortices • There is a mean circulation of the atmosphere which is known as the general circulation. • What does this do? • The atmosphere has two dominate balances, at least away from the tropics: • Hydrostatic balance • Geostrophic balance • It is the deviations from this balance which we are most interested in.

6. Links for today • http://www.wunderground.com/maps/ • http://www.ecmwf.int/products/forecasts/d/charts/medium/deterministic/msl_uv850_z500!Geopotential%20500%20hPa%20and%20Temperature%20at%20850%20hPa!0!North%20hemisphere!pop!od!oper!public_plots!2013120500!!/ • http://www.weather-forecast.com/maps/Arctic

7. Outline • Analysis of equations of atmospheric motion scaled for large-scale middle latitude dynamics / Quasi-geostrophic formalism • Review of large-scale formalism • Long and short waves / Barotropic waves • Examine ageostrophic wind • Cyclone development • Occlusion • Baroclinic  Barotropic /// energy conversion • Vertical motion

8. Vorticity Equation DIVERGENCE TILTING SOLENOIDAL or BAROCLINIC • Changes in relative vorticity are caused by: • Divergence • Tilting • Gradients in density on a pressure surface • Advection

9. QG Theory: Assumptions • Assume the horizontal wind is approximately geostrophic • Scale the material derivative • Neglect the vertical advection • Horizontal advection due to geostrophic wind • Assume the north-south variation of the coriolis parameter is constant • Divergence in the continuity equation only due to ageostrophic wind • Modify the thermodynamic equation • Advection by the geostrophic wind • Assume hydrostatic balance • Vertical velocity acts on a mean static stability profile

10. The quasi-geostrophic (QG) equations momentum equation geostrophic wind continuity equation thermodynamicequation

11. Vorticity Advection Thickness Advection Geopotential tendency equation • Linear partial differential equation for geopotential tendency. • Given a geopotential distribution at an initial time, can compute geopotential distribution at a later time. • The right hand side is like a forcing.

12. First major set of conclusions from the quasi-geostrophic system We see that • Geostrophic advection of geostrophic vorticity causes waves to propagate • The vertical difference in temperature (thickness) advection causes waves to amplify

13. Remember our scaled vorticity equation? We see that the QG vorticity equation is very similar to the scaled vorticity equation we developed before …with a few additional assumptions

14. Long and Short Waves • In this discussion these are both “large-scale” meaning that rotation is important. • Therefore, this is a “short” large-scale wave versus a “long” large-scale wave.

15. Barotropic Wave Dispersion • Look at the barotropicwave equation

16. Consider a barotropicfluid Think here

17. Assume a “wave like solution”(get used to this…) Dispersion relation. Relates frequency and wave number to flow. Must be true for waves.

18. Stationary wave? Wind must be positive, from the west, for a wave.

19. Consider a more specific form of a wave solution

20. Assume that the geopotential takes the form of a wave Mean Wave Mean Ф Gradient in y Horizontal wavenumber

21. Remember the relationship between geostrophic wind and geopotential Plug in the wave solution for the geopotential height

22. Divide the geostrophic wind into mean and perturbation Divide wind into mean and perturbation, assume no mean north/south wind Perturbation only mean perturbation

23. Plug into the advection of relative vorticity

24. Plug into the advection of planetary vorticity

25. Compare advection of planetary and relative vorticity

26. Advection of vorticity ζ < 0; anticyclonic  Advection of ζ tries to propagate the wave this way  ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0  Advection of f tries to propagate the wave this way  ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic

27. Advection of vorticity ζ < 0; anticyclonic  Short waves  ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0 • Long waves  ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic

28. More about the ageostrophic wind • Review ageostrophic wind and implications for vertical motion and cyclone development • Use all that we know to describe development of a mid-latitude cyclone

29. A closer look at the ageostrophic wind • Start with our momentum equation • Just for kicks, take • and see what happens

30. A closer look at the ageostrophic wind • Now, by the right hand rule: • and remember • so we can write

31. A closer look at the ageostrophic wind • We end up with • …the ageostrophic wind! • Knowing that the divergence of the ageostrophic wind leads to vertical motion, let’s explore the implications of this…

32. Acceleration Ageostrophic wind D C Where do we find acceleration? • Curvature ΔΦ > 0 Φ0 - ΔΦ Φ0 y, north Φ0 + ΔΦ x, east D = Divergence and C = Convergence

33. Where do we find acceleration? • Along-flow speed change

34. Acceleration Ageostrophic wind C D D C Where do we find acceleration? • Along-flow speed change J D = Divergence and C = Convergence

35. Where do we find acceleration? • Along-flow speed change

36. Vorticity at upper and lower levels • Continue to examine the divergence of the wind  Which is a proxy for the vertical velocity

37. One more application… • Start with the identity • Now, consider the divergence of the ageostrophic wind • Use the identity, and we have

38. What is this? • Formally, this is • Which is how we derived the vorticity equation

39. What can we do with this? • Plug in our QG assumptions • Let’s think about the difference in divergence (of the ageostrophic wind) between two levels • By the continuity equation, this means that mass is either increasing in the column (net convergence) or decreasing in the column (net divergence) • This should tell us whether low pressure or high pressure is developing at the surface…

40. Column Net Convergence/Divergence • Subtract vorticity equation at 1000 hPa from vorticity equation at 500 hPa • Gives us the net divergence between 1000 and 500 hPa • It can be shown that

41. Examine each term • “Steering term” • Low-level centers of vorticity propagate in the direction of the thermal wind • (Along the gradient of thickness)

42. Examine each term • “Development term” • A bit complicated • Remember, thermal wind is the vertical change in the geostrophic wind • This term indicates the influence of a tilt with height of the location of the maximum (minimum) in vorticity • Fundamentally: if the location of the maximum in vorticity shifts westward with height, the low will develop.

43. Development term Combine terms Definition of thermal wind Definition of geostrophic vorticity

44. Development term • Remember the barotropic height tendency equation? • How about the omega equation?

45. Development term • Development term is the vertical change in barotropic advection of vorticity • This is the same as the stretching term in the omega equation • If the upper-level wave propagates faster than the surface wave, the system decays • Otherwise, the system may develop…

46. Implications • Surface low and high pressure systems (centers of maximum/minimum vorticity) propagate along lines of constant thickness • If there is a vertical tilt westward with height of the vorticity, then a surface low pressure system can intensify (increase in low-level positive vorticity)

47. Look at Cyclones

48. Mid-latitude cyclone development

49. Mid-latitude cyclone development From: http://www.srh.weather.gov/jetstream/synoptic/cyclone.htm

50. Mid-latitude cyclone development From: http://www.srh.weather.gov/jetstream/synoptic/cyclone.htm