1 / 131

Atmospheric Waves: Perturbation Theory

Based on Chapter 7 of Holton ’ s An Introduction to Dynamic Meteorology. Atmospheric Waves: Perturbation Theory. Wavelike behavior commonly observed Wave solutions to conservation laws help us understand physical interactions and energy propagation

wbowers
Download Presentation

Atmospheric Waves: Perturbation Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Based on Chapter 7 of Holton’s An Introduction to Dynamic Meteorology Atmospheric Waves: Perturbation Theory

  2. Wavelike behavior commonly observed • Wave solutions to conservation laws help us understand physical interactions and energy propagation • As first approximation, one can superimpose wave solutions of different scales to depict atmospheric flow Waves in Atmosphere

  3. Full equations too complicated for physical insight - need simplified models • Chapter 6: Primitive equations simplified to quasi-geostrophic system • Chapter 7: Q-G equations simplified to linearized equations. Simplification Needed

  4. Assume: One can view much important atmospheric behavior as perturbations about a basic state, e.g., Basic state is given (known), but it must be a solution to the governing equations Perturbations much smaller than basic state, e.g., or Applying 1- 3 gives linearized equations Perturbation Method: Assumptions

  5. Class Slide

  6. Class Slide

  7. Class Slide

  8. Class Slide

  9. Class Slide

  10. Class Slide 7.1 Perturbation Method

  11. Resulting equation is linear in ( )' variables • Since basic state is given, applying same method to all of our conservation laws gives a set of linearized equations in ( )' variables. • Linear equations are much easier to solve than nonlinear equations. • Linear equations often give wave solutions. • Typically assume ( ) ~ sinusoidal waves. Class Slide 7.1 Perturbation Method

  12. Class Slide

  13. Look to find specific properties: • Phase speed • Energy propagation • Vertical structure • Conditions for existence, growth and decay of waves (when & where we might expect to see physical interactions represented by the waves) Solving for Waves

  14. Class Slide (Holton gives another example.) 7.2 Wave Properties

  15. Class Slide 7.2 Wave Properties

  16. Possible solution: Class Slide Can test: substitute into equation. Note that 2ond derivatives of trig functions return -(original function). E.g.: d2cos(t)/dt2 = -2cos(t) 7.2 Wave Properties

  17. Class Slide 7.2 Wave Properties

  18. Class Slide 7.2 Wave Properties

  19.  = frequency of oscillations One wave period or cycle = 2/  is independent of Xo (amplitude) Phase of oscillation is  = t - o 7.2 Wave Properties

  20. Example is stationary oscillator. Propagating oscillations? Propagating Waves Similarity: characterization by amplitude & phase Phase now function of time and space: e.g., in 1-D:  = kx - t + o k = 2/Lx (wavenumber) Phase speed: c = /k Speed observer must move for phase of wave to be constant (e.g., speed of trough/crest movement) 7.2 Wave Properties

  21. Class Slide 7.2 Wave Properties

  22. Class Slide 7.2 Wave Properties

  23. If observer is moving with the wave, then phase is constant. Thus: Class Slide This gives the change in position x in time t, hence speed, for point maintaining constant phase with respect to wave. 7.2 Wave Properties

  24. In 2 or More Dimensions Lines of constant  k = (phase-change in x-direction)/(unit-length) |K| =(phase-change)/(unit-length) l = (phase-change in y-direction)/(unit-length) 7.2 Wave Properties

  25. |K| = ( phase) (unit-length) Wavelength in 2 or More Dimensions Lines of constant  Then ( phase) = (length-moved) x { ( phase)/(unit-length) } If ( phase) = 2, then wavelength =  = 2/|K| Wavelength = distance for wave form to repeat (e.g., crest-to-crest distance)

  26. Move with point of constant phase - e.g., crest Phase Speed in 2 or More Dimensions By analogy with 1-D, for phase speed C, perpendicular to lines of constant  Lines of constant  7.2 Wave Properties

  27. Move with point of constant phase - e.g., crest Move only in x-direction: Phase Speed in Coordinate Directions Similarly, looking at phase change only in y direction (e.g., crest movement in y) 7.2 Wave Properties

  28. C Is Not A Vector! - 1 Cx is rate of phase advance in x-direction (e.g., rate of advance of point P on crest) Cxincreases with decreasing projection of K vector onto x axis: P P Cx 7.2 Wave Properties

  29. C Is Not A Vector! - 2 Cxincreases with decreasing projection of K vector onto x axis. Thus: As angle   90˚, Cx  ! Cx thus > speed of light => not a physical velocity Rather, this is location change of a geometric point Thus, phase “speed”, not“velocity”  P 7.2 Wave Properties

  30. A Physical Vector: Group Velocity Class Slide The group velocity describes energy propagation. 7.2 Wave Properties

  31. Class Slide 7.2 Wave Properties

  32. Class Slide (See also figures shown in class) 7.2 Wave Properties

  33. Class Slide 7.2 Wave Properties

  34. Class Slide 7.2 Wave Properties

  35. Class Slide 7.2 Wave Properties

  36. Class Slide 7.2 Wave Properties

  37. Class Slide 7.2 Wave Properties

  38. Simple Wave Types 7.3 Wave Types

  39. Class Slide 7.3 Wave Types

  40. Class Slide 7.3 Wave Types

  41. Class Slide 7.3 Wave Types

  42. Conservation Laws Class Slide 7.3 Wave Types

  43. Class Slide 7.3 Wave Types

  44. Class Slide 7.3 Wave Types

  45. Class Slide 7.3 Wave Types

  46. Class Slide 7.3 Wave Types

  47. Class Slide 7.3 Wave Types

  48. Class Slide 7.3 Wave Types

  49. Class Slide 7.3 Wave Types

  50. Class Slide 7.3 Wave Types

More Related