Atmospheric Waves: Perturbation Theory

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

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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

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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

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48. Class Slide 7.3 Wave Types

49. Class Slide 7.3 Wave Types

50. Class Slide 7.3 Wave Types