phase and group velocity

2. phase and group velocity Note: NOT vector components! (“trace speeds”)

4. Simplest case: Ignoring f, why not align x axis with waves (no equation 2a). Form a vorticity equation from 1a, 3a (eliminating p’) Use mass continuity (4a) to eliminate u. (wxx + wzz)tt = N2 wxx Assume wavelike solutions exp(i (kx+mz-st))  w2 = N2 k2 (k2 + m2)-2 w = N cos q

5. freq = N cosq • (slope of parcel motion paths) • http://www.atmos.ucla.edu/~fovell/DTDM/ • http://www.atmos.ucla.edu/~fovell/AOS101/sgw.html Note: momentum is being fluxed (u’w’)

6. Mountain waves

7. Courtesy: Geraint Vaughan

8. Inertio-gravity waves • These rather forbidding equations hide a pleasing elegance to the gravity-wave solutions which become more evident if x is taken to be the direction of propagation of the wave. Then ℓ=0 and V = -iUf/ω. V and U are in quadrature, causing the wind vector to rotate elliptically – clockwise for upward energy propagation and anticlockwise for downward. For oscillations near to the inertial frequency (which are common in the lower stratosphere) the ellipses are nearly circles, as shown in the example below measured by a VHF radar at Aberystwyth, Wales. Geraint Vaughan

9. Taylor-Goldstein eq • Don’t assume wavelike in z, only in x & t • Permit U(z), and r(z) (Anelastic)

10. Taylor-Goldstein eq • Don’t assume wavelike in z, only in x & t • Permit U(z), and r(z) (Anelastic)

11. Critical level, where u = cor intrinsic frequency  0 • dumps westerly momentum below critical level • think QBO (downward propagation of u)

12. “overreflection”: ideal for wave trapping/ducting

13. “Importance” of gravity waves • For middle atmopshere: Systematic upward flux of zonal momentum [u’w’] • goes as (wind amplitude) x (tilt of motions) • tilt goes with frequency (w = N cosf) • high frequencies are highly weighted in this importance metric • systematic  E/W asymmetry • asymmetric sources • form drag in flow over obstacles (mtns or cloud tops) • asymmetric filtering on the way up (critical levels etc.) • even w/ symmetric sources, but base state flows/ shears • detectability an issue for global assessment • vertical wavelengths for satellite kernels, etc.

14. “Importance” of gravity waves • Notes: w’T’ = 0 so they don’t carry heat flux vertically • often viewed in terms of w field which emphasizes short wavelength and high frequency waves • These ideas can almost close off from view another important role for long (hydrostatic) waves: as an adjustment mechanism • carrying heat away from convection horizontally • adjusting stratification profile • to moist adiabatic

15. “Importance” of gravity waves 2. • adjusting stratification: vertical displacement • strong interactions with convection • goes as (w amplitude) x (period or duration) • low frequency components especially important • like from net heating events (zero frequency limit) • a whole different slice of (k,l,m,w) space • which is only really 3D since dispersion relation holds:

16. M&R Bores: horizontally moving wavefronts • Parsons ppt on 663 course web page

17. Back to simple waves, in resting Boussinesq fluid.Hydrostatic, or low frequency limit (small k & l, i.e. m >> k,l) • Aligning x with the wave (l=0), • w = Nk/m • c = cg = N/m = vert. wavelength/ BV period • “Horizontally propagating vertical modes”

18. Vertical modes • w=0 BC at top and bottom of stratified layer discretizes allowable vertical wavenumbers m • w = sin(kz) • wave numbers ½, 1, 1.5, ... of the layer • waveguide or duct • no vertical propagation, or upward + downward propagation are equal (reflections) yielding standing oscillations (vertical modes) • The higher you place the lid, the closer together the wavelengths permitted

19. Example: response to deep convective heat sources • Boussinesq, hydrostatic, constant-N, nonrotating, resting basic state • w = Nk/m is dispersion relation for waves • Heating is forcing, confined to ‘troposphere’ • lower portion of deep stratified fluid under lid • Heating turns on at t=0 and then maintained • 2 modes of tropospheric layer • convective (sin(z) all positive) • stratiform (sin(2z) vertical dipole)

20. w solution 20 hours later • radiating upward • slope of ray path0 set by 20h timescale since heating began acts like virtual dipole source in infinte fluid since BC is a symmetry line (w=0) Mapes JMSJ 1998

21. Lids cause errors: interference w/artificially reflected waves • Here steep rays are those set by k,m • Their arrival peapods w solution Mapes JMSJ 1998

23. displacement (T’) field is far less wrong than w’ field, even for 10km lid! Vertical propagation and reflection error misses the point.

24. Vertically propagating waves don’t carry heat upward • but they did carry it outward Mapes JMSJ 1998

25. Let’s just look at tropopase lid case • here for f=0, 6h heating event 16 hours ago w field at t=16h, 10h after a 6h heating event happened at the origin. T field for same case

26. What does f do? • Traps heat within a Rossby deformation radius of the origin, held back by thermal wind shear inertio-gravity waves b trapped inertio-gravity waves (zero-sum waves) escape

27. Solution procedure Governing equations Unknowns: u(x,z,t), w(x,z,t), T(x,z,t), p(x,z,t) Known: Q(x,z,t) Fourier transform in z along with B.C.s Since N is const. Shallow water system (1 for each Fourier mode: n = 1, 2,3,..) Stefan Tulich ppt

28. Sketch of solution* Vertical velocity w at times ti < t < tf -cn cn Stefan Tulich ppt *References: Nicholls et al. 1991; Mapes 1998

29. Sketch of solution Horizontal velocity u at time t < tf -50 m/s 50 m/s Stefan Tulich ppt

30. Sketch of solution Temperature T at time t < tf -50 m/s 50 m/s Warm Stefan Tulich ppt

31. Sketch of solution Times after the heating is switched off (t > tf) -50 m/s 50 m/s Warm Warm Stefan Tulich ppt

32. -50 m/s -50 m/s -50 m/s -50 m/s Warm Warm Warm Warm 50 m/s 50 m/s 50 m/s 50 m/s Warm Warm Warm Warm Sketch of solution Times after the heating is switched off (t > tf) Warm Warm

33. “Modes”? Convective and Stratiform Example: 2 radar echo (rain) maps (w/ VAD circles) 200 km

34. Convective & stratiform “modes” In pure simplest theory case Con: sin(z) Strat: sin(2z) Strat Con Strat Con Houze 1997 BAMS

35. warming QDC0 QST0 cooling tDCi tStf tDCf = tSti time Addition of a stratiform-like source Spatial structure: Temporal structure: Stefan Tulich ppt

36. warm 50 m/s warm cold Response to stratiform heating Just before the heating is switched off 25 m/s Stefan Tulich ppt

37. Longer vertical wavelengths travel faster horizontally early A complex convective event in a salt-stratified tank excites many vertical wavelengths in the surrounding fluid (photo inverted to resemble a cloud). Strobe-illuminated dye lines are displaced horizontally, initially in smooth, then more sharply with time. late Mapes 1993 JAS

38. Glimpses of invisible env. flow

39. Fourier spectra of QDC and QSt for different lid heights positive coeff. implies warming response near the surface Ztop = D Ztop = 2D Ztop = 4D Ztop ∞ negative coeff. implies cooling response near the surface Lidded solutions are crude approximations to continuous solution Multiple wave packets are excited Stefan Tulich ppt

40. Summary of wave/mode background • The flow of stratified clear air outside convective clouds is dispersive • longer vertical wavelength components expand faster/farther away from source horizontally • Any vertical profile, e. g. divergence, can be expressed as a spectrum, w/ axis labeled by phase speed. • lid discretizes spectrum; bands robust

41. Is all this sin(z) ghost/mode stuff realistic? (or kinda kooky?) • Need: modes of a realistic atmosphere (actual  stratification profiles) • Ready: Fulton and Schubert 1985 • Need: realistic heating (divergence) profiles • Ready: many many VAD measurements

42. What about for real atmospheres where N is not constant? We can still use the same procedure but a more general (“vertical mode”) transform must be used: The vertical structure functions n (and their associated phase speeds cn) are obtained as numerical solutions to the vertical structure problem: an eigenvalue problem with dependence on N Stefan Tulich ppt

43. Vertical modes associated with the CRM’s basic state atmosphere* cos(2z/Lzn) Lzn ≈ 28 km Lzn ≈ 14 km“ Lzn” ≈ 11 km“ Structure functions of both u and  *calculated using the algorithm of Fulton and Schubert (1985)

44. Hey -- what’s this? Spectrum of average VAD divergencefrom many profiles in tropical rain different lid pressures -> different discretizations, bands robust Mapes 1998

45. Top-heavy C+S: spectrum & response T response when observed mean VAD divergence is used as a mass source in observed mean stratification Shallow water solved for each mode, then sum it up Mapes and Houze 1995

46. Melting mode Melting: forcing is localized in z, response is localized in wavenumber! Mapes and Houze 1995

47. Raw data:Snow melts, whole troposphere shivers(wavelength set by melting layer thickness?) spectral view not quite so kooky?

48. m=3/2 Does this exist? Re: kookinessAre convective and stratiform really dynamical modes? m=1 m=1/2

49. (great data quality) Rare, but compelling Jialin Lin

50. (5h of data, from front to back of storm) Rare, but compelling Aboard the R/V Brown JASMINE project considerable front-back cancellation