1. Sound speed in air: C Ｓ ∝ [T] 1/2 T[K] C Ｓ [m/s] 273 331 300 347 373 383 Conv. Div. tendency of pressure & density >0 <0 wave propagation CＳCＳ velocity 3. Waves 3.1 Sound waves  compressional wave wave equation

2.  observation of infrasonic waves Yamamoto (1954) Pressure variations due to nuclear-bomb experiment at Bikini observed in Japan with a microbarogram

3. 3.2 Gravity waves  surface (external) gravity wave  gravity waves in a rotating shallow-water system wave equation dispersion relation for gravity waves geostrophic adjustment problem  separation of component z z x buoyancy H0H0

4.  static stability a parcel motion in a stratified fluid Brunt-Vaisala frequency Sakai (1997) GFD Experiments on internal gravity waves http://www.gfd-dennou.org/library/gfd_exp/index.htm z buoyancy

5.  propagation of internal gravity waves density perturbation heavy light pressure perturbation high low high pressure grad. force total force buoyancy force wave propagation

6.  some considerations on waves (1) linear vs. nonlinear  small perturbation to a basic field  linearization  finite amplitude  nonlinear world local vs. global  boundary conditions for infinite or finite domain “global” mode “local” mode

7.  observations gravity waves visualized by clouds over Scotland XXX(Weather, 2000?)

8. 3.3 Rossby waves  conservation of potential vorticity Rossby waves on a beta-plane  the meridional variation in Coriolis effect topographic Rossby waves  horizontal (alongshore) variation of fluid depth Ishioka et al. (1999) Pattern formation from two- dimensional decaying turbulence on a rotating sphere. NAGARE Multimedia http://www.nagare.or.jp/mm/99/ishioka/ http://www.nagare.or.jp/mm/99/ishioka/

9.  dynamics quasi-geostrophic potential vorticity (QG-PV) equation propagation of Rossby Waves  basic state: monotonic increase of PV  perturbation: wave-like meridional displacement W E NSNS Induced flow small PV wave propagation PV perturbation large PV - + - PV of basic state NSNS

10.  some considerations on waves (2) neutral vs. unstable  monotonic increase of PV in the basic field  neutral wave motion  negative gradient of PV  barotropic instability neutral waves  free traveling waves  forced waves stationary in some cases (e.g., topographically forced) unstable waves  growth of perturbation  mixing of PV  dissolution of unstable condition  when an unstable basic field is maintained, what will happen? PV(y) stable unstable basic flow field y

11.  Rossby waves in a 2-D barotropic fluid wave equation dispersion relation with a mean flow U 0  westward propagation to the mean flow  stationary wave (c =0) may exist only for the westerly wind (0<U 0 ) Seasonal mean height fields of 30 hPa in the NH [solid line, km] (Holton, 1975) H L winter summer

12. Potential vorticity distribution on 850 K isentropic surface in September 2002 in the SH (Baldwin et al., 2003)  observations Transient Rossby waves (CP ≠0) can be observed in the animation of PV maps

13. 3.4 Some other waves in GFD  tidal waves  equatorial waves  coastal Kelvin waves  solitary waves ..... Rossby- gravity wave Rossby wave Kelvin wave ω Westward propagating Eastward propagating k : n=1 Rossby wave : n=0 Rossby-gravity wave : n= – 1 Kelvin wave Equator Dispersion of equatorial waves Cushman-Roisin(1994; Fig.19.2) Matsuno (1966; Figs.4, 6, 8)

14. 4. Instabilities 4.1 Parcel methods  Static stability density stratification in the gravity field  Inertial instability meridional shear of the mean zonal flow

15. 4.2 Thermal convection  Rayleigh-Benard problem heat conduction solution linear stability of the heat conduction solution  Rayleigh number: structure of the growing perturbation energetics  [T*w*] > 0  conversion: PE  KE  some GFD applications Moist convection Mantle convection z T D ΔT g

16. 4.3 Barotropic instability  Rayleigh-Kuo-Fjortoft problem integral theorems  linear stability of a basic zonal flow eigenvalue problem structure of the growing perturbation nonlinear phase of the instability  some GFD applications meander of African jet (?) Kuroshio meander PV(y) stable unstable basic flow field y Cushman-Roisin(1994; Fig.7.2~2)

17. 4.4 Baroclinic instability  Eady problem, Charney problem linear stability of a basic zonal flow structure of the growing perturbation  rotating annulus experiments basic flow field U(z) z vertical shear ~ meridional temperature gradient C WC W L H × Axisymmetric Steady wave Turbulent flow Cold Warm

18. Ogura (2000; Fig.7.2) L cold & dry warm & humid heat flux  extratropical cyclones Salby (1996; Fig.1.9)

19. 4.5 Some other instability in GFD  Kelvin-Helmholtz instability  CISK (conditional instability of the second kind) http://www.cira.colostate.edu/ramm/rmsdsol/isabel-web.html Colson (1954; Weatherwise, 7) http://www.gfd-dennou.org/library /gfd_exp/index.htm

20. 5. Nonlinear phenomena 5.1 Breaking waves  finite amplitude  chaotic mixing 5.2 Wave-mean flow interaction  QBO (quasi-biennial oscillation)  stratospheric vacillation 5.3 Chaotic phenomena in GFD  Lorenz chaos application to numerical weather predictions (NWPs) http://www-mete.kugi.kyoto-u.ac.jp/mete/ J/benkyo/QBO/tzsection-grad.png