3 Waves 3 1 Sound waves v compressional

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

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

3. 2 Gravity waves v surface (external) gravity wave z z buoyancy H 0 x v gravity waves in a rotating shallow-water system l wave equation l dispersion relation for gravity waves l geostrophic adjustment problem Øseparation of component

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

v propagation of internal gravity waves density perturbation heavy buoyancy force light e av w total force n io at ag op pr pressure perturbation high low pressure grad. force high

v some considerations on waves (1) l linear vs. nonlinear Øsmall perturbation to a basic field linearization Øfinite amplitude nonlinear world l local vs. global Øboundary conditions for infinite or finite domain “local” mode “global” mode

v observations l gravity waves visualized by clouds over Scotland XXX(Weather, 2000? )

3. 3 Rossby waves v conservation of potential vorticity Rossby waves on a beta-plane Øthe meridional variation in Coriolis effect l topographic Rossby waves Øhorizontal (alongshore) variation of fluid depth l l Ishioka et al. (1999) Pattern formation from twodimensional decaying turbulence on a rotating sphere. NAGARE Multimedia http: //www. nagare. or. jp/mm/99/ishioka/

v dynamics l quasi-geostrophic potential vorticity (QG-PV) equation l propagation of Rossby Waves Øbasic state: monotonic increase of PV Øperturbation: wave-like meridional displacement N N - PV perturbation + - large PV Induced flow small PV S wave propagation S PV of basic state W E

v some considerations on waves (2) l neutral vs. unstable Ømonotonic increase of PV in the basic field neutral wave motion basic flow field y Ønegative gradient of PV stable barotropic instability l neutral waves Øfree traveling waves Øforced waves unstable PV(y) stationary in some cases (e. g. , topographically forced) l unstable waves Øgrowth of perturbation mixing of PV dissolution of unstable condition Øwhen an unstable basic field is maintained, what will happen?

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

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

3. 4 Some other waves in GFD Cushman-Roisin(1994; Fig. 19. 2) v tidal waves Dispersion of equatorial waves ω Westward propagating Eastward propagating v equatorial waves v coastal Kelvin waves gravity wave v solitary waves v. . . Rossbygravity wave Kelvin wave Rossby wave Matsuno (1966; Figs. 4, 6, 8) Equator : n=1 Rossby wave : n=0 Rossby-gravity wave : n= – 1 Kelvin wave k

4. Instabilities 4. 1 Parcel methods v Static stability l density stratification in the gravity field v Inertial instability l meridional shear of the mean zonal flow

4. 2 Thermal convection v Rayleigh-Benard problem z D heat conduction solution l linear stability of the heat conduction solution ΔT ØRayleigh number: l structure of the growing perturbation l energetics Ø[T*w*] > 0 Øconversion: PE KE l v some GFD applications l Moist convection l Mantle convection g T

4. 3 Barotropic instability v Rayleigh-Kuo-Fjortoft problem l y basic flow field stable integral theorems v linear stability of a basic zonal flow l eigenvalue problem l structure of the growing perturbation l nonlinear phase of the instability unstable PV(y) Cushman-Roisin(1994; Fig. 7. 2~2) v some GFD applications l meander of African jet (? ) l Kuroshio meander

4. 4 Baroclinic instability v Eady problem, Charney problem linear stability of a basic zonal flow l structure of the growing perturbation l z basic flow field vertical shear ~ meridional temperature gradient U(z) v rotating annulus experiments Cold Warm × L H CW Axisymmetric Steady wave Turbulent flow

v extratropical cyclones Salby (1996; Fig. 1. 9) dr y heat flux & um id d L &h co l wa rm Ogura (2000; Fig. 7. 2)

4. 5 Some other instability in GFD v Kelvin-Helmholtz instability Colson (1954; Weatherwise, 7) http: //www. gfd-dennou. org/library /gfd_exp/index. htm v CISK (conditional instability of the second kind) http: //www. cira. colostate. edu/ramm/rmsdsol/isabel-web. html

5. Nonlinear phenomena 5. 1 Breaking waves v finite amplitude v chaotic mixing 5. 2 Wave-mean flow interaction v QBO (quasi-biennial oscillation) http: //www-mete. kugi. kyoto-u. ac. jp/mete/ J/benkyo/QBO/tzsection-grad. png v stratospheric vacillation 5. 3 Chaotic phenomena in GFD v Lorenz chaos l application to numerical weather predictions (NWPs)