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

C Ｓ

velocity

3. Waves

3.1 Sound waves compressional wave

wave equation

observation of infrasonic wavesYamamoto (1954)

Pressure variations due tonuclear-bomb experimentat Bikini observed in Japanwith a microbarogram

3.2 Gravity waves surface (external) gravity wave

gravity waves in a rotating shallow-water systemwave equation

dispersion relation for gravity waves

geostrophic adjustment problemseparation of component

z z

x

buoyancy

H0

static stabilitya 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

propagation of internal gravity waves

density 　　　　 　 　 perturbation heavy light

pressure perturbation high low high

pressure grad. force

total force

buoyancy

forcewave propagation

some considerations on waves (1) linear vs. nonlinear

small perturbation to a basic field linearization

finite amplitude nonlinear world

local vs. globalboundary conditions for infinite or finite

domain

“global” mode“local” mode

observationsgravity waves visualize

d by clouds over Scotland

XXX(Weather, 2000?)

3.3 Rossby waves conservation of potential vorticity

Rossby waves on a beta-planethe meridional variation in Coriolis effect

topographic Rossby waveshorizontal (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/ 　　　　

　　　

dynamicsquasi-geostrophic potential vorticity (QG-PV)

equation

propagation of Rossby Wavesbasic state: monotonic increase of PVperturbation: wave-like meridional

displacement

W E

N

S

Induced flow

small PVwave propagation

PV perturbation

large PV- + -

PV of basic state

N

S

some considerations on waves (2)neutral vs. unstable

monotonic increase of PV in the basic field neutral wave motionnegative gradient of PV barotropic instability

neutral wavesfree traveling wavesforced waves

stationary in some cases (e.g., topographically forced)

unstable wavesgrowth of perturbation mixing of PV

dissolution of unstable conditionwhen an unstable basic field is maintained, what

will happen?

PV(y)

stable

unstable

basic flow fieldy

Rossby waves in a 2-D barotropic fluidwave equation

dispersion relation

with a mean flow U0

westward propagation to the mean flowstationary wave (c =0) may exist only for the westerly wind (0<U0 )

Seasonal mean height fields of 30 hPa in the NH

[solid line, km] (Holton, 1975)

HL

winter summer

Potential vorticitydistribution on 850 Kisentropic surfacein September 2002in the SH (Baldwin et al., 2003)

observationsTransient Rossby waves (CP ≠0) can be

observed in the animation of PV maps

3.4 Some other waves in GFD tidal waves equatorial waves coastal Kelvin waves solitary waves .....

Rossby-gravity wave

gravity wave

Rossby wave

Kelvin wave

ωWestward propagating Eastward propagating

k

: n=1 Rossby wave : n=0 Rossby-gravity wave : n= –1 Kelvin wave

Equator

Dispersion of equatorial wavesCushman-Roisin(1994; Fig.19.2)

Matsuno (1966; Figs.4, 6, 8)

4. Instabilities

4.1 Parcel methods Static stability

density stratification in the gravity field

Inertial instabilitymeridional shear of the mean zonal flow

4.2 Thermal convection Rayleigh-Benard problem

heat conduction solution linear stability of the heat conduction solution

Rayleigh number:

structure of the growing perturbationenergetics

[T*w*] > 0 conversion: PE KE

some GFD applications Moist convectionMantle convection

z

T

D

ΔT

g

4.3 Barotropic instability Rayleigh-Kuo-Fjortoft problem

integral theorems linear stability of a basic zonal flow

eigenvalue problemstructure of the growing perturbationnonlinear phase of the instability

some GFD applicationsmeander of African jet (?)Kuroshio meander

PV(y)

stable

unstable

basic flow fieldy

Cushman-Roisin(1994; Fig.7.2~2)

4.4 Baroclinic instability Eady problem, Charney problem

linear stability of a basic zonal flowstructure of the growing perturbation

rotating annulus experiments

basic flow field

U(z)

z

vertical shear~ meridionaltemperaturegradient

C W

L H×

Axisymmetric Steady wave Turbulent flow

Cold Warm

Ogura (2000; Fig.7.2) Lcold & dryw

arm

& h

umid

hea

t fl

ux

extratropical cyclones

Salby (1996; Fig.1.9)

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

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