1. Atmospheric Gravity Waves
References:
1 “Atmospheric Physics” Andrews, Chapter 5 (5.5)
2 “Introduction to Dynamic Meteorology”, Holton, Chapter 7

2. Examples for Waves

3. Atmospheric Waves

4. Evidence for GW in the Thermosphere of Jupiter
Galileo probe T-profile
Electron density profile
1200
200
400
600
800
1000
300
500
700
900
1400
600
800
1000
1200
4e+10
8e+10 T Ne
Altitude [km]
[K]
[m-3]
High stratospheric T-profile
240
280
320
360 -4 4 8 12
In situ measurements
dT/dz
Radio occultations
Altitude [km]
Stellar occultations
[K/km]

5. Transverse and Longitudinal Waves
Compression
The oscillation is perpendicular the phase lines
Example: Sound wave
Transverse wave
No compression
The oscillation is along the phase lines
Example: Gravity wave
Wave propagation
Wave propagation
Particle oscillation
Particle oscillation

6. Basic parameters General oscillation: A – amplitude
lx, ly, lz – wavelength in x, y, z direction respectively
k, l, m - wavenumbers
w=2p/T – frequency of the wave
f – phase of the oscillation
f=constant – phase line (plane)

7. Phase lines
Consider 2D motion (xz-plane). z
x,z2
x,z1
x,z0

8. Atmospheric Oscillations
Environmental Lapse Rate
Colder
Altitude
Air parcel
Adiabatic Lapse Rate z0
Warmer
Temperature

9. Simple gravity wave model
Consider a corrugated sheet pulled horizontally at a speed c through a stratified fluid. z q

10. Example: Wind, c, is blowing over a mountain.
Wave motion will be expected in the air. It is a stationary wave (cp=0 wrt the mountain).
The horizontal phase speed of the wave with respect to the wind will be -c.
The horizontal wavelength of the wave is lx and is set by the wavelength of the topography (mountain).
The horizontal and the vertical wavenumbers of the wave are:

11. General solution and some properties
Displacement eq.:
General solution:
Wave frequency:
Large frequency - small q
Short periods - small q
The largest w is for a vertical phase line (infinitely large vertical wavelength lz)
These waves are stationary with respect to the mountain (cp=0)

12. Propagation of AGW Particle motion Altitude [km]
Distance from the source [km]
Particle
motion
Alexander CEDAR

13. Dependence on the wind speed
Strong wind
Week wind

14. Dependence on horizontal scale of the topography
Short wavelengths (large k) q
Long wavelengths (small k)

15. Dispersion relation lx l lzq
For a vertically propagating wave m needs to be real.
Therefore w<N.
Internal GW have periods longer than ~5 min in the Earth troposphere.

16. Phase and Group velocity
Dispersion relation
Phase velocities: Group velocity:

17. Phase and group velocity
Phase velocity shows the speed and the direction of phase propagation.
Group velocity shows the speed and the direction of energy propagation.
The AGW are dispersive: the phase speed is a function of the wavenumber and the phase speed is equal the group velocity.
In contrast, sound waves are not dispersive. All sound waves propagate at the speed of sound independent of their wavelengths or frequencies and the phase and group velocities are equal.

18. Direction of energy propagation
Homework problem: show that the group velocity vector is perpendicular to the wavenumber vector.
Downward propagating phase lines correspond to upward propagating energy!
Energy flow
Phase propagation

19. Why Do We Care About GW? Vertical momentum and energy transport;
Meridional circulation and heat transport;
Transport of chemical species;
Atmospheric mixing end eddy diffusion;
Quasi-biennial oscillations;
Traveling ionospheric disturbances (TID);
Sporadic ionospheric instabilities: communications;
Air-breaking of satellites - Mars; …

20. THE END