1. ATMOSPHERIC WAVES

2. DEFINITION
An atmospheric wave is a periodic disturbance in the fields of atmospheric variables (like surface pressure or geopotential height, temperature, or wind velocity) which may either propagate (traveling wave) or not (standing wave). Atmospheric waves range in spatial and temporal scale from large-scale planetary waves (Rossby waves) to minute sound waves. Atmospheric waves with periods which are harmonics of 1 solar day (e.g. 24 hours, 12 hours, 8 hours... etc.) are known as atmospheric tides.

3. CAUSES AND EFFECTS
The mechanism for the forcing of the wave, for example the generation of the initial or prolonged disturbance in the atmospheric variables, can vary. Generally, waves are either excited by heating or dynamic effects, for example the obstruction of the flow by mountain ranges like the Rocky Mountains in the U.S. or the Alps in Europe.

4. CAUSES AND EFFECTS
Heating effects can be small-scale (like the generation of gravity waves by convection) or large-scale (the formation of Rossby waves by the temperature contrasts between continents and oceans in the Northern hemisphere winter).

5. CAUSES AND EFFECTS
Atmospheric waves transport momentum, which is fed back into the background flow as the wave dissipates. This wave forcing of the flow is particularly important in the stratosphere, where this momentum deposition by planetary scale Rossby waves gives rise to sudden stratospheric warmings and the deposition by gravity waves gives rise to the quasi-biennial oscillation.

6. CAUSES AND EFFECTS
In the mathematical description of atmospheric waves, spherical harmonics are used. When considering a section of a wave along a latitude circle, this is equivalent to a sinusoidal shape.

7. TYPES OF WAVES
Because the propagation of the wave is fundamentally caused by an imbalance of the forces acting on the air (which is often thought of in terms of air parcels when considering wave motion), the types of waves and their propagation characteristics vary latitudinally, principally because the Coriolis effect on horizontal flow is maximal at the poles and zero at the equator

8. TYPES OF WAVES The different wave types are:
sound waves (usually eliminated from the atmospheric equations of motion due to their high frequency)
These are longitudinal or compression waves. The sound wave propagates in the atmosphere through a series of compressions and expansions parallel to the direction of propagation.

9. TYPES OF WAVES
internal gravity waves (require stable stratification of the atmosphere)
inertio-gravity waves (also include a significant Coriolis effect as opposed to "normal" gravity waves)
Rossby waves (can be seen in the troughs and ridges of 500 hPa geopotential caused by midlatitude cyclones and anticyclones)
At the equator, mixed Rossby-gravity and Kelvin waves can also be observed.

10. OVERVIEW OF WAVE TYPES SNO. NAME BALANCING TERM 1.
ACOUSTIC OR SOUND WAVES
(ESSENTIALLY COMPRESSIBLE)
PRESSURE VARIATIONS 2.
KELVIN WAVES
(INCOMPRESSIBLE , BAROTROPIC)
GRAVITY AND CORIOLIS ALONG A FLUID BOUNDARY 3.
POINCARE WAVES
GRAVITY AND CORIOLIS 4.
ROSSBY WAVES (OR PLANETARY WAVES)
CORIOLIS PARAMETER VARIATIONS 5.
TOPOGRAPHIC WAVES
FLOW DEPTH VARIATIONS 6.
INTERNAL (GRAVITY BUOYANCY WAVES)
(INCOMPRESSIBLE , BAROCLINIC)
DENSITY DIFFERENCES

11. KELVIN WAVE
A Kelvin wave is a wave in the ocean or atmosphere that balances the Earth's Coriolis force against a topographic boundary such as a coastline, or a waveguide such as the equator. A feature of a Kelvin wave is that it is non-dispersive, i.e., the phase speed of the wave crests is equal to the group speed of the wave energy for all frequencies. This means that it retains its shape in the alongshore direction over time.

12. COASTAL KELVIN WAVE
In a stratified ocean of mean depth H, free waves propagate along coastal boundaries (and hence become trapped in the vicinity of the coast itself) in the form of internal Kelvin waves on a scale of about 30 km. These waves are called coastal Kelvin waves, and have propagation speeds of approximately 2 m/s in the ocean.

13. KELVIN WAVE
A Kelvin wave (fluid dynamics) is also a long scale perturbation mode of a vortex in superfluid dynamics; in terms of the meteorological or oceanographical derivation, one may assume that the meridional velocity component vanishes (i.e. there is no flow in the north–south direction, thus making the momentum and continuity equations much simpler).

14. EQUATORIAL KELVIN WAVE
The equatorial zone essentially acts as a waveguide, causing disturbances to be trapped in the vicinity of the equator, and the equatorial Kelvin wave illustrates this fact because the equator acts analogously to a topographic boundary for both the Northern and Southern Hemispheres, making this wave very similar to the coastally-trapped Kelvin wave.

15. Because these waves are equatorial, the Coriolis parameter vanishes at 0 degrees; therefore, it is necessary to use the equatorial beta plane approximation that states:
         
where β is the variation of the Coriolis parameter with latitude. This equatorial Beta plane assumption requires a geostrophic balance between the eastward velocity and the north-south pressure gradient. The phase speed is identical to that of coastal Kelvin waves, indicating that the equatorial Kelvin waves propagate toward the east without dispersion (as if the earth were a non-rotating planet).[1] For the first baroclinic mode in the ocean, a typical phase speed would be about 2.8 m/s, causing an equatorial Kelvin wave to take 2 months to cross the Pacific Ocean between New Guinea and South America; for higher ocean and atmospheric modes, the phase speeds are comparable to fluid flow speeds.

18. Phase difference is the difference, expressed in electrical degrees or time, between two waves having the same frequency and referenced to the same point in time.
Two oscillators that have the same frequency and different phases have a phase difference, and the oscillators are said to be out of phase with each other.
The amount by which such oscillators are out of step with each other can be expressed in degrees from 0° to 360°, or in radians from 0 to 2π.
If the phase difference is 180 degrees (π radians), then the two oscillators are said to be in antiphase.
If two interacting waves meet at a point where they are in antiphase, then destructive interference will occur.

19. The phase velocity of a wave is the rate at which the phase of the wave propagates in space.
This is the speed at which the phase of any one frequency component of the wave travels.
For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity.

20. The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space.
For example, imagine what happens if a stone is thrown into the middle of a very still pond. When the stone hits the surface of the water, a circular pattern of waves appears. It soon turns into a circular ring of waves with a quiescent center. The ever expanding ring of waves is the wave group, within which one can discern individual wavelets of differing wavelengths traveling at different speeds. The longer waves travel faster than the group as a whole, but they die out as they approach the leading edge. The shorter waves travel slower and they die out as they emerge from the trailing boundary of the group.

21. The solution to these equations yields the following phase speed: c
The solution to these equations yields the following phase speed: c*c = gH, which is the same speed as for shallow-water gravity waves without the effect of Earth’s rotation. It is important to note that for an observer traveling with the wave, the coastal boundary (maximum amplitude) is always to the right in the northern hemisphere and to the left in the southern hemisphere (i.e. these waves move equatorward/southward – negative phase speed – on a western boundary and poleward/northward – positive phase speed – on an eastern boundary; the waves move cyclonically around an ocean basin).

22. When the motion at the equator is to the east, any deviation toward the north is brought back toward the equator because the Coriolis force acts to the right of the direction of motion in the Northern Hemisphere, and any deviation to the south is brought back toward the equator because the Coriolis force acts to the left of the direction of motion in the Southern Hemisphere.
Equatorial Kelvin waves are only possible for eastward motion. Both atmospheric and oceanic equatorial Kelvin waves play an important role in the dynamics of El Nino-Southern Oscillation, by transmitting changes in conditions in the Western Pacific to the Eastern Pacific.

23. ATMOSPHERIC GRAVITY WAVES
Gravity waves are the oscillations of air parcels by the lifting force of buoyancy and the restoring force of gravity. These waves propagate vertically as well as horizontally, and actively transport energy and momentum from the troposphere to the middle and upper atmosphere. Gravity waves are caused by a variety of sources, including the passage of wind across terrestrial landforms, interaction at the velocity shear of the polar jet stream and radiation incident from space. They are found to affect atmospheric tides in the middle atmosphere and terrestrial weather in the lower atmosphere.

25. GRAVITY WAVES
Gravity waves are investigated using pressure sensors, in situ aircraft measurement, and imaging data. Gravity waves are often seen in the lower atmosphere (~10 km) by thin bands of cloud and sky. Higher in the atmosphere (80-100km), gravity waves can be 'seen' in moving bands of atmospheric air glow. The airglow emits spectra from chemiluminescence of atmospheric molecules. By analyzing the motion of different spectra mesospheric gravity waves can be roughly plotted.

29. TERRESTRIAL OBSERVATION OF GRAVITY WAVES

30. TERRESTRIAL OBSERVATION OF GRAVITY WAVES

31. OBSERVATIONS FROM THE AIR
Waves in Airglow measured in the Space Shuttle wake

32. GRAVITY WAVES AS SEEN FROM SPACE
Airglow images were taken by a camera on board the Space Shuttle looking back into the wake of the flight path. These images show the airglow layer at about 93 km altitude up to the horizon of the local observer. Traveling along its flight path the camera observes new parts of the airglow layer which can be considered stationary compared to the velocity of the shuttle ( ~7 km/s).

33. GRAVITY WAVES AS SEEN FROM AIR

34. ROSSBY-GRAVITY WAVES
Rossby-gravity waves are equatorially-trapped waves (much like Kelvin waves), meaning that they rapidly decay as their distance increases away from the equator. These waves have the same trapping scale as Kelvin waves, more commonly known as the equatorial Rossby deformation radius.They always carry energy eastward, but, oddly, their 'crests' and 'troughs' may propagate westward if their periods are long enough.

35. The eastward speed of propagation of these waves can be derived for an inviscid slowly moving layer of fluid of uniform depth H.
Because the Coriolis parameter (f = 2Ω sin(θ) where Ω is the angular velocity of the earth,  × 10−5 rad/s, and θ is latitude) vanishes at 0 degrees latitude (equator), the “equatorial beta plane” approximation must be made.
This approximation states that “f” is approximately equal to βy, where “y” is the distance from the equator and "β" is the variation of the coriolis parameter with latitude,

36. Once the frequency relation is formulated in terms of ω, the angular frequency, the problem can be solved with 3 distinct solutions.
These three solutions correspond to the equatorially-trapped gravity wave, the equatorially-trapped Rossby wave and the mixed Rossby-gravity wave (which has some of the characteristics of the former two) .

37. VERTICALLY-PROPAGATING ROSSBY-GRAVITY WAVES
The mixed Rossby-gravity waves are equatorially-trapped waves unless the buoyancy frequency remains constant, introducing an additional vertical wave number to complement the zonal wave number and angular frequency. If this Brunt–Vaisala frequency does not change, then these waves become vertically-propagating solutions.

38. These vertically-propagating mixed Rossby-gravity waves were first observed in the stratosphere as westward-propagating mixed waves by M. Yanai.
They had the following characteristics: periods of 4–5 days, horizontal wave numbers of 4 (four waves circling the earth, corresponding to wavelengths of 10,000 km), vertical wavelengths of 4–8 km, and upward group velocity.