ATMS 310 Rossby Waves

Properties of Waves in the Atmosphere

Waves – Oscillations in field variables that propagate in space and time.

There are several aspects of waves that we can use to characterize their nature:

1) Period – The amount of time it takes to complete one oscillation of the wave\ 2) Wavelength (λ) – Distance between two peaks of troughs 3) Amplitude – The distance between the peak and the trough of the wave 4) Phase – Where the wave is in a cycle of amplitude change

For a 1-D wave moving in the x-direction, the phase is defined by:

φ ),( = −υtkxtx − α (1)

2π where φ is the phase, k is the wave number = , υ = frequency of oscillation (s-1), and λ α = constant determined by the initial conditions. If the observer is moving at the phase υ speed of the wave ( c ≡ ), then the phase of the wave is constant. k

For simplification purposes, we will only deal with linear sinusoidal wave motions.

Dispersive vs. Non-dispersive Waves

When describing the velocity of waves, a distinction must be made between the group velocity and the phase speed. The group velocity is the velocity at which the observable disturbance (energy of the wave) moves with time. The phase speed of the wave (as given above) is how fast the constant phase portion of the wave moves.

A dispersive wave is one in which the pattern of the wave changes with time. In dispersive waves, the group velocity is usually different than the phase speed. A non- dispersive wave is one in which the patterns of the wave do not change with time as the wave propagates (“rigid” wave). Rossby waves are dispersive waves.

Some Types of Waves

Internal Gravity (Buoyancy) Waves

These waves can occur in the atmosphere only when the atmosphere is stably stratified. That is, if a parcel of air is lifted or forced to descend from its equilibrium level, it will return to that level. This occurs when the potential temperature increases with height, or Γ < Γd (environmental lapse rate is less than the dry adiabatic lapse rate).

Technically these types of waves are called buoyancy waves, since the buoyancy force is the restoring force (force that causes the parcel to return to the equilibrium level). However, these waves are commonly referred to as gravity waves.

In the atmosphere, gravity waves may propagate both vertically and horizontally. In a vertically propagating wave, the phase is a function of height. These waves are called internal waves. Internal waves have important implications at scales smaller than the synoptic scale.

Rossby (“Planetary”) Waves

Rossby waves are the most important type of wave for large-scale meteorological processes. In a baroclinic atmosphere, Rossby waves conserve potential vorticity.

Movement of Rossby waves is due to this conservation aspect. It is easiest described for a barotropic atmosphere with constant depth. In this case, Rossby waves conserve absolute vorticity. If a closed chain of air parcels around a circle of latitude is displaced, the new relative vorticity at a later time (ζ ) is: t1

ζ = − ttt 101 = −βδyff (1)

∂f where ft0 = Coriolis parameter at time 0, ft1 = Coriolis parameter at time 1, and β ≡ ∂y (planetary vorticity gradient at the original latitude).

Equation (1) describes a sinusoidal wave, where a southward displacement of a parcel results in a positive (cyclonic) vorticity perturbation and a northward displacement results in a negative (anti-cyclonic) vorticity perturbation. This results in a simple wave pattern shown below:

The direction of movement for Rossby waves depends upon two factors: the speed of the westerly flow and the number of troughs/ridges around a latitude circle. A Rossby wave will be stationary if there is westerly flow of 15 m/s with three troughs and ridges. Speeds less than this “critical” speed will result in a westerly drift of the Rossby wave, while higher speeds result in an easterly propagation.

Examine the 500 mb height analysis from Monday, 10-April 2006:

There are 5 distinct trough/ridge pairs in the Rossby wave at this time. Some of these may be due to more transient short-wave phenomena. In this circumstance, if each trough/ridge pair were considered part of a Rossby wave, the critical speed would be lower than 15 m/s.

It is possible to examine the propagation speed of Rossby waves with a bit of mathematics. If we have a Rossby wave in a barotropic atmosphere, the zonal phase speed (c) of the wave is:

β uc −= (2) K 2

where:

u = mean zonal wind speed ∂f β ≡ ∂y K2 = k2 + l2 where k = zonal wave number and l = meridional wave number. A wavenumber is just the number of waves over a given distance. So K2 is the total horizontal wavenumber squared.

Eq. (2) shows that Rossby waves always propagate westward relative to the mean zonal flow. We can develop an equation for when the Rossby wave is stationary. When c = 0, equation (2) becomes:

β K 2 = (3) u

Eq. (3) defines the relationship between the mean zonal flow and the wavenumber when the phase speed is zero.

Note that even though the phase speed of the wave may be zero, the group velocity of the Rossby wave does not have to be since they are dispersive waves. The group velocity of Rossby waves is eastward relative to the ground. If Rossby waves were non-dispersive (no changes in amplitude along the wave), the group velocity = phase speed.

See jump rope demonstration in class.

Equation (2) can be simplified. It is sometimes assumed that l (meridional wave number) = 0, since Rossby waves do not typically tilt in the meridional direction. Equation (2) becomes:

β uc −= (4) k 2

2π Furthermore, we know that the zonal wavenumber (k) = . Eq. (4) becomes: λ β uc −= (5) 2π [ ]2 λx