Chapter 8: Simple Wave Motions

Chapter 8: Simple Wave Motions

Weather systems can be understood physically as waves with particular wavelengths that are excited by various forcings acting on the atmosphere.

Properties of Waves

Example: Pendulum

What is the physical interpretation of each term in this equation?

Propagating Wave

The phase of this wave is given by:

The phase depends not only on time (t) but also position (x,y,z).

k, l, and m are wavenumbers in the x, y, and z directions

Wavenumber : Vector given by components k, l, and m

Wavelength (): Distance between wave crests

We can identify wavelengths in each direction (x, y, and z)

Wavelength and wavenumber are related by:

Phase speed (c): The rate at which the phase of the wave propagates in each direction (e.g. the rate at which the crest of the wave propagates)

This relationship implies that waves with different wavenumbers (k, l, and m), and thus different wavelengths (x, y, and z), will propagate with different phase speeds (cx, cy, and cz) and will spread out (disperse) with time.

Dispersion relationship: The relationship between the frequency () and wavenumber () for a wave

If the phase speed (c) is not a function of wavenumber (k) and the wave is non-dispersive.

Group velocity (): The rate at which the energy of a wave travels

The components of the group velocity are given by:

Example: Atmospheric gravity wave

A gravity wave is a wave in the atmosphere in which buoyancy is the restoring force.

For a two-dimensional (x,z) gravity wave the frequency is given by:

where N is the Brunt-Väisälä frequency

What is the phase speed of this wave in each direction (cx and cz)?

What is the total phase speed (c) of this wave?

Note that the zonal (cx) and vertical (cz) phase speeds are not components of a vector whose magnitude is the total phase speed (c).

What is the group velocity of this wave in the zonal and vertical directions?

Perturbation Analysis

How can we avoid the nonlinearities in the atmospheric equations of motion?

Assume that all variables are the sum of a basic state and a small departure from that basic state.

Require that the basic state of all variables satisfy the governing equations.

Assume that the perturbations are sufficiently small that products of perturbations can be neglected.

Example: x-component of the Quasi-geostrophic (QG) momentum equation

Define the basic state and perturbation of each term in the equation as:

Substitute these definitions into the QG momentum equation:

We can drop all terms that include products of perturbations to give:

By definition the basic state satisfies the governing equation, so:

Using this gives a governing equation for the perturbation variables:

This equation is a linear equation – there are no products of dependent variables.

Application of perturbation analysis to the QG vorticity equation

For this analysis we will assume that the vertical velocity (w) is zero (which also implies that the flow is non-divergent), so:

Beta-plane approximation: Assume that the Coriolis parameter, f, varies linearly with latitude

where  is given by:

0 = reference latitude

f0 = 2sin(0)

a = radius of Earth (=6.37x106 m)

Define the basic state and perturbations as:

where we have assumed that the basic state has no meridional flow () (i.e. the basic state is purely zonal flow)

Using this definitions and the beta-plane approximation gives:

A wave-like solution of this equation is:

This solution is non-divergent () as required by the assumption of w = 0.

u'g and v’gcan be used to find ’g:

The dispersion relation (the relationship between  and k) for this wave can be found by substituting ’g and v’g into the perturbation form of the governing equation:

From this dispersion relation we note that the frequency is a function of wavenumber.

The phase speed is given by:

This equation is known as Rossby’s formula.

The zonal phase speed is a function of wavenumber (and wavelength) and waves with different wavenumber will propagate at different speeds.

Therefore, this wave is a dispersive wave.

What is the direction of propagation of these waves for ?

How does the zonal phase speed of longer wavelength waves compare to shorter wavelength waves?

Planetary Waves

The assumption of non-divergent flow used when deriving Rossby’s formula is not very realistic for mid-latitude weather systems, but is a more reasonable assumption for larger, planetary-scale, flows known as planetary waves or Rossby waves.

Example of planetary waves on upper air weather map

The phase speed of Rossby waves:

is determined in part by the advection of the wave by the mean flow and by the propagation of the wave.

We can consider three cases for Rossby waves:

1. Short wavelength waves (large k):

2. Medium wavelength waves:

3. Long wavelength waves (small k):

What is the direction of motion of the Rossby waves for each of these cases?

The critical wavenumber, ks, at which the Rossby wave will be stationary is given by:

The wavelength of a Rossby wave can be calculated as:

where n is the number of waves present at latitude 

Once we know the wavelength we can calculate the wavenumber and phase speed of the wave.

At  = 45° this gives:

We can also determine the period, T, of the wave at  = 45° (noting that ):

By definition  = 2/day, so T = n days.

Example: Calculate the wavelength, wavenumber, phase speed, and period of a Rossby wave from an upper air weather map

What speed zonal flow is required for this wave to be stationary?

Forcing of Planetary Waves

Three primary mechanisms for generating planetary waves:

1. Topographic forcing

2. Thermal forcing due to distribution of oceans and continents

3. Non-linear interactions with smaller scale disturbances

Long-term means of 500 mb geopotential heights indicate that planetary waves are located preferentially downstream of meridionally oriented mountain ranges.

Troughs are found downstream of the mountains and ridges are located upstream of the mountains.

What does this imply about the force extered on the Earth by these waves?

Equation for topographically forced Rossby wave:

Start with QG vorticity equation:

In a barotropic fluid the wind, and hence vorticity, does not vary with height, so we can integrate the QG voriticity equation from the surface of the Earth [h(x,y)] to the height of the tropopause, H, which is assumed to be a constant height.

For simplicity we will assume that the surface of the Earth is made up of a sinusoidal mountain:

We will also assume that the height of the mountain is much less than the depth of the tropopause (h<H).

A stationary wave solution (i.e. it does not depend on time) is given by:

The amplitude of this solution, A, can be found by substituting the solution into the governing equation:

Using gives:

What is the amplitude of this solution for ?