Rossby Waves

If the spinning earth's crust was completely uniform in height and character (i.e. no mountains, seas etc.), but a surface made of the same substance, it is likely that the differential heating between poles and the equator would still produce a west-to-east airflow in the lower atmosphere [ i.e. the troposphere and lower stratosphere ], ( assuming of course in this unlikely situation we had an atmosphere), but without the very variable wave-like motion that can be seen on both daily and time-average maps at any particular level in the atmosphere. (N.B. there would be detectable wave-trains due to the inertial forces on the atmospheric mass on a rotating sphere.)

The major (or 'long') waves (also known as Planetary Waves) we see on weather maps for the upper air are produced largely because the atmosphere in motion encounters barriers to its progress, and is forced to ascend (by the changing surface level), then allowed to descend (under gravitational influence), and the resultant "squashing" and "stretching" respectively of the air columns involved lead to alterations to the rates of "spin" of the air flow (vorticity). These variations in the rate of spin must be balanced on a rotating earth for the system to remain stable -- assuming there are no other forces at work: see later. 

The principle is known as CAV, the Conservation of Absolute Vorticity, and was investigated by Carl-Gustav Rossby and others in the late 1930's. When considering the northern hemisphere, air that is forced to ascend tends to turn to the right, and as it descends again, it tends to turn to the left, inducing a ridge/trough pattern to the broadscale westerlies. These long waves are often known as Rossby waves, after the person who did so much to investigate their character. 

Major mountain chains provide obvious sources of such deflection, and the Rockies and the Andes, which lie astride the westerly flow in each hemisphere, provide good examples. These long-waves are key elements in the atmospheric circulation, and can be traced well into the stratosphere. At any one time, there are between 3 and 7 such waves, the number in any particular latitude band dependent upon a fine balance between the speed of the airflow through the trough/ridge system and the wavelength. Rossby, following his investigation of these long-waves, derived the relationship: The Rossby Equation

From this equation, we see:

  1. At any particular latitude (phi), and for a given wavelength(L), the governing factor which forces change is U, which is the mean west-to-east airflow through the system. This is in turn governed by short-term developments in the synoptic pattern; however, the development/vigour of such "synoptic-scale" features (i.e. frontal or baroclinic systems), is directly influenced by the position and strength of the aforementioned long-wave pattern. In general, a strengthening zonal flow drives the major long-waves apart; a weakening zonal flow allows a 'closing-up' of the wave pattern. There is therefore a delicate feedback loop at work ... ideal for super-computers running sophisticated NWP packages to get their teeth into.
  2. For a particular latitude band, there is a combination of U and L which will give C=0, i.e. the wave-train is stationary. These conditions often apply downstream of the so-called 'anchor troughs', which form due to the principles of CAV outlined above. From this, we can predict how many long-waves might be found in any latitude band for a particular zonal component of wind, and the wavelength of same...the table below summarises the various variables and results:
  3. The wave speed decreases with the square of the wave-length, i.e. very long waves are slow-moving, and this is commonly observed in the real atmosphere. The average west-to-east movement of a major long wave is around 2 to 5 degrees of longitude per 24 hr.
Table for Rossby long-waves
latitude zonal wind speed        
>>>> knots 10 20 30 40 50
60 deg.(N/S) 76 deg long. 108 deg long. 132 deg long. 152 deg long. 170 deg long.
50 deg.(N/S) 52 deg long. 74 deg long. 90 deg long. 104 deg long. 116 deg long.
40 deg.(N/S) 40 deg long. 57 deg long. 69 deg long. 80 deg long. 90 deg long.
approx. number of waves (at 50 degN/S) 7 5 4 3 to 4 3

 

All the foregoing assumes that other convergent/divergent effects due for example to cyclonic development, are negligible; this is unrealistic for the real atmosphere except perhaps at the level of non-divergence (LND) around 600 mbar, and operational forecasting models attempt to take such factors into account. The same principles will apply though.

In Summary

The circulation in the "upper air" (say 700 mbar [or hPa]) upwards, consists of a large circumpolar vortex system -- on this broad flow are superimposed a system of variable, but usually slow-moving long waves, sometimes stationary, or very occasionally retrogressive (i.e. they move east-to-west). Down the scale again, there are a series of short-waves running quickly (circa 15 degrees of longitude per 24 hr: +/- 5 degrees) through the flow, driving, and being in turn modified, by the familiar synoptic scale disturbances in the lower troposphere (i.e. frontal systems), which in turn feed-back energy (in all forms) up the scale to influence the broadscale pattern. 

It is no wonder that L.F. Richardson, postulated in 1922, that computing power of some weight would be needed to properly solve the equations governing atmospheric motion, and much endeavour in operational and research meteorology has been devoted to this end ... and the story goes on!