Rossby Wave Speed Equation
The Rossby Wave Speed Equation can provide powerful insight into the evolution of flow patterns in the mid troposphere (and, thus, using synoptic-scale arguments, the whole free troposphere from 700 mb to the Tropopause). But it can only be used judiciously:
- the portion of the pattern that is geostrophic (away from strong short waves with their strong divergence/convergence patterns and temperature advection fields)
- the portion of the pattern for which the winds are not affected by short waves or jet streaks
The steps would be as follows
- isolate a long wave in the flow pattern...be careful not to be distracted by the short waves moving around the pattern, as they will have their own curvature fields and very amplified wind speeds;
- compute the wavelength...if you have a perfectly shaped long wave with few or no short waves moving around it, then this is simple...more commonly you will have to compute a half-wavelength and multiply by 2;
- find the mean latitude of the long wave. This may be a bit more difficult than you think. That's because some map projections expand out the latitude as one goes northward;
- diagnose the mean wind speeds (on NWS maps in knots)in the long wave. To do this, first ignore all winds associated with short waves or jet streaks. Second, find a wind speed at the axes of the trough, ridge and down stream trough, or a mean wind, AWAY from short waves or curvature maxima/jet streaks. Add those up and divide by the number of wind observations you are using to obtain a mean wind.
- Insert all these values in the Rossby Wind Equation to determine the phase speed of the wave.
The result often gives you great insight into the forecast issues for a given area. A strongly progressive or retrogressive pattern will bring changing weather conditions to given longitudes. A stationary Rossby wave will produce repetitive similar weather events at given longitudes.
How do the constraints on the long wave pattern change? Unfortunately, by not considering the vorticity, temperature and momentum advection associated with short waves and jet streaks, we eliminate the very factors that might change the constraints. For example, strong short waves can produce strong warm or cold advection that will change the height field aloft...in such a manner that the wavelength of the Rossby Wave will change, and or the mean winds will change.
As an exercise, let's do this for the long wave pattern over the eastern Pacific today.
By the way, here is the sea level pressure and sea-level pressure and 1000-500 mb thickness fields for the Pacific at the same time.