Rossby Waves

1. Rossby Waves Prof. Alison Bridger MET 205A October, 2008

2. Review • The Rossby wave analysis in Holton’s Chapter 7 is set in a simple, barotropic atmosphere. • We are able show that the waves exist, and that they propagate westward. • In a slightly more complicated analysis these waves can be shown to propagate north-south, as well as east-west. • Karoly and Hoskins (1982) looked at Rossby wave propagation in a spherical barotropic model, and showed that from a source region, waves propagate away following a great circle.

3. Continued... • By this mechanism, disturbances can be spread to remote regions of Earth (e.g., from the tropics to mid-latitudes, for example as a consequence of El Nino). • These simple Rossby waves do not propagate in the vertical.

4. Rossby Waves in a Stratified Atmosphere • In a stratified atmosphere, the BVE is no longer the appropriate equation to study. • Instead we must use the QGPVE (Cht 6). • The analysis can get a lot more complicated! • As usual, we linearized the QGPVE to study waves, and we assume a non-zero background wind U. • If U=constant, we can solve analytically. • If not, we cannot!!!

5. continued... • With U=constant, we get (Holton 12.11): • where: • And  is the eddy streamfunction.

6. continued... • To solve, we assume the usual: • The “(z)” term is the amplitude as a function of height, and substitution shows that (z) satisfies the 2nd order ODE:

7. continued... • In solving this, we find the vertical propagation characteristics! • m2 > 0  propagation • m2 < 0  evanescent wave • And

8. continued... • So for a stationary wave (c=0) we have… • Rossby waves canpropagate in this case providedthe prevailing background wind has these properties: FIRST, The prevailing background wind U must be positive! • This means that Rossby waves will only propagate upward (e.g., from tropospheric sources into the stratosphere) when the background winds are positive (westerly), as they are in winter.

9. continued... • This explains why in winter (U > 0) we observe large-scale waves in the stratosphere, whereas in summer (U < 0) they are absent! • Here, we are talking about stationary planetary waves, rather than travelling waves.

10. continued... • SECOND, the westerly winds cannot be too strong, and the critical strength depends on the scale of the wave. • The scale-dependence is such that wave one can most effectively propagate upward, wave two somewhat less effectively, wave three even less, etc. • This explains why we see large-amplitudes in waves one and two in the stratosphere, but much smaller amplitudes in waves three and upward.

11. continued... • A NEXT STEP is to let U be linear in z. • At this point (already!) the resulting equation (the vertical structure equation) becomes difficult to solve. • The above analysis was first performed by Charney & Drazin (1961). This is a paper that is often referred to!

12. continued... • If you then proceed to assume U(y,z) - as is more realistic - you leave the realm of being able to solve the exact equation on paper. Instead we solve analytically. • In this case, we can develop a second order PDE for the (complex) wave amplitude, (y,z), having assumed a solution of the usual wave-like form.

13. continued... • The governing equation is then: • Here, and • qbar is the basic state potential vorticity – refer back to Eq. 6.25.

14. continued... • d(qbar)/dy depends upon U and its first and second order derivatives in the vertical and in the horizontal. • The quantity n2 is the equivalent of a (refractive index) 2 - just as in the propagation of light! • Thus, Rossby waves will tend to propagate into regions of high n2, and will avoid regions of negative n2.

15. continued... • Plots of n2 - when compared with plots of wave amplitude (both as functions of y and z) - help us understand the distribution of wave amplitude. • We note that n2 depends also on wavenumber squared. • The impact of this is that if wave one can propagate for a certain wind structure, wave two may not be able to (etc. for waves three, etc.)

16. continued... • This is a generalization of the result above for constant U (and again explains why we see waves one and two in the stratosphere, but not so much three onward). • The first numerical solution of the problem is due to Matsuno (1970), who solved the structure equation on a hemispheric yz-grid. • He assumed a background wind state U=U(y,z) that was analytical but a fair representation of observed wintertime stratospheric winds.

17. continued... • He solved for the steady wave structure (amplitude and phase as functions of y and z for waves one and two). • The forcing was provided via specification of wave amplitude at the lower boundary - simulating the upward propagation of wave energy from the troposphere. • Matsuno also computed the (refractive index) 2 quantity, thus demonstrating the link between (refractive index) 2 and wave amplitude distribution.

18. continued... • The results compared well with observations in a general sense (meaning that they might not look like a specific winter, but might generally look like observations). • Subsequent work has looked at: • how variations in wind profiles (sometimes subtle) impact wave structures • the role of different forcing mechanisms (topographic vs thermal) • solution of the full primitive equation problem (Matsuno solved the QG version) • detailed calculations of travelling Rossby modes with realistic background states

19. continued... • for example, there is a 5-day wave both observed and theoretically predicted • it has these characteristics: wave one (east-west), symmetric about the equator, westward propagating with a period of 5 days • the simulated wave has the same characteristics and these do not depend strongly upon the background wind details • there are other modes that are very sensitive to the background winds