1. Prof. Alison Bridger MET 205A October, 2007
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Rossby Waves
Prof. Alison Bridger
MET 205A
October, 2007

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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.

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

6. continued... To solve, we assume the usual:
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To solve, we assume the usual:
The quantity “(z)” is the amplitude, and in this case can be a function of height. Substitution shows that (z) satisfies this 2nd order ODE:

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In solving this, we find the vertical propagation characteristics (just like with internal gravity waves):
m2 > 0  propagation
m2 < 0  evanescent wave (not propagating)
And

8. So for a stationary wave (c=0) we have…
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So for a stationary wave (c=0) we have…
Rossby waves can propagate in this case provided the 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.
This explains why in winter (U > 0) we observe large-scale waves in the stratosphere, whereas in summer (U < 0) they are absent!

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Here, we are talking about stationary planetary waves, rather than travelling waves.
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.

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This explains why we see large-amplitudes in waves one and two in the stratosphere, but much smaller amplitudes in waves three and upward.
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.
This analysis was first performed by Charney & Drazin (1961) – a paper that is often referred to!

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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 computationally.
When U(y,z), we can develop a second order PDE for the (complex) wave amplitude, (y,z), having assumed a solution of the usual wave-like form.

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

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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.

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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.)

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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.

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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.

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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

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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