Rossby%20Waves

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Rossby Waves

• Prof. Alison Bridger
• MET 205A
• October, 2007

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Review

• The Rossby wave analysis in Holtons 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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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.

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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 Uconstant, we can solve analytically.
• If not, we cannot!!!

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continued...

• With Uconstant, we get (Holton 12.11)
• where
• And ?? is the eddy streamfunction.

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continued...

• 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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continued...

• In solving this, we find the vertical propagation
  characteristics (just like with internal gravity
  waves)
• m2 gt 0 ? propagation
• m2 lt 0 ? evanescent wave (not propagating)
• And

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• So for a stationary wave (c0) 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 gt 0) we observe
  large-scale waves in the stratosphere, whereas in
  summer (U lt 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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continued...

• 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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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 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.

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continued...

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

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continued...

• 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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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.)

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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 UU(y,z) that
  was analytical, but a fair representation of
  observed wintertime stratospheric winds.

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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.

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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

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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