Rossby wave propagation

1
Rossby wave propagation
2
Propagation

• Three basic concepts
• Propagation in the vertical
• Propagation in the y-z plane
• Propagation in the x-y plane

3

• 1. Vertical propagation
• Reference back to Charney Drazin (1961)
• Recall the QG equations from MET 205A
• QGVE
• QGTE

4

• Vertical propagation
• These were combined as follows
• We defined a tendency ? and developed the
  tendency equation (and used it to diagnose height
  tendencies)
• We also developed the omega equation.
• A further equation not much emphasized was
  the quasi-geostrophic potential vorticity
  equation (QGPVE), found by elimination of
  vertical velocity.

5

• Vertical propagation
• If we use log-pressure vertical coordinates, the
  result is
• Where

6

• Vertical propagation
• Here, ? is streamfunction.
• We now linearize this in the usual way, assuming
  a constant basic state wind U (see Holton pp
  421-422).
• We next assume the usual wave-like solution to
  the linearized equation
• Where z is log-pressure height, H is scale
  height, and everything else is as usual.

7

• Vertical propagation
• Upon substitution, we get the vertical structure
  equation for ?(z)
• Where

8

• Vertical propagation
• Obviously, the quantity m2 is crucial as it was
  in the vertical propagation (or not!) of gravity
  waves.
• When m2 gt 0, the wave can propagate in the
  vertical, and ?(z) is wave-like.
• When m2 lt 0, the wave does not propagate, and the
  solution decays expenentially with height (given
  the normal upper BCs).

9

• Vertical propagation
• Specializing to stationary waves, where c0, we
  have
• Stationary waves WILL propagate in the vertical
  if the mean wind U satisifies
• i.e., we require 0 lt U lt Uc U must be westerly
  but not too strong.

10

• Vertical propagation
• If the mean wind is easterly (U lt 0), stationary
  waves cannot propagate in the vertical.
• Likewise, if mean winds are westerly but too
  strong, there is no propagation.
• What does this tell us about the the observed
  atmosphere?

11

• Vertical propagation
• Observations? See ppt slide
• Obs show the presence of stationary
  planetary-scale (Rossby) waves in winter (Ugt0)
  but not in summer (Ult0).
• The theory above helps us to understand this.
• Further the results are wavenumber-dependent.
• Consider winter and assume 0ltUltUc.

12

• Vertical propagation
• Note that m depends on zonal scale (Lx) thru k
• Consider zonal waves N1, 2, and 3.
• Lx decreases as N increases, which means that k
  increases as N increases, which means that Uc
  decreases as N increases.
• So propagation becomes more difficult as N
  increasesthe window of opportunity 0ltUltUc
  shrinks as N increases.

13

• Vertical propagation
• This means that we are MOST LIKELY to see wave 1
  in the stratosphere, less likely to see wave 2,
  and even less likely to see waves 3 etc.
  precisely as observed!
• Thus, stratospheric dynamics (at least for
  stationary waves) is dominated by large-scale
  waves.

14

• Vertical propagation - summary
• For stationary waves, theory verifies
  observations (or vice versa) that the largest
  waves can propagate vertically when flow is
  westerly, but not easterly.
• Thus we expect large-scale waves, but not
  transient eddy-scale waves to propagate upward
  (smaller waves are trapped in the troposphere).
• Theory gets more complicated if we let UU(z)
  see Charney Drazin.

15

• 2. Propagation in the y-z plane
• Reference back to Matsuno (1970)
• Matsuno extended these ideas to 2D (y-z)
• These ideas were also developed in part II of the
  EP paper.

16

• Propagation in the y-z plane
• Matsuno again considered a QG atmosphere, this
  time in spherical coordinates (Charney Drazin
  beta plane).
• He also considered the linearized QGPVE, and this
  time assumed a more general solution of the form
• He allowed UU(y,z) now, and thus the amplitude
  of the eddy ?(y,z) is also a function of y and
  z this is to be solved for.
• Overall this is more realistic (than Uconstant).

17

• Propagation in the y-z plane
• Matsuno thus obtained a PDE for the amplitude
• A second order PDE for amplitude, which was
  solved numerically.
• The only thing to be prescribed was the mean
  wind, U, which was taken from an analytical
  expression to be representative of the observed
  atmosphere.

18

• Propagation in the y-z plane
• In the equation, we have
• Here, an important term is s zonal wavenumber
  (integer).
• The quantity ns2 acts as a refractive index, as
  we will see, and note here that it depends on the
  mean wind (U) and on wavenumber (s).

19

• Propagation in the y-z plane
• The results? Matsuno computed structures
  (amplitude and phase? is assumed complex) for
  waves 1 and 2.
• Matsuno found qualitatively good agreement
  between his results and observations, in both
  phase and amplitude.
• In particular, in regions where ns2 is negative,
  wave amplitudes are small, indicating that Rossby
  waves propagate away from these regions.
• Conversely, in regions where ns2 is positive and
  large, wave amplitudes are also large.

20

• Propagation in the y-z plane

21

• Propagation in the y-z plane

22

• Propagation in the y-z plane

23

• Propagation in the y-z plane
• In fact, we can develop based on the EP paper
  a quantity called the Eliassen-Palm Flux vector
  (F) and use it to show wave propagation.
• Without going deep into details, we can write for
  the QG case
• It can be shown that the direction of F is the
  same as the direction of wave propagation (F //
  cg), and also that div(F) indicates the wave
  forcing on the mean flow.
• See Holton Cht 10, 12 for more.

24

• Propagation in the y-z plane

25

• Propagation in the y-z plane

26

• Summary
• Planetary-scale (stationary Rossby) waves can
  propagate both vertically and meridionally
  through a background flow varying with latitude
  and height.
• The ability to propagate can be measured in terms
  of both a refractive index, and the EP flux
  vector.
• Both will be used in the next section on
  propagation in the x-y plane.