The Eliassen-Palm (EP) paper

1
The Eliassen-Palm (EP) paper

• These days, the EP paper is referenced for its
  work on large-scale wave propagation in the
  vertical and meridional directionchapter II in
  the paper.
• Chapter I deals with Gwaves and their vertical
  propagation lets review this here.

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Some reminders

• Gwave propagation depends strongly on mean wind
  profile and on background stability N2.
• The case with constant wind and N2 are easy to
  solve, but less relevant.
• This paper looks at the more complicated cases of
  wind U(z) and N2(z).
• Remember also that GWs can transport energy
  momentum (which are related see below) in the
  vertical.

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Chapter I part 2

• Assumptions
• Adiabatic (piezotropic!!)
• Mean wind is a function of height
• Mean state is hydrostatic (2.2)
• Brunt-Vaisala frequency is given by ?2 (2.4)
• Motion in x-z plane
• Thus the eddy/perturbation equations are the
  (familiar!!!) u-momentum (2.6), w-momentum (2.7)
  and continuity equations (2.8).

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Continued

• Note that we have
• Where ? is some vertical displacement.
• We use the equations to get an Energy Equation
  (2.10), with energy

5
Continued

• This is KE PE IE.
• Wave energy has components (pw) which is vertical
  wave energy flux, and (pu) (horizontal).
• Eq (2.10) says that the divergence of wave energy
  flux depends on both the mean wind shear (Uz) and
  on the product (uw), which is almost ( ).

6
Continued

• If we integrate (2.10) over x, and use our
  notation, we get
• If we also manipulate (2.6), we can get
• Comparingit must be that is independent
  of height so long as U?? 0.

7
Continued

• Also, note that wave energy and vertical momentum
  fluxes are in opposite directions (from Eq
  (2.13)).
• Section 3
• Next, we take the standard wave-like approach,
  which is why the overbars appear.
• So (3.1) and (3.2) are equivalent to (2.12) and
  (2.13), and (3.3) is the new (2.14).

8
Continued

• By manipulating the governing equations, we
  develop (3.5) which allows us to eliminate u and
  write everything in terms of w.
• (3.6) is a good approximation to (3.5), so we can
  now say that does not vary with
  height.
• Assuming the usual wave-like form for the
  solution (3.8), we can now develop the usual diff
  eq for the unknown, w

9
Continued

• Which is sometimes called the Scorer Equation.
  Here,
• So as usual, propagation depends on whether l2 gt
  k2 (good) or vice versa (bad).

10
Continued

• We will later use
• Section 4 constant l2
• Eqs (4.1) (4.3) are the external (boring) wave
  case, as seen in 205A.
• Eqs (4.4) (4.6) are in (interesting) internal
  case.
• Note that in this case, from (4.6), the A piece
  is the upward-propagating piece, while the B
  piece is the downward propagating component.

11
Continued

• We can define a reflection coefficient, r, with
• Section 5 layered atmospheres
• Choose l2 constant in each layer.
• Demand also that w and wz are continuous across
  the layer boundaries.

12
Continued

• Always have the uppermost layer be infinite in
  extent.
• Two-layer casefirst suppose

ExternalW2B2exp(-?z)
internalW1A1exp(i??z)B1exp(-i??z)
13
Continued

• In this case, we get r1.
• Thus, the wave propagates in the lower layer, not
  in the upper layer, and is 100 reflected from
  the upper layer apparently through the entire
  layer (see text).
• Next suppose both layers are internal (different
  ls).
• In the upper layer, we insist on upward energy
  propagation, so that the solution there is
  W2A2exp(i??2z), which is then matched with the
  solution in the lower layer (see previous
  figure).

14
Continued

• In this case, we get
• So that any value of 0 ? r ? 1 is possible.
• When r1, there is total reflection at the
  interface.

15
Continued

• Three-layer case

internalW3A3exp(i??3z)B3exp(-i??3z)
Internal or external
internalW1A1exp(i??z)B1exp(-i??z)
16
Continued

• The equations are (5.11) for the bottom thru
  (5.13) for the top.
• We can use them to compute r (5.14 and 5.15,
  which, bothugh).
• Everything is shown graphically on page 14.
• According to the text, the quantity
• is the main thing that determines the power
  of the middle layer to transmit energy upwards.

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Continued

• For b?? 1, reflection is small (when Xlt0,
  internal middle layer).
• Difficult to draw major conclusions here!
• One thing to noteeven if the middle layer is
  external, the reflection coefficient r lt1,
  meaning that some energy leaks through into the
  upper layer!

18
Continued

• Section 6 a real example
• Fig 5 shows the observed wind and stability
  profiles for the day in question, plus the
  assumed layering (four total layers).
• Results are in Table 1

Lx (km) 63 26 17 7
r 0.65 0.82 0.97 1.0
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Results

• The shortest waves (Lx lt 17 km) are almost
  completely reflected (r gt 0.97) and do not make
  it into the stratosphere.
• Even waves a bit longer (Lx ?? 26 km) have r
  0.82 (82 reflected).
• Waves with Lx gt 26 km (note that this cutoff is
  for this day only, and depends on wind, shear,
  and stability), do propagate up into the
  stratosphere, although even some of their energy
  is reflected back down into the troposphere.

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Results

• We see then that the troposphere acts as a
  filter, removing short-wavelength gravity waves
  from the spectrum of waves propagating energy up
  into the stratosphereand beyond.
• What happens next to these waves is the breaking
  discussed first by Lindzen.

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Results

• One final note from the bottom of page 17
• For stationary GWs (with phase speed c0), as
  soon as they encounter an elevation where the
  mean wind U0, there is a singularity in the
  equation, and this is where the wave stops.
• Observations tell us that winds aloft are
  westerly in winter (Ugt0) and easterly in summer
  (Ult0).
• Thus there will be differences in the spectrum of
  waves propagating upwards into the higher
  atmosphere in winter versus summer.

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Results

• For waves with c?? 0, this is not a problem.
• More precisely, when c?? 0, the singularity
  occurs at different altitudes, where c U.
• In a westerly wind regime, GWs with c gt0 will
  (may) find an elevation at which c U, but GWs
  with c lt 0 will not! Easterly vice versa.
• Again there is filtering! Observations should
  be able to confirm this.