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 direction…chapter II in the paper. Chapter I deals with Gwaves and their vertical propagation – let’s review this here.

Some reminders…  Gwave propagation depends strongly on mean wind profile [ ] and on background stability [N 2 ].  The case with constant wind and N 2 are easy to solve, but less relevant.  This paper looks at the more complicated cases of wind U(z) and N 2 (z).  Remember also that GWs can transport energy & momentum (which are related – see below) in the vertical.

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).

Continued…  Note that we have:  Where  is some “vertical displacement”.  We use the equations to get an Energy Equation (2.10), with energy

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 (U z ) and on the product (uw), which is almost ( ).

Continued…  If we integrate (2.10) over x, and use our notation, we get:  If we also manipulate (2.6), we can get  Comparing…it must be that is independent of height so long as U 0.

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).

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’:

Continued…  Which is sometimes called the Scorer Equation. Here,  So as usual, propagation depends on whether l 2 > k 2 (good) or vice versa (bad).

Continued…  We will later use: Section 4 …constant l 2  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.

Continued…  We can define a reflection coefficient, r, with: Section 5 …layered atmospheres  Choose l 2 constant in each layer.  Demand also that w’ and w z ’ are continuous across the layer boundaries.

Continued…  Always have the uppermost layer be infinite in extent.  Two-layer case…first suppose: External…W 2 =B 2 exp(-z) internal…W 1 =A 1 exp(iz)+B 1 exp(-iz)

Continued…  In this case, we get r=1.  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 “l’s”).  In the upper layer, we insist on upward energy propagation, so that the solution there is W 2 =A 2 exp(i 2 z), which is then matched with the solution in the lower layer (see previous figure).

Continued…  In this case, we get  So that any value of 0  r  1 is possible.  When r=1, there is total reflection at the interface.

Continued…  Three-layer case: Internal or external internal…W 1 =A 1 exp(iz)+B 1 exp(-iz) internal…W 3 =A 3 exp(i 3 z)+B 3 exp(-i 3 z)

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, both…ugh).  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.

Continued…  For b 1, reflection is small (when X<0, internal middle layer).  Difficult to draw major conclusions here!  One thing to note…even if the middle layer is external, the reflection coefficient r <1, meaning that some energy “leaks” through into the upper layer!

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 L x (km) r

Results…  The shortest waves (L x 0.97) and do not make it into the stratosphere.  Even waves a bit longer ( L x  26 km) have r = 0.82 (82% reflected).  Waves with L x > 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.

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 stratosphere…and beyond.  What happens next to these waves is the breaking discussed first by Lindzen.

Results… One final note from the bottom of page 17  For stationary GWs (with phase speed c=0), as soon as they encounter an elevation where the mean wind U=0, there is a singularity in the equation, and this is where the wave stops.  Observations tell us that winds aloft are westerly in winter (U>0) and easterly in summer (U<0).  Thus there will be differences in the spectrum of waves propagating upwards into the higher atmosphere in winter versus summer.

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 >0 will (may) find an elevation at which c = U, but GWs with c < 0 will not! Easterly – vice versa.  Again – there is filtering! Observations should be able to confirm this.