Rossby wave propagation

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

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

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.

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

Vertical propagation Here,  is streamfunction. We now linearize this in the usual way, assuming a constant basic state wind U (see Holton pp ). 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.

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

Vertical propagation Obviously, the quantity m 2 is crucial – as it was in the vertical propagation (or not!) of gravity waves. When m 2 > 0, the wave can propagate in the vertical, and  (z) is wave-like. When m 2 < 0, the wave does not propagate, and the solution decays expenentially with height (given the normal upper BCs).

Vertical propagation Specializing to stationary waves, where c=0, we have: Stationary waves WILL propagate in the vertical if the mean wind U satisifies: i.e., we require 0 < U < U c … U must be westerly but not too strong.

Vertical propagation If the mean wind is easterly (U < 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?

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

Vertical propagation Note that “m” depends on zonal scale (L x ) thru k: Consider zonal waves N=1, 2, and 3. L x decreases as N increases, which means that k increases as N increases, which means that U c decreases as N increases. So propagation becomes more difficult as N increases…the “window of opportunity” [0<U<U c ] shrinks as N increases.

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.

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 U=U(z) – see Charney & Drazin.

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.

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 U=U(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 U=constant).

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.

Propagation in the y-z plane In the equation, we have Here, an important term is s = zonal wavenumber (integer). The quantity n s 2 acts as a “refractive index”, as we will see, and note here that it depends on the mean wind (U) and on wavenumber (s).

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 n s 2 is negative, wave amplitudes are small, indicating that Rossby waves propagate away from these regions. Conversely, in regions where n s 2 is positive and large, wave amplitudes are also large.

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 // c g ), and also that div(F) indicates the wave forcing on the mean flow. See Holton Cht 10, 12 for more.

Propagation in the y-z plane

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.