Rossby wave propagation

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 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.

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

Vertical propagation • Obviously, the quantity m2 is crucial – as it was in the vertical propagation (or not!) of gravity waves. • When m2 > 0, the wave can propagate in the vertical, and (z) is wave-like. • When m2 < 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 < Uc … 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<Uc.

Vertical propagation • Note that “m” depends on zonal scale (Lx) thru k: • Consider zonal waves N=1, 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 increases…the “window of opportunity” [0<U<Uc] 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 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).

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.

Propagation in the y-z plane

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.

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.

Rossby wave propagation