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Rossby wave propagation. Propagation in the x-y plane Hoskins & Karoly Section 5. The linearized equations reduce to (5.9) where ′ is the usual perturbation streamfunction. In this case, And which is basically (from 205A). Propagation in the x-y plane
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Propagation in the x-y plane • Hoskins & Karoly Section 5. • The linearized equations reduce to (5.9) • where ′ is the usual perturbation streamfunction. • In this case, • And • which is basically (from 205A)
Propagation in the x-y plane • Assuming a wave-like solution, we get the dispersion equation • We can define a group velocity vector:
Propagation in the x-y plane • The orientation of the vector cg then describes the direction in which the wave packet (i.e., the wave’s energy) propagates. • This defines a ray in general. • An important property from (5.11) is that bothkand must be constant along a ray (cannot vary in x and/or t). • Thus, only l can vary along a ray as the background properties (mostly mean wind) vary. • For the special case of a a stationary wave, we set =0, and compute the ray structure from dy/dx = k / l . • Energy propagates along a ray with speed |cg|.
Propagation in the x-y plane • HK define Ks2=(k2+l2), with • So, as a wave propagates, it encounters varying background winds. • As the background wind varies (through 5.10), M varies, which changes Ks (5.16) and thus l (5.15), the meridional wavenumber. • Next HK derive a WKJB solution for the wave’s amplitude: • Assuming: • We get that P(y) satisfies:
Propagation in the x-y plane • Note that l is a function of y (so the equation is not straightforward to solve) • The WKJB solution gives: • where C is an integration constant. • So here, note that as l decreases/increases, the amplitude increases/decreases.
Propagation in the x-y plane • Thus, as we travel down the ray, we encounter varying background conditions (through (5.10)). • From this we compute a new value of M (from 5.10). • From this we compute a new value of Ks (from 5.16). • From this we compute a new value of l (from 5.15, where we have decided, for example, to track wavenumber one, which gives the value of k). • From this we compute the new amplitude (from 5.23).
Some background information (from James’ book): • The group velocity gives the direction and rate at which “wave action” is spread away from the source region. • James defines the “wave action density” as: • The quantity cgyA is conserved along a ray (as long as the background varies “slowly” only in y). • It can be shown that • And thus as l decreases as a wave propagates poleward, it’s amplitude increases. • Thus a relatively “small” forcing a lower latitudes can give a large response in midlatitudes.
Propagation in the x-y plane • Back to HK… • Using the relations developed, one can compute the progress of a ray as time evolves, and one can compute the wave amplitude along the ray. • Figures 14 & 15 give examples. • NOTE THAT in section 5.c, they examine the case of a constant angular velocity background flow. • In this case, they make the point that the equation of the ray (5.27) is the same as the equation for a great circle (citing Longuet-Higgins)!
Propagation in the x-y plane • Note that for some waves, l 0 at some latitude, and thus the wave is ‘refracted” back to lower latitudes (and is evanescent poleward of there).
Propagation in the x-y plane - summary • HK solved the linearized equations to find the structure on the sphere of a Rossby wave train forced by an isolated heat source. • The results bear a strong resemblance to observed teleconnections patterns (i.e., in character). • The theory indicates that we expect rays to propagate away from the source region following a great circle, and with speed and direction and wave amplitude given in terms of basic wave and background properties (including wavenumber and the mean wind profile).
Propagation in the x-y plane - summary • Based on this work (and other studies), there is plenty of reason to believe that teleconnections owe their existence to Rossby wave trains that have propagated considerable distances around the globe. • Next – let’s look at and review some of the more prominent teleconnection patterns.
