Les ondes atmosphériques considérées comme associées aux discontinuités du tourbillon
Authors: Queney Paul 
Published: January 2012

The atmospheric waves considered as associated with the vorticity discontinuities
It is suggested that the jet stream at about 12 km-level may be idealized as a system of two or three zonal currents with uniform but different values of absolute vertical vorticity ζ, and such that the wind velocity itself is continuous at each boundary.
The approximation of a horizontal non-divergent motion is made for the jet-stream’s waves (chapter II), and a proper system of equations is derived for these waves (by using the meridional coordinate γ defined by tanh γ = sin φ (φ = latitude), the equations are particularly simple even when the earth’s curvature is taken into account, and in the case of constant ζ the problem is reduced to the resolution of the very classical equation ∇2 ѱ = 0, where ѱ is the local disturbance of the stream function.
The case of a simple jet (two currents), then that of a double jet (three currents) are successively considered in the next chapter, the wave motion being assumed to vanish at each pole, and it is also shown how some of the results can be generalized for a larger number of currents. (In any case the dispersion equation, giving the angular phase-velocity α in terms of the angular wave-length λ (inverse of the wave number, which must be an integer), is an algebraic one in α, and its degree equals the number of internal boundaries.) The waves of a simple jet are therefore all stable (their dispersion equation is α = ω0 – σ0 λ, if ω0 is the angular wind-velocity at the axis, and 2σ0 denotes the ζ-discontinuity, positive northwards). On the contrary, a system of more than two currents may have unstable (amplified) waves, if one discontinuity 2σ0 is negative while the other ones are all positive. If in addition σ0 is small, the wave lengths of the unstable waves form a definite series of spectral lines, the maximum number of which equals the number of the positive discontinuities, and the phase velocity of these waves is approximately the wind velocity at the latitude of the negative discontinuity: in the special case of a double jet, there is therefore at most one unstable wave length λ0, given by λ0 = (ω1–ω0)/σ1, where the subscripts 0 and 1 are relative to the negative and positive discontinuity, respectively. For all the waves of a simple or double jet, there is a maximum of the amplitude at a definite latitude and the wave motion becomes negligible at a distance of the order of the wave length.
The effect of horizontal eddy-viscosity is discussed in chapter IV. If the kinematical coefficient is of the order of 106 C. G. S. at most, it is found that the wave motion is generally modified only in the vicinity of the boundaries. The conclusions are however different in the special case where one discontinuity is very small: some of the stable waves are then changed to damped waves, while the unstable waves are practically unmodified. The problem involves Bessel functions of order 1/3.
Chapter V contains a tentative theory of the semi-permanent centers of action, based on the properties of the stationary waves of a simple or double jet. For any given wave number, stationary waves are found to be possible in a westerly simple jet with a maximum windvelocity U0 along the axis, provided U0 = σ0L/2σ (resonance condition), where L is the wave length at the latitude of the axis, and if σ0 is assumed to have the constant value 3.5 × 10–5 not differ essentially from those derived from Rossby’s theory. But this agreement disappears in the case of a double jet, since therc are generally at least two very different such jets corresponding to the same given wave number, and that may explain why the correlation between the zonal index and the number of centers of action is often rather loose. It is also possible to have two simultaneous different stationary wave lengths in the same jet-stream.
The last chapter is similarly devoted to a tentative theory of cyclone waves, based on the properties of the unstable waves appearing in a simple or double jet, when such a jet is slightly deformed by an additional negative ζ-discontinuity. It is suggested that the agent of this deformation may be the vertical convection created in the lower atnlosphere by thermal instability, since this convection tends to reduce the wind velocity in a relatively narrow zone above the instability area. If the assumptioil is made that the unstable waves can represent cyclone waves provided their wave length is larger than 1,000 km and smaller than 4,000 km, with an amplification coefficient larger than 10–6 C. G. S., it is found that cyclone waves can develop in a simple jet if the deformation takes place in either one of two “sensitive zones” located at less than 300 km on each side of the axis, and not larger than 500 km about. For a double jet there are normally four such sensitive zones. All the properties of these theoretical cyclone waves agree quite well with that of the actual cyclone waves of the polar fronts, except for the phase velocity which is somewhat too large, but this discrepancy is not surprising since the dynamic31 effect of the lower atmosphere has been disregarded. Finally it is suggested that the same theory can also apply to tropical cyclones and easterly waves, if the assumption is made that these features are produced when one of the sensitive zones of the jet stream happens to penetrate into a monsoon area for instance.