Les ondes atmosphériques considérées comme associées aux discontinuités du tourbillon

Abstract

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 ? (? = 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 ? = o, where ? 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 – σ0λ, 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.