An analysis is made of the three-dimensional periodic oscillations in a jet at high Reynolds numbers. It is shown that there are two physically significant solutions: one is a symmetric and the other an antisymmetric function of the lateral coordinate; both involve a Lommel function. In the case of the symmetric solution, the maximum amplitude of the vertical velocity occurs near the inflection points of the velocity profile of the jet, whereas in the case of the antisymmetric solution, it occurs near the center as well as the inflection points of the jet.

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To study the growth and decay of the atmospheric wave motions in middle latitudes, we have analyzed the mechanism for the kinetic energy change in wavenumber domain, and computed the composite average of each term in the kinetic energy equation at various stages in the life cycle of the atmospheric waves. It is found that for the first 1 to 2 days the extra‐long waves of wavenumbers 1 and 3 grow by receiving kinetic energy from other finite amplitude waves through nonlinear interactions; in the next 1 to 2 days, they grow by gaining energy through nonlinear interactions and converting available potential to kinetic energy. These waves then maintained their peak energy for 3 to 4 days through the balance between the energy supply from conversion of available potential energy to kinetic energy and nonlinear interactions, and the energy lost by dissipation. The contribution of nonlinear interactions then changes to negative; and in the next 3 to 4 days, the waves decay by losing energy through nonlinear interactions and dissipation. The average life cycle of these extra‐long waves is about 10 to 11 days. Similar results have been found with regard to the synoptic‐scale waves of wavenumbers 4 to 8. They also intensify by receiving energy and decay by losing energy through nonlinear interactions among finite amplitude waves. Nevertheless, the conversion of the available potential energy to kinetic energy plays a more important role in the growing stage of the synoptic‐scale waves than for the extra‐long waves. The average life cycle for the synoptic‐scale waves is about 6 to 8 days. It is interesting to note that the characteristics of waves of wavenumber 2 are quite different from those of the other waves. In all stages of its life cycle, the contribution of nonlinear interactions is negative and the conversion term always has large positive values. We have also analyzed the intensification and decay of the subtropical jet stream, and found that it is greatly affected by the convergence and divergence of eddy momentum flux. The fluctuations of the net momentum flux are mainly contributed by the synoptic‐scale waves of wavenumbers 4 to 8.

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