ON THE PROPAGATION OF FREQUENCIES AND ENERGY IN CERTAIN TYPES OF OCEANIC AND ATMOSPHERIC WAVES

Abstract

In this paper the energy and frequency propagation in certain types of plane waves in the atmosphere and in the ocean are investigated, these wave trains being characterized by the property that frequency and wave number are slowly varying functions of the space coordinate and time. The assumption is made that wave crests are conserved, and on this basis a simple kinematic relationship between frequency and wave number is established. Wave trains for which the physical theory results in a frequency equation of such a character that phase velocity and wave length uniquely determine each other are then found to possess a characteristic group velocity, and it is shown that frequencies (and hence group velocities) are propagated with this characteristic group velocity. In the more general case the frequency equation resulting from the physical theory is found to contain a correction term which is an explicit function of the space coordinate and time. In that case observers travelling with the conventionally defined group velocity (obtained by disregarding the correction term) will observe gradual changes in frequency, but it is shown that these changes, at least in the wave types here investigated, are insignificant except in the leading and trailing boundaries of the wave trains. Energy propagation equations are derived for three different types of wave motion. It is found that the relative flow of energy across planes which move with the conventionally defined group velocity (hereafter referred to as group-velocity planes), is nondivergent in those regions where the frequencies associated with individual group velocity planes remain constant. The particular limited wave trains analyzed in this paper possess the property that the boundary regions of the wave trains, where frequencies no longer are conserved, serve as sources and sinks for the energy. These results indicate that the classical energy propagation equation, which is found to be valid in the central portion of the wave trains, must not be interpreted to mean that energy is propagated with group velocity, but rather that the energy transport relative to the group-velocity planes is nondivergent. This conclusion is verified through a computation of the accumulation of energy in the extreme forerunner of a train of surface waves in deep water. The main portion of this wave train is characterized by a non-divergent relative flux of energy across the group-velocity planes. It is shown that the equations which determine the propagation of group-velocity and the nondivergent propagation of energy together form a closed system, the form of which is independent of the special dynamic characteristics of the wave train under consideration. If the distributions of energy and group velocity for one particular time are known, it is possible to compute the distribution of these elements for any subsequent time, provided the relative flux of energy remains nondivergent. As a test of the theory for frequency and energy propagation developed in this paper, exact solutions are given for the train of long gravitational waves following a limited progressive perturbation in a sheet of water on a rotating disc and for a spreading “forerunner” wave train of horizontal, plane, nondivergent shear waves generated by a limited progressive perturbation in the belt of west winds in middle latitudes. These exact solutions are compared with indications furnished by energy and group-velocity considerations. The analysis of wave trains in the westerlies suggests a simple explanation for the rapid-moving isallobaric waves which seem to emanate from the principal cyclonic centers of action in the Northern Hemisphere. Such isallobaric waves normally spread quite rapidly eastward, at the same time losing their intensity, and breaking up into ever more irregular patterns. These observed characteristics appear to be in good accord with the characteristics of theoretical “forerunner” wave trains obtained from the theoretical analysis of waves in the westerlies. It is finally suggested that the well-known retrograde “blocking” action observed in the west wind belt of middle latitudes might be interpreted as the effect of a convergent distribution of the group velocity in the long, quasistationary waves in the westerlies.