Studies of Atmospheric Disturbance (Theory of Vortical Waves) Part I. (Introductory Discussion)

Abstract

The problem whether cyclones are waves or vortices is one of the fundamental problems in meteorology. The wave theroy of cyclones is the leading idea of Norwegian school. But inspite of her brilliant success in finding unstable cyclone waves, much difficulties lie on her way from both physical and mathematical points of view. It is obvious that cyclones are vortices or of vortical nature in hydrodynamical sense. Further, it is a possible solution of cyclone formation that extratropical cyclones bear itself on the surface of polar front and grow there. But at the present state of development of Norwegian wave-theory, the connexion between wave and vortex is impossible and there is no hope of giving connexion between them in the near future⁽¹⁾.
In our present investigation, the author starts from another point of view, introducing an idea of vortical waves, and intends to treat the whole evolution of cyclones mathematically. The vortical wave is defined by where v_x v_y; u, v are velocities of propagation of the vortical wave and those of the air respectively.
In Chapter I, the author derives the fundamental equation of the problem. Namely, from the equations of motion on the rotating earth, the following equation is obtained: where x, y, z are cartesian coordinates taken southwards, eastwards and upwards respectively, u, v, w, the velocity components, (horizontal divergence), (vertical components of vorticity), θ, latitude, a, radius of the earth, k, kinematic coefficieut of viscosity, p, pressure, p, density, ω, angular velocity of rotation of the earth, and
In Chapter II are discussed the vortical waves in a uniform and nonvariable fundamental flow for following special cases:
In the first case, following results are obtained: When there is no viscosity and the solenoidal field vanishes, there exist free waves which propagate with the velocity of the fundamental flow. The effect of coriolis' force on the velocity is very small, when the wave-length of the vortical wave is of the same order with cyclones. From the above solution of sinnusoidal type can be easily constructed a vortex of solitary type which represents a cyclone by Fourier double integral theorem, provided that the term due to the latitudinal variation of coriolis' force is neglected. When the viscosity is taken into consideration, vortical waves of dissipating nature are obtained, as it may be expected. The time requisite for that the amplitude becomes 1/e of its initial value is given for various values of viscosity-coefficient and wave-length. Finally, the baroclinic term is discussed and the general solution is obtained for the case: From the general solution, it follows that the vortical wave is constant as for time when j(t) is constant, i. e. the bounded vortical wave is constant when the baroclinic field is constant. The above result is interpreted as follows: If there is an invariable baroclinic field in the fundamental flow, there are vortical waves with constant amplitude associated with the field, so that the baroclinic field does not contribute to the development of cyclones. Next the author gets the solution for the case where the baroclinic field mores with a certain velocity. When the velocity is different from that of the fundamental flow, the bounded vortical wave moves only with the velocity of the baroclinic field, but when the velocity of the solenoidal field is the same as that of the fundamental flow, the vortical wave develops with the time linearly.