Analogy between density stratification and rotation effects

Abstract

7. L. T. Chernyi, "Uniform flow of electrified aerosols carrying a unipolar charge in an accelerating electric field," Dokl. Akad. Nauk SSSR, 273, No. 3 (1983). 8. L. I. Sedov, Mechanics of Continuous Media [in Russian]T 3rd ed., Vols. 1 and 2, Nauka, Moscow (1976). 9. P. V. Vorob'ev, "Method for investigating the operating efficiency of aerosol charging devices," in: Reports at the Scientificotechnical Conference on the NIR, Results for 1966-1967 [in Russian], Moscow Power Institute (MEI), Moscow (1967). i0. L. M. Levin, Investigations in the Physics of Coarsely Dispersed Aerosols [in Russian], Izd. Akad. Nauk SSSR, Moscow (1961). ii. S. P. Belyaev, N. K. Nikiforova, V. V. Smirnov, and G. I. Shchelchkov, Electron-Optical Methods of Investigating Aerosols [in Russian], Energoizdat, Moscow (1981). ANALOGY BETWEEN DENSITY STRATIFICATION AND ROTATION EFFECTS V. A. Vladimirov UDC 532.5.51+532.5.527 The resemblance (analogy) between the properties of rotating and density-stratified flows was first noted by Rayleigh in 1916 [i]. Since that time, a whole series of studies have been published in which this analogy is successfully employed to solve problems of wave theory and stability theory and to describe secondary regimes and turbulence. Some of these achievements are reviewed in [2, 3]. Although the successes achieved in using the analogy to obtain new results are impor- tant, our general understanding of the question is unsatisfactory. One problem is the dis- connectedness of the examples with references to which the analogy has been demonstrated. The degree of proximity on the basis of which results from the two domains are considered to be analogous varies from identicalness to very distant similarity. There has been no classi- fication of the examples of the analogy on the basis of general principles. The limits of applicability of the analogy remain unclear. The present study is an attempt to clarify these points. From the most general standpoint, the analogy between stratification and rotation ef- fects is a consequence of the known principle of mechanics which states that following tran- sition to the corresponding moving frame of reference any part of the true acceleration of an object can be regarded as a "body force" field. This approach is attractive because of its simplicity and universality. However, it turns out that in all nontrivial cases it is useless owing to the velocity dependence of the "body force" field. A good example is pro- vided by the equations of motion of a fluid written in a rotating coordinate system. Here the Coriolis force has to be taken as the "body force." Clearly, the introduction of "body forces" of this sort cannot give any basis for transposing the known results for a uniform gravitational field to a new domain. At the same time, there are more subtle and also more productive means of explaining the analogy. At present, the only possible way of unifying the theories is mathematical. The motions of a rotating and a stratified fluid will be analogous if they are governed by equations of similar form. The degree of similarity must be such that the description of a certain class of motions in one field makes an important contribution to the solution of a related problem in the other. Given this approach, the analogy question reduces to the prob- lem of classifying the corresponding differential equations. In general form this problem is extremely complex. The present study offers several examples illustrating the possi- bility of progressing along this path. Two levels of analogy, differing considerably with respect to the rigorousness of the requirements, are examined: i) the level of similarity of the initial nonlinear equations of motion of the rotating and stratified fluids; 2) the similarity of the linearized equations Of motion or their corollaries (e.g., spectral problems for linear waves and stability theory). Comparison of the equations makes it possible to state that the properties of the mo- tions of a rotating fluid are, generally speaking, much more complex than those of a strati- Novosibirsk. Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 58-68, May-June, 1985. Original article submitted March 22, 1984. 0021-8944/85/2603-0353509.50 9 1985 Plenum Publishing Corporation 353