Almost horizontal turbulence

In this note an attempt is made to create the necessary prerequisites for the development of a theory of ‘almost horizontal turbulence’. A system of simplified equations of ‘almost horizontal motions’ is proposed. These equations are affinely invariant, namely, one can independently modify the vertical and horizontal scales. It is shown that the simplification used does not prevent us from obtaining a correct theory of internal waves whose frequencies are significantly less than the Brent–Väisälä frequency. 1. The mathematical model of ‘two-dimensional turbulence’ can in fact be applied to the motion of layers of some finite thickness. For instance, it is assumed that two-dimensional considerations of this kind lead to a reasonable model of formation of meanders with horizontal sizes of hundreds of kilometers on a flow involving a layer of hundreds of meters, with subsequent disintegration of these meanders into vortices gradually decreasing in size to several kilometers. If one has in mind a layer of constant density with intensive vertical mixing and if this layer can slide rather freely along the underlying layer with a strong steady-state stratification, then these concepts seem sufficiently justified. However, data have gradually accumulated about the detailed structure of vertical profiles of temperature, salinity, density, and velocity in layers with steady-state stratification, together with data about turbulized ‘pancakes’ and the The paper is published without modifications. Kolmogorov’s manuscript was apparently prepared during his participation in one of expeditions of the ship “D. Mendeleev” to the Atlantic Ocean (1969) or in a circumnavigation of the world (1971) organized by the Institute for Oceanology led at the time by A. S. Monin. As Kolmogorov himself wrote, the choice of the topic was stimulated by observations concerning “ . . . meanders with horizontal sizes of hundreds of kilometers on a flow involving a layer of hundreds of meters, with subsequent disintegration of these meanders into vortices gradually decreasing in size to several kilometers.” In modern terminology, the paper is devoted to the problem of intensive mixing in pycnoclines, that is, thin layers of stratified fluid, caused by internal waves whose frequencies are less than the Brent–Väisälä frequency. Here I would like to note two circumstances. The first is the scientific insight characteristic for Kolmogorov; this very approachwas later reflected in numerous publications (see, for instance, the monograph by V. S. Modevich, V. I. Nikulin, and A. G. Stetsenko “Dynamics of internal mixing in a stratified medium,” Institute for Hydromechanics, Academy of Sciences of Ukraine, Naukova Dumka, Kiev 1988). The second, the more significant in my opinion, is the genuine intellectual curiosity and breadth of thought of this great thinker, who studied not only the most abstract mathematical constructions but also got his head out of the clouds with great interest to solve concrete applied problems. (F. V. Dolzhanskii’s note.) AMS 2000 Mathematics Subject Classification. Primary 76F02. 198 A. N. Kolmogorov non-turbulized interlayers (in the sense of small-scale three-dimensional turbulence) that separate the pancakes, data forcing us to assume that in this case, at the corresponding scales, the horizontal displacements of rather thin layers are independent of one another to a great extent. An ‘almost horizontal turbulence’ must occur such that its structure functions, Dij(∆ x) = ∆vi ∆vj, increase under a vertical displacement much more rapidly than under a horizontal displacement. In this note we make an attempt to create the necessary prerequisites for the development of a theory of an ‘almost horizontal turbulence’ of this kind. We propose a system of simplified equations of ‘almost horizontal motions’, and note that these equations are affinely invariant: one can independently modify the vertical and horizontal scales. It is shown that our simplification does not prevent us from obtaining a correct theory of internal waves whose frequencies are significantly less than the Brent–Väisälä frequency. The proposed simplified system of equations is of the form du dt + fv + ∂p ∂x = Φx, (1.1) dv dt − fv + ∂p ∂y = Φy, (1.2) ∂p ∂z − b = 0, (1.3) ∂u ∂x + ∂v ∂y + ∂w ∂z = 0, (1.4)