HYDRODYNAMIC WAKES and MINIMAL SURFACES with FRACTAL BOUNDARIES

The observable features of hydrodynamic wakes can be put into correspondence with those characteristic surfaces of tangential discontinuities upon which the solutions to the evolutionary equations of hydrodynamics are not unique. Only the robust minimal surface subset, associated with a harmonic vector field, will be persistent and of minimal dissipation. Surprisingly, those minimal surfaces generated by iterates of complex holomorphic curves in four dimensions are related to fractal sets. Introduction A remarkable feature of hydrodynamic wakes and coherent structures in stratified flows is that their associated "instability" patterns seem to belong to two broad equivalence classes of spiral shapes. These two patterns are epitomized by the Kelvin-Helmholtz instability pattern (Figure 1a) and the Rayleigh-Taylor instability pattern (Figures 1b). The two basic experimental patterns are often replicated and deformed by the fluid motion, but otherwise they have vividly sharp visible boundaries and remarkably long persistent lifetimes in what otherwise would be considered to be a diffusive and dissipative environment. Note that the Kelvin-Helmholtz instability pattern is characterized by a replication of the primitive pattern of a Cornu spiral. The RayleighTaylor instability pattern is characterized by a replication of the primitive pattern of a Mushroom spiral. Although an analytic description of the Cornu spiral has been known for more than 100 years, only recently has the present author become aware of a closed form analytic description for the mushroom spiral. This work was presented at the August 1992, IUTAM meeting at Poitier, and is summarized in section 4. The essential questions are: Why do these spiral patterns appear almost universally in wakes? Why do they persist for such substantial periods of time? Why are they so sharply defined? What are the details of their creation? As H.K.Browand said [Browand, 1986] "There does not exist a satisfactory theoretical explanation for these wake patterns." Although the theory and ideas presented in this article were initiated and motivated by topological arguments, the presentation will utilize only the most fundamental of topological notions to describe the basic physical phenomena. 2. Two Remarkable Observations. Although the ubiquitous mushroom pattern and its topological features have fascinated this author for many years, it is important to note that there were two (relatively recent) experiments, Figure 1a. The pattern of the KelvinHelmholtz instability. Figure 1b. The pattern of the RayleighTaylor Instability.