04-281 Andrew Neate, Aubrey Truman

Real and Complex Turbulence for the Stochastic Burgers Equation (852K, pdf) Sep 10, 04

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Abstract. The inviscid limit of Burgers equation, with body forces white noise in time, is discussed in terms of the level surfaces of the minimising Hamilton-Jacobi function and the classical mechanical caustic. Presurfaces and precaustics are introduced by using the classical mechanical flow map. When the prelevel surface touches the precaustic, the geometry (number of cusps) on the level surface changes infinitely rapidly causing `real turbulence' (Davies, Truman and Zhao). Using an idea of Felix Klein, it is shown that the geometry (number of swallowtails) on the caustic also changes infinitely rapidly when the real part of the precaustic touches its complex counterpart, which we call `complex turbulence'. These two new kinds of turbulence are both inherently stochastic in nature. A complete analysis of this problem is given in terms of a reduced (one dimensional) action function. This characterises which parts of the original caustic are singular - an old problem in applied mathematics relevant for our `elementary formula' with Elworthy and Zhao. It also determines when this turbulence is intermittent in terms of the recurrent behaviour of two processes.

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