Statistical Dynamics of Chaotic Flows

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Abstract
A cascade model for chaotic flows proposed in a previous note is further developed to determine the Hausdorff dimension D of some strange attractors and to relate the turbulence statistics to local vortex dynamics. It thus turns out that the Lorenz attractor has D=2.06 at r=40, σ=16, b=4, and the Smale solenoid has D=1+[ln 2/ln (1/ε)], (ε≪1/2)and the Hénon mapping has D=1.26 at a=1.4, b=0.3. The statistics of fully-developed fluid turbulence is determined by the intermittency exponent µ which is related to the Hausdorff dimension D of the dissipative structure of vorticity by µ=3-D. It is shown that µ can be expressed in terms of a statistical average of the local expansion rates of an active region of vorticity over an orbit of the chaotic flow in fluid space and over an ensemble of extrinsic randomness, and represents how fast spatial spottiness is generated by the vortex stretching. Thus a generalized β-model is formulated in terms of local vortex dynamics without the assumption of self-similarity. It also turns out that the validity of the universality and the scaling laws depends on the speed of the mixing of flow due to the vortex stretching.

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