We study dynamics of nearly elastic particles constrained to move on a line with energy input from the boundaries. We find that for typical initial conditions, the system evolves to an ``extraordinary'' state with particles separated to two groups: The majority of the particles get clamped into a small region of space and move with very slow velocities; a few remaining particles travel between the boundaries at much higher speeds. Such a state clearly violates equipartition of energy. The simplest hydrodynamic approach fails to give a correct description of the system.
Turbulent quantities such as vorticity, which oscillate in sign on very fine scales, have recently been characterized by sign-singular measures [E. Ott, Y. Du, K. R. Sreenivasan, A. Juneja, and A. K. Suri, Phys. Rev. Lett. 69, 2654 (1992)] and quantified by the so-called cancellation exponent. Here, the connection between the cancellation exponent and other known exponents for velocity structure functions and multifractal spectrum of the energy dissipation field is discussed. Comparison with high-Reynolds-number experimental data in one dimension and direct measurements of vorticity in a plane in moderate-Reynolds-number flows reveals excellent internal consistency. Estimates for second-order cancellation exponent are presented.
It is shown that the exponential growth rate of the fast kinematic dynamo instability can be related to the Lagrangian stretching properties of the underlying chaotic flow. In particular, a formula is obtained relating the growth rate to the finite time Lyapunov numbers of the flow and the cancellation exponent κ. (The latter quantity characterizes the extremely singular nature of the magnetic field with respect to fine-scale spatial oscillation in orientation.) The growth rate formula is illustrated and tested on two examples: an analytically soluble model, and a numerically solved spatially smooth temporally periodic flow.
It is shown that signed measures (i.e., measures that take on both positive and negative values) may exhibit an extreme form of singularity in which oscillations in sign occur everywhere on arbitrarily fine scale. A cancellation exponent is introduced to characterize such measures quantitatively, and examples of significant physical situations which display this striking type of singular behavior are discussed.
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