Spectral, diffusive and convective properties of fractal and spiral fields

Abstract

Spectral, diffusive and convective properties of one-dimensional pulse fields displaying a well-defined Kolmogorov capacity D ε [0, 1] are investigated. The energy spectrum of fractal or spiral alternating pulse fields scales as k D . The energy spectrum of homogeneous fractal non-alternating pulse fields scales as k −D . Both these scaling laws hold in a range of wavenumbers between η −1 and L −1, where η is the smallest distance between pulses and L (≫ η) is a characteristic large scale of the structure. The space-filling geometry, which is quantified by the Kolmogorov capacity D, makes the field less autocorrelated (more singular) in the alternating case, whereas it makes it more autocorrelated (less singular) in the non-alternating case. Significant quantitative differences between the spectral properties of homogeneous fractals and of spirals exist. The energy spectrum of a spiral non-alternating pulse field scales as k −1 between x −1 N and L −1, where x N ∼ η( L/η) D ≫ η characterizes the inhomogeneity of the structure. The spectrum is flat outside this wavenumber range. When submitted to the action of molecular diffusion (molecular diffusivity v) the energy of these fields decays as follows. Energy decay is accelerated in the case of fractal or spiral alternating pulse fields: E ( t ) ~ ( v t η ) − 1 − D for η 2 v ≪ t ≪ L 2 v and is delayed (“trapped”) in the case of non-alternating homogeneous fractal pulse fields: E ( t ) ~ ( v t η ) − 1 + D for η 2 v ≪ t ≪ L 2 v This energy trapping manifests itself in a different manner in the case of spiral non-alternating pulse fields. In this case energy decays only logarithmically for η 2/ v ≪ t ≪ x 2 N / v, then decays like t −1/2 for longer times. When submitted to the combined action of convection and diffusion (Burgers equation) the energy of these fields of N pulses each of integral m ≫ v decays as follows. It is independent of D in the case of alternating pulse fields, and is delayed in the case of non-alternating pulse fields. For homogeneous fractal non-alternating pulse fields energy decays as E ( t ) ~ ( m t η 2 ) − ( D − 1 ) / ( D − 2 ) for η 2 m ≪ t ≪ L 2 N m For spiral non-alternating pulse fields energy decays as t −1/2 for x 2 N / Nm ≪ t ≪ L 2/ Nm, and the decay is much slower for t ≪ x 2 N / Nm. This delay of energy decay is due to an anomalous collision rate between shocks which manifests itself differently according to whether the structure is homogeneous or not.