Spectral, diffusive and convective properties of fractal and spiral fields

Spectral, diffusive and convective properties of one-dimensional pulse fiel ds displaying a well-defined Kolmogorov capacity D is an element of [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 fie lds scales as k(-D). Both these scaling laws hold in a range of wavenumbers between eta(-1) and L-1, where eta is the smallest distance between pulses and L (much greater than eta) is a characteristic large scale of the struc ture. The space-filling geometry, which is quantified by the Kolmogorov cap acity D, makes the field less autocorrelated (more singular) in the alterna ting case, whereas it makes it more autocorrelated (less singular) in the n on-alternating case. Significant quantitative differences between the spectral properties of hom ogeneous fractals and of spirals exist. The energy spectrum of a spiral non -alternating pulse field scales as k(-1) between x(N)(-1) and L-1, where x( N) similar to eta(L/eta)(D) much greater than eta characterizes the inhomog eneity of the structure. The spectrum is flat outside this wavenumber range . When submitted to the action of molecular diffusion (molecular diffusivity nu) the energy of these fields decays as follows. Energy decay is accelerat ed in the case of fractal or spiral alternating pulse fields: E(t) similar to (root vt/eta)(-1-D) for eta(2/)v much less than t much less than L(2/)v, and is delayed ("trapped") in the case of non-alternating homogeneous fract al pulse fields: E(t) similar to (root vt/eta)(-1+D) for eta(2)/v much less than t much less than 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 logari thmically for eta(2)/v much less than t much less than x(N)(2)/v, then deca ys 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 much greater than v decays as follows. It is ind ependent of D in the case of alternating pulse fields, and is delayed in th e case of non-alternating pulse fields. For homogeneous fractal non-alterna ting pulse fields energy decays as E(t) similar to (mt/eta(2))(-(D-1)/(D-2)) for eta(2)/m much less than t muc h less than L-2/Nm. For spiral non-alternating pulse fields energy decays as t(-1/2) for x(N)(2 )/Nm much less than t much less than L-2/Nm, and the decay is much slower f or t much less than x(N)(2/)Nm. This delay of energy decay is due to an ano malous collision rate between shocks which manifests itself differently acc ording to whether the structure is homogeneous or not. (C) 1998 Elsevier Sc ience B.V.