Proceedings with Extended Abstracts (single PDF file) - Radio ...

individual particles in a statistical flow? A detailed investigation on the diffusionalrelaxation of quantities based on several random walk models was made available byTalkner [4]. We think that this diffusional relaxation behavior of characteristic functionswill provide an important framework for obtaining a theoretically strong formulation ofthe backscatter spectra from steady-state turbulent mediums.In the next section, we start by a definition of the characteristic function and explain thebasis how it can be related to the scattered field amplitude. Next, we describe phaserelaxation function and how it can be identified as the generalized susceptibility of thefluctuation-dissipation theorem. Then, a simple relation is obtained to connect the spectraof the scattered field from steady-state fluctuations to the phase relaxation function.Finally, we will show several random walk models that may result in anomalous spectra.Phase diffusionThe characteristic function is defined as the time behavior of phase density for awavenumber k,∫ ∞ −∞ikxφ ( k,t)= dxe ρ(x,t)(1)where x is the extended space variable along the radar line of sight. If the observationpoint is far from the whole scattering system the characteristic function yields thecomplex amplitude of the scattered field for an incident electromagnetic wave withwavenumber k/2 at time t. The halving of the wavenumber comes from the Braggcondition. Hereinafter, we will refer to φ( , t)as the phase relaxation function, wherekBis the Bragg wavenumber.k BRandom walk concept comes here in modeling turbulent scattering processes so that theobserved backscatter spectra can be related to the phase relaxation function. If a randomwalk model is successful in modeling a turbulent process, the radar backscatter spectrawill have an analytical solution through the corresponding phase relaxation function. Onesimilar study by Balescu [1] shows that the modelling of anomalous diffusion influctuating magnetic fields by continuous time random walks is successful to a certaindegree. While the physical meaning of a random walk model is beyond the scope of ourdiscussion here, we find it useful here to show that such modeling has significantimplication in interpreting coherent radar spectra.Connection between the phase relaxation function and the steady-state turbulenceWe have discussed that the characteristic function at half the electromagneticwavenumber yields the measured scattered field. Consider for example an inertial rangeturbulence where energy is cascading from lower wavenumbers to higher wavenumbersand that we are scattering from this turbulent flow at the Bragg wavenumber k B. Wedefine that in the absence of energy input from scales larger than kB, the system is in astate of subsiding turbulence. This is the state where the complex amplitude of thescattered field will decay toward zero. Our main assumption will be that this form of thedecay curve of the scattered field is the phase relaxation function.We now consider that the turbulent system is in the steady state and responding an111