The wave-number–frequency dependent spectral function, S(k, ω), and the space–time correlation function, C(r, t), are considered in a turbulent flowing plasma. The decay mechanisms are associated with either velocity fluctuations about the mean convection velocity or diffusion effects or attachment, or combinations of these, including the Brownian motion model. The ψ(k, ω) function, which is the ratio of S(k, ω) to its frequency-integrated value, depends on the mechanism and exhibits a profile which can be Gaussian, Lorentzian, a Z function, a Hermite polynomial modification of the Gaussian, or a confluent hypergeometric function. Anisotropic forms are also considered.The function C(r, t), obtained by convolving ψ (r, t) with C(r), the space autocorrelation function, is next considered. Adopting a Gaussian or an exponential model (which may be anisotropic) for C(r), we illustrate C(r, t) forms, which can readily be manipulated. Furthermore, letting r = 0, we derive two conditions for the applicability of Taylor's hypothesis. The assumption of frozen flow is not necessary, only that the root-mean-square Lagrangian displacement in a given time, associated with the decay, be much smaller than both the flow distance and the characteristic size of blobs having maximum energy.
Spatial and space-time correlations in systems of subpopulations with genetic drift and migration.