The effect of forcing on the spatial structure and spectra of chaotically advected passive scalars

The stationary distribution of passive tracers chaotically advected by a tw o-dimensional large-scale flow is investigated. The value of the tracer is conserved following each fluid element except when the element enters certa in localized regions. The tracer value is then instantenously reset to a va lue associated with the region entered. This resetting acts as a forcing fo r the tracer field. This problem is mathematically equivalent to advection in open flows and results in a fractal tracer structure. The spectral expon ent of the tracer field is different from that for a passive tracer with th e usual additive forcing (the so-called Batchelor spectrum) and is related to the fractal dimension of the set of points that have never visited the f orcing regions. We illustrate this behavior by considering a time-periodic flow whose effect is equivalent to a simple two-dimensional area-preserving map. We also show that similar structure in the tracer field is found when the flow is aperiodic in time. (C) 2000 American Institute of Physics. [S1 070-6631(00)01010-2].