On the initial streakness of a dispersing tracer in two- and three-dimensional turbulence

If a spot of tracer is released into a turbulent flow, the peak concentration at some subsequent time will initially be much greater than that implied by a solution for the ensemble average concentration at fixed points. For two-dimensional turbulence three areas may be defined: (1) an area A_d related to the ensemble average concentration field; (2) an area A_p defined in terms of the relative dispersion of particles seeded into the patch after a short initial diffusion time; and (3) the area A_t occupied by tracer. It is argued that A_d grows linearly with time, whereas A_p and A_t grow exponentially; A_p faster than A_t. Thus, the concentration field is significantly streaky, even within the particle domain, until A_t becomes comparable with A_d. The time taken for this to occur is estimated; after this time, fluctuations about the ensemble average concentration field should not be greater than those given by a simple mixing length argument. In three-dimensional turbulence the volume V_t of the tracer domain grows much more rapidly than the volume V_p of the particle domain if the merging of streaks is ignored. However, V_t cannot be greater than V_p so streaks must merge and V_p can be used to provide a rough estimate of peak concentration, or concentration variance.