  We examine the dispersion of a passive scalar released in an incompressible
fluid flow in an unbounded domain. The flow is assumed to be spatially
periodic, with zero spatial average, and random in time, in the manner of the
random-phase alternating sine flow which we use as an exemplar. In the
long-time limit, the scalar concentration takes the same, predictable form for
almost all realisations of the flow, with a Gaussian core characterised by an
effective diffusivity, and large-deviation tails characterised by a rate
function (which can be evaluated by computing the largest Lyapunov exponent of
a family of random-in-time partial differential equations). We contrast this
single-realisation description with that which applies to the average of the
concentration over an ensemble of flow realisations. We show that the
single-realisation and ensemble-average effective diffusivities are identical
but that the corresponding rate functions are not, and that the
ensemble-averaged description overestimates the concentration in the tails
compared with that obtained for single-flow realisations. This difference has a
marked impact for scalars reacting according to the
Fisher--Kolmogorov--Petrovskii--Piskunov (FKPP) model. Such scalars form an
expanding front whose shape is approximately independent of the flow
realisation and can be deduced from the single-realisation large-deviation rate
function. We test our predictions against numerical simulations of the
alternating sine flow.
