  We study the scale dependence of effective diffusion of fluid tracers,
specifically, its dependence on the P\'{e}clet number, a dimensionless
parameter of the ratio between advection and molecular diffusion. Here, we
address the case that length and time scales on which the effective diffusion
can be described are not separated from those of advection and molecular
diffusion. For this, we propose a new method for characterizing the effective
diffusivity without relying on the scale separation. For a given spatial domain
inside which the effective diffusion can emerge, a time constant related to the
diffusion is identified by considering the spatio-temporal evolution of a test
advection-diffusion equation, where its initial condition is set at a pulse
function. Then, the value of effective diffusivity is identified by minimizing
the $L_\infty$ distance between solutions of the above test equation and the
diffusion one with mean drift. With this method, for time-independent gyre and
time-periodic shear flows, we numerically show the scale dependence of the
effective diffusivity and its discrepancy from the classical limits that were
derived on the assumption of the scale separation. The kinematic origins of the
discrepancy are revealed as the development of the molecular diffusion across
flow cells of the gyre and as the suppression of the drift motion due to a
temporal oscillation in the shear.
