  Particle pair (relative) diffusion in a field of homogeneous turbulence with
generalised power-law energy spectra, $E(k)\sim k^{-p}$ for $1< p\le 3$ and
$k_1\le k\le k_\eta$ with $k_\eta/k_1=10^6$, is investigated numerically using
Kinematic Simulation (Kraichnan 1970, Fung et al 1992). If $\Delta=|{\bf
x_2}(t)-{\bf x}_1(t)|$ and $\sigma_\Delta=\sqrt{\langle\Delta^2\rangle}$, (the
angled brackets is the ensemble average over particle pairs), then we find
that:
  (1) The pair diffusivity scales like $D_p\sim \sigma_\Delta^{\gamma_p}$ with
$\gamma_p$ obtained from the simulations such that $\gamma^l_p<\gamma_p<2$,
where $\gamma^l_p =(1+p)/2$ is the Richardson locality scaling. The range of
$\Delta$ over which these scalings are observed diminishes from above and from
below as $p$ increases towards $3$.\\ (2) $M(p)=\gamma_p/\gamma^l_p>1$ in the
range $1<p<3$, and $M$ has a peak at $p_m\approx 1.8$. This suggests that for
spectra close to this in the range $1.5<p<2$, which includes Kolmogorov
turbulence, local and non-local correlations play comparable roles in the pair
diffusion process.\\ (3) The mean square separation scales like
$\langle\Delta^2\rangle_p \sim \tau_p^{\chi_p}$ where $\chi_p=1/(1-\gamma_p/2)$
and $\tau_p$ is an adjusted travel time.\\ (4) For Kolmogorov turbulence
$p=5/3$, we observe $D_{5/3}\sim \sigma_\Delta^{1.53}$, and
$\langle\Delta^2\rangle_{5/3}\sim\tau^{4.2}$. \\ (5) At $p_*\approx 1.4$
($E(k)\sim k^{-1.4}$) we observe $D_{p_*}\sim\sigma_\Delta^{4/3}$, and
$\langle\Delta^2\rangle_{p_*}\sim\tau^3$; these are different to the Richardson
$4/3$-law and Richardson-Obukov $t^3$-regime which occur for $p=5/3$.
  These results are consistent with Malik's (2014) theory, and supports the
principle upon which the theory is based that both local and non-local
correlations are effective in the pair diffusion process inside the inertial
subrange.
