  The motion of solid particles embedded in gaseous protoplanetary disks is
influenced by turbulent fluctuations. Consequently, the dynamics of moderately
to weakly coupled solids can be distinctly different from the dynamics of the
gas. Additionally, gravitational perturbations from an embedded planet can
further impact the dynamics of solids. In this work, we investigate the
combined effects of turbulent fluctuations and planetary dust stirring in a
protoplanetary disk on three-dimensional dust morphology and on synthetic ALMA
continuum observations. We carry out 3D radiative two-fluid (gas+1-mm-dust)
hydrodynamic simulations in which we explicitly model the gravitational
perturbation of a Jupiter-mass planet. We derived a new momentum-conserving
turbulent diffusion model that introduces a turbulent pressure to the
pressureless dust fluid to capture the turbulent transport of dust. The model
implicitly captures the effects of orbital oscillations and reproduces the
theoretically predicted vertical settling-diffusion equilibrium. We find a
Jupiter-mass planet to produce distinct and large-scale three-dimensional flow
structures in the mm-size dust, which vary strongly in space. We quantify these
effects by locally measuring an effective vertical diffusivity (equivalent
alpha) and find azimuthally averaged values in a range
$\delta_\mathrm{eff}\sim5\cdot 10^{-3} - 2\cdot 10^{-2}$ and local peaks at
values of up to $\delta_\mathrm{eff}\sim3\cdot 10^{-1}$. In synthetic ALMA
continuum observations of inclined disks, we find effects of turbulent
diffusion to be observable, especially at disk edges, and effects of planetary
dust stirring in edge-on observations.
