  We develop a McKean-Vlasov interpretation of Navier-Stokes equations with
external force field in the whole space, by associating with local mild
$L^p$-solutions of the 3d-vortex equation a generalized nonlinear diffusion
with random space-time birth that probabilistically describes creation of
rotation in the fluid due to nonconservativeness of the force. We establish a
local well-posedness result for this process and a stochastic representation
formula for the vorticity in terms of a vector-weighted version of its law
after its birth instant. Then we introduce a stochastic system of 3d vortices
with mollified interaction and random space-time births, and prove the
propagation of chaos property, with the nonlinear process as limit, at an
explicit pathwise convergence rate. Convergence rates for stochastic
approximation schemes of the velocity and the vorticity fields are also
obtained. We thus extend and refine previous results on the probabilistic
interpretation and stochastic approximation methods for the nonforced equation,
generalizing also a recently introduced random space-time-birth particle method
for the 2d-Navier-Stokes equation with force.
