  Local nonlinear approximations to the growth of cosmic perturbations are
developed, resulting in relations, at a given epoch, between the peculiar
velocity and gravity fields and their gradients. Only the equation of motion is
approximated, while mass conservation and the computation of the gravitational
field are treated exactly. The second-order relation is derived for arbitrary
geometry and cosmological parameters. Solutions are developed to fourth order
for laminar spherical perturbations in an Einstein-de Sitter universe, but the
gain in accuracy for higher orders is modest. All orders become comparable when
the peculiar kinetic energy per unit mass equals the peculiar potential,
typically at relative density perturbations, $\delta \sim 4$. The general
second-order relation, while implicit, is simple to solve. \nbody\ simulations
show that it provides moderate gains in accuracy over other local
approximations. It can therefore be easily applied in the comparison of
large-scale structures and velocities in the quasi-linear regime, $\delta \sim
1 - 4$, as well as in the reconstruction of the primordial perturbations from
which they grew.
