  We present a new approach to gauge-invariant cosmological perturbations at
second order, which is also covariant. We examine two cases in particular for a
dust Friedman-Lemaitre-Robertson-Walker model of any curvature: we investigate
gravity waves generated from clustering matter, that is, induced tensor modes
from scalar modes; and we discuss the generation of density fluctuations
induced by gravity waves -- scalar modes from tensor perturbations. We derive a
linear system of evolution equations for second-order gauge-invariant variables
which characterise fully the induced modes of interest, with a source formed
from variables quadratic in first-order quantities; these we transform into
fully-fledged second-order gauge-invariant variables. Both the invariantly
defined variables and the key evolution equations are considerably simpler than
similar gauge-invariant results derived by other methods. By finding analytical
solutions, we demonstrate that non-linear effects can significantly amplify or
dampen modes present in standard linearised cosmological perturbation theory,
thereby providing an important source of potential error in, and refinement of,
the standard model. Moreover, these effects can dominate at late times, and on
super-Hubble scales.
