  We study the motion of a grain boundary that separates two sets of mutually
perpendicular rolls in Rayleigh-B\'enard convection above onset. The problem is
treated either analytically from the corresponding amplitude equations, or
numerically by solving the Swift-Hohenberg equation. We find that if the rolls
are curved by a slow transversal modulation, a net translation of the boundary
follows. We show analytically that although this motion is a nonlinear effect,
it occurs in a time scale much shorter than that of the linear relaxation of
the curved rolls. The total distance traveled by the boundary scales as
$\epsilon^{-1/2}$, where $\epsilon$ is the reduced Rayleigh number. We obtain
analytical expressions for the relaxation rate of the modulation and for the
time dependent traveling velocity of the boundary, and especially their
dependence on wavenumber. The results agree well with direct numerical
solutions of the Swift-Hohenberg equation. We finally discuss the implications
of our results on the coarsening rate of an ensemble of differently oriented
domains in which grain boundary motion through curved rolls is the dominant
coarsening mechanism.
