  The Majda-Biello system models the interaction of Rossby waves. It consists
of two coupled KdV equations one of which has a parameter $\alpha$ as
coefficient of its dispersion. This work studies this system on the half line
with Robin, Neumann, and Dirichlet boundary data. It shows that for
$0<\alpha<1$ or $1<\alpha<4$ all these problems are well-posed for initial data
in Sobolev spaces $H^s$, $s\ge 0$. For $\alpha=1$ or $\alpha>4$ well-posedness
holds for Dirichlet data if $s>-3/4$, while for Neumann and Robin data it
depends on the sign of the parameters involved in the data. For $\alpha=4$
well-posedness of all problems holds for $s\ge 3/4$. The Robin and Neumann
boundary data are in $H^{s/3}$ while the Dirichlet boundary data are in
$H^{(s+1)/3}$. These are consistent with the time regularity of the Cauchy
problem for the corresponding linear system. The proof is based on linear
estimates in Bourgain spaces derived by utilizing the Fokas solution formula
for the forced linear system, and appropriate bilinear estimates suggested by
the coupled nonlinearities. These show that the iteration map defined via the
Fokas formula is a contraction in appropriate solution spaces. All the
well-posedness results obtained here are optimal.
