  In this paper we examine triad resonances in a rotating shallow water system
when there are two free interfaces. This allows for an examination in a
relatively simple model of the interplay between baroclinic and barotropic
dynamics in a context where there is also a geostrophic mode. In contrast to
the much-studied one-layer rotating shallow water system, we find that as well
as the usual slow geostrophic mode, there are now two fast waves, a barotropic
mode and a baroclinic mode. This feature permits triad resonances to occur
between three fast waves, with a mixture of barotropic and baroclinic modes, an
aspect which cannot occur in the one-layer system. There are now also two
branches of the slow geostrophic mode with a repeated branch of the dispersion
relation. The consequences are explored in a derivation of the full set of
triad interaction equations, using a multi-scale asymptotic expansion based on
a small amplitude parameter. The derived nonlinear interaction coefficients are
confirmed using energy and enstrophy conservation. These triad interaction
equations are explored with an emphasis on the parameter regime with small
Rossby and Froude numbers.
