  Starting from the two-dimensional Boussinesq equation without rotation, we
derive a kinetic equation for weak interaction of internal waves using
non-canonical variables. We follow a formalism introduced by P. Ripa in the
80's. The advantage of this formalism is that it describes the system in terms
of the natural linear eigenfunctions of eastward and westward propagating
internal waves. Using properties of orthogonality of the eigenfunctions with
respect to a (pseudo) metric set by the energy we can write non perturbative
theory for the interaction of waves given in terms of the expansion amplitudes.
The evolution is controlled by a system of equations, with quadratic
nonlinearity, which is an exact representation of the original model equations.
The dynamics is constrained by the conservation of energy and pseudo-momentum,
which can be written simply as a linear combination of the squared absolute
value of the amplitudes. The possibility of a generalization of the Fjortoft's
argument to internal gravity waves and observation of a non trivial double
cascade of energy and pseudo-momentum is discussed.
