In this thesis, Rossby waves are considered within the onelayer CharneyHasegawaMima (CHM) equation and twolayer quasigeostrophic (QG) model. They are studied from a wave turbulence (WT) perspective. Since nonlinearity is quadratic, interactions take place between triplets of waves known as triads. A triad is said to be resonant if its wave vectors and frequencies satisfy k1 + k2  k3 = 0 and w(k1)+w(k2)w(k3) = 0 respectively. These triads can then be joined together to form resonant clusters of various sizes. The wave vectors can be continuous, in an unbounded domain, or discrete, in a bounded domain. Continuous, otherwise known as kinetic, WT has been extensively studied in the onelayer case. It is known that three quadratic invariants exist and they take part in a triple cascade in kspace. This thesis is interested in �finding quadratic invariants, of which there can be many, in the discrete regime. It begins by considering discrete clusters of resonant triads arising from a Hamiltonian threewave equation. A cluster consists of N modes forming a total of M connected triads. It is shown that that finding quadratic invariants is equivalent to a basic linear algebra problem, consisting of fi�nding the null space of a rectangular M x N matrix A with entries 1, 1 and 0. An algorithm is then formulated for decomposing large clusters into smaller ones to show how the quadratic invariants are related to topological parts of the cluster. Speci�c examples of clusters arising in the CHM wave model are considered. The second part of this thesis focusses on the largescale limit of the CHM equation. This limit has been studied the least; however, it would appear to be more relevant since Rossby waves in the ocean are largescale. Recently a new quadratic invariant, known as semiaction, has been discovered in this limit. Its density is one in the meridional region ky< /3kx and zero in the zonal region ky> /3kx: As a consequence of the conservation of semiaction, conditions are placed on the triad interactions involving zonal (Z) and meridional (M) modes. In this thesis it is proved directly, without appealing to conservation, that the following triad interactions are prohibited: M > M +M,M > Z + Z,Z > M + Z and Z > M +M: The cascade directions are studied of the three invariants, the energy, enstrophy and, depending whether the initial spectrum is in the meridional or zonal sector, the semiaction or zonsotrophy respectively. The results are interpreted to explain the formation of unisotropic turbulence with dominating zonal scales. In the �nal part of this thesis, a symmetric form of the twolayer kinetic equation for Rossby waves is derived using canonical variables, allowing the turbulent cascade of energy between the barotropic and baroclinic modes to be studied. It turns out that energy is transferred via local triad interactions from largescale baroclinic modes to the baroclinic and barotropic modes at the Rossby deformation scale. From there it is transferred into largescale barotropic modes via a nonlocal inverse transfer.
