The role of sign indefinite invariants in shaping turbulent cascades

We highlight a non-canonical yet natural choice of variables for an efficient derivation of a kinetic equation for the energy density in non-isotropic systems, including internal gravity waves on a vertical plane, inertial and Rossby waves. The existence of a second quadratic invariant simplifies the kinetic equation and leads to extra conservation laws for resonant interactions. We analytically determine the scaling of the radial turbulent energy spectrum. Our findings suggest the existence of an inverse energy cascade of internal gravity waves, from small to large scales, in practically relevant scenarios.