Sign-Indefinite Invariants Shape Turbulent Cascades

We highlight a noncanonical yet natural choice of variables for an efficient derivation of a kinetic equation for the energy density in nonisotropic 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.