from small to large wavenumbers, as long as the viscosity coefficient of the small-scale dissipation term is non-zero. We have proposed that this small “energy leak” should be viewed as a downscale energy cascade that coexists with the dominant downscale enstrophy cascade. To support this theory, in @article:Tung:2005 [@article:Tung:2005:1] we noted that the triad interactions responsible for the enstrophy cascade are independent from those responsible for the downscale energy cascade. This is, in fact, an immediate but unstated consequence of the original argument by @article:Kraichnan:1967:1, as noted in section 3.2 of @article:Tung:2005. We have also theorized that the two cascades can be viewed as two independent homogeneous solutions of the governing statistical theory that can be linearly superposed on each other. @article:Davidson:2008 confirmed the validity of the linear superposition principle for the 3rd-order structure functions, thereby adding further detail to a corresponding proof sketch given in section 3.1 of @article:Tung:2005. Some of the details of my statistical theory of two-dimensional turbulence was given in @article:Gkioulekas:2008:1 and @article:Gkioulekas:p14, and further development of this theory is currently in progress.

To elaborate further, our claim is that the energy spectrum of the downscale cascade is given by the linear combination of a dominant 