Solution of functional equations and reduction of dimension in the local energy transfer theory of incompressible, three-dimensional turbulence

It is shown that the set of integrodifferential and algebraic functional equations of the local energy transfer theory may be considerably reduced in dimension for the case of isotropic turbulence. This is achieved without restricting the solution space. The basis for this is a complete analytical solution to the functional equations Q(k;t,t^′)=H(k;t,t^′)Q(k;t^′,t^′) and H(k;t,s)H(k;s,t^′)=H(k;t,t^′). The solution is proved to depend only on a single function φ(k;t) solely determining Q and H. Hence the dimension of both the dependent and the independent variables is reduced by one. From the latter, the corresponding two integrodifferential equations are lowered to a single integrodifferential equation for φ(k;t), extended by an integral side condition on the k dependence of φ(k;t). In the limit ν⃗0, a partial solution to the reduced set of equations is presented in the Appendix.