Conditional-averaging procedure for problems with mode-mode coupling

A field is represented by its Fourier modes u(k,t) on the wave-number interval 0≤k≤${\mathit{k}}_{\max }$. We seek to reduce the number of degrees of freedom needed to describe the system by eliminating those modes in the band of wave numbers ${\mathit{k}}_{\mathit{c}}$≤k≤${\mathit{k}}_{\max }$, the ${\mathbf{u}}^{+}$, while retaining their average effect on the remaining modes, the ${\mathbf{u}}^{\mathrm{-}}$. Because of mode-mode coupling, this requires, in principle, a conditional average over the ${\mathbf{u}}^{+}$, in a subensemble in which the ${\mathbf{u}}^{\mathrm{-}}$ are held (approximately) constant. The conditional average can be related to the full ensemble average by means of the decomposition ${\mathbf{u}}^{+}$=${\mathbf{v}}^{+}$+${\mathrm{\Delta }}^{+}$, where ${\mathbf{v}}^{+}$ is independent of ${\mathbf{u}}^{\mathrm{-}}$ but has the same global mean properties as ${\mathbf{u}}^{+}$, while ${\mathrm{\Delta }}^{+}$ represents the effect of mode coupling on the conditional average. Implementation of the formalism for any particular physical system requires an ansatz for the relationship between ${\mathbf{v}}^{+}$ and ${\mathbf{u}}^{+}$ such that the conditional average of ${\mathrm{\Delta }}^{+}$ is small. This procedure is illustrated by its application to the Navier-Stokes equation, where ${\mathbf{v}}^{+}$ is taken to be the first-order expansion of ${\mathbf{u}}^{+}$ in a Taylor series about k=${\mathit{k}}_{\max }$, thus permitting a renormalization-group calculation of the turbulent effective viscosity, as reported previously [Phys. Rev. Lett. 65, 3281 (1990)]. For systems like this, with deterministic time evolution, the conditional average must be based on the weak, or imprecise, condition that ${\mathbf{u}}^{\mathrm{-}}$+${\mathrm{\Phi }}^{\mathrm{-}}$ is held constant, where ${\mathrm{\Phi }}^{\mathrm{-}}$ is small (but not zero) and has zero mean under the conditional average, in order to ensure chaotic ${\mathbf{u}}^{+}$. We show that our formalism, relating conditional and full ensemble averages, is independent of ${\mathrm{\Phi }}^{\mathrm{-}}$ to second order...