D-dimensional turbulence

Abstract

$d$-dimensional homogeneous isotropic incompressible turbulence is defined, for arbitrary nonintegral $d$, by analytically continuing the Taylor expansion in time of the energy spectrum ${E}_{k}(t)$, assuming Gaussian initial conditions. If $dl2$, the positivity of the energy spectrum is not necessarily preserved in time. For $d\ensuremath{\ge}2$ all steady-state and initial-value calculations have been made with a realizable second-order closure, the eddy-damped quasi normal Markovian approximation. Near two dimensions the enstrophy (mean square vorticity) conservation law is weakly broken, enough to allow ultraviolet singularities to develop in a finite time but not enough to prevent energy from cascading in the infrared direction. A systematic investigation is made of zero-transfer (inertial) steady-state scaling solutions ${E}_{k}\ensuremath{\propto}{k}^{\ensuremath{-}m}$ and of their stability. Energy-inertial solutions with $m=\frac{5}{3}$ exist for arbitrary $d$; the direction of the energy cascade reverses at $d={d}_{c}\ensuremath{\simeq}2.05$. For $dl{{d}^{\ensuremath{'}}}_{c}\ensuremath{\simeq}2.06$ there are in addition, as in the cascade studied by Bell and Nelkin, inertial solutions with zero energy flux; their exponents $m(d)$ are given by a roughly parabolic curve in the ($m, d$) plane, linking enstrophy cascade ($m=3$, $d=2$) to enstrophy equipartition ($m=1$, $d=2$) For any point in the ($m, d$) plane such that the transfer integral is finite and negative, a steady-state scaling solution ${E}_{k}\ensuremath{\propto}{k}^{\ensuremath{-}\mathrm{m}}$ is obtained when the fluid is subject to random forces with spectrum ${F}_{k}\ensuremath{\propto}{k}^{\frac{3(m\ensuremath{-}1)}{2}}$. A special case is the model B [$m=\ensuremath{-}1=\frac{2}{3}\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$, $d=4\ensuremath{-}\ensuremath{\epsilon}$] obtained by Forster, Nelson, and Stephen using a dynamical renormalization-group procedure. Forced steady-state solutions are actually not resticted to the neighborhood of $m=\ensuremath{-}1$, $d=4$; they are amenable to renormalization-group calculations on the primitive equations for arbitrary $dg2$ when $m$ is close to the crossover -1 and, perhaps, also near the crossover +3.