d-dimensional turbulence

$d$-dimensional homogeneous isotropic incompressible turbulence is defined, for arbitrary nonintegral $d$, by analytically continuing the Taylor expansion in time of the energy spectrum $E_k\mathrm{}\left(t\right)\mathrm{}$, assuming Gaussian initial conditions. If $d<\mathrm{}2\mathrm{}$, the positivity of the energy spectrum is not necessarily preserved in time. For $d\ge 2$ 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 $E_k\propto k^{-m}$ and of their stability. Energy-inertial solutions with $m=\frac{5}{3}$ exist for arbitrary $d$; the direction of the energy cascade reverses at $d=d_c\simeq 2.05$. For $d<\mathrm{}{d^{\prime }}_c\simeq 2.06$ there are in addition, as in the cascade model studied by Bell and Nelkin, inertial solutions with zero energy flux; their exponents $m\left(d\right)\mathrm{}$ are given by a roughly parabolic curve in the ($m, d$) plane, linking enstrophy cascade ($m=3\mathrm{}$, $d=2\mathrm{}$) to enstrophy equipartition ($m=1\mathrm{}$, $d=2\mathrm{}$) For any point in the ($m, d$) plane such that the transfer integral is finite and negative, a steady-state scaling solution $E_k\propto k^{-\mathrm{m}}$ is obtained when the fluid is subject to random forces with spectrum $F_k\propto k^{\frac{3(m-1\mathrm{})\mathrm{}}{2}}$. A special case is the "model B" [$m=-1=\frac{2}{3}\epsilon +O\left({\epsilon }^2\right)$, $d=4-\epsilon$] obtained by Forster, Nelson, and Stephen using a dynamical renormalization-group procedure. Forced steady-state solutions are actually not resticted to the neighborhood of $m=-1\mathrm{}$, $d=4\mathrm{}$; they are amenable to renormalization-group calculations on the primitive equations for arbitrary $d>\mathrm{}2\mathrm{}$ when $m$ is close to the crossover -1 and, perhaps, also near the crossover +3.