A correspondence between the multifractal model of turbulence and the Navier-Stokes equations

We study a correspondence between the multifractal model of turbulence and the Navier-Stokes equations in $d$ spatial dimensions by comparing their respective dissipation length scales. In Kolmogorov’s 1941 theory the key parameter $h$, which is an exponent in the Navier-Stokes invariance scaling, is fixed at $h=1\mathrm{/}3$ but is allowed a spectrum of values in multifractal theory. Taking into account all derivatives of the Navier-Stokes equations, it is found that for this correspondence to hold the multifractal spectrum $C(h)$ must be bounded from below such that $C(h)\ge 1-3h$, which is consistent with the four-fifths law. Moreover, $h$ must also be bounded from below such that $h \geq (1-d)/3$\mathrm{.}When$d=3$theallowedrangeof$h$isgivenby$h \geq -2/3$therebybounding$h$awayfrom$h=-1$. The implications of this are discussed.