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The dissipation range contains scales at which kinematic viscosity appreciably affects velocity gradients and converts kinetic energy into heat. In a high-Reynolds-number three-dimensional cascade its characteristic length and time are the Kolmogorov microscales. Intense vortex cores can be intermittent within this range.

In the locally isotropic equilibrium range, the small-scale velocity increment statistics depend on the separation, mean viscous dissipation and kinematic viscosity, independently of large-scale forcing details. Dimensional analysis leaves the Kolmogorov microscales as normalization scales and gives the displayed second-order longitudinal structure function. Universality of $F$ is part of the hypothesis, not something proved by dimensional analysis.

The Kolmogorov length scale is the length among the Kolmogorov microscales. Kinematic viscosity $\nu$ and mean energy dissipation per unit mass $ϵ$ give the displayed dimensional scale. The associated speed is $(\nu ϵ{)}^{1/4}$, so the Reynolds number formed at this length is one.

Equating an inertial eddy turnover time to the inverse buoyancy frequency gives the scale separating buoyancy-affected motions from a smaller approximately isotropic cascade. It is distinct from the viscous Kolmogorov microscales. In very stable local surface-layer scaling it is proportional to the Obukhov length, up to closure constants.

Assume statistically steady, three-dimensional turbulence at large Reynolds number, with local statistical homogeneity and statistical isotropy at scales small compared with the forcing scale $L$. The Kolmogorov 1941 theory additionally assumes a local energy cascade, constant mean energy flux per unit mass $ϵ$, and negligible viscous dissipation within the inertial range. Its dimensional form neglects corrections from internal intermittency.

At separation $l$, let $\delta u_l$ be a typical velocity increment. The eddy turnover time is ${\tau }_l\sim l/\delta u_l$. Constant transfer rate then gives $ϵ\sim \delta u_l^2/{\tau }_l\sim \delta u_l^3/l$, henceHere $E(k)$ is the one-dimensional turbulent energy spectrum, obtained from $\delta u_l^2\sim kE(k)$ with $k\sim l^{-1}$. The scaling holds only forThe Kolmogorov length scale follows by setting the scale-dependent Reynolds number $\delta u_{l}l/\nu$ to one. The associated Kolmogorov microscales give ${\tau }_{\eta }=(\nu /ϵ{)}^{1/2}$. At larger scales the forcing matters; at smaller scales kinematic viscosity matters.

Local isotropy of turbulence means that joint small-scale velocity increment statistics are approximately unchanged by rotations, even when the large-scale flow is anisotropic. The universal equilibrium range contains separations much smaller than the integral scale of turbulence: their adjustment times are short compared with the outer eddy turnover time, and their normalized statistics are assumed independent of the detailed large-scale forcing. It includes the inertial range and the dissipation range.

The Kolmogorov first similarity hypothesis says that these locally isotropic statistics depend only on $r$, mean viscous dissipation $ϵ$ and kinematic viscosity $\nu$. Dimensional analysis therefore expresses the second-order longitudinal structure function in terms of the Kolmogorov microscales:The universality of $F$ is an assumption of the theory, not a consequence of dimensional analysis alone.

The Kolmogorov second similarity hypothesis removes dependence on kinematic viscosity when $\eta \ll r\ll \ell$. Only $ϵ$ and $r$ remain, so the Kolmogorov two-thirds law isThe same reasoning gives the predicted order-$P$ longitudinal structure function:These are signed moments for integer $P$; for noninteger orders, use $⟨|\Delta v{|}^P⟩$. In particular, the signed third-order coefficient is negative, ${\beta }_3=-4/5$, in the Kolmogorov four-fifths law.

The zeroth law of turbulence states that at fixed outer velocity $u$ and integral scale of turbulence $\ell$, the mean viscous dissipation per unit mass approaches a nonzero finite value as the Reynolds number $\mathrm{Re}=u\ell /\nu$ becomes large. In an established forward energy cascade, this gives $ϵ\sim C_{ϵ}{u}^3/\ell$, with $C_{ϵ}$ of order one. The small-scale velocity gradients become large enough to offset decreasing kinematic viscosity; this is the turbulent dissipation anomaly.

At the Kolmogorov microscales, the local Reynolds number is of order one, $v\eta /\nu \sim 1$, and the viscous dissipation is $ϵ\sim \nu (v/\eta {)}^2$. Solving these balances givesSubstituting the outer-scale estimate, and omitting order-one constants, givesThe associated eddy turnover time is $\eta /v=(\nu /ϵ{)}^{1/2}$.

The first similarity hypothesis of Kolmogorov 1941 theory states that at sufficiently large Reynolds number, the small-scale statistics in the equilibrium range are locally universal and determined by the mean turbulent kinetic energy dissipation rate $ϵ$ and kinematic viscosity $\nu$, together with the separation or wavenumber being measured. Large-scale forcing details are excluded only at scales well below the integral scale of turbulence.

These parameters define the Kolmogorov microscalesDimensional analysis gives universal functions $F$ and $G$ withThe dimensional factors follow because $S_2$ has units of velocity squared, whereas $E$ has units of velocity squared times length. The universal equilibrium forms include viscous effects; they do not impose a power law through the dissipation range.

The second similarity hypothesis asserts independence of kinematic viscosity at scales large compared with $\eta$ but still small compared with the integral scale. In this inertial range, $S_2$ can depend only on $ϵ$ and $r$. Balancing time and length dimensions forces powers $2/3$ and $2/3$. The same calculation for $E$, using $k$ instead of $r$, forces powers $2/3$ and $-5/3$. HenceThe constants $C_2$ and $C_K$ are universal within the hypotheses and the stated normalization. They are related by the exact structure-function spectral filter, rather than by assigning all energy to one sharp scale.

For unforced incompressible flow, multiply the vorticity equation by ${\omega }_i$. The antisymmetric part of the velocity gradient tensor drops out of ${\omega }_i{\omega }_j{\partial }_j{u}_i$, leavingStatistical homogeneity removes mean divergences. Integration by parts, together with $\nabla \cdot \mathbf{\omega }=0$, givesThe last equality holds for the averages, not pointwise. It follows from the usual curl/divergence identity after the divergence terms have averaged to zero. Vortex stretching produces enstrophy, while viscous diffusion destroys it.

Let $T=\ell /u$, $ϵ\sim u^3/\ell$, and use the Kolmogorov microscales $\eta =({\nu }^3/ϵ{)}^{1/4}$ and ${\tau }_{\eta }=(\nu /ϵ{)}^{1/2}$. Since $⟨{\omega }^2⟩=ϵ/\nu$, a slowly changing mean enstrophy has a time derivative of order $ϵ/(\nu T)$. The usual small-scale estimate givesThe relative imbalance is consequently ${\tau }_{\eta }/T\sim (\nu /u\ell {)}^{1/2}={\mathrm{Re}}^{-1/2}$, yieldingThe estimate assumes an established high-Reynolds-number cascade and integral-time evolution of the mean, not a rapid initial transient. In a forced flow the exact balance includes $⟨\mathbf{\omega }\cdot \nabla \times \mathbf{f}⟩$; forcing smooth on the integral scale of turbulence is smaller than the leading small-scale terms.

For very stable continuously turbulent conditions, $z\gg {\ell }_O>0$, local eddies lose leading dependence on their distance from the wall. In very stable surface-layer similarity, the dimensionless functions must therefore scale as ${\varphi }_m\sim c_m\zeta$ and ${\varphi }_h\sim c_h\zeta$, with positive closure constants. Using ${\ell }_O=u_{\ast }^3/(\kappa |B_0|)$ givesThus the mean velocity is approximately linear in height and the real buoyancy frequency is approximately constant, $N\sim \sqrt{c_h}|B_0|/u_{\ast }^2$. Under the Reynolds analogy, $c_h=c_m$.

The local turbulent kinetic energy balance givesPositive dissipation requires $c_m>1$. The turbulent energy-turnover length is of order $u_{\ast }^3/|B_0|$, hence of order the Obukhov length, independent of observation height. A related buoyancy-limited length is $u_{\ast }/N$. The Ozmidov length follows by equating an eddy turnover time ${\ell }^{2/3}{ϵ}^{-1/3}$ with $N^{-1}$:These are turbulent turnover or buoyancy-transition scales. If “the scale at which dissipation occurs” means the actual viscous cutoff, molecular viscosity is also required: the Kolmogorov microscales giveWithout $\nu$, the viscous length cannot be fixed by the supplied stress and flux alone. Separating it from the turbulent turnover length avoids confusing an energy-budget estimate with the molecular cutoff.

Finally,They become height-independent constants. The commonly used continuation $c_m=c_h=5$ gives ${\mathrm{Ri}}_f={\mathrm{Ri}}_g\simeq 0.2$ and $ϵ\simeq 4|B_0|$. This numerical value is a closure choice, not a deduction from units; setting $c_m=1$ would incorrectly leave zero dissipation. Extremely stable intermittent or collapsed turbulence need not satisfy this local-equilibrium model. The relation between the Obukhov and buoyancy-dissipation lengths in the local regime is also analyzed in Grachev and colleagues' similarity study.