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At separations well above the integral scale of turbulence, two fluid velocities can decorrelate spatially. If the relative-velocity autocorrelation $R_{\mathrm{rel}}$ is stationary and integrable, the displacement identity used in Taylor turbulent dispersion gives the displayed long-time linear mean-square growth. With speed $U$ and Lagrangian integral time of order $L/U$, its relative diffusivity is of order $UL$. The characteristic separation grows as $t^{1/2}$, unlike inertial-range Richardson pair dispersion.

The integral scale of turbulence characterizes energy-containing velocity correlations; in isotropic turbulence one definition is $\ell ={\int }_0^{\infty }f(r)dr$. Its turnover time is of order $\ell /u$. Large-scale intensity fluctuations can change the energy supplied to the equilibrium range.

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 concern is that the rapid-decay assumptions need not be preserved by the dynamics. In a constant-density incompressible flow, the divergence of the Navier-Stokes equation gives the pressure Poisson equation for incompressible flow, with $\pi =p/\rho$:For a localized eddy, inversion with the free-space Green function and two integrations by parts yieldAt large separation the differentiated kernel has leading form $(3n_i{n}_j-{\delta }_{ij})/r^3$. Thus an anisotropic stress moment of an individual eddy produces a quadrupolar fluid pressure of order $r^{-3}$, even if the velocity source is spatially localized. Correlating that fluid pressure with local $u_x^2$ allows $⟨u_x^2{p}^{\prime }⟩=O(r^{-3})$. Isotropy may cancel a mean multipole but does not automatically cancel such a mixed fourth-order statistic.

The triple-correlation evolution contains a pressure-gradient term of the form $-⟨u_x^2{\partial }_{x^{\prime }}{\pi }^{\prime }⟩$, so these long-range pressure correlations in turbulence can generate a longitudinal triple-correlation tail $K\sim a{r}^{-4}$. With $L=0$, part (ii) then gives $\dot{I}=8\pi u^{3}a$, which need not vanish. This is the substance of the objection to deducing exact conservation from initially short velocity correlations. Batchelor and Proudman's original long-range calculation concerned anisotropic turbulence; extending it to isotropic decay requires the appropriate higher-order correlations and allows cancellations. A permitted tail is not a proof that its coefficient stays appreciable in fully developed turbulence.

This possibility appealed to workers constructing heuristic turbulence closures. A closure that generated a changing low-wavenumber $k^4$ coefficient and a decay exponent different from $10/7$ could then be viewed as representing a pressure-mediated effect, rather than as automatically violating a universal invariant. The closure's far-field coefficient still needs physical validation; closure-generated nonconservation does not itself prove the true dynamics nonconserving.

For evidence available before this exam, the 2006 simulations of Ishida, Davidson and Kaneda found that $I$ approached an approximately constant value after a transient when the box was much larger than the integral scale of turbulence and the Reynolds number was large. The energy decay was correspondingly close to $u^2\sim t^{-10/7}$. The observed conclusion was approximate late-time conservation in that regime, indicating weak net long-range triple-correlation transfer, rather than exact conservation for every initial condition, box size or Reynolds number.

Measure the cloud radius about its moving centre, so uniform sweeping does not count as spreading. In the inertial range, an eddy of size $R$ has velocity difference $u_R\sim (ϵR{)}^{1/3}$ and turnover time $t_R\sim R/u_R$. A scale-local relative diffusivity is thereforegiving the estimateIt requires separation from the molecular and viscous cutoffs and from the integral scale of turbulence, high Reynolds number, approximately steady energy supply, and scale-local relative motion. Mean shear, walls, particle inertia or coherent large-scale strain can invalidate this simple diffusivity model. At very short times a pair retains its initial relative velocity, so a diffusion closure is not yet appropriate.

Let $s(t)=⟨|\delta \mathbf{x}(t){|}^2⟩$. A self-similar pair-separation distribution, together with the relative-diffusivity estimate, givesSelf-similarity is needed to replace a separation-dependent moment such as $⟨r^{4/3}⟩$ by a fixed multiple of $s^{2/3}$; stationarity alone does not do that. Integrating this model givesAfter loss of initial-separation memory, it reduces to the Richardson pair dispersion lawThe required time window is $t\gg t_0\sim (r_0^2/ϵ{)}^{1/3}$ while the separation remains far below the integral scale. The stated stationarity and inertial-range separation conditions alone cannot imply the displayed $t^3$ law at every time: a nonzero release separation already gives $s(0)=r_0^2$, whereas $gϵt^3$ vanishes at zero. At sufficiently short times, $\delta \mathbf{x}(t)-\delta {\mathbf{x}}_0\simeq \delta {\mathbf{u}}_{0}t$, so its mean-square change is ballistic and remembers $r_0$.

The Richardson constant might be universal if the inertial-range relative dynamics, including the normalized separation distribution, become independent of release details and forcing at sufficiently high Reynolds number. This is an additional universality hypothesis. As evidence available before 2008, Boffetta and Sokolov's numerical study estimated a coefficient near $0.55$ and reported agreement with experimental estimates, alongside small deviations from ideal Richardson scaling. Bourgoin and collaborators' 2006 experiment showed substantial dependence on initial separation in the accessible laboratory regime. Values of order $0.5$ are supported in suitable regimes, but finite-time fits do not establish one universally observed $t^3$ coefficient for all releases.

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.

The fluctuation speed $q$ and the integral scale of turbulence $L_u$ give a turnover time $L_u/q$. Dividing the velocity-variance scale $q^2$ by that time gives a turbulent kinetic energy dissipation rate proportional to $q^3/L_u$. Likewise mass density variance $\overline{{\rho }^{\prime 2}}$ removed over scalar mixing time $L_{\rho }/q$ gives the density-variance dissipation rate proportional to $\overline{{\rho }^{\prime 2}}q/L_{\rho }$. Absorbing dimensionless proportionality coefficients into the specified length scales givesThese are integral-scale closure estimates, not exact consequences of dimensional analysis alone. In particular $q^2=2k$ for the turbulent-energy convention in Question 3. The normalization of $\chi$ is the one fixed by the printed scalar balance; if the budget is written for the full variance rather than one half of it, both production and dissipation terms acquire a factor two.

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.

Internal intermittency is the uneven distribution of activity within an already turbulent fluid: intense gradients and viscous dissipation occupy relatively small, fluctuating regions separated by much quieter regions. It concerns the internal field, rather than a turbulent/non-turbulent interface intermittently passing a probe.

Integral-scale intermittency is the variability of the energy-containing eddies and their local or time-dependent energy supply. Its modulation occurs on lengths and times of order $\ell$ and $\ell /u$, can depend strongly on forcing or initial conditions, and changes the background intensity experienced by smaller eddies.

Equilibrium-range intermittency concerns the increasing fluctuations of local transfer and viscous dissipation as the averaging scale is reduced below the integral scale of turbulence. It appears as non-Gaussian velocity-gradient statistics, fluctuating coarse-grained viscous dissipation and anomalous longitudinal structure function exponents. It includes the inertial range and its continuation into the dissipation range, where kinematic viscosity ultimately limits gradients. Large-scale modulation varies the local energy supply; small-scale intermittency concentrates and redistributes that supply. These effects can coexist, so they should not be identified simply because both broaden a distribution.

Let the two-dimensional enstrophy cascade spectrum be $E(k)=C{k}^{-3}$ between $k_0$ and $k_d$, and choose a separation well inside the range: $k_0\ll k_c=\pi /r\ll k_d$. The high-wavenumber energy isThe low-wavenumber correction instead givesIts ratio to the preceding energy contribution is $({\pi }^2/4)\log (k_c/k_0)$, which is large when the separation is sufficiently far below the integral scale of turbulence. Consequentlyup to the stated piecewise-filter approximation and subleading contributions. Equivalently, if $Z_{<k_c}={\int }_0^{k_c}{k}^{2}E(k)dk$ is the cumulative enstrophy in motions larger than $r$, then $S_2\simeq r^2{Z}_{<k_c}/4$.

The leading increment variance is controlled by the accumulated enstrophy at scales larger than the separation, rather than the kinetic energy at smaller scales. For a $k^{-3}$ spectrum, each logarithmic wavenumber interval contributes the same amount to that enstrophy integral, so the conclusion does not mean that only the largest eddy supplies it. The logarithm is a marginal failure of a pure local $r^2$ estimate. The stipulated wide power-law range is essential, and the dominance need not hold at its upper spatial edge where $k_c/k_0$ is only of order one.

Let $\mathcal{E}=⟨|\mathbf{u}{|}^2⟩/2$ be the turbulent kinetic energy per unit mass. In addition to $L>0$ and high Reynolds number, use the usual self-preserving large-eddy decay assumption: the normalized large-scale correlation has a fixed shape, sowhere $C_L>0$ is constant. For freely decaying high-Reynolds-number turbulence, the cascade viscous dissipation law is $ϵ=C_{ϵ}{\mathcal{E}}^{3/2}/\ell$. The kinetic-energy balance therefore givesPut $A=C_{ϵ}{C}_L^{1/3}{L}^{-1/3}$. Integrating,The constant $t_{\ast }$ is a virtual time origin. This is the Saffman decay law. The energy decreases while the integral scale of turbulence grows so that the conserved large-volume momentum variance remains constant.

The exponent does not follow from the invariant alone: it also uses unforced decay, one evolving large-eddy length and an asymptotically constant cascade viscous dissipation coefficient. A changing normalized spectral shape or a confining boundary can invalidate that closure. Moreover $\mathrm{Re}\sim {\mathcal{E}}^{1/2}\ell /\nu \propto (t-t_{\ast }{)}^{-1/5}$, so at fixed kinematic viscosity the high-Reynolds-number decay regime eventually ends; the $-6/5$ law is not a claim about the ultimate viscous limit.

With nonzero invariant $L$, a self-preserving integral scale of turbulence gives $L\propto \mathcal{E}{\ell }^3$. Combining this with freely decaying high-Reynolds-number viscous dissipation $ϵ\propto {\mathcal{E}}^{3/2}/\ell$ gives $\mathcal{E}\propto (t-t_{\ast }{)}^{-6/5}$ and $\ell \propto (t-t_{\ast }{)}^{2/5}$. The closure assumptions, not the invariant alone, determine this exponent.

For positive initial separation below the Kolmogorov length scale, a phenomenological pair-dispersion model has smooth-flow exponential stretching, inertial-range Richardson pair dispersion, and diffusive large-scale pair dispersion beyond the integral scale of turbulence. A positive Lyapunov exponent is needed for the first regime, scale locality and initial-memory loss for the second, and integrable relative-velocity autocorrelation for the third. The inertial-range crossing time is of order $L^{2/3}{ϵ}^{-1/3}$, one outer eddy turnover time.

At fixed outer velocity and integral scale of turbulence, the mean viscous dissipation per unit mass approaches a positive finite value as the Reynolds number becomes large. The dimensionless coefficient $C_{ϵ}=ϵ\ell /u^3$ is then of order one. This is a phenomenological property of established three-dimensional turbulence, not a consequence of taking kinematic viscosity to zero in an arbitrary smooth flow.