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Let $E(k)$ be the one-dimensional turbulent energy spectrum, normalized so that $\frac{1}{2}⟨|{\mathbf{u}}^{\prime }{|}^2⟩={\int }_0^{\infty }E(k)dk$. Large energy-containing eddies receive energy from the forcing. In the inertial range, nonlinear transfer passes this energy towards smaller scales at a mean rate $ϵ$ per unit mass. Dimensional analysis of Kolmogorov 1941 theory givesThe dissipation range begins near the inverse of the Kolmogorov length scale, where kinematic viscosity removes the transferred kinetic energy. The large-scale spectrum depends on the forcing; the rising low-$k$ branch in this sketch is illustrative, rather than a universal power law.

The tube has neither homogeneous nor isotropic large-scale turbulence: gravity selects a vertical direction, its walls constrain the motion, and its density gradient varies with height. The width $W$ bounds the transverse size of the energy-containing eddies. At sufficiently high Reynolds number, much smaller eddies may become approximately locally isotropic and exhibit an inertial range, even though the full tube flow does not satisfy the homogeneous-isotropic assumptions.

To obtain the Reynolds stress, write $u_i=U_i+u_i^{\prime }$ and $p=P+p^{\prime }$, with $⟨u_i^{\prime }⟩=0$. Averaging the incompressible Navier-Stokes equations uses $⟨u_i{u}_j⟩=U_i{U}_j+⟨u_i^{\prime }{u}_j^{\prime }⟩$ and gives the Reynolds-averaged momentum equationHere the final term is the mean buoyancy force, with the hydrostatic reference removed. Thus averaging the quadratic momentum flux introduces the covariance $⟨u_i^{\prime }{u}_j^{\prime }⟩$; multiplied by ${\rho }_0$, it has the dimensions of a stress. Its divergence transfers momentum between the mean flow and the fluctuations.

An eddy viscosity model represents this Reynolds stress byThe isotropic term can be absorbed into the mean pressure, while the deviatoric term gives a diffusion of mean momentum with turbulent viscosity ${\nu }_T\sim u_{\ast }\ell$, where $u_{\ast }$ and $\ell$ are representative turbulent speed and mixing length. Unlike kinematic viscosity, this coefficient depends on the flow. The closure assumes a local relation between stress and mean strain; anisotropic, buoyancy-driven large eddies may require a more detailed Reynolds stress model.

Work with an incompressible Newtonian fluid of constant kinematic viscosity. The vorticity equation is $D\mathbf{\omega }/Dt=(\mathbf{\omega }\cdot \nabla )\mathbf{u}+\nu {\nabla }^2\mathbf{\omega }$. In the axisymmetric Burgers vortex the swirl does not advect the axial vorticity, and the stretching rate is ${\partial }_z{u}_z=\alpha$. Put $s(t)={\delta }^2(t)$ and $W={\omega }_z$. The Gaussian profile hasSubstitution into ${\partial }_{t}W+u_r{\partial }_{r}W=\alpha W+\nu {\nabla }^{2}W$ matches both the constant and $r^2$ terms precisely whenFor every finite positive initial $s_0$,The strain compresses the cross-section and stretches its vorticity, while kinematic viscosity spreads the core. The attracting radius is the balance of these effects; its relaxation time is ${\alpha }^{-1}$. Total circulation stays fixed because stretching increases the vorticity as the cross-sectional area decreases.

At the steady core radius, the swirl speed is of order ${\Gamma }_0/\delta$, and its velocity gradient tensor is of order ${\Gamma }_0/{\delta }^2$. Its viscous dissipation integrated over a cross-section therefore scales asThis is per unit length and per unit density. More precisely, for the swirl contribution, integration by parts relates the integrated strain viscous dissipation to $\nu \int W^{2}dA$; the boundary term vanishes for the $1/r$ swirl velocity. ThusMultiply by density for power per physical length. Here the azimuthal velocity's circulation $\Gamma$ is ${\Gamma }_0$, as follows by integrating $W$ over the cross-section. The result is finite as $\nu \to 0$ with strain and circulation held fixed. Local core viscous dissipation grows like ${\nu }^{-1}$ while its area shrinks like $\nu$. For ${\Gamma }_0/\nu \gg 1$, the swirl gradients dominate the background strain within the core.

This finite quantity is the excess viscous dissipation due to the vortex. The uniform straining flow alone dissipates $3\nu {\alpha }^2$ per unit volume, and integrating that over an infinite cross-section diverges. The total viscous dissipation of the complete idealized flow is therefore not a finite tube power unless the background is subtracted or spatially bounded.

A Burgers vortex sheet instead smooths a fixed velocity jump $U$ across a layer of thickness ${\delta }_s\sim \sqrt{\nu /\alpha }$. Its viscous dissipation per sheet area is of orderA tube with fixed circulation supports finite viscous dissipation in a vanishingly small core; a sheet with fixed velocity jump does not. This makes strained vortex tubes plausible intense dissipative structures at high Reynolds number. It does not assert that sheets are absent from turbulence: sheets can carry inertial range organization and become unstable, producing finer vortical structures.

In the Townsend-Betchov cascade cartoon, bi-axial strain sweeps distributed vorticity into extended sheets at the upper part of the inertial range. These are predominantly inviscid structures: their shape and instability arise from advection and strain rather than local viscous smoothing. Strong velocity variation across a thin sheet permits Kelvin-Helmholtz instability, roll-up and three-dimensional deformation. The resulting small vortex tubes are stretched into still finer cores.

At the dissipative end, stretching and kinematic viscosity balance in Burgers-like cores. A core of strain rate $s$ has radius of order $(\nu /s{)}^{1/2}$; taking $s\sim (ϵ/\nu {)}^{1/2}$ gives the Kolmogorov length $({\nu }^3/ϵ{)}^{1/4}$. Circulation and shrinking cross-section allow finite integrated tube viscous dissipation. Thus sheet formation and tube formation play different roles, rather than the entire cascade being a succession of identical geometrical eddies.

The cartoon challenges a literal, uniformly self-similar, step-by-step cascade with independent interactions only between neighboring scales. Large-scale sheet formation can generate much smaller scales through an instability, and the resulting structures retain coherent spatial organization. A branching model that assumes every scale-transfer step has the same geometry and loses information about the parent flow can miss this mechanism. Closures that erase the relevant phase correlations also have difficulty representing its concentrated gradients and intermittency. The objection is to these dynamical assumptions, not to conservation of energy or the existence of a mean energy flux through scales. The cartoon alone is not a proof that all spectral transfer is nonlocal.

Let ${ϵ}_{\mathrm{loc}}(\mathbf{x})=2\nu S_{ij}{S}_{ij}$ and define a coarse-grained energy dissipation at scale $r$, for example by a ball average,This is the random variable denoted ${ϵ}_{\mathrm{AV}}(r)$. The refined similarity hypothesis uses the local supply of energy available to motions of size $r$, rather than replacing it everywhere by a global mean. In its inertial range form,At high local Reynolds number, $V$ has a universal conditional distribution independent of ${ϵ}_r$, forcing details and $r$ within the inertial range. This givesFor signed odd orders the coefficients carry the appropriate sign; absolute moments use their own coefficients. Universality is retained in the normalized conditional velocity statistics, while the distribution of ${ϵ}_r$ can retain integral-scale information. This distinction resolves the particular averaging objection: its nonuniversal viscous dissipation moments are now explicitly present.

The quoted second viscous dissipation moment does not by itself determine every longitudinal structure function exponent. The extra lognormal intermittency model assumes $Y_r=\log ({ϵ}_r/ϵ)$ is Gaussian. Write its variance as ${\sigma }_r^2$. Normalization of the mean requires $⟨Y_r⟩=-{\sigma }_r^2/2$, soThe second-moment estimate gives $\exp ({\sigma }_r^2)=B(\ell /r{)}^{\mu }$. Hence, with $q=p/3$,In particular ${\zeta }_3=1$, ${\zeta }_2=2/3+\mu /9$ and ${\zeta }_6=2-\mu$.

The most problematic step is the extrapolation of a lognormal viscous dissipation law to rare events and all moment orders. It is not entailed by a measured second moment. Its quadratic exponents eventually decrease with $p$ and become negative for $p>3+6/\mu$. If the velocity has finite uniformly bounded moments, the bound $⟨|\Delta v{|}^p⟩\le 2^p⟨|u_x{|}^p⟩$ prohibits such divergence as $r\to 0$. Thus the model cannot be an unrestricted high-order inertial range law. This is a criticism of the lognormal closure, rather than an algebraic contradiction in refined similarity itself.

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.

Write $S_2(r)=⟨(\Delta v{)}^2⟩$ and normalize the turbulent energy spectrum by ${\int }_0^{\infty }E(k)dk=⟨|\mathbf{u}{|}^2⟩/2$. In isotropic incompressible turbulence the longitudinal velocity correlation isCombining $S_2=2[Q_{LL}(0)-Q_{LL}(r)]$ with $Q_{LL}(0)=2\int Edk/3$ gives the exact spectral filterThe expression has a smooth removable limit at $x=0$. Its asymptotic forms areAn eddy much larger than $r$ contributes nearly the same velocity at the two points and mostly cancels in the increment. An eddy much smaller than $r$ contributes decorrelated velocities and does not cancel. Replacing the smooth transition of $H$ by a step at $kr=\pi$ therefore gives the common estimate $3S_2/4\simeq {\int }_{\pi /r}^{\infty }E(k)dk$. The cutoff corresponds roughly to a half wavelength equal to the point separation; its exact numerical location is conventional.

Keeping the leading nonzero low-wavenumber term gives the improved estimateIt remains imperfect because $kr$ is not small throughout the whole low-wavenumber interval and the exact kernel oscillates above the transition. The data sheet's alternative coefficient $1/{\pi }^2$ is a nearby approximate matching prescription; $1/10$ is the exact Taylor coefficient. Neither piecewise filter is an exact identity.

Higher-order longitudinal structure functions also retain large-scale information. Decompose $\mathbf{u}=\mathbf{U}+{\mathbf{u}}_s$, where $\mathbf{U}$ varies on a length much larger than $r$. Its longitudinal increment is $\Delta U_L\simeq r{\partial }_L{U}_L$, which is small but not zero. Expanding $(\Delta U_L+\Delta u_{s,L}{)}^p$ gives its own moments and mixed moments with the small-scale increment. Rare large gradients or correlated amplitude modulation can matter particularly strongly at large $p$.

Thus a velocity increment is not an exact scale-local bandpass filter. Its moments may retain large-scale strain and forcing information, undermining an unqualified claim of universal small-scale statistics determined by one local parameter. This observation alone does not refute an asymptotic inertial range theory: for a three-dimensional $k^{-5/3}$ spectrum, the low-wavenumber correction is mainly supplied by wavenumbers of order $1/r$, while an isolated integral-scale smooth contribution behaves as $r^2$ and is subleading to $r^{2/3}$. The issue is whether large-scale contamination and mixed moments actually become negligible in the regime and moment order being used.

Richardson pair dispersion concerns relative separation of two marked fluid elements, rather than their centre-of-mass motion. In an inertial range, scale-local relative diffusivity is of order ${ϵ}^{1/3}{r}^{4/3}$. A self-similar closure gives $d⟨r^2⟩/dt=A{ϵ}^{1/3}⟨r^2{⟩}^{2/3}$ and hence a $t^3$ regime after loss of memory of initial separation. Stationarity and inertial-range separation alone do not imply this law at arbitrarily short times.

The Townsend-Betchov cascade cartoon organizes inertial range vorticity into sheets by bi-axial strain, then uses sheet instability and roll-up to generate finer vortex tubes. Dissipative tube cores balance stretching with kinematic viscosity. Such differing mechanisms across scales challenge literal identical-step, memoryless cascade models without contradicting a mean energy flux.