3.1.1. Ideal turbulence, renormalisation group and large-eddy simulation

The most basic conclusion that can be inferred from the empirical observations of $Re$-independence in (2.1) or (2.8) is that the non-dimensionalised velocity gradients, if interpreted as ordinary classical derivatives, must diverge as $|\hat {\boldsymbol {\nabla }}\hat {{\boldsymbol {u}}}|\to \infty$ when $Re\to \infty.$ This was the starting point of Onsager's own line of argument, who began the short abstract in Onsager (Reference Onsager1945a) with the observation: ‘The dissipation of energy by turbulence is regarded as primarily a “violet catastrophe”.’ In the language of modern field theory, turbulence exhibits ultraviolet (UV) divergences of velocity gradients in the limit $Re\to \infty.$ As a consequence, the equations of motion (2.4) (or alternatively (2.5)) cannot remain valid in a naïve sense in the infinite-$Re$ limit. Just as in field theory, to obtain a dynamical description of turbulence as $Re\to \infty$, one must somehow regularise these UV divergences. The simple but effective approach of Constantin et al. (Reference Constantin, Weinan and Titi1994) was to perform what is called ‘low-pass filtering’ in engineering, ‘spatial coarse-graining’ in physics and ‘mollifying’ in mathematics. This method involves the use of a smooth kernel $G$ for space dimension $d$ that satisfies:

1. (i) $G(\boldsymbol {r}) \geq 0$;

2. (ii) $G(\boldsymbol {r}) \rightarrow 0 \quad \text {rapidly for} \ |\boldsymbol {r}| \rightarrow \infty$;

3. (iii) $\int \,{\rm d}^d r \,G(\boldsymbol {r}) =1$.

It is understood that $G$ is centred at $\boldsymbol {r} = {\boldsymbol 0}$, $\int \,{\rm d}^d r \,\boldsymbol {r} \,G(\boldsymbol {r}) = \boldsymbol {0}$ and that $\int \,{\rm d}^d r |\boldsymbol {r}|^2 G(\boldsymbol {r}) \approx 1$. We can then set

(3.1)\begin{equation} G_\ell (\boldsymbol{r}) \equiv \ell^{-d} G(\boldsymbol{r}/\ell) \end{equation}

so that all of the above properties hold, except that now $\int \,{\rm d}^d r |\boldsymbol {r}|^2 G_\ell (\boldsymbol {r}) \approx \ell ^2$, with $\ell >0$ the regularisation length scale. Finally, one defines a coarse-grained velocity at length scale $\ell$ by the following formula involving a convolution integral:

(3.2)\begin{equation} \bar{{\boldsymbol{u}}}_\ell(\boldsymbol{x},t)=(G_\ell*{\boldsymbol{u}})(\boldsymbol{x},t)=\int \,{\rm d}^dr\,G_\ell(\boldsymbol{r})\, {\boldsymbol{u}}(\boldsymbol{x}+\boldsymbol{r},t), \end{equation}

spatially averaging over the eddies of size $<\ell.$

This coarse-grained field is roughly analogous to a ‘block spin’ in critical phenomena and field theory. An identical operation is also employed in the ‘filtering approach’ to turbulence advocated by Leonard (Reference Leonard1974) and Germano (Reference Germano1992), but with a different motivation than regularisation of divergences. For Onsager's theory, the important point is that the coarse-graining operation (3.2) regularises gradients, so that $\boldsymbol {\nabla }\bar {{\boldsymbol {u}}}_\ell$ remains finite as $\nu \to 0$ for any fixed length $\ell >0.$ This may be shown using the simple integration-by-parts identity

(3.3)\begin{equation} \boldsymbol{\nabla}\bar{{\boldsymbol{u}}}_\ell(\boldsymbol{x},t)=-\frac{1}{\ell}\int\, {\rm d}^dr (\boldsymbol{\nabla} G)_\ell(\boldsymbol{r})\, {\boldsymbol{u}}(\boldsymbol{x}+\boldsymbol{r},t), \end{equation}

which by Cauchy–Schwartz inequality yields the bound $|\boldsymbol {\nabla }\bar {{\boldsymbol {u}}}_\ell (\boldsymbol {x},t)|\leq (1/\ell ) \sqrt {C_\ell \int \,{\rm d}^dr |{\boldsymbol {u}}(\boldsymbol {r},t)|^2}$ with constant $C_\ell =\int \,{\rm d}^dr |(\boldsymbol {\nabla } G)_\ell (\boldsymbol {r})|^2.$ Thus, the coarse-grained gradient is bounded as long as the total kinetic energy remains finite as $\nu \to 0$ (which is necessarily true for freely decaying turbulence with no stirring). The price of this regularisation, as for quantum field-theory divergences, is that a new, arbitrary regularisation scale $\ell$ has been introduced.

Because divergences have been eliminated, one may now seek a dynamical description in terms of the coarse-grained field defined in (3.2). The equation which is satisfied is easily found to be

(3.4)\begin{equation} \partial_t\bar{{\boldsymbol{u}}}_\ell+\boldsymbol{\nabla}\boldsymbol{\cdot}\overline{({\boldsymbol{u}}{\boldsymbol{u}})_\ell}=-\boldsymbol{\nabla}\bar{p}_\ell+\nu\Delta\bar{{\boldsymbol{u}}}_\ell, \quad \boldsymbol{\nabla}\boldsymbol{\cdot}\bar{{\boldsymbol{u}}}_\ell=0, \end{equation}

because the coarse-graining operation commutes with all space- and time-derivatives, and one may inquire about its limit as $Re\to \infty.$ For this purpose, one should non-dimensionalise the above equation and introduce hats everywhere, but doing so simply replaces the physical viscosity with the dimensionless viscosity $\hat {\nu }=1/Re.$ Thus, as customary in the mathematical literature, one may omit the hats on non-dimensionalised variables and consider the limit $Re\to \infty$ as a ‘zero-viscosity limit’ limit $\nu \to 0.$ I shall do so hereafter, when this causes no confusion. Because the quantity $\Delta \bar {{\boldsymbol {u}}}_\ell$ in (3.4) remains bounded by a similar estimate as (3.3), one can easily show rigorously that $\nu \Delta \bar {{\boldsymbol {u}}}_\ell \to 0$ as $\nu \to 0$ for fixed $\ell.$ If one indexes the solutions ${\boldsymbol {u}}^\nu$ of the Navier–Stokes equation (2.4) by viscosity $\nu,$ then the above results suggest that a limiting field ${\boldsymbol {u}}=\lim _{\nu \to 0}{\boldsymbol {u}}^\nu$ will satisfy the equation

(3.5)\begin{equation} \partial_t\bar{{\boldsymbol{u}}}_\ell+\boldsymbol{\nabla}\boldsymbol{\cdot}\overline{({\boldsymbol{u}}{\boldsymbol{u}})}_\ell=-\boldsymbol{\nabla}\bar{p}_\ell, \quad \boldsymbol{\nabla}\boldsymbol{\cdot}\bar{{\boldsymbol{u}}}_\ell=0. \end{equation}

Since one is dealing with velocity fields in function spaces, a careful statement of this hypothesis requires a suitable notion of convergence ${\boldsymbol {u}}^\nu \to {\boldsymbol {u}}.$ It is not hard to show that strong $L^2$ convergence suffices, which is the condition that $\lim _{\nu \to 0}\|{\boldsymbol {u}}^\nu -{\boldsymbol {u}}\|_2=0$ where

(3.6)\begin{equation} \|{\boldsymbol{u}}\|_p:=\left[\frac{1}{T}\int_0^T \,{\rm d}t \frac{1}{|\varOmega|}\int_\varOmega \,{\rm d}^d x |{\boldsymbol{u}}(\boldsymbol{x},t)|^p\right]^{1/p},\quad p\geq 1. \end{equation}

As I shall discuss later (see § 3.2.3), such strong $L^2$-convergence is not guaranteed a priori, although it is in fact implied by some relatively mild assumptions on the energy spectrum which can be tested empirically. For the moment, I shall simply assume such convergence so that (3.5) directly follows, but I return to this issue later.

An important observation is that the validity of the coarse-grained equations (3.5) for all lengths $\ell >0$ is equivalent to the condition in mathematical PDE theory that the velocity ${\boldsymbol {u}}$ is a weak solution of the incompressible Euler equations. Readers may be more familiar with the standard formulation of weak solutions by smearing the dynamical equations with smooth space–time test functions, so that the equations hold in the sense of distributions or generalised functions. A simple proof that ‘coarse-grained solutions’, satisfying (3.5) for all $\ell >0,$ are equivalent to standard weak solutions is given in § 2 of Drivas & Eyink (Reference Drivas and Eyink2018). Furthermore, these notions of weak solution are equivalent to what Onsager (Reference Onsager1949) meant when he wrote that ‘the ordinary formulation of the laws of motion in terms of differential equations becomes inadequate and must be replaced by a more general description; for example, the formulation (15) in terms of FOURIER series will do’. Indeed, the incompressible Euler equations written as an infinite-dimensional set of ordinary differential equations (ODEs) for the Fourier coefficients of the velocity field yield also standard weak solutions, as discussed by Eyink (Reference Eyink1994) and De Lellis & Székelyhidi (Reference De Lellis and Székelyhidi2013). All of these equivalent formulations of weak solutions make sense even when spatial derivatives of velocity no longer exist in the classical sense, but only in the sense of distributions.

It may appear odd to some readers that (3.5) can be interpreted as ‘Euler equations’, since the filtered equations (3.5) are generally regarded as unclosed whereas the Euler equations are closed PDEs. It is indeed true that (3.5) are not closed equations for the coarse-grained velocity $\bar {{\boldsymbol {u}}}_\ell$ itself. This fact is usually underlined by introducing the turbulent/subscale stress

(3.7)\begin{equation} \boldsymbol{\tau}_\ell({\boldsymbol{u}},{\boldsymbol{u}}) := \overline{({\boldsymbol{u}}{\boldsymbol{u}})}_\ell - \bar{{\boldsymbol{u}}}_\ell\bar{{\boldsymbol{u}}}_\ell, \end{equation}

so that (3.5) can be rewritten as

(3.8)\begin{equation} \partial_t\bar{{\boldsymbol{u}}}_\ell+\boldsymbol{\nabla}\boldsymbol{\cdot}\left[ \bar{{\boldsymbol{u}}}_\ell \bar{{\boldsymbol{u}}}_\ell + \boldsymbol{\tau}_\ell({\boldsymbol{u}},{\boldsymbol{u}})\right] =-\boldsymbol{\nabla}\bar{p}_\ell, \quad \boldsymbol{\nabla}\boldsymbol{\cdot}\bar{{\boldsymbol{u}}}_\ell=0 \end{equation}

with the non-closed term $\boldsymbol {\tau }_\ell$ distinguished. However, the coarse-grained velocity $\bar {{\boldsymbol {u}}}_\ell =G_\ell *{\boldsymbol {u}}$ and the subscale stress $\boldsymbol {\tau }_\ell ({\boldsymbol {u}},{\boldsymbol {u}})$ are both explicit functions of the fine-grained velocity field ${\boldsymbol {u}},$ as emphasised by my notation, and thus (3.8) are explicit conditions on that field. I may note that it is quite standard to consider that the large-scales $>\ell$ in a turbulent flow must satisfy the Euler fluid equations. For example, Landau & Lifshitz (Reference Landau and Lifshitz1959, § 31), wrote that: ‘We therefore conclude that, for the large eddies which are the basis of any turbulent flow, the viscosity is unimportant and may be equated to zero, so that the motion of these eddies obeys Euler's equation.’ What needs to be stressed is that the proper interpretation of such commonplace remarks is not that the coarse-grained velocity $\bar {{\boldsymbol {u}}}_\ell$ satisfies the Euler equations in the usual naïve sense, but instead that ${\boldsymbol {u}}$ is described by a weak Euler solution at those scales, in the sense that (3.8) holds. Although this point is elementary, it is frequently misunderstood and the source of common errors.

There is another very important conceptual point which must be emphasised about the coarse-grained equations (3.8). Some critics have argued that these equations are unphysical and not appropriate as the basis for a fundamental theory of turbulence because both the length scale $\ell$ and the filter kernel $G$ are arbitrary. For example, Tsinober (Reference Tsinober2009) wrote ‘The filter decomposition is formally more general than the Reynolds decomposition. However, the former is one among many decompositions, so to say, of a technical nature’ (p. 379) and also ‘After all Nature may and likely does not know about our decompositions’ (p. 114). These are very shrewd remarks. In fact, I agree completely with these concerns of Tsinober (Reference Tsinober2009) that Nature should not care about such arbitrary choices. Interestingly, this same problem has arisen in another area of physics, relativistic quantum field theory, which is also plagued with similar UV divergences. In that case also, arbitrary regularisations are required to eliminate the divergences and these introduce a new arbitrary length scale, equivalent to $\ell$ or, more commonly, a related energy scale $\mu =c\hbar /\ell,$ with $c$ the speed of light and $\hbar$ Planck's constant. In the renormalised field theory, fundamental parameters of the theory such as coupling constants $\lambda (\mu )$ become dependent on this arbitrary scale, in the same manner that $\boldsymbol {\tau }_\ell$ becomes dependent upon $\ell$. Elementary particle physicists in the 1950s were also worried that predictions of the theory should not depend upon such arbitrary choices and the concept of renormalisation group (RG) invariance arose as the commonsense demand that any observable consequence of the theory should be independent of $\mu.$ For a very clear discussion, see Gross (Reference Gross1976, § 4.1). What makes RG invariance interesting and important is that non-trivial consequences can be deduced precisely by varying $\mu$ and demanding that physical results be $\mu$-independent. What we shall see is that Onsager anticipated such arguments in the 1940s and that his 1/3 Hölder claim for turbulent velocities is an exact non-perturbative consequence of such RG invariance.

To underscore the nature of the argument, I emphasise that coarse-graining is a purely passive operation – ‘removing one's spectacles’ – which changes no physical process. The effect of the coarse-graining operation in (3.2) is illustrated by figure 3, which shows the velocity field observed at successively coarser spatial resolutions $\ell$. Although the dynamics of the velocity field resolved at the Kolmogorov scale $\ell =\eta$ is described rather well by the Navier–Stokes equation (2.4), the coarse-grained velocity field $\bar {{\boldsymbol {u}}}_\ell$ at scales $\ell \gg \eta$ is described by the highly non-Newtonian equation (3.8). While the description of the dynamics changes with resolution $\ell,$ nevertheless objective facts cannot depend upon the ‘eyesight’ of the observer. Thus, an energy dissipation rate which is non-vanishing in the limit as $\nu \to 0$ implies that kinetic energy will decrease over a fixed interval of time $[0,t],$ as observed by experiment, and this fact cannot depend upon the arbitrary scale $\ell.$ An observation at resolution $\ell$ can only miss some kinetic energy of smaller eddies, since by convexity

(3.9)\begin{equation} \tfrac{1}{2}|\bar{{\boldsymbol{u}}}_\ell(\boldsymbol{x},t)|^2\leq \tfrac{1}{2}\overline{\left(\left|{\boldsymbol{u}}(\boldsymbol{x},t)\right|^2\right)}_\ell, \end{equation}

and it then follows that $E_\ell (t):=(1/2)\int \,{\rm d}^d x |\bar {{\boldsymbol {u}}}_\ell (\boldsymbol {x},t)|^2\leq (1/2)\int \,{\rm d}^d x |{\boldsymbol {u}}(\boldsymbol {x},t)|^2=E(t).$ If kinetic energy continues to decay even in the limit as $\nu \to 0,$ then such persistent energy decay must also be seen by the ‘myopic’ observer who observes fluid features only at space-resolution $\ell.$ As I now show, however, the persistent energy decay observed at the fixed length scale $\ell$ with $\nu \to 0$ is not due to molecular viscosity acting directly at those scales.

Figure 3. Schematic illustration of spatial coarse-graining for turbulent flow past a grid: (a) fine-grained flow resolved to the dissipation scale $\eta$; (b) flow coarse-grained at a length scale $\ell >\eta$; (c) flow coarse-grained further at scale $\ell '>\ell.$ At each stage, eddies smaller than the coarsening scale are unresolved and ignored. Panel (a) is reproduced from Laizet et al. (Reference Laizet, Fortuné, Lamballais and Vassilicos2012) and panels (b,c) modified by image filtering, with permission from Elsevier.

The local kinetic energy balance at length scales $\ell$ in the inertial-range obtained for the limit $\nu \to 0$ is calculated straightforwardly from (3.8) to be

(3.10)\begin{equation} \partial_t\left(\tfrac{1}{2}|\bar{{\boldsymbol{u}}}_\ell|^2\right) +\boldsymbol{\nabla}\boldsymbol{\cdot}\left[\left( \tfrac{1}{2}|\bar{{\boldsymbol{u}}}_\ell|^2+\bar{p}_\ell\right)\bar{{\boldsymbol{u}}}_\ell +\boldsymbol{\tau}_\ell\boldsymbol{\cdot}\bar{{\boldsymbol{u}}}_\ell\right] =-\varPi_\ell, \end{equation}

where the quantity on the right-hand side of the equation is given (with ${\boldsymbol A}\boldsymbol {:}{\boldsymbol B}={\sum }_{ij} A_{ij}B_{ij}$) by

(3.11)\begin{equation} \varPi_\ell(\boldsymbol{x},t) =-\boldsymbol{\nabla}\bar{{\boldsymbol{u}}}_\ell(\boldsymbol{x},t)\boldsymbol{:}\boldsymbol{\tau}_\ell(\boldsymbol{x},t), \end{equation}

and represents the ‘deformation work’ of the large-scale strain acting against small-scale stress, or the ‘energy flux’ from resolved scales $>\ell$ to unresolved scales $<\ell.$ The mechanism of loss of energy by the inertial-range eddies is thus ‘energy cascade’, a term first used in this connection by Onsager (Reference Onsager1945a). A key observation is that the stress-tensor $\boldsymbol {\tau }_\ell ({\boldsymbol {u}},{\boldsymbol {u}})$ may be rewritten in terms of velocity-increments $\delta {\boldsymbol {u}}(\boldsymbol {r};\boldsymbol {x},t)={\boldsymbol {u}}(\boldsymbol {x}+\boldsymbol {r},t)-{\boldsymbol {u}}(\boldsymbol {x},t),$ as

(3.12)\begin{equation} \boldsymbol{\tau}_\ell({\boldsymbol{u}},{\boldsymbol{u}})=\langle \delta {\boldsymbol{u}}\delta {\boldsymbol{u}}\rangle_\ell -\langle \delta {\boldsymbol{u}}\rangle_\ell\langle \delta {\boldsymbol{u}}\rangle_\ell, \end{equation}

where $\langle\, f\,\rangle _\ell (\boldsymbol {x},t):=\int \,{\rm d}^dr\, G_\ell (r) f(\boldsymbol {r};\boldsymbol {x},t).$ This formula was originally obtained by Constantin et al. (Reference Constantin, Weinan and Titi1994) in a slightly different form, and as above by Eyink (Reference Eyink1995b) as a physical re-interpretation of their result. Equation (3.12) is easy to verify by direct calculation, but it can be simply understood as due to the invariance of the second-order cumulant $\boldsymbol {\tau }_\ell ({\boldsymbol {u}},{\boldsymbol {u}})$ to shifts of ${\boldsymbol {u}}$ by vectors that are ‘non-random’ with respect to the average $\langle {\cdot }\rangle _\ell$ over displacements $\boldsymbol {r},$ i.e. that are independent of $\boldsymbol {r}.$ This allows ${\boldsymbol {u}}(\boldsymbol {x}+\boldsymbol {r},t)$ in the definition (3.7) of $\boldsymbol {\tau }_\ell ({\boldsymbol {u}},{\boldsymbol {u}})$ to be replaced with $\delta {\boldsymbol {u}}(\boldsymbol {r};\boldsymbol {x},t),$ yielding (3.12). Similarly, one may rewrite (3.3) for coarse-grained velocity-gradients in terms of increments as

(3.13)\begin{equation} \boldsymbol{\nabla}\bar{{\boldsymbol{u}}}_\ell(\boldsymbol{x},t)=-\frac{1}{\ell}\int \,{\rm d}^dr (\boldsymbol{\nabla} G)_\ell(\boldsymbol{r}) \delta{\boldsymbol{u}}(\boldsymbol{r};\boldsymbol{x},t), \end{equation}

using the fact that $\int \,{\rm d}^dr (\boldsymbol {\nabla } G)_\ell (\boldsymbol {r})=\boldsymbol {0}.$

As an immediate application of these formulae, one can rederive the prediction of Onsager (Reference Onsager1949) that Hölder singularities $h\leq 1/3$ are required in the velocity field for energy dissipation to persist in the limit $\nu \to 0.$ Indeed, assuming for some constant $C>0$ that

(3.14)\begin{equation} |\delta {\boldsymbol{u}}(\boldsymbol{r};\boldsymbol{x},t)|\leq C|\boldsymbol{r}|^h, \end{equation}

then it is straightforward to show using (3.11), (3.12) and (3.13) that

(3.15)\begin{equation} \varPi_\ell(\boldsymbol{x},t)=O\left(\ell^{3h-1}\right). \end{equation}

As is clear from (3.10), persistent energy decay at resolution length $\ell$ can only occur if $\int \, {\rm d}^dx\, \varPi _\ell (\boldsymbol {x},t)>0$ as $\nu \to 0.$ However, the resolution scale $\ell$ is completely arbitrary. For any fixed $\ell$, one can take $\nu$ sufficiently small so that the ‘ideal equations’ (3.5) or (3.8) hold to any desired accuracy at that scale, and then sequentially further decreasing $\ell$, one can correspondingly decrease $\nu$. If the Hölder regularity (3.14) held for all $(\boldsymbol {x},t)$ with $h>1/3,$ then clearly by (3.15), it would follow that $\int \, {\rm d}^d x \varPi _\ell (\boldsymbol {x},t)\to 0$ as $\ell \to 0.$ This is a contradiction, since the rate of decay of energy must be independent of the arbitrary length scale of resolution $\ell$ as $\ell \to 0.$ Just as done by Onsager (Reference Onsager1949), we thus infer that there must appear Hölder singularities $h\leq 1/3$ in the limit as $\nu \to 0,$ or $Re\to \infty$. As I shall discuss in § 3.1.2, Onsager had given a very similar argument to that above in his unpublished work.

As an aside, I remark that the RG character of this argument can be made even more explicit by a somewhat different technical approach of Johnson (Reference Johnson2020, Reference Johnson2022), where it is observed for a Gaussian kernel $G(\boldsymbol {r})=\exp (-r^2/2)/(2{\rm \pi} )^{3/2}$ that $({\partial }/{\partial \ell ^2}) \bar {{\boldsymbol {u}}}_\ell =\tfrac {1}{2}\Delta \bar {{\boldsymbol {u}}}_\ell$ and thus

(3.16)\begin{equation} \frac{\partial}{\partial \ell^2} \boldsymbol{\tau}_\ell=\frac{1}{2}\Delta\boldsymbol{\tau}_\ell +(\boldsymbol{\nabla}\bar{{\boldsymbol{u}}}_\ell)^\top \boldsymbol{\nabla}\bar{{\boldsymbol{u}}}_\ell. \end{equation}

In this approach, the subscale stress that ‘renormalises’ the bare Navier–Stokes dynamics can in fact be obtained by solving an equation that evolves $\boldsymbol {\tau }_\ell$ in the scale parameter $\ell$, analogous to the RG flow equations in high-energy physics and critical phenomena (Wilson Reference Wilson1975; Gross Reference Gross1976). Solving (3.16) with the initial data $\boldsymbol {\tau }_\ell |_{\ell =0}=\boldsymbol {0},$ Johnson (Reference Johnson2020) obtained

(3.17)\begin{equation} \boldsymbol{\tau}_\ell = \int_0^{\ell^2} \,{\rm d}\lambda^2 \, \overline{((\boldsymbol{\nabla}\bar{{\boldsymbol{u}}}_\lambda)^\top \boldsymbol{\nabla}\bar{{\boldsymbol{u}}}_\lambda )}_{\sqrt{\ell^2-\lambda^2}}. \end{equation}

Assuming the Hölder condition (3.14) then gives $\boldsymbol {\nabla }\bar {{\boldsymbol {u}}}_\lambda =O(\lambda ^{h-1})$ as before and, from the identity (3.17), $\boldsymbol {\tau }_\ell =O(\ell ^{2h}).$ Thus, the same bound (3.15) is deduced once again and this implies the 1/3 Hölder claim of Onsager. See also Isett & Oh (Reference Isett and Oh2016) for a similar approach.

The paper of Constantin et al. (Reference Constantin, Weinan and Titi1994) derived in fact a stronger result, by replacing the Hölder regularity originally assumed by Onsager (Reference Onsager1949) with a weaker assumption of Besov regularity. As discussed by Eyink (Reference Eyink1995b), Besov regularity can be understood in terms of deterministic $p$th-order ‘velocity-structure functions’ defined for absolute velocity increments and for space-averages over the flow domain $\varOmega :$

(3.18)\begin{equation} S_p(\boldsymbol{r})= \frac{1}{|\varOmega|}\int_\varOmega \,{\rm d}^d x |\delta {\boldsymbol{u}}(\boldsymbol{r};\boldsymbol{x})|^p. \end{equation}

Although power laws in the separation distance $|\boldsymbol {r}|$ are generally observed empirically for turbulent fluid flows, Besov regularity requires only an upper bound

(3.19)\begin{equation} S_p(\boldsymbol{r})\leq C_p|\boldsymbol{r}|^{\zeta_p}, \quad |\boldsymbol{r}|\leq 1. \end{equation}

If this inequality holds along with the modest assumption of finite $p$th-order moments

(3.20)\begin{equation} \frac{1}{|\varOmega|}\int_\varOmega \,{\rm d}^d x |{\boldsymbol{u}}(\boldsymbol{x})|^p <\infty, \end{equation}

then the velocity field ${\boldsymbol {u}}$ is said to belong to the Besov space $B^{\sigma,\infty }_p(\varOmega )$ with $\sigma =\sigma _p:=\zeta _p/p.$ Note that the optimal constant $C_p = {\sup }_{|\boldsymbol {r}|\leq 1} {S_p(\boldsymbol {r})}/{|\boldsymbol {r}|^{\zeta _p}}$ in the inequality (3.19) is related to the so-called ‘Besov semi-norm’ in the mathematical literature by the formula ${\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \boldsymbol {u} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{B^{\sigma,q}_p}=C_p^{1/p},$ so that the abstract semi-norm is given by the prefactor in expected power scaling laws (Drivas & Eyink Reference Drivas and Eyink2019). Note further that the Hölder condition (3.14) is equivalent to the Besov condition (3.19) for $p=\infty,$ with $\sigma _\infty =h.$

The result of Constantin et al. (Reference Constantin, Weinan and Titi1994) was that energy dissipation non-vanishing in the limit as $\nu \to 0$ requires $\zeta _p\leq p/3$ for all $p\geq 3$ or, equivalently, $\sigma _p\leq 1/3$ for $p\geq 3.$ The argument generalises that given previously for $p=\infty$ and uses the simple bound from the Hölder inequality valid for any $p\geq 3$,

(3.21)\begin{equation} \left| \frac{1}{|\varOmega|}\int_\varOmega \,{\rm d}^d x\, \varPi_\ell(\boldsymbol{x},t)\right| \leq \|\varPi_\ell\|_{p/3} \leq \|\boldsymbol{\nabla}\bar{{\boldsymbol{u}}}_\ell\|_{p} \|\boldsymbol{\tau}_\ell\|_{p/2}=O\left(\ell^{3\sigma_p-1}\right), \end{equation}

since $\|\boldsymbol {\nabla }\bar {{\boldsymbol {u}}}_\ell \|_{p}=O(\ell ^{\sigma _p-1})$ and $\|\boldsymbol {\tau }_\ell \|_{p/2}=O(\ell ^{2\sigma _p}).$ For mathematical details, see Constantin et al. (Reference Constantin, Weinan and Titi1994), Eyink (Reference Eyink1995b) or Eyink (Reference Eyink2007). Just as before, if the Besov regularity (3.19) held with $\sigma _p>1/3,$ then it would follow that $\int \,{\rm d}^d x, \varPi _\ell (\boldsymbol {x},t)\to 0$ as $\ell \to 0.$ This would yield the same contradiction as previously, since the rate of decay of energy must be independent of the arbitrary length scale of resolution $\ell$ as $\ell \to 0.$ Generalising the statement of Onsager (Reference Onsager1949), one can infer that in fact $\sigma _p\leq 1/3$ for all $p\geq 3$ in the limit as $\nu \to 0.$

Onsager's ideas anticipate the ‘multifractal model’ proposed by Parisi & Frisch (Reference Parisi and Frisch1985) for the turbulent velocity field; see also Frisch (Reference Frisch1995) for a comprehensive exposition. In fact, it is worth recalling the first sentence of Parisi & Frisch (Reference Parisi and Frisch1985): ‘A simple way of explaining power law structure function is to invoke singularities of the Euler equations considered as limit of the Navier–Stokes equations as the viscosity tends to zero.’ Parisi & Frisch (Reference Parisi and Frisch1985) proposed that the turbulent velocity field possesses a spectrum of Hölder exponents $h\in [h_{min},h_{max}]$ with each exponent $h$ occurring on a set $S(h)\subset \varOmega$ with Hausdorff dimension $D(h).$ The velocity scaling exponents are then obtained by the formula

(3.22)\begin{equation} \zeta_p=\inf_h \{ hp+ (3-D(h))\}, \end{equation}

thus accounting for the observed deviations from the prediction $\zeta _p=p/3$ of Kolmogorov (Reference Kolmogorov1941a,Reference Kolmogorovb). The velocity scaling exponent relevant in this context can be understood as the ‘maximal Besov exponent’ $\sigma _p=\zeta _p/p$ for any $p\geq 0$ with

(3.23)\begin{equation} \zeta_p:=\liminf_{|\boldsymbol{r}|\to 0} \frac{\log S_p(\boldsymbol{r})}{\log|\boldsymbol{r}|}, \end{equation}

a concept which is meaningful even without any power-law scaling (Eyink Reference Eyink1995a). As we shall see later, Onsager had arrived at similar conclusions already by 1945 (but without the modern concept of fractals). Note that $h_p=\mathrm {d}\zeta _p/\mathrm {d}p$ is the Hölder exponent $h$ that yields the infimum in (3.22) for each $p\geq 0.$ It then follows from (3.22) or even directly from the concavity of $\zeta _p$ in $p$ (Frisch Reference Frisch1995; Eyink Reference Eyink2007) that $h_p\leq \sigma _p$ for all $p\geq 0.$ Thus, the theorem of Constantin et al. (Reference Constantin, Weinan and Titi1994) implies also that $h_p\leq 1/3$ for $p\geq 3$ and the original result of Onsager (Reference Onsager1949) is equivalent to the statement that $h_\infty =h_{\min }\leq 1/3.$ It is important to emphasise that these are predictive statements which have received subsequent support from empirical determination of the multifractal dimension spectrum $D(h)$ both from numerical simulations (Kestener & Arneodo Reference Kestener and Arneodo2004) and from hot-wire experiments (Lashermes et al. Reference Lashermes, Roux, Abry and Jaffard2008). As I shall discuss later, the ‘ideal turbulence’ theory makes connection also with the alternative multifractal theory for the energy dissipation rate (Kolmogorov Reference Kolmogorov1962; Mandelbrot Reference Mandelbrot1974, Reference Mandelbrot1989; Meneveau & Sreenivasan Reference Meneveau and Sreenivasan1991).

Another successful prediction made by Onsager (Reference Onsager1945a, Reference Onsager1949) concerned the locality of the energy cascade, which is the statement that the nonlinear flux $\varPi _\ell$ defined in (3.11) as a cubic function of the velocity field ${\boldsymbol {u}}$ is determined predominantly by velocity modes or ‘eddies’ of scale near $\ell.$ In the words of Onsager (Reference Onsager1945a): ‘The modulation of a given Fourier component of the motion is mostly due to those others which belong to wavenumbers of comparable magnitude.’ In his 1945 letter to von Kármán and Lin (reproduced in Eyink & Sreenivasan Reference Eyink and Sreenivasan2006, appendix B), Onsager stated more precisely that ‘With a hypothesis slightly stronger than (14) the motion which belongs to wave-numbers of the same order of magnitude as $\underline {k}$ itself will furnish the greater part of the effective rate of shear’, where the mentioned condition (14) asserts that ${\boldsymbol {u}}$ is square-integrable but that $\boldsymbol {\nabla }{\boldsymbol {u}}$ is not. A condition of exactly this type is the Hölder condition (3.14), which may be shown for any $0< h<1$ to imply locality. In fact, scale locality of energy cascade holds assuming only the $p$th-order Besov condition (3.19) for any $0<\sigma _p<1$ (Eyink Reference Eyink2005; Cheskidov et al. Reference Cheskidov, Constantin, Friedlander and Shvydkoy2008). These predictions have been confirmed in a number of numerical studies, for example, Domaradzki & Carati (Reference Domaradzki and Carati2007), Aluie & Eyink (Reference Aluie and Eyink2009) and Cardesa et al. (Reference Cardesa, Vela-Martín, Dong and Jiménez2015).

It is crucial that these locality properties hold instantaneously and deterministically for individual flow realisations (Kraichnan Reference Kraichnan1974). The cited mathematical analyses and numerical simulations verify that averaging over velocity realisations improves the degree of locality but also that averaging is not necessary for local-in-scale interactions to dominate in energy transfer. Locality for individual realisations is necessary for another important feature of turbulence, the universality of small-scale statistics. The usual argument for such universal statistics has to do with the scale invariance of the Euler equations and the scale-locality of the energy transfer by which small scales are excited. It is by the latter property that, in the words of Onsager (Reference Onsager1949), ‘we are led to expect a cascade such that the wavenumbers increase typically in a geometric series, by a factor of the order 2 per step’. If each step in the cascade is chaotic and also of comparable nature to the previous steps, because of scale-invariance, then it is reasonable to expect that the details of large-scale flow features will be lost and superseded by motions intrinsic to the local Euler dynamics. It is further quite obvious that scale-locality for energy transfer only in the mean sense would be inadequate for such universality, because then long-range communication between large and small scales could exist instantaneously which is cancelled only by averaging over time or initial data.

This universality is one of the many features that connects Onsager's ‘ideal turbulence’ with the LES method of modelling turbulence. In LES, the large-scale motions (large eddies) of turbulent flow are computed directly and only small-scale (sub-grid scale; SGS) motions are modelled, resulting in a significant reduction in computational cost compared with DNS (Piomelli Reference Piomelli1999; Meneveau & Katz Reference Meneveau and Katz2000). The justification for the LES scheme is that the turbulent small-scales are expected to be statistically similar for every flow and thus may be universally modelled, while the non-universal large scales are explicitly computed at far lower cost. LES originated in the pioneering work of Smagorinsky (Reference Smagorinsky1963) and Lilly (Reference Lilly1967), and it has developed historically with little connection to Onsager's theory of ‘ideal turbulence’, although the two are quite intimately related. At the most superficial level, the analysis used by Constantin et al. (Reference Constantin, Weinan and Titi1994) to prove Onsager's 1/3 result is the same as the filtering approach advocated by Leonard (Reference Leonard1974) and Germano (Reference Germano1992), and widely used in LES. Thus, the filtered equations (3.8) used to derive Onsager's theorem are the same as those employed in LES. However, filtering/coarse-graining is just one specific form of regularisation and other schemes are possible (e.g. Onsager used also Fourier series as a regulariser). The specific regularisation by spatial filtering is not intrinsic to either method. More important is that both approaches aim to describe individual flow realisations, unlike Reynolds-averaged Navier–Stokes (RANS) modelling which resigns itself to calculation of time- or ensemble-averaged velocities only. Furthermore, Onsager's theory justifies the truism that an LES model should run even with molecular viscosity set to zero and thus should describe turbulent dissipation at infinite Reynolds number.

In concluding this section, I want to emphasise the conceptual and practical importance of a deterministic description of turbulent flow for individual flow realisations. In much of the scientific literature on fluid turbulence, it is often reflexively assumed that any mathematical description must be statistical in nature. However, this instinctive reaction does not permit one to discuss even such a commonplace flow as your morning cup of coffee (see figure 4). This flow has a decent Reynolds number that may be estimated of order $Re=UD/\nu =3\times 10^4$ (assuming stirring velocity $U=20\ {\rm cm}\ {\rm s}^{-1}$, cup diameter $D=6\ {\rm cm}$ and kinematic viscosity $\nu =0.4\ {\rm mm}^2\ {\rm s}^{-1}$) and is thus modestly turbulent. However, there is no ensemble in sight! The coffee is stirred only a few times to mix the cream and thus the flow is decaying, so that one cannot average over time. Also, the size of the largest eddies is of the order of the cup diameter and thus one cannot average over many integral lengths of the flow. Nevertheless, the fluid flow in the cup is turbulent each time you stir it and one must be able to describe the specific, individual flow. This need is common in many areas of science, such as geophysics and astrophysics, where some specific hurricane or some specific supernova must be understood. These remarks are not intended to deny the intrinsic stochasticity of turbulent flow. It is also true that when you stir your morning cup of coffee, each time, you will see a different pattern of the cream, no matter how carefully you try to repeat your actions each day. However, in many discussions of turbulence, the applications of probabilistic methods are entirely gratuitous. It is only by eliminating the superficial and unnecessary resorts to a statistical description that one can uncover where probability theory is truly essential.

Figure 4. Mixing of cream in a mug of coffee, moderately turbulent from mild stirring. Reproduced on license from Shutterstock.

3.1.2. Local deterministic $4/5$th law and dissipative anomaly

One of the landmark results of the statistical approach to turbulence is the celebrated 4/5th law derived by Kolmogorov (Reference Kolmogorov1941a), who invoked statistical hypotheses of local homogeneity and isotropy of the flow. In a remarkable work developing Onsager's ideas, Duchon & Robert (Reference Duchon and Robert2000) have shown that the 4/5th law holds not only deterministically for individual flow realisations, not requiring any statistical hypotheses, but also that it holds in a much stronger space–time local form. As we shall see, these developments were also foreshadowed by Onsager's own unpublished work. Here I shall sketch the essential ideas in the work of Duchon & Robert (Reference Duchon and Robert2000) and discuss their various ramifications. For a much more detailed explanation, see the online course notes of Eyink (Reference Eyink2007, § III.C).

The analysis of Duchon & Robert (Reference Duchon and Robert2000) again attempts to understand the zeroth-law of turbulence or the ‘inertial-dissipation’ of kinetic energy. The basic problem remains that UV divergences must appear in the inviscid limit, so that the fluid equations can no longer be understood as PDEs in the naïve sense. The starting point of Duchon & Robert (Reference Duchon and Robert2000) is a balance equation for a point-split kinetic energy density $\tfrac {1}{2}{\boldsymbol {u}}(\boldsymbol {x},t)\boldsymbol{\cdot} {\boldsymbol {u}}(\boldsymbol {x}+ \boldsymbol {r},t)$:

(3.24) $$\begin{gather} \partial_t \left(\tfrac{1}{2} {\boldsymbol{u}} \boldsymbol{\cdot} {\boldsymbol{u}}'\right) + \boldsymbol{\nabla} \boldsymbol{\cdot} \left[\left(\tfrac{1}{2}{\boldsymbol{u}} \boldsymbol{\cdot} {\boldsymbol{u}}'\right){\boldsymbol{u}} + \tfrac{1}{2}(\,p {\boldsymbol{u}}' + p' {\boldsymbol{u}}) + \tfrac{1}{4}|{\boldsymbol{u}}'|^2 \delta {\boldsymbol{u}} - \nu \boldsymbol{\nabla}\left(\tfrac{1}{2}{\boldsymbol{u}} \boldsymbol{\cdot}{\boldsymbol{u}}'\right)\right]\nonumber\\ = \tfrac{1}{4} \boldsymbol{\nabla}_{\boldsymbol{r}}\boldsymbol{\cdot} [\delta {\boldsymbol{u}} |\delta {\boldsymbol{u}}|^2] - \nu \boldsymbol{\nabla} {\boldsymbol{u}} : \boldsymbol{\nabla} {\boldsymbol{u}}' + \tfrac{1}{2}(\boldsymbol{f} \boldsymbol{\cdot}{\boldsymbol{u}}' + \boldsymbol{f'} \boldsymbol{\cdot} {\boldsymbol{u}}), \end{gather}$$

where I have introduced the abbreviated notation

(3.25)\begin{equation} \left. \begin{aligned} {\boldsymbol{u}} & = {\boldsymbol{u}}(\boldsymbol{x},t),\quad p = p(\boldsymbol{x},t),\quad {\boldsymbol f} = {\boldsymbol f}(\boldsymbol{x},t), \\ {\boldsymbol{u}}' & = {\boldsymbol{u}}(\boldsymbol{x}+ \boldsymbol{r},t), \quad p' = p(\boldsymbol{x}+ \boldsymbol{r},t),\quad {\boldsymbol f}' = {\boldsymbol f}(\boldsymbol{x}+ \boldsymbol{r},t),\\ \delta {\boldsymbol{u}} & = {\boldsymbol{u}}' - {\boldsymbol{u}} = {\boldsymbol{u}}(\boldsymbol{x}+ \boldsymbol{r},t) - {\boldsymbol{u}}(\boldsymbol{x},t), \end{aligned} \right\} \end{equation}

and I have permitted a body force ${\boldsymbol f}$ in the Navier–Stokes equation, for greater generality. Similar exact equations were introduced somewhat later by Hill (Reference Hill2001, Reference Hill2002) as deterministic versions of the Kármán–Howarth–Monin relations. However, the balance equation (3.24) does not yet eliminate UV divergences by point-splitting alone, which requires further multiplying through by a filter kernel $G_\ell (\boldsymbol {r})$ and integrating over $\boldsymbol {r}.$ This yields a corresponding balance equation for the regularised energy density $\tfrac {1}{2} {\boldsymbol {u}} \boldsymbol{\cdot} \bar {{\boldsymbol {u}}}_\ell$:

(3.26) \begin{align} &\partial_t \left(\frac{1}{2} {\boldsymbol{u}} \boldsymbol{\cdot} \bar{{\boldsymbol{u}}}_\ell\right) + \boldsymbol{\nabla} \boldsymbol{\cdot} \left[\left(\frac{1}{2}{\boldsymbol{u}} \boldsymbol{\cdot} \bar{{\boldsymbol{u}}}_\ell\right){\boldsymbol{u}} + \frac{1}{2}(\,p \bar{{\boldsymbol{u}}}_\ell + \bar{p}_\ell {\boldsymbol{u}}) + \frac{1}{4} \overline{(|{\boldsymbol{u}}|^2 {\boldsymbol{u}})_\ell}\right. \notag\\ &\qquad - \left. \frac{1}{4} \overline{(|{\boldsymbol{u}}|^2)_\ell} {\boldsymbol{u}} - \nu \boldsymbol{\nabla}\left(\frac{1}{2}{\boldsymbol{u}} \boldsymbol{\cdot} \bar{{\boldsymbol{u}}}_\ell\right)\right]\nonumber\\ &\quad =-\frac{1}{4 \ell} \int \,{\rm d}^d \boldsymbol{r} (\boldsymbol{\nabla} G)_\ell(\boldsymbol{r}) \boldsymbol{\cdot} \delta {\boldsymbol{u}}(\boldsymbol{r}) \,|\delta {\boldsymbol{u}}(\boldsymbol{r})|^2 - \nu \boldsymbol{\nabla} {\boldsymbol{u}} : \boldsymbol{\nabla} \bar{{\boldsymbol{u}}}_\ell + \frac{1}{2}(\,{\boldsymbol f} \boldsymbol{\cdot} \bar{{\boldsymbol{u}}}_\ell + \bar{{\boldsymbol f}}_\ell \boldsymbol{\cdot} {\boldsymbol{u}}). \end{align}

It is now easy to check that all terms remain bounded in the limit as $\nu \to 0.$

The next step of Duchon & Robert (Reference Duchon and Robert2000) was to study the convergence as $\nu \to 0$ of the various terms in the regularised energy balance (3.26), under the assumption of strong $L^3$-convergence of Navier–Stokes solutions ${\boldsymbol {u}}^\nu$ to some ${\boldsymbol {u}},$ i.e. $\lim _{\nu \to 0}\|{\boldsymbol {u}}^\nu -{\boldsymbol {u}}\|_3=0.$ Note that ${\boldsymbol {u}}$ is then necessarily a weak Euler solution. As with the prior assumption of strong $L^2$-convergence, this even stronger convergence assumption is not guaranteed a priori, but we shall see that it is implied by other empirically observed properties. Taking strong $L^3$-convergence for granted, it is easy to check that all of the terms proportional to $\nu$ in (3.26) in fact disappear in the limit. Furthermore, all of the remaining terms are found to converge in the sense of distributions to the same expression but with Navier–Stokes velocity ${\boldsymbol {u}}^\nu$ replaced with the limiting field ${\boldsymbol {u}}$, yielding a modified energy balance for the inviscid Euler solution:

(3.27)$$\begin{gather} \partial_t \left(\frac{1}{2} {\boldsymbol{u}} \boldsymbol{\cdot} \bar{{\boldsymbol{u}}}_\ell\right) + \boldsymbol{\nabla} \boldsymbol{\cdot} \left[\left(\frac{1}{2}{\boldsymbol{u}} \boldsymbol{\cdot} \bar{{\boldsymbol{u}}}_\ell\right){\boldsymbol{u}} + \frac{1}{2}(\,p \bar{{\boldsymbol{u}}}_\ell + \bar{p}_\ell {\boldsymbol{u}}) + \frac{1}{4} \overline{(|{\boldsymbol{u}}|^2 {\boldsymbol{u}})_\ell} - \frac{1}{4} \overline{(|{\boldsymbol{u}}|^2)_\ell} {\boldsymbol{u}} \right] \nonumber\\ =-\frac{1}{4 \ell} \int \,{\rm d}^d \boldsymbol{r} (\boldsymbol{\nabla} G)_\ell(\boldsymbol{r}) \boldsymbol{\cdot} \delta {\boldsymbol{u}}(\boldsymbol{r}) |\delta {\boldsymbol{u}}(\boldsymbol{r})|^2 + \frac{1}{2}(\,{\boldsymbol f} \boldsymbol{\cdot} \bar{{\boldsymbol{u}}}_\ell + \bar{{\boldsymbol f}}_\ell \boldsymbol{\cdot} {\boldsymbol{u}}). \end{gather}$$

Note, in particular, that validity in the sense of distributions means that the above balance equation is implicitly smeared with a smooth test function $\varphi (\boldsymbol {x},t)$ in space–time and that all derivatives acting on ${\boldsymbol {u}}$ or related solution fields can be moved over to the test function. From these remarks, it is then easy to check further that the limit as $\ell \to 0$ of all terms exist directly, except for the term containing $(\boldsymbol {\nabla } G)_\ell (\boldsymbol {r})$. However, because all other terms in the equation converge, this term must also converge. Duchon & Robert (Reference Duchon and Robert2000) thus concluded that a local kinetic energy balance holds in the sense of distributions for the limiting Euler solution

(3.28)\begin{equation} \partial_t \left(\tfrac{1}{2}|{\boldsymbol{u}}|^2\right) + \boldsymbol{\nabla} \boldsymbol{\cdot} \left[\left(\tfrac{1}{2}|{\boldsymbol{u}}|^2 + p\right){\boldsymbol{u}}\right] \equiv- D({\boldsymbol{u}}) +{\boldsymbol f}\boldsymbol{\cdot}{\boldsymbol{u}}, \end{equation}

where

(3.29)\begin{equation} D({\boldsymbol{u}}) = \lim_{\ell \rightarrow 0} \frac{1}{4 \ell} \int \,{\rm d}^d \boldsymbol{r} (\boldsymbol{\nabla} G)_\ell (\boldsymbol{r}) \boldsymbol{\cdot} \delta {\boldsymbol{u}}(\boldsymbol{r}) |\delta {\boldsymbol{u}}(\boldsymbol{r})|^2 := \lim_{\ell \rightarrow 0} D_\ell({\boldsymbol{u}}). \end{equation}

The naïve kinetic energy balance of the Euler equations is broken by the term $D({\boldsymbol {u}}),$ which can be easily shown to vanish unless the velocity has Besov regularity exponent $\sigma _p\leq 1/3$ for $p\geq 3.$ Note in fact that the same final equation (3.28) can be obtained also by considering the coarse-grained energy balance (3.10) with both factors of ${\boldsymbol {u}}$ filtered and then taking the limit $\ell \to 0$ in the same manner yields

(3.30)\begin{equation} D({\boldsymbol{u}}) = \lim_{\ell \rightarrow 0} \varPi_\ell \end{equation}

so that the ‘inertial dissipation’ in fact represents nonlinear energy cascade to infinitesimally small length scales.

Physically, this loss of energy must arise from viscous dissipation. Notice, however, (3.27)–(3.30) can be derived for any weak Euler solution, under the sole requirement that ${\boldsymbol {u}}\in L^3.$ Because the notion of weak Euler solution is time-reversible but $D({\boldsymbol {u}})$ is cubic in the velocity field, it follows that any Euler solution with $D({\boldsymbol {u}})>0$ yields by time-reversal another Euler solution with $D({\boldsymbol {u}})<0.$ It is only Euler solutions obtained in the inviscid limit, or so-called ‘viscosity solutions’ of Euler, which can be expected to satisfy the physical expectation of positive dissipation. Because coarse-graining is purely optional, one should be able to obtain the limiting energy balance (3.28) directly in the limit $\nu \to 0.$ This is what Duchon & Robert (Reference Duchon and Robert2000) did, starting with the familiar kinetic energy balance for Navier–Stokes solutions ${\boldsymbol {u}}^\nu$:

(3.31)\begin{align} \partial_t\left(\tfrac{1}{2} |{\boldsymbol{u}}^\nu|^2\right) + \boldsymbol{\nabla} \boldsymbol{\cdot} \left[\left(\tfrac{1}{2}|{\boldsymbol{u}}^\nu|^2 + p^\nu\right) {\boldsymbol{u}}^\nu - \nu \boldsymbol{\nabla}\left(\tfrac{1}{2} |{\boldsymbol{u}}^\nu|^2\right) \right]=- \nu |\boldsymbol{\nabla} {\boldsymbol{u}}^\nu|^2 +{\boldsymbol f}\boldsymbol{\cdot}{\boldsymbol{u}}^\nu. \end{align}

(Here I note in passing that Duchon & Robert (Reference Duchon and Robert2000) considered a more general situation where the Navier–Stokes solutions are themselves singular and must be interpreted distributionally à la Leray (Reference Leray1934), in which case, the ‘=’ in (3.31) is replaced by ‘$\leq$’. For reasons discussed later, I am not interested physically in this mathematical generality.) Now assuming once more the strong $L^3$ convergence ${\boldsymbol {u}}^\nu \to {\boldsymbol {u}},$ it is easy to see that all of the terms on the left-hand side converge distributionally as $\nu \to 0$ to the expected limit on the left-hand side of (3.28) and likewise ${\boldsymbol f}\boldsymbol{\cdot} {\boldsymbol {u}}^\nu \to {\boldsymbol f}\boldsymbol{\cdot}{\boldsymbol {u}}.$ The only term which cannot be shown directly to converge is the viscous dissipation $\nu |\boldsymbol {\nabla } {\boldsymbol {u}}^\nu |^2,$ but, since all other terms in the Navier–Stokes balance (3.31) converge, so also does this term. One thus derives the same local energy balance (3.28) as before for the limiting Euler solution, but one obtains furthermore the new expression

(3.32)\begin{equation} D({\boldsymbol{u}}) = \lim_{\nu \rightarrow 0} \nu |\boldsymbol{\nabla} {\boldsymbol{u}}^\nu|^2 \geq 0. \end{equation}

This result can be understood as a matching relation for the ‘viscosity solutions’ of Euler equations, which equates the energy carried by nonlinear cascade to infinitesimally small scales and the usual energy dissipation by molecular viscosity.

A final important result obtained by Duchon & Robert (Reference Duchon and Robert2000) exploited additional freedom in the regularisation of UV divergences. In fact, not only is the length scale $\ell$ arbitrary, but so also is the precise choice of the kernel $G.$ This freedom can be exploited by choosing a rotationally symmetric kernel for which

(3.33)\begin{equation} \boldsymbol{\nabla} G(\boldsymbol{r}) = \hat{\boldsymbol{r}} G'(r). \end{equation}

In that case, the $d$-dimensional integral over $\boldsymbol {r}$ in (3.29) for $D({\boldsymbol {u}})$ can be transformed into hyperspherical coordinates so that

(3.34)\begin{equation} D_\ell({\boldsymbol{u}}) = \frac{1}{4 \ell} \int^\infty_0 r^{d-1} \,{\rm d}r \int_{S^{d-1}} \,{\rm d} \omega_d (\hat{\boldsymbol{r}}) \hat{\boldsymbol{r}} \boldsymbol{\cdot} \delta {\boldsymbol{u}}(\boldsymbol{r}) |\delta {\boldsymbol{u}}(\boldsymbol{r})|^2 (G')_\ell (r), \end{equation}

where $S^{d-1}$ is the unit hypersphere and $d \omega _d$ is the measure on in $d$-dimensional solid angles. Using the integration by parts identity $\int _0^\infty r^d G'(r)\,\textrm {d}r=d,$ one can infer a new identity

(3.35)\begin{equation} \lim_{r \rightarrow 0} \frac{\langle \delta u_L(\boldsymbol{r}) |\delta {\boldsymbol{u}}(\boldsymbol{r})|^2\rangle_{ang}}{r} =- \frac{4}{d} D({\boldsymbol{u}}), \end{equation}

where $\langle \boldsymbol{\cdot}\rangle _{ang}$ denotes for any function of $\boldsymbol {r}$ the average with respect to $\omega _d$ over the direction $\hat {\boldsymbol {r}}$, and $\delta u_L(\boldsymbol {r})=\hat {\boldsymbol {r}}\boldsymbol{\cdot}\delta {\boldsymbol {u}}(\boldsymbol {r})$ is the longitudinal velocity increment. Combined with the previous expression (3.32) for $D({\boldsymbol {u}})$ in terms of the viscous dissipation, one can see that (3.35) for $d=3$ has the form of the Kolmogorov–Yaglom 4/3rd law (Antonia et al. Reference Antonia, Ould-Rouis, Anselmet and Zhu1997). It is well known in the standard derivation by ensemble averages that the Kolmogorov 4/5th and 4/15th laws are equivalent to the 4/3rd law under the assumption of statistical isotropy. Since the angle average in the present derivations provides isotropy without any statistical hypotheses, some further manipulation (Eyink Reference Eyink2003; Novack Reference Novack2023) yields

(3.36)\begin{gather} \lim_{r \rightarrow 0} \frac{\langle\delta u^3_L(\boldsymbol{r})\rangle_{ang}}{r} =-\frac{12}{d(d+2)} D({\boldsymbol{u}}), \end{gather}

(3.37)\begin{gather} \lim_{r \rightarrow 0} \frac{\langle\delta u_L(\boldsymbol{r}) \delta u^2_T(\boldsymbol{r})\rangle_{ang}}{r} =-\frac{4}{d(d+2)} D({\boldsymbol{u}}), \end{gather}

where $\delta {\boldsymbol {u}}_T(\boldsymbol {r})=\delta {\boldsymbol {u}}(\boldsymbol {r})-\hat {\boldsymbol {r}}\delta u_L(\boldsymbol {r})$ is the full transverse velocity increment and $\delta u_T(\boldsymbol {r})$ is the magnitude of any particular (fixed) component. One can see for $d=3$ that (3.36) has the form of the usual 4/5th law and (3.37) has the form of the 4/15th law.

Note, however, that the relations (3.35)–(3.37) are deterministic, holding for individual flow realisations, and are furthermore space–time local in the sense of distributions. The latter statement simply means that they hold with both sides smeared by an arbitrary space–time test function $\varphi (\boldsymbol {x},t),$ taking first $\nu \to 0$ and then $r\to 0.$ Thus, these results represent a considerable strengthening of the original ensemble-average relations of Kolmogorov (Reference Kolmogorov1941a). An effort was made by Taylor, Kurien & Eyink (Reference Taylor, Kurien and Eyink2003) to test these deterministic local relations in pseudospectral numerical simulations, but the $512^3$ resolution available at the time was insufficient. It was possible to show that angle-averaging greatly improved the validity of the observed $4/5$th law, attaining results comparable to those at $1024^3$ resolution even without time-averaging. Thus, the local 4/5th law appeared in fact to hold instantaneously, without smearing in time, beyond what could be proved mathematically. Note that it is only with simulations of $16\,384^3$ resolution that the 4/5th law has recently been demonstrated convincingly, exploiting both angle-averaging and time-averaging (Iyer, Sreenivasan & Yeung Reference Iyer, Sreenivasan and Yeung2020). This brings us close to being able to check the local $4/5$th law in simulations of $32\,768^3$ resolution by dividing the computational cube into eight $16\,384^3$ subcubes, and verifying that the relation holds with the structure functions and mean dissipation calculated by space-averages individually over each subcube. Of course, the result should hold with any space–time test function $\varphi$ held fixed in the limit first $\nu \to 0$ and then $r\to 0,$ but current computational limitations seem to make characteristic functions of octants the only choice consistent with the stringent condition that the local Reynolds number must be high. It is worth remarking that the analogue of the deterministic, local 4/5th law can be rigorously derived for the ‘viscosity solutions’ of the inviscid Burgers equation, $\partial _t u +\partial _x (\tfrac {1}{2}u^2)=0.$ In that case, the local energy balance holds

(3.38)\begin{equation} \partial_t \left(\tfrac{1}{2}u^2\right) + \partial_x \left(\tfrac{1}{3} u^3\right) =- D(u) \end{equation}

with the ‘inertial dissipation’ given by

(3.39)\begin{equation} \lim_{r \rightarrow 0} \frac{\langle\delta u^3_L(r)\rangle_{ang}}{|r|} =-12 D(u), \end{equation}

where $\delta u_L(r):=\textrm {sign}(r) \delta u(r)$ and $\langle \delta u^3_L(r)\rangle _{ang} = [\delta u^3(+ |r|) - \delta u^3(-|r|)]/2$. See Eyink (Reference Eyink2007, § III) and Novack (Reference Novack2023). These relations correspond to the ensemble-averaged ‘$12$th law’ for Burgulence, but valid in space–time local form with ‘inertial dissipation’ arising entirely from shock singularities.

It should be emphasised again that the deterministic versions of the 4/5th law (3.36) and the 4/15th law (3.37), and all of the earlier results in this section, require no statistical assumptions whatsoever. Neither local homogeneity, local stationarity, isotropy nor any other hypothesis about the flow statistics was invoked in their derivation. In particular, these relations will apply in the ‘non-equilibrium decay regime’ highlighted in the review of Vassilicos (Reference Vassilicos2015) on turbulent energy dissipation. Vassilicos (Reference Vassilicos2015) reported observations in a near wake region of various grids and bluff bodies that the time-averaged quantity $C_\epsilon := \langle D\rangle =\langle \varepsilon \rangle /(u^{\prime 3}/L)$ scales as $Re_I^m/Re_L^n$, where in his notation, $Re_I=UL_b/\nu$ is the ‘global’ or ‘inlet Reynolds number’ based on the inlet flow speed $U$ and a length $L_b$ giving the overall size of the grid or body, whereas $Re_L=u'L/\nu$ is the ‘local Reynolds number’. In the wake flows considered by Vassilicos (Reference Vassilicos2015), both the r.m.s. velocity fluctuation $u'$ and integral length $L$ are statistical quantities that depend upon the longitudinal distance $x$ of the observation point from the grid/body. Since wake flows involve interactions with solid walls, I shall treat the energy cascade in such flows in § 4.2.4, but here I note that all of our previous results carry over, with possible modifications only directly at the solid surface. As I shall discuss in § 4.1, for flows with walls or solid bodies, I always consider the global Reynolds number $Re_I,$ which appears in the non-dimensionalisation of the Navier–Stokes equation. As noted already by Vassilicos (Reference Vassilicos2015, p. 108), the quantity $C_\epsilon$ at each $x$-location is ‘more or less independent of the Reynolds number’ $Re_I$, even in the non-equilibrium decay region because ‘$m\approx 1\approx n$, and $Re_L$ increases linearly with $Re_I$’. In fact, $m=n$ corresponds exactly to the ‘dissipative anomaly’ conjectured by Onsager (Reference Onsager1949) and all of our previous conclusions follow in the limit $Re_I\to \infty.$ In particular, the local 4/5th law (3.36) and the 4/15th law (3.37) will hold in the non-equilibrium decay region, not only for time-/ensemble-averages but even for individual flow realisations. Vassilicos (Reference Vassilicos2015) has observed that ‘the Richardson–Kolmogorov cascade does not seem to be the $\ldots$interscale energy transfer mechanism at work’, but the dissipative anomaly for $Re_I\gg 1$ must occur by Onsager's local cascade mechanism, which generalises Kolmogorov's equilibrium picture. I agree with the observation of Vassilicos (Reference Vassilicos2015) that the 4/5th law of Kolmogorov (Reference Kolmogorov1941a) need not be valid in general for $r\simeq L.$ In fact, our local, deterministic version (3.36) will generally only hold for $r\ll L_\varphi,$ where $L_\varphi$ measures the spatial diameter of the support of the test function $\varphi$ required to make (3.36) meaningful.

Amazingly enough, most of the previous developments were foreseen by Onsager in his unpublished work in the 1940s. When I was on sabbatical at Yale in 2000, I was able to examine Onsager's private research notes available there on microfiche and I was astonished to discover that Onsager's own proof of his 1/3 Hölder claim was essentially identical to that given by Duchon & Robert (Reference Duchon and Robert2000). These notes are now all available online through the Onsager Archive hosted by the NTNU in Trondheim and the reader can peruse them at https://ntnu.tind.io/record/121183. The essential calculations are on pp. 14–19 of Onsager's notes in this folder, where he derives the identity (3.27) but in a space integrated form and with the filtering function $G_\ell (\boldsymbol {r})$ denoted $F(\boldsymbol {r}).$ It is interesting that on p. 15, Onsager begins the calculation by writing an overbar, his notation for ensemble-average, but then scratches out the overbar in the second line, apparently realising that volume-averaging was sufficient to derive the result. This seems to have been the moment that the theme of my essay was born. Onsager communicated his identity to von Kármán and Lin in his letter of June 1945 (reproduced as appendix B in Eyink & Sreenivasan Reference Eyink and Sreenivasan2006) where he gave an argument for $1/3$-scaling by taking $F(\boldsymbol {r})$ to be a spherical tophat filter of radius $a$ and then letting $a\to 0.$ It is further interesting that Onsager in this same letter also foresaw that inertial-range intermittency could lead to an energy spectrum having a steeper slope than $-5/3,$ writing that, ‘As far as I can make out, a more rapid decrease of $\overline {a_k^2}$ with increasing $\underline {k}$ would require a “spotty” distribution of the regions in which the velocity varies rapidly between neighboring points.’ Onsager then notes that all velocity increments supported in discontinuities at vortex sheets would produce $S_2(r)\propto r,$ corresponding to a $k^{-2}$ energy spectrum as for Burgers equation. Such sheets would also be consistent with $S_3(r)\propto r$, but Onsager remarks that they would not agree well with experimental observations on velocity traces. It is worth remarking that vortex sheets cannot, in fact, be the origin of anomalous energy dissipation (De Rosa & Inversi Reference De Rosa and Inversi2024). For more discussion of Onsager's insights on inertial-range intermittency, see Eyink & Sreenivasan (Reference Eyink and Sreenivasan2006, § IV.C).

The local kinetic energy balance (3.28) derived by Duchon & Robert (Reference Duchon and Robert2000) for weak Euler solutions is the most succinct formulation of Onsager's proposal for turbulent dissipation ‘without the final assistance by viscosity’. This result has a very familiar appearance to a modern field-theorist, because it is reminiscent of anomalies in classical conservation laws due to UV divergences in the quantum field theory. This analogy seems to have been first pointed out by Polyakov (Reference Polyakov1992, Reference Polyakov1993), who did not know about Onsager's work, of course, but who based his discussion on the statistical theory of Kolmogorov. Polyakov pointed out the essential similarity between turbulent cascades and chiral anomalies in quantum Yang–Mills theory, as both involve a constant flux through wavenumbers which vitiates a naïve conservation law. He pointed out also that the first derivation of the chiral anomaly in quantum electrodynamics by Schwinger (Reference Schwinger1951) using a point-splitting regularisation was very similar to the derivation of the 4/5th law by Kolmogorov (Reference Kolmogorov1941a). It is interesting that Schwinger seemingly did not fully appreciate the significance of his own calculation and did not point out its implication that conservation of axial charge is violated (Adler Reference Adler2005). However, we now see that Onsager had used a similar point-splitting regularisation already in the 1940s and had fully appreciated its consequence that naïve conservation of kinetic energy in the inviscid limit is broken by the ‘violet catastrophes’ in turbulent flows. This phenomenon is thus now commonly called a turbulent dissipative anomaly, because of the close connection with quantum field theory anomalies. For more detailed discussion of this analogy, see Eyink & Sreenivasan (Reference Eyink and Sreenivasan2006, § IV.B).

I would like to make just a brief remark why the term ‘anomaly’ is apt for the turbulent phenomenon, independent of the connection with quantum field theory. It should be emphasised that no fundamental microscopic conservation laws for total energy, total linear momentum, total angular momentum, etc. are ever violated by such anomalies. However, there is obviously no microscopic law of ‘conservation of kinetic energy’! This conservation law is an emergent property of inviscid hydrodynamics due to its Hamiltonian character, in which the kinetic energy of the fluid is itself the conserved Hamiltonian (Salmon Reference Salmon1988; Morrison, Francoise & Naber Reference Morrison, Francoise and Naber2006). It is because of this Hamiltonian structure of the Euler equations that kinetic energy conservation is formally expected and thus the breakdown of this conservation is ‘anomalous’. The same is true more generally for turbulent dissipative anomalies. For example, conservation of helicity and conservation of circulation on arbitrary loops are not sacrosanct microscopic laws but are instead emergent laws connected with ‘relabelling symmetry’, an infinite-dimensional symmetry group of the action for three-dimensional ideal Euler equations. It is such emergent conservation laws of the ideal fluid which may be afflicted with dissipative anomalies in turbulent flows. It is interesting that some of these symmetries are preserved in a formulation of the viscous Navier–Stokes via a stochastic least-action principle (Eyink Reference Eyink2010), suggesting that not all of these emergent conservation laws will be explicitly broken by dissipative anomalies but may instead be realised in an unconventional stochastic form. See further discussion in § 3.1.3.2.

Finally, I point out that the work of Duchon & Robert (Reference Duchon and Robert2000) makes a connection between the ‘ideal turbulence’ theory and the multifractal model of energy dissipation. After Kolmogorov (Reference Kolmogorov1962) had proposed his lognormal model of turbulent energy dissipation to account for intermittency corrections, Mandelbrot (Reference Mandelbrot1974, Reference Mandelbrot1989) introduced a more general description of the energy dissipation as a multifractal measure with a spectrum of singularities. Subsequent experimental studies of Meneveau & Sreenivasan (Reference Meneveau and Sreenivasan1991) supported these predicted scaling properties of turbulent energy dissipation. The result (3.32) of Duchon & Robert (Reference Duchon and Robert2000) shows that the dissipative anomaly term $D({\boldsymbol {u}})$ in the inviscid energy balance coincides exactly with this multifractal measure. In fact, the analysis of Duchon & Robert (Reference Duchon and Robert2000) implies that the inviscid limit of the viscous energy dissipation $\varepsilon ^\nu =2\nu |\boldsymbol {S}^\nu |^2$ exists and coincides with $D({\boldsymbol {u}}),$ which is a non-negative distribution and, thus, a measure. Recent rigorous results relate the fractal dimension of the support of this measure to the inertial-range intermittency of the velocity increments (De Rosa & Isett Reference De Rosa and Isett2024).

3.1.3. Implications and open questions

There are many questions raised by the original analysis of Onsager (Reference Onsager1945a, Reference Onsager1949) and also many subsequent developments extending and elaborating his ideas. Here I shall briefly review and discuss such further implications and open issues.

3.1.3.2 Lagrangian spontaneous stochasticity

One of the most remarkable consequences of Onsager's Hölder singularity result was first discovered in the work of Bernard, Gawȩdzki & Kupiainen (Reference Bernard, Gawȩdzki and Kupiainen1998), who studied Lagrangian fluid particle trajectories that solve the initial-value problem:

(3.40)\begin{equation} \dot{\boldsymbol{x}} ={\boldsymbol{u}}(\boldsymbol{x},t), \quad \boldsymbol{x}(0)=\boldsymbol{x}_0. \end{equation}

When ${\boldsymbol {u}}$ is the limiting Euler solution at infinite-$Re$ then, as recalled by Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998), the maximal Hölder regularity derived by Onsager (Reference Onsager1949) is insufficient to guarantee a unique solution of (3.40) and there is generally a continuous infinity of solutions with exactly the same initial data. It was realised by Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998) that this non-uniqueness permits stochasticity to persist in the high-Reynolds-number limit. For example, the position of a Brownian particle such as a small colloid or a dye molecule advected by a turbulent flow will satisfy instead a stochastic ODE:

(3.41)\begin{equation} {\rm d}\tilde{\boldsymbol{x}} ={\boldsymbol{u}}(\tilde{\boldsymbol{x}},t)\,{\rm d}t+\sqrt{2D}\, {\rm d}\tilde{\boldsymbol{W}}(t) , \quad \boldsymbol{x}(0)=\boldsymbol{x}_0, \end{equation}

where $D$ is the molecular diffusivity of the particle and $\tilde {{\boldsymbol W}}(t)$ is a Wiener process modelling the Brownian motion. To study the high-$Re$ limit, one may again non-dimensionalise by defining $\hat {{\boldsymbol {u}}}={\boldsymbol {u}}/u',$ $\hat {\boldsymbol {x}}=\boldsymbol {x}/L,$ $\hat {t}=t/(L/u')$ and then $D$ is replaced by $1/Pe,$ where $Pe=u'L/D$ is the Péclet number. In the joint limit $Re\gg 1,$ $Pe\gg 1,$ the Lagrangian particle trajectories $\tilde {\boldsymbol {x}}(t)$ solve the limiting deterministic ODE (3.40), but Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998) showed that they may remain random in the limit with a non-trivial transition probability density $p_{\boldsymbol {u}}(\boldsymbol {x},t|\boldsymbol {x}_0,0)$ to arrive at position $\boldsymbol {x}$ at time $t$ (see figure 5). The essential physics was implicit already in the work of Richardson (Reference Richardson1926) on two-particle turbulent dispersion, according to which initial particle separations are ‘forgotten’ at sufficiently long times. It is important to note however that there is no averaging over velocities in the prediction of Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998) and that the transition probability represents stochastic particle evolution in a fixed realisation ${\boldsymbol {u}}$ of the turbulent velocity field. Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998) furthermore showed that this ‘spontaneous stochasticity’ of Lagrangian particles provides a mechanism for anomalous dissipation of a concentration field $c$ of such tracers, which satisfies the advection–diffusion equation

(3.42)\begin{equation} \partial_t c(\boldsymbol{x},t)+{\boldsymbol{u}}(\boldsymbol{x},t)\boldsymbol{\cdot}\boldsymbol{\nabla} c(\boldsymbol{x},t)= D\Delta c(\boldsymbol{x},t), \quad c(\boldsymbol{x},0)=c_0(\boldsymbol{x}). \end{equation}

Since the solution of this equation (in non-dimensionalised form) has the exact Feynman–Kac representation

(3.43)\begin{equation} c(\boldsymbol{x},t) = \int p_{\boldsymbol{u}}^{Re,Pe}(\boldsymbol{x},t|\boldsymbol{x}_0,0) c_0(\boldsymbol{x}_0) \, {\rm d}^dx_0, \end{equation}

the existence of a non-trivial limit $\lim _{Re,Pe\to \infty } p_{\boldsymbol {u}}^{Re,Pe}(\boldsymbol {x},t|\boldsymbol {x}_0,0) =p_{\boldsymbol {u}}(\boldsymbol {x},t|\boldsymbol {x}_0,0)$ implies by convexity that $\int \tfrac {1}{2} c^2(\boldsymbol {x},t)\, \textrm {d}^d x < \int \tfrac {1}{2} c^2_0(\boldsymbol {x}_0)\, \textrm {d}^dx_0$ in the limit $Re,Pe\to \infty.$

Figure 5. Sketch of the solutions of a deterministic ODE $\textrm {d}\kern 0.06em \boldsymbol {x}/\textrm {d}t={\boldsymbol {u}}(\boldsymbol {x},t)$ for deterministic initial data $\boldsymbol {x}_0$ but with singular velocity ${\boldsymbol {u}}$. Unlike traditional unique solutions, the trajectories spread randomly, like a plume of smoke. Reproduced with permission from Falkovich, Gawȩdzki & Vergassola (Reference Falkovich, Gawȩdzki and Vergassola2001). Copyright (2001) by the American Physical Society.

The original work of Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998) was carried out for the synthetic turbulence model of Kraichnan (Reference Kraichnan1968), who considered advection by Gaussian random velocity fields that are delta-correlated in time. Note that for suitable power-law spatial correlations to mimic inertial-range turbulent velocity fields, the realisations of the Kraichnan velocity ensemble are Hölder continuous with probability one. A very important feature of this model is that the limiting probability distributions

(3.44)\begin{equation} p_{\boldsymbol{u}}(\boldsymbol{x},t|\boldsymbol{x}_0,0)=\lim_{Re,Pe\to\infty} p_{\boldsymbol{u}}^{Re,Pe}(\boldsymbol{x},t|\boldsymbol{x}_0,0) \end{equation}

are independent of the particular subsequences $Re_k,$ $Pe_k\to \infty$ and, in fact, are independent of the particular form of regularisation and noise, e.g. the Brownian motion $\tilde {\boldsymbol {W}}(t)$ in (3.41) might be replaced with a fractional Brownian motion and the limit would be the same. This strong result for the Kraichnan model follows from a rigorous study by Le Jan & Raimond (Reference Le Jan and Raimond2002) who gave a direct construction of a Markov transition probability $p_{\boldsymbol {u}}(\boldsymbol {x},t|\boldsymbol {x}_0,0)$ for the zero-regularisation, zero-noise problem by a Weiner chaos expansion in the white-noise advecting velocity field ${\boldsymbol {u}}(\boldsymbol {x},t).$ It is a consequence of this construction that any reasonable spatial regularisation and noisy perturbation will converge to this same transition probability in the zero-regularisation, zero-noise limit. It was also shown by Le Jan & Raimond (Reference Le Jan and Raimond2002) that the realisations $\tilde {\boldsymbol {x}}(t)$ of the limiting Markov process with transition densities $p_{\boldsymbol {u}}(\boldsymbol {x},t|\boldsymbol {x}_0,0)$ are solutions (in a generalised sense) of the deterministic initial-value problem (3.40) for each fixed realisation ${\boldsymbol {u}}$ of the Kraichnan velocity ensemble. Thus, in this case, the Markov process $\tilde {\boldsymbol {x}}(t)$ should be regarded as the proper solution of the above initial-value problem (3.40), which is then well-posed in the sense of Hadamard. Spontaneous stochasticity in the strong sense should be understood to include such universality of the limiting random process, that is, the robustness of the stochastic solution to different regularisations and noisy perturbations.

Extensions of the work of Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998) and further developments on the Kraichnan model are described by Falkovich et al. (Reference Falkovich, Gawȩdzki and Vergassola2001). This comprehensive review makes clear that many of the results have validity extending beyond the Kraichnan model and, in fact, Drivas & Eyink (Reference Drivas and Eyink2017a) showed that spontaneous stochasticity is the only possible mechanism of scalar anomalous dissipation away from solid walls, for both passive and active scalars advected by a general incompressible velocity field. Note that universality of the limiting spontaneous statistics is not required in this case, so that different limiting random processes yielding anomalous dissipation may result from different subsequences and/or different regularisations. There are, however, only a handful of model velocity fields for which Lagrangian spontaneous stochasticity can be demonstrated from first principles (Drivas & Mailybaev Reference Drivas and Mailybaev2021; Drivas, Mailybaev & Raibekas Reference Drivas, Mailybaev and Raibekas2023). Renormalisation group methods from statistical physics apply to velocity fields with suitable scaling properties and yield universality of the limit (Eyink & Bandak Reference Eyink and Bandak2020), but the underlying ergodic properties of the dynamical flows that are required for this method are difficult to establish in general. One interesting example of a velocity field solving a PDE is the solution of the inviscid Burgers equation, where Lagrangian spontaneous stochasticity holds backwards in time at shock locations and explains anomalous energy dissipation (Eyink & Drivas Reference Eyink and Drivas2015). There is no such spontaneous stochasticity for particles advected forwards in time by the Burgers velocity field and it has been speculated that similar time asymmetry may be a more general feature of Lagrangian spontaneous stochasticity in dissipative weak solutions of PDEs obtained as inviscid limits.

As far as Navier–Stokes turbulence is concerned, there is so far no compelling evidence of Lagrangian spontaneous stochasticity from laboratory experiments (Bourgoin et al. Reference Bourgoin, Ouellette, Xu, Berg and Bodenschatz2006; Tan & Ni Reference Tan and Ni2022). However, numerical simulations of homogeneous, isotropic turbulence in a periodic domain provide reasonable evidence for the ‘forgetting’ of initial separations of deterministic Lagrangian particles (Bitane, Homann & Bec Reference Bitane, Homann and Bec2013; Buaria, Sawford & Yeung Reference Buaria, Sawford and Yeung2015) and ‘forgetting’ of both initial separations and molecular diffusivity of stochastic Lagrangian trajectories (Eyink Reference Eyink2011; Buaria, Yeung & Sawford Reference Buaria, Yeung and Sawford2016). Setting aside the important issue of empirical evidence, another crucial question is what implications Lagrangian spontaneous stochasticity might have for incompressible Navier–Stokes turbulence. A Feynman–Kac representation of Navier–Stokes solutions was derived by Constantin & Iyer (Reference Constantin and Iyer2008) which is similar to (3.43) for the advection–diffusion equation but more non-trivial, because of the nonlinearity of the equations. It was pointed out by Eyink (Reference Eyink2010) that the Constantin–Iyer representation corresponds to a stochastic principle of least-action, thus making connection with Hamiltonian fluid mechanics. In particular, the stochastic action functional for incompressible Navier–Stokes is invariant under the infinite-dimensional symmetry group of particle-relabelling just as is the action for deterministic Euler. The remarkable properties of vorticity under Euler dynamics that follow from that symmetry therefore carry over to Navier–Stokes in a stochastic fashion. For example, the Kelvin theorem of conservation of circulation on an arbitrary Lagrangian loop for Euler holds also for Navier–Stokes in the sense that the circulation on stochastically advected loops is a backwards-in-time martingale (Constantin & Iyer Reference Constantin and Iyer2008; Eyink Reference Eyink2010). This property prescribes the ‘arrow-of-time’ of the irreversible Navier–Stokes dynamics and it is plausible that a similar property carries over to dissipative weak Euler solutions obtained in the inviscid limit (Eyink Reference Eyink2006). The dissipative anomaly for Navier–Stokes turbulence may be characterised also in terms of the time asymmetry of separation of stochastic Lagrangian particle (Drivas Reference Drivas2019; Cheminet et al. Reference Cheminet2022), which in three dimensions separate faster backwards in time than they do forwards in time (Eyink Reference Eyink2011; Buaria et al. Reference Buaria, Yeung and Sawford2016).

3.1.3.3 Finite Reynolds numbers

A frequently made criticism of Onsager's ‘ideal turbulence’ theory is that it applies only to the unrealistic limit $Re=\infty$ and is thus inapplicable to real-world turbulence which is, of course, always at a finite Reynolds number. As an example, I may quote from the monograph of Tsinober (Reference Tsinober2009, § 10.3.2): ‘$\ldots$it is not clear why results for finite $Re$ (i.e. for NSE having no singularities or extremely “intermittent” ones) are relevant for the limit (if such exists) $Re\to \infty$ (e.g. for Euler equation with space-filling singularities)’. In some respects, such criticisms are a simple misunderstanding of the elementary concept of limit. Indeed, the predictions of a theory for $Re\to \infty$ are experimentally falsifiable, as they must be valid to arbitrary accuracy by taking Reynolds numbers larger (but finite). There is, however, a legitimate question regarding any such predictions of an ‘ultimate regime’ concerning how large $Re$ must be chosen to observe the predictions of the theory. This is especially important for Onsager's theory since, as has often been observed, the strict requirement for its validity is that the number of cascade steps must be large, that is, $\log _2 Re\gg 1$ and this condition is hard to satisfy in even the highest Reynolds number terrestrial turbulent flows. There are two responses to this very important question.

First, there are explicit error estimates in the mathematics of the Onsager theory which provide bounds on the correction terms at large but finite $Re.$ For example, in the coarse-grained energy balance equation (3.10), there is at finite Reynolds number a viscous dissipation term $\nu |\boldsymbol {\nabla }\bar {{\boldsymbol {u}}}_\ell |^2$ in addition to the inertial flux term $\varPi _\ell.$ It is not difficult to derive bounds of the form

(3.45)\begin{equation} \nu |\boldsymbol{\nabla}\bar{{\boldsymbol{u}}}_\ell|^2 = O(\nu \delta u^2(\ell)/\ell^2) \end{equation}

locally in space–time with $\delta u(\ell )=\sup _{|\boldsymbol {r}|<\ell } |\delta {\boldsymbol {u}}(\boldsymbol {r})|,$ or similar bounds in terms of Besov norms for space-integrated dissipation. These estimates provide concrete upper estimates on the finite-$Re$ corrections, which can explain departures from the $Re=\infty$ theory.

Second, Onsager's RG-type arguments can also be applied directly to the Navier–Stokes equations at large but finite $Re$ and do not require any assumptions about existence of weak Euler solutions in the limit $Re\to \infty.$ For example, Drivas & Eyink (Reference Drivas and Eyink2019) study the balance of the subscale kinetic energy $k_\ell :=(1/2)\textrm {Tr}\,\boldsymbol {\tau }_\ell$ for the forced Navier–Stokes equation, which takes the form

(3.46)\begin{equation} \partial_t k_\ell + \boldsymbol{\nabla}\boldsymbol{\cdot} {\boldsymbol J}_\ell =-\overline{(\nu|\boldsymbol{\nabla}{\boldsymbol{u}}|^2)}_\ell +\nu |\boldsymbol{\nabla}\bar{{\boldsymbol{u}}}_\ell|^2 +\varPi_\ell + \tau_\ell(\,{\boldsymbol f};{\boldsymbol{u}}), \end{equation}

where ${\boldsymbol J}_\ell$ is a suitable spatial flux of the subscale kinetic energy and $\tau _\ell (\,{\boldsymbol f};{\boldsymbol {u}}):=\overline {({\boldsymbol {u}}\boldsymbol{\cdot}{\boldsymbol f})}_\ell -\bar {{\boldsymbol {u}}}_\ell \boldsymbol{\cdot}\bar {\boldsymbol f}_\ell$ is the direct power input by the force into unresolved scales. If one assumes suitable Besov regularity of the Navier–Stokes solutions, ${\boldsymbol {u}}^\nu \in B^{\sigma,\infty }_p,$ $p\geq 3,$ uniform in the Reynold number, then one may derive bounds of the form

(3.47)\begin{equation} D=(1/Re)|\hat{\boldsymbol{\nabla}}\hat{{\boldsymbol{u}}}|^2=O(Re^{({1-3\sigma})/({1+\sigma})}). \end{equation}

These bounds follow by the same basic principle of independence of the physics on the arbitrary coarse-graining scale $\ell,$ which allows one to choose $\ell /L\propto Re^{-1/(1+\sigma )}$ to optimise the estimates. Following the same ideas of Onsager's argument, one can then deduce the existence of ‘quasi-singularities’ in the Navier–Stokes solutions from the observation of energy dissipation vanishing slowly with $Re$:

(3.48)\begin{equation} D=(1/Re)|\hat{\boldsymbol{\nabla}}\hat{{\boldsymbol{u}}}|^2\propto Re^{-\alpha}, \end{equation}

which will require $\sigma \leq \sigma _{\alpha }:=({1+\alpha })/({3-\alpha }).$ Note that when $\alpha =0,$ one recovers Onsager's critical value $\sigma =\tfrac {1}{3},$ but when $\alpha >0$ instead, $\sigma _\alpha >\tfrac {1}{3}.$ This is a significant strengthening of the Onsager's original result, because empirical observations alone could never allow one to distinguish $\alpha =0$ from a very tiny value of $\alpha.$ Nevertheless, Onsager's conclusions about ‘quasi-singularities’ are robust, remaining valid even under the weaker condition (3.48).

The condition (3.48) on the dimensionless dissipation rate for $\alpha >0$ has been evocatively termed a ‘weak dissipative anomaly’ by Bedrossian et al. (Reference Bedrossian, Coti Zelati, Punshon-Smith and Weber2019). Note that whenever $\alpha <1$ in (3.48), then $|\hat {\boldsymbol {\nabla }}\hat {{\boldsymbol {u}}}|^2\propto Re^{1-\alpha }\to \infty$ as $Re\to \infty$ and thus classical solutions of the PDEs can no longer exist in the limit of infinite $Re$. Although no assumption about the existence of weak solutions is necessary to derive the bound (3.47), such weak Euler solutions nevertheless emerge under the natural assumption of some Besov regularity with $\sigma <\sigma _\alpha$ that is uniform in $Re$ (Drivas & Eyink Reference Drivas and Eyink2019). Note furthermore that Bedrossian et al. (Reference Bedrossian, Coti Zelati, Punshon-Smith and Weber2019) have proved that the Kolmorogov 4/5th law remains valid under the assumption of a weak dissipative anomaly only, unlike the original derivation of Kolmogorov (Reference Kolmogorov1941a) which assumed a strong dissipative anomaly ($\alpha =0$). It is easy to check that the Taylor microscale defined by $\lambda ^2=15\nu u^{\prime 2}/\varepsilon$ satisfies $\lim _{Re\to \infty } \lambda /L=0$ precisely when $\alpha <1$ in (3.48) and then it is proved in Bedrossian et al. (Reference Bedrossian, Coti Zelati, Punshon-Smith and Weber2019) that the maximum difference between ${\langle \delta u_L^3(r)\rangle }/{r}$ and $-\tfrac {4}{5}\varepsilon ^\nu$ becomes vanishingly small over an increasing range of length scales $\lambda \ll r \ll L.$ This important result shows that validity of the 4/5th law cannot be taken as evidence for a strong anomaly. The proof by Bedrossian et al. (Reference Bedrossian, Coti Zelati, Punshon-Smith and Weber2019) considers the statistical steady-state of turbulence in a periodic domain driven by a body force which is a Gaussian random field, delta-correlated in time and takes advantage of simplifications of that stochastic forcing. More recently, analogous results have been derived by Novack (Reference Novack2023) for incompressible Navier–Stokes equations with a deterministic forcing and are valid for individual flow realisations.

These results have gained some added significance by a recent surprising observation that the scaling exponent $\zeta _3$ of absolute third-order structure functions defined by (3.19) or (3.23) for $p=3$ in fact satisfies $\zeta _3>1$ in some high-$Re$ numerical simulations of forced turbulence in a periodic domain (Iyer et al. Reference Iyer, Drivas, Eyink and Sreenivasan2024). This observation is, of course, inconsistent with the bound $\zeta _3\leq 1$ derived by Constantin et al. (Reference Constantin, Weinan and Titi1994) under the assumption of a strong dissipative anomaly. Iyer et al. (Reference Iyer, Drivas, Eyink and Sreenivasan2024) appear to have found instead that the normalised dissipation rate $D$ very slowly decays at high $Re$ according to the bound (3.47) derived by Drivas & Eyink (Reference Drivas and Eyink2019). Therefore, only a weak dissipative anomaly appears to occur in this particular simulation of forced homogeneous and isotropic turbulence in a periodic box. Consistent with the results of Bedrossian et al. (Reference Bedrossian, Coti Zelati, Punshon-Smith and Weber2019) and Novack (Reference Novack2023), however, the Kolmogorov 4/5th law is still observed to hold in this simulation, in agreement with earlier observations of Iyer et al. (Reference Iyer, Sreenivasan and Yeung2020). These observations need to be confirmed with other forcing schemes and extended to even higher Reynolds numbers, but they raise additional doubts about the simplification of forced turbulence in a periodic box as a valid paradigm for incompressible fluid turbulence more generally.

3.1.3.4 Extensions

One of the great virtues of Onsager's ‘ideal turbulence’ theory is that it extends readily to turbulent flows in other physical systems than incompressible fluids. For example, consequences have been worked out for turbulence in compressible fluids, both barotropic (Feireisl et al. Reference Feireisl, Gwiazda, Świerczewska-Gwiazda and Wiedemann2017) and with a general thermodynamic equation of state (Drivas & Eyink Reference Drivas and Eyink2018; Eyink Reference Eyink2018a), in quantum superfluids (Tanogami Reference Tanogami2021), in relativistic fluids (Eyink & Drivas Reference Eyink and Drivas2018), in magnetohydrodynamics, both incompressible (Caflisch, Klapper & Steele Reference Caflisch, Klapper and Steele1997; Galtier Reference Galtier2018; Faraco, Lindberg & Székelyhidi Reference Faraco, Lindberg and Székelyhidi2022) and compressible (Lazarian et al. Reference Lazarian, Eyink, Jafari, Kowal, Li, Xu and Vishniac2020, § IV.B), and in kinetic plasmas at low collisionality (Eyink Reference Eyink2015, Reference Eyink2018a; Bardos, Besse & Nguyen Reference Bardos, Besse and Nguyen2020). In many of these examples, the dimensional and statistical reasoning of Kolmogorov (Reference Kolmogorov1941a,Reference Kolmogorovb) does not obviously apply and the Onsager theory provides the only first-principles formulation. Almost all of the just-listed examples have applications in astrophysics and space science and, in particular, most astrophysical fluids and plasmas are compressible. Astrophysics provides, in many ways, the optimal arena for Onsager's theory. For one thing, the Reynolds numbers in astrophysics are often much larger than the highest Reynolds numbers attainable in terrestrial turbulence. In addition, problems such as the physical origin of fluid singularities are much easier to address in compressible fluids. A well-advanced mathematical theory already exists for generic development of shocks in smooth solutions of compressible Euler equations and the consequent production of vorticity (Buckmaster, Shkoller & Vicol Reference Buckmaster, Shkoller and Vicol2023). Shocks and baroclinic generation of vorticity are expected to be a common source of astrophysical turbulence, e.g. supernovae-driven shocks in the interstellar medium (Gatto et al. Reference Gatto2015).

Other recent results on the $Re\to \infty$ limit involve incompressible fluids, but fully incorporating the usually neglected effect of thermal fluctuations. Starting with the fluctuating hydrodynamic equations (2.5) of Landau & Lifshitz (Reference Landau and Lifshitz1959) for a low-Mach incompressible fluid (see also Forster, Nelson & Stephen Reference Forster, Nelson and Stephen1977; Donev et al. Reference Donev, Nonaka, Sun, Fai, Garcia and Bell2014), Eyink & Peng (Reference Eyink and Peng2024) have studied the limiting dynamics in the inertial range. The earlier work of Bandak et al. (Reference Bandak, Mailybaev, Eyink and Goldenfeld2024), which shall be discussed more below, already considered the standard large-scale non-dimensionalisation of the equations via $\hat {{\boldsymbol {u}}}={\boldsymbol {u}}/u',$ $\hat {\boldsymbol {x}}=\boldsymbol {x}/L,$ $\hat {t}=u't/L,$ which takes the form

(3.49)\begin{equation} \partial_{\hat{t}}\hat{{\boldsymbol{u}}} + (\hat{{\boldsymbol{u}}}\boldsymbol{\cdot}\hat{\boldsymbol{\nabla}})\hat{{\boldsymbol{u}}} =-\hat{\boldsymbol{\nabla}}\hat{p} + \frac{1}{Re}\hat{\Delta}\hat{{\boldsymbol{u}}} + \sqrt{\frac{2\theta_\eta}{Re^{15/4}}} \hat{\boldsymbol{\nabla}}\boldsymbol{\cdot} \hat{\boldsymbol{\xi}} + \digamma \hat{{\boldsymbol f}}, \end{equation}

where the dimensionless quantity $\theta _\eta =k_BT/\rho u_\eta ^2\eta ^3$ is the thermal energy relative to the energy of an eddy of Kolmogorov scale $\eta$ and with Kolmogorov velocity $u_\eta =(\nu \varepsilon )^{1/4},$ with $\theta _\eta$ generally of order $10^{-6}$ or even smaller. In deriving (3.49), Bandak et al. (Reference Bandak, Mailybaev, Eyink and Goldenfeld2024) assumed the Taylor relation $\varepsilon \sim u^{\prime 3}/L$ and an external body force with dimensionless magnitude $\digamma =f_{rms}L/u^{\prime 2}$ has also been included to generate turbulence. It is naïvely clear that, on inertial-range scales, the direct effect of the thermal noise term must be negligible, even smaller than the viscous diffusion. In fact, the noise term leads to a mathematically ill-defined dynamics, unless the stochastic noise field $\tilde {\boldsymbol {\xi }}$ and velocity field $\tilde {{\boldsymbol {u}}}$ are both cut off at some high wavenumber $\varLambda,$ or $\hat {\varLambda }=\varLambda L$ after non-dimensionalisation. In the statistical physics literature, such a cutoff is standard and represents physically a coarse-graining length $\ell =\varLambda ^{-1}$ of the measured velocity field, which is chosen somewhere between $\eta$ and the molecular mean-free path length $\ell _{mfp}$. In the limit $Re\to \infty$ and $\hat {\varLambda }\to \infty,$ Eyink & Peng (Reference Eyink and Peng2024) prove under natural conditions that are observed in experiment (see § 3.2.3 below) that the limiting velocity fields are weak solutions of incompressible Euler, as assumed in the Onsager (Reference Onsager1949) theory.

It is important to stress that Onsager's ‘ideal turbulence’ description does not apply in the dissipation range and, indeed, Onsager never discussed dissipation-range turbulence in any of his published or unpublished writings, to my knowledge. This is perhaps one reason why he never discussed the possible interactions of thermal fluctuations and turbulence. This issue was raised approximately a decade later, however, by Betchov (Reference Betchov1957, Reference Betchov1961), who pointed out that the kinetic energy spectrum of thermal fluid fluctuations

(3.50)\begin{equation} E(k) \sim \frac{k_BT}{\rho} \frac{4{\rm \pi} k^2}{(2{\rm \pi})^3} \end{equation}

must surpass the spectrum of turbulent fluctuations at a wavenumber of order $k_\eta \sim 1/\eta.$ This same idea was rediscovered much later by Bandak et al. (Reference Bandak, Goldenfeld, Mailybaev and Eyink2022) who provided numerical evidence for the conjecture in a shell model, before full verification by Bell et al. (Reference Bell, Nonaka, Garcia and Eyink2022) in a numerical simulation of the Landau–Lifschitz fluctuating hydrodynamic equations (2.5). Confirmation by laboratory experiment is crucial but appears much more difficult at this time. In principle, all turbulent processes at sub-Kolmogorov scales should be strongly affected by thermal noise, including droplet formation, combustion, biolocomotion, etc. but it is unclear whether such probes of the small-scale physics can show the clear signature of thermal noise at large scales. A prototypical example is high-Schmidt-number turbulent advection, where the classical theories of Batchelor (Reference Batchelor1959) and Kraichnan (Reference Kraichnan1968) that ignore thermal fluctuations miss power-law scaling of the concentration spectrum below the Batchelor dissipation scale (Eyink & Jafari Reference Eyink and Jafari2022). However, the classical prediction of a $k^{-1}$ spectrum in the viscous-convective range remains intact, because the strong effect of thermal hydrodynamic fluctuations in renormalising the molecular diffusivity in that range is exactly the same as in laminar flows (Onsager Reference Onsager1945b) and thus hidden phenomenologically. There are effects of thermal noise even in the turbulent inertial range (Bandak et al. Reference Bandak, Mailybaev, Eyink and Goldenfeld2024), but these are much more subtle and will be discussed further below.

3.1.3.5 Physicality of weak solutions

A frequent objection made by physicists and fluid mechanicians to the ideal turbulence theory is that the mathematical concept of a ‘weak solution’ is unphysical. In fact, such criticisms are directed not only at Onsager's proposed Euler solutions, but also against the weak solutions of the incompressible Navier–Stokes equation constructed by Leray (Reference Leray1934). To quote again from Tsinober (Reference Tsinober2009, § 10.3.1), regarding Leray's work: ‘An important point is that if one looks at real turbulence at finite Reynolds numbers (however large) there seems to be no need for weak solutions at all.’ Here the presumption is that smooth, strong solutions of Navier–Stokes must exist, as there is no obvious empirical evidence for the severe singularities required to overcome the viscous regularisation. This issue of the physical meaning of weak solutions is quite important, since it is at the centre of the empirical testability of Onsager's theory. Thus, I must discuss the matter briefly here. Furthermore, my own views may not coincide with those of most mathematicians who work on Onsager's theory, and these differences must be carefully explained. The issues are subtle and intimately related with the famous Sixth Problem of Hilbert (Reference Hilbert1900), which was ‘the problem of developing mathematically the limiting processes $\ldots$which lead from the atomistic view to the laws of motion of continua’ and which remains still largely unresolved.

As I have already emphasised, ‘weak solutions’ are mathematically equivalent to ‘coarse-grained solutions’ such as (3.4) for Navier–Stokes and (3.5) for Euler, when those equations are imposed for all $\ell >0$ (Drivas & Eyink Reference Drivas and Eyink2018). In reality, the ‘coarse-grained solution’ of Euler equation (3.5) is an accurate physical description of a turbulent flow only in the inertial range of scales for $\ell \gg \eta.$ The most important fact about the formulation in (3.5) of ‘weak Euler solutions’, or its equivalent (3.8), is that it involves only the coarse-grained fields $\bar {{\boldsymbol {u}}}_\ell$ and $\bar {p}_\ell$ for $\ell \gg \eta,$ since the subscale stress $\boldsymbol {\tau }_\ell ({\boldsymbol {u}},{\boldsymbol {u}})$ depends as well only upon $\bar {{\boldsymbol {u}}}_{\ell '}$ with $\ell '\lesssim \ell$ by the fundamental property of scale locality. To state my view succinctly, it is the coarse-grained fields such as $\bar {{\boldsymbol {u}}}_\ell$ and $\bar {p}_\ell$ which are physical, because they correspond to what is experimentally measurable. In fact, every experiment has some spatial resolution $\ell,$ such that only averaged properties for length scales $>\ell$ are obtained. It is instead the fine-grained/bare fields such as ${\boldsymbol {u}}(\boldsymbol {x})$ which are unphysical, because they are unobservable objects corresponding to a mathematical idealisation $\bar {{\boldsymbol {u}}}_\ell \to {\boldsymbol {u}}$ as $\ell \to 0,$ which goes beyond the validity of a hydrodynamic description and is physically unachievable.

The above views are probably not the same as those held currently by the majority of mathematicians and fluid mechanicians who work on turbulence, who mostly have followed von Neumann (Reference von Neumann1949) in believing that

‘Turbulent’ and ‘molecular’ disorder are in general distinct. A macroscopic and yet turbulent ${\boldsymbol {u}}$ (and, more generally, a complete system of system of fluid dynamics with such characteristics) can be defined. (von Neumann Reference von Neumann1949, § 3.2, p. 446)

According to this common view, the deterministic Navier–Stokes equations are assumed valid for scales $\ell \gtrsim \lambda _{mfp},$ the molecular mean free path length, with $\bar {{\boldsymbol {u}}}_\ell \approx {\boldsymbol {u}},$ the Navier–Stokes solution, for all $\ell$ in the range $\eta \gtrsim \ell \gtrsim \lambda _{mfp}.$ There is, however, no rigorous mathematical theory that justifies this assumption and there are no experimental measurements available at scales $\ell <\eta$ to confirm it. A major problem with this view is that it omits the physical effects of thermal fluctuations, which leads to deviations from the predictions of deterministic Navier–Stokes at scales $\ell \gg \lambda _{mfp}$ that are experimentally well observed in laminar flows (de Zarate & Sengers Reference de Zarate and Sengers2006). In fact, deterministic Navier–Stokes is a physically inconsistent set of equations, because it incorporates molecular dissipation without including the corresponding molecular fluctuations. It thus violates the fundamental fluctuation-dissipation theorem of statistical physics (Onsager & Machlup Reference Onsager and Machlup1953). I therefore do not regard solutions of deterministic Navier–Stokes, either weak or strong, as having fundamental physical significance. Deterministic Navier–Stokes is instead just a reasonably accurate ‘large-eddy simulation’ model of turbulence for resolved scales down to $\ell \sim \eta$.

A more fundamental model of molecular fluids are the fluctuating hydrodynamic equations of Landau & Lifshitz (Reference Landau and Lifshitz1959) which incorporate the fluctuation-dissipation relation and which explain the experiments on thermal fluctuations in laminar flows (de Zarate & Sengers Reference de Zarate and Sengers2006). It was first pointed out by Betchov (Reference Betchov1961) that fluctuating hydrodynamics is an appropriate model of the turbulent dissipation range and, in fact, the incompressible, low-Mach-number equations (2.5) were proposed in his work independently of Landau & Lifshitz (Reference Landau and Lifshitz1959). However, as observed in the previous subsection, fluctuating hydrodynamics is not a continuum theory but instead involves an explicit cutoff $\varLambda =\ell ^{-1}$ and it is believed to describe the evolution of the coarse-grained fields $\bar {{\boldsymbol {u}}}_\ell$ for all scales $\ell \gtrsim \lambda _{mfp}$ (or, in fact, for the incompressible version (2.5), only at scales where $|\bar {{\boldsymbol {u}}}_\ell |\lesssim c_s,$ the sound speed Donev et al. Reference Donev, Nonaka, Sun, Fai, Garcia and Bell2014). Furthermore, the viscosity $\nu _\ell$ in the model depends upon the scale $\ell,$ as shown, for example, by the renormalisation group analysis of Forster et al. (Reference Forster, Nelson and Stephen1977). Thus, even in laminar flows, the observed viscosity $\nu _\ell$ at length scale $\ell$ is an ‘eddy-viscosity’, although the fluctuating eddies arise there from thermal fluctuations rather than turbulent fluctuations. In the language of modern physics, the Landau–Lifschitz fluctuating hydrodynamics equations are low-wavenumber ‘effective field theories’ (Schwenk & Polonyi Reference Schwenk and Polonyi2012; Liu & Glorioso Reference Liu and Glorioso2018).

An important consequence of these considerations is that velocity-gradients are $\ell$-dependent at all length scales $\ell$ in a turbulent flow and not only for $\ell >\eta.$ However, it should be true to a good approximation that, for typical thermal fluctuations,

(3.51)\begin{equation} \boldsymbol{\nabla}\bar{{\boldsymbol{u}}}_\ell\sim \boldsymbol{\nabla}{\boldsymbol{u}}+O\left(\left(\frac{k_BT}{\rho \ell^5}\right)^{1/2}\right), \quad \ell\lesssim \eta, \end{equation}

where ${\boldsymbol {u}}$ is again the solution of deterministic Navier–Stokes. It then easily follows that

(3.52)\begin{equation} \nu_\ell\langle |\boldsymbol{\nabla}\bar{{\boldsymbol{u}}}_\ell|^2\rangle \simeq \nu \langle |\boldsymbol{\nabla}{\boldsymbol{u}}|^2\rangle, \quad \nu=\nu_\eta, \end{equation}

where $\langle {\cdot }\rangle$ denotes a space–time average and $\ell /\eta$ spans approximately 2–3 decades of scales. For more details, see Eyink (Reference Eyink2007, § II.E). In this sense, typical thermal fluctuations should matter little for mean energy dissipation. However, this statement is presumably not true for large, rare thermal fluctuations. Indeed, in a simple particle model of hydrodynamics, the ‘totally asymmetric exclusion process’, dissipative weak solutions of inviscid Burgers equation arise with overwhelming probability (law of large numbers) but rare fluctuations (large deviations) can lead to weak solutions with local energy production in some space–time regions rather than dissipation everywhere (Jensen Reference Jensen2000; Varadhan Reference Varadhan2004). Furthermore, higher-order gradients $\boldsymbol {\nabla }^k\bar {{\boldsymbol {u}}}_\ell$ with $k>1$ will be much more strongly affected by thermal noise and, indeed, the numerical simulation of Bell et al. (Reference Bell, Nonaka, Garcia and Eyink2022) found that such higher-order gradients are completely dominated by thermal noise already for $\ell \lesssim \eta.$

I therefore fundamentally disagree with a common view, clearly expressed by Tsinober (Reference Tsinober2009), that hypothetical fine-grained fields ${\boldsymbol {u}}$ and their gradients are physically more ‘objective’ than coarse-grained fields $\bar {{\boldsymbol {u}}}_\ell$:

There is a generic ambiguity in defining the meaning of the term small scales (or more generally scales) and consequently the meaning of the term cascade in turbulence research. As mentioned in chapter 5, the specific meaning of this term and associated inter-scale energy exchange/cascade (e.g. spectral energy transfer) is essentially decomposition/representation dependent. Perhaps, the only common element in all decompositions/ representations (D/R) is that the small scales are associated with the field of velocity derivatives. Therefore, it is natural to look at this field as the one objectively (i.e. D/R independent) representing the small scales. Indeed, the dissipation is associated precisely with the strain field, $s_{ij}$, both in Newtonian and non-Newtonian fluids. (Tsinober Reference Tsinober2009, § 6.2, p. 127)

This point of view is based on the belief, without evidence, that there is some hypothetical fine-grained field ${\boldsymbol {u}} =\lim _{\ell \to 0} \bar {{\boldsymbol {u}}}_\ell$ and that objectivity is achieved by ‘convergence’ as $\ell \to 0.$ The existence of such a fine-grained velocity in turbulent flow was probably first questioned by the visionary scientist, Lewis Fry Richardson, whose famous paper on turbulent pair-dispersion contained a section entitled ‘Does the Wind possess a Velocity?’ He wrote there:

This question, at first sight foolish, improves on acquaintance. A velocity is defined, for example, in Lamb's “Dynamics” to this effect: Let $\Delta x$ be the distance in the $x$ direction passed over in a time $\Delta t,$ then the $x$-component of velocity is the limit of $\Delta x/\Delta t$ as $\Delta t\to 0$. But for an air particle it is not obvious that $\Delta x/\Delta t$ attains a limit as $\Delta t\to 0.$ (Richardson Reference Richardson1926, § 1.2, p. 709)

Indeed, each individual air molecule is moving at approximately 346 m s$^{-1}$, the speed of sound in that fluid, with the mean wind velocity a small correction, and the local measured velocity will depend entirely upon the resolution scale $\ell$ which is adopted.

I certainly agree that science should be concerned with objective facts. However, in contrast to the 19th-century continuum mechanics perspective that objectivity is achieved by convergence, the modern tool to achieve objectivity is the renormalisation group. I may cite here the physics Nobel laureate Kenneth G. Wilson:

A procedure is now being developed to understand the statistical continuum limit. The procedure is called the renormalisation group. It is the tool that one uses to study the statistical continuum limit in the same way that the derivative is the basic procedure for studying the ordinary continuum limit. (Wilson Reference Wilson1975, p. 774)

The fundamental understanding that arose in physics was that for systems with strong fluctuations at all length scales – a class which includes both quantum field theories and turbulent flows – the effective description varies with the length scale $\ell$ of resolution. In that case, the proper goal is not to seek for some idealised and unobservable ‘truth’ at $\ell \to 0$, but instead to attain objectivity by understanding how the description changes as $\ell$ is varied. The application of this approach to hydrodynamics has mostly been restricted to flows much simpler than turbulence, cf. Forster et al. (Reference Forster, Nelson and Stephen1977), where weak-coupling perturbation expansions are applicable, and only recently has some progress been made on non-perturbative application to turbulence by functional renormalisation group methods (Canet Reference Canet2022). It is conceptually important, however, to understand that LES models resolved at length scale $\ell$ are not really continuum models. Indeed, it is a truism in the LES community that the true model is not the ‘continuum’ LES model but instead the discretisation of the model used to solve it numerically. It therefore makes sense to combine the steps of coarse-graining and numerical discretisation, as in some approaches to modelling thermal fluctuations (Español, Anero & Zúñiga Reference Español, Anero and Zúñiga2009). I shall return to this theme later in discussing the relation of Onsager's ‘ideal turbulence’ theory with LES modelling.

3.2.1. Existence of dissipative Euler solutions

The program which has successfully led to a mathematical construction of the weak Euler solutions conjectured by Onsager (Reference Onsager1949) in breakthrough work by Isett (Reference Isett2018) was initiated by the works of De Lellis & Székelyhidi (Reference De Lellis and Székelyhidi2009) and De Lellis & Székelyhidi (Reference De Lellis and Székelyhidi2010), which both appeared as preprints already in 2007. A very good review of this early work is given by De Lellis & Székelyhidi (Reference De Lellis and Székelyhidi2013) and an updated review is to be found in De Lellis & Székelyhidi (Reference De Lellis and Székelyhidi2019). These authors pointed out a remarkable connection of Onsager's conjecture with famous work of the mathematician John Nash (Reference Nash1954) on $C^1$ isometric embeddings. I give here a very succinct review, following closely the discussions in the previous references.

Nash (Reference Nash1954) addressed a classical problem of differential geometry, whether a smooth manifold $M$ of dimension $n\geq 2$ with Riemannian metric $g$ may be isometrically imbedded in $m$-dimensional Euclidean space ${\mathbb {R}}^m$, i.e. whether a $C^1$ embedding map ${\boldsymbol u}: M \to {\mathbb {R}}^m$ exists so that the Riemannian metric induced by the embedding agrees with $g,$ or

(3.53)\begin{equation} \partial_i{\boldsymbol u}\boldsymbol{\cdot}\partial_j{\boldsymbol u}= g_{ij}. \end{equation}

To answer this question, Nash considered a more general problem of short embeddings which do not preserve lengths of curves on $M$ but can only decrease lengths, so that

(3.54)\begin{equation} \partial_i \bar{{\boldsymbol u}}\boldsymbol{\cdot}\partial_j\bar{{\boldsymbol u}}\leq g_{ij} \end{equation}

in the matrix sense. The startling result obtained by Nash, with some improvement due to Kuiper (Reference Kuiper1955), is the following:

This result is surprising for two reasons. First, the condition (3.53) is a set of $n(n+1)/2$ equations in $m$ unknowns. A reasonable guess would be that the system is solvable, at least locally, when $m\geq n(n+1)/2$ and this indeed was a classical conjecture of Schläfli. However, for $n \geq 3$ and $m = n + 1$, the system (3.53) is hugely overdetermined! It is not obvious that there should be any solutions at all, but the Nash–Kuiper theorem shows that there exists an enormous set of $C^1$-solutions that are $C^0$-dense in the set of short embeddings. I shall not discuss here the details of Nash's proof of his astonishing result, but just remark that his construction of the isometry ${\boldsymbol u}$ was in a series of stages, by adding at each stage a new small, high-frequency perturbation. The construction is not an ‘abstract nonsense’ argument that requires the Axiom of Choice or some other non-constructive element, and, in fact, the entire procedure can be implemented to any finite stage of iteration, in principle, by a computer algorithm. For example, see the work of Borrelli et al. (Reference Borrelli, Jabrane, Lazarus and Thibert2012) which calculates numerically a $C^1$-isometric embedding of the flat 2-torus ${\mathbb {T}}^2$ into ${\mathbb {R}}^3$ and figure 6 for the first four stages of iteration. It is interesting to note that Nash himself regarded his 1954 paper as a ‘sidetrack’ on the problem of isometric embeddings (Raussen & Skau Reference Raussen and Skau2016) and attached greater importance to his later work that constructed $C^\infty$ embeddings in higher dimensions (Nash Reference Nash1956). The latter paper is notable for its introduction of the Nash–Moser implicit function theorem and also for its exploitation of a smooth mollification scheme with a continuous parameter $\ell,$ involving a careful study of variations with $\ell$ similar to an RG approach (L. Székelyhidi, Jr., private communication). A very intriguing recent stochastic formulation of Menon (Reference Menon2021) and Inauen & Menon (Reference Inauen and Menon2023) makes this analogy more clear.

Figure 6. First four stages in the iterative construction of a $C^1$-isometric embedding of the flat 2-torus ${\mathbb {T}}^2$ into ${\mathbb {R}}^3.$ The initial map is corrugated along the meridians to increase their length. Corrugations are then applied repeatedly in various directions to produce a sequence of maps. Each successive map is strictly short, with reduced isometric default. Reproduced from Borrelli et al. (Reference Borrelli, Jabrane, Lazarus and Thibert2012), with permission of PNAS.

The fundamental contribution of De Lellis and Székelyhidi Jr. was to realise that there is a very close mathematical analogy between the problem of isometrically embedding a smooth manifold by a map of low regularity and the problem of solving the Cauchy initial-value problem for incompressible Euler equations by a velocity field of low regularity, and that Nash's method of construction can be carried over to the latter. The analogue of a ‘short mapping’ for the Euler system is what De Lellis and Székelyhidi Jr. call a smooth subsolution, i.e. a smooth triple $(\bar {{\boldsymbol {u}}}, \bar {p},\bar {\boldsymbol {\tau }})$ with $\bar {\boldsymbol {\tau }}$ a symmetric, positive-definite tensor such that

(3.55)\begin{equation} \partial_t\bar{{\boldsymbol{u}}}+\boldsymbol{\nabla}\boldsymbol{\cdot}(\bar{{\boldsymbol{u}}}\ \bar{{\boldsymbol{u}}}+\bar{\boldsymbol{\tau}})=-\boldsymbol{\nabla}\bar{p}, \quad \boldsymbol{\nabla}\boldsymbol{\cdot}\bar{{\boldsymbol{u}}}=0. \end{equation}

Everyone from the turbulence modelling community will recognise at once that this has the form of an LES model equation incorporating a positive-definite ‘turbulent stress’ tensor $\bar {\boldsymbol {\tau }}.$ Approximation results can be obtained for subsolutions entirely analogous to the Nash–Kuiper theorem on approximation of short embeddings. As an example, a theorem of Buckmaster et al. (Reference Buckmaster, De Lellis, Székelyhidi and Vicol2018) states the following result.

Theorem (Buckmaster et al. Reference Buckmaster, De Lellis, Székelyhidi and Vicol2018)

Let $(\bar {{{\boldsymbol {u}}}},\bar {p},\bar {\boldsymbol {\tau }})$ be any smooth, strict subsolution of the Euler equations on ${\mathbb {T}}^3 \times [0,T]$ and let $h< 1/3$. Then there exists a sequence $({\boldsymbol {u}}_k,p_k)$ of weak Euler solutions such that ${\boldsymbol {u}}_k\in C^h({\mathbb {T}}^3 \times [0,T])$ satisfy, as $k\to \infty,$

(3.56)\begin{equation} \int_{{\mathbb{T}}^3} \,{\rm d}^3x\, f \, {\boldsymbol{u}}_k\rightarrow \int_{{\mathbb{T}}^3} \,{\rm d}^3x\, f \bar{{\boldsymbol{u}}}, \quad \int_{{\mathbb{T}}^3} \,{\rm d}^3x\, f \, {\boldsymbol{u}}_k{\boldsymbol{u}}_k\rightarrow \int_{{\mathbb{T}}^3} \,{\rm d}^3x\ f (\bar{{\boldsymbol{u}}}\ \bar{{\boldsymbol{u}}}+\bar{\boldsymbol{\tau}}) \end{equation}

for all $f\in L^1({\mathbb {T}}^3)$ uniformly in time, and furthermore for all $t\in [0, T ]$ and all $k$,

(3.57)\begin{equation} \int_{{\mathbb{T}}^3}\, {\rm d}^3x \tfrac{1}{2}|{\boldsymbol{u}}_k|^2 =\int_{{\mathbb{T}}^3} \,{\rm d}^3x\, \tfrac{1}{2}(|\bar{{\boldsymbol{u}}}|^2+{\rm Tr}\, \bar{\boldsymbol{\tau}}). \end{equation}

The notion of convergence in this theorem can be made more physically transparent by taking $f(\boldsymbol {r})=\tilde {G}_\delta (\boldsymbol {x}+\boldsymbol {r})$ for $\delta >0,$ in which case the approximation property implies that pointwise,

(3.58)\begin{equation} \tilde{{\boldsymbol{u}}}_{k,\delta}\to\tilde{\bar{{\boldsymbol{u}}}}_\delta, \quad \tilde{\tau}_\delta({\boldsymbol{u}}_k,{\boldsymbol{u}}_k)\to\tilde{\tau}_\delta(\bar{{\boldsymbol{u}}},\bar{{\boldsymbol{u}}}) +\tilde{\bar{\boldsymbol{\tau}}}_\delta. \end{equation}

Thus, if the subsolution $(\bar {{{\boldsymbol {u}}}},\bar {p},\bar {\boldsymbol {\tau }})$ arose from a coarse-grained Euler solution in the sense of Drivas & Eyink (Reference Drivas and Eyink2018), so that $\bar {{{\boldsymbol {u}}}}=\bar {{{\boldsymbol {u}}}}_\ell$, $\bar {p}=\bar {p}_\ell$, $\bar {\boldsymbol {\tau }}=\bar {\boldsymbol {\tau }}_\ell ({\boldsymbol {u}},{\boldsymbol {u}}),$ then the simple identity of Germano (Reference Germano1992) for the convolution filter $\tilde {\bar {G}}_{\ell,\delta }=\bar {G}_\ell *\tilde {G}_\delta$,

(3.59)\begin{equation} \tilde{\bar{\boldsymbol{\tau}}}_{\ell,\delta}({\boldsymbol{u}},{\boldsymbol{u}})= \tilde{\tau}_\delta(\bar{{\boldsymbol{u}}}_\ell,\bar{{\boldsymbol{u}}}_\ell) + \widetilde{(\bar{\boldsymbol{\tau}}_\ell({\boldsymbol{u}},{\boldsymbol{u}}))}_\delta, \end{equation}

implies that for any $\delta >0$,

(3.60)\begin{equation} \tilde{{\boldsymbol{u}}}_{k,\delta}\to\tilde{\bar{{\boldsymbol{u}}}}_{\ell,\delta}, \quad \tilde{\tau}_\delta({\boldsymbol{u}}_k,{\boldsymbol{u}}_k)\to\tilde{\bar{\tau}}_{\ell,\delta}({\boldsymbol{u}},{\boldsymbol{u}}) \end{equation}

pointwise in space. When $\delta \ll \ell,$ this yields the remarkable statement that a coarse-grained Euler solution $(\tilde {\bar {{\boldsymbol {u}}}}_{\ell,\delta },\tilde {\bar {p}}_{\ell,\delta }) \simeq (\bar {{\boldsymbol {u}}}_{\ell },\bar {p}_{\ell })$ at scale $\ell$ can be well approximated pointwise in space by a sequence $(\tilde {{\boldsymbol {u}}}_{k,\delta },\tilde {p}_{k,\delta })$ of coarse-grained Euler solutions at the much smaller scale $\delta.$

The constructions of such weak Euler solutions follow a strategy similar to that of Nash, by a sequence of stages ${\boldsymbol {u}}_0, {\boldsymbol {u}}_1, {\boldsymbol {u}}_2, \ldots\, $. At stage $n$, one has, after coarse-graining, a subsolution

(3.61)\begin{equation} \partial_t {\boldsymbol{u}}_n +\boldsymbol{\nabla}\boldsymbol{\cdot}({\boldsymbol{u}}_n{\boldsymbol{u}}_n+\boldsymbol{\tau}_n)=-\boldsymbol{\nabla} p_n, \end{equation}

which is supported on wavenumbers $<\varLambda _n.$ By adding a small-scale carefully chosen perturbation, one can succeed to cancel a large part of the stress $\boldsymbol {\tau }_n$ so that, in the limit, $\boldsymbol {\tau }_n\to {\boldsymbol 0}$ weakly and one obtains a weak limit ${\boldsymbol {u}}$ which is a distributional Euler solution. A number of different models of the small scales have been employed, such as Beltrami flows (De Lellis & Székelyhidi Reference De Lellis and Székelyhidi2013) and Mikado flows (Buckmaster et al. Reference Buckmaster, De Lellis, Székelyhidi and Vicol2018; Isett Reference Isett2018), together with other operations, such as evolving under smooth Euler dynamics locally in time and ‘gluing’ the different time-segments. In the language of LES modelling, the constructions can be regarded as a sort of iterative ‘defiltering’, in which unresolved scales are successively restored. A physicist might prefer to call this an ‘inverse renormalisation group’, the reverse of the successive coarse-graining employed by Wilson (Reference Wilson1975). Clearly, there is a huge number of subsolutions, since one may adopt any positive definite tensor $\bar {\boldsymbol {\tau }}$ whatsoever. A consequence is therefore results such as the following theorem:

Theorem (Buckmaster et al. Reference Buckmaster, De Lellis, Székelyhidi and Vicol2018)

Let $e:[0,T]\to {\mathbb {R}}^+$ be any strictly positive, smooth function. Then for any $0< h<1/3$, there exists a weak Euler solution ${\boldsymbol {u}}\in C^h({\mathbb {T}}^3 \times [0,T])$ such that

(3.62)\begin{equation} \int_{{\mathbb{T}}^3} \,{\rm d}^3x\ \tfrac{1}{2}|{\boldsymbol{u}}|^2 =e(t). \end{equation}

In particular, one may take $e(t)$ to be any function strictly decreasing in time and then the Euler solution of the above theorem (globally) dissipates kinetic energy.

The details of the constructions are complex and rapidly evolving, so that they may be best gleaned from the journal articles themselves. I therefore make here just a few remarks. First, in addition to solutions that satisfy physical constraints, one can construct by the same methods solutions with obviously unphysical behaviours (e.g. kinetic energy increasing in time for unforced 3-D incompressible flows!) Thus, it is clear that the methods suffice to answer some very fundamental PDE questions, but they still miss much essential physics. Another important point is that the existence proofs of the weak Euler solutions are constructive, exactly as are the Nash constructions of isometric embeddings. Frisch, Székelyhidi & Matsumoto (Reference Frisch, Székelyhidi and Matsumoto2018) have attempted to compute numerically the weak Euler solutions constructed by Mikado flows (Buckmaster et al. Reference Buckmaster, De Lellis, Székelyhidi and Vicol2018; Isett Reference Isett2018). Visualisation, however, appears difficult with current constructions because of the very tiny scales of the added eddies. These weak Euler solutions can be regarded as a sophisticated sort of ‘synthetic turbulence’ (Juneja et al. Reference Juneja, Lathrop, Sreenivasan and Stolovitzky1994), manufactured space–time fields with many of the same properties as a physical turbulent flow velocity, but in addition solving exactly the inviscid equations of motion. Such solutions have been constructed which satisfy additional observed properties such as spatial intermittency (Novack & Vicol Reference Novack and Vicol2023; De Rosa & Isett Reference De Rosa and Isett2024). We might hope in coming years to see weak Euler solutions with features in ever closer agreement with physical turbulent flows. As in the field of artificial intelligence and the Turing test, it would then be crucial to design tests of increasing sensitivity that could distinguish a true turbulent velocity field and such artificially constructed flows. This exercise could help to clarify better the essential properties that characterise physical turbulence.

3.2.2. Non-uniqueness for the initial-value problem

One very fundamental feature of the result of Nash (Reference Nash1954) which I have not yet duly emphasised is the profligate non-uniqueness of the $C^1$ isometric embeddings produced by his construction. This stands in stark contrast to embeddings with higher smoothness $C^n,$ $n\geq 2$ for which the the Gauss–Codazzi equations relate the second fundamental form of the embedded surface to the Riemann curvature tensor. In fact, it is known that a compact Riemannian surface with positive Gauss curvature may be isometrically embedded into ${\mathbb {R}}^3$ by a $C^2$ map in just one way, up to a rigid-body motion. Thus it is clear that isometric embeddings have very different qualitative behaviour at low and high regularity, i.e. $C^1$ versus $C^2$, often referred to as a dichotomy between rigidity at high regularity versus flexibility at low regularity. This type of wild non-uniqueness at low regularity is a central aspect of the $h$-principle introduced by mathematician Mikhail Gromov (Reference Gromov1971, Reference Gromov1986), with the isometric embedding problem as a primary example. The theorem of Buckmaster et al. (Reference Buckmaster, De Lellis, Székelyhidi and Vicol2018) that I cited in the previous subsection, as emphasised by those authors, is an extension of the $h$-principle to low-regularity weak solutions of the Euler equations, which exhibit a similar ‘flexibility’. The name convex integration for the discussed techniques to construct these Euler solutions, by the way, goes back also to the work of Gromov (Reference Gromov1971, Reference Gromov1986), who introduced it as one of several methods to prove the $h$-principle. The origin of the name has to do with differential relations of the type $Df\in A,$ which satisfy the $h$-principle when the convex hull of $A$ contains a small neighbourhood of the origin.

Convex integration methods have fundamental implications for non-uniqueness of weak solutions to the Cauchy initial-value problem for the Euler equations, as discussed already by De Lellis & Székelyhidi (Reference De Lellis and Székelyhidi2010). Theorem 1 from that paper shook many previous expectations:

Theorem (De Lellis & Székelyhidi Reference De Lellis and Székelyhidi2010)

Let $d\geq 2$. There exist compactly supported divergence-free vector fields ${\boldsymbol {u}}_0\in L^\infty$ for which there are infinitely many weak Euler solutions with that initial data, satisfying both the global energy equality

(3.63)\begin{equation} \int_{{\mathbb{R}}^d} |{\boldsymbol{u}}(\boldsymbol{x},t)|^2\,{\rm d}^nx = \int_{{\mathbb{R}}^d} |{\boldsymbol{u}}(\boldsymbol{x},s)|^2\,{\rm d}^d x \quad \text{for all }t>s \end{equation}

and the local energy equality

(3.64)\begin{equation} \partial_t\left(\tfrac{1}{2}|{\boldsymbol{u}}|^2\right)+\boldsymbol{\nabla}\boldsymbol{\cdot}\left[\left(\tfrac{1}{2}|{\boldsymbol{u}}|^2+p\right){\boldsymbol{u}}\right]=0. \end{equation}

Furthermore, there are infinitely many weak Euler solutions with that initial data satisfying the global energy inequality:

(3.65)\begin{equation} \int_{{\mathbb{R}}^d} |{\boldsymbol{u}}(\boldsymbol{x},t)|^2\,{\rm d}^d x < \int_{{\mathbb{R}}^n} |{\boldsymbol{u}}(\boldsymbol{x},s)|^2\,{\rm d}^d x \quad \text{for all }t>s. \end{equation}

This result showed that one cannot add a local energy inequality

(3.66)\begin{equation} \partial_t\left(\tfrac{1}{2}|{\boldsymbol{u}}|^2\right)+\boldsymbol{\nabla}\boldsymbol{\cdot}\left[\left(\tfrac{1}{2}|{\boldsymbol{u}}|^2+p\right){\boldsymbol{u}}\right]\leq 0 \end{equation}

(or even an equality) and obtain a unique weak solution to the Euler equations for certain $L^\infty$ initial data. Furthermore, non-uniqueness occurs even if total energy is strictly decreasing. This is in stark contrast to simpler equations such as the inviscid Burgers model (or general hyperbolic scalar conservation laws) where it is known that insisting on such a local energy inequality selects a unique weak solution, which also coincides with the ‘viscosity solution’ obtained by incorporating viscosity and passing to the inviscid limit. The above theorem shows that such a simple selection principle for ‘physical weak solutions’ cannot succeed for the incompressible Euler equations.

Note that such initial data with non-unique solutions cannot have smoothness such as $C^{1+\epsilon }$ (velocity-gradients existing, with $C^\epsilon$ Hölder regularity) or higher smoothness, because this would violate the following important type of result.

Theorem Let ${\boldsymbol {u}}\in L^\infty ((0,T);L^2({\mathbb {T}}^d))$ be a weak Euler solution, $\boldsymbol {U}\in C^1({\mathbb {T}}^d\times [0,T])$ a strong solution, and assume that ${\boldsymbol {u}}$ and $\boldsymbol {U}$ share the same initial datum ${\boldsymbol {u}}_0$. Assume moreover that

(3.67)\begin{equation} \int_{{\mathbb{T}}^d} |{\boldsymbol{u}}(\boldsymbol{x},t)|^2\,{\rm d}^d x \leq \int_{{\mathbb{T}}^d} |{\boldsymbol{u}}_0(\boldsymbol{x})|^2\,{\rm d}^d x \end{equation}

for almost every $t\in (0,T)$. Then ${\boldsymbol {u}}(\boldsymbol {x},t) =\boldsymbol {U}(\boldsymbol {x},t)$ for almost every $(\boldsymbol {x},t).$

Results of this type go by the name of strong-weak uniqueness. For a clear and lucid review, see Wiedemann (Reference Wiedemann2018). The conclusion of such results is that any ‘admissible’ weak Euler solution, i.e. satisfying the global kinetic energy inequality (3.67), must coincide with a classical Euler solution, as long as that exists. Since local-in-time existence of a classical solution is guaranteed for ${\boldsymbol {u}}_0\in C^{1+\epsilon },$ then the results in the cited theorem of De Lellis & Székelyhidi (Reference De Lellis and Székelyhidi2010) cannot hold for such initial data, at least for some finite time interval. Note, by the way, that strong-weak uniqueness applies even to ‘measure-valued weak Euler solutions’ such as those constructed by DiPerna & Majda (Reference DiPerna and Majda1987). See Wiedemann (Reference Wiedemann2018).

These non-uniqueness results have since been considerably extended and the question is still currently under active investigation. Some very important results are contained in the more recent paper of Daneri, Runa & Szekelyhidi (Reference Daneri, Runa and Szekelyhidi2021), who prove the following theorem:

This theorem shows that the non-uniqueness holds right up to the critical Onsager 1/3 exponent and for a dense set of initial data. Thus, non-uniqueness is in some sense ‘typical’. In a recent remarkable work of Giri, Kwon & Novack (Reference Giri, Kwon and Novack2023), it has been shown by an $L^3$-generalisation of convex integration methods that weak Euler solutions exist, with ${\boldsymbol {u}}\in C([0,T],B^\sigma _3({\mathbb {T}}^3)$ for any $\sigma <1/3,$ which are even locally dissipative, satisfying the local energy inequality (3.66) and not merely the global inequality (3.67). This result suggests that non-uniqueness may hold also for locally dissipative Euler solutions up to Onsager's $1/3$ exponent. To my knowledge, such results have been proved in the $C^h$-setting so far only for $h<1/15$ by Isett (Reference Isett2022) and subsequently for $h<1/7$ by De Lellis & Kwon (Reference De Lellis and Kwon2022). Very intriguingly, De Lellis & Székelyhidi (Reference De Lellis and Székelyhidi2022) have argued heuristically that $h=1/3$ should be the critical Hölder exponent not only for conservation of kinetic energy, but also for convex integration constructions. For a physicist, this remarkable coincidence smacks of a turbulent ‘fluctuation-dissipation relation’.

These startling results on the ‘flexibility’ or non-uniqueness of the dissipative weak Euler solutions that were conjectured by Onsager pose a fascinating problem for their physical interpretation. One possible attitude is that additional conditions should be added to select the ‘physically correct’ weak solution obtained in the inviscid limit, presumed also to be unique. It is certainly true that the Euler solutions constructed by current convex integration methods have unphysical features and thus solutions obtained in the inviscid limit should plausibly have additional properties, e.g. the ‘martingale property’ of fluid circulations conjectured by Eyink (Reference Eyink2006). However, I believe myself that the non-uniqueness or ‘flexibility’ uncovered by the convex integration theory is probably physically real and that, even if sufficient conditions are identified to select physical solutions, infinitely many, non-unique solutions for the same initial-data will remain. One motivation for this conjecture is the ‘spontaneous stochasticity’ phenomenon discovered by Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998), which was proposed by Eyink (Reference Eyink2014) to manifest not only in Lagrangian particle trajectories but also in the Eulerian evolution of the turbulent velocity field itself. Independently and much more concretely, Mailybaev (Reference Mailybaev2016) presented strong evidence from numerical simulations that such Eulerian spontaneous stochasticity appears in the Sabra shell model in the high-$Re$ limit. Mailybaev (Reference Mailybaev2016) used another evocative term ‘stochastic anomaly’ for this phenomenon, stressing the close formal relation with the ‘dissipative anomaly’, and pointed out also connections with the classic paper of Lorenz (Reference Lorenz1969) on turbulence predictability. Indeed, the pioneering work of Lorenz (Reference Lorenz1969) nearly anticipated the modern concept of Eulerian spontaneous stochasticity, as is clear from the first sentences in the abstract of that paper:

Stated differently, the claim of Lorenz (Reference Lorenz1969) was that deterministic fluid systems at high Reynolds numbers with ‘many scales of motion’ may exhibit stochastic solutions, even as random errors in the initial data are taken to zero. This is the essence of spontaneous stochasticity. The only thing missing from the concept of Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998) is the clear understanding that such stochastic behaviour is made possible by non-uniqueness of the solutions of the initial-value problem for the limiting ideal evolution equation as $Re\to \infty.$

Mailybaev and his collaborators have gone on to provide numerical evidence for Eulerian spontaneous stochasticity in some prototypical fluid instabilities, such as the Rayleigh–Taylor mixing layer (Biferale et al. Reference Biferale, Boffetta, Mailybaev and Scagliarini2018) and a Kelvin–Helmholtz unstable vortex sheet (Thalabard, Bec & Mailybaev Reference Thalabard, Bec and Mailybaev2020). It is noteworthy that for both of these problems, Rayleigh–Taylor (Gebhard, Kolumbán & Székelyhidi Reference Gebhard, Kolumbán and Székelyhidi2021) and Kelvin–Helmholtz (Székelyhidi Reference Székelyhidi2011; Mengual & Székelyhidi Reference Mengual and Székelyhidi2023), non-unique solutions of the Cauchy problem for the limiting inviscid equations have been demonstrated by convex integration methods. These examples suggest that spontaneous stochasticity must be a fairly widespread phenomenon in geophysics and astrophysics. Mailybaev & Raibekas (Reference Mailybaev and Raibekas2023b) discuss also some simple multiscale models with known ergodic mixing properties, based on the Arnold cat map or irrational rotations of the circle, where Eulerian spontaneous stochasticity can be rigorously proved a priori. Furthermore, Mailybaev & Raibekas (Reference Mailybaev and Raibekas2023a) have set up a general renormalisation group approach which can demonstrate Eulerian spontaneous stochasticity by identifying a suitable RG fixed point and establish also universality properties of the limiting spontaneous statistics. Finally, Bandak et al. (Reference Bandak, Mailybaev, Eyink and Goldenfeld2024) provide evidence that the Landau–Lifschitz fluctuating Navier–Stokes equations (2.5) should exhibit spontaneously stochastic solutions in the limit $Re\to \infty$ for turbulent initial data, triggered just by thermal noise, by means of shell-model simulations.

I therefore believe that the non-uniqueness or ‘flexibility’ phenomenon uncovered by convex integration theory is manifested by the essential unpredictability of individual turbulent flow realisations, but whose statistics are universal and predictable. The ground-breaking work of Lorenz (Reference Lorenz1969) identified a physical mechanism for such turbulent unpredictability through an ‘inverse error cascade’, in which dissipation-scale errors propagate up to the largest scales of the flow in a time of the order of the large-eddy turnover time (Bandak et al. Reference Bandak, Mailybaev, Eyink and Goldenfeld2024). The ideas of Lorenz (Reference Lorenz1969) are paradigm-altering, fundamentally changing the methods and aims of scientific prediction. I believe that the revolutionary nature of the work by Lorenz (Reference Lorenz1969) was obscured by his use of a quasi-normal closure to enable numerical calculations, but which hid the underlying mathematical foundations of his ideas. The paper of Palmer, Döring & Seregin (Reference Palmer, Döring and Seregin2014) has recalled attention to the ‘real butterfly effect’ of Lorenz (Reference Lorenz1969) but, in my opinion, mis-identified the origin of the phenomenon in the non-uniqueness of solutions of the viscous Navier–Stokes dynamics. However, their proposal does not explain the observations of Thalabard et al. (Reference Thalabard, Bec and Mailybaev2020) on the 2-D vortex sheet, for example, since the 2-D Navier–Stokes equation has unique solutions. I believe that the fundamental unpredictability identified by Lorenz (Reference Lorenz1969) has its origin instead in the non-uniqueness of solutions of the ideal Euler equations obtained in the limit $Re\to 0$ and the Eulerian spontaneous stochasticity associated with statistical distributions over those non-unique solutions (Bandak et al. Reference Bandak, Mailybaev, Eyink and Goldenfeld2024). I hope that proper understanding of the mathematical foundations will help to accelerate progress on this important problem.

We see furthermore that probability re-enters Onsager's ‘ideal turbulence’ theory, as an essential aspect. Although your stirred morning cup of coffee, or similar turbulent flows at higher Reynolds numbers, are described by a single velocity realisation, the same is not true if one considers the ensemble of flows that arise from all of the cups prepared over a long sequence of mornings. No matter how carefully one arranges to stir the coffee in the same manner each day, the individual flows in the cup and the patterns and whorls of cream will differ on each individual day. In fact, the coarse-grained Euler equation (3.8) becomes a stochastic equation when considered as an initial-value problem for $\bar {{\boldsymbol {u}}}_\ell$ alone, since the initial data for the unresolved eddies ${\boldsymbol {u}}_\ell ':={\boldsymbol {u}}-\bar {{\boldsymbol {u}}}_\ell$ are unknown. In that case, the turbulent stress term $\boldsymbol {\tau }_\ell$ becomes a random quantity (Eyink Reference Eyink1996), rationalising the introduction of ‘eddy noise’ (Rose Reference Rose1977) or ‘stochastic backscatter’ (Leith Reference Leith1990) in large-eddy simulation modelling. What spontaneous stochasticity tells us is that such randomness persists even in the idealised limit $\ell \to 0.$ It is in the prediction of the future from present initial data that probability enters turbulence theory intrinsically.

3.2.3. The infinite-Reynolds-number limit

None of the weak Euler solutions discussed so far are obtained by taking infinite-Reynolds-number limits of the solutions of fluid equations valid at dissipation-range scales, such as the incompressible Navier–Stokes equation (2.4) (valid down to approximately the Kolmogorov scale $\eta$) or the incompressible Landau–Lifschitz equations (2.5) (valid down to approximately the mean-free-path length $\ell _{mfp}$). The Euler solutions of physical relevance are just approximations to these more basic equations in the inertial range of length scales ($\ell \gg \eta$) and such solutions must possess very special properties not shared by general weak Euler solutions, including those manufactured by convex integration methods. Several very challenging problems remain in the investigation of the infinite-$Re$ limits, which include the existence and uniqueness of the limits themselves. Another important issue is the selection problem, which is the question whether suitable constraints or conditions can be formulated which fully characterise a priori the Euler solutions obtained in the inviscid limit, without actually passing to the limit. The latter problem is important for potential practical applications, since one of the great hopes of Onsager's ‘ideal turbulence’ theory is that it might provide a shortcut to describe and compute the infinite-$Re$ limit directly, avoiding the great expense to resolve the small dissipation-range scales. I provide here a very brief review of a few significant mathematical results that bear on these questions.

A basic problem in the theory of the inviscid limit is that the principal a priori bounds on the velocity field follow from the decay of kinetic energy and such ‘$L^2$-bounds’ alone provide insufficient compactness to guarantee that inviscid limits are standard weak Euler solutions, even along suitable subsequences. Without additional conditions, the inviscid limits are more general objects such as ‘measure-valued Euler solutions’ (DiPerna & Majda Reference DiPerna and Majda1987; Wiedemann Reference Wiedemann2018) or clumsy ‘dissipative Euler solutions’ in the sense of Lions (Reference Lions1996). There are some researchers who take measure-valued solutions as the proper physical description and who investigate numerical schemes converging to such solutions (Fjordholm et al. Reference Fjordholm, Käppeli, Mishra and Tadmor2017). However, an important observation was made by Chen & Glimm (Reference Chen and Glimm2012) and Isett (Reference Isett2022) (see also Drivas & Eyink Reference Drivas and Eyink2019; Drivas & Nguyen Reference Drivas and Nguyen2019) who observed that empirical observations on scaling laws of energy spectra and velocity structure functions guarantee strong inviscid limits that are standard weak Euler solutions. The criterion can be stated in terms of instantaneous structure functions of absolute velocity increments, where, as in (3.19), strict power-law scaling is not required but only a power-law upper bound of the form

(3.68)\begin{equation} S_p(\boldsymbol{r},t)\leq C_p(t)U^p |\boldsymbol{r}/L|^{\zeta_p}, \quad \eta_p\leq |\boldsymbol{r}|\leq L, \end{equation}

where $\eta _p/L\to 0$ as $Re\to \infty.$ If the $p$th-order moment condition (3.20) holds and if also

(3.69)\begin{equation} \bar{C}_p:=\frac{1}{T}\int_0^T C_p(t)\, {\rm d}t <\infty \end{equation}

for some $p\geq 2$ and for some $Re$-independent values $\zeta _p>0$ and $\bar {C}_p>0,$ then it follows that strong limits of Navier–Stokes solutions exist in $L^p([0,T]\times {\mathbb {T}}^d)$ along subsequences $Re_k\to \infty$ and the limiting velocities are standard weak Euler solutions. Of course, these solutions may not be unique for the same initial data ${\boldsymbol {u}}_0$ and different subsequences $Re_k\to \infty$ may yield different limiting Euler solutions. These results have recently been extended to the Landau–Lifschitz equations (2.5) in a probabilistic setting (Eyink & Peng Reference Eyink and Peng2024), where the argument now yields a probability distribution ${\mathbb {P}}$ on space–time velocity fields in the limit $Re_k\to \infty$ whose realisations, with probability one, are standard weak Euler solutions with the specified initial data ${\boldsymbol {u}}_0$ (either deterministic or random).

Perhaps the most impressive results that have been achieved to date are on the problem of scalar turbulence, in particular, for the viscous-convective regime described by the theories of Batchelor (Reference Batchelor1959) and Kraichnan (Reference Kraichnan1968) for the limit of high Schmidt number. In the breakthrough work of Bedrossian et al. (Reference Bedrossian, Blumenthal and Punshon-Smith2022), the Onsager ‘ideal turbulence’ predictions have been derived from first principles for the vanishing diffusivity limit of a passive scalar advected by a 2-D Navier–Stokes flow (or a 3-D hyperviscous Navier–Stokes flow) at fixed Reynolds number, when both the scalar advection–diffusion equation (3.42) and the Navier–Stokes equation (2.4) are forced by low-wavenumber random sources, Gaussian and white-in-time. Bedrossian et al. (Reference Bedrossian, Blumenthal and Punshon-Smith2022) study the unique stationary measures $\mu ^{Re,Sc}$ of the joint velocity-scalar fields $({\boldsymbol {u}},c)$ in the limit as $Sc=\nu /D\to \infty$ and establish the existence of limiting measures $\mu ^{Re,\infty }$ for which the scalar concentration field $c$ has the Batchelor $k^{-1}$ spectrum and a constant scalar flux $\chi$ to infinite wavenumbers. Furthermore, Bedrossian et al. (Reference Bedrossian, Blumenthal and Punshon-Smith2022) show that the scalar advection equation obtained from (3.42) in the formal limit $D\to 0$ still has unique weak solutions in a natural integral form and the limiting measures $\mu ^{Re,\infty }$ are invariant under this limiting joint dynamics of $({\boldsymbol {u}},c)$ for $Sc\to \infty.$ The analogue of Onsager's $1/3$ criterion for dissipative weak solutions of the scalar in the Batchelor range advected by a smooth velocity field ${\boldsymbol {u}}$ is the statement that $c\notin B^{\sigma,\infty }_2$ for any $\sigma >0$ and Bedrossian et al. (Reference Bedrossian, Blumenthal and Punshon-Smith2022) prove that the scalar fields $c$ for the limiting measures $\mu ^{Re,\infty }$ indeed cannot possess such regularity. This is the only example I know where, in a quite physical setting, the analogue of Onsager's conjectures can be rigorously derived from first principles. Although probabilistic methods are not ‘intrinsic’ to this problem in the same sense as for spontaneous stochasticity, this is a beautiful example where a probabilistic approach is able to achieve physical results where purely deterministic techniques are still inadequate. Extending such results appears quite difficult. The enstrophy cascade of two-dimensional turbulence (Kraichnan Reference Kraichnan1967; Batchelor Reference Batchelor1969) is physically very similar to the Batchelor regime, but here the vorticity $\omega$ is an active scalar with $Sc=1$. The analogue of the Onsager theory for the 2-D enstrophy cascade (Eyink Reference Eyink2001; Lopes Filho, Mazzucato & Nussenzveig Lopes Reference Lopes Filho, Mazzucato and Nussenzveig Lopes2006) closely resembles the results derived by Bedrossian et al. (Reference Bedrossian, Blumenthal and Punshon-Smith2022) for the Batchelor regime, but now the limit $Re\to \infty$ must be tackled directly to prove anything rigorously.

It seems very difficult to extend the results of Bedrossian et al. (Reference Bedrossian, Blumenthal and Punshon-Smith2022) even to the passive scalar in the inertial-convective range, where likewise $Sc$ is held fixed and $Re\to \infty$ or, equivalently, Péclet number $Pe=Sc Re\to \infty.$ However, there has been interesting progress on this problem by Colombo, Crippa & Sorella (Reference Colombo, Crippa and Sorella2023), and especially by Armstrong & Vicol (Reference Armstrong and Vicol2023) and Burczak, Székelyhidi & Wu (Reference Burczak, Székelyhidi and Wu2023). None of these authors consider an advecting velocity field which satisfies the Navier–Stokes equations, but instead they construct ‘synthetic turbulence’ velocity fields which are divergence-free and such that the passive scalar satisfying the advection–diffusion equation (3.42) exhibits anomalous dissipation. The velocity fields in these constructions all have (in a suitable sense) the spatial Hölder regularity $C^h$ with $0< h<1$ that is expected for an inertial-range velocity field and the advected scalar exhibits anomalous dissipation in the sense that, with dimensionless diffusivity $\hat {D}=1/Pe,$

(3.70)\begin{equation} \limsup_{\hat{D}\to 0} \int_0^{\hat{T}} \,{\rm d}\hat{t} \int_{{\mathbb{T}}^2} \,{\rm d}^2\hat{x}\, \hat{D}|\hat{\boldsymbol{\nabla}}\hat{c}|^2 >0, \end{equation}

where $\widehat {({\cdot })}$ denotes outer-scale non-dimensionalisation. A striking result of these constructions is that there are distinct limit points $\hat {c}_*$ for the scalar field obtained along different subsequences $\hat {D}_k\to 0,$ which give different weak solutions of the ideal advection equation for the same initial data $\hat {c}_0.$ Similar non-uniqueness was found also by Huysmans & Titi (Reference Huysmans and Titi2023) for a passive scalar advected by a rougher divergence-free velocity field that is only $L^\infty$ in space, and here the non-unique solutions of the ideal advection equation conserve the scalar energy. Such non-uniqueness is somewhat surprising for passive scalar advection. Recall that the scalar advection equation for the Batchelor regime at $Sc\to \infty$ was shown by Bedrossian et al. (Reference Bedrossian, Blumenthal and Punshon-Smith2022) to have unique solutions and likewise the white-noise advection model of Kraichnan (Reference Kraichnan1968) in the inertial-convective regime is known to have unique weak solutions of the scalar advection equations (Lototskii & Rozovskii Reference Lototskii and Rozovskii2004; note that the latter solutions are ‘weak’ in the PDE sense but strong in the stochastic sense, with exactly one solution for each realisation of the advecting random velocity field). The recent results raise the intriguing possibility of Eulerian spontaneous stochasticity also for passive scalar advection in the inertial-convective range.

There are important differences between the methods in the above papers which may be appreciated by reading the originals, but which deserve to be emphasised briefly here. See also the extensive review by Armstrong & Vicol (Reference Armstrong and Vicol2023, § 1.1). The construction of Colombo et al. (Reference Colombo, Crippa and Sorella2023) combines alternating shear flows that focus in a singular manner at the final time step. A drawback of this approach is that the anomalous scalar dissipation occurs only at the final time and the velocity field has only one active scale at each time, both features very uncharacteristic of physical turbulence. Armstrong & Vicol (Reference Armstrong and Vicol2023) overcome these problems, synthesising a true multiscale velocity field for $h<1/3$ that produces anomalous scalar dissipation at all times. Their approach is an iterative homogenisation technique which has much in common with renormalisation group methods in physics, generating an eddy-diffusivity at each scale by integrating over the smaller scales. Furthermore, the synthetic turbulent field which is constructed by Armstrong & Vicol (Reference Armstrong and Vicol2023) has a very fundamental feature of true turbulent flows that small eddies are advected by larger eddies, which turns out to be essential in obtaining anomalous dissipation for the scalar at all times. To my knowledge, this is the first example of a synthetic turbulent velocity which captures the sweeping properties of a physical Navier–Stokes or Euler solution. Burczak et al. (Reference Burczak, Székelyhidi and Wu2023) have extended this approach by combining it with convex integration techniques to ensure that the advecting velocity field is in fact a weak solution of the Euler equations.

The methods of Colombo et al. (Reference Colombo, Crippa and Sorella2023) have recently been applied to prove the existence of kinetic energy dissipation anomaly for the forced 3-D Navier–Stokes equation (Bruè et al. Reference Bruè, Colombo, Crippa, De Lellis and Sorella2023; Bruè & De Lellis Reference Bruè and De Lellis2023). For a sequence of body forces ${\boldsymbol f}^{Re}$ that satisfy some modest uniform smoothness properties, such as

(3.71)\begin{equation} \sup_{Re > 0} \|\,{\boldsymbol f}^{Re}\|_{C([0,T],C^h({\mathbb{T}}^3))}<\infty, \end{equation}

for any $h\in (0,1),$ it has been proved that the viscous dissipation by the Navier–Stokes solution is anomalous for $\hat {\nu }=1/Re\to 0$ in the sense that

(3.72)\begin{equation} \limsup_{\hat{\nu}\to 0} \int_0^{\hat{T}} \,{\rm d}\hat{t} \int_{{\mathbb{T}}^3} \,{\rm d}^3\hat{x}\, \hat{\nu} |\hat{\boldsymbol{\nabla}}\hat{{\boldsymbol{u}}}|^2 >0. \end{equation}

Although the body force is not a smooth, large-scale forcing as usually assumed in turbulence theory, it is regular enough that the previous lim-sup vanishes for the linear Stokes equation, which lacks nonlinear energy cascade. Even more interestingly, the inviscid limits are found to be non-unique exactly as in the passive scalar case, with different subsequences of Reynolds numbers $Re_k\to \infty$ yielding distinct weak solutions of the Euler equations with the same initial data ${\boldsymbol {u}}_0$ and the same body force ${\boldsymbol f}=\lim _{Re\to \infty } {\boldsymbol f}^{Re}$ (Bruè et al. Reference Bruè, Colombo, Crippa, De Lellis and Sorella2023).

4.2.1. Regularisation of ultraviolet divergences

A key conclusion that may be be drawn from the brief summary of empirical observations in the previous section is that new UV divergences appear at the solid interfaces in wall-bounded turbulence for $Re\gg 1$. In fact, whenever the non-dimensionalised wall shear stress $\hat {\boldsymbol {\tau }}_w=\boldsymbol {\tau }_w/U^2=({1}/{Re})({\partial \hat {{\boldsymbol {u}}}}/{\partial \hat {n}})$ vanishes more slowly than the laminar rate $\propto 1/Re,$ then the non-dimensionalised velocity-gradient at the wall diverges:

(4.5)\begin{equation} \lim_{Re\to\infty} \frac{\partial \hat{{\boldsymbol{u}}}}{\partial \hat{n}}= \infty. \end{equation}

Note that such divergences appear at the wall even if there is only a ‘weak anomaly’ in the turbulent momentum balance. To obtain a dynamical description that remains valid in the limit $Re\to \infty$, these new divergences at the wall must be regularised along with the divergences of the non-dimensionalised gradients $\hat {\boldsymbol {\nabla }}\hat {{\boldsymbol {u}}}$ in the bulk of the flow.

A recent paper of Bardos & Titi (Reference Bardos and Titi2018) has initiated the mathematical investigation of Onsager's ‘ideal turbulence’ theory for wall-bounded flows, followed already by several works with improvements (Drivas & Nguyen Reference Drivas and Nguyen2018; Bardos et al. Reference Bardos, Titi and Wiedemann2019; Chen, Liang & Wang Reference Chen, Liang and Wang2022). These works employ as regulariser a modification of the spatial coarse-graining of the velocity defined in (3.2). It is convenient to assume that the filter kernel $G$ is supported in a ball of radius 1, so that the definition of $\bar {{\boldsymbol u}}_\ell ({\boldsymbol x},t)$ makes sense for points ${\boldsymbol x}\in \varOmega$ with distance at least $\ell$ from the boundary $\partial \varOmega.$ To eliminate also the divergences of the velocity-gradients at the wall and to obtain a well-defined coarse-grained velocity, one may also smoothly ‘window out’ eddies at distances $< h$ to the wall, with $h>\ell.$ This can be accomplished by taking a smooth windowing function $\theta _{h,\ell }(\delta )$ with the properties that

(4.6)\begin{equation} \theta_{h,\ell}(\delta)=\left\{\begin{array}{@{}ll} 0, & \delta< h, \\ 1, & \delta>h+\ell \end{array}\right. \end{equation}

and $\theta _{h,\ell }(\delta )$ monotone increasing on the interval $[h,h+\ell ]$ (see figure 7). Finally, one defines $\eta _{\ell,h}({\boldsymbol x}):=\theta _{\ell,h}(d({\boldsymbol x}))$ and for all ${\boldsymbol x}\in \varOmega$,

(4.7)\begin{equation} \tilde{\boldsymbol{u}}_{\ell,h}({\boldsymbol x},t)=\eta_{\ell,h}({\boldsymbol x}) \bar{\boldsymbol{u}}_\ell({\boldsymbol x},t), \end{equation}

where $d({\boldsymbol x})$ measures the distance of ${\boldsymbol x}\in \varOmega$ to the boundary:

(4.8)\begin{equation} {\rm d}({\boldsymbol x})=\inf_{{\boldsymbol y}\in\partial\varOmega}|{\boldsymbol x}-{\boldsymbol y}|. \end{equation}

With this definition, $\boldsymbol {\nabla } \,\textrm {d}({\boldsymbol x}) = {\boldsymbol n}(\kern 1.5pt {\boldsymbol y}_{\boldsymbol x}) :={\boldsymbol n}({\boldsymbol x}),$ where ${\boldsymbol n}(\kern 1.5pt {\boldsymbol y})$ is the inward-pointing unit normal vector at a point ${\boldsymbol y}\in \partial \varOmega$ and ${\boldsymbol y}_{\boldsymbol x}\in \partial \varOmega$ is the point at which the infimum in (4.8) is achieved for each ${\boldsymbol x}\in \varOmega.$ See Bardos & Titi (Reference Bardos and Titi2018). The coarse-grained velocity defined by (4.7) may be described picturesquely as the fluid velocity seen by an observer who is myopic and who also has tunnel vision, with parameter $\ell$ characterising the blurriness of their eyesight and $h$ their loss of peripheral vision.

Figure 7. Window function $\theta _{h,\ell }$ used to screen from observation fluid eddies near the wall.

It is important to emphasise here that this specific choice of regularisation is not essential to the theory. Just as in quantum field theory, there are many possible regularisers to eliminate UV divergences (Gross Reference Gross1976), and it is ultimately a matter of convenience and ease of application which to choose for a particular problem. Onsager in his unpublished research notes in fact explored a different regularisation method for plane-parallel channel flow based on expansion of the velocity field in eigenmodes of the linear Stokes operator. See the folder of notes in the online Onsager Archive at https://ntnu.tind.io/record/121186, pp. 4–8, where Onsager solves the eigenproblem for the Stokes operator ${\mathit A}_D\boldsymbol {w}:={P}(-\Delta )\boldsymbol {w}$ with Dirichlet b.c. $\boldsymbol {w}=\boldsymbol {0},$ where ${P}$ is the Leray projection onto divergence-free fields:

(4.9)\begin{equation} {A}_D\boldsymbol{w}_n = \lambda_n \boldsymbol{w}_n, \quad n=0,1,2,\ldots. \end{equation}

Identical results were published later by Rummler (Reference Rummler1997). Onsager clearly had the idea to expand the turbulent velocity field in the channel into such eigenmodes, thus regularising all UV divergences. With a smooth exponential cutoff, this would correspond to taking

(4.10)\begin{equation} \bar{{\boldsymbol{u}}}_\ell = \sum_{n=0}^\infty e^{-\lambda_n\ell^2} c_n \boldsymbol{w}_n \end{equation}

for some regularisation scale $\ell,$ where ${\boldsymbol {u}}={\sum }_{n=0}^\infty c_n\boldsymbol {w}_n.$ With this approach, $\boldsymbol {\nabla }\boldsymbol{\cdot}\bar {{\boldsymbol {u}}}_\ell =0$ in $\varOmega$ exactly and $\bar {{\boldsymbol {u}}}_\ell =\boldsymbol {0}$ on $\partial \varOmega.$ Note that the above expansion is equivalent to defining the regularised velocity field by the auxiliary initial-value problem for the Stokes equation:

(4.11)\begin{equation} \frac{\partial\bar{{\boldsymbol{u}}}_\ell}{\partial\ell^2} =-{A}_D\bar{{\boldsymbol{u}}}_\ell, \quad \left.\bar{{\boldsymbol{u}}}_\ell\right|_{\ell=0}={\boldsymbol{u}}. \end{equation}

Exactly this PDE approach to spatial filtering in wall-bounded domains was proposed recently by Johnson (Reference Johnson2022), which can be regarded as a version of the heat-kernel regularisation of Isett & Oh (Reference Isett and Oh2016), adapted to manifolds with boundaries. This PDE approach is more flexible than the original eigenmode-expansion of Onsager, since it is difficult for general domains to calculate the eigenfunctions explicitly. Note that one may apply this Stokes regularisation also with the operator ${A}_N$ for Neumann b.c. $\partial \boldsymbol {w}/\partial n=\boldsymbol {0}$ on $\partial \varOmega,$ so that $\int _\varOmega \bar {{\boldsymbol {u}}}_\ell \, \textrm {d}V=\int _\varOmega {\boldsymbol {u}}\, \textrm {d}V.$

In general, the necessity of some spatial filtering or coarse-graining closely connects the Onsager theory with the turbulence modelling method of LES. Note that wall-bounded turbulence is currently one of the most challenging areas of LES research and the subject of intense effort (Bose & Park Reference Bose and Park2018). Many other types of spatial filtering or regularisation have been explored in that context, including regularisations specified by elliptic PDE problems such as

(4.12)\begin{equation} \bar{{\boldsymbol{u}}}_\ell - \frac{\partial}{\partial x_k} \left(\ell^2 \frac{\partial\bar{{\boldsymbol{u}}}_\ell}{\partial x_k}\right)={\boldsymbol{u}}, \quad \text{in }\varOmega, \end{equation}

for some chosen length scale $\ell (\boldsymbol {x}),$ which may now be chosen to depend on position. See Germano (Reference Germano1986a,Reference Germanob), Bose & Moin (Reference Bose and Moin2014) and Bae et al. (Reference Bae, Lozano-Durán, Bose and Moin2019). As stated in the quote of Onsager (Reference Onsager1949), the integral length in wall-bounded turbulence is generally proportional to the distance from the wall, or $L(\boldsymbol {x})\sim \kappa \,\textrm {d}(\boldsymbol {x})$ with $\kappa$ the so-called Kármán constant. Thus, one might be tempted to take $\ell (\boldsymbol {x})\propto L(\boldsymbol {x}),$ which would correspond to integrating out all eddies smaller than the local integral length. This so-called ‘wall-resolved LES’ is, however, almost as computationally expensive as DNS of the full Navier–Stokes equation (Yang & Griffin Reference Yang and Griffin2021), so that one is generally interested in taking $\ell (\boldsymbol {x})\gg L(\boldsymbol {x})$ for practical applications. This problem of coarse-graining and LES modelling in the presence of solid walls is an important application area where mathematicians and engineers can very productively interact.

4.2.2. Coarse-grained equations

The filtering procedures employed in LES lead to effective coarse-grained equations which contain quantities unclosed in terms of the coarse-grained velocity itself, which may then be modelled. The same is true also of the regularisation procedures used by mathematicians to study the infinite-Reynolds limit. For example, the combined spatial filtering and windowing introduced by Bardos & Titi (Reference Bardos and Titi2018) when applied to the Navier–Stokes equation (2.4) leads to an exact regularised equation for the coarse-grained field $\tilde {\boldsymbol {u}}_{\ell,h}$ of the form

(4.13)\begin{equation} \frac{\partial \tilde{\boldsymbol{u}}_{\ell,h} }{\partial t} +\boldsymbol{\nabla} \boldsymbol{\cdot} \left[\tilde{\tau}_{\ell,h}(\boldsymbol{u},\boldsymbol{u}) + \tilde{\boldsymbol{u}}_{\ell,h}\tilde{\boldsymbol{u}}_{\ell,h} + \tilde{p}_{\ell,h} \boldsymbol{I} - \nu\widetilde{\boldsymbol{\nabla} \boldsymbol{u}}_{\ell,h}\right]=-{\boldsymbol f}_{\ell,h}. \end{equation}

Here, the turbulent (or subgrid) stress due to eddies of size $<\ell$ may be defined as usual by

(4.14)\begin{equation} \tilde{\tau}_{\ell,h}(\boldsymbol{u},\boldsymbol{u}) = (\widetilde{\boldsymbol{uu}})_{\ell,h}-\tilde{\boldsymbol{u}}_{\ell,h}\tilde{\boldsymbol{u}}_{\ell,h}. \end{equation}

In addition, a new inertial drag force appears associated with the eliminated near-wall eddies

(4.15)\begin{equation} {\boldsymbol f}_{\ell,h}=-\theta_{\ell,h}'(d({\boldsymbol x}))\,{\boldsymbol n}({\boldsymbol x}) \boldsymbol{\cdot} \left[ (\overline{\boldsymbol{uu}})_\ell + \bar{p}_\ell \boldsymbol{I}-\nu \boldsymbol{\nabla}\bar{\boldsymbol{u}}_\ell \right], \end{equation}

and represents momentum transfer to the unresolved near-wall eddies. It is non-vanishing only for $h< d(\boldsymbol {x})< h+\ell.$ This force includes resolved viscous momentum diffusion, but nonlinear transfer dominates for $Re\gg 1$ at fixed $\ell,$ $h.$ Mathematically, the regularised equation (4.13), when considered for all possible choices of $h>\ell$, is equivalent to the standard weak formulation of the incompressible Navier–Stokes equation in a domain with boundary (see Boyer & Fabrie Reference Boyer and Fabrie2012, Ch. V.1.2).

If (4.13) were considered for the purpose of LES modelling, both the subgrid stress $\tilde {\boldsymbol {\tau }}_{\ell,h}$ and the inertial drag force ${\boldsymbol f}_{\ell,h}$ would need to be somehow closed or parametrised. Indeed, one can expect that these quantities have universal statistical properties independent of the small-scale dissipation and of the detailed properties of the wall, if the distance $h$ is chosen in the inertial sublayer and length scale $\ell$ is chosen also sufficiently small. In addition, however, another quantity must be modelled which appears in the coarse-grained mass balance:

(4.16)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot} \tilde{{\boldsymbol u}}_{\ell,h}=\tilde{\sigma}_{\ell,h}, \end{equation}

which I call the inertial mass source

(4.17)\begin{equation} \tilde{\sigma}_{\ell,h}:=\theta_{h,\ell}'(d({\boldsymbol x})) {\boldsymbol n}({\boldsymbol x}) \boldsymbol{\cdot}\bar{\boldsymbol u}_\ell. \end{equation}

This quantity measures the mass-exchange with the unresolved near-wall eddies and it is also non-vanishing only for $h< d({\boldsymbol x})< h+\ell,$ with furthermore a vanishing space integral $\int _{h< d< h+\ell } \tilde {\sigma }_{\ell,h} \, \textrm {d}V=0.$ The windowing operation has thus introduced effective ‘compressibility’ so that the Poisson equation for coarse-grained pressure becomes

(4.18)\begin{equation} -\!\triangle\tilde{p}_{\ell,h}= \partial_t \tilde{\sigma}_{\ell,h} +\boldsymbol{\nabla}\boldsymbol{\nabla}:\left[\tilde{\boldsymbol{u}}_{\ell,h}\tilde{\boldsymbol{u}}_{\ell,h} +\tilde{\boldsymbol{\tau}}_{\ell,h} -\nu\widetilde{\boldsymbol{\nabla} \boldsymbol{u}}_{\ell,h}\right] -\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{f}_{h,\ell}, \end{equation}

which involves derivatives of all three modelled quantities but which can yield the resolved pressure by applying standard Poisson solvers. Similar effects appear with other coarse-graining methods, e.g. the velocity field from the elliptic filtering (4.12) satisfies an equation $\boldsymbol {\nabla }\boldsymbol{\cdot}\bar {{\boldsymbol {u}}}_\ell =\bar {\sigma }_\ell$ analogous to (4.16) with

(4.19)\begin{equation} \bar{\sigma}_\ell = \overline{\left(\frac{\partial}{\partial x_k} \left[\boldsymbol{\nabla}\ell^2 \boldsymbol{\cdot}\frac{\partial\bar{{\boldsymbol{u}}}_\ell}{\partial x_k}\right]\right)}_\ell. \end{equation}

In this case, however, it is not true that $\int _{\varOmega } \bar {\sigma }_{\ell } \, \textrm {d}V=0$ unless $\partial \bar {{\boldsymbol {u}}}_\ell /\partial n=0$ at $\partial \varOmega.$

The regularised equation (4.13) involves two arbitrary lengths $\ell$ and $h$ and, as well, three arbitrary functions $G,$ $\eta$ and $d.$ Other choices of distance function might be more useful for some purposes, e.g. $d({\boldsymbol x})=\min \{|y-H|,|y+H|\}$ could be useful in an iterative RG analysis of a rough-wall channel flow, with $H$ the channel half-width, to establish universal statistics in the ‘inertial sublayer’. A similar type of ‘time windowing’ was employed in the recent RG analysis of Lagrangian spontaneous stochasticity (Eyink & Bandak Reference Eyink and Bandak2020), where it corresponds to ignoring non-universal initial times of the particle position histories. The ‘principle of renormalisation group invariance’ is that no objective physics can depend upon these arbitrary quantities introduced for the purpose of regularisation (Gross Reference Gross1976; Eyink Reference Eyink2018b). The present example is a case of a several-parameter renormalisation group involving changes of the entire regularisation scheme, which was encountered already in quantum field theory (Stückelberg & Petermann Reference Stückelberg and Petermann1953; Stevenson Reference Stevenson1981; Peterman Reference Peterman1982) and which has been applied since to PDEs, including boundary-value problems (Chen, Goldenfeld & Oono Reference Chen, Goldenfeld and Oono1996; Kovalev & Shirkov Reference Kovalev and Shirkov1999). A key idea in RG methods is that arbitrariness in regularisation parameters may be exploited by choosing them in some optimal way to deduce non-trivial consequences. I describe some applications of that principle in the following section.

4.2.3. Momentum cascade in space

Most of the prior work on the Onsager theory for wall-bounded flows has considered energy cascade and energy dissipation (Bardos & Titi Reference Bardos and Titi2018; Drivas & Nguyen Reference Drivas and Nguyen2018; Bardos et al. Reference Bardos, Titi and Wiedemann2019; Chen et al. Reference Chen, Liang and Wang2022), but I prefer to begin first with the momentum anomaly suggested by Taylor (Reference Taylor1915). I shall employ the regularisation by spatial filtering and windowing introduced by Bardos & Titi (Reference Bardos and Titi2018), which leads to the important concept of spatial momentum cascade. To avoid any confusion, I emphasise at the outset that the notion of momentum cascade arising in the Onsager theory does not coincide with the one which is now standard in the literature on wall-bounded turbulence (Tennekes & Lumley Reference Tennekes and Lumley1972; Jiménez Reference Jiménez2012). I shall discuss the essential differences below. My analysis shall follow closely the work of Quan & Eyink (Reference Quan and Eyink2022a), where momentum balance in the limit $Re\to \infty$ is treated. It is also possible to extend this analysis to finite Reynolds numbers and some beginning steps in this direction have been taken by Eyink, Kumar & Quan (Reference Eyink, Kumar and Quan2022), following the ideas of Drivas & Eyink (Reference Drivas and Eyink2019) for energy cascade, but I consider here only the regime $Re\gg 1.$

The work of Quan & Eyink (Reference Quan and Eyink2022a) followed the same RG strategy as in prior works on the Onsager theory such as Duchon & Robert (Reference Duchon and Robert2000), by considering the limit $Re\to \infty$ for both the fine-grained description and the coarse-grained description with regularisation scales $\ell,$ $h.$ Non-trivial consequences can be deduced simply by requiring that both descriptions lead to the same observable conclusions and exploiting the freedom to vary $h,\ell$ by taking $h,$ $\ell \to 0.$ The specific example analysed by Quan & Eyink (Reference Quan and Eyink2022a) was external flow around a finite, smooth body $B,$ as considered in the famous paradox of d'Alembert (Reference d'Alembert1749, Reference d'Alembert1768). As seen from the empirical results reviewed in § 4.1, this is probably the simplest flow which provides evidence for a strong dissipation anomaly. However, the analysis of Quan & Eyink (Reference Quan and Eyink2022a) carries over straightforwardly to related more complex flows, such as turbulent flows through pipes with hydraulically rough (but mathematically smooth) walls. Since only mathematically smooth surfaces are considered, we are investigating the limit $Re=UL/\nu \gg 1$ as achieved by taking $L$ very large (where $L$ would be a length such as the diameter of the body or the radius of the pipe). In the case of a hydraulically rough wall, we keep the ratio $\hat {k}=k/L$ fixed, so that the body remains geometrically similar as $Re$ increases.

The analysis of Quan & Eyink (Reference Quan and Eyink2022a) (with some pedagogical simplifications introduced by Eyink (Reference Eyink2023)) begins with the fine-grained momentum balance/ Navier–Stokes equation smeared against a smooth space–time test function $\boldsymbol {\varphi }(\boldsymbol {x},t):$

(4.20)$$\begin{gather} \int_0^T\int_{\varOmega} \left[\left(\partial_t\boldsymbol{\varphi}+\nu \Delta \boldsymbol{\varphi}\right) \boldsymbol{\cdot} \boldsymbol{u}^{\nu} + \boldsymbol{\nabla}\boldsymbol{\varphi}\boldsymbol{:}\left(\boldsymbol{u}^{\nu}\otimes\boldsymbol{u}^{\nu} + p^{\nu}\boldsymbol{I}\right) \right] \, {\rm d}V\, {\rm d}t\nonumber\\ = \int_0^T\int_{\partial\varOmega} \left[\boldsymbol{\tau}_w^\nu\boldsymbol{\cdot}\boldsymbol{\varphi} - p_w^\nu(\boldsymbol{n}\boldsymbol{\cdot}\boldsymbol{\varphi}) \right] \,{\rm d}S\,{\rm d}t. \end{gather}$$

Note that I am using the standard non-dimensionalisation with $\hat {\nu }=1/Re,$ but I drop hats $\widehat {({\cdot })}$ to keep notation simple. In contrast to usual discussions of weak solutions in domains $\varOmega$ with boundaries, however, the test functions adopted in this study need not vanish near the surface $\partial \varOmega,$ and thus the smeared balance equation gets surface contributions both from the skin friction $\boldsymbol {\tau }_w$ and also from $p_w,$ the pressure at the wall. Under the assumption of strong $L^2$-convergence in the space–time domain $[0,T]\times \varOmega$ as $\nu \to 0$ (see § 3.2.3), we see that the left-hand side converges and thus also the limits exist, $\boldsymbol {\tau }_w=\lim _{\nu \to 0} \boldsymbol {\tau }_w^\nu,$ $p_w=\lim _{\nu \to 0} p_w^\nu,$ as distributions on the surface $\partial \varOmega,$ so that

(4.21)$$\begin{align} &\int_0^T\int_{\varOmega} \left[ \partial_t\boldsymbol{\varphi} \boldsymbol{\cdot} \boldsymbol{u} + \boldsymbol{\nabla}\boldsymbol{\varphi}\boldsymbol{:}(\boldsymbol{u}\otimes\boldsymbol{u} + p\boldsymbol{I}) \right] \, {\rm d}V \,{\rm d}t\nonumber\\ &\quad = \int_0^T\int_{\partial\varOmega} \left[\boldsymbol{\tau}_w \boldsymbol{\cdot}\boldsymbol{\varphi} - p_w (\boldsymbol{n}\boldsymbol{\cdot}\boldsymbol{\varphi}) \right] \,{\rm d}S\,{\rm d}t. \end{align}$$

The limiting fields $({\boldsymbol {u}},p)$ in $[0,T]\times \varOmega$ thus constitute a weak Euler solution, as seen by restricting (4.21) to test functions $\boldsymbol {\varphi }$ whose supports vanish on $\partial \varOmega,$ but (4.21) gives additional information at the surface inherited from the Navier–Stokes solutions.

The key step is now to consider the corresponding smeared version of the coarse-grained momentum balance (4.13):

(4.22)$$\begin{gather} \int_0^T\int_{\varOmega}\partial_t\boldsymbol{\varphi}\boldsymbol{\cdot}(\eta_{h,\ell}\bar{\boldsymbol{u}}_{\ell}^\nu)\,{\rm d}V\,{\rm d}t +\int_0^T\int_{\varOmega}\boldsymbol{\nabla}\boldsymbol{\varphi}\boldsymbol{:}\eta_{h,\ell}(\bar{\boldsymbol{T}}_{\ell}^\nu+\bar{p}_{\ell}^\nu\boldsymbol{I}-\boldsymbol{\nabla}\bar{\boldsymbol{u}}_{\ell}^\nu) \,{\rm d}V\,{\rm d}t\nonumber\\ =-\int_0^T\int_{\varOmega}\boldsymbol{\varphi}\boldsymbol{\cdot}(\bar{\boldsymbol{T}}_{\ell}^\nu \boldsymbol{\cdot} \boldsymbol{\nabla}\eta_{h,\ell} + \bar{p}_{\ell}^\nu\boldsymbol{\nabla}\eta_{h,\ell})\,{\rm d}V\,{\rm d}t, \end{gather}$$

where I have defined $\bar {\boldsymbol {T}}_{\ell }^\nu :=\overline {({\boldsymbol {u}}^\nu {\boldsymbol {u}}^\nu )}_\ell$ and also I have recalled the definition $\tilde {a}_{h,\ell }=\eta _{h,\ell }\bar {a}_\ell$ for any quantity $a.$ Taking the double limits, first $\nu \to 0$ and subsequently $h,\ell \to 0,$ then it is easy to see that the left-hand side of (4.22) converges exactly to the left-hand side of (4.21). In that case, however, the right-hand side of (4.22) must converge also to the right-hand side of (4.21). We thereby get results on spatial cascade of wall-parallel momentum:

(4.23)\begin{equation} \lim_{h,\ell\to 0} -\boldsymbol{n}\times \bar{\boldsymbol{T}}_{\ell}\boldsymbol{\cdot}\boldsymbol{\nabla}\eta_{h,\ell} =\boldsymbol{n}\boldsymbol{\times} \boldsymbol{\tau}_w \end{equation}

with $\bar {\boldsymbol {T}}_{\ell }:=\overline {({\boldsymbol {u}}{\boldsymbol {u}})}_\ell$, and on spatial cascade of wall-normal momentum:

(4.24)\begin{equation} \lim_{h,\ell\to 0} \boldsymbol{n}\boldsymbol{\cdot} \left[ \bar{\boldsymbol{T}}_{\ell}\boldsymbol{\cdot}\boldsymbol{\nabla}\eta_{h,\ell}+ \bar{p}_{\ell}\boldsymbol{\nabla}\eta_{h,\ell}\right]=p_w, \end{equation}

both convergence statements interpreted in the sense of distributions on $[0,T]\times \partial \varOmega.$ To obtain these results, I had to make suitable choices for test functions $\boldsymbol {\varphi }$ and, in particular, that allowed us to neglect $\lim _{h,\ell \to 0} -\boldsymbol {n}\times \bar {p}_{\ell }\boldsymbol {\nabla }\eta _{h,\ell } =0$ in the first statement. The physical meaning of these results are straightforward. The first result (4.23) simply states that the spatial cascade of transverse momentum to the wall via turbulent nonlinear advection in the limiting Euler solution must match onto the direct infinite-$Re$ limit of the viscous skin friction at the wall. Recall that $\boldsymbol {\nabla }\eta _{h,\ell }(\boldsymbol {x})=\boldsymbol {n}(\boldsymbol {x}) \theta _{h,\ell }'(d(\boldsymbol {x})) \simeq \boldsymbol {n}(\boldsymbol {x}) \delta (d(\boldsymbol {x})-h)$, so that the quantity on the left-hand side of (4.23) indeed represents advective momentum flux towards the wall at distance $h$ from the wall. Likewise, the second result (4.24) states that the spatial cascade of wall-normal momentum away from the wall (note the change of sign) in the limiting Euler solution must match onto the direct infinite-$Re$ limit of the kinematic pressure at the wall. These two results are exact analogues of that derived for scale-cascade by Duchon & Robert (Reference Duchon and Robert2000), i.e. the matching of the expressions (3.29), (3.30) for nonlinear energy cascade in the limiting Euler solution and the direct infinite-$Re$ limit of viscous energy dissipation (3.32).

Unlike the scale-cascade of kinetic energy studied by Duchon & Robert (Reference Duchon and Robert2000), which can be understood very roughly as a deterministic version of the cascade of Kolmogorov (Reference Kolmogorov1941a,Reference Kolmogorovb), in our case, the meaning of ‘momentum cascade’ is distinctly different from the traditional one in the literature on wall-bounded turbulence. I may recall that Tennekes & Lumley (Reference Tennekes and Lumley1972) and Jiménez (Reference Jiménez2012), and many others, posit the existence in canonical wall-bounded flows (pipe, channel, boundary-layer) of an ‘inertial layer’ where the Reynolds stress or mean momentum flux $-\langle u_x u_y\rangle$ ($x$ wall-parallel, $y$ wall-normal) is approximately constant and equal to $u_\tau ^2=\langle \tau _{w,xy}\rangle,$ the mean skin friction. This looks superficially similar to (4.23), except that the latter is a deterministic result for the weak Euler solutions obtained in the infinite-$Re$ limit. However, the result (4.23) apparently becomes trivial in all of these canonical flows where there is only a weak anomaly and $\boldsymbol {\tau }_w=\boldsymbol {0}!$ Conversely, both the traditional and the deterministic cascades occur in a turbulent pipe flow with a hydraulically rough wall, but at completely different wall normal distances. Indeed, the standard mean momentum cascade then holds in the traditional inertial layer $k\ll y \ll H,$ whereas result (4.23) holds in the range $\nu /u_\tau \ll h\ll k,$ i.e. within the roughness sublayer! Recall that our limit is one in which $Re_k\to \infty$ so that our flows are in the fully rough regime and completely inertial-dominated turbulence exists between the roughness elements.

So far, I have not established a result equivalent to Onsager's $1/3$ Hölder exponent result. As glimpsed already by Taylor (Reference Taylor1915), a corresponding condition has to do with continuity of the velocity at the wall. In fact, Bardos & Titi (Reference Bardos and Titi2018); Bardos et al. (Reference Bardos, Titi and Wiedemann2019) understood that the essential condition has to do with the vanishing of the normal velocity component approaching the wall, since it is the normal velocity which transports momentum and other conserved quantities to and from the wall. The specific condition that I consider is that of Drivas & Nguyen (Reference Drivas and Nguyen2018):

(4.25)\begin{equation} \lim_{\delta\to 0} \|\boldsymbol{n}\boldsymbol{\cdot}{\boldsymbol{u}}\|_{L^2([0,T],L^\infty(\varOmega_\delta))} = 0, \end{equation}

which means that the maximum wall-normal velocity in the $\delta$-neighbourhood $\varOmega _\delta =\{\boldsymbol {x}\in \varOmega : d(\boldsymbol {x})<\delta \}$ of the boundary $\partial \varOmega$ must vanish as $\delta \to 0.$ With this assumption, the advective transport of momentum to the wall can be shown without much difficulty to vanish:

(4.26)\begin{equation} \lim_{h,\ell\to 0} \bar{\boldsymbol{T}}_{\ell}\boldsymbol{\cdot}\boldsymbol{\nabla}\eta_{h,\ell}=\boldsymbol{0}, \end{equation}

see Quan & Eyink (Reference Quan and Eyink2022a) and Eyink (Reference Eyink2023) for details. In that case, the previous results simplify drastically. The first result (4.23) reduces to

(4.27)\begin{equation} \boldsymbol{\tau}_w=\boldsymbol{0}, \end{equation}

which is the statement that condition (4.25) rules out a strong momentum anomaly. The second result (4.24) furthermore becomes

(4.28)\begin{equation} \lim_{h,\ell\to 0} \bar{p}_\ell\theta_{h,l}'(d(\boldsymbol{x})) = p_w. \end{equation}

This is essentially a statement about commutation of two limits: the same pressure field at the wall is obtained either by taking the $Re\to \infty$ limit $p_w$ of the pressure $p_w^{Re}$ directly at the wall or instead by taking $Re\to \infty$ first and then taking the limit of the pressure $p$ of the Euler solution in the flow interior at points approaching to the wall. This commutation of limits may be loosely termed ‘continuity of pressure at the wall’.

The last two results seem very satisfactory. As I have already discussed in § 4.1, the mean skin friction $\langle \boldsymbol {\tau }\rangle$ (e.g. a long-time average) seems to vanish quite generally. Thus, the absence of a strong momentum anomaly (4.27) looks consistent with experiment. The second result (4.28) on the continuity of pressure at the wall is very good news for LES modelling. As I have discussed in § 4.1, the limiting drag force on a moving body at high-$Re$ seems to arise entirely from pressure forces (form drag) with negligible drag arising from viscous skin friction. Thus, the result (4.28) means that the asymptotic drag on a body can be successfully calculated from the weak Euler solution, with the limiting pressure on the body correctly calculated from the Euler pressure in the flow interior after taking the limit approaching the body. However, it is a bit premature to take this as good news! First, there is currently no consensus how to determine the pressure uniquely for the weak Euler solution and in particular what boundary condition should be imposed to replace the Neumann condition in (4.3) for the Navier–Stokes pressure (Bardos & Titi Reference Bardos and Titi2022; Bardos, Boutros & Titi Reference Bardos, Boutros and Titi2023; De Rosa, Latocca & Stefani Reference De Rosa, Latocca and Stefani2023, Reference De Rosa, Latocca and Stefani2024). Second, and more seriously, we shall see in § 4.2.6 that the assumption that $\boldsymbol {\tau }_w\equiv \boldsymbol {0}$ identically leads to the paradoxical conclusion that all drag vanishes in the limit $Re\to 0$ and that the Navier–Stokes solution converges to the smooth potential Euler solution of d'Alembert (Reference d'Alembert1749, Reference d'Alembert1768) with no drag.

4.2.4. Energy cascade in space and in scale

Kinetic energy balance can be analysed in a very similar manner. Quan & Eyink (Reference Quan and Eyink2022b) considered external flow past a body $B$ with fluid velocity constant at infinity, so that the total kinetic energy was infinite. The flow was thus divided into a potential part ${\boldsymbol {u}}_\phi$ and a rotational part ${\boldsymbol {u}}_\omega ={\boldsymbol {u}}-{\boldsymbol {u}}_\phi,$ so that the ‘relative energy’ $E_{rel}(t)=\int _\varOmega [{\boldsymbol {u}}_\phi \boldsymbol{\cdot}{\boldsymbol {u}}_\omega +\tfrac {1}{2}|{\boldsymbol {u}}_\omega |^2] \, \textrm {d}V$ (heuristically, total energy minus energy of the potential flow) remains finite. To keep notation simple, however, I consider here internal flow in a bounded domain $\varOmega$ as did Bardos & Titi (Reference Bardos and Titi2018), Bardos et al. (Reference Bardos, Titi and Wiedemann2019), Drivas & Nguyen (Reference Drivas and Nguyen2018) and Chen et al. (Reference Chen, Liang and Wang2022).

The fine-grained kinetic energy balance for incompressible Navier–Stokes when smeared with a scalar test function $\varphi$ is easily calculated to be

(4.29)$$\begin{gather} \int_0^T\int_{\varOmega}(\partial_t+\nu\Delta) \varphi \times \tfrac{1}{2}|\boldsymbol{u}^\nu|^2\,{\rm d}V\, {\rm d}t + \int_0^T\int_{\varOmega}\boldsymbol{\nabla}\varphi\boldsymbol{\cdot} \left(\tfrac{1}{2}|\boldsymbol{u}^\nu|^2+p^\nu\right)\boldsymbol{u}^\nu \, {\rm d}V\,{\rm d}t\nonumber\\ + \int_0^T\int_{\varOmega} \nu(\boldsymbol{\nabla}\boldsymbol{\nabla}\varphi)\boldsymbol{:} {\boldsymbol{u}}^\nu{\boldsymbol{u}}^\nu \, {\rm d}V\, {\rm d}t = \int_0^T\int_{\varOmega}\varphi \, \varepsilon^\nu\,{\rm d}V\,{\rm d}t, \end{gather}$$

with $\varepsilon ^\nu =2\nu |\boldsymbol {S}^\nu |^2$ the viscous energy dissipation. As in the previous section, I allow the smooth test function $\varphi$ to be non-vanishing on the boundary $\partial \varOmega$ but no boundary terms arise after integration by parts because of the stick b.c. Assuming strong convergence of ${\boldsymbol {u}}^\nu,p^\nu$ in the bulk to ${\boldsymbol {u}},p$ as $\nu \to 0,$ it is easy just as in a periodic domain to obtain the inviscid limit

(4.30) $$\begin{align} &\int_0^T\int_{\varOmega} \partial_t\varphi \times \tfrac{1}{2}|\boldsymbol{u}|^2\,{\rm d}V\, {\rm d}t + \int_0^T\int_{\varOmega}\boldsymbol{\nabla}\varphi\boldsymbol{\cdot} \left(\tfrac{1}{2}|\boldsymbol{u}|^2+p\right)\boldsymbol{u} \, {\rm d}V\,{\rm d}t\nonumber\\ &\quad \quad = \int_0^T\int_{\varOmega}\varphi \, D \,{\rm d}V\,{\rm d}t, \end{align}$$

where

(4.31)\begin{equation} \lim_{\nu\to 0} \int_0^T\int_{\varOmega}\varphi \varepsilon^\nu\,{\rm d}V\,{\rm d}t = \int_0^T\int_{\varOmega}\varphi \, D \,{\rm d}V\,{\rm d}t. \end{equation}

One can also write the inviscid coarse-grained kinetic energy balance (3.10), window out the near-wall region and then smear with the test function $\varphi$ to obtain

(4.32) $$\begin{align} &\int_0^T\int_{\varOmega} \partial_t\varphi \times \eta_{h,\ell} \tfrac{1}{2}|\bar{{\boldsymbol{u}}}_\ell|^2\,{\rm d}V\, {\rm d}t + \int_0^T\int_{\varOmega}\boldsymbol{\nabla}\varphi\boldsymbol{\cdot} \eta_{h,\ell} \bar{\boldsymbol{J}}_e \, {\rm d}V\,{\rm d}t\nonumber\\ &\quad \quad = \int_0^T\int_{\varOmega}\varphi \eta_{h,\ell} \varPi_\ell\,{\rm d}V\,{\rm d}t - \int_0^T\int_{\varOmega}\varphi \bar{\boldsymbol{J}}_\ell \boldsymbol{\cdot}\boldsymbol{\nabla}\eta_{h,\ell} \,{\rm d}V\,{\rm d}t, \end{align}$$

where I have introduced the spatial flux of coarse-grained kinetic energy:

(4.33)\begin{equation} \bar{\boldsymbol{J}}_e =\left(\tfrac{1}{2}|\bar{{\boldsymbol{u}}}_\ell|^2+\bar{p}_\ell\right)\bar{{\boldsymbol{u}}}_\ell + \boldsymbol{\tau}_\ell\boldsymbol{\cdot}\bar{{\boldsymbol{u}}}_\ell. \end{equation}

Taking the limit $h,\ell \to 0,$ the left-hand side of the coarse-grained balance (4.32) coincides with the left-hand side of the fine-grained balance (4.30). Thus, the right-hand sides must also coincide and one obtains

(4.34)\begin{equation} \lim_{h,\ell\to 0} \left( \eta_{h,\ell} \varPi_\ell - \bar{\boldsymbol{J}}_\ell \boldsymbol{\cdot}\boldsymbol{\nabla}\eta_{h,\ell}\right)= D. \end{equation}

In contrast to the result (3.30) obtained in periodic domains where anomalous dissipation $D$ matches to inertial energy flux $\varPi _\ell$ alone, in wall-bounded domains, there is an additional contribution from $- \bar {\boldsymbol {J}}_\ell \boldsymbol{\cdot}\boldsymbol {\nabla }\eta _{h,\ell }$, which represents spatial energy flux to the wall. I mention in passing that at any finite distance from the wall, the expression (3.29) of Duchon & Robert (Reference Duchon and Robert2000) can also be derived here for $D$, as well as the 4/5th law (3.36) and 4/15th law (3.37).

A condition on vanishing of the wall-normal velocity analogous to (4.25) (but $L^3$ in time) suffices to show that the additional spatial flux contribution must vanish. However, the potential presence of this additional term raises the question whether the anomalous dissipation measure might have some contribution from the boundary, so that $D(\partial \varOmega )>0.$ This question recalls the famous theorem of Kato (Reference Kato1984) (see also Sueur Reference Sueur2012; Bardos & Titi Reference Bardos and Titi2013, among others), according to which vanishing of the viscous dissipation in the boundary strip $\varOmega _{c\nu }=\{\boldsymbol {x}\in \varOmega : d(\boldsymbol {x})< c\nu \}$ for any constant $c>0,$ or

(4.35)\begin{equation} \lim_{\nu\to 0}\int_0^T \int_{\varOmega_{c\nu}} \varepsilon^\nu \, {\rm d}V \, {\rm d}t = 0 \end{equation}

is necessary and sufficient that Navier–Stokes solution ${\boldsymbol {u}}^\nu$ converges strongly in $L^2([0,T]\times \varOmega )$ to the smooth Euler solution ${\boldsymbol {u}}$ with the same initial data ${\boldsymbol {u}}_0,$ over any time period $[0,T]$ for which this smooth Euler solution exists. This type of result is closely allied to the strong-weak uniqueness theory which I shall discuss in § 4.2.6. A consequence is that vanishing of the energy dissipation in the ‘Kato layer’ $\varOmega _{c\nu }$ implies vanishing energy dissipation everywhere in $\varOmega,$ unless the smooth Euler solution ${\boldsymbol {u}}$ develops a singularity before time $T.$ Bardos & Titi (Reference Bardos and Titi2013) have shown furthermore that the vanishing of energy dissipation in $\varOmega _{c\nu }$ implies also that $\boldsymbol {\tau }_w=\boldsymbol {0}.$ Here I may recall the numerical study by Nguyen van yen et al. (Reference Nguyen van yen, Waidmann, Klein, Farge and Schneider2018) of a compact quadrupole vortex in two dimensions impacting on a flat, solid wall. At very high Reynolds numbers, these authors find evidence for anomalous energy dissipation arising in the Kato layer from maximum vorticity values scaling as $\omega ^\nu \sim 1/\nu.$ These high values at the wall imply that for some points on the boundary, at least, skin friction $\boldsymbol {\tau }_w=\nu \boldsymbol {\omega }^\nu \boldsymbol {\times }\boldsymbol {n}\not \to \boldsymbol {0}$ as $\nu \to 0.$ I return to these issues again in § 4.2.6.

4.2.5. Vorticity cascade in space

The same methods as discussed previously can be applied also to vorticity and, in some respects, the analysis is much easier, because the vorticity balance is just the curl of the momentum balance already treated. I describe here only a few key results, following the more extended treatment of Eyink (Reference Eyink2023).

The fine-grained balance of vorticity for the incompressible Navier–Stokes equation is, of course, the viscous Helmholtz equation:

(4.36)\begin{equation} \partial_t\boldsymbol{\omega}^\nu=\boldsymbol{\nabla}\boldsymbol{\times}({\boldsymbol{u}}^\nu\boldsymbol{\times}\boldsymbol{\omega}^\nu-\nu\boldsymbol{\nabla}\boldsymbol{\times}\boldsymbol{\omega}^\nu):=-\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{\varSigma}^\nu, \end{equation}

where $\boldsymbol {\varSigma }^\nu$ is a suitable anti-symmetric tensor representing vorticity flux through the flow volume (Huggins Reference Huggins1971, Reference Huggins1994). We might adopt this as the starting point to derive the fine-grained vorticity balance in the limit $\nu \to 0$, but it is easier just to take $\boldsymbol {\varphi }\to \boldsymbol {\nabla }\boldsymbol {\times }\boldsymbol {\varphi }$ in the ideal momentum balance equation (4.21). This yields directly

(4.37) $$\begin{align} &\int_0^T\int_{\varOmega} \left[ \partial_t (\boldsymbol{\nabla}\boldsymbol{\times}\boldsymbol{\varphi}) \boldsymbol{\cdot} \boldsymbol{u} + \boldsymbol{\nabla}(\boldsymbol{\nabla}\boldsymbol{\times}\boldsymbol{\varphi})\boldsymbol{:} \boldsymbol{u}\otimes\boldsymbol{u}\right] \, {\rm d}V \,{\rm d}t\nonumber\\ &\quad \quad = \int_0^T\int_{\partial\varOmega} \left[\boldsymbol{\tau}_w \boldsymbol{\cdot}(\boldsymbol{\nabla}\boldsymbol{\times}\boldsymbol{\varphi}) - \boldsymbol{\sigma}_w\boldsymbol{\cdot}\boldsymbol{\varphi} \right] \,{\rm d}A\,{\rm d}t, \end{align}$$

where surface integration by parts $\int _{\partial \varOmega } p_w (\boldsymbol {n}\boldsymbol {\times }\boldsymbol {\nabla })\boldsymbol{\cdot}\boldsymbol {\varphi } \,\textrm {d}A= -\int _{\partial \varOmega } (\boldsymbol {n}\boldsymbol {\times }\boldsymbol {\nabla })p_w \boldsymbol{\cdot}\boldsymbol {\varphi } \,\textrm {d}A$ was used in the last line to introduce the vorticity source $\boldsymbol {\sigma }_w=-\boldsymbol {n}\boldsymbol {\times }\boldsymbol {\nabla } p_w$ of Lighthill (Reference Lighthill1963). The terms in this equation have mostly transparent physical meaning. Note in particular that the $\boldsymbol {\tau }_w$-term arises from viscous diffusion of vorticity away from the surface in the limit $\nu \to 0$, through the integration-by-parts identity

(4.38)\begin{equation} \int_\varOmega(\boldsymbol{\nabla}\boldsymbol{\times}\boldsymbol{\varphi})\boldsymbol{\cdot}\nu\boldsymbol{\nabla}\boldsymbol{\times}\boldsymbol{\omega}^\nu= \nu\int_\varOmega \boldsymbol{\nabla}\boldsymbol{\times} (\boldsymbol{\nabla}\boldsymbol{\times}\boldsymbol{\varphi})\boldsymbol{\cdot}\boldsymbol{\omega}^\nu+\int_{\partial\varOmega}(\boldsymbol{\nabla}\boldsymbol{\times}\boldsymbol{\varphi})\boldsymbol{\cdot}\boldsymbol{\tau}_w^\nu\, {\rm d}A. \end{equation}

We thus see that anomalous viscous diffusion of vorticity may remain as $\nu \to 0$ if $\boldsymbol {\tau }_w\neq \boldsymbol {0}.$

The corresponding coarse-grained vorticity balance is obtained by taking the curl of the coarse-grained momentum balance equation (4.13) and can be written most simply as

(4.39)\begin{equation} \partial_t (\eta_{h,\ell} \bar{\boldsymbol{\omega}}_{\ell}+\boldsymbol{\nabla}\eta_{h,\ell}\boldsymbol{\times}\bar{{\boldsymbol{u}}}_\ell) +\boldsymbol{\nabla}\boldsymbol{\times}\left[\boldsymbol{\nabla}\boldsymbol{\cdot}(\eta_{h,\ell}\bar{{\boldsymbol{T}}}_\ell)\right] =-\boldsymbol{\nabla}\boldsymbol{\times}{\boldsymbol f}_{\ell,h}. \end{equation}

In the time-derivative term, I used the identity $\boldsymbol {\nabla }\boldsymbol {\times }\tilde {{\boldsymbol {u}}}_{h,\ell }=\eta _{h,\ell } \bar {\boldsymbol {\omega }}_{\ell }+\boldsymbol {\nabla }\eta _{h,\ell }\boldsymbol {\times }\bar {{\boldsymbol {u}}}_\ell,$ which makes clear that the balance equation is for the sum of a vorticity $\boldsymbol {\omega }=\boldsymbol {\nabla }\boldsymbol {\times }{\boldsymbol {u}}$ in the flow interior and a vortex sheet $\boldsymbol {n}\boldsymbol {\times }{\boldsymbol {u}}$ on the solid surface. Note that ${\boldsymbol f}_{\ell,h}$ is just the inviscid limit of the inertial-drag force defined in (4.15), which may also be written as ${\boldsymbol f}_{\ell,h}= -(\bar {{\boldsymbol {T}}}_\ell + \bar {p}_\ell \boldsymbol {I} ) \boldsymbol{\cdot} \boldsymbol {\nabla }\eta _{h,\ell }.$ The term $-\boldsymbol {\nabla }\boldsymbol {\times }{\boldsymbol f}_{\ell,h}$ in the coarse-grained vorticity balance rationalises the force field method used in aerodynamics and wake flows behind complex bodies, where the solid surface is replaced by a suitable body force (van Kuik Reference van Kuik2022). Smearing the coarse-grained vorticity balance (4.39) with a vector test function $\boldsymbol {\varphi }$ and taking the limit $h,\ell \to 0$ recovers the same (4.37) obtained from the fine-grained balance. Note in particular that

(4.40)\begin{align} -\lim_{h,\ell\to 0} \int_0^T \int_\varOmega \boldsymbol{\varphi}\boldsymbol{\cdot} \boldsymbol{\nabla}\boldsymbol{\times}{\boldsymbol f}_{\ell,h} \, {\rm d}V\, {\rm d}t = \int_0^T\int_{\partial\varOmega} \left[-\boldsymbol{\tau}_w \boldsymbol{\cdot}(\boldsymbol{\nabla}\boldsymbol{\times}\boldsymbol{\varphi}) + \boldsymbol{\sigma}_w\boldsymbol{\cdot}\boldsymbol{\varphi} \right] \,{\rm d}A\,{\rm d}t, \end{align}

which relates $-\boldsymbol {\nabla }\boldsymbol {\times }{\boldsymbol f}_{\ell,h}$ to inviscid generation of vorticity at the body surface and possible anomalous viscous diffusion out of the surface. Equation (4.40) follows directly from (4.23) and (4.24).

If the strong continuity condition for wall-normal velocity (4.25) holds so that $\boldsymbol {\tau }_w\equiv \boldsymbol {0},$ then one may derive another simple relation for the surface vorticity source of Lighthill (Reference Lighthill1963). In that case (Quan & Eyink Reference Quan and Eyink2022a), the pressure-continuity result (4.28) implies that

(4.41)\begin{align} -\int_0^T \,{\rm d}t\int_{\partial\varOmega} \,{\rm d}A\, \boldsymbol{\varphi}\boldsymbol{\cdot}(\boldsymbol{n}\boldsymbol{\times}\boldsymbol{\nabla})p_w &=\int_0^T \,{\rm d}t\int_{\partial\varOmega} \,{\rm d}A\, (\boldsymbol{n}\boldsymbol{\times}\boldsymbol{\nabla})\boldsymbol{\cdot}\boldsymbol{\varphi} p_w\nonumber\\ &=\lim_{h,\ell\to 0} \int_0^T \,{\rm d}t\int_\varOmega \,{\rm d}V \, \boldsymbol{\nabla}\eta_{h,\ell}\boldsymbol{\times}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{\varphi} \bar{p}_\ell\nonumber\\ &=\lim_{h,\ell\to 0} \int_0^T \,{\rm d}t\int_\varOmega \,{\rm d}V \, -\boldsymbol{\varphi}\boldsymbol{\cdot}\boldsymbol{\nabla}\eta_{h,\ell}\boldsymbol{\times}\boldsymbol{\nabla}\bar{p}_\ell\nonumber\\ &=\lim_{h,\ell\to 0} \int_0^T \,{\rm d}t\int_\varOmega \,{\rm d}V \,\boldsymbol{\varphi}\boldsymbol{\cdot}\left(\boldsymbol{\nabla}\eta_{h,\ell}\boldsymbol{\times}\partial_t\bar{{\boldsymbol{u}}}_\ell +\boldsymbol{\nabla}\eta_{h,\ell}\boldsymbol{\cdot}\boldsymbol{\varSigma}_{\ell}\right), \end{align}

where in the last line, I used the coarse-grained momentum balance in the flow interior, written as

(4.42)\begin{equation} \partial_t\bar{{\boldsymbol{u}}}_{\ell} =-\boldsymbol{\nabla}\boldsymbol{\cdot}\bar{{\boldsymbol{T}}}_\ell-\boldsymbol{\nabla}\bar{p}_\ell = \boldsymbol{\varSigma}_\ell^*-\boldsymbol{\nabla}\bar{p}_\ell, \end{equation}

where $\boldsymbol {\varSigma }_\ell ^*$ is the Hodge dual to anti-symmetric vorticity flux tensor $\boldsymbol {\varSigma }_\ell,$ or in components $\boldsymbol {\varSigma }_{\ell,i}^*=\frac {1}{2}\epsilon _{ijk} \boldsymbol {\varSigma }_{\ell,jk}.$ Equation (4.41) states that under the condition (4.25), tangential pressure gradients at the body surface act in the inviscid limit as a local source of vorticity, which either accumulates in the surface vortex sheet or else flows away via turbulent advection.

Vorticity flux from the body surface can be directly related to drag. Relations of this type have been derived by many authors, including Burgers, Lighthill and many others, as discussed in the very comprehensive review by Biesheuvel & Hagmeijer (Reference Biesheuvel and Hagmeijer2006). Eyink (Reference Eyink2021) derives such a relation for the power consumed by the instantaneous drag force ${\boldsymbol F}_B(t)$ on a body $B$ held fixed in a flow with velocity ${\boldsymbol V},$ making connection with ideas in quantum superfluids. This Josephson–Anderson relation,

(4.43)\begin{align} {\boldsymbol F}_B^\nu(t)\boldsymbol{\cdot} {\boldsymbol V} &=-\rho\int_\varOmega {\boldsymbol{u}}_\phi\boldsymbol{\cdot}({\boldsymbol{u}}^\nu\boldsymbol{\times}\boldsymbol{\omega}^\nu-\nu\boldsymbol{\nabla}\boldsymbol{\times}\boldsymbol{\omega}^\nu)\, {\rm d}V\nonumber\\ &=-\rho \int_{\varOmega} \boldsymbol{\nabla}{\boldsymbol{u}}_\phi\boldsymbol{:}{\boldsymbol{u}}_\omega^\nu {\boldsymbol{u}}_\omega^\nu \, {\rm d}V +\rho\int_{\partial\varOmega} {\boldsymbol{u}}_\phi\boldsymbol{\cdot}\boldsymbol{\tau}_w^\nu\, {\rm d}A \end{align}

is in fact a special case of a relation derived by Howe (Reference Howe1995). In the last line of (4.43), integration by parts has been used to rewrite the relation in a form which should remain valid in the limit $\nu \to 0,$ as pointed out by Eyink (Reference Eyink2021). This inviscid limit has been rigorously derived by Quan & Eyink (Reference Quan and Eyink2022b) when the body surface $\partial B$ is mathematically smooth, i.e. when the body is very large. The Josephson–Anderson relation and the drag on the body are related to Onsager's dissipation anomaly by the balance equation of the kinetic energy $(1/2)|{\boldsymbol {u}}_\omega |^2$ in the rotational flow ${\boldsymbol {u}}_\omega ={\boldsymbol {u}}-{\boldsymbol {u}}_\phi,$ where ${\boldsymbol {u}}_\phi$ is the stationary potential flow of d'Alembert (Reference d'Alembert1749, Reference d'Alembert1768) with no drag. This balance takes the form

(4.44)\begin{align} & -\int_0^T\int_{\varOmega}\tfrac{1}{2}\partial_t\varphi|\boldsymbol{u}_{\omega}|^2\,{\rm d}V\, {\rm d}t -\int_0^T\int_{\varOmega}\boldsymbol{\nabla}\varphi\boldsymbol{\cdot}\left[\tfrac{1}{2}|\boldsymbol{u}_{\omega}|^2\boldsymbol{u}+p_{\omega}\boldsymbol{u}_{\omega}\right] \,{\rm d}V\,{\rm d}t\nonumber\\ & \qquad =\int_0^T\int_{\varOmega} \varphi {\boldsymbol{u}}_\phi\boldsymbol{\cdot} \boldsymbol{\tau}_w \, {\rm d}A\, {\rm d}t -\int_0^T\int_{\varOmega}\varphi\boldsymbol{\nabla}{\boldsymbol{u}}_{\phi}\boldsymbol{:} \boldsymbol{u}_{\omega}\otimes\boldsymbol{u}_{\omega}\,{\rm d}V\,{\rm d}t\nonumber\\ &\qquad\quad -\int_0^T\int_{\varOmega} \varphi \, D({\boldsymbol{u}}) \, {\rm d}V\, {\rm d}t, \end{align}

with $\varphi$ a smooth test function that may not vanish at the boundary (Quan & Eyink Reference Quan and Eyink2022b). The Josephson–Anderson relation represents a transfer of kinetic energy into rotational fluid motions, which is then disposed by the dissipation $D$ into heat energy.

4.2.6. Weak-strong uniqueness and extreme near-wall events

A fact of fundamental importance for the theory of incompressible turbulence is that weak-strong uniqueness, as reviewed in § 3.1.3, is not valid unconditionally in flows with solid walls, as it is for incompressible flows in periodic domains or in infinite space. Bardos, Székelyhidi & Wiedemann (Reference Bardos, Székelyhidi and Wiedemann2014) (and see also Bardos & Titi Reference Bardos and Titi2013; Wiedemann Reference Wiedemann2018) have noted that additional conditions are required in wall-bounded flows to obtain weak-strong uniqueness and the first work uses convex integration methods to construct a counterexample showing that uniqueness indeed fails without such added hypotheses. One sufficient condition for weak-strong uniqueness is that the inviscid limit of the skin friction, $\boldsymbol {\tau }_w=\lim _{\nu \to 0} \boldsymbol {\tau }_w^\nu,$ should be vanishing as a distribution at all times on the solid surface, i.e. $\boldsymbol {\tau }_w\equiv \boldsymbol {0}$ when smeared with any smooth space–time test function (Bardos & Titi Reference Bardos and Titi2013). Another condition which guarantees the later is the uniform vanishing of wall-normal velocity (4.25), which I have already noted in (4.27) implies vanishing skin friction (Quan & Eyink Reference Quan and Eyink2022b).

The gist of the weak-strong uniqueness result can be easily understood from the following sketch of a proof taken from Quan & Eyink (Reference Quan and Eyink2024), which shows in the context of flow around a smooth body $B$ that even the weaker condition $\boldsymbol {\tau }_w\boldsymbol{\cdot}{\boldsymbol {u}}_\phi \equiv 0$ on $[0,T]\times \partial B$ implies weak-strong uniqueness. The argument is based on the inviscid balance of global kinetic energy in the rotational flow, $E_\omega (\tau )=\tfrac {1}{2}\int _\varOmega |{\boldsymbol {u}}_\omega (\cdot,\tau )|^2\, \textrm {d}V,$ which is

(4.45)\begin{align} E_\omega(\tau) = E_\omega(0) + \int_0^\tau \int_{\partial\varOmega} {\boldsymbol{u}}_\phi\boldsymbol{\cdot}\boldsymbol{\tau}_w \, {\rm d}A\, {\rm d}t -\int_0^\tau \int_\varOmega \boldsymbol{\nabla}{\boldsymbol{u}}_\phi\boldsymbol{:}{\boldsymbol{u}}_\omega{\boldsymbol{u}}_\omega \, {\rm d}V \, {\rm d}t -\int_0^\tau \int_\varOmega D\, {\rm d}V\, {\rm d}t. \end{align}

This result can be obtained formally by taking $\varphi \equiv \chi _{[0,\tau ]\times \varOmega },$ the characteristic function of the set $[0,\tau ]\times \varOmega,$ in the local balance (4.44) and can be obtained rigorously by a careful limiting argument. Using the condition $\boldsymbol {\tau }_w\boldsymbol{\cdot}{\boldsymbol {u}}_\phi \equiv 0$ and the fact that $D\geq 0,$ one obtains

(4.46)\begin{equation} E_\omega(\tau) \leq E_\omega(0) -\int_0^\tau \int_\varOmega \boldsymbol{\nabla}{\boldsymbol{u}}_\phi\boldsymbol{:}{\boldsymbol{u}}_\omega{\boldsymbol{u}}_\omega \, {\rm d}V\, {\rm d}t. \end{equation}

Since ${\boldsymbol {u}}_\phi$ is smooth and globally bounded, application of Cauchy–Schwartz inequality gives

(4.47)\begin{equation} E_\omega(\tau) \leq E_\omega(0) + C\|\boldsymbol{\nabla}{\boldsymbol{u}}_\phi\|_\infty \int_0^\tau \,{\rm d}t\, E_\omega(t) \end{equation}

and then the Gronwall inequality finally yields

(4.48)\begin{equation} E_\omega(\tau) \leq E_\omega(0) \exp\left(C\|\boldsymbol{\nabla}{\boldsymbol{u}}_\phi\|_\infty\times\tau\right). \end{equation}

If $E_\omega (0)=0$ or equivalently ${\boldsymbol {u}}(0)\equiv {\boldsymbol {u}}_\phi,$ then $E_\omega (\tau )\equiv 0$ for all $\tau >0$ and thus ${\boldsymbol {u}}\equiv {\boldsymbol {u}}_\phi$ identically for all time. Careful examination of the proof shows that this result remains valid even if $\|{\boldsymbol {u}}^\nu (0)-{\boldsymbol {u}}_\phi \|_{L^2(\varOmega )}\to 0$ as $\nu \to 0,$ which allows for a viscous boundary layer in the Navier–Stokes initial data, as long as the energy $E_\omega ^\nu (0)$ in that layer shrinks to zero as $\nu \to 0.$ Note that the estimate (4.48) allows for instability of ${\boldsymbol {u}}_\phi$ in the dynamical systems sense, so that a small perturbation may grow exponentially. However, the inequality (4.48) shows that if ${\boldsymbol {u}}_\phi \boldsymbol{\cdot} \boldsymbol {\tau }_w\equiv 0,$ then the smooth potential Euler solution ${\boldsymbol {u}}_\phi$ is stable in the sense of Hadamard well-posedness and that any Navier–Stokes solution whose initial data ${\boldsymbol {u}}^\nu (0)$ converges to ${\boldsymbol {u}}_\phi$ in $L^2$ as $\nu \to 0$ will converge globally to ${\boldsymbol {u}}_\phi$ in the inviscid limit.

In my opinion, a plausible conclusion is that weak-strong uniqueness is an inappropriate condition to select physically correct weak solutions of the Euler equations, unlike what has been often proposed in the mathematics literature. The clearest example is the constant flow past a smooth body, the subject of the famous paradox of d'Alembert (Reference d'Alembert1749, Reference d'Alembert1768), where the potential Euler solution is smooth and stationary in time, thus existing globally without blow-up. Nevertheless, drag does not seem to vanish in the $\nu \to 0$ limit, as shown by laboratory experiments like that of Achenbach (Reference Achenbach1972) for flow past a sphere and by numerical simulations such as that of Chatzimanolakis, Weber & Koumoutsakos (Reference Chatzimanolakis, Weber and Koumoutsakos2022) for 2-D flow past a disk of diameter $D$ impulsively accelerated to velocity $U.$ In the latter case, the initial conditions are exactly ${\boldsymbol {u}}(0)={\boldsymbol {u}}_\phi$ at $t=0$ and, after a short period of a few mean-free-times when a kinetic description is required, the dynamics is well described by incompressible Navier–Stokes. In principle, empirical observations like those above might be explained by instability of the potential Euler solution in a dynamical systems sense, so that small perturbations, e.g. due to external flow disturbances, molecular viscosity or even thermal noise, grow exponentially in time. However, excluding for the moment external and thermal noise, there is an asymptotically exact solution of the 2-D incompressible Navier–Stokes equation for the case of the impulsively accelerated disk as $\nu \to 0$ (Bar-Lev & Yang Reference Bar-Lev and Yang1975) and although its time of validity is expected to be only $\sim D/U,$ this nevertheless suffices to show that $E_\omega ^\nu (0+)\to 0$, where $t=0+$ is the initial time of validity of the Navier–Stokes description. In that case, the bound (4.48) expressing weak-strong uniqueness, if valid, would imply that ${\boldsymbol {u}}=\lim _{\nu \to 0}{\boldsymbol {u}}^\nu$ coincides with ${\boldsymbol {u}}_\phi$ identically. Although I have explained this result in the case of the impulsively accelerated body via the energy balance (4.45), similar arguments apply when the body is accelerated over a finite time-interval. However, the empirical observations suggest instead that the limit ${\boldsymbol {u}}$ is a dissipative weak Euler solution with initial condition ${\boldsymbol {u}}(0)={\boldsymbol {u}}_\phi$ but distinct from ${\boldsymbol {u}}_\phi$ and co-existing with that smooth, stationary Euler solution.