2 Lagrangian FDR

We consider in this paper turbulent fluid flows in finite domains without walls. A relevant numerical example is direct numerical simulation (DNS) of turbulence in a periodic box. A set of examples from nature is provided by large-scale flows in thin planetary atmospheres, which can be modelled as 2D flows on a sphere. Mathematically speaking, the results in this section apply to fluid flows on any compact Riemannian manifold without boundary, merely replacing the Wiener process with the Brownian motion on the manifold whose infinitesimal generator is the Laplace–Beltrami operator (Ikeda & Watanabe Reference Ikeda and Watanabe1989). For simplicity of presentation, we derive the relation only for periodic domains.

Scalar fields $\unicode[STIX]{x1D703}$ (such as temperature, dye or pollutants) transported by a fluid with velocity $\boldsymbol{u}$ are described by the advection–diffusion equation

(2.1) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D703}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}=\unicode[STIX]{x1D705}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}+S,\end{eqnarray}$$

where $S(\boldsymbol{x},t)$ is a source field and $\unicode[STIX]{x1D705}>0$ is the molecular diffusivity of the scalar. The work of Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998) employed a stochastic representation of the solutions of this equation, which is known as the Feynman–Kac representation in the mathematics literature (Oksendal Reference Oksendal2013) and as a stochastic Lagrangian representation in the turbulence modelling field (Sawford Reference Sawford2001). This stochastic approach is the natural extension to diffusive scalars of the Lagrangian description developed for ‘ideal’ scalar fields without diffusion advected by smooth velocities. We presently discuss this representation only for domains $\unicode[STIX]{x1D6FA}$ without boundaries, as assumed also by Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998), and in Part II we describe the extension to wall-bounded domains. We shall further discuss in these papers only advection by an incompressible fluid satisfying

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0,\end{eqnarray}$$

so that the ideal advection term formally conserves all integrals of the form $I_{h}(t)=\int _{\unicode[STIX]{x1D6FA}}\text{d}^{d}x~h(\unicode[STIX]{x1D703}(\boldsymbol{x},t))$ for any continuous function $h(\unicode[STIX]{x1D703})$ . It should be noted that the representation applies in any space dimension $d$ , with most immediate physical interest for $d=2,3$ , of course.

The stochastic representation of non-ideal scalar dynamics involves stochastic Lagrangian flow maps $\widetilde{\unicode[STIX]{x1D743}}_{t,s}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x})$ describing the motion of particles labelled by their positions $\boldsymbol{x}$ at time $t$ to random positions at earlier times $s<t$ . The physical relevance of the backward-in-time particle trajectories can be anticipated from the fact that the advection–diffusion equation (2.1) mixes (averages) the values of the scalar field given in the past and not, of course, the future values. Mathematically, the relevant stochastic flows are governed by the backward Itô stochastic differential equations,

(2.3) $$\begin{eqnarray}\hat{\text{d}}_{s}\widetilde{\unicode[STIX]{x1D743}}_{t,s}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x})=\boldsymbol{u}^{\unicode[STIX]{x1D708}}(\widetilde{\unicode[STIX]{x1D743}}_{t,s}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}),s)\text{d}s+\sqrt{2\unicode[STIX]{x1D705}}\hat{\text{d}}\widetilde{\boldsymbol{W}}_{s},\quad \widetilde{\unicode[STIX]{x1D743}}_{t,t}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x})=\boldsymbol{x}.\end{eqnarray}$$

Here, $\widetilde{\boldsymbol{W}}_{s}$ is a standard Brownian motion and $\hat{\text{d}}_{s}$ denotes the backward Itô stochastic differential in the time $s$ . For detailed discussions of backward Itô equations and stochastic flows, see Kunita (Reference Kunita1997) and Friedman (Reference Friedman2006). For those who are familiar with the more standard forward Itô equations, the backward equations are simply the time reverses of the forward ones. Thus, a backward Itô equation in the time variable $s$ is equivalent to a forward Itô equation in the time ${\hat{s}}=t_{r}-s$ reflected around a chosen reference time $t_{r}$ . (We note that the difference between forward and backward Itô equations is not essentially the direction of time in which they are integrated. Rather, the difference has to do with the time direction in which those equations are adapted (Kunita Reference Kunita1997; Friedman Reference Friedman2006). Thus, a forward Itô differential $b(\widetilde{\boldsymbol{W}}_{t})\,\text{d}\widetilde{\boldsymbol{W}}_{t}$ is discretized in time as $b(\widetilde{\boldsymbol{W}}_{t_{n}})(\widetilde{\boldsymbol{W}}_{t_{n+1}}-\widetilde{\boldsymbol{W}}_{t_{n}})$ for $t_{n+1}>t_{n}$ , with the increment $\widetilde{\boldsymbol{W}}_{t_{n+1}}-\widetilde{\boldsymbol{W}}_{t_{n}}$ statistically independent of $\widetilde{\boldsymbol{W}}_{t}$ for $t\leqslant t_{n}$ . Instead, a backward Itô differential $b(\widetilde{\boldsymbol{W}}_{t})\hat{\text{d}}\widetilde{\boldsymbol{W}}_{t}$ is discretized as $b(\widetilde{\boldsymbol{W}}_{t_{n}})(\widetilde{\boldsymbol{W}}_{t_{n}}-\widetilde{\boldsymbol{W}}_{t_{n-1}})$ for $t_{n}>t_{n-1}$ , with $\widetilde{\boldsymbol{W}}_{t_{n}}-\widetilde{\boldsymbol{W}}_{t_{n-1}}$ statistically independent of $\widetilde{\boldsymbol{W}}_{t}$ for $t\geqslant t_{n}$ . The distinction only matters when, as in our (2.4), the differential of $\widetilde{\boldsymbol{W}}_{s}$ is multiplied by a stochastic function of $\widetilde{\boldsymbol{W}}$ .) The noise term involving the Brownian motion in (2.3) is proportional to the square root of the molecular diffusivity $\unicode[STIX]{x1D705}$ . The velocity field $\boldsymbol{u}^{\unicode[STIX]{x1D708}}$ is assumed to be smooth so long as the parameter $\unicode[STIX]{x1D708}>0$ . In the case of greatest physical interest when $\boldsymbol{u}^{\unicode[STIX]{x1D708}}$ is a solution of the incompressible Navier–Stokes equation, then $\unicode[STIX]{x1D708}$ represents the kinematic viscosity and we assume, for simplicity of presentation, that there is no blowup in these solutions. (See Rezakhanlou (Reference Rezakhanlou2014) for weak solutions.) Because (2.3) involves both $\unicode[STIX]{x1D708}$ and $\unicode[STIX]{x1D705}$ , its random solutions $\tilde{\unicode[STIX]{x1D743}}_{t,s}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}$ have statistics that depend upon these parameters, represented by the superscripts. To avoid too heavy a notation, we omit these superscripts and write simply $\tilde{\unicode[STIX]{x1D743}}_{t,s}$ unless it is essential to refer to the dependence upon $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}$ . It should be noted that when $\unicode[STIX]{x1D705}=0$ and $\boldsymbol{u}^{\unicode[STIX]{x1D708}}$ remains smooth, then $\unicode[STIX]{x1D743}_{t,s}^{\unicode[STIX]{x1D708},0}(\boldsymbol{x})$ is no longer stochastic and gives the usual reverse Lagrangian flow from time $t$ backward to the earlier time $s<t$ .

The stochastic representation of the solutions of the advection–diffusion equation follows from the backward differential

(2.4) $$\begin{eqnarray}\displaystyle \hat{\text{d}}_{s}\unicode[STIX]{x1D703}(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s) & = & \displaystyle [(\unicode[STIX]{x2202}_{s}+\boldsymbol{u}^{\unicode[STIX]{x1D708}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}-\unicode[STIX]{x1D705}\unicode[STIX]{x1D6E5})\unicode[STIX]{x1D703}](\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\text{d}s+\sqrt{2\unicode[STIX]{x1D705}}\hat{\text{d}}\widetilde{\boldsymbol{W}}_{s}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\nonumber\\ \displaystyle & = & \displaystyle S(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\text{d}s+\sqrt{2\unicode[STIX]{x1D705}}\hat{\text{d}}\widetilde{\boldsymbol{W}}_{s}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s),\end{eqnarray}$$

using the backward Itô formula (Kunita Reference Kunita1997; Friedman Reference Friedman2006) in the first line and (2.1) in the second. Integration over time $s$ from 0 to $t$ gives

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D703}(\boldsymbol{x},t)=\unicode[STIX]{x1D703}_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))+\int _{0}^{t}S(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\text{d}s+\sqrt{2\unicode[STIX]{x1D705}}\int _{0}^{t}\hat{\text{d}}\widetilde{\boldsymbol{W}}_{s}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s),\end{eqnarray}$$

where $\unicode[STIX]{x1D703}_{0}$ represents the initial data for the scalar at time $0.$ Because the backward Itô integral term in (2.5) averages to zero, one obtains

(2.6) $$\begin{eqnarray}\unicode[STIX]{x1D703}(\boldsymbol{x},t)=\mathbb{E}\left[\unicode[STIX]{x1D703}_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))+\int _{0}^{t}S(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\text{d}s\right],\end{eqnarray}$$

where $\mathbb{E}$ denotes the average over the Brownian motion. Equation (2.6) is the desired stochastic representation of the solution of the advection–diffusion equation (2.1). It should be noted that the reverse statement is also true, that the field $\unicode[STIX]{x1D703}(\boldsymbol{x},t)$ defined a priori by (2.6) is the solution of (2.1) for the initial data $\unicode[STIX]{x1D703}_{0}$ . For a simple proof, see Eyink & Drivas (Reference Eyink and Drivas2015b , § 4.1), which gives the analogous argument for the Burgers equation.

To see that this stochastic representation naturally generalizes the standard Lagrangian description to non-ideal scalars, we observe that the scalar values along stochastic Lagrangian trajectories $\unicode[STIX]{x1D703}(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)$ are, for $S\equiv 0,$ martingales backward in time. This means that

(2.7) $$\begin{eqnarray}\mathbb{E}[\unicode[STIX]{x1D703}(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)|\{\widetilde{\boldsymbol{W}}_{\unicode[STIX]{x1D70F}},r<\unicode[STIX]{x1D70F}<t\}]=\unicode[STIX]{x1D703}(\tilde{\unicode[STIX]{x1D743}}_{t,r}(\boldsymbol{x}),r),\quad s<r<t,\end{eqnarray}$$

where the expectation is conditioned upon knowledge of the Brownian motion over the time interval $[r,t]$ . Thus, the conditional average value is the last known value (going backward in time). This is the property for diffusive flow, which corresponds to the statement for diffusionless smooth advection that $\unicode[STIX]{x1D703}$ is conserved along Lagrangian trajectories, or that $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D743}_{t,s}(\boldsymbol{x}),s)$ is constant in $s$ . The proof is obtained by integrating the differential (2.4) over the time interval $[s,t]$ to obtain

(2.8) $$\begin{eqnarray}\unicode[STIX]{x1D703}(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)=\unicode[STIX]{x1D703}(\boldsymbol{x},t)-\sqrt{2\unicode[STIX]{x1D705}}\int _{s}^{t}\hat{\text{d}}\widetilde{\boldsymbol{W}}_{\unicode[STIX]{x1D70F}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}(\tilde{\unicode[STIX]{x1D743}}_{t,\unicode[STIX]{x1D70F}}(\boldsymbol{x}),\unicode[STIX]{x1D70F})\end{eqnarray}$$

and then exploiting the corresponding martingale property of the backward Itô integral (Kunita Reference Kunita1997; Friedman Reference Friedman2006). It is important to emphasize that a martingale property like (2.7) does not hold forward in time, which would instead give a solution of the negative-diffusion equation with $\unicode[STIX]{x1D705}$ replaced by $-\unicode[STIX]{x1D705}<0$ . Thus, the backward-in-time martingale property (2.7) expresses the arrow of time arising from the irreversibility of the diffusion process.

The main result of this paper is a new exact FDR between scalar dissipation due to molecular diffusivity and fluctuations associated with stochastic Lagrangian trajectories. To state the result, we introduce a stochastic scalar field

(2.9) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)\equiv \unicode[STIX]{x1D703}_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))+\int _{0}^{t}S(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\text{d}s,\end{eqnarray}$$

which, according to (2.6), satisfies $\unicode[STIX]{x1D703}(\boldsymbol{x},t)=\mathbb{E}[\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)]$ when averaged over Brownian motions. (It should be noted that, for all $s<t$ , the quantity $\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t;s)=\unicode[STIX]{x1D703}(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)+\int _{s}^{t}S(\tilde{\unicode[STIX]{x1D743}}_{t,r}(\boldsymbol{x}),r)\,\text{d}r$ is a martingale backward in time, by the same argument as used above for $S=0$ .) Thus, $\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)$ in (2.9) represents the contribution to $\unicode[STIX]{x1D703}(\boldsymbol{x},t)$ from an individual stochastic Lagrangian trajectory as it samples the initial data $\unicode[STIX]{x1D703}_{0}$ and scalar source $S$ backward in time. Using this definition and (2.6), we can rewrite (2.5) as

(2.10) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)-\mathbb{E}[\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)]=-\sqrt{2\unicode[STIX]{x1D705}}\int _{0}^{t}\hat{\text{d}}\widetilde{\boldsymbol{W}}_{s}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s).\end{eqnarray}$$

Squaring this equation and averaging over the Brownian motion gives

(2.11) $$\begin{eqnarray}\text{Var}[\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)]=2\unicode[STIX]{x1D705}\int _{0}^{t}\,\text{d}s\,\mathbb{E}[|\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)|^{2}],\end{eqnarray}$$

where ‘Var’ on the left-hand side denotes the stochastic scalar variance in the average over the Brownian motion, and on the right-hand side we have used the Itô isometry (see Oksendal Reference Oksendal2013, § 3.1) to evaluate the mean square of the backward Itô integral. If we now average in $\boldsymbol{x}$ over the flow domain $\unicode[STIX]{x1D6FA}$ , use the fact that the stochastic flows $\tilde{\unicode[STIX]{x1D743}}_{t,s}$ with condition (2.2) preserve volume, and divide by $1/2$ we obtain

(2.12) $$\begin{eqnarray}\frac{1}{2}\langle \text{Var}\,\tilde{\unicode[STIX]{x1D703}}(t)\rangle _{\unicode[STIX]{x1D6FA}}=\unicode[STIX]{x1D705}\int _{0}^{t}\,\text{d}s\langle |\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}(s)|^{2}\rangle _{\unicode[STIX]{x1D6FA}}.\end{eqnarray}$$

This is our exact FDR. The quantity on the right-hand side is just the volume-averaged and cumulative (time-integrated) scalar dissipation and the quantity on the left-hand side is (half) the stochastic scalar variance. The relation (2.12) thus represents a balance between scalar dissipation and the input of scalar fluctuations from the initial scalar field and the scalar sources, as sampled by stochastic Lagrangian trajectories backward in time.

It is important to emphasize that the origin of statistical fluctuations in our relation (2.12) is not that assumed in most traditional discussions of turbulence, i.e. random ensembles of initial scalar fields, of advecting velocity fields or of stochastic scalar sources. Our FDR (2.12) is valid for fixed realizations of all of these quantities. The fluctuating quantity $\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)$ which is defined in (2.9) and which appears in our (2.12) is an entirely different object from the conventional ‘turbulent’ scalar fluctuation $\unicode[STIX]{x1D703}^{\prime }(\boldsymbol{x},t)$ . The latter is usually defined by $\unicode[STIX]{x1D703}^{\prime }:=\unicode[STIX]{x1D703}-\langle \unicode[STIX]{x1D703}\rangle$ , where the scalar mean $\langle \unicode[STIX]{x1D703}\rangle$ is taken to be an ensemble or space/time average. Instead, the origin of randomness in $\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)$ is the Brownian motion in the stochastic flow equation (2.3). In special cases, e.g. a dye passively advected by a turbulent flow, this mathematical Wiener process has direct significance as the description of a physical Brownian motion of individual dye molecules in the liquid (Saffman Reference Saffman1960; Buaria et al. Reference Buaria, Yeung and Sawford2016). In general, however, the Wiener process is simply a means to model the effects of diffusion in a Lagrangian framework. For example, for a temperature field, there are no ‘thermal molecules’ undergoing physical Brownian motion.

Because our FDR is valid for fixed realizations of initial scalar fields, of advecting velocity fields or of scalar sources, we are free to average subsequently over random ensembles of these objects. In this manner, we recover from (2.12) as special cases some known results. For example, when the scalar source is a random field with zero mean and delta-correlated in time,

(2.13) $$\begin{eqnarray}\langle \tilde{S}(\boldsymbol{x},t)\tilde{S}(\boldsymbol{x}^{\prime },t^{\prime })\rangle =2C_{S}(\boldsymbol{x},\boldsymbol{x}^{\prime })\unicode[STIX]{x1D6FF}(t-t^{\prime }),\end{eqnarray}$$

then we recover the steady-state balance equation for the scalar dissipation,

(2.14) $$\begin{eqnarray}\langle \unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}|^{2}\rangle _{\unicode[STIX]{x1D6FA},\infty ,S}=\frac{1}{V}\int _{\unicode[STIX]{x1D6FA}}\text{d}^{d}x\,C_{S}(\boldsymbol{x},\boldsymbol{x}),\end{eqnarray}$$

where the average on the left-hand side is over space domain $\unicode[STIX]{x1D6FA}$ , an infinite time interval and the random source $\tilde{S}$ . This is the standard result usually derived for Gaussian random source fields as an application of the Furutsu–Donsker–Novikov theorem (Novikov Reference Novikov1965; Frisch Reference Frisch1995). We derive it instead as a consequence of a general steady-state FDR,

(2.15) $$\begin{eqnarray}\langle \unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}|^{2}\rangle _{\unicode[STIX]{x1D6FA},\infty }=\int _{-\infty }^{0}\,\text{d}t\langle \langle \tilde{S}_{L}(0)\tilde{S}_{L}(t)\rangle _{\mathbb{E},\unicode[STIX]{x1D6FA}}^{\text{T}}\rangle _{\infty },\end{eqnarray}$$

where the random variable $\tilde{S}_{L}(\boldsymbol{x},s)=S(\tilde{\unicode[STIX]{x1D743}}_{0,s}(\boldsymbol{x}),s)$ arises by sampling a single realization of the source $S$ along stochastic Lagrangian trajectories, $\langle \cdot \rangle _{\mathbb{E},\unicode[STIX]{x1D6FA}}^{\text{T}}$ denotes the truncated correlation function (covariance) in the average over Brownian motion and space domain, and $\langle \cdot \rangle _{\infty }$ an infinite-time average with respect to the release time $0$ of stochastic particles. Further averaging of (2.15) over random ensembles of $S$ with delta covariance (2.13) then gives the steady-state balance (2.14). For details, see appendix B. Similar relations hold for freely decaying scalars with no sources but random initial scalar fields. For example, when the initial scalar has a uniform random space gradient, $\tilde{\unicode[STIX]{x1D703}}_{0}(\boldsymbol{x})=\tilde{\boldsymbol{G}}\boldsymbol{\cdot }\boldsymbol{x}$ with isotropic statistics

(2.16) $$\begin{eqnarray}\langle \tilde{\boldsymbol{G}}\tilde{\boldsymbol{G}}^{\text{T}}\rangle _{G}=G^{2}\boldsymbol{I},\end{eqnarray}$$

then we recover a relation of Sawford et al. (Reference Sawford, Yeung and Borgas2005) and Buaria et al. (Reference Buaria, Yeung and Sawford2016),

(2.17) $$\begin{eqnarray}\unicode[STIX]{x1D705}\int _{0}^{t}\,\text{d}s\langle |\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}(s)|^{2}\rangle _{\unicode[STIX]{x1D6FA},\unicode[STIX]{x1D703}_{0}}=\frac{1}{4}G^{2}\mathbb{E}^{1,2}\langle |\tilde{\unicode[STIX]{x1D743}}_{t,0}^{(1)}-\tilde{\unicode[STIX]{x1D743}}_{t,0}^{(2)}|^{2}\rangle _{\unicode[STIX]{x1D6FA}},\end{eqnarray}$$

where the $1,2$ averages are taken over two independent ensembles of Brownian motion. We delay the derivation of the special cases (2.14), (2.15), (2.17) to appendix B, since the proofs require additional material which will be introduced in subsequent sections.

It should be noted, finally, that the result (2.11) provides a spatially local FDR, which we may write in the form

(2.18) $$\begin{eqnarray}\frac{1}{2t}\text{Var}[\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)]=\langle \mathbb{E}[\unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)|^{2}]\rangle _{t},\end{eqnarray}$$

where, on the right-hand side, $\langle \cdot \rangle _{t}$ denotes an average over $s$ in the time interval $[0,t],$ carried out along stochastic Lagrangian trajectories moving backward in time from space–time point $(\boldsymbol{x},t)$ . It follows that at short times the local scalar variance exactly recovers the local scalar dissipation,

(2.19) $$\begin{eqnarray}\lim _{t\rightarrow 0}\frac{1}{2t}\text{Var}[\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)]=\unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}(\boldsymbol{x},0)|^{2}.\end{eqnarray}$$

A substantial spatial correlation between $(1/2t)\text{Var}[\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)]$ and $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)=\unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}(\boldsymbol{x},t)|^{2}$ should persist for relatively short times $t$ . On the other hand, in the long-time limit, the local scalar variance becomes space–time-independent and equals

(2.20) $$\begin{eqnarray}\lim _{t\rightarrow \infty }\frac{1}{2t}\text{Var}[\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)]=\langle \unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}|^{2}\rangle _{\unicode[STIX]{x1D6FA},\infty }\quad \text{for all }\boldsymbol{x}\in \unicode[STIX]{x1D6FA}.\end{eqnarray}$$

To see that (2.20) should be true, one can note that the random variables $\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x})\in \unicode[STIX]{x1D6FA}$ for each fixed $\boldsymbol{x}$ are ergodic random processes in the time variable $s$ for $\unicode[STIX]{x1D705}>0$ . Because of incompressibility of the velocity field and the ergodicity of the stochastic Lagrangian flow, the variables $\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x})$ will be nearly uniformly distributed over $\unicode[STIX]{x1D6FA}$ at times $s\leqslant t-\unicode[STIX]{x1D70F}$ , where $\unicode[STIX]{x1D70F}$ is a characteristic scalar mixing time. This time $\unicode[STIX]{x1D70F}$ will be at most of the order $L^{2}/\unicode[STIX]{x1D705},$ where $L$ is the diameter of the domain, and thus finite for $\unicode[STIX]{x1D705}>0$ , but usually much shorter because of advective mixing by the velocity field. For any positive integer $n$ ,

(2.21) $$\begin{eqnarray}\lim _{t\rightarrow \infty }\frac{1}{2t}\text{Var}[\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)]=\lim _{t\rightarrow \infty }\frac{1}{t}\int _{0}^{t-n\unicode[STIX]{x1D70F}}\,\text{d}s\,\mathbb{E}[\unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)|^{2}],\end{eqnarray}$$

since the corrections are vanishing as $O(n\unicode[STIX]{x1D70F}/t)$ . By choosing an $n$ sufficiently large but fixed as $t\rightarrow \infty$ , we can make the right-hand side arbitrarily close to

(2.22) $$\begin{eqnarray}\lim _{t\rightarrow \infty }\frac{1}{t-n\unicode[STIX]{x1D70F}}\int _{0}^{t-n\unicode[STIX]{x1D70F}}\,\text{d}s\langle \unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}(\unicode[STIX]{x1D743},s)|^{2}\rangle _{\unicode[STIX]{x1D6FA}}=\langle \unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}(\unicode[STIX]{x1D743},s)|^{2}\rangle _{\unicode[STIX]{x1D6FA},\infty },\end{eqnarray}$$

where the space–time average $\langle \cdot \rangle _{\unicode[STIX]{x1D6FA},\infty }$ on the right-hand side is over $\unicode[STIX]{x1D743}\in \unicode[STIX]{x1D6FA}$ and $s\in [0,\infty )$ . Since $\lim _{t\rightarrow \infty }(1/2t)\text{Var}[\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)]$ is independent of the choice of $n$ , we obtain (2.20). Of course, here we have assumed that all of the various infinite-time averages exist, as they shall (at least along subsequences of times $t_{k}\rightarrow \infty$ ) if the space-averaged scalar dissipation remains a bounded function of time.

It should be noted that if the scalar is freely decaying from bounded initial data $\unicode[STIX]{x1D703}_{0}$ , then the variance on the left-hand side of (2.20) is also bounded. In this case, the long-time-averaged scalar dissipation rate tends to zero, which comes as no surprise. In order to have a non-vanishing long-time dissipation, the scalar must be continually supplied to the system so that the variance of $\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)$ grows linearly in time. For example, a scalar source $S(\boldsymbol{x},t)$ within the flow domain can provide the necessary scalar input. In such a case, the variance of $\tilde{\unicode[STIX]{x1D703}}(\boldsymbol{x},t)$ grows proportionally to time $t$ at long times because of the cumulative contribution from the scalar source $S$ in the time integral $\int _{0}^{t}S(\tilde{\unicode[STIX]{x1D743}}_{t,s},s)\,\text{d}s$ , and the long-time average scalar dissipation rate matches the mean input rate of the scalar. In fact, the expression on the right-hand side of (2.15) arises after dividing by $t$ the variance of this time integral of $S$ and then taking the limit $t\rightarrow \infty$ . The linear growth of the variance and the expression in (2.15) are central-limit-theorem results based on the statistical independence of the flow maps $\tilde{\unicode[STIX]{x1D743}}_{t,s}$ over widely separated intervals of time $[t,s]$ .

3 Spontaneous stochasticity of Lagrangian trajectories

We now specialize in this section to the sourceless case $S\equiv 0$ , in order to make contact with the work of Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998) on spontaneous stochasticity and anomalous scalar dissipation. The stochastic representation (2.6) simplifies in this case to

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D703}(\boldsymbol{x},t)=\mathbb{E}[\unicode[STIX]{x1D703}_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}))]=\int \,\text{d}^{d}x_{0}\,\unicode[STIX]{x1D703}_{0}(\boldsymbol{x}_{0})p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}_{0},0|\boldsymbol{x},t),\end{eqnarray}$$

where we have introduced the backward-in-time transition probability

(3.2) $$\begin{eqnarray}p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}^{\prime },t^{\prime }|\boldsymbol{x},t)=\mathbb{E}[\unicode[STIX]{x1D6FF}^{d}(\boldsymbol{x}^{\prime }-\tilde{\unicode[STIX]{x1D743}}_{t,t^{\prime }}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}))],\quad t^{\prime }<t\end{eqnarray}$$

for the stochastic flow. As already noted, the stochastic flow preserves volume when the velocity field is divergence-free. In terms of the transition probability, this means that

(3.3) $$\begin{eqnarray}\int p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}^{\prime },t^{\prime }|\boldsymbol{x},t)\,\text{d}^{d}x=1,\end{eqnarray}$$

where $\text{det}(\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D743}}_{t,t^{\prime }}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x})/\unicode[STIX]{x2202}\boldsymbol{x})=1$ is used to write $\unicode[STIX]{x1D6FF}^{d}(\boldsymbol{x}^{\prime }-\tilde{\unicode[STIX]{x1D743}}_{t,t^{\prime }}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}))=\unicode[STIX]{x1D6FF}^{d}(\boldsymbol{x}-(\tilde{\unicode[STIX]{x1D743}}_{t,t^{\prime }}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}})^{-1}(\boldsymbol{x}^{\prime }))$ and perform the integral over $\boldsymbol{x}$ . It should be noted that if the limit $\unicode[STIX]{x1D705}\rightarrow 0$ is taken with $\unicode[STIX]{x1D708}$ fixed (infinite-Prandtl-number limit), then the stochastic flow (3.2) becomes deterministic and

(3.4) $$\begin{eqnarray}p^{\unicode[STIX]{x1D708},0}(\boldsymbol{x}^{\prime },t^{\prime }|\boldsymbol{x},t)=\unicode[STIX]{x1D6FF}^{d}(\boldsymbol{x}^{\prime }-\unicode[STIX]{x1D743}_{t,t^{\prime }}^{\unicode[STIX]{x1D708},0}(\boldsymbol{x})),\quad t^{\prime }<t,\end{eqnarray}$$

which corresponds to a single deterministic Lagrangian trajectory passing through position $\boldsymbol{x}$ at time $t$ .

In the Kraichnan model of turbulent advection, it was shown by Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998) that the joint limit $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ with $Pr=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}$ fixed is non-deterministic and corresponds to more than one Lagrangian trajectory passing through space–time point $(\boldsymbol{x},t)$ . We remind the reader that the Kraichnan model of turbulent advection replaces the Navier–Stokes solution with a realization drawn from an ensemble of Gaussian random fields $\boldsymbol{u}^{\unicode[STIX]{x1D708}}$ with mean zero $\langle \boldsymbol{u}^{\unicode[STIX]{x1D708}}\rangle =\mathbf{0}$ and covariance satisfying

(3.5) $$\begin{eqnarray}\langle [u_{i}^{\unicode[STIX]{x1D708}}(\boldsymbol{x}+\boldsymbol{r},t)-u_{i}^{\unicode[STIX]{x1D708}}(\boldsymbol{x},t)][u_{j}^{\unicode[STIX]{x1D708}}(\boldsymbol{x}+\boldsymbol{r},t^{\prime })-u_{j}^{\unicode[STIX]{x1D708}}(\boldsymbol{x},t^{\prime })]\rangle =D_{ij}^{\unicode[STIX]{x1D708}}(\boldsymbol{r})\unicode[STIX]{x1D6FF}(t-t^{\prime })\end{eqnarray}$$

for a spatial covariance function satisfying $D_{ij}^{\unicode[STIX]{x1D708}}(\boldsymbol{r})=D_{ji}^{\unicode[STIX]{x1D708}}(\boldsymbol{r})$ , $\unicode[STIX]{x2202}D_{ij}^{\unicode[STIX]{x1D708}}(\boldsymbol{r})/\unicode[STIX]{x2202}r_{j}=0,$ and

(3.6) $$\begin{eqnarray}D_{ii}(\boldsymbol{r})\sim \left\{\begin{array}{@{}ll@{}}D_{1}r^{\unicode[STIX]{x1D709}},\quad & \ell _{\unicode[STIX]{x1D708}}\ll r\ll L,\\ D_{2}r^{2},\quad & r\ll \ell _{\unicode[STIX]{x1D708}},\end{array}\right.\end{eqnarray}$$

for some $0<\unicode[STIX]{x1D709}<2$ , with the effective ‘dissipation length’

(3.7) $$\begin{eqnarray}\ell _{\unicode[STIX]{x1D708}}=(D_{1}/D_{2})^{1/(2-\unicode[STIX]{x1D709})}.\end{eqnarray}$$

It should be noted that $D_{2}\propto \langle |\unicode[STIX]{x1D735}\boldsymbol{u}^{\unicode[STIX]{x1D708}}|^{2}\rangle$ and in real turbulence would be proportional to $\unicode[STIX]{x1D700}/\unicode[STIX]{x1D708},$ where $\unicode[STIX]{x1D700}$ is the viscous energy dissipation. Hence, $D_{2}\rightarrow \infty$ or $\ell _{\unicode[STIX]{x1D708}}\rightarrow 0$ with $D_{1}$ fixed is the analogue for the Kraichnan model of the infinite-Reynolds-number limit for Navier–Stokes turbulence. In fact, one can introduce a ‘viscosity’ parameter $\unicode[STIX]{x1D708}$ for the Kraichnan model with units of $(\text{length})^{2}/(\text{time})$ , so that $\ell _{\unicode[STIX]{x1D708}}=(\unicode[STIX]{x1D708}/D_{1})^{1/\unicode[STIX]{x1D709}}$ . For any $\unicode[STIX]{x1D708}>0$ , the velocity realizations are spatially smooth, but in the limit $\unicode[STIX]{x1D708}\rightarrow 0$ , they are only Hölder continuous in space with exponent $0<\unicode[STIX]{x1D709}/2<1$ . It is well known that for such ‘rough’ limiting velocity fields the solutions of the deterministic initial-value problem

(3.8) $$\begin{eqnarray}\text{d}\unicode[STIX]{x1D743}(s)/\text{d}s=\boldsymbol{u}(\unicode[STIX]{x1D743}(s),s),\quad \unicode[STIX]{x1D743}(t)=\boldsymbol{x}\end{eqnarray}$$

need not be unique and, if not, form a continuum of solutions (e.g. see Hartman Reference Hartman2002). In the Kraichnan model, it has been proved in the double limit with both $\unicode[STIX]{x1D708}\rightarrow 0$ and $\unicode[STIX]{x1D705}\rightarrow 0$ that the transition probabilities tend to a limiting form

(3.9) $$\begin{eqnarray}p^{\ast }(\boldsymbol{x}^{\prime },t^{\prime }|\boldsymbol{x},t)=\lim _{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0}p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}^{\prime },t^{\prime }|\boldsymbol{x},t).\end{eqnarray}$$

It is important to stress here that no average is taken over $\boldsymbol{u}$ in defining these transition probabilities, but only an average over Brownian motions in the stochastic flow equations (2.3), while the velocity realization is held fixed. (It would be less ambiguous to write them as $p_{\boldsymbol{u}}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}^{\prime },t^{\prime }|\boldsymbol{x},t)$ , with $\boldsymbol{u}$ denoting the fixed flow realization, but this would lead to an even heavier notation.) Most importantly, the limiting transition probabilities for the Kraichnan model are not delta distributions of the form (3.4), but non-trivial probabilities over an ensemble of non-unique solutions of the limiting ordinary differential equation (3.8). This remarkable phenomenon is called spontaneous stochasticity. See Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998) and the later papers of Vanden-Eijnden & Vanden-Eijnden (Reference Vanden-Eijnden and Vanden-Eijnden2000, Reference Vanden-Eijnden and Vanden-Eijnden2001), Gawȩdzki & Vergassola (Reference Gawȩdzki and Vergassola2000), Falkovich et al. (Reference Falkovich, Gawȩdzki and Vergassola2001) and Le Jan & Raimond (Reference Le Jan and Raimond2002, Reference Le Jan and Raimond2004).

As shown in these works, spontaneous stochasticity occurs because of the analogue of Richardson (Reference Richardson1926) dispersion in the Kraichnan model, which leads to a loss of influence of the molecular diffusivity $\unicode[STIX]{x1D705}$ on the separation of the perturbed Lagrangian trajectories after a short time of order $(\unicode[STIX]{x1D705}^{2-\unicode[STIX]{x1D709}}/D_{1})^{1/\unicode[STIX]{x1D709}}$ . It is important to emphasize that this result does not mean that randomness in the Lagrangian trajectories suddenly ‘appears’ only for $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}=0$ , but instead that the randomness persists even as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ . It is thus a phenomenon that can be observed with sequences of positive values, $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}>0$ , for which the velocity field is smooth. For the case of a divergence-free velocity that we discuss here, it is furthermore known that the result does not depend upon the order of limits $\unicode[STIX]{x1D708}\rightarrow 0$ and $\unicode[STIX]{x1D705}\rightarrow 0$ , which can be taken in either order or together. (The only delicate case is when $\unicode[STIX]{x1D705}\rightarrow 0$ first, so that the Prandtl number goes to infinity, and then $\unicode[STIX]{x1D708}\rightarrow 0$ subsequently. Since the Brownian motion disappears from the stochastic equation (2.3) while the velocity field remains smooth, the limiting Lagrangian trajectories are deterministic. To observe spontaneous stochasticity in that limit, one must additionally allow the initial condition to be random, e.g. with $\tilde{\unicode[STIX]{x1D743}}(t)=\boldsymbol{x}+\unicode[STIX]{x1D716}\tilde{\unicode[STIX]{x1D746}}$ for a stochastic perturbation $\tilde{\unicode[STIX]{x1D746}}$ drawn from some fixed distribution $P(\unicode[STIX]{x1D746})$ . In that case, spontaneous stochasticity appears in the double limit with $\unicode[STIX]{x1D716}\rightarrow 0$ and $\unicode[STIX]{x1D708}\rightarrow 0$ together, and, for a divergence-free velocity $\boldsymbol{u}$ , the limiting transition probabilities are identical to those obtained for the other limits involving $\unicode[STIX]{x1D705}\rightarrow 0$ . This infinite-Prandtl case is discussed carefully by Vanden-Eijnden & Vanden-Eijnden (Reference Vanden-Eijnden and Vanden-Eijnden2000) and Gawȩdzki & Vergassola (Reference Gawȩdzki and Vergassola2000).) See Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998), Vanden-Eijnden & Vanden-Eijnden (Reference Vanden-Eijnden and Vanden-Eijnden2000, Reference Vanden-Eijnden and Vanden-Eijnden2001) Gawȩdzki & Vergassola (Reference Gawȩdzki and Vergassola2000), Falkovich et al. (Reference Falkovich, Gawȩdzki and Vergassola2001) and Le Jan & Raimond (Reference Le Jan and Raimond2002, Reference Le Jan and Raimond2004) for discussions of this point.

There is empirical evidence for such phenomena also in Navier–Stokes turbulence obtained from numerical studies of two-particle dispersion. Eyink (Reference Eyink2011) studied stochastic Lagrangian particles whose motion is governed by (2.3) in a $1024^{3}$ DNS at $Re_{\unicode[STIX]{x1D706}}=433$ and found that the mean-square dispersion becomes independent of $\unicode[STIX]{x1D705}$ after a short time of order $(\unicode[STIX]{x1D705}/\unicode[STIX]{x1D700})^{1/2}$ . Bitane, Homann & Bec (Reference Bitane, Homann and Bec2013) studied dispersion of deterministic Lagrangian trajectories $(\unicode[STIX]{x1D705}=0)$ in a $2048^{3}$ DNS at $Re_{\unicode[STIX]{x1D706}}=460$ and a $4096^{3}$ DNS at $Re_{\unicode[STIX]{x1D706}}=730$ , and found that the mean-square dispersion becomes independent of the initial separation $r_{0}$ of particle pairs in a short time of order $r_{0}^{2/3}/\unicode[STIX]{x1D700}^{1/3}$ . The results of these studies provide evidence of Lagrangian spontaneous stochasticity for Navier–Stokes solutions. In particular, Bitane et al. (Reference Bitane, Homann and Bec2013) found consistent Richardson-dispersion statistics for the two Reynolds numbers studied there. The principal limitation of these previous studies is that they averaged over the release points $\boldsymbol{x}$ of the particles. A particle dispersion averaged over release points which remains non-vanishing in the joint limit $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ is enough to infer spontaneous stochasticity for a set of points $\boldsymbol{x}$ of non-zero volume measure (Bernard et al. Reference Bernard, Gawȩdzki and Kupiainen1998). However, averaging over $\boldsymbol{x}$ removes information about the effects of spatial intermittency and the local fluid environment on the limiting behaviour of the particle distributions $p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}^{\prime },t^{\prime }|\boldsymbol{x},t)$ for specific release locations $\boldsymbol{x}$ . There was some previous study of such spatial intermittency in pair dispersion by Biferale et al. (Reference Biferale, Boffetta, Celani, Lanotte and Toschi2005, Reference Biferale, Lanotte, Scatamacchia and Toschi2014) but they studied only deterministic Lagrangian particles at small (Kolmogorov scale) initial separations, and not the stochastic Lagrangian particles relevant to our FDR.

We present here new data obtained from numerical experiments on a high-Reynolds-number turbulence simulation in a $2\unicode[STIX]{x03C0}$ -periodic box, for a couple of representative release points. We use simulation data from the homogeneous isotropic dataset in the Johns Hopkins Turbulence Database (Li et al. Reference Li, Perlman, Wan, Yang, Burns, Meneveau, Chen, Szalay and Eyink2008; Yu et al. Reference Yu, Kanov, Perlman, Graham, Frederix, Burns, Szalay, Eyink and Meneveau2012), publicly available online at http://turbulence.pha.jhu.edu. It is ideal for our purposes, since the entire time history of the velocity is stored for a full large-scale eddy-turnover time, allowing us to integrate the flow equations (2.3) backward in time. A significant limitation, however, is that only one Reynolds number is available, $Re\simeq 5058$ . One should consider together with the limit $\unicode[STIX]{x1D705}\rightarrow 0$ also a limit $\unicode[STIX]{x1D708}\rightarrow 0$ , so that the Navier–Stokes solution $\boldsymbol{u}^{\unicode[STIX]{x1D708}}$ converges to a fixed velocity $\boldsymbol{u}$ that is some sort of weak solution of Euler (as always occurs along a suitable subsequence $\unicode[STIX]{x1D708}_{k}\rightarrow 0$ ; see Lions (Reference Lions1996, § 4.4)). (The necessity of extracting such a subsequence makes the empirical study of spontaneous stochasticity quite difficult, as a matter of principle. The compactness argument of Lions (Reference Lions1996) establishes existence of subsequences of $\unicode[STIX]{x1D708}_{k}\rightarrow 0$ such that Navier–Stokes solutions $\boldsymbol{u}^{\unicode[STIX]{x1D708}_{k}}(\boldsymbol{x},t)$ converge to a fixed limiting velocity field $\boldsymbol{u}(\boldsymbol{x},t)$ that is a ‘dissipative Euler solution’, in a suitable sense. Unfortunately, the proof is not constructive and therefore there is currently no concrete computational algorithm to generate any specific convergent subsequence. Amusingly, the direct experimental observation of spontaneous stochasticity in such a joint limit may be easier in quantum mechanics than in turbulent fluids. See Eyink & Drivas (Reference Eyink and Drivas2015a ).) Since no such joint limit $\unicode[STIX]{x1D708}$ , $\unicode[STIX]{x1D705}\rightarrow 0$ can be considered within the given database at one fixed value of viscosity, our study of spontaneous stochasticity is based on the assumption that the Reynolds numbers is already ‘sufficiently large’. More precisely, we assume that an inertial-range superballistic Richardson-type dispersion of particle pairs released at space–time point $(\boldsymbol{x},t)$ will occur at times $|t^{\prime }-t|>t_{c}$ , with a crossover time $t_{c}=\max \{(\unicode[STIX]{x1D700}/\unicode[STIX]{x1D708})^{1/2},(\unicode[STIX]{x1D700}/\unicode[STIX]{x1D705})^{1/2}\}$ , and then $p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}^{\prime },t^{\prime }|\boldsymbol{x},t)\simeq p^{\ast }(\boldsymbol{x}^{\prime },t^{\prime }|\boldsymbol{x},t)$ for $|t^{\prime }-t|\gg t_{c}$ . It should be noted that, for $Pr<1$ , mean-square dispersion grows diffusively $\propto \unicode[STIX]{x1D705}|t^{\prime }-t|$ up to a time $t_{\unicode[STIX]{x1D705}}=(\unicode[STIX]{x1D700}/\unicode[STIX]{x1D705})^{1/2}$ , when relative advection begins to dominate at the length scale $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D705}}=(\unicode[STIX]{x1D705}^{3}/\unicode[STIX]{x1D700})^{1/4}$ within the inertial range. Instead, for $Pr>1$ , particle pairs also separate diffusively initially but then transition to exponential divergence ${\sim}\text{exp}(t/t_{\unicode[STIX]{x1D702}})\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D705}}$ , with Kolmogorov time $t_{\unicode[STIX]{x1D702}}=(\unicode[STIX]{x1D700}/\unicode[STIX]{x1D708})^{1/2}$ , until the particles separate to the Kolmogorov dissipation scale $\unicode[STIX]{x1D702}=(\unicode[STIX]{x1D708}^{3}/\unicode[STIX]{x1D716})$ at time ${\sim}t_{\unicode[STIX]{x1D702}}\log Pr$ , when superballistic dispersion commences. The particle distributions $p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}^{\prime },t^{\prime }|\boldsymbol{x},t)$ at $|t^{\prime }-t|\simeq t_{c}$ will be distinct in these different cases and will presumably also depend upon the particular values of $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}$ even as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ . However, Richardson dispersion leads to a very rapid ‘forgetting’ of the precise initial data, and thus it is reasonable to expect that $p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}^{\prime },t^{\prime }|\boldsymbol{x},t)\simeq p^{\ast }(\boldsymbol{x}^{\prime },t^{\prime }|\boldsymbol{x},t)$ for $|t^{\prime }-t|\gg t_{c}$ . With this assumption, we may study the limiting particle distributions $p^{\ast }(\boldsymbol{x}^{\prime },t^{\prime }|\boldsymbol{x},t)$ in the database at large but finite Reynolds number. A check on this assumption is provided by the fact that, for incompressible flows, the limiting distributions are also expected to be independent of the Prandtl number $Pr$ (Vanden-Eijnden & Vanden-Eijnden Reference Vanden-Eijnden and Vanden-Eijnden2000, Reference Vanden-Eijnden and Vanden-Eijnden2001; Gawȩdzki & Vergassola Reference Gawȩdzki and Vergassola2000). By varying $\unicode[STIX]{x1D705}$ for the fixed $\unicode[STIX]{x1D708}$ in the database, we can change $Pr$ and verify to what extent the Prandtl independence of limiting distributions holds for our numerical results.

Figure 1. (a,c,e) Plots for release at $\boldsymbol{x}=(4.9637,3.1416,3.8488)$ in the background region; (b,d,f) plots for release at $\boldsymbol{x}=(0.2610,3.1416,1.4617)$ near a strong vortex. (a,b) Plots of 30 representative stochastic trajectories for $Pr=0.1$ (green, light), 1.0 (blue, medium) and 10 (red, heavy) together with isosurfaces of coarse-grained vorticity $|\bar{\unicode[STIX]{x1D74E}}|T_{L}=15$ at time $s=(2/3)T_{L}$ . (c,d) Plots of particle dispersions (heavy) and short-time results $12\unicode[STIX]{x1D705}{\hat{s}}$ (light) for each $Pr$ , with $Pr=0.1$ (green, dot, $\cdots \cdots$ ), 1.0 (blue, dash–dot, — ⋅ —) and 10 (red, dash, – – –), and a plot in (solid, ——) of $g\unicode[STIX]{x1D700}{\hat{s}}^{3}$ with $g=0.7$ (c) and $g=4/3$ (d). (e,f) Plots of $p_{y}(y^{\prime },0|\boldsymbol{x},t_{f})$ for the three $Pr$ values with the same line styles as in (c,d).

We consider two release points $\boldsymbol{x}$ at time $t_{f}=2.048$ , the final database time, one chosen in a typical turbulent ‘background’ region and the other in the vicinity of a strong large-scale vorticity. We study stochastic trajectories with diffusivities $\unicode[STIX]{x1D705}$ corresponding to three values of the Prandtl number, $Pr=0.1$ , 1 and 10. See appendix C for details about the numerical methods employed in our analysis. Figure 1(a,b) shows 30 representative particle trajectories for the two release points and for each of the three Prandtl numbers. To illustrate the local fluid environment, we also plot isosurfaces of the vorticity filtered with a box filter of width $L/4$ (where $L$ is the integral scale) at the time $s=(2/3)T_{L}$ (where $T_{L}$ is the large-scale turnover time). The isosurfaces are for magnitudes of filtered vorticity equal to $15/T_{L}$ . Figure 1(a,c,e) shows the particles released in a typical ‘background’ region with spottier weaker vortices and figure 1(b,d,f) shows particles released near a strong vortex. (In the weak background region in figure 1(a), $\overline{\unicode[STIX]{x1D714}}_{rms}T_{L}=2.98$ , and thus the isosurface level is $|\overline{\unicode[STIX]{x1D74E}}|\approx 5.0\overline{\unicode[STIX]{x1D714}}_{rms}$ , with 0.75 % of the volume in that box carrying filtered vorticity above this threshold. In the strong vortex region in figure 1(b), instead, $\overline{\unicode[STIX]{x1D714}}_{rms}T_{L}=4.25$ , so that the isosurface there is $|\overline{\unicode[STIX]{x1D74E}}|\approx 3.5\overline{\unicode[STIX]{x1D714}}_{rms}$ , with 2.9 % of the volume above.) The three colourings of the trajectories (green/blue/red) represent the three values of the Prandtl number, $Pr=0.1$ , 1, 10 respectively. The clearly observable ‘splitting’ of the bundle of stochastic trajectories into sub-bundles at specific times recalls one proposed mechanism for Richardson dispersion, via a sequence of smooth transport and rapid ‘flight-like’ departures at fluid separatrices (Shlesinger, West & Klafter Reference Shlesinger, West and Klafter1987; Davila & Vassilicos Reference Davila and Vassilicos2003; Thalabard, Krstulovic & Bec Reference Thalabard, Krstulovic and Bec2014). Most importantly, as one can see by eye, the ensembles of trajectories are quite similar for the three $Pr$ values.

To make the latter observation more quantitative, we plot in figure 1(c,d) the mean-square dispersion of pairs of stochastic Lagrangian particles with different realizations of the noise, for the two release points and the three Prandtl numbers. The error bars (almost too small to be observed) represent the standard error of the mean (s.e.m.) for averages over $N=1024$ sample trajectories. For both release points, there is an initial period (going backward in time) where the dispersion grows diffusively as $12\unicode[STIX]{x1D705}{\hat{s}}$ with ${\hat{s}}=t_{f}-s$ , but which then crosses over to a regime of superballistic separation that is close to the ${\hat{s}}^{3}$ growth predicted by Richardson (Reference Richardson1926) and is approximately independent of $Pr$ . An essential observation from figure 1 is that ${\hat{s}}\approx t_{c}$ is indeed the time of crossover to a roughly Richardson $t^{3}$ growth. The two release points shown here illustrate behaviour that we have observed also in many other points of the turbulent fluid, where we find that the Richardson ${\hat{s}}^{3}$ law is surprisingly robust (albeit imperfect), without the necessity of averaging over release points $\boldsymbol{x}$ . This is especially so for points $\boldsymbol{x}$ in ‘background’ regions, and is at least approximately observed for $\boldsymbol{x}$ located in more intermittent regions.

Finally, we plot in figure 1(e,f) particle transition probabilities, which provide even more information about the limiting behaviour. We plot at time 0, in the approximate Richardson range, the one-dimensional probability density functions (PDFs) of the $y$ -coordinate, or

(3.10) $$\begin{eqnarray}p_{y}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(y^{\prime },0|\boldsymbol{x},t_{f})=\int \,\text{d}x^{\prime }\,\text{d}z^{\prime }p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}^{\prime },0|\boldsymbol{x},t_{f}),\end{eqnarray}$$

for each of the two release points $\boldsymbol{x}$ and three Prandtl numbers. We observe very similar behaviour also for the $x$ - and $z$ -coordinates. In order to minimize the number of samples required to construct the PDFs numerically, we employed kernel density estimator techniques, which gave us good results with only $N=6144$ samples. See Silverman (Reference Silverman1986) and appendix C, where our numerical procedures are completely described. The error bars represent both the s.e.m.s for the $N$ -sample averages and the effects of variation in the kernel density bandwidth. Consistent with the dispersion plots, we see that the transition PDFs are approximately independent of $Pr$ for times in the superballistic dispersion range. This is especially true for the release point $\boldsymbol{x}$ in the ‘background’ region, and for the strong vorticity region, such independence holds better for the two largest values of $Pr$ (smallest $\unicode[STIX]{x1D705}$ ), when the Richardson-like superballistic range is the longest. This approximate independence of the PDFs from the Prandtl number gives some support to the conjecture that $p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}^{\prime },0|\boldsymbol{x},t_{f})\simeq p^{\ast }(\boldsymbol{x}^{\prime },0|\boldsymbol{x},t_{f})$ and that the infinite- $Re$ limit is already achieved for such particle transition kernels in the database at finite $Re$ . These numerical studies illustrate the present quality of direct evidence for Lagrangian spontaneous stochasticity in high-Reynolds-number Navier–Stokes turbulence, which is suggestive but far from compelling. As we shall now demonstrate, observations of anomalous scalar dissipation provide further evidence, as the two phenomena are essentially related.

4 Spontaneous stochasticity and anomalous dissipation

The phenomenon of spontaneous stochasticity leads to a simple explanation of anomalous dissipation in a turbulent flow, as was first pointed out by Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998) for decaying scalars (no sources) in the Kraichnan model of random advection. This connection can be understood more directly and more generally using our FDR. In fact, it is intuitively clear from the FDR (2.12) that there can be scalar dissipation that is non-vanishing in the limit $\unicode[STIX]{x1D705}\rightarrow 0$ only if there is a non-vanishing variance in that same limit, implying that Lagrangian trajectories must remain stochastic. This argument holds in the presence of scalar sources and for a scalar advected by any velocity field $\boldsymbol{u}^{\unicode[STIX]{x1D708}}$ whatsoever. In particular, the argument holds when $\boldsymbol{u}^{\unicode[STIX]{x1D708}}$ is a Navier–Stokes solution. Thus, spontaneous stochasticity is the only possible mechanism of anomalous dissipation, for both passive and active scalars, away from walls. Furthermore, we shall show for a passive scalar that does not react back on the flow that spontaneous stochasticity also makes anomalous scalar dissipation possible. Thus, for passive scalars, the two phenomena are completely equivalent. In this section, we shall deduce these conclusions, assuming only that the flow domain is compact (closed and bounded) and without any bounding walls.

We first discuss the technically simpler case with $S\equiv 0$ and then show that the same argument extends easily to the case with a non-zero scalar source. When $S\equiv 0$ , we can rewrite the left-hand side of the FDR (2.12) using

(4.1) $$\begin{eqnarray}\displaystyle \text{Var}[\unicode[STIX]{x1D703}_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))] & = & \displaystyle \int \,\text{d}^{d}x_{0}\int \,\text{d}^{d}x_{0}^{\prime }\,\unicode[STIX]{x1D703}_{0}(\boldsymbol{x}_{0})\unicode[STIX]{x1D703}_{0}(\boldsymbol{x}_{0}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \times \,[p_{2}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}_{0},0;\boldsymbol{x}_{0}^{\prime },0|\boldsymbol{x},t)-p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}_{0},0|\boldsymbol{x},t)p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}_{0}^{\prime },0|\boldsymbol{x},t)],\qquad\end{eqnarray}$$

where we have introduced the two-time (backward-in-time) transition probability density

(4.2) $$\begin{eqnarray}p_{2}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{y},s;\boldsymbol{y}^{\prime },s^{\prime }|\boldsymbol{x},t)=\mathbb{E}[\unicode[STIX]{x1D6FF}^{d}(\boldsymbol{y}-\tilde{\unicode[STIX]{x1D743}}_{t,s}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}))\unicode[STIX]{x1D6FF}^{d}(\boldsymbol{y}^{\prime }-\tilde{\unicode[STIX]{x1D743}}_{t,s^{\prime }}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}))],\quad s,s^{\prime }<t,\end{eqnarray}$$

which gives the joint probability for the particle to end up at $\boldsymbol{y}$ at time $s<t$ and at $\boldsymbol{y}^{\prime }$ at time $s^{\prime }<t,$ given that it started at $\boldsymbol{x}$ at the final time $t$ (moving backward from final to earlier times). At equal times $s=s^{\prime }$ ,

(4.3) $$\begin{eqnarray}p_{2}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{y},s;\boldsymbol{y}^{\prime },s|\boldsymbol{x},t)=\unicode[STIX]{x1D6FF}^{d}(\boldsymbol{y}-\boldsymbol{y}^{\prime })p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{y},s|\boldsymbol{x},t).\end{eqnarray}$$

We now consider the limit $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ , so that the transition probabilities approach limiting values $p^{\ast }(\boldsymbol{y},s;\boldsymbol{y}^{\prime },s|\boldsymbol{x},t)$ , $p^{\ast }(\boldsymbol{y},s|\boldsymbol{x},t)$ . Such limits exist, at least along suitably chosen subsequences $\unicode[STIX]{x1D708}_{n},\unicode[STIX]{x1D705}_{n}\rightarrow 0$ , whenever the flow domain is compact. This can be shown using Young measure methods similar to those that have been employed previously to study statistical equilibria for 2D Euler solutions (Robert Reference Robert1991; Robert & Sommeria Reference Robert and Sommeria1991; Sommeria, Staquet & Robert Reference Sommeria, Staquet and Robert1991). Because the proof of this result is a little technical, we give it in appendix A.1. When the Lagrangian particles move according to a deterministic flow $\unicode[STIX]{x1D743}_{t,s}^{\ast }$ , one easily sees that the two-time transition probability factorizes as

(4.4) $$\begin{eqnarray}p_{2}^{\ast }(\boldsymbol{y},s;\boldsymbol{y}^{\prime },s^{\prime }|\boldsymbol{x},t)=\unicode[STIX]{x1D6FF}^{d}(\boldsymbol{y}-\unicode[STIX]{x1D743}_{t,s}^{\ast }(\boldsymbol{x}))\unicode[STIX]{x1D6FF}^{d}(\boldsymbol{y}^{\prime }-\unicode[STIX]{x1D743}_{t,s^{\prime }}^{\ast }(\boldsymbol{x}))=p^{\ast }(\boldsymbol{y},s|\boldsymbol{x},t)p^{\ast }(\boldsymbol{y}^{\prime },s^{\prime }|\boldsymbol{x},t).\end{eqnarray}$$

Hence, non-factorization in the limit $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ is an unequivocal sign of spontaneous stochasticity. The variance on the left-hand side of the FDR (2.12) can only be non-vanishing in the limit if factorization fails, so that anomalous dissipation clearly requires spontaneous stochasticity. In the other direction, if there is spontaneous stochasticity and thus factorization fails for some positive-measure set of $\boldsymbol{x}\in \unicode[STIX]{x1D6FA}$ , then the contribution to the volume-integrated variance from that subset must be positive for some suitable smooth choice of $\unicode[STIX]{x1D703}_{0}$ , which implies a positive lower bound to the cumulative volume-integrated scalar dissipation. In short, anomalous scalar dissipation and Lagrangian spontaneous stochasticity are seen to be equivalent. This argument is given as a formal mathematical proof in appendix A.2.

The sufficiency argument works only for a passive scalar. For active scalars, the initial datum $\unicode[STIX]{x1D703}_{0}$ partially determines the velocity field $\boldsymbol{u}$ and so is not free to vary. In order to conclude sufficiency in this case, one needs to assume that the resulting velocity field does not ‘conspire’ with the initial scalar to cause the variance to vanish, i.e. for the random trajectories to sample only points on a single level set of $\unicode[STIX]{x1D703}_{0}$ . If this remarkable behaviour did happen to occur for some choice of $\unicode[STIX]{x1D703}_{0}$ , then one would not expect it to persist for a small perturbation of $\unicode[STIX]{x1D703}_{0}$ . Thus, it is highly likely also for active scalars that spontaneous stochasticity implies anomalous dissipation, but we have not proved that with the FDR. We can, however, conclude rigorously both for passive and for active scalars that anomalous dissipation implies spontaneous stochasticity. The above proposition shows that any evidence for anomalous scalar dissipation in the free decay of an active or passive scalar (no sources) obtained from DNS in a periodic box is also evidence for spontaneous stochasticity. The argument in this section is a strong motivation to perform DNS studies to verify anomalous dissipation in the free decay of a scalar, since this would provide additional confirmation of spontaneous stochasticity. All of the DNS cited by Yeung, Donzis & Sreenivasan (Reference Yeung, Donzis and Sreenivasan2005, § 2.1) employed sources (e.g. a mean scalar gradient coupled to the velocity field) that maintained a statistical steady state for the scalar fluctuations.

Inclusion of a non-zero scalar source involves only minor changes to the previous argument. First, it should be noted that

(4.5) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{Var}\left[\unicode[STIX]{x1D703}_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))+\int _{0}^{t}S(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\text{d}s\right]=\text{Var}[\unicode[STIX]{x1D703}_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))]\nonumber\\ \displaystyle & & \displaystyle \quad +\,2\,\text{Cov}\left[\unicode[STIX]{x1D703}_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x})),\int _{0}^{t}S(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\text{d}s\right]+\text{Var}\left[\int _{0}^{t}S(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\text{d}s\right].\end{eqnarray}$$

Furthermore, one has for the variance of the time-integrated source sampled along the stochastic particle trajectory that

(4.6) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{Var}\left[\int _{0}^{t}S(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\text{d}s\right]=\int _{0}^{t}\,\text{d}s\int _{0}^{t}\,\text{d}s^{\prime }\int \,\text{d}^{d}y\int \,\text{d}^{d}y^{\prime }S(\boldsymbol{y},s)S(\boldsymbol{y}^{\prime },s^{\prime })\nonumber\\ \displaystyle & & \displaystyle \quad \times \,[p_{2}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{y},s;\boldsymbol{y}^{\prime },s^{\prime }|\boldsymbol{x},t)-p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{y},s|\boldsymbol{x},t)p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{y}^{\prime },s^{\prime }|\boldsymbol{x},t)]\end{eqnarray}$$

and for the covariance between the sampled initial data and the integrated source that

(4.7) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{Cov}\left[\unicode[STIX]{x1D703}_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x})),\int _{0}^{t}S(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\text{d}s\right]=\int _{0}^{t}\,\text{d}s\int \,\text{d}^{d}x_{0}\int \,\text{d}^{d}y\,\unicode[STIX]{x1D703}_{0}(\boldsymbol{x}_{0})S(\boldsymbol{y},s)\nonumber\\ \displaystyle & & \displaystyle \quad \times \,[p_{2}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}_{0},0;\boldsymbol{y},s|\boldsymbol{x},t)-p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}_{0},0|\boldsymbol{x},t)p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{y},s|\boldsymbol{x},t)].\end{eqnarray}$$

Clearly, anomalous scalar dissipation requires spontaneous stochasticity. For a passive scalar, we can also argue in the other direction. Indeed, we can repeat the previous argument to conclude that, if there is spontaneous stochasticity for a positive-measure set of $\boldsymbol{x}$ , then not only is there a smooth choice of $\unicode[STIX]{x1D703}_{0}$ so that the variance associated with the initial condition in (4.1) is positive when integrated over this set of $\boldsymbol{x}$ , but also there is a smooth choice of source field $S$ so that the contribution of the variance (4.6) is positive. This is already enough to conclude that there must be anomalous dissipation for the scalar with initial condition 0 and with the chosen source $S$ . We can also conclude that there is anomalous dissipation for the initial condition $\unicode[STIX]{x1D703}_{0}$ and the source $S$ . Indeed, if the total variance contribution in (4.5) is not positive, then it must vanish, which implies that the covariance term in (4.7) provides a negative contribution. In this case, we simply take $S\rightarrow -S$ to make the contributions of all three terms (4.1), (4.6), (4.7) positive. We thus conclude that, also for the passive scalar rejuvenated by a source, there is equivalence between anomalous scalar dissipation and Lagrangian spontaneous stochasticity. The argument is given more carefully in appendix A.2.

It has not been generally appreciated that similar conclusions can be reached in the special case of sourceless scalars using the arguments of Bernard et al. (Reference Bernard, Gawȩdzki and Kupiainen1998), which are not at all restricted to the Kraichnan model. To underline this point and, also, to give additional insight, we here briefly summarize their reasoning. It should be noted that the stochastic representation (3.1) of the advected scalar in the limit $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ becomes, using (3.9),

(4.8) $$\begin{eqnarray}\unicode[STIX]{x1D703}^{\ast }(\boldsymbol{x},t)=\int \,\text{d}^{d}x_{0}\,\unicode[STIX]{x1D703}_{0}(\boldsymbol{x}_{0})p^{\ast }(\boldsymbol{x}_{0},0|\boldsymbol{x},t).\end{eqnarray}$$

It is worth noting that $\unicode[STIX]{x1D703}^{\ast }(\boldsymbol{x},t)$ is a kind of ‘weak solution’ of the ideal advection equation, $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D703}^{\ast }+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}^{\ast }=0$ , although this fact is not needed for the argument. It follows from (4.8) that for any strictly convex function $h(\unicode[STIX]{x1D703})$ , e.g. $h(\unicode[STIX]{x1D703})=\unicode[STIX]{x1D703}^{2}/2$ ,

(4.9) $$\begin{eqnarray}h(\unicode[STIX]{x1D703}^{\ast }(\boldsymbol{x},t))\leqslant \int \,\text{d}^{d}x_{0}\,h(\unicode[STIX]{x1D703}_{0}(\boldsymbol{x}_{0}))p^{\ast }(\boldsymbol{x}_{0},0|\boldsymbol{x},t),\end{eqnarray}$$

and equality holds if and only if the transition probability is a delta distribution of type (3.4). This is the so-called Jensen inequality (e.g. see Itô Reference Itô1984). Since the limiting transition probabilities are not delta distributions in the Kraichnan model, the inequality in (4.9) is strict. Furthermore, the limiting transition probabilities for $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ inherit the volume-preservation property (3.3), so that

(4.10) $$\begin{eqnarray}\int p^{\ast }(\boldsymbol{x}^{\prime },t^{\prime }|\boldsymbol{x},t)\,\text{d}^{d}x=1.\end{eqnarray}$$

In this case, integrating (4.9) over $\boldsymbol{x}$ gives

(4.11) $$\begin{eqnarray}\int h(\unicode[STIX]{x1D703}^{\ast }(\boldsymbol{x},t))\,\text{d}^{d}x<\int h(\unicode[STIX]{x1D703}_{0}(\boldsymbol{x}_{0}))\,\text{d}^{d}x_{0},\end{eqnarray}$$

so that the $h$ -integral is decaying (dissipated) even in the limit $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ . The anomalous scalar dissipation in the Kraichnan model thus has an elegant Lagrangian mechanism. Essentially, the molecular diffusivity is replaced by a ‘turbulent diffusivity’ associated with the persistent stochasticity of the Lagrangian trajectories, which continues to homogenize the scalar field even as the molecular diffusivity vanishes. We give rigorous details of this argument in appendix A.3, where, in the absence of sources, we obtain necessary and sufficient conditions for anomalous dissipation identical to those derived from the FDR.

5 Summary and discussion

This paper has derived a Lagrangian FDR for scalars advected by an incompressible fluid. Our relation expresses an exact balance between molecular dissipation of scalar fluctuations and the input of scalar fluctuations from the initial scalar values and internal sources as these are sampled by stochastic Lagrangian trajectories backward in time. We have exploited this relation to give a simple proof (in domains without walls) that spontaneous stochasticity of Lagrangian trajectories is necessary and sufficient for anomalous dissipation of passive scalars, and necessary (but possibly not sufficient) for anomalous dissipation of active scalars.

An important outstanding question is the extent to which the results of this paper can be carried over to provide a Lagrangian picture of anomalous energy dissipation in Navier–Stokes turbulence. (The most direct application of our scalar results to Navier–Stokes might appear to be to analyse the viscous dissipation of enstrophy in freely decaying 2D turbulence, where the vorticity is an active (pseudo)scalar field. Unfortunately, all of our analysis assumes that the initial scalar field is square-integrable or $L^{2}$ , but it has been shown by Eyink (Reference Eyink2001) and Tran & Dritschel (Reference Tran and Dritschel2006) that there can be no anomalous enstrophy dissipation for a freely decaying 2D Navier–Stokes solution with finite initial enstrophy. It may still be the case that there is anomalous enstrophy dissipation for more singular infinite-enstrophy initial data and that this dissipation is associated with spontaneous stochasticity (see the further discussion in Eyink Reference Eyink2001). However, we cannot investigate this delicate issue using the FDR of the present paper.) We briefly comment upon this issue here. The formal extension of our FDR to viscous energy dissipation is straightforward. We can exploit the stochastic Lagrangian representation for the incompressible Navier–Stokes equation

(5.1) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x2202}_{t}\boldsymbol{u}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}=-\unicode[STIX]{x1D735}p+\unicode[STIX]{x1D708}\unicode[STIX]{x0394}\boldsymbol{u}, & \displaystyle\end{eqnarray}$$

(5.2) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

recently elaborated by Constantin & Iyer (Reference Constantin and Iyer2008, Reference Constantin and Iyer2011), which is valid both for flows in domains without boundaries and for wall-bounded flows. Their results can be most simply derived using a backward stochastic particle flow $\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x})$ and a corresponding ‘momentum’ $\tilde{\unicode[STIX]{x1D745}}_{t,s}(\boldsymbol{x})\equiv \boldsymbol{u}(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)$ , which together satisfy the backward Itô equations

(5.3) $$\begin{eqnarray}\displaystyle & \hat{\text{d}}\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x})=\tilde{\unicode[STIX]{x1D745}}_{t,s}(\boldsymbol{x})\,\text{d}s+\sqrt{2\unicode[STIX]{x1D708}}\,\hat{\text{d}}\widetilde{\boldsymbol{W}}_{s}, & \displaystyle\end{eqnarray}$$

(5.4) $$\begin{eqnarray}\displaystyle & \hat{\text{d}}\tilde{\unicode[STIX]{x1D745}}_{t,s}(\boldsymbol{x})=-\unicode[STIX]{x1D735}p(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\text{d}s+\sqrt{2\unicode[STIX]{x1D708}}\,\hat{\text{d}}\widetilde{\boldsymbol{W}}_{s}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s). & \displaystyle\end{eqnarray}$$

These are a stochastic generalization of Hamilton’s particle equations, making contact with traditional methods of Hamiltonian fluid mechanics (Salmon Reference Salmon1988). See the more detailed discussion of Eyink (Reference Eyink2010) and Rezakhanlou (Reference Rezakhanlou2014). By integrating the second of these Hamilton equations from 0 to $t$ and taking expectations over the Brownian motion, one readily obtains

(5.5) $$\begin{eqnarray}\boldsymbol{u}(\boldsymbol{x},t)=\mathbb{E}\left[\boldsymbol{u}_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))-\int _{0}^{t}\unicode[STIX]{x1D735}p(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\text{d}s\right],\end{eqnarray}$$

using the fact that the stochastic integral $\sqrt{2\unicode[STIX]{x1D708}}\int _{0}^{t}\hat{\text{d}}\widetilde{\boldsymbol{W}}_{s}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}^{\unicode[STIX]{x1D708}}(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)$ is a backward martingale and so vanishes under expectation. The formula (5.5) was previously derived by Albeverio & Belopolskaya (Reference Albeverio and Belopolskaya2010). Moreover, by exploiting the same Itô-isometry argument as applied earlier for scalars, one can derive

(5.6) $$\begin{eqnarray}\unicode[STIX]{x1D708}\int _{0}^{t}\,\text{d}s\langle |\unicode[STIX]{x1D735}\boldsymbol{u}(s)|^{2}\rangle _{\unicode[STIX]{x1D6FA}}=\frac{1}{2}\left\langle \text{Var}\left[\boldsymbol{u}_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0})-\int _{0}^{t}\unicode[STIX]{x1D735}p(\tilde{\unicode[STIX]{x1D743}}_{t,s},s)\,\text{d}s\right]\right\rangle _{\unicode[STIX]{x1D6FA}}.\end{eqnarray}$$

This can be considered as an ‘FDR’ for viscous energy dissipation in a Navier–Stokes solution.

Unfortunately, this relation does not appear to be particularly useful for analysing the high-Reynolds-number (or inviscid) limit. It has a mixed Eulerian–Lagrangian character, since it involves both the particle trajectories $\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x})$ and the Eulerian pressure-gradient field $\unicode[STIX]{x1D735}p(\boldsymbol{x},t)$ . The latter field is furthermore a dissipation-range object, which grows increasingly singular as $\unicode[STIX]{x1D708}\rightarrow 0$ . For example, using the classical K41 scaling estimates (Obukhov Reference Obukhov1949; Yaglom Reference Yaglom1949; Batchelor Reference Batchelor1951), one expects a root-mean-squared value of the pressure gradient $(\unicode[STIX]{x1D735}p)_{rms}\sim (\unicode[STIX]{x1D700}^{3}/\unicode[STIX]{x1D708})^{1/4}$ , and intermittency effects will make this field even more singular. Mathematically speaking, the pressure gradient cannot be expected to exist as an ordinary function in the limit $\unicode[STIX]{x1D708}\rightarrow 0$ but only as a distribution. Because of these facts, we cannot derive from (5.6) any relation between anomalous energy dissipation and spontaneous stochasticity for Navier–Stokes turbulence. In particular, even if there were anomalous energy dissipation, the limiting stochastic particle trajectories might become deterministic as $\unicode[STIX]{x1D708}\rightarrow 0$ . In that case, the variance on the right-hand side of (5.6) could remain non-vanishing, because the smaller fluctuations due to vanishing stochasticity could be compensated by the diverging magnitude of the pressure gradient.

More fundamentally, we believe that (5.6) misses essential physics. It should be noted that this relation holds for freely decaying Navier–Stokes turbulence both in 2D and in 3D, but in the former case there is certainly no anomalous energy dissipation. Furthermore, in forced steady-state 2D turbulence, there is evidence in the inverse energy cascade range for Richardson dispersion and Lagrangian spontaneous stochasticity (Boffetta & Sokolov Reference Boffetta and Sokolov2002; Faber & Vassilicos Reference Faber and Vassilicos2009), but this is associated not with small-scale energy dissipation by viscosity but instead with large-scale energy dissipation by Ekman-type damping. A possibly important clue is provided by the fact that Richardson dispersion is faster backward in time for 3D forward energy cascade Sawford et al. (Reference Sawford, Yeung and Borgas2005), Berg et al. (Reference Berg, Lüthi, Mann and Ott2006), Eyink (Reference Eyink2011), but faster forward in time for 2D inverse energy cascade (Faber & Vassilicos Reference Faber and Vassilicos2009). By a comparison of these observations for 2D and 3D Navier–Stokes turbulence and by means of exact results for Burgers turbulence, Eyink & Drivas (Reference Eyink and Drivas2015b ) have argued that anomalous energy dissipation for Navier–Stokes turbulence should be related not simply to the presence of spontaneous stochasticity but instead to time asymmetry of the stochastic Lagrangian trajectories. This is reminiscent of so-called ‘fluctuation theorems’ in non-equilibrium statistical mechanics, which imply exponential asymmetry in the probability of entropy production with positive and negative signs. See Gawȩdzki (Reference Gawȩdzki2013) and Schuster et al. (Reference Schuster, Klages, Just and Jarzynski2013) for recent reviews. These results are deeply related to traditional fluctuation–dissipation theorems in statistical physics, but we have been unable to discover any connection with our Lagrangian FDR. More recently, a time asymmetry has been established in the very-short-time dispersion of nearby Lagrangian trajectories by Falkovich & Frishman (Reference Falkovich and Frishman2013) and Jucha et al. (Reference Jucha, Xu, Pumir and Bodenschatz2014). However, these results hold only for times of order ${\sim}(r_{0}^{2}/\unicode[STIX]{x1D700})^{1/3}$ and therefore cannot explain the long-time Richardson behaviour or the observed time asymmetry therein.

The most important implication of the present work is the additional support provided to the concept of Lagrangian spontaneous stochasticity. Exploiting our Lagrangian FDR, we have shown that any empirical evidence for anomalous scalar dissipation, either for passive or for active scalars, and away from walls, must be taken as evidence also for spontaneous stochasticity. There are profound implications of this phenomenon for many Lagrangian aspects of turbulent flows. For example, Constantin & Iyer (Reference Constantin and Iyer2008) have shown that the classical Kelvin–Helmholtz theorems for vorticity dynamics in smooth solutions of the incompressible Euler equations generalize within their stochastic framework to solutions of the incompressible Navier–Stokes equation with a positive viscosity. In fact, similarly to the case of the advected scalars discussed in the present work, Constantin & Iyer (Reference Constantin and Iyer2008) proved that circulations around stochastically advected loops are martingales backward in time for the Navier–Stokes solution and also proved that this property completely characterizes these solutions. This ‘stochastic Kelvin theorem’ demonstrates again that the stochastic Lagrangian approach is the natural generalization to non-ideal fluids of the Lagrangian methods for ideal fluids. Furthermore, if there is spontaneous stochasticity, then vortex motion must remain stochastic for arbitrarily high Reynolds numbers. Contrary to the traditional arguments of Taylor & Green (Reference Taylor and Green1937), vortex lines in the ideal limit will not be ‘frozen into’ the turbulent fluid flow in the usual sense. Similar results holds also for magnetic-field-line motion in resistive magnetohydrodynamics (Eyink Reference Eyink2009), and spontaneous stochasticity then implies the possibility of fast magnetic reconnection in astrophysical plasmas for arbitrarily small electrical conductivity (Eyink et al. Reference Eyink, Vishniac, Lalescu, Aluie, Kanov, Bürger, Burns, Meneveau and Szalay2013). In Parts II and III, we extend the derivation of our Lagrangian FDR to wall-bounded flows, and derive similar relations between anomalous scalar dissipation and spontaneous stochasticity, as well as new Lagrangian relations for Nusselt–Rayleigh scaling in turbulent convection.