2.1 Pseudo-dissipation as an exponentiated Ornstein–Uhlenbeck process

As a building block of their stochastic model for the velocity gradients, Girimaji & Pope (Reference Girimaji and Pope1990) have considered for pseudo-dissipation $\unicode[STIX]{x1D711}$ the exponential of an Ornstein–Uhlenbeck process. This model is justified by the observations made in early numerical simulations of Yeung & Pope (Reference Yeung and Pope1989) that the shape of the one-point PDF of $\log \unicode[STIX]{x1D711}$ is close to a Gaussian and, furthermore, the autocorrelation of $\log \unicode[STIX]{x1D711}(t)$ follows an exponential decrease over a typical time scale given by the integral time scale $T$ of turbulence (Pope Reference Pope1990). At this point, we may remark that the small-scale quantity $\unicode[STIX]{x1D711}$ is correlated over the large energy containing scale of turbulence, which is characteristic of the absence of scale decoupling of turbulence. This was already recognized in the Eulerian framework where it is observed that dissipation is correlated in space over the large integral length scale $L$ (Gagne & Hopfinger Reference Gagne and Hopfinger1979; Antonia, Phan Thien & Satyaprakash Reference Antonia, Phan Thien and Satyaprakash1981). From the theoretical side, this long-range correlated nature of the small scales is at the heart of the models of intermittency of Kolmogorov (Reference Kolmogorov1962) and Obukhov (Reference Obukhov1962) (see also Novikov Reference Novikov1989, Reference Novikov1990 for discussions in the Lagrangian framework), although, at that stage, nothing is said about how the correlation function decreases over $T$ , and as we will see, multifractal phenomenology requires a logarithmic decrease (see § 2.2) which is not reproduced by the proposition of Pope (Reference Pope1990).

The statistical properties of this model are reviewed in appendix A. Its causal dynamics is given in (A 1) and its unique solution, given an initial condition, is given explicitly in (A 2). Let us call $\hat{a}$ , as in Girimaji & Pope (Reference Girimaji and Pope1990), the free parameter of this model. It can be conveniently related to the variance of the logarithm of pseudo-dissipation as $\mathbb{E}\log ^{2}\unicode[STIX]{x1D711}-\mathbb{E}^{2}\log \unicode[STIX]{x1D711}=2\hat{a}^{2}T$ , or equivalently, to the moments $\mathbb{E}(\unicode[STIX]{x1D711}^{q})=(1/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{2q})\text{e}^{\hat{a}^{2}Tq(q-1)}$ (A 3). Whatever the value of $\hat{a}$ , this model fulfils the basics of Kolmogorov’s phenomenology, namely $\mathbb{E}(\unicode[STIX]{x1D711})=1/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{2}$ (1.7). As is explained and observed in Yeung & Pope (Reference Yeung and Pope1989), $\hat{a}^{2}$ is expected to depend logarithmically on the Reynolds number (see (A 9)) in order to be consistent with both empirical observations and the phenomenology of Kolmogorov (Reference Kolmogorov1962) and Obukhov (Reference Obukhov1962) (from now on referred to as KO62) such as $\hat{a}^{2}\sim \log {\mathcal{R}}_{e}$ when ${\mathcal{R}}_{e}\rightarrow \infty$ .

To build a more precise connection with KO62 phenomenology, and more generally with the multifractal (i.e. intermittent) phenomenology (Borgas Reference Borgas1993; Frisch Reference Frisch1995), let us consider the average value of pseudo-dissipation over a time interval of extension $\unicode[STIX]{x1D70F}$ , i.e.

(2.1) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}}(t)=\frac{1}{\unicode[STIX]{x1D70F}}\int _{s\in [t-\unicode[STIX]{x1D70F},t]}\unicode[STIX]{x1D711}(s)\,\text{d}s.\end{eqnarray}$$

This quantity is the Lagrangian analogue of the (pseudo-)dissipation averaged over a ball of size $\ell$ as considered by Kolmogorov (Reference Kolmogorov1962) and Obukhov (Reference Obukhov1962) in the refined similarity hypothesis relating velocity structure functions and dissipation field (Borgas Reference Borgas1993; Frisch Reference Frisch1995). It is shown in appendix A (see (A 8)) how to compute the high-order moments of $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}}$ , and it is easy to be convinced that if $\hat{a}^{2}T$ is taken proportional to $\log {\mathcal{R}}_{e}$ , as is required by the intermittent phenomenology formerly described, then, for any $\unicode[STIX]{x1D70F}>0$ , we get diverging high-order moments of the coarse-grained pseudo-dissipation, i.e. for $q>1$ , $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{2q}\mathbb{E}[\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}}^{q}]\rightarrow \infty$ when ${\mathcal{R}}_{e}\rightarrow \infty$ , which is not consistent with the refined similarity hypothesis (see the discussion on the Reynolds number dependence of the model in appendix A). Thus, taking as a stochastic model for pseudo-dissipation the exponential of an Ornstein–Uhlenbeck process prevents us from fulfilling the statistical properties required by the KO62 phenomenology. We will see in the following section that, on the other hand, a pseudo-dissipation model given by the multiplicative chaos allows one to obtain (i) moments $\mathbb{E}[\unicode[STIX]{x1D711}^{q}]$ that scale as a power of the Reynolds number, and (ii) bounded moments $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}}$ for any finite $\unicode[STIX]{x1D70F}>0$ when ${\mathcal{R}}_{e}\rightarrow \infty$ .

2.2 Pseudo-dissipation as a causal and stationary multiplicative chaos

Multiplicative chaos was introduced by Mandelbrot (Reference Mandelbrot, Rosenblatt and Van Atta1972) in the context of turbulence, following the work of Yaglom (Reference Yaglom1966), and formalized by Kahane (Reference Kahane1985) in order to give a stochastic meaning to the dissipation field as depicted in the KO62 phenomenology (see recent mathematical developments in Rhodes & Vargas (Reference Rhodes and Vargas2014)). In the Eulerian framework, for which we need to define spatial fields, dissipation is taken as the exponential of a isotropic Gaussian field logarithmically correlated over the integral length scale $L$ . As is briefly reviewed in Pereira, Garban & Chevillard (Reference Pereira, Garban and Chevillard2016), we can give a clear meaning to the dissipation field as depicted in KO62 while proposing a stochastic representation of it. At a given finite Reynolds number, implying a finite Kolmogorov length scale $\unicode[STIX]{x1D702}$ , this representation reads

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D711}(\boldsymbol{x})=\frac{1}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{2}}\text{e}^{\sqrt{\unicode[STIX]{x1D707}^{e}}X_{\unicode[STIX]{x1D702}}(\boldsymbol{x})-(\unicode[STIX]{x1D707}^{e}/2)\mathbb{E}[X_{\unicode[STIX]{x1D702}}^{2}]},\end{eqnarray}$$

where $X_{\unicode[STIX]{x1D702}}$ is a homogeneous and isotropic Gaussian field whose variance diverges logarithmically with $\unicode[STIX]{x1D702}$ and is logarithmically correlated over the integral length scale $L$ . Being Gaussian, the field $X_{\unicode[STIX]{x1D702}}$ can be conveniently expressed as a linear operation (i.e. a convolution) on a white measure as

(2.3) $$\begin{eqnarray}X_{\unicode[STIX]{x1D702}}(\boldsymbol{x})=\frac{1}{4\unicode[STIX]{x03C0}}\int _{|\boldsymbol{x}-\boldsymbol{y}|\leqslant L}\frac{1}{|\boldsymbol{x}-\boldsymbol{y}|_{\unicode[STIX]{x1D702}}^{3/2}}W(\text{d}y),\end{eqnarray}$$

where a regularized norm $|\boldsymbol{x}|_{\unicode[STIX]{x1D702}}$ is introduced, and whose exact expression has little importance at this stage, since the statistical properties of $X_{\unicode[STIX]{x1D702}}$ become independent of its precise form in the limit of infinite Reynolds number (i.e.  $\unicode[STIX]{x1D702}\rightarrow 0$ ). We will take for instance, to illustrate our purpose, the homogeneous and isotropic small-scale cutoff $|\boldsymbol{x}|^{2}=|\boldsymbol{x}|^{2}+\unicode[STIX]{x1D702}^{2}$ for any $\boldsymbol{x}\in \mathbb{R}^{3}$ . See Rhodes & Vargas (Reference Rhodes and Vargas2014) and Pereira et al. (Reference Pereira, Garban and Chevillard2016) for a discussion on this subject.

It is then possible to show (see for example Pereira et al. (Reference Pereira, Garban and Chevillard2016) for detailed computations and Rhodes & Vargas (Reference Rhodes and Vargas2014) for a general approach) that

(2.4) $$\begin{eqnarray}\mathbb{E}[X_{\unicode[STIX]{x1D702}}^{2}]\underset{\unicode[STIX]{x1D702}\rightarrow 0}{{\sim}}\log \left(\frac{L}{\unicode[STIX]{x1D702}}\right).\end{eqnarray}$$

Furthermore, locally, we have the convergence, for $|\boldsymbol{h}|>0$ ,

(2.5) $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D702}\rightarrow 0}\mathbb{E}[X_{\unicode[STIX]{x1D702}}(\boldsymbol{x})X_{\unicode[STIX]{x1D702}}(\boldsymbol{x}+\boldsymbol{h})]=\log ^{+}\left(\frac{L}{|\boldsymbol{h}|}\right)+g(|\boldsymbol{h}|),\end{eqnarray}$$

where $g$ is a continuous and bounded function of its argument, and $\log ^{+}(|\boldsymbol{x}|)=\log (|\boldsymbol{x}|)$ for $|\boldsymbol{x}|\geqslant 1$ and vanishes elsewhere. Doing so, we can derive the statistical properties of the modelled (Eulerian) pseudo-dissipation field (2.2) that fulfils the axiomatics of KO62, namely

(2.6) $$\begin{eqnarray}\mathbb{E}[\unicode[STIX]{x1D711}^{q}]\underset{\unicode[STIX]{x1D702}\rightarrow 0}{{\sim}}\frac{\text{e}^{(\unicode[STIX]{x1D707}^{e}/2)q(q-1)g(0)}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{2q}}\left(\frac{L}{\unicode[STIX]{x1D702}}\right)^{(\unicode[STIX]{x1D707}^{e}/2)q(q-1)}.\end{eqnarray}$$

Recalling that the Kolmogorov scale $\unicode[STIX]{x1D702}$ behaves with the Reynolds number ${\mathcal{R}}_{e}$ as $\unicode[STIX]{x1D702}\propto L{\mathcal{R}}_{e}^{-3/4}$ , we see that modelling the pseudo-dissipation as a multiplicative chaos (2.2) gives moments proportional to a power of the Reynolds number (2.6) as required by KO62. Furthermore, the locally averaged pseudo-dissipation

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{\ell }(\boldsymbol{x})=\frac{1}{\frac{4}{3}\unicode[STIX]{x03C0}\ell ^{3}}\int _{|\boldsymbol{x}-\boldsymbol{y}|\leqslant \ell }\unicode[STIX]{x1D711}(\boldsymbol{y})\,\text{d}y,\end{eqnarray}$$

is such that its moments remain bounded when $\unicode[STIX]{x1D702}\rightarrow 0$ and we have the following behaviour at small scales:

(2.8) $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D702}\rightarrow 0}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{2q}\mathbb{E}[\unicode[STIX]{x1D711}_{\ell }^{q}]\underset{\ell \rightarrow 0}{{\sim}}C_{q}\left(\frac{\ell }{L}\right)^{(\unicode[STIX]{x1D707}^{e}/2)q(1-q)},\end{eqnarray}$$

where $C_{q}$ is a scale and Reynolds number independent constant, that can be eventually calculated.

We can see from the proposition of Mandelbrot (Reference Mandelbrot, Rosenblatt and Van Atta1972) that taking the pseudo-dissipation (2.2) as the exponential of a log-correlated field (2.3) ensures that its moments behave as power laws of the Reynolds number (2.6), in particular its average is consistent with Kolmogorov’s prediction (1.7). Moreover, moments of its locally averaged version over a ball of radius $\ell$ (2.7) become independent of the Reynolds number and behave as power laws at small scales (2.8).

In the Lagrangian context of velocity gradient dynamics we are interested in (1.2), we would like to build up a one-dimensional version of the multiplicative chaos used in the Eulerian framework (2.2) to model pseudo-dissipation as seen by a fluid particle along its trajectory. It has to be stationary, its variance particularly should be time independent, and, furthermore, it has to be causal. Causality is crucial here, not only as a physical requirement but, technically speaking, because the so-obtained multiplicative chaos has to be defined as a solution of a stochastic differential equation that will be coupled to the dynamics of $\unicode[STIX]{x1D63C}$ (1.3), as done by Girimaji & Pope (Reference Girimaji and Pope1990), who imposed that $\text{tr}(\unicode[STIX]{x1D63C}\unicode[STIX]{x1D63C}^{\top })$ follows the statistics of an exponentiated Ornstein–Uhlenbeck process, as already depicted in § 2.1. Thus, instead of the three-dimensional spatial field considered in (2.2), we will consider the following one-dimensional temporal stochastic field,

(2.9) $$\begin{eqnarray}\unicode[STIX]{x1D711}(t)=\frac{1}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{2}}\text{e}^{\sqrt{\unicode[STIX]{x1D707}^{l}}X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)-(\unicode[STIX]{x1D707}^{l}/2)\mathbb{E}[X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}^{2}]},\end{eqnarray}$$

where now $X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)$ is a zero-average causal and stationary Gaussian field whose variance $\mathbb{E}[X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}^{2}]$ blows up logarithmically with the Kolmogorov time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ (instead of (2.4)), and whose covariance is independent of the Reynolds number and decreases logarithmically over the integral time scale $T$ of turbulence (in contrast to (2.5)). As proved in appendix B, the unique solution $X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)$ of the following stochastic dynamics fulfils these two requirements,

(2.10) $$\begin{eqnarray}\text{d}X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)=\left[-\frac{1}{T}X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)+\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)\right]\,\text{d}t+\frac{1}{\sqrt{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}}W(\text{d}t),\end{eqnarray}$$

$W$ being a Gaussian white noise and $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(s)$ a random function known in the literature as a ( $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ -regularized) fractional Gaussian noise with Hurst exponent $H=0$ (Mandelbrot & Van Ness Reference Mandelbrot and Van Ness1968):

(2.11) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)=-\frac{1}{2}\int _{s=-\infty }^{t}\frac{1}{(t-s+\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}})^{3/2}}W(\text{d}s).\end{eqnarray}$$

We note that (2.10) is linear and so can be integrated in a simple fashion such that the solution $X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)$ can be equivalently defined as

(2.12) $$\begin{eqnarray}X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)=\int _{s=-\infty }^{t}\text{e}^{-(t-s)/T}\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(s)\,\text{d}s+\frac{1}{\sqrt{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}}\int _{s=-\infty }^{t}\text{e}^{-(t-s)/T}W(\text{d}s),\end{eqnarray}$$

where we have chosen formally the initial condition $X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(-\infty )=0$ . As we can see, the process $X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ is the sum of two Gaussian causal fields. The first process on the right-hand side of (2.12) is similar, in a certain sense, to the one in the formulation of the Eulerian modelling of pseudo-dissipation (2.3). It can be seen as an exponentially smoothed version of a fractional Brownian noise of vanishing Hurst exponent, and is known in the literature as a fractional Ornstein–Uhlenbeck process (Cheridito, Kawaguchi & Maejima Reference Cheridito, Kawaguchi and Maejima2003). The second process on the right-hand side of (2.12) is a usual Ornstein–Uhlenbeck process whose variance blows up as $1/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ . Notice that the same white measure $W$ appears in both these Gaussian fields, making them correlated. This correlation will be responsible for a strong cancellation taking place between these two processes. Actually, this cancellation, formalized in appendix B, will make the variance of $X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}\sim \log (T/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}})$ way smaller than that of each of these formerly described Gaussian processes. The idea of adding a correlated process to a fractional Gaussian noise can be understood in practice with the heuristics developed in Schmitt (Reference Schmitt2003). Our contribution to this matter is to properly regularize the dynamics proposed by Schmitt (Reference Schmitt2003), injecting the large-scale $T$ dependence not in the fractional Brownian noise, but through a linear damping, allowing us to rigorously define $X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ (appendix B).

It is then possible to show that the pseudo-dissipation defined as the exponential of $X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ fulfils the KO62 requirements, namely, high-order moments behave as power laws of the Reynolds number

(2.13) $$\begin{eqnarray}\mathbb{E}[\unicode[STIX]{x1D711}^{q}]\underset{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}\rightarrow 0}{\propto }\frac{1}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{2q}}\left(\frac{T}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}\right)^{(\unicode[STIX]{x1D707}^{l}/2)q(q-1)},\end{eqnarray}$$

where the multiplicative constant (given explicitly in (B 54)) is independent of the Reynolds number, and moments of its coarse-grained version along the trajectory (2.1) behaving as power laws of the time scale $\unicode[STIX]{x1D70F}$ in the asymptotic limit of infinite Reynolds number, according to

(2.14) $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}\rightarrow 0}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{2q}\mathbb{E}[\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}}^{q}]\underset{\unicode[STIX]{x1D70F}\rightarrow 0}{\propto }\left(\frac{T}{\unicode[STIX]{x1D70F}}\right)^{(\unicode[STIX]{x1D707}^{l}/2)q(q-1)}\text{e}^{(\unicode[STIX]{x1D707}^{l}/2)q(q-1)\tilde{g}(0)}\int _{[0,1]^{q}}\mathop{\prod }_{i<j}^{q}\frac{1}{|s_{i}-s_{j}|^{\unicode[STIX]{x1D707}^{l}}}\mathop{\prod }_{i=1}^{q}\,\text{d}s_{i},\end{eqnarray}$$

where again the exact expression of the multiplicative constant, which is both scale and Reynolds number independent, is given in (B 57). The constant $\tilde{g}(0)$ has the same status as $g(0)$ in (2.6) and is precisely defined alongside detailed computations in § B.4.

2.3 Differences between stochastic models of pseudo-dissipation and comparison with numerical flows

As we have seen, modelling pseudo-dissipation as an exponentiated Ornstein–Uhlenbeck process, as done by Girimaji & Pope (Reference Girimaji and Pope1990) and discussed in § 2.1, or as a multifractal measure, as discussed in Borgas (Reference Borgas1993) and considered by Schmitt (Reference Schmitt2003), who proposed in the context of multiplicative chaos a causal dynamics based on heuristics arguments that we defined rigorously in § 2.2, are very different matters. First of all, the first model is not consistent with the multifractal picture, which asks for (i) moments of $\unicode[STIX]{x1D711}$ to behave as power laws of the Reynolds number, and (ii) moments of $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}}$ to behave, in the asymptotic limit of infinite Reynolds number, as a power laws of the coarse-graining scale $\unicode[STIX]{x1D70F}$ . We have seen that indeed the second model, based on the multiplicative chaos, results in such statistical behaviour.

One may wonder whether these two different statistical behaviours could be tested against experiments and/or numerical simulations. Testing the behaviour of high-order moments can be difficult since the different scalings are rather similar, and moreover, statistical convergence may be an issue. Instead, we could check the behaviour of the correlation function of the logarithm of pseudo-dissipation. Indeed, these two stochastic models have different predictions for the correlation of $\log \unicode[STIX]{x1D711}$ along trajectories. On the one hand, the model of Girimaji & Pope (Reference Girimaji and Pope1990) predicts that $\log \unicode[STIX]{x1D711}$ is an Ornstein–Uhlenbeck process and so $\mathbb{E}[\log \unicode[STIX]{x1D711}(t)\log \unicode[STIX]{x1D711}(t+\unicode[STIX]{x1D70F})]$ should decrease exponentially in time (over the integral time scale $T$ ). On the other hand, multifractal modelling asks for a logarithmic decrease over the time scale $T$ . The analysis of Huang & Schmitt (Reference Huang and Schmitt2014), based on Lagrangian trajectories extracted from a numerical simulation among the highest Reynolds numbers available from modern computers (with Taylor based Reynolds number of the order of 400) shows that data, and more precisely the correlation function of $\log \unicode[STIX]{x1D711}$ , are consistent both with exponential and logarithmic decreases. In other words, even at the highest accessible Reynolds numbers it is very difficult to make a distinction between the two models, both being able to reproduce in a quantitative way numerical data. In the following, we will prefer to work with the multifractal model, because (i) it has a proper behaviour in the asymptotic limit of infinite Reynolds numbers and (ii) the free parameter of the models, $\unicode[STIX]{x1D707}^{l}$ , which appears in both models (see (A 9) for the first model, and directly in (2.9) for the second model) has a clear meaning only in a multifractal framework, where it is denominated the intermittency coefficient.

3.1 General formulation and isotropic conditions

We consider (3.1) as a modelled stochastic evolution of the velocity gradient tensor $\unicode[STIX]{x1D63C}$ . We are also interested in the respective dynamics of the pseudo-dissipation $\unicode[STIX]{x1D711}=\text{tr}(\unicode[STIX]{x1D63C}\unicode[STIX]{x1D63C}^{\top })$ which is obtained from (3.1) after applying the chain rule (Ito’s lemma). Doing so, we obtain

(3.2) $$\begin{eqnarray}\text{d}\unicode[STIX]{x1D711}=\text{d}\text{tr}(\unicode[STIX]{x1D63C}\unicode[STIX]{x1D63C}^{\top })=[2\unicode[STIX]{x1D608}_{ij}\unicode[STIX]{x1D61D}_{ij}+\unicode[STIX]{x1D60B}_{ijkl}\unicode[STIX]{x1D60B}_{ijkl}]\,\text{d}t+2\unicode[STIX]{x1D608}_{ij}\unicode[STIX]{x1D60B}_{ijkl}\unicode[STIX]{x1D61E}_{kl}(\text{d}t).\end{eqnarray}$$

Assume now that the diffusion term $\unicode[STIX]{x1D60B}_{ijkl}$ has a isotropic form, namely

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D60B}_{ijkl}=a\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}_{kl}+b\unicode[STIX]{x1D6FF}_{ik}\unicode[STIX]{x1D6FF}_{jl}+c\unicode[STIX]{x1D6FF}_{il}\unicode[STIX]{x1D6FF}_{kj},\end{eqnarray}$$

where $a$ , $b$ , $c$ are unknown and, in general, scalar functionals of $\unicode[STIX]{x1D63C}$ and time. The incompressibility condition imposes

(3.4) $$\begin{eqnarray}3a+b+c=0.\end{eqnarray}$$

Computing the spurious drift term $\unicode[STIX]{x1D60B}_{ijkl}\unicode[STIX]{x1D60B}_{ijkl}$ entering the dynamics of $\unicode[STIX]{x1D711}$ (3.2) and using the incompressibility condition $\text{tr}(\unicode[STIX]{x1D63C})=A_{ii}=0$ results in the following dynamics for $\unicode[STIX]{x1D711}$ :

(3.5) $$\begin{eqnarray}\text{d}\unicode[STIX]{x1D711}=\text{d}\text{tr}(\unicode[STIX]{x1D63C}\unicode[STIX]{x1D63C}^{\top })=[2\unicode[STIX]{x1D608}_{ij}\unicode[STIX]{x1D61D}_{ij}+9(a^{2}+b^{2}+c^{2})+6(ab+ac+bc)]\,\text{d}t+2(b\unicode[STIX]{x1D608}_{kl}+c\unicode[STIX]{x1D608}_{lk})\unicode[STIX]{x1D61E}_{kl}(\text{d}t).\end{eqnarray}$$

3.2 Mean-square estimation of the diffusion term

We want to impose a dynamics for $\unicode[STIX]{x1D711}$ (3.5) as close as possible to that of a multiplicative chaos, derived in § B.4 and expressed in (B 51), which we recall for convenience:

(3.6) $$\begin{eqnarray}\displaystyle \text{d}\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}} & = & \displaystyle \unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)\left[-\frac{1}{T}\left(\log [\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{2}\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)]+\frac{\unicode[STIX]{x1D707}^{l}}{2}\mathbb{E}[(X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}})^{2}]\right)+\sqrt{\unicode[STIX]{x1D707}^{l}}\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)+\frac{\unicode[STIX]{x1D707}^{l}}{2\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}\right]\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle +\,\sqrt{\frac{\unicode[STIX]{x1D707}^{l}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}}\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)W(\text{d}t).\end{eqnarray}$$

This can be done at the cost of an approximation since, as explained, the true dynamics of $\unicode[STIX]{x1D711}$ (3.5), comprising a tensorial white noise $\unicode[STIX]{x1D652}$ , involves more degrees of freedom than the multiplicative chaos one (3.6), where only a scalar white noise $W$ appears.

We follow the approximation of Girimaji & Pope (Reference Girimaji and Pope1990) that consists in adopting a mean-square approximation of the diffusion term. We thus take the diffusion term from (3.2) such that its variance conditioned to $\unicode[STIX]{x1D63C}$ equals the variance of the diffusion term from (3.6) conditioned to $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}=\unicode[STIX]{x1D711}=\unicode[STIX]{x1D608}_{ij}\unicode[STIX]{x1D608}_{ij}$ . Since we are dealing with random distributions $\unicode[STIX]{x1D652}$ and $W$ , the meaning of their square is a priori not clear. Nonetheless, this can be clarified while considering the square of their integral over a finite range and then taking a limit as depicted in Girimaji & Pope (Reference Girimaji and Pope1990). Doing so, we obtain, for any time $t$ ,

(3.7) $$\begin{eqnarray}4[(b^{2}+c^{2})\unicode[STIX]{x1D711}+2bc\,\text{tr}(\unicode[STIX]{x1D63C}^{2})]=\frac{\unicode[STIX]{x1D707}^{l}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}\unicode[STIX]{x1D711}^{2}.\end{eqnarray}$$

As in Girimaji & Pope (Reference Girimaji and Pope1990), we remark that the equality in (3.7) provides an underdetermined problem in relation to the free parameters $b$ and $c$ (one equation for two unknowns). In this regard, we assume that the dynamics of $\unicode[STIX]{x1D711}=\text{tr}(\unicode[STIX]{x1D63C}\unicode[STIX]{x1D63C}^{\top })$ only depends on itself, and not on the contraction $\text{tr}(\unicode[STIX]{x1D63C}^{2})$ . Using furthermore the incompressibility condition (3.4) we arrive at

(3.8a-c ) $$\begin{eqnarray}a=-\frac{1}{6}\sqrt{\frac{\unicode[STIX]{x1D707}^{l}\unicode[STIX]{x1D711}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}},\quad b=\frac{1}{2}\sqrt{\frac{\unicode[STIX]{x1D707}^{l}\unicode[STIX]{x1D711}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}}\quad \text{and}\quad c=0.\end{eqnarray}$$

It would be interesting to study the dependence of the model on the choice $c\neq 0$ , particularly if we want to build up separately dynamics for dissipation $\text{tr}(\unicode[STIX]{x1D64E}^{2})$ and enstrophy $-\text{tr}(\unicode[STIX]{x1D6FA}^{2})$ . We keep these developments for further investigations. The dynamics of $\unicode[STIX]{x1D711}$ (3.5) becomes

(3.9) $$\begin{eqnarray}\text{d}\unicode[STIX]{x1D711}=\text{d}\text{tr}(\unicode[STIX]{x1D63C}\unicode[STIX]{x1D63C}^{\top })=\left[2\unicode[STIX]{x1D608}_{ij}\unicode[STIX]{x1D61D}_{ij}+2\frac{\unicode[STIX]{x1D707}^{l}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}\unicode[STIX]{x1D711}\right]\,\text{d}t+\sqrt{\frac{\unicode[STIX]{x1D707}^{l}\unicode[STIX]{x1D711}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}}\unicode[STIX]{x1D608}_{kl}\unicode[STIX]{x1D61E}_{kl}(\text{d}t).\end{eqnarray}$$

We can see that this mean-square procedure suggests the replacement of the scalar white noise $W$ in the dynamics of $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ (3.6) by the scalar noise $\unicode[STIX]{x1D608}_{kl}\unicode[STIX]{x1D61E}_{kl}/\sqrt{\unicode[STIX]{x1D711}}$ that emerges from the $\unicode[STIX]{x1D711}$ dynamics (3.9).

3.3 Determination of the drift term and introduction of the recent fluid deformation closure

Analogously to Girimaji & Pope (Reference Girimaji and Pope1990), we would like to impose on $\unicode[STIX]{x1D711}$ (3.9) a dynamics as close as possible to that fulfilled by the multiplicative chaos (3.6). To do so, we need to provide an expression for the tensor $\unicode[STIX]{x1D651}$ of the drift part of (3.9). This tensor is the only one able to give a flavour of the Navier–Stokes equation (1.2), and should be closed in terms of $\unicode[STIX]{x1D63C}$ . The very first idea would be to take into account only the self-stretching term $\unicode[STIX]{x1D651}=\unicode[STIX]{x1D651}^{RE}$ (1.4). This would be done disregarding all the interesting physics of the joint action of the pressure Hessian and viscous term which compose the true dynamics of $\unicode[STIX]{x1D63C}$ (1.2). So instead, we will adopt the more sophisticated closure $\unicode[STIX]{x1D651}=\unicode[STIX]{x1D651}^{RFD}$ (1.5) given by the recent fluid deformation (RFD) approximation and discussed in Chevillard & Meneveau (Reference Chevillard and Meneveau2006), Chevillard et al. (Reference Chevillard, Meneveau, Biferale and Toschi2008) and Meneveau (Reference Meneveau2011). Although it reproduces some of the physics governed by the pressure and viscosity terms, this closure is not consistent with the constraint we want to impose for the pseudo-dissipation (3.6) when inserted in (3.9). To make it so, we will furthermore consider an additional damping term proportional to $\unicode[STIX]{x1D63C}$ and show that the associated multiplicative factor can be chosen such that the resulting dynamics for $\unicode[STIX]{x1D711}$ is similar, in a certain sense, to the one fulfilled by the multiplicative chaos (3.6). This is similar to the procedure made by Girimaji & Pope (Reference Girimaji and Pope1990). Accordingly, we choose for the drift tensorial term $\unicode[STIX]{x1D651}$ (3.1) the following model,

(3.10) $$\begin{eqnarray}\unicode[STIX]{x1D651}(t)=\unicode[STIX]{x1D651}^{RFD}(t)+f(t)\unicode[STIX]{x1D63C}(t),\end{eqnarray}$$

where the multiplicative factor $f(t)$ is chosen so that the respective drift term of the dynamics of $\unicode[STIX]{x1D711}$ is as close as possible to the one of the multiplicative chaos (3.6). With this aim, and to find an expression for $f(t)$ , we must identify the drift parts of the dynamics of $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ (3.6) with the those of the $\unicode[STIX]{x1D711}$ dynamics (3.9).

As we have seen in § 3.2, the application of a mean-square procedure to the diffusion term entering in the dynamics of $\unicode[STIX]{x1D711}$ (3.9) suggests replacing the scalar noise $W$ with $\unicode[STIX]{x1D608}_{kl}\unicode[STIX]{x1D61E}_{kl}/\sqrt{\unicode[STIX]{x1D711}}$ . We will follow this suggestion and replace accordingly the fractional noise $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ entering (3.6) and defined in (2.11), built from the scalar noise $W$ , by its respective version $\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ built on the noise $\unicode[STIX]{x1D608}_{kl}\unicode[STIX]{x1D61E}_{kl}\sqrt{\unicode[STIX]{x1D711}}$ . This new noise reads

(3.11) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)=-\frac{1}{2}\int _{s=-\infty }^{t}\frac{1}{(t-s+\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}})^{3/2}}\frac{\unicode[STIX]{x1D608}_{ij}(s)}{\sqrt{\unicode[STIX]{x1D711}(s)}}\unicode[STIX]{x1D61E}_{ij}(\text{d}s).\end{eqnarray}$$

Several reasons pushed us to consider this noise $\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ . First of all, as mentioned, the mean-square procedure used in § 3.2 to determine the diffusion term $\unicode[STIX]{x1D60B}_{ijkl}$ suggests the replacement of $W$ by $\unicode[STIX]{x1D608}_{kl}\unicode[STIX]{x1D61E}_{kl}/\sqrt{\unicode[STIX]{x1D711}}$ . One could try to replace the scalar noise $W$ by a another scalar such as $\text{tr}(\unicode[STIX]{x1D652})$ , but in doing so one loses the correlation between $\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ and the random term $\unicode[STIX]{x1D608}_{kl}\unicode[STIX]{x1D61E}_{kl}$ from the dynamics of $\unicode[STIX]{x1D711}$ (3.9). This correlation, as shown in appendix B, is crucial in order to achieve multifractal statistics. One could then wonder that the statistics of $\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ might be completely different from the initial Gaussian noise $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ that forms the multiplicative chaos dynamics (3.6) and leads to exact multifractal properties of the positive field $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ . It is to be checked numerically, a posteriori, that their imprints on the final dynamics have similar consequences. And indeed, as we verify in § 5, dedicated to numerical results, the model we are constructing with the noise $\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ culminate in a pseudo-dissipation whose statistics is remarkably similar to the exact multifractal model $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ from (3.6). This being said, it is possible to verify numerically that the variances of $\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ and $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ are actually the same, i.e.

(3.12) $$\begin{eqnarray}\mathbb{E}[\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}^{2}]\approx \mathbb{E}[\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}^{2}].\end{eqnarray}$$

Thus, as far as the variance is concerned, the noise components $\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ (3.11) and $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ (2.11) behave similarly. Higher-order moments will be different, however, since $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ is a Gaussian process and $\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ is not.

Again analogously to Girimaji & Pope (Reference Girimaji and Pope1990), we equate the drift term of the multifractal process (3.6) with the drift term of pseudo-dissipation (3.9) using as a model for $\unicode[STIX]{x1D651}$ the sum of the RFD closure’s drift term and a (nonlinear) damping term (3.10), while replacing the noise $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ by the new noise $\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ , to deduce an expression for the functional $f(t)$ :

(3.13) $$\begin{eqnarray}f(t)=-\frac{1}{2T}\left(\log [\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{2}\unicode[STIX]{x1D711}(t)]+\frac{\unicode[STIX]{x1D707}^{l}}{2}\mathbb{E}[(X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}})^{2}]\right)+\frac{\sqrt{\unicode[STIX]{x1D707}^{l}}}{2}\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)-\frac{3\unicode[STIX]{x1D707}^{l}}{4\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}-\frac{\unicode[STIX]{x1D608}_{ij}(t)\unicode[STIX]{x1D61D}_{ij}^{RFD}(t)}{\unicode[STIX]{x1D711}(t)}.\end{eqnarray}$$

Let us note that the quantity $\mathbb{E}[(X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}})^{2}]$ , a function of $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ only (see (B 6)), already present in the dynamics of $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ (3.6), also appears in (3.13).

3.4 Assembling the model

By inserting the various terms discussed above in the dynamics of $\unicode[STIX]{x1D63C}$ (3.1), we forge a closed dynamics along a trajectory. For clarity and convenience, we recall here all the pieces gathered to build the model. For the drift term (3.10), $\unicode[STIX]{x1D651}=\unicode[STIX]{x1D651}^{RFD}+f(t)\unicode[STIX]{x1D63C}$ , there are two contributions. The RFD part is given by (1.5), that is,

(3.14) $$\begin{eqnarray}\unicode[STIX]{x1D651}^{RFD}=-\unicode[STIX]{x1D63C}^{2}+\frac{\text{tr}(\unicode[STIX]{x1D63C}^{2})}{\text{tr}(\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}^{-1})}\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}^{-1}-\frac{\text{tr}(\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}^{-1})}{3T}\unicode[STIX]{x1D63C},\end{eqnarray}$$

$\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ being the short-time Cauchy–Green tensor (1.6) and $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ and $T$ the Kolmogorov and integral time scales respectively. The damping part is formed with the scalar function $f(t)$ derived just above in (3.13), which comprises the intermittency coefficient $\unicode[STIX]{x1D707}^{l}$ , the quantity $\mathbb{E}[(X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}})^{2}]$ , which is function of $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ only, and justified in (B 6), that reads

(3.15) $$\begin{eqnarray}\mathbb{E}[(X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}})^{2}]=\int _{\mathbb{ R}^{+}}\frac{\text{e}^{-h/T}}{h+\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}+\sqrt{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}(h+\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}})}}\,\text{d}h,\end{eqnarray}$$

and can be eventually expressed with special functions, as given by a symbolic calculation software. A modification to the regularized fractional Gaussian noise of vanishing Hurst exponent (2.11) denoted as $\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ (3.11), which reads

(3.16) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)=-\frac{1}{2}\int _{s=-\infty }^{t}\frac{1}{(t-s+\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}})^{3/2}}\frac{\unicode[STIX]{x1D608}_{ij}(s)}{\sqrt{\unicode[STIX]{x1D711}(s)}}\unicode[STIX]{x1D61E}_{ij}(\text{d}s),\end{eqnarray}$$

enters also in the definition of the scalar function $f(t)$ . Together with the isotropic diffusion term $\unicode[STIX]{x1D60B}_{ijkl}$ (3.3) and (3.8), constructed by a mean-square estimation procedure (§ 3.2), one finally arrives at

(3.17) $$\begin{eqnarray}\text{d}\unicode[STIX]{x1D608}_{ij}(t)=[\unicode[STIX]{x1D61D}_{ij}^{RFD}(t)+f(t)\unicode[STIX]{x1D608}_{ij}(t)]\,\text{d}t+\frac{1}{2}\sqrt{\frac{\unicode[STIX]{x1D707}^{l}\unicode[STIX]{x1D711}(t)}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}}\left[\unicode[STIX]{x1D61E}_{ij}(\text{d}t)-\frac{1}{3}\text{tr}[\unicode[STIX]{x1D652}(\text{d}t)]\unicode[STIX]{x1D6FF}_{ij}\right].\end{eqnarray}$$

This proposed dynamics is closed. It yields, starting for instance at time $t=0$ , the time evolution of $\unicode[STIX]{x1D63C}(t)_{t\geqslant 0}$ once some history $\unicode[STIX]{x1D63C}(t)_{t<0}$ is given. This boundary condition is required by the modification of the fractional Gaussian noise $\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ (3.16) that seeks in the past to set its current value. In this sense the dynamics proposed in (3.17) is non-Markovian. We expect this noise to decorrelate as fast as $1/t^{2}$ because its Gaussian version $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ does decorrelate this way (see developments in appendix B). So in practice, the noise $\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)$ (3.16) is well approximated by a truncation over a finite interval, say for instance $t\in [t-T,t]$ , and this approximation is more accurate as the small time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ gets smaller compared to $T$ .

5.1 Parameters of the simulations

In the following simulations we rescale time by $T$ and take $T=1$ , with no loss of generality. As for the intermittency coefficient in the Lagrangian framework $\unicode[STIX]{x1D707}^{l}$ , we use the recently estimated value on high Reynolds number numerical turbulent data of Huang & Schmitt (Reference Huang and Schmitt2014), who found $\unicode[STIX]{x1D707}^{l}=0.3$ . Notice that this is consistent with the commonly accepted value of the intermittency coefficient in the Eulerian framework $\unicode[STIX]{x1D707}^{e}=0.2$ , as may be checked with the phenomenological theory of Borgas (Reference Borgas1993), which relates $\unicode[STIX]{x1D707}^{l}$ and $\unicode[STIX]{x1D707}^{e}$ through a nonlinear relation (relating more precisely the respective singular spectra, using the vocabulary of the multifractal formalism). If one neglects the fluctuating nature of the spatial and temporal dissipative scales, an approximative linear relation can be drawn in a straightforward way: we observe that the moments of the pseudo-dissipation field $\unicode[STIX]{x1D711}$ can be expressed in the Eulerian (2.6) and Lagrangian (2.13) frameworks, and, recalling that $\unicode[STIX]{x1D702}/L\propto {\mathcal{R}}_{e}^{-3/4}$ and $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T\propto {\mathcal{R}}_{e}^{-1/2}$ , one is led to $3\unicode[STIX]{x1D707}^{e}=2\unicode[STIX]{x1D707}^{l}$ by equating their Reynolds number dependence.

The positive quantity $\mathbb{E}[(X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}})^{2}]$ contained in the dynamics of $\unicode[STIX]{x1D63C}$ (3.17) is given explicitly as an integral in (B 6). This integral could be numerically evaluated, but it may be analytically expressed in terms of error and generalized hypergeometric functions, which in turn must be numerically evaluated as well, but whose handling by popular math softwares can be quite efficient and controllable. We follow this path to compute $\mathbb{E}[(X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}})^{2}]$ for the various values of $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ considered.

Using $\unicode[STIX]{x0394}t=2\times 10^{-3}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ , or $\unicode[STIX]{x0394}t=10^{-3}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ for higher ${\mathcal{R}}_{e}$ , we numerically integrate numerous realizations of the $\unicode[STIX]{x1D63C}$ dynamics over time intervals of dozens or hundreds of $T$ , such that at least a total time of $3\times 10^{3}\,T$ is covered. We observe then, for the explored range $2.48\times 10^{-3}\leqslant \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T\leqslant 2.64\times 10^{-1}$ , or equivalently $-6.0\leqslant \log (\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T)\leqslant -1.33$ , the stationary statistics of $\unicode[STIX]{x1D63C}$ and its contraction $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D608}_{ij}\unicode[STIX]{x1D608}_{ij}$ .

It is worth mentioning that extensive tests were successfully done in order to establish not only the stationarity of the generated processes but also its independence of the initial conditions. In this perspective, we checked (data not shown) that an ensemble of highly anisotropic initial conditions indeed relaxes to an isotropic state, at least with the same level of success as in § 5.3.

5.2 Variance and covariance of the velocity gradients, and the Reynolds number dependence

5.2.1 Variance

For illustrative purposes, let us first begin by showing some temporal profiles of a diagonal element $\unicode[STIX]{x1D608}_{11}$ (figure 1 a), an off-diagonal element $\unicode[STIX]{x1D608}_{12}$ (figure 1 b) and the related pseudo-dissipation $\unicode[STIX]{x1D711}$ (figure 1 c) along a Lagrangian trajectory, at a given Reynolds number quantified by $\log (\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T)=-4.67$ . We observe an erratic behaviour of gradients that is expected in a highly turbulent flow. We may also notice the non-symmetrical behaviour of the diagonal element $\unicode[STIX]{x1D608}_{11}$ fluctuations (figure 1 a), reminiscent of the skewness phenomenon, and the highly intermittent $\unicode[STIX]{x1D711}$ fluctuations. We invite the reader to take a look at § 5.6 for a more in-depth study of the non-Gaussian nature of these fluctuations.

Figure 1.

In arbitrary units, samples of (a) a diagonal element $\unicode[STIX]{x1D608}_{11}$ , (b) an off-diagonal element $\unicode[STIX]{x1D608}_{12}$ and (c) the implied pseudo-dissipation $\unicode[STIX]{x1D711}$ along a trajectory for $\log (\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T)=-4.67$ showing characteristic erratic and intermittent fluctuations. (d) Variance of diagonal (i.e.  $\unicode[STIX]{x1D608}_{11}$ , ▫) and off-diagonal (i.e.  $\unicode[STIX]{x1D608}_{12}$ , ○) components and expectation value of pseudo-dissipation $\unicode[STIX]{x1D711}$ (▴) as a function of the ratio of the two characteristic scales $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ . The solid line shows in this representation the power law $1/15(T/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}})^{2}$ .

The second behaviour we would like to check is whether the proposed stochastic model (3.17) is consistent with basic Kolmogorov phenomenology, such as the Reynolds number (or $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ ) dependence of the variance of the elements of $\unicode[STIX]{x1D63C}$ . In particular, we would like the average pseudo-dissipation $\mathbb{E}(\unicode[STIX]{x1D711})=\mathbb{E}[\text{tr}(\unicode[STIX]{x1D63C}\unicode[STIX]{x1D63C}^{\top })]$ to be proportional to $(T/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}})^{2}$ , according to (1.7). Let us note that we expect this since we are essentially imposing this property by design. Indeed, we use as a constraint for building the dynamics of $\unicode[STIX]{x1D63C}$ (3.17) that it fulfils as closely as possible the dynamics of a multiplicative chaos $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D702}}$ (3.6), which is such that $\mathbb{E}(\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D702}})=1/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{2}$ . It remains to check numerically that this behaviour is actually observed for $\unicode[STIX]{x1D711}$ . More precisely, we would like to check whether, as required by isotropy (see for instance Pope Reference Pope2000),

(5.1) $$\begin{eqnarray}\mathbb{E}(\unicode[STIX]{x1D608}_{11}^{2})=\frac{1}{2}\mathbb{E}(\unicode[STIX]{x1D608}_{12}^{2})=\frac{1}{15}\mathbb{E}(\unicode[STIX]{x1D711})=\frac{1}{15}\left(\frac{T}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}\right)^{2}.\end{eqnarray}$$

We represent in figure 1(d) in a logarithmic scale the behaviour of the variances of components $\unicode[STIX]{x1D608}_{11}$ and $\unicode[STIX]{x1D608}_{12}$ and the average of $\unicode[STIX]{x1D711}$ as a function of the ratio $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ . We see that the isotropic relations (5.1) are verified on our numerical simulations of the process $\unicode[STIX]{x1D63C}$  (3.17) for a wide interval of $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ . This shows that, in this sense, statistics of $\unicode[STIX]{x1D63C}$ are isotropic and behave in a consistent way with $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ .

5.2.2 Autocovariance

As far as second-order statistics are concerned, another important property that should be checked in the simulated trajectories of $\unicode[STIX]{x1D63C}$ is the covariance of its elements. Kolmogorov’s phenomenology (Frisch Reference Frisch1995) suggests that, along Lagrangian trajectories, elements of $\unicode[STIX]{x1D63C}$ (such as $\unicode[STIX]{x1D608}_{11}$ or $\unicode[STIX]{x1D608}_{12}$ ) are correlated over the small time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ . We remark that if indeed this short-time correlation is observed in the present stochastic model, it is not clear to us how to derive it directly from the dynamics given in (3.17) (see dedicated works on this subject by Afonso & Meneveau Reference Afonso and Meneveau2010; Yu & Meneveau Reference Yu and Meneveau2010) since we are only imposing that pseudo-dissipation (which is a positive quantity) is correlated over the large time scale $T$ . We will come back to this long-range correlation behaviour in the next section. As we are dealing with a highly nonlinear dynamics and not imposing in a rigorous and definitive way that the elements of $\unicode[STIX]{x1D63C}$ should be correlated over the short time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ , it needs to be precisely quantified in our simulations. We represent in figure 2 the estimation of the correlation function of the diagonal element $\unicode[STIX]{x1D608}_{11}$ as a function of (i) the rescaled time $\unicode[STIX]{x1D70F}/T$ (figure 2 a), and (ii) the rescaled time $\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ (figure 2 b). Since the variance of $\unicode[STIX]{x1D608}_{11}$ increases as $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ decreases, we focus on the normalized covariance, namely

(5.2) $$\begin{eqnarray}\mathbb{E}_{n}[\unicode[STIX]{x1D608}_{11}(t)\unicode[STIX]{x1D608}_{11}(t+\unicode[STIX]{x1D70F})]=\frac{1}{\mathbb{E}[\unicode[STIX]{x1D608}_{11}^{2}]}\mathbb{E}[\unicode[STIX]{x1D608}_{11}(t)\unicode[STIX]{x1D608}_{11}(t+\unicode[STIX]{x1D70F})],\end{eqnarray}$$

which is independent of $t$ by stationarity, and tends towards unity as $\unicode[STIX]{x1D70F}\rightarrow 0$ . Note also that we do not subtract the square of the average of the diagonal element since we have $\mathbb{E}(\unicode[STIX]{x1D608}_{11})=0$ . A comparison between figures 2(a) and 2(c) shows clearly that the diagonal element $\unicode[STIX]{x1D608}_{11}$ is indeed correlated over the Kolmogorov time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ . Similar observations can be made on the off-diagonal element $\unicode[STIX]{x1D608}_{12}$ (data not shown).

Figure 2.

Autocorrelation functions of the diagonal element $\unicode[STIX]{x1D608}_{11}$ , as a function of the time lag. We represent in (a,c) the correlation function $\mathbb{E}_{n}[\unicode[STIX]{x1D608}_{11}(t)\unicode[STIX]{x1D608}_{11}(t+\unicode[STIX]{x1D70F})]$ (defined in (5.2)), and in (b,d) the correlation functions of the absolute values $\mathbb{E}_{nc}[|\unicode[STIX]{x1D608}_{11}(t)\unicode[STIX]{x1D608}_{11}(t+\unicode[STIX]{x1D70F})|]$ (defined in (5.3)). We show data for $\log (\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T)=-1.33,-2.0,-2.67,-3.33,-4.0,-4.67,-5.33,-6.0$ (from lighter to darker). In (a,b) the time lag is normalized by $T$ , and in (c,d) by $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ .

As we have seen, the diagonal element $\unicode[STIX]{x1D608}_{11}$ is correlated over the small time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ , as it should be. One may wonder what is the characteristic correlation time scale of the amplitude of the diagonal element $\unicode[STIX]{x1D608}_{11}$ , or equivalently the absolute value of the diagonal element $|\unicode[STIX]{x1D608}_{11}|$ . To do so, we define the following (normalized and centred) autocorrelation function:

(5.3) $$\begin{eqnarray}\mathbb{E}_{nc}[|\unicode[STIX]{x1D608}_{11}(t)\unicode[STIX]{x1D608}_{11}(t+\unicode[STIX]{x1D70F})|]=\frac{\mathbb{E}[(|\unicode[STIX]{x1D608}_{11}(t)|-\mathbb{E}[|\unicode[STIX]{x1D608}_{11}|])(|\unicode[STIX]{x1D608}_{11}(t+\unicode[STIX]{x1D70F})|-\mathbb{E}[|\unicode[STIX]{x1D608}_{11}|])]}{\mathbb{E}[(|\unicode[STIX]{x1D608}_{11}|-\mathbb{E}[|\unicode[STIX]{x1D608}_{11}|])^{2}]},\end{eqnarray}$$

which, by stationarity, should be time independent.

We represent in figure 2 the behaviour of this correlation function (5.3) both as a function of $\log (\unicode[STIX]{x1D70F}/T)$ (figure 2 b) and $\log (\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}})$ (figure 2 d). We indeed observe two type of behaviours. When the lag $\unicode[STIX]{x1D70F}$ is rescaled by the large scale $T$ , figure 2(b) reveals that the short-time portion of the correlations, typically when $\unicode[STIX]{x1D70F}$ is of the order of $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ , does not coincide, whereas for large times (i.e. when $\unicode[STIX]{x1D70F}\gg \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ ) the correlations vanish around the large scale $T$ . This analysis is confirmed in figure 2(d) where it is seen that the correlations superimpose over the short time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ but not over larger time lags.

This study shows that the (signed) diagonal elements decorrelate over the short time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ , whereas their absolute values decorrelate over the large time scale $T$ . This is a hallmark of intermittency: the sign of fluctuations decorrelates very fast, whereas the amplitude remains correlated over the large time scale $T$ . This is consistent with saying that intermittency and high values of gradients appear as bursts, that can be correlated over possibly large time scales (Frisch Reference Frisch1995).

5.3 Comparisons between the dynamics of $\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$

In this section, we would like to study numerically the statistical properties of the pseudo-dissipation $\unicode[STIX]{x1D711}=\text{tr}(\unicode[STIX]{x1D63C}\unicode[STIX]{x1D63C}^{\top })$ as obtained from the stochastic process (3.17), which are difficult to obtain analytically. We are also interested in comparing them to those given by the model $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ (3.6), that can be carried on analytically (appendix B). The statistics of $\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ are indeed expected to be very similar, since their dynamics are themselves driven by similar equations. To justify this, compare the dynamics of $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ (3.6) to that of $\unicode[STIX]{x1D711}$ , which can be obtained while inserting (3.13) in the dynamics of $\unicode[STIX]{x1D63C}$ (3.17) and applying the chain rule, which reads

(5.4) $$\begin{eqnarray}\displaystyle \text{d}\unicode[STIX]{x1D711} & = & \displaystyle \unicode[STIX]{x1D711}(t)\left[-\frac{1}{T}\left(\log [\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{2}\unicode[STIX]{x1D711}(t)]+\frac{\unicode[STIX]{x1D707}^{l}}{2}\mathbb{E}[(X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}})^{2}]\right)+\sqrt{\unicode[STIX]{x1D707}^{l}}\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)+\frac{\unicode[STIX]{x1D707}^{l}}{2\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}\right]\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle +\,\sqrt{\frac{\unicode[STIX]{x1D707}^{l}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}}\sqrt{\unicode[STIX]{x1D711}(t)}\unicode[STIX]{x1D608}_{kl}\unicode[STIX]{x1D61E}_{kl}(\text{d}t).\end{eqnarray}$$

We see that (3.6) and (5.4) are different for two reasons: (i) the noise $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ (2.11) is replaced by the noise $\hat{\unicode[STIX]{x1D6FD}}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ (3.12) and (ii) the noise $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}W$ is replaced by $\sqrt{\unicode[STIX]{x1D711}(t)}\unicode[STIX]{x1D608}_{kl}\unicode[STIX]{x1D61E}_{kl}$ .

Figure 3.

(a) Logarithmic representation of the resulting PDFs of $\log \unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ (lines) and $\log \unicode[STIX]{x1D711}$ (points), centred at the origin, normalized to unit variance and vertically shifted arbitrarily for clarity. Values of $\log (\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T)$ range from $-$ 2.67 to $-$ 5.67 in unit steps (from lighter to darker). (b) PDFs of dissipation (lines) and enstrophy (points) for the same ratios $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ . (c) Averages $\mathbb{E}[\unicode[STIX]{x1D711}]$ (○), $\mathbb{E}[\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}]$ (▪), $2\mathbb{E}[\text{tr}(\unicode[STIX]{x1D64E}\unicode[STIX]{x1D64E}^{\top })]$ (▿) and $2\mathbb{E}[\text{tr}(\unicode[STIX]{x1D6FA}\unicode[STIX]{x1D6FA}^{\top })]$ (▵), as functions of $\log (\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T)$ . (d) Autocovariance $\mathbb{E}_{c}$ (5.5) as a function of time lag for $\log \unicode[STIX]{x1D711}$ (lines) and $\log \unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ (circles). Values of $\log (\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T)=$ span from $-$ 1.33 to $-$ 6.0 (lighter to darker) just as in figure 2. The asymptotic logarithm behaviour (5.6) fulfilled by $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ is also shown for comparison (dashed line).

Continuing our analysis, we represent in figure 3(a) the estimation of the PDFs of the realizations of the random variables $\log \unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ and $\log \unicode[STIX]{x1D711}$ , once centred (we subtract their respective means) and normalized (we divide them by their respective standard deviations) for different values of the ratio $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ . We recall that $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ is obtained while exponentiating $X_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ , accordingly with (2.9), which is itself given by a linear stochastic differential equation (2.10). We see that, as expected, the statistics of $\log \unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ is indeed normal for any $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ . As for $\log \unicode[STIX]{x1D711}$ , its PDFs closely follow those of $\log \unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ , and hence their fluctuations are indeed observed to be very similar.

As also studied by Yeung & Pope (Reference Yeung and Pope1989), Pope (Reference Pope1990) and Huang & Schmitt (Reference Huang and Schmitt2014), we reproduce in figure 3(b) a similar estimation of the PDFs of two other quantities derived from $\unicode[STIX]{x1D63C}$ , namely dissipation $\text{tr}(\unicode[STIX]{x1D64E}\unicode[STIX]{x1D64E}^{\top })$ and enstrophy $\text{tr}(\unicode[STIX]{x1D6FA}\unicode[STIX]{x1D6FA}^{\top })$ , where $\unicode[STIX]{x1D64E}$ and $\unicode[STIX]{x1D6FA}$ are respectively the symmetric and antisymmetric parts of the decomposition of $\unicode[STIX]{x1D63C}$ . We see that whereas the statistics of $\unicode[STIX]{x1D711}$ is very close to that of a lognormal process, those of dissipation and enstrophy differ significantly from such a process. This fact also holds true in direct numerical simulations (DNS) of turbulence as shown in Yeung & Pope (Reference Yeung and Pope1989), Pope (Reference Pope1990) and Huang & Schmitt (Reference Huang and Schmitt2014).

Let us now focus on the averages of these fields. We represent in figure 3(c) the average values of the pseudo-dissipation fields $\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ . As expected, since this is imposed by design, it is observed that $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}^{2}\mathbb{E}[\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}]\approx 1$ for all studied values of $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ . Up to statistical fluctuations, we also observe that $\mathbb{E}[\unicode[STIX]{x1D711}]\approx \mathbb{E}[\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}]$ for all $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ . Here, numerous tests with smaller time steps and bigger ensembles were necessary to establish this result, since systematic differences may appear at high Reynolds numbers if statistical convergence is not properly attained. In fact, the mild increases in the averages one observes in figure 3(c) for the smaller ratios $\unicode[STIX]{x1D70F}/T$ (higher Reynolds) seem to be mainly due to numerical errors, since they are much more acute if greater time steps are used for integration or smaller ensembles are considered (data not shown). We also observe – and this is a limitation of the present model – that statistics are not fully consistent with those expected from isotropic turbulence. Indeed, for a homogeneous and isotropic velocity field, we expect that $\mathbb{E}[\unicode[STIX]{x1D711}]=2\mathbb{E}[\text{tr}(\unicode[STIX]{x1D64E}\unicode[STIX]{x1D64E}^{\top })]=2\mathbb{E}[\text{tr}(\unicode[STIX]{x1D6FA}\unicode[STIX]{x1D6FA}^{\top })]$ (Pope Reference Pope2000), a relation which is not perfectly met by the model. Figure 3(c) shows that the average enstrophy is a bit too small, with $2\mathbb{E}[\text{tr}(\unicode[STIX]{x1D6FA}\unicode[STIX]{x1D6FA}^{\top })]\approx 0.88\mathbb{E}[\unicode[STIX]{x1D711}]$ , while the average dissipation a bit too high, $2\mathbb{E}[\text{tr}(\unicode[STIX]{x1D64E}\unicode[STIX]{x1D64E}^{\top })]\approx 1.17\mathbb{E}[\unicode[STIX]{x1D711}]$ , compared to the average pseudo-dissipation. We discuss possible improvements of the model in order to correct this bias in § 6.

Next, we study in figure 3(d) the correlation structure of the fields $\unicode[STIX]{x1D711}(t)$ and $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)$ . With this aim, we estimate the autocovariance

(5.5) $$\begin{eqnarray}\mathbb{E}_{c}[\log \unicode[STIX]{x1D711}(t)\log \unicode[STIX]{x1D711}(t+\unicode[STIX]{x1D70F})]=\mathbb{E}[(\log \unicode[STIX]{x1D711}(t)-\mathbb{E}[\log \unicode[STIX]{x1D711}])(\log \unicode[STIX]{x1D711}(t+\unicode[STIX]{x1D70F})-\mathbb{E}[\log \unicode[STIX]{x1D711}])],\end{eqnarray}$$

which, by stationarity, should be time independent. We know for sure, in the asymptotic regime $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}\rightarrow 0$ , that when it comes to the field $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ this correlation function behaves logarithmically with the time lag $\unicode[STIX]{x1D70F}$ , i.e.

(5.6) $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T\rightarrow 0}\mathbb{E}_{c}[\log \unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t)\log \unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}(t+\unicode[STIX]{x1D70F})]\underset{\unicode[STIX]{x1D70F}/T\rightarrow 0}{{\sim}}\unicode[STIX]{x1D707}^{l}\log \left(\frac{T}{\unicode[STIX]{x1D70F}}\right).\end{eqnarray}$$

We compare in figure 3(d) the estimation of the autocovariance $\mathbb{E}_{c}$ (5.5) for the field $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ with the asymptotic logarithmic behaviour (5.6). It is seen that, as $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ gets smaller, the autocovariance becomes more consistent with the expected logarithmic behaviour. As for $\unicode[STIX]{x1D711}$ , it is found that its corresponding centred correlation function follow very closely those of $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}$ , for all the ratios $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ considered here. We may nonetheless remark that, once again, improved time resolutions and larger ensembles were decisive in establishing this numerical trend.

5.4 Joint PDF of the two invariants $R$ and $Q$

As an important characterization of the velocity gradient tensor $\unicode[STIX]{x1D608}_{ij}$ , we study its two non-vanishing invariants. The second invariant $Q$ is given by

(5.7) $$\begin{eqnarray}Q=-{\textstyle \frac{1}{2}}\text{tr}(\unicode[STIX]{x1D63C}^{2})={\textstyle \frac{1}{4}}|\unicode[STIX]{x1D74E}|^{2}-{\textstyle \frac{1}{2}}\text{tr}(\unicode[STIX]{x1D64E}^{2}),\end{eqnarray}$$

where $\unicode[STIX]{x1D74E}$ is the vorticity vector and $\unicode[STIX]{x1D64E}$ the symmetric part of $\unicode[STIX]{x1D63C}$ , and can be interpreted as the competition between enstrophy and dissipation (per unit viscosity). Therefore, positive $Q$ represents rotation-dominated regions and negative $Q$ dissipation-dominated regions. Analogously, the third invariant $R$ is given by

(5.8) $$\begin{eqnarray}R=-{\textstyle \frac{1}{3}}\text{tr}(\unicode[STIX]{x1D63C}^{3})=-{\textstyle \frac{1}{4}}\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D61A}_{ij}\unicode[STIX]{x1D714}_{j}-{\textstyle \frac{1}{3}}\text{tr}(\unicode[STIX]{x1D64E}^{3}),\end{eqnarray}$$

representing competition between enstrophy production and dissipation production (we refer to Tsinober Reference Tsinober2001; Wallace Reference Wallace2009; Meneveau Reference Meneveau2011, for discussions on this topic). A numerical estimation of the PDF of the invariants $Q$ and $R$ is represented in figure 4(a). One may see that it is more elongated along the right tail of the Vieillefosse line and in the upper-left quadrant, precisely as observed in direct numerical simulations.

Figure 4.

(a) Logarithmic representation of the joint PDF ${\mathcal{P}}(Q^{\ast },R^{\ast })$ of $R^{\ast }=R/\langle \unicode[STIX]{x1D61A}_{ij}\unicode[STIX]{x1D61A}_{ij}\rangle ^{3/2}$ and $Q^{\ast }=Q/\langle \unicode[STIX]{x1D61A}_{ij}\unicode[STIX]{x1D61A}_{ij}\rangle$ computed from the statistics of $\unicode[STIX]{x1D63C}$ at $\log (\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T)=-5.67$ . Contour lines logarithmically spaced by factors of $1/10$ , starting at 1 near the origin. The thick line represents the zero-discriminant (or Vieillefosse) line: $(27/4)R^{2}+Q^{3}=0$ . (b) PDF of the cosine of the angle between vorticity and the eigendirections of strain associated with the most compressive, intermediate and most extensive eigenvalues (lighter to darker), for the same $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ .

5.5 Alignments of vorticity with the eigenframe of the deformation rate

Another striking property of turbulence is the preferential alignment of vorticity with the strain eigendirection associated with the intermediate eigenvalue. We refer again to Tsinober (Reference Tsinober2001), Wallace (Reference Wallace2009) and Meneveau (Reference Meneveau2011) for further discussions. Figure 4(b) displays the probability density function of the cosine of the angle $\unicode[STIX]{x1D703}$ between vorticity and the strain eigenvectors. It indicates the preferential alignment of vorticity with the intermediate eigendirection of deformation, as is observed in DNS. We can add that the alignment PDFs (figure 4 b) do not depend significantly on the $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ value in the studied range, as well as the $RQ$ plane from figure 4(a), where the contour lines just become a bit more spaced as ${\mathcal{R}}_{e}$ increases due to additional non-Gaussianity (data not shown).

5.6 Higher moments of the gradients and multifractal properties

We now characterize and quantify the intermittent properties of the statistics of $\unicode[STIX]{x1D63C}$ in a sense that will become more precise.

Figure 5.

(a) PDFs of diagonal elements $\unicode[STIX]{x1D608}_{11}$ in logarithm scale renormalized to unity variance and arbitrarily vertically shifted for clarity. Spanned $\log (\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T)$ values range from $-$ 1.33 to $-$ 5.67 with increments of $-$ 1/3 (from lighter to darker). (b) Similar study for the off-diagonal elements $\unicode[STIX]{x1D608}_{12}$ . (c) Skewness and (d) flatness of both diagonal (○) and off-diagonal (▪) components. Predicted power laws (5.9) and (5.10) are shown (dotted curves) for comparison.

First we estimate the PDF of the diagonal and off-diagonal elements of $\unicode[STIX]{x1D63C}$ . They are portrayed respectively in figure 5(a,b). All PDFs are normalized to unit variance and are arbitrarily vertically shifted for the sake of clarity. It is clear that in both cases, as the ratio $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ decreases, the PDF undergoes a continuous shape deformation, from a Gaussian shape at large ratios towards large tails at small ratios. It is additionally noticeable that the cores of the diagonal elements PDFs (figure 5 a) are not symmetrical. This continuous shape deformation of the PDFs as the Reynolds number increases is a hallmark of multifractal behaviour.

To further characterize the intermittency phenomenon and the non-Gaussian behaviour of fluctuations, we compute the skewness $\mathbb{E}[\unicode[STIX]{x1D608}_{ij}^{3}]/(\mathbb{E}[\unicode[STIX]{x1D608}_{ij}^{2}])^{3/2}$ of the diagonal and off-diagonal elements. It is seen from figure 5(c) that, up to statistical uncertainties, only diagonal elements are skewed, as allowed by isotropy. Moreover, we contemplate that the skewness of diagonal elements exhibits a Reynolds number dependence (through the ratio $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ ), as expected from multifractal phenomenologies. We superimpose in figure 5(c) the power law (dotted line) observed in experimental and numerical data (see equation 4.4 in Ishihara et al. Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007):

(5.9) $$\begin{eqnarray}\frac{\mathbb{E}[\unicode[STIX]{x1D608}_{11}^{3}]}{(\mathbb{E}[\unicode[STIX]{x1D608}_{11}^{2}])^{3/2}}\propto -\left(\frac{T}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}\right)^{0.11},\end{eqnarray}$$

which is very close to the prediction of the multifractal formalism (see for instance Frisch Reference Frisch1995; Chevillard et al. Reference Chevillard, Castaing, Arneodo, Lévêque, Pinton and Roux2012). It is worth noting that in figure 5(c) the power law (5.9) has been fitted beforehand on a log–log plot in order to fix the suitable multiplicative constant (not shown).

In the same spirit, we pursue the quantification of the level of intermittency through an estimate of the velocity gradient flatness, i.e.  $\mathbb{E}[\unicode[STIX]{x1D608}_{ij}^{4}]/(\mathbb{E}[\unicode[STIX]{x1D608}_{ij}^{2}])^{2}$ , again for both diagonal and off-diagonal elements of $\unicode[STIX]{x1D63C}$ . The resulting behaviours are shown in figure 5(d). It is evident that the depicted flatnesses exceed the Gaussian value $3$ . We superimpose in this representation (dotted lines) the power law observed in numerical and experimental flows (Ishihara et al. Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007):

(5.10) $$\begin{eqnarray}\frac{\mathbb{E}[\unicode[STIX]{x1D608}_{11}^{4}]}{(\mathbb{E}[\unicode[STIX]{x1D608}_{11}^{2}])^{2}}\propto \left(\frac{T}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}\right)^{0.34},\end{eqnarray}$$

which is also very close to what is predicted by the multifractal formalism (Frisch Reference Frisch1995; Chevillard et al. Reference Chevillard, Castaing, Arneodo, Lévêque, Pinton and Roux2012). For our simulated process (3.17), the power law (5.10) seems to be representative of the statistics only for the smallest values of $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ tested.

We emphasize that these numerically observed results for the velocity gradients’ higher-order moments illustrate the self-consistency of the present approach, once its single free parameter $\unicode[STIX]{x1D707}^{l}$ is given. Indeed, the power-law exponents as given in (5.9) and (5.10) can be readily related to the intermittency coefficient $\unicode[STIX]{x1D707}^{l}$ of the Lagrangian framework since it governs the statistics of the overall amplitude of the gradients, and more precisely pseudo-dissipation. They can also be derived, in the Eulerian framework, from the respective intermittency coefficient $\unicode[STIX]{x1D707}^{e}$ (see (2.2)) as is traditionally done in the literature (Frisch Reference Frisch1995). We see that arguments developed by Borgas (Reference Borgas1993), as recalled in § 5.1, which relate both intermittency coefficients as $3\unicode[STIX]{x1D707}^{e}=2\unicode[STIX]{x1D707}^{l}$ , give a consistent picture of the power laws announced in (5.9) and (5.10).

5.7 Rotation rate of anisotropic particles

Many numerical and theoretical studies employ the Jeffery equation (Jeffery Reference Jeffery1922) to predict the time evolution of the orientation of axisymmetric ellipsoidal particles as they are advected and influenced by a turbulent velocity field. Specifically, Jeffery’s equation for the unit orientation vector $\boldsymbol{p}(t)$ in the ellipsoid major axis reads

(5.11) $$\begin{eqnarray}\frac{\text{d}p_{n}}{\text{d}t}=\unicode[STIX]{x1D6FA}_{nj}p_{j}+\unicode[STIX]{x1D706}(\unicode[STIX]{x1D61A}_{nj}p_{j}-p_{n}p_{k}\unicode[STIX]{x1D61A}_{kl}p_{l}),\end{eqnarray}$$

where $\unicode[STIX]{x1D64E}$ and $\unicode[STIX]{x1D6FA}$ are the strain and rotation rate tensors respectively, and $\unicode[STIX]{x1D706}=(\unicode[STIX]{x1D6FC}^{2}-1)/(\unicode[STIX]{x1D6FC}^{2}+1)$ , $\unicode[STIX]{x1D6FC}$ being the particle’s aspect ratio. In this convention, the limit $\unicode[STIX]{x1D6FC}\rightarrow \infty$ corresponds to rods (elongated particles along the axis of symmetry), whereas discs are given in the limit $\unicode[STIX]{x1D6FC}\rightarrow 0$ . See for instance Chevillard & Meneveau (Reference Chevillard and Meneveau2013) or Gustavsson, Einarsson & Mehlig (Reference Gustavsson, Einarsson and Mehlig2014) for details, and Voth & Soldati (Reference Voth and Soldati2017) for a recent review on the subject. It was proposed by Parsa et al. (Reference Parsa, Calzavarini, Toschi and Voth2012) to quantify, in experiments and numerical simulations, the variance of the rotation rate of anisotropic particles $\mathbb{E}[|\dot{\boldsymbol{p}}|^{2}]$ as a function of the aspect ratio $\unicode[STIX]{x1D6FC}$ , which turns out to be a precise and demanding way to test models for the velocity gradient tensor $\unicode[STIX]{x1D63C}$ (Chevillard & Meneveau Reference Chevillard and Meneveau2013).

Figure 6.

Variance of the rotation rate of anisotropic particles as a function of the aspect ratio $\unicode[STIX]{x1D6FC}$ (see (5.11)). Solid line: DNS (Li et al. Reference Li, Perlman, Wan, Yang, Burns, Meneveau, Burns, Chen, Szalay and Eyink2008). Dotted line: RFD approximation (Chevillard & Meneveau Reference Chevillard and Meneveau2013). Symbols: predictions from the present model (3.17). Displayed $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ ratios correspond to $\log (\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T)=-3.67$ (○), $-$ 4.67 (▵) and $-$ 5.67 (▫).

We reproduce in figure 6 the behaviour of the normalized variance of the rotation rate $\mathbb{E}[|\dot{\boldsymbol{p}}|^{2}]/\mathbb{E}[\unicode[STIX]{x1D711}]$ as estimated in DNS (Li et al. Reference Li, Perlman, Wan, Yang, Burns, Meneveau, Burns, Chen, Szalay and Eyink2008) and in trajectories built from the RFD approximation (see Chevillard & Meneveau Reference Chevillard and Meneveau2013, for details), along with the outcome of the present stochastic model (3.17) for different $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ . Let us recall some of the interpretations that can be drawn from the behaviour of the rotation rate as a function of the aspect ratio $\unicode[STIX]{x1D6FC}$ . At high aspect ratios $\unicode[STIX]{x1D6FC}\gg 1$ , corresponding to elongated particles along the axis of symmetry (or rods), the dynamics of anisotropic particles resembles that of material lines (see for instance Guala et al. Reference Guala, Lüthi, Liberzon, Tsinober and Kinzelbach2005, for a review on this matter). Therefore, a strong correlation between vorticity and the orientation vector is expected, leading in particular to a preferential alignment between them, as observed in numerical simulations (Pumir & Wilkinson Reference Pumir and Wilkinson2011). At low aspect ratios $\unicode[STIX]{x1D6FC}\ll 1$ , corresponding to disc shaped particles in the plane perpendicular to the orientation vector, it was observed that, contrary to rods, the orientation vector becomes preferentially perpendicular to vorticity (Chevillard & Meneveau Reference Chevillard and Meneveau2013; Gustavsson et al. Reference Gustavsson, Einarsson and Mehlig2014). As a consequence of this preferential alignment it is expected that discs will tumble, implying a higher rotation rate than in the rod case. This is what is observed in DNS (Parsa et al. Reference Parsa, Calzavarini, Toschi and Voth2012): $\mathbb{E}[|\dot{\boldsymbol{p}}|^{2}]$ increases monotonically as the aspect ratio $\unicode[STIX]{x1D6FC}$ decreases from rods ( $\unicode[STIX]{x1D6FC}\gg 1$ ) to discs ( $\unicode[STIX]{x1D6FC}\ll 1$ ).

It was shown in Chevillard & Meneveau (Reference Chevillard and Meneveau2013) that the stochastic model for velocity gradients coined with the RFD approximation is only able to satisfactorily reproduce the behaviour of the rotation rate observed in DNS for aspect ratios larger than unity. In this regard, the present model (3.17) exhibits an even more striking agreement for these aspect ratios. For particles with smaller aspect ratios ( $\unicode[STIX]{x1D6FC}<1$ ), it is seen that the RFD approximation predicts a decrease of the rotation rate, whereas in DNS it keeps increasing up to a saturation value. The present model (3.17) shows a similar saturation structure of the fluctuations with the aspect ratio $\unicode[STIX]{x1D6FC}$ , an improvement in the modelling of the rotation rate of anisotropic particles compared to the RFD approximation, but overall, even if the present model (3.17) exhibits many aspects of the statistical properties of the velocity gradient tensor $\unicode[STIX]{x1D63C}$ , it still fails at providing a satisfying picture of the tumbling motions of discs in turbulence. Let us finally remark that the predictions of the present model on the rotation rate are almost independent of the Reynolds number (or equivalently $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}/T$ ), whereas it is shown in former sections that the statistics of $\unicode[STIX]{x1D63C}$ do have a realistic dependence. This is consistent with the trends observed from DNS studies (Parsa et al. Reference Parsa, Calzavarini, Toschi and Voth2012).