3.1. Analytical estimations

Our first level of analysis is the analytical one. We intend to estimate the change of the diffusion coefficient induced by non-Gaussian features of turbulence. In order to do that, we start by considering a simplified case of frozen turbulence, when the potential is time independent $\varphi (\boldsymbol {x},t) \equiv \varphi (\boldsymbol {x})$. This case can be obtained setting $\tau _c\to \infty$. Due to the Hamiltonian structure of the equations (2.1), the trajectories are closed and both the fictitious and the real potentials are conserved $\varphi (\boldsymbol {x}(t)) = \varphi (\boldsymbol {x}(0)) = \varphi (\boldsymbol {0})\implies \phi (\boldsymbol {x}(t)) = \phi (\boldsymbol {x}(0)) = \phi (\boldsymbol {0})$. For further purposes, let us denote the following function:

(3.3)\begin{equation} A[\varphi]=\frac{1+2\alpha \varphi+3\beta \varphi^2}{\sqrt{1+2\alpha^2+3\beta (2+5\beta)}}. \end{equation}

With this notation, it turns out that the equations of motion in the Gaussian and non-Gaussian ($0$ index) cases become

(3.4)\begin{gather} \frac{{\rm d}\boldsymbol{x}_0(t)}{{\rm d}t} = \hat{e}_z\times\boldsymbol{\nabla}\varphi(\boldsymbol{x}_0(t)), \end{gather}

(3.5)\begin{gather}\frac{{\rm d}\boldsymbol{x}(t)}{{\rm d}t} = A[\varphi(0)]\hat{e}_z\times\boldsymbol{\nabla}\varphi(\boldsymbol{x}(t)). \end{gather}

It can be easily shown, via a variable transformation, that $\boldsymbol {x}(t) = \boldsymbol {x}_0(A[\varphi (\boldsymbol {0})]t)$. This exact relation allows us to relate the running diffusion coefficients between these two cases, non-Gaussian $D(t)$ and the Gaussian limit $D_0(t)$. In order to do that, let us denote the ‘conditional diffusion’ for trajectories starting at equal potential values as

(3.6)\begin{equation} d_0(t;\varphi(\boldsymbol{0})) = \frac{1}{2}\frac{{\rm d}}{{\rm d}t}\langle \boldsymbol{x}_0^2(t) \rangle_{\varphi(\boldsymbol{0})} \end{equation}

for which it holds true that $d(t;\varphi (\boldsymbol {0}))= A[\varphi (\boldsymbol {0})]d_0(A[\varphi (\boldsymbol {0})]t;\varphi (\boldsymbol {0}))$. Finally, we write

(3.7)\begin{gather} D_0(t) = \int {\rm d}\varphi(0) P[\varphi(\boldsymbol{0})]d_0(t;\varphi(\boldsymbol{0})), \end{gather}

(3.8)\begin{gather}D(t) = \int {\rm d}\varphi(0) P[\varphi(\boldsymbol{0})]d_0(A[\varphi(\boldsymbol{0})]t;\varphi(\boldsymbol{0}))A[\varphi(\boldsymbol{0})]. \end{gather}

Without any proof, we assume that some sort of generalized mean value theorem is valid and these two integrals can be related through an effective potential

(3.9)\begin{equation} D(t) = A[\varphi_{{\rm eff}}(t)] D_0(A[\varphi_{{\rm eff}}(t)]t). \end{equation}

On the other hand, the anomalous feature of transport is reflected in the asymptotic behaviour of running Lagrangian averages $L(t)$ which algebraically decay $L_0(t)\sim t^{-\gamma }$ (Isichenko Reference Isichenko1992; Ottaviani Reference Ottaviani1992; Reuss & Misguich Reference Reuss and Misguich1996; Reuss, Vlad & Misguich Reference Reuss, Vlad and Misguich1998; Vlad et al. Reference Vlad, Spineanu, Misguich and Balescu1998a, Reference Vlad, Spineanu, Misguich, Reuss, Balescu, Itoh and Itoh2004). At small times $t\ll \tau _{{\rm fl}}$ the dependence of $L(t)$ can usually be analytically computed. The presence of a decorrelation mechanism (finite $\tau _c$ in our case) tends to saturate asymptotically all Lagrangian quantities at values which can be estimated (Vlad et al. Reference Vlad, Spineanu, Misguich and Balescu1998a, Reference Vlad, Spineanu, Misguich, Reuss, Balescu, Itoh and Itoh2004) as $\lim _{t\to \infty } L(t) =L^\infty \approx L(\tau _c)$. We assume this to be true both for diffusion $D(t)$ and $\varphi _{{\rm eff}}(t)$.

Combining all these behaviours, the algebraic decay $D(t)\sim t^{-\gamma }, \varphi _{{\rm eff}}(t)\sim t^{-\zeta }$, the approximate saturation at the decorrelation time $D^\infty \approx D(\tau _c), \varphi ^\infty _{{\rm eff}} \approx \varphi _{{\rm eff}}(\tau _c)$ and the assumed relation between Gaussian and non-Gaussian diffusion (3.9), one can show that

(3.10)\begin{equation} \begin{cases} D^\infty/D_0^\infty = A[\varphi_{{\rm eff}}(\tau_c;\alpha,\beta)]^2 \approx 1 + 2 \alpha^2 + 12 \beta^2, & K_\star{\ll} 1\\ D^\infty/D_0^\infty = A[\varphi_{{\rm eff}}(\tau_c;\alpha,\beta)]^{1-\gamma} \approx 1 +3 \beta({-}1 + \gamma), & K_\star{\gg} 1 . \end{cases} \end{equation}

These estimations suggest that $\alpha$ has only a quadratic effect on the diffusion coefficient. This aspect can be understood from another perspective, analysing how the Lagrangian correlation of velocities $L_v(t) = \langle v_x(0)v_x(t)\rangle$ (the time derivative of $D(t)$) varies with $\alpha$ up to the first order

(3.11)\begin{equation} \frac{\partial }{\partial\alpha}L_v(t) = \frac{\partial}{\partial\alpha}\langle v_x(\boldsymbol{0},0)v_x(\boldsymbol{x}(t),t)\rangle \propto \langle \varphi(0)\partial_y\varphi(0)\partial_y\varphi(t)\rangle+\langle \varphi(t)\partial_y\varphi(0)\partial_y\varphi(t)\rangle \approx 0. \end{equation}

Note that $\varphi (0)$ and $\partial _i\varphi (0)$ are uncorrelated Gaussian quantities. Moreover, Lumley's theorem (Monin & Yaglom Reference Monin and Yaglom1973) assures us that the space derivatives $\partial _i\varphi (t)$ remain Gaussian quantities at all times. Only the distribution of Lagrangian potentials $\varphi (t)$ might depart from Gaussianity in the case of finite $\tau _c$, but only slightly. Thus, the derivative $\partial _\alpha L_v(t)$ is roughly made up of averages of products of three Gaussian quantities and, therefore, is zero. This means that $L_v(t)$ is roughly independent of $\alpha$ up to first order. The same goes for the diffusion.

Following the above reasoning and estimations (3.10), we expect that the non-Gaussian diffusion will vary roughly as $\mathcal {O}(\beta ), \mathcal {O}(\alpha ^2)$. For this reason, we define a response function (susceptibility $\chi$) to quantify the possible linear dependency between diffusion variation and turbulence excess kurtosis (as a measurable quantity)

(3.12)\begin{equation} \chi = \lim_{\delta\kappa\to 0 }\frac{1}{\delta\kappa}\left(1-\frac{D^\infty(\delta\kappa)}{D^\infty(0)}\right). \end{equation}

3.2. Numerical results

We intend to test further if the analytical estimations found above bear any meaning in real situations. For that, we use the statistical methods described previously: DNS and DTM. We underline that DNS is an exact-in-principle method which is hindered in practice only by the numerical resolution, thus, it requires a large amount of CPU resources. DTM is an approximation which provides only qualitative results and it is easy to implement numerically. The purpose of DTM in this work is to serve as a supplementary test for the results of DNS which might be plagued with a small, but uncertain, degree of numerical inaccuracy.

We perform numerical simulations of the running diffusion coefficients $D(t)$ for the incompressible motion (2.1) where the potential $\phi (\boldsymbol {x},t)$ is described via a fictitious field $\varphi (\boldsymbol {x},t)$ ( 2.6) with known Eulerian correlation ( 3.1). The numerical method for trajectory propagation both in ( 2.18) (DNS) and ( 2.11) (DTM) is a fourth-order Runge–Kutta method with a fixed time step $\Delta t\sim 10^{-1}\min (\tau _c,\lambda _c^2)$. The simulation time is $t_{\max }\sim 5\tau _c$. The number of trajectories simulated with DNS is routinely $N_p \sim 10^5$ while the number of sub-ensembles used in DTM $N_s\sim 10^5$. These resolutions are chosen for numerical accuracy and statistical precision. Using dedicated programming procedures, typical simulations on personal computers require in terms of CPU time $t_{{\rm CPU}}^{{\rm DTM}}\sim 1\,\min$ and $t_{{\rm CPU}}^{{\rm DNS}}\sim 10\,\min$.

Beyond diving into the matter of non-Gaussianity, let us have a look at a typical running diffusion coefficient $D(t)$ obtained in the case $\alpha = \beta = 0$. The results of both methods are shown in figure 3 at $\tau _c = 10$ and $\tau _c\to \infty$. In the case of frozen turbulence one can see the algebraic decay of diffusion $D(t)\sim t^{-\gamma }$. The effect of finite $\tau _c$ is the saturation of diffusion to a constant value $D^\infty = D(t\to \infty )$. Note that DTM reproduces the qualitative behaviour of trapping and decorrelation at all times. Yet, the results are quantitatively different in the asymptotic region $t\gg \tau _{{\rm fl}}$. This is a due to the overestimation of trapping in the DTM approximation. Consequently, DTM overestimates the $\gamma$ exponent too and we expect it will overestimate the effect of intermittency in the strong/low-frequency turbulence regime.

Figure 3.

Running diffusion coefficient $D(t)$ obtained for the correlation $\mathcal {E}_1, V_p = 0$ in the case $\tau _c\to \infty$ (blue) and $\tau _c = 10$ (red) with the use of DNS (full line) and DTM (dashed line).

We investigate further the dependence of the diffusion on $\alpha$ and find that both methods, DTM and DNS, predict a negligible variation (figure 4). The results are in line with our analytical estimation that $\alpha$ affects the transport only at second order. A very weak linear dependence supplemented by a weak quadratic one can be observed.

Figure 4.

Asymptotic values of diffusion at $\beta = 0, \tau _c = 10, V_p = 0$ vs the skewness $s$ obtained with the use of DTM (blue line) and DNS (red line).

Given this fact, let us set $\alpha = 0$ and look further at how $\beta$ affects the diffusion. The mechanism can be seen at work in figure 5, where several running diffusion profiles $D(t)$ are shown for different $\beta$ values. As expected, the effect is visible only at larger times, at least at the order of $\tau _{{\rm fl}}$, and it results in a decrease of the diffusion coefficient. This behaviour can be quantified further by inspecting the variation of the asymptotic diffusion $D^\infty$ with $\beta$.

Figure 5.

Running diffusion profiles $D(t)$ obtained with DNS for the correlation $\mathcal {E}_1$ with $\tau _c =10$ and $V_p = 0$ at different $\beta$ values and $\alpha = 0$.

We plot in figure 6 asymptotic diffusion coefficients computed for different values of $\beta$ with DNS (blue) for the correlation function $\mathcal {E}_1$ and with DTM (red) for $\mathcal {E}_2$. Our expectation that the $\beta$ parameter drives a linear change in the diffusion is confirmed. Also, it must be emphasized how close the results of DTM to those of DNS are, given the fact that they use two distinct correlations.

Figure 6.

The relative asymptotic diffusion dependence on the excess kurtosis obtained with DTM for $\mathcal {E}_2$ (blue line) and with DNS for $\mathcal {E}_1$ (red line).

Going further into understanding the effects of intermittency, we plot in figure 7 profiles of asymptotic diffusion coefficients vs the Kubo number at different values of the $\beta$ parameter. As expected from the results of figure 6 the effect of $\beta$ is to inhibit the overall values of diffusion. However, something supplementary must be underlined: in the high $K_\star$ region, that of trajectory trapping, the profiles are approximately parallel to each other and to the $\sim K_\star ^{-0.3}$ line. This tells us that the Kubo number scaling is universal $\gamma = 0.3$ and it is unaffected by non-Gaussianity.

Figure 7.

Asymptotic diffusion coefficient vs the Kubo number at different $\beta$ values. The results are obtained with DNS for $\mathcal {E}_1$ and $V_p = 0$.

After confirming the linear behaviour in $\beta$, we look into the dependence of the susceptibility $\chi$ on the Kubo number $K_\star$. We have performed extensive numerical simulations both with DTM and DNS varying both the Kubo number (through $\tau _c$) and the $\beta$ parameter. Final results are shown in figure 8, where we plot $\chi (K_\star )$ obtained with DNS at several distinct $V_p$ values. As one can see, at small correlation times, this quantity is null. It increases only quadratically with $K_\star$, until around $\tau _c$ of the order of the time of flight and saturates for $V_d=0$. An interesting effect of the average velocity can be seen in figure 8, which consists in the strong attenuation of the effect of the kurtosis. The susceptibility has a strong algebraic decay in these conditions.

Figure 8.

Susceptibility $\chi$ as a function of Kubo number $K_\star$ obtained with DNS for $\mathcal {E}_1$ at distinct $V_p$ values.

In figure 9 we show the profile of $\chi$ obtained with DTM in the case of $V_p = 0$. The method is able to confirm what was found with DNS: the susceptibility grows to a maximum around the time of flight and decays at larger values of $K_\star$. As expected, since DTM overestimates the trapping, it also overestimates the decay of $\chi$.

Figure 9.

Susceptibility $\chi$ as a function of Kubo number $K_\star$ obtained with DTM for $\mathcal {E}_2$ at $V_p = 0$.

We underline that high Kubo number regimes $K_\star \gg 1$ are less relevant for a realistic tokamak plasma. Yet, we have considered in our simulations a wide parametric region $K_\star \in (0,100)$ in order to offer a complete, quantitative, representation of the effects found.

3.3. Microscopic analysis

The effect of non-Gaussianity on transport can be understood from a microscopic perspective, following how individual trajectories change, or how their statistics is modified.

One can start the analysis from the limiting case of frozen turbulence $\tau _c\to \infty$. While the trajectories remain unchanged (see § 3.1) the velocity is changed by a factor $A[\varphi (\boldsymbol {0})]$. The consequence is that the diffusion across that particular equipotential line becomes $d(t;\varphi (\boldsymbol {0})) = A[\varphi (\boldsymbol {0})]d_0(A[\varphi (\boldsymbol {0})]t;\varphi (\boldsymbol {0}))$. This is equivalent to a change of the trajectory's period by a factor $A^{-1}[\varphi (\boldsymbol {0})]$. The statistical effects can be seen in figure 10 where the PDF of the periods $P(T)$ is plotted at different $\beta$ values with $\alpha = 0$.

Figure 10.

PDF $P(T)$ of the trajectory periods at different $\beta$ values in frozen turbulence $\tau _c\to \infty$ and no poloidal velocity $V_p = 0$.

One can notice that the intermittency lowers the general values of the periods. This is a natural consequence of the fact that $A[\varphi (0)]>1$. The effect is more pronounced at small and intermediate values $T\sim \tau _{{\rm fl}}$ since the low-frequency trajectories, i.e. $T\gg \tau _{{\rm fl}}$, are those with low values of the potential $\varphi (\boldsymbol {0})\sim 0$. For the latter, the factor $A[\varphi (\boldsymbol {0})]\sim 1$.

Although at small times $t\ll \tau _{{\rm fl}}$ all trajectories contribute to the diffusion, their average is only slightly dependent on $\beta$. The effect becomes more pronounced at times of the order of the time of flight. The fact that the distributions of slow trajectories $T\gg \tau _{{\rm fl}}$ are almost unchanged explains why the asymptotic behaviour of diffusion in the nonlinear regime $K_\star \gg 1$ is universal: the scaling law $D\sim K^{1-\gamma }$ is invariant to non-Gaussianity at $K_\star \gg 1$.

On the other hand, the distribution of potentials is changed. Thus, the effect of non-Gaussianity on diffusion results both from a change in the weight of each equipotential line and from the distortion of time periods.

Furthermore, we underline that, if the turbulence is frozen, both the distribution of Lagrangian potentials and that of Lagrangian velocities (Lumley's theorem, due to the divergenceless property of the Eulerian velocity field) are invariant in time. Under these two strong constrains, it is clear that the only microscopic effect of non-Gaussianity is the redistribution of trajectory frequencies. Other dynamical phenomena which might affect the transport are not present.

Finally, we note how the distribution of small-valued potentials $\phi \ll \varPhi$ is virtually unchanged due to the shape of $f$ (the nonlinear mapping between $\varphi$ and $\phi$). These small values are linked to long low-frequency trajectories, thus, to the behaviour of diffusion in the nonlinear regime $K_\star \gg 1, \tau _c\gg \tau _{{\rm fl}}$. This explains why the scaling behaviour ($\gamma$) is unchanged by intermittency.