Symmetry breaking in MAST plasma turbulence due to toroidal flow shear

Before discussing statistical properties of turbulence, let us inspect the instantaneous spatial structure of the fluctuating density field during the time windows being studied. In figure 2, example snapshots are plotted of the BES-measured intensity amplitude at certain chosen times. Each channel has been normalised by its own mean, so these are essentially snapshots of the relative density perturbation $\delta n/n$.

While we know (and it will also become very obvious in section 2.3.2) that the BLM case (figures 2(a) and (b)) has flow shear and the DLM case (figures 2(c) and (d)) does not, it is not easy (indeed, well-nigh impossible) to deduce that by inspection of such snapshots or by watching the full time-evolution sequences for these two cases. Indeed, comparing figures 2(a) and (c), we see that similar untilted (and so seemigly unsheared) structures occur in both the BLM and the DLM cases; conversely, comparing figures 2(b) and (d), we also see that similar tilted structures also occur in both cases¹⁰ . Therefore, the morphology of particular realisations of the turbulence does not reveal in any especially glaring way that the turbulence is different between the two cases. Statistically, however, it is very different, as we are about to discover.

We now chracterise the average spatial structure of turbulence using the spatial correlation function of the BES-measured intensity signal. Denoting by I_i(t) the time-dependent intensity measured by the BES detector channel i, we split this signal into mean (averaged over time) and fluctuating parts,

$$\begin{eqnarray}&&{I}_{i}(t)=\langle {I}_{i}\rangle +\delta {I}_{i}(t).\end{eqnarray} \tag{ 10 }$$

The two-point correlation matrix of the fluctuating intensity field is then

$$\begin{eqnarray}&&{C}_{{ij}}=\displaystyle \frac{\langle \delta {I}_{i}({r}_{i},{Z}_{i})\delta {I}_{j}({r}_{j},{Z}_{j})\rangle }{\sqrt{\langle \delta {I}_{i}^{2}({r}_{i},{Z}_{i})\rangle \langle \delta {I}_{j}^{2}({r}_{j},{Z}_{j})\rangle }},\end{eqnarray} \tag{ 11 }$$

where r_i, and Z_i are the radial and poloidal (vertical) locations of the detector channel i, respectively, and the angle brackets again denote time averages (the auto-correlation functions for each detector, calculated for i = j, are corrected for photon-noise effects as described in [11] and at the beginning of section 2.4). Assuming the turbulence is (approximately) spatially homogeneous across the BES subarray used for our analysis (see section 2.1), its two-point correlation function depends only on the relative distances between channels ${\rm{\Delta }}r={r}_{i}-{r}_{j}$ and ${\rm{\Delta }}Z={Z}_{i}-{Z}_{j}$. We reconstruct this correlation function from the correlation matrix (11) by considering the relevant ranges of values of ${\rm{\Delta }}r$ and ${\rm{\Delta }}Z$ and using the following fitting function

$$\begin{eqnarray}C({\rm{\Delta }}r,{\rm{\Delta }}Z) & = & p+(1-p)\exp \left[-\left(\displaystyle \frac{{\rm{\Delta }}{r}^{2}}{{{\ell }}_{r}^{2}}+\displaystyle \frac{{\rm{\Delta }}{Z}^{2}}{{{\ell }}_{Z}^{2}}\right)\right]\\ & & \times \cos [{k}_{r}{\rm{\Delta }}r+{k}_{Z}{\rm{\Delta }}Z].\end{eqnarray} \tag{ 12 }$$

The parameter p is used to account for global modes [11], as well as for fluctuations in the neutral beam [37]. As we have selected times during which there is no MHD activity (see section 2.1), the value of p should be small. Indeed, in all measured cases it is at most a few percent, therefore also indicating that neutral-beam fluctuations do not affect the measurements.

The fit (12) allows us to extract the spatial correlation parameters of the turbulence: the radial, ${{\ell }}_{r}$, and poloidal, ${{\ell }}_{Z}$, correlation lengths and the radial, ${k}_{r}$, and poloidal, ${k}_{Z}$, wave numbers. Operationally, in order to constrain the fit (12), we fix the product ${k}_{Z}{{\ell }}_{Z}$, which is determined from the two-time, single-point correlation function of the intensity field (see section 2.4 and appendix A of [28]). To obtain correct values of the spatial correlation parameters of the true density field, one must take account of the finite spatial resolution of the BES system, which is quantified in terms of the point-spread functions (PSFs) of its detector channels. How to do this systematically is worked out in [28]; here ${{\ell }}_{r}$, ${{\ell }}_{Z}$, ${k}_{r}$, and ${k}_{Z}$ are all corrected for PSF effects using this technique.

Physically, we would like to work with the correlation function in terms of point separations in the plane (x, y) perpendicular to the mean magnetic field, where x is the coordinate perpendicular to the flux surface (so, in the outboard midplane of the tokamak, the same as the radial coordinate) and y is the 'binormal' coordinate, perpendicular both to the magnetic field and to the x direction. Assuming that the correlation length of the fluctuating density field along the magnetic field is long compared to ${{\ell }}_{Z}$, we then convert the correlation function (12) to

$$\begin{eqnarray}C({\rm{\Delta }}x,{\rm{\Delta }}y) & = & \exp \left[-\left(\displaystyle \frac{{\rm{\Delta }}{x}^{2}}{{{\ell }}_{x}^{2}}+\displaystyle \frac{{\rm{\Delta }}{y}^{2}}{{{\ell }}_{y}^{2}}\right)\right]\\ & & \times \cos [{k}_{x}{\rm{\Delta }}x+{k}_{y}{\rm{\Delta }}y],\end{eqnarray} \tag{ 13 }$$

where ${{\ell }}_{x}={{\ell }}_{r}$, ${{\ell }}_{y}={{\ell }}_{Z}\cos \alpha$, ${k}_{x}={k}_{r}$, ${k}_{y}={k}_{Z}/\cos \alpha$, and $\alpha =\arcsin ({B}_{{\rm{p}}}/B)$. These correlation lengths and wave numbers are given in table 2. The spatial correlation function (13) is plotted in figures 3(a) and (b) for the BLM and DLM cases, respectively.

The difference between the BLM case (with flow shear) and the DLM case (without flow shear) is manifest: the BLM correlation function is highly tilted whereas the DLM one is very nearly even in ${\rm{\Delta }}x$. The tilt can be defined in terms of the tilt angle (7), where we now use the ${k}_{x}$ and ${k}_{y}$ parameters of the fitting function (13). The values of Θ are given in table 2. Thus, in line with the theoretical discussion in section 1.2, the statistical symmetry of the plasma turbulence under radial reflection is broken in the presence of flow shear.

As promised in section 1, we may calculate the effective 'life time' of a sheared structure, defined, in view of (6), by

$$\begin{eqnarray}&&{\tau }_{\mathrm{life}}\equiv \left|\displaystyle \frac{{k}_{x}}{{\gamma }_{E}{k}_{y}}\right|=\left|\displaystyle \frac{\tan {\rm{\Theta }}}{{\gamma }_{E}}\right|.\end{eqnarray} \tag{ 14 }$$

The values of this quantity for our experimental cases are given in table 2. We also give there the correlation time ${\tau }_{{\rm{c}}}$ of the fluctuating density field determined from its time-correlation function using the cross-correlation time delay (CCTD) method [38, 39]. We see that the values of ${\tau }_{\mathrm{life}}$ and ${\tau }_{{\rm{c}}}$, while not wildly disparate, cannot really be declared to agree. We are going to be relaxed about this problem: first, because both the CXRS value of ${\gamma }_{E}$ and the CCTD value of ${\tau }_{{\rm{c}}}$ are subject to large errors (regarding the reliability of the CCTD method in the presence of non-negligible PSFs, see section 9 of [28]), and secondly, because (6) is obviously a relation allowing an arbitrary constant of order unity. However, if the measurement of ${\gamma }_{E}$ is viewed as reliable, the life time defined by (14) can be meanigfully used as an effective measure of the (Lagrangian¹¹ ) correlation time for comparative studies of turbulence at different equilibrium-parameter values, as it will be in sections 3.1.1 and 3.3.1.

Correlation functions of MAST turbulence generically exhibit a tilt when flow shear is present, with a range of values of Θ—we have checked this in a number of experimental cases, not shown here. This is a reliable feature, to the extent that it could be thought of as a way to find out whether a flow shear is present¹² . Given that the values of ${\gamma }_{E}$ might not be computed very precisely from CXRS measurements of the rotation profiles, one might wonder whether in fact a more reliable (and dynamically relevant) measure of the local flow shear would be obtained from the tilt angle of the spatial correlation function and the correlation time ${\tau }_{{\rm{c}}}$ computed from the time-correlation function, viz, ${\gamma }_{E}^{(\mathrm{eff})}=\tan {\rm{\Theta }}/{\tau }_{{\rm{c}}}$ (ignoring here the difference between Lagrangian and Eulerian correlation times; note that ${\gamma }_{E}^{(\mathrm{eff})}$ could also be the local shear associated with zonal flows, as will be discussed in section 3.1.2).

The measurement uncertainties discussed above and a relatively limited range of values of ${\gamma }_{E}$ for which well-resolved BES data is available¹³ preclude us from being able to provide here a straightforward Θ versus ${\gamma }_{E}$ parameter study. Generally speaking, larger ${\gamma }_{E}$ will give larger Θ, but the spread is large (in section 3.3, we will argue that this is because Θ is a function not just of ${\gamma }_{E}$ but also of how far from the stability threshold the system is). The case of relatively low, but measurable, values of shear is perhaps an interesting benchmark as in this case we expect the tilt to be small. We have found such a case, case IFS, described in section 2.2 and documented in tables 1 and 2. This has normalised flow shear (9) that is $\sim 40 \%$ smaller than that in the BLM case (with its other equilibrium parameters not differing in any particularly remarkable way from the BLM ones).

The correlation function for the IFS case is shown in figure 4. Its tilt is in between that of the BLM and DLM cases and its ${\tau }_{\mathrm{life}}$ is shorter than for the stronger-sheared BLM case (see table 2). Clearly, two points a parameter scan do not make, but this is the best one can do with current data. The salient message is that the transition from stronger flow shear (BLM) to no flow shear (DLM) is gradual in parameter space, with the tilt angle passing through a range of values—and a more comprehensive parameter study ought to be on experimentalists' agenda. The most interesting outcome from such a study would be the dependence of ${\tau }_{\mathrm{life}}$ on ${\gamma }_{E}$, telling us how the dynamical properties of turbulence change with shear (see section 3.3.1).

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In figure 5, we show the probability distribution of the normalised fluctuating intensities $\delta {I}_{i}/\delta {I}_{i}^{\mathrm{std}}$ for the BLM and DLM cases. In the case without flow shear (DLM), the distribution is even with respect to positive and negative $\delta I$ (and very nearly normal). In contrast, the case with shear (BLM) exhibits a relatively small, but measurable preponderance of positive $\delta I$, i.e., overdensities are statistically somewhat more common than underdensities. This can be quantified as a positive value of the skewness of the distribution¹⁴

$$\begin{eqnarray}&&S=\displaystyle \frac{1}{N}\displaystyle \sum _{i}^{N}\displaystyle \frac{\langle \delta {I}_{i}^{3}(t)\rangle }{{(\delta {I}_{i}^{\mathrm{std}})}^{3}}.\end{eqnarray} \tag{ 17 }$$

For the BLM case, $S=0.21;$ for the DLM case, S = 0.01, but this is similar to the skewness in the background emission signal, as well as to the size of the error in the determination of the skewness that occurs due to PSF effects (see appendix A.2), so cannot be considered significant.

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In figure 6, we show the fluctuating-intensity distribution for the IFS case, with flow shear lower than BLM, already discussed in the context of its lower tilt in section 2.3.5. We see that the distribution is again skewed, with its skewness lower than for the BLM case (but still measurable), just as its tilt was (to identify a quantitative dependence between the skewness and flow shear, we will need to resort to simulations, in section 3.2, as we do not have a sufficient range of data for a proper parameter scan). What is more interesting is that the sign of the skew is negative, i.e., here, underdensities are privileged compared to overdensities. This emphasises the fact that, while the presence of a flow shear allows such asymmetric distributions, theory (at least in its current state) does not predict the sign of the asymmetry—we do not know how the sign of the skew depends on local equilibrium parameters.

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Finally, we must stress a very strong caveat. The fact that the symmetry (1) is formally broken by flow shear does not necessarily imply that a skewed distribution should result, it merely allows it (in contrast to the case of zero flow shear, when a skewed distribution would clash with theory [12, 13]). Here we have demonstrated, as a proof of principle, that skewed distributions can indeed be found experimentally when flow shear is present. However, skewness is not as robust nor as inevitable an effect of flow shear as tilted correlation functions appear to be and we have found many experimental cases in which flow shear is present according to CXRS measurements but the skewness of the fluctuating-intensity distribution is so small that, in view of measurement uncertainties, it cannot be declared present beyond reasonable doubt. In section 3.3, we will see, by analysing numerical simulations, that how skewed the distribution is depends quite strongly on how far from the nonlinear stability threshold the system finds itself, with only near-marginal cases showing significant skewness. Thus, while a tilt (possibly small) in the correlation function might be viewed as a signature of the presence of flow shear (as argued in section 2.3.4), a noticeable skewness of the fluctuating-density distribution should perhaps be viewed as a signature of a sheared turbulent state being close to threshold. The physical reason for this is probably that, close to the threshold, the turbulent state in MAST is dominated (as shown in [18] and discussed in section 3.2.1) by a finite number of long-lived, finite-amplitude coherent structures, whereas farther from the threshold, a more (evenly) distributed sea of fluctuations emerges.

The above discussion and the greater robustness of the tilted correlation functions than of the skewed distributions as signatures of flow shear suggest that these features describe the response to the flow shear of different types of turbulent fluctuations. Indeed, intuitively, as perturbations are sheared by the flow, their radial wave number increases according to (5) and so then does ${k}_{\perp }{\rho }_{i}={({k}_{x}^{2}+{k}_{y}^{2})}^{1/2}{\rho }_{i}$. This will push them into a damped region of wave-number space (see, e.g., [20]), quickening their demise. It is these perturbations, which are on their way out, that would statistically contribute to the tilt in the correlation function. In contrast, one might imagine that rare but particularly strong perturbations whose presence skews the distribution of the fluctuating field, might be less vulnerable to (or simply not yet affected by) such destruction by shear and so would turn out to be less tilted. This view is perhaps reinforced by the observation, which we will be in a position to make when we consider the numerically simulated turbulence in section 3.2, that the rms fluctuation level $\overline{\delta n/n}$ is significantly lower in the turbulence with flow shear than without it and so the tail of the distribution represents not some additional population appearing alongside the 'core turbulence' but rather the 'survivors'—sufficiently large structures that can create sufficiently large local velocity gradients to overcome the suppressing effect of the shear; these large structures would diffuse and disappear rapidly in the strong unsheared turbulence.

These speculations can be experimentally tested, if in a rather tentative manner, by constructing a correlation function conditional on the fluctuation amplitude. Indeed, the turbulent density field is continuously (and quickly) advected by the flow past our, relatively small, BES subarray, which can thus be thought of as sampling different types of turbulent structures at different times, rather than capturing a fully statistically representative spatial domain containing all types of structures at all times. Thus, we collect all snapshots of the turbulent field (such as those discussed in section 2.3.1) in which the fluctuating intensities at all detector channels are below a certain threshold, specifically, $| \delta {I}_{i}/\delta {I}_{i}^{\mathrm{std}}| \lt 2.75$, and calculate the spatial correlation function of this collection, representing the 'core' of the distribution, in the way described in section 2.3.2. We repeat the same procedure for a complementary set of snapshots, in which at least one detector channel sees $\delta {I}_{i}/\delta {I}_{i}^{\mathrm{std}}\gt 3$, to extract the correlation parameters of the 'tail' of the distribution. This analysis is necessarily crude, because the reduced number of snapshots (especially for the tail) reduces the quality of the statistics and the size of the BES window may have an effect that we cannot study systematically. Proceeding with it nevertheless, we find that the tilt angle for the core of the distribution in the BLM case is noticeably larger than in the tail (and also that the radial correlation length ${{\ell }}_{x}$ in the tail is much larger than in the core; see table 3). For the DLM case, there is no significant difference between the tail and the core, the tilt angle is essentially zero for both. This result appears to support the line of reasoning proposed above, although obviously a much more extensive study of a large number of well-resolved cases with good statistics is called for.

Table 3.  Correlation parameters of the fitting function (13) for the BLM and IFS core and tail conditional correlation functions (for the BLM case, they are shown in figure 7). The errors given here are for the fitting procedure only. The parameters for the overall correlation functions can be found in table 2. Note that the number of snapshots does not represent the number of statistically independent instances: neighbouring snapshots are separated by $0.5\,\mu {\rm{s}}$, so it takes approximately 20 snapshots for a given structure to pass through the BES subarray (due to the apparent poloidal motion caused by toroidal rotation [39]).

Case	BLM core	BLM tail	IFS core	IFS tail
Condition	$| \delta {I}_{i}/\delta {I}_{i}^{\mathrm{std}}| \lt 2.75$	$\delta {I}_{i}/\delta {I}_{i}^{\mathrm{std}}\gt 3$	$| \delta {I}_{i}/\delta {I}_{i}^{\mathrm{std}}| \lt 2.75$	$\delta {I}_{i}/\delta {I}_{i}^{\mathrm{std}}\lt -3$
No. of snapshots	1434	123	3065	206
${{\ell }}_{x}/{\rho }_{i}$	1.9 ± 0.2	8.4 ± 0.3	2.9 ± 0.1	4.8 ± 0.3
${{\ell }}_{y}/{\rho }_{i}$	9.2 ± 0.1	9.2 ± 0.3	10.4 ± 0.1	11.0 ± 0.2
${k}_{x}{\rho }_{i}$	−1.00 ± 0.14	−0.36 ± 0.01	−0.18 ± 0.02	−0.35 ± 0.02
${k}_{y}{\rho }_{i}$	0.44 ± 0.01	0.43 ± 0.01	0.46 ± 0.1	0.43 ± 0.01
Tilt: Θ/deg	67 ± 3	40 ± 1	22 ± 3	39 ± 2

The need for such a study to make our argument fully convincing (or otherwise) is accentuated by the disconcerting failure of the IFS case to agree with the core versus tail distinction seen in the BLM case: in fact, we find that the tilt angle in the tail is larger than in the core (see table 3). We do not know why the IFS case has this behaviour: whether, for example, it might be related to the lower (dynamically unimportant?) flow shear, or to the preponderance of under-, rather than overdensities in the distribution of the fluctuating field, or to the presence of zonal flows¹⁵ . In section 3.2.2, we will see that the trend in numerical simulations appears to be broadly consistent with the BLM case.

Cognizant of all the uncertainties and of the clearly insufficient amount of experimental evidence on offer, we conclude this section with a cautious conjecture that turbulence under the action of flow shear consists of a tilted, sheared, statistically more ubiquitous, but smaller-amplitude component and less tilted, rarer, but stronger perturbations, which skew the distribution function of the fluctuating field. We shall test this idea further in the next section.

Quantitatively, we find that the tilt $\tan {\rm{\Theta }}={k}_{x}/{k}_{y}$ increases approximately linearly with ${\hat{\gamma }}_{E}$, except near the stability threshold. Recalling the discussion at the end of section 1.2 and in section 2.3.4, we conclude that, except near the stability threshold, the life time (14) of the turbulent perturbations is independent of the flow shear—presumably because dynamically the flow shear is not very important away from the threshold. This point is illustrated by figure 9(a). The data for other values of the ion temperature gradient also shown in this figure supports this conclusion and will be further discussed in section 3.3.1.

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As we saw in figure 8(d), when the mean flow shear is zero, the correlation function of the fluctuating density field is untilted, in agreement with the symmetry (1). We stress, however, that this is only true if the correlation function is calculated over a sufficiently large spatial domain. Because drift-wave turbulence is prone to developing zonal flows [43, 44], locally a flow shear can be present—and indeed is, in the simulations that we are analysing. As these simulations have kinetic electrons, they can, and do, develop a zonal density perturbation on ion scales, so the presence of a zonal field is detectable directly from the measured density field, in the form of a y-independent, long-lived component. Because zonal fields have a long life time, compared to the life time of the turbulence, they can only be seen if we remove the high-pass frequency filter effectively imposed on our fluctuating density field by subtracting the running time average (as stated at the beginning of section 3.1). This is illustrated in figure 10, where we show the y-integrated perturbed density field for the zero-flow-shear run GKa5 with and without this filter.

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Experimentally, the high-pass frequency filter may be difficult to abandon (see appendix A.1), however, the zonal shear can be detected indirectly via correlation functions of the fluctuating density field¹⁹ : if those are calculated over a spatial domain that is smaller (radially) than the wave length of the zonal flow, they will be tilted by the same mechanism as in the case of a mean flow shear. This is illustrated in figure 11, which shows correlation functions for the zero-flow-shear run GKa5 restricted to regions of positive and negative zonal density, as marked in figure 10—as is obvious from the tilts of the correlation functions, these regions turn out also to be regions of positive and negative zonal shear (in this context, we should perhaps recall the presence of tilted instantaneous structures in the zero-flow-shear DLM case, noted in section 2.3.1). The overall correlation function in figure 8(d) is an average over several such regions—the opposite tilts average out and the overall radial-reflection symmetry is preserved.

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Since turbulence with mean flow shear will also develop zonal flows, a similar analysis applied to it will show a certain spread in the tilt angle of the correlation function depending on whether the zonal shear locally subtracts from or adds to the mean flow shear. This can matter if the mean flow shear is sufficiently weak.

Besides encouraging the physical interpretation of the experimental evidence in terms of symmetry breaking associated with flow shear, numerical simulations give us a crucial insight as to how, dynamically, this symmetry breaking occurs. In [18], for which these simulations were originally carried out, it was shown that the transition to turbulence in these flow-sheared MAST configurations occurs via emergence and accumulation (as equilibrium parameters move deeper into the unstable regime) of long-lived, intense coherent structures, which occupy a small fraction of the volume. This is because turbulence in these equilibria is subcritical—while the system is formally linearly stable, perturbations do grow transiently and, for finite perturbations, this can lead to non-zero nonlinearly saturated turbulent fluctuation levels and heat fluxes. Since finite amplitudes are required for such a state to persist, the only way for the system to have small overall heat fluxes as the threshold is approached is by restricting finite amplitudes to a small fraction of the volume, hence the spatially sparse, but intense structures. Clearly, in terms of the probability distribution of the amplitudes, the presence of such structures and the fact that they give rise to a non-zero overall heat flux, implies skewed distributions. Away from the threshold, the structures become more numerous, overlap, interact, break up and dissolve into a more volume-filling turbulent sea of 'eddies,' characteristic of standard, unsheared drift-wave turbulence—resulting in approximately symmetric distributions.

Another important observation afforded us by the numerical simulations but unavailable experimentally (because equilibrium parameters in experiments cannot in general be changed independently) is that, other things being equal, increasing flow shear does have a suppressing effect on the rms amplitude of the turbulent density field: this is documented by the $\overline{\delta n/n}$ values in table 4 (it is not evident in figure 12, because these are distributions of $\delta n$ normalised to its own standard deviation). Thus, as we anticipated in section 2.4.2, the tail of the skewed distributions found for the near-threshold, sheared turbulence is, in a sense, a remnant of a wider unsheared/far-from-threshold distribution, containing large-amplitude perturbations that might have existed in a more unstable local equilibrium configuration (although, as argued above, these perturbations have a rather distinct character once only a dwindling number of them are left with a sole responsibility of maintaining turbulent transport; see also section 3.2.2).

In section 2.4.2, we conjectured that turbulence in the presence of flow shear is a two-component mix of strong but rare perturbations giving rise to the skew in the distribution and a wider sea of weaker ones, which constitute the core of the distribution and are tilted, more or less passively, by the flow shear. In other words, the tilt and the skew represent symmetry breaking by flow shear occurring in different types of perturbations.

To test this picture, we introduced conditional correlation functions, designed to measure the spatial correlation parameters associated with the core and the tail of the distribution. Here we perform the same analysis on the numerically simulated fluctuating density fields, with the same condition that frames containing at least one instance of $\delta n$ larger than 3 standard deviations are declared to belong to the tail (for the purposes of this selection, the full artificially large $40\,{\rm{c}}{\rm{m}}\times 80\,\mathrm{cm}$ GS2 computational domain is split into $7.48\,{\rm{c}}{\rm{m}}\times 9.19\,\mathrm{cm}$ subdomains—this is comparable to the $10\,{\rm{c}}{\rm{m}}\times 8\,\mathrm{cm}$ BES subarray that we used in section 2.4.2 and necessary in order for conditioning on the maximum value of $\delta n$ within a subdomain to pick out statistically different patches of turbulence). The results are documented in table 4 and conditional correlation functions for the core and tail of two of the runs with flow shear are shown in figures 13 (run GKa1) and 14 (Marginal run). We see that fluctuations in the tail of the distribution are always less tilted than those in the core. This is similar to the behaviour exhibited by the experimental BLM case in section 2.4.2.

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Another qualitative similarity between simulated and real sheared turbulence is that the perturbations in the tail of the distribution, in both cases, have longer radial correlation lengths ${{\ell }}_{x}$, not just lower tilt (i.e., smaller ${k}_{x}$). Table 3 shows this for both BLM and IFS cases (even despite the failure of the latter to follow the trend as far as its tilt angle is concerned) and table 4 confirms the same trend for our numerical runs with varying flow shear; it is also quite obvious from the plots of the conditional correlation functions in figures 13 and 14.

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Comparing these two figures, we note also a sizeable increase of the poloidal correlation length in the core of the distribution as the threshold is approached (see table 4). Interestingly, in the Marginal run, the core fluctuations are so weak, that the zonal density perturbation (in our definition of the fluctuating density field, it is removed by time averaging/high-pass frequency filtering; see section 3.1.2) becomes comparable to them—so much so that, as we see in figure 14, it dominates the conditional core correlation function (although not the total one or the tail) if the high-pass frequency filter is removed (suggesting what is perhaps a promising experimental direction of enquiry).

It is worth discussing a little further how the effective restoration of symmetry happens for the tilt of the correlation functions. We argued at the end of section 1.2 and again in section 3.1.1 that the flow shear did not need to be dynamically important in order to induce a tilt in the correlation function. In such a 'passive' tilting scenario, the tilt $\tan {\rm{\Theta }}={k}_{x}/{k}_{y}$ is simply proportional to ${\gamma }_{E}$, with the constant of proportionality being the life time ${\tau }_{\mathrm{life}}$ of the perturbations, as per (8) or (14). This life time, defined by (14) and given in table 5, becomes shorter as the ITG is increased. This is also manifest in figures 9(a) and (b)—the latter figure is a scatter plot of ${\tau }_{\mathrm{life}}$ for a large set of values of ${\hat{\gamma }}_{E}$ and $a/{L}_{{T}_{i}}$ against the gyro-Bohm-normalised ion heat flux ${Q}_{i}/{Q}_{\mathrm{gB}}$, which is a good measure of distance to threshold (it is zero at the threshold and increases steeply away from it [18]). The reason correlation times shorten is that larger ITG implies stronger drive and so higher turbulent amplitudes: 'eddies' turn over faster²² . Shorter ${\tau }_{\mathrm{life}}$ then implies smaller tilt angle ($\tan {\rm{\Theta }}\sim {\gamma }_{E}{\tau }_{\mathrm{life}}$) farther from marginality.

In physical terms, the flow shear ceases to be important if it is smaller than the local, fluctuating velocity shear acting on a turbulent perturbation and caused by its fellow perturbations. This nonlinear fluctuating shear must, in saturation, be comparable to some appropriate linear rate at which the perturbations are replenished by the ITG drive. In this sense, the competition between the life time of the turbulence and the flow shear as the threshold is approached is reminiscent of the so-called Waltz rule [48]: that in order to affect (suppress) the turbulence, the flow shear must be similar to, or greater than, the linear growth rate (we recall, however, that the turbulence that we are dealing with here is subcritical [18] and so formally there is no linear growth rate; what effective quantity from the linear theory should be used instead is a matter of current research interest [10, 20, 41]).

Figures 15(a) and (b) show the tilt angle of the correlation function and skewness of the distribution of the fluctuating density field for a range of values of the flow shear ${\hat{\gamma }}_{E}$ and ITG $a/{L}_{{T}_{i}}$ around the nonlinear stability threshold and departing from it—this extends the ${\hat{\gamma }}_{E}$ and $a/{L}_{{T}_{i}}$ scans documented in tables 4 and 5. The conclusion remains the same: away from the threshold, symmetry is effectively restored—particularly quickly for the distribution functions (as we argued in section 3.2.1, the restoration of symmetry of the distribution of fluctuating densities far from the threshold occurs because intense coherent structures that dominate the transition to turbulence and are responsible for the skewed distributions become too numerous for individual survival, interact and break each other apart [18]).

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As we already did in section 3.3.1 for the life time of the perturbations, we quantify the dependence of the measures of asymmetry (tilt and skewness) on the distance to threshold in terms of their dependence on the gyro-Bohm-normalised heat flux ${Q}_{i}/{Q}_{\mathrm{gB}}$, which vanishes at the threshold but increases strongly and monotonically away from it [18] and thus constitutes a good order parameter. Scatter plots of the tilt and the skewness versus ${Q}_{i}/{Q}_{\mathrm{gB}}$ are shown in figures 15(c) and (d), respectively, for the entire data base of numerical simulations originally carried out in [18]. We see that, modulo some scatter, both the tilt and the skewness are approximately functions of the distance to the threshold (to be precise, figure 9(b) suggests that it is the life time ${\tau }_{\mathrm{life}}=\tan {\rm{\Theta }}/{\gamma }_{E}$, rather than the tilt, that is a nearly single-valued function of this distance, at least when measured sufficiently far from the threshold). The case of ${\hat{\gamma }}_{E}=0$ is the exception to this rule, as expected theoretically, being subject to the exact symmetry (1): namely, at zero flow shear, both the tilt and the skew are essentially zero regardless of how marginal, or otherwise, the system is.