# Published Data

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### Publication Figures

Publication Date:

2016-09-30

First Author:

Michael Fox

Title:

Symmetry breaking in MAST plasma turbulence due to toroidal flow shear

Paper Identifier:

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Figure Reference | Title | Description | Number of Figure Data Items | Identifier | Download Figure Details | ||
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Figure 1 | Cubic-spline-fitted equilibrium profiles for the BLM, DLM and IFS cases | Cubic-spline-fitted equilibrium profiles for the BLM, DLM and IFS cases (described in Section 2.2): (a) electron density, measured using Thomson scattering (TS) [32, 33], (b) ion temperature, measured using Charge-Exchange Recombination Spectroscopy (CXRS) [34], (c) electron temperature (from TS), (d) toroidal velocity (from CXRS), and (e) q-profile, reconstructed by EFIT constrained by the motional-Stark-effect (MSE) [35] measurements. The vertical solid lines indicate, for each of the three cases, the centre of the BES viewing location for the subarray being used (the width of the subarray is 10 cm). The vertical dashed lines indicate the position of the magnetic axis in each case. The black curve in the inset to (d) is the cubic-spline fit used for the calculation of (9). | 0 | CF/16/268 | Download | ||

Table 1 | Local equilibrium parameters | Local equilibrium parameters for each of the experimental cases analysed (Section 2.2). The BLM and DLM cases are the main comparison cases discussed in Sections 2.3 and 2.4; the IFS case is first introduced in Section 2.3.5. The EGK case gives the parameters for the gyrokinetic simulations analysed in Section 3 and is also discussed there. Profiles of ne, Ti, Te, v?, and q are plotted in Figure 1 for the BLM, DLM and IFS cases. The numerical values for these three cases are average quantities over the width of the BES subarray, whilst the values for the EGK case are taken at the centre of the BES array (those were the values used in simulations). | 0 | CF/16/269 | Download | ||

Figure 2 | Snapshots of the raw BES fluctuating-intensity signal | Snapshots of the raw (but 2D-spline-interpolated) BES fluctuating-intensity signal ?Ii/?Ii? from shot #28155 at: (a) t = 0.1409345 s, (b) t = 0.1410565 s, (c) t = 0.3696070 s, (d) t = 0.3692290 s. Times (a) and (b) occur during the BLM time period (significant flow shear) and times (c) and (d) occur during the DLM time period (no flow shear). The correlation functions of these two cases are given in Figure 3, which include these snapshots in the temporal average. | 0 | CF/16/270 | Download | ||

Figure 3 | Spatial two-point correlation function BLM and DLM cases | Spatial two-point correlation function (13) of the fluctuating density field for (a) the case before the locked mode (BLM, with flow shear) and (b) the case during the locked mode (DLM, no flow shear), both described in Section 2.2. Table 2 shows the parameters of these correlation functions calculated by fitting (12) to the spatial correlation function (11), correcting for PSF effects, and transforming into the (x, y) coordinates perpendicular to the magnetic field. These correlation functions should be compared to the correlation functions of the numerically simulated turbulence in Figure 8. | 0 | CF/16/271 | Download | ||

Table 2 | Correlation parameters for BLM, DLM, and IFS cases | Correlation parameters for BLM, DLM, and IFS cases, all described in Section 2.2. The correlation functions (13) with these parameters are shown in Figures 3 and 4. All lengths are normalised by the ion gyroradius ?i = vthi/?i (vthi is the ion thermal speed, ?i the ion cyclotron frequency), all times to a/vthi; the values of both of these can be found in Table 1. The rms amplitude ?n/n of the fluctuating density field is calculated from the rms amplitude ?I/I of the fluctuating intensity field, as defined by (16), using the method described in Section 7.1 of [28]. The skewness is calculated using (17) for the distribution of ?I. The life time ?life is defined in (14); the correlation time ?c was calculated using the CCTD method [38, 39]. | 0 | CF/16/272 | Download | ||

Figure 4 | Spatial two-point correlation function for the IFS case | Same as Figure 3, but for the IFS case, whose normalised flow shear (9) is ? 40% smaller than in the BLM case—the tilt of the correlation function is also smaller (see Table 2). | 0 | CF/16/273 | Download | ||

Figure 5 | Distribution of the fluctuating intensity field for BLM and DLM cases | Distribution of the fluctuating intensity field for (a) the case before the locked mode (BLM, with flow shear) and (b) the case during the locked mode (DLM, no flow shear), both described in Section 2.2. For each detector channel, ?Ii is normalised by its own root-mean-square (standard deviation) value (15). The black-dashed line in each case gives the unit normal distribution. The distribution function and its skewness for the DLM case (b) were calculated from 2 (rather than 5, as in all other cases) sets of 4 poloidal BES channels, located at R = 1.33 m and R = 1.35 m, where the toroidal rotation profile is completely flat, as seen in Figure 1(d). | 0 | CF/16/274 | Download | ||

Figure 6 | Distribution of the fluctuating intensity field for IFS case | Same as Figure 5, but for the IFS case (lower flow shear than for BLM; see Table 2). | 0 | CF/16/275 | Download | ||

Figure 7 | Conditional correlation function BLM case | Same correlation function as in Figure 3(a), for the BLM case, but this time conditioned on (a) lower intensities (core of the distribution, |?Ii/?Istd| < 2.75) and (b) higher intensities (tail of the distribution, max ?Ii/?Istd > 3), in the way described in Section 2.4.2. The parameters of the i fitting function (13) in each case are given in Table 3. The tail is manifestly less tilted than the core and also has a larger radial correlation length. These correlation functions should be compared to the ones for numerically simulated turbulence in Figure 13. | 0 | CF/16/276 | Download | ||

Table 3 | Conditional correlation function parameters for BLM and IFS cases | 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 ?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]). | 0 | CF/16/277 | Download | ||

Table 4 | Flow shear scan table | Various statistical characteristics of turbulence in a series of simulations corresponding to the equilibrium parameters of the EGK case (Table 1, except a/LTi = 4.8 in these runs) and a sequence of values of flow shear ?ˆE. The Marginal case is the EGK case itself. The rms fluctuation amplitude ?n/n is calculated using (16), but replacing ?Ii(t) with ?ni(t); the summation is over all grid points i. Similarly, the skewness is calculated for the fluctuating density field using (17). The spatial correlation functions for the GKa5, GKa3, GKa1, and Marginal cases are plotted in Figure 8. The fluctuating-density distributions for the GKa5, GKa1, and Marginal runs (the latter two skewed) are shown in Figure 12. The conditional correlation functions (CORE and TAIL) are discussed in Section 3.2.2 and plotted in Figures 13 (run GKa1) and 14 (Marginal run). | 0 | CF/16/278 | Download | ||

Figure 8 | Spatial correlation functions from GS2 flow shear scan | Directly calculated spatial correlation functions of the fluctuating density field for 4 of the runs documented in Table 4, with, from (a) to (d), value of the flow shear decreasing from the experimental (EGK) value to ?ˆE = 0. The spatial domain of the simulation is regularly gridded, therefore multiple values of the correlation function (11) (with ?ni instead of ?Ii, i being the grid point) occur for each (?r,?Z) pair; these values are then averaged to produce the spatial correlation function that is plotted (it is also averaged over time, typically several hundred ?s, i.e., tens of correlation times). These correlation functions are to be compared with experimental correlation functions with and without flow shear in Figure 3. The correlation parameters for the fitting function (13), approximating the true correlation functions plotted here, are given in Table 4. The red contours in these plots correspond to the fitting function (13) with these parameters, showing the quality of the fit (and thus supporting its use for experimental data; see [28] for further discussion). | 0 | CF/16/279 | Download | ||

Figure 9 | Lifetime vs. Flow shear plots | (a) Life time of perturbations, defined by (14), ?life = | tan ?/?E |, and normalised by a/vthi, vs. flow shear ?ˆE for a number of values of the ion-temperature gradient a/LTi (distinguished by different shapes of the data points); the data points are coloured according to the value of the gyro-Bohm-normalised heat flux Qi/QgB (see discussion in Section 3.3). (b) Same, but here ?life is plotted vs. Qi/QgB with data points coloured according to the value of a/LTi (this plot includes data from the full set of simulations carried out in [18] and so covers a larger number of values of ITG than (a)); the vertical dashed line indicates the experimental value of Qi/QgB for the EGK case (Table 1). | 0 | CF/16/280 | Download | ||

Figure 10 | Perturbed density field w and w/o filtering | Perturbed density field from run GKa5 (?ˆE = 0) integrated over y shown vs. x and time (a) without subtracting moving time average and (b) with moving time average subtracted (as for all fluctuating density fields used in Section 3). The presence of a zonal density component is manifest. The red boxes show regions for which conditional correlation functions shown in Figure 11 and discussed in Section 3.1.2 were calculated. | 0 | CF/16/281 | Download | ||

Figure 11 | Spatial correlation function on zonal density perturbations | Same correlation function as Figure 8(d), for run GKa5 (?ˆE = 0), but calculated for two spatial subdomains: (a) with positive zonal density and (b) with negative zonal density, as discussed in Section 3.1.2. and seen in Figure 10. | 0 | CF/16/282 | Download | ||

Figure 12 | Distribution of fluctuating density field from GS2 | Distribution of the fluctuating density field ?n, normalised by its own standard deviation, for (a) GKa5 (no flow shear), (b) GKa1 (with flow shear, close but not at the nonlinear stability threshold), (c) Marginal (at the threshold) runs (see Table 4). The black-dashed line in each case gives the unit normal distribution. These distributions should be compared with experimental ones with and without flow shear in Figure 5. | 0 | CF/16/283 | Download | ||

Figure 13 | Conditional correlation function GKa1 | Same correlation function as Figure 8(b), for run GKa1, but this time conditioned on (a) lower densities (core of the distribution, |?n| < 2.75 standard deviations) and (b) higher densities (tail of the distribution, max ?n > 3 standard deviations) in the way described in Section 3.2.2. The correlation parameters for each case are given in Table 4. These correlation functions can be compared to Figure 7 for the experimental BLM case. | 0 | CF/16/284 | Download | ||

Figure 14 | Conditional correlation function for Marginal case | Same as Figure 13, but for the Marginal run (see Table 4 and Figure 8(a)). The bottom row has correlation functions calculated in the standard way from the high-pass-filtered fluctuating density field (as explained at the beginning of Section 3.1); the top row has correlation functions of an unfiltered field, i.e., including the zonal density perturbation. The latter manifestly dominates the core population but not the tail. | 0 | CF/16/285 | Download | ||

Table 5 | Scan in ion temperature gradient | Same as Table 4, but for ?ˆE = 0.16 and a sequence of values of the ion-temperature gradient a/LTi . | 0 | CF/16/286 | Download | ||

Figure 15 | Skewness and Tilt vs. distance from marginal boundary | (a) Tilt angle of the correlation function and (b) skewness of the distribution of the fluctuating density field vs. flow shear ?ˆE and ITG a/LTi in a range of values of these parameters around the nonlinear stability threshold. The tilt and the skewness are replotted in (c) and (d), respectively, vs. the gyro-Bohm-normalised ion heat flux, with data points coloured according to the value of ?ˆE . These plots contain data for ?ˆE ? [0, 0.19] and ITG a/LTi ? [4.3, 8], covering the entire database of the runs carried out in [18]. In (d), inverted triangles mark the cases where skewness is negative. | 0 | CF/16/287 | Download | ||

Figure 16 | Accounting for alternative sources of skewness | Probability distribution of the intensity perturbation ?Ii (unnormalised) for (a) the DLM case (see Section 2.2) for 2 × 4 BES channels, as in in Figure 5(b); (b) the same as (a) but spike-filtered; (c) the same as (a) but spike- and then bandpass-filtered in the range 20 ? 100 kHz; (d) the detected signal of the BES diagnostic when the shutter is closed, measured during shot #27367 at t ? [250, 252] ms, using the 5 × 4 channels inner BES subarray; (e) the same as (d) but spike-filtered; (f) same as (d) but spike- and then bandpass-filtered in the range 20 ? 100 kHz. Note the different x-axis scales in (a-c) and (d-f). | 0 | CF/16/288 | Download | ||

Figure 17 | PSF effects on skewness | Effect of PSFs on the distribution of amplitudes from the GKa5, GKa1, and Marginal simulations. The intensity fields (d-l) are calculated from the density fields (a-c) by evaluating (18). The dashed line in each plot is the normal distribution. | 0 | CF/16/289 | Download | ||

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