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Edge turbulence
77
Journal of Nuclear Materials 176 62 177 (1990) 77-88 North-Holland
Edge turbulence A.J. Wootton Fusion Research Center, University of Texas at Austin, Austin, TX 78712, USA
This review concentrates on tokamak results, summarizing some of the characteristics (e.g. amplitudes, frequencies, wavelengths) and comparing them with theoretical expectations. With ohmic heating the measured electrostatic fluctuations are shown to explain the total anomalous edge fluxes, but no useful scaling law yet exists. Magnetic turbulence may become important with additional heating (perhaps higher beta). Although an adequate theory to explain all the measurements does not exist, there is an understanding of some of the basic drives and damping mechanisms, and these are discussed. Finally, a teat particle approach to including the effects of the electrostatic turbulence induced transport in complicated geometry (e.g. with stochastic magnetic fidds) is presented.
1. Analysis techniques Plasma parameters fluctuate about their mean values, and these fluctuations can cause transport. Fig. 1 depicts how waves, driven unstable by the gradients of plasma parameters, can partly determine the gradients themselves. Processes such as radiation and charge exchange, as well as local sources (e.g. a gas puff) and sinks (a limiter) will affect the gradients; they may also affect the turbulence directly. One of the purposes of turbulence studies is to identify the free energy sources, and derive the transport matrix which relates the transport coefficients to the gradients themselves. For very
340.00
340.10 time (ms)
Fig. 2. Two time histories illustrating the random nature of the signals. The results show the density fluctuations obtained from a heavy ion beam probe on TEXT, with the two sample volumes separated poloidally by 2.3 cm, at a radial position r/a = 0.96.
Fig. 1. The connection between the free energy sources, turbulence and transport processes.
unstable modes there is a simpler situation; the condition for marginal stability should determine the gradients directly. Fig. 2 shows typical ~u~tuations signals obtained simultaneously with a heavy ion beam probe [l], data digitized at 1 MHz, from two sample volumes. The volumes (= 2 mm x 2 mm x 2 cm) are separated poloidally by &, = 2.3 cm; both are just inside the main plasma of TEXT [2] (r/u = 0.96, with a the limiter radius). The fluctuations in signal amplitude are caused by density fluctuations ii. Correlation analysis can be applied to this data, and fig. 3 shows the auto and cross correlation functions [3]. There is an apparent velocity
0022-3115/90/%03.50 Q 1990 - Elsevier Science Publishers B.V. (North-Holland)
78
A.J. Woorton / Edge rurhulence
v, = &JT, = 4.6 km SC’. Two processes are assumed to occur; a drift of random patterns past the sample volumes with a velocity v,, and changes to the patterns themselves with a velocity v,. LIP is a combination of both. In terms of the times indicated in fig. 2, v, = &/(27z) = 4.5 km SC’, and v, = ((0, - u,)v,)“* = 0. For this typical case, the correlation time of the turbulence in the drifting frame of reference is much longer (> 10 times) than that observed in the laboratory frame (- 3
-1
Ps). This model does not cover all the possible types of moving patterns which can occur [4]. For example, if the movement of the pattern is caused by some kind of wave progression, and if the waves involved are dispersive, the velocity of a given Fourier component will be a function of its wavelength. To test for this type of behavior the pattern is Fourier analyzed into sinusoidal components of different wavelength and a velocity calculated for each component. This is called dispersion analysis. The amplitude and phase of each Fourier component are found; it is assumed that this Fourier component is produced by the passage over the sample volumes of a sinusoidal wave of wavelength h and velocity vPh, such that up,, = fX. The wave normal k = 27r/h and the velocity are calculated from the phase difference associated with the records; this calculation is repeated for each frequency. Fig. 4 shows the results of applying this dispersion analysis to the data of fig. 2, to give the spectral power density function S(k, f), a n”*. A poloidal phase velocity v,,,, = 6 km SC’ is found. Most of the power is at f< 100 kHz, k I 1.5 cm-’ (i.e. poloidal mode number m = rk I 40 or X 2 4 cm). The spectrum is broadband with Af=f and Ak = k (i.e. turbulent). This method depends upon the assumption that a given frequency is produced by a single wave passing over the recording system. This need not be the case,
0
1
2
3
4
5
wavenumber (cm-l) Fig. 4. The power spectrum S(k, f) obtained from the data shown in fig. 2.
because waves with the same wavelength but different directions might be superimposed. The analysis will always give an apparent velocity, but if several waves are superimposed, the value will be meaningless [4]. The situation is considerably improved if the whole pattern can be observed, allowing a spatial Fourier analysis to be performed, rather than a temporal one. Two dimensional radial-poloidal arrays have been used [5,6] and show correlation lengths both poloidally and radially = 1 cm (I 1 wavelength). Toroidally separated sample volumes show that along the field lines the correlation distance is much longer, > vR [7]. Localized “blobs” of plasma are seen, which move irregularly. The lifetime and area distributions of these blobs are consistent with those expected from the measured conelation times and lengths. As such the blobs are consistent with the crests of an average ii spectrum, rather than being coherent structures [6]. This picture of randomly changing filamentary structures resembles those observed photographically in e.g. TFTR [8].
2. Phase velocity
auto-correlatm
p(0.r)
cross-correlation. p&r)
0
-1.0
1
I 10
6
I
1t
t
1
0
I
I
10
r, T2 delay time z (us) Fig. 3. The auto correlation and cross correlation obtained from the data shown in fig. 2.
functions
Both correlation and dispersion analysis provide a phase velocity vph, which helps identify the turbulence and provides information on the radial electric field E,. For example, some modes are expected to propagate in the electron drift direction (u,& while others should propagate in the ion drift direction ( udi). Because we make our observations in the laboratory frame we must account for Doppler shifts. With the ion momentum balance equation we can relate the mass velocities to E,: niqi [ Er + uoB+ - u+Be] - Vpi = 0,
(1)
A.J. Woofton / Edge
with PI;, qi, pi the ion density, charge, pressure, u the mass rotation velocities, subscript 8, rp referring to poloidal, toroidal. When we do this, we expect f9]: * l$h = vph
Er/B+,
(2)
with vph the measured (lab. frame) poloidal phase velocity and vp;l the phase velocity ignoring E,; for example *( e 1n,B+) for drift waves. Here %h - v,, = - Qn,/ar/( e is the electron charge. Therefore, we must measure E, = - a@,/& (@ is the plasma potential) before we can measuring r+ provides indetermine up;l: alternatively formation on E,. Fig. 5 shows Q, in TEXT, both for a normal discharge and for a discharge where a set of external coils has been energized with a current I, to perturb the edge magnetic field [10,11,12], creating a stochastic edge region, or ergodic magnetic limiter (EML), discussed later. Normally q~> 0, E, > 0 for r 2 a, and @ < 0, E,
I
300
/
c
200 _ VI c
100 0
9 v
I
,0"l,
+ -100 1 -200 -300
1 /’
t 1 0.6
d
//’
ii 0
/
L 1 .o
0.8
P Fig. 5. The radial dependence of plasma potential Cp, with (I, = 4 lc.4, solid symbols) and without (I, = 0, open symbols) externally applied perturbing magnetic fields (the ergodic magnetic limiter, or EML). Circles represent heavy ion beam probe data and triangles represent Langmuir probe data.
19
turbulence W,thout
5
electron
4
perturbrng
mognetlc
f&d
drift 1
(aI
xw
z!!___:; ion drift -2
0.6
i limiter 0.8
Wilh perturbrng 6
; I.2
I.0 r/o -+ magnetic
fmid
(It,= PkA)
/ (bt
-4
,; llmlter
1
0,8
1.0
1
I.2
r/o +
Fig. 6. The measured radial dependence of the fluctuation phase velocities up,, (a) without and (b) with externally applied perturbing magnetic fields (the EML, I,= 4 kA case). Also shown are the expectations assuming oph= u,,, - I&/B,, and the component u,, alone. Data from Langmuir probes (0). the heavy ion beam probe (x), FIR scattering (I) and an m = 2 tearing mode ( + ) are shown.
within one spatial correlation length might do the same thing. With a stochastic magnetic field we expect 1161 E, =j jr,(L,’ + 0.5L~e’) > 0, where I,,, LC are scale lengths of density and electron temperature. A positive E, is set up to restrain the electrons which try to escape along stochastic field lines to the limiter faster than the ions. Therefore, with I, = 4 kA in fig. 5 we expect and see the region with @ > 0 expanded; the width of the velocity shear layer is increased but the amplitude of the velocity shear is reduced. If ey. (2) is correct, this should modify the measured vPh. Fig. 6 shows the measured turbulence phase velocity uph for cases (a) without (Zh = 0) and (b) with (Zh = 4 kA) the EML and thus modified E, [11,12]. Data from different diagnostics shows that normally (fig. 6a) a@, changes direction from ion (v < 0) to electron (v > 0) drift at r = a, but with the modified E, (fig. 6b) this reversal is moved to smaller r. In both cases eq. (2) (solid line) approximately describes the results. The broken line shows u,,; generally the term E,/B,
80
A.J. Wootton / Edge turbulence 0.8
0.0 0.6
0.7
0.8
0.9
1.0
1.1
1.2
r/a
Fig. 7. The radial dependences of the normalized rms fluctuating amplitudes of density Z/n, potential 4/(/c&), temperature %/r, and magnetic field 6,/B+, in TEXT.
Boltzmann relationship is influenced by changing the impurity concentrations (discussed later). A simple mixing length model predicts that H/n should saturate when the local density profile is flattened (implying no further drive), i.e. when A/n = (k,L,)-‘. There is a strong trend in the direction expected [18], but with two discrepancies. First, the amplitude fi/n = 2(kL,)-‘, larger than expected (locally the gradient can reverse). Second, within one machine (e.g. TEXT) the scaling is not adhered to. A factor - 5 variation can occur, suggesting there are important processes not considered in the simple model. Table 1 shows that similar fluctuation amplitudes are found in stellarators [19] and reversed field pinches (RFP’s) [20] except for the much larger magnetic fluctuation level in RFP’s.
4. Wave numbers dominates, so that upi, = v, - veBB/B+ = ve •t v,k,/ke, i.e. the phase velocity in the laboratory frame gives us information on the mass rotation velocity directly.
3. Amplitudes
Fig. 7 [17] shows the rms values of C/n, G/k&), Fe/T, and &/Be for a particular discharge in TEXT as a function of r/a (kb is Boltzmann’s constant). The values shown are characteristic of all tokamak edges. Note that A/n # ;/(k,T,); this departure from the
Table 1 Edge turbulence parameters at r = a parameter
TEXT
ATF
ZT-40
cl(a) n (lo’* me3)
3 4
1
0.1
1
5
T. (ev)
30 2 5
20 3 5 0.05-0.1 0.1-0.2
30 1.5 8 0.3-0.5 0.4-0.6
L, (cm) + (cm) ii/n G/(k,T,) C/T,
0.1-0.2
s/B(o)
k (cm-‘) r Q Q, IQ,,
a)
0.15-0.3 0.05-0.1 10-S 3
0.1
0.2-0.5
10-6 1
0.01-0.02 0.1
t rr.E I q’.E
r rf.E I q’.E
r
Cl
(1
>1
z rf.E Q&b
Parameters which are significantly different are in bold type. ‘) Width of h spectrum is tabulated.
Typical wave numbers observed near the edge [17] k, 3 1 cm-’ (i.e. - ke) and are k B - 1 to 3 cm-‘, k,, - l/(qR), with q the safety factor. If the turbulence is drift wave like we expect keps = 1 (211, with ps the ion gyroadius using T,, so k, a B&eo.5. Alternatively if the magnetic geometry is defining the structure then we might expect k, to be given by the lowest (most unstable) value of m in a correlation volume around the rational surface where q = m/n. Some evidence for both properties exists. We know [18] k a Be. However, for TEXT k,p, = 0.05; i.e. the wavelengths are > 10 times longer than expected from drift wave theories. Also there is a very large variation in k, at a given B,, larger than any variation in Te-‘.‘. On the other hand, comparing tokamaks, stellarators and RFPs (table 1) suggests that k is dependent on q, i.e. the magnetic geometry is important. We have not uncovered all the important physics yet.
5. Particle confinement
and electrostatic turbulence
We categorize the fluctuations as either electrostatic (fluctuation density ii, potential 6 = -i/k, temperature f) and magnetic (fluctuating magnetic field 6). In reality both will exist simultaneously. The electrostatic fluctuations produce a fluctuating radial velocity fir = &/Be. Ignoring poloidal and toroidal asymmetries, and assuming w K wCi, the ion cyclotron frequency, the electrostatic fluctuation driven radial particle flux (denoted by superscript f, E) is [22] Ff,E = (&i)/B,.
(3)
A.J. Wootton / Edge turbulence
81
Subscripts r, 6, 4, I], I refer to radial, poloidal, toroidal, parallel and perpendicular to the magnetic field, and ( * . . ) denotes an ensemble average (an average in space and time). We can recast eq. (3) so that FfgE is given as a function of frequency 0/(2rr) over a spectral width dw in terms of the root mean square (rms) fluctuation levels due to the spectral components of width do centered at o [23]: rfvE( w) = ii rms(+&&G
lY,*(W) I Ma)
xsin(c,+(w))/B.+.
(4)
Now we see that both the coherence y,,+ and phase angle a,,+ between ii and I$ are important. It is not sufficient just to measure the fluctuating amplitudes. Ff,E has now been measured in many ohmically heated plasmas [17]. A comparison with the total flux, measured spectroscopically, is shown in fig. 8 (TEXT). A simple model for parallel flow to the limiter has been used to obtain the total perpendicular component F’ [24]. The approximate agreement demonstrates that the measured turbulent flux accounts for an important part, if not all, of the total particle flux in the edge. This agreement is found for a variety of Be, Ip (plasma current) and ii, [25]. A similar conclusion (that I’f*Ecan explain particle transport) is drawn for stellarators and RFPs (see table 1). Experimental results from ISX-B [26] with neutral beam heating, and DITE [27] and TEXT with electron cyclotron heating, show an increased FfsE explains the decreased particle confinement time with additional (L-mode) heating. No results are available with the H-mode [28], in which improved confinement with additional heating is found, but it seems reasonable to suppose that the improvement in
analysis of fluctuation driven rf,E in TEXT on local density n and toroidal field B+.
Fig. 9. A regression
particle confinement
is associated with a decrease in
rfsE. A useful result would be a scaling law (understood or otherwise!) for the flux FfVEor the diffusion coefficient D ‘SE = - I’f,E/vn; this could be used in predictive transport codes. Because we expect FfsE to be a complicated function of more than one gradient (e.g. vn, 8T, VU), it is better to seek a scaling of the flux directly, rather than Df*E. Fig. 9 shows the results of a regression analysis on TEXT data for r > a; FfsE a d3B;‘.‘, the significant deviations show not all the dependencies have been found. A different dependance is found for r < u, suggesting that the limiter itself partly determines the fluxes [29]. One method to demonstrate that FfsE is important is to show that it can predict the measured density e-folding length X, behind the limiter. To do this we recast the scaling as I’f*E= KTIB-‘.~ [29], which is almost as good (or bad) a fit as that shown in fig. 9. By balancing parallel and perpendicular fluxes in slab geometry [24], and ignoring sources, we have:
a/ar(rf,E) = KB;%~/ar = -PIU,,/L~,
‘.
04 . 1 t 0.6
0.9
b .
I
1 .o
1 .l
I 1.2
r/a Fig. 8. The radial dependence of the &asured fluctuation driven rf*E (solid line) and total r’ (broken line) particle fluxes (TEXT). Solid symbols represent Langmuir probe data; open symbols represent heavy ion beam probe data.
(5)
with u,, a parallel sound velocity (taken as 0.5 times the sound speed cs), L, a typical distance along a field line to a material surface, and K = 3.8 x lo4 (units T’.’ cm s-‘) the constant of proportionality. This predicts an exponential fall-off behind the limiter, with X, = XrxT = 2 KL,c;‘B; ‘.‘. Fig. 10 shows a comparison of the measured X, with the prediction XrxT for TEXT and other tokamaks (data taken from [30]). For TEXT (horizontal bars) the agreement is good, considering the may approximations used to derive eq. (5) [24]. However, to explain all the data base results ( X ) requires a factor between 0.1 and 10 to be applied to the scaled
A.J. Wootton / Edge turbulence
Velocity shear Where aE,/ar or shear layer, any ferential velocity
has been discussed in explaining up,,. au/& (- au,,,/&) is high, i.e. in the turbulent “blobs” experience a difAu = I,( au/ar - u/r) across a radial correlation length I,. This will tear the blob apart in a time TV= IO/Au, with I, ( = I,) the poloidal correlation length, and suppress the turbulence [34]. Such a mechanism explains the measured auto-correlation times in TEXT [35], and may explain the H-mode. 0
2
4
6
8
10
*m= 12KL, 1,-m (cs Bm’5 Fig. 10. The measured density e-folding length X, compared to that predicted (XTXT) based on TEXT scaling of rf.n. Results from a data base (x), and TEXT (horizontal bars represent the spread due to T, variations) are shown. ASDEX results (0) are connected by arrows, illustrating the change from ohmic heating (OH) to additional heating low mode (L) to H-mode
6. Energy confinement
and electrostatic turbulence
From rf,E we can derive the electrostatic fluctuation driven convected energy flux Q,, = 5/2k,T,Tf*E. The total electrostatic fluctuation driven energy flux is given by [22]: Qf.E = 3/2k,n(gof)/B,
+ 3/2k,T(&,ii)/B,.
(H).
from TEXT. This show again that not all the important parameters have been identified. Although this comparison (fig. 10) is poor, it is no worse than assuming a Bohm diffusion coefficient D,, which predicts A, = (2 D,L&‘)‘/* [24]. Again a factor between 0.1 and 10 must be applied to D, to explain all the data base results. However there is one situation where X is better than XTXT,and that is in predicting what ha;pens at the transitions from ohmic (OH) to Lto H-mode in tokamaks, marked in fig. 10 for ASDEX [28] (0) by arrows (OH to L; T, increases and X, increases; L to H: T, decreases and h, decreases). Because XrxT a Te-'-' it predicts exactly the opposite behavior, whereas X, a To.25 does show the general trend. Again not all the important parameters determining rf*E have been accounted for. Two important parameters not identified in the scaling for rf*E discussed above are the impurity concentration nr and the mass velocity shear au,%. Experimentally, adding impurities can increase G/(k,T,) [31]. Impurities imply a radiative loss Prad = n,n,Z,(T). Regions will exist where 8Z,/Uc 0, i.e. a local temperature decrease will increase Z,, so that T, will decrease still further. This thermal drive can turn an otherwise stable mode into an instability, and has been used to explain solar flares [32] and MARFEs [e.g. 331. Further effects can occur by “condensation”; if there is some mechanism trying to enforce parallel pressure balance,then decreasing T, will increase n,, increase the local radiation, and further reduce T,. rf*E
(6)
Therefore, from the measurements of &, f and their correlations [36] we can also deduce the conducted electrostatic fluctuation driven energy flux Qcd = ( Qf,E - Q,). The radial variation of both Qcd and Q,, is shown for electrons in fig. 11, together with the total conducted plus convected energy flux Qtot, deduced from a power balance and confirmed by limiter thermography [37]. Convection dominates conduction, and the total energy flux is reasonably well explained by the measured electrostatic fluctuation driven energy fluxes. The measured magnetic fluctuations are too small to contribute in ohmic heated tokamaks (Qf.b < 1 x I,-- 3QtOt; see next section). Similar conclusions are drawn from results on other machines [17]. Because of
0.8
1 .o r/a
Fig. 11. The radial dependence of the total electron energy flux Q’“’ (solid line with error bars), and the measured electrostatic fluctuation driven convected (0) and conducted (0) electron energy fluxes.
A.J. Wootton
the large uncertainties in the power balance analysis, all of Q”’ could be explained by convection. No results are as yet available for either L- or H-mode discharges. RFP’s are clearly different (table l), with magnetic fluctuations dominating the energy transport process. The measurements of .!& f and their correlations are difficult, and any supporting evidence for the relative importance of conduction to convection is important. One check that can be made is to derive the ratio of e-folding lengths of T, (X,) and particle flux F (A, = 2hnhrc/(2hre + h,)) in terms of measured fluctuation parameters, and compare the predictions with the experimental observations. Considering only the electrons, assuming all losses are to the limiter, with no other sources or sinks, and writing aFJ3r=F,,/&,
aQ,/ar=
Q, =y,kJr, we predict
h/h,=
Q,,L> Q,,=qh,TI;,
83
Edge turbulence
4 island k%Effx island
b) intermediate islands appear
and
9
d
[38]:
Y*/(Y,,- Y,>.
From eq. (6) and the definition
(7)
x points are stochastic
of y,, we can write:
(8) Taking y,, = 4.5 [24] we predict h,/X, = 0.5 if F= 0 (i.e. yI = 3/2), and X,/X, = 1.5 with the measured value yI = 3. Experimental values are X,/X, = 1 to 2, so that the measured e-folding lengths are consistent with the measured fluctuation characteristics. That is, to fluctuaexplain the measured X, finite temperature tions are required. Preliminary results from stellarators and RFP’s show similar values of F (see table 1).
7. Magnetic turbulence Perturbing magnetic fields, produced by internal instabilities or error fields from misaligned coils (in principle avoidable), destroy magnetic surfaces. Internally produced magnetic fluctuations are themselves categorized as either “low frequency (= 10 kHz) MHD”, which at low j3 can be avoided by a suitable choice of operating conditions, and higher frequency (> 50 kHz) “ turbulence”, which appears to be unavoidable. It is these higher frequency components which we are intersted in here. When such magnetic fluctuations exist the field line and particle orbits can become stochastic, or chaotic [e.g. 391. At surfaces where q = m/n a perturbation of the form b, = b,,, cos(m0 - n#) will split the surface
4 large region of chaos secondary islands appear
Fig. 12. The growth of magnetic islands and stochasticity with overlap parameter y from the standard map.
into magnetic islands (b without a - is used to represent a time independent magnetic perturbation, unfortunately n is commonly used both for density and toroidal mode number). When these islands overlap we get chaos or stochasticity, requiring a statistical description for the field line and particle behavior. Fig. 12 shows in radius and poloidal angle the intersection of field lines with a given toroidal plane, a PoincarC map, in the presence of various perturbing harmonics. The calculation shown is for the standard map of Chirikov and Taylor [40], which approximates many physical situations, including waves in sheared flow (Kelvin’s
A.J. Woorton / Edge furbulence
84
“cat’s eyes” [NJ). “Islands” appear at each rational surface where q = m/n. An overlap parameter y is defined in terms of the radial separation Ai between two adjacent surfaces with q = m/n and m’/nf separated by Aq = Aii3q/ar, and the island widths a,,,, and 6m’mf [e.g. 42): (9)
Y = (a,, + %~,~)/‘(2A,f, with S,, = 2(4RqZb,,,/(
mB*aq/ar))1’2.
(10)
R is the major radius of the system. For y = 0.6 coherent islands separated by good flux surfaces are found. With y = 0.8 additional islands appear, and with y = 1 stochastic regions appear, especially around the X points. With y = 1.2 large areas of stochastic fields (chaos) are found. As an example of the magnitudes of brmn required to get stochasticity, consider the case of two equal size islands at q = m/n and q’ = (m + 1)/n (not necessarily the nearest islands). Then stochasticity occurs ( y - 1) when: (b,,,/B+)
> (16RmCiq/i%)
-‘.
We associate the electron radial thermal diffusivity xE with the calculated test electron particle diffusion coefficient, and particle radial diffusion coefficient D with the test ion particle diffusion coefficient. Although the models are for stationary perturbations, they are applicable to time dependent 6 if the electrons become decorrelated with a particular field line faster than the magnetic structure changes in time. In the usual collisionless case the electron conducted power is:
(11)
As we are considering the plasma edge we can approximate Zlg/i3r = 24/r. Thus, typically we need (b,,, /B+) = 10m3/m for stochasticity to onset. We can derive the transport coefficients of test particles in a stochastic field using a statistical description which contains three ingredients [43]. First, the field lines (or particle trajectories) diverge exponentially with respect to one another over a characteristic length L,, and follow a random walk in space, After following a field line a distance L, it has moved radially a distance Ar, leading to a random walk “magnetic” diffusion coefficient 0, = Ar2/(2L). Second, the particles diffuse along these chaotic field lines with the classical diffusion coefficient Di,, covering a distance L in a time t = L2/(2D,,). Thirdly, the particles diffuse across the field lines, as a result of both classical collisions and the anomalous electrostatic fluxes discussed previously. If at every effective collision the particles with velocity u,, have traveled a distance L, > L, (the collisionless case) in a time TV, then an infinitesimally small radial displacement finds them on a new field line uncorrelated with the first field line. Thus the particles themselves behave in a random walk manner with a diffusion = D,,,v,,, coefficient D = DC = Ar2/(2rC) = D,,,L,/r, with 0, = qR(br/B+)2. If the particles are collisional (Lc =c Lk) we expect D = Df = D,,(b&J2. Other formulae apply in “strong turbulence” regimes 143). These formulae are not self consistent; the particle motion is not allowed to affect the perturbed field structure.
with the uthe the electron thermal velocity, n the density. The predicted particle flux rf*b is small, because D/xc = Di/D, = (m,/mi)‘/2. Therefore, the convected losses are small, and QLb = QCd. However, it should be pointed out that there is a controversy as to whether or not, when self consistent calculations are performed, internally produced magnetic fluctuations will lead to any particle transport and potential changes at all 1441. That externally produced perturbing magnetic fields significantly increase transport and modify the plasma potential is verified. For example, fig. 13 shows the increase in electron thermal diffusi~ty Ax, produced on TEXT using external (to the plasma) conductors to purposely produce a stochastic edge [lo]. The comparison with the theoretical expressions (already discussed) for xe, shown plotted only in their relevant parameter
0.75
0.80
0.85
0.90
0.95
1.00
1.05
r/a
Fig. 13. The measured incremental and theoretical (collisionless xc and collisional/fluid x’) electron thermal diffusivity Ax_ in the presence of externally produced stochastic magnetic fields (TEXT).
85
A.J. Wootton / Edge turbulence
regime (collisionless and collisional), is rather good. The two sets of lines plotted for each case (experiment, theory) represent different interpretations of the 7”(r) data. This good agreement between model and experiment is even true for perturbing fields below the threshold for stochasticity (eq. (11)). We have already seen (fig. 5) that the same externally produced stochastic fields can increase the edge potential, even providing a region with E,> 0 consistent with expectations, and changing the direction of the turbulence phase velocity (fig. 6). It is more difficult to asses the effects of internally produced fields. This is because most measurements are not made in the plasma, but rather at or even outside the vacuum vessel. For tokamaks with ohmic heating alone probes have been used inside the plasma, and the measured &,,, used directly in eq. (12). For example, a perturbation level (&./B+) = 10m3 is needed to conduct the measured total electron energy flux Q’“’ = 1 W cmm2 in TEXT (see fig. 11). This is small, but still larger than observed (fig. 7). At higher T,,smaller (&/B,) are needed. In RFP’s (see table 1) the larger 8 can explain the electron energy fluxes. To interpret most available data we have to extrapoat r > a to a value of &,,, where late a measured $,,, we want to estimate the transport. If we want to know whether or not the fields are stochastic, we also have to distribute the interpreted b, over a range of m and n at various resonant surfaces r,,,, where q = m/n. For this purpose, note the results obtained on TEXT, shown in fig. 14 [45]. A magnetic probe signal (a/at(&)) ar r > a is only correlated with a Langmuir probe signal (if) at r < a, i.e. inside the limiter, and then only for frequen-
ties 50 to 150 kHz. The radial correlation length of n’ is represents an short, = 5 mm, while the 6 measurement integral over J’(r, 0) with significant contributions over several cm around the probe. Thus the correlation between 6 and ii at one radius suggests the measured d at the edge in the range 50 to 150 kHz is a result of J' inside the main plasma. With this information we can attempt to interpret the measured &,,, at the edge as &,,,, at various surfaces. The frequency information provides an estimate of m through eq. (2); 25 < m -C 75. Radial dependencies b( r )/b( r,,,,,) a (r/rmn)-(“‘+‘) can be assumed, with r,,,, - a. The result, accounting for the many assumptions, is that the required h,,, a the measured &%nS* but the typical values in ohmic plasmas are only marginally sufficient to produce stochasticity at r = a. However, the increase observed with additional heating (higher beta poloidal) suggests that magnetic fluctuations may then become important; under these conditions correlations between decreasing energy confinement and increasing 6 are seen [17]. There may be ohmic tokamak plasmas where 6 is sufficient to produce stochasticity. Typically $/Be and thus &,,,,/B+ and y are high at low densities and low toroidal fields. Under these conditions measurements show the region in the edge where the plasma potential @ > 0 and E,> 0 is larger than usual [46]. The measured Q(r) is similar to that obtained with externally imposed magnetic stochasticity (see fig. 5) suggesting that under these conditions (of low density and field) the intrinsic islands have overlapped. If this is the case, we would expect the fluctuation phase velocity up,, to be in the ion diamagnetic direction (see fig. 6) but this has
o
150
0-FREQUENCY
(kHz)
200
250
,I
I.
0
I,
50
_‘.
*
100 FREQUENCY
18,
150
-d 200
250
(kllz)
Fig. 14. The frequency spectrum of poloidal magnetic field a/i%(&) and density A fluctuations in TEXT and the coherence between the two. The Langmuir probe was situated 1 cm inside the limiter radius (r = 25 cm) and the magnetic probe 2 cm behind the limiter radius (r = 27 cm), separated approximately along 4 = 27.
X6
A.J. Woorron
not been verified. If the edge of RFP’s are magnetically stochastic as thought then we also expect uph in the ion drift direction (unless L, is very small), but this is not the case. One possible escape is that only externally imposed magnetic stochasticity, not intrinsic stochasticity, can affect @. However, another explanation must then be found for the tokamak result (of E, > 0 at low density and field).
Edgeturbulence still some problems to solve. First, the drift wave limit applied to ATF predicts stability for n, = 0, so that other drive mechanisms (such as a particle source) are being studies. Second, scaling results suggest that the limiter itself is important, and limiter effects such as line tying are seldom included in theoretical models. Third, the model applied to TEXT is expected to be stable for T, > 100 eV, so that it is likely inappropriate for the edge of machines such as JET.
8. What is it? 9. Test particle calculations Theory, modeling and experiments in edge turbulence are still progressing, and as an example this section describes the progress towards explaining the TEXT data. Initially collisional drift waves looked promising [21], but they are stabilized by OT,. The rippling (or the generalized resistivity gradient driven) mode [47,48] had been found to explain Macrotor results quite well, but when applied to TEXT the model predicted too small fluctuation levels, the wrong k, and a sensitivity to &, or I, which is not found experimentally (note that the currentless ATF results in table 1 are remarkably similar to those of TEXT). This resistivity gradient model, damped by parallel electron thermal conduction, was extended to allow impurities to produce a thermal and condensation drive [49]; the latter is now thought inapplicable. The same model applied to a stellarator predicts thermally driven drift waves. Now reasonable fluctuation levels (including g), k values (and the ratio of k values between TEXT and ATF), and some aspects of the sensitivity to impurities, are explained. There are
Having presented all this information, one might ask how to use it in a practical manner for predicting, for example, density profiles in complicated situations near the plasma edge. We know that 3-dimensional phenomena occur [24], making analytic equations intractable. One approach around this problem is to use the concept of test particles [50], as we used to discuss the effects of magnetic turbulence. The idea is to place particles, either eletrons or ions, in a background density and temperature which determines their collisionality. These test particles approximately follow field lines, suffering Coulomb collisions and “anomalous” collisions. The Coulomb collisions determine the parallel diffusion, i.e. how fast the particles travel along the field line. The anomalous transport is included as a random walk; every coherence time rc the particle suffers an “anomalous” collision, and is displaced radially by p, = (27,0)“2, with D = -Tf’E/~n the experimental anomalous diffusivity. For rc the decorrelation time
A
n/n
-90
POLOIDAL
ANGLE
8 c
Fig. 15. Test particle (TP) simulations in a stochastic magnetic field (the EML) incorporating the measured anomalous transport as a random walk. (a) The predictions for the radial and poloidal dependence of density, using an artificial reflecting boundary at 21 cm; (b) the measured and predicted poloidal dependence of density at r = 27 cm for the case shown in (a). The predictions are for the relative change in density (An/n, broken line) while the measurements are of the relative change in ion saturation current (AI+/i’, solid line).
A.J. Wootton / Edge turbulence
between the turbulence and a particle is taken (typically 3 ps in TEXT). Particles hitting material barriers are reintroduced to simulate recycling and ionization. By expressing the particle trajectories between the turbulent steps in an analytic form (actually a map), = lo6 particles can be followed even on a small computer. Fig. 15a shows the predicted ion density distribution in radius and poloidal angle for TEXT with externally applied stochastic fields (the EML). Strong poloidal asymmetries are predicted, with a poloidal mode number dictated by the magnetic topology. There is even experimental evidence that such structure exists; fig. 15b shows a section taken from fig. 15a at r = 27 cm, together with the poloidal dependence of the ion saturation current I+ to a poloidal array of single electrode probes. The measured poloidal structure is similar to that computed; the different amplitudes may be explained by Z+a nT’/‘, whereas the model result is for density only. With the stochastic magnetic fields the variation in I+ along what was a flux surface is extreme!
10. Summary and conclusions (1) The data can be interpreted as “blobs” which are both changing in time and drifting, or as a set of dispersive waves in packets with known correlation lengths and times. These two pictures are the same if the blobs are thought of as crests of the distribution function of, e.g., ii(k, w). (2) Frequencies (= 100 kHz) are largely determined by the radial electric field. Amplitudes are large ( = 30%) decrease towards the plasma interior, and are approximately predicted by a mixing length formula. k values are smaller than expected, = 1 to 3 cm-’ and increase with increasing magnetic field. (3) Concerning confinement and turbulence: (a) It is not sufficient to seek correlations between amplitudes (e.g. B or A) and transport coefficients or fluxes. Correlations and phase angles are important. (b) Tokamaks: electrostatic turbulence determines edge particle and energy fluxes in ohmic and additional heated (L-mode, presumably H-mode?) discharges. No useful scaling laws have been identified yet. Magnetic fluctuations usually increase, possibly becoming important for energy fluxes, with reduced parameter ohmic discharges and with additional heating (L-mode). (c) Stellarators look like tokamaks but the k values are smaller. In RFP’s magnetic fluctuations are important in determining energy fluxes. (4) While we don’t know all the driving and damp-
87
ing terms, the resistivity gradient (rippling) model looks good for tokamaks, together with a thermal drive (impurities) and sheared velocity flow decorrelation. (5) The test particle mapping approach allows anomalous transport effects to be included in complicated geometries.
Ackowledgements I thank Paul Schoch for unpublished HIBP data, Terry Rhodes for use of his data base, Phil Morrison for the standard map calculations, and Jack Daniels for inspiration. This work was performed under DOE grant DEFGO588ER53267.
References [l] P. Schoch et al., Rev. Sci. Instr. 9 (1988) 1646. [2] K.W. Gentle, Nuclear Technol./Fusion 1 (1981) 479. [3] B.H. Briggs, G.J. Phillips and D.H. Shinn, Proc. Phys. Sot. B63 (1950) 106. [4] B.H. Briggs, J. Atm. Terr. Phys. 30 (1968) 1789. [5] S.J. Zweben and R.W. Gould, Nucl. Fusion 25 (1985) 171. [6] S.J. Zweben and R.W. Gould, Phys. Fluids 28 (1985) 974. [7] Ch.P. Ritz et al., Rev. Sci. Instr. 9 (1988) 1739. [8] S.J. Zweben and S.S. Medley, Phys. Fluid 81 (1989) 2058. 191 R.A. Koch and W.M. Tang, Phys. Fluids 21 (1978) 1236. [lo] S.M. McCool et al., Nucl. Fusion 29 (1989) 547. [ll] S.M. McCool et al., Nucl. Fusion 30 (1989) 167. [12] P. Schoch et al., in: Proc. 15th Eur. Conf. on Controlled Fusion and Plasma Physics (European Physical Society, Petit-Lancy, Switzerland, 1988) Vol. 1, p. 191. [13] Ch.P. Ritz, R.D. Bengtson, S.J. Levinson and E.J. Powers, Phys. Fluids 27 (1984) 2956. [14] G.A. Hallock et al., Phys. Rev. Lett. 56 (1986) 1248. [15] R.D. Hazeltine, Phys. Fluids Bl (1990) 2031. 116) R.W. Harvey, M.G. McCoy, J.Y. Hsu and A.A. Mirin, Phys. Rev. Lett. 47 (1981) 102. [17] A.J. Wootton et al., University of Texas Fusion research Center Report, Fluctuations and Anomalous Transport in Tokamaks. FRCR 340 and DOE/ER 53267-52 (1989) submitted to Phys. Fluids. [18] C.M. Surko, in: Turbulence and Anomalous Transport in Magnetized Plasmas, Cargese Workshop 1986, Eds. D. Gresillon and M. Dubois (Ecole Polytechnique, Palaissau, France, 1986) p. 93. [19] C. Hidalgo, J.D. Bell, B.A. Carreras, J.L. Dunlap, R. Dyer, Ch.P. Ritz, A.J. Wootton, K. Carter and T.L. Rhodes, in: Proc. 17th Eur. Conf. on Controlled Fusion and Plasma Physics, Amsterdam (European Physical Society, Petit-Lancy, Switzerland, 1990). [20] G. Miller, J.C. Ingraham, C.P. Munson, K.F. Schoenberg,
88
[21]
[22] [23] [24]
(251 [26] [27]
[28] [29]
(301 [31]
[32] [33] [34]
A.J. Wootton / Edge turbulence P.G. Weber, H.Y.W. Tsui and Ch.P. Ritz, in: Proc. 17th Eur. Conf. on Controlled Fusion and Plasma Physics, Amsterdam (European Physical Society, Petit-Lancy, Switzerland, 1990). W. Horton, in: Basic Plasma Physics, II, Eds. A.A. Galeev and R.N. Sudan (North-Holland, Amsterdam, 1985) p. 383. D.W. Ross, Comm. Plasma Phys. and Contr. Fusion XII (1989) 155. E.J. Powers, Nucl. Fusion 14 (1974) 749. P.C. Stangeby and G.M. McCracken, Institute for Aerospace Studies Report, University of Toronto, Plasma Boundary Phenomena in Tokamaks, (1988) submitted to Nucl. Fusion. W.L. Rowan et al., Nucl. Fusion 27 (1987) 1105. G.H. Hallock, A.J. Wootton and R.L. Hickok, Phys. Rev. Lett. 59 (1987) 1301. P. Mantica, S. Cirant, J. Hugill, G.F. Matthews, R.A. Pitts and G. Vayakis, in: Proc. 16th Eur. Conf. on Controlled Fusion and Plasma Physics (European Physical Society, Petit-Lancy, Switzerland, 1989) Vol. 3, p. 967. The ASDEX Team, Nucl. Fusion 29 (1989) 1959. T. Rhodes, University of Texas at Austin Thesis Collection (1989) and Fusion Research Center Report, Experiments on Turbulence and Transport in the Edge Plasma of the TEXT Tokamak, FRC 352 (1989). G.M. McCracken and P.C. Stangeby, Plasma Phys. Contr. Fusion 27 (1985) 1411. D.R. Thayer et al., in: Proc. Int. Conf. on Plasma Physics and Controlled Nuclear Fusion Research, 1988, Nice, Vol. IAEA-CN-50/D-4-10 (IAEA, Vienna, 1989). E.N. Parker, Astrophys. J. 117 (1953) 431. J.F. Drake, Phys. Fluids 30 (1987) 2429. H. Biglari, P.H. Diamond and P.W. Terry, Phys. Fluids B2 (1990) 1.
[35] Ch. Ritz, H. Lin, R.D. Bengtson, S.C. McCool, T.L. Rhodes and A.J. Wootton, in: Proc. 17th Eur. Conf. on Controlled Fusion and Plasma Physics, Amsterdam (European Physical Society, Petit-Lancy, Switzerland, 1990). [36] H. Lin, R.D. Bengtson and Ch.P. Ritz, Phys. Fluids Bl (1989) 2027. [37] Ch. Ritz et al., Phys. Rev. Lett. 62 (1989) 1844. [38] R.D. Bengtson et al., in: Proc. 17th Eur. Conf. on Controlled Fusion and Plasma Physics, Amsterdam (European Physical Society, Petit-Lancy, Switzerland, 1990). [39] A.B. Rechester, M.N. Rosenbluth and R.B. White, in: Intrinsic Stochasticity in Plasmas, Cargese Workshop 1979, Eds. D. Gresillon and M. Dubois (Ecole Polytechnique, Palaissau, France, 1979) p. 239. [40] B.V. Chirikov, Phys. Rep. 52 (1979) 263. [41] G.I. Taylor, Proc. Roy. Sot. A CXXXII (1931) 499. 1421 P.H. Rebut and M. Brusati, Plasma Phys. Contr. Fusion 28 (1986) 113. [43] J.A. Krommes, C. Obermann and R.G. Kleva, J. Plasma Phys. 30 (1983) 11. [44] J. Krommes and C-B Kim, Phys. Fluids 31 (1988) 869. [45] Y.J. Kim, K.W. Gentle, C.P. Ritz, T.L. Rhodes and R.D. Bengtson, Nucl. Fusion 29 (1989) 99. [46] V.I. Bugarya, Nucl. Fusion 25 (1985) 1707. [47] L. Garcia, P.H. Diamond, B.A. Carreras and J.D. Callen, Phys. Fluids 28 (1985) 2147. [48] T.S. Hahm, P.H. Diamond, P.W. Terry, L. Garcia and B.A. Carreras, Phys. Fluids 30 (1987) 1452. [49] D.R. Thayer and P.H. Diamond, Phys. Fluids 30 (1987) 3724. [50] A.J. Wootton and S.B. Zheng, University of Texas Fusion Research Center Report, Test Particle Calculations in TEXT with Resonant Magnetic Fields, FRCR 325 (1988).
Journal of Nuclear Materials 176 62 177 (1990) 77-88 North-Holland
Edge turbulence A.J. Wootton Fusion Research Center, University of Texas at Austin, Austin, TX 78712, USA
This review concentrates on tokamak results, summarizing some of the characteristics (e.g. amplitudes, frequencies, wavelengths) and comparing them with theoretical expectations. With ohmic heating the measured electrostatic fluctuations are shown to explain the total anomalous edge fluxes, but no useful scaling law yet exists. Magnetic turbulence may become important with additional heating (perhaps higher beta). Although an adequate theory to explain all the measurements does not exist, there is an understanding of some of the basic drives and damping mechanisms, and these are discussed. Finally, a teat particle approach to including the effects of the electrostatic turbulence induced transport in complicated geometry (e.g. with stochastic magnetic fidds) is presented.
1. Analysis techniques Plasma parameters fluctuate about their mean values, and these fluctuations can cause transport. Fig. 1 depicts how waves, driven unstable by the gradients of plasma parameters, can partly determine the gradients themselves. Processes such as radiation and charge exchange, as well as local sources (e.g. a gas puff) and sinks (a limiter) will affect the gradients; they may also affect the turbulence directly. One of the purposes of turbulence studies is to identify the free energy sources, and derive the transport matrix which relates the transport coefficients to the gradients themselves. For very
340.00
340.10 time (ms)
Fig. 2. Two time histories illustrating the random nature of the signals. The results show the density fluctuations obtained from a heavy ion beam probe on TEXT, with the two sample volumes separated poloidally by 2.3 cm, at a radial position r/a = 0.96.
Fig. 1. The connection between the free energy sources, turbulence and transport processes.
unstable modes there is a simpler situation; the condition for marginal stability should determine the gradients directly. Fig. 2 shows typical ~u~tuations signals obtained simultaneously with a heavy ion beam probe [l], data digitized at 1 MHz, from two sample volumes. The volumes (= 2 mm x 2 mm x 2 cm) are separated poloidally by &, = 2.3 cm; both are just inside the main plasma of TEXT [2] (r/u = 0.96, with a the limiter radius). The fluctuations in signal amplitude are caused by density fluctuations ii. Correlation analysis can be applied to this data, and fig. 3 shows the auto and cross correlation functions [3]. There is an apparent velocity
0022-3115/90/%03.50 Q 1990 - Elsevier Science Publishers B.V. (North-Holland)
78
A.J. Woorton / Edge rurhulence
v, = &JT, = 4.6 km SC’. Two processes are assumed to occur; a drift of random patterns past the sample volumes with a velocity v,, and changes to the patterns themselves with a velocity v,. LIP is a combination of both. In terms of the times indicated in fig. 2, v, = &/(27z) = 4.5 km SC’, and v, = ((0, - u,)v,)“* = 0. For this typical case, the correlation time of the turbulence in the drifting frame of reference is much longer (> 10 times) than that observed in the laboratory frame (- 3
-1
Ps). This model does not cover all the possible types of moving patterns which can occur [4]. For example, if the movement of the pattern is caused by some kind of wave progression, and if the waves involved are dispersive, the velocity of a given Fourier component will be a function of its wavelength. To test for this type of behavior the pattern is Fourier analyzed into sinusoidal components of different wavelength and a velocity calculated for each component. This is called dispersion analysis. The amplitude and phase of each Fourier component are found; it is assumed that this Fourier component is produced by the passage over the sample volumes of a sinusoidal wave of wavelength h and velocity vPh, such that up,, = fX. The wave normal k = 27r/h and the velocity are calculated from the phase difference associated with the records; this calculation is repeated for each frequency. Fig. 4 shows the results of applying this dispersion analysis to the data of fig. 2, to give the spectral power density function S(k, f), a n”*. A poloidal phase velocity v,,,, = 6 km SC’ is found. Most of the power is at f< 100 kHz, k I 1.5 cm-’ (i.e. poloidal mode number m = rk I 40 or X 2 4 cm). The spectrum is broadband with Af=f and Ak = k (i.e. turbulent). This method depends upon the assumption that a given frequency is produced by a single wave passing over the recording system. This need not be the case,
0
1
2
3
4
5
wavenumber (cm-l) Fig. 4. The power spectrum S(k, f) obtained from the data shown in fig. 2.
because waves with the same wavelength but different directions might be superimposed. The analysis will always give an apparent velocity, but if several waves are superimposed, the value will be meaningless [4]. The situation is considerably improved if the whole pattern can be observed, allowing a spatial Fourier analysis to be performed, rather than a temporal one. Two dimensional radial-poloidal arrays have been used [5,6] and show correlation lengths both poloidally and radially = 1 cm (I 1 wavelength). Toroidally separated sample volumes show that along the field lines the correlation distance is much longer, > vR [7]. Localized “blobs” of plasma are seen, which move irregularly. The lifetime and area distributions of these blobs are consistent with those expected from the measured conelation times and lengths. As such the blobs are consistent with the crests of an average ii spectrum, rather than being coherent structures [6]. This picture of randomly changing filamentary structures resembles those observed photographically in e.g. TFTR [8].
2. Phase velocity
auto-correlatm
p(0.r)
cross-correlation. p&r)
0
-1.0
1
I 10
6
I
1t
t
1
0
I
I
10
r, T2 delay time z (us) Fig. 3. The auto correlation and cross correlation obtained from the data shown in fig. 2.
functions
Both correlation and dispersion analysis provide a phase velocity vph, which helps identify the turbulence and provides information on the radial electric field E,. For example, some modes are expected to propagate in the electron drift direction (u,& while others should propagate in the ion drift direction ( udi). Because we make our observations in the laboratory frame we must account for Doppler shifts. With the ion momentum balance equation we can relate the mass velocities to E,: niqi [ Er + uoB+ - u+Be] - Vpi = 0,
(1)
A.J. Woofton / Edge
with PI;, qi, pi the ion density, charge, pressure, u the mass rotation velocities, subscript 8, rp referring to poloidal, toroidal. When we do this, we expect f9]: * l$h = vph
Er/B+,
(2)
with vph the measured (lab. frame) poloidal phase velocity and vp;l the phase velocity ignoring E,; for example *( e 1n,B+) for drift waves. Here %h - v,, = - Qn,/ar/( e is the electron charge. Therefore, we must measure E, = - a@,/& (@ is the plasma potential) before we can measuring r+ provides indetermine up;l: alternatively formation on E,. Fig. 5 shows Q, in TEXT, both for a normal discharge and for a discharge where a set of external coils has been energized with a current I, to perturb the edge magnetic field [10,11,12], creating a stochastic edge region, or ergodic magnetic limiter (EML), discussed later. Normally q~> 0, E, > 0 for r 2 a, and @ < 0, E,
I
300
/
c
200 _ VI c
100 0
9 v
I
,0"l,
+ -100 1 -200 -300
1 /’
t 1 0.6
d
//’
ii 0
/
L 1 .o
0.8
P Fig. 5. The radial dependence of plasma potential Cp, with (I, = 4 lc.4, solid symbols) and without (I, = 0, open symbols) externally applied perturbing magnetic fields (the ergodic magnetic limiter, or EML). Circles represent heavy ion beam probe data and triangles represent Langmuir probe data.
19
turbulence W,thout
5
electron
4
perturbrng
mognetlc
f&d
drift 1
(aI
xw
z!!___:; ion drift -2
0.6
i limiter 0.8
Wilh perturbrng 6
; I.2
I.0 r/o -+ magnetic
fmid
(It,= PkA)
/ (bt
-4
,; llmlter
1
0,8
1.0
1
I.2
r/o +
Fig. 6. The measured radial dependence of the fluctuation phase velocities up,, (a) without and (b) with externally applied perturbing magnetic fields (the EML, I,= 4 kA case). Also shown are the expectations assuming oph= u,,, - I&/B,, and the component u,, alone. Data from Langmuir probes (0). the heavy ion beam probe (x), FIR scattering (I) and an m = 2 tearing mode ( + ) are shown.
within one spatial correlation length might do the same thing. With a stochastic magnetic field we expect 1161 E, =j jr,(L,’ + 0.5L~e’) > 0, where I,,, LC are scale lengths of density and electron temperature. A positive E, is set up to restrain the electrons which try to escape along stochastic field lines to the limiter faster than the ions. Therefore, with I, = 4 kA in fig. 5 we expect and see the region with @ > 0 expanded; the width of the velocity shear layer is increased but the amplitude of the velocity shear is reduced. If ey. (2) is correct, this should modify the measured vPh. Fig. 6 shows the measured turbulence phase velocity uph for cases (a) without (Zh = 0) and (b) with (Zh = 4 kA) the EML and thus modified E, [11,12]. Data from different diagnostics shows that normally (fig. 6a) a@, changes direction from ion (v < 0) to electron (v > 0) drift at r = a, but with the modified E, (fig. 6b) this reversal is moved to smaller r. In both cases eq. (2) (solid line) approximately describes the results. The broken line shows u,,; generally the term E,/B,
80
A.J. Wootton / Edge turbulence 0.8
0.0 0.6
0.7
0.8
0.9
1.0
1.1
1.2
r/a
Fig. 7. The radial dependences of the normalized rms fluctuating amplitudes of density Z/n, potential 4/(/c&), temperature %/r, and magnetic field 6,/B+, in TEXT.
Boltzmann relationship is influenced by changing the impurity concentrations (discussed later). A simple mixing length model predicts that H/n should saturate when the local density profile is flattened (implying no further drive), i.e. when A/n = (k,L,)-‘. There is a strong trend in the direction expected [18], but with two discrepancies. First, the amplitude fi/n = 2(kL,)-‘, larger than expected (locally the gradient can reverse). Second, within one machine (e.g. TEXT) the scaling is not adhered to. A factor - 5 variation can occur, suggesting there are important processes not considered in the simple model. Table 1 shows that similar fluctuation amplitudes are found in stellarators [19] and reversed field pinches (RFP’s) [20] except for the much larger magnetic fluctuation level in RFP’s.
4. Wave numbers dominates, so that upi, = v, - veBB/B+ = ve •t v,k,/ke, i.e. the phase velocity in the laboratory frame gives us information on the mass rotation velocity directly.
3. Amplitudes
Fig. 7 [17] shows the rms values of C/n, G/k&), Fe/T, and &/Be for a particular discharge in TEXT as a function of r/a (kb is Boltzmann’s constant). The values shown are characteristic of all tokamak edges. Note that A/n # ;/(k,T,); this departure from the
Table 1 Edge turbulence parameters at r = a parameter
TEXT
ATF
ZT-40
cl(a) n (lo’* me3)
3 4
1
0.1
1
5
T. (ev)
30 2 5
20 3 5 0.05-0.1 0.1-0.2
30 1.5 8 0.3-0.5 0.4-0.6
L, (cm) + (cm) ii/n G/(k,T,) C/T,
0.1-0.2
s/B(o)
k (cm-‘) r Q Q, IQ,,
a)
0.15-0.3 0.05-0.1 10-S 3
0.1
0.2-0.5
10-6 1
0.01-0.02 0.1
t rr.E I q’.E
r rf.E I q’.E
r
Cl
(1
>1
z rf.E Q&b
Parameters which are significantly different are in bold type. ‘) Width of h spectrum is tabulated.
Typical wave numbers observed near the edge [17] k, 3 1 cm-’ (i.e. - ke) and are k B - 1 to 3 cm-‘, k,, - l/(qR), with q the safety factor. If the turbulence is drift wave like we expect keps = 1 (211, with ps the ion gyroadius using T,, so k, a B&eo.5. Alternatively if the magnetic geometry is defining the structure then we might expect k, to be given by the lowest (most unstable) value of m in a correlation volume around the rational surface where q = m/n. Some evidence for both properties exists. We know [18] k a Be. However, for TEXT k,p, = 0.05; i.e. the wavelengths are > 10 times longer than expected from drift wave theories. Also there is a very large variation in k, at a given B,, larger than any variation in Te-‘.‘. On the other hand, comparing tokamaks, stellarators and RFPs (table 1) suggests that k is dependent on q, i.e. the magnetic geometry is important. We have not uncovered all the important physics yet.
5. Particle confinement
and electrostatic turbulence
We categorize the fluctuations as either electrostatic (fluctuation density ii, potential 6 = -i/k, temperature f) and magnetic (fluctuating magnetic field 6). In reality both will exist simultaneously. The electrostatic fluctuations produce a fluctuating radial velocity fir = &/Be. Ignoring poloidal and toroidal asymmetries, and assuming w K wCi, the ion cyclotron frequency, the electrostatic fluctuation driven radial particle flux (denoted by superscript f, E) is [22] Ff,E = (&i)/B,.
(3)
A.J. Wootton / Edge turbulence
81
Subscripts r, 6, 4, I], I refer to radial, poloidal, toroidal, parallel and perpendicular to the magnetic field, and ( * . . ) denotes an ensemble average (an average in space and time). We can recast eq. (3) so that FfgE is given as a function of frequency 0/(2rr) over a spectral width dw in terms of the root mean square (rms) fluctuation levels due to the spectral components of width do centered at o [23]: rfvE( w) = ii rms(+&&G
lY,*(W) I Ma)
xsin(c,+(w))/B.+.
(4)
Now we see that both the coherence y,,+ and phase angle a,,+ between ii and I$ are important. It is not sufficient just to measure the fluctuating amplitudes. Ff,E has now been measured in many ohmically heated plasmas [17]. A comparison with the total flux, measured spectroscopically, is shown in fig. 8 (TEXT). A simple model for parallel flow to the limiter has been used to obtain the total perpendicular component F’ [24]. The approximate agreement demonstrates that the measured turbulent flux accounts for an important part, if not all, of the total particle flux in the edge. This agreement is found for a variety of Be, Ip (plasma current) and ii, [25]. A similar conclusion (that I’f*Ecan explain particle transport) is drawn for stellarators and RFPs (see table 1). Experimental results from ISX-B [26] with neutral beam heating, and DITE [27] and TEXT with electron cyclotron heating, show an increased FfsE explains the decreased particle confinement time with additional (L-mode) heating. No results are available with the H-mode [28], in which improved confinement with additional heating is found, but it seems reasonable to suppose that the improvement in
analysis of fluctuation driven rf,E in TEXT on local density n and toroidal field B+.
Fig. 9. A regression
particle confinement
is associated with a decrease in
rfsE. A useful result would be a scaling law (understood or otherwise!) for the flux FfVEor the diffusion coefficient D ‘SE = - I’f,E/vn; this could be used in predictive transport codes. Because we expect FfsE to be a complicated function of more than one gradient (e.g. vn, 8T, VU), it is better to seek a scaling of the flux directly, rather than Df*E. Fig. 9 shows the results of a regression analysis on TEXT data for r > a; FfsE a d3B;‘.‘, the significant deviations show not all the dependencies have been found. A different dependance is found for r < u, suggesting that the limiter itself partly determines the fluxes [29]. One method to demonstrate that FfsE is important is to show that it can predict the measured density e-folding length X, behind the limiter. To do this we recast the scaling as I’f*E= KTIB-‘.~ [29], which is almost as good (or bad) a fit as that shown in fig. 9. By balancing parallel and perpendicular fluxes in slab geometry [24], and ignoring sources, we have:
a/ar(rf,E) = KB;%~/ar = -PIU,,/L~,
‘.
04 . 1 t 0.6
0.9
b .
I
1 .o
1 .l
I 1.2
r/a Fig. 8. The radial dependence of the &asured fluctuation driven rf*E (solid line) and total r’ (broken line) particle fluxes (TEXT). Solid symbols represent Langmuir probe data; open symbols represent heavy ion beam probe data.
(5)
with u,, a parallel sound velocity (taken as 0.5 times the sound speed cs), L, a typical distance along a field line to a material surface, and K = 3.8 x lo4 (units T’.’ cm s-‘) the constant of proportionality. This predicts an exponential fall-off behind the limiter, with X, = XrxT = 2 KL,c;‘B; ‘.‘. Fig. 10 shows a comparison of the measured X, with the prediction XrxT for TEXT and other tokamaks (data taken from [30]). For TEXT (horizontal bars) the agreement is good, considering the may approximations used to derive eq. (5) [24]. However, to explain all the data base results ( X ) requires a factor between 0.1 and 10 to be applied to the scaled
A.J. Wootton / Edge turbulence
Velocity shear Where aE,/ar or shear layer, any ferential velocity
has been discussed in explaining up,,. au/& (- au,,,/&) is high, i.e. in the turbulent “blobs” experience a difAu = I,( au/ar - u/r) across a radial correlation length I,. This will tear the blob apart in a time TV= IO/Au, with I, ( = I,) the poloidal correlation length, and suppress the turbulence [34]. Such a mechanism explains the measured auto-correlation times in TEXT [35], and may explain the H-mode. 0
2
4
6
8
10
*m= 12KL, 1,-m (cs Bm’5 Fig. 10. The measured density e-folding length X, compared to that predicted (XTXT) based on TEXT scaling of rf.n. Results from a data base (x), and TEXT (horizontal bars represent the spread due to T, variations) are shown. ASDEX results (0) are connected by arrows, illustrating the change from ohmic heating (OH) to additional heating low mode (L) to H-mode
6. Energy confinement
and electrostatic turbulence
From rf,E we can derive the electrostatic fluctuation driven convected energy flux Q,, = 5/2k,T,Tf*E. The total electrostatic fluctuation driven energy flux is given by [22]: Qf.E = 3/2k,n(gof)/B,
+ 3/2k,T(&,ii)/B,.
(H).
from TEXT. This show again that not all the important parameters have been identified. Although this comparison (fig. 10) is poor, it is no worse than assuming a Bohm diffusion coefficient D,, which predicts A, = (2 D,L&‘)‘/* [24]. Again a factor between 0.1 and 10 must be applied to D, to explain all the data base results. However there is one situation where X is better than XTXT,and that is in predicting what ha;pens at the transitions from ohmic (OH) to Lto H-mode in tokamaks, marked in fig. 10 for ASDEX [28] (0) by arrows (OH to L; T, increases and X, increases; L to H: T, decreases and h, decreases). Because XrxT a Te-'-' it predicts exactly the opposite behavior, whereas X, a To.25 does show the general trend. Again not all the important parameters determining rf*E have been accounted for. Two important parameters not identified in the scaling for rf*E discussed above are the impurity concentration nr and the mass velocity shear au,%. Experimentally, adding impurities can increase G/(k,T,) [31]. Impurities imply a radiative loss Prad = n,n,Z,(T). Regions will exist where 8Z,/Uc 0, i.e. a local temperature decrease will increase Z,, so that T, will decrease still further. This thermal drive can turn an otherwise stable mode into an instability, and has been used to explain solar flares [32] and MARFEs [e.g. 331. Further effects can occur by “condensation”; if there is some mechanism trying to enforce parallel pressure balance,then decreasing T, will increase n,, increase the local radiation, and further reduce T,. rf*E
(6)
Therefore, from the measurements of &, f and their correlations [36] we can also deduce the conducted electrostatic fluctuation driven energy flux Qcd = ( Qf,E - Q,). The radial variation of both Qcd and Q,, is shown for electrons in fig. 11, together with the total conducted plus convected energy flux Qtot, deduced from a power balance and confirmed by limiter thermography [37]. Convection dominates conduction, and the total energy flux is reasonably well explained by the measured electrostatic fluctuation driven energy fluxes. The measured magnetic fluctuations are too small to contribute in ohmic heated tokamaks (Qf.b < 1 x I,-- 3QtOt; see next section). Similar conclusions are drawn from results on other machines [17]. Because of
0.8
1 .o r/a
Fig. 11. The radial dependence of the total electron energy flux Q’“’ (solid line with error bars), and the measured electrostatic fluctuation driven convected (0) and conducted (0) electron energy fluxes.
A.J. Wootton
the large uncertainties in the power balance analysis, all of Q”’ could be explained by convection. No results are as yet available for either L- or H-mode discharges. RFP’s are clearly different (table l), with magnetic fluctuations dominating the energy transport process. The measurements of .!& f and their correlations are difficult, and any supporting evidence for the relative importance of conduction to convection is important. One check that can be made is to derive the ratio of e-folding lengths of T, (X,) and particle flux F (A, = 2hnhrc/(2hre + h,)) in terms of measured fluctuation parameters, and compare the predictions with the experimental observations. Considering only the electrons, assuming all losses are to the limiter, with no other sources or sinks, and writing aFJ3r=F,,/&,
aQ,/ar=
Q, =y,kJr, we predict
h/h,=
Q,,L> Q,,=qh,TI;,
83
Edge turbulence
4 island k%Effx island
b) intermediate islands appear
and
9
d
[38]:
Y*/(Y,,- Y,>.
From eq. (6) and the definition
(7)
x points are stochastic
of y,, we can write:
(8) Taking y,, = 4.5 [24] we predict h,/X, = 0.5 if F= 0 (i.e. yI = 3/2), and X,/X, = 1.5 with the measured value yI = 3. Experimental values are X,/X, = 1 to 2, so that the measured e-folding lengths are consistent with the measured fluctuation characteristics. That is, to fluctuaexplain the measured X, finite temperature tions are required. Preliminary results from stellarators and RFP’s show similar values of F (see table 1).
7. Magnetic turbulence Perturbing magnetic fields, produced by internal instabilities or error fields from misaligned coils (in principle avoidable), destroy magnetic surfaces. Internally produced magnetic fluctuations are themselves categorized as either “low frequency (= 10 kHz) MHD”, which at low j3 can be avoided by a suitable choice of operating conditions, and higher frequency (> 50 kHz) “ turbulence”, which appears to be unavoidable. It is these higher frequency components which we are intersted in here. When such magnetic fluctuations exist the field line and particle orbits can become stochastic, or chaotic [e.g. 391. At surfaces where q = m/n a perturbation of the form b, = b,,, cos(m0 - n#) will split the surface
4 large region of chaos secondary islands appear
Fig. 12. The growth of magnetic islands and stochasticity with overlap parameter y from the standard map.
into magnetic islands (b without a - is used to represent a time independent magnetic perturbation, unfortunately n is commonly used both for density and toroidal mode number). When these islands overlap we get chaos or stochasticity, requiring a statistical description for the field line and particle behavior. Fig. 12 shows in radius and poloidal angle the intersection of field lines with a given toroidal plane, a PoincarC map, in the presence of various perturbing harmonics. The calculation shown is for the standard map of Chirikov and Taylor [40], which approximates many physical situations, including waves in sheared flow (Kelvin’s
A.J. Woorton / Edge furbulence
84
“cat’s eyes” [NJ). “Islands” appear at each rational surface where q = m/n. An overlap parameter y is defined in terms of the radial separation Ai between two adjacent surfaces with q = m/n and m’/nf separated by Aq = Aii3q/ar, and the island widths a,,,, and 6m’mf [e.g. 42): (9)
Y = (a,, + %~,~)/‘(2A,f, with S,, = 2(4RqZb,,,/(
mB*aq/ar))1’2.
(10)
R is the major radius of the system. For y = 0.6 coherent islands separated by good flux surfaces are found. With y = 0.8 additional islands appear, and with y = 1 stochastic regions appear, especially around the X points. With y = 1.2 large areas of stochastic fields (chaos) are found. As an example of the magnitudes of brmn required to get stochasticity, consider the case of two equal size islands at q = m/n and q’ = (m + 1)/n (not necessarily the nearest islands). Then stochasticity occurs ( y - 1) when: (b,,,/B+)
> (16RmCiq/i%)
-‘.
We associate the electron radial thermal diffusivity xE with the calculated test electron particle diffusion coefficient, and particle radial diffusion coefficient D with the test ion particle diffusion coefficient. Although the models are for stationary perturbations, they are applicable to time dependent 6 if the electrons become decorrelated with a particular field line faster than the magnetic structure changes in time. In the usual collisionless case the electron conducted power is:
(11)
As we are considering the plasma edge we can approximate Zlg/i3r = 24/r. Thus, typically we need (b,,, /B+) = 10m3/m for stochasticity to onset. We can derive the transport coefficients of test particles in a stochastic field using a statistical description which contains three ingredients [43]. First, the field lines (or particle trajectories) diverge exponentially with respect to one another over a characteristic length L,, and follow a random walk in space, After following a field line a distance L, it has moved radially a distance Ar, leading to a random walk “magnetic” diffusion coefficient 0, = Ar2/(2L). Second, the particles diffuse along these chaotic field lines with the classical diffusion coefficient Di,, covering a distance L in a time t = L2/(2D,,). Thirdly, the particles diffuse across the field lines, as a result of both classical collisions and the anomalous electrostatic fluxes discussed previously. If at every effective collision the particles with velocity u,, have traveled a distance L, > L, (the collisionless case) in a time TV, then an infinitesimally small radial displacement finds them on a new field line uncorrelated with the first field line. Thus the particles themselves behave in a random walk manner with a diffusion = D,,,v,,, coefficient D = DC = Ar2/(2rC) = D,,,L,/r, with 0, = qR(br/B+)2. If the particles are collisional (Lc =c Lk) we expect D = Df = D,,(b&J2. Other formulae apply in “strong turbulence” regimes 143). These formulae are not self consistent; the particle motion is not allowed to affect the perturbed field structure.
with the uthe the electron thermal velocity, n the density. The predicted particle flux rf*b is small, because D/xc = Di/D, = (m,/mi)‘/2. Therefore, the convected losses are small, and QLb = QCd. However, it should be pointed out that there is a controversy as to whether or not, when self consistent calculations are performed, internally produced magnetic fluctuations will lead to any particle transport and potential changes at all 1441. That externally produced perturbing magnetic fields significantly increase transport and modify the plasma potential is verified. For example, fig. 13 shows the increase in electron thermal diffusi~ty Ax, produced on TEXT using external (to the plasma) conductors to purposely produce a stochastic edge [lo]. The comparison with the theoretical expressions (already discussed) for xe, shown plotted only in their relevant parameter
0.75
0.80
0.85
0.90
0.95
1.00
1.05
r/a
Fig. 13. The measured incremental and theoretical (collisionless xc and collisional/fluid x’) electron thermal diffusivity Ax_ in the presence of externally produced stochastic magnetic fields (TEXT).
85
A.J. Wootton / Edge turbulence
regime (collisionless and collisional), is rather good. The two sets of lines plotted for each case (experiment, theory) represent different interpretations of the 7”(r) data. This good agreement between model and experiment is even true for perturbing fields below the threshold for stochasticity (eq. (11)). We have already seen (fig. 5) that the same externally produced stochastic fields can increase the edge potential, even providing a region with E,> 0 consistent with expectations, and changing the direction of the turbulence phase velocity (fig. 6). It is more difficult to asses the effects of internally produced fields. This is because most measurements are not made in the plasma, but rather at or even outside the vacuum vessel. For tokamaks with ohmic heating alone probes have been used inside the plasma, and the measured &,,, used directly in eq. (12). For example, a perturbation level (&./B+) = 10m3 is needed to conduct the measured total electron energy flux Q’“’ = 1 W cmm2 in TEXT (see fig. 11). This is small, but still larger than observed (fig. 7). At higher T,,smaller (&/B,) are needed. In RFP’s (see table 1) the larger 8 can explain the electron energy fluxes. To interpret most available data we have to extrapoat r > a to a value of &,,, where late a measured $,,, we want to estimate the transport. If we want to know whether or not the fields are stochastic, we also have to distribute the interpreted b, over a range of m and n at various resonant surfaces r,,,, where q = m/n. For this purpose, note the results obtained on TEXT, shown in fig. 14 [45]. A magnetic probe signal (a/at(&)) ar r > a is only correlated with a Langmuir probe signal (if) at r < a, i.e. inside the limiter, and then only for frequen-
ties 50 to 150 kHz. The radial correlation length of n’ is represents an short, = 5 mm, while the 6 measurement integral over J’(r, 0) with significant contributions over several cm around the probe. Thus the correlation between 6 and ii at one radius suggests the measured d at the edge in the range 50 to 150 kHz is a result of J' inside the main plasma. With this information we can attempt to interpret the measured &,,, at the edge as &,,,, at various surfaces. The frequency information provides an estimate of m through eq. (2); 25 < m -C 75. Radial dependencies b( r )/b( r,,,,,) a (r/rmn)-(“‘+‘) can be assumed, with r,,,, - a. The result, accounting for the many assumptions, is that the required h,,, a the measured &%nS* but the typical values in ohmic plasmas are only marginally sufficient to produce stochasticity at r = a. However, the increase observed with additional heating (higher beta poloidal) suggests that magnetic fluctuations may then become important; under these conditions correlations between decreasing energy confinement and increasing 6 are seen [17]. There may be ohmic tokamak plasmas where 6 is sufficient to produce stochasticity. Typically $/Be and thus &,,,,/B+ and y are high at low densities and low toroidal fields. Under these conditions measurements show the region in the edge where the plasma potential @ > 0 and E,> 0 is larger than usual [46]. The measured Q(r) is similar to that obtained with externally imposed magnetic stochasticity (see fig. 5) suggesting that under these conditions (of low density and field) the intrinsic islands have overlapped. If this is the case, we would expect the fluctuation phase velocity up,, to be in the ion diamagnetic direction (see fig. 6) but this has
o
150
0-FREQUENCY
(kHz)
200
250
,I
I.
0
I,
50
_‘.
*
100 FREQUENCY
18,
150
-d 200
250
(kllz)
Fig. 14. The frequency spectrum of poloidal magnetic field a/i%(&) and density A fluctuations in TEXT and the coherence between the two. The Langmuir probe was situated 1 cm inside the limiter radius (r = 25 cm) and the magnetic probe 2 cm behind the limiter radius (r = 27 cm), separated approximately along 4 = 27.
X6
A.J. Woorron
not been verified. If the edge of RFP’s are magnetically stochastic as thought then we also expect uph in the ion drift direction (unless L, is very small), but this is not the case. One possible escape is that only externally imposed magnetic stochasticity, not intrinsic stochasticity, can affect @. However, another explanation must then be found for the tokamak result (of E, > 0 at low density and field).
Edgeturbulence still some problems to solve. First, the drift wave limit applied to ATF predicts stability for n, = 0, so that other drive mechanisms (such as a particle source) are being studies. Second, scaling results suggest that the limiter itself is important, and limiter effects such as line tying are seldom included in theoretical models. Third, the model applied to TEXT is expected to be stable for T, > 100 eV, so that it is likely inappropriate for the edge of machines such as JET.
8. What is it? 9. Test particle calculations Theory, modeling and experiments in edge turbulence are still progressing, and as an example this section describes the progress towards explaining the TEXT data. Initially collisional drift waves looked promising [21], but they are stabilized by OT,. The rippling (or the generalized resistivity gradient driven) mode [47,48] had been found to explain Macrotor results quite well, but when applied to TEXT the model predicted too small fluctuation levels, the wrong k, and a sensitivity to &, or I, which is not found experimentally (note that the currentless ATF results in table 1 are remarkably similar to those of TEXT). This resistivity gradient model, damped by parallel electron thermal conduction, was extended to allow impurities to produce a thermal and condensation drive [49]; the latter is now thought inapplicable. The same model applied to a stellarator predicts thermally driven drift waves. Now reasonable fluctuation levels (including g), k values (and the ratio of k values between TEXT and ATF), and some aspects of the sensitivity to impurities, are explained. There are
Having presented all this information, one might ask how to use it in a practical manner for predicting, for example, density profiles in complicated situations near the plasma edge. We know that 3-dimensional phenomena occur [24], making analytic equations intractable. One approach around this problem is to use the concept of test particles [50], as we used to discuss the effects of magnetic turbulence. The idea is to place particles, either eletrons or ions, in a background density and temperature which determines their collisionality. These test particles approximately follow field lines, suffering Coulomb collisions and “anomalous” collisions. The Coulomb collisions determine the parallel diffusion, i.e. how fast the particles travel along the field line. The anomalous transport is included as a random walk; every coherence time rc the particle suffers an “anomalous” collision, and is displaced radially by p, = (27,0)“2, with D = -Tf’E/~n the experimental anomalous diffusivity. For rc the decorrelation time
A
n/n
-90
POLOIDAL
ANGLE
8 c
Fig. 15. Test particle (TP) simulations in a stochastic magnetic field (the EML) incorporating the measured anomalous transport as a random walk. (a) The predictions for the radial and poloidal dependence of density, using an artificial reflecting boundary at 21 cm; (b) the measured and predicted poloidal dependence of density at r = 27 cm for the case shown in (a). The predictions are for the relative change in density (An/n, broken line) while the measurements are of the relative change in ion saturation current (AI+/i’, solid line).
A.J. Wootton / Edge turbulence
between the turbulence and a particle is taken (typically 3 ps in TEXT). Particles hitting material barriers are reintroduced to simulate recycling and ionization. By expressing the particle trajectories between the turbulent steps in an analytic form (actually a map), = lo6 particles can be followed even on a small computer. Fig. 15a shows the predicted ion density distribution in radius and poloidal angle for TEXT with externally applied stochastic fields (the EML). Strong poloidal asymmetries are predicted, with a poloidal mode number dictated by the magnetic topology. There is even experimental evidence that such structure exists; fig. 15b shows a section taken from fig. 15a at r = 27 cm, together with the poloidal dependence of the ion saturation current I+ to a poloidal array of single electrode probes. The measured poloidal structure is similar to that computed; the different amplitudes may be explained by Z+a nT’/‘, whereas the model result is for density only. With the stochastic magnetic fields the variation in I+ along what was a flux surface is extreme!
10. Summary and conclusions (1) The data can be interpreted as “blobs” which are both changing in time and drifting, or as a set of dispersive waves in packets with known correlation lengths and times. These two pictures are the same if the blobs are thought of as crests of the distribution function of, e.g., ii(k, w). (2) Frequencies (= 100 kHz) are largely determined by the radial electric field. Amplitudes are large ( = 30%) decrease towards the plasma interior, and are approximately predicted by a mixing length formula. k values are smaller than expected, = 1 to 3 cm-’ and increase with increasing magnetic field. (3) Concerning confinement and turbulence: (a) It is not sufficient to seek correlations between amplitudes (e.g. B or A) and transport coefficients or fluxes. Correlations and phase angles are important. (b) Tokamaks: electrostatic turbulence determines edge particle and energy fluxes in ohmic and additional heated (L-mode, presumably H-mode?) discharges. No useful scaling laws have been identified yet. Magnetic fluctuations usually increase, possibly becoming important for energy fluxes, with reduced parameter ohmic discharges and with additional heating (L-mode). (c) Stellarators look like tokamaks but the k values are smaller. In RFP’s magnetic fluctuations are important in determining energy fluxes. (4) While we don’t know all the driving and damp-
87
ing terms, the resistivity gradient (rippling) model looks good for tokamaks, together with a thermal drive (impurities) and sheared velocity flow decorrelation. (5) The test particle mapping approach allows anomalous transport effects to be included in complicated geometries.
Ackowledgements I thank Paul Schoch for unpublished HIBP data, Terry Rhodes for use of his data base, Phil Morrison for the standard map calculations, and Jack Daniels for inspiration. This work was performed under DOE grant DEFGO588ER53267.
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[35] Ch. Ritz, H. Lin, R.D. Bengtson, S.C. McCool, T.L. Rhodes and A.J. Wootton, in: Proc. 17th Eur. Conf. on Controlled Fusion and Plasma Physics, Amsterdam (European Physical Society, Petit-Lancy, Switzerland, 1990). [36] H. Lin, R.D. Bengtson and Ch.P. Ritz, Phys. Fluids Bl (1989) 2027. [37] Ch. Ritz et al., Phys. Rev. Lett. 62 (1989) 1844. [38] R.D. Bengtson et al., in: Proc. 17th Eur. Conf. on Controlled Fusion and Plasma Physics, Amsterdam (European Physical Society, Petit-Lancy, Switzerland, 1990). [39] A.B. Rechester, M.N. Rosenbluth and R.B. White, in: Intrinsic Stochasticity in Plasmas, Cargese Workshop 1979, Eds. D. Gresillon and M. Dubois (Ecole Polytechnique, Palaissau, France, 1979) p. 239. [40] B.V. Chirikov, Phys. Rep. 52 (1979) 263. [41] G.I. Taylor, Proc. Roy. Sot. A CXXXII (1931) 499. 1421 P.H. Rebut and M. Brusati, Plasma Phys. Contr. Fusion 28 (1986) 113. [43] J.A. Krommes, C. Obermann and R.G. Kleva, J. Plasma Phys. 30 (1983) 11. [44] J. Krommes and C-B Kim, Phys. Fluids 31 (1988) 869. [45] Y.J. Kim, K.W. Gentle, C.P. Ritz, T.L. Rhodes and R.D. Bengtson, Nucl. Fusion 29 (1989) 99. [46] V.I. Bugarya, Nucl. Fusion 25 (1985) 1707. [47] L. Garcia, P.H. Diamond, B.A. Carreras and J.D. Callen, Phys. Fluids 28 (1985) 2147. [48] T.S. Hahm, P.H. Diamond, P.W. Terry, L. Garcia and B.A. Carreras, Phys. Fluids 30 (1987) 1452. [49] D.R. Thayer and P.H. Diamond, Phys. Fluids 30 (1987) 3724. [50] A.J. Wootton and S.B. Zheng, University of Texas Fusion Research Center Report, Test Particle Calculations in TEXT with Resonant Magnetic Fields, FRCR 325 (1988).