Scalings and Plasma Profile Parameterisation of ASDEX High Density Ohmic Discharges
P.J. Mc Carthy, K.S. Riedel, O.J.W.F. Kardaun, H. Murmann, K. Lackner
SSCALINGS AND PLASMA PROFILE PARAMETERISATIONOF ASDEX HIGH DENSITY OHMIC DISCHARGES
P.J. Mc Carthy , K.S. Riedel , O.J.W.F. Kardaun,H. Murmann, K. Lackner, the ASDEX TeamMax-Planck-Institut f¨ur Plasmaphysik, EURATOM Association,D-8046 Garching bei M¨unchen, Fed. Rep. Germany On attachment from University College, Cork, Ireland New York University, 251 Mercer St., New York, USA
Abstract
A database of high density ( . 1. INTRODUCTION In most high magnetic field tokamaks, the experimental Ohmic energy con-finement time increases roughly linearly with plasma density over a largerange of densities. In general, moderate field devices observe a similar butweaker increase in confinement time at small to moderate densities. Unfor-tunately, the Ohmic energy confinement time saturates at higher values ofthe Murakami parameter n e R/B t [2]. This phenomenon is termed densityrollover and, under usual operating conditions in ASDEX, occurs at line-averaged densities around . × m − and magnetic fields on the order oftwo Tesla. (Pellet fuelled discharges and the recently discovered ImprovedOhmic Confinement regime on ASDEX [3] are not considered here.) Wepresent a statistical analysis of the bulk parameter scalings and profile shapesin this saturated high density regime. For a detailed discussion of the the-oretical aspects of profile shape determination, the reader is referred to [1],which we abbreviate as KRML.In our datasets, neither of which extends into the linear regime, no signifi-cant change was observed in the scaling of the total kinetic plasma energy asthe rollover density was crossed and exceeded. Plasma energy and the Ohmicpower scalings nearly cancel, leaving a moderate q a dependence ( τ E ∼ q . a ) inthe confinement time.To analyse the profile shapes, we fit all the measured profiles simultane-ously by means of a radial spline function each of whose coefficients dependson the plasma parameters q a , I p and n e . This powerful technique enablesus to quantify the various parametric dependencies as a function of radius.For the electron temperature profile, we find that the profile shape variationconsists almost exclusively of a q a dependence confined to r/a ≤ . 60. Thedensity profile also exhibits a q a dependence, though it is weaker than thatof the temperature. Unlike the temperature, however, we find a broadeningof the density profile near the plasma edge with increasing plasma current.Our research systematises and validates earlier graphical and qualitativestudies of “profile consistency”. (See [4] for a review of this topic.) The statis-tical approach enables us to make quantitative statements about the relative2trength of the interior q a dependence and the weaker exterior parametricdependencies of the profile shape.Previous authors have reported a variety of different results on the peakingfactor and the interior domain scalings. The multiplicity of results arises froma combination of tokamak to tokamak differences, different physics regimes,systematic measurement errors, and sometimes also from the use of relativelyunsophisticated analysis methods. We find, for instance, that normalising by 5. ASDEX discharges have a fixed circular geome-try with R plasma = 1 . 65m and a minor = . . . × m − s − for 10 out the 50discharges, which accounted for (cid:39) 50% of the datapoints. For the affecteddischarges, this implies a typical density variation about the mean of ± . 8T or 2 . . Nevertheless, the distinction between q a , B t and I p scaling willbe quite apparent. Bolometric measurements showed that radiation lossesfor these discharges (10% < P rad /P Ohmic < , increasing with density)were strongly localised at the plasma edge.For comparison, we present a complementary analysis of a second database(38 compressed datapoints from 38 deuterium discharges) of moderate den-sity ( n e (cid:39) . × m − ), moderate q a ( q a (cid:39) . n e , q a , and wall condition(gettered / non-gettered). Thus we can view the combined database as beingclustered in two cells of eight possible combinations. Indeed, some scalingdifferences between the two sets of data were visually apparent in prelim-inary efforts at fitting the combined dataset. By analysing each databaseseparately, we are able to estimate secondary, weaker effects that are notimmediately apparent in a joint analysis. In our case, these effects are weakcurrent and density dependencies of the outer section of the normalised den-sity profiles. If the two data sets were combined, these weaker effects wouldbe obscured by artificial secondary scalings arising from the clustering men-tioned above.As is typical of single machine databases, geometric parameters such as theplasma position and cross-sectional area vary very little (about 1%) and, oncethe channel positions have been mapped onto the normalised flux surfacecoordinate (0 ≤ r ≤ n e and two of the three parameters I p , q a and B t . Theseparameters represent the major control parameters which the experimentalistutilises to vary the plasma state in an Ohmic plasma.The condition of the plasma wall may significantly influence plasma per-formance. In an effort to include these effects, an additional plasma variable5uch as the effective ion charge Z eff , or the total Ohmic power P Ω is some-times used as an extra independent variable. Both Z eff and P Ω depend onthe control parameters n e , I p and q a , and therefore are not purely measuresof the condition of the plasma wall. To examine the extent to which the Ohmic power and Z eff vary indepen-dently of the control parameters n e , I p and q a , we carry out a principalcomponent analysis (PCA) [6]: The correlation matrix of the logarithms of n e , I p , q a , Z eff , and P Ω is calculated and diagonalised (see Table II). Eacheigenvalue is interpreted as the sample variance of the corresponding prin-cipal component over the database, and a small eigenvalue indicates a nearcollinearity between the original variables. Since the principal components,by construction, are statistically uncorrelated, the sum of a subset of eigen-values gives the cumulative variance explained by the corresponding subsetof principal components. We find that, for both databases, over of the(standardised) data variation can be explained by the first three components. The residual 3% of the total variance is attributed to noise in the tempera-ture measurements which affects Z eff and noise in U loop which affects both Z eff and P Ω . Hence, we discard the principal components having the twosmallest eigenvalues.The remaining eigenvectors span a three-dimensional subspace. This sub-space can be efficiently represented by linear combinations of any well-conditionedset of three of the five variables. That n e , I p and q a constitute such a well-conditioned set is checked by re-doing the PCA for these three alone. Theeigenvalues of the 3 × n e , B t , and I p because the correlation between I p and B t wasmuch less than that of either variable with q a (see Table II(c)). However,some scalings are given in terms of n e , q a and BI , where the latter parameterwas chosen to maintain a set of nearly uncorrelated bulk variables. Thedominance of q a and the weak I p influence in determining plasma profileshapes motivated us to use n e , I p and q a for the profile shape analysis.6 BULK SCALINGS In this section, we examine the scalings of the bulk plasma variables. Letus briefly summarise our results: The scaling of the total plasma energy asdetermined by kinetic measurements is virtually the same in both databases,i.e. no transition was observed going from the rollover to the saturatedOhmic confinement (SOC) regime. However, the diamagnetic and equi-librium mhd estimates of this parameter show strong differences betweenthe two databases. The plasma energy and the Ohmic heating power haveroughly similar parametric dependencies, which results in a confinement timescaling consisting mainly of a rather moderate q a dependence. Power law scal-ings for Z eff as well as for Z eff − /n e , indicating that the impurity density is independent of n e .We use the Spitzer value for the effective ion charge Z eff , which is cal-culated from the electron temperature profile assuming resistive equilibrium(well satisfied for the current flat-top profiles selected) and Spitzer conduc-tivity. We note that it cannot be used as an independent parameter in thetemperature profile regressions below. Recent Bremstrahlung measurementson ASDEX [7] show that for the SOC regime (to which our high densitydatabase belongs), Z eff is very flat over most of the plasma radius, thoughtending to rise strongly near the boundary. We adopt the conventional as-sumption of zero radial dependence here. The ion density is calculated withthe assumption that the sole impurity is oxygen. The plasma kinetic energy,Wp kin = (cid:82) ( n e T e + n i T i ) dv , is calculated by assuming the ion temperatureis equal to the electron temperature, T i = T e . This assumption is justifiedwhen the electron-ion energy exchange time is much shorter than the energyconfinement time. In the main database, the typical values are τ ei = 5 msecand τ E = 75 msec. In some of the moderate density discharges, this conditionis not satisified.As an independent measure of the plasma energy content we also make useof the diamagnetic flux measurement on ASDEX whose interpretation is notaffected by the uncertainty in T i and is further simplified, in the case of Ohmicplasmas, by the absence of pressure anisotropy. The extreme sensitivity of the7easurement ( ψ dia ψ tor (cid:39) − ) to such factors as slight mechanical displacementsof the diamagnetic loop means, however, that the typical error associatedwith the derived value for beta poloidal is δ ( β p ( dia ) ) = ± . 05 . For β p ( dia ) (cid:39) . ± (cid:39) ± β pol + l i / β pol , we need an estimate for l i / 2, which, as is well known,cannot be determined from equilibrium data in the case of circular plas-mas. From an earlier investigation, we use the following empirical relationfrom a parameterisation of current density profiles derived from experimen-tal temperature profiles and the assumption of Spitzer resistive equilibrium: l i / . 332 + . 199 ln q a .The regression models considered are of the form y = (cid:80) i α i x i + (cid:15) where y and x denote the logarithms of the dependent and independent bulk plasmavariables respectively. The root mean square error (RMSE) of the fit is (cid:113)(cid:80) Ni =1 ( y i − ˆ y i ) / ( N − p ) where p is the number of independent variables(including the intercept) and ˆ y i is the fitted value of y i . As our responsevariables are natural logarithms, the RMSE corresponds to a relative errorin the physical variable. We also quote the squared multiple correlationcoefficient R , which represents the fraction of total variance about the meanaccounted for by the fit: R = (cid:80) i (ˆ y i − y ) / (cid:80) i ( y i − y ) . In the bulk scalingresults that follow, all physical variables are expressed in the units of TableI. Total Plasma Energy Scaling The plasma energy content from kinetic data for the main database satisfiesWp kin = 133( ± n e . ± . I p. ± . B t. ± . RMSE = . 04 R = . kin = 128( ± n e . ± . I p. ± . B t. ± . RMSE = . 04 R = . dia = 130( ± n e . ± . I p. ± . B t. ± . RMSE = . 04 R = . dia = 83( ± n e . ± . I p. ± . B t. ± . RMSE = . 06 R = . mhd = 156( ± n e . ± . I p. ± . B t − . ± . RMSE = . 03 R = . mhd = 255( ± n e . ± . I p . ± . B t − . ± . RMSE = . 05 R = . is discussed.From the indicated uncertainties (1 standard deviation) of the regressioncoefficients, we see, for instance, that the B t scaling for Wp kin is insignificantfor the main database, while in the secondary database it is (just about)statistically significant. In both cases, omission of B t affects the goodnessof fit only slightly in absolute terms (eq. (A7) gives ∆(R ) (cid:39) . , . n e , I p and B t are small (see Table II(c)). This results in small correlations between theregression estimates of the exponents of n e , I p and B t . In our regressions, allthese correlations are less than 0.15 for the main and less than 0.35 for thesecondary database. For assessing the significance of the difference betweena postulated scaling law and our empirical scalings, such low correlationsbetween the estimates can be neglected in practice.The toroidal field shows the most extreme variation in the six Wp scalingspresented above. There is also, however, a strong tendency for the I p coeffi-cient to move in the opposite direction to B t when going from one scaling toanother. This suggests a representation with q a . To preserve the near inde-pendence of the regression estimates, we choose BI as the conjugate variable[8]. In the new representation, we have the following regression coefficientsfor the six cases (same order as above) B t I p : ( . , . , . , . , . , . , a : ( − . , − . , − . , − . , − . , − . . The errors in the coefficients are roughly the same as in the ( I p , B t ) scal-ing. Obviously, to a good approximation, the simple transformation α BI = ( α B + α I ) and α q = ( α B − α I ) holds. We roughly summarise: Wp (cid:39) ( BI ) . ± . q − . ± . a , where here the approximate ranges are indicated forthe regression coefficients in the six cases above. Hence, at constant n e , the BI scaling is nearly constant, but the q a scaling varies considerably with thetype of measurement and with the database.The reasons for the observed q a scaling differences may be partly due tophysics, and partly to systematic errors. Wp mhd is particularly vulnerableto errors in the (cid:39) l i / β pol + l i / 2. To get an idea of theinfluence of this error, we performed a sensitivity analysis. Using l i / . 332 + δ + ( . 199 + δ ) ln q a , we made a number of Wp mhd regressions fordifferent choices of δ and δ in the range -.1 to +.1. (Note that δ candescribe an error in β pol + l i / l i / . ) We found that the exponentfor q a in the Wp mhd regression varied as − . 47 + 1 . δ − . δ for the maindatabase and − . 75 + . δ − . δ for the secondary database. Thus an errorof − . q a coefficient for l i / q a exponents of − . 31 and − . 55 for main and secondary databasesrespectively. We conclude that this error source cannot reconcile the scalingof Wp mhd with q a for the secondary database with the remaining five Wpscalings.Now we turn to Wp dia . Suppose we have an offset δ β in β pol,dia . A sen-sitivity analysis showed that this roughly gives an offset of − δ β in the q a coefficient (for both data bases). The density dependence is offset by − δ β forthe main data base and by − . δ β for the secondary database. It is notedthat the observed 6 kJ systematic difference between the Wp kin and Wp dia measurements corresponds (at I p = . β pol of 0 . , which is consistent with the observed difference in q a dependence betweenWp kin and Wp dia for either database. Similarly, it explains half of the differ-ence in n e dependence. It does not, however, offer a satisfactory explanationfor the Wp dia or Wp mhd scaling differences between the databases, for whichwe must appeal to the already referred to differences in operating regimes10gettered/non–gettered walls, high densities/rollover densities, etc.). Volume Averaged Electron Temperature Both databases are in rough agreement with the Pfeiffer-Waltz [9] and JETOhmic scalings [10]. The main database satisfies < T e > = 0 . ± . n e − . ± . I p. ± . B t − . ± . RMSE = . 04 R = . < T e > = 0 . ± . n e − . ± . I p. ± . B t. ± . RMSE = . 05 R = . B t is insignificant for both databases and its omission has a very small( (cid:39) . . Though a natural candidate for determining the tem-perature, we have not included Z eff in the list of regressors, since, in ourcase, it is derived, assuming Spitzer resistivity, from the temperature profileitself: Z eff ∝ (cid:104) T (cid:105) area V loop I p R plas (neglecting variations in the Coulomb logarithm). Loop Voltage/Ohmic Power We present the loop voltage scalings. The Ohmic power scalings differ fromthese by exactly one power of I p . For the main database, the loop voltagescales as V loop = 2 . ± . n e . ± . I p. ± . B t − . ± . RMSE = . 04 R = . V loop = 2 . ± . n e . ± . I p. ± . B t − . ± . RMSE = . 03 R = . I p on both sides, we get necessarily higher R values (.935 and .984) for the Ohmic power regressions (The RMSE valuesare unchanged). Spitzer Z effective For the main database, the Spitzer Z eff scales as Z eff = 2 . ± . n e − . ± . I p. ± . B t − . ± . RMSE = 0 . 05 R = . Z eff gives an entirely different scal-ing: 11 eff − . ± . n e − . ± . I p . ± . B t − . ± . RMSE = . 18 R = . Z eff scales as Z eff = 2 . ± . n e − . ± . I p. ± . B t − . ± . RMSE = . 09 R = . Z eff − . ± . n e − . ± . I p . ± . B t − . ± . RMSE = . 16 R = . n H ; D = n e − (cid:80) i n i Z i (i-summation over impurity speciesonly), Z eff ≡ (cid:80) j n j Z j /n e (j-summation over all species) can be re-expressedas Z eff = 1 + (cid:80) i n i ( Z i − Z i ) n e With this representation, we see that the n e exponents in the impurity scalingssuggest an impurity density (almost) independent of the line density. Thestrong I p and B t scalings are not so readily interpretable. Energy Confinement Time The I p and n e dependencies of the total plasma energy and the Ohmic powerapproximately cancel to leave a relatively weak q a dependence in τ E . Thismakes the energy confinement scalings less pronounced. The main databaseyielded the following scaling for the kinetic τ E : τ Ekin = 56( ± n e . ± . I p − . ± . B t. ± . RMSE = . 07 R = . τ Ekin = 53( ± n e . ± . I p − . ± . B t. ± . RMSE = . 05 R = . B t I p and q a : τ Ekin = 68( ± n e . ± . ( BI ) . ± . q a. ± . (High density database) τ Ekin = 72( ± n e . ± . ( BI ) . ± . q a. ± . (Moderate density database)The confinement time derived from the diamagnetic measurement of theenergy content for the main database scales as τ Edia = 69( ± n e . ± . ( BI ) . ± . q a. ± . RMSE = . 06 R = . τ Edia = 50( ± n e − . ± . ( BI ) . ± . q a. ± . RMSE = . 06 R = . τ Emhd = 77( ± n e . ± . ( BI ) . ± . q a. ± . RMSE = . 06 R = . τ Emhd = 103( ± n e − . ± . ( BI ) − . ± . q a − . ± . RMSE = . 06 R = . q a scaling between the two databases suggests the pos-sibility of a quadratic (i.e. ln q a ln q a ) dependence [8]. No curvature inthe ln q a dependence was apparent, however, in a plot of ln τ Edia for bothdatabases with their respective n e and BI dependencies removed. Fig. 3shows ln τ Edia / ( BI/. . ( n e /. . for the main database and ln τ Edia / ( BI/. . ( n e /. − . for the secondary database. A similar picture holds for ln τ Ekin , whereas forln τ Emhd there appeared to be a deterioration at high q a (figures not shown).Three points at low q a seem to be somewhat outlying in Fig. 3. However,their removal does not change the ln q a regression coefficient by more than onestandard deviation. Unlike some previous reported results (see, e.g., [13] sec-tion 4) we found no significant confinement time deterioration with densityin the SOC regime . Fig. 4 shows τ Ekin (with I p and B t dependencies re-moved) versus n e for both databases, i.e. ln τ Ekin / ( I p /. − . ( B t / . for themain database and ln τ Ekin / ( I p /. − . ( B t / . for the secondary database.Similar behaviour is observed for both τ Edia and τ Emhd . For the difference in q a scaling, the reader is referred to the sensitivity discussion in the subsectionon total plasma energy. 4. CENTRAL TEMPERATURE AND PEAKING FACTORS4.1 Electron Temperature, Density and Pressure at r = 0.2 To enable reconstruction of absolute profiles from the profile shape scalingspresented later, we regressed T e , n e and p e = 1 . n e T e at the 20% flux radius.This normalisation radius lies inside the inversion radius for our q a range andalso has data points on either side of it. (The YAG channel closest to themagnetic axis lies typically on the 14% flux radius.) The high density low q a database satisfies 13 e = . ± . n e − . ± . I p . ± . B t . ± . RMSE = . 04 R = . n e = . ± . n e . ± . I p − . ± . B t . ± . RMSE = . 02 R = . p e = . ± . n e . ± . I p . ± . B t . ± . RMSE = . 05 R = . T e = . ± . n e − . ± . I p . ± . B t . ± . RMSE = . 05 R = . n e = . ± . n e . ± . I p − . ± . B t . ± . RMSE = . 03 R = . p e = . ± . n e . ± . I p − . ± . B t . ± . RMSE = . 06 R = . √ n e B t – like dependence of p e for bothdatabases. On regressing p e at each of the five most central YAG channels,which typically lie between the 14% and 25% flux radii, we found a similarabsence of an I p dependence (coefficients ranged from 0. to .05 with a typicalstandard deviation of .05 for the main database; -.13 to .18 with a typicalstandard deviation of .075 for the secondary database). The remaining 10channels all exhibited strong I p dependencies (coefficients up to 2.0). Hence,we note that the central electron pressure is independent of the total plasmacurrent. Since the onset of I p dependence occurs for those channels whoseradii roughly correspond to minimum values of the sawtooth inversion radius( r inv ( min ) (cid:39) /q a ( max ) = . 30 and .24 for the main and secondary databasesrespectively), we speculate that this I p independence is coupled to sawtoothstability and sawtooth induced transport. Though we later analyse, as a function of radius, the bulk parameter de-pendencies of the local shape parameter L − T e , we present here results for theusual single-parameter measure of the temperature profile shape, namely thetemperature profile peaking factor. We show that the observed peaking factoris very close to that expected assuming a Spitzer resistive equilibrium .14e first express the local cylindrical safety factor q ( r ) as q ( r ) = rB tor RB pol ( r ) = 2 πr B tor µ RI ( r ) = 2 B tor µ R < J > r (1)where < J > r = πr (cid:82) rr (cid:48) =0 J ( r (cid:48) )2 πr (cid:48) d r (cid:48) is the current density averaged up toradius r .Assuming a classical resistive equilibrium, we immediately have that < T / > r < T / > r =1 (cid:39) q r =1 q r (2)where the uncertainty arises from the fact that we neglect radial variations(assumed weak) in Z eff or the Coulomb logarithm. Since all profiles aresawtoothing, we have q ( r 06 R = . 04 R = . T / 10 R = . 06 R = . n e and I p coefficients are at best marginallysignificant at the 95% level, the results indeed strongly indicate that thepeaking factor (eq. (2)) is determined solely by q a . We now assume a sole q a T / 06 R = . 04 R = . T / 10 R = . 07 R = . q a exponent is unity to within two standard deviations.Similarly, the constant factors are within two standard deviations of unity(with the marginal exception of the second regression: . ≤ const ≤ . q / a ≤ T (0) < T > ≤ q a (3)derived by Waltz et al [15] using classical resistivity plus sawtoothing. Sincewe use 5. PROFILE SHAPE ANALYSIS The next two sections are devoted to a detailed description of profile pa-rameterisation techniques and the ensuing methods of data analysis, tailoredto our case. For a more theoretical background, the reader is referred toKRML. The experimental results are presented in section 7.We assume that the logarithm of the temperature satisfies ln T ( r, q a , I p , n e ) = µ ( r, (cid:126)x ) + (cid:15) where (cid:15) is a random error. The deterministic part, µ ( r, (cid:126)x ), is rep-resented as a spline with reasonably high resolution in the radial direction,16nd a simple (polynomial-type) dependence on the plasma-parameters. Thecoefficients of the representation are determined by fitting all profiles simul-taneously in a weighted least squares regression. The temperature and density measurements are obtained using the AS-DEX sixteen channel YAG Thomson scattering diagnostic [5] with a sam-pling rate of 60 Hz. This system consists of sixteen spatial channels locatedin the vertical plane at R = 1 . 63 m. They are spaced at approximately 4 cmintervals from Z = . 200 m to Z = − . 394 m. We did not use the 16 th channelwhich lies very close to or on the separatrix, as the measurement failed forthe majority of profiles in this database. The radius (averaged over all pro-files) of the flux surface passing through each channel is presented in column2 of Tables III and V for the main and secondary databases respectively. In this subsection, we discuss continuous radial representations of plasmaprofiles. We consider data consisting of n separate compressed profiles of aspatially varying plasma variable such as temperature or density, at p distinctradial points. Each compressed profile is the average of m = 12 consecutivemeasurements taken at 17 msec intervals. We do, however, make use of theuncompressed profiles for the purpose of estimating the channel-by-channelraw measurement fluctuations within each discharge. Thus our temperaturedata can be described by T i,j ( r (cid:48) l ), where i = 1 , . . . , m labels the uncompressedtimepoint, j = 1 , . . . , n is the compressed profile index, and l = 1 , . . . , p denotes the radial channel number. We make a preliminary transformationof the physical measurement locations r (cid:48) l to the corresponding flux surfaceradii r l . Continuous representations have the following characteristics: a) A largenumber of dependent variables, represented by point data, is replaced by asmall number of coefficients which nevertheless will be sufficient to representall significant features of the profiles. b) Profiles measured at two differentsets of radial locations may be compared. This is relevant, e.g., where wewish to compare YAG temperature measurements with electron cyclotronemission (ECE) data measured at different spatial locations. c) Smoothnessis imposed in the belief that the profiles are in diffusive equilibrium.17nstead of fitting the profile itself, we choose to fit its natural logarithm Y .Minimising the residual of the logarithm of the plasma profile corresponds tominimising the relative rather than the absolute error. Preliminary compar-isons with low order spline or polynomial fits to the actual profiles revealedthat the logarithmic fit tended to have not only smaller residual errors on thelogarithmic scale but also on the usual physical scale. This indicates that the‘exponentiated form’ of the logarithmic fit is a better approximation to theactual plasma profiles than a comparable low order fit to the linear profile.We note that the difference between logarithmic and linear fits decreases asmore regression parameters (either spline knots or higher order polynomialterms) are added. Logarithmic fits have several other advantages. Firstly,the predicted profile can never be negative. Secondly, well-known power–lawtype scalings reduce to linear models. Finally, if the noise level is propor-tional to the absolute value of the measurement (an admittedly idealisedsituation), then, on the logarithmic scale, unweighted least squares may beused. Spline representations, which we employ here for profile parameterisation,give flexibility in choosing between local resolution and compact global repre-sentation. The profile parameterisations presented in the present work arebased on twice continuously differentiable splines with a selectable numberof knots, ν. The profile is forced to be parabolic inside the first knot, i.e.the region enclosing the magnetic axis. The radius is decomposed into ν + 1regions with knots at r , r , . . . , r ν . Such a profile may be parameterised18xplicitly by: µ ( r ) = µ ( r ) + µ (cid:48)(cid:48) (0)( r − r ) / ≤ r ≤ r (Inner Region) µ ( r ) + µ (cid:48)(cid:48) (0)( r − r ) / c ( r − r ) for r < r ≤ r ( Region 1) µ ( r ) + µ (cid:48)(cid:48) (0)( r − r ) / c ( r − r ) + c ( r − r ) for r < r ≤ r (Region 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . µ ( r ) + µ (cid:48)(cid:48) (0)( r − r ) / c ( r − r ) + c ( r − r ) + . . . + c ν ( r − r ν ) for r ν < r ≤ µ ( r ) as µ ( r ) = ν (cid:88) k = − α k H ( r k ) ϕ k ( r ) (5)where ν is the number of knots used, α k represent the spline parameters: µ ( r ) , µ (cid:48)(cid:48) (0) , c , c , . . . , c ν ; ϕ k ( r ) are the polynomials: ϕ − ( r ) = 1, ϕ ( r ) = ( r − r ) / ϕ ( r ) = ( r − r ) , .. . . , ϕ ν ( r ) = ( r − r ν ) , and H ( r k ) = (cid:26) r < r k r ≥ r k (6)is the Heaviside unit stepfunction ( r k = 0 , , r , r , . . . r ν ) .As well as the parabolic restriction near the axis, a ‘natural’ spline bound-ary condition, µ (cid:48)(cid:48) (1) = 0, was imposed in practice. The spline fits werecarried out using the SAS REG procedure [17] which contains convenientpossibilities for restricted regression. A similar spline model for transportanalysis of individual profiles has been used in [19]. Since the plasma profile shapes depend on the bulk plasma variables suchas q a , the spline coefficients will be functions of these parameters. Since19ur database is not expected to contain sharp transitions in behaviour inparameter space, a low order parametric representation is expected to beadequate. Therefore, we approximate the smooth parametric dependencies ofthe profile shape by linear or possibly quadratic polynomials in the logarithmsof the bulk variables. Let (cid:126)x = ( x , x , x ) be the vector consisting of the bulk plasma variables q a , I p , n e . We define the linear basis functions g ( (cid:126)x ) = 1 (intercept), g j ( (cid:126)x ) =ln x j /x ∗ j , j = 1 , , ; where x ∗ j is a representative value of the variable x j in thedatabase of interest. For ease of comparison, we choose for both databasesthe same normalising values q ∗ a = 2 . I ∗ p = . n e ∗ = . × m − ,although these do not constitute a typical parameter set for the secondarydatabase. By normalising the bulk variables to x ∗ j , the value of the responsevariable at the intercept in the regression becomes the predicted value at (cid:126)x = (cid:126)x ∗ .The full profile representation can be written as Y ( r, (cid:126)x ) = (cid:88) k,j α k,j g j ( (cid:126)x ) H ( r k ) ϕ k ( r ) + (cid:15) ≡ µ ( r, (cid:126)x ) + (cid:15) (7)where (cid:15) is an error term, whose structure will be discussed in the next sub-section. The basis functions are as defined before. Note that higher orderterms can be included, if necessary, by extending the set of possible basisfunctions: g j,l ( (cid:126)x ) = ln x j /x ∗ j ln x l /x ∗ l , etc. In this subsection, we present and motivate the splitting-up of the totalregression error into contributions that are attributable to different physicalsources. An efficient analysis should take into account the particular featuresof such an error structure. In the next subsection, we discuss the methods ofestimation we applied in our case.We distinguish between several categories of random profile variations. Weuse the term ‘internal’ variations, to denote fluctuations on a time scale atleast as fast as that of the diagnostic sampling rate. These include statisticalnoise from the measuring process and plasma fluctuations arising, in partic-ular, from the m = 1 sawtooth instability. Discharge-to-discharge variations20re changes in the plasma profiles not observed within a single discharge.These discharge variations include effects such as impurity accumulation onthe diagnostic windows and the condition of the plasma wall. These impu-rity and plasma wall effects tend to vary to an even larger extent from oneexperimental operating period to the next. In addition, discharges separatedby a recalibration of the YAG system can exhibit systematic differences inthe measured profiles.As discussed in [12] appendix B, this hierarchy of temporal scales forplasma variation generates a compound error structure which can be treatedstatistically. We give a simplified discussion here. For convenience, we as-sume normally distributed errors, although this assumption can be relaxedin most of the discussion. Since the profiles in our database are already aver-aged over twelve consecutive time samples (see introduction), they no longerpossess the same variance as the original uncompressed observations. Fortime averaged datapoints, Y ( r, (cid:126)x ) = m (cid:80) mi =1 Y i ( r, (cid:126)x ), the total unexplainedvariance can be decomposed into: σ tot = σ int m + σ dis (8)where σ int is the variance of the internal or ‘within-discharge’ fluctuationsof the uncompressed profiles, m is the number of timepoints in the compressedprofile (in our case m = 12) , and σ dis denotes the variance due to discharge-to-discharge variations.To estimate the within-discharge variance ˆ σ int , we analysed the originaluncompressed data and, for each channel, calculated the empirical variancefor each 12-point set separately. We estimated ˆ σ tot by regressing the set of n (compressed) datapoints for each individual measurement channel againstthe bulk variables and noting the unexplained variance. The difference, asgiven by the third term in eq. (8), is an estimate of the discharge-to-dischargevariance.In Tables III(b) – VI(b) columns 3 and 4, one can see the estimates of σ int (scaled for compressed profiles) and σ tot for each of the 15 channels. It isclear from the large channel to channel variation displayed in these tables,21hat it would not be justified to make model assumptions that σ int and/or σ tot are the same for all channels. We determine the spline coefficients, including parametric dependencies,by fitting all profiles simultaneously in a weighted least-squares regressionwhere the weights, W ( r l ), depend on the channel location.We wish to determine that vector of coefficients α which minimises (cid:88) j,l (cid:16) Y j,obs ( r l ) − Y j,fit ( r l , α ) (cid:17) W ( r l ) (9)where W ( r l ) , l = 1 , ..., 15 are appropriately chosen weights for each of the15 YAG channels. We investigated two approaches for determining W ( r l ). In the first method, we rely on the total unexplained variance for eachchannel (as discussed in the previous subsection) as a measure of the channelweighting: W ( r l ) = ˆ σ − tot ( r l ). A second approach to the selection of theregression weights, is the iterative estimation of the residual variance of thespline fit at each channel. At the k th iteration we have:ˆ σ k +1) ( r l ) = 1 n n (cid:88) j =1 (cid:16) Y j,obs ( r l ) − Y j,fit ( k ) ( r l , ˆ α k ) − δY k ( r l ) (cid:17) (10)where δY ( r l ) is a possible systematic bias in fitting the l th measurementchannel which can be estimated by including an indicator dummy variablefor each of the 15 channels in the regression. The inclusion of this termprevents undue downweighting of channels where the parameterised profilemay consistently fail to match the observed data. For the first iteration, thevariances ˆ σ ( r l ), are initialised to unity (equal weights). The iteration isterminated after the third iteration. We then have W ( r l ) = ˆ σ − last iter. ) ( r l ).If the regression model and the assumed error structure is correct, this is alikely to be a consistent and efficient estimate (see e.g. [20]). Nevertheless,we regard ˆ σ − tot ( r l ) as a more robust estimate than ˆ σ − last iter. ) ( r l ) since it de-pends only on the fit by the bulk variables whereas the latter estimate alsodepends on the spline model and has the additional problem of the strong22anti)correlation of outer channel residuals. Accordingly, we preferred to use W ( r l ) = ˆ σ − tot ( r l ) . The dominant effect of this reweighting is to decrease theinfluence of the channels near the plasma boundary where the relative erroris largest. In the context of profile parameterisation, some relevant statistical testsfor the significance of including additional variables are discussed in KRML.In the case of independent errors, these criteria are given by the F test [6]and Mallows C p statistic [24]. Mallows C p statistic is the sum of the totalbias in the regression and the total variance of the predicted values. As morefree parameters are added, the bias decreases but the variance increases. Todetermine whether to add another parameter, one can look at the change inthe C p statistic or apply the F test.It should be noted that in practice these statistical approaches may eitherunder- or overestimate the significance of the additional variables since thecorrelations in the errors are neglected. In addition, these tests neglect sys-tematic errors and assume that the ideal data, without measurement errors,is exactly describable by the regression equation under consideration. Thusmany spurious dependencies may be included and real dependencies missedby unthoughtful or automated use of these methods. 6. PROFILE PARAMETERISATION TECHNIQUES In this section, we discuss a number of practical aspects encountered dur-ing our investigation, which are expected to be useful in any profile analysis.Logarithmic representations are employed throughout, for the reasons out-lined in subsection 5.1. Initial efforts concentrated on fitting polynomial representations of theform: T ( r ) = T exp ( ar + br + cr )This model was successful in reproducing the general properties of the AS-DEX profiles, but not detailed features. Sharp gradients and local flattenings23due, perhaps, to magnetic islands) were poorly modelled. A disadvantageof the above model, exacerbated by the addition of higher order polynomialterms, is the insensitivity of the inner region to r and higher powers as wellas the high degree of stiffness of these polynomials. To enable us to fit eachregion of the profile in moderate powers of the radial coordinate, we turnedto spline representations. Following [21] we first used a five parameter, two knot Hermitian spline,i.e. one with no continuity requirement on the second derivative. This turnedout to be clearly better in parameterising steep gradients and abrupt spatialtransitions in profile shape than the polynomial model. By experimentation,we found that a total of four knots, requiring seven regression parameters,gave a practical balance between fitting accuracy and significance of thespline coefficients.A serious disadvantage to the Hermitian spline emerged, however. Byallowing discontinuous second derivatives at the knots, continuous transitionsin plasma behaviour were modelled as sharp jumps across knot boundaries.This effect was especially prominent in the slope of the inverse fall-off lengthas a function of q a . After some investigation, we opted instead for the twicecontinuously differentiable spline model, eq. (4). The knot positions are chosen such that the measuring channels are dis-tributed roughly equally in the various regions between and outside of theknot positions. Too many knots result in spuriously oscillatory fits. Theknot locations were varied manually to achieve a near ‘optimal’ fit as de-termined by the balance between goodness of fit and significance of the fitcoefficients. We decided on a set of five knots at the following locations: r knot = . , . , . , . , . . The innermost channel is typically located at r = . 14 and the outermostchannel at r = . 89 (Table III column 2). When third degree polynomialswere used in the innermost and outermost regions, the extrapolated curves(to r = 0 and r = 1 respectively) had unphysical oscillations. These oscil-lations were eliminated by reducing the number of free parameters for these24egions. Near the origin, the profile was forced to be parabolic (this is alreadyenforced in eq. (4)). The so-called natural boundary condition, µ (cid:48)(cid:48) (1) = 0,was applied at r = 1. We investigated the regression fits with the naturalboundary conditions applied to (a) all spline coefficients (b) bulk parameter–dependent spline coefficients only. Case (a) resulted in considerably higherfitting errors than case (b) for, in particular, the outer channels of the maindatabase density profile fit. However, we concluded that the improvement incase (b) was at the expense of overfitting of the outer channels and we presenthere only the results of the fully applied boundary conditions. In a prelimi-nary version of this work [12], the results of case (b) are presented. Thus our5-knot set yields a model of seven spline coefficients with one boundary con-straint per bulk parameter basis function used in the fit. The profile param-eterisations presented later were carried out using the linear basis functions g j ( (cid:126)x ) , j = 0 , , , , only. Some quadratic and cross terms were very signifi-cant in preliminary regressions involving both databases simultaneously. Forreasons given in the introduction, however, the results we present come fromseparate profile shape analyses for each database. For these regressions, sec-ond order terms were rarely significant and the goodness of fit was scarcelyaffected by the restriction to linear terms. Using the three bulk parameters q a , I p and n e , we have a regression model with a total of (intercept + 3 bulkparameters) × (7 radial coefficients - 1 boundary condition) = 24 degreesof freedom to fit (e.g. for the main database) 15 × 105 = 1575 individualtemperature (or density) data. With the foregoing boundary conditions, thisspline representation tended to be stable in extrapolating profile behaviourinto regions where there were no measurement channels. The goal of our profile analysis is to determine the dependence of the pro-file shapes, i.e. the functions L − T e ( r ) = T e ddr T e (r) and L − n e ( r ) = n e ddr n e (r),0 < r < 1, on the plasma parameters. This brings up the problem of fittingthe profile size. We found that when we made a simultaneous fit of the sizeand the shape of the profile, the residual sum of squares was dominated bythe uncertainty in fit of the profile size. Therefore, a model was fitted whichprovided a free parameter to fit each individual profile size. This normal-25sation procedure, justified because the profile size scales out of the profileshape definition above, causes very significant reductions in the residual sumof squares of the shape regressions.Originally, we normalised each profile by its line-average, calculated fromthe spline fit. This greatly reduced, but did not minimise, the residual errorin the profile parameterisation since the line-average is itself a function ofthe profile shape. Instead, we estimated the profile size parameters, usingthe SAS procedure GLM [17], by treating the profile index as an indicatorvariable. This yielded as normalising factor the radially independent term inour spline repesentation, i.e. µ ( r ), the profile value at the first knot which issited at r = . 2. Normalisation has the effect of reducing by one the numberof degrees of freedom for each individual spline. Hence, the total numberof degrees of freedom for the 5-knot spline with coefficients dependent on 3bulk parameters (as described in the previous subsection) is reduced from 24to 20. In the course of determining ˆ σ tot ( r l ) for each YAG channel, plots of residu-als versus shot number revealed that the secondary database residuals, whosedischarges spanned a period of over six months in contrast to the one weekspan of the main database, fell into four distinct groupings which we ascribeto four distinct experimental operating periods (see introduction). This 4-cluster formation was observed for all 15 channels, although the patternformed by the clusters differed for each channel. To remove this operat-ing period contribution to the overall unexplained channel variances for thisdatabase, and hence to enable a comparison to be made between the twodatabases, separate indicator variables for each operating period were addedin the individual channel regressions used to determine ˆ σ tot . These indica-tor variables were not, however, included in the profile shape regressions.To do so would have required an additional 60 independent variables (4 foreach channel) which, in our judgement, would have led to overfitting of theprofiles. Apart from the operating period effects mentioned above, plots of raw26ersus fitted data for the same channel-by-channel regressions revealed thata small number of individual channel measurements from both databasesproduced strongly outlying residuals (the worst case was one of 8.5 stan-dard deviations). To arrive at a quantitative criterion for identifying suspectdata, we analysed the Studentised residuals. A Studentised residual is thedifference between the observed and fitted value, normalised to the RMSE.For normally distributed errors, they have approximately a standard normaldistribution. If we consider a single Studentised residual, the probabilitythat it lies outside ± c is (cid:15), where (cid:15) (cid:39) − (2 π ) − (cid:82) + c − c e − x / dx. Consideringnow n uncorrelated residuals together, we have that the probability of all n residuals lying inside ± c is (1 − (cid:15) ) n . Hence the probability of at least oneamong n Studentised residuals lying outside ± c is given by 1 − (1 − (cid:15) ) n = β ,say. Provided the correct model is used to fit the data, we suspect any out-lier whose Studentised residual exceeds ± c β for a suitably small β (using (cid:15) = 1 − (1 − β ) /n (cid:39) β/n for β << , we invert the probability integral todetermine c β ). We chose β = 1% which, for n = 105 and n = 38 , yields c = 3 . 90 and c = 3 . 64 for the main and secondary databases respectively. Us-ing this criterion, we identified 6 suspect outliers from the main database and18 from the secondary, amounting to 0 . 2% and 1 . 5% of the data respectively.Profiles containing any suspect observations were now examined individ-ually. In most cases it was visually obvious that the affected channel wasinconsistent with the rest of the profile, and such observations were markedas bad data. One discharge accounted for the majority of the suspect data inthe secondary database. On inspection, it was clear that the quality of theprofile data for this discharge was so poor, that it was excluded entirely fromthe subsequent analysis, thereby reducing the number of discharges from 38to 37 for this database. On the other hand, several suspect observations froma single profile in the main database were not visually inconsistent with therest of that profile’s data. On investigating further, it turned out that thisdischarge had the highest B t value (2.73 T) in the database. This highlightsthe need to examine all suspect outliers individually, since the influential po-sition of this data suggests an inadequacy in the model used to fit the datarather than in the data itself, and its rejection would be quite unjustified.27he small number of observations finally deemed to be faulty (a total of 4points affecting 3 profiles in the main database; 12 points affecting 9 profilesin the secondary database) were deleted from the regression. Ten of the fifteen YAG channels in ASDEX are located in nearly symmetricpositions with respect to the horizontal midplane. By examining the residualerrors for each channel separately, an up-down asymmetry was found in thedensity profile (up-down difference (cid:39) . µm ≤ λ scatt ≤ . µm ), the impurity accumulation causes a spectrallyuniform reduction in transmission, then the temperature, which is calculatedfrom the ratio of the scattering signals of two spectral channels at the samespatial location, is insensitive to such asymmetries and only the density isaffected.To estimate and correct for this density profile asymmetry, we expandedthe set of bulk parameter dependent, radial basis functions to include a singleasymmetry indicator variable which takes the value of +1 for the channelsabove the midplane and − 7. EXPERIMENTAL RESULTS7.1. Temperature Profile In this subsection, we present a detailed graphical representation of our28esults. The dominant shape dependency is a peaking of the profile inside r = . q a . The profile shape shows very little dependence onany bulk parameter outside r = . 6. Thus our result is similar to Fredrickson’s[18] (profile shape invariant for r ≥ . 4) and Murmann’s [22] (profile shapeinvariant for region outside influence of q = 1 surface), though our moreelaborate model enables us to examine more sensitively the extent of the q a dependent region.In Figs. 5 and 6 we display, as a function of radius, the reference pro-files and the parametric dependencies of the negative inverse fall-off length(IFOL), normalised to the inverse minor radius. Dashed curves indicate lo-cal 95% confidence bands. The corresponding global 95% confidence bandsare, in this case, approximately 1.5 times wider. See KRML for a discus-sion on local and global confidence bands. By the term reference profile wemean the evaluation of the parameterised spline fit at the representative setof parameter values (cid:126)x = (cid:126)x ∗ as discussed in section 5.2. Since the vector ofbasis functions (cid:126)g ( (cid:126)x ) = (1 , ln x /x ∗ , ln x /x ∗ , ln x /x ∗ ) reduces in this case to (cid:126)g ( (cid:126)x ∗ ) = (1 , , , q a dependencewhich, using eq. (7) (with g ( (cid:126)x ) = ln q a − ln q ∗ a ) , is given by ∂ ( − L − T e ) ∂ ln q a = − ∂ µ ( r, (cid:126)x ) ∂g ∂r = − ∂∂r (cid:88) k α k, H ( r k ) ϕ k ( r ) (11)We see that at r (cid:39) . 35, where the profile shape is most sensitive to q a , a unitchange in ln q a causes a change of (cid:39) . − for an ASDEX minor radius of .4 m. Parametric dependenciesof ‘experimental’ point values are also displayed. These are calculated bydifferencing the measurement values of pairs of neighbouring channels: L − i = T i +1 − T i . T i +1 + T i )( r i +1 − r i ) and regressing these approximate IFOL’s on q a , I p and n e . Such a point value has the advantage that it is more local than the29ontinuous function represented by eq. (11), but the disadvantage that thesignal to noise ratio will be lower. The mean value will also be particularlyaffected by systematic errors in one or both of the adjacent channels.The negative IFOL profiles for minimum, reference, and maximum q a ,displayed in Fig. 5(a), show that the temperature profile shape for themain database is remarkably invariant outside r = . 6. This behaviour isbroadly similar for the profiles of the secondary database. The larger errorbands of the latter, reflecting the substantially higher regression error forthis database, are due in large measure to the already discussed problem ofmultiple operating periods for this database. The parametric dependenciesof the IFOL profiles are detailed in the remaining sub-figures.The q a dependence of the inverse fall-off length of both databases increasesrapidly and reaches a maximum near r = . . The strength of this q a de-pendence then decreases sharply, and outside r = . q a . Between . ≤ r ≤ . 25 the point estimates of the ‘ex-perimental’ IFOL suggest a radially uniform q a dependence. However, thelarge error bars allow for a slope between roughly -10 and 10 on the scale ofthe plot. Inside the first knot, our radial spline model consists of a parabolawith T (cid:48) = 0 . This requires the IFOL as well as each individual parametricdependency to describe a straight line through the origin in this region.The I p and n e dependencies are much weaker. Although, over some por-tions of the radius, rejection at the 5% level is marginally avoided, tem-perature profile invariance with respect to I p and n e is generally seen tohold, within the experimental error limits, for all radii. Figs. 7 and 8 showreference profiles and parametric radial dependencies for the normalised pro-files ln ˆ T e ( r ) (where ˆ T e ( r ) = T e ( r ) /T e ( r = . T e ( r = . 2) = 1 . The invariance of the profile shape outside r = . T e ( r ) has a Gaussian shape. Since,to a high approximation, the profile shape has been shown to be solely a30unction of q a , the most general family of admissible Gaussian profiles is: T e ( r ) = T e, ( n e , I p , q a )exp( − f ( q a ) r ) (12)where f ( q a ) is a positive function. The negative IFOL would then satisfy − L − T e = − ddr ln T e ( r ) = 2 rf ( q a ) (13)which describes a family of straight lines through the origin. It is clearfrom Fig. 5(a) (and Fig. 6(a)) that this hypothesis is false for our data. It is also visually obvious that even the q a -independent part of the pro-file ( r > . 6) does not lie on a Gaussian. To quantify this, we made aregression of channels 12 – 15 (i.e. those roughly satisfying r ≥ . T e ( r ≥ . 6) = 3 . T line av.e ( r ≥ . exp ( − . r ) with a RMS relative error of 5 . . Adding a cubic term already strongly reduces the error: T e ( r ≥ . 6) =1 . T line av.e ( r ≥ . exp (1 . r − . r ) with a RMS relative error of 3 . 8% (F valuefor cubic term (cid:39) . Thus, even for the outlying portion of the profile, wecan categorically rule out a Gaussian shape.Figs. 9 and 10 show a sample experimental T e profile from each database,representing extremes of the q a range covered by our data. In each case, thepredicted profile with 95% global confidence bands is also shown. Since weare concerned here with both profile size and shape recovery, these predictionscome from a parameterised spline regression of unnormalised experimentalprofiles. Hence the confidence bands are wider (by (cid:39) We now examine how well the fitted profiles describe the data both interms of root mean squared error and the fraction of data variance explainedby the model. We also give a break-down of the profile fluctuations intowithin-discharge and discharge-to-discharge contributions.Table III(a) presents some descriptive statistics for the temperature profilesin the main database on a channel-by-channel basis. Table V(a) displaysthe equivalent information for the secondary database. The channels are31umbered according to their vertical position with channel 1 at Z = . 200 m,channel 6 at Z = 0 m, and channel 15 at Z = − . 353 m. For each channel, themean normalised flux radius is presented, followed by the mean temperatureand the spread in keV. The spread (this term is chosen to avoid possibleconfusion with standard deviation in the sense of ‘regression error’) is justthe ‘standard deviation from the mean’, i.e. spread = (cid:113) N − (cid:80) Ni =1 ( y i − ¯ y ) ,where ¯ y is the sample mean. Recall that before the regression, the profileswere normalised by the size parameter from the SAS GLM procedure, i.e. T e ( r = . σ int / √ ≡ ˆ σ int, comp. ; the estimatedstandard deviation of the within-discharge noise scaled for time-compressedprofiles. Column 4 tabulates ˆ σ tot , the total noise level (of the compressedprofiles), estimated by regressing ln T e for each channel on the bulk parame-ters. The differences of the squares of the entries in column 4 and column 3are an estimate of the discharge-to-discharge variance.The ratio of the two noise estimates lies in the range . ≤ ˆ σ int, comp. / ˆ σ tot ≤ . σ tot . Using eq. (8) and the RMS values (over the 15 channels) forˆ σ int, comp. and ˆ σ tot , we find that for the main database, ˆ σ dis ( rms ) (cid:39) . . Forthe secondary database, . ≤ ˆ σ int, comp. / ˆ σ tot ≤ . σ dis ( rms ) (cid:39) . σ spline ( rms ) = . σ channel ( rms ) = . − . 3% and +2 . 9% respectively; the nextlargest channel bias was 1 . . In contrast to the maindatabase, the spline regression errors in the secondary database (see TableV(b) column 6) are, in general, much larger than the channel-by-channelerrors in column 5 (ˆ σ spline ( rms ) = . σ channel ( rms ) = . . This is ex-plained by the fact that, unlike the channel-by-channel regressions, we didnot use operating period indicator variables in the spline parameterisationof the secondary database profiles (subsection 6.5). For both databases, theregression errors generally increase for outlying channels, reflecting a pro-gressively deteriorating signal-to-noise ratio. This is due to the decrease inscattered laser light signal intensity with decreasing electron density. Notethat channel 12 is an exception, with errors similar to channel 15. This isconsistent with the fact that, whereas all other channels had three distinctspectral filters (normally offering the choice of the less noisy of two inde-pendent determinations of the temperature), channel 12, at the time thedischarges for our databases were made, had only two. Since many of the results presented in the last section apply to the densityprofiles as well, we only mention the differences. Tables IV and VI containthe density statistics for the main and secondary databases respectively.Figs. 11 and 12 portray graphically our parameterisation of the densityprofile local shape parameter − L − n e = − d ln n e / d r for each database. Figs.13 and 14 show the integrals of Figs. 11 and 12, i.e. the normalised densityprofile parameterisations. Figs. 15 and 16 show sample experimental n e pro-files (with prediction profiles and confidence bands) for the same discharges33hich provided the sample T e profiles shown in Figs. 9 and 10. Note the‘jump’ in the predicted profiles at r = 0 , which arises from the presence inthe regression of the density up-down asymmetry variable (subsection 6.7).The dominant feature of the density IFOL profiles (Figs. 11 and 12) isa q a dependence closely mirroring that of the temperature IFOL, though ata reduced magnitude ( − ∂ ( L − T ) ∂ ln q a ( max ) (cid:39) − ∂ ( L − n ) ∂ ln q a ( max ) (cid:39) . < r < . 55 for the main database) issmaller than the equivalent region for the temperature. Thus the variationof density profile shape with q a , while not as dramatic as that of the tem-perature, is nonetheless considerable, as is evident in the contrast betweenFigs. 15 and 16. The magnitude of the density IFOL remains smaller thanthat of the temperature over the entire profile. At r = . . r = . L − (cid:39) . a = . I p and n e shapedependencies are weaker than that of q a . Near the edge, however, there is astatistically significant broadening of the density profile shape with increasingcurrent . (We note with caution, however, that the ‘experimental’ IFOL dat-apoints suggest that this current dependence is due solely to the outermostchannel.) In addition, some flattening of the density profile with increasing n e occurs in the region . < r < . σ int, comp. andˆ σ tot , the discharge-to-discharge variance ( ˆ σ dis ( rms ) (cid:39) . 2% for the maindatabase, and (cid:39) . 2% for the secondary) forms the largest contribution tothe total variance, as was the case for the temperature profiles. The overallspline regression RMS relative error for each database (.032 and .065) is verysimilar to the corresponding temperature value.A number of density profiles in both databases are slightly hollow in theregion . < r < . . Since the set of reference profile parameters ( q ∗ a = 2 . I ∗ p = . n e ∗ = . × m − ) was not very typical for the secondarydatabase, this feature looks somewhat exaggerated for that database (seeFigs. 12(a) and 14(a)). 34 .4. Electron Pressure The analysis of the (logarithmic) electron pressure profiles, defined asln P e = . T e + ln n e , offers additional insight, as can be seen fromthe parametric dependencies shown in Figs. 17-20. For the main database,the most striking feature is that the I p dependence is significant over mostof the radius. In the outer region of the plasma, there is a clear broadeningof the pressure profile with the current, while peaking occurs in the region . < r < . . The q a dependence of the inner half of the profile is very strong(temperature and density profile dependencies reinforce each other) whereasthe n e dependence is little changed from that of the density profile. 8. DISCUSSION AND SUMMARY Bulk scalings In the first part of this work, we presented and compared the scalings ofvarious global plasma parameters for two complementary ohmic datasets.For the main, high density database, the volume-averaged temperature andthree independent measurements of the total plasma energy depend on theplasma current I p and line-averaged density n e , but, at constant I p and n e , are practically unaffected by the toroidal magnetic field. For the secondarydatabase, Wp dia and Wp mhd show strong q a scaling differences, while theWp kin and < T e > scalings are almost the same as in the main database.The nearly linear current dependence for both the temperature and the totalenergy is reminiscent of L mode scaling.The Spitzer Z eff depends on all three control variables. Regression of Z eff − n e , but strongly dependent on both I p and B t . The elec-tron temperature profile peaking factor T / / A careful statistical analysis is necessary to determine the radially varyingparametric dependencies of the profile shapes on the bulk plasma variables.By simultaneously fitting all profiles with spline coefficients which dependon the plasma variables, we have been able to examine profile dependencieson a detailed, quantitative level. Based on this spline model, a convenientgraphical represention has been used to inspect visually the influence of thevarious plasma parameters on the profile shapes.An earlier study of ASDEX temperature profile shapes [22] (for bothOhmic and neutral beam heated discharges) revealed that the shape dependsstrongly on q a inside the sawtooth mixing radius, but is almost independentof plasma parameters outside ‘the influence of the q = 1 surface’. The resultsof our profile parameterisation are roughly consistent with, and constitute arefinement of this analysis, for Ohmic profiles.Except for a dependency of the outer region of the density profile shape onplasma current (and to a lesser extent on n e ), the I p and n e dependencies ofboth the temperature and density profile shapes are rather weak in general.In most cases the current and density dependencies are not significant giventhe error bars of the datasets.In the interior, q a is the dominant bulk plasma parameter in determiningthe temperature shape. By r = . 5, however, this dependence has weakenedconsiderably and outside r = . 6, as is clear from Fig. 5, the IFOL profile36as an invariant shape. We note that the extent of the q a -sensitive regionis reasonably consistent with the widest sawtooth inversion radius in eachdatabase ( r inv ( max ) (cid:39) /q a ( min ) = . 54 and .42 for the main and secondarydatabases respectively).Comparing Figs. 11 and 5, we see that the variation in the density profileshape, while significant, is much weaker than that of the temperature. Thisfollows from the result that, over the inner half of the radius, the sensitivityof the density IFOL to q a is only (cid:39) 40% that of the temperature. The q a dependence is only significant for . ≤ r ≤ . 5. In contrast to the temper-ature shape, which is unique outside r = . 6, the density profile broadenssignificantly near the edge with increasing current.The electron pressure IFOL exhibits a very strong q a dependence in theinner half of the profile, while increasing I p causes a broadening of the outerhalf, a tendency which intensifies approaching the plasma boundary.Our findings are well described in terms of profile invariance [1, 21] andin quantitative agreement with important criteria for profile consistency asdescribed by [4] and developed by many authors. However, we have notaddressed the relative merits of profile consistency versus local transportmodels [10, 11] containing sawtooth effects. This issue could be addressedby a statistical comparison of experimental profile dependencies with thedependencies predicted by local transport models. ACKNOWLEDGEMENTS The work of PJM was performed under a EURATOM supported reciprocalresearch agreement between IPP and University College, Cork. The workof KSR was supported by the U.S. Department of Energy, Grant No DE-FG02-86ER53223. We are indebted to the referees, who made several criticalsuggestions from which the manuscript has profited considerably. APPENDIX AON STATISTICAL SIGNIFICANCE OF REGRESSION VARI-ABLES The significance of a regressor x j in the least squares model can be inter-37reted in terms of ˆ α j / ˆ σ ( ˆ α j ), the ratio of the fitted coefficient to its standarderror estimate. Under standard least squares assumptions, including (a) thecorrectness of the regression model and (b) normally distributed errors in thedependent variable, the ratio ˆ α j / ˆ σ ( ˆ α j ) has a Student’s t distribution underthe null-hypothesis that α j = 0 . For any statistic T that has a t distributionwith f degrees of freedom, the following relation between the critical valuet f,(cid:15) and the ‘exceedence probability’ or significance level (cid:15) holds: P (cid:110) | T | > t f,(cid:15) (cid:111) = (cid:15) ( A α j = 0 is rejected, and the regressor is consideredsignificant, if | ˆ α j / ˆ σ ( ˆ α j ) | > t f,(cid:15) for some small value of (cid:15) , say 5%. For manydegrees of freedom ( f > 30 usually suffices) as in the present case, Stu-dent’s t can be well approximated by the normal distribution. Thus we havet f, . (cid:39) . 0, and the significance criterion is | ˆ α/ ˆ σ ( ˆ α ) | > . from each independent variable. Without loss of generality, we consider themultiple linear regression problem y = α x + α x + . . . + α p x p + (cid:15) ( A < x j , x k > ≡ (cid:80) Ni =1 x i,j x i,k = (cid:107) x j (cid:107) δ j,k , it iseasily shown that the least squares solution reduces toˆ α j = < x j , y > (cid:107) x j (cid:107) ; ˆ σ ( ˆ α j ) = ˆ σ (cid:107) x j (cid:107) ( A σ ( ˆ α j ) is the estimate of the standard error for the coefficient estimateˆ α j and ˆ σ = (cid:107) y − ˆ y (cid:107) / ( N − p ) is the mean square regression error. From thedefinition of the t statistic, we have that38 j ≡ ˆ α j ˆ σ ( ˆ α j ) = < x j , y > (cid:107) x j (cid:107) ˆ σ = √ N − p (cid:107) y − ˆ y (cid:107) < x j , y > (cid:107) x j (cid:107) ( A < x j , y > / (cid:107) x j (cid:107) is the projection of y onto x j where y and x j are vectors in (cid:60) N . Hence, adding x j to the regression model makes afractional contribution to the total variance of∆(R ) j = < x j , y > (cid:107) x j (cid:107) (cid:107) y (cid:107) ( A − R = (cid:107) y − ˆ y (cid:107) (cid:107) y (cid:107) ( A x j or y in eq.(A5) and we finally obtain∆(R ) j = t j N − p (1 − R ) ( A if the j th regressor is removedfrom the model. Note this relationship strictly holds only for uncorrelatedregressors. If we now sum up all contributions, we obtain, using Pythagoras’theorem, (cid:80) pj =1 < x j , y > / (cid:107) x j (cid:107) = (cid:107) ˆ y (cid:107) which leads to the equality p (cid:88) j =1 t j = ( N − p ) (cid:107) ˆ y (cid:107) (cid:107) y − ˆ y (cid:107) = ( N − p ) R − R ( A C p [23]. It provides auseful practical check on the applicability of eq. (A7) when the regressorsare correlated, which is usually the case. APPENDIX BCOMPOUND ERROR STRUCTURES: TEMPORAL HIERAR-CHY 39o efficiently estimate the spline coefficients, α , we try to model the actualcovariance matrix for the errors. The closer the assumed or estimated Σ is tothe actual covariance error structure, the more accurate the ensuing estmatesfor α are.The assumption of independent errors is not always justified. In general,tokamaks possess a compound error structure. The first level of errors arestatistical fluctuations which vary from time point to time point within agiven discharge. The next level consists of those errors which vary fromdischarge-to-discharge (we assume here that there is only one compresseddatapoint per discharge) but remain constant within a given discharge. Fi-nally, there are variations which only change between operating periods of atokamak. We denote the covariance matrices of for the radial fluctuations ofeach of these three types of errors by Σ int , Σ disch , Σ op respectively.We use a triple index, ( p, i, t ) to denote a given profile timepoint where p indexes the operating period, i the discharge number, and t the time.Within a single profile timepoint, the individual radial measurements aredenoted by a fourth index, l . The cross-correlation of any two pairs of profilemeasurements, ( p, i, t ) and ( p (cid:48) , i (cid:48) , t (cid:48) ) is given by a 15 × 15 matrix, Σ p,i,t,p (cid:48) ,i (cid:48) ,t (cid:48) .We assume that the errors do not depend on the plasma parameters andthat the covariance structure does not vary between different blocks of dataat each level. The most general error structure of this form isΣ p,i,t,p (cid:48) ,i (cid:48) ,t (cid:48) = Σ int δ p,p (cid:48) δ i,i (cid:48) δ t,t (cid:48) + Σ disch δ p,p (cid:48) δ i,i (cid:48) + Σ op δ p,p (cid:48) ( B int δ i,i (cid:48) δ t,t (cid:48) + Σ disch δ i,i (cid:48) . For simplicity, weassume that each discharge consists of n t timepoints.We estimate the within-discharge variance, Σ int empirically by calculat-ing the time point average and the time point variance for each dischargeseparately:ˆΣ intk,l = 1 n d ( n t − n d (cid:88) i =1 n t (cid:88) t =1 ( Y i,t ( r k ) − Y i ( r k ))( Y i,t ( r l ) − Y i ( r l )) ( B Y i ( r k ) = 1 n t (cid:88) t Y i,t ( r k ) ( B n t times the dischargevariation, a statistical analysis based on structured covariance matrices isdesirable.Within a single operating period, the total variation between datapointsis estimated byˆΣ k,l = 1 n d n t − f n d (cid:88) i =1 n t (cid:88) t =1 ( Y i,t ( r k ) − Y fit ( r k ; α ))( Y i,t ( r l ) − Y fit ( r l ; α )) ( B f denotes the number of fitted parameters. The fitted values, Y fit ( r l ; α ) depend on the values of the plasma parameters and thereforeimplicitly on the indices, i and t . Y fit ( r l ; α ) may be estimated either byregressing each measurement channel separately or by fitting all channelssimultaneously using the spline representation. The latter method will in-flate the variance if the profiles cannot be well approximated by the splinerepresentation.The discharge variance is computed by subtracting the within-dischargevariance as defined in eqn. (B2) from the total datapoint variance defined ineqn. (B4).In describing nested error structures of this form, statisticians use theterms “within discharge variation” to refer to the time point to timepointvariation and “between discharge variation” for the discharge variation.Several caveats must be placed on this procedure. First, using too manyor too few terms in the regression analysis will artificially inflate the varianceestimates. Second, the errors in the estimates of the variances tend to berather larger unless a substantial number of profiles are available.41 PPENDIX CANALYSIS FOR RADIALLY CORRELATED ERRORS When the random errors are correlated, the weighted least squares estima-tor is consistent but not efficient, i.e. as the number of datapoints approachesinfinity, the estimates of the regression coefficients converge to their true valuebut the rate of convergence is not optimal.To increase the precision of the estimate, one tries to model the actualcovariance matrix for the errors. The closer the assumed or estimated Σ isto the actual error structure, the more accurate the ensuing estimates for α are. We continue to assume that the statistical fluctuations are temporallyuncorrelated and neglect the parametric dependencies. However we nowallow radial correlations in the fluctuations. Since the dataset consists of 15 n datapoints, the entire covariance matrix is 15 n × n . However we assume ablock diagonal form for Σ of the form:Σ toti,k,i (cid:48) ,l = Σ k,l δ i,i (cid:48) ( C i, i (cid:48) index the time-averaged profile and k, l index the channel num-ber.This covariance matrix of the residual radial errors may be estimated by:ˆΣ k,l = 1 n (cid:88) i ( Y i ( r k ) − Y i,fit ( r k , α ))( Y i ( r l ) − Y i,fit ( r l , α )) ( C θ ) to model the observed covariance. We can simultaneously esti-mate ¡ α and θ using maximum liklihood estimates [1].42 PPENDIX DDoes Radius of q a -dependent Region Contract as q a Increases? Since the sawtooth inversion radius decreases with increasing q a , ( r inv (cid:39) /q a [ ? ]), a natural hypothesis is that r tran , the ‘transition radius’ where thetemperature profile shape becomes independent of q a , also scales with 1 /q a .This question cannot be decided on the evidence of Figs. 2 and 3, beyondthe assertion that for r ≥ . q a - independent for allvalues of q a in the two databases. To make this analysis as radially localisedas possible, while still attempting to satisfy the conflicting requirement thatindividual channel noise be smoothed out, we discarded the spline model usedgenerally in this paper in favour of a moving average - type local quadraticfit (3 fit parameters) to successive groups of four neighbouring channels. Toeliminate the possibility that the higher density of channels for r ≤ . radius (only that radial interval lying between the second and third channelin each group was used), a set of smoothed IFOLs was accumulated for eachof the 142 experimental profiles in the combined database. These formed theworking data for the following analysis.To identify a possible q a dependence of r trans , we assume that r trans issome simple function of q a , e.g. r trans = α + β q a , where the shape of thetemperature profile is postulated to be q a dependent inside r trans only. Thedetermination of the parameters α and β is a nonlinear problem which washandled by the following procedure:(i) Choose candidates α and β from a grid of possible values.(ii) For each ( α, β ) fit all profiles simultaneously with the scheme L − ( r, q a ) = (cid:26) u ( r, q a ) for 0 ≤ r < r trans v ( r ) for r ≥ r trans (20)where r trans = α + βq a varies with each individual value of q a , and u ( r, q a )and v ( r ) are quadratic functions of r : u ( r, q a ) = a + a q a + ( b + b q a ) r +( c + c q a )r and v ( r ) = d + e r + f r with zeroth order continuity imposed43t r = r trans .(iii) Store α , β , and the mean square error (MSE) for each fit.(iv) Locate ˆ α and ˆ β giving the best fit (i.e. the global minimum MSEvalue).(v) Make a contour plot of MSE as a function of ( α, β ) and hence determinethe 95% confidence contour for ˆ α and ˆ β (vi) If β = 0 lies inside the 95% confidence contour, then the null-hypothesis β = 0 is compatible with the data.The contour plot obtained using the above procedure is shown in Fig. 14.The global minimum MSE, ˆ σ min = . r trans = . − . q a .To determine the 95% confidence contour for β where α is arbitrary, weuse the result that the relative difference in the sum of squares has asymp-totically a χ distribution with, in our case, one degree of freedom (see [ ? ]Chap. 5). For this analysis, because of the restrictions used in generatingthe input IFOL values, we have effectively 9 independent measurements perprofile, giving a total (for 142 profiles) of n = 1278. From [ ? ] Table C.1we have χ (95%) = 3 . . This gives ∆(RSS) = 3 . × ˆ σ min = . σ = . . / . rd contour in Fig. 14 which leads toa (slightly conservative) 95% confidence band for β of ( − . , − . r trans = ˆ α + ˆ β q a we have the extreme values r trans ( q a = 1 . 9) = . 60 and r trans ( q a = 4 . 2) = . 57 . A search over the entire region bounded by the95% confidence contour yielded r trans ( q a = 1 . max = . 61 and r trans ( q a =4 . min = . 57. Thus, although the null-hypothesis β = 0 was not satisfied,we have established ¡ a 95% confidence interval ( . , . 61) for r trans for thecombined database where q a varies from 1.9 to 4.2. This small variationin r trans ( r trans,max /r trans,min (cid:39) . 1) clearly contadicts the hypothesis that r trans ∝ /q a since we would expect, for our database, that r trans would varyby a factor of . . = 2 . . We can offer no convincing explanation for thisresult. 44 eferences [1] KARDAUN, O.J.W.F., RIEDEL, K.S., MCCARTHY, P.J., LACKNER,K., Max-Planck-Institut f¨ur Plasma Physik, Report No. 5/35[2] MURAKAMI, M., CALLEN, J.D., BERRY, B.A., Nucl. Fusion (1976) 347.[3] M ¨ULLER, E.R., S ¨OLDNER, F.X., JANESCHITZ, G., MURMANN,H., FUSSMANN, G., KORNHERR, M., POSCHENRIEDER, W.,WAGNER, F., W ¨URZ, et al., in Controlled Fusion and Plasma Physics(Proc. 15th Eur. Conf. Dubrovnik, 1988) Europhys. Conf. Abst. Vol. , European Physical Society (1988) 19.[4] ARUNASALAM, V.D., BRETZ, N.L., EFTIMION, P.C., et. al.,Nuclear Fusion, , (1990) 2111.[5] R ¨OHR, H., STEUER, K.-H., MURMANN H., MEISEL D., Max-Planck-Institut f¨ur Plasma Physik, Report No. 3/121 (1987) (also Nucl.Fusion (1982) 1099).[6] Mardia, K.V., Kent J.T., Bibby,J.M., Multivariate Analysis AcademicPress Inc., London (1979).[7] STEUER, K.-H., R ¨OHR, H., ROBERTS, D.E., et al., in ControlledFusion and Plasma Physics (Proc. 15th Eur. Conf. Dubrovnik, 1988)Europhys. Conf. Abst. Vol. , European Physical Society (1988)31.[8] KARDAUN, O., THOMSEN, K., CORDEY, J., et al., in ControlledFusion and Plasma Physics¡ (Proc. 17th Eur. Conf. Amsterdam 1990)45ol. , European Physical Society (1990) 110.[9] PFEIFFER W., WALTZ R., Nucl. Fusion (1979) 51.[10] BARTLETT D.V., BICKERTON R.J., BRUSATI M., et al., Nucl.Fusion (1988) 73.[11] CHRISTIANSEN, J.P., CALLEN, J.D., CORDEY, J.G., THOMSEN,K., Nucl. Fusion (1988) 817.[12] MC CARTHY, P.J., RIEDEL, K.S., KARDAUN, O., MURMANN, H.,LACKNER, K., et al., Scalings and Plasma Profile Parameterisationof ASDEX High Density Ohmic Discharges , Max-Planck-Institut f¨urPlasmaphysik, Report No. 5/34.[13] MERTENS, V., BESSENRODT-WEBERPALS, M., DODEL, G., etal., in: ASDEX Contributions to the 17th European Conference onControlled Fusion and Plasma Physics, Max-Planck-Institut f¨ur PlasmaPhysik, Report No. 3/164 (1990) 1.[14] WAGNER, F., GRUBER, O., BARTIROMO, R. et al., in: ASDEXContributions to the 12th European Conference on Controlled Fusionand Plasma Physics, Max-Planck-Institut f¨ur Plasma Physik, ReportNo. 3/107 (1985) 16.[15] WALTZ, R.E., WONG, S.K., GREENE, J.M., DOMINGUEZ, R.R.,Nucl. Fusion (1986) 1729.[16] EJIMA, S., PETRIE, T.W., RIVIERE, A.C. et aL., Nucl. Fusion (1982) 1627. 4617] SAS, User’s guide: Statistics (6 th ed.), SAS Institute Inc., Cary, NC,(1989).[18] FREDRICKSON, E.D., MCGUIRE, K.M., GOLDSTON, R.J., et. al.,Nuclear Fusion, , p. 1897, (1987).[19] ST. JOHN, H., JAHNS, G.I., BURELL, K.H., DEBOO, J.C., Bull.Am. Phys. Soc. (1987) 9.[20] POORTEMA, K., On the statistical analysis of growth, PhD thesis,Groningen (1989).[21] KARDAUN, O., MC CARTHY, P.J., LACKNER, K., RIEDEL, K.,GRUBER, O., in Theory of Fusion Plasmas, Varenna (1987) 435.[22] MURMANN H., WAGNER F., et al., in Controlled Fusion and PlasmaPhysics (Proc. 13th Eur. Conf. Schliersee 1986) Vol. , EuropeanPhysical Society (1986) 216.[23] WEISBERG, S., Technometrics (1981) 27.[24] DRAPER, H., SMITH, N., Applied Regression Analysis , 2nd ed., Wiley(1981). 47 c:: (') rm > "' 'Tl < 2. z - °' - UI I 0 >, .0 "C 11) .':! ..s - . ,. • (o) T -l -2-3 -4,...... __ ....,... ___ ,...... _____________ ,...... __ ....,... ______ � (c) J -I -2-3 0.3 0.4 0.5 0. 6 Normalized Flux Surface Radius ,,- / / ----- --- -4,......---,--�------,----,-.---,----.,-. ---,-- -----� 0. I Normalized Flux Surface Radius • (b) -- --- ---- - l --- -2-3 -4----,----,----,--- -----,- ---,-.---,----- --� 0. 7 Normalized Flux Surface Radius (d) -2 -3-4 ...... -------r----,--------,----,-.--.-.-----,---__,. Normalized Flux Surface Radius FIG. Reference negative /FOL profiles for the main (high density) database T,, and parametric dependence profiles for q •. IP and 'ii, with local 95% confidence bands. Reference values for discrete (two-channel) /FOLs and the corresponding parametric dependences are also plotted as individual data points. The central solid curve in {a) is the predicted negative /FOL at the reference parameter values q. = 2.5, I P = = x m·1• The local 95% confidence bands are plotted as dashed curves. The flattest and steepest predicted negative /FOL profiles in the database (i.e. those for minimum (q. = I. 9) and maximum (q. = 3. 4) values of q,J art! plottt!d as solid curves without confidt!nce bands. Tht! individual data points are predictions for discretely calculated /FOLs (Section 7./), again evaluated at the reference parameter values and accompanied by 95% confidence bands. (b) Parametric dependence of L:;.: on q •. The solid curve is the radial profile of tht! change (in units of I /minor radius) in the negative /FOL per unit chonge in In q. (su Eq. (11)). The individual data pains give similar information on the discute /FOLs. (c) Parametric dependence of Li on I P and ii,. (d) Parametric dept!ndence of Li-: on 'ii,. c C A R T HY e t a l . / ( / / ‘ D - : C ) U U , x D - E — , I . - o ° b ug / a . L ’) a ’ t D U , < _ I I D - o C ) D U ’ D • C ) E z -- G — ° i ( q p C ) z i ( . O u ) u NU C LE A R F U S I ON , V o l . , N o . ( ) —o —2 0.0. —o ‘0 - . C‘0 —1—2 V —2 O - . C’) d a) E —1 CC FC‘0 . . . . . . . ? Normalized Flux Surface Radius ( c ) — . . . —2.5 . / V rr . . . . . . . . ? . . . Normalized Flux Surface Radius . . —0 .5 —1.0 - . —2 .0—2 ( d ) z CC zz —0 .5 C‘0F—C‘0 - . —2 .0—2.5 — . . . . . . . . . ? . . . Normalized Flux Surface Radius . . . . . . . . . . Normalized Flux Surface Radius FIG. Z Reference In profiles (normalized at r = f o r the main database, and p a r a m e t r i c dependence profiles f o r q0. I,, and h . ,. .,, C: j � z ;c, � (a) R ci -0.5 II -� -1 . 0 E � . -1 . 5 ,-.. .E -2.0 -2. 5 ,._��--�--......--.....,...-- ......... �---,.------,---.,.......---r---. 1. 0 Normalized Flux Surface Radius (c) ---- -0.5 "' -..... -1. 0 ,-..• .E "' -1 . 5 -2 . 0 -2. 51,.-....-......,.---..---......,....--..----.......---..---...-,---....--...-,.---,. 0. 1 0. 7 0.8 0.9 1. 0 Normalized Flux Surface Radius o.� ---.... (bJ / -0.5 / / ,, tT .,,. ,, .E "' -1. 0 -..... --- ------ ,-..• .E "' -1 . 5-2 . 0-2.5r-----.--..-,----.----.---......--........ ---.----.-----� (d) -0.5 le" .'.:'._-1.0 . ,-.. £ "' -1 . 5 -2.0 0.2 1. 0 Normalized Flux Surface Radius t ---f------- --- -------- - .... .... .... .... -2 . 5 ,._......,.........,.........,.........,....,.............,.........,.........,.........,....,---......,....--..---......,.---..,......- ........... ---,. 0. 0 0. 1 1. 0 Normalized flux Surface Radius FIG. 8. Reference In T, pro.files (nonnaliud at r = P and n,. cCARTHY et al. .3 .2 I. 1. 0 .8 > 0. 60. * * * * ** ** * * * * * * 0. 2 * 0. 10.0 -1.0 -0. 8 -0. 6 -0.4 -0. 2 Normalized Flux Surface Radius FIG. 9. Sample experimental and predicted temperature profile with global 95% confidence bands from the main database. Parameters: q. = 1. 94, I P = n, = 0. 704 X m-1 • . 3 2 1.1 * ** .0 * * 0. 8 * * % * :=, * * * * * * 0. 10.0 -1 .0 -0.8-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Normalized Flux Surface Radius FIG. 10. Sample experimental and predicted temperature profile with global 95 % confidence bands from the secondary database. Parameters: q. = P = n, = x m-3. c C A R T HY e t a l . / ‘I t o C ) U t r s C t o C I) - C t o E — , z t o t o ) C l ) t o • ) N E H C C • . U CC ) ) E C C C ° b - o ) U C U ) x3 ) C F z ) - C t o CC ) t o - t o t a t N I I I / I / — t o I I / I ii ’ / / ‘I \ // -- / ‘ / •• \\\ ( _o ( q po z il o w i o C I) — U d1 u I F / — NU C LE A R F U S t ON , V o l . . N o . ( ) SDEX SCALING AND PLASMA PROFILE PARAMETERJATJON (0 -o U) 0 o .t -.,- (I) (0 J L , i (&) ., : t c U, D -o a) D Cl, (0 D L ‘0U) S z°b \ \ C (,_o (q pazilowiou) ti— U d1 ul/(9_NUCLEAR FUSION. Vol.31, No.9 t1991) c C A R T HY e t a t . U , D o a ) U S o X , U U , a , E - z , U a , U , U • a , E I i h i //i . III - D : U I / UU U I - C -‘ S ‘ S — a . ’ I U , h \\ a , , C o C , U ., ( J o X ’ U - - o — a , E - , z C U ( o = o p az il o w J ou ) U U I d1 u I o / a u U l o NU C LE A R F U S I ON . V o I . I . N o . ( ) —o a V E CC C . a C ‘ C C ‘ z n z C z C U i N o r m a li ze d F l ux S u rf ace R a d i u s . . . —2 — . . . . . . . . . . . . . . . . . . . . . . N o r m a li ze d F l ux S u rf ace R a d i u s . C - . C C ‘ — . . c ‘ - . C - . — . ( a ) C ‘ . - . . . . . . . . - . N o r m a li ze d F l ux S u rf ace R a d i u s . — . — . r d ) Cl) C I F I G . . R e f e r e n ce I n n , p r o f il e s ( no r m a li ze d a t r = . ) f o r t h e s ec ond a r y d a t a b a s e , a nd p a r a m e t r i c d e p e nd e n ce p r o f il e s f o r q ,. , a nd . . . . . . . . . . N o r m a li ze d f l ux S u rf ace R od i u s cCARTHY et al. 0. 0.8 0. 70.6� 0. 5 � 0. 4 . C 0. 2 * * * * * * * * * ** * * * * O.O----....... --......... ---..,....-,---....--.......-if--.--�------ - I . 0 -0. 8 -0.6 -0. 4 -0.20.0 0.2 0.4 0.6 Normalized Flux Surface Radius FIG. 15. Sample experimental and predicted densiry profile with global 95% confidence bands from the main database. Parameters: q = I. 94, I P = n, = 0. 704 x 10 m-3. 0. 90.80.70.6 'E O. 5 � 0 * ** ** ** * c• .3 * * * * * * * 0. 2 Normalized Flux Surface Radius FIG. 16. Sample experimental and predicted density profile with global 95% confidence bands from the secondary database. Parameters: q = P = n, = m-3. J L - A S D E X S C A L I NG AND P L A S M A P R O F I LE P A R A M ETE R I Z A T I ON ‘ : ‘I / , / , — / - / - N N N NN N \ N - o C ) D U , z V E z aa I’ I’ II II — - c D - D ) c i D U , x L c ) E z D - o V c i U , x • V E z ‘ ’ V cc V - cc L V E V I • D a - o VU cc - , D U , ‘ C - o V a E . -‘ a z / I / I I! // / // /i I , / / , \ N N N / —— J R / \ , _ — / / E : , j J ‘I — ( L - ° q po z i J o w i ou ) U NU C LE A R F U S I ON . V o I . l . N o . ( I) c C A R T HY e t a l . c U DU ) S U - a ) N E — , • z J —— / // / \ , \ \\ \ \ -- S a U U ) x S U - • - o E — a • z ° b u I @ / L d7_ - D I I C o - VU C ’- , U ) S U F C - , — U S a , U C ’- , o S U ) S U - • F — S • z ( _o X q po z i ( o w l ou ) d1 u ( / , NU C LE A R F U S I ON , V o l . , N o . ( ) p t .. . - • - . — — -- - . — - — . - -- —— F -- NU C LE A R F U S I ON . V o l . , N o . ( ) A S D E X S C A L I NG AND P L A S M A P R O F I LE P A R A M ETE R I Z AON I t -‘ t ‘ ‘ , L i , ‘ - t L i t t L ii -‘ L L ii E ‘ ‘ , -‘ t ‘ __________ L ii N ‘ -‘ ‘ (I (I) F I / F ; ’ N D - o C t o o C D o D - o N ’ N o E C , n z C II II C N ‘ - C C ) C ‘ o X ‘ U - - o N - ) N E C - o dug1 t a . j g p p I - o ° b u i g / ° d U I Q C )‘ - D C o ) C ‘ o X ‘ U - - o o E C - D Z ( o ( C P O Z II O W J C U ) a d u I U ) Q / ° d U ) Q ’ E a C L O ( c ) . . —0 5 — - C‘0 - . ‘0 —2 5 —3 0 — . — . — . . — . - . C‘00. C ‘0 I i w . ‘I - . — . C.’ C z C m CM C z z . . . . . . . . ? . . . N o r m a li ze d F l ux S u rf ace R a d i u s —3—4 Z :: P . . . . . . . . ? . . . N o r m a li ze d F l ux S u rf ace R a d i u s ( a ) _ L j . . . —0 . I c ’ . C ‘0 . a . ‘0 —2 5 — . — . . . . . . . . ? . . N o r m a li ze d F l ux S u rf ace R a d i u s — - . . - . . . . . . . . . . . . N o r m a li ze d F l ux S u rf ace R a d i u s F I G . . R e f e r e n ce I n p p r o f il e s ( no r m a li ze d a t r = . ) f o r t h e s ec ond a r y d a t a b a s e , a nd p a r a m e t r i c d e p e nd e n ce p r o f il e s f o r q5 , a nd ii.,.ii.,.