Remarks to the H-mode workshop paper

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MargaretMargaret Hopkins
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1 2 nd ITPA Confinement Dtbse nd Modeling Topicl Group Meeting, Mrch 11-14, 2002, Princeton Remrks to the H-mode workshop pper The development of two-term model for the confinement in ELMy H-modes using the Globl Confinement nd Pedestl Dtbses A. Chudnovskiy RRC Kurchtov Institute The totl plsm therml energy is presented in the two-term model of confinement s sum of nd pedestl energies: = + th The pedestl component is developed ssuming some expression for the pedestl energy dependence on the plsm prmeters nd fitting this expression to the pedestl dtbse. The result is ped M = exp( 4.61)I R q 2 sh ε (1) nr ped 1

2 The procedure of the scling development is s follows: It is ssumed tht energy is difference: =, where th ped th - is therml energy tken from H-mode dtbse nd - is clculted by mens of expression (1). ped Stndrd power low expression for energy is fitted using the extended H-mode stndrd dtset 1, consisting of 2678 pulses. The result is: = exp( 2.73)I B R n P ε M (2) 0.2 And conclusion is: The hs pure gyro-bohm dependence with only wek β dependence. The remrks re concerned the development of the component (2) of the two-term scling. 1 O. Krdun, et l., 18 th IAEA Fusion Energy Conference, Sorrento, Itly (2000) IAEA-CN-77/ITERP/04] 2

3 Usul power low expression for energy through the engineering prmeters is XI XB XR Xn XP X ε X XM = CI B R n P ε M. (3) The energy confinement time, expressed through the physicl prmeters, is D B τ ρ β q τ = τ ρ ν νβ q ε ε M M. (4) Scling (4) is dimensionlly correct nd coefficient C is dimensionless if τ = 1. This is the Kdomtsev constrint. Scling (4) is of gyro-bohm type if τ = 1 nd ρ = -1. The exponents of the physicl prmeters re ssocited with the exponents of engineering prmeters by the known expressions. 3

4 The physicl scling (4) exponents clculted for the exponents of the energy scling (2), re: τ = 1, ρ = -1, β = 0, ν = 0, q = The numbers re not rounded. This is result of direct clcultions. hen reder will found tht the exponents τ, ρ, β nd ν re exctly equl to the integer vlues he cn suppose tht gyro-bohm constrints ( τ = 1 nd ρ = -1) together with the constrints β = 0 nd ν = 0 were used in the scling (2) development. Then the reder cn continue If this constrints were relly used then the conclusion The hs pure gyro-bohm dependence with only wek β dependence is result of the constrints ppliction nd it is not feture of the experimentl confinement dt. Inclusion in the pper of the some detils of the scling (2) development would llow voiding the reder confusion. Some clcultions show tht the procedure of the scling (2) development contins likely interesting detils, which re worth mentioning. 4

5 ped th H = th / H(y,2) ASDEX AUG CMOD Compss DIII-D JET JFT-2M JT-60U PBX-M PDX TCV TFTR TDEV START Frction of the pedestl energy ped in totl therml energy th s function of the enhncement fctor H. ped is clculted in ccordnce with the pedestl scling (1). th is tken from H-mode DB3v10. There re some points with ped > th shded region. Most of these points hve th less thn energy H(y,2) predicted by the ITERH-98P(y,2) scling, however there re points with th > H(y,2). The points with ped / th > 1 cnnot be used in the usul log-liner fitting procedure, since = th ped is negtive for these points. The restriction level of ped / th is quit rbitrry. Let we consider results of fitting the to the subsets of points with different level of restriction on ped / th. The horizontl lines on the figure show the restriction levels. 5

6 Restriction level η Number RMSE of points τ ρ β ν (%) The pper vlues Restriction level is ped / th < η. The exponents of XI XB XR Xn XP X ε X XM = CI B R n P ε M re fitted to the restricted dtsets nd then the exponents of the physicl prmeters re clculted. (The log-liner fit is used). N i 1 2 The RMSE = ln th is clculted for the totl set of N i i= 1 scl N = 2678 observtions. scl = + ped Dt in the tble show: the exponent τ of the Bohm time is close to 1 t lest for 3 fist restriction levels (no specil constrint ws pplied), the exponent ρ of the normlized gyro-rdius vries pproximtely from 1 to 1, the exponent β vries from positive vlue to strong negtive vlue, the exponent ν is close to 0, the restriction level η = 0.65 provides the minimum of RMSE, the restriction level η = 0.65 provides closeness of the physicl prmeter exponents to tht in the pper. 6

7 The fit result with η = 0.65 expressed in engineering prmeters is = 0.057I B R n P ε M (5) The scling (5) is close to scling from the pper: = 0.652I 0.45 B 0.35 R 2.55 n 0.6 P 0.4 ε M 0.2 Vrition of constnt nd prticulr exponents of the component of the two-term scling shows tht RMSE minimum lies close to vlues specified in the pper. For exmple, the influence of the constnt C vritions on RMSE is presented on the next figure RMSE (%) C The constnt C = = exp(-2.73) provides locl minimum for RMSE. It is interesting tht scling, providing equlity of 4 exponents to integer vlues, gives stisfctory description of the dtset. 7

8 CONCLUSION An ddition of the some intermedite results of the scling development would llow voiding the reder confusion. The ddition could lso help the reder in understnding the interesting fetures of the scling. It is difficult to indicte specific detils, which could be dded, since the exct procedure of the scling development is unknown. For exmple, the intermedite scling (like (5)) with indiction of its dtset could be dded. The trnsition from the intermedite scling to the finl version could be explined. Editoril remrk. The second line from the down on pge 5. nd the fit to of the complete two-term model to dtbse 8

Contents. Fitting Pedestal data to a) Thermal Conduction Model. b) MHD Limit Model. Determining scaling of Core.

Contents. Fitting Pedestal data to a) Thermal Conduction Model. b) MHD Limit Model. Determining scaling of Core. Contents An Analysis of the Multi-machine Pedestal and Core Databases by J.G. Cordey, O. Kardaun, D.C. McDonald and members of the ITPA Pedestal and Global database groups Fitting Pedestal data to a) Thermal