A Hierarchy of Empirical Models of Plasma Profiles and Transport

Two families of statistical models are presented which generalize global confinement expressions to plasma profiles and local transport coefficients. The temperature or diffusivity is parameterized as a function of the normalized flux radius, $\bar{\psi}$, and the engineering variables, ${\bf u} = (I_p,B_t,\bar{n},q_{95})^\dagger$. The log-additive temperature model assumes that $\ln [T(\bar{\psi}, {\bf u})] =$ $f_0 (\bar{\psi}) + f_I (\bar{\psi})\ln[I_p]$ $+ f_B (\bar{\psi}) \ln [B_t]$ $+ f_n (\bar{\psi}) \ln [ \bar{n}] + f_{q}\ln[q_{95}]$. The unknown $f_i (\bar{\psi})$ are estimated using smoothing splines. A 43 profile Ohmic data set from the Joint European Torus is analyzed and its shape dependencies are described. The best fit has an average error of 152 eV which is 10.5 \% percent of the typical line average temperature. The average error is less than the estimated measurement error bars. The second class of models is log-additive diffusivity models where $\ln [ \chi (\bar{\psi}, {\bf u})] $ $=\ g_0 (\bar{\psi}) + g_I (\bar{\psi}) \ln[I_p]$ $+ g_B (\bar{\psi}) \ln [B_t ]$ $+ g_n (\bar{\psi}) \ln [ \bar{n} ]$. These log-additive diffusivity models are useful when the diffusivity is varied smoothly with the plasma parameters. A penalized nonlinear regression technique is recommended to estimate the $g_i (\bar{\psi})$. The physics implications of the two classes of models, additive log-temperature models and additive log-diffusivity models, are different. The additive log-diffusivity models adjust the temperature profile shape as the radial distribution of sinks and sources. In contrast, the additive log-temperature model predicts that the temperature profile depends only on the global parameters and not on the radial heat deposition.