Alcator C-Mod Predictive Modeling

Alexei Pankin, G. Bateman, A. H. Kritz,

Lehigh University Physics Department
16 Memorial Drive East, Bethlehem, PA 18015

and the Alcator C-Mod team

MIT

American Physical Society DPP Meeting
Québec City, Canada, 23-27 October 2000

Abstract

Predictive simulations are carried out using the BALDUR 1-1/2-D transport code for L- and H-mode discharges in the Alcator C-Mod tokamak. The plasma densities in these C-Mod discharges (from 0.9 to 3.8·10²⁰ m^-3) are higher than the densities in other tokamaks that we have simulated. Three transport models are used in these simulations: the standard Multi-Mode model (MMM95), a new version of the Multi-Mode model (MMM99), and the Mixed-Bohm/gyro-Bohm (JET) model. It is found that the Bohm term dominates in the Mixed-Bohm/gyro-Bohm model, while the Multi-Mode models have gyro-Bohm scaling. The simulated density profiles are found to be systematically flatter than the experimental data. The calculated temperature and density profiles for each of the three models are found to match experimental data with an average rms deviation of 18% (normalized by the corresponding central experimental values).

DoE contracts DE-FG02-92-ER-54141 and DE-FC02-99ER54512.

The BALDUR code: 

Predicts temperature and density profiles in tokamaks using
theoretically derived transport models

input		computes
	$\small \begin{array}{c} \underbrace{% \begin{array}{c}... ...\;Bohm/gyroBohm\\ Multi-Mode\;Model\;99 \end{array} } \end{array}$	$\left\{ \small \begin{array}{l} \circ\ evolution\ of\ te... ...stabilities\ (e.g.,\ sawtooth\\ instabilities). \end{array} \right.$

The Multi-Mode Model (MMM 95)^{1 2}

Multi-Mode Model - combination of theory based transport models

• used to predict the temperature and density profiles in tokamaks

Multi-Mode Model consists of three components:

• Weiland model for the Ion Temperature Gradient (ITG) modes and the Trapped Electron Modes (TEM)
• Guzdar-Drake model for the drift-resistive ballooning modes
• model for Kinetic ballooning mode

The Multi-Mode Model: 
Weiland contribution for ITG and TEM modes

Principal Contribution to Transport in the Most of Plasma Core

• based on linearized fluid equations with magnetic drift for each particle spices

• includes electromagnetic effects, electron-ion collisions, finite trapped electrons, impurities, and fast ions

• diffusion matrix is computed as a derivative of the heat and particle fluxes, obtained in a quasilinear approximation, with respect to the temperature and density gradients; convective velocities are computed from the difference between the total fluxes and the diffusive fluxes

New Multi-Mode Model (MMM 99)³

• Drift Alfvén mode turbulence simulation by Bruce Scott:
  • for transport near edge scales like q²[R(dp/dr)/p]⁴
  • strong elongation scaling e^-3.1236k
• Improvements to the Weiland model for ITG and TEM turbulence:
  • eigenfunctions extended along field lines for better
    approximation of high elongation effect
  • flow shear rate substracted from growth rates
• New kinetic ballooning mode model:
  • contributes transport mainly near the magnetic axis

The Mixed Bohm/gyro-Bohm (JET) Model⁴

Empirically based model

Combines three major terms:

${\alpha_{e,i}^{B}\frac{cT_{e}}{eB}\left(L_{p_{e}}^{\ast }\... ... ^{-1}q^{2}\left\langle L_{T_{e}}^{\ast}\right\rangle _{\Delta V}^{-1}}$	${+\alpha _{e,i}^{gB}\frac{cT_{e}}{eB}\left( L_{T_{e}}^{\ast }\right) ^{-1}\rho ^{\ast }}$	Bohm term: originally developed to fit JET data
	${+n_{i}\delta ^{1/2}\frac{\rho _{i\theta }^{2}}{\tau _{ii}}k_{2}^{neo}}$	gyro-Bohm term: required to fit discharges in smaller tokamaks
	$L_{T_e}^{\ast }=\frac{T_{e}}{a}\left( \frac{dT_e}{dr}\right) ^{-1}$	neoclassical term: dominates ion thermal transport near the magnetic axis

where 
$L_{p_{e}}^{\ast }=\frac{p_{e}}{a}\left( \frac{dp_{e}}{dr}\right) ^{-1}$, $\rho ^{\ast }=\frac{M^{1/2}cT_{e}^{1/2}}{aZ_{i}eB_{T}}$,
$\delta=\frac{r}{R}$, $k_{2}^{neo}=\frac{0.66+1.88\delta^{1/2}-1.54\delta }{1+1.03\nu _{\ast ii}^{1/2}+0.31\nu _{\ast ii}}\left\langle\frac{B_{0}^{2}}{B^{2}}\right\rangle$
$\left\langle \frac{B_{0}^{2}}{B^{2}}\right\rangle =\frac{1+1.5\delta \left( \delta +\left( 1+0.25\delta ^{2}\right) dR_{0}/dr\right) }{1+0.5\delta dR_{0}/dr}$,
$\rho _{i\theta }$  
r_iq is poloidal gyroradius, n_*ii is ion collision frequency;

constants 

$\alpha _{e}^{gB}=2\alpha _{i}^{gB}=3.5\times 10^{-2}$   and  $R=0.68$
are empirical parameters.

Alcator C-Mod Tokamak⁵

Compact, High-field, High-density Divertor Tokamak

Typical Parameters R=0.68 m a=0.22 m BT<9 T IP<2 MA k<1.85 d<0.6 ne< 1.5·1020 m-3 Ti~Te<6 keV ICRF Auxiliary Heating

C-Mod Discharges Considered

L-Mode Discharges

C-MOD shot	950407013	960229042	960126007	960301009
R (m)	0.673	0.673	0.673	0.672
a (m)	0.210	0.219	0.217	0.218
k	1.64	1.64	1.65	1.61
d	0.42	0.42	0.41	0.44
BT (T)	5.38	5.38	5.24	5.42
IP (MA)	1.01	1.00	0.80	0.83
<ne>·1020 (m-3)	1.52	1.82	0.93	1.50
Zeff	2.02	2.02	2.38	1.85
dr*(0) (m)	0.0094	0.010	0.0085	0.011
Paux (MW)	2.7	2.2	1.4	2.8

H-Mode Discharges

C-MOD shot	960116024	960116027	960214017
R (m)	0.676	0.676	0.677
a (m)	0.221	0.219	0.222
k	1.65	1.65	1.59
d	0.41	0.42	0.41
BT (T)	5.22	5.22	5.21
IP (MA)	1.01	1.02	1.02
<ne>·1020 ( m-3)	3.17	3.86	3.36
Zeff	2.40	1.41	1.49
dr*(0) (m)	0.0099	0.0096	0.0092
Paux (MW)	2.35	2.7	2.5

All the discharges above have much higher densities,
than it was simulated before with BALDUR.

Statistical Analysis
Criteria for comparison of the BALDUR results with experiment

• Deviation of  j^th experimental point (X_j^(k)exp) and corresponding simulation result ( X_j^(k)sim) for k^th discharge is defined as:

$\sigma ^{\left( k\right) }$

• Root-Mean-Square (rms) deviation (s^(k)) and offset (f(k)) allow estimation of model validity:
  $f^{\left( k\right) }\equiv \frac{1}{N}\sum\limits_{j=1}^{N}\epsilon _{j}^{\left( k\right) }$      and $\bar{\sigma}$
• Average rms deviation ($\sigma$) and the rms of  (s_s) are:
  
  $\sigma _{\sigma }\equiv \sqrt{\frac{1}{K-1} \sum_{k=1}^{K}\left( \sigma ^{\left( k\right) }-\bar{\sigma}\right) ^{2}}$        $K$
  where  K is the number of discharges.

Statistical Results

	960116024		960116027		960214017
	MMM95	JET	MMM95	JET	MMM95	JET
fne	0.025	0.002	-0.006	-0.002	-0.050	-0.037
fTe	0.062	0.031	0.100	0.071	0.057	-0.032
fTi	0.139	0.111	0.147	0.116	0.130	0.046
snerel	0.075	0.064	0.072	0.060	0.101	0.068
sTerel	0.045	0.030	0.115	0.041	0.096	0.102
sTirel	0.095	0.074	0.167	0.061	0.062	0.046

 

	960126007		950407013		960301009		960229042
	MMM95	JET	MMM95	JET	MMM95	JET	MMM95	JET
fne	-0.067	-0.080	0.027	0.043	-0.043	-0.060	-0.035	-0.049
fTe	-0.492	-0.241	-0.065	-0.137	0.088	-0.098	-0.033	-0.058
fTi	0.069	0.268	-0.087	-0.179	0.070	0.091	-0.295	-0.303
snerel	0.081	0.040	0.048	0.042	0.074	0.076	0.123	0.095
sTerel	0.514	0.307	0.057	0.079	0.156	0.152	0.034	0.050
sTirel	0.048	0.178	0.145	0.195	0.042	0.049	0.253	0.265