XGC-0 anomalous transport verification | Integrated Modeling of Tokamak Plasmas

Smoothing algorithm is modified

Smoothing procedure adopted from the ASTRA code on Friday (see previous post) is tested for set of 200 random numbers. The plots are generated for the smoothness parameter $alpha=0.001$ (left figure), $alpha=0.001$ (center figure), and $alpha=0.001$ (left figure):

In order to introduce more smoothing in the plasma core which is needed to compensate additional numerical noise in the core introduced by volume effect, $alpha$ dependence on $rho$ is introduced:

$alpha(rho) = alpha_0 (1-rho/rho_{edge})^beta$

where $beta$ is the coefficient that controls region where smoothing is applied. The following function is selected for the next test:

$f(rho)=displaystyle{3-2rho^3-50exp^{-displaystylefrac{(rho-rho_b)^2}{Delta}} + 40 frac{R}{1+9rho^2}}$

where R is a random number in the range from 0 to 1, $rho_0$ and $Delta$ are the coefficient that are set to 0.95 and 0.001 correspondingly. This function reproduce XGC-0 results for the radial electric field with more noise in the plasma core and large potential well in the plasma edge. The goal of smoothing is to remove noise in the plasma core and to preserve the details of potential well in the plasma edge. The results on the figure below are obtained for smoothness parameters $alpha_0=0.01$ and $beta=4$ .

The level of smoothness in the plasma core is controlled by $alpha_0$ and the region where soothing is applied is controlled by the coefficient $beta$ . Smaller values of $beta$ should result in more extended region where the smoothing is applied. Test below show results for the same $alpha_0$ , but $beta$ set to 2.

3 – 2 rho^3

New idea for smoothing of plasma profiles is being tested

The idea for smoothing of plasma profiles is based on routine implemented in the ASTRA code. The resulting function is based on finding of the minimum of the functional

$intleft(alpha frac{dU}{dx}^2+(U-F)^2right) dx$

with respect to U. The functional is design for the cylindrical case and the resulting function has zero derivative for x=0 and preserve boundary condition at x=1. The description of the method can be found at:

B. K. P. Horn and B. G. Schunck, Artifical Intelligence, 17:185-203, 1981.
N. P. Galatsanos and A. K. Katsaggelos. IEEE transaction on image processing, 1(3):322-336, 1992.

XGC-212 case

New simulation for the whole plasma profile with XGC-0 standard anomalous transport model is submitted on FRANKLIN. The FMCFM interface is not used. All other settings are the same.

XGC-209 case

Error in the initialization of the case XGC-209 is discovered. The case is resubmitted with the following additional changes:

fm_nsmooth = 0   ! number of applications of smoothing in fmcfm_call
fm_nsmooth_er = 4 ! number of smoothing for radial electric field in the core region
fm_smooth_er_rhob = 0.95 ! rho, where transition between different level of smoothing is applied
fm_smooth_er_coef = 50. ! coefficient that controls sharpness of transition between different level of smoothness for Er
fm_nsmooth_chi = 2 ! number of applications of diffusivity smoothing in fmcfm_call
fm_wexb   = 1.0d-6

Note that the flow shear stabilization coefficient is reduced from 1e-5 to 1e-6.

Averaging over three neighboring points is used in this case.

New two short simulations to test smoothing are submitted

New short XGC-0 simulations are submitted on FRANKLIN in the debug mode. The case XGC-210 is based on the case XGC-209. The case XGC-211 uses modified source code where smoothing for the radial electric field is done for three neighboring grid points instead of one (as in the case XGC-210).

IF ( fm_nsmooth_er > 0 ) THEN
  DO i=1, fm_nsmooth_er
    CALL smoothb0( z_e_radial, diag_flow_npsi, 3 )
  ENDDO
ENDIF

The radial electric field profiles as well as flow shear rates are significantly smoother comparing to the case where is smoothing is done over one neighboring grid point.

Results that use the same smoothing in the plasma core, but use averaging over one neighboring points are given below for comparison.

Particle losses in the outer region of dicharge are analyzed

Significant particle losses noted in the last two posts are studied by analyzing the diffusivity profiles.

The particle diffusivity remains too high after 13999 time steps, which correspond to approximately 28 toroidal ion transit times (13999×0.002). Thermal diffusivity remains very high as well. However, the ion and electron temperatures continue to rise with very high auxiliary heating power.

Smoothing of thermal diffusivity profiles (fm_nsmooth_chi=2) improves numerical stability.

The flow shear factor is zero in the plasma core. Also, the flow shear factor remains relatively noisy.

Simulations to test different smoothing algorithms are submitted on FRANKLIN

Two new cases are submitted on FRANKLIN in order to investigate different settings for smoothing.

Case XGC-208 is based on the case XGC-206, except of fm_nsmooth_chi that is increased from 1 to 2.
Case XGC-209 is submitted to test new procedure to smooth the radial electric field profiles. In addition to the parameter fm_nsmooth that is kept 1 (as in the previous cases), the following settings are used:

fm_nsmooth_er = 4 ! number of smoothing for radial electric field in the core refion
fm_smooth_er_rhob = 0.95 ! rho where transition between different level of smoothing is applied
fm_smooth_er_coef = 75. ! coefficient that controls sharpness of transition between different level of smoothness for Er
fm_nsmooth_chi = 2 ! number of applications of diffusivity smoothing in fmcfm_call

Cases XGC-205 and XGC-206 are completed on FRANKLIN

Set of simulations where particle bounce on the inner boundary was disabled is complete. The results indicate reduction of particle near the magnetic axis. The radial electric field has significant oscillations near the magnetic axis.

New smoothing procedure for radial electric field is implemented

New procedure for smoothing of radial electric field profile is implemented in the fmcfm_call routine. Different levels of smoothing is applied in the plasma core and in the plasma edge. The radial electric field profile in the plasma edge is smoothed fm_nsmooth times together other plasma profiles such as electron and ion temperatures and plasma density profiles. The radial electric field in the plasma core is smoothed fm_nsmooth_er times. The new smoothed profiles are merge together using the following formula:

${Huge E_r} = displaystylefrac{1}{2} left[ E_r^{rm core} left( 1 - tanh(C_{er} (rho-rho_b)) ) + E_r^{rm edge} ( 1 + tanh(C_{er} (rho-rho_b)) right) right]$

and used in the computation of the $ExB$ flow shear. Here, $C_{er}$ is the coefficient that describes sharpness of the transition between two smoothed profiles The fm_smooth_er_coef is used to set this coefficient in the XGC-0 code through the namelist. The $rho_b$ parameter controls the location of the transition. The fm_smooth_er_rhob variable is used to set this parameter in the code. The default values for these parameters are: $C_{er}=75$ and $rho_b=0.95$ .