XGC-0 code | Integrated Modeling of Tokamak Plasmas

Neoclassical transport in the Alcator C-Mod discharge 1120815027

Following a request from Ahmed Diallo, I have started the analysis of the Alcator C-Mod discharge 1120815027. The objective if this study is to compute the neoclassical transport at the plasma edge of this discharge and to compare the neoclassical coefficients with some known analytical formulas. After usual several short XGC0 simulations, I was able to reconstruct the anomalous transport coefficients that are needed to match the simulated density profiles with the experimental profiles. There is still some work that is needed to match the temperature profiles. Here is the summary of plasma profiles that are computed so far (case xcmod.a27):
Electron density / Electron and ion temperatures

Toroidal and poloidal velocities

Effective neoclassical diffusivites

Thermal fluxes

Radial electric field and bootstrap current

Particle fluxes

Here, HH stands for the Hinton-Hazeltine formulas for the bootstrap current and the radial electric field.

These results were discussed with Ahmed and it has been decided that the toroidal and possible poloidal rotation profiles need to be compared with the experimental data from Alcator C-Mod. However, because most of measurements are for impurities, it will be necessary to rerun the XGC0 code with impurities. I will need to update the code either from Devon or somebody else from the CS group.

Initial equilibrium for PB study for the KSTAR discharge 7328

Based on I have created an initial equilibrium for the PB studies. In addition to the information in the Minwoo’s post, I have also used some information from our e-mail correspondence. In particular, I have set the line average density as $3.6times10^{19} m^{-3}$ and the central electron temperature as 1.3 keV. I have also made the following assumptions:

The pedestal width is the same for the temperature and density profiles. It is equal to 0.06 of the normalized poloidal flux;
The pedestal temperature is 400 eV;
The pedestal density is $2.6times 10^{19} m^{-3}$ ;
The ion temperature is equal to the electron temperature;
I assumed that the central q is slightly above unity.

I have also used relatively standard geometry parameters for KSTAR such as the major radius is 1.77 m, the minor radius is 0.47 m, elongation is 1.9, and triangularity is 0.8.

The q95 value that I am getting using the above settings is 5.06. The plasma pressure and q profiles are shown below.

Minwoo indicated that they observed the ELM mode structures around 222~223 (cm) in major radius. In the equilibrium that I have generated the H-mode pedestal extends from R=221.8 cm to 224 cm. The location of maximum pressure gradient in my equilibrium is at R=223 cm.I think these settings are consistent with your experimental observations.

PB diagram of KSTAR discharge

Target KSTAR discharge is #7328 since we have clear ECEI observation data and toroidal mode analysis data from toroidal Mirnov coil array and toroidally separated two ECEI system.

At the current flat top, plasma current, I_p, is 750 (kA), toroidal magnetic field, B_T, is 2.25 (T), safety factor at 95% poloidal flux, q₉₅, is ~ 5 and the plasma is heated by ~3 (MW) neutral beam only.

At first, bootstrap current is calculated using Sauter’s formula.

Generating a new equilibrium for the KSTAR discharge 6123 simulated with the XGC0 code

I have used the TEQ code in CALTRANS to generate a new equilibrium for the case T85Y that I have simulated using the XGC0 code last week. The plasma pressure and the bootstrap current profiles in the pedestal region are updated and a new equilibrium is generated. In order to ensure that the plasma profiles are monotonic, I have introduced a transition region where the profiles are modified. The plasma pressure in the transition region is computed using the following expression:

$p(psi) = p_{rm eq}(psi) + dp left(tanhleft(displaystylefrac{psi-0.83}{0.2}right)+1right),$

where $p_{rm eq}$ is the plasma pressure in the original equilibrium files that is updated with the plasma pressure $p(psi)$ , $dp$ is the gap between the equilibrium pressure $p_{rm eq}$ and the plasma pressure found by the XGC0 code at $psi=0.83$ . This formula is applied for $psi$ from 0 to 0.83. The characteristic width of the transition region is set to 0.2. The bootstrap current profile is updated in the same way.

The plasma profiles after update for the cases T85Y and T87Y looks like:

Plasma profiles in the original equilibrium and updated plasma profiles for the case T85Y	Plasma profiles in the original equilibrium and updated plasma profiles for the case T87Y

There is still change in the plasma pressure gradient in the transition region for the case T87Y. The transition width might need to be increased.

KSTAR peeling-ballooning color map

Following Dr. Park’s request, I have added a schematic color map for various instabilities to the peeling ballooning stability diagram:

Stability diagram for the KSTAR discharge 6123

Bootstrap current in the KSTAR plasma collisionality scan

I have compared the bootstrap currents for the first three cases. Due to additional particle sources in the last case, I have excluded this case from comparison.

Case XGC-T85Y

Bootstrap current for the case XGC-T85Y

Case XGC-T86Y

Bootstrap current for the case XGC-T86Y

Case XGC-T87Y

Bootstrap current for the case XGC-T87Y

Comparison of bootstrap current after 40 ion transit times

Comparison of bootstrap currents

Modeling of the H-mode pedestal buildup in KSTAR using the XGC0 code

Four cases has been created based on the equilibrium for the KSTAR discharge 06123. The original KSTAR equilibrium does not have the H-mode pedestal. The H-mode pedestal structure was selected to have the pedestal width approximately 0.08 of the normalized poloidal flux so that the pedestal top was located at $psi=0.92$ . These assumptions are used only for the initial profiles. The XGC0 code computes the pedestal structure using the initial plasma profiles as initial conditions. Four cases that I have studied differ by the plasma collisionality. The plasma density and temperatures in each case has been modified so that the ratio $n_e / T_e^2$ for the initial profiles would change, but the plasma pressure $sum_{j=i,e}(n_j T_j)$ remains constant. Differences in initial plasma profiles are summarized in the table below.

Case #	$n_{ped} (10^{19} m^{-3})$	$T_{ped} (keV)$	ptl_charge	Collisionality factor
XGC-T85Y	1.54	1.43	1	1
XGC-T86Y	3.14	0.7	1	8.5
XGC-T87Y	4.00	0.55	1	17.56
XGC-T88Y	4.92	0.447	2	65.39

For all four cases, I used the following assumption for the anomalous transport:

! particle transport
tbl_D_coeff_in=0.7D0 ! m^2/sec, inside (pedestal top) value of D_turb
tbl_D_coeff=0.04D0 ! m^2/sec, outside (pedestal bottom) value of D_turb
tbl_D_coeff2=1.3D0 ! m^2/sec, D_turb for outside of separatrix (SOL+private flux region)
! ion thermal diffusivities
tbl_therDi_coeff_in=0.8D0 ! m^2/sec, inside (pedestal top) value of thermal D_turb for ions
tbl_therDi_coeff=1D-8 ! m^2/sec, outside (pedestal bottom) value of thermal D_turb for ions
tbl_therDi_coeff2=4D0 ! m^2/sec, thermal D_turb for outside of separatrix (SOL+private flux region) for ions
! electron thermal diffusivities
tbl_therDe_coeff_in=1.2D0 ! m^2/sec, inside (pedestal top) value of thermal D_turb for electrons
tbl_therDe_coeff=1D-8 ! m^2/sec, outside (pedestal bottom) value of thermal D_turb for electrons
tbl_therDe_coeff2=4D0 ! m^2/sec, thermal D_turb for outside of separatrix (SOL+private flux region) for electrons

The electron and ion temperature profiles and electron density profiles for all four cases are given below.

Case XGC-T85Y


Electron temperature profile for the case XGC-T85Y	Ion temperature profile for the case XGC-T85Y
Electron density profile for the case XGC-T85Y	Plasma pressure profile for the case XGC-T85Y

Case XGC-T86Y


Electron temperature profile for the case XGC-T86Y	Ion temperature profile for the case XGC-T86Y
Electron density profile for the case XGC-T86Y	Plasma pressure profile for the case XGC-T86Y

Case XGC-T87Y


Electron temperature profile for the case XGC-T87Y	Ion temperature profile for the case XGC-T87Y
Electron density profile for the case XGC-T87Y	Plasma pressure profile for the case XGC-T87Y

Case XGC-T88Y


Electron temperature profile for the case XGC-T88Y	Ion temperature profile for the case XGC-T88Y
Electron density profile for the case XGC-T88Y	Plasma pressure profile for the case XGC-T88Y

Some observations:

The pedestal width is smaller than the width that is used for the initial profiles. The computed pedestal width is around 0.6-0.7 of the normalized poloidal flux for all cases.
During the XGC0 runs (44-80 ion transit times), the pedestal continue to grow only for the first two cases. The pedestal height decreases for two other cases. This effect is most notable for the last case, that has the largest collisionality. I had to increase internal particle source in this case in order to run it to 44 ion transit times. Event with increased particle sources, the simulations for the case XGC-T88Y crashes after 44 ion transit times.
The slop of the plasma pressure profiles continue to change for all four cases.
There is a density depletion in the SOL region for the second and third cases at t=80. This effect results in an artificial increase of ion temperatures in the SOL region.

KSTAR Stability Analysis

The stability map for the KSTAR geometry is generated. The base equilibrium parameters in TOQ are the following:

rmax=47.000000
rzero=177.000000
eshape=2.00
xshape=.64000
totcur=600000.000000
baxis0=20000.000000

The base temperature and density profiles were set using parametric expressions in TOQ using the following parameters:

modelp=27
nedge13=0.7
nped13=2.0
ncore13=5.7
nfactor=1.0
tedgeEV=100.
tpedEV= 500.0
tcoreEV=2000.
widthp=.03
xphalf=.97

The corresponding density profile has the following shape:

The stability map is generated by varying the nped13, tpedEV and cboot parameters the ranges [0.6:5.5], [100:1600], and [0.1:9] correspondingly. Over 150 new equilibria have been generated. The pedestal stability in these equilbria were tested using the DCON and BALOO codes. The resulting stability map is shown below:

j-alpha stability diagram for KSTAR

Here $alpha$ is the normalized plasma pressure gradient defined as

$alpha = -displaystyle{frac{mu_0}{2pi^2} frac{partial p}{partial psi} frac{partial V}{partial psi} sqrt{frac{V}{2pi^2 R}}},$

where $p$ is the plasma pressure, $psi$ is the poloidal flux, $V$ is the plasma volume, and $R$ is the plasma major radius.

The results are valid for a particular choice of plasma parameters such as $I_{rm pl}=0.6 MA$ , $B_{rm T}=2.0 T$ , $kappa=2$ , and $delta=0.64$ . The plasma shaping is defined according to the Bondeson’s Dee Formula. The pedestal width has been also fixed in my scan and it has been set to 0.6. The location of the H-mode pedestal top for density and temperature profiles was set to 0.94 of the normalized poloidal flux.

Summary of anomalous diffusivity parameters that are used in the XGC0 modeling of four DIII-D discharges

DIII-D 132014

&tbl_param             ! (anomalous) turbulence diffusion
tbl_diffusion_on=1     ! 0: no turbulence diffusion, 1: simple turbulence diffusion
tbl_diffusion_mode=3   ! 1 = random walk model, 2 = advection-diffusion model
! mode=2 enables separate control of particle and heat diffusion
tbl_stop_time=500000   ! Stop the turbulence diffusion
tbl_period = 1       ! frequency of calling turbulence diffusion and fmcfm_call

tbl_d_profile_on=-1 ! 1: spatially varying D_turb profile (tanh), 0: spatially constant profile in pedestal region
tbl_slope_on=0 ! D_turb turned on only within finite poloidal angles (+45 deg) around outside midplane
tbl_d_profile_psicen=0.93D0 ! D_turb profile center
tbl_d_profile_psiwid=0.02D0 ! D_turb profile full width
tbl_d_profile_psicen2=0.999D0 ! D_turb profile center
tbl_d_profile_psiwid2=-0.01D0 ! D_turb profile full width

! particle diffusivities (random walk diffusion coefficients)
tbl_D_coeff_in=0.2D0     ! m^2/sec, inside (pedestal top) value of D_turb
tbl_D_coeff=0.03D0      ! m^2/sec, outside (pedestal bottom) value of D_turb
tbl_D_coeff2=0.6D0     ! m^2/sec, D_turb for outside of separatrix (SOL+private flux region)

! ion thermal diffusivities
tbl_therDi_coeff_in=.070D0    ! m^2/sec, inside (pedestal top) value of thermal D_turb for ions
tbl_therDi_coeff=-0.050D0    ! m^2/sec, outside (pedestal bottom) value of thermal D_turb for ions
tbl_therDi_coeff2=0.9D0    ! m^2/sec, thermal D_turb for outside of separatrix (SOL+private flux region) for ions

! electron thermal diffusivities
tbl_therDe_coeff_in=0.8D0    ! m^2/sec, inside (pedestal top) value of thermal D_turb for electrons
tbl_therDe_coeff=-0.1D0    ! m^2/sec, outside (pedestal bottom) value of thermal D_turb for electrons
tbl_therDe_coeff2=0.9D0    ! m^2/sec, thermal D_turb for outside of separatrix (SOL+private flux region) for electrons

DIII-D 132016

tbl_d_profile_on=-1 ! 1: spatially varying D_turb profile (tanh), 0: spatially constant profile in pedestal region
tbl_slope_on=0 ! D_turb turned on only within finite poloidal angles (+45 deg) around outside midplane
tbl_d_profile_psicen=0.94D0 ! D_turb profile center
tbl_d_profile_psiwid=0.02D0 ! D_turb profile full width
tbl_d_profile_psicen2=0.999D0 ! D_turb profile center
tbl_d_profile_psiwid2=-0.01D0 ! D_turb profile full width

! particle diffusivities (random walk diffusion coefficients)
tbl_D_coeff_in=0.13D0     ! m^2/sec, inside (pedestal top) value of D_turb
tbl_D_coeff=0.04D0      ! m^2/sec, outside (pedestal bottom) value of D_turb
tbl_D_coeff2=0.3D0     ! m^2/sec, D_turb for outside of separatrix (SOL+private flux region)

! ion thermal diffusivities
tbl_therDi_coeff_in=0.1D0    ! m^2/sec, inside (pedestal top) value of thermal D_turb for ions
tbl_therDi_coeff=-0.05D0    ! m^2/sec, outside (pedestal bottom) value of thermal D_turb for ions
tbl_therDi_coeff2=0.7D0    ! m^2/sec, thermal D_turb for outside of separatrix (SOL+private flux region) for ions

! electron thermal diffusivities
tbl_therDe_coeff_in=0.25D0    ! m^2/sec, inside (pedestal top) value of thermal D_turb for electrons
tbl_therDe_coeff=-0.05D0    ! m^2/sec, outside (pedestal bottom) value of thermal D_turb for electrons
tbl_therDe_coeff2=0.7D0    ! m^2/sec, thermal D_turb for outside of separatrix (SOL+private flux region) for electrons
/

DIII-D 132017

tbl_d_profile_on=-1 ! 1: spatially varying D_turb profile (tanh), 0: spatially constant profile in pedestal region
tbl_slope_on=0 ! D_turb turned on only within finite poloidal angles (+45 deg) around outside midplane
tbl_d_profile_psicen=0.91D0 ! D_turb profile center
tbl_d_profile_psiwid=0.02D0 ! D_turb profile full width
tbl_d_profile_psicen2=0.995D0 ! D_turb profile center
tbl_d_profile_psiwid2=-0.01D0 ! D_turb profile full width

! particle diffusivities (random walk diffusion coefficients)
tbl_D_coeff_in=0.13D0     ! m^2/sec, inside (pedestal top) value of D_turb
tbl_D_coeff=0.015D0      ! m^2/sec, outside (pedestal bottom) value of D_turb
tbl_D_coeff2=0.3D0     ! m^2/sec, D_turb for outside of separatrix (SOL+private flux region)

! ion thermal diffusivities
tbl_therDi_coeff_in=.1D0    ! m^2/sec, inside (pedestal top) value of thermal D_turb for ions
tbl_therDi_coeff=0.001D0    ! m^2/sec, outside (pedestal bottom) value of thermal D_turb for ions
tbl_therDi_coeff2=0.5D0    ! m^2/sec, thermal D_turb for outside of separatrix (SOL+private flux region) for ions

! electron thermal diffusivities
tbl_therDe_coeff_in=0.35D0    ! m^2/sec, inside (pedestal top) value of thermal D_turb for electrons
tbl_therDe_coeff=-0.1D0    ! m^2/sec, outside (pedestal bottom) value of thermal D_turb for electrons
tbl_therDe_coeff2=0.9D0    ! m^2/sec, thermal D_turb for outside of separatrix (SOL+private flux region) for electrons
/

DIII-D 132018

&tbl_param ! (anomalous) turbulence diffusion
tbl_diffusion_on=1 ! 0: no turbulence diffusion, 1: simple turbulence diffusion

tbl_diffusion_mode=3   ! 1 = random walk model, 2 = advection-diffusion model
! mode=2 enables separate control of particle and heat diffusion
tbl_stop_time=500000   ! Stop the turbulence diffusion
tbl_period = 1       ! frequency of calling turbulence diffusion and fmcfm_call

tbl_d_profile_on=-1 ! 1: spatially varying D_turb profile (tanh), 0: spatially constant profile in pedestal region
tbl_slope_on=0 ! D_turb turned on only within finite poloidal angles (+45 deg) around outside midplane
tbl_d_profile_psicen=0.89D0 ! D_turb profile center
tbl_d_profile_psiwid=0.02D0 ! D_turb profile full width
tbl_d_profile_psicen2=0.995D0 ! D_turb profile center
tbl_d_profile_psiwid2=-0.01D0 ! D_turb profile full width

! particle diffusivities (random walk diffusion coefficients)
tbl_D_coeff_in=0.05D0     ! m^2/sec, inside (pedestal top) value of D_turb
tbl_D_coeff=0.002D0      ! m^2/sec, outside (pedestal bottom) value of D_turb
tbl_D_coeff2=1D0     ! m^2/sec, D_turb for outside of separatrix (SOL+private flux region)

! ion thermal diffusivities
tbl_therDi_coeff_in=0.5D0    ! m^2/sec, inside (pedestal top) value of thermal D_turb for ions
tbl_therDi_coeff=0.001D0    ! m^2/sec, outside (pedestal bottom) value of thermal D_turb for ions
tbl_therDi_coeff2=2.0D0    ! m^2/sec, thermal D_turb for outside of separatrix (SOL+private flux region) for ions

! electron thermal diffusivities
tbl_therDe_coeff_in=.8D0    ! m^2/sec, inside (pedestal top) value of thermal D_turb for electrons
tbl_therDe_coeff=-0.07D0    ! m^2/sec, outside (pedestal bottom) value of thermal D_turb for electrons
tbl_therDe_coeff2=0.2D0    ! m^2/sec, thermal D_turb for outside of separatrix (SOL+private flux region) for electrons
/

Effect of neutrals is studied for the discharge DIII-D 132016

The effect of neutrals is studied for the discharge DIII-D 132016. The heat load width is relatively weekly affected by the neutral collisions.

Summary, the effect of neutrals are rather small comparing to the anomalous effects.