August, 2013 | Integrated Modeling of Tokamak Plasmas

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 plasma collisionality studies

The MMM8.1 model in TRANSP reproduces the experimental dependence of the energy confinement times on the plasma collisionality. However, the values of the predicted confinement times are systematically larger than the confinement times derived from experiments. Most likely reason is missing transport from modes that are not part of MMM8.1 such as microtearing modes.

Experimental energy confinement times

Predicted energy confinement times
Plasma collisionality scan using standalone MMM8.1 model revealed that most components in MMM8.1 can not explain the dependence that is observed in predicted TRANSP simulations and in experiment. The Weiland component that computes fluxes associated with most drift-wave instabilities such as ITG modes in the plasma core shows that the electron and ion thermal diffusivities are stabilized by the plasma collisionality. This effect is expected because of known dumping effect of plasma collisionality on the drift-wave modes. There are two possible explanations of the discrepancies between predictive and standalone MMM8.1 runs.
The dependence of electron thermal diffusivity on the collision frequency for DIII-D

The resistive mode in DRIBM that is found to be destabilized with the plasma collisionality is typically has large contribution in the plasma edge region near the separatrix. Note that the paleoclassical model that is distributed together with the MMM8.1 model is expected to have a similar dependence on the resistivity. However, the paleoclasical model is still being validated. Usually, it is not used in the predictive runs with the MMM8.1 model.

Direct comparison of energy confinement times for the NSTX discharges

The predictive simulations with the MMM8.1 model show the same trend as observed in experiments. The confinement times for the low plasma collisionality discharge are larger. However, the values of confinement time are systematically higher comparing to the confinement times derived from the experimental data. Most likely reason is missing transport from instabilities that are not part of MMM8.1 (such as microtearing modes).

Experimental energy confinement times	Predicted energy confinement times

Why the thermal diffusivity increases with plasma collisionality when the experimental profiles are used but not in the independent tests wih the transport model?

It is found that the safety factor q profile and the magnetic shear s profile can change the $chi(nu_star)$ dependence. When the q and s profiles for the NSTX discharge 141031 are replaced with the corresponding profiles for the discharge 141040, the experimental $chi(nu_star)$ dependence is partially recovered. The electron thermal diffusivity profiles on the figure below that is marked as hybrid31c is generated using the Weiland model in MMM8.1 with the plasma profiles for the discharge 141031 with the exception of the q and s profiles that are taken from the discharge 141040.

Electron thermal diffusivity

Varying the collisionality for the NSTX discharges 141031 and 141040

In order keep the plasma equilibrium unchanged, the density and temperature profiles should be changed with a scaling factor $C_\nu$ :

$n_{e,i,z,f} \rightarrow C_\nu n_{e,i,z,f} , T_{e,i} \rightarrow T_{e,i} / C_\nu,$

where the indices $e,i,z$ , and $f$ stand for electrons, ions, impurities and fast particles correspondingly. The plasma pressure, $Z_{\rm eff}$ , normalized gradients and scale lengths as well as ratio of the electron to ion temperatures will remain unchanged.

The plasma collisionality for the NSTX discharge 141031 is varied from 0.2 to 20 from the original plasma collisionality. The density and temperature profiles in these scans are shown below:

The electron density profiles for different collisionalities	The electron temperature profiles for different collisionalities

It has been found that increased collisionality results in smaller values of anomalous diffusivities computed with the Weiland model in MMM8.1:

Electron thermal diffusivities	Ion thermal diffusivities

The diffusivity is monotonically decreases with the collisionality. The results for the NSTX discharge 141040 are similar:

Electron thermal diffusivity	Ion thermal diffusivity

When used standalone, the Weiland model in MMM8.1 can not explain the experimental trend $\tau_E \propto 1/\nu_\star$ . However, when used in the whole device integrated modeling code, the MMM8.1 model can reproduce the experimental profiles and $\nu_\star$ trend reasonably well. The main remaining question is what are the changes in other profiles (e.g., safety factor and magnetic shear) can change the $\chi(\nu_\star)$ dependence.

Additional analysis of NSTX discharges with different collisionality

The NSTX discharges 141031 and 141040 have comparable density profiles, but their temperature profiles are distinguishably different:

If collisionality is evaluated using the expression $nu_starpropto n_e/T_e^{3/2}$ , the difference in collisionality between two NSTX discharges is several percents in the plasma core and increases to about 25% at r = 0.5 m. The plasma collisionality in the NSTX discharge 141031 is above the collisionality in the discharge 141040 for all radii:

Collisionality in the NSTX discharges 141031 and 141040.

Being tested directly in the MMM8.1 model, the anomalous diffusivities for the NSTX discharge 141031 are found above the anomalous diffusivities for the NSTX discharge 141040. This statement is not true in the H-mode pedestal region where the results from anomalous transport models are not reliable. Smaller anomalous diffusivities are likely to result in better energy confinement times. The predicted energy confinement for the NSTX discharge 141040 is likely to be better than the predicted energy confinement for the NSTX discharge 141031. This is in agreement with the experimental trends for these two discharges.

Electron thermal diffusivity	Ion thermal diffusivity

The observed dependence of the anomalous diffusivities on the collisionality is not necessary comes from the model, but might be associated with the differences in other profiles and plasma parameters that are difficult to maintain constant between discharges. In particular, in order to isolate the collisionality and ExB flow shear effects, the ExB flow shear profiles were set to zero in these tests. The ExB flow shear profiles between the two NSTX discharge are different:

The ExB flow shear profiles

The ExB flow shear rates for the NSTX discharge 141031 are above the ExB flow shear rates for the discharge 141040 in the plasma region from r=0.18m to r=0.5m and below the ExB flow shear rates in the outer plasma region from r=0.8m. In particular, the ExB flow shear rates are large in the region from r=0.35m to r=0.45m. While neglected in the direct tests with the MMM8.1 model, the ExB flow shear rates were used in the TRANSP predictive simulations last week.

In order to eliminate possible effects from other plasma parameters that are different for the NSTX discharges 141031 and 141040, independent collisionality scans for each discharge are necessary.