Expression for the ExB flow shear rate is improved in the XGC-0 code

The ExB flow shear rate has been implemented as simple derivative of the radial electric field with respect to minor radius:

omega_{rm ExB}equivfrac{partial E_r}{partial r}.

New implementation is according to Ref. [T.S. Hahm and K.H. Burrell Phys. Plasmas 2, 1648 (1995)]:

omega_{rm ExB}equiv left|frac{R B_theta}{B_phi}frac{partial}{partial r} left(frac{E_r}{R B_theta}right)right|,

where B_theta and B_phi are the poloidal and toroidal components of the magnetic field.

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

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

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.