Adiabatic electron/ion model for nprocs=1

When one runs a single-species simulation with setting nprocs =1, GKV employs adiabatic model for electrons or ions. In both case, electrostatic limit is assumed ( \(\tilde{A}_\parallel = 0\) ), and lambda_i and beta in gkvp_namelist are neglected. Setting of kinetic electrons with adiabatic ion model is nprocs =1 in src/gkvp_header.f90 , and Anum =1.d0, Znum =1.d0, fcs =1.d0, sgn =-1.d0 in run/gkvp_namelist . Then the Poisson eq. with adiabatic ion model is

(1) \[\left[ \frac{e^2 n_\mathrm{0}}{T_\mathrm{i}} + \frac{e^2 n_\mathrm{0}}{T_\mathrm{e}} \left( 1 - \Gamma_{0\mathrm{e}\bm{k}} \right) \right] \tilde{\phi}_{\bm{k}} = - e \int dv^3 J_{0\mathrm{e}\bm{k}} \tilde{f}_{\mathrm{e}\bm{k}}\]

Density, temperature and mass are normalized electrons’ value. Then the normalized Poisson eq. is

(2) \[\left[ \frac{T_\mathrm{e}}{T_\mathrm{i}} + 1 - \bar{\Gamma}_{0\mathrm{e}\bm{k}} \right] \bar{\phi}_{\bm{k}} = - \int d\bar{v}^3 \bar{J}_{0\mathrm{e}\bm{k}} \bar{f}_{\mathrm{e}\bm{k}}.\]

The temperature ratio \(T_\mathrm{e}/T_\mathrm{i}\) is given by tau_ad in run/gkvp_namelist . Setting of kinetic ions with adiabatic electron model is nprocs =1 in src/gkvp_header.f90 , and Anum =1.d0, Znum =1.d0, fcs =1.d0, sgn =1.d0 in run/gkvp_namelist . Then the Poisson eq. with adiabatic electron model is

(3) \[\frac{e^2 n_\mathrm{0}}{T_\mathrm{i}} \left( 1 - \Gamma_{0\mathrm{i}\bm{k}} \right) \tilde{\phi}_{\bm{k}} = - \frac{e^2n_\mathrm{0}}{T_\mathrm{e}} \left( \tilde{\phi}_{\bm{k}} - \langle \tilde{\phi}_{\bm{k}} \rangle \delta_{k_y,0} \right) + e \int dv^3 J_{0\mathrm{i}\bm{k}} \tilde{f}_{\mathrm{i}\bm{k}},\]

where \(\langle \cdots \rangle\) denotes the flux-surface average, and \(\delta_{i,j}\) is the Kronecker’s delta. Density, temperature and mass are normalized ions’ value. Then the normalized Poisson eq. is

(4) \[\left( 1 - \bar{\Gamma}_{0\mathrm{i}\bm{k}} \right) \bar{\phi}_{\bm{k}} + \frac{T_\mathrm{i}}{T_\mathrm{e}} \left( \bar{\phi}_{\bm{k}} - \langle \bar{\phi}_{\bm{k}} \rangle \delta_{k_y,0} \right) = \int dv^3 \bar{J}_{0\mathrm{i}\bm{k}} \bar{f}_{\mathrm{i}\bm{k}},\]

The temperature ratio \(T_\mathrm{i}/T_\mathrm{e}\) is given by tau_ad in run/gkvp_namelist .