$\eta_i$ Modes

The $\eta_i$ and trapped electron mode model by Weiland et al[5] is implemented when ${\tt lswitch(1)}$ is set greater than 1. When ${\tt lswitch(1)} = 2$, only the hydrogen equations are used (with no trapped electrons or impurities) to compute only the $\eta_i$ mode. When ${\tt lswitch(1)} = 4$, trapped electrons are included, but not impurities. When ${\tt lswitch(1)} = 6$, a single species of impurity ions is included as well as trapped electrons. When ${\tt lswitch(1)} = 7$, the effect of collisions is included. When ${\tt lswitch(1)} = 8$, parallel ion (hydrogenic) motion and the effect of collisions are included. When ${\tt lswitch(1)} = 9$, finite beta effects and collisions are included. When ${\tt lswitch(1)} = 10$, parallel ion (hydrogenic) motion, finite beta effects, and the effect of collisions are included. When ${\tt lswitch(1)} = 11$, parallel ion (hydrogenic and impurity) motion, finite beta effects, and the effect of collisions are included. Finite Larmor radius corrections are included in all cases. Values of lswitch(1) greater than 11 are reserved for extensions of this Weiland model.

The mode growth rate, frequency, and effective diffusivities are computed in subroutine weiland14. Frequencies are normalized by $\omega_{De}$ and diffusivities are normalized by $\omega_{De} / k_y^2$. The order of the diffusivity equations is $T_H$, $n_H$, $T_e$, $n_Z$, $T_Z$, ... Note that the effective diffusivities can be negative.

The diffusivity matrix $D = {\tt difthi(j1,j2)}$ is given above.

The impurity density gradient scale length is defined as

$$g_{nz}=-R{{d\ }\over {dr}}\left(Zn_z\right)/(Zn_z)$$

The electron density gradient scale length is defined as

$$g_{ne}=(1-Zf_z-f_s)g_{nH}+Zf_zg_{nz}+f_sg_{ns}$$

where $f \equiv n_Z / n_e$ and $n_e = n_H + Z n_Z +n_s$. For this purpose, all the impurity species are lumped together as one effective impurity species and all the hydrogen isotopes are lumped together as one effective hydrogen isotope.