Kinetic Ballooning

For transport due to the kinetic ballooning mode, we compute $D^{KB}$ and the thermal diffusivities in terms of the pressure gradient $\beta'$.

$$D^{KB} = \frac{c_s \rho_s^2}{p}\left(-{{dp}\over{dr}}\right) ... ... \frac{\beta '} {\beta_{c1}'} -1 \right) \right] \eqno{\tt zdk}$$

and where $\beta_{cl}'$ is the ideal pressure gradient threshold for the onset of the ideal ballooning mode in $s-\alpha$ geometry,

$$\beta_{c1}' = c_{8} \hat{s}/(1.7 q^{2}R_{o}) \eqno{\tt zbc1}$$

Here, $c_8=$ cswitch(8)=1 by default, but is included for flexibility. The coefficent $c_2$ in the expression for $f_{\beta th}$ is set equal to cswitch(2).

The diffusivities are then given as:

$$D_{a}^{KB}=D^{KB}F^{KB}_{1} \kappa^{c_{5}} \eqno{\tt thdkb}$$

$$Q_{e}^{KB}\frac{L_{Te}}{n_{e}T_{e}}=D^{KB}F^{KB}_{2} \kappa^{c_{5}} \eqno{\tt thekb}$$

$$Q_{i}^{KB}\frac{L_{Ti}}{n_{i}T_{i}}=D^{KB}F^{KB}_{3} \kappa^{c_{5}} \eqno{\tt thikb}$$

where $F_1^{KB}=$ fkb(1), $F_2^{KB}=$ fkb(2), $F_3^{KB}=$ fkb(3), and $c_5=$ cswitch(5). Note that the new version does not include the (5/2) factor in the thermal diffusivities.

The relevant coding is: