Safety factor profile

The magnetic shear for a concentric circular configuration is defined by

$$\hat{s} = \frac{d q}{d r} \frac{r}{q},$$ (363)

where $r$ is the minor radius of a magnetic surface. The above expression can be re-arranged as

$$\hat{s} \frac{d r}{r} = \frac{d q}{q}$$ (364)

Integrating the above equation over $r$ and assuming $\hat{s}$ is a constant, we obtain

$$\hat{s} \int_{r_0}^r \frac{d r}{r} = \int_{r_0}^r \frac{d q}{q},$$ (365)

Performing the integration, the abvoe equation is written as

$$\hat{s} (\ln r - \ln r_0) = \ln q - \ln q_0,$$ (366)

where $q_0 = q (r_0)$. Equation (366) can be finally written as

$$q = q_0 \left( \frac{r}{r_0} \right)^{\hat{s}} .$$ (367)

This is a profile with a constant magnetic shear $s$. In Ben's toroidal ITG simulation, the following $q$ profile is used:

$$q = q_0 + (r - r_0) q' (r_0),$$ (368)

with $q' (r_0) = \hat{s} q_0 / r_0$. This is a linear profile over $r$, with the values of $q$ and the shear at $r = r_0$ being $q_0$ and $\hat{s}$, respectively.