Local safety factor

Local safety factor $\hat{q}$ is defined by

$$\hat{q} = \frac{\mathbf{B} \cdot \nabla \phi}{\mathbf{B} \cdot \nabla \theta},$$ (178)

which characterizes the local pitch angle of a magnetic field line on a magnetic surface (i.e. on $(\theta, \phi)$ plane). Substituting the contravariant representation of the magnetic field, Eq. (176), into the above equation, the local safety factor is written

$$\hat{q} (\psi, \theta) = - \frac{g}{R^2} \frac{\mathcal{J}}{\Psi'} .$$ (179)

Note the expression $\hat{q}$ in Eq. (179) depends on the Jacobian $\mathcal{J}$. This is because the definition of $\hat{q}$ depends on the definition of $\theta$, which in turn depends on the the Jacobian $\mathcal{J}$. [In passing, note that in terms of $\hat{q}$, the contravariant form of the magnetic field, Eq. (176), is written

$$\mathbf{B}= - \Psi' (\nabla \phi \times \nabla \psi + \hat{q} \nabla \psi \times \nabla \theta) .$$ (180)

] The global safety factor is defined as the poloidal average of the local safety factor,

$$q (\psi) \equiv \frac{1}{2 \pi} \int_0^{2 \pi} \hat{q} d \theta$$ (181)

$$= - \frac{1}{2 \pi} \frac{g}{\Psi'} \int_0^{2 \pi} \frac{\mathcal{J}}{R^2} d \theta .$$ (182)

The physical meaning of $\hat{q}$ is obvious: it represents the number of toroidal circles a magnetic field line travels when the line travels a complete poloidal circle. Note that $\hat{q}$ and $\hat{q}$ defined this way can be negative, which depends on the choice of the positive direction of $\phi$ and $\theta$ coordinates (note that the safety factor given in G-eqdsk file is always positive, i.e. it is the absolute value of the safety factor defined here).

(In passing, let us consider the numerical calculation of $q$. Using the relation $d \ell_p$ and $d \theta$ [Eq. (188)], equation (182) is further written

$$q (\psi) = - \frac{1}{2 \pi} \frac{g}{\Psi'} \oint \ensuremath{\operatorname{sign}} (\mathcal{J}) \frac{d \ell_p}{R \vert \nabla \psi \vert}$$

$$= - \frac{1}{2 \pi} g \frac{\ensuremath{\operatorname{sign}} (\math... ...operatorname{sign}} (\Psi')} \oint \frac{d \ell_p}{R \vert \nabla \Psi \vert} .$$ (183)

Equation (183) is used in the GTAW code to numerically calculate the value of $q$ on magnetic surfaces, which agrees with the value specified in the G-eqdsk file. Using $B_p = \vert \nabla \Psi \vert / R$ and $B_{\phi } = g / R$, the absolute value of $q$ is written

$$\vert q\vert = \frac{1}{2 \pi} \oint \frac{1}{R} \frac{\vert B_{\phi} \vert}{B_p} d \ell_p,$$ (184)

which is the familiar formula we see in textbooks.)