$(s, \alpha )$ parameters

The normalized pressure gradient, $\alpha$, which appears frequently in tokamak literature, is defined by[2]

$$\alpha = - R_0 q^2 \frac{1}{B^2_0 / 2 \mu_0} \frac{d p}{d r},$$ (504)

which can be further written

$$\alpha = - R_0 q^2 \frac{d \overline{p}}{d r},$$ (505)

where $\overline{p} = p / (B_0^2 / 2 \mu_0)$. Equation (505) can be further written as

$$\alpha = - \frac{1}{\varepsilon_a} q^2 \frac{d \overline{p}}{d \overline{r}},$$ (506)

where $\varepsilon_a = a / R_0$, $\overline{r} = r / a$, and $a$ is the minor radius of the boundary flux surface. (Why is there a $q^2$ factor in the definition of $\alpha$?)

The global magnetic shear $s$ is defined by

$$s = \frac{r}{q} \frac{d q}{d r},$$ (507)

which can be written

$$s = \frac{\overline{r}}{q} \frac{d q}{d \overline{r}} .$$ (508)

In the case of large aspect ratio and circular flux surface, the leading order equation of the Grad-Shafranov equation in $(r, \theta )$ coordinates is written

$$\frac{1}{r} \frac{d}{d r} r \frac{d \Psi}{d r} = - \mu_0 R_0 J_{\phi} (r),$$ (509)

which gives concentric circular flux surfaces centered at $(R = R_0, Z = 0)$. Assume that $J_{\phi}$ is uniform distributed, i.e., $\vert J_{\phi} \vert = I / (\pi a^2)$, where $I$ is the total current within the flux surface $r = a$. Further assume the current is in the opposite direction of $\nabla \phi$, then $J_{\phi} = - I / (\pi a^2)$. Using this, Eq. (509) can be solved to give

$$\Psi = \frac{\mu_0 I}{4 \pi a^2} R_0 r^2 .$$ (510)

Then it follows that the normalized radial coordinate $\rho \equiv (\Psi - \Psi_0) / (\Psi_b - \Psi_0)$ relates to $\overline{r}$ by $\overline{r} = \sqrt{\rho}$ (I check this numerically for the case of EAST discharge #38300). Sine in my code, the radial coordinate is $\Psi$, I need to transform the derivative with respect to $\overline{r}$ to one with respect to $\Psi$, which gives

$$\alpha = - \frac{1}{\varepsilon} q^2 \frac{d \overline{p}}{d \ove... ...} q^2 \frac{d \overline{p}}{d \Psi} \frac{1}{2 \sqrt{\rho}} (\Psi_b - \Psi_0) .$$ (511)

$$s = \frac{\overline{r}}{q} \frac{d q}{d \overline{r}} = \frac{\sq... ...\rho}} (\Psi_b - \Psi_0) = \frac{1}{2 q} \frac{d q}{d \Psi} (\Psi_b - \Psi_0) .$$ (512)

The necessary condition for the existence of TAEs with frequency near the upper tip of the gap is given by[2]

$$\alpha < - s^2 + \varepsilon,$$ (513)

which is used in my paper on Alfvén eigenmodes on EAST tokamak[8]. Equations (511) and (512) are used in the GTAW code to calculate $s$ and $\alpha$.