Expression of normal curvature $\kappa _{\psi }$

The component $\kappa _{\psi }$ is defined by $\kappa_{\psi} =\ensuremath{\boldsymbol{\kappa}} \cdot \nabla \Psi$. Using Eq. (262), $\kappa _{\psi }$ is written as

$$\kappa_{\psi} = - \frac{\Psi'^3}{B_0} \mathcal{J}^{- 1} \frac{\partial}{\partial ... ... + 0 - \frac{\Psi' g^2}{B_0^2} \frac{1}{R^3} \hat{\mathbf{R}} \cdot \nabla \psi$$

$$= - \frac{\Psi'^3}{B_0} \mathcal{J}^{- 1} \left\{ \left[ \frac{\par... ...mathcal{J}} (Z_{\theta} \hat{\mathbf{R}} - R_{\theta} \hat{\mathbf{Z}}) \right]$$

$$= - \frac{\Psi'^3}{B_0} \mathcal{J}^{- 1} \frac{R}{\mathcal{J}} \le... ...ight\} + \frac{\Psi' g^2}{B_0^2} \frac{1}{R^2} \frac{1}{\mathcal{J}} Z_{\theta}$$

$$= - \frac{\Psi'^3}{B_0} \mathcal{J}^{- 1} \frac{R}{\mathcal{J}} \fr... ...heta}) + \frac{\Psi' g^2}{B_0^2} \frac{1}{R^2} \frac{1}{\mathcal{J}} Z_{\theta}$$

$$= - \Psi'^3 R \frac{\mathcal{J}^{- 3}}{B_0^2} (R_{\theta \theta} Z_... ...heta}) + \frac{\Psi' g^2}{B_0^2} \frac{1}{R^2} \frac{1}{\mathcal{J}} Z_{\theta}$$ (266)

$$= - \Psi'^3 R \frac{\mathcal{J}^{- 3}}{B_0^2} Z_{\theta}^2 \frac{\p... ...ght) + \frac{\Psi' g^2}{B_0^2} \frac{1}{R^2} \frac{1}{\mathcal{J}} Z_{\theta} .$$ (267)

Equation (266) is used in GTAW code to calculate $\kappa _{\psi }$. Equation (267) is not suitable for numerical calculation because $Z_{\theta}$, which appears both in numerator and denominator, can be very small, leading to significant errors in the numerical results. [My notes: the bad results calculated by Eq. (267) in my code reminded me that Eq. (266) may be better. I switch back to adopt Eq. (266) and the results clearly show that the results given by Eq. (266) are indeed better than those of Eq. (267), as shown in Fig. 30.]

Figure 30: The normal curvature $\kappa _{\psi }$ calculated by Eq. (267) (left) and Eq. (266) (right) as a function of the poloidal angle. The different lines corresponds to different magnetic surfaces. The stars correspond to the values of $\kappa _{\psi }$ on the boundary magnetic surface while the plus signs correspond to the value on the innermost magnetic surface (the magnetic surface adjacent to the magnetic axis). The equilibrium is a Solovev equilibrium.