Magnetic curvature

In this section, we derive formulas for calculating the geodesic curvature and normal curvature in magnetic surface coordinate system $(\psi, \theta, \phi)$. The derivation looks tedious but the final results are compact (especially for the geodesic curvature $\kappa _s$). The magnetic curvature is defined by $\ensuremath{\boldsymbol{\kappa}}=\mathbf{b} \cdot \nabla \mathbf{b}$, which can be further written as

$$\ensuremath{\boldsymbol{\kappa}} = \frac{1}{B_0} \mathbf{B}_0 \cdot \nabla \frac{\mathbf{B}_0}{B_0}$$

$$= \frac{1}{B_0} (\nabla \Psi \times \nabla \phi + g \nabla \phi) \cdot \nabla \frac{\mathbf{B}_0}{B_0}$$ (261)

In magnetic surface coordinate system $(\psi, \theta, \phi)$, equation (261) is written as

$$\ensuremath{\boldsymbol{\kappa}} = \frac{1}{B_0} (\Psi' \triangledown \psi \times \triangledown \phi + g \triangledown \phi) \cdot \nabla \frac{\mathbf{B}_0}{B_0}$$

$$= \frac{\Psi'}{B_0} (\triangledown \psi \times \triangledown \phi) ... ...}{B_0} + \frac{g}{B_0} \triangledown \phi \cdot \nabla \frac{\mathbf{B}_0}{B_0}$$

$$= - \frac{\Psi'}{B_0} \mathcal{J}^{- 1} \frac{\partial}{\partial \t... ...{1}{R^2} \frac{\partial}{\partial \phi} \left( \frac{\mathbf{B}_0}{B_0} \right)$$

$$= - \frac{\Psi'}{B_0} \mathcal{J}^{- 1} \frac{\partial}{\partial \t... ...riangledown \psi \times \triangledown \phi + g \triangledown \phi}{B_0} \right)$$

$$= - \frac{\Psi'^2}{B_0} \mathcal{J}^{- 1} \frac{\partial}{\partial ... ...i \times \triangledown \phi) - \frac{g^2}{B_0^2} \frac{1}{R^3} \hat{\mathbf{R}}$$ (262)

Using $\nabla \psi = - \frac{R}{\mathcal{J}} (Z_{\theta} \hat{\mathbf{R}} - R_{\theta} \hat{\mathbf{Z}})$, we obtain $\nabla \psi \times \triangledown \phi = - \frac{1}{\mathcal{J}} (Z_{\theta} \hat{\mathbf{Z}} + R_{\theta} \hat{\mathbf{R}})$. Using this, the three partial derivatives in the above equation are written respectively as

$$\frac{\partial}{\partial \theta} \left( \frac{\triangledown \psi \times \triangledown \phi}{B_0} \right) = - (Z_{\theta} \hat{\mathbf{Z}} + R_{\theta} \hat{\mathbf{R}}) \fr... ...{B_0} (Z_{\theta \theta} \hat{\mathbf{Z}} + R_{\theta \theta} \hat{\mathbf{R}})$$

$$= - \left[ \frac{\partial}{\partial \theta} \left( \frac{\mathcal{J... ...ta} + \frac{\mathcal{J}^{- 1}}{B_0} Z_{\theta \theta} \right] \hat{\mathbf{Z}},$$ (263)

$$\frac{\partial}{\partial \theta} \left( \frac{\triangledown \phi}... ...al \theta} \left( \frac{1}{R B_0} \right) \hat{\ensuremath{\boldsymbol{\phi}}},$$ (264)

$$\frac{\partial}{\partial \phi} (\triangledown \psi \times \triang... ...hi) = - \frac{1}{\mathcal{J}} R_{\theta} \hat{\ensuremath{\boldsymbol{\phi}}} .$$ (265)

Using these, $\ensuremath{\boldsymbol{\kappa}}$ is written as

$$\ensuremath{\boldsymbol{\kappa}} = \frac{\Psi'^2}{B_0} \mathcal{J}^{- 1} \left\{ \left[ \frac{\parti... ...rac{\mathcal{J}^{- 1}}{B_0} Z_{\theta \theta} \right] \hat{\mathbf{Z}} \right\}$$

$$- \frac{g \Psi'}{B_0} \mathcal{J}^{- 1} \frac{\partial}{\partial ... ...suremath{\boldsymbol{\phi}}} - \frac{g^2}{B_0^2} \frac{1}{R^3} \hat{\mathbf{R}}$$