Weight functions used in Fourier integration

In the GTAW code, the weight functions appearing in the Fourier integration are numbered as follows:

$$W_1 (\psi, \theta) \equiv \frac{\mathcal{J}^{- 2}}{B_0^2}, W_2 (\... ...\frac{\partial}{\partial \theta} \left( \frac{\mathcal{J}^{- 1}}{B_0^2} \right)$$ (199)

$$W_3 \equiv \frac{\vert \nabla \Psi \vert^2 \mathcal{J}^{- 2}}{B_0... ...theta} \left( \frac{\vert \nabla \Psi \vert^2 \mathcal{J}^{- 1}}{B_0^2} \right)$$ (200)

$$W_5 \equiv \frac{\vert \nabla \Psi \vert^2}{B_0^2}, W_6 \equiv \kappa_s$$ (201)

$$W_7 \equiv \frac{1}{B_0^2}, W_8 = \frac{\mathcal{J}^{- 1}}{B_0^2}, W_9 ={\textmu}_0 \sigma \mathcal{J}^{- 1}$$ (202)

$$W_{10} =\mathcal{J}^{- 1} \frac{\vert \nabla \Psi \vert^2 S}{B^2_... ...tial}{\partial \theta} \left( \frac{\vert \nabla \Psi \vert^2 S}{B^2_0} \right)$$ (203)

$$W_{12} = \frac{\kappa_{\psi}}{\vert \nabla \Psi \vert^2}, W_{13} = \vert \nabla \Psi \vert^2$$ (204)

$$W_{14} = \vert \nabla \Psi \vert^2 \mathcal{J}^{- 1} \frac{\parti... ... \theta} \left( \frac{\mathcal{J}^{- 1}}{B_0^2} \right), W_{15} = \kappa_{\psi}$$ (205)

$$W_{16} = \frac{\Psi_R}{R} + \Psi_{R R} + \Psi_{Z Z} - \frac{1}{\v... ..._R \Psi_R \Psi_{R R} + 4 \Psi_R \Psi_Z \Psi_{Z R} + 2 \Psi_Z \Psi_Z \Psi_{Z Z})$$ (206)

$$W_{17} = (\vert \nabla \Psi \vert^2 S - B_0^2 {\textmu}_0 \sigma) \frac{\vert \nabla \Psi \vert^2 S}{B^2_0}$$ (207)

$$W_{18} = (\vert \nabla \Psi \vert^2 S - B_0^2 {\textmu}_0 \sigma) \frac{\vert \nabla \Psi \vert^2 \mathcal{J}^{- 1}}{B^2_0}$$ (208)

$$W_{19} = \nabla \theta \cdot \nabla \psi, W_{20} = \nabla \zeta \cdot \nabla \psi$$ (209)

$$W_{21} = \vert \nabla \Psi \vert^2 \mathcal{J}^{- 1} \frac{\parti... ...mathcal{J}^{- 1}}{\vert \nabla \Psi \vert^2} \right), W_{22} =\mathcal{J}^{- 2}$$ (210)

$$W_{23} = \frac{\vert \nabla \Psi \vert^2 \mathcal{J}^{- 1}}{B_0^2}, W_{24} = \vert \nabla \Psi \vert^2 \kappa_s$$ (211)

The formulas for calculating the equilibrium quantities, such as the geodesic curvature $\kappa _s$, normal curvature $\kappa _{\psi }$, and the local magnetic shear $S$, are given in Sec. 8.4.