Formula for matrix elements $C_{22}$

Next, consider the calculation of matrix elements $C_{22}$. In cylindrical coordinates, we have

$$\nabla \cdot \left( \frac{\nabla \Psi}{\vert \nabla \Psi \vert^2} \right) = \nabla \cdot \left( \frac{\Psi_R}{\vert \nabla \Psi \vert^2} \mathbf{e}_R + \frac{\Psi_Z}{\vert \nabla \Psi \vert^2} \mathbf{e}_Z \right)$$

$$= \frac{1}{R} \frac{\partial}{\partial R} \left( R \frac{\Psi_R}{\v... ...c{\partial}{\partial Z} \left( \frac{\Psi_Z}{\vert \nabla \Psi \vert^2} \right)$$

$$= \frac{\Psi_R}{R \vert \nabla \Psi \vert^2} + \frac{\partial}{\par... ...{\vert \nabla \Psi \vert^4} (2 \Psi_R \Psi_{R Z} + 2 \Psi_Z \Psi_{Z Z}) \right]$$

$$= \frac{\Psi_R}{R \vert \nabla \Psi \vert^2} + \Psi_{R R} \left( \f... ...{\vert \nabla \Psi \vert^4} (2 \Psi_R \Psi_{R Z} + 2 \Psi_Z \Psi_{Z Z}) \right]$$

$$= \frac{\Psi_R}{R \vert \nabla \Psi \vert^2} + \frac{\Psi_{R R} + \... ..._R \Psi_R \Psi_{R R} + 4 \Psi_R \Psi_Z \Psi_{Z R} + 2 \Psi_Z \Psi_Z \Psi_{Z Z})$$ (300)

Using this, $C_{22}$ is written

$$C_{22} = - \vert \nabla \Psi \vert^2 \nabla \cdot \left( \frac{\nabla \Psi}{\vert \nabla \Psi \vert^2} \right)$$

$$= - \frac{\Psi_R}{R} - (\Psi_{R R} + \Psi_{Z Z}) + \frac{1}{\vert \... ... \Psi_R \Psi_{R R} + 4 \Psi_R \Psi_Z \Psi_{Z R} + 2 \Psi_Z \Psi_Z \Psi_{Z Z}) .$$ (301)

Equation (301) is used in GTAW to calculate $C_{22}$.

$C_{22}$ can also be calculated in the magnetic surface coordinates.

$$\frac{C_{22}}{\Psi' \vert \nabla \psi \vert^2} = - \Psi' \nabla \cdot \left( \frac{\nabla \Psi}{\vert \nabla \Psi \vert^2} \right),$$ (302)

the term on the r.h.s of the above equation is written as

$$\nabla \cdot \left( \frac{\nabla \Psi}{\vert \nabla \Psi \vert^2} \right) = \nabla \cdot \left( \frac{\nabla \Psi}{R^2} \frac{R^2}{\vert \nabla \Psi \vert^2} \right)$$

$$= \frac{R^2}{\vert \nabla \Psi \vert^2} \nabla \cdot \left( \frac{\... ... \Psi}{R^2} \cdot \nabla \left( \frac{R^2}{\vert \nabla \Psi \vert^2} \right) .$$ (303)

The first term on the r.h.s of Eq. (303) is written as

$$\frac{R^2}{\vert \nabla \Psi \vert^2} \nabla \cdot \left( \frac{\nabla \Psi}{R^2} \right) = \frac{1}{\vert \nabla \Psi \vert^2} \triangle^{\ast} \Psi$$

$$= \frac{1}{\vert \nabla \Psi \vert^2} \left[ -{\textmu}_0 R^2 \frac{d p_0}{d \Psi} - \frac{d g}{d \Psi} g (\Psi) \right] .$$ (304)

The second term on the r.h.s of Eq. (303) is written as

$$\frac{\nabla \Psi}{R^2} \cdot \nabla \left( \frac{R^2}{\vert \nabla \Psi \vert^2} \right) = \frac{\Psi'}{R^2} \left[ \vert \nabla \psi \vert^2 \frac{\partial... ...}{\partial \theta} \left( \frac{R^2}{\vert \nabla \Psi \vert^2} \right) \right]$$ (305)

Using these, Eq. (302) is finally written as

$$\frac{C_{22}}{\Psi' \vert \nabla \psi \vert^2} = \Psi' \left[ {\textmu}_0 \frac{d p_0}{d \Psi} \frac{R^2}{\vert \n... ...}{\partial \theta} \left( \frac{R^2}{\vert \nabla \Psi \vert^2} \right) \right]$$ (306)