Local magnetic shear

The negative local magnetic shear is defined by

$$S = \left( \nabla \times \frac{\mathbf{B}_0 \times \nabla \Psi}{\... ...ht) \cdot \frac{(\mathbf{B}_0 \times \nabla \Psi)}{\vert \nabla \Psi \vert^2} .$$ (284)

There are two ways of calculating $S$. The first way is to calculate $S$ in cylindrical coordinate system; the second one is in flux coordinate system. Next, consider the first way. We have

$$\frac{\mathbf{B}_0 \times \nabla \Psi}{\vert \nabla \Psi \vert^2} = \frac{(\nabla \Psi \times \nabla \phi + g \nabla \phi) \times \nabla \Psi}{\vert \nabla \Psi \vert^2}$$

$$= \frac{\vert \nabla \Psi \vert^2 \nabla \phi + g \nabla \phi \times \nabla \Psi}{\vert \nabla \Psi \vert^2}$$

$$= \nabla \phi + g \frac{\nabla \phi \times \nabla \Psi}{\vert \nabla \Psi \vert^2} .$$ (285)

Using this, Eq. (284) is written as

$$S = \left[ \nabla \times \left( \nabla \phi + g \frac{\nabla \phi \ti... ...hi + g \frac{\nabla \phi \times \nabla \Psi}{\vert \nabla \Psi \vert^2} \right]$$

$$= \left[ \nabla \times \left( g \frac{\nabla \phi \times \nabla \Ps... ... + g \frac{\nabla \phi \times \nabla \Psi}{\vert \nabla \Psi \vert^2} \right] .$$ (286)

Next, we work in cylindrical coordinates and obtain

$$g \frac{\nabla \phi \times \nabla \Psi}{\vert \nabla \Psi \vert^2} = \frac{1}{\vert \nabla \Psi \vert^2} \frac{g}{R} \Psi_Z \hat{\mathbf{R}} - \frac{1}{\vert \nabla \Psi \vert^2} \frac{g}{R} \Psi_R \hat{\mathbf{Z}}$$ (287)

and

$$\nabla \times \left[ g \frac{\nabla \phi \times \nabla \Psi}{\ver... ...rt^2} \frac{g}{R} \Psi_R \right) \right] \hat{\ensuremath{\boldsymbol{\phi}}} .$$ (288)

Using Eq. (288), Eq. (286) is written as

$$S = \left[ \frac{\partial}{\partial Z} \left( \frac{1}{\vert \nab... ...{1}{\vert \nabla \Psi \vert^2} \frac{g}{R} \Psi_R \right) \right] \frac{1}{R} .$$ (289)

The two partial derivatives appearing in the above equation can be calculated to give

$$\frac{\partial}{\partial Z} \left( \frac{1}{\vert \nabla \Psi \vert^2} \frac{g}{R} \Psi_Z \right) = \frac{1}{\Psi_R^2 + \Psi_Z^2} \frac{1}{R} (g \Psi_Z)_Z - \frac{2 ... ...R \Psi_{R Z} + 2 \Psi_Z \Psi_{Z Z}}{[\Psi_R^2 + \Psi_Z^2]^2} \frac{g}{R} \Psi_Z$$

$$= \frac{1}{\Psi_R^2 + \Psi_Z^2} \frac{g \Psi_{Z Z} + g' \Psi_Z^2}{R... ...\Psi_{R Z} + 2 \Psi_Z \Psi_{Z Z}}{[\Psi_R^2 + \Psi_Z^2]^2} \frac{g}{R} \Psi_Z .$$ (290)

$$\frac{\partial}{\partial R} \left( \frac{1}{\vert \nabla \Psi \vert^2} \frac{g}{R} \Psi_R \right) = \frac{1}{\Psi_R^2 + \Psi_Z^2} \left( \frac{g}{R} \Psi_R \right)_R... ...R \Psi_{R R} + 2 \Psi_Z \Psi_{Z R}}{[\Psi_R^2 + \Psi_Z^2]^2} \frac{g}{R} \Psi_R$$

$$= \frac{1}{\Psi_R^2 + \Psi_Z^2} \left( \frac{g}{R} \Psi_{R R} + \Ps... ...\Psi_{R R} + 2 \Psi_Z \Psi_{Z R}}{[\Psi_R^2 + \Psi_Z^2]^2} \frac{g}{R} \Psi_R .$$ (291)

Using these, we obtain

$$S = \frac{1}{\Psi_R^2 + \Psi_Z^2} \frac{1}{R^2} \left( g \Psi_{Z ... ... (4 \Psi_R \Psi_{R Z} \Psi_Z + 2 \Psi_Z^2 \Psi_{Z Z} + 2 \Psi_R^2 \Psi_{R R}) .$$ (292)

(The above results for $S$ has been verified by using Mathematica Software.) The results calculated by using Eq. (292) are plotted in Fig. 32.

Figure 32: The local magnetic shear $S$ as a function of the poloidal angle. The different lines corresponds to the shear on different magnetic surfaces. The stars correspond to the values of the shear 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.

Next, we consider the calculation of $S$ in the flux coordinates system $(\psi, \theta, \phi)$. The $\mathbf{B}_0 \times \nabla \Psi$ term can be written as

$$\mathbf{B}_0 \times \nabla \Psi = [\nabla \Psi \times \nabla \phi + g \nabla \phi] \times \nabla \Psi$$

$$= \Psi'^2 \vert \nabla \psi \vert^2 \nabla \phi + g \Psi' \nabla \phi \times \nabla \psi$$

$$= \Psi'^2 \vert \nabla \psi \vert^2 \nabla \phi + g \Psi' \mathcal{... ...psi + \frac{\mathcal{J}^2 \vert \nabla \psi \vert^2}{R^2} \nabla \theta \right)$$

$$= - \Psi' \frac{g\mathcal{J} \nabla \psi \cdot \nabla \theta}{R^2} ... ...psi \vert^2}{R^2} \nabla \theta + \Psi'^2 \vert \nabla \psi \vert^2 \nabla \phi$$ (293)

$$\Rightarrow \frac{\mathbf{B}_0 \times \nabla \Psi}{\vert \nabla \... ...t^2} \nabla \psi + \frac{g\mathcal{J}}{\Psi' R^2} \nabla \theta + \nabla \phi .$$ (294)

By using the curl formula in generalized coordinates $(\psi, \theta, \phi)$, we obtain

$$\nabla \times \frac{\mathbf{B}_0 \times \nabla \Psi}{\vert \nabla \Psi \vert^2} = \nabla \times \left( - \frac{g\mathcal{J}}{\Psi' R^2} \frac{\nabl... ...nabla \psi + \frac{g\mathcal{J}}{\Psi' R^2} \nabla \theta + \nabla \phi \right)$$

$$= \left[ \frac{\partial}{\partial \psi} \left( \frac{g\mathcal{J}}{... ...ta}{\vert \nabla \psi \vert^2} \right) \right] \nabla \psi \times \nabla \theta$$ (295)

Using Eqs. (294) and (295), the negative local magnetic shear [Eq. (284)] is written as

$$S = \left\{ \left[ \frac{\partial}{\partial \psi} \left( \frac{g\math... ...nabla \psi + \frac{g\mathcal{J}}{\Psi' R^2} \nabla \theta + \nabla \phi \right)$$

$$= \left[ \frac{\partial}{\partial \psi} \left( \frac{g\mathcal{J}}{... ...psi \vert^2} \right) \right] \nabla \psi \times \nabla \theta \cdot \nabla \phi$$

$$= \left[ \frac{\partial}{\partial \psi} \left( \frac{g\mathcal{J}}{... ...t \nabla \theta}{\vert \nabla \psi \vert^2} \right) \right] \mathcal{J}^{- 1} .$$ (296)

Note that the partial derivatives $\partial / \partial \psi$ and $\partial / \partial \theta$ in Eq. (296) is taken in the $(\psi, \theta, \phi)$ coordinates and they are usually different from their counterparts in $(\psi, \theta, \zeta)$ coordinates. In Eq. (296), the partial derivatives are operating on equilibrium quantity, which is independent of $\phi$ and $\zeta$. In this case, the partial derivatives in the two sets of coordinates are equal to each other. Figure 33 plots the poloidal dependence of local magnetic shear $S$ and $\vert \nabla \Psi \vert^2 S$.

Figure 33: The local magnetic shear $S$ (left) and $\vert \nabla \Psi \vert^2 S$ (right) as a function of the poloidal angle. The different lines corresponds to the shear on different magnetic surfaces. The stars correspond to the values of the shear 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.

Next, let us examine the flux surface average of $S$, which is written as

$$\langle S \rangle = \frac{\int_0^{2 \pi} S\mathcal{J}d \theta}{\int_0^{2 \pi} \mathcal{J}d \theta}$$

$$= \frac{1}{\int_0^{2 \pi} \mathcal{J}d \theta} \int_0^{2 \pi} \left... ...a \psi \cdot \nabla \theta}{\vert \nabla \psi \vert^2} \right) \right] d \theta$$

$$= \frac{1}{\int_0^{2 \pi} \mathcal{J}d \theta} \int_0^{2 \pi} \left... ...}{\partial \psi} \left( \frac{g\mathcal{J}}{\Psi' R^2} \right) \right] d \theta$$

$$= \frac{1}{\int_0^{2 \pi} \mathcal{J}d \theta} \frac{\partial}{\partial \psi} \int_0^{2 \pi} \frac{g\mathcal{J}}{\Psi' R^2} d \theta$$ (297)

Note that the global safety factor is given by

$$q (\psi) = - \frac{1}{2 \pi} \int_0^{2 \pi} \frac{g\mathcal{J}}{\Psi' R^2} d \theta .$$ (298)

Using this, equation (297) is written as

$$\langle S \rangle = - \frac{2 \pi}{\int_0^{2 \pi} \mathcal{J}d \theta} \frac{d q (\psi)}{d \psi},$$ (299)

Equation (299) provides a way to verify the correctness of the numerical implementation of $S$. Figure 34 compares $d q / d \psi$ with $- \langle S \rangle \left( \int_0^{2 \pi} \mathcal{J}d \theta \right) / (2 \pi)$, which shows that the two results agree with each other well.

Figure 34: $d q / d \psi$ (solid line) and $- \langle S \rangle \int_0^{2 \pi} \mathcal{J}d \theta / (2 \pi)$ (plus mark) as a function of the radial coordinate $\overline {\Psi }$ ( $\overline {\Psi }$ is the normalized poloidal magnetic flux). The equilibrium is a Solovev equilibrium.