Expression of geodesic curvature $\kappa _s$

Next, consider the calculation of the surface component of $\ensuremath{\boldsymbol{\kappa}}$, the geodesic curvature $\kappa _s$, which is defined by

$$\kappa_s \equiv \ensuremath{\boldsymbol{\kappa}} \cdot \frac{\mathbf{B}_0 \times \nabla \Psi}{B_0^2} .$$ (268)

Using

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

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

$$= \Psi'^2 \vert \nabla \psi \vert^2 \frac{1}{R} \hat{\ensuremath{\b... ...eta} \hat{\mathbf{Z}} + \frac{g \Psi'}{\mathcal{J}} R_{\theta} \hat{\mathbf{R}}$$ (269)

and Eq. (263), we obtain

$$\frac{\partial}{\partial \theta} \left( \frac{\triangledown \psi ... ...ngledown \phi}{B_0} \right) \cdot \frac{\mathbf{B}_0 \times \nabla \Psi}{B_0^2} = \left\{ - \left[ \frac{\partial}{\partial \theta} \left( \frac{\m... ...{B_0^2 \mathcal{J}} (Z_{\theta} \hat{\mathbf{Z}} + R_{\theta} \hat{\mathbf{R}})$$

$$= - \frac{g \Psi'}{B_0^2 \mathcal{J}} \left\{ \left[ \frac{\partial... ...} + \frac{\mathcal{J}^{- 1}}{B_0} R_{\theta \theta} \right] R_{\theta} \right\}$$ (270)

$$\frac{\partial}{\partial \theta} \left( \frac{\triangledown \phi}{B_0} \right) \cdot \frac{\mathbf{B}_0 \times \nabla \Psi}{B_0^2} = \frac{\partial}{\partial \theta} \left( \frac{1}{R B_0} \right) \frac{\Psi'^2 \vert \nabla \psi \vert^2}{B_0^2 R} .$$ (271)

$$\frac{\partial}{\partial \phi} (\triangledown \psi \times \triangledown \phi) \cdot \frac{\mathbf{B}_0 \times \nabla \Psi}{B_0^2} = - \frac{1}{\mathcal{J}} R_{\theta} \frac{\Psi'^2 \vert \nabla \psi \vert^2}{B_0^2 R} .$$ (272)

Using Eqs. (270), (271), and (272), equation (268) is written as

$$\kappa_s = - \frac{\Psi'^2}{B_0} \mathcal{J}^{- 1} \left( - \frac{g \Psi'}{B... ...} + \frac{\mathcal{J}^{- 1}}{B_0} R_{\theta \theta} \right] R_{\theta} \right\}$$

$$- \frac{g \Psi'}{B_0} \mathcal{J}^{- 1} \frac{\partial}{\partial ... ... - \frac{g^2}{B_0^2} \frac{1}{R^3} \frac{g \Psi'}{B_0^2 \mathcal{J}} R_{\theta}$$ (273)

The terms in the first line of Eq. (273) is written as

$$\frac{g \Psi'^3}{B_0^3} \mathcal{J}^{- 2} \left[ \frac{\partial}{... ...{\theta}^2 + \frac{\mathcal{J}^{- 1}}{B_0} R_{\theta \theta} R_{\theta} \right]$$

$$= \frac{g \Psi'^3}{B_0^3} \mathcal{J}^{- 2} \left[ \frac{\partial... ... 1}}{B_0} (Z_{\theta \theta} Z_{\theta} + R_{\theta \theta} R_{\theta}) \right]$$ (274)

Noting that

$$\vert \nabla \psi \vert^2 = \frac{R^2}{\mathcal{J}^2} (Z_{\theta}^2 + R_{\theta}^2),$$ (275)

and

$$\frac{\partial}{\partial \theta} \left( \frac{\mathcal{J}^2 \vert... ...{R^2} \right) = 2 Z_{\theta} Z_{\theta \theta} + 2 R_{\theta} R_{\theta \theta}$$ (276)

expression (274) is written as

$$\frac{g \Psi'^3}{B_0^3} \mathcal{J}^{- 2} \left[ \frac{\partial}{... ...eta} \left( \frac{\mathcal{J}^2 \vert \nabla \psi \vert^2}{R^2} \right) \right]$$

$$= \frac{g \Psi'^3}{B_0^3} \frac{\vert \nabla \psi \vert^2}{R^2} \... ...tial \theta} \left( \frac{\mathcal{J}^2 \vert \nabla \psi \vert^2}{R^2} \right)$$ (277)

The first two terms on the second line of Eq. (273) can be written as

$$- \frac{g \Psi'^3}{B_0^3} \vert \nabla \psi \vert^2 \mathcal{J}^{... ...eft( \frac{1}{R B_0} \right) \frac{1}{R} + \frac{1}{B_0 R^3} R_{\theta} \right]$$ (278)

The sum of the expression (278) and the first term of expression (277) is written as

$$\frac{g \Psi'^3}{B_0^3} \frac{\vert \nabla \psi \vert^2}{R^2} \fr... ...eft( \frac{1}{R B_0} \right) \frac{1}{R} + \frac{1}{B_0 R^3} R_{\theta} \right]$$

$$= \frac{g \Psi'^3}{B_0^3} \vert \nabla \psi \vert^2 \left[ \frac{... ...B_0} \right) \frac{1}{R} - \frac{\mathcal{J}^{- 1}}{B_0 R^3} R_{\theta} \right]$$

$$= \frac{g \Psi'^3}{B_0^3} \vert \nabla \psi \vert^2 \left[ \frac{... ...B_0} \right) \frac{1}{R} - \frac{\mathcal{J}^{- 1}}{B_0 R^3} R_{\theta} \right]$$

$$= \frac{g \Psi'^3}{B_0^3} \vert \nabla \psi \vert^2 \left[ \frac{... ...B_0} \right) \frac{1}{R} - \frac{\mathcal{J}^{- 1}}{B_0 R^3} R_{\theta} \right]$$

$$= \frac{g \Psi'^3}{B_0^3} \vert \nabla \psi \vert^2 \left[ \frac{... ...c{\partial}{\partial \theta} \left( \frac{1}{R B_0} \right) \frac{1}{R} \right]$$

$$= \frac{g \Psi'^3}{B_0^3} \vert \nabla \psi \vert^2 \left\{ \frac... ...l}{\partial \theta} \left( \frac{1}{R B_0} \right) \frac{1}{R} \right] \right\}$$

$$= \frac{g \Psi'^3}{B_0^3} \vert \nabla \psi \vert^2 \left\{ \frac... ...- 1} \frac{\partial}{\partial \theta} \left( \frac{1}{R^2 B_0} \right) \right\}$$

$$= \frac{g \Psi'^3}{B_0^3} \vert \nabla \psi \vert^2 \left\{ \frac{1}{B_0 R^2} \frac{\partial}{\partial \theta} (\mathcal{J}^{- 1}) \right\}$$ (279)

Using the above results, $\kappa _s$ is written as

$$\kappa_s = \frac{g \Psi'^3}{B_0^4} \mathcal{J}^{- 3} \frac{1}{2} \frac{\part... ... \vert \nabla \psi \vert^2 \frac{\partial}{\partial \theta} (\mathcal{J}^{- 1})$$

$$= \frac{g \Psi'^3}{B_0^4} \frac{1}{\mathcal{J}^2} \left[ \frac{1}{2... ...al{J}^{- 1}) \right] - \frac{\Psi' g^3}{R^3 B_0^4} \mathcal{J}^{- 1} R_{\theta}$$ (280)

$$= \frac{g \Psi'^3}{B_0^4} \frac{1}{\mathcal{J}^2} \left[ \frac{1}{2... ...al{J}^{- 1}) \right] - \frac{\Psi' g^3}{R^3 B_0^4} \mathcal{J}^{- 1} R_{\theta}$$

$$= \frac{g \Psi'}{B_0^4} \frac{1}{\mathcal{J}^2} \left[ \frac{1}{2} ... ...al{J}^{- 1}) \right] - \frac{\Psi' g^3}{R^3 B_0^4} \mathcal{J}^{- 1} R_{\theta}$$

$$= \frac{g\mathcal{J}^{- 1} \Psi'}{B_0^3} \left[ \frac{1}{B_0 \mathc... ...}{\partial \theta} (\mathcal{J}^{- 1}) - \frac{g^2}{R^3 B_0} R_{\theta} \right]$$ (281)

Using

$$\frac{\partial B_0}{\partial \theta} = \frac{\partial}{\partial \theta} \sqrt{\left( \frac{\vert \nabla \Psi \vert}{R} \right)^2 + \left( \frac{g}{R} \right)^2}$$

$$= \frac{\nabla \Psi}{B_0 R} \frac{\partial}{\partial \theta} \left( \frac{\vert \nabla \Psi \vert}{R} \right) - \frac{g^2}{R^3 B_0} R_{\theta},$$

equation (281) is written

$$\kappa_s = \frac{g\mathcal{J}^{- 1} \Psi'}{B_0^3} \left[ \frac{1... ...al}{\partial \theta} \left( \frac{\vert \nabla \Psi \vert}{R} \right) \right]$$

$$\$$

Excluding the $\partial B_0 / \partial \theta$ term, the other terms on the r.h.s of the above equation are written

$$\frac{1}{2} \mathcal{J}^{- 2} \frac{\partial}{\partial \theta} \l... ...\partial}{\partial \theta} \left( \frac{\vert \nabla \Psi \vert^2}{R^2} \right)$$

$$= - \frac{1}{2} \frac{\mathcal{J}^2 \vert \nabla \Psi \vert^2}{R^... ...\nabla \Psi \vert^2 }{R^2} \frac{\partial}{\partial \theta} (\mathcal{J}^{- 1})$$

$$= \frac{\vert \nabla \Psi \vert^2}{\mathcal{J}R^2} \frac{\partial... ...\nabla \Psi \vert^2 }{R^2} \frac{\partial}{\partial \theta} (\mathcal{J}^{- 1})$$

$$= 0$$

Therefore equation (8.4.2) is written

$$\kappa_s = \frac{g\mathcal{J}^{- 1} \Psi'}{B_0^3} \left( \frac{\partial B_0}{\partial \theta} \right),$$ (282)

which agrees with the formula given in Ref. [6]. Equation (282) takes a very simple form, and provides a clear physical meaning for the geodesic curvature: $\kappa _s$ is proportional to the poloidal derivative of the magnetic field strength. Equation () indicates that the geodesic curvature is zero for an equilibrium configuration that is uniform in poloidal direction. Note that this formula for $\kappa _s$ is valid for arbitrary Jacobian. (Remarks: When I derived the formula of $\kappa _s$ for the first time, I found that $\kappa _s$ can be written in the simple form given by Eq. (282) for the equa-arc length Jacobian. Later I found that $\kappa _s$ can also be written in the simple form given by Eq. (282) for the Boozer Jacobian. This makes me realize that the simple form given by Eq. (282) may be universally valid for arbitrary Jacobian. However, I did not verify this then. About two years later, I reviewed this notes and succeeded in providing the derivation given above. The derivation given above seems to be tedious and may be greatly simplified in some aspects. But, at present, the above derivation is the only one that I can provide.)

If we choose the equal-arc Jacobian, then Eq. (280) becomes relatively simple:

$$\kappa_s = \frac{g \Psi'^3}{B_0^4} \frac{1}{\mathcal{J}^2} \left[... ...{J}^{- 1}) \right] - \frac{\Psi' g^3}{R^3 B_0^4} \mathcal{J}^{- 1} R_{\theta} .$$ (283)

This form is implemented in GTAW code. Figure 31 gives the results for $\kappa _s$ calculated by using Eq. (283). I have verified numerically that the results given by Eqs. (282) and (283) agree with each other.

Figure 31: The geodesic magnetic curvature $\kappa _s$ (calculated by Eq. (283)) as a function of the poloidal angle. The different lines corresponds to different magnetic surfaces. The stars correspond to the values of $\kappa _s$ 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.