Metric Terms in Pyrokinetics

H.G. Dudding

Email: harry.dudding@ukaea.uk

Toroidal coordinate system

Metric terms

The transformations from cylindrical coordinates \(\left \{R, \Phi, Z \right \}\) to toroidal coordinates \(\left \{r, \theta, \zeta \right \}\) are

\[\begin{split}\begin{aligned} r&=r \left(R, Z \right) & R &=R \left(r, \theta \right) \\ \theta&=\theta\left(R, Z \right) & \Phi&=- \zeta \\ \zeta&= - \Phi & Z&=Z\left(r, \theta \right) \end{aligned}\end{split}\]

where the exact relations are defined through the flux-surface parameterisation chosen. Here \(\Phi\) is oriented anti-clockwise when viewed from above, \(\zeta\) is oriented clockwise, and \(\theta\) is oriented anti-clockwise around the flux surface, such that \(\{R, \Phi, Z \}\) and \(\{r, \theta, \zeta \}\) form right-handed coordinate systems.

The covariant metric components (\(g_{i j} = \partial \mathbf{r} / \partial \xi^i \cdot \partial \mathbf{r} / \partial \xi^ j\), for position vector \(\mathbf{r}\) and general coordinates \(\xi^i\)) are

\[g_{r r} = \left(\frac{\partial R}{\partial r}\right)^2 + \left(\frac{\partial Z}{\partial r }\right)^2\]

\[g_{r \theta} = \frac{\partial R}{\partial r}\frac{\partial R}{\partial \theta} + \frac{\partial Z}{\partial r}\frac{\partial Z}{\partial \theta}\]

\[g_{\theta \theta} = \left(\frac{\partial R}{\partial \theta}\right)^2 + \left(\frac{\partial Z}{\partial \theta}\right)^2\]

\[g_{r \zeta} = g_{\theta \zeta} = 0\]

\[g_{\zeta \zeta} = R^2\]

with the Jacobian

\[\mathcal{J}_{r \theta \zeta} = R\left(\frac{\partial R}{\partial r} \frac{\partial Z}{\partial \theta} - \frac{\partial R}{\partial \theta}\frac{\partial Z}{\partial r} \right). \label{eq:toroidal Jacob}\]

Note that in general \(g_{r \theta} \neq 0\) and thus the toroidal coordinate system is non-orthogonal.

The contravariant metric components (\(g^{i j} = \nabla \xi^i \cdot \nabla \xi^j\)) are

\[g^{r r} = \frac{g_{\theta \theta} g_{\zeta \zeta}}{\left(\mathcal{J}_{r \theta \zeta}\right)^2}\]

\[g^{r \theta} = - \frac{g_{r \theta} g_{\zeta \zeta }}{\left(\mathcal{J}_{r \theta \zeta}\right)^2}\]

\[g^{\theta \theta} = \frac{g_{r r} g_{\zeta \zeta}}{\left(\mathcal{J}_{r \theta \zeta}\right)^2}\]

\[g^{r \zeta} = g^{\theta \zeta} = 0\]

\[g^{\zeta \zeta} = \frac{1}{g_{\zeta \zeta}}.\]

Physical quantities

The magnetic field can be written

\[\begin{split}\begin{split} \mathbf{B} & = \psi^{\prime} \nabla r \times \nabla \left[ q\left(r\right) \theta - \zeta + G_0\left(r, \theta\right) \right] \\ & = \psi' \nabla \zeta \times \nabla r + B_{\zeta}\left(r\right) \nabla \zeta \end{split}\end{split}\]

where \(\psi^{\prime} = \mathrm{d} \psi / \mathrm{d} r\) is the radial derivative of the poloidal magnetic flux divided by \(2 \pi\), \(q(r)\) is the safety factor, \(B_{\zeta}(r)\) is the current function

\[B_{\zeta}\left( r \right) = \frac{q \psi^{\prime}}{\left \langle \mathcal{J}_{r \theta \zeta} g^{\zeta \zeta} \right \rangle} \label{eq: B zeta def}\]

where \(\left \langle f \right \rangle = \frac{1}{2 \pi} \int_{- \pi}^{\pi} f \, \mathrm{d} \theta\) denotes a poloidal average and \(G_0(r, \theta)\) is a periodic function in \(\theta\) which satisfies

\[\frac{\partial G_0}{\partial \theta} = \frac{B_{\zeta}}{\psi^{\prime}} \mathcal{J}_{r \theta \zeta} g^{\zeta \zeta} - q. \label{eq:G0 deriv}\]

The current density is

\[\mu_0 \mathbf{J} = \frac{\mathrm{d} B_{\zeta}}{\mathrm{d} r} \nabla r \times \nabla \zeta - \frac{1}{\psi^{\prime}} \left(B_{\zeta} \frac{\mathrm{d} B_{\zeta}}{\mathrm{d} r} + g_{\zeta \zeta} \mu_0 \frac{\mathrm{d}P}{\mathrm{d} r} \right) \nabla \zeta\]

where, from the local Grad-Shafranov solution, we have

\[\begin{split}\begin{split} \frac{\mathrm{d} B_{\zeta}}{\mathrm{d} r} = \frac{B_{\zeta}}{H} \Bigg(\frac{q'}{q} \left \langle \mathcal{J}_{r \theta \zeta} g^{\zeta \zeta} \right \rangle & - \left \langle \mathcal{J}_{r \theta \zeta} \frac{\partial g^{\zeta \zeta}}{\partial r} \right \rangle - \left[ \frac{\mu_0}{\left( \psi^{\prime} \right)^2} \frac{\mathrm{d} P}{\mathrm{d} r} \right] \left \langle \frac{\left(\mathcal{J}_{r \theta \zeta}\right)^3 g^{\zeta \zeta}}{g_{\theta \theta}} \right \rangle \\ & + \left \langle \frac{\mathcal{J}_{r \theta \zeta} g^{\zeta \zeta}}{g_{\theta \theta}} \left( \frac{\partial g_{r \theta}}{\partial \theta} - \frac{\partial g_{\theta \theta}}{\partial r} - \frac{g_{r \theta}}{\mathcal{J}_{r \theta \zeta}} \frac{\partial \mathcal{J}_{r \theta \zeta}}{\partial \theta} \right) \right \rangle \Bigg ) \end{split} \label{eq:dBzetadr}\end{split}\]

where \(P = P\left( r \right)\) is the equilibrium plasma pressure, \(q^{\prime} = \mathrm{d}q/\mathrm{d} r\) is the radial derivative of the safety factor and

\[H = \left \langle \mathcal{J}_{r \theta \zeta} g^{\zeta \zeta} \right \rangle + \left(\frac{B_{\zeta}}{\psi^{\prime}}\right)^2 \left \langle \frac{\left( \mathcal{J}_{r \theta \zeta}\right)^3 \left(g^{\zeta \zeta}\right)^2}{g_{\theta \theta} } \right \rangle. \label{eq:H}\]

The Grad-Shafranov solution also constrains the radial derivative of the Jacobian to be

\[\begin{split}\begin{split} \frac{\partial \mathcal{J}_{r \theta \zeta}}{\partial r} = \, & \mathcal{J}_{r \theta \zeta} \frac{\psi''}{\psi'} - \frac{\mathcal{J}_{r \theta \zeta}}{g_{\theta \theta}} \left( \frac{\partial g_{r \theta}}{\partial \theta} - \frac{\partial g_{\theta \theta}}{\partial r} - \frac{g_{r \theta}}{\mathcal{J}_{r \theta \zeta}} \frac{\partial \mathcal{J}_{r \theta \zeta}}{\partial \theta} \right) \\ & + \frac{\left(\mathcal{J}_{r \theta \zeta}\right)^3}{g_{\theta \theta}} \left[\frac{\mu_0}{\left( \psi^{\prime} \right)^2} \frac{\mathrm{d} P}{\mathrm{d} r} \right] + \frac{\left(\mathcal{J}_{r \theta \zeta}\right)^3 g^{\zeta \zeta}}{g_{\theta \theta}} \left[ \frac{B_{\zeta}}{\left(\psi' \right)^2} \frac{\mathrm{d} B_{\zeta}}{\mathrm{d} r} \right]. \end{split} \label{eq:Radial_Deriv_Jacobian}\end{split}\]

where \(\psi^{\prime \prime} = \mathrm{d}^2 \psi / \mathrm{d} r^2\) is arbitrary, due to it always appearing as part of the combination \(\partial \mathcal{J}_{r \theta \zeta} / \partial r - \mathcal{J}_{r \theta \zeta} \left( \psi'' / \psi' \right)\) in quantities of interest for local equilibria.

Global field-aligned coordinates

Metric terms

To obtain field-aligned coordinates we transform from the toroidal system to the global field-aligned system, \(\{r, \theta, \zeta \} \rightarrow \{r, \alpha, \theta \}\), defined by

\[\begin{split}\begin{aligned} r&=r & r &= r \\ \alpha &= \sigma_{\alpha} \left[q\left( r \right) \theta - \zeta + G_0 \left(r, \theta \right) \right] & \theta & = \theta \\ \theta &=\theta & \zeta &=q\left( r \right) \theta - \sigma_{\alpha} \alpha + G_0 \left(r, \theta \right) \end{aligned}\end{split}\]

where \(\sigma_{\alpha}\) takes values of either \(1\) or \(-1\). For \(\sigma_{\alpha} = 1\), \(\{r, \alpha, \theta \}\) forms a right-handed system, such as in GENE, whereas with \(\sigma_{\alpha} = -1\) then \(\{r, \theta, \alpha\}\) forms a right-handed system, such as in CGYRO. The covariant metric components are, using \(\partial \zeta / \partial r = \sigma_{\alpha} \partial \alpha / \partial r\), \(\partial \zeta / \partial \theta = \sigma_{\alpha} \partial \alpha / \partial \theta\) and using a tilde to denote the field-aligned system,

\[\tilde{g}_{rr} = g_{r r} + \left(\frac{\partial \alpha}{\partial r} \right)^2 g_{\zeta \zeta}\]

\[\tilde{g}_{r \alpha} = - \frac{\partial \alpha}{\partial r} g_{\zeta \zeta}\]

\[\tilde{g}_{r \theta} = g_{r \theta } + \frac{\partial \alpha}{\partial r} \frac{\partial \alpha}{\partial \theta} g_{\zeta \zeta}\]

\[\tilde{g}_{\alpha \alpha} = g_{\zeta \zeta}\]

\[\tilde{g}_{\alpha \theta} = - \frac{\partial \alpha}{\partial \theta} g_{\zeta \zeta}\]

\[\tilde{g}_{\theta \theta } = g_{\theta \theta } + \left( \frac{\partial \alpha}{\partial \theta}\right)^2 g_{\zeta \zeta}.\]

where \(\partial \alpha / \partial r\) and \(\partial \alpha / \partial \theta\) are calculated below. Note that the Jacobian remains unchanged from the toroidal system. The contravariant metric components are

\[\tilde{g}^{r r} = g^{r r}\]

\[\tilde{g}^{r \alpha} = \frac{\partial \alpha}{\partial r} g^{r r} + \frac{\partial \alpha}{\partial \theta} g^{r \theta} \label{eq:eqgralpha}\]

\[\tilde{g}^{\alpha \alpha} = \left( \frac{\partial \alpha}{\partial r} \right)^2 g^{r r} + 2 \frac{\partial \alpha}{\partial r} \frac{\partial \alpha}{\partial \theta} g^{r \theta} + \left( \frac{\partial \alpha}{\partial \theta} \right)^2 g^{\theta \theta} + g^{\zeta \zeta}. \label{eq:g^alphaalpha}\]

\[\tilde{g}^{r \theta} = g^{r \theta}\]

\[\tilde{g}^{\theta \theta} = g^{\theta \theta}\]

\[\tilde{g}^{\alpha \theta} = \frac{\partial \alpha}{\partial r} g^{r \theta} + \frac{\partial \alpha}{\partial \theta} g^{\theta \theta}.\]

Evaluating \(\partial \alpha / \partial \theta\), we find

\[\begin{split}\begin{split} \frac{\partial \alpha}{\partial \theta} & = \sigma_{\alpha} \left[ q + \frac{\partial G_0}{\partial \theta} \right] \\ & = \sigma_{\alpha} \left [\frac{B_{\zeta}}{\psi^{\prime}} \mathcal{J}_{r \theta \zeta} g^{\zeta \zeta} \right]. \end{split} \label{eq:dalphadtheta}\end{split}\]

where we have used equation [eq:G0 deriv]. To then calculate \(\partial \alpha / \partial r\), we differentiate equation [eq:dalphadtheta] with respect to \(r\) and integrate over \(\theta\). First differentiating, we get

\[\begin{split}\begin{split} \frac{\partial^2 \alpha}{\partial r \partial \theta} & = \sigma_{\alpha} \left[ \frac{\mathrm{d} q}{\mathrm{d} r} + \frac{\partial^2 G_0}{\partial r \partial \theta} \right] \\ & = \sigma_{\alpha} \left[ \frac{\mathrm{d} B_{\zeta}}{\mathrm{d} r} \frac{\mathcal{J}_{r \theta \zeta} g^{\zeta \zeta}}{\psi^{\prime}} + \frac{B_{\zeta}}{\psi^{\prime}} g^{\zeta \zeta} \left(\frac{\partial \mathcal{J}_{r \theta \zeta}}{\partial r} - \frac{\psi^{\prime \prime}}{ \psi^{\prime} } \mathcal{J}_{r \theta \zeta} \right) + \frac{B_{\zeta}}{\psi^{\prime}} \frac{\partial g^{\zeta \zeta} }{\partial r}\mathcal{J}_{r \theta \zeta} \right]. \end{split}\end{split}\]

where the forms of \(\mathrm{d}B_{\zeta} / \mathrm{d} r\) and \(\partial \mathcal{J}_{r \theta \zeta} / \partial r\) are given by equations [eq:dBzetadr] and [eq:Radial_Deriv_Jacobian]. Now integrating over \(\theta\),

\[\begin{split} \frac{\partial \alpha}{\partial r} = \sigma_{\alpha} \int_{0}^{\theta} \left[ \frac{\mathrm{d} B_{\zeta}}{\mathrm{d} r} \frac{\mathcal{J}_{r \theta \zeta} g^{\zeta \zeta}}{\psi^{\prime}} + \frac{B_{\zeta}}{\psi^{\prime}} g^{\zeta \zeta} \left(\frac{\partial \mathcal{J}_{r \theta \zeta}}{\partial r} - \frac{\psi^{\prime \prime}}{ \psi^{\prime} } \mathcal{J}_{r \theta \zeta} \right) + \frac{B_{\zeta}}{\psi^{\prime}} \frac{\partial g^{\zeta \zeta} }{\partial r}\mathcal{J}_{r \theta \zeta} \right] \, \mathrm{d} \theta^{\prime} \label{eq:dalphadr} \end{split}\]

where \(\left. \partial \alpha / \partial r \right\vert_{\theta = 0} = 0\) is assumed. Note that \(\partial \alpha / \partial r\) is not periodic, but satisfies

\[\left . \frac{\partial \alpha}{\partial r} \right\vert_{\theta + 2 M \pi} = \left . \frac{\partial \alpha}{\partial r} \right\vert_{\theta} + \sigma_{\alpha} \frac{\mathrm{d} q}{\mathrm{d} r} 2 M \pi.\]