Basic Cylindrical Coordinate System

Please also look at the Notation Conventions.

The basic cylindrical coordinate system $(u^1,u^2,u^3)=(R,Z,\phi)$ of JOREK is given by \(\begin{align*} x &= R~\mathrm{cos} \phi \\ y &= -R~\mathrm{sin} \phi \\ z &= Z, \end{align*}\) where $(x,y,z)$ denotes Cartesian coordinates. Thus, $\phi$ goes clockwise if looked at from above! Note this is not always the same for different tokamaks

Note: In element_matrix routines, x and y are used as synonyms for R and Z which must not be confused with the Cartesian coordinates.

Basis Vectors

Covariant basis vectors $ \mathbf{a}_\alpha = \partial\mathbf{X}/\partial u^{\alpha} $ are:

\[\begin{align*} \mathbf{a}_1 &= \begin{pmatrix} \mathrm{cos}\phi \\ -\mathrm{sin}\phi \\ 0\end{pmatrix}, \quad \mathbf{a}_2 &= \begin{pmatrix} 0 \\ 0 \\ 1\end{pmatrix}, \quad \mathbf{a}_3 &= \begin{pmatrix} -R~\mathrm{sin}\phi \\ -R~\mathrm{cos}\phi \\ 0\end{pmatrix} \end{align*}\]

Cross products between the basis vectors:

\[\begin{align*} \mathbf{a}_1 \times \mathbf{a}_2 &= \mathbf{a}_3/R \\ \mathbf{a}_1 \times \mathbf{a}_3 &= -R~\mathbf{a}_2 \\ \mathbf{a}_2 \times \mathbf{a}_3 &= R~\mathbf{a}_1 \\ \end{align*}\]

and of course: $ \mathbf{a}\alpha \times \mathbf{a}\alpha = 0 $ as well as $ \mathbf{a}\alpha \times \mathbf{a}\beta = -\mathbf{a}\beta \times \mathbf{a}\alpha $. The contravariant basis vectors are given by:

\[\begin{align*} \mathbf{a}^1 &= \mathbf{a}_1 \\ \mathbf{a}^2 &= \mathbf{a}_2 \\ \mathbf{a}^3 &= \mathbf{a}_3 / R^2 \end{align*}\]

Of course, $ \mathbf{a}\alpha \cdot \mathbf{a}^\beta = \delta\alpha^\beta $. Also:

\[\begin{align*} \mathbf{a}^1 &= \nabla R \qquad\qquad \mathbf{a}_1 = J \nabla Z \times \nabla \phi \\ \mathbf{a}^2 &= \nabla Z \qquad\qquad \mathbf{a}_2 = J \nabla \phi \times \nabla R \\ \mathbf{a}^3 &= \nabla \phi \qquad\qquad \mathbf{a}_3 = J \nabla R \times \nabla Z \\ \end{align*}\]

and

\[\begin{align*} \mathbf{a}^1 \times \mathbf{a}^2 &= R\mathbf{a}^3 \\ \mathbf{a}^1 \times \mathbf{a}^3 &= -\mathbf{a}^2/R \\ \mathbf{a}^2 \times \mathbf{a}^3 &= \mathbf{a}^1/R \\ \end{align*}\]

With J the Jacobian:

\[\begin{equation*} J= \mathbf{a}_1\cdot(\mathbf{a_2}\times\mathbf{a}_3) = R \end{equation*}\]

Normalized basis vectors are given by:

\[\begin{align*} \mathbf{e}_1 \equiv \mathbf{e}^1 &= \mathbf{a}_1 = \mathbf{a}^1 \\ \mathbf{e}_2 \equiv \mathbf{e}^2 &= \mathbf{a}_2 = \mathbf{a}^2 \\ \mathbf{e}_3 \equiv \mathbf{e}^3 &= \mathbf{a}_3 / R = R~\mathbf{a}^3 \end{align*}\]

Metric Tensor

The co- and contravariant metric tensors are given by: \(\begin{align*} g_{\alpha\beta} = \mathbf{a}_\alpha\cdot\mathbf{a}_\beta &= \mathrm{diag}(1,~ 1, ~ R^2) \\ g^{\alpha\beta} = \mathbf{a}^\alpha\cdot\mathbf{a}^\beta &= \mathrm{diag}(1,~ 1, ~ 1/R^2) \end{align*}\)

The determinant of the covariant metric tensor is \(\begin{equation*} g \equiv J^2 =\det{\left(g_{\alpha\beta}\right)}=R^2 \end{equation*}\)

Differential Operators

General expressions

\[\begin{align*} \nabla U &= \partial_i U \mathbf{a}^i \\ \nabla \cdot \mathbf{V} &= \frac{1}{J} \partial_i (J V^i) \\ \nabla \times \mathbf{V} &= \frac{1}{J}(\partial_2 V_3 - \partial_3 V_2)\mathbf{a}_1 + \frac{1}{J}(\partial_3 V_1 - \partial_1 V_3)\mathbf{a}_2 + \frac{1}{J}(\partial_1 V_2 - \partial_2 V_1)\mathbf{a}_3 \end{align*}\]

Cylindrical coordinates

\[\begin{align*} \nabla U &=\partial_1 U~\mathbf{a}^1 + \partial_2 U~\mathbf{a}^2 + \partial_3 U~\mathbf{a}^3 \\ \nabla_\text{pol} U &=\partial_1 U~\mathbf{a}^1 + \partial_2 U~\mathbf{a}^2 \\ \nabla\cdot\mathbf{V} &=\frac{1}{R}\partial_1 (R V^1) + \partial_2 V^2 + \partial_3 V^3 \\ \nabla \times \mathbf{V} &= \frac{1}{R}\left(\partial_2 V_3 - \partial_3 V_2 \right)~\mathbf{a}_1 + \frac{1}{R}\left(\partial_3 V_1 - \partial_1 V_3 \right)~\mathbf{a}_2 + \frac{1}{R}\left(\partial_1 V_2 - \partial_2 V_1 \right)~\mathbf{a}_3 \\ \Delta U &= \nabla\cdot\nabla U = \frac{1}{R}\partial_1(R \partial_1 U) + \partial_{2,2}U + \frac{1}{R^2}\partial_{3,3}U \\ \Delta_\text{pol}U &= \nabla\cdot\nabla_\text{pol} U = \frac{1}{R}\partial_1(R \partial_1 U) + \partial_{2,2}U \\ \Delta^*U &= R^2 \nabla\cdot\left(\frac{1}{R^2}\nabla_\text{pol} U\right)=R~\partial_1\left(\frac{1}{R}\partial_1 U\right)+\partial_{2,2} U \\ [A,B] &= e_3 \cdot (\nabla A \times\nabla B) = \partial_1 A~\partial_2 B - \partial_2 A~\partial_1 B\\ \nabla \cdot \mathbf{\Pi} &= \left[\partial_R \Pi^{RR}+\partial_Z \Pi^{ZR}+\frac{1}{R}\partial_\phi \Pi^{\phi R}+\frac{1}{R}\left(\Pi^{RR}-\Pi^{\phi\phi}\right)\right] \mathbf{e}_R\\ &+ \left[\partial_R \Pi^{RZ}+\partial_Z \Pi^{ZZ}+\frac{1}{R}\partial_\phi \Pi^{\phi Z}+\frac{1}{R}\Pi^{RZ}\right]\mathbf{e}_Z\\ &+\left[\partial_R\Pi^{R\phi}+\partial_Z \Pi^{Z\phi}+\frac{1}{R}\partial_\phi\Pi^{\phi\phi}+\frac{1}{R}\Pi^{\phi R}+\frac{1}{R}\Pi^{R\phi}\right]\mathbf{e}_\phi \end{align*}\]

Christoffel symbols

General Expressions

\[\begin{align*} \mathbf{A} \cdot \nabla \mathbf{B} &= A^i \partial_i (B^j a_j) = A^i (\partial B^j)a_j + A^i B^j (\partial_i a_j ) = A^i(\partial_i B^j + \Gamma^j_{ik} B^k) a_j\\ (\mathbf{A} \cdot \nabla \mathbf{B})^j &= A^i (\partial_i B^j + \Gamma^j_{ik} B^k) \end{align*}\]

Cylindrical coordinates

\[\begin{align*} \partial_3 \mathbf{a}^1 &= R \nabla \phi = R \mathbf{a}^3 \\ \partial_3 \mathbf{a}^3 &= -\frac{1}{R} \mathbf{a}^1 \\ \partial_1 \mathbf{a}^3 &= -\frac{1}{R} \mathbf{a}^3 \\ \end{align*}\]

COCOS convention

The tokamak coordinate convention COCOS used by JOREK is 8. This implies the following COCOS coefficients: \(\begin{align*} e_{\mathrm{Bp}} &= 0 \\ \sigma_{\mathrm{Bp}} &= -1 \\ \sigma_{R\phi Z} &= -1 \\ \sigma_{\rho \theta \phi} &= 1 \end{align*}\)

For details about the COCOS convention please see Sauter, O., Medvedev, S.Yu, Computer Physics Communications 184 (2), 293-302 (2013).