# Geometry and Differential Operator#

Author:

X. Q. Xu

## Geometry#

In a axisymmetric toroidal system, the magnetic field can be expressed as

${\bf B}=I(\psi)\nabla\zeta+\nabla\zeta\times\nabla\psi,$

where $$\psi$$ is the poloidal flux, $$\theta$$ is the poloidal angle-like coordinate, and $$\zeta$$ is the toroidal angle. Here, $$I(\psi)=RB_t$$. The two important geometrical parameters are: the curvature, $$\bf \kappa$$, and the local pitch, $$\nu(\psi,\theta)$$,

$\nu(\psi,\theta)= {I(\psi){\bf \cal J}/R^2}.$

The local pitch $$\nu(\psi,\theta)$$ is related to the MHD safety q by $$\hat q(\psi)={2\pi}^{-1}\oint\nu(\psi,\theta) d\theta$$ in the closed flux surface region, and $$\hat q(\psi)={2\pi}^{-1}\int_{inboard}^{outboard}\nu(\psi,\theta) d\theta$$ in the scrape-off-layer. Here $${\bf \cal J}=(\nabla\psi\times\nabla\theta\cdot\nabla\zeta)^{-1}$$ is the coordinate Jacobian, $$R$$ is the major radius, and $$Z$$ is the vertical position.

## Geometry and Differential Operators#

In a axisymmetric toroidal system, the magnetic field can be expressed as $${\bf B}=I(\psi)\nabla\zeta+\nabla\zeta\times\nabla\psi$$, where $$\psi$$ is the poloidal flux, $$\theta$$ is the poloidal angle-like coordinate, and $$\zeta$$ is the toroidal angle. Here, $$I(\psi)=RB_t$$. The two important geometrical parameters are: the curvature, $$\bf \kappa$$, and the local pitch, $$\nu(\psi,\theta)$$, and $$\nu(\psi,\theta)= {I(\psi){\bf \cal J}/R^2}$$. The local pitch $$\nu(\psi,\theta)$$ is related to the MHD safety q by $$\hat q(\psi)={2\pi}^{-1}\oint\nu(\psi,\theta) d\theta$$ in the closed flux surface region, and $$\hat q(\psi)={2\pi}^{-1}\int_{inboard}^{outboard}\nu(\psi,\theta) d\theta$$ in the scrape-off-layer. Here $${\bf \cal J}=(\nabla\psi\times\nabla\theta\cdot\nabla\zeta)^{-1}$$ is the coordinate Jacobian, $$R$$ is the major radius, and $$Z$$ is the vertical position.

### Differential Operators#

For such an axisymmetric equilibrium the metric coefficients are only functions of $$\psi$$ and $$\theta$$. Three spatial differential operators appear in the equations given as: $${\bf v_E}\cdot\nabla_\perp$$, $$\nabla_\|$$ and $$\nabla_\perp^2$$.

\begin{split}\begin{aligned} \nabla_\|&=&{\bf b_0}\cdot\nabla={1\over {\cal J}B}{\partial\over\partial\theta}+{I\over BR^2}{\partial\over\partial\zeta}={B_p\over hB}{\partial\over\partial\theta}+{B_t\over RB}{\partial\over\partial\zeta}, \\ {\cal J}\nabla^2&=& {\partial\over\partial\psi}\left({\cal J}J_{11}{\partial\over\partial\psi}\right) +{\partial\over\partial\psi}\left({\cal J}J_{12}{\partial\over\partial\theta}\right) \nonumber\\ &+&{\partial\over\partial\theta}\left({\cal J}J_{22}{\partial\over\partial\theta}\right) +{\partial\over\partial\theta}\left({\cal J}J_{12}{\partial\over\partial\psi}\right) \nonumber\\ &+&{1\over R^2}{\partial^2\over\partial\zeta^2}. \\ \nabla_\|^2&=&{\bf b}_0\cdot\nabla({\bf b}_0\cdot\nabla)={1\over {\cal J}B}{\partial\over\partial\theta}\left({1\over {\cal J}B}{\partial\over\partial\theta}\right) +{1\over {\cal J}B}{\partial\over\partial\theta}\left({B_t\over RB}{\partial\over\partial\zeta}\right) \\ &+&{B_t\over {\cal J}RB^2}{\partial^2\over\partial\theta\partial\zeta} +\left({B_t\over {\cal J}RB}\right)^2{\partial^2\over\partial\zeta^2}, \\ \nabla_\perp^2\Phi&=&-\nabla\cdot[{\bf b}\times({\bf b}\times\nabla\Phi)]=\nabla^2\Phi-(\nabla\cdot{\bf b})({\bf b}\cdot\nabla\Phi)-\nabla_\|^2\Phi\end{aligned}\end{split}

where the coordinate Jacobian and metric coefficients are defined as following:

\begin{split}\begin{aligned} {\cal J}&=&\nabla\psi\times\nabla\theta\cdot\nabla\zeta={h\over B_p}, \\ h&=&\sqrt{Z_\theta^2+R_\theta^2}, \\ J_{11}&=&|\nabla\psi|^2={R^2\over {\cal J}^2}(Z_\theta^2+R_\theta^2), \\ J_{12}&=&J_{21}=\nabla\psi\cdot\nabla\theta=-{R^2\over {\cal J}^2}(Z_\theta Z_\psi+R_\psi R_\theta), \\ J_{13}&=&J_{31}=0, \\ J_{22}&=&|\nabla\theta|^2={R^2\over {\cal J}^2}(Z_\psi^2+R_\psi^2), \\ J_{23}&=&J_{32}=0, \\ J_{33}&=&|\nabla\zeta|^2={1\over R^2}.\end{aligned}\end{split}

### Concentric circular cross section inside the separatrix without the SOL#

For concentric circular cross section inside the separatrix without the SOL, the differential operators are reduced to:

$\begin{split}R &= R_0+r\cos\theta, \\ Z &= r\sin\theta, \\ B_t &= {B_{t0}R_0\over R}, \\ B_p &= {1\over R}{\partial\psi\over\partial r}, \\ R_\psi &= {\cos\theta\over RB_p}, \\ R_\theta &= -r\sin\theta, \\ Z_\psi &= {\sin\theta\over RB_p}, \\ Z_\theta &= r\cos\theta, \\ {\cal J} &= {r\over B_p}, \\ h &= r, \\ J_{11} &= |\nabla\psi|^2=r^2B_p^2, \\ J_{12} = J_{21} &= \nabla\psi\cdot\nabla\theta=0,\\ J_{13} = J_{31} &= 0, \\ J_{22} &= |\nabla\theta|^2={1\over r^2}, \\ J_{23} = J_{32} &= 0, \\ J_{33} &= |\nabla\zeta|^2={1\over R^2},\\ \nabla^2 &\simeq {1\over r}{\partial\over\partial r}\left(r{\partial\over\partial r}\right)+{1\over r^2}{\partial^2\over\partial \theta^2}+{1\over R^2}{\partial^2\over\partial \zeta^2}\end{split}$

### Field-aligned coordinates with $$\theta$$ as the coordinate along the field line#

A suitable coordinate mapping between field-aligned ballooning coordinates ($$x$$, $$y$$, $$z$$) and the usual flux coordinates ($$\psi$$, $$\theta$$, $$\zeta$$) is

\begin{split}\begin{aligned} x&=&\psi-\psi_s, \nonumber \\ y&=&\theta, \nonumber \\ z&=&\zeta-\int_{\theta_0}^\theta \nu(x,y)dy.\end{aligned}\end{split}

as shown in Fig. 1. The covering area given by the square ABCD in the usual flux coordinates is the same as the parallelogram ABEF in the field-aligned coordinates. The magnetic separatrix is denoted by $$\psi=\psi_s$$. In this choice of coordinates, $$x$$ is a flux surface label, $$y$$, the poloidal angle, is also the coordinate along the field line, and $$z$$ is a field line label within the flux surface.

The coordinate Jacobian and metric coefficients are defined as following:

\begin{split}\begin{aligned} {\cal J}&=&\nabla\psi\times\nabla\theta\cdot\nabla\zeta={h\over B_p}, \\ h&=&\sqrt{Z_\theta^2+R_\theta^2}, \\ {\cal J}_{11}&=&|\nabla x|^2={R^2\over {\cal J}^2}(Z_\theta^2+R_\theta^2), \\ {\cal J}_{12}&=&{\cal J}_{21}=\nabla x\cdot\nabla y=-{R^2\over {\cal J}^2}(Z_\theta Z_\psi+R_\psi R_\theta), \\ {\cal J}_{22}&=&|\nabla y|^2={R^2\over {\cal J}^2}(Z_\psi^2+R_\psi^2), \\ {\cal J}_{13}&=&{\cal J}_{31}=\nabla x\cdot\nabla z=-\nu\nabla x\cdot\nabla y-|\nabla x|^2\left(\int_{y_0}^y {\partial \nu(x,y)\over\partial\psi}dy\right)=-|\nabla x|^2I_s, \\ {\cal J}_{23}&=&{\cal J}_{32}=\nabla y\cdot\nabla z=-\nu|\nabla y|^2-\nu\nabla x\cdot\nabla y\left(\int_{y_0}^y {\partial \nu(x,y)\over\partial\psi}dy\right), \\ {\cal J}_{33}&=&|\nabla z|^2=\left |\nabla\zeta-\nu\nabla \theta-\nabla\psi\left(\int_{y_0}^y {\partial \nu(x,y)\over\partial\psi}dy\right)\right |^2, \\ I_s &=& {{\cal J}_{12}\over|\nabla\psi|^2}\nu(x,y)+\left(\int_{y_0}^y {\partial \nu(x,y)\over\partial\psi}dy\right).\end{aligned}\end{split}

Here $$h$$ is the local minor radius, $$I_s$$ is the integrated local shear, and $$y_0$$ is an arbitrary integration parameter, which, depending on the choice of Jacobian, determines the location where $$I_s=0$$. The disadvantage of this choice of coordinates is that the Jacobian diverges near the X-point as $$B_p\rightarrow 0$$ and its effect spreads over the entire flux surafces near the separatrix as the results of coordinate transform $$z$$. Therefore a better set of coordinates is needed for X-point divertor geometry. The derivatives are obtained from the chain rule as follows:

\begin{split}\begin{aligned} {d\over d\psi}&=&{\partial\over \partial x} - \left(\int_{y_0}^y {\partial \nu(x,y)\over\partial\psi}dy\right){\partial\over \partial z}, \\ {d\over d\theta}&=&{\partial\over \partial y} - \nu(x,y){\partial\over \partial z}, \\ {d\over d\zeta}&=&{\partial\over \partial z}.\end{aligned}\end{split}

In the field-aligned ballooning coordinates, the parallel differential operator is simple, involving only one coordinate $$y$$

\begin{aligned} \partial_\|^0 &=& {\bf b}_0\cdot\nabla_\|=\left({B_p\over hB}\right){\partial\over\partial y}.\end{aligned}

which requires a few grid points. The total axisymmetric drift operator becomes

The perturbed $${\bf E}\times {\bf B}$$ drift operator becomes

\begin{split}\begin{aligned} {\delta\bf v_E}\cdot\nabla_\perp&=& {c\over BB_\|^*}\left\{ -{I\over J}{\partial\langle\delta\phi\rangle\over\partial\theta} +{B_p^2} {\partial\langle\delta\phi\rangle\over\partial z} \right\}{\partial\over\partial\psi} \nonumber\\ &+&{c\over BB_\|^*}\left\{{I\over{\cal J}} {\partial\langle\delta\phi\rangle\over\partial\psi} +{{\cal J}_{12}\over R^2} {\partial\langle\delta\phi\rangle\over\partial z} \right\}{\partial\over\partial\theta} \nonumber\\ &-&{c\over BB_\|^*}\left\{B_p^2 {\partial\langle\delta\phi\rangle\over\partial\psi} +{{\cal J}_{12}\over R^2} {\partial\langle\delta\phi\rangle\over\partial\theta} \right\}{\partial\over\partial z},\end{aligned}\end{split}

when the conventional turbulence ordering ($$k_\|\ll k_\perp$$) is used, the perturbed $${\bf E}\times {\bf B}$$ drift operator can be further reduced to a simple form

\begin{aligned} {\delta\bf v_E}\cdot\nabla_\perp&=& {cB\over B_\|^*}\left( {\partial\langle\delta\phi\rangle\over\partial z}{\partial\over\partial x} -{\partial\langle\delta\phi\rangle\over\partial x}{\partial\over\partial z}\right)\end{aligned}

where $$\partial/\partial\theta\simeq -\nu\partial/\partial z$$ is used. In the perturbed $${\bf E}\times {\bf B}$$ drift operator the poloidal and radial derivatives are written in the usual flux $$(\psi,\theta,\zeta)$$ coordinates in order to have various options for valid discretizations. The general Laplacian operator for potential is

\begin{align}\begin{aligned}\begin{split} \begin{aligned} {\cal J}\nabla^2\Phi&=&{\partial\over\partial x}\left({\cal J}{\cal J}_{11}{\partial\Phi\over\partial x} +{\cal J}{\cal J}_{12}{\partial\Phi\over\partial y} +{\cal J}{\cal J}_{13}{\partial\Phi\over\partial z}\right) \nonumber\\ &+&{\partial\over\partial y}\left({\cal J}{\cal J}_{21}{\partial\Phi\over\partial x} +{\cal J}{\cal J}_{22}{\partial\Phi\over\partial y} +{\cal J}{\cal J}_{23}{\partial\Phi\over\partial z}\right) \nonumber\\ &+&{\partial\over\partial z}\left({\cal J}{\cal J}_{31}{\partial\Phi\over\partial x} +{\cal J}{\cal J}_{32}{\partial\Phi\over\partial y} +{\cal J}{\cal J}_{33}{\partial\Phi\over\partial z}\right).\end{aligned}\end{split}\\The general perpendicular Laplacian operator for potential is\end{aligned}\end{align}
\begin{split}\begin{aligned} {\cal J}\nabla_\perp^2\Phi&=&{\partial\over\partial x}\left({\cal J}{\cal J}_{11}{\partial\Phi\over\partial x} +{\cal J}{\cal J}_{12}{\partial\Phi\over\partial y} +{\cal J}{\cal J}_{13}{\partial\Phi\over\partial z}\right) \nonumber\\ &+&{\partial\over\partial y}\left({\cal J}{\cal J}_{21}{\partial\Phi\over\partial x} +{\cal J}{\cal J}_{22}{\partial\Phi\over\partial y} +{\cal J}{\cal J}_{23}{\partial\Phi\over\partial z}\right) \nonumber\\ &+&{\partial\over\partial z}\left({\cal J}{\cal J}_{31}{\partial\Phi\over\partial x} +{\cal J}{\cal J}_{32}{\partial\Phi\over\partial y} +{\cal J}{\cal J}_{33}{\partial\Phi\over\partial z}\right) \nonumber\\ &-&\left({B_p\over hB}\right){\partial\over\partial y} \left[\left({B_p\over hB}\right){\partial\Phi\over\partial y}\right] \nonumber\\ &-&\left({B_p\over hB}\right)^2{\partial\ln B\over\partial y}{\partial\Phi\over\partial y}.\end{aligned}\end{split}

The general perpendicular Laplacian operator for axisymmetric potential $$\Phi_0(x,y)$$ is

\begin{split}\begin{aligned} {\cal J}\nabla_\perp^2\Phi_0&=&{\partial\over\partial x}\left({\cal J}{\cal J}_{11}{\partial\Phi_0\over\partial x} +{\cal J}{\cal J}_{12}{\partial\Phi_0\over\partial y}\right) \nonumber\\ &+&{\partial\over\partial y}\left({\cal J}{\cal J}_{21}{\partial\Phi_0\over\partial x} +{\cal J}{\cal J}_{22}{\partial\Phi_0\over\partial y}\right) \nonumber\\ &-&\left({B_p\over hB}\right){\partial\over\partial y} \left[\left({B_p\over hB}\right){\partial\Phi_0\over\partial y}\right] \nonumber\\ &-&\left({B_p\over hB}\right)^2{\partial\ln B\over\partial y}{\partial\Phi\over\partial y}.\end{aligned}\end{split}

For the perturbed potential $$\delta\phi$$, we can drop the $$\partial/\partial y$$ terms in Eq. (69) due to the elongated nature of the turbulence ($$k_\|/k_\perp\ll1$$). The general perpendicular Laplacian operator for perturbed potential $$\delta\phi$$ reduces to

\begin{split}\begin{aligned} {\cal J}\nabla_\perp^2\delta\phi&=& {\partial\over\partial x}\left({\cal J}{\cal J}_{11}{\partial\delta\phi\over\partial x} +{\cal J}{\cal J}_{13}{\partial\delta\phi\over\partial z}\right) \nonumber\\ &+&{\partial\over\partial z}\left({\cal J}{\cal J}_{31}{\partial\delta\phi\over\partial x} +{\cal J}{\cal J}_{33}{\partial\delta\phi\over\partial z}\right).\end{aligned}\end{split}

If the non-split potential $$\Phi$$ is a preferred option, the gyrokinetic Poisson equation (18) and the general perpendicular Laplacian operator Eq. (69) have to be used. Then the assumption $$k_\|/k_\perp\ll1$$ is not used to simplify the perpendicular Laplacian operator.