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