Magnetic Geometry

We take the magnetic field $\mathbf{B}$ to have the form

\[\begin{equation} \mathbf{B} = B_z \hat{\mathbf{z}} + B_\zeta \hat{\mathbf{\zeta}}, \end{equation}\]

with $B_\zeta = B(r,z) b_\zeta$, $B_z = B(r,z) b_z$ and $b_z$ and $b_\zeta$ the direction cosines of the magnetic field vector. Here the basis vectors are those of cylindrical geometry $(r,z,\zeta)$, i.e., $\hat{\mathbf{r}} = \nabla r $, $\hat{\mathbf{z}} = \nabla z$, and $\hat{\mathbf{\zeta}} = r \nabla \zeta$. The unit vectors $\hat{\mathbf{r}}$, $\hat{\mathbf{z}}$, and $\hat{\mathbf{\zeta}}$ form a right-handed orthonormal basis.

Supported options

To choose the type of geometry, set the value of "option" in the geometry namelist. The namelist will have the following appearance in the TOML file.

[geometry]
option="constant-helical" # ( or "1D-mirror" )
pitch = 1.0
rhostar = 1.0
DeltaB = 0.0

If rhostar is not set then it is computed from reference parameters.

[geometry] option = "constant-helical"

Here $b_\zeta = \sqrt{1 - b_z^2}$ is a constant, $b_z$ is a constant input parameter ("pitch") and $B$ is taken to be 1 with respect to the reference value $B_{\rm ref}$.

[geometry] option = "1D-mirror"

Here $b_\zeta = \sqrt{1 - b_z^2}$ is a constant, $b_z$ is a constant input parameter ("pitch") and $B = B(z)$ is taken to be the function

\[\begin{equation} \frac{B(z)}{B_{\rm ref}} = 1 + \Delta B \left( 2\left(\frac{2z}{L_z}\right)^2 - \left(\frac{2z}{L_z}\right)^4\right) \end{equation}\]

where $\Delta B $ is an input parameter ("DeltaB") that must satisfy $\Delta B > -1$. Recalling that the coordinate $z$ runs from $z = -L_z/2$ to $L_z/2$, if $\Delta B > 0$ than the field represents a magnetic mirror which traps particles, whereas if $\Delta B < 0$ then the magnetic field accelerates particles by the mirror force as they approach the wall. Note that this field does not satisfy $\nabla \cdot \mathbf{B} = 0$, and is only used to test the implementation of the magnetic mirror terms. 2D simulations with a radial domain and$\mathbf{E}\times\mathbf{B}$ drifts are supported in the "1D-mirror" geometry option.

Geometric coefficients

Here, we write the geometric coefficients appearing in the characteristic equations explicitly.

The $z$ component of the $\mathbf{E}\times\mathbf{B}$ drift is given by

\[\begin{equation} \frac{\mathbf{E}\times\mathbf{B}}{B^2} \cdot \nabla z = \frac{E_r B_\zeta}{B^2} \nabla r \times \hat{\mathbf{\zeta}} \cdot \nabla z = - J \frac{E_r B_\zeta}{B^2}, \end{equation}\]

where we have defined $J = r \nabla r \times \nabla z \cdot \nabla \zeta$. Note that $J$ is dimensionless. The $r$ component of the $\mathbf{E}\times\mathbf{B}$ drift is given by

\[\begin{equation} \frac{\mathbf{E}\times\mathbf{B}}{B^2} \cdot \nabla r = \frac{E_z B_\zeta}{B^2} \nabla z \times \hat{\mathbf{\zeta}} \cdot \nabla r = J \frac{E_z B_\zeta}{B^2}. \end{equation}\]

Due to the axisymmetry of the system, the differential operator $\mathbf{b} \cdot \nabla (\cdot) = b_z \partial {(\cdot)}{\partial z}$, and the convective derivative

\[\begin{equation} \frac{d B}{d t} = \frac{d z}{d t} \frac{\partial B}{ \partial z} + \frac{dr}{dt}\frac{\partial B}{\partial r}. \end{equation}\]