Toroidal coordinates

A simple toroidal coordinate system

Coordinate systems used in toroidal systems:

Cartesian coordinates

(X, Y, Z)

Cylindrical coordinates

$(R,\phi ,Z)$, where

• $R^{2}=X^{2}+Y^{2}$, and
• $\tan \phi =Y/X$.

$\phi$ is called the toroidal angle (and not the cylindrical angle, at least not in the context of magnetic confinement).

Cylindrical symmetry (symmetry under rotation over $\phi$) is referred to as axisymmetry.

Simple toroidal coordinates

$(r,\phi ,\theta )$, where

• $R=R_{0}+r\cos \theta$, and
• $Z=r\sin \theta$

$R_{0}$ corresponds to the torus axis and is called the major radius, while $0\leq r\leq a$ is called the minor radius, and θ the poloidal angle. The ratio $R_{0}/a$ is called the aspect ratio of the torus.

In order to adapt this simple system better to the magnetic surfaces of an axisymmetric MHD equilibrium, it may be enhanced by

• letting $R_{0}/a$ depend on $r$ (to account for the Shafranov shift of flux surfaces)
• adding ellipticity ($\kappa$), triangularity ($\delta$), and squareness ($\zeta$) to account for non-circular flux surface cross sections. A popular simple expression for shaped flux surfaces is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(r,\theta) = R_0(r) + r \cos(\theta + \arcsin \delta \sin \theta)\\ Z(r,\theta) = Z_0(r) + \kappa(r) r \sin(\theta + \zeta \sin 2 \theta) }

Toroidal coordinates

$(\zeta ,\eta ,\phi )$, where

$R=R_{p}{\frac {\sinh \zeta }{\cosh \zeta -\cos \eta }}$

$Z=R_{p}{\frac {\sin \eta }{\cosh \zeta -\cos \eta }}$

where $R_{p}$ is the pole of the coordinate system. Surfaces of constant $\zeta$ are tori with major radii $R=R_{p}/\tanh \zeta$ and minor radii $r=R_{p}/\sinh \zeta$. At $R=R_{p}$, $\zeta =\infty$, while at infinity and at $R=0,\zeta =0$. The coordinate $\eta$ is a poloidal angle and runs from 0 to $2\pi$. This system is orthogonal.

The Laplace equation separates in this system of coordinates, thus allowing an expansion of the vacuum magnetic field in toroidal harmonics.

Magnetic coordinates

Magnetic coordinates are a particular type of flux coordinates in which the magnetic field lines are straight lines. Magnetic coordinates adapt to the magnetic field, and therefore to the MHD equilibrium (also see Flux surface). Magnetic coordinates simplify the description of the magnetic field. In 3 dimensions (not assuming axisymmetry), the most commonly used coordinate systems are:

• Hamada coordinates. In these coordinates, both the field lines and current lines corresponding to the MHD equilibrium are straight.
• Boozer coordinates. In these coordinates, the field lines corresponding to the MHD equilibrium are straight and so are the diamagnetic lines , i.e. the integral lines of $\nabla \psi \times \mathbf {B}$.

These two coordinate systems are related.