Toroidal coordinates

A toroidal coordinate system

Coordinate systems used in toroidal systems:

Cartesian

(X, Y, Z)

Cylindrical

(R, φ, Z), where

• R² = X² + Y², and
• tan φ = Y/X.

φ 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 φ) is referred to as axisymmetry.

Simple toroidal

(r, φ, θ), where

• R = R₀ + r cos θ, and
• Z = r sin θ

R₀ corresponds to the torus axis and is called the major radius, while r is called the minor radius, and θ the poloidal angle.

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

• letting R₀ depend on r (to account for the Shafranov shift of flux surfaces)
• adding ellipticity (ε), triangularity (κ), etc. (to account for non-circular flux surface cross sections)

Toroidal

(ζ, η, φ), 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 ζ are tori with major radii R = R_p/tanh ζ and minor radii r = R_p/sinh ζ. At R = R_p, ζ = ∞, while at infinity and at R = 0, ζ = 0. The coordinate η is a poloidal angle and runs from 0 to 2π. 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.

General Curvilinear Coordinates

Here we briefly review the basic definitions of a general curvilinear coordinate system for later convenience when discussing toroidal flux coordinates and magnetic coordinates.

Function coordinates and basis vector

Given the spatial dependence of a coordinate set $(\psi (\mathbf {x} ),\theta (\mathbf {x} ),\phi (\mathbf {x} ))$ we can calculate the contravariant basis vectors

$\mathbf {e} ^{i}=\{\nabla \psi ,\nabla \theta ,\nabla \phi \}$

and the dual covariant basis defined as

$\mathbf {e} _{i}={\frac {\partial \mathbf {x} }{\partial {u^{i}}}}\to \mathbf {e} _{i}\cdot \mathbf {e} ^{j}=\delta _{i}^{j}\to \mathbf {e} _{i}={\frac {\mathbf {e} ^{j}\times \mathbf {e} ^{k}}{|\mathbf {e} ^{i}\cdot \mathbf {e} ^{j}\times \mathbf {e} ^{k}|}}={\sqrt {g}}\;\mathbf {e} ^{j}\times \mathbf {e} ^{k}~,$

where $(i,j,k)$ are cyclic permutations of $(1,2,3)$ and we have used the notation $(u^{1},u^{2},u^{3})=(\psi ,\theta ,\phi )$. The Jacobian ${\sqrt {g}}$ is defined below.

Any vector field $\mathbf {B}$ can be represented as

$\mathbf {B} =(\mathbf {B} \cdot \mathbf {e} ^{i})\mathbf {e} _{i}=B^{i}\mathbf {e} _{i}$

or

$\mathbf {B} =(\mathbf {B} \cdot \mathbf {e} _{i})\mathbf {e} ^{i}=B_{i}\mathbf {e} ^{i}~.$

In particular any basis vector $\mathbf {e} _{i}=(\mathbf {e} _{i}\cdot \mathbf {e} _{j})\mathbf {e} ^{j}$. The metric tensor is defined as

$g_{ij}=\mathbf {e} _{i}\cdot \mathbf {e} _{j}\;;\;g^{ij}=\mathbf {e} ^{i}\cdot \mathbf {e} ^{j}\;;\;g_{i}^{j}=\mathbf {e} _{i}\cdot \mathbf {e} ^{j}=\delta _{i}^{j}~.$

Jacobian

The Jacobian of the coordinate transformation $\mathbf {x} (\psi ,\theta ,\phi )$ is defined as

$J=\det \left({\frac {\partial (x,y,z)}{\partial (\psi ,\theta ,\phi )}}\right)={\frac {\partial \mathbf {x} }{\partial {\psi }}}\cdot {\frac {\partial \mathbf {x} }{\partial {\theta }}}\times {\frac {\partial \mathbf {x} }{\partial {\phi }}}$

and that of the inverse transformation

$J^{-1}=\det \left({\frac {\partial (\psi ,\theta ,\phi )}{\partial (x,y,z)}}\right)=\nabla {\psi }\cdot \nabla {\theta }\times \nabla {\phi }$

It can be seen that $g\equiv \det(g_{ij})=J^{2}\Rightarrow J={\sqrt {g}}$

Magnetic

All of the above coordinate systems are fixed and axisymmetric (except the Cartesian system). By contrast, 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.

These two coordinate systems are related.