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

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_p \frac{\sinh \zeta}{\cosh \zeta - \cos \eta}}

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 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 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 (\psi(\mathbf{x}),\theta(\mathbf{x}),\phi(\mathbf{x}))} we can calculate the contravariant basis vectors

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

and the dual covariant basis defined as

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

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

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 \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 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 \mathbf{e}_i = (\mathbf{e}_i\cdot\mathbf{e}_j)\mathbf{e}^j} . The metric tensor is defined as

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 g_{ij} = \mathbf{e}_i\cdot\mathbf{e}_j \; ; \; g^{ij} = \mathbf{e}^i\cdot\mathbf{e}^j \; ; \; g^j_i = \mathbf{e}_i\cdot\mathbf{e}^j = \delta_i^j ~. }

Jacobian

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

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 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 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 g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}}

Flux Coordinates

A flux coordinate set is one that includes a flux surface label as a coordinate. A flux surface label is a function that is constant and single valued on each flux surface. In our naming of the general curvilinear coordinates we have already adopted the usual flux coordinate convention for toroidal equilibrium with nested flux surfaces with 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 \psi} being the flux surface label and 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 \theta, \phi} are 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 2\pi} -periodic poloidal and toroidal-like angles.

Different flux surface labels can be chosen like toroidal or poloidal magnetic fluxes or the volume contained within the flux surface. By single valued we mean to ensure that any flux label $\psi _{1}=f(\psi _{2})$ is a monotonous function of any other flux label 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 \psi_2} , so that the function $f$ is invertible at least in a volume containing the region of interest.

Flux Surface Average

The flux surface average of a function 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 \Phi} is defined as the limit

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 \langle\Phi\rangle = \lim_{\delta V \to 0}\frac{1}{\delta V}\int_{\delta V} \Phi\; dV }

where 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 \delta V} is the volume confined between two flux surfaces. It is therefore a volume average over an infinitesimal spatial region rather than a surface average.

Introducing the differential volume element 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 dV = \sqrt{g} d\psi d\theta d\phi}

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 \langle\Phi\rangle = \lim_{\delta V \to 0} \frac{1}{\delta V}\int_{\delta V} \Phi\; \sqrt{g} d\psi d\theta d\phi = \frac{d\psi}{d V}\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi }

or, noting that 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 \langle 1\rangle = 1} , we have 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 \frac{dV}{d\psi} = \int_0^{2\pi}\int_0^{2\pi} \sqrt{g} d\theta d\phi} and we get to a more practical form of the Flux Surface Average

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 \langle\Phi\rangle = \frac{\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi} {\int_0^{2\pi}\int_0^{2\pi} \sqrt{g} d\theta d\phi} }

Note that 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 dS = |\nabla\psi|\sqrt{g}d\theta d\phi} , so the FSA is a surface integral weighted by 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 |\nabla V|^{-1}}  :

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 \langle\Phi\rangle = \frac{d\psi}{d V}\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi = \frac{d\psi}{d V}\int_{S(\psi)}\frac{\Phi}{|\nabla\psi|}\; dS = \int_{S(\psi)}\frac{\Phi}{|\nabla V|}\; dS }

Applying Gauss' theorem to the definition of FSA we get to the identity

$\langle \nabla \cdot \Gamma \rangle =\lim _{\delta V\to 0}{\frac {1}{\delta V}}\int _{\delta V}\nabla \cdot \Gamma \;dV=\lim _{\delta V\to 0}{\frac {1}{\delta V}}\int _{S(\delta V)}\Gamma \cdot {\frac {\nabla V}{|\nabla V|}}dS=\lim _{\delta V\to 0}{\frac {1}{\delta V}}\int _{0}^{2\pi }\int _{0}^{2\pi }\Gamma \cdot \nabla V\;{\sqrt {g}}d\theta d\phi ={\frac {d}{dV}}\langle \Gamma \cdot \nabla V\rangle ~.$

Useful properties of FSA

Some useful properties of the FSA are

• 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 \langle\nabla\cdot\Gamma\rangle = \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle = \frac{1}{V'}\frac{d}{d\psi}V'\langle\Gamma\cdot\nabla \psi\rangle}
• 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 \int_{V}\nabla\cdot\Gamma\; dV = \langle\Gamma\cdot\nabla V\rangle = V'\langle\Gamma\cdot\nabla \psi\rangle}
• 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 \langle \sqrt{g}^{-1}\rangle = \frac{4\pi^2}{V'} }
• 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 \langle \mathbf{B}\cdot\nabla f \rangle = 0~,\qquad \forall~ \mathrm{single~valued~} f(\mathbf{x}), ~ \mathrm{if}~ \nabla\cdot\mathbf{B} = 0 ~\mathrm{and}~ \nabla V\cdot\mathbf{B} = 0 }
• 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 \langle \mathbf{B}\cdot\nabla \theta\rangle =2\pi\frac{d\Psi_{pol}}{dV} \qquad (\mathrm{Note:}~ \theta(\mathbf{x})~\mathrm{is~not~single~valued}) }
• $\langle \mathbf {B} \cdot \nabla \phi \rangle =2\pi {\frac {d\Psi _{tor}}{dV}}\qquad (\mathrm {Note:} ~\phi (\mathbf {x} )~\mathrm {is~not~single~valued} )$

where 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 V' = \frac{dV}{d\psi}} .

Magnetic field representation in flux coordinates

Contravariant From

Any solenoidal vector field 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 \mathbf{B}} can be written as 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 \mathbf{B} = \nabla\alpha\times\nabla\nu } called its Clebsch representation. For a magnetic field with flux surfaces 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 (\psi = \mathrm{const}\; , \; \nabla\psi\cdot\mathbf{B} = 0)} we can choose, say, $\alpha$ to be the flux surface label 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 \psi}

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 \mathbf{B} = \nabla\psi\times\nabla\nu }

Field lines are then given as the intersection of the constant-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 \psi} and constant-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 \nu} surfaces. This form provides a general expression for 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 \mathbf{B}} in terms of the covariant basis vectors of a flux coordinate system

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 \mathbf{B} = \frac{\partial\nu}{\partial\theta}\nabla\psi\times\nabla\theta + \frac{\partial\nu}{\partial\phi}\nabla\psi\times\nabla\phi = \frac{1}{\sqrt{g}}\frac{\partial\nu}{\partial\theta}\mathbf{e}_\phi -\frac{1}{\sqrt{g}}\frac{\partial\nu}{\partial\phi}\mathbf{e}_\theta = B^\phi\mathbf{e}_\phi + B^\theta\mathbf{e}_\theta~. }

in terms of the function 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 \nu} , sometimes referred to as the magnetic field's stream function.

It is worthwhile to note that the Clebsch form of 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 \mathbf{B} } corresponds to a magnetic vector potential 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 \mathbf{A} = \nu\nabla\psi } (or 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 \mathbf{A} = \psi\nabla\nu } as they differ only by the Gauge transformation 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 \mathbf{A} \to \mathbf{A} - \nabla (\psi\nu)} ).

The general form of the stream function 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 \nu(\psi,\theta,\phi) = \frac{1}{2\pi}(\dot{\Psi}_{tor}\theta - \dot{\Psi}_{pol}\phi) + \tilde{\nu}(\psi,\theta,\phi) }

where 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 \tilde{\nu}} is a differentiable function periodic in the two angles. This general form can be derived by using the fact that 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 \mathbf{B}} is a physical function (hence singe-valued). The specific form for the coefficients in front of the secular terms (i.e. the non-periodic terms) can be obtained from the FSA properties .

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.