Toroidal and poloidal

This article is about toroidal and poloidal coordinates. For toroidal and poloidal vectorfields, see Poloidal–toroidal decomposition.
A diagram depicting the poloidal (\theta) direction, represented by the red arrow, and the toroidal (\zeta or \phi) direction, represented by the blue arrow.

The earliest use of these terms cited by the Oxford English Dictionary (OED) is by Walter M. Elsasser (1946) in the context of the generation of the Earth's magnetic field by currents in the core, with "toroidal" being parallel to lines of latitude and "poloidal" being in the direction of the magnetic field (i.e. towards the poles).

The OED also records the later usage of these terms in the context of toroidally confined plasmas, as encountered in magnetic confinement fusion. In the plasma context, the toroidal direction is the long way around the torus, the corresponding coordinate being denoted by z in the slab approximation or \zeta or \phi in magnetic coordinates; the poloidal direction is the short way around the torus, the corresponding coordinate being denoted by y in the slab approximation or \theta in magnetic coordinates. (The third direction, normal to the magnetic surfaces, is often called the "radial direction", denoted by x in the slab approximation and variously \psi, \chi, r, \rho, or s in magnetic coordinates.)

Toroidal and poloidal coordinates

As a simple example from the physics of magnetically confined plasmas, consider an axisymmetric system with circular, concentric magnetic flux surfaces of radius r (a crude approximation to the magnetic field geometry in an early Tokamak but topologically equivalent to any toroidal magnetic confinement system with nested flux surfaces) and denote the toroidal angle by \zeta and the poloidal angle by \theta. Then the Toroidal/Poloidal coordinate system relates to standard Cartesian Coordinates by these transformation rules:

x = (R_0 +r \cos \theta) \cos\zeta \,
y = s_\zeta (R_0 + r \cos \theta) \sin\zeta \,
z = s_\theta r \sin \theta. \,

where s_\theta = \pm 1, s_\zeta = \pm 1.

The natural choice geometrically is to take s_\theta = s_\zeta = +1, giving the toroidal and poloidal directions shown by the arrows in the figure above, but this makes r,\theta,\zeta a left-handed curvilinear coordinate system. As it is usually assumed in setting up flux coordinates for describing magnetically confined plasmas that the set r,\theta,\zeta forms a right-handed coordinate system, \nabla r\cdot\nabla\theta\times\nabla\zeta > 0, we must either reverse the poloidal direction by taking s_\theta = -1, s_\zeta = +1, or reverse the toroidal direction by taking s_\theta = +1, s_\zeta = -1. Both choices are used in the literature.

Kinematics in toroidal and poloidal coordinates

To study single particle motion in toroidally confined plasma devices, velocity and acceleration vectors must be known. Considering the natural choice s_\theta = s_\zeta = +1, the unit vectors of toroidal and poloidal coordinates system \left(r,\theta,\zeta\right) can be expressed as:

\mathbf{e}_r = \begin{pmatrix} \cos\theta \cos\zeta \\ \cos\theta \sin\zeta \\ \sin\theta \end{pmatrix} \quad \mathbf{e}_\theta = \begin{pmatrix} -\sin\theta \cos\zeta \\ -\sin\theta \sin\zeta \\ \cos\theta \end{pmatrix} \quad \mathbf{e}_\zeta = \begin{pmatrix} -\sin\zeta \\ \cos\zeta \\ 0 \end{pmatrix}

according to Cartesian coordinates. The position vector is expressed as:

\mathbf{r} = \left( r + R_0 \cos\theta \right) \mathbf{e}_r - R_0 \sin\theta \mathbf{e}_\theta

The velocity vector is then given by:

\mathbf{\dot{r}} = \dot{r} \mathbf{e}_r + r\dot{\theta} \mathbf{e}_\theta + \dot{\zeta} \left( R_0 + r \cos\theta \right) \mathbf{e}_\zeta

and the acceleration vector is:

\mathbf{\ddot{r}} = \left( \ddot{r} - r \dot{\theta}^2 - r \dot{\zeta}^2 \cos^2\theta - R_0 \dot{\zeta}^2 \cos\theta \right) \mathbf{e}_r

+ \left( 2\dot{r}\dot{\theta} + r\ddot{\theta} + r\dot{\zeta}^2\cos\theta\sin\theta + R_0 \dot{\zeta}^2 \sin\theta \right) \mathbf{e}_\theta

+ \left( 2 \dot{r}\dot{\zeta}\cos\theta - 2 r \dot{\theta}\dot{\zeta} \sin\theta + \ddot{\zeta} \left( R_0 + r\cos\theta \right) \right) \mathbf{e}_\zeta

See also

Look up toroidal or poloidal in Wiktionary, the free dictionary.

References

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