Toroidal and poloidal coordinates

Example

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

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:

: \begin{align} \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 \[5pt] & {} + \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 \[5pt] & {} + \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 \end{align}

References