Toroidal and poloidal

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 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 ζ {displaystyle zeta } or ϕ {displaystyle 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 θ {displaystyle heta } 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 ψ {displaystyle psi } , χ {displaystyle chi } , r, ρ {displaystyle ho } , or s in magnetic 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 {displaystyle r} (a crude approximation to the magnetic field geometry in an early but topologically equivalent to any toroidal magnetic confinement system with nested flux surfaces) and denote the toroidal angle by ζ {displaystyle zeta } and the poloidal angle by θ {displaystyle heta } .Then the Toroidal/Poloidal coordinate system relates to standard Cartesian Coordinates by these transformation rules: where s θ = ± 1 , s ζ = ± 1 {displaystyle s_{ heta }=pm 1,s_{zeta }=pm 1} . The natural choice geometrically is to take s θ = s ζ = + 1 {displaystyle s_{ heta }=s_{zeta }=+1} , giving the toroidal and poloidal directions shown by the arrows in the figure above, but this makes r , θ , ζ {displaystyle r, heta ,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 , θ , ζ {displaystyle r, heta ,zeta } forms a right-handed coordinate system, ∇ r ⋅ ∇ θ × ∇ ζ > 0 {displaystyle abla rcdot abla heta imes abla zeta >0} , we must either reverse the poloidal direction by taking s θ = − 1 , s ζ = + 1 {displaystyle s_{ heta }=-1,s_{zeta }=+1} , or reverse the toroidal direction by taking s θ = + 1 , s ζ = − 1 {displaystyle s_{ heta }=+1,s_{zeta }=-1} . Both choices are used in the literature. To study single particle motion in toroidally confined plasma devices, velocity and acceleration vectors must be known. Considering the natural choice s θ = s ζ = + 1 {displaystyle s_{ heta }=s_{zeta }=+1} , the unit vectors of toroidal and poloidal coordinates system ( r , θ , ζ ) {displaystyle left(r, heta ,zeta ight)} can be expressed as: