The terms toroidal and poloidal refer to directions relative to a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
of reference. They describe a three-dimensional
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
in which the poloidal direction follows a small circular ring around the surface, while the toroidal direction follows a large circular ring around the torus, encircling the central void.
The earliest use of these terms cited by the
Oxford English Dictionary
The ''Oxford English Dictionary'' (''OED'') is the first and foundational historical dictionary of the English language, published by Oxford University Press (OUP). It traces the historical development of the English language, providing a com ...
is by
Walter M. Elsasser (1946) in the context of the generation of the
Earth's magnetic field
Earth's magnetic field, also known as the geomagnetic field, is the magnetic field that extends from Earth's interior out into space, where it interacts with the solar wind, a stream of charged particles emanating from the Sun. The magnetic f ...
by currents in the core, with "toroidal" being parallel to lines of
latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
and "poloidal" being in the direction of the magnetic field (i.e. towards the
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...
).
The OED also records the later usage of these terms in the context of toroidally confined plasmas, as encountered in
magnetic confinement fusion
Magnetic confinement fusion is an approach to generate thermonuclear fusion power that uses magnetic fields to confine fusion fuel in the form of a plasma. Magnetic confinement is one of two major branches of fusion energy research, along with i ...
. In the plasma context, the toroidal direction is the long way around the torus, the corresponding coordinate being denoted by in the slab approximation or or in magnetic coordinates; the poloidal direction is the short way around the torus, the corresponding coordinate being denoted by in the slab approximation or in magnetic coordinates. (The third direction, normal to the magnetic surfaces, is often called the "radial direction", denoted by in the slab approximation and variously , , , , or in magnetic 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
(a crude approximation to the magnetic field geometry in an early
tokamak
A tokamak (; russian: токамáк; otk, 𐱃𐰸𐰢𐰴, Toḳamaḳ) is a device which uses a powerful magnetic field to confine plasma in the shape of a torus. The tokamak is one of several types of magnetic confinement devices being d ...
but topologically equivalent to any toroidal magnetic confinement system with nested flux surfaces) and denote the toroidal angle by
and the poloidal angle by
.
Then the Toroidal/Poloidal coordinate system relates to standard
Cartesian Coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
by these transformation rules:
:
:
:
where
.
The natural choice
geometrically is to take
, giving the toroidal and poloidal directions shown by the arrows in the figure above, but this makes
a left-handed curvilinear coordinate system. As it is usually assumed in setting u
''flux coordinates''for describing magnetically confined plasmas that the set
forms a ''right''-handed coordinate system,
, we must either reverse the poloidal direction by taking
, or reverse the toroidal direction by taking
. 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
, the unit vectors of toroidal and poloidal coordinates system
can be expressed as:
:
according to Cartesian coordinates. The position vector is expressed as:
:
The velocity vector is then given by:
:
and the acceleration vector is:
:
See also
*
Toroidal coordinates
Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci F_1 and F_2 in bipolar coordinates ...
*
Torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
*
Zonal and poloidal In magnetic confinement fusion the zonal direction primarily connotes the poloidal direction (i.e. the short way around the torus), the corresponding coordinate being denoted by ''y'' in the slab approximation or ''θ'' in magnetic coordinates. Howe ...
*
Poloidal–toroidal decomposition In vector calculus, a topic in pure and applied mathematics, a poloidal–toroidal decomposition is a restricted form of the Helmholtz decomposition. It is often used in the spherical coordinates analysis of solenoidal vector fields, for example, m ...
*
Zonal flow (plasma) In toroidally confined fusion plasma experiments the term zonal flow means a plasma flow within a magnetic surface primarily in the poloidal direction. This usage is inspired by the analogy between the quasi-two-dimensional nature of large-scale a ...
References
*
*{{cite journal , last=Elsasser , first=W. M. , author-link=Walter M. Elsasser , year=1946 , title=Induction Effects in Terrestrial Magnetism, Part I. Theory , journal=Phys. Rev. , volume=69 , pages=106–116 , url=http://prola.aps.org/abstract/PR/v69/i3-4/p106_1 , access-date=2007-08-10 , doi=10.1103/PhysRev.69.106 , issue=3–4
Coordinate systems