In
vector calculus
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a ...
, a topic in pure and applied
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a poloidal–toroidal decomposition is a restricted form of the
Helmholtz decomposition
In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector fields can be resolved into the sum of an irrotational ( curl-free) vector field and a sole ...
. It is often used in the
spherical coordinates
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are
* the radial distance along the line connecting the point to a fixed point ...
analysis of
solenoidal vector field
In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field:
\nabla \cdot \mathbf ...
s, for example,
magnetic fields
A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
and
incompressible fluids.
Definition
For a three-dimensional
vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
F with zero
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...
:
this
can be expressed as the sum of a toroidal field
and poloidal vector field
:
where
is a radial vector in
spherical coordinates
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are
* the radial distance along the line connecting the point to a fixed point ...
. The toroidal field is obtained from a
scalar field
In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...
,
, as the following
curl
cURL (pronounced like "curl", ) is a free and open source computer program for transferring data to and from Internet servers. It can download a URL from a web server over HTTP, and supports a variety of other network protocols, URI scheme ...
,
:
and the poloidal field is derived from another scalar field
, as a twice-iterated curl,
:
This
decomposition
Decomposition is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. The process is a part of the nutrient cycle and is ess ...
is symmetric in that the curl of a toroidal field is poloidal, and the curl of a poloidal field is toroidal, known as
Chandrasekhar–Kendall function
Chandrasekhar–Kendall functions are the eigenfunctions of the curl operator derived by Subrahmanyan Chandrasekhar and P. C. Kendall in 1957 while attempting to solve the force-free magnetic fields. The functions were independently derived by both ...
.
Geometry
A toroidal vector field is tangential to spheres around the origin,
:
while the curl of a poloidal field is tangential to those spheres
:
The poloidal–toroidal decomposition is unique if it is required that the average of the scalar fields Ψ and Φ vanishes on every sphere of radius ''r''.
Cartesian decomposition
A poloidal–toroidal decomposition also exists in
Cartesian coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
, but a mean-field flow has to be included in this case. For example, every solenoidal vector field can be written as
:
where
denote the unit vectors in the coordinate directions.
See also
*
Toroidal and poloidal
The terms toroidal and poloidal refer to directions relative to a torus of reference. They describe a three-dimensional coordinate system in which the poloidal direction follows a small circular ring around the surface, while the toroidal direct ...
*
Chandrasekhar–Kendall function
Chandrasekhar–Kendall functions are the eigenfunctions of the curl operator derived by Subrahmanyan Chandrasekhar and P. C. Kendall in 1957 while attempting to solve the force-free magnetic fields. The functions were independently derived by both ...
Notes
References
''Hydrodynamic and hydromagnetic stability'' Chandrasekhar, Subrahmanyan; International Series of Monographs on Physics, Oxford: Clarendon, 1961, p. 622.
Decomposition of solenoidal fields into poloidal fields, toroidal fields and the mean flow. Applications to the boussinesq-equations Schmitt, B. J. and von Wahl, W; in ''The Navier–Stokes Equations II — Theory and Numerical Methods'', pp. 291–305; Lecture Notes in Mathematics, Springer Berlin/ Heidelberg, Vol. 1530/ 1992.
Anelastic Magnetohydrodynamic Equations for Modeling Solar and Stellar Convection Zones Lantz, S. R. and Fan, Y.; The Astrophysical Journal Supplement Series, Volume 121, Issue 1, Mar. 1999, pp. 247–264.
* Plane poloidal-toroidal decomposition of doubly periodic vector fields
an
G. D. McBain
ANZIAM J.
* .
* .
* .
{{DEFAULTSORT:Poloidal-toroidal decomposition
Vector calculus