Poloidal–toroidal decomposition

The Toroidal - poloidal decomposition divides a divergence- free vector field, such as the velocity field of an incompressible flow or a magnetic field, such as the earth's magnetic field into two parts, each depending only on a unique scalar field. The decomposition is based on a vector potential A of the vector field F. It can be in a radial Ar and split a tangential component at.

It provides a convenient for the geometry of the problem unit vector dar. For spherical geometry provides the unit vector in the radial direction. By suitable choice of the poloidal vector potential A2 it can be derived from a divergence field without the magnetic field is changed.

The vector field F is thus the shape

Toroidal field

The toroidal field resulting from the rotation of the vector potential:

Multiplying out the curl in spherical coordinates one sees that the field has no radial shares:

The field is divergence-free on the spherical surface. Clearly this means that the field has no sources and sinks. On the other hand, there is no radial flow area. A toroidal magnetic field may thus be different from zero only within matter. This means in geophysics that generated by the Earth's core toroidal fields at the surface can not be measured.

The term derives from the toroidal donut shape of these fields into rotationally symmetric systems. For the description of general toroidal fields it is therefore misleading.

Poloidal field

The poloidal field resulting from the rotation of the vector potential.

It has both radial and tangential components. The term derives from the dipole form from the Earth's magnetic field. Since the toroidal magnetic field occurs only in matter, the poloidal field completely describes the geomagnetic field above the earth's surface.

Examples

Central dipole

The field of a magnetic dipole at the origin has a vector potential

Which is immediately recognizable as a poloidal field. The associated potential χ results from the multipole expansion to

If dipoles are located outside the coordinate origin, the field also contains other multipole orders.

Radial dipole

When the above-described radial dipole magnetic moment along the shifts (that is, the magnetic moment in the radial direction ), then the vector potential changes in

The associated potential is χ

Tangential dipole

That means of radial dipole moments only a poloidal field can be generated. For the generation of toroidal components tangential magnetic moments must be involved.

The first term represents the poloidal part of the field, the second the toroidal component.

Sets to the magnetic moment on the Z-axis and the dipole itself to the X axis, we obtain

Radial and tangential dipoles can serve as basis for the assembly of the magnetic field. That together with spatial rotations of the basic elements can be created, each magnetic configuration. If, therefore, for both dipole potentials ψ and χ determined, one can calculate for each configuration, the total potential.

Application

• Toroidal fields are used in nuclear fusion devices for the confinement of the fusion plasma.
• In the aquatic or Helio dynamics poloidal and toroidal fields find use in the calculation of the magnetic field of the Earth's core and the sun.
• In the numerical solution of the MHD equations in spherical limits to spectral methods offer based on poloidal and toroidal functions.