Toroidal-poloidal decomposition

The toroidal-poloidal decomposition is a decomposition of a three-dimensional, divergence-free vector field , e.g. B. the earth's magnetic field , into two parts, each of which only depends on a unique scalar field .

Disassembly

In this diagram, the poloidal direction ( ) is shown in red. The toroidal direction ( or ) is shown in blue. ${\displaystyle \theta }$ ${\displaystyle \zeta }$ ${\displaystyle \phi }$

The decomposition is based on a vector potential of the three-dimensional divergence-free vector field . Divergence-free means that it is, or clearly, that the field has no sources and sinks . An example of a divergence-free field is the magnetic flux density . The vector potential of the vector field is defined in such a way that the rotation of the potential results in the vector field: ${\displaystyle {\vec {A}}}$ ${\displaystyle {\vec {F}}}$ ${\displaystyle \operatorname {div} {\vec {F}}=0}$

${\displaystyle {\vec {F}}=\operatorname {rot} {\vec {A}}}$

The potential can be divided into a radial and a tangential component . ${\displaystyle {\vec {A}}_{r}}$ ${\displaystyle {\vec {A}}_{t}}$

${\displaystyle {\vec {A}}={\vec {A}}_{r}+{\vec {A}}_{t}=\Psi {\vec {e}}_{r}+{\vec {e}}_{r}\times {\vec {A}}_{2}}$

It represents a unit vector suitable for the geometry of the problem. The unit vector in the radial direction is suitable for spherical geometry. By suitable choice of the poloidal vector potential , it can be derived from a divergence field without changing the magnetic field. ${\displaystyle {\vec {e}}_{r}}$ ${\displaystyle {\vec {A}}_{2}}$

${\displaystyle {\vec {A}}_{2}=-\nabla \chi }$

This gives the vector field its shape ${\displaystyle {\vec {F}}}$

${\displaystyle {\vec {F}}=\operatorname {rot} (\Psi {\vec {e}}_{r})+\operatorname {rot} \operatorname {rot} (\chi {\vec {e}}_{r})}$

and can thus be described by the two scalar potentials and . ${\displaystyle \Psi }$ ${\displaystyle \chi }$

Toroidal field

The toroidal field results from the rotation of the vector potential : ${\displaystyle T}$ ${\displaystyle {\vec {A}}_{r}}$

${\displaystyle T=\operatorname {rot} {\vec {A}}_{r}=\operatorname {rot} \Psi {\vec {e}}_{r}.}$

By multiplying the rotation in spherical coordinates you can see that the field has no radial components:

${\displaystyle T=\operatorname {rot} {\vec {A}}_{r}={\frac {1}{\sin \theta }}{\frac {\partial \Psi }{\partial \varphi }}{\vec {e}}_{\theta }-{\frac {\partial \Psi }{\partial \theta }}{\vec {e}}_{\varphi }}$

The field is divergence-free on the spherical surface. There is no radial field component. Toroidal magnetic fields can be driven by poloidal currents and vice versa. In geophysics, this means that the toroidal magnetic fields generated by eddy currents flowing in the earth and oceans are zero on the earth's surface.

The term toroidal is derived from the toroidal shape of these fields in rotationally symmetrical systems - the field lines run in circles. It is therefore misleading for the description of general toroidal fields.

Poloidal field

The poloidal field arises from the rotation of the vector potential . ${\displaystyle P}$ ${\displaystyle {\vec {A}}_{t}}$

${\displaystyle P=\operatorname {rot} {\vec {A}}_{t}=-\operatorname {rot} ({\vec {e}}_{r}\times \nabla \chi )}$

It has both radial and tangential components. The term is derived from the dipole shape of the earth's magnetic field. Since the earth's toroidal magnetic field only occurs in it, the poloidal field completely describes the earth's magnetic field above the earth's surface.

Examples

Central dipole

The field of a magnetic dipole with a dipole moment in the coordinate origin has a vector potential ${\displaystyle {\vec {m}}}$

${\displaystyle {\vec {A}}({\vec {r}})={\frac {\mu _{0}}{4\pi }}\,{\frac {{\vec {m}}\times {\vec {r}}}{r^{3}}}}$ ,

which is immediately recognizable as a poloidal field. Here is the magnetic field constant . The associated potential results from the multipole development to ${\displaystyle \mu _{0}}$ ${\displaystyle \chi }$

${\displaystyle \chi ({\vec {r}})=-{\frac {\mu _{0}}{4\pi }}\,{\frac {{\vec {m}}\cdot {\vec {r}}}{r^{3}}}}$.

If dipoles are outside of the origin, the field also contains multipole moments of other orders.

Radial dipole

If the radial dipole described above is shifted along the magnetic moment (i.e. the magnetic moment is in the radial direction), the vector potential changes in

${\displaystyle {\vec {A}}_{t}({\vec {r}})={\frac {\mu _{0}}{4\pi }}\,{\frac {{\vec {m}}\times \left({\vec {r}}-{\vec {c}}_{m}\right)}{\left|{\vec {r}}-{\vec {c}}_{m}\right|^{3}}}={\frac {\mu _{0}}{4\pi }}\,{\frac {{\vec {m}}\times {\vec {r}}}{\left|{\vec {r}}-{\vec {c}}_{m}\right|^{3}}}}$.

The associated potential is ${\displaystyle \chi }$

${\displaystyle \chi ({\vec {r}})=-{\frac {\mu _{0}}{4\pi }}\,{\frac {{\vec {m}}\left({\vec {r}}-{\vec {c}}_{m}\right)}{\left|{\vec {r}}-{\vec {c}}_{m}\right|^{3}}}}$.

Tangential dipole

This means that only a poloidal field can be generated from radial dipole moments. Tangential magnetic moments must be involved in generating toroidal components.

${\displaystyle {\vec {A}}({\vec {r}})={\frac {\mu _{0}}{4\pi }}\,{\frac {{\vec {m}}\times \left({\vec {r}}-{\vec {c}}_{\perp m}\right)}{\left|{\vec {r}}-{\vec {c}}_{\perp m}\right|^{3}}}={\frac {\mu _{0}}{4\pi }}\,{\frac {{\vec {m}}\times {\vec {r}}-\left({\vec {m}}\times {\vec {c}}_{\perp m}-\left(\left({\vec {m}}\times {\vec {c_{\perp }m}}\right){\vec {e_{r}}}\right){\vec {e_{r}}}\right)}{\left|{\vec {r}}-{\vec {c_{\perp m}}}\right|^{3}}}-{\frac {\mu _{0}}{4\pi }}{\frac {\left(\left({\vec {m}}\times {\vec {c_{\perp m}}}\right){\vec {e_{r}}}\right){\vec {e}}_{r}}{\left|{\vec {r}}-{\vec {c}}_{\perp m}\right|^{3}}}}$.

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

${\displaystyle {\vec {A}}_{t}({\vec {r}})=-{\frac {\mu _{0}}{4\pi }}{\frac {\left(\left({\vec {m}}\times {\vec {c}}_{\perp m}\right){\vec {e_{r}}}\right){\vec {e}}_{r}}{\left|{\vec {r}}-{\vec {c}}_{\perp m}\right|^{3}}}}$

If you put the magnetic moment on the z -axis and the dipole itself on the x -axis, you get

${\displaystyle {\vec {B}}_{t}=\operatorname {rot} {\vec {A}}_{r}=\operatorname {rot} \left(\Psi {\vec {e}}_{r}\right)={\frac {\mu _{0}m\,c_{\perp m}}{4\pi }}\operatorname {rot} \left({\frac {y}{r^{2}\left|{\vec {r}}-{\vec {c_{\perp m}}}\right|^{3}}}{\vec {r}}\right)}$

Radial and tangential dipoles can serve as the basis for building up the magnetic field. I.e. Together with spatial rotations of the basic elements, any magnetic configuration can be created. So if you determine potentials and for both dipoles , you can calculate the total potentials for each configuration. ${\displaystyle \psi }$ ${\displaystyle \chi }$

application

• Toroidal magnetic fields are used in nuclear fusion systems to contain the fusion plasma, see fusion by means of magnetic containment or tokamak .
• In geo- or heliodynamics and in real terms in earth and sun , poloidal and toroidal magnetic fields occur.
• For the numerical solution of the MHD equations with spherical limits, spectral methods based on poloidal and toroidal functions can be used.
• Transmitter masts transmit toroidal magnetic fields and poloidal electric fields in the near field, while ferrite antennas ( magnetic antennas ) send out poloidal magnetic fields and toroidal electric fields.

literature

• Eric W. Weisstein : Toroidal Field . In: MathWorld (English).
• Eric W. Weisstein : Poloidal Field . In: MathWorld (English).
• Eric W. Weisstein : Divergenceless Field . In: MathWorld (English).
• toroidal field (english)
• Exercise sheet of the LMU Munich (PDF; 847 kB)

Individual evidence

1. ^ DD Joseph: Stability of Fluid Motions I , 1st edition. Springer, 1976, ISBN 3-642-80993-6 .
2. ↑ Jan Dostal: Predication of oceanic tidal signals in satellite observations of the earth's magnetic field (PDF; 10.7 MB) p. 21ff ISSN  1610-0956