The kinematic dynamo problem, part I: analytical treatment with the Bullard–Gellman formalism

  • Sebastian Glane
  • Felix A. Reich
  • Wolfgang H. Müller
Original Article
  • 10 Downloads

Abstract

This paper is dedicated to the description of kinematic dynamo action in a sphere and its analytical treatment with the BullardGellman formalism. One goal of dynamo theory is to answer the question: Can magnetic fields of stellar objects be generated or sustained due to (fluid) motion in the interior? Bullard and Gellman were among the first to study this question, leading the way for many subsequent studies, cf. Bullard (Philos Trans R Soc A 247(928):213–278, 1954). In their publication the differential equations resulting from a toroidal–poloidal decomposition of the velocity and magnetic field are stated without an in-depth discussion of the employed methods and computation steps. This study derives the necessary formalism in a compact and concise manner by using an operator-based approach. The focus lies on the mathematical steps and necessary properties of the considered formalism. Prior to that a derivation of the induction equation is presented based on rational continuum electrodynamics. As an example of the formalism the decay of two magnetic fields is analyzed.

Keywords

Magnetohydrodynamics Toroidal–poloidal decomposition Kinematic dynamo Induction equation 

List of symbols

General quantities

c

Speed of light in vacuum

\(\mathcal {B}_R\)

Spherical material body of radius R

\(\varvec{v}\)

Velocity

\(\varvec{n}\)

Surface unit normal vector

\(\varvec{e}\)

Unit normal vector of a singular surface or material interface

\(w_\perp \)

Normal component of the velocity of a singular surface or material interface

\(\,\mathrm {d}A\)

Area element

\(\,\mathrm {d}\varOmega \)

\(r^{-2}\,\mathrm {d}A\). Normalized area element

\(\,\mathrm {d}V\)

Volume element

\(\mathcal {O}(\bullet )\)

Order of magnitude of the considered field

\(\mathbf {A}\)

Dimension matrix

\(\mathbf {k}_i\)

Null space vectors of \(\mathbf {A}\)

\(\mathrm {\Pi }_i\)

Dimensionless products

\( Re _\text {mag.}\)

Magnetic Reynolds number

\(\lambda \)

Eigenvalue of the induction equation

Electromagnetic quantities

\(\varvec{B}\)

Magnetic flux density

\(\varvec{E}\)

Electric field

\(\varvec{H}\)

Total current potential

\(\varvec{\mathfrak {H}}\)

Free current potential

\(\varvec{M}\)

Magnetization or bond current potential

\(\varvec{D}\)

Total current potential

\(\varvec{\mathfrak {D}}\)

Free current potential

\(\varvec{P}\)

Magnetization or bond charge potential

q, \(q^{\text {f}}\), \(q^{\text {r}}\)

Total, free and bond charge density

\(\varvec{J}\), \(\varvec{J}^{\text {f}}\), \(\varvec{J}^{\text {r}}\)

Total, free and bond current density

\(V_\mathrm {m}\)

Magnetic scalar potential

\(\sigma \)

Electric conductivity

\(\varepsilon _0\), \(\mu _0\)

Vacuum permittivity and permeability

Special functions

\(Y^m_n\)

Complex-valued spherical harmonics

\(C^m_n\)

Real-valued cosine spherical harmonics

\(S^m_n\)

Real-valued sine spherical harmonics

\(P^m_n\)

Associated Legendre polynomials

\(J_n\)

Bessel functions of 1\(\mathrm {st}\) kind

Differential operators

\(\nabla _S\)

\((\varvec{1}-\varvec{n}\otimes \varvec{n})\cdot \nabla \) Surface del-operator

\(\nabla _{\theta ,\varphi }\)

\(r\nabla _S=\varvec{e}_\theta \frac{\partial }{\partial \theta }+\varvec{e}_\varphi \frac{1}{\sin (\theta )}\frac{\partial }{\partial \varphi }\). Angular part of the spherical surface del-operator

\(\Delta _S\)

\(\nabla _S\cdot \nabla _S\). Surface Laplace-operator

\(\Delta _{\theta ,\varphi }\)

\(r^2\Delta _S=\frac{1}{\sin (\theta )}\frac{\partial }{\partial \theta }\big (\sin (\theta )\frac{\partial }{\partial \theta }\big )+\frac{1}{\sin ^2(\theta )}\frac{\partial ^2 }{\partial \varphi ^2}\). Angular part of the spherical surface Laplace-operator

\(\varvec{\mathcal {D}}_{\theta ,\varphi }\)

\(\varvec{e}_\theta \frac{1}{\sin (\theta )}\frac{\partial }{\partial \varphi }-\varvec{e}_\varphi \frac{\partial }{\partial \theta }\). Differential operator related to toroidal vector fields

\( \mathcal {D}_{\theta ,\varphi }[f,g] \)

\(\nabla _{\theta ,\varphi }f\cdot \varvec{\mathcal {D}}_{\theta ,\varphi }[g] \). Bilinear antisymmetric differential operator

\(D_{r}^{1}{[}{f}{]}\)

\(\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}(r f(r))\)

Toroidal and poloidal fields—Stratton expansion

\(\varvec{T}^{i}_{j}\)

Complex-valued toroidal vector fields

\(\varvec{P}^{i}_{j}\)

Complex-valued poloidal vector fields

\(t^i_j(r)\)

Radial functions of \(\varvec{T}^{i}_{j}\)

\(p^i_j(r)\)

Radial functions of \(\varvec{P}^{i}_{j}\)

\(\varvec{U}^{k}_{l}\)

Complex-valued toroidal velocity fields

\(\varvec{Q}^{k}_{l}\)

Complex-valued poloidal velocity fields

\(u^k_l(r)\)

Radial functions of \(\varvec{U}^{k}_{l}\)

\(q^k_l(r)\)

Radial functions of \(\varvec{Q}^{k}_{l}\)

\(\varvec{\mathcal {T}}^{m}_{n}\)

Complex-valued toroidal projectors

\(\varvec{\mathcal {P}}^{m}_{n}\)

Complex-valued poloidal projectors

Toroidal and poloidal fields—Bullard and Gellman expansion

\(\tilde{\varvec{T}}\phantom {\varvec{T}}^{i}_{j}\)

Complex-valued toroidal vector fields

\(\tilde{\varvec{P}}\phantom {\varvec{P}}^{i}_{j}\)

Complex-valued poloidal vector fields

\(\tilde{t}^i_j(r)\)

Radial functions of \(\tilde{\varvec{T}}\phantom {\varvec{T}}^{i}_{j}\)

\(\tilde{p}^i_j(r)\)

Radial functions of \(\tilde{\varvec{P}}\phantom {\varvec{P}}^{i}_{j}\)

\(\tilde{\varvec{U}}\phantom {\varvec{U}}^{k}_{l}\)

Complex-valued toroidal velocity fields

\(\tilde{\varvec{Q}}\phantom {\varvec{Q}}^{k}_{l}\)

Complex-valued poloidal velocity fields

\(\tilde{\varvec{\mathcal {T}}}\phantom {\varvec{\mathcal {T}}}^{m}_{n}\)

Complex-valued toroidal projectors

\(\tilde{\varvec{\mathcal {P}}}\phantom {\varvec{\mathcal {P}}}^{m}_{n}\)

Complex-valued poloidal projectors

Dynamo integrals

\(K^{ikm}_{jln}\)

AdamsGaunt integral

\(L^{ikm}_{jln}\)

Elsasser integral

Abbreviations

w(ijk)

\(i(i+1)+j(j+1)-k(k+1)\)

ODE

Ordinary differential equation

PDE

Partial differential equation

Semantic subscripts

ref.

Reference scale

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Sebastian Glane
    • 1
  • Felix A. Reich
    • 1
  • Wolfgang H. Müller
    • 1
  1. 1.Institut für Mechanik, Kontinuumsmechanik und MaterialtheorieTechnische Universität BerlinBerlinGermany

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