Continuum Mechanics and Thermodynamics

, Volume 30, Issue 4, pp 731–774

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

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

## 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

## 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

$$K^{ikm}_{jln}$$

$$L^{ikm}_{jln}$$

Elsasser integral

## Abbreviations

w(ijk)

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

ODE

Ordinary differential equation

PDE

Partial differential equation

ref.

Reference scale

## References

1. 1.
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Applied Mathematics Series, vol. 55, 10th edn. National Bureau of Standards (1972). http://people.math.sfu.ca/~cbm/aands/
2. 2.
Arfken, G.B., Weber, H.J.: Mathematical Methods for Physicists, 6th edn. Elsevier Academic Press, Amsterdam (2005)
3. 3.
Bronstein, I.N., Semendjajew, K.A., Musiol, G., Mühlig, H.: Handbook of Mathematics, 6th edn. Springer, Berlin (2015)
4. 4.
Bullard, E., Gellman, H.: Homogeneous dynamos and terrestrial magnetism. Philos. Trans. R. Soc. A 247(928), 213–278 (1954)
5. 5.
Collatz, L.: Eigenwertaufgaben mit technischen Anwendungen, 2nd edn. Akademische Verlagsgesellschaft, Leipzig (1963)
6. 6.
Dudley, M.L., James, R.W.: Time-dependent kinematic dynamos with stationary flows. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 425, 407–429 (1989)
7. 7.
Dziubek, A.: Equations for two-phase flows: a primer. Meccanica 47(8), 1819–1836 (2012).
8. 8.
Elsasser, W.M.: On the origin of the earth’s magnetic field. Phys. Rev. 55, 489–498 (1939).
9. 9.
Elsasser, W.M.: Induction effects in terrestrial magnetism part I: theory. Phys. Rev. 69(3–4), 106 (1946a).
10. 10.
Elsasser, W.M.: Induction effects in terrestrial magnetism part II: the secular variation. Phys. Rev. 70(3–4), 202 (1946b)
11. 11.
Elsasser, W.M.: Induction effects in terrestrial magnetism: part III. Electric modes. Phys. Rev. 72(9), 821 (1947)
12. 12.
Eringen, A.C., Maugin, G.A.: Electrodynamics of Continua I, 1st edn. Springer, Berlin (1990).
13. 13.
Gauß, C.F.: Allgemeine Theorie des Erdmagnetismus. In: Werke, Fünfter Band, Mathematische Physik, vol. 5. Dieterich, pp. 121–193 (1838)Google Scholar
14. 14.
Glatzmaier, G.A., Roberts, P.H.: A three-dimensional self-consistent computer simulation of a geomagnetic field reversal. Nature 377, 203–208 (1995).
15. 15.
Gubbins, D.: Numerical solutions of the kinematic dynamo problem. Philos. Trans. R. Soc. A 274(1241), 493–521 (1973).
16. 16.
Gubbins, D., Barber, C.N., Gibbons, S., Love, J.J.: Kinematic dynamo action in a sphere. I. Effects of differential rotation and meridional circulation on solutions with axial dipole symmetry. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 456(1998), 1333–1353 (2000a)
17. 17.
Hansen, W.W.: A new type of expansion in radiation problems. Phys. Rev. 47, 139–143 (1935).
18. 18.
Hutter, K., Jöhnk, K.: Continuum Methods of Physical Modeling. Springer, Berlin (2004).
19. 19.
Hutter, K., van de Ven, A.A.F., Ursescu, A.: Electromagnetic Field Matter Interactions in Thermoelastic Solids and Viscous Fluids. Lecture Notes in Physics, vol. 710. Springer, Berlin (2006)Google Scholar
20. 20.
Itskov, M.: Tensor Algebra and Tensor Analysis for Engineers. Springer, Berlin (2009).
21. 21.
Jackson, J.D.: Classical Electrodynamics, 3rd edn. Wiley, New York (1998)
22. 22.
Kovetz, A.: Electromagnetic Theory. Oxford University Press, Oxford (2000)
23. 23.
Kumar, S., Roberts, P.H.: A three-dimensional kinematic dynamo. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 344(1637), 235–258 (1975).
24. 24.
Larmor, J.: How could a rotating body such as the sun become a magnet? Report of the British Association for the Advancement of Science 87th Meeting, pp. 159–160 (1919)Google Scholar
25. 25.
Merrill, M.W., McElhinny, M.W., McFadden, P.L.: The Magnetic Field of the Earth: Paleomagnetism, the Core, and the Deep Mantle, vol. 63, 1st edn. Academic Press, Boca Raton (1998)Google Scholar
26. 26.
Müller, I.: Thermodynamics, 1st edn. Pitman Publishing, Boston (1985)
27. 27.
Müller, W.H.: An expedition to continuum theory. In: Gladwell, G.M.L. (ed.) Solid Mechanics and its Applications, vol. 210. Springer, Berlin (2014). Google Scholar
28. 28.
Roberts, P.H.: Kinematic dynamo models. Philos. Trans. R. Soc. A 272(1230), 663–698 (1972).
29. 29.
Roberts, P.H.: Theory of the geodynamo (chap. 3). In: Schubert, G., Olson, P. (eds.) Core Dynamics: Treatise on Geophysics, vol. 8, pp. 67–105. Elsevier, Amsterdam (2009)Google Scholar
30. 30.
Simpson, J., Weiner, E. (eds.): The Oxford English dictionary, 2nd edn. Oxford University Press, Oxford (1989)Google Scholar
31. 31.
Slattery, J.C., Sagis, L., Oh, E.S.: Interfacial Transport Phenomena, 2nd edn. Springer, Berlin (2007)
32. 32.
Stratton, J.A.: Electromagnetic Theory. The IEEE Press Series on Electromagnetic Wave Theory. McGraw-Hill, New York (1941)Google Scholar
33. 33.
Winch, D., James, R.: Computations with spherical harmonics and Fourier series in geomagnetism. In: Bolt, B.A. (ed.) Geophysics: Methods in Computational Physics: Advances in Research and Applications, vol. 13, pp. 93–132. Elsevier, Amsterdam (1973). Google Scholar
34. 34.
Wolfram Research, Inc.: Mathematica (Version 10.1). Champaign, Illinois (2015)Google Scholar

© 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