ASTR/ATOC-5410: Fluid Instabilities, Waves, and Turbulence

Project description October 24, 2016, Axel Brandenburg

Project on oscillatory α^{2} dynamos

Background

Textbook knowledge (e.g. Parker, 1979) suggests that αΩ dynamos tend to be oscillatory while α^{2} dynamos tend to be non-oscillatory. That this does not have to be true was clear for some time (Baryshnikova & Shukurov, 1987; Rädler & Bräuer, 1987), but only recently it became clear that such dynamos can also have migratory wave solutions similar to what is seen in the Sun (Mitra et al., 2010; Masada & Sano, 2014). The goal of this project is to present a “textbook example” of an analytically solvable α^{2} dynamo in 1-D. Particular aims of this project include a better understanding of

• the excitation condition,

• the migration properties, and the

• dependence on boundary conditions.

Project details

The one-dimensional α^{2} dynamo is governed by the equations

(∂_{t} − ∂_{z}^{2})A_{x} + α∂_{z}A_{y} = 0,
(∂_{t} − ∂_{z}^{2})A_{y} − α∂_{z}A_{x} = 0,  (1)

with four boundary conditions

\begin{matrix}A_{x} = A_{y} = 0 & on z = 0, \\ ∂_{z}A_{x} = ∂_{z}A_{y} = 0 & on z = π/2. \end{matrix}  (2)

1. Show that Equations (1) and (1) can be combined into a fourth order equation that is obeyed both by A_{x} and by A_{y}, so we drop in the following subscripts x and y and write the equation in the form

[(∂_{t} − ∂_{z}^{2})^{2} + α^{2}∂_{z}^{2}] A = 0,  (3)

with four boundary conditions

\begin{matrix}A = (∂_{z}^{2} + α^{2})∂_{z}A = 0 & on z = 0, \\ ∂_{z}A = (∂_{t} − ∂_{z}^{2})A = 0 & on z = π/2. \end{matrix}  (4)

2. Seek the solution in the form

A = ∑_{i=0}^{4} C_{i}e^{q_{i}z}e^{−ωt},  (5)

where the q_{i} obey

[(iω + q_{i}^{2})^{2} + α^{2}q_{i}^{2}] A = 0.  (6)

3. Show that this leads to the dispersion relation

q_{i}^{4} + (2iω + α^{2})q_{i}^{2} − ω^{2} = 0  (7)

with the solutions

q_{i} = ± [−(iω + α^{2}/2) ± (iωα^{2} + α^{4}/4)^{1/2}]^{1/2} ,  (8)

where i = 1, 2, 3, 4 correspond to the signs ++, +−, −+, −− in Equation (8).

4. Argue that the four boundary conditions can be satisfied by solving the following eigenvalue problem MC = 0 with C = (C_{1}, C_{2}, C_{3}, C_{4})^{T} and

M =( \begin{matrix}1 & 1 & 1 & 1 \\ (q_{1}^{2} + α^{2})q_{1} & (q_{2}^{2} + α^{2})q_{2} & (q_{3}^{2} + α^{2})q_{3} & (q_{4}^{2} + α^{2})q_{4} \\ (iω + q_{1}^{2}) eq̃_{1} & (iω + q_{2}^{2}) eq̃_{2} & (iω + q_{3}^{2}) eq̃_{3} & (iω + q_{4}^{2}) eq̃_{4} \\ q_{1}eq̃_{1} & q_{2}eq̃_{2} & q_{3}eq̃_{3} & q_{4}eq̃_{4} \end{matrix} ) .  (9)

where we have defined q̃_{i} ≡ q_{i}π/2 as a shorthand.

5. Find solutions and compare with numerical solutions using, for example, the setup prepared

in the Pencil Code exercises; http://www.nordita.org/~brandenb/teach/PencilCode/
MeanFieldCartesian_1D.html

References

Baryshnikova, Y. & Shukurov, A. M., “Oscillatory α^{2}-dynamo: numerical investigation,” Astron. Nachr. 308, 89-100 (1987).

Masada, Y., & Sano, T., “Mean-field modeling of an α^{2} dynamo coupled with direct numerical simulations of rigidly rotating convection,” Astrophys. J. Lett. 794, L6 (2014).

Mitra, D., Tavakol, R., Käpylä, P. J., & Brandenburg, A., “Oscillatory migrating magnetic fields in helical turbulence in spherical domains,” Astrophys. J. Lett. 719, L1-L4 (2010).

Parker, E. N. Cosmical magnetic fields. Clarendon Press, Oxford (1979).

Rädler, K.-H., & Bräuer, H.-J., “On the oscillatory behaviour of kinematic mean-field dynamos,” Astron. Nachr. 308, 101-109 (1987).