Small-scale dynamo in Riemannian spaces of constant curvature

Abstract

We compare temporal growth of the mean magnetic energy ℰ driven by a small-scale dynamo in Euclidean and Lobachevsky spaces. The governing parameters of the dynamo, such as the rms turbulent velocity and the correlation scale of turbulence, are presumed to vary randomly in space so the dynamo growth rate is a Gaussian random field. Since such a field is unbounded in unbounded space and can achieve, with a low probability, very large values of ℰ, it can grow super-exponentially in both cases. The super-exponential growth of ℰ in Euclidian space, known since the 1980s, can be considered as a statement that the mean energy growth rate is determined up to a weakly growing factor proportional to . We demonstrate that the super-exponential growth of ℰ in Lobachevsky space is a much more radical phenomenon, where ℰ grows as exp (const × t ^5/3). We stress that extrapolating the properties of small-scale dynamos in Euclidean space to curved geometries such Lobachevsky space is not straightforward and requires some care. The effects under discussion becomes however important only if the spatial scale of domain in which the small-scale magnetic field is excited exceeds the radius of curvature.

Keywords:

• Small-scale dynamo
• Early Universe
• Lobachevsky space

Acknowledgements

Financial support from ISTC under grant 3726 is acknowledged. We are grateful to Andrew Fletcher, Newcastle, for careful reading of the manuscript.

Notes

†Parameters a and μ can be introduced in Euclidean space as well, however, there they do not contribute to the leading term of the asymptotical expansion for the growth rate.