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K.-H. Rädler and A. Brandenburg

Remarks on kinetic helicity, tex2html_wrap_inline26-effect and dynamo action

Time: October 29, 2003, 3:00 pm
Place: Building 19, Room 19.415


The relevance of kinetic helicity tex2html_wrap_inline28 of a flow of an electrically conducting fluid for dynamo action is discussed.

All dynamo mechanisms revealed by mean-field dynamo theory require some deviation of the small-scale motion from reflectional symmetry. In the simple case of isotropic turbulence an tex2html_wrap_inline26-effect and thus the possibility of a dynamo occurs if, in the high-conductivity limit, the averaged kinetic helicity tex2html_wrap_inline32 is non-zero. This result is often overinterpreted in the sense that tex2html_wrap_inline32 is the crucial quantity for any tex2html_wrap_inline26-effect dynamo or even any mean-field dynamo. However, in the case of an anisotropic turbulence and thus an anisotropic tex2html_wrap_inline26-effect it is no longer exactly this but some related quantity which is of interest for dynamo action. For isotropic turbulence and low-conductivity limit the tex2html_wrap_inline26-effect is proportional to tex2html_wrap_inline42, where tex2html_wrap_inline44 is a vector potential of tex2html_wrap_inline46, and in special cases tex2html_wrap_inline42 may well be non-zero even if tex2html_wrap_inline32 vanishes. Apart from this, there are mean-field dynamos without any tex2html_wrap_inline26-effect, e.g., due to a combination of an electromotive force proportional to tex2html_wrap_inline54, resulting from some anisotropic turbulence, with a differential rotation, where tex2html_wrap_inline56 is an axial vector, e.g., the angular velocity of the rotation. Such dynamos as well as related ones work even if tex2html_wrap_inline32 is equal to zero.

These results support the now well known finding that also dynamos in the general sense (`laminar dynamos') are well possible if the kinetic helicity tex2html_wrap_inline28 vanishes everywhere. Some examples for that are given. In this context also a few comments concerning the geodynamo mechanism are made.

Particular attention is focused on the Roberts dynamo as it has been realized in the Karlsruhe dynamo experiment. For the original Roberts flow, and so in the experiment, tex2html_wrap_inline28 is non-zero. It has been shown both analytically and numerically that the dynamo works with modified flows, too, for which tex2html_wrap_inline28 vanishes everywhere.

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Udo Schwarz
Tue Oct 14 12:29:11 MEST 2003