Alladi Ramakrishnan Hall

Growth rate distribution and intermittency in kinematic turbulent dynamos: Which moment predicts the dynamo onset?

Kannabiran Seshasayanan

CEASaclay,France

The dynamo effect is an instability that converts kinetic energy of an electrically conducting fluid into magnetic energy. It is the source of the magnetic field observed in most astrophysical objects, Earth and most planets, Sun and other stars, galaxies. This effect was identified by Larmor a hundred years ago and yet many questions are still unanswered, in particular concerning the effect of turbulent fluctuations on the dynamo process. Dynamo instability occurring in nature and in laboratory experiments are driven by a highly turbulent flow.

An analytical model that takes into account the turbulent fluctuations was proposed by Kazantsev [1]. Around the same time, Kraichnan [2] proposed a similar model for the passive scalar advection. The Kazantsev model considers a velocity field that is white in time process with a Gaussian distribution. With this hyposthesis the induction equation that governs the evolution of the magnetic field B can be solved. The induction equation is linear in B and its retro-action on the velocity field is not considered (kinematic dynamo). The magnetic field driven by the velocity field grows in a fluctuation manner, i.e., the growth is not monotonic. The growth rate of the n-th moment of the magnetic field is a nonlinear function of n. This implies that each moment of B predicts a different threshold of the instability. Thus the question we ask is, which of these moments predict the actual threshold in the nonlinear system of equations (with the retro-action)?

To respond to this question we develop an analytical model using some results of the theory of large deviations [3]. This allows us to find the pdf of the magnetic field for the linear problem. We show that the growth rate of the different moments of B is a nonlinear function of n. We find the theoretical threshold of the different moments of the magnetic field. Then, we use numerical simulations to solve the nonlinear system of equations. We show that the threshold is predicted by the zeroth moment of the magnetic field (the log of B) of the linear problem which is the correct threshold of the system. Notably higher moments like the magnetic energy overestimate the threshold. We verify this for the dynamo instability driven by different types of turbulent flows.