Saturation of Zeldovich Stretch-Twist-Fold Map Dynamos

TL;DR

This paper investigates the saturation behavior of simplified dynamo models based on Zeldovich's STF process, revealing how magnetic fields reach saturation at high magnetic Reynolds numbers through different mechanisms.

Contribution

It introduces a detailed analysis of Baker's map dynamo models, exploring their saturation mechanisms and how the magnetic field structure evolves at high Reynolds numbers.

Findings

Magnetic field amplifies only above a critical R_M of about 4.

Saturation can occur via decreased effective R_M or reduced stretching efficiency.

Different saturation behaviors lead to magnetic fields resembling eigenfunctions at critical or intermediate R_M.

Abstract

Zeldovich's stretch-twist fold (STF) dynamo provided a breakthrough in conceptual understanding of fast dynamos, including fluctuation or small scale dynamos. We study the evolution and saturation behaviour of two types of Baker's map dynamos, which have been used to model Zeldovich's STF dynamo process. Using such maps allows one to analyze dynamos at much higher magnetic Reynolds numbers $R_M$ as compared to direct numerical simulations. In the 2-strip map dynamo there is constant constructive folding while the 4-strip map dynamo also allows the possibility of field reversal. Incorporating a diffusive step parameterised by $R_M$, we find that the magnetic field $B(x)$ is amplified only above a critical $R_M=R_{crit} \sim 4$ for both types of dynamos. We explore the saturation of these dynamos in 3 ways; by a renormalized decrease of the effective $R_M$ (Case I) or due to a decrease in the efficiency of field amplification by stretching (Case II), or a combination of both effects (Case III). For Case I, we show that $B(x)$ in the saturated state, for both types of maps, goes back to the marginal eigenfunction, which is obtained for the critical $R_M=R_{crit}$. This is independent of the initial $R_M=R_{M0}$. On the other hand in Case II, for the 2-strip map, we show that $B(x)$ now saturates preserving the structure of the kinematic eigenfunction. Thus the energy is transferred to larger scales in Case I but remains at the smallest resistive scales in Case II. For the 4-strip map, the $B(x)$ oscillates with time, although with a structure similar to the kinematic eigenfunction. Interestingly, the saturated state for Case III shows an intermediate behaviour, with $B(x)$ now similar to the kinematic eigenfunction for an intermediate $R_M=R_{sat}$, with $R_{M0}>R_{sat}>R_{crit}$. These saturation properties are akin to the ones discussed in the context of fluctuation dynamos.