Saturation of Zeldovich Stretch-Twist-Fold Map Dynamos
  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.
