Searching for the fastest dynamo: Laminar ABC flows -
Alexakis, AlexandrosPHYSICAL REVIEW E 84,
(2011) LPS
Abstract : The growth rate of the dynamo instability as a function of the magnetic
Reynolds number R-M is investigated by means of numerical simulations
for the family of the Arnold-Beltrami-Childress (ABC) flows and for two
different forcing scales. For the ABC flows that are driven at the
largest available length scale, it is found that, as the magnetic
Reynolds number is increased: (a) The flow that results first in a
dynamo is the 2 1/2-dimensional flow for which A = B and C = 0 (and all
permutations). (b) The second type of flow that results in a dynamo is
the one for which A = B similar or equal to 2C/5 (and permutations). (c)
The most symmetric flow, A = B = C, is the third type of flow that
results in a dynamo. (d) As R-M is increased, the A = B = C flow stops
being a dynamo and transitions from a local maximum to a third-order
saddle point. (e) At larger R-M, the A = B = C flow reestablishes itself
as a dynamo but remains a saddle point. (f) At the largest examined R-M,
the growth rate of the 2 1/2-dimensional flows starts to decay, the A =
B = C flow comes close to a local maximum again, and the flow A = B
similar or equal to 2C/5 (and permutations) results in the fastest
dynamo with growth rate gamma similar or equal to 0.12 at the largest
examined R-M. For the ABC flows that are driven at the second largest
available length scale, it is found that (a) the 2 1/2-dimensional flows
A = B, C = 0 (and permutations) are again the first flows that result in
a dynamo with a decreased onset. (b) The most symmetric flow, A = B = C,
is the second type of flow that results in a dynamo. It is, and it
remains, a local maximum. (c) At larger R-M, the flow A = B similar or
equal to 2C/5 (and permutations) appears as the third type of flow that
results in a dynamo. As R-M is increased, it becomes the flow with the
largest growth rate. The growth rates appear to have some correlation
with the Lyapunov exponents, but constructive refolding of the field
lines appears equally important in determining the fastest dynamo flow.