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Fluctuation dynamos at finite correlation times using renewing flows

Part of: Macroscopic Randomness

Published online by Cambridge University Press:  13 July 2015

Pallavi Bhat  and
Kandaswamy Subramanian
Pallavi Bhat*
Affiliation:
IUCAA, Post Bag 4, Ganeshkhind, Pune 411 007, India
Kandaswamy Subramanian
Affiliation:
IUCAA, Post Bag 4, Ganeshkhind, Pune 411 007, India
*
†Email address for correspondence: palvi@iucaa.ernet.in
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Abstract

Fluctuation dynamos are generic to turbulent astrophysical systems. The only analytical model of the fluctuation dynamo, due to Kazantsev, assumes the velocity to be delta-correlated in time. This assumption breaks down for any realistic turbulent flow. We generalize the analytic model of fluctuation dynamos to include the effects of a finite correlation time, ${\it\tau}$, using renewing flows. The generalized evolution equation for the longitudinal correlation function $M_{L}$ leads to the standard Kazantsev equation in the ${\it\tau}\rightarrow 0$ limit, and extends it to the next order in ${\it\tau}$. We find that this evolution equation also involves third and fourth spatial derivatives of $M_{L}$, indicating that the evolution for finite-${\it\tau}$ will be non-local in general. In the perturbative case of small-${\it\tau}$ (or small Strouhal number), it can be recast using the Landau–Lifschitz approach, to one with at most second derivatives of $M_{L}$. Using both a scaling solution and the WKBJ approximation, we show that the dynamo growth rate is reduced when the correlation time is finite. Interestingly, to leading order in ${\it\tau}$, we show that the magnetic power spectrum preserves the Kazantsev form, $M(k)\propto k^{3/2}$, in the large-$k$ limit, independent of ${\it\tau}$.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 5 , October 2015 , 395810502
Copyright
© Cambridge University Press 2015 

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