Proving dynamo growth by analyzing the stretch-folder-shear operator #

Farhana Akond Pramy

Poster session

Abstract #

I will present the stretch-fold-shear mechanism, a process by which magnetic fields can be generated and amplified in the interiors of planets, stars, and other celestial objects such as Earth. The source of the magnetic field of a celestial body is called the dynamo. My research concerns the stretch-fold-shear (SFS) operator, $S_\alpha$, which arises from a stylised model in kinematic dynamo theory where the magnetic field growth of Earth corresponds to an eigenvalue of modulus greater than 1. This research aims to prove the existence of such an eigenvalue, thereby confirming an outstanding conjecture on dynamo growth.

Though the SFS operator is easy to write down, its spectral properties are more challenging to demonstrate using the usual analytic methods. Consequently, we have used a computer-assisted proof to find rigorous bounds on the leading eigenvalue for positive shear ($0<\alpha \le 5$) to show where $S_\alpha$ has an eigenvalue of modulus greater than $1$. Although previously, the spectrum of $S_\alpha$ has been approximated numerically, the computer-assisted results in this research are the first rigorous results to be obtained. We have also used generating function techniques to find the spectrum of $S_\alpha$ for zero shear ($\alpha =0$) related to the Bernoulli polynomials.