A Computer-Assisted Proof of Dynamo Growth in the Stretch-Fold-Shear Operator

Pramy, Farhana Akond (2024). A Computer-Assisted Proof of Dynamo Growth in the Stretch-Fold-Shear Operator. PhD thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.00017614

Abstract

This thesis concerns a functional linear operator called the Stretch-Fold-Shear (SFS) operator, which arises from a stylized model in kinematic dynamo theory. A dynamo is the source of the magnetic field of a celestial body such as the Earth, the Sun and, indeed, any planet or star. The existence of an eigenvalue of the SFS operator of magnitude greater than one ensures dynamo growth. This research aims to prove the existence of such an eigenvalue using a computer-assisted proof, thereby confirming a conjecture of Andrew Gilbert.

We have extensively studied the cases of zero shear and positive shear for this operator. We determined the exact eigenvalues and eigenfunctions in the case of zero shear. However, when it came to positive shear, the spectral properties proved challenging to demonstrate using conventional analytic methods. As a result, we employed the generating function method for zero shear and relied on a computer-assisted proof to rigorously establish the properties for positive shear.

Computer-assisted proof is a powerful mathematical technique that utilizes computer technology to develop and verify mathematical proofs. While the computational requirements of these methods can be time-consuming, we have successfully proven Andrew Gilbert's conjecture for certain positive shear values. While previous numerical approximations have been made for the spectrum of the SFS operator, the computer-assisted results presented in this research are the first rigorous results to be obtained.

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