Optimizing ABC kinematic dynamo in the Lagrangian coordinates

We consider the kinematic dynamo problem with the Arnold--Beltrami--Childress (ABC) flow. A well-known goal of this problem is to understand the exponential growth rate of the magnetic energy. Here we first consider the ideal MHD case and use the Cauchy integral formula for the magnetic field. We calculate numerically the Jacobi matrix of the Lagrangian map at each grid point along with the eigenvalues and eigenvectors of the Cauchy--Green tensor. With this information, we present a numerical way to construct an initial magnetic field that yields possibly the maximum growth rate for a given time. This given time cannot be set to a large value due to a numerical limitation, however. Then we explore a diffusive extension of the Cauchy integral formula and discuss how our optimization method in the ideal case can be carried over to a finite magnetic Reynolds number case.