## Abstract

Given a passive tracer distribution *f *(*x*, *y*), what is the simplest unstirred pattern that may be reached under incompressible advection? This question is partially motivated by recent studies of three-dimensional (3-D) magnetic reconnection, in which the patterns of a topological invariant called the field line helicity greatly simplify until reaching a relaxed state. We test two approaches: a variational method with minimal constraints, and a magnetic relaxation scheme where the velocity is determined explicitly by the pattern of *f*. Both methods achieve similar convergence for simple test cases. However, the magnetic relaxation method guarantees a monotonic decrease in the Dirichlet seminorm of *f*, and is numerically more robust. We therefore apply the latter method to two complex mixed patterns modelled on the field line helicity of 3-D magnetic braids. The unstirring separates *f *into a small number of large-scale regions determined by the initial topology, which is well preserved during the computation. Interestingly, the velocity field is found to have the same large-scale topology as *f*. Similarity to the simplification found empirically in 3-D magnetic reconnection simulations supports the idea that advection is an important principle for field line helicity evolution.

Original language | English |
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Article number | A30 |

Number of pages | 21 |

Journal | Journal of Fluid Mechanics |

Volume | 911 |

Early online date | 28 Jan 2021 |

DOIs | |

Publication status | Published - 25 Mar 2021 |

## Keywords

- Variational methods
- Topological fluid dynamics