Optimal unstirred state of a passive scalar

TY - JOUR

T1 - Optimal unstirred state of a passive scalar

AU - Chen, L.

AU - Yeates, A. R.

AU - Russell, A. B. J.

N1 - Funding. This work was supported by Leverhulme Trust grant PRG-2017-169

PY - 2021/3/25

Y1 - 2021/3/25

N2 - Given a passive tracer distribution, 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. Both methods achieve similar convergence for simple test cases. However, the magnetic relaxation method guarantees a monotonic decrease in the Dirichlet seminorm of, 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 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. 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.

AB - Given a passive tracer distribution, 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. Both methods achieve similar convergence for simple test cases. However, the magnetic relaxation method guarantees a monotonic decrease in the Dirichlet seminorm of, 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 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. 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.

KW - Key words variational methods

KW - topological fluid dynamics

UR - https://www.scopus.com/pages/publications/85100066532

U2 - 10.1017/jfm.2020.1154

DO - 10.1017/jfm.2020.1154

M3 - Article

SN - 0022-1120

VL - 911

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

M1 - A30

ER -