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We study structure formation in two-dimensional turbulence driven by an external force, interpolating between linear instability forcing and random stirring, subject to nonlinear damping. Using extensive direct numerical simulations, we uncover a rich parameter space featuring four distinct branches of stationary solutions: large-scale vortices, hybrid states with embedded shielded vortices (SVs) of either sign, and two states composed of many similar SVs. Of the latter, the first is a dense vortex gas where all SVs have the same sign and diffuse across the domain. The second is a hexagonal vortex crystal forming from this gas when the linear instability is sufficiently weak. These solutions coexist stably over a wide parameter range. The late-time evolution of the system from small-amplitude initial conditions is nearly self-similar, involving three phases: initial inverse cascade, random nucleation of SVs from turbulence and, once a critical number of vortices is reached, a phase of explosive nucleation of SVs, leading to a statistically stationary state. The vortex gas is continued in the forcing parameter, revealing a sharp transition towards the crystal state as the forcing strength decreases. This transition is analysed in terms of the diffusivity of individual vortices using ideas from statistical physics. The crystal can also decay via an inverse cascade resulting from the breakdown of shielding or insufficient nonlinear damping acting on SVs. Our study highlights the importance of the forcing details in two-dimensional turbulence and reveals the presence of non-trivial SV states in this system, specifically the emergence and melting of a vortex crystal.
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A new canonical reduction of three-vortex motion and its application to vortex-dipole scattering
We introduce a new reduction of the motion of three point vortices in a two-dimensional ideal fluid. This proceeds in two stages: a change of variables to Jacobi coordinates and then a Nambu reduction. The new coordinates demonstrate that the dynamics evolve on a two-dimensional manifold whose topology depends on the sign of a parameter κ2 that arises in the reduction. For κ2>0, the phase space is spherical, while for κ2<0, the dynamics are confined to the upper sheet of a two-sheeted hyperboloid. We contrast this reduction with earlier reduced systems derived by Gröbli, Aref, and others in which the dynamics are determined from the pairwise distances between the vortices. The new coordinate system overcomes two related shortcomings of Gröbli's reduction that have made understanding the dynamics difficult: their lack of a standard phase plane and their singularity at all configurations in which the vortices are collinear. We apply this to two canonical problems. We first discuss the dynamics of three identical vortices and then consider the scattering of a propagating dipole by a stationary vortex. We show that the points dividing direct and exchange scattering solutions correspond to the locations of the invariant manifolds of equilibria of the reduced equations and relate changes in the scattering diagram as the circulation of one vortex is varied to bifurcations of these equilibria.
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- Award ID(s):
- 2206016
- PAR ID:
- 10534886
- Publisher / Repository:
- AIP Publishing
- Date Published:
- Journal Name:
- Physics of Fluids
- Volume:
- 36
- Issue:
- 6
- ISSN:
- 1070-6631
- Page Range / eLocation ID:
- 067110
- Subject(s) / Keyword(s):
- fluid mechanics vortex dynamics Hamiltonian systems
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0208538
