Generalized Hamiltonian point vortex dynamics on arbitrary domains using the method of fundamental solutions

Ashbee, T. L.; Esler, J. G.; McDonald, N. Robb

doi:10.1016/j.jcp.2013.03.044

Cited by 8 publications

(6 citation statements)

References 34 publications

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“…Some properties of this dynamical system are given in Section 3 of [5] and in [2]. Moreover, there are particular initial data z i (0), values of N , and values of a i , for which |z i (t) − z j (t)| go to infinity as t → ∞.…”

Section: Improvementsmentioning

confidence: 99%

“…In [5,14,15,26], some rigorous results about the connection between the mSQG equation and the dynamics of pseudo-vortices have been proved (the so-called localization result). Some papers have been devoted to the analysis of this dynamical system, we refer for example to [2] and [3]. Moreover, in [10] the point vortex approximaton in the mSQG equation has been studied from the stochastic point of view.…”

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confidence: 99%

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Long time localization of modified surface quasi-geostrophic equations

Cavallaro¹,

Garra²,

Marchioro³

2021

DCDS-B

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Section: Improvementsmentioning

confidence: 99%

mentioning

confidence: 99%

Long time localization of modified surface quasi-geostrophic equations

Cavallaro¹,

Garra²,

Marchioro³

2021

DCDS-B

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“…are given in [1]. Here, very long integrations with N = 100 vortices are performed (length 10 5 Γ −1 0 ), and vortex positions recorded at interval Γ −1 0 in order to obtain converged statistics.…”

Section: Comparison With Direct Numerical Simulationsmentioning

confidence: 99%

Statistics of Point Vortex Turbulence in Non-neutral Flows and in Flows with Translational and Rotational Symmetries

Esler

2017

J Stat Phys

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A theory [Esler and Ashbee, J. Fluid Mech., 779, 275, 2015] describing the statistics of N freely-evolving point vortices in a bounded twodimensional domain is extended. First, the case of a non-neutral vortex gas is addressed, and it is shown that the density of states function can be identified with the probability density function of an infinite sum of independent non-central chi-squared random variables, the details of which depend only on the shape of the domain. Equations for the equilibrium energy spectrum and other statistical quantities follow, the validity of which are verified against direct numerical simulations of the equations of motion. Second, domains with additional conserved quantities associated with a symmetry (e.g. circle, periodic channel) are investigated, and it is shown that the treatment of the non-neutral case can be modified to account for the additional constraint.

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“…(36).] PL76 used equation (25) both to obtain an exact solution in the special situation of a doubly-periodic domain and to derive an equation for the thermodynamic curve in the limit β → ∞, corresponding to large negative energies. Taking the divergence of (25) results in…”

Section: B the Vorticity Fluctuation Equationmentioning

confidence: 99%

“…An adaptive time-stepping algorithm, with an advancement criterion based upon convergence of final vortex positions over a fixed time interval, is used to update vortex positions. Further details of the integration method and numerical parameters used are given elsewhere [25]. For each of the two Neumann oval domains, integrations are performed for 11 different energy levels, evenly spaced betweenẼ = −0.15 andẼ = 0.1.…”

Section: Verification By Direct Numerical Simulationmentioning

confidence: 99%

Statistical mechanics of a neutral point-vortex gas at low energy

Esler

Ashbee²,

McDonald³

2013

Phys. Rev. E

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The statistics of a neutral point-vortex gas in an arbitrary two-dimensional simply connected and bounded container are investigated in the framework of the microcanonical ensemble, following the cumulant expansion method of Pointin and Lundgren [Phys. Fluids 19, 1459 (1976)]. The equation for vorticity fluctuations, obtained when a thermodynamic scaling limit is taken, is solved explicitly. The solution depends on an infinite sequence of negative "domain inverse temperatures," determined by the domain shape, which are obtained from solutions of a "vorticity mode" eigenvalue problem. An explicit expression for the thermodynamic curve relating inverse temperature and energy is found and is shown to depend on the geometry and not on the scale of the domain. Explicit formulas are then obtained for the time variance of the projection of the vorticity field onto each vorticity mode. The results are verified by two methods. First, for a chosen single-parameter family of domains, direct sampling of the microcanonical ensemble is used to demonstrate the accuracy of the formula for the thermodynamic curve. Second, direct numerical simulations are used to verify the formulas for the variance of the projections of the vorticity field, with convincing results.

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Generalized Hamiltonian point vortex dynamics on arbitrary domains using the method of fundamental solutions

Cited by 8 publications

References 34 publications

Long time localization of modified surface quasi-geostrophic equations

Long time localization of modified surface quasi-geostrophic equations

Statistics of Point Vortex Turbulence in Non-neutral Flows and in Flows with Translational and Rotational Symmetries

Statistical mechanics of a neutral point-vortex gas at low energy

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