A geometric look at MHD and the Braginsky dynamo

Gilbert, Andrew D.; Vanneste, Jacques

doi:10.1080/03091929.2020.1839896

Cited by 5 publications

(4 citation statements)

References 53 publications

(102 reference statements)

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“…The averaging of vectors, differential forms and more general tensors is, however, of interest in applications. For instance, the Lagrangian means of the momentum 1-form (the integrand in Kelvin's circulation) and of the magnetic flux 2-form play crucial roles in the theory of wave–mean-flow interactions in fluid dynamics and magnetohydrodynamics (Soward 1972; Andrews & McIntyre 1978; Holm 2002; Gilbert & Vanneste 2018, 2021). The derivations in § 3 generalise straightforwardly to tensors when the language of push-forwards, pull-backs and Lie derivatives is employed.…”

Section: Discussionmentioning

confidence: 99%

Computing Lagrangian means

Kafiabad

Vanneste

2023

J. Fluid Mech.

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Lagrangian averaging plays an important role in the analysis of wave–mean-flow interactions and other multiscale fluid phenomena. The numerical computation of Lagrangian means, e.g. from simulation data, is, however, challenging. Typical implementations require tracking a large number of particles to construct Lagrangian time series, which are then averaged using a low-pass filter. This has drawbacks that include large memory demands, particle clustering and complications of parallelisation. We develop a novel approach in which the Lagrangian means of various fields (including particle positions) are computed by solving partial differential equations (PDEs) that are integrated over successive averaging time intervals. We propose two strategies, distinguished by their spatial independent variables. The first, which generalises the algorithm of Kafiabad (J. Fluid Mech., vol. 940, 2022, A2), uses end-of-interval particle positions; the second uses directly the Lagrangian mean positions. The PDEs can be discretised in a variety of ways, e.g. using the same discretisation as that employed for the governing dynamical equations, and solved on-the-fly to minimise the memory footprint. We illustrate the new approach with a pseudo-spectral implementation for the rotating shallow-water model. Two applications to flows that combine vortical turbulence and Poincaré waves demonstrate the superiority of Lagrangian averaging over Eulerian averaging for wave–vortex separation.

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Section: Discussionmentioning

confidence: 99%

Computing Lagrangian means

Kafiabad

Vanneste

2023

J. Fluid Mech.

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“…Finally we consider magnetohydrodynamics (MHD) and outline a derivation of the conservation form of the governing equation of ideal MHD which generalises (3.5) by including the Lorentz force; see the classic study by Newcomb (1962) and also Gilbert and Vanneste (2019). The general procedure is already established, but because the flow u and magnetic field b have distinct transport properties, there are notable differences, and one effect is that a magnetic pressure term emerges from the analysis.…”

Section: Magnetohydrodynamicsmentioning

confidence: 99%

A Geometric Look at Momentum Flux and Stress in Fluid Mechanics

Gilbert

Vanneste

2023

J Nonlinear Sci

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We develop a geometric formulation of fluid dynamics, valid on arbitrary Riemannian manifolds, that regards the momentum-flux and stress tensors as 1-form-valued 2-forms, and their divergence as a covariant exterior derivative. We review the necessary tools of differential geometry and obtain the corresponding coordinate-free form of the equations of motion for a variety of inviscid fluid models—compressible and incompressible Euler equations, Lagrangian-averaged Euler-$$\alpha $$ α equations, magnetohydrodynamics and shallow-water models—using a variational derivation which automatically yields a symmetric momentum flux. We also consider dissipative effects and discuss the geometric form of the Navier–Stokes equations for viscous fluids and of the Oldroyd-B model for visco-elastic fluids.

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“…Finally we consider magnetohydrodynamics (MHD) and outline a derivation of the conservation form of the governing equation of ideal MHD which generalises (3.5) by including the Lorentz force; see also [27]. The general procedure is already established, but because the flow u and and magnetic field b have distinct transport properties, there are notable differences, and one effect is that a magnetic pressure term emerges from the analysis.…”

Section: Magnetohydrodynamicsmentioning

confidence: 99%

A geometric look at momentum flux and stress in fluid mechanics

Gilbert¹,

Vanneste²

2019

Preprint

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We develop a geometric formulation of fluid dynamics, valid on arbitrary Riemannian manifolds, that regards the momentum-flux and stress tensors as 1-form valued 2-forms, and their divergence as a covariant exterior derivative. We review the necessary tools of differential geometry and obtain the corresponding coordinate-free form of the equations of motion for a variety of inviscid fluid models -compressible and incompressible Euler equations, Lagrangian-averaged Euler-α equations, magnetohydrodynamics and shallow-water models -using a variational derivation which automatically yields a symmetric momentum flux. We also consider dissipative effects and discuss the geometric form of the Navier-Stokes equations for viscous fluids and of the Oldroyd-B model for visco-elastic fluids.

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A geometric look at MHD and the Braginsky dynamo

Cited by 5 publications

References 53 publications

Computing Lagrangian means

Computing Lagrangian means

A Geometric Look at Momentum Flux and Stress in Fluid Mechanics

A geometric look at momentum flux and stress in fluid mechanics

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