Self-Organization in Three-Dimensional Hydrodynamic Turbulence Self-Organization in Three-Dimensional Hydrodynamic Turbulence

Knorr, G.; Lynov, Jens-Peter; Pécseli, H. L.

doi:10.1515/zna-1990-9-1002

Cited by 7 publications

(2 citation statements)

References 27 publications

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“…We define an 'inverse cascade' as a process in which a fraction of the energy in a spectrum migrates either consistently or temporally to smaller rather than larger wavenumbers. Inverse cascades have been observed in two-dimensional hydrodynamics (Frisch et al 1975) and two-dimensional guiding-centre plasmas (Knorr 1974;Seyler et al 1975;Knorr, Lynov & Pecseli 1990), even those with a finite Larmor radius (Knorr & Pecseli 1989). In three-dimensional hydrodynamics it has been shown (Knorr et al 1990) that some basic interactions represent an inverse cascade, whereas others allow a migration of all energy to larger wavenumbers.…”

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confidence: 99%

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Relaxed states of an ideal MHD plasma with external magnetic field

1995

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We study an ideal MHD plasma with the non-vanishing invariants energy, crosshelicity and magnetic helicity, confined in a cylinder with infinitely conducting walls and an externally applied magnetic field B0. The magnetic and velocity fields are expanded in base vector fields, satisfying Δ × Bλ = Bλ. Boundary conditions are imposed to make the curl a self-adjoint operator. The three invariants depend on the time-dependent coefficients of the base vector fields, and are used to construct the partition function to gather statistical information about the equlibrium thermodynamic state to which the plasma relaxes after a turbulent transition. For zero external magnetic field but large magnetic helicity, the energy resides preferentially in magnetic field fluctuations. A sizeable fraction of the kinetic energy initially present is transformed into magnetic energy. The energy condenses via an inverse cascade predominantly to the lowest energy eigenstate, in agreement with results obtained by Taylor. However, since we consider the whole spectrum of eigenstates, the energy does not exclusively occupy the lowest eigenstate. If the eigenvalues are densely spaced (as in a thin torus), the higher eigenmodes also contain appreciable amounts of energy, resulting in a finite pressure of the plasma. For constant and finite external magnetic field, the average induced magnetic field exactly cancels the external field. This indicates that, on a statistical average, the plasma is diamagnetic or superconducting. Superimposed on the average statistical state are fluctuations that may become large if the magnetic helicity is large.

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confidence: 99%

“…This boundary condition corresponds to an infinitely conducting cylinder. We shall follow the hydrodynamic analysis of Knorr et al (1990) and use the eigenfields of V xB x = kB x {r,<p,z).…”

Section: Expansion Of the Vector Fieldsmentioning

confidence: 99%

Relaxed states of an ideal MHD plasma with external magnetic field

1995

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Kinetic Helicity in the Earth's Atmosphere

Chkhetiani,

Kurgansky

2023

Geophysical Monograph Series

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Symmetries in hydrodynamic turbulence and MHD dynamo theory

Knorr

1996

J. Plasma Phys.

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The three-dimensional equations of ideal hydrodynamics and ideal MHD are expanded in eigenfunctions of the curl, and the resulting basic interactions of these nonlinear systems are analysed. As the equations are invariant under time and amplitude reversal, a criterion defining the arrow of time is introduced. A new parameter, the center of energy, serves to characterize a basic interaction. In the 3D Euler equations we find four different interactions and their mirror images, two of which can transport energy to smaller wavenumbers. This can lead to the appearance of structures in turbulent flow, and throws doubt on a derivation of Kolmogorov's law based on a cascading of energy to higher wavenumbers.In energy the corresponding two-dimensional equations, which are isomorphic to the guiding centre model in plasma physics, only one interaction exists, with a strong inverse cascade, which can lead to accumulation of energy in the spatially largest accessible modes. In MHD theory it is possible to separate magnetic from kinetic interactions. The former give again four basic interactions, two being regular and two being inverse cascades. One of these is quite strong, and can lead to the MHD dynamo effect. Kinetic energy can be transferred into magnetic energy. The dynamo effect is is accompanied by alignment of velocity and magnetic fields. We show that stationary velocity fields may lead to exponentially growing magnetic fields and we give an explicit criterion for this instability.

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Self-Organization in Three-Dimensional Hydrodynamic Turbulence Self-Organization in Three-Dimensional Hydrodynamic Turbulence

Cited by 7 publications

References 27 publications

Relaxed states of an ideal MHD plasma with external magnetic field

Relaxed states of an ideal MHD plasma with external magnetic field

Kinetic Helicity in the Earth's Atmosphere

Symmetries in hydrodynamic turbulence and MHD dynamo theory

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