Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD

Caflisch, Russel E.; Klapper, Isaac; Steele, Gregory V.

doi:10.1007/s002200050067

Cited by 256 publications

(256 citation statements)

References 8 publications

Supporting

Mentioning

249

Contrasting

Order By: Relevance

“…If, further, the vortex viscosity χ = 0, the velocity u does not depend on the micro-rotation field ω, and the first equation reduces to the classical Navier-Stokes equation which has been greatly analyzed, see, for example, the classical books by Ladyzhenskaya [13], Lions [15] or Lemarié-Rieusset [14]. If we ignore the micro-rotation of particles, it reduces to the viscous incompressible magnetohydrodynamic equations, which has also been studied extensively [1][2][3]9,20]. It is worthy to note that He and Xin [9], Zhou [25,26] proved the regularity criteria of weak solutions to the magneto-hydrodynamic equations, which only need the velocity u or its gradient ∇u or the vorticity ∇ × u or p and b or ∇p and b to satisfy some conditions.…”

mentioning

confidence: 99%

Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey–Campanato space

Gala

2009

Nonlinear Differ. Equ. Appl.

View full text Add to dashboard Cite

show abstract

mentioning

confidence: 99%

Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey–Campanato space

Gala

2009

Nonlinear Differ. Equ. Appl.

View full text Add to dashboard Cite

show abstract

“…In MHD [4], one deals with the Elsässer fields z ± = v±b and ω ± = ω±j = ∇×(v±b), with ω the vorticity, v the velocity, j the current density and b = ∇×A the induction in dimensionless Alfvenic units, A being the vector potential. Intense current sheets are known to form at either magnetic nulls (b ≡ 0) or when one or two (but not all) components of the magnetic field go to zero or have strong gradients.…”

mentioning

confidence: 99%

Small-Scale Structures in Three-Dimensional Magnetohydrodynamic Turbulence

2006

View full text Add to dashboard Cite

We investigate using direct numerical simulations with grids up to 1536 3 points, the rate at which small scales develop in a decaying three-dimensional MHD flow both for deterministic and random initial conditions. Parallel current and vorticity sheets form at the same spatial locations, and further destabilize and fold or roll-up after an initial exponential phase. At high Reynolds numbers, a self-similar evolution of the current and vorticity maxima is found, in which they grow as a cubic power of time; the flow then reaches a finite dissipation rate independent of Reynolds number.Magnetic fields are ubiquitous in the cosmos and play an important dynamical role, as in the solar wind, stars or the interstellar medium. Such flows have large Reynolds numbers and thus nonlinear mode coupling leads to the formation of strong intermittent structures. It has been observed that such extreme events in magnetohydrodynamics (MHD) are more intense than for fluids; for example, wings of Probability Distribution Functions of field gradients are wider and one observes a stronger departure from purely self-similar linear scaling with the order of the anomalous exponents of structure functions [1]. Since Reynolds numbers are high but finite, viscosity and magnetic resistivity play a role, tearing mode instabilities develop and reconnection takes place. The question then becomes at what rate does dissipation occur, as the Reynolds number increases? What is the origin of these structures, and how fast are they formed? This is a long-standing problem in astrophysics, e.g. in the context of reconnection events in the magnetopause, or of heating of solar and stellar corona. In such fluids, many other phenomena may have to be taken into account, such as finite compressibility and ionization, leading to a more complex Ohm's law with e.g. a Hall current or radiative or gravitational processes to name a few. Many aspects of the two-dimensional (2D) case are understood, but the three-dimensional (3D) turbulent case remains more obscure. Pioneering works [2] show that the topology of the reconnecting region, more complex than in 2D, can lead to varied behavior.The criterion for discriminating between a singular and a regular behavior in the absence of magnetic fields follows the seminal work by Beale, Kato and Majda (hereafter BKM) [3] where, for a singularity to develop in the Euler case, the time integral of the supremum of the vorticity must grow as (t−t * ) −α with α ≥ 1 and t * the singularity time. In MHD [4], one deals with the Elsässer fields z ± = v±b and ω ± = ω±j = ∇×(v±b), with ω the vorticity, v the velocity, j the current density and b = ∇×A the induction in dimensionless Alfvenic units, A being the vector potential. Intense current sheets are known to form at either magnetic nulls (b ≡ 0) or when one or two (but not all) components of the magnetic field go to zero or have strong gradients. In two dimensional configurations, a vortex quadrupole is also associated with these structures. The occurrence of singularities in MH...

show abstract

“…If both v = 0 and c = 0, then [22][23][24][25][26][27][28][29][30][31][32][33][34]). In this paper, we consider the magneto-micropolar fluid equations (1.1) with partial viscosity, i.e., μ = c = 0.…”

Section: Introductionmentioning

confidence: 99%

Blow-up criterion of smooth solutions for magneto-micropolar fluid equations with partial viscosity

Wang

2011

Bound Value Probl

View full text Add to dashboard Cite

show abstract

Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD

Cited by 256 publications

References 8 publications

Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey–Campanato space

Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey–Campanato space

Small-Scale Structures in Three-Dimensional Magnetohydrodynamic Turbulence

Blow-up criterion of smooth solutions for magneto-micropolar fluid equations with partial viscosity

Contact Info

Product

Resources

About