scholarly journals On quasisymmetric plasma equilibria sustained by small force

2021 ◽  
Vol 87 (1) ◽  
Author(s):  
Peter Constantin ◽  
Theodore D. Drivas ◽  
Daniel Ginsberg
Keyword(s):  
Vector Field ◽  
Euclidean Space ◽  
Euler Equations ◽  
Three Dimensional ◽  
Flat Space ◽  
Smooth Structure ◽  
Shafranov Equation ◽  
Small Force ◽  
Plasma Equilibria ◽  
Incompressible Euler

We construct smooth, non-symmetric plasma equilibria which possess closed, nested flux surfaces and solve the magnetohydrostatic (steady three-dimensional incompressible Euler) equations with a small force. The solutions are also ‘nearly’ quasisymmetric. The primary idea is, given a desired quasisymmetry direction $\xi$ , to change the smooth structure on space so that the vector field $\xi$ is Killing for the new metric and construct $\xi$ –symmetric solutions of the magnetohydrostatic equations on that background by solving a generalized Grad–Shafranov equation. If $\xi$ is close to a symmetry of Euclidean space, then these are solutions on flat space up to a small forcing.

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

1990 ◽  
Vol 45 (9-10) ◽  
pp. 1059-1073 ◽  
Author(s):  
G. Knorr ◽  
J. P. Lynov ◽  
H. L. Pécseli
Keyword(s):  
Euler Equations ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Self Organization ◽  
Two Dimensional ◽  
Hydrodynamic Turbulence ◽  
Available Energy ◽  
Wave Numbers ◽  
Negative Helicity ◽  
Incompressible Euler

Abstract The three-dimensional incompressible Euler equations are expanded in eigenflows of the curl operator, which represent positive and negative helicity flows in a particularly simple and convenient way. Four different basic types of interactions between eigenflows are found. Two represent an "inverse cascade", the interaction familiar from the two-dimensional Euler equations, in which only modes of the same sign of the helicity interact. The other two interactions mix positive and negative helicity modes. Only these interactions can transport all of the available energy to higher wave numbers. Initial conditions, which lead to the appearance of structures and self-organization, are discussed.

scholarly journals Explicit Solutions of the Wave Equation on Three Dimensional Space-Times: Two Examples with Dirichlet Boundary Conditions on a Disk

2006 ◽  
Vol 4 (4) ◽  
Author(s):  
Daniel Boykis ◽  
Patrick Moylan
Keyword(s):  
Wave Equation ◽  
Euclidean Space ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Flat Space ◽  
Dirichlet Boundary Conditions ◽  
Dimensional Euclidean Space ◽  
Curved Space

We study solutions of the wave equation with circular Dirichlet boundary conditions on a flat two-dimensional Euclidean space, and we also study the analogous problem on a certain curved space which is a Lorentzian variant of the 3-sphere. The curved space goes over into the usual flat space-time as the radius R of the curved space goes to infinity. We show, at least in some cases, that solutions of certain Dirichlet boundary value problems are obtained much more simply in the curved space than in the flat space. Since the flat space is the limit R → ∞ of the curved space, this gives an alternative method of obtaining solutions of a corresponding problem in Euclidean space.

scholarly journals Spherical Images of W-Direction Curves in Euclidean 3-Space

2020 ◽  
Vol 12 (3) ◽  
pp. 39
Author(s):  
Ìlkay Arslan Güven ◽  
Semra Kaya Nurkan ◽  
Ìpek Agaoglu Tor
Keyword(s):  
Vector Field ◽  
Euclidean Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Darboux Vector ◽  
Slant Helix ◽  
Spherical Images

In this paper, we study the spherical indicatrices of W-direction curves in three dimensional Euclidean space which were defined by using the unit Darboux vector field W of a Frenet curve. We obtain the Frenet apparatus of these spherical indicatrices and the characterizations of being general helix and slant helix. Moreover we give some properties between the spherical indicatrices and their associated curves.

scholarly journals The onset of thermalization in finite-dimensional equations of hydrodynamics: insights from the Burgers equation

2017 ◽  
Vol 473 (2197) ◽  
pp. 20160585 ◽  
Author(s):  
Divya Venkataraman ◽  
Samriddhi Sankar Ray
Keyword(s):  
Energy Spectrum ◽  
Numerical Simulations ◽  
Euler Equations ◽  
Burgers Equation ◽  
Three Dimensional ◽  
Direct Numerical Simulations ◽  
Infinite Dimensional ◽  
Localized Structures ◽  
Finite Dimensional ◽  
Incompressible Euler

Solutions to finite-dimensional (all spatial Fourier modes set to zero beyond a finite wavenumber K G ), inviscid equations of hydrodynamics at long times are known to be at variance with those obtained for the original infinite dimensional partial differential equations or their viscous counterparts. Surprisingly, the solutions to such Galerkin-truncated equations develop sharp localized structures, called tygers (Ray et al. 2011 Phys. Rev. E 84 , 016301 ( doi:10.1103/PhysRevE.84.016301 )), which eventually lead to completely thermalized states associated with an equipartition energy spectrum. We now obtain, by using the analytically tractable Burgers equation, precise estimates, theoretically and via direct numerical simulations, of the time τ c at which thermalization is triggered and show that τ c ∼ K G ξ , with ξ = − 4 9 . Our results have several implications, including for the analyticity strip method, to numerically obtain evidence for or against blow-ups of the three-dimensional incompressible Euler equations.

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