Results 1  10
of
11
A quaternionic structure in the threedimensional Euler and ideal magnetohydrodynamics equation
 PHYSICA D
, 2002
"... By considering the threedimensional incompressible Euler equations, a 4vector ζ is constructed out of a combination of scalar and vector products of the vorticity ω and the vortex stretching vector ω ·∇u = Sω. The evolution equation for ζ can then be cast naturally into a quaternionic Riccati equa ..."
Abstract

Cited by 17 (10 self)
 Add to MetaCart
By considering the threedimensional incompressible Euler equations, a 4vector ζ is constructed out of a combination of scalar and vector products of the vorticity ω and the vortex stretching vector ω ·∇u = Sω. The evolution equation for ζ can then be cast naturally into a quaternionic Riccati equation. This is easily transformed into a quaternionic zeroeigenvalue Schrödinger equation whose potential depends on the Hessian matrix of the pressure. With minor modifications, this system can alternatively be written in complex notation. An infinite set of solutions of scalar zeroeigenvalue Schrödinger equations has been found by Adler and Moser, which are discussed in the context of the present problem. Similarly, when the equations for ideal magnetohydrodynamics (MHD) are written in Elsasser variables, a pair of 4vectors ζ ± are governed by coupled
Quaternions and particle dynamics in Euler fluid equations
 NONLINEARITY 19 1969–83
"... Vorticity dynamics of the threedimensional incompressible Euler equations is cast into a quaternionic representation governed by the Lagrangian evolution of the tetrad consisting of the growth rate and rotation rate of the vorticity. In turn, the Lagrangian evolution of this tetrad is governed by a ..."
Abstract

Cited by 16 (7 self)
 Add to MetaCart
Vorticity dynamics of the threedimensional incompressible Euler equations is cast into a quaternionic representation governed by the Lagrangian evolution of the tetrad consisting of the growth rate and rotation rate of the vorticity. In turn, the Lagrangian evolution of this tetrad is governed by another that depends on the pressure Hessian. Together these form the basis for a direction of vorticity theorem. Moreover, in this representation, fluid particles carry orthonormal frames whose Lagrangian evolution in time are shown to be directly related to the FrenetSerret equations for a vortex line. The frame dynamics suggest an elegant Lagrangian relation regarding the pressure Hessian tetrad. The equations for ideal MHD are similarly considered.
An Exact Mapping from NavierStokes Equation to Schrödinger Equation via Riccati Equation
 PROGRESS IN PHYSICS
, 2008
"... In the present article we argue that it is possible to write down Schrödinger representation of NavierStokes equation via Riccati equation. The proposed approach, while differs appreciably from other method such as what is proposed by R. M. Kiehn, has an advantage, i.e. it enables us extend further ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
In the present article we argue that it is possible to write down Schrödinger representation of NavierStokes equation via Riccati equation. The proposed approach, while differs appreciably from other method such as what is proposed by R. M. Kiehn, has an advantage, i.e. it enables us extend further to quaternionic and biquaternionic version of NavierStokes equation, for instance via Kravchenko’s and Gibbon’s route. Further observation is of course recommended in order to refute or verify this proposition.
A geometric interpretation of coherent structures in NavierStokes Flows
 In preparation
, 2008
"... The pressure in the incompressible threedimensional NavierStokes and Euler equations is governed by a Poisson equation: this equation is studied using the geometry of three forms in six dimensions. By studying the linear algebra of the vector space of threeforms Λ 3 W ∗ where W is a sixdimension ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
The pressure in the incompressible threedimensional NavierStokes and Euler equations is governed by a Poisson equation: this equation is studied using the geometry of three forms in six dimensions. By studying the linear algebra of the vector space of threeforms Λ 3 W ∗ where W is a sixdimensional real vector space, we relate the characterization of nondegenerate elements of Λ 3 W ∗ to the sign of the Laplacian of the pressure — and hence to the balance between the vorticity and the rate of strain. When the Laplacian of the pressure, ∆p, satisfies ∆p> 0, the threeform associated with the Poisson equation is the real part of a decomposable complex form and an almostcomplex structure can be identified. When ∆p < 0 a real decomposable structure is identified. These results are discussed in the context of coherent structures in turbulence. 1 Equations for an Incompressible Fluid It is rare in fluid dynamics for highly technical abstract geometrical criteria to have a direct correspondence with experimental observations. In a seminal paper, Douady et al.
Kähler Geometry and Burgers’ Vortices
, 2008
"... We study the NavierStokes and Euler equations of incompressible hydrodynamics in two spatial dimensions. Taking the divergence of the momentum equation leads, as usual, to a Poisson equation for the pressure: in this paper we study this equation using MongeAmpère structures. In two dimensional flo ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We study the NavierStokes and Euler equations of incompressible hydrodynamics in two spatial dimensions. Taking the divergence of the momentum equation leads, as usual, to a Poisson equation for the pressure: in this paper we study this equation using MongeAmpère structures. In two dimensional flows where the laplacian of the pressure is positive, a Kähler geometry is described on the phase space of the fluid; in regions where the laplacian of the pressure is negative, a product structure is described. These structures can be related to the ellipticity and hyperbolicity (respectively) of a MongeAmpère equation. We then show how this structure can be extended to a class of canonical vortex structures in three dimensions.
HyperKähler geometry and semigeostrophic theory
"... We use the formalism of MongeAmpère operators to study the geometric properties of the MongeAmpère equations arising in semigeostrophic theory and related models of geophysical fluid dynamics. We show how Kähler and hyperKähler structures arise, and the Legendre duality arising in semigeostroph ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We use the formalism of MongeAmpère operators to study the geometric properties of the MongeAmpère equations arising in semigeostrophic theory and related models of geophysical fluid dynamics. We show how Kähler and hyperKähler structures arise, and the Legendre duality arising in semigeostrophic theory is generalized to other models of nearly geostrophic flows. Keywords: Semigeostrophic theory, Balanced models, MongeAmpère equations, hyperKähler geometry
In the present article we argue that it is possible to write down Schrödinger representation
"... ..."
(Show Context)
1 Kähler Geometry and the NavierStokes Equations
, 2005
"... We study the NavierStokes and Euler equations of incompressible hydrodynamics in two and three spatial dimensions and show how the constraint of incompressiblility leads to equations of Monge–Ampère type for the stream function, when the Laplacian of the pressure is known. In two dimensions a Kähle ..."
Abstract
 Add to MetaCart
We study the NavierStokes and Euler equations of incompressible hydrodynamics in two and three spatial dimensions and show how the constraint of incompressiblility leads to equations of Monge–Ampère type for the stream function, when the Laplacian of the pressure is known. In two dimensions a Kähler geometry is described, which is associated with the Monge–Ampère problem. This Kähler structure is then generalised to ‘twoandahalf dimensional ’ flows, of which Burgers ’ vortex is one example. In three dimensions, we show how a generalized Calabi–Yau structure emerges in a special case. 1 Equations for an Incompressible Fluid Flow visualization methods, allied to largescale computations of the threedimensional incompressible NavierStokes equations, vividly illustrate the fact that vorticity has a tendency to accumulate on ‘thin sets ’ whose morphology is characterized by quasi onedimensional tubes or filaments and quasi twodimensional sheets. This description is in itself approximate as these thin structures undergo dramatic morphological changes in time and space. The topology is highly complicated; sheets tend to rollup into tubelike structures, while tubes tangle and knot like spaghetti boiling in a pan (Vincent & Meneguzzi 1994). Moreover, vortex tubes usually have short lifetimes, vanishing at one place and reforming at another. The behaviour of NavierStokes flows diverge in behaviour from Euler flows once viscosity has taken effect in reconnection processes. Nevertheless, the creation and early/intermediate evolution of their vortical sets appear to be similar. No adequate mathematical theory has been forthcoming explaining why thin sets tend to be favoured. The purpose of this paper is to investigate this enduring question in the light of the recent advances made in in the geometry of Kähler and other complex manifolds. While many difficult questions remain to be solved and explored, we believe that sufficient evidence exists that suggests that threedimensional turbulent vortical dynamics may be governed by geometric principles. The incompressible NavierStokes equations, in two or three dimensions, are ∂u ∂t + u · ∇u +
Complex Contact and Lift Transformations
, 2012
"... We study mappings from sets of real variables into complex variables, which extend features of lift and contact transformations between real variables that we explored in a previous paper. In particular the relationship between lifts in R2n+1 and the CauchyRiemann equations for functions of n compl ..."
Abstract
 Add to MetaCart
We study mappings from sets of real variables into complex variables, which extend features of lift and contact transformations between real variables that we explored in a previous paper. In particular the relationship between lifts in R2n+1 and the CauchyRiemann equations for functions of n complex variables is discussed. Explicit examples are given to illustrate the anatomy of such transformations, including the occurrence of singularities. Applications to nonlinear partial differential equations arising in fluid mechanics are presented.