The gradient of potential vorticity, quaternions and an orthonormal frame for fluid particles

The gradient of potential vorticity (PV) is an important quantity because of the way PV (denoted as $q$) tends to accumulate locally in the oceans and atmospheres. Recent analysis by the authors has shown that the vector quantity \bdB = \bnabla q\times \bnabla\theta for the three-dimensional incompressible rotating Euler equations evolves according to the same stretching equation as for \bom the vorticity and \bB, the magnetic field in magnetohydrodynamics (MHD). The \bdB-vector therefore acts like the vorticity \bom in Euler's equations and the \bB-field in MHD. For example, it allows various analogies, such as stretching dynamics, helicity, superhelicity and cross helicity. In addition, using quaternionic analysis, the dynamics of the \bdB-vector naturally allow the construction of an orthonormal frame attached to fluid particles\,; this is designated as a quaternion frame. The alignment dynamics of this frame are particularly relevant to the three-axis rotations that particles undergo as they traverse regions of a flow when the PV gradient \bnabla q is large.Comment: Dedicated to Raymond Hide on the occasion of his 80th birthda

Lagrangian analysis of alignment dynamics for isentropic compressible magnetohydrodynamics

After a review of the isentropic compressible magnetohydrodynamics (ICMHD) equations, a quaternionic framework for studying the alignment dynamics of a general fluid flow is explained and applied to the ICMHD equations.Comment: 12 pages, 2 figures, submitted to a Focus Issue of New Journal of Physics on "Magnetohydrodynamics and the Dynamo Problem" J-F Pinton, A Pouquet, E Dormy and S Cowley, editor

Optical control of solar sails using distributed reflectivity

The dynamics of solar sails with a variable surface reflectivity distribution are investigated. When changing the reflectivity across the sail film, which can be achieved using electro-chromic coatings, the solar radiation pressure forces and torques across the sail film can be controlled without changing the attitude of the spacecraft relative to the Sun and without using mechanical systems. The paper presents two approaches. First, a continuous reflectivity distribution is presented to control the sail attitude under the influence of, for example, gravity gradient torques in Earth orbit. The second approach assumes discrete on/o reflectivity regions across the surface. Both concepts of `optical reconfiguration' of solar sails enable a more flexible steering of the spacecraft and minimize actuation effort

Açılabilir Regle Yüzeylerin Küresel Görüntüleri Üzerine

Konferans Bildirisi -- Teorik ve Uygulamalı Mekanik Türk Milli Komitesi, 2013Conference Paper -- Theoretical and Applied Mechanical Turkish National Committee, 2013Açılabilir yüzeyler CAGD de yaygın bir şekilde kullanılmaktadır. Bu makalede, açılabilir yüzeylerin küresel görüntüsü SLERP vasıtasıyla oluşturulmuş, açılabilir yüzeyler yerine küresel eğriler üzerinde çalışmanın avantajı ifade edilmiştir. Ayrıca açılabilir yüzeylerin özellikleri ile SLERP hakkında gerekli hatırlatmalar yapılmıştır. Anahtar kelimeler- Açılabilir yüzeyler, kuaterniyon, SLERP, Gauss map.Developable surfaces are used widely in CAGD. In this paper, the spherical image of developable surfaces via SLERP is formed. Also, the advantage of working on the spherical curves instead of developable surfaces is expressed. In addition to them, required reminders about properties of developable surfaces and SLERP are given. Key words- Developable surfaces, Gauss map, quaternion, SLERP

Lagrangian particle paths and ortho-normal quaternion frames

Experimentalists now measure intense rotations of Lagrangian particles in turbulent flows by tracking their trajectories and Lagrangian-average velocity gradients at high Reynolds numbers. This paper formulates the dynamics of an orthonormal frame attached to each Lagrangian fluid particle undergoing three-axis rotations, by using quaternions in combination with Ertel's theorem for frozen-in vorticity. The method is applicable to a wide range of Lagrangian flows including the three-dimensional Euler equations and its variants such as ideal MHD. The applicability of the quaterionic frame description to Lagrangian averaged velocity gradient dynamics is also demonstrated.Comment: 9 pages, one figure, revise