Asymmetric transmission: a generic property of two-dimensional periodic patterns

The theory of three‐dimensional gravity instability of multilayers is developed with particular application to salt structures. It is shown that three‐dimensional solutions are immediately obtained without further numerical work from the solution of the corresponding two‐dimensional problem. Application to a number of typical three‐dimensional structures yields the characteristic distance between peaks and crests and shows that this distance does not differ significantly from the wavelength of the two‐dimensional solution. Various periodic patterns are examined corresponding to rectangular and hexagonal cells. The time history of nonperiodic structures corresponding to initial deviations from perfect horizontality is also derived. The method is applied to the three‐dimensional problem of generation of salt structures when the time‐history of sedimentation is taken into account with variable thickness and compaction of the overburden and establishes the general validity of the geological conclusions derived from the previous two‐dimensional treatment of the same problem (Biot and Odé, 1965). The present method of deriving three‐dimensional solutions, which is developed here in the special context of gravity instability, is valid for a wide variety of problems in theoretical physics.

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Theoretical analysis of the zigzag instability of a vertical columnar vortex pair in a strongly stratified fluid

Journal of Fluid Mechanics ◽

10.1017/s0022112000001166 ◽

2000 ◽

Vol 419 ◽

pp. 29-63 ◽

Cited By ~ 48

Author(s):

PAUL BILLANT ◽

JEAN-MARC CHOMAZ

Keyword(s):

Froude Number ◽

Evolution Equations ◽

Vortex Pair ◽

Horizontal Velocity ◽

Phase Variable ◽

Two Dimensional ◽

Stratified Turbulence ◽

Periodic Patterns ◽

Phase Dynamics ◽

Columnar Vortex

A general theoretical account is proposed for the zigzag instability of a vertical columnar vortex pair recently discovered in a strongly stratified experiment.The linear inviscid stability of the Lamb–Chaplygin vortex pair is analysed by a multiple-scale expansion analysis for small horizontal Froude number (Fh = U/LhN, where U is the magnitude of the horizontal velocity, Lh the horizontal lengthscale and N the Brunt–Väisälä frequency) and small vertical Froude number (Fv = U/LvN, where Lv is the vertical lengthscale) using the scaling of the equations of motion introduced by Riley, Metcalfe & Weissman (1981). In the limit Fv = 0, these equations reduce to two-dimensional Euler equations for the horizontal velocity with undetermined vertical dependence. Thus, at leading order, neutral modes of the flow are associated, among others, to translational and rotational invariances in each horizontal plane. To each broken invariance is related a phase variable that may vary freely along the vertical. Conservation of mass and potential vorticity impose at higher order the evolution equations governing the phase variables that we derive for Fh [Lt ] 1 and Fv [Lt ] 1 in the spirit of phase dynamics techniques established for periodic patterns. In agreement with the experimental observations, this asymptotic analysis shows the existence of an instability consisting of a vertically modulated rotation and a translation of the columnar vortex pair perpendicular to the travelling direction. The dispersion relation as well as the spatial eigenmode of the zigzag instability are determined. The analysis predicts that the most amplified vertical wavelength should scale as U/N and the maximum growth rate as U/Lh.Our main finding is thus that the typical thickness of the ensuing layers will be such that Fv = O(1) and not Fv [Lt ] 1 as assumed by Riley et al. (1981) and Lilly (1983). This implies that such strongly stratified flows are not described by two- dimensional horizontal equations. These results may help to understand the layering commonly observed in stratified turbulence and the fundamental differences with strictly two-dimensional turbulence.

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Phase separation in the two-dimensional electron liquid in MOSFETs

Journal de Physique IV (Proceedings) ◽

10.1051/jp4:20020432 ◽

2002 ◽

Vol 12 (9) ◽

pp. 337-341

Author(s):

B. Spivak

Keyword(s):

Phase Separation ◽

Intermediate Phase ◽

Electron Gas ◽

Generic Property ◽

Wigner Crystal ◽

Two Dimensional ◽

Zero Temperature ◽

Dimensional Electron ◽

Physical Reason ◽

Supersolid Phase

We show that the existence of an intermediate phase between the Fermi liquid and the Wigner crystal phases is a generic property of the two-dimensional pure electron liquid in MOSFET's at zero temperature. The physical reason for the existence of the phases is a partial separation of the uniform phases. We discuss properties of these phases and a possible explanation of experimental results on transport properties of low density electron gas in MOSFET's. We also argue that in certain range of parameters the partial phase separation corresponds to a supersolid phase discussed in [25].

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