Lagrangian Properties Research Articles

Mass Mixing Effect and Oblique Radiative Corrections in ExtendedSU(2)R×SU(2)L×U(1)Effective Theory

We analyze the properties of electroweak chiral effective Lagrangian with an extendedSU(2)Rgauge group. Right-handedSU(2)Rgauge bosons affect electroweak observables by mixing with electroweak gauge bosonsWL,μandBμ. We discuss all possible mass mixing terms and calculate the exact physical mass eigenvalues by diagonalization of mixing matrix without any approximate assumptions. The contributions to oblique radiative corrections parameters STU fromSU(2)Rfields are also presented.

Lagrangian transport properties of pulmonary interfacial flows

Disease states characterized by airway fluid occlusion and pulmonary surfactant insufficiency, such as respiratory distress syndrome, have a high mortality rate. Understanding the mechanics of airway reopening, particularly involving surfactant transport, may provide an avenue to increase patient survival via optimized mechanical ventilation waveforms. We model the occluded airway as a liquid-filled rigid tube with the fluid phase displaced by a finger of air that propagates with both mean and sinusoidal velocity components. Finite-time Lyapunov exponent (FTLE) fields are employed to analyse the convective transport characteristics, taking note of Lagrangian coherent structures (LCSs) and their effects on transport. The Lagrangian perspective of these techniques reveals flow characteristics that are not readily apparent by observing Eulerian measures. These analysis techniques are applied to surfactant-free velocity fields determined computationally, with the boundary element method, and measured experimentally with micro particle image velocimetry (μ-PIV). We find that the LCS divides the fluid into two regimes, one advected upstream (into the thin residual film) and the other downstream ahead of the advancing bubble. At higher oscillatory frequencies particles originating immediately inside the LCS experience long residence times at the air-liquid interface, which may be conducive to surfactant transport. At high frequencies a well-mixed attractor region is identified; this volume of fluid cyclically travels along the interface and into the bulk fluid. The Lagrangian analysis is applied to velocity data measured with 0.01 mg ml(-1) of the clinical pulmonary surfactant Infasurf in the bulk fluid, demonstrating flow field modifications with respect to the surfactant-free system that were not visible in the Eulerian frame.

On the Lagrangian formulation of multidimensionally consistent systems

Multidimensional consistency has emerged as a key integrability property for partial difference equations (PΔEs) defined on the ‘space–time’ lattice. It has led, among other major insights, to a classification of scalar affine-linear quadrilateral PΔEs possessing this property, leading to the so-called Adler–Bobenko–Suris (ABS) list. Recently, a new variational principle has been proposed that describes the multidimensional consistency in terms of discrete Lagrangian multi-forms. This description is based on a fundamental and highly non-trivial property of Lagrangians for those integrable lattice equations, namely the fact that on the solutions of the corresponding PΔE the Lagrange forms are closed, i.e. they obey a closure relation . Here, we extend those results to the continuous case: it is known that associated with the integrable PΔEs there exist systems of partial differential equations (PDEs), in fact differential equations with regard to the parameters of the lattice as independent variables, which equally possess the property of multidimensional consistency. In this paper, we establish a universal Lagrange structure for affine-linear quad-lattices alongside a universal Lagrange multi-form structure for the corresponding continuous PDEs, and we show that the Lagrange forms possess the closure property.

A quantitative characterization of the seasonal Lagrangian circulation of the Gulf of California from a three-dimensional numerical model

A set of four indices that quantify Lagrangian properties of the Gulf of California seasonal circulation were implemented from outputs of a three-dimensional numerical model. From trajectories of particles seeded over the entire Gulf, we calculated for 12 one-month periods the following indices: net and total distance traveled by the particles, the number of particles that are found within an area centered on the release positions after one month, and time taken by particles to escape from a 50-km-radius circle. These indices can be used for studies on transport of inert properties and passive planktonic organisms such as eggs and early-stage larvae; their use is illustrated for typical summer and winter conditions in the Gulf of California. These indices show the potential for connecting areas separated by a few hundreds of km along the eastern side of the Gulf, due to the strong seasonal up-gulf and down-gulf current. In the Northern Gulf, large displacements occur at the borders of the basin-wide seasonally reversing eddy that dominates the large-scale circulation (cyclonic in summer, anticyclonic in winter). On the other hand, the potential for self-recruitment areas is found as particles can be trapped for longer than one month within these eddies, as well as in smaller ones in the Northern Gulf, and near the coast of the peninsular side of the Southern Gulf, where current speeds are slow and many small capes and islands are present.

Nontwist symplectic maps in tokamaks

We review symplectic nontwist maps that we have introduced to describe Lagrangian transport properties in magnetically confined plasmas in tokamaks. These nontwist maps are suitable to describe the formation and destruction of transport barriers in the shearless region (i.e., near the curve where the twist condition does not hold). The maps can be used to investigate two kinds of problems in plasmas with non-monotonic field profiles: the first is the chaotic magnetic field line transport in plasmas with external resonant perturbations. The second problem is the chaotic particle drift motion caused by electrostatic drift waves. The presented analytical maps, derived from plasma models with equilibrium field profiles and control parameters that are commonly measured in plasma discharges, can be used to investigate long-term transport properties.

Lagrangian properties of turbulent diffusion with passive chemical reaction in the framework of the premixed combustion theory

In this paper, we analyze two effects caused by the Lagrangian nature of turbulent transfer which are usually ignored in the theory of turbulent premixed combustion. These effects are (i) the nonequilibrium behavior of the turbulent diffusion coefficient, which is important for modeling the initial stage of combustion (for example, in the spark ignition engine), and (ii) the existence of a traveling front of turbulent diffusion with finite speed, which controls the velocity of the steady state flame in strong turbulence. However, in order to derive simple and exact results, the hydrodynamical and the combustion subproblems are stated to be independent by assuming a constant density so that a passive chemical reaction is actually considered. First, we derive a parabolic diffusion equation with both diffusion and chemical source terms expressed by Lagrangian characteristics of turbulence. We show that, in general, the diffusivity of product particles is not zero in the moment of their generation by chemical...

Non-convex self-dual Lagrangians: New variational principles of symmetric boundary value problems

We study the concept and the calculus of Non-convex self-dual ( Nc-SD) Lagrangians and their derived vector fields which are associated to many partial differential equations and evolution systems. They indeed provide new representations and formulations for the superposition of convex functions and symmetric operators. They yield new variational resolutions for large class of Hamiltonian partial differential equations with variety of linear and nonlinear boundary conditions including many of the standard ones. This approach seems to offer several useful advantages: It associates to a boundary value problem several potential functions which can often be used with relative ease compared to other methods such as the use of Euler–Lagrange functions. These potential functions are quite flexible, and can be adapted to easily deal with both nonlinear and homogeneous boundary value problems. Additionally, in most cases the solutions generated using this new method have greater regularity than the solutions obtained using the standard Euler–Lagrange function. Perhaps most remarkable, however, are the permanence properties of Nc-SD Lagrangians; their calculus is relatively manageable, and their applications are quite broad.

Dispersion of surface drifters and model-simulated trajectories

From a data set encompassing the years 1990–2008 pairs of surface drifters with maximum initial separations of 5, 10 and 25 km have been identified. Model trajectories have been calculated using the same initial positions and times as the selected pairs of surface drifters. The model trajectories are based on the TRACMASS trajectory code and driven by the ocean general circulation model NEMO. The trajectories are calculated off-line, i.e. with the stored velocity fields from the circulation model. The sensitivity of the trajectory simulations to the frequency of the stored velocity fields was tested for periods of 3 and 6 h as well as 5 days. The relative dispersion of the surface-drifter and model trajectories has been compared, where the latter was found to be too low compared to the relative dispersion of the drifters. Two low-order trajectory sub-grid parameterisations were tested and successfully tuned so that the total amplitude of the relative dispersion of the model trajectories is similar to that associated with the drifter trajectories. These parameterisations are, however, too simple for a correct simulation of Lagrangian properties such as the correlation time scales and the variance of the eddy kinetic energy. The importance of model-grid resolution is quantified by comparing the relative dispersion from an eddy-permitting and a coarse-resolution model, respectively. The dispersion rate is halved with the coarse grid. The consequences of the two-dimensionality of the trajectories is evaluated by comparing the results obtained with the 2D and the Lagrangian 3D trajectories. This shows that the relative dispersion is 15% stronger when the trajectories are freely advected with the 3D velocity field.

Duality Gap Estimation of Linear Equality Constrained Binary Quadratic Programming

We investigate in this paper the Lagrangian duality properties of linear equality constrained binary quadratic programming. We derive an underestimation of the duality gap between the primal problem and its Lagrangian dual or SDP relaxation, using the distance from the set of binary integer points to certain affine subspace, while the computation of this distance can be achieved by the cell enumeration of hyperplane arrangement. Alternative Lagrangian dual schemes via the exact penalty and the squared norm constraint reformulations are also discussed.

Properties of an augmented Lagrangian for design optimization

We consider the task of design optimization, where the constraint is a state equation that can only be solved by a typically rather slowly converging fixed point solver. This process can be augmented by a corresponding adjoint solver, and based on the resulting approximate reduced derivatives, also an optimization iteration that updates the design variables simultaneously. To coordinate the three iterative processes, we use an exact penalty function of a doubly augmented Lagrangian type that should be consistently reduced. Some numerical experiments on a variant of the Bratu problem are presented.

Looking for the best constant in a Sobolev inequality: a numerical approach

A numerical method for the computation of the best constant in a Sobolev inequality involving the spaces H 2(?) and $C^{0}(\overline{\Omega})$ is presented. Green's functions corresponding to the solution of Poisson problems are used to express the solution. This (kind of) non-smooth eigenvalue problem is then formulated as a constrained optimization problem and solved with two different strategies: an augmented Lagrangian method, together with finite element approximations, and a Green's functions based approach. Numerical experiments show the ability of the methods in computing this best constant for various two-dimensional domains, and the remarkable convergence properties of the augmented Lagrangian based iterative method.

Lagrangian experiment and solution for irrotational finite-amplitude progressive gravity waves at uniform depth

This study investigates the theory for predicting Lagrangian properties including particle orbit, Lagrangian mean level, Lagrangian wave frequency, mass transport velocity, wave profile, velocity distribution and wave pressure in progressive gravity water waves at uniform depth. A series of laboratory experiments are performed to measure the trajectories of particles and the wave pressure. Asymptotic solutions up to fifth order that describe irrotational finite amplitude progressive gravity water waves are derived in completely Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid satisfies the irrotational condition, the dynamic boundary condition and the zero pressure at the free surface. The explicit fifth-order parametric solution highlights the trajectory of a water particle and the wave kinematics above the mean water level and within a vertical water column, which were calculated previously by an approximation method using an Eulerian approach. Mass transport up to fourth order associated with a particle displacement can now be obtained directly in Lagrangian form. In particular, the Lagrangian wave frequency and the Lagrangian mean level of particle motion can also be obtained, which are different from those in an Eulerian description. By comparing the present fifth-order asymptotic solution with data from laboratory experiments, it is found that theoretical results show good agreement with experimental data.

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions

We apply the quadratic penalization technique to derive strong Lagrangian duality property for an inequality constrained invex program. Our results extend and improve the corresponding results in the literature.

The Analysis of Lagrangian and Hamiltonian Properties of the Classical Relativistic Electrodynamics Models and Their Quantization

The Lagrangian and Hamiltonian properties of classical electrodynamics models and their associated Dirac quantizations are studied. Using the vacuum field theory approach developed in (Prykarpatsky et al. Theor. Math. Phys. 160(2): 1079–1095, 2009 and The field structure of a vacuum, Maxwell equations and relativity theory aspects. Preprint ICTP) consistent canonical Hamiltonian reformulations of some alternative classical electrodynamics models are devised, and these formulations include the Lorentz condition in a natural way. The Dirac quantization procedure corresponding to the Hamiltonian formulations is developed. The crucial importance of the rest reference systems, with respect to which the dynamics of charged point particles is framed, is explained and emphasized. A concise expression for the Lorentz force is derived by suitably taking into account the duality of electromagnetic field and charged particle interactions. Finally, a physical explanation of the vacuum field medium and its relativistic properties fitting the mathematical framework developed is formulated and discussed.

Lyapunov exponents and information dimension of the mass distribution in turbulent compressible flows

Turbulent density fluctuations in isothermal highly compressible turbulent flows are highly clumped and can be quantified by the scaling properties of powers of the mass distribution. This Eulerian quantity can be related to Lagrangian properties of the system given by the Lyapunov exponents of tracer particles advected with the flow. Using highly resolved numerical simulations, we show that the Kaplan–Yorke conjecture holds within numerical uncertainties.

Experimental investigation of Lagrangian structure functions in turbulence

Lagrangian properties obtained from a particle tracking velocimetry experiment in a turbulent flow at intermediate Reynolds number are presented. Accurate sampling of particle trajectories is essential in order to obtain the Lagrangian structure functions and to measure intermittency at small temporal scales. The finiteness of the measurement volume can bias the results significantly. We present a robust way to overcome this obstacle. Despite no fully developed inertial range, we observe strong intermittency at the scale of dissipation. The multifractal model is only partially able to reproduce the results.

Polar night vortex breakdown and large-scale stirring in the southern stratosphere

The present paper examines the vortex breakdown and large-scale stirring during the final warming of the Southern Hemisphere stratosphere during the spring of 2005. A unique set of in situ observations collected by 27 superpressure balloons (SPBs) is used. The balloons, which were launched from McMurdo, Antarctica, by the Strateole/VORCORE project, drifted for several weeks on two different isopycnic levels in the lower stratosphere. We describe balloon trajectories and compare them with simulations obtained on the basis of the velocity field from the GEOS-5 and NCEP/NCAR reanalyses performed with and without VORCORE data. To gain insight on the mechanisms responsible for the horizontal transport of air inside and outside the well-isolated vortex we examine the balloon trajectories in the framework of the Lagrangian properties of the stratospheric flow. Coherent structures of the flow are visualized by computing finite-time Lyapunov exponents (FTLE). A combination of isentropic analysis and FTLE distributions reveals that air is stripped away from the vortex’s interior as stable manifolds eventually cross the vortex’s edge. It is shown that two SPBs escaped from the vortex within high potential vorticity tongues that developed in association with wave breaking at locations along the vortex’s edge where forward and backward FTLE maxima approximately intersect. The trajectories of three SPBs flying as a group at the same isopycnic level are examined and their behavior is interpreted in reference to the FTLE field. These results support the concept of stable and unstable manifolds governing transport of air masses across the periphery of the stratospheric polar vortex.

Eulerian indicators for predicting and optimizing mixing quality

Many different methods exist for measuring, quantifying and predicting the quality of a mixture resulting from a fluid mixing device. In general, these rely on Lagrangian information from the system, as they involve the evolution of individual trajectories. In this paper, we propose tools that are formulated using only Eulerian information, and yet can predict the Lagrangian mixing properties of a mixing device. The proposed Eulerian indicators are motivated by the recent linked twist map (LTM) formulation of fluid mixing. We describe their features and application in the context of four different fluid flows: the blinking vortex flow and the partitioned pipe mixer (two paradigm flows from the early studies of chaotic mixing), a model of a channel flow containing ‘separatrices’ in the cross-sectional flow (which is a common flow feature in a large number of micromixers), and a flow generated by pulsed source–sink pairs (which has been used as a model for mixing in DNA microarrays).

Lagrangian studies in convective turbulence

We present high-resolution direct numerical simulations of turbulent three-dimensional Rayleigh-Bénard convection with a focus on the Lagrangian properties of the flow. The volume is a Cartesian slab with an aspect ratio of four bounded by free-slip planes at the top and bottom and with periodic side walls. The turbulence is inhomogeneous with respect to the vertical direction. This manifests in different lateral and vertical two-particle dispersion and in a dependence of the dispersion on the initial tracer position for short and intermediate times. Similar to homogeneous isotropic turbulence, the dispersion properties depend in addition to the initial pair separation and yield a short-range Richardson-like scaling regime of two-particle dispersion for initial separations close to the Kolmogorov dissipation length. The Richardson constant is about half the value of homogeneous isotropic turbulence. The multiparticle statistics is very close to the homogeneous isotropic case. Clusters of four Lagrangian tracers show a clear trend to form flat, almost coplanar, objects in the long-time limit and deviate from the Gaussian prediction. Significant efforts have been taken to resolve the statistics of the acceleration components up to order four correctly. We find that the vertical acceleration is less intermittent than the lateral one. The joint statistics of the vertical acceleration with the local convective and conductive heat flux suggests that rising and falling thermal plumes are not associated with the largest acceleration magnitudes. It turns out also that the Nusselt number which is calculated in the Lagrangian frame converges slowly in time to the standard Eulerian one.

The Lagrangian and Hamiltonian analysis of some relativistic electrodynamics models and their quantization

The work is devoted to the study of the Lagrangian and Hamiltonian properties of some relativistic electrodynamics models and is a continuation of our previous investigations. Based on the vacuum field theory approach, the Lagrangian and Hamiltonian reformulation of some classical electrodynamics models is devised. The Dirac type quantization procedure, based on the canonical Hamiltonian formulation, is developed. Within the approach proposed in the work a possibility of the combined description both of electrodynamics and gravity is analyzed.