Euler-Lagrange Transformation Research Articles

Automatic Euler—Lagrange Transformation of Nonlinear Progressive Waves in Water of Uniform Depth

In this paper, we provide a technical note for the implementation of a symbolic Euler-Lagrange transformation on nonlinear progressive water waves. By the implementation, the solution of nonlinear steady waves in the Eulerian description can be obtained and then transformed to the corresponding solution in the Lagrangian description automatically. We consider four closure conditions in a single symbolic implementation to derive both the Eulerian and Lagrangian solutions. In order to accomplish an automatically symbolic computation, we propose trigonometry-polynomial transformations to convert the hierarchical system of differential equations of different orders into a system of algebraic ones. During the symbolic computation, the solutions of a user-prescribed order can be automatically obtained via multiple loops. For example, the fifth- and seventh-order Euler-Lagrange solutions are obtained within 7 min and 3.5 h, respectively.

A modified Euler–Lagrange transformation for particle orbits in nonlinear progressive waves

This paper presents a modified Euler–Lagrange transformation method to obtain the third-order trajectory solution in a Lagrangian form for the water particles in nonlinear water waves. We impose the assumption that the Lagrangian wave frequency is a function of wave steepness and an arbitrary vertical position for each water particle. Expanding the unknown function in a small perturbation parameter and using a successive expansion in a Taylor series for the water particle path and the period of a particle motion, the third-order asymptotic expressions for the Lagrangian particle trajectories, the mass transport velocity and the period of particle motion can be derived directly in Lagrangian form. The wave frequency and mean level of the particle motion in Lagrangian form differ from those of the Eulerian. Finally, the third-order asymptotic solution obtained is uniformly valid in contrast with early works containing resonant terms presented by Wiegel [1964. Oceanographical Engineering. Prentice-Hall, New Jersey, pp. 37–40] (Eqs. (B.1) and (B.2) in Appendix B) or Chen et al.[2006. Theoretical analysis of surface waves shoaling and breaking on a sloping bottom. Part 2 nonlinear waves. Wave motion, 43, 356–369] based on a straightforward expansion for two-dimensional progressive waves.

NONLINEAR WATER WAVES ON UNIFORM CURRENT IN LAGRANGIAN COORDINATES

This paper presents a new third-order trajectory solution in Lagrangian form for the water particles in a wave-current interaction flow based on an Euler–Lagrange transformation. The explicit 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 Eulerian approach. Mass transport associated with a particle displacement can now be obtained directly in Lagrangian form. The angular frequency and Lagrangian mean level of the particle motion in Lagrangian form differ from those of the Eulerian. The variations in the wave profile and the water particle orbits resulting from the interaction with a steady uniform current of different magnitudes are also investigated. Comparison on the wave profiles given by the Eulerian and Lagrangian solution to a third-order reveals that the latter is more accurate than the former in describing the shape of the wave profile. Moreover, the influence of a following current is found to increase the relative horizontal distance traveled by a water particle, while the converse is true in the case of an opposing current.

EULER–LAGRANGE CORRESPONDENCE OF GENERALIZED BURGERS CELLULAR AUTOMATON

Recently, we have proposed a Euler–Lagrange transformation for cellular automata (CA) by developing new transformation formulas. Applying this method to the Burgers CA (BCA), we have succeeded in obtaining the Lagrange representation of the BCA. In this paper, we apply this method to multi-value generalized Burgers CA (GBCA) which include the Fukui–Ishibashi model and the quick-start model associated with traffic flow. As a result, we have succeeded in clarifying the Euler–Lagrange correspondence of these models. It turns out moreover that the GBCA can naturally be considered as a simple model of a multi-lane traffic flow.

Euler-lagrange correspondence of cellular automaton for traffic-flow models.

We propose a Euler-Lagrange transformation for cellular automata (CA) by developing new explicit transformation formulas. This transformation is done in the fully discrete level of variables, and corresponds to the well-known continuous version of it which appears in continuous mechanics such as fluid dynamics and plasma physics. Applying this method to the traffic problem, we have obtained the Lagrange representation of a traffic model, and also succeeded in clarifying the relation between different types of traffic models. It is shown that the Burgers CA, which is a corresponding CA of the continuous Burgers equation, plays a central role in considering this relation.

Detection of curved edges at subpixel accuracy using deformable models

One approach to the detection of curves at subpixel accuracy involves the reconstruction of such features from subpixel edge data points. A new technique is presented for reconstructing and segmenting curves with subpixel accuracy using deformable models. A curve is represented as a set of interconnected Hermite splines forming a snake generated from the subpixel edge information that minimises the global energy functional integral over the set. While previous work on the minimisation was mostly based on the Euler-Lagrange transformation, the authors use the finite element method to solve the energy minimisation equation. The advantages of this approach over the Euler-Lagrange transformation approach are that the method is straightforward, leads to positive m-diagonal symmetric matrices, and has the ability to cope with irregular geometries such as junctions and corners. The energy functional integral solved using this method can also be used to segment the features by searching for the location of the maxima of the first derivative of the energy over the elementary curve set.

The tidal whirlpool: A review of horizontal dispersion by tidal and residual currents

Horizontal dispersion by tidal and residual currents is discussed in terms of two different mechanisms, viz. a “cascade” of shear-dispersion processes and “Lagrangian chaos”. In the first one, turbulence interacts with vertical shear, producing horizontal dispersion. In turn, that mechanism interacts with horizontal shear of residual currents, giving rise to a large effective dispersion in the residual current direction. In “Lagrangian chaos”, genuine turbulence is absent and the Eulerian velocity is supposed to be a simple periodic function of time and, whether or not, of the horizontal position. Particle-trajectories may then become random functions of time, either by spatial randomness of the Eulerian field, or by the appearance of deterministic chaos in the Euler-Lagrange transformation.

On the Euler-Lagrange transformation and the stokes' drift in the presence of oscillatory and residual currents

It is shown that in an Eulerian velocity field consisting of a spatially varying component on which a uniform oscillatory component is superimposed, the residual mass transport velocity at a certain place is in general different from the residual Lagrangian velocity of a parcel labeled at that place. The approximation of the rigorous Euler-Lagrange transformation leading to the familiar derivation of the Stokes' drift is examined. The latter is a two-fold truncation of expansions of the former in a series of equations and a Taylor series of derivatives. The truncation of the latter series breaks down in cases of considerable spatial variability of the current velocity field, whereas the truncation of the former series may still be used if the uniform current velocity component energetically dominates the spatially varying component. It is then shown that the error introduced by substituting the local mass transport velocity for the residual Lagrangean velocity depends on the ratio of the amplitude of horizontal displacement, due to the uniform oscillatory component, to the integral length scale of the spatially varying component of the velocity field. Errors are largest if this ratio is of order 1. Such a situation may occur particularly in shallow tidal areas of complicated geometry.

On the Euler-Lagrange transformation and the Stokes drift in the prescence of oscillatory and residual currents : Zimmerman, J. T. F., 1979. Deep-Sea Res., 26(5A): 505–520

On the Euler-Lagrange transformation and the Stokes drift in the prescence of oscillatory and residual currents : Zimmerman, J. T. F., 1979. Deep-Sea Res., 26(5A): 505–520