Nonpotential Effects Research Articles

Energy Dissipation and Nonpotential Effects in Wave Breaking

This paper presents a comparative numerical study of the energy dissipation process in the breaking of focused waves by using a potential flow model and a coupled potential/viscous flow model. An empirical eddy viscosity term is introduced to the fully-nonlinear potential (FNP) flow model to account for the breaking wave energy dissipation. The FNP model is further coupled with an incompressible two-phase NavierStokes (NS) flow solver through a one-way linkage to generate and propagate focused waves in the domain. Numerical absorbing regions are placed in front of the outlet boundaries to dampen wave reflection. The standalone FNP model and the coupled FNP+NS model are applied to deal with each scenario comparatively. This enables the accurate quantification and comparison of the energy loss of breaking waves calculated by the two numerical models. The velocity field is decomposed into potential component, which is reconstructed from the corresponding free surface elevation computed in the coupled model by using the weakly-nonlinear wave theory, and non-potential rotational component. Detailed analysis of the numerical results shows that: (1) energy loss is closely related to wave steepness, (2) mild rotational motion produced by a non-breaking wave is local in time with a short life-span, (3) strong non-potential motion triggered by wave breaking is not local in time but persists in the flow for dozens of or even many more wave periods.

ВЛИЯНИЯ ВОЗДЕЙСТВИЙ ПРИКАТЫВАЮЩИХ КАТКОВ НА ОБОБЩЕННЫЕ СИЛЫ МАЛОГАБАРИТНОЙ ПОЧВООБРАБАТЫВАЮЩЕЙ МАШИНЫ

Increasing the efficiency of agricultural production, in particular the production of crop products in personal subsidiary plots, depends on the development and implementation of high-tech machines and their working units. The need for small-sized tillage equipment is increasing from year to year. But in order to increase the functionality of the equipment and the quality of soil preparation for sowing, it is necessary to study the possibility of using various additional working units.
 The purpose of the work is theoretical studies of the movement and kinematic connection of a trailed slatted-spiral roller with a walk-behind tractor, which make it possible to substantiate rational design and technological parameters of a small-sized tillage tool. The study used the provisions of classical mechanics and analytical geometry, methods of equilibrium and motion of mechanical systems based on differential and integral principles of mechanics. The design of active and passive rollers for a walk-behind tractor is considered, which allows to qualitatively prepare the soil for sowing at the depth of seeding, the influence of potential and non-potential effects on their generalized forces is revealed, the angular velocity and their acceleration are determined, as well as the dynamic characteristics of the moment of inertia of the rollers relative to the axes of rotation X4 and Z4 and their frames relative to the axes of rotation X3 and Z3. The difference in the generalized force for a passive roller relative to the angle φ 5.49 N∙m was obtained. The angular speed of the active roller is 23.0 rad/s higher than that of the passive roller, and as a result of research it was revealed that the moments of inertia of the active roller and its frame relative to the axles are significantly higher than that of the passive roller. Generalized forces for an active roller relative to the angle q = 2.58 N∙m and relative to the angle φ = 1.98 N∙m, for a passive roller - relative to the angle q = 2.32 N∙m and relative to the angle φ = 7.47 N∙m. The generalized forces for the potential effects of an active roller are Qθa=1.58 N∙m, Qφa=2.26 N∙m, for a passive roller Qθn=1.32 N∙m, Qφn= 4.60 N∙m. Mθa = 1 N∙m, Mφa = 114.63 N∙m; passive roller - respectively Mθn = 1 N∙m, Mφn = 178.9 N∙m

Measurement of the open magnetic flux in the inner heliosphere down to 0.13 AU

Context.Robustly interpreting sets of in situ spacecraft data of the heliospheric magnetic field (HMF) for the purpose of probing the total unsigned magnetic flux in the heliosphere is critical for constraining global coronal models as well as understanding the large scale structure of the heliosphere itself. The heliospheric flux (ΦH) is expected to be a spatially conserved quantity with a possible secular dependence on the solar cycle and equal to the measured radial component of the HMF weighted by the square of the measurement’s heliographic distance (BRR2). It is also expected to constitute a direct measurement of the total unsigned magnetic flux escaping the corona (Φopen). Previous work indicates that measurements of ΦHexceed the value predicted by standard coronal models (the “open flux problem”). However, the value of the open flux derived from in situ measurements remains uncertain because it depends on the method employed to derive it. Past derivations also pointed towards an increase in ΦHwith heliocentric distance, although this may also be related to its method of computation.Aims.In this work, we attempt to determine a more robust estimate of the heliospheric magnetic flux (ΦH) using data from the FIELDS instrument on board Parker Solar Probe (PSP), to analyse how susceptible it is to overestimation and a dependence on time and space, as well as considering how it compares to simple estimates of Φopenfrom potential field source surface (PFSS) models.Methods.We compared computations of the heliospheric magnetic flux using different methods of data processing on magnetic field data from PSP, STEREO A, and Wind. Measured radial trends in fluctuations and background magnetic structure were used to generate synthetic data to analyse their effect on the estimate ofBRR2. The resulting best estimates were computed as a function of time and space and then compared to estimates from PFSS models.Results.Radially varying fluctuations of the HMF vector as well as large-scale variations in the inclination of the Parker spiral angle are shown to have a non-trivial effect on the 1D distributions ofBRR2. This causes the standard statistical metrics of the mean and mode (the most probable values) to evolve with radius, independently of the central value about which the vector fluctuates. In particular, the mean systematically underestimates ΦHforR< 0.8 AU and increases close to 1 AU. We attempt to mitigate for this by using the “Parker spiral method” of projecting the vector onto the background Parker spiral direction (which requires vector fluctuations to be evenly distributed about a central value). Even with this method, we find evidence of a small enhancement in flux close to 1 AU. The fraction of field which is locally inverted in a given time interval grows with radial distance from the Sun which remains a possible physical reason for this excess but is essentially negligible at PSP’s perihelia distances where the impact of fluctuations in general is also much reduced. The Parker spiral method (PSM) and most probable values converge close to the Sun. Our derived best estimate for the time interval studied is ~2.5−0.6+0.3nT AU2. To the extent probed by PSP, no strong dependence on latitude or longitude is apparent, although at 1 AU, the spread of measured values appears to grow at the highest latitudes. The best estimate of the heliospheric flux is significantly larger than estimates from PFSS models studied here, which predict values from 1.2–1.8 nT AU2, depending on the choice of magnetogram or source surface height.Conclusions.Of the methods for computing the heliospheric flux over a wide range of heliocentric distances using only magnetic field data considered in this work, the most robust choice is to use the PSM. The decay of fluctuations and weakening importance of local flux inversions at smaller heliocentric distances indicate that the measurement is most accurate close to the sun and that it is justified for us to consider that ΦH~ Φopenfor these measurements. The determined value is too high to be explained via PFSS models. Contemporary magnetohydrodynamic models with the same photospheric input are unlikely to close this gap. Therefore, the most likely solutions remain in improvements of coronal models, for example, through improved boundary conditions via the direct measurement of the photospheric field in the solar polar regions or through the inclusion of missing physical processes such as time-dependent or non-potential effects, which can produce a contribution to the open flux that is not rooted in obvious coronal holes.

Asymptotic behaviour of a dynamical system governed by non-monotone potential and non-potential operators

We consider the following second order equation where is quasi-nonexpansive and Lipschitz continuous on bounded sets and is a continuously differentiable quasiconvex function such that is Lipschitz continuous on bounded sets. We study the asymptotic behaviour of solutions to this equation. Assuming some mild conditions on the operators, we prove weak and strong convergence of solutions to some point in . We also obtain similar results for the asymptotic behaviour of solutions to the discrete version of the above equation. Finally, we apply our results to solving a minimization problem and approximating a common fixed point of two mappings. Our work is motivated by the papers of H. Attouch and P. E. Maingé [Asymptotic behaviour of second-order dissipative evolution equations combining potential with non-potential effects. ESAIM Control Optim Calc Var. 2011;17:836–857.], X. Goudou and J. Munier [The gradient and heavy ball with friction dynamical systems: the quasiconvex case. Math Program Ser B. 2009;116:173–191.], and F. Alvarez and H. Attouch [An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping. Well-posedness in optimization and related topics. Set Valued Anal. 2001;9:3–11.], and extends some of their results.

Nonhomogeneous nonlinear oscillator with damping: asymptotic analysis in continuous and discrete time

Abstract We consider the following second order evolution equation modelling a nonlinear oscillator with damping ü ( t ) + γ u ˙ ( t ) + A u ( t ) = f ( t ) , ( SEE ) $$\ddot{u} (t) + \gamma \dot u(t) + Au\left( t \right) = f\left( t \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\rm{SEE}}} \right)$$ where A is a maximal monotone and α-inverse strongly monotone operator in a real Hilbert space H. With suitable assumptions on γ and f(t) we show that A −1(0) ≠ ∅, if and only if (SEE) has a bounded solution and in this case we provide approximation results for elements of A −1(0) by proving weak and strong convergence theorems for solutions to (SEE) showing that the limit belongs to A −1(0). As a discrete version of (SEE), we consider the following second order difference equation u n + 1 - u n - α n ( u n - u n - 1 ) + λ n A u n + 1 ∋ f ( t ) , $${u_{n + 1}} - {u_n} - {\alpha _n}\left( {{u_n} - {u_{n - 1}}} \right) + {\lambda _n}A{u_{n + 1}\ni} f\left( t \right),$$ where A is assumed to be only maximal monotone (possibly multivalued). By using the results in [Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411–417], we prove ergodic, weak and strong convergence theorems for the sequence un , and show that the limit is the asymptotic center of un and belongs to A −1(0). This again shows that A −1(0) ≠ ∅ if and only if un is bounded. Also these results solve an open problem raised in [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3–11], namely the study of the convergence results for the inexact inertial proximal algorithm. Our paper is motivated by the previous results in [Djafari Rouhani B., Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl., 1990, 147, 465–476; Djafari Rouhani B., Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl., 1990, 151, 226–235; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Math., 2010, 40, 1289–1311; Djafari Rouhani B., Khatibzadeh H., A strong convergence theorem for solutions to a nonhomogeneous second order evolution equation, J. Math. Anal. Appl., 2010, 363, 648–654; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to a class of second order nonhomogeneous evolution equations, Nonlinear Anal., 2009, 70, 4369–4376; Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411–417] and significantly improves upon the results of [Attouch H., Maingé P. E., Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects, ESAIM Control Optim. Calc. Var., 2011, 17(3), 836–857], and [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3–11].

Amplification of Nonpotential Waves in a Magnetized Beam–Plasma System Taking into Account the Evolution of the Plasma

We have analyzed the beam instability taking into account nonpotential effects in a completely magnetized plasma with a density varying slowly with time. It is shown that a frequency shift appears in the amplified signal (during the propagation in such an electrodynamic system); analytic relations determining this shift have been obtained.

Disentangling the timescales behind the nonperturbative heavy quark potential

The static part of the heavy quark potential has been shown to be closely related to the spectrum of the rectangular Wilson loop. In particular the lowest lying positive frequency peak encodes the late time evolution of the two-body system, characterized by a complex potential. While initial studies assumed a perfect separation of early and late time physics, where a simple Lorentian (Breit-Wigner) shape suffices to describe the spectral peak, we argue that scale decoupling in general is not complete. Thus early time, i.e. non-potential effects, significantly modify the shape of the lowest peak. We derive on general grounds an improved peak distribution that reflects this fact. Application of the improved fit to non-perturbative lattice QCD spectra now yields a potential that is compatible with a transition to a deconfined screening plasma.

Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects

In the setting of a real Hilbert space H, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolution equations ¨(t )+ γ ˙ u(t )+ ∇φ(u(t)) + A(u(t)) = 0, where ∇φ is the gradient operator of a convex differentiable potential function φ : H→ R, A : H→H is a maximal monotone operator which is assumed to be λ-cocoercive, and γ> 0i s a damping parameter. Potential and non-potential effects are associated respectively to ∇φ and A. Under condition λγ 2 > 1, it is proved that each trajectory asymptotically weakly converges to a zero of ∇φ + A. This condition, which only involves the non-potential operator and the damping parameter, is sharp and consistent with time rescaling. Passing from weak to strong convergence of the trajectories is obtained by introducing an asymptotically vanishing Tikhonov-like regularizing term. As special cases, we recover the asymptotic analysis of the heavy ball with friction dynamic attached to a convex potential, the second-order gradient-projection dynamic, and the second-order dynamic governed by the Yosida approximation of a general maximal monotone operator. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of constrained optimization, dynamical approach to Nash equilibria for noncooperative games, and asymptotic stabilization in the case of a continuum of equilibria.

Electromagnetic waves in cylindrical plasma-dielectric structures

The theory of electromagnetic waves in cylindrical structures (waveguides) with plasma and plasma-dielectric fillings is presented in a systematic form. The case of isotropic cold electron plasma with sharp transverse plasma/vacuum, plasma/dielectric, plasma/plasma, and plasma/metal interfaces is considered. Particular attention is given to surface plasma waves and their interaction with bulk electromagnetic modes. The coupling of E and B waveguide modes that is due to the transverse inhomogeneity is investigated. The important question of the applicability of the potential (electrostatic) approximation in the theory of plasma waves and the role of nonpotential effects are discussed.

Coarsening in potential and nonpotential models of oblique stripe patterns

We study the coarsening of two-dimensional oblique stripe patterns by numerically solving potential and nonpotential anisotropic Swift-Hohenberg equations. Close to onset, all models exhibit isotropic coarsening with a single characteristic length scale growing in time as t1/2. Further from onset, the characteristic lengths along the preferred directions x and ŷ grow with different exponents, close to 1/3 and 1/2, respectively. In this regime, one-dimensional dynamical scaling relations hold. We draw an analogy between this problem and model A in a stationary, modulated external field. For deep quenches, nonpotential effects produce a complicated dislocation dynamics that can lead to either arrested or faster-than-power-law growth, depending on the model considered. In the arrested case, small isolated domains shrink down to a finite size and fail to disappear. A comparison with available experimental results for electroconvection in nematics is presented.

NONPOTENTIAL EFFECTS IN NONLINEAR DYNAMICS OF MARANGONI CONVECTION

Nonpotential effects in nonlinear evolution of Marangoni convection patterns are investigated analytically and numerically. Three manifestations of nonpotential effects are considered: (i) spatial modulations of hexagonal patterns; (ii) interaction between a short-scale hexagonal pattern and a long-scale slow deformational (Goldstone) mode; (iii) generation of the mean flow by the free-surface deformation in a large-scale Marangoni convection with poorly conducting boundaries. Nonpotential effects are shown to cause various secondary instabilities leading to skewed hexagonal structures, coexisting u- and d-hexagons, oscillating hexagonal patterns, spatially irregular cellular patterns, etc.

Nonpotential effects in dynamics of fronts between convection patterns

The dynamics of fronts between hexagonal convection patterns and a mechanical equilibrium state are investigated in the cases where the hexagons are stable and the equilibrium is either stable or unstable. The Newell-Whitehead-Segel, equations with additional nonpotential terms are used. The velocity of the domain boundary between two competing convection patterns and the wave number selected in the presence of the front are calculated.

Theory of Rayleigh-Bénard convection in planar nematic liquid crystals.

A rigorous three-dimensional linear and weakly nonlinear analysis of Rayleigh-B\'enard convection in planarly aligned nematic liquid crystals is presented. We use realistic boundary conditions and allow for a stabilizing magnetic field. The analysis includes a determination of all parameters of the amplitude equation in different regimes of the magnetic field where normal, oblique, and parallel rolls occur. It turns out that there exists a range at intermediate fields were the primary transition is subcritical for the standard material (4-methoxybenzylidene-4'-n-butylaniline). The (``nonlinear'') stability of rolls is determined to second order in the amplitude so that nonpotential effects (especially mean flow) are included. In the low-field supercritical range the roll solutions are destabilized at a secondary threshold via a skewed-varicose instability leading presumably to spatiotemporal chaos. In the high-field supercritical normal-roll range a secondary transition can lead to oblique rolls that are ultimately destabilized by a short-wavelength instability.