Set Of Dynamic Equations Research Articles

Three-pattern decomposition of global atmospheric circulation: part II—dynamical equations of horizontal, meridional and zonal circulations

The three-pattern decomposition of global atmospheric circulation (TPDGAC) partitions three-dimensional (3D) atmospheric circulation into horizontal, meridional and zonal components to study the 3D structures of global atmospheric circulation. This paper incorporates the three-pattern decomposition model (TPDM) into primitive equations of atmospheric dynamics and establishes a new set of dynamical equations of the horizontal, meridional and zonal circulations in which the operator properties are studied and energy conservation laws are preserved, as in the primitive equations. The physical significance of the newly established equations is demonstrated. Our findings reveal that the new equations are essentially the 3D vorticity equations of atmosphere and that the time evolution rules of the horizontal, meridional and zonal circulations can be described from the perspective of 3D vorticity evolution. The new set of dynamical equations includes decomposed expressions that can be used to explore the source terms of large-scale atmospheric circulation variations. A simplified model is presented to demonstrate the potential applications of the new equations for studying the dynamics of the Rossby, Hadley and Walker circulations. The model shows that the horizontal air temperature anomaly gradient (ATAG) induces changes in meridional and zonal circulations and promotes the baroclinic evolution of the horizontal circulation. The simplified model also indicates that the absolute vorticity of the horizontal circulation is not conserved, and its changes can be described by changes in the vertical vorticities of the meridional and zonal circulations. Moreover, the thermodynamic equation shows that the induced meridional and zonal circulations and advection transport by the horizontal circulation in turn cause a redistribution of the air temperature. The simplified model reveals the fundamental rules between the evolution of the air temperature and the horizontal, meridional and zonal components of global atmospheric circulation.

Decoding early myelopoiesis from dynamics of core endogenous network.

A decade ago mainstream molecular biologists regarded it impossible or biologically ill-motivated to understand the dynamics of complex biological phenomena, such as cancer genesis and progression, from a network perspective. Indeed, there are numerical difficulties even for those who were determined to explore along this direction. Undeterred, seven years ago a group of Chinese scientists started a program aiming to obtain quantitative connections between tumors and network dynamics. Many interesting results have been obtained. In this paper we wish to test such idea from a different angle: the connection between a normal biological process and the network dynamics. We have taken early myelopoiesis as our biological model. A standard roadmap for the cell-fate diversification during hematopoiesis has already been well established experimentally, yet little was known for its underpinning dynamical mechanisms. Compounding this difficulty there were additional experimental challenges, such as the seemingly conflicting hematopoietic roadmaps and the cell-fate inter-conversion events. With early myeloid cell-fate determination in mind, we constructed a core molecular endogenous network from well-documented gene regulation and signal transduction knowledge. Turning the network into a set of dynamical equations, we found computationally several structurally robust states. Those states nicely correspond to known cell phenotypes. We also found the states connecting those stable states. They reveal the developmental routes-how one stable state would most likely turn into another stable state. Such interconnected network among stable states enabled a natural organization of cell-fates into a multi-stable state landscape. Accordingly, both the myeloid cell phenotypes and the standard roadmap were explained mechanistically in a straightforward manner. Furthermore, recent challenging observations were also explained naturally. Moreover, the landscape visually enables a prediction of a pool of additional cell states and developmental routes, including the non-sequential and cross-branch transitions, which are testable by future experiments. In summary, the endogenous network dynamics provide an integrated quantitative framework to understand the heterogeneity and lineage commitment in myeloid progenitors.

Waves in non-conducting continuum with frozen-in magnetization

The report reviews the new approach to the ferrohydrodynamics and magnetoelasticity, based on the concept of frozen-in magnetization. Up till now, there were well-known two frozen-in vector fields. The first one is the vorticity field was introduced by Helmgoltz. The second frozen-in field is a magnetic field in a perfectly conducting fluid was introduced by Alfven. The acoustic approximation of the ferrohydrodynamic equations with frozen-in magnetization allowed us to describe the experimental results on the anisotropy of the ultrasonic velocity in magnetized magnetic nanofluids and predict new waves: Alfven-type hydrodynamic wave and slow magnetosonic one. Alfven-type wave is accompanied by oscillations of the magnetization. It is shown that, in a non-conducting solid with a frozen-in magnetization, the propagation of three types of waves are possible: the longitudinal wave, pure shear wave, and new wave which have mixed form of Alfven-type and shear waves. The theoretical results are found to agree well with the experimental data on the dependence of the velocity of longitudinal and transverse waves in polycrystalline nickel on the magnetizing field strength. The set of dynamical equations that we have derived can be used to tackle many problems in ferrohydrodynamics and magnetoelasticity.

Modeling the network dynamics of pulse-coupled neurons.

We derive a mean-field approximation for the macroscopic dynamics of large networks of pulse-coupled theta neurons in order to study the effects of different network degree distributions and degree correlations (assortativity). Using the ansatz of Ott and Antonsen [Chaos 18, 037113 (2008)], we obtain a reduced system of ordinary differential equations describing the mean-field dynamics, with significantly lower dimensionality compared with the complete set of dynamical equations for the system. We find that, for sufficiently large networks and degrees, the dynamical behavior of the reduced system agrees well with that of the full network. This dimensional reduction allows for an efficient characterization of system phase transitions and attractors. For networks with tightly peaked degree distributions, the macroscopic behavior closely resembles that of fully connected networks previously studied by others. In contrast, networks with highly skewed degree distributions exhibit different macroscopic dynamics due to the emergence of degree dependent behavior of different oscillators. For nonassortative networks (i.e., networks without degree correlations), we observe the presence of a synchronously firing phase that can be suppressed by the presence of either assortativity or disassortativity in the network. We show that the results derived here can be used to analyze the effects of network topology on macroscopic behavior in neuronal networks in a computationally efficient fashion.

Dynamic characteristics of an equal-thickness screen with a variable amplitude and screening analysis

Screening is the key unit in coal processing and utilization. Recently, equal-thickness screens characterized by a large capacity and high efficiency have been extensively used. Conventionally, equal-thickness screening is achieved by changing the inclinations of each stage or adjusting the exciting force. In this paper, a single-deck equal-thickness vibrating screen (ETVS) driven externally by an unbalanced two-axle excitation with a large span is proposed, and a set of dynamic equations governing the motion of the screen are also presented. The vibration data was obtained using a vibration test and analysis unit, and the bed stratification and particle behavior on the screen were obtained by image acquisition using a high-speed camera and data analysis unit. Screening experiments were conducted to evaluate the classification performance of the ETVS with a variable amplitude. The amplitude of the ETVS showed a decreasing trend, and a constant bed thickness was obtained throughout the screen deck. The ETVS has advantages of a high screen deck utilization and an efficient classification. Compared to the normal vibrating screen (NVS), the screening efficiency of the ETVS increased by 3.05%–8.76%. The screening performance of the ETVS is obviously better than that of the NVS, especially when dealing with a large amount of materials and those with high moisture.

Refractory period of an excitable semiconductor laser with optical injection.

Injection-locked semiconductor lasers can be brought to a neuronlike excitable regime when parameters are set close to the unlocking transition. Here we study experimentally the response of this system to repeated optical perturbations and observe the existence of a refractory period during which perturbations are not able to elicit an excitable response. The results are analyzed via simulations of a set of dynamical equations which reproduced adequately the experimental results.

Consensus-based multi-piezoelectric microcantilever for scanning probe microscopy

A new multi-piezoelectric microcantilever sensor is introduced in this article for replacing laser sensors in atomic force microscopes. This microcantilever consists of multiple piezoelectric layers over its surface, and a consensus algorithm is designed to provide a robust and accurate estimation of the deflections at the tip of the microcantilever. The dynamic equation set of the microcantilever is developed first and then the consensus observer is designed. A set of Riccati equations is used to obtain the optimal gains for the observer, and the robustness of the microcantilever is considered by designing a H∞ norm constraint. A set of numerical simulations is conducted to evaluate the performance of the microcantilever. Results show that the consensus-based multi-piezoelectric microcantilever can successfully provide an accurate estimation of the deflections at the tip of the microcantilever. It is also shown that the robustness of the design can positively improve the estimation performance in the presence of noise. Additionally, a comparison with a single-layered design shows the advantages of the new sensor.

Arbitrary amplitude electrostatic wave propagation in a magnetized dense plasma containing helium ions and degenerate electrons

The obliquely propagating arbitrary amplitude electrostatic wave is studied in a dense magnetized plasma having singly and doubly charged helium ions with nonrelativistic and ultrarelativistic degenerate electrons pressures. The Fermi temperature for ultrarelativistic degenerate electrons described by N. M. Vernet [(Cambridge University Press, Cambridge, 2007), p. 57] is used to define ion acoustic speed in ultra-dense plasmas. The pseudo-potential approach is used to solve the fully nonlinear set of dynamic equations for obliquely propagating electrostatic waves in a dense magnetized plasma containing helium ions. The upper and lower Mach number ranges for the existence of electrostatic solitons are found which depends on the obliqueness of the wave propagation with respect to applied magnetic field and charge number of the helium ions. It is found that only compressive (hump) soliton structures are formed in all the cases and only subsonic solitons are formed for a singly charged helium ions plasma case with nonrelativistic degenerate electrons. Both subsonic and supersonic soliton hump structures are formed for doubly charged helium ions with nonrelativistic degenerate electrons and ultrarelativistic degenerate electrons plasma case containing singly as well as doubly charged helium ions. The effect of propagation direction on the soliton amplitude and width of the electrostatic waves is also presented. The numerical plots are also shown for illustration using dense plasma parameters of a compact star (white dwarf) from literature.

Kinetic theory for interacting Luttinger liquids

We derive a closed set of equations for the kinetics and non-equilibrium dynamics of interacting Luttinger Liquids with cubic resonant interactions. In the presence of these interactions, the Luttinger phonons become dressed but still well defined quasi-particles, characterized by a life-time much larger then the inverse energy. This enables the separation of forward time dynamics and relative time dynamics into slow and fast dynamics and justifies the so-called Wigner approximation, which can be seen as a "local-time approximation" for the relative dynamics. Applying field theoretical methods in the Keldysh framework, i.e. kinetic and Dyson-Schwinger equations, we derive a closed set of dynamic equations, describing the kinetics of normal and anomalous phonon densities, the phonon self-energy and vertex corrections for an arbitrary non-equilibrium initial state. In the limit of low phonon densities, the results from self-consistent Born approximation are recaptured, including Andreev's scaling solution for the quasi-particle life-time in a thermal state. As an application, we compute the relaxation of an excited state to its thermal equilibrium. \red{ While the intermediate time dynamics displays exponentially fast relaxation, the last stages of thermalization are governed by algebraic laws. This can be traced back to the importance of energy and momentum conservation at the longest times, which gives rise to dynamical slow modes.

Exact model of a nongeostrophic baroclinic instability

A nongeostrophic version of the classical problem of zonal flow instability with constant shear (the Eady problem) is considered. The linearized set of dynamic equations for two-dimensional disturbances is reduced to a single wave-type second-order equation relative to modified pressure (a linear combination of pressure and stream function). Dynamic features of disturbances with zero potential vorticity are studied in the framework of the equations formulated. Asymptotic solutions of the spectral problem of hydrodynamic stability theory are derived. The initial-value problem at large Richardson numbers is considered using multiple-time-scale expansions. The solution to the problem is represented as the sum of fast (wave) and slow (quasi-geostrophic) components. In the unstable regime, the slow component describes baroclinic waves (cyclones and anticyclones) generated by inhomogeneous initial buoyancy (potential temperature) distributions at the boundaries.

On the dynamics and kinematics of two‐fluid‐phase flow in porous media

AbstractA model formulated in terms of both conservation and kinematic equations for phases and interfaces in two‐fluid‐phase flow in a porous medium system is summarized. Macroscale kinematic equations are derived as extensions of averaging theorems and do not rely on conservation principles. Models based on both conservation and kinematic equations can describe multiphase flow with varying fidelity. When only phase‐based equations are considered, a model similar in form to the traditional model for two‐fluid‐phase flow results. When interface conservation and kinematic equations are also included, a novel formulation results that naturally includes evolution equations that express dynamic changes in fluid saturations, pressures, the capillary pressure, and the fluid‐fluid interfacial area density in a two‐fluid‐system. This dynamic equation set is unique to this work, and the importance of the modeled physics is shown through both microfluidic experiments and high‐resolution lattice Boltzmann simulations. The validation work shows that the relaxation of interface distribution and shape toward an equilibrium state is a slow process relative to the time scale typically allowed for a system to approach an apparent equilibrium state based upon observations of fluid saturations and external pressure measurements. Consequently, most pressure‐saturation data intended to denote an equilibrium state are likely a sampling from a dynamic system undergoing changes of interfacial curvatures that are not typically monitored. The results confirm the importance of kinematic analysis in combination with conservation equations for faithful modeling of system physics.

Turbulent convection and pulsation stability of stars – I. Basic equations for calculations of stellar structure and oscillations

Starting from hydrodynamic equations, we have established a set of hydrodynamic equations for average flow and a set of dynamic equations of auto- and cross-correlations of turbulent velocity and temperature fluctuations, following the classic Reynold's treatment of turbulence. The combination of the two sets of equations leads to a complete and self-consistent mathematical expressions ready for the calculations of stellar structure and oscillations. In this paper, non-locality and anisotropy of turbulent convection are concisely presented, together with defining and calibrating of the three convection parameters ($c_1$, $c_2$ and $c_3$) included in the algorithm. With the non-local theory of convection, the structure of the convective envelope and the major characteristics of non-adiabatic linear oscillations are demonstrated by numerical solutions. Great effort has been exercised to the choice of convection parameters and pulsation instabilities of the models, the results of which show that within large ranges of all three parameters ($c_1$, $c_2$ and $c_3$) the main properties of pulsation stability keep unchanged.

Tolman–Oppenheimer–Volkoff equations in nonlocal f(R) gravity

Nonlocal f(R) gravity was proposed as a powerful alternative to general relativity (GR). This theory has potentially adverse implications for infrared (IR) regime as well as ultraviolet (UV) early epochs. However, there are a lot of powerful features, making it really user-friendly. A scalar–tensor frame comprising two auxiliary scalar fields is used to reduce complex action. However, this is not the case for the modification complex which plays a distinct role in modified theories for gravity. In this work, we study the dynamics of a static, spherically symmetric object. The interior region of space–time had rapidly filled the perfect fluid. However, it is possible to derive a physically based model which relates interior metric to nonlocal f(R). The Tolman–Oppenheimer–Volkoff (TOV) equations would be a set of first-order differential equations from which we can deduce all mathematical (physical) truths and derive all dynamical objects. This set of dynamical equations govern pressure p, density ρ, mass m and auxiliary fields {ψ, ξ}. The full conditional solutions are evaluated and inverted numerically to obtain exact forms of the compact stars Her X-1, SAX J 1808.4-3658 and 4U 1820-30 for nonlocal Starobinsky model of f(◻-1 R) = ◻-1 R+α(◻-1 R)2. The program solves the differential equations numerically using adaptive Gaussian quadrature. An ascription of correctness is supposed to be an empirical equation of state [Formula: see text] for star which is informative in so far as it excludes an alternative nonlocal approach to compact star formation. This model is most suited for astrophysical observation.

Dynamical equations of multibody systems on Lie groups

The Euler–Poinaré principle is a reduced Hamilton’s principle under Lie group framework. In this article, it is applied to derive a hybrid set of dynamical equations of rigid multibody systems, which include four parts: the classical Euler–Lagrange equations of rigid bodies in their translational coordinates of mass center; Euler–Poinaré equations via orientation matrices and their related angular velocities; the constraint equations due to different joints in Cartesian coordinates and Lie groups; and the reconstruction equations between special orthogonal groups and their Lie algebras. The generalized mass matrices of dynamical equations are constant, which is computationally efficient. All the equations can be constructed systematically and can be solved easily. The construction equations can be used to design Lie group integrators of multibody system dynamics. The procedure presented in this article can be extended easily to flexible multibody systems, systems with non-holonomic constraints, and so on.

Cosmic expansion in extended quasidilaton massive gravity

Quasidilaton massive gravity offers a physically well-defined gravitational theory with non-zero graviton mass. We present the full set of dynamical equations governing the expansion history of the universe, valid during radiation domination, matter domination, and a late-time self-accelerating epoch related to the graviton mass. The existence of self-consistent solutions constrains the amplitude of the quasi-dilaton field and the graviton mass, as well as other model parameters. We point out that the effective mass of gravitational waves can be significantly larger than the graviton mass, opening the possibility that a single theory can explain both the late-time acceleration of the cosmic expansion and modifications of structure growth leading to the suppression of large-angle correlations observed in the cosmic microwave background.

Fully non-linear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids

Many composite materials, including biological tissues, are modeled as non-linear elastic materials reinforced with elastic fibers. In the current paper, the full set of dynamic equations for finite deformations of incompressible hyperelastic solids containing a single fiber family are considered. Finite-amplitude wave propagation ansätze compatible with the incompressibility condition are employed for a generic fiber family orientation. Corresponding non-linear and linear wave equations are derived. It is shown that for a certain class of constitutive relations, the fiber contribution vanishes when the displacement is independent of the fiber direction.Point symmetries of the derived wave models are classified with respect to the material parameters and the angle between the fibers and the wave propagation direction. For planar shear waves in materials with a strong fiber contribution, a special wave propagation direction is found for which the non-linear wave equations admit an additional symmetry group. Examples of exact time-dependent solutions are provided in several physical situations, including the evolution of pre-strained configurations and traveling waves.

Cooperative behavior in [formula omitted]-person evolutionary snowdrift games with punishment

We generalize the N-person evolutionary snowdrift game to incorporate the effects of costly punishment in a well-mixed population. A set of dynamical equations that account for the evolution of the frequencies of the three strategies under replicator dynamics is formulated. At long time, the system evolves into one of two phases with different properties consisting only of two strategies, and three-strategy coexistence is not allowed. Small cost-to-benefit ratio, big competing group size, and severe punishment tend to suppress non-cooperators, and lead to a cooperative system with a mixture of cooperators and punishers. The resulting composition depends on the initial conditions as the dynamics is frozen once non-cooperators extinct. Large cost-to-benefit ratio, small competing group size, and light punishment tend to be self-destructive for the punishers, and lead to a mixture of cooperators and non-cooperators with composition independent of initial conditions and a continual dynamics. The frozen phase and dynamical phase correspond to a line of fixed points and a single fixed point on different axes in the phase space, respectively. A simulation algorithm that mimics the replicator dynamics exactly is proposed. Results of the dynamical equations and numerical simulations are found to be in exact agreement.

A multidimensional pendulum in a nonconservative force field

INTRODUCTIONThe nonconservative force field in the dynamics ofa multidimensional solid is constructed according tothe results from the dynamics of real solids under theaction of a force field in a medium. In this case, thegeneralization of equations of motion for a multidimensional solid in a similarly constructed force fieldand the receipt of, generally speaking, a full list of thetranscendental first integrals, which are expressedthrough a final combination of elementary functions,become possible. The obtained results are importantin the sense of the presence of a nonconservative forcefield in the system; previously, they were used mainlyin the potentialforce field (see, for example, [1–3]).In [4, 5], we showed the integrability of equationsof planeparallel motion of a fixed pendulum in anincidentmedium flow, when the first integral, whichis the transcendental function (in the sense of complexanalysis) of quasivelocities, was found explicitly forthe set of dynamic equations. In this case, the wholeinteraction of the medium with the solid is concentrated on that part of the surface that has the form ofthe 1D plate. Then in [5–8], the problem was generalized to the spatial case (the spherical pendulum); inthis case, the full set of transcendental first integralswas found explicitly. Here already the whole interaction of the medium with the solid is concentrated onthat part of its surface that has the shape of the plane(2D) disk. Further in [9–11], the equations of motionof 4D solids of various types of dynamic symmetry,where the force field is concentrated on that part of thesurface of the solid that has the shape of the 2D (3D)disk, were investigated; in this case, the force action isconcentrated on the 2D plane (the 1D straight line)perpendicular to this disk.FORMULATION OF PROBLEM AND THE GROUP OF DYNAMIC EQUATIONS ON LIE ALGEBRA so(

Integration of Bayesian analysis for eutrophication prediction and assessment in a landscape lake.

Eutrophication models have been widely used to assess water quality in landscape lakes. Because flow rate in landscape lakes is relatively low and similar to that of natural lakes, eutrophication is more dominant in landscape lakes. To assess the risk of eutrophication in landscape lakes, a set of dynamic equations was developed to simulate lake water quality for total nitrogen (TN), total phosphorous (TP), dissolve oxygen (DO) and chlorophyll a (Chl a). Firstly, the Bayesian calibration results were described. Moreover, the ability of the model to reproduce adequately the observed mean patterns and major cause-effect relationships for water quality conditions in landscape lakes were presented. Two loading scenarios were used. A Monte Carlo algorithm was applied to calculate the predicated water quality distributions, which were used in the established hierarchical assessment system for lake water quality risk. The important factors affecting the lake water quality risk were defined using linear regression analysis. The results indicated that the variations in the landscape lake receiving recharge water quality caused considerable landscape lake water quality risk in the surrounding area. Moreover, the Chl a concentration in lake water was significantly affected by TP and TN concentrations; the lake TP concentration was the limiting factor for growth of plankton in lake water. The lake water TN concentration provided the basic nutritional requirements. Lastly, lower TN and TP concentrations in the receiving recharge water caused increased lake water quality risk.

Vibration of Discharge Filaments in a Dielectric Barrier Discharge

The vibration of discharge filaments in a concentric spotring pattern in a dielectric barrier discharge is investigated by high-speed camera. The pattern, consisting of several sparsely and densely arranged-spot rings, bifurcates from a vibrating hexagonal superlattice. The discharge spots vibrate between two neighboring densely arranged-spot rings. The vibration period, generally ranging from 12 to 24 ms, increases with the increase of radius and argon concentration. The instantaneous images taken by a high-speed framing camera show that most of the vibrating spots and static spots discharge asynchronously. Based on the forces exerted on one vibrating spot by two neighboring static spots, a set of dynamical equations are developed to explain the vibration.