Multicomponent Wave Research Articles

Magnetospheric Line Radiation: Temporal Modulation Corresponding to a Bouncing Wave

AbstractMagnetospheric line radiation (MLR) is a peculiar type of whistler‐mode emissions formed by several similarly spaced spectral lines, whose frequencies may vary with time. These emissions typically occur at frequencies between about 1 and 8 kHz and are observed in both ground‐based and satellite measurements. However, their origin and source locations are not yet understood, as their detection by spacecraft instruments is challenging due to the limited frequency resolution of onboard wave measurements. We use high‐resolution multicomponent wave data obtained by the Van Allen Probes to demonstrate that, in addition to the frequency modulation, MLR events possess a temporal modulation with periods on the order of seconds, which may be related to the wave bouncing between hemispheres. Observed modulation periods are used to estimate a tentative source L‐shell. The modulation periods are shorter for events with larger frequency spacing, and the estimated source L‐shells correlate with model plasmapause locations.

An Application of 3D Cross-Well Elastic Reverse Time Migration Imaging Based on the Multi-Wave and Multi-Component Technique in Coastal Engineering Exploration

Precise surveys are indispensable in coastal engineering projects. The extensive presence of sand in the coastal area leads to significant attenuation of seismic waves within unsaturated loose sediments. As a result, it becomes challenging for seismic waves to penetrate the weathered zone and reach the desired depth with significant amount of energy. In this study, the application of three-dimensional (3D) cross-well elastic reverse time migration (RTM) imaging based on multi-wave and multi-component techniques in coastal engineering exploration is explored. Accurate decomposition of vector compressional (P) and shear (S) waves is achieved through two wavefield decoupling algorithms without any amplitude and phase distortion. Additionally, compressional wave pressure components are obtained, which facilitates subsequent independent imaging. This study discusses and analyzes the imaging results of four imaging strategies under cross-correlation imaging conditions in RTM imaging. The analysis leads to the conclusion that scalarizing vector wavefields imaging yields superior imaging of P- and S-waves. Furthermore, the imaging results obtained through this approach are of great physical significance. In order to validate the efficacy of this method in 3D geological structure imaging in coastal areas, RTM imaging experiments were performed on two representative models. The results indicate that the proposed 3D elastic wave imaging method effectively generates accurate 3D cross-well imaging of P- and S-waves. This method utilizes the multi-wave and multi-component elastic wave RTM imaging technique to effectively leverage the Earth’s elastic medium without increasing costs. It provides valuable information about the distribution of subsurface rock layers, interfaces, and other structures in coastal engineering projects. Importantly, this can be achieved without resorting to extensive excavation or drilling operations. This method addresses the limitations of current cross-well imaging techniques, thereby providing abundant and accurate geological and geophysical information for the analysis and interpretation of 3D geological structures in coastal engineering projects. It has important theoretical and practical significance in real-world production, as well as for the study of geological structures in coastal engineering.

Geometric localization of waves on thin elastic structures.

We consider the localization of elastic waves in thin elastic structures with spatially varying curvature profiles, using a curved rod and a singly curved shell as concrete examples. Previous studies on related problems have broadly focused on the localization of flexural waves on such structures. Here, using the semiclassical WKB approximation for multicomponent waves, we show that in addition to flexural waves, extensional and shear waves also form localized, bound states around points where the absolute curvature of the structure has a minimum. We also see excellent agreement between our numerical experiments and the semiclassical results, which hinges on the vanishing of two extra phases that arise in the semiclassical quantization rule. Our findings open up novel ways to fine-tune the acoustic and vibrational properties of thin elastic structures and raise the possibility of introducing new phenomena not easily captured by effective models of flexural waves alone.

Magnetospheric Line Radiation Observed Close to the Source: Properties and Propagation

AbstractMagnetospheric Line Radiation (MLR) is a type of whistler mode electromagnetic wave phenomenon observed in the inner magnetosphere at frequencies of a few kilohertz, that is characterized by a frequency modulation of the wave intensity. Although such events are quite regularly observed by ground‐based stations and low‐altitude spacecraft, their observations in the equatorial region at larger radial distances (i.e., close to tentative source regions) are extremely limited, likely due to the generally low frequency resolution of available measurements. A systematic search for MLR in continuous intervals of high‐resolution multicomponent wave data obtained by the Van Allen Probes spacecraft detects 15 events. They occur primarily on the dayside at frequencies between about 1 and 5 kHz, propagating with oblique wave normals away from the geomagnetic equator. For one event, simultaneous ground‐based observations are available, providing limits on the spatial extent of the event: it does not extend beyond the high‐density plasmasphere region. An electrostatic wave at a frequency corresponding to the modulation frequency of MLR is observed in three events. This is likely linked to the event formation mechanism and has not been observed before. Our results can lead to an understanding of the formation mechanism of MLR.

On research of dispersion characteristics of multi-component surface waves from traffic-induced seismic ambient noise

As one of the commonly used sources in passive surface wave exploration, traffic-induced seismic ambient noise is widely used in urban underground velocity structure detection at present. However, characteristics of the dispersion curve extracted from such kind of ambient noise lack systematical research, especially for the multi-component surface wave. Due to the complex character of the real-world underneath structures, dispersion characteristics concluded from real-world traffic-induced seismic ambient noise that is hard to verify. The numerical simulation technique is a good choice for systematical research of characteristics of dispersion curves extracted from multi-component traffic-induced seismic ambient noise. Most of the numerical examples that appeared in the present literature, however, are based on the dispersion-curve-based numerical simulation technique. Hence, most of these can only simulate the vertical component. In this paper, the two-dimensional finite difference method is used to simulate multi-component traffic seismic ambient noise. The characteristics of original traffic noise are analyzed, and the effect of trace intervals on surface wave imaging of different types of traffic-induced noise is studied. Rayleigh wave dispersion energy image is extracted by the passive surface-wave frequency wavenumber transformed method. The differences in Rayleigh-wave dispersion images in traffic noise with different components are analyzed and compared. The results show that Rayleigh wave dispersion energy images appearing different characteristics in different components. The surface wave dispersion information presented by different types of traffic noise is frequency-dependent. The trace spacing also strongly affect the surface-wave dispersion energy images extracted from different types of traffic-induced noise. The numerical results are significant for guiding the traffic-induced seismic noise data acquisition and processing.

Hydrodynamics of plastic deformations in electronic crystals

We construct a new hydrodynamic framework describing plastic deformations in electronic crystals. The framework accounts for pinning, phase, and momentum relaxation effects due to translational disorder, diffusion due to the presence of interstitials and vacancies, and strain relaxation due to plasticity and dislocations. We obtain the hydrodynamic mode spectrum and correlation functions in various regimes in order to identify the signatures of plasticity in electronic crystal phases. In particular, we show that proliferation of dislocations de-pins the spatially resolved conductivity until the crystal melts, after which point a new phase of a pinned electronic liquid emerges. In addition, the mode spectrum exhibits a competition between pinning and plasticity effects, with the damping rate of some modes being controlled by pinning-induced phase relaxation and some by plasticity-induced strain relaxation. We find that the recently discovered damping-attenuation relation continues to hold for pinned-induced phase relaxation even in the presence of plasticity and dislocations. We also comment on various experimental setups that could probe the effects of plasticity. The framework developed here is applicable to a large class of physical systems including electronic Wigner crystals, multicomponent charge density waves, and ordinary crystals.

From ray tracing to waves of topological origin in continuous media

Continuous media commonly support a discrete number of wave modes that are trapped along interfaces defined by spatially varying parameters. In the case of multicomponent wave problems, those trapped modes fill a frequency gap between different wave bands. When they are robust against continuous deformations of parameters, such waves are said to be of topological origin. It has been realized over the last decades that waves of topological origin can be predicted by computing a single topological invariant, the first Chern number, in a dual bulk wave problem that is much simpler to solve than the original wave equation involving spatially varying coefficients. The correspondence between the simple bulk problem and the more complicated interface problem is usually justified by invoking an abstract index theorem. Here, by applying ray tracing machinery to the paradigmatic example of equatorial shallow water waves, we propose a physical interpretation of this correspondence. We first compute ray trajectories in the phase space given by position and wavenumber of the wave packet, using Wigner-Weyl transforms. We then apply a quantization condition to describe the spectral properties of the original wave operator. This bridges the gap between previous work by Littlejohn and Flynn showing manifestation of Berry curvature in ray tracing equations, and more recent studies that computed the Chern number of flow models by integrating the Berry curvature over a closed surface in parameter space. We find that an integral of Berry curvature over this closed surface emerges naturally from the quantization condition, which allows us to recover the bulk-interface correspondence.

Two-component three-dimensional atomic Bose-Einstein condensates supporting complex stable patterns

We report the computational discovery of complex, topologically charged, and spectrally stable states in three-dimensional multicomponent nonlinear wave systems of nonlinear Schr\"odinger type. While our computations relate to two-component atomic Bose-Einstein condensates in parabolic traps, our methods can be broadly applied to high-dimensional nonlinear systems of partial differential equations. The combination of the so-called deflation technique with a careful selection of initial guesses enables the computation of an breadth of patterns, including ones combining vortex lines, rings, stars, and vortex labyrinths. Despite their complexity, they may be dynamically robust and amenable to experimental observation, as confirmed by Bogoliubov--de Gennes spectral analysis and numerical evolution simulations.

A 3D crustal shear speed model of the Auckland volcanic field, New Zealand, from multi-component ambient noise tomography

We present a 3D shear-wave speed model of the crust in the intra-plate Auckland Volcanic Field (AVF), that provides new depth information on the basement geology and is a step towards determining lithospheric controls on volcanism of the AVF, within the Zealandia continent. Beneath the AVF, our model finds a mid-crustal contact of the upturning ophiolite-rich Dun Mountain-Maitai Terrane (DMT), and the western end of the accretionary Waipapa (Caples) Terranes which are dominated by terrigenous sedimentary rocks. Westward and at depth, the DMT presents as a high-speed, thick, and horizontal part of the Gondwana-ward verging syncline seen in the DMT elsewhere in New Zealand. Along the western side of the Junction Magnetic Anomaly (which indicates the presence of the DMT), a major vertical discontinuity probably marks the contact between the horizontal DMT and the Waipapa (Caples) Terranes. The Waipapa (Caples) Terrane contain a large wedge-shaped low-speed zone tapering eastward and extending NNW-SSE alongside the DMT. In addition, the model identifies a sharp vertical discontinuity at the eastern edge of the AVF that extends from the near-surface through the low-speed wedge. We interpret this discontinuity as a relatively young subvertical fault with a significant downthrow to the west. It appears to be a reactivation of a fault zone inherited from the break-up of Gondwana. The model also imaged the largest massif of crystalline ultramafic rocks in the DMT that causes the Takapuna Gravity Anomaly and the NE striking Port Waikato-Mangatangi cross-fault system. The strike directions of the faults mirror a rhombic-rectangular fault grid inherited from the breakaway of Zealandia from Gondwana.The shear speed model was produced using frequency-time analysis of multi-component Rayleigh wave Green's functions, reversible-jump Markov chain Monte Carlo inversion of group arrival times, and the inversion of dispersion relations for shear speed as a function of depth. Plain language summaryA 3D model of shear wave speeds in the crust provides new information on the geological structure beneath the Auckland Volcanic Field (AVF). The field is positioned above the contact between ocean floor rocks of the Dun Mountain-Maitai Terrane (DMT) in the west with fast wave speeds and a prominent wedge-shaped low-speed zone in the sandstone-dominated Waipapa/Caples Terrane in the east. The model images a fold structure of the DMT similar to that seen throughout New Zealand and at depth delineates the contact as a major sub-vertical discontinuity. A second vertical discontinuity just east of the AVF is very abrupt, is also oriented NNW-SSE and is probably caused by faulting. The model has also imaged the NE-oriented Port Waikato-Mangatangi cross-fault system in the south of the region and the large massif of coarse-grained intrusive rocks in the DMT that causes the previously recognised Takapuna gravity anomaly.

Synergistic regulation of dielectric-magnetic dual-loss and triple heterointerface polarization via magnetic MXene for high-performance electromagnetic wave absorption

• NiCo/CeO 2 /Ti 3 C 2 T x composites with multilayer structure were prepared. • The effective bandwidth of the composite is up to 6.32 GHz at 1.9 mm. • The modulation of electromagnetic parameters was achieved by controlling the amount of Ce 3+ introduced. • The mechanism of wave absorption of composite materials is explored and explained in depth. The single dielectric/magnetic loss mechanism and invalid polarization loss are the crucial issues which limit the exploitation of multi-component electromagnetic wave (EMW) absorption materials, especially Ti 3 C 2 T x MXene based absorption materials. In this work, a series of loosely packed accordion-like 3D NiCo/CeO 2 /Ti 3 C 2 T x with tunable and high-efficient EMW absorption were fabricated by introducing self-assembled NiCo alloys on the surface of Ti 3 C 2 T x and subsequent anchoring of CeO 2 . On the one hand, the magnetic NiCo particles and highly conductive Ti 3 C 2 T x achieved the dielectric-magnetic dual-loss. On the other hand, the triple interfacial polarization among NiCo/CeO 2 /Ti 3 C 2 T x further enhanced the relaxation loss. In addition, the effects of cerium ions on tuning the compositional structure, electromagnetic parameters and impedance behavior of the composites are explored. The NiCo/CeO 2 /Ti 3 C 2 T x -0.1 performed a wide absorption bandwidth of 6.32 GHz at 1.9 mm, as well as, the minimum reflection loss value of -42.48 dB at 2.0 mm. This work putted forward a new synergistic regulation strategy of dielectric-magnetic dual-loss and triple heterointerface polarization via magnetic MXene for high-performance EMW absorption, which may shed new light on the fabrication of multiple-component synergistic loss absorption materials.

Power Line Harmonic Radiation Observed by the Van Allen Probes Spacecraft

AbstractWe present a systematic analysis of Power Line Harmonic Radiation (PLHR) electromagnetic wave events observed by the Van Allen Probes spacecraft over the entire duration of the mission (2012–2019). PLHR events are radiated by electric power systems on the ground at harmonics of the base power system frequency (50 or 60 Hz, depending on the region). We analyze wave intensities at frequencies corresponding to the first few harmonics and we demonstrate that they are significantly enhanced at the times when the field‐aligned magnetic footprint of the spacecraft is above industrialized areas. Considering the spacecraft locations close to the equatorial plane, this represents experimental evidence that PLHR emissions may propagate up to radial distances as large as about 4 Earth radii. Average frequency spectra above selected industrialized regions reveal increased wave intensities at the frequencies corresponding to the first and the third harmonics, in particular during the nighttime. Multicomponent wave measurements allow us to perform a detailed wave analysis of polarization and propagation properties of the events. The waves are typically right‐handed elliptically or nearly circularly polarized, propagating with oblique wave normal angles from low latitude sources and more field‐aligned wave vectors from high latitude sources. The sign of the parallel component of the Poynting flux confirms that the events indeed propagate from the anticipated ground regions. Few events observed above the United States region propagate in the opposite direction, which is interpreted as wave bouncing between the hemispheres.

Multi-Component Electroacoustic Waves (Mceaw) In Piezo Crystalline Textures: Applied Opportunities

In this paper is shown, that in each sagittal plane of the piezoelectric texture, both the one-component elastic shear wave and the two-component elastic plane deformation wave can be accompanied by either oscillations of the transverse component of the electric field or oscillations of the plane electric field. Four such packets of multicomponent electroelastic waves are formed.Necessary and sufficient conditions are obtained that allow the formulation of a two-dimensional problem of electroelasticity in a vibrated sagittal plane of a piezocrystal, where separate excitation and propagation of a multicomponent quasistatic electroelastic wave is possible.

Crustal Structure across the West Antarctic Rift System from Multicomponent Ambient Noise Surface Wave Tomography

Abstract Approximately 2 yr (2010–2011) of continuous seismic records from a subset of the Antarctic component of the Polar Earth Observing Network (POLENET-ANET) seismic network deployed in West Antarctica are used to compute the nine components of the correlation tensor between each pair of stations in the network. Rayleigh wave velocity information from the vertical and radial components was extracted in the form of group and phase velocity dispersion curves, whereas the transverse component provided complimentary Love wave velocity information. The multicomponent Rayleigh wave measurements (ZZ, RR, ZR, and RZ) were averaged and used to infer the measurement uncertainties. The Rayleigh and Love wave group and phase velocities were then regionalized in space using a 2D deterministic tomography. A transect that spans the West Antarctic rift system was extracted from the tomography at individual periods between 7 and 60 s for the four types of surface wave velocities (i.e., Rayleigh and Love phase and group velocities). A transdimensional Bayesian joint inversion algorithm was used to invert these four datasets for a 1D model of isotropic shear-wave velocity versus depth at each point along the transect. In this way, surface wave dispersion curves from multicomponent noise correlations were used to build a 2D isotropic shear-wave velocity model down to ∼55 km depth. In this model, the top of the large low-velocity zone beneath Marie Byrd Land was imaged (up to a 5% decrease in velocity at ∼50 km depth), which provides further evidence for a mantle hot spot beneath the crust that supports the high topography in this region. We also observed a large velocity contrast in the lower crust beneath Marie Byrd Land at a depth where previous long-period seismicity has been observed. This strong contrast occurs more shallow than in previous crustal models, which compared to our model identify a deeper Moho (∼5–10 km deeper) beneath Marie Byrd Land. This new model has implications for interpreting earthquake locations in this region and perhaps necessitates that we revisit past hypocenter estimation studies using updated velocity models for the region.

Strong and weak interactions of rational vector rogue waves and solitons to anyn-component nonlinear Schrödinger system with higher-order effects

The higher-order effects play an important role in the wave propagations of ultrashort (e.g. subpicosecond or femtosecond) light pulses in optical fibres. In this paper, we investigate anyn-component fourth-order nonlinear Schrödinger (n-FONLS) system with non-zero backgrounds containing then-Hirota equation and then-Lakshmanan–Porsezian–Daniel equation. Based on the loop group theory, we find the multi-parameter family of novel rational vector rogue waves (RVRWs) of then-FONLS equation starting from the plane-wave solutions. Moreover, we exhibit the weak and strong interactions of some representative RVRW structures. In particular, we also find that the W-shaped rational vector dark and bright solitons of then-FONLS equation as the second- and fourth-order dispersion coefficients satisfy some relation. Furthermore, we find the higher-order RVRWs of then-FONLS equation. These obtained rational solutions will be useful in the study of RVRW phenomena of multi-component nonlinear wave models in nonlinear optics, deep ocean and Bose–Einstein condensates.

Gravity–capillary multi-component wave patterns generated by a single-frequency wave-maker oscillation and subsequent resonances

Cross-waves are standing waves with crests perpendicular to a wave-maker; they are subharmonic waves excited by parametric instability. The modulational and chaotic behaviours of nonlinear cross-waves have been studied widely since the 1970s. Most of the previous work has focused on gravity waves where surface tension can be neglected. In this work we study cross-waves that are highly dependent on surface tension as well as gravity. By oscillating a planar wave-maker either vertically or horizontally with frequencies of 25 Hz through 40 Hz at one end of a rectangular basin, two-dimensional multi-component surface patterns are realized. Using the free-surface synthetic Schlieren technique to measure the surface elevations, multi-dimensional Fourier transforms are utilized to track the evolutionary spectrum of the water surface in both the temporal and spatial domains. Wavelet transforms are implemented to show the development of the various frequency components. Three-wave resonances with and without first subharmonics are observed for small nonlinearity. Three-dimensional oblique propagating cross-waves are generated at higher nonlinearity; unlike most previous cross-wave experiments, this staggered pattern propagates far downstream. Experimental evidence shows that two oblique propagating waves form a two-dimensional short-crested pattern, and that the lateral component of the waves develops into parametric sloshing modes corresponding to the width of the tank. Two regimes of nonlinear wave patterns, resonant triads and oblique propagating cross-waves, are delineated.

Manifestation of the Berry curvature in geophysical ray tracing

Geometrical phases, such as the Berry phase, have proven to be powerful concepts to understand numerous physical phenomena, from the precession of the Foucault pendulum to the quantum Hall effect and the existence of topological insulators. The Berry phase is generated by a quantity named the Berry curvature, which describes the local geometry of wave polarization relations and is known to appear in the equations of motion of multi-component wave packets. Such a geometrical contribution in ray propagation of vectorial fields has been observed in condensed matter, optics and cold atom physics. Here, we use a variational method with a vectorial Wentzel–Kramers–Brillouin ansatz to derive ray- tracing equations for geophysical waves and to reveal the contribution of the Berry curvature. We detail the case of shallow-water wave packets and propose a new interpretation of their oscillating motion around the equator. Our result shows a mismatch with the textbook scalar approach for ray tracing, by predicting a larger eastward velocity for Poincaré wave packets. This work enlightens the role of the geometry of wave polarization in various geophysical and astrophysical fluid waves, beyond the shallow-water model.

The Equations of Dirac and Maxwell as a Result of Combining Minkowski Space and the Space of Orientations into Seven-Dimensional Space-Time

The multicomponent wave function of the spin and vector fields is presented as a one-component function depending on the position and orientation of a zero-size moving rotating observer. It is shown that the Dirac and Maxwell equations and the fine structure constant are the result of the connection of two spaces: the Minkowski space and the space of orientations (the observer), and this relationship is not mathematical, but physical in nature.

Development of nuclear basis sets for multicomponent quantum chemistry methods.

The nuclear-electronic orbital (NEO) framework provides a practical approach for directly incorporating nuclear quantum effects and non-Born-Oppenheimer effects of specified nuclei, typically protons, into quantum chemistry calculations. Multicomponent wave function based methods, such as NEO coupled cluster singles and doubles, and multicomponent density functional theory (DFT), such as NEO-DFT, require the appropriate selection of electronic and nuclear basis sets. Although a wide array of electronic basis sets are available, systematically developed nuclear basis sets that balance accuracy and efficiency have been lacking. Herein, a series of nuclear basis sets are developed and shown to be accurate and efficient for describing both ground and excited state properties of multicomponent systems in which electrons and specified protons are treated quantum mechanically. Three series of Gaussian-type nuclear basis sets, denoted PB4, PB5, and PB6, are developed with varying levels of angular momentum. A machine-learning optimization procedure relying on the Gaussian process regression method is utilized to accelerate the optimization process. The basis sets are validated in terms of predictions of ground state energies, proton densities, proton affinities, and proton vibrational excitation energies, allowing the user to select the desired balance between accuracy and efficiency for the properties of interest. These nuclear basis sets will enhance the tractability of NEO methods for applications to a wide range of chemical systems.

Nuclear-electronic all-particle density matrix renormalization group.

We introduce the Nuclear-Electronic All-Particle Density Matrix Renormalization Group (NEAP-DMRG) method for solving the time-independent Schrödinger equation simultaneously for electrons and other quantum species. In contrast to the already existing multicomponent approaches, in this work, we construct from the outset a multi-reference trial wave function with stochastically optimized non-orthogonal Gaussian orbitals. By iterative refining of the Gaussians' positions and widths, we obtain a compact multi-reference expansion for the multicomponent wave function. We extend the DMRG algorithm to multicomponent wave functions to take into account inter- and intra-species correlation effects. The efficient parameterization of the total wave function as a matrix product state allows NEAP-DMRG to accurately approximate the full configuration interaction energies of molecular systems with more than three nuclei and 12 particles in total, which is currently a major challenge for other multicomponent approaches. We present the NEAP-DMRG results for two few-body systems, i.e., H2 and H3 +, and one larger system, namely, BH3.

A wave propagation approach for reduced dynamic modeling of distillation columns: Optimization and control

Reduced models enable real-time optimization of large-scale processes. We propose a reduced model of distillation columns based on multicomponent nonlinear wave propagation (Kienle 2000). We use a nonlinear wave equation in dynamic mass and energy balances. We thus combine the ideas of compartment modeling and wave propagation. In contrast to existing reduced column models based on nonlinear wave propagation, our model deploys a hydraulic correlation. This enables the column holdup to change as load varies. The model parameters can be estimated solely based on steady-state data. The new transient wave propagation model can be used as a controller model for flexible process operation including load changes. To demonstrate this, we implement full-order and reduced dynamic models of an air separation process and multi-component distillation column in Modelica. We use the open-source framework DyOS for the dynamic optimizations and an Extended Kalman Filter for state estimation. We apply the reduced model in-silico in open-loop forward simulations as well as in several open- and closed-loop optimization and control case studies, and analyze the resulting computational speed-up compared to using full-order stage-by-stage column models. The first case study deals with tracking control of a single air separation distillation column, whereas the second one addresses economic model predictive control of an entire air separation process. The reduced model is able to adequately capture the transient column behavior. Compared to the full-order model, the reduced model achieves highly accurate profiles for the manipulated variables, while the optimizations with the reduced model are significantly faster, achieving more than 95% CPU time reduction in the closed-loop simulation and more than 96% in the open-loop optimizations. This enables the real-time capability of the reduced model in process optimization and control.