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At least 361 records · Page 20

Magnetic and thermodynamic structure of the solar corona during sunspot minimum

A model of the solar corona during sunspot minimum was constructed. It was suggested that pole-equator asymmetry is caused by interaction of global azimuthal electric currents in the corona and potential magnetic fields originating below the corona. The problem was reduced to a nonlinear ordinary differential equation for a structural function. Taking the difference in electron density between the equatorial plane and the polar direction from observations, the basic equation was solved and the global electric current distribution around the Sun along with the magnetic and thermodynamic structures of the solar corona was derived. The asymmetry between the two hemispheres, which can be due to the magnetic quadrupole, is considered in detail. It is shown that an additional quadrupole term in the magnetic field representation affects the position of polar coronal hole boundaries differently in the two hemispheres. As a result, the outflow from the two polar regions is found to be different.

Osherovich, V. A.↗

Semidiscrete Galerkin modelling of compressible viscous flow past a circular cone at incidence

A numerical study of the laminar and compressible boundary layer, about a circular cone in a supersonic free stream, is presented. It is thought that if accurate and efficient numerical schemes can be produced to solve the boundary layer equations, they can be joined to numerical codes that solve the inviscid outer flow. The combination of these numerical codes is competitive with the accurate, but computationally expensive, Navier-Stokes schemes. The primary goal is to develop a finite element method for the calculation of 3-D compressible laminar boundary layer about a yawed cone. The proposed method can, in principle, be extended to apply to the 3-D boundary layer of pointed bodies of arbitrary cross section. The 3-D boundary layer equations governing supersonic free stream flow about a cone are examined. The 3-D partial differential equations are reduced to 2-D integral equations by applying the Howarth, Mangler, Crocco transformations, a linear relation between viscosity, and a Blasius-type of similarity variable. This is equivalent to a Dorodnitsyn-type formulation. The reduced equations are independent of density and curvature effects, and resemble the weak form of the 2-D incompressible boundary layer equations in Cartesian coordinates. In addition the coordinate normal to the wall has been stretched, which reduces the gradients across the layer and provides high resolution near the surface. Utilizing the parabolic nature of the boundary layer equations, a finite element method is applied to the Dorodnitsyn formulation. The formulation is presented in a Petrov-Galerkin finite element form and discretized across the layer using linear interpolation functions. The finite element discretization yields a system of ordinary differential equations in the circumferential direction. The circumferential derivatives are solved by an implicit and noniterative finite difference marching scheme. Solutions are presented for a 15 deg half angle cone at angles of attack of 5 and 10 deg. The numerical solutions assume a laminar boundary layer with free stream Mach number of 7. Results include circumferential distribution of skin friction and surface heat transfer, and cross flow velocity distributions across the layer.

Meade, Andrew James, Jr.↗

Relaxation methods in fluid mechanics

The present work considers the iterative solution of a coupled set of difference equations and examines methods that carry successive approximates to a state that is invariant with further iteration and independent of the initial guess. Methods are studied with regard to their efficiency and economy of computer resources. The basic principles of classical relaxation are set forth, with attention confined to linear elliptic equations. This discussion involves the evaluation of the spectral radius that is the magnitude of the eigenvalue with largest modulus. The subject of relaxation is then related to the study of ordinary differential equations and hyperbolic partial differential equations. Problems that occur when linearly dependent eigenvectors appear in the relaxation matrix are discussed, leading to multiply connected eigenvalues in the Jordan canonical form. Finally, a brief survey of relaxation methods used in aerodynamics is given.

Lomax, H.↗

Legendre-tau approximations for functional differential equations

The numerical approximation of solutions to linear retarded functional differential equations are considered using the so-called Legendre-tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time-differentiation. The approximate solution is then represented as a truncated Legendre series with time-varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximation is made.

Ito, K.↗

Modeling and control of flexible space stations (slew maneuvers)

Large orbiting space structures are expected to experience mechanical vibrations arising from several disturbing forces such as those induced by shuttle takeoff or docking and crew movements. The problem is considered of modeling and control of large space structures subject to these and other disturbing forces. The system consists of a (rigid) massive body, which may play the role of experimental modules located at the center of the space station and flexible configurations, consisting of several beams, forming the space structure. A complete dynamic model of the system was developed using Hamilton's principle. This model consists of radial equations describing the translational motion of the central body, rotational equations describing the attitude motions of the body and several beam equations governing the vibration of the flexible members (platform) including appropriate boundary conditions. In summary, the dynamics of the space structure is governed by a complex system of interconnected partial and ordinary differential equations. Using Lyapunov's approach the asymptotic stability of the space structure is investigated. For asymptotic stability of the rest state (nominal trajectory), feedback controls are suggested. In the investigation, stability of the slewing maneuvers is also considered. Several numerical results are presented for illustration of the impact of coupling and the effectiveness of the stabilizing controls. Some insight is provided into the complexity of modeling, analysis and stabilization of actual space structures.

Ahmed, N. U.↗

Application of a one-strip integral method to the unsteady supersonic aerodynamics of an inclined flat surface

The method of integral relations is applied in a one-strip approximation to the perturbation equations governing small motions of an inclined, sharp-edged, flat surface about the mean supersonic steady flow. Algebraic expressions for low reduced-frequency aerodynamics are obtained and a set of ordinary differential equations are obtained for general oscillatory motion. Results are presented for low reduced-frequency aerodynamics and for the variation of the unsteady forces with frequency. The method gives accurate results for the aerodynamic forces at low reduced frequency which are in good agreement with available experimental data. However, for cases in which the aerodynamic forces vary rapidly with frequency, the results are qualitatively correct, but of limited accuracy. Calculations indicate that for a range of inclination angles near shock detachment such that the flow in the shock layer is low supersonic, the aerodynamic forces vary rapidly both with inclination angle and with reduced frequency.

Bennett, R. M.↗

Analysis of some fully-discrete algorithms for the one-dimensional heat equation

The present investigation is concerned with a fully discrete accuracy and stability analysis of the one-dimensional heat equation, taking into account the evaluation of two-pass explicit schemes which simultaneously employ lumped and coupled capacity matrices. Schemes of the considered characteristics are not amenable to uncoupled semidiscrete and ordinary differential equation analyses. The obtained results illustrate that superior behavior may be achieved by schemes of the employed type when compared with the performance of the standard one-pass explicit schemes. The key idea in the considered approach is related to the utilization of a reduced-quadrature capacity matrix in the evaluation of the right-hand-side residual.

Hughes, T. J. R.↗

Stability of distributed parameter systems.

A theorem is derived for the stability of solutions of general linear partial differential equations. A norm of the state space in the form of multiple integrals over the spatial domain is used for the Liapunov functional. Theorems and lemmas are also given for linear time-invariant constant coefficient distributed parameter systems, a class of nonlinear distributed parameter systems and for others with a Lure-type nonlinearity. The theorem conditions are similar to those known for corresponding ordinary differential equations but with operators replacing matrices.

Park, K. E.↗

Using computational singular perturbation as a diagnostic tool in ODE and DAE systems: a case study in heterogeneous catalysis

We have extended the computational singular perturbation (CSP) method to differential algebraic equation (DAE) systems and demonstrated its application in a heterogeneous-catalysis problem. The extended method obtains the CSP basis vectors for DAEs from a reduced Jacobian matrix that takes the algebraic constraints into account. Here we use a canonical problem in heterogeneous catalysis, the transient continuous stirred tank reactor (T-CSTR), for illustration. The T-CSTR problem is modelled fundamentally as an ordinary differential equation (ODE) system, but it can be transformed to a DAE system if one approximates typically fast surface processes using algebraic constraints for the surface species. We demonstrate the application of CSP analysis for both ODE and DAE constructions of a T-CSTR problem, illustrating the dynamical response of the system in each case. We also highlight the utility of the analysis in commenting on the quality of any particular DAE approximation built using the quasi-steady state approximation (QSSA), relative to the ODE reference case.

97 MATHEMATICS AND COMPUTING↗

Linear and nonlinear stability of the Blasius boundary layer

Two new techniques for the study of the linear and nonlinear instability in growing boundary layers are presented. The first technique employs partial differential equations of parabolic type exploiting the slow change of the mean flow, disturbance velocity profiles, wavelengths, and growth rates in the streamwise direction. The second technique solves the Navier-Stokes equation for spatially evolving disturbances using buffer zones adjacent to the inflow and outflow boundaries. Results of both techniques are in excellent agreement. The linear and nonlinear development of Tollmien-Schlichting (TS) waves in the Blasius boundary layer is investigated with both techniques and with a local procedure based on a system of ordinary differential equations. The results are compared with previous work and the effects of non-parallelism and nonlinearity are clarified. The effect of nonparallelism is confirmed to be weak and, consequently, not responsible for the discrepancies between measurements and theoretical results for parallel flow.

Bertolotti, F. P.↗

Towards an Enhanced Droplet Activation Scheme for Multi-Moment Bulk Microphysics Schemes

Initial droplet spectra produced upon activation impact the ensuing chain of microphysical processes andtherefore play a crucial role in cloud evolution. This work re-examines dependencies of newly formed clouddroplet size distribution (CDSD) characteristics on environmental and aerosol properties via parcel model simulationsthat serve as the basis for a multi-moment bulk microphysics droplet activation scheme suitable for acloud-resolving model (CRM). It is found that applying a fixed size threshold to define activated droplets versusemploying physical considerations can lead to erroneous activation and overly broad CDSDs for high aerosolconcentration and weak updraft conditions. Aerosol distributions characterized by larger median sizes and/orincreased solubility can result in greater activated droplet numbers, whereas impacts of these parameters onCDSD spectral width depend on both aerosol number concentration and updraft velocity. An expansion of theactivation scheme to include CDSD spectral width is proposed to aid efforts to extend high-order momentprediction to cloud droplet categories in CRMs as well as better represent variability in the activation process onthe cloud scale.simulations to investigate the regime dependence of the relative dispersion(d)1 of newly activated CDSDs, where d is the ratio of dropletradius standard deviation (σ) to the mean radius (r ). C16 demonstratedthat increasing Na resulted in increasing (decreasing) d values via reducedcondensational narrowing (spectral broadening) rates within theAL (UL) regime, with d values peaking in the TR regime. Their findingssuggest a similar regime dependence for d as R09 noted for Nc and helpexplain reportedly conflicting relationships between Na and CDSDspectral characteristics (cf. Hudson and Noble, 2014; Liu et al., 2014),although the applicability of these results within bulk microphysicalschemes was not addressed.Simulating aerosol-cloud interactions with CRMs employing bulkmicrophysics requires that the scheme minimally predict two CDSDparameters, namely mass and number concentrations, and represent thedroplet activation process. Various activation schemes aim to determineNc from aerosol and environmental properties and include analyticalexpressions (e.g., Abdul-Razzak et al., 1998; Morrison et al., 2005) aswell as lookup tables (LUTs) based on detailed parcel model calculations(e.g., Saleeby and Cotton, 2004, hereafter SC04; Segal and Khain,2006; Thompson and Eidhammer, 2014). Expressions to diagnose CDSDspectral width from Nc (Grabowski, 1998; Liu et al., 2006; Morrison andGrabowski, 2007) or cloud water content (Geoffroy et al., 2010) havealso been developed, although more robust methods to obtain CDSDspectral width upon activation are presently lacking. This latter point isrelevant for triple-moment (3 M) bulk microphysics that aim to predictdistribution spectral width alongside number and mass concentrations(e.g., Loftus et al., 2014; Milbrandt and Yau, 2005).The current work extends the findings of C16 to the current LUTbasedaerosol activation scheme used in the Regional AtmosphericModeling System (RAMS) (Cotton et al., 2003; SC04; Saleeby and vanden Heever, 2013, hereafter SvdH13) and additionally examinesaerosol size and solubility impacts on newly activated CDSD properties.Because early cloud development processes such as condensationalgrowth, evaporation, and droplet self-collection depend on and impactCDSD spectral width (Hudson and Yum, 1997; Seifert and Beheng 2001;Lu and Seinfeld, 2006; Igel and van den Heever, 2017), an expansion ofthe activation LUTs to include CDSD spectral width is proposed as apreliminary step for extending 3M prediction to CDSDs in CRMs forimproved simulations of aerosol-cloud interactions.2. MethodologyThe current RAMS two-moment microphysics module determinesthe fractional number of aerosol particles that activate to cloud dropletsfrom five-dimensional LUTs based on model predicted air temperature(T), w, Na, and the geometric median radius (rg) and soluble fraction (ε)of the aerosol size distribution (SvdH13). These LUTs are created offlineusing a one-dimensional Lagrangian adiabatic parcel model (Feingoldand Heymsfield, 1992; Heymsfield and Sabin, 1989; SC04) to simulateexplicit droplet activation and initial CDSD growth for a range of ambientatmospheric conditions [T, w] and binned lognormal aerosol sizedistributions given by= ⎡⎣ ⎢− ⎤⎦ ⎥N r Nr π σr rσ( )2 lnexp[ln( / )]2(ln )aggg22 (1)where r is the dry aerosol particle bin radius and σg is the geometricstandard deviation of the distribution. As the parcel model simulationsfocus on the activation process, other processes such as coalescence,sedimentation, and mixing are not considered. Details of the parcelmodel can be found in SC04 and SvdH13, and only a brief description isprovided here. At the onset of parcel model calculations, the initiallydry aerosol particles in all bins first deliquesce and reach theirequilibrium diameters in a sub-saturated environment based on theKöhler equation for solution droplets. The parcel is then lifted at a fixedupward velocity w, and particle growth by vapor diffusion, along withconcurrent changes in the ambient environment, are iteratively computedusing the Variable-coefficient Ordinary Differential Equation(VODE) solver (Brown et al., 1989). The time resolution of these calculationsis determined within the VODE solver, and the frequency atwhich the solver is called is controlled by a longer model time stepbased on fixed upward parcel displacement increments (Δz) at thespecified w (Δt=Δz/w). Model calculations proceed until the parcelreaches a height 50m beyond the level of maximum saturation ratio(Smax) or total parcel displacement exceeds 2 km. Upon model termination,Smax and the fractional number of aerosols (factv) resulting innewly formed cloud droplets, defined as particles having diameters of atleast 2 μm, are cataloged in the LUTs according to the specified T, w, Na,rg, and ε parameter values.A critical point regarding the creation of these LUTs is the use of afixed minimum diameter (Dmin) to define cloud droplets in the parcelmodel, which can produce erroneous CDSD characteristics, particularlywithin the UL regime. For aerosol distributions with large rg valuesunder low SS conditions, for example, deliquesced aerosols within thelarge tail of the distribution can exceed 2 μm in diameter yet remainunactivated as ‘haze’ particles (Levin and Cotton, 2009; McFigganset al., 2006). For this study, aerosol particles activate to cloud dropletsbased on the critical diameter Dcrit as a function of parcel supersaturationratio (Sr) as in R09:D = σ MS RTρ83 ln( ) critsol wr w (2)where σsol is the surface tension of a solution droplet, Mw and ρw are themolar mass and density of liquid water, respectively, and R is theuniversal gas constant. Additionally, at relatively large w values withinthe AL regime, Nc stabilizes shortly after reaching supersaturation.However, parcel ascent and condensational growth continue beyondthe level of Smax, potentially causing additional narrowing of the CDSD.In the current work, model calculations terminate upon reaching Smaxas changes in Nc are negligible with continued ascent (Peng et al., 2007;R09).Parcel model simulations are performed to examine the sensitivitiesof CDSD characteristics to w, Na, rg, and ε, with the ranges for theseparameters listed in Table 1. Aerosols are assumed to be a mix of solubleand insoluble material of equal density, specified by ε, where fullysoluble aerosols correspond to ammonium sulfate with hygroscopicityparameter κ=0.61 (Petters and Kreidenweis, 2007). FollowingSvdH13, aerosol geometric standard deviation is fixed at σg=1.8, andaerosol distributions (Eq. 1) are partitioned into 100 logarithmicallyspacedbins spanning a size range specific to each rg value. For all simulations,Δz=1 m, and initial values of relative humidity, air temperatureand pressure are set to RH=0.99, T=10 °C and p=900 hPa,respectively.

Loftus, Adrian M.↗

Estimation of elastic aircraft parameters using the maximum likelihood method

The application of the maximum likelihood method to estimate the aerodynamic parameters of elastic flight vehicles in a symmetric flight condition is discussed. In this application, particular attention is directed toward the center of mass, elastic deformation, and sensor equations of motion. It is shown that the two major computational problems to be overcome are the inversion of large-sized matrices and the time-wise integration of a large number of linear, ordinary, differential equations.

Schwanz, R. C.↗

Time and frequency domain analysis of sampled data controllers via mixed operation equations

Specification of the mathematical equations required to define the dynamic response of a linear continuous plant, subject to sampled data control, is complicated by the fact that the digital components of the control system cannot be modeled via linear ordinary differential equations. This complication can be overcome by introducing two new mathematical operations; namely, the operation of zero order hold and digial delay. It is shown that by direct utilization of these operations, a set of linear mixed operation equations can be written and used to define the dynamic response characteristics of the controlled system. It also is shown how these linear mixed operation equations lead, in an automatable manner, directly to a set of finite difference equations which are in a format compatible with follow on time and frequency domain analysis methods.

Frisch, H. P.↗

Latent Twins

Over the past decade, scientific machine learning has transformed the development of mathematical and computational frameworks for analyzing, modeling, and predicting complex systems. From inverse problems to numerical partial differential equations (PDEs), dynamical systems, and model reduction, these advances have pushed the boundaries of what can be simulated. Yet they have often progressed in parallel, with representation learning and algorithmic solution methods evolving largely as separate pipelines. With Latent Twins, we propose a unifying mathematical framework that creates a hidden surrogate in latent space for the underlying equations. Whereas digital twins mirror physical systems in the digital world, Latent Twins mirror mathematical systems in a learned latent space governed by operators. Through this lens, classical modeling, inversion, model reduction, and operator approximation all emerge as special cases of a single principle. We establish the fundamental approximation properties of Latent Twins for both ordinary differential equations (ODEs) and PDEs and demonstrate the framework across three representative settings: (i) canonical ODEs, capturing diverse dynamical regimes; (ii) a PDE benchmark using the shallow-water equations, contrasting Latent Twin simulations with deep operator network and forecasts with a four-dimensional variational method baseline; and (iii) a challenging real-data geopotential reanalysis dataset, reconstructing and forecasting from sparse, noisy observations. Latent Twins provide a compact, interpretable surrogate for solution operators that evaluate across arbitrary time gaps in a single-shot, while remaining compatible with scientific pipelines such as assimilation, control, and uncertainty quantification. Looking forward, this framework offers scalable, theory-grounded surrogates that bridge data-driven representation learning and classical scientific modeling across disciplines.

Latent Twins↗

Machine-learned closure of URANS for stably stratified turbulence: connecting physical timescales & data hyperparameters of deep time-series models

Stably stratified turbulence (SST), a model that is representative of the turbulence found in the oceans and atmosphere, is strongly affected by fine balances between forces and becomes more anisotropic in time for decaying scenarios. Moreover, there is a limited understanding of the physical phenomena described by some of the terms in the Unsteady Reynolds-Averaged Navier–Stokes (URANS) equations—used to numerically simulate approximate solutions for such turbulent flows. Rather than attempting to model each term in URANS separately, it is attractive to explore the capability of machine learning (ML) to model groups of terms, i.e. to directly model the force balances. We develop deep time-series ML for closure modeling of the URANS equations applied to SST. We consider decaying SST which are homogeneous and stably stratified by a uniform density gradient, enabling dimensionality reduction. We consider two time-series ML models: long short-term memory and neural ordinary differential equation. Both models perform accurately and are numerically stable in a posteriori (online) tests. Furthermore, we explore the data requirements of the time-series ML models by extracting physically relevant timescales of the complex system. We find that the ratio of the timescales of the minimum information required by the ML models to accurately capture the dynamics of the SST corresponds to the Reynolds number of the flow. The current framework provides the backbone to explore the capability of such models to capture the dynamics of high-dimensional complex dynamical system like SST flows.

97 MATHEMATICS AND COMPUTING↗

A new formulation of electromagnetic wave scattering using an on-surface radiation boundary condition approach

A new formulation of electromagnetic wave scattering by convex, two-dimensional conducting bodies is reported. This formulation, called the on-surface radiation condition (OSRC) approach, is based upon an expansion of the radiation condition applied directly on the surface of a scatterer. It is now shown that application of a suitable radiation condition directly on the surface of a convex conducting scatterer can lead to substantial simplification of the frequency-domain integral equation for the scattered field, which is reduced to just a line integral. For the transverse magnetic case, the integrand is known explicitly. For the transverse electric case, the integrand can be easily constructed by solving an ordinary differential equation around the scatterer surface contour. Examples are provided which show that OSRC yields computed near and far fields which approach the exact results for canonical shapes such as the circular cylinder, square cylinder, and strip. Electrical sizes for the examples are ka = 5 and ka = 10. The new OSRC formulation of scattering may present a useful alternative to present integral equation and uniform high-frequency approaches for convex cylinders larger than ka = 1. Structures with edges or corners can also be analyzed, although more work is needed to incorporate the physics of singular currents at these discontinuities. Convex dielectric structures can also be treated using OSRC.

Kriegsmann, Gregory A.↗

An adaptive grid algorithm for one-dimensional nonlinear equations

Richards' equation, which models the flow of liquid through unsaturated porous media, is highly nonlinear and difficult to solve. Step gradients in the field variables require the use of fine grids and small time step sizes. The numerical instabilities caused by the nonlinearities often require the use of iterative methods such as Picard or Newton interation. These difficulties result in large CPU requirements in solving Richards equation. With this in mind, adaptive and multigrid methods are investigated for use with nonlinear equations such as Richards' equation. Attention is focused on one-dimensional transient problems. To investigate the use of multigrid and adaptive grid methods, a series of problems are studied. First, a multigrid program is developed and used to solve an ordinary differential equation, demonstrating the efficiency with which low and high frequency errors are smoothed out. The multigrid algorithm and an adaptive grid algorithm is used to solve one-dimensional transient partial differential equations, such as the diffusive and convective-diffusion equations. The performance of these programs are compared to that of the Gauss-Seidel and tridiagonal methods. The adaptive and multigrid schemes outperformed the Gauss-Seidel algorithm, but were not as fast as the tridiagonal method. The adaptive grid scheme solved the problems slightly faster than the multigrid method. To solve nonlinear problems, Picard iterations are introduced into the adaptive grid and tridiagonal methods. Burgers' equation is used as a test problem for the two algorithms. Both methods obtain solutions of comparable accuracy for similar time increments. For the Burgers' equation, the adaptive grid method finds the solution approximately three times faster than the tridiagonal method. Finally, both schemes are used to solve the water content formulation of the Richards' equation. For this problem, the adaptive grid method obtains a more accurate solution in fewer work units and less computation time than required by the tridiagonal method. The performance of the adaptive grid method tends to degrade as the solution process proceeds in time, but still remains faster than the tridiagonal scheme.

Gutierrez, William E.↗

Tutorial: Langevin Dynamics methods for aerosol particle trajectory simulations and collision rate constant modeling

The Langevin Dynamics (LD) method (also known in the literature as Brownian Dynamics) is routinely used to simulate aerosol particle trajectories for transport rate constant calculations as well as to understand aerosol particle transport in internal and external fluid flows. This tutorial intends to explain the methodological details of setting up a LD simulation of a population of aerosol particles and to deduce rate constants from an ensemble of classical trajectories. We discuss the applicability and limitations of the translational Langevin equation to model the combined stochastic and deterministic motion of particles in fields of force or fluid flow. The drag force and stochastic “diffusion” force terms that appear in the Langevin equation are discussed elaborately, along with a summary of common forces relevant to aerosol systems (electrostatic, gravity, van der Waals, …); a commonly used first order and a fourth order Runge-Kutta time stepping schemes for linear stochastic ordinary differential equations are presented. A MATLAB® implementation of a LD code for simulating particle settling under gravity using the first order scheme is included for illustration. Scaling analysis of aerosol transport processes and the selection of timestep and domain size for trajectory simulations are demonstrated through two specific aerosol processes: particle diffusion charging and coagulation. Fortran® implementations of the first order and fourth order time-stepping schemes are included for simulating the 3D motion of a particle in a periodic domain. Lastly, potential applications and caveats to the usage of LD are included as a summary.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗