Curl Operator Research Articles

Modeling Tangential Vector Fields on a Sphere

ABSTRACTPhysical processes that manifest as tangential vector fields on a sphere are common in geophysical and environmental sciences. These naturally occurring vector fields are often subject to physical constraints, such as being curl-free or divergence-free. We start with constructing parametric models for curl-free and divergence-free vector fields that are tangential to the unit sphere through applying the surface gradient or the surface curl operator to a scalar random potential field on the unit sphere. Using the Helmholtz–Hodge decomposition, we then construct a class of simple but flexible parametric models for general tangential vector fields, which are represented as a sum of a curl-free and a divergence-free components. We propose a likelihood-based parameter estimation procedure, and show that fast computation is possible even for large datasets when the observations are on a regular latitude–longitude grid. Characteristics and practical utility of the proposed methodology are illustrated through extensive simulation studies and an application to a dataset of ocean surface wind velocities collected by satellite-based scatterometers. We also compare our model with a bivariate Matérn model and a non-stationary bivariate global model. Supplementary materials for this article are available online.

Lower bound of Riesz transform kernels and commutator theorems on stratified nilpotent Lie groups

We provide a study of the Riesz transforms on stratified nilpotent Lie groups, and obtain a certain version of the pointwise lower bound of the Riesz transform kernel. Then we establish the characterisation of the BMO space on stratified nilpotent Lie groups via the boundedness of the commutator of the Riesz transforms and the BMO function. This extends the well-known Coifman, Rochberg, Weiss theorem from Euclidean space to the setting of stratified nilpotent Lie groups. In particular, these results apply to the well-known example of the Heisenberg group. As an application, we also study the curl operator on the Heisenberg group and stratified nilpotent Lie groups, and establish the div–curl lemma with respect to the Hardy space on stratified nilpotent Lie groups.

Densely defined non‐closable curl on carpet‐like metric measure spaces

AbstractThe paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on 1‐forms on metric measure spaces. The main examples we consider are the non self‐similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one‐dimensional, they may have positive two‐dimensional Lebesgue measure and carry nontrivial 2‐forms. We prove that in this case the curl operator (and therefore also the exterior derivative on 1‐forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abstract result. It states that for spaces that are, in a certain way, structurally similar to Sierpinski carpets, the exterior derivative operator taking 1‐forms into 2‐forms cannot be closable if the martingale dimension is larger than one.

Are There the Pure TE and TM Modes in the Closed Waveguide Filled With a Homogeneous, Anisotropic and Lossless Medium?

Using the Abelian group theory and the classic electromagnetic waveguide theory, we propose a sufficient condition for the existence of the pure TE and TM modes in the closed waveguide filled with a homogeneous, anisotropic, and lossless medium. This sufficient condition is the relation between the permittivity and permeability tensors. Four eigenvalue problems that can be used to simulate the pure TE and TM modes in the anisotropic waveguide are derived from source-free Maxwell’s equations. Two of them are eigenvalue problems of the elliptic differential operator, and the other two are eigenvalue problems of the curl–curl operator. Moreover, we solve these four eigenvalue problems by the finite element method. Finally, we carry out some numerical experiments to verify our theoretical results.

Applications of a formula on Beltrami flow

In this note, we obtain uniqueness results for Beltrami flow in both bounded and unbounded domain with nonempty boundary by establishing an elementary but useful formula involving operators div and curl. We also use this formula to deal with Maxwell and Stokes eigenvalue problems.

Inner Structure of CME Shock Fronts Revealed by the Electromotive Force and Turbulent Transport Coefficients in Helios-2 Observations

Abstract Electromotive force is an essential quantity in dynamo theory. During a coronal mass ejection (CME), magnetic helicity gets decoupled from the Sun and advected into the heliosphere with the solar wind. Eventually, a heliospheric magnetic transient event might pass by a spacecraft, such as the Helios space observatories. Our aim is to investigate the electromotive force, the kinetic helicity effect (α term), the turbulent diffusion (β term), and the cross-helicity effect (γ term) in the inner heliosphere below 1 au. We set up a one-dimensional model of the solar wind velocity and magnetic field for a hypothetic interplanetary CME. Because turbulent structures within the solar wind evolve much slower than this structure needs to pass by the spacecraft, we use a reduced curl operator to compute the current density and vorticity. We test our CME shock-front model against an observed magnetic transient that passes by the Helios-2 spacecraft. At the peak of the fluctuations in this event we find strongly enhanced α, β, and γ terms, as well as a strong peak in the total electromotive force. Our method allows us to automatically identify magnetic transient events from any in situ spacecraft observations that contain magnetic field and plasma velocity data of the solar wind.

Elastic 3D PS converted-wave Gaussian beam migration

Elastic-wave migration is a desirable technique because it can image the structure of the earth more accurately. We develop a new elastic Gaussian beam migration method with 3D three component (3D-3C) seismic data that focuses on a complex PS-converted wave. Based on the elastic-wave equations and complete boundary conditions, we derive effective work formulas for an accurate multimode wave downward continuation for the free-space, ocean-bottom, and free-surface models. We separate the PS-wave into linear-polarized P-S1 and P-S2 waves to simplify the expression and derivation of the migration. To image the vectorial wave directly and solve the reverse-polarity issue, we use the crosscorrelations of P-wave divergence and PS-wave curl operators as the 3D P- and PS-imaging conditions, and we develop a unit vector to define the rotation direction of the PS-wave. With our approach, 3D-3C multimode waves are automatically decomposed to P- and PS-waves during the migration without the need for prior data separation, which not only reduces the crosstalk noise caused by inaccurate multimode wave decomposition but also decreases the processing cost. Applications of this method to two 3D-3C synthetic examples indicate successful PS-wave migration. Also, confirming that the PS-image can be constructed by summing the P-S1 and P-S2 images and is independent of the choice of the ray-centered local coordinates validates this method.

Closed-Form Inverses of the Weakly Singular and Hypersingular Operators on Disks

We establish new explicit expressions and variational forms of boundary integral operators that provide the exact inverses of the weakly singular and hypersingular operators for \(-\Delta \) on flat disks. We derive closed-form formulas for their singular kernels. We also show that the inverse of the weakly singular operator can be obtained by composing surface curl operators and the inverse of the hypersingular operator.

Finite Element Approximation of the Spectrum of the Curl Operator in a Multiply Connected Domain

In this paper we are concerned with two topics: the formulation and analysis of the eigenvalue problem for the $$\mathop {\mathbf {curl}}\nolimits $$ operator in a multiply connected domain and its numerical approximation by means of finite elements. We prove that the $$\mathop {\mathbf {curl}}\nolimits $$ operator is self-adjoint on suitable Hilbert spaces, all of them being contained in the space for which $$\mathop {\mathbf {curl}}\nolimits \varvec{v}\cdot \varvec{n}=0$$ on the boundary. Additional constraints must be imposed when the physical domain is not topologically trivial: we show that a viable choice is the vanishing of the line integrals of $$\varvec{v}$$ on suitable homological cycles lying on the boundary. A saddle-point variational formulation is devised and analyzed, and a finite element numerical scheme is proposed. It is proved that eigenvalues and eigenfunctions are efficiently approximated and some numerical results are presented in order to assess the performance of the method.

The Brezis–Nirenberg problem for the curl–curl operator

We look for solutions E:Ω→R3 of the problem{∇×(∇×E)+λE=|E|p−2Ein Ων×E=0on ∂Ω on a bounded Lipschitz domain Ω⊂R3, where ∇× denotes the curl operator in R3. The equation describes the propagation of the time-harmonic electric field ℜ{E(x)eiωt} in a nonlinear isotropic material Ω with λ=−μεω2≤0, where μ and ε stand for the permeability and the linear part of the permittivity of the material. The nonlinear term |E|p−2E with p>2 is responsible for the nonlinear polarisation of Ω and the boundary conditions are those for Ω surrounded by a perfect conductor. The problem has a variational structure and we deal with the critical value p, for instance, in convex domains Ω or in domains with C1,1 boundary, p=6=2⁎ is the Sobolev critical exponent and we get the quintic nonlinearity in the equation. We show that there exist a cylindrically symmetric ground state solution and a finite number of cylindrically symmetric bound states depending on λ≤0. We develop a new critical point theory which allows to solve the problem, and which enables us to treat more general anisotropic media as well as other variational problems.

A theoretical and practical clarification on the calculation of reflection loss for microwave absorbing materials

Reflection loss is usually calculated and reported as a function of the thickness of microwave absorption material. However, misleading results are often obtained since the principles imbedded in the popular methods contradict the fundamental facts that electromagnetic waves cannot be reflected in a uniform material except when there is an interface and that there are important differences between the concepts of characteristic impedance and input impedance. In this paper, these inconsistencies have been analyzed theoretically and corrections provided. The problems with the calculations indicate a gap between the background knowledge of material scientists and microwave engineers and for that reason a concise review of transmission line theory is provided along with the mathematical background needed for a deeper understanding of the theory of reflection loss. The expressions of gradient, divergence, Laplacian, and curl operators in a general orthogonal coordinate system have been presented including the concept of reciprocal vectors. Gauss’s and Stokes’s theorems have been related to Green’s theorem in a novel way.

SP- and SS-imaging for 3D elastic reverse time migration

Three-dimensional elastic reverse time migration has been confronted with the problem of generating scalar images with vector S-waves. The underlying principle for solving this problem is to convert the vector S-waves into scalars. Previous methods were mainly focused on PS-imaging, but they usually cannot work properly on SP- and SS-cases. The complexity of SP- and SS-imaging arises from the fact that the incident S-wave has unpredictable relationship with the raypath plane. We have suggested that S-wave should be treated separately as SV- and SH-waves, which keep predictable relationships with the raypath plane. First, the elastic wavefield is separated into P- and S-waves using the Helmholtz decomposition. Then, we evaluate the normal direction of the raypath plane at each imaging grid. Next, we separate the vector S-wave obtained with curl operator into SH- and SV-waves, both of which are scalars. Finally, correlation imaging conditions are implemented to those scalar wave modes to produce scalar SV-P, SV-SV, and SH-SH images.

A consistent framework of material configurational mechanics in piezoelectric materials

The fundamental concepts of material configurational mechanics are formulated in piezoelectric materials. A consistent thermodynamic framework is outlined to develop the corresponding theory of material configurational stresses and forces associated with the \(J_{k}\)-, M- and L-integrals by the gradient, divergence and curl operation of the electric enthalpy density, respectively. The physical interpretation of material configurational stresses is explored, and they can be explained as the energy release rates due to the translation of material point along \(x_{k}\)-direction, the self-similar expansion, the rotation of material element, respectively. The path independence or path dependence of variant integrals such as the \(J_{k}\)-, M- and L-integrals is examined in piezoelectric material. As an application of material configurational mechanics in piezoelectric materials, an explicit method is derived to calculate the change of the J-integral as a dominant fracture parameter for a crack interaction with domain switching near the crack tip. It is concluded that domain switching has an obvious shielding effect on the fracture toughness in piezoelectrics by the present explicit forms. The present framework of material configurational mechanics is expected to provide an effective and convenient tool to deal with various crack or damage problems in piezoelectric materials.

Compatible Maxwell solvers with particles I: conforming and non-conforming 2D schemes with a strong Ampere law

This article is the second of a series where we develop and analyze structure-preserving finite element discretizations for the time-dependent 2D Maxwell system with long-time stability properties, and propose a charge-conserving deposition scheme to extend the stability properties in the case where the current source is provided by a particle method. The schemes proposed here derive from a previous study where a generalized commuting diagram was identified as an abstract compatibility criterion in the design of stable schemes for the Maxwell system alone, and applied to build a series of conforming and non-conforming schemes in the 3D case. Here the theory is extended to account for approximate sources, and specific charge-conserving schemes are provided for the 2D case. In this second article we study two schemes which include a strong discretization of the Faraday law. The first one is based on a standard conforming mixed finite element discretization and the long-time stability is ensured by the natural $L^2$ projection for the current, also standard. The second one is a new non-conforming variant where the numerical fields are sought in fully discontinuous spaces. In this 2D setting it is shown that the associated discrete curl operator coincides with that of a classical DG formulation with centered fluxes, and our analysis shows that a non-standard current approximation operator must be used to yield a charge-conserving scheme with long-time stability properties, while retaining the local nature of $L^2$ projections in discontinuous spaces. Numerical experiments involving Maxwell and Maxwell-Vlasov problems are then provided to validate the stability of the proposed methods.

ELASTIC REVERSE TIME MIGRATION BASED ON WAVEFIELD SEPARATION

AbstractCompared with other imaging algorithms (e.g., ray‐based, one‐way wave equation), reverse time migration (RTM) based on the two‐way wave equation exhibits greater superiority, especially in handling steeply dipping structures. However, imaging with conventional single‐component seismic data is unsuited for some complicated structures (e.g., gas clouds). Elastic RTM, which is based on the elastodynamic equation and uses multi‐component seismic data to extract PP and PS reflectivity and subsurface information, can more consistently reproduce the characteristics of elastic wave propagation in real Earth media, resulting in seismic images that more accurately characterize the subsurface. To begin with, we exploit the first order stress‐velocity equations to extrapolate the elastic vector wavefield, then the P‐ and S‐wavefields are separated by computing the divergence and curl operator of the extrapolated particle‐velocity wavefield. Then, imaging profiles with pure wave modes are computed by applying the source normalized cross‐correlation imaging condition, thus avoiding crosstalk between unseparated wave modes. To address the polarity reversal problem of the converted image, we propose an alternative method in the common‐shot domain. We also develop an efficient method that reconstructs the source wavefield in the reverse time direction to save storage in the GPU and to avoid large input/output in the elastic reverse time migration. During the forward modeling, the method only saves the particle‐velocity wavefield of all time intervals within an efficient absorbing boundary and the total wavefields in the final time interval. When we extrapolate the receiver wavefield in the reverse time direction, we simultaneously reconstruct the total source wavefields via the saved wavefields. Numerical examples performed with the graben and Marmousi2 models have shown that the polarity reversal correction method works, and elastic reverse time migration can accurately characterize complicated structures.

Fast Explicit and Unconditionally Stable FDTD Method for Electromagnetic Analysis

In this paper, a fast explicit and unconditionally stable finite-difference time-domain (FDTD) method is developed, which does not require a partial solution of a global eigenvalue problem. In this method, a patch-based single-grid representation of the FDTD algorithm is developed to facilitate both theoretical analysis and efficient computation. This representation results in a natural decomposition of the curl–curl operator into a series of rank-1 matrices, each of which corresponds to one patch in a single grid. The relationship is then theoretically analyzed between the fine patches and unstable modes, based on which an accurate and fast algorithm is developed to find unstable modes from fine patches with a bounded error. These unstable modes are then upfront eradicated from the numerical system before performing an explicit time marching. The resultant simulation is absolutely stable for the given time step irrespective of how large it is, the accuracy of which is also ensured. In addition, both lossless and general lossy problems are addressed in the proposed method. The advantages of the proposed method are demonstrated over the conventional FDTD and the state-of-the-art explicit and unconditionally stable FDTD methods by numerical experiments.

About vortex equations of two dimensional flows

A method to obtain a time-independent vortex solution of a nonlinear differential equation describing two-dimensional flow is investigated. In the usual way, starting from the Navier–Stokes equation the vortex equation is derived by taking a curl operation. After rearranging the equation of the vortex, we get a continuity equation or a divergence-free equation: \(\partial _1V_1+\partial _2V_2=0\). Additional irrotationality of \(V_1\) and \(V_2\) leads us to the Cauchy–Riemann condition satisfied by a newly introduced stream function \(\Psi\) and velocity potential \(\Phi\). As a result, if we know \(V_1\) and \(V_2\) or a combination of two, the differential equation is mapped to a lower-order partial differential equation. This differential equation is the one satisfied by the stream function \(\psi\) where the vorticity vector \(\omega\) is given by \(-(\partial _1^2+\partial _2^2) \psi\). A simple solution is discussed for the two different limits of viscosity.

Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium

We find solutions E:Ω→R3 of the problem{∇×(μ(x)−1∇×E)−ω2ε(x)E=∂EF(x,E)in Ων×E=0on ∂Ω on a bounded Lipschitz domain Ω⊂R3 with exterior normal ν:∂Ω→R3. Here ∇× denotes the curl operator in R3. The equation describes the propagation of the time-harmonic electric field ℜ{E(x)eiωt} in an anisotropic material with a magnetic permeability tensor μ(x)∈R3×3 and a permittivity tensor ε(x)∈R3×3. The boundary conditions are those for Ω surrounded by a perfect conductor. It is required that μ(x) and ε(x) are symmetric and positive definite uniformly for x∈Ω, and that μ,ε∈L∞(Ω,R3×3). The nonlinearity F:Ω×R3→R is superquadratic and subcritical in E, the model nonlinearity being of Kerr-type: F(x,E)=|Γ(x)E|p for some 2<p<6 with Γ(x)∈GL(3) invertible for every x∈Ω and Γ,Γ−1∈L∞(Ω,R3×3). We prove the existence of a ground state solution and of bound states if F is even in E. Moreover if the material is uniaxial we find two types of solutions with cylindrical symmetries.

Efficient wave-mode separation in vertical transversely isotropic media

Wave-mode separation can be achieved by projecting elastic wavefields onto mutually orthogonal polarization directions. In isotropic media, because the P-wave’s polarization vectors are consistent with wave vectors, the isotropic separation operators are represented by divergence and curl operators, which are easy to realize. In anisotropic media, polarization vectors deviate from wave vectors based on local anisotropic strength and separation operators lose their simplicity. Conventionally, anisotropic wave-mode separation is implemented either by direct filtering in the wavenumber domain or nonstationary filtering in the space domain, which are computationally expensive. Moreover, in conventional anisotropic separation, correcting for amplitude and phase changes of waveforms by applying separation operators is also more difficult than in an isotropic case. We have developed new operators for efficient wave-mode separation in vertical transversely isotropic (VTI) media. Our separation operators are constructed by local rotation of wave vectors to directions where the quasi-P (qP) wave is polarized. The deviation angles between the wave vectors and the qP-wave’s polarization vectors are explicitly estimated using the Poynting vectors. Obtaining polarization directions by rotating wave vectors yields separation operators in VTI media with the same forms as divergence and curl operators, except that the spatial derivatives are now rotated to implement wavefield projections in accurate polarization directions. The main increase in computational cost relative to isotropic separation operators is the estimation of the Poynting vectors, which is relatively small within elastic-wave extrapolation. As a result, applying the proposed operators is efficient. In the meantime, the waveforms corrections for divergence and curl operators can be directly extended for our new operators due to the similarities between these operators. By numerical exercises, we have determined that wave modes can be well-separated with small numerical cost using the present separation operators. The conservation of energy in wave-mode separation by applying waveform corrections was also verified.

Elastic wavefield separation based on the Helmholtz decomposition

Divergence and curl operators used for the decomposition of P- and S-wave modes in elastic reverse time migration (RTM) change the amplitudes, units, and phases of extrapolated wavefields. I separate the P- and S-waves in elastic media based on the Helmholtz decomposition. The decomposed wavefields based on this approach have the same amplitudes, units, and phases as the extrapolated wavefields. To avoid expensive multidimensional integrals in the Helmholtz decomposition, I introduce a fast Poisson solver to efficiently solve the vector Poisson’s equation. This fast algorithm allows us to reduce computational complexity from [Formula: see text] to [Formula: see text], where [Formula: see text] is the total number of grid points. Because the decomposed P- and S-waves are vector fields, I use vector imaging conditions to construct PP-, PS-, SS-, and SP-images. Several 2D numerical examples demonstrate that this approach allows us to accurately and efficiently decompose P- and S-waves in elastic media. In addition, elastic RTM images based on the vector imaging conditions have better quality and avoid polarity reversal in comparison with images based on the divergence and curl separation or direct component-by-component crosscorrelation.