On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives

In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.

Electromagnetic Scattering from a Graphene Disk: Helmholtz-Galerkin Technique and Surface Plasmon Resonances

The surface plasmon resonances of a monolayer graphene disk, excited by an impinging plane wave, are studied by means of an analytical-numerical technique based on the Helmholtz decomposition and the Galerkin method. An integral equation is obtained by imposing the impedance boundary condition on the disk surface, assuming the graphene surface conductivity provided by the Kubo formalism. The problem is equivalently formulated as a set of one-dimensional integral equations for the harmonics of the surface current density. The Helmholtz decomposition of each harmonic allows for scalar unknowns in the vector Hankel transform domain. A fast-converging Fredholm second-kind matrix operator equation is achieved by selecting the eigenfunctions of the most singular part of the integral operator, reconstructing the physical behavior of the unknowns, as expansion functions in a Galerkin scheme. The surface plasmon resonance frequencies are simply individuated by the peaks of the total scattering cross-section and the absorption cross-section, which are expressed in closed form. It is shown that the surface plasmon resonance frequencies can be tuned by operating on the chemical potential of the graphene and that, for orthogonal incidence, the corresponding near field behavior resembles a cylindrical standing wave with one variation along the disk azimuth.

Anisotropic Statistics of Lagrangian Structure Functions and Helmholtz Decomposition

We present a new method to estimate second-order horizontal velocity structure functions, as well as their Helmholtz decomposition into rotational and divergent components, from sparse data collected along Lagrangian observations. The novelty compared to existing methods is that we allow for anisotropic statistics in the velocity field and also in the collection of the Lagrangian data. Specifically, we assume only stationarity and spatial homogeneity of the data and that the cross covariance between the rotational and divergent flow components is either zero or a function of the separation distance only. No further assumptions are made and the anisotropy of the underlying flow components can be arbitrarily strong. We demonstrate our new method by testing it against synthetic data and applying it to the Lagrangian Submesoscale Experiment (LASER) dataset. We also identify an improved statistical angle-weighting technique that generally increases the accuracy of structure function estimations in the presence of anisotropy.

The rotational and divergent components of atmospheric circulation on tidally locked planets

Tidally locked exoplanets likely host global atmospheric circulations with a superrotating equatorial jet, planetary-scale stationary waves, and thermally driven overturning circulation. In this work, we show that each of these features can be separated from the total circulation by using a Helmholtz decomposition, which splits the circulation into rotational (divergence-free) and divergent (vorticity-free) components. This technique is applied to the simulated circulation of a terrestrial planet and a gaseous hot Jupiter. For both planets, the rotational component comprises the equatorial jet and stationary waves, and the divergent component contains the overturning circulation. Separating out each component allows us to evaluate their spatial structure and relative contribution to the total flow. In contrast with previous work, we show that divergent velocities are not negligible when compared with rotational velocities and that divergent, overturning circulation takes the form of a single, roughly isotropic cell that ascends on the day side and descends on the night side. These conclusions are drawn for both the terrestrial case and the hot Jupiter. To illustrate the utility of the Helmholtz decomposition for studying atmospheric processes, we compute the contribution of each of the circulation components to heat transport from day side to night side. Surprisingly, we find that the divergent circulation dominates day–night heat transport in the terrestrial case and accounts for around half of the heat transport for the hot Jupiter. The relative contributions of the rotational and divergent components to day–night heat transport are likely sensitive to multiple planetary parameters and atmospheric processes and merit further study.

Anisotropic statistics of Lagrangian structure functions and Helmholtz decomposition

<p>Second-order velocity structure functions are commonly estimated from Lagrangian tracer trajectories.  A Helmholtz decomposition of these structure functions, which separates their divergent and rotational components, can indicate the robustness of geostrophic balance at different scales, and serves as a building block for analysis of scale-dependent energy distributions. We present a new method to estimate second-order horizontal velocity structure functions, as well as their Helmholtz decomposition, from sparse data collected by Lagrangian observations.   The novelty compared to existing methods is that we allow for anisotropic statistics in the velocity field as well as in the distribution of the Lagrangian trackers. We conduct the analysis through the lens of azimuthal Fourier expansions, and find Helmholtz decomposition formulae targeted at individual Fourier modes. We also identify an improved statistical angle-weighting technique that generally increases the accuracy of structure function estimations in the presence of anisotropy. The new methods are tested against synthetic data and applied to surface drifter data sets such as LASER and GLAD. Importantly, the new method does not require extra measurements compared to existing methods based on isotropy.</p>

Helmholtz’s decomposition for compressible flows and its application to computational aeroacoustics

The Helmholtz decomposition, a fundamental theorem in vector analysis, separates a given vector field into an irrotational (longitudinal, compressible) and a solenoidal (transverse, vortical) part. The main challenge of this decomposition is the restricted and finite flow domain without vanishing flow velocity at the boundaries. To achieve a unique and $$L_2$$ L 2 -orthogonal decomposition, we enforce the correct boundary conditions and provide its physical interpretation. Based on this formulation for bounded domains, the flow velocity is decomposed. Combining the results with Goldstein’s aeroacoustic theory, we model the non-radiating base flow by the transverse part. Thereby, this approach allows a precise physical definition of the acoustic source terms for computational aeroacoustics via the non-radiating base flow. In a final simulation example, Helmholtz’s decomposition of compressible flow data using the finite element method is applied to an overflowed rectangular cavity at Mach 0.8. The results show a reasonable agreement with the source data and illustrate the distinct parts of the Helmholtz decomposition.