Equations Of Geometrical Optics Research Articles

УСВОЕНИЕ ДАННЫХ РАДИОЗАТМЕННЫХ НАБЛЮДЕНИЙ В ЧИСЛЕННОМ ПРОГНОЗЕ ПОГОДЫ

The paper discusses the principles of the assimilation of radio occultation observations into numerical weather prediction models. The most widely used approach is assimilation of the bending angle profiles obtained from the observations by the wave optics methods. The observation operators are constructed from the geometric optics equations and can be three-dimensional (3D), taking into account the actual field of meteorological parameters, or one-dimensional (1D), using only one vertical profile. In the approximation of weak gradients, the 3D operator is reduced to the two-dimensional (2D) one, which takes into account the section of the fields of meteorological parameters by the vertical plane of the occultation event. Another approach is the assimilation of the retrieved atmospheric refractivity profiles by means of a simplest 1D or more accurate 2D operator. Some examples of statistical comparisons of the profiles of temperature, pressure, and specific humidity obtained by variational assimilation with the GFS reanalysis data are given.

A hybrid semi-Lagrangian DG and ADER-DG solver on a moving mesh for Liouville's equation of geometrical optics

Liouville's equation describes light propagation through an optical system. It governs the evolution of an energy distribution, the basic luminance, on phase space. The basic luminance is discontinuous across optical interfaces, as is described by non-local boundary conditions at each optical interface. Curved optical interfaces manifest themselves as moving boundaries on phase space. A common situation is that an optical system is described by a piecewise constant refractive index field. Away from optical interfaces, the characteristics of Liouville's equation reduce to straight lines. This property is exploited in the novel solver developed in this paper by employing a semi-Lagrangian discontinuous Galerkin (SLDG) scheme away from optical interfaces. Close to optical interfaces we apply an ADER discontinuous Galerkin (ADER-DG) method on a moving mesh. The ADER-DG method is a fully discrete explicit scheme which must obey a CFL condition that restricts the stepsize, whereas the SLDG scheme can be CFL-free. A moving mesh is used to align optical interfaces with the mesh. Very small elements cannot always be avoided, even when applying mesh refinement. Local time stepping is introduced in the solver to ensure these very small elements only have a local impact on the stepsize. By construction we allow elements of SLDG type to run at a stepsize independent of these small elements. The proposed SLDG scheme uses the exact evolution of the solution, as is described by the characteristics, to update the numerical solution. We impose the condition that no characteristic emanating from an SLDG element can cross an optical interface. In the novel hybrid SLDG and ADER-DG solver this condition is used to naturally divide the spatio-temporal domain into different regions, describing where the SLDG scheme and where the ADER-DG scheme need to be used. An intermediate element is introduced to efficiently couple an SLDG region with an ADER-DG region. Numerical experiments validate that the hybrid solver obeys energy conservation up to machine precision and numerical convergence results show the expected order of convergence. The performance of the hybrid solver is compared to a pure ADER-DG scheme with global time stepping to show the efficiency of the hybrid solver. In particular, a speed-up of 1.6 to 10, in favour of the hybrid solver, for computation times up to 4 minutes was seen in an example. Finally, the hybrid solver and the pure ADER-DG scheme are compared to quasi-Monte Carlo ray tracing. In the examples considered, amongst the three methods the hybrid solver is shown to converge the fastest to high accuracies.

Radio Occultation Data Analysis With Analytical Ray‐Tracing

AbstractRadio occultation experiments are a sensing technique dedicated to the remote sounding of planetary atmospheres. The technique exploits the frequency shift of a radio signal due to refraction in a planetary atmosphere. The aim is to infer the physical properties of the neutral atmosphere (e.g., pressure and temperature) and ionosphere (e.g., the electron number density). For one‐way occultations, the data processing usually relies on Abel transform algorithms when the atmosphere is spherically symmetric. For two‐way occultations, such techniques require the introduction of approximate relationships for the bending experienced by the signal to be obtained. In this context, we introduce a new method to process two‐way occultations data by spherically symmetric atmospheres using a ray‐tracing approach. However, the numerical integration of the geometrical optics equation through the atmosphere requires a significant computational time due to initial pointing issues. For this reason, our novel algorithm exploits a closed‐form solution to the equations of geometrical optics (Bourgoin et al., A&A, 624, A41, 2019, https://doi.org/10.1051/0004-6361/201834962) applied to a spherically symmetric atmosphere. Within this approach, the bending is directly provided by the analytical solution and no numerical integration is required. In addition, we develop a procedure enabling us to disentangle the contributions from dispersive and neutral media in the frequency shift. This procedure is validated by comparing our vertical profiles to those obtained using Abel inversion or numerical ray‐tracing for Mars and Titan occultation experiments. We show that our algorithm provides similar results to purely numerical ray‐tracing algorithms while significantly decreasing the computational time.

An ADER discontinuous Galerkin method on moving meshes for Liouville's equation of geometrical optics

Liouville's equation describes light propagation through an optical system. It governs the evolution of an energy distribution on phase space. This energy distribution is discontinuous across optical interfaces. Curved optical interfaces manifest themselves as moving boundaries on phase space. In this paper, an ADER discontinuous Galerkin (DG) method on a moving mesh is applied to solve Liouville's equation. In the ADER approach a temporal Taylor series is computed by replacing temporal derivatives with spatial derivatives using the Cauchy-Kovalewski procedure. The result is a fully discrete explicit scheme of arbitrary high order of accuracy. A moving mesh is not sufficient to be able to solve Liouville's equation numerically for the optical systems considered in this article. To that end, we combine the scheme with a new method we refer to as sub-cell interface method. When dealing with optical interfaces in phase space, non-local boundary conditions arise. These are incorporated in the DG method in an energy-preserving manner. Numerical experiments validate energy-preservation up to machine precision and show the high order of accuracy. Furthermore, the DG method is compared to quasi-Monte Carlo ray tracing for two examples showing that the DG method yields better accuracy in the same amount of computational time.

Wave–Tide Interaction for a Strongly Modulated Wave Field

Abstract Observations from Coastal Data Information Program (CDIP) moored buoys off the coast of Florida reveal tidally driven wave–current interactions that modify significant wave heights by up to 25% and shift peak periods by up to a second. A case study at Fernandina Beach, Florida, shows surface waves steepening on following tidal currents and becoming less steep on opposing tidal currents, with the largest modulations occurring in the long-period swell band. To better understand tidal modulations as a function of the phase of the tide, we use simplified analytical and numerical solutions to the equations of geometrical optics and conservation of wave action under the assumption of a one-dimensional tide acting as a progressive shallow-water wave. The theoretical frameworks allow us to identify parameters that characterize the magnitude of variation in surface waves due to tidally induced currents and changes in water depth. We compute modulations to the omnidirectional and directional wave spectrum (between 0.05 and 0.15 Hz), as well as characteristic bulk parameters such as significant wave height and peak period. The theory is corroborated using directional wave and surface current observations from the Fernandina Beach CDIP station (located in water of average depth of 16 m). We find that the numerical results reproduce the observed wave modulations due to tidal currents and changes in water depth. Specifically, surface waves traveling in the direction of the tide are strongly modulated, and the relative speeds between the tide and surface waves set the sign and magnitude of these modulations. Given knowledge of tidal currents, water-depth variations, and wave climatology, theoretical and numerical predictions may be used to provide both statistical and instantaneous estimates of wave-height variations due to tides. Because operational forecasts and nowcasts do not routinely include tides or currents, these findings can help to accurately represent nearshore surface wave variability.

Correction to: An Energy Conservative hp-method for Liouville’s Equation of Geometrical Optics

Correction to: An Energy Conservative hp-method for Liouville’s Equation of Geometrical Optics

An Ader Discontinuous Galerkin Method on Moving Meshes for Liouville's Equation of Geometrical Optics

An Ader Discontinuous Galerkin Method on Moving Meshes for Liouville's Equation of Geometrical Optics

A discontinuous Galerkin method to solve Liouville’s equation of geometrical optics

We present an alternative method to ray tracing that is based on a phase space description of light propagation. Liouville’s equation of geometrical optics describes the evolution of the basic luminance on phase space. At an optical interface, the laws of optics describe non-local boundary conditions for the basic luminance. A discontinuous Galerkin method is employed to solve Liouville’s equation for a dielectric total internal reflection concentrator.

The response of laser interferometric gravitational wave detectors beyond the eikonal equation

The response of Michelson interferometers to weak plane gravitational waves is computed at one order of accuracy beyond the eikonal equation. The modulation of the electromagnetic field amplitude and polarisation are taken into account by solving the transport equations of geometrical optics with boundary conditions adapted to laser interferometry. Considering both DC and balanced homodyne readout schemes, explicit formulae for the interferometer output signals are derived. These signals comprise perturbations of the optical path length, frequency and amplitude, and are shown to be insensitive to polarisation perturbations.

Numerical-analytical method for calculating the refraction of radio waves in a chaotic upper atmosphere

An operational method is proposed for calculating the refraction of decameter radio waves in a randomly-inhomogeneous upper atmosphere. The method is based on the numerical-analytical solution of stochastic equations of geometric optics. An integral expression is obtained for the dispersion of the refraction angle of a radio wave on the atmospheric path using the approximation of the perturbation method. For a quick calculation of the statistical moment of the refraction angle, the integral expression is reduced to an ordinary first-order differential equation. Joint numerical solution of the unperturbed ray equations and the equations for the statistical moment allows an operational estimate of beam width of radio waves arriving at the observation point. The results of numerical calculations of the standard deviations of the refraction angles of radio waves on paths of various lengths are presented.

Analytical ray-tracing in planetary atmospheres

Context. Ground-based astro-geodetic observations and atmospheric radio occultations are two examples of observational techniques requiring a scrutiny analysis of atmospheric refraction. In both cases, the measured changes of the observables are geometrically related to changes in the photon path through the refractive profile of the crossed medium. Therefore, having a clear knowledge of how the refractivity governs the photon path evolution is of prime importance to clearly understand observational features. Aims. We analytically performed the integration of the photon path and the light time of rays traveling across a non-spherically symmetric planetary atmosphere. Methods. Assuming that the atmospheric refraction evolves linearly with the Newtonian potential, we derived an exact solution to the equations of geometrical optics. By varying the solution’s arbitrary constants of integration, we reformulated the equation of geometrical optics into a new set of osculating equations describing the constants’ evolution following any changes in the refractive profile. We have highlighted the capabilities of the formalism, carrying out five realistic applications in which we derived analytical expressions. Finally, we assessed the accuracy by comparing the solution to results from a numerical integration of the equations of geometrical optics in the presence of a quadrupolar moment (J2). Results. Analytical expressions for the light time and the refractive bending are given with relative errors at the level of one part in 108 and one part in 105, for typical values of the refractivity and J2 at levels of 10−4 and 10−2, respectively. Conclusions. The establishment of the osculating equations for the ray propagation has two main advantages. Firstly, it provides an easy and comprehensive geometrical picture for interpreting the photon path. Secondly, it allows the analytical solving of the ray propagation in the presence of non-radial dependencies in the refractive profile.

Elliptical Gaussian beam propagation in nonlinear fibres with focusing and defocusing refractive profiles

The paper shows systematic analysis of elliptical Gaussian beam (GB) propagating in self-focusing/self-defocusing media of Kerr type with focusing/defocusing linear refractive profiles. Instead of solving commonly accepted Nonlinear Schrödinger Equation (NLS) we propose equations of geometrical optics: complex eikonal equation and complex transport one. Eikonal equation let us derive immediately the ordinary differential equations for principle widths omitting this way complicated variational process and virial theory used in nonlinear optics. From the transport equation we obtain first order ordinary differential equation for complex amplitude evolution and conservation principle for energy flux in the elliptical GB cross-section. For combined effects of diffraction, nonlinear refraction and focusing/defocusing graded-index factor we observe three types of the solutions for the elliptical radius of principle beam widths: self-focusing or self-defocusing together with diffraction widening, stationary and oscillatory ones under and without collapse effect. We prove here that principle curvatures of the wave front can introduce into our description the additional ellipticity of the beam, which can effectively limit collapse effect. Moreover, we discuss the evolution of elliptical Gaussian beam in nonlinear self-defocusing fibre with focusing refractive profile presenting conditions for existence of elliptical spatial solitons. We discuss here also the influence of initial wave front curvatures on elliptical Gaussian beam evolution in nonlinear inhomogeneous fibres of Kerr-type.

Light propagation and optical scalars in torsion theories of gravity

We investigate the propagation of light rays and evolution of optical scalars in gauge theories of gravity where torsion is present. Recently, the modified Raychaudhuri equation in the presence of torsion has been derived. We use this result to derive the basic equations of geometric optics for several different interesting solutions of the Poincaré gauge theory of gravity. The results show that the focusing effects for neighboring light rays will be different than general relativity. This in turn has practical consequences in the study of gravitational lensing effects and also in determining the angular diameter distance for cosmological objects.

U‐Shaped Spectrograms Registered by the DEMETER Satellite: Observational Features and Formation Mechanism

AbstractU‐shaped spectrograms observed by the DEMETER satellite in the equatorial region of the upper ionosphere are presented and explained for the first time. These spectra are characterized by the latitude‐dependent upper cutoff frequency that increases with the latitude during nighttime. The observations suggest that this very low frequency wave phenomenon is closely connected with the anomalous increase in cold plasma density at and below the satellite. We show that restricting the analysis to propagation effects is insufficient to explain U‐shaped spectra, and other effects, in particular, the collisional wave damping should be taken into consideration. To calculate the wave damping, we use the method of successive approximations. In the first approximation, we calculate the ray trajectory and the wave normal vector along it, using the equations of geometrical optics neglecting collisions. In the second approximation we calculate the collisional wave damping along the precalculated trajectory. We show that the effects of very low frequency wave propagation and attenuation in the equatorial region of the upper ionosphere explain the main features of the phenomenon under discussion.

Vanishing Viscosity Limit for Riemann Solutions to a Class of Non-Strictly Hyperbolic Systems

By the vanishing viscosity approach, a class of non-strictly hyperbolic systems of conservation laws that contain the equations of geometrical optics as a prototype are studied. The existence, uniqueness and stability of solutions involving delta shock waves and generalized vacuum states are discussed completely.

Whistler Mode Waves Below Lower Hybrid Resonance Frequency: Generation and Spectral Features

AbstractEquatorial noise in the frequency range below the lower hybrid resonance frequency, whose structure is shaped by high proton cyclotron harmonics, has been observed by the Cluster spacecraft. We develop a model of this wave phenomenon which assumes (as, in general, has been suggested long ago) that the observed spectrum is excited due to loss cone instability of energetic ions in the equatorial region of the magnetosphere. The wavefield is represented as a sum of constant frequency wave packets which cross a number of cyclotron resonances while propagating in a highly oblique mode along quite specific trajectories. The growth (damping) rate of these wave packets varies both in sign and magnitude along the raypath, making the wave net amplification, but not the growth rate, the main characteristic of the wave generation process. The growth rates and the wave amplitudes along the ray paths, determined by the equations of geometrical optics, have been calculated for a 3‐D set of wave packets with various frequencies, initial L shells, and initial wave normal angles at the equator. It is shown that the dynamical spectrum resulting from the proposed model qualitatively matches observations.

Effect of a small dispersion on self-focusing in a spatially one-dimensional case

The effect of the small dispersion on the self-focusing of solutions of the equations of nonlinear geometric optics in one-dimensional case is investigated. In the main order this influence is described by means of the universal special solution of the nonlinear Schr\"odinger equation, which is isomonodromic. Analytic and asymptotic properties of this solution are described.

Extending geometrical optics: A Lagrangian theory for vector waves

Even when neglecting diffraction effects, the well-known equations of geometrical optics (GO) are not entirely accurate. Traditional GO treats wave rays as classical particles, which are completely described by their coordinates and momenta, but vector-wave rays have another degree of freedom, namely, their polarization. The polarization degree of freedom manifests itself as an effective (classical) “wave spin” that can be assigned to rays and can affect the wave dynamics accordingly. A well-known manifestation of polarization dynamics is mode conversion, which is the linear exchange of quanta between different wave modes and can be interpreted as a rotation of the wave spin. Another, less-known polarization effect is the polarization-driven bending of ray trajectories. This work presents an extension and reformulation of GO as a first-principle Lagrangian theory, whose effective Hamiltonian governs the aforementioned polarization phenomena simultaneously. As an example, the theory is applied to describe the polarization-driven divergence of right-hand and left-hand circularly polarized electromagnetic waves in weakly magnetized plasma.

Generation of auroral kilometric radiation in inhomogeneous magnetospheric plasma

The generation of auroral kilometric radiation in a narrow 3D plasma cavity, in which a weakly relativistic electron flow is propagated along the magnetic field against a low-density cold background plasma, is studied. The time dynamics of the propagation and intensification of waves are analyzed using geometric optics equations. The waves have different wave vector components and start from the cavity center at an altitude of about the Earth’s radius at plasma parameters typical for the auroral zone at this altitude. It is shown that the global inhomogeneity of the Earth’s magnetic field is of key importance in shaping the auroral kilometric radiation spectra.

Conservation of energy for non-dissipative water waves

In this paper Reynolds transport theory is applied to obtain energy conservation equation (ECE). Depth integrated ECE is derived considering a control volume extended over water depth. It is demonstrated that mild slope equation (MSE) is derivable from depth integrated ECE for linear waves. Two equations are derived by separating real and imaginary parts of depth integrated ECE for multidirectional wave fields that can properly deal with reflecting waves. Geometrical optics equations, derived for non-reflecting waves, can be obtained from aforementioned equations as well. Depth integrated ECE is averaged over a wave period for monochromatic waves. The result is similar in form to energy transport equation (ETE) governing spectral wave models. The terms of energy density and energy flux, obtained here, are more accurate expressions, comparing to those of ETE. It should be mentioned that ETE is also obtained from ECE, the general form derived here. The strength of newly derived equations in dealing with reflective wave fields has been illustrated by simple numerical computations. In this research MSE has been derived by the new presented method. Besides, coupled equations for multidirectional waves as well as time averaged depth integrated ECE have also been obtained through mentioned new approach.