Angular Momentum Approach Research Articles

From Black Hole Spectral Instability to Stable Observables.

The quasinormal mode spectrum of black holes is unstable under small perturbation of the potential and has observational consequences in time signals. Such signals might be experimentally difficult to observe and probing this instability will be a technical challenge. Here, we investigate the spectral instability of time-independent data. This leads us to study the Regge poles (RPs), the counterparts to the quasinormal modes in the complex angular momentum plane. We present evidence that the RP spectrum is unstable but that not all overtones are affected equally by this instability. In addition, we reveal that behind this spectral instability lies an underlying structure. The RP spectrum is perturbed in such a way that one can still recover stable scattering quantities using the complex angular momentum approach. Overall, the study proposes a novel and complementary approach on the black hole spectral instability phenomena that allows us to reveal a surprising and unexpected mechanism at play that protects scattering quantities from the instability.

Decomposition of the wall-heat flux of compressible boundary layers

We use the method developed by Elnahhas and Johnson [“On the enhancement of boundary layer skin friction by turbulence: An angular momentum approach,” J. Fluid Mech. 940, A36 (2022)] and Xu et al. [“Decomposition of the skin-friction coefficient of compressible boundary layers,” Phys. Fluids 35, 035107 (2023)] for the decomposition of the skin-friction coefficient to integrate the mean temperature equation for high-Reynolds-number compressible boundary layers and arrive at an identity for the decomposition of the wall-heat flux. The physical interpretation of the identity and the limitations of this approach are discussed. We perform an integration on the mean temperature equation to obtain an identity that is the heat-transfer analog to the compressible von Kármán momentum integral equation for the skin-friction coefficient. This identity is applied to numerical data for laminar and turbulent compressible boundary layers, revealing that the mean-flow dissipation and production of turbulent kinetic energy given by the Favre–Reynolds stresses dominate the thermal-energy balance. The term related to the growth of the turbulent boundary layer opposes the wall cooling. Other identities for the wall-heat flux, inspired by the method of Fukagata et al. [“Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows,” Phys. Fluids 14(11), L73–L76 (2002)], are studied numerically and by asymptotic methods. The terms of these identities depend spuriously on the upper integration bound because this bound is a mathematical quantity used in the derivation. When the bound is asymptotically large, the integral identities simplify to the heat-transfer analog to the von Kármán momentum equation. We also prove that an existing multiple-integration identity reduces to the definition of the wall-heat flux when the number of integrations is asymptotically large. No information about the wall-heat transfer is extracted because the impact of the integration number is nonphysical.

Scattering and conversion of electromagnetic and gravitational waves by Reissner-Nordström black holes: The Regge pole description

We investigate the problem of scattering and conversion of monochromatic planar gravitational and electromagnetic waves impinging upon a Reissner-Nordstr\"om black hole using a Regge pole description, i.e., a complex angular momentum approach. For this purpose, we first compute numerically the Regge pole spectrum for various charge-to-mass ratio configurations. We then derive an asymptotic expressions for the lowest Regge poles, and by considering Bohr-Sommerfeld-type quantization conditions, obtain the spectrum of weakly damped quasinormal frequencies from the Regge trajectories. Next, we construct the scattering and conversion amplitudes as well as the total differential cross sections for different processes using both a complex angular momentum representation and a partial wave expansion method. Finally, we provide an analytical approximation of the scattering and conversion cross sections of different processes from asymptotic expressions for the lowest Regge poles and the associated residues based on the correspondence Regge poles, ``surface waves'' propagating close to the photon (graviton) sphere. This allows us to extract the physical interpretation encoded in the partial wave expansions in the high-frequency regime (i.e., in the short-wavelength regime), and to describe semiclassically with very good agreement both black hole glory and a large part of the orbiting oscillations, thus unifying these two phenomena from a purely wave point of view.

Regge pole description of scattering by dirty black holes

We study the problem of plane monochromatic scalar waves impinging upon a Schwarzschild dirty black hole -- a Schwarzschild black hole surrounded by a thin spherical shell of matter -- using the complex angular momentum approach. We first recall general results concerning the differential scattering cross section in the classical limit through a null geodesic analysis by exploring different configurations of the shell. In particular, we show that dirty black hole spacetimes may exhibit various critical effects for geometrical optics. We compute the Regge pole spectrum for various shell configurations and show that it exhibits two or three distinct branches of poles, labelled inner surface waves, broad resonances and outer surface waves. In the latter, two sub-families have been identified, the surface waves associated with the outer light-ring and the creeping modes associated with the surface of the shell. We show, using WKB analysis, that the position of the shell sets the real part of the broad resonances while its energy-momentum and the discontinuity of the potential at the shell's surface set their imaginary part. Next, we provide the complex angular momentum representation of the differential scattering cross section and examine the role of the different Regge pole branches. We compute the differential scattering cross section for various configurations at several frequencies and show a very good agreement with the partial-wave calculations. Finally, we highlight the role of the critical effects, i.e., orbiting, glory, grazing and rainbow scattering, and their impact on the differential scattering cross section.

Decomposition of the skin-friction coefficient of compressible boundary layers

We derive an integral formula for the skin-friction coefficient of compressible boundary layers by extending the formula of Elnahhas and Johnson [“On the enhancement of boundary layer skin friction by turbulence: An angular momentum approach,” J. Fluid Mech. 940, A36 (2022)] for incompressible boundary layers. The skin-friction coefficient is decomposed into the sum of the contributions of the laminar coefficient, the change of the dynamic viscosity with the temperature, the Favre–Reynolds stresses, and the mean flow. This decomposition is applied to numerical data for laminar and turbulent boundary layers, and the role of each term on the wall-shear stress is quantified. We also show that the threefold integration identity of Gomez et al. [“Contribution of Reynolds stress distribution to the skin friction in compressible turbulent channel flows,” Phys. Rev. E 79(3), 035301 (2009)] and the twofold integration identities of Wenzel et al. [“About the influences of compressibility, heat transfer and pressure gradients in compressible turbulent boundary layers,” J. Fluid Mech. 930, A1 (2022)] and Xu et al. [“Skin-friction and heat-transfer decompositions in hypersonic transitional and turbulent boundary layers,” J. Fluid Mech. 941, A4 (2022)] for turbulent boundary layers all simplify to the compressible von Kármán momentum integral equation when the upper limit of integration is asymptotically large. The dependence of these identities on the upper integration bound is studied. By using asymptotic methods, we prove that the multiple-integration identity of Wenzel et al. [“About the influences of compressibility, heat transfer and pressure gradients in compressible turbulent boundary layers,” J. Fluid Mech. 930, A1 (2022)] degenerates to the definition of the skin-friction coefficient when the number of integrations is asymptotically large.

Revisiting the Cassini States of synchronous satellites with an angular momentum approach

AbstractLike our Moon, the large icy satellites of Jupiter are thought to be in a Cassini State, an equilibrium rotation state characterized by a synchronous rotation rate and a precession rate of the rotation axis equal to that of the normal to the orbit. In these equilibrium states (up to four Cassini States are possible for a solid and rigid satellite), the spin axis of the satellite, the normal to its orbit and the normal to the inertial plane remain coplanar with an obliquity that remains theoretically constant. However, as the gravitational torque exerted on the satellite shows small periodic variations, the orientation of the rotation axis will also vary with time and nutations in obliquity will appear.Here we present a dynamical model for the study of the Cassini States. This model includes the coupling between the polar motion and the spin axis precession/nutation which is neglected in the classical studies. We study the influence of the triaxiality of Ganymede on its four possible Cassini States, use a Toy model of the Moon to illustrate the nutations in obliquity obtained with the dynamical model, and investigate the influence of the presence of a subsurface ocean on the Cassini State I of Europa.

On the enhancement of boundary layer skin friction by turbulence: an angular momentum approach

Turbulence enhances the wall shear stress in boundary layers, significantly increasing the drag on streamlined bodies. Other flow features such as free stream pressure gradients and streamwise boundary layer growth also strongly influence the local skin friction. In this paper, an angular momentum integral (AMI) equation is introduced to quantify these effects by representing them as torques that alter the shape of the mean velocity profile. This approach uniquely isolates the skin friction of a Blasius boundary layer in a single term that depends only on the Reynolds number most relevant to the flow's engineering context, so that other torques are interpreted as augmentations relative to the laminar case having the same Reynolds number. The AMI equation for external flows shares this key property with the so-called FIK relation for internal flows (Fukagata et al., Phys. Fluids, vol. 14, 2002, pp. L73–L76). Without a geometrically imposed boundary layer thickness, the length scale in the Reynolds number for the AMI equation may be chosen freely. After a brief demonstration using Falkner–Skan boundary layers, the AMI equation is applied as a diagnostic tool on four transitional and turbulent boundary layer direct numerical simulation datasets. Regions of negative wall-normal velocity are shown to play a key role in limiting the peak skin friction during the late stages of transition, and the relative strengths of terms in the AMI equation become independent of the transition mechanism a very short distance into the fully turbulent regime. The AMI equation establishes an intuitive, extensible framework for interpreting the impact of turbulence and flow control strategies on boundary layer skin friction.

Angular-Momentum Biased Circulator With Locally Generated Modulation

This article presents a novel type of time-modulated magnetless circulators, where modulation is generated locally at each modulated element through a network of local oscillators instead of a central source and a complex distribution network based on phase shifters and filters. The design follows the angular-momentum approach, and the modulation signals are generated from three cross-coupled oscillators interconnected through transmission lines in a ring topology to phase lock and attain the necessary phase difference of 120°. The oscillators are interwound with the circulator circuit by designing them to operate in a differential mode, in contrast to the common mode of the circulator circuit, allowing natural isolation between the RF and modulation paths. Two prototypes were built to test the design. An initial preliminary printed-circuit-board prototype was fabricated to test the common-differential mode topology with modulation signals generated off the board. The design was tested at 1.78 GHz, showing isolation of more than 20 dB, insertion loss of 5.5 dB, 2.6% bandwidth, and power handling of 21 dBm. A final prototype was fabricated to test a fully autonomous circulator, with the modulation signals generated locally at each modulated tank. The final design was tested at 1.65 GHz, showing isolation greater than 20 dB, insertion loss of 5.3 dB, 4.8% bandwidth, and power handling of 25.52 dBm. The results open a path for a new class of autonomous time-modulated devices without the need for complicated modulation networks and with an ability to incorporate a large number of modulated elements.

Multimode communication through the turbulent atmosphere.

A central question in free-space optical communications is how to improve the transfer of information between a transmitter and a receiver. The capacity of the communication channel can be increased by multiplexing of independent modes using either: (1) the multiple-input-multiple-output (MIMO) approach, where communication is done with modes obtained from the singular value decomposition of the transfer matrix from the transmitter array to the receiver array, or (2) the orbital angular momentum (OAM) approach, which uses vortex beams that carry angular momenta. In both cases, the number of usable modes is limited by the finite aperture of the transmitter and receiver, and the effect of the turbulent atmosphere. The goal of this paper is twofold: first, we show that the MIMO and OAM multiplexing schemes are closely related. Specifically, in the case of circular apertures, the leading singular vectors of the transfer matrix, which are useful for communication, are essentially the same as the commonly used Laguerre-Gauss vortex beams, provided these have a special radius that depends on the wavelength, the distance from the transmitter to the receiver, and the ratio of the radii of their apertures. Second, we characterize the effect of atmospheric turbulence on the communication modes using the phase screen method put in the mathematical framework of beam propagation in random media.

Review on hermiticity of the volume operators in Loop Quantum Gravity

The aim of this article is to provide a rigorous-but-simple steps to prove the hermiticity of the volume operator of Rovelli-Smolin and Ashtekar-Lewandowski using the angular momentum approach, as well as pointing out some subleties which have not been given a lot of attention previously. Besides of being hermitian, we also prove that both volume operators are real, symmetric, and positive semi-definite, with respect to the inner product defined on the Hilbert space over SU(2). Other special properties follows from this fact, such as the possibility to obtain real orthonormal eigenvectors. Moreover, the matrix representation of the volume operators are degenerate, such that the real positive eigenvalues always come in pairs for even dimension, with an additional zero if the dimension is odd. As a consequence, one has a freedom in choosing the orthonormal eigenvectors for each 2-dimensional eigensubspaces. Furthermore, we provide a formal procedure to obtain the spectrum and matrix representation of the volume operators. In order to compare our procedure with the earlier ones existing in the literature, we give explicit computational examples for the case of monochromatic quantum tetrahedron, where the eigenvalues agrees with the standard earlier procedure.

Coupling between the spin precession and polar motion of a synchronously rotating satellite: application to Titan

We here develop, in an angular momentum approach, a consistent model that integrates all rotation variables and considers forcing both by the central planet and a potential atmosphere. Existing angular momentum approaches for studying the polar motion, precession, and libration of synchronously rotating satellites, with or without an internal global fluid layer (e.g., a subsurface ocean) usually focus on one aspect of rotation and neglect coupling with the other rotation phenomena. The model variables chosen correspond most naturally with the free modes, although they differ from those of Earth rotation studies, and facilitate a comparison with existing decoupled rotation models that break the link between the rotation motions. The decoupled models perform well in reproducing the free modes, except for the Free Ocean Nutation in the decoupled polar motion model. We also demonstrate the high accuracy of the analytical forced solutions of decoupled models, which are easier to use to interpret observations from past and future space missions. We show that the effective decoupling between the polar motion and precession implies that the spin precession and its associated mean obliquity are mainly governed by the external gravitational torque by the parent planet, whereas the polar motion of the solid layers is mainly governed by the angular momentum exchanges between the atmosphere (e.g., for Titan) and the surface. To put into perspective the difference between rotation models for a synchronously rotating icy moon with a thin ice shell and classical Earth rotation models, we also consider the case of the Moon, which has a thick outer layer above a liquid core. We also show that for non-synchronous rotators, the free precession of the outer layer in space degenerates into the tilt-over mode.

Integrated Optics Modules Based Proposal for Quantum Information Processing, Teleportation, QKD, and Quantum Error Correction Employing Photon Angular Momentum

To address key challenges for both quantum communication and quantum computing applications in a simultaneous manner, we propose to employ the photon angular momentum approach by invoking the well-known fact that photons carry both the spin angular momentum (SAM) and the orbital angular momentum (OAM). SAM is associated with polarization, while OAM is associated with azimuthal phase dependence of the complex electric field. Given that OAM eigenstates are mutually orthogonal, in principle, an arbitrary number of bits per single photon can be transmitted. The ability to generate/analyze states with different photon angular momentum, by using either holographic or interferometric methods, allows the realization of quantum states in multidimensional Hilbert space. Because OAM states provide an infinite basis state, while SAM states are 2-D only, the OAM can also be used to increase the security for quantum key distribution (QKD) applications and improve computational power for quantum computing applications. The goal of this paper is to describe photon angular momentum based deterministic universal quantum qudit gates, namely, {generalized-X, generalized-Z, generalized-CNOT} qudit gates, and different quantum modules of importance for various applications, including (fault-tolerant) quantum computing, teleportation, QKD, and quantum error correction. For instance, the basic quantum modules for quantum teleportation applications include the generalized-Bell-state generation module and the QFT-module. The basic quantum module for quantum error correction and fault-tolerant computing is the nonbinary syndrome calculator module. The basic module for entanglement assisted QKD is either the generalized-Bell-state generation module or the Weyloperator-module. The possibility of implementing all these modules in integrated optics is discussed as well. Finally, we provide security analysis of entanglement assisted multidimensional QKD protocols, employing the proposed qudit modules, by taking into account the imperfect generation of OAM modes.

Quasinormal modes of nearly extremal Kerr spacetimes: Spectrum bifurcation and power-law ringdown

We provide an in-depth investigation of quasinormal-mode oscillations of Kerr black holes with nearly extremal angular momenta. We first discuss in greater detail the two distinct types of quasinormal mode frequencies presented in a recent paper (arXiv:1212.3271). One set of modes, that we call "zero-damping modes", has vanishing imaginary part in the extremal limit, and exists for all corotating perturbations (i.e., modes with azimuthal index m being nonnegative). The other set (the "damped modes") retains a finite decay rate even for extremal Kerr black holes, and exists only for a subset of corotating modes. As the angular momentum approaches its extremal value, the frequency spectrum bifurcates into these two distinct branches when both types of modes are present. We discuss the physical reason for the mode branching by developing and using a bound-state formulation for the perturbations of generic Kerr black holes. We also numerically explore the specific case of the fundamental l=2 modes, which have the greatest astrophysical interest. Using the results of these investigations, we compute the quasinormal mode response of a nearly extremal Kerr black hole to perturbations. We show that many superimposed overtones result in a slow power-law decay of the quasinormal ringing at early times, which later gives way to exponential decay. This exceptional early-time power-law decay implies that the ringdown phase is long-lived for black holes with large angular momentum, which could provide a promising strong source for gravitational-wave detectors.

Recent estimates of Earth‐atmosphere interaction torques and their use in studying polar motion variability

The use of Earth‐atmosphere interaction torques is a potential but generally less addressed alternative to the classical angular momentum approach for modeling variations in Earth rotation. We present an update on this subject for the purpose of explaining seasonal and intraseasonal polar motion variability based on the output of the most recent meteorological reanalysis systems of the ECMWF (European Centre for Medium‐Range Weather Forecasts) and NASA's Global Modeling and Assimilation Office. The agreement of both models in terms of the three prime torque constituents is shown to be far superior to that of the conventionally deployed wind term of atmospheric angular momentum (AAM). A sufficiently good closure of the equatorial AAM budget equation within the ECMWF reanalysis provides additional endorsement for the use of atmospheric torques as excitation measures. When used as such, polar motion residuals after reduction of the AAM pressure term, as well as oceanic and hydrological excitation, are considerably better modeled by the torque‐based quantities than by the standard wind term of AAM, in particular at intraseasonal periodicities. This finding is obtained by means of a newly proposed, hybrid excitation formalism, which derives the AAM counterparts of torque terms from inversion of the AAM budget equation in the frequency domain.

Multiple inner core wobbles in a simple Earth model with inviscid core

The inner core wobble (ICW) is the chandler wobble of the inner core. Its predicted period for the PREM model is about 7.5 years, based upon the resolution of the Liouville equations of conservation of angular momentum. Here, solving the local equation of conservation of linear momentum with a truncated chain that couples the toroidal and spheroidal displacement fields, the ICW is computed for a model made up of three homogeneous layers: an incompressible liquid outer core and rigid mantle and inner core. Contrary to the angular momentum approach, as implemented up to now, that provides a single ICW, the linear momentum approach shows that the dynamics of the neutrally stratified outer core may generate a family of ICWs with periods ranging from a few dozens to thousands of days. The mode with the largest wobble amplitude in the inner core has a period close to that obtained with the angular momentum approach.

Basic Relations for Studying the Influence of Geophysical Processes on the Earth’s Rotation: The Angular Momentum Approach

Relations to study the influence of geophysical processes on the temporally varying rotation of the Earth are considered. Liouville’s equations of rotational motion are derived for a two-component Earth model (consisting of a solid mantle and a fluid core) and suitably simplified for calculations of the influence of mass redistributions on the Earth’s rotational behaviour. Excitation functions, or effective angular momentum functions, describing the influence of mass redistributions on the equations of rotational motion are derived, and their calculation is elucidated by some examples. Relations between temporally varying second degree Stokes coefficients of the gravity field and excitation functions are discussed. Different solutions of the equations of rotational motion are described. The identification of exciting geophysical processes by the kinematics of the inverse calculated excitation function is portrayed.

Nuclear bremsstrahlung and its radiation effects in fusion reactors

Nucleons, i.e. protons or neutrons, are emitted in many fusion processes of light nuclei. The fusion-generated nucleons can in turn fuse with or be captured by an un-reacted nuclear fuel, for example deuterium. The fusion reaction occurs at an average center of mass (COM) energy of 10 keV (thousand electron volts) or more in a typical fusion reactor. At such a relative low COM energy, the proton and deuteron are in a state where the relative angular momentum approaches zero, or an s-wave state. The single gamma radiation process is thus suppressed due to the conservation of parity. Instead, the gamma ray (typically of the order of a few MeV or more) released is likely to be accompanied by soft x-ray photons from a nuclear bremsstrahlung process, which generates continuous x-ray radiation peaked around a few hundred eV to a few keV. The average photon energy and spectrum properties of such a process is calculated with a semiclassical approach. This phenomenon may cause additional parasitic issues to a fusion reactor, while also opening up the possibility of new plasma diagnostics.

Influence of liquid core dynamics on rotational modes

SUMMARY We investigate the influence of the structure and dynamics of the liquid outer core on the Earth’s rotational modes through the squared Brunt–V¨¨¨ frequency N 2 . The frequencies of the rotational modes are embedded into the continuous spectrum of inertia-gravity modes, which is governed in the complex domain by N 2 and the Earth’s rotation speed. By solving the equations for the normal modes of a rotating ellipsoidal elastic earth model and varying the N 2 parameter, we show interactions between pseudo-modes of the liquid core and three rotational modes: the Chandler Wobble (CW), Free Inner Core Nutation (FICN) and Free Core Nutation (FCN). The interaction between pseudo-modes of the outer core and the CW gives rise to avoided crossings, which result in two modes sharing similar displacements, that is, an almost rigid wobble of the mantle and oscillations in the outer core having roughly the same amplitude. The corresponding eigenperiods in a corotating frame of reference are separated by a few days. Avoided crossings are also the only kind of interaction that occurs between core pseudo-modes and the FICN. The coupling is stronger than for the CW, the eigenperiods being a few hundred days apart in an inertial frame of reference. The eigenfunctions are mixed in such a way that the amplitude of the nutation of the inner core is an order of magnitude bigger than the oscillations in the outer core. The FCN shows weaker interactions with the pseudomodes of the core. However, the shift of its nutation period can reach up to 15 d. Consequently, our results show that the angular momentum approach through Liouville’s equations is not sufficient to describe fully the nutational modes.

The complex angular momentum (CAM) route to reactive scattering resonances: from a simple model to the F+H2→HF+H reaction

We advocate the use of the complex angular momentum (CAM) approach for quantitative analysis of the resonance effects in reactive differential and integral cross sections. We start by reviewing a simple potential scattering model and progress to analyse the F+H2(v=0,j=0)→HF(v′ =2,j′ =0)+H reaction recently considered by Cavalli and De Fazio (2007 Phys. Scr. 76 C21) within the conventional complex energy plane approach. We argue that although the two approaches have the same origin, preference may be given to the CAM method whenever a quantity of interest is obtained by summing a partial wave series.

Regge oscillations in electron-atom elastic cross sections

We consider a system trapped in a resonance state, whose decay at zero scattering angle can be related, through the optical theorem, to the total cross section (TCS). We show that for the resonance to contribute to the TCS a peak structure the resonance conditions must be satisfied: (i) Several rotations of the complex (the Regge trajectory---viz., imaginary part versus the real part of the complex angular momentum---stays close to the real axis) and (ii) coherent addition of forward-scattering subamplitudes (the real part of the Regge pole is close to an integer). We exploit the recent complex angular momentum approach of Macek et al. [Phys. Rev. Lett. 93, 183203 (2004)], used to analyze low-energy oscillations observed in the elastic TCS for proton-H scattering, for a detailed analysis of Regge trajectories and their contributions to the TCS in electron-atom scattering for the case of $Z=75$ using the model Thomas-Fermi potential. We conclude by demonstrating through comparison with existing theory and measurements that the Thomas-Fermi potential when used with the appropriate parameters captures the essential physics (Ramsauer-Townsend minima and the Wigner threshold law) in the near-threshold $e\text{\ensuremath{-}}\mathrm{Ar}$ and $e\text{\ensuremath{-}}\mathrm{Kr}$ elastic scattering.