Equatorial Motion Research Articles

Analytical Theory for the Quasi-Steady and Low-Frequency Equatorial Ocean Response to Wind Forcing: The “Tilt” and “Warm Water Volume” Modes

Abstract Analytical theory is used to examine the linear response of a meridionally unbounded stratified ocean to large-scale, low-frequency wind forcing. The following results, applied mainly to the equatorial Pacific, were obtained. (i) Provided that the wind stress curl vanishes at large distance from the equator, a general Sverdrup solution is valid in the quasi-steady (frequency ω → 0) limit. The meridionally averaged zonal flow toward the western boundary layer is zero so that there is no net mass flow into the boundary layer and the large-scale boundary condition is therefore satisfied. This solution predicts a zero pycnocline response in the eastern equatorial Pacific. It therefore predicts that, for the eastern equatorial Pacific, a slow weakening of the equatorial trade winds will not lead to long-term El Niño conditions there. (ii) Consistent with observations and other previous work, for finite but small frequencies there are two modes of equatorial motion. One is a “tilt” mode in which the equatorial sea level and thermocline are tilted by the in-phase zonal wind stress and the other is an equatorial warm water volume (WWV) mode in which the discharge of equatorial warm water (negative WWV anomaly) lags the wind stress forcing by a quarter of a period. (iii) The amplitude of the WWV mode approaches zero like ω1/2. Therefore, as ω → 0, the equatorial solution reduces to the tilt mode. (iv) The WWV mode is not due to a dominant meridional divergence driven by the wind, as suggested by some previous work. Meridional and zonal divergence approximately cancel. Reflection of energy at both ocean boundaries together with the strong dependence of long Rossby wave speed on latitude is crucial to the existence of the disequilibrium WWV mode. Because higher-latitude Rossby waves travel so much more slowly, the Rossby waves reflecting from the western ocean boundary are not in phase. This gives rise to a reflected equatorial Kelvin wave and a WWV that is not in phase with the wind stress forcing. (v) Observations from past work have shown that much low-frequency wave energy, particularly westward propagating Rossby wave energy poleward of about 5°N and 5°S, is damped out before it reaches the western ocean boundary. In this way dissipation likely has a strong influence on the equatorial Kelvin wave reflection and hence the disequilibrium WWV.

An equatorial oscillation in Saturn’s middle atmosphere

The middle atmospheres of planets are driven by a combination of radiative heating and cooling, mean meridional motions, and vertically propagating waves (which originate in the deep troposphere). It is very difficult to model these effects and, therefore, observations are essential to advancing our understanding of atmospheres. The equatorial stratospheres of Earth and Jupiter oscillate quasi-periodically on timescales of about two and four years, respectively, driven by wave-induced momentum transport. On Venus and Titan, waves originating from surface-atmosphere interaction and inertial instability are thought to drive the atmosphere to rotate more rapidly than the surface (superrotation). However, the relevant wave modes have not yet been precisely identified. Here we report infrared observations showing that Saturn has an equatorial oscillation like those found on Earth and Jupiter, as well as a mid-latitude subsidence that may be associated with the equatorial motion. The latitudinal extent of Saturn's oscillation shows that it obeys the same basic physics as do those on Earth and Jupiter. Future highly resolved observations of the temperature profile together with modelling of these three different atmospheres will allow us determine the wave mode, the wavelength and the wave amplitude that lead to middle atmosphere oscillation.

Periapsis and gravitomagnetic precessions of stellar orbits in Kerr and Kerr–de Sitter black hole spacetimes

The exact solution for the motion of a test particle in a non-spherical polar orbit around a Kerr black hole is derived. Exact novel expressions for frame dragging (Lense–Thirring effect), periapsis advance and the orbital period are produced. The resulting formulae are expressed in terms of Appell's first hypergeometric function F1, Jacobi's amplitude function and Appell's F1 and Gauss hypergeometric function, respectively. The exact expression for frame dragging is applied for the calculation of the Lense–Thirring effect for the orbits of S-stars in the central arcsecond of our Galaxy assuming that the galactic centre is a Kerr black hole, for various values of the Kerr parameter including those supported by recent observations. In addition, we apply our solutions for the calculation of frame dragging and periapsis advance for stellar non-spherical polar orbits in regions of strong gravitational field close to the event horizon of the galactic black hole, e.g. for orbits in the central milliarcsecond of our galaxy. Such orbits are the target of the GRAVITY experiment. We provide examples with orbital periods in the range of 100 min–54 days. Detection of such stellar orbits will allow the possibility of measuring the relativistic effect of periapsis advance with high precision at the strong field realm of general relativity. Further, an exact closed form formula for the orbital period of a test particle in a non-circular equatorial motion around a Kerr black hole is produced. We also derive exact expressions for the periapsis advance and the orbital period for a test particle in a non-circular equatorial motion in the Kerr field in the presence of the cosmological constant in terms of Lauricella's fourth hypergeometric function FD.

Dynamics of extended spinning masses in a gravitational field

We develop a first-order approximation method for the influence of spin on the motion of extended spinning test masses in a gravitational field. This approach is illustrated for approximately circular equatorial motion in the exterior Kerr spacetime. In this case, the analytic results for the first-order approximation are compared to the numerical integration of the exact system and the limitations of the first-order results are pointed out. Furthermore, we employ our analytic results to illustrate the gravitomagnetic clock effect for spinning particles.

Equatorial dynamics of charged particles in planetary magnetospheres

The dynamics of a charged particle under axisymmetric gravitational and electromagnetic forces is studied. The symmetry allows a phase space reduction from three to two degrees of freedom. The reduced system is a function of two parameters (essentially: the ratio “charge to mass” and the z -component of the angular momentum) and it has an invariant plane associated to the equatorial motion. This paper contains: a global bifurcation chart of equatorial phase-portraits, including stability of equilibria with respect to out of equatorial plane perturbations; an analysis of stability of separatrices and nearby periodic orbits with respect to out of equatorial plane perturbations; an explanation for a lower bound found on the size of particles in the so called “spokes” in Saturn’s B ring; and the new concept of “Kepler caustic”, which is a distinguished curve in the two-parameter space related to the way dust is generated in real planetary rings.

Balanced tropical data assimilation based on a study of equatorial waves in ECMWF short-range forecast errors

SUMMARY This paper seeks to represent the tropical short-range forecast error covariances of the European Centre for Medium-Range Weather Forecasts (ECMWF) model in terms of equatorial waves. The motivation for undertaking this investigation is increasing observational evidence indicating that a substantial fraction of the tropical largescale variability can be explained by equatorially trapped wave solutions known from shallow-water theory. Shortrange forecast differences from a data-assimilation ensemble were taken to serve as a proxy for background errors. It was found that the equatorial waves coupled to convection can explain on average 60–70% of the error variance in the tropical free atmosphere. The largest part of this explained variance is represented by the equatorial Rossby (ER) modes, and a significant percentage pertains to theequatorial inertio-gravity (EIG) modes. Eastwardpropagating EIG modes have maximum variance in the stratosphere, where the short-wave variance in westwardmoving waves is particularly small. This feature is most likely related to the phase of the quasi-biennial oscillation during the study period, suggesting that significant temporal variations could be present in longer-term time series of such statistics. The vertical correlations for ER modes display characteristics similar to those of their extratropical counterparts: correlations narrow towards shorter scales and in the stratosphere. However, the present statistics do not display the significant increase with altitude of the horizontal correlation scale for the height field which is typical for global, quasi-geostrophic statistics commonly used in current data-assimilation schemes. Furthermore, tropospheric ER correlations are vertically asymmetric and deeper for the n = 1 mode than for higher modes. Most likely, deep convection, acting as a generator of equatorial wave motion, is the dominant mechanism underlying these results. In spite of its relatively small contribution to the tropospheric variance, the Kelvin-wave coupling plays a decisive role for determining the characteristics of the horizontal correlation near the equator. EIG modes also play an important role for the tropical mass–wind coupling; these waves have a major impact by reducing the meridional correlation scales and the magnitudes of the balanced height-field increments.

Pseudo-Newtonian potential for charged particle in Kerr–Newman geometry

We consider the equatorial circular motion of a test particle of specific charge q≪m in the Kerr–Newman geometry of a rotating charged black hole. We find the particle's conserved energy and conserved projection of the angular momentum on the black hole's axis of rotation as corrections, in leading order of q/m, to the corresponding energy and angular momentum of a neutral particle. We determine the centripetal force acting on the test particle and, consequently, we find a classical pseudo-Newtonian potential with which one can mimic this general relativistic problem.

Equatorial circular orbits in the Kerr–de Sitter spacetimes

Equatorial motion of test particles in Kerr--de Sitter spacetimes is considered. Circular orbits are determined, their properties are discussed for both black-hole and naked-singularity spacetimes, and their relevance for thin accretion disks is established. The circular orbits constitute two families that coalesce at the so-called static radius. The orientation of the motion along the circular orbits is, in accordance with case of asymptotically flat Kerr spacetimes, defined by relating the motion to the locally nonrotating frames. The minus-family orbits are all counterrotating, while the plus-family orbits are usually corotating relative to these frames. However, the plus-family orbits become counterrotating in the vicinity of the static radius in all Kerr--de Sitter spacetimes, and they become counterrotating in the vicinity of the ring singularity in Kerr--de Sitter naked-singularity spacetimes with a low enough rotational parameter. In such spacetimes, the efficiency of the conversion of the rest energy into heat energy in the geometrically thin plus-family accretion disks can reach extremely high values exceeding the efficiency of the annihilation process. The transformation of a Kerr--de Sitter naked singularity into an extreme black hole due to accretion in the thin disks is briefly discussed for both the plus-family and minus-family disks. It is shown that such a conversion leads to an abrupt instability of the innermost parts of the plus-family accretion disks that can have strong observational consequences.

The iron K α-line diagnostics of the rotating black hole metric in Seyfert galaxies

Observations of Seyfert galaxies in X-ray region reveal the wide emissive lines in their spectra, which can arise in inner parts of accretion disks, where the effects of General Relativity (GR) must be taken into account. A spectrum of a solitary emission line of a hot spot in Kerr accretion disk is simulated depending on the radial coordinate r and the angular momentum a = J/ M of a black hole, under the assumption of equatorial circular motion of a hot spot. It is shown that the characteristic two-peak line profile with the sharp edges arises at a large distance (about r ≈ (3–10) r g). The inner regions emit the line, which is observed with one maximum and extremely wide red wing. We present results of simulations for the isothermal and Shakura–Sunayev disks.

Dynamical response of equatorial waves in four-dimensional variational data assimilation

In this study we question the relative importance of direct wind measurements in the tropics by investigating limits of four-dimensional variational assimilation (4D-Var) in the tropics when only wind or mass field observations are available. Typically observed equatorial wave motion fields (Kelvin, mixed Rossby-gravity and n = 1 equatorial Rossby waves) are assimilated in a non-linear shallow water model. Perfect observations on the full model grid are utilized and no background error term is used. The results illustrate limits of 4D-Var with only one type of information, in particular mass field information. First, there is a limit of information available through the internal model dynamics. This limit is defined by the length of the assimilation window, in relation to the characteristics of the motion being assimilated. Secondly, there is a limit related to the type of observations used. In all cases of assimilation of wind field data, two or three time instants with observations are sufficient to recover the mass field, independent of the length of the assimilation time window. Assimilation of mass field data, on the other hand, although capable of wind field reconstruction, is much more dependent on the dynamical properties of the assimilated motion system. Assimilating height information is less efficient, and the divergent part of the wind field is always recovered first and more completely than its rotational part.

The Shape of the Kα Line as the Evidence for the Black Hole Existence

Observations of Seyfert galaxies in X-ray region reveal the wide emissive lines in their spectra, which can arise in inner parts of accretion disks, where the effects of General Relativity (GR) must be taken into account. A spectrum of a solitary emission line of a hot spot in Kerr accretion disk is simulated, depending on the radial coordinate r and the angular momentum a = J/M of a black hole, under the assumption of equatorial circular motion of a hot spot. It is shown that the characteristic two-peak line profile with the sharp edges arises at a large distance, (about r ≈ (3 − 10) rg). The inner regions emit the line, which is observed with one maximum and extremely wide red wing. We present results of simulations for the isothermal and Shakura – Sunayev disks.

Celestial mechanics in Kerr spacetime

The dynamical parameters conventionally used to specify the orbit of a test particle in Kerr spacetime are the energy E, the axial component of the angular momentum, Lz, and Carter's constant Q. These parameters are obtained by solving the Hamilton–Jacobi equation for the dynamical problem of geodesic motion. Employing the action-angle variable formalism, on the other hand, yields a different set of constants of motion, namely, the fundamental frequencies ωr, ωθ and ωϕ associated with the radial, polar and azimuthal components of orbital motion, respectively. These frequencies, naturally, determine the time scales of orbital motion and, furthermore, the instantaneous gravitational wave spectrum in the adiabatic approximation. In this paper, it is shown that the fundamental frequencies are geometric invariants and explicit formulae in terms of quadratures are derived. The numerical evaluation of these formulae in the case of a rapidly rotating black hole illustrates the behaviour of the fundamental frequencies as orbital parameters, such as the semi-latus rectum p, the eccentricity e or the inclination parameter θ− are varied. The limiting cases of circular, equatorial and Keplerian motion are investigated as well and it is shown that known results are recovered from the general formulae.

On the secondary meridional circulation associated with the quasi-biennial oscillation

Concentrations and distributions of stratospheric aerosol, hydrogen fluoride and ozone from the Halogen Occultation Experiment (HALOE) on the Upper Atmosphere Research Satellite (UARS) are used to investigate features associated with transport by the secondary meridional circulation induced by the quasi-biennial oscillation (QBO). The points of maxima in the divergence and convergence of the QBO-induced meridional velocity at the equator are identified from the meridional gradients of the tracers. Such points can be identified from the tracer fields in the westerly shear zones but not in the easterly shear zones. The temporal variation of tracer concentration at the equator is determined mainly by vertical advection, which is significantly larger during the westerly shear phase of the QBO than during the easterly shear phase, since the QBO-induced equatorial sinking motion amplifies the vertical gradient. Thus, the vertical advection associated with the secondary circulation has a stronger influence on the equatorial tracer variation during the westerly shear phase than during the easterly shear phase.

Mid‐latitude reflection of ion upflows during substorm dipolarization

Reconfiguration of the magnetospheric current system during the expansion phase of substorms leads to dipolarization of the magnetic field lines in the inner plasma sheet. This process is characterized by inward field line motion in the equatorial region and outward motion at mid‐latitudes. We investigate the parallel motion of ions during such dynamical reconfigurations. We demonstrate that, in contrast to equatorial latitudes where ions are centrifugally jetted toward the midplane and experience prominent energization, ions at mid‐latitudes are subjected to a downward (earthward) oriented acceleration. We accordingly show that ions with small parallel speeds such as those originating from the topside ionosphere may be decelerated upon reaching mid‐latitudes and return to low altitudes. Field line dipolarization during substorms may thus go together with substantial downward flow of low‐energy ions, as evidenced from measurements of the low‐altitude orbiting Interball‐Auroral spacecraft.

Equatorial photon motion in the Kerr-Newman spacetimes with anon-zero cosmological constant

Discussion of the equatorial photon motion in Kerr-Newman black-holeandnaked-singularity spacetimes with a non-zero cosmological constant ispresented. Both repulsive and attractive cosmological constants areconsidered. An appropriate `effective potential' governing the photonradial motion is defined, circular photon orbits are determined, andtheir stability with respect to radial perturbations is established. Thespacetimes are divided into separated classes according to the propertiesof the `effective potential'. There is a special class ofKerr-Newman-de Sitter black-holespacetimes with the restricted repulsive barrier.In such spacetimes, photons withhigh positive and all negative values of their impact parameter cantravel freely between the outer black-hole horizon and the cosmologicalhorizon due to an interplay between the rotation of the source and thecosmological repulsion. It is shown that this type of behaviour of thephoton motion is connected to an unusual relation between the values ofthe impact parameters of the photons and their directional anglesrelative to the outward radial direction as measured in the locallynon-rotating frames. Surprisingly, some photons counterrotating in theseframes havea positive impact parameter. Such photons can be both escapingor captured in the black-hole spacetimes with the restricted repulsivebarrier. For theblack-hole spacetimes with a standard, divergent repulsive barrier of theequatorial photon motion, the counterrotating photons with positiveimpact parameters must all be captured from the region near theblack-hole outer horizon as in the case of Kerr black holes, while theyall escape from the region near the cosmological horizon. Further, theazimuthal motion is discussed and photon trajectories are given intypical situations. It is shown that for some photons with negativeimpact parameter turning points of theirazimuthal motion can exist.

The equatorial bulge, angular momentum and planetary wave motion

The equatorial bulge, angular momentum and planetary wave motion

The equatorial bulge, angular momentum and planetary wave motion

The impact of the equatorial bulge on the equatorial angular momentum of planetary flows isinvestigated by considering both the situation on earth and on a spherical planet. First, arigorous proof of the suggestion of Bell is given that both the standard nonlinear and linearprimitive equations capture the impact of the bulge almost correctly, despite the fact that aspherical earth is assumed in these equations. Linear eigenmodes of shallow water flow withequivalent depth H are discussed both for the spheroid with a corresponding torque and aspherical planet without torque. Surprisingly, it is found that Rossby–Haurwitz waves as rotationalmodes for H→∞ are not affected by the bulge. For finite H, however, there are bothgravitational and rotational modes with equatorial angular momentum which are affected bythe bulge. In contrast, there is only one mode with equatorial angular momentum on thespherical planet. This mode is stationary.

Kerr–Newman–de Sitter black holes with a restricted repulsive barrier of equatorial photon motion

A restricted repulsive barrier exists for equatorial photon motion in some Kerr--Newman--de Sitter black-hole spacetimes, due to an interplay between the rotation of the source and the cosmological repulsion. Quite surprisingly, photons with high positive and all negative values of their impact parameter can travel freely between the black-hole horizon and the cosmological horizon in these spacetimes. The special class of the Kerr--Newman--de Sitter spacetimes with restricted repulsive barriers is represented in the space of the parameters of the geometry. The area of the parameter space corresponding to this class is largest in the case of Kerr--de Sitter spacetimes. With the charge parameter of the spacetime growing, the area shrinks. In all of these cases, the mass parameter of the spacetime must be so high that the sizes of the black-hole and cosmological horizons are comparable.

Low-Frequency Equatorial Waves in Vertically Sheared Zonal Flow. Part I: Stable Waves

Abstract The mechanism by which a vertically sheared zonal flow affects large-scale, low-frequency equatorial waves is investigated with two-level equatorial,β-plane and spherical coordinates models. Vertical shears couple barolinic and barotropic components of equatorial wave motion, affecting significantly the Rossby wave and westward propagating Yanai wave but not the Kelvin wave. This difference results from the fact that the barotropic component is a modified Rossby mode and can be resonantly excited only by westward propagating internal waves. The barotropic components emanate poleward into the extratropics with a pronounced amplitude, while the baroclinic components remain equatorially trapped. A westerly vertical shear favors the trapping of Rossby and Yanai waves in the upper troposphere, whereas an easterly shear tends to confine them in the lower troposphere. As such, their westward propagation is slowed down by both westerly and easterly shears. When the strength of the vertical shear varies w...

The analysis of star catalogs. 2: an intercomparison among the GC, the N30, and the FK5(B1950.0/FK4)

We use the general method of compiled data set analysis developed in our first paper (Bucciarelli et. al. 1994) and extend our analysis of fundamental star catalogs to include the two American ones. The problems found in the FK3/FK4/FK5 sequence are not present in either the GC or the N30. This lends support to the hypothesis advanced in our previous paper that the FK3, and to a lesser extent the FK4 declination system, are distorted by high angular frequency systematics. On the other hand, both the GC and N30 equatorial coordinate and proper motion distributions nicely follow the predictions of mathematical statistics. As with the FK3/FK4/FK5 analysis, the new results are as seen by the FK5 (on the FK4/B1950.0 system).