Exact Orbits Research Articles

GEOSAT Gulf Stream region synthetic geoid available

Total dynamic sea surface topography can be recovered from radar altimeter data given an accurate estimate of the local level surface or geoid. A new technique, the “synthetic geoid,” has emerged from research using data from the GEOSAT 17‐day exact repeat orbit (Eos, July 11, 1989, p. 698). The geoid estimate, useful for mesoscale topography, is constructed by subtracting an independent estimate of the mean mesoscale oceanography from the main sea surface as measured by the altimeter.The GEOSAT Data Validation effort—a Navy‐sponsored collaboration between the Johns Hopkins University Applied Physics Laboratory, Harvard University, and Rutgers University—has completed the construction and validation of a synthetic geoid that allows the total dynamic topography of the Gulf Stream front and eddy system to be determined from GEOSAT, the future GEOSAT Follow‐On, or any future system in the SEASAT/GEOSAT ground track. The synthetic geoid covers the region between Cape Hatteras and the Grand Banks. The independent estimate of the mean oceanographic dynamic topography was obtained from the Harvard University Gulfcast data base which includes 1 year of daily realizations of the region's velocity field. The synthetic geoid was validated by comparing absolute dynamic topography derived from individual GEOSAT passes with concurrent topography from Gulfcast and AXBT underflights.

Testing the infrared method on a mechanical model.

The strong-coupling infrared or eikonal approximation, introduced in quantum field theory and heavy-ion-collision analysis, is tested by comparison with the large-scale shape of the exact orbits of a classical, anharmonic oscillator. In an appendix, a joint, nonuniform time-rescaling procedure is suggested as a method for reintroducing previously suppressed high-frequency effects.

Long-time behaviour of numerically computed orbits: Small and intermediate timestep analysis of one-dimensional systems

Abstract The long-time behaviour of numerically computed orbits in one-dimensional systems is studied by deriving a continuous-time “pseudo-dynamics” equivalent to the discrete-time numerical dynamics. The derivation applies to any numerical algorithm which conserves phase-space volume. A conservation law of the continuous-time system (conservation of the “pseudo-Hamiltonian”) guarantees that the numerical orbits are close to the exact orbits, even after an unlimited number of timesteps. The equivalence between the discrete-time and continuous-time dynamics holds only for sufficiently small of the timestep Δ. For intermediate values of Δ (sufficiently large that the conservation law does not hold, but sufficiently small that the numerical orbits are not chaotic) a new “super-adiabatic” invariant Δ is derived, and it is shown that conservation of Δ forces the numerical orbits to lie on smooth closed curves. If the potential energy varies rapidly over a small region, it is shown that very high-order resonances between the timestep and the orbital period T , (i.e., T / Δ = n , where n is a large integer) produce large deviations of these closed curves from the exact orbit. Such resonances also cause extreme sensitivity of the numerical orbit to the timestep.

Equatorial Kelvin and Rossby waves evidenced in the Pacific Ocean through Geosat sea level and surface current anomalies

Equatorial Kelvin and Rossby waves are comprehensively demonstrated over most of the equatorial Pacific basin, through their signatures in sea level and zonal surface geostrophic current anomalies. This was made possible with altimeter data pertaining to the first year of the Geosat (Geodetic Satellite) 17‐day exact repeat orbit (November 8, 1986, to November 8, 1987). To this end, along‐track corrected Geosat sea level anomalies (SLAs), relative to the time period of interest, were first smoothed using nonlinear and linear filters. The original 17‐day time step was then reduced by combining all ascending and descending tracks within 10° longitudinal bands. Finally, SLAs were gridded onto a regular grid, and low‐pass filters were applied in latitude and time in order to smooth out remaining high‐frequency noise. Anomalies of zonal surface geostrophic current were calculated using the first and second derivatives of the SLA meridional gradient, off and on the equator, respectively. Sea level and surface current anomalies are validated in the western equatorial Pacific with in situ data gathered during seven hydrographic cruises at 165°E, and through expendable bathythermograph and mooring measurements. Following their chronological appearance along the 165°E meridian, the major low‐frequency SLAs and zonal surface current anomalies are described and explained in terms of the equatorial wave theory. An equatorial downwelling Kelvin wave, known to be the main oceanic signal of the 1986‐1987 El Niño, is generated in December 1986, concomitant with a strong westerly wind anomaly occurring west of the dateline. The associated propagating equatorial SLAs correspond to an elevation of 15 cm. Independent estimates of this Kelvin wave phase speed are obtained through time‐lag correlation matrix analysis (2.82 ± 0.96 m s−1) and the least squares fit of the SLA meridional structures to theoretical Kelvin wave shape (2.26 ± 1.02 m s−1). Both estimates indicate that the Kelvin wave has the characteristic of a first baroclinic mode. An equatorial upwelling Kelvin wave is then detectable in June 1987. It is characterized by a 10‐cm sea level drop, propagating only from the western to the central equatorial Pacific. A first meridional mode (m = 1) equatorial upwelling Rossby wave crossing the entire Pacific basin from March 1987 (eastern part) to September 1987 (western part) shows up in SLAs and zonal surface current anomalies. Such a Rossby wave corresponds to propagating sea level drops which are extreme (−12 cm) at about 4°N and 4°S latitudes. The consequences on zonal surface geostrophic current are very important since, in the case of the upwelling, it dramatically decreases the three major surface currents (the North and South Equatorial Countercurrents, and South Equatorial Current) by an amplitude similar to their mean annual velocity values. The least squares fit of the Rossby wave SLA meridional structures to its theoretical m = 1 form cogently suggests the dominance of the first baroclinic mode (c = 2.59 ± 0.65 m s−1). This dominance is corroborated by an estimate of the Rossby wave phase speed (1.02 ± 0.37 m s−1), which roughly corresponds to the theoretical phase speed (c/2m + 1) of the m = 1 equatorial Rossby wave. It is suggested that the equatorial upwelling Rossby wave is mostly due to a reflection of an equatorial upwelling Kelvin wave generated in January 1987 near the dateline. Whether or not the overall propagating features are part of the 1986–1987 El Niño or belong to the “normal” seasonal cycle cannot be decided in the absence of longer altimeter sea level time series.

Collisionless tearing instability of a bi-Maxwellian neutral sheet: An integrodifferential treatment with exact particle orbits

The integrodifferential equation describing the linear tearing instability in the bi-Maxwellian neutral sheet is solved without approximating the particle orbits or the eigenfunction ψ. Results of this calculation are presented. Comparison between the exact solution and the three-region approximation [Phys. Fluids 27, 1198 (1984)] motivates the piecewise-straight-line approximation, a simplification that allows faster solution of the integrodifferential equation, yet retains the important features of the exact solution.

Periodic orbits in atomic hydrogen exposed to circularly polarised laser light

Exact circular periodic orbits for a hydrogen atom in a strong circularly polarised laser field are derived. A stability analysis shows one orbit to be stable or unstable depending on the laser parameters, whereas the other orbit is always unstable. These orbits are expected to manifest themselves as resonances in laser assistede − −H + scattering. The binding mechanism for these resonances is provided by the trapping of the quantum mechanical wave function onto the neighbourhood of a classical periodic orbit. This mechanism is analogous to that for a Wannier ridge resonance in a doubly excited two-electron atom.

A functional integral approach to plasma turbulence

The study of weakly turbulent plasmas uses perturbative methods, in contrast to the strong turbulence case which is investigated numerically. A functional integral method is proposed as a unifying approach based on the examination of the structure of the space of exact phase-space particle orbits. The method is nonperturbative and easily obtains basic results of the weak turbulence theory.

The calculation of resonance separatrices for the near-integrable case of the standard mapping

The standard mapping arises in many physical applications, including the analysis of nonlinear resonant acoustic oscillations in a closed tube. A perturbation expansion, in powers of the amplitude parameter, is given for the calculation of the fixed points of various orders and the associated separatrices. It is shown how exact homoclinic orbits can be calculated numerically. Explicit analytic expressions are given for the separatrices associated with the first four resonances when the perturbation parameter is small.

The structure of the sideband spectrum in free electron lasers

The one-dimensional, fast time averaged Hamiltonian is derived in a free electron laser (FEL) for electrons passing through a constant parameter wiggler and a radiation field. The exact unperturbed orbits without sidebands are obtained for all particles and arbitrary strength of the main signal. Integration, in action-angle variables, of the linearized kinetic equation with perturbing sidebands over the unperturbed orbits yields the sideband growth rate including both trapped and untrapped particles. The structure and scaling of the unstable spectrum are different from previous results obtained for electrons localized at the bottom of the ponderomotive well. It is found that upper and lower sidebands that are symmetric relative to the FEL frequency have opposite growth rates. There is no differentiation in the magnitude of the gain between upper and lower sidebands. The stability is determined by the sign of df0/dωb, the relative population of quantized oscillators with energy quantum ℏωb, where ωb is the synchrotron frequency in resonance with the sideband. The shear dωb/dJ, where J is the action variable, is stabilizing and distributions with gradients df0/dJ localized near the separatrix have the minimum growth rates.

Computer modeling of test particle acceleration at oblique shocks

We review the basic techniques and results of numerical codes used to model the acceleration of charged particles at oblique, fast-mode, collisionless shocks. The emphasis is upon models in which accelerated particles (ions) are treated as test particles, and particle dynamics is calculated by numerically integrating along exact phase-space orbits. We first review the case where ions are sufficiently energetic so that the shock can be approximated by a planar discontinuity, and where the electromagnetic fields on both sides of the shock are defined at the outset of each computer run. When the fields are uniform and static, particles are accelerated by the scatter-free drift acceleration process at a single shock encounter. We review the characteristics of scatter-free drift acceleration by considering how an incident particle distribution is modified by interacting with a shock. Next we discuss drift acceleration when magnetic fluctuations are introduced on both sides of the shock, and compare these results with those obtained under scatter-free conditions. We describe the modeling of multiple shock encounters, discuss specific applications, and compare the model predictions with theory. Finally, we review some recent numerical simulations that illustrate the importance of shock structure to both the ion injection process and to the acceleration of ions to high energies at quasi-perpendicular shocks.

Ergodic properties of computer orbits for simple piecewise monotonic transformations

Using a general model for normalized floating point arithmetic, we study the computer generated orbit of simple piecewise monotonic transformations on [0, 1] of the form τ( x) = kx(mod 1). Conditions under which the computer orbit is eventually exact are established. Moreover it is shown that for sufficiently high precision of computation the computer orbits display the same ergodic properties as the exact orbits of the transformations.

The free electron laser sideband instability reconsidered

The one-dimensional fast-time averaged Hamiltonian is derived in a free electron laser (FEL) for electrons passing through a constant parameter, linear wiggler and a radiation field. The exact unperturbed orbits without sidebands are obtained for all particles and arbitrary separatrix height δγ. Integration in action-angle variables of the linearized Vlasov equation with perturbing sidebands over the unperturbed orbits yields the sideband growth rate for both trapped and untrapped particles. The unperturbed distribution f 0 is in an equilibrium with the main signal field. It is found that upper and lower sidebands that are symmetric relative to the FEL frequency have opposite growth rates. There is no differentiation in the magnitude of the gain between upper and lower sidebands. The stability is determined by the sign of df 0 dω b , i.e., the relative population of oscillation quanta hω b, where ω b is the bounce frequency around resonance. The shear dω b dJ , where J is the action variable, is stabilizing and distributions with gradients df 0 dJ localized near the separatrix have the minimum growth rates. The structure and scaling of the unstable spectrum are different from previous results obtained for electrons localized at the bottom of the ponderomotive well.

Chaos in a quartic dynamical model

We study the orbital characteristics of a time independent, two dimensional quartic dynamical model with two exact periodic orbits that displays always closed zero velocity curves. It is shown that the stability of the periodic orbits depends on the value of the coupling parameter α. Computer calculations suggest that the degree of stochasticity is small for the values of α in the range 1 3. We also compute the Lyapunov characteristic exponents for different values of the coupling parameter.

GEOSAT data available soon

GEOSAT altimeter data collected after November 8, 1986, will be made available to the general research community by the National Oceanic and Atmospheric Administration (NOAA) beginning in early 1987. Although GEOSAT has been operating since April 1985, observations from the first 18 months are classified. In October 1986 the satellite was maneuvered into a 17‐day exact repeat orbit whose ground track coincides with the previous Seasat altimeter tracks, allowing new GEOSAT data to be unclassified. Under agreement reached with the U.S. Navy and the Johns Hopkins University Applied Physics Laboratory (Laurel, Md.), NOAA will assume responsibility for generating the unclassified data sets from this Exact Repeat Mission (ERM). Doppler tracking data will first be evaluated by the Naval Astronautics Group (NAG) to ensure that the GEOSAT ground track has deviated by no more than 1 km (cross track) from the exactrepeat orbit. On‐board thrusters will be fired when necessary (approximately monthly) to maintain colinearity. Raw data in, the form of sensor data records (SDRs) will then be transmitted to a NOAA processing facility in Rockville, Md., where they will be converted to finished geophysical data records (GDRs). SDR‐to‐GDR production consists of merging the altimeter data with an ephemeris provided by NAG and adding correctionfields for tides, troposphere (wet and dry components), and ionosphere. Completed GDRs will be sent to NOAA National Environmental Satellite Data and Information Service (NESDIS) in Washington, D.C., where they will be made avilable to the public. (GEOSAT data will initially be distributed to U.S. institutions only; foreign institutions are advised to seek access through formal embassy channels.)

Modeling of ion acceleration through drift and diffusion at interplanetary shocks

We describe a test particle simulation designed to study energetic charged particle acceleration at oblique fast‐mode shocks when magnetic fluctuations exist upstream and downstream of the shock. The technique consists of integrating along exact particle orbits in a system where the angle θ1 between the shock normal and mean upstream magnetic field, the level of magnetic fluctuations, and the energy of injected particles can assume a range of values. This allows us to study time‐dependent shock acceleration under conditions not amenable to analytical techniques. To illustrate the capability of the numerical model, we consider proton acceleration under conditions appropriate for interplanetary shocks near 1 AU, including large‐amplitude transverse magnetic fluctuations derived from power spectra of both ambient and shock‐associated MHD waves. With all other parameters held fixed, protons injected at θ1 = 0° and θ1 = 60° shocks are accelerated from 10 keV to ∼100 keV and ∼ 1 MeV, respectively, within 300 gyroperiods by a combination of the shock drift and first‐order Fermi processes. The energy spectrum downstream of the 60° shock can be fit with two power laws, with spectral exponent γ = 1.7 from 10 keV to ∼80 keV and γ = 2.6 from ∼ 80 keV to 800 keV. Results are also presented for the situation where the variance and upstream extent of the shock‐associated waves at the 60° shock are reduced relative to the values used at the 0° shock.

Collisionless tearing instability in a non-Maxwellian neutral sheet: An integrodifferential formulation

An integro-differential formalism is developed to study the collisionless tearing instability in a non-Maxwellian neutral sheet. The exact unperturbed particle orbits are used analytically in the orbit integrals. The treatment is linear, self-consistent, and kinetic for both ions and electrons. The analysis is carried out for low-frequency, purely growing electromagnetic perturbations (‖ω‖≪ωci). Using the Galerkin method, the integro-differential equation is solved to obtain the dispersion relation and the eigenmode structure. A sufficient condition for instability is given on the basis of a quadratic form and the eigenvalues of a self-adjoint integro-differential operator. The formalism is applied to a specific model distribution. For the case where the electrons and ions are both non-Maxwellian, it is found that the instability is dominated by the axis-crossing electrons and that the eigenmode is strongly localized to a region of the order of ( ρezp)1/2 at the null plane, where ρe is a measure of the electron gyroradius in the asymptotic magnetic field and zp is the sheet thickness. It is shown that the growth rate can be enhanced by several orders of magnitude over the isotropic case and that short wavelength perturbations are strongly preferred. The dispersion relation has the general form γ/kve∥=const where k∥B0 and ve∥ is the electron thermal velocity along B0, the equilibrium magnetic field.

Exact periodic orbits in an asymmetric dynamical system

Exact periodic orbits are found analytically in a time-independent, two-dimensional potential field. This potential is considered to represent stellar motion near the centre of a real elliptical galaxy, that shows small asymmetries due to the effect of companions. The theoretical results coincide with those given by numerical integration.

On exact orbit sequences

On exact orbit sequences

Electron orbits and stability in realizable and unrealizable wigglers of free-electron lasers

Magnetostatic wiggler fields $\stackrel{^}{x}coskz+\stackrel{^}{y}sin\mathrm{kz}$, commonly used to model the undulators of free-electron lasers, violate Maxwell's equations and are unrealizable. Realizable wigglers that approximate the unrealizable ones near the axis have a radial variation and an axial field component, both of which affect electron motion. Exact helical equilibrium orbits are given for relativistic electrons in a combined uniform guide field and realizable wiggler, in cylindrical geometry. The parameter $\mathrm{ka}$ that measures the size of the helix also measures the imparted quiver motion, on which the gain of the laser depends. Hence, wigglers that impart substantial quiver motion necessarily have electrons far from the axis, for which the unrealizable wiggler model is not valid. A linearized stability analysis shows that the equilibrium helical orbits are either strongly unstable or else exhibit a secular growth, linear in time. The trajectory of an electron that starts from given position and velocity in combined guide field and wiggler is also found from the perturbation analysis, with corrections for realizability and for harmonics of practical wigglers, such as a bifilar winding. Although the helical orbits are either strongly or weakly unstable, a class of nonhelical bounded orbits is found when the secular behavior is suppressed.

The structure of periodic solutions in the restricted problem of three bodies

In this work we consider four families of plane periodic orbits direct around the Sun which approach Jupiter but they are sufficiently far from it so as to be considered as predominantly two body orbits of the Sun-asteroid system. We study their horizontal and vertical stabilities and we give the exact orbits of bifurcations of these families with three-dimensional families of the same multiplicity or twice the multiplicity of the above families of plane symmetric periodic orbits. Moreover, we give the first segments of the three dimensional families of symmetric periodic orbits which emanate from these plane bifurcations and we study their stability relating it with the stability of the plane bifurcations.