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NASA NTRS1981

The nonlinear wave equation for higher harmonics in free-electron lasers

The nonlinear wave equation and self-consistent pendulum equation are generalized to describe free-electron laser operation in higher harmonics; this can significantly extend their tunable range to shorter wavelengths. The dynamics of the laser field's amplitude and phase are explored for a wide range of parameters using families of normalized gain curves applicable to both the fundamental and harmonics. The electron phase-space displays the fundamental physics driving the wave, and this picture is used to distinguish between the effects of high gain and Coulomb forces.

Colson, W. B.↗

NASA NTRS1982

Zonal harmonic model of Saturn's magnetic field from Voyager 1 and 2 observations

An analysis of the magnetic field of Saturn is presented which takes into account both the Voyager 1 and 2 vector magnetic field observations. The analysis is based on the traditional spherical harmonic expansion of a scale potential to derive the magnetic field within 8 Saturn radii. A third-order zonal harmonic model fitted to Voyager 1 and 2 observations is found to be capable of predicting the magnetic field characteristics at one encounter based on those observed at another, unlike models including dipole and quadrupole terms only. The third-order model is noted to lead to significantly enhanced polar surface field intensities with respect to dipole models, and probably represents the axisymmetric part of a complex dynamo field.

Connerney, J. E. P.↗

NASA NTRS1983

Transformations of the perturbed two-body problem to unperturbed harmonic oscillators

Singular, nonlinear, and Liapunov unstable equations are made regular and linear through transformations that change the perturbed planar problem of two bodies into unperturbed and undamped harmonic oscillators with constant coefficients, so that the stable solution may be immediately written in terms of the new variables. The use of arbitrary and special functions for the transformations allows the systematic discussion of previously introduced and novel anomalies. For the case of the unperturbed two-body problem, it is proved that if transformations are power functions of the radial variable, only the eccentric and the true anomalies (with the corresponding transformations of the radial variable) will result in harmonic oscillators. The present method significantly reduces computation requirements in autonomous space operations.

Szebehely, V.↗

NASA NTRS1983

Observations of odd-half cyclotron harmonic emissions in a shell-Maxwellian laboratory plasma

During the last 14 years, the subject of odd-half electron cyclotron harmonic emission from plasma has been given considerable attention. It was apparently first reported to occur in a beam plasma machine. The existence of the emission in space has been well documented through observations made with satellites. Because wavelengths are difficult to observe in space, no wave number spectrum has ever been obtained for the half-odd harmonic emission. Such a spectrum together with a frequency spectrum might provide the basis for a successful modeling of the instability. The present investigation is concerned with the design of a laboratory experiment in which a plasma with an anisotropic velocity distribution is produced and measured. In addition, the dispersion relation is directly measured, and a noise analysis is conducted. The obtained plasma, a mixture of shell and Maxwellian distributions, is found to emit waves within the Bernstein wave branches. By correlation measurements, a mode is found which is essentially an absolute instability in a narrow frequency band.

Urrutia, J. M.↗

NASA NTRS1984

Auroral kilometric radiation: Wave modes, harmonic and source region electron density structures

A change from extraordinary (X) mode to ordinary (0) mode dominance is observed in the auroral kilometric radiation (AKR) detected on ISIS 1 topside sounder ionograms as the source region plasma to gyrofrequency ratio fN/fH varies from 0.1 to 1.3. The X and 0 mode AKR, Z (the slow branch of the X mode) and whistler (W) mode are also observed. The Z mode is typically slightly less intense than the 0-mode. Thw W-mode is confined to frequencies less than fH/2, suggesting that it is the result of field aligned ducted signals reaching the satellite from a source at lower altitudes. Harmonic AKR bands are commonly observed and the 2nd harmonic appears to be due to propagating signals. The deduced (fN/fH) at the bottom of the AKR source region is always less than 0.4 and is typically less than 0.2 during the generation of X-mode AKR, but approaches 0.9 for 0-mode AKR. No large density enhancements were observed within AKR source region density cavities. It is suggested that the observed INTENSE AKR IS cyclotron X-mode radiation rather than plasma frequency 0-mode radiation.

Benson, R. F.↗

NASA NTRS1983

Autoregressive harmonic analysis of the earth's polar motion using homogeneous International Latitude Service data

The homogeneous set of 80-year-long (1900-1979) International Latitude Service (ILS) polar motion data is analyzed using the autoregressive method (Chao and Gilbert, 1980), which resolves and produces estimates for the complex frequency (or frequency and Q) and complex amplitude (or amplitude and phase) of each harmonic component in the data. The ILS data support the multiple-component hypothesis of the Chandler wobble. It is found that the Chandler wobble can be adequately modeled as a linear combination of four (coherent) harmonic components, each of which represents a steady, nearly circular, prograde motion. The four-component Chandler wobble model 'explains' the apparent phase reversal during 1920-1940 and the pre-1950 empirical period-amplitude relation. The annual wobble is shown to be rather stationary over the years both in amplitude and in phase, and no evidence is found to support the large variations reported by earlier investigations. The Markowitz wobble is found to be marginally retrograde and appears to have a complicated behavior which cannot be resolved because of the shortness of the data set.

Chao, B. F.↗

NASA NTRS1984

Electron-cyclotron maser emission - Relative growth and damping rates for different modes and harmonics

The temporal growth rate and the number of e-folding growths are calculated and compared for the following wave modes due to a loss-cone-driven cyclotron maser: fundamental x, o, and z modes and second harmonic x and o modes. The dominant mode of the maser should be the fastest growing mode for a saturated maser and should be the mode with the greatest number of e-folding growths for an unsaturated maser; this mode is the fundamental x mode) for a plasma frequency to cyclotron frequency ratio of less than about 0.3; it is the z mode (or perhaps the fundamental o mode) for ratios between 0.3 and 1.0, and the z mode (or perhaps the second harmonic x mode) for ratios between 1.0 and 1.3. Two main points are made: the dominance of the z mode over the range of ratios considered and the very weak effect of cyclotron damping. Electron-cyclotron maser emission is seen as responsible for auroral kilometric radiation, decametric radio emission from Jupiter and Saturn, solar microwave spike bursts, and microwave emission from some flare stars.

Melrose, D. B.↗

NASA NTRS1984

An analytic solution for the orbital perturbations of the Venus Radar Mapper due to gravitational harmonics

Hill's variational equations are solved analytically for the orbital perturbations of a spacecraft nominally in an elliptic orbit around a non-spherical body. The rotation of the central planet about its spin-axis is not considered in the analysis. The perturbations are restricted to the planetary gravitational harmonics only. An extremely simple algorithm is derived to transform the spherical harmonic potentials to the orbital coordinate system, and the resulting accelerations are shown to be simply trigonometric functions of the true anomaly. With the principal matrix solution for the differential equations of the adjoint system given in closed form, the orthogonality of the trigonometric functions makes it possible to obtain an analytic solution for the non-homogeneous problem, at intervals of 2 pi in true anomaly. The solution for orbital perturbations can be extended over several revolutions by applying well-known results from Floquet's theory. The technique is demonstrated with results presented on the spacecraft periapsis altitude for the forthcoming Venus Radar Mapper Mission.

Vijayaraghavan, A.↗

NASA NTRS1984

The importance of nonlinearity on the higher harmonic control of helicopter vibration

The effect of nonlinearity on the higher harmonic control (HHC) of helicopter vibration is investigated using a nonlinear aeroelastic simulation. A nonlinear solution is proposed which relates the HHC inputs to vibration outputs on the basis of a Volterra functional series. The Volterra series solution is shown to reduce to a vector polynomial equation relating HHC inputs to vibration outputs at any harmonic frequency. The nonlinear transfer relationship is identified from a nonlinear vibration analysis computer program, and the identification model is examined in detail. Improvements to current HHC algorithms are presented, and several Kalman filter divergence problems are quantified.

Molusis, J. A.↗

NASA NTRS1984

Development and applications of algorithms for calculating the transonic flow about harmonically oscillating wings

A finite difference method to solve the unsteady transonic flow about harmonically oscillating wings was investigated. The procedure is based on separating the velocity potential into steady and unsteady parts and linearizing the resulting unsteady differential equation for small disturbances. The differential equation for the unsteady velocity potential is linear with spatially varying coefficients and with the time variable eliminated by assuming harmonic motion. An alternating direction implicit procedure was investigated, and a pilot program was developed for both two and three dimensional wings. This program provides a relatively efficient relaxation solution without previously encountered solution instability problems. Pressure distributions for two rectangular wings are calculated. Conjugate gradient techniques were developed for the asymmetric, indefinite problem. The conjugate gradient procedure is evaluated for applications to the unsteady transonic problem. Different equations for the alternating direction procedure are derived using a coordinate transformation for swept and tapered wing planforms. Pressure distributions for swept, untaped wings of vanishing thickness are correlated with linear results for sweep angles up to 45 degrees.

Ehlers, F. E.↗

NASA NTRS1985

Characteristics of cosmic ray pole-equator anisotropy derived from spherical harmonic analysis of neutron monitor data

The spherical harmonic analysis of cosmic ray neutron data from the worldwide network neutron monitor stations during the years, 1966 to 1969 was carried out. The second zonal harmonic component obtained from the analysis corresponds to the Pole-Equator anisotropy of the cosmic ray neutron intensity. Such an anisotropy makes a semiannual variation. In addition to this, it is shown that the Pole-Equator anisotropy makes a variation depending on the interplanetary magnetic field (IMF) sector polarities around the passages of the IMF sector boundary. A mechanism to interpret these results is also discussed.

Takahashi, H.↗

NASA NTRS1984

Type III bursts in interplanetary space - Fundamental or harmonic?

ISEE-3 spacecraft observation of 120 relatively simple, isolated bursts in the 30-1980 kHz range are the basis of the present study of Type III bursts in the solar wind. Several characteristics are identified for many of these bursts which imply that the mode of emission changes from predominantly fundamental plasma radiation during the rise phase to predominantly second harmonic during decay. The fundamental emission begins in time coincidence with the start of Langmuir waves, confirming the conventional belief in these waves' causation of Type III bursts. Attention is given to the characteristics of fundamental components, by comparison to harmonics, at km-wavelengths.

Dulk, G. A.↗

NASA NTRS1985

Auroral kilometric radiation - Wave modes, harmonics, and source region electron density structures

A change from extraordinary (X) mode to ordinary (O) mode dominance is observed in the auroral kilometric radiation (AKR) detected on ISIS 1 topside sounder ionograms as the source region plasma to gyrofrequency ratio fN/fH varies from 0.1 to 1.3. The X and O mode AKR, Z (the slow branch of the X mode) and whistler (W) mode are also observed. The Z mode is typically slightly less intense than the O-mode. The W-mode is confined to frequencies less than fH/2, suggesting that it is the result of field aligned ducted signals reaching the satellite from a source at lower altitudes. Harmonic AKR bands are commonly observed and the 2nd harmonic appears to be due to propagating signals. The deduced (fN/fH) at the bottom of the AKR source region is always less than 0.4 and is typically less than 0.2 during the generation of X-mode AKR, but approaches 0.9 for O-mode AKR. No large density enhancements were observed within AKR source region density cavities. It is suggested that the observed intense AKR is cyclotron X-mode radiation rather than plasma frequency O-mode radiation.

Benson, R. F.↗

NASA NTRS1985

Venus gravity - A harmonic analysis and geophysical implications

An improved theoretical model of Venusian global gravity has been obtained by fitting a tenth degree spherical harmonic series to 78 orbital arcs of Doppler tracking data from the Pioneer Venus Orbiter. Maps of the free-air anomaly and its formal error are presented. Isostatic anomaly and 'Geoid' maps are also presented, and their geophysical implications are discussed in details. Comparison with equivalent resolution topographic models reveals a strong correlation between long wavelength gravity and the topography of Venus. Analysis of the second degree harmonics showed two aspects of the orientation of the inertial axes of Venus: (1) a significant (about three degrees) departure of the axis of greatest inertia from the rotational axis; and (2) a near alignment of the axis of least inertia with the location of the subterrestrial point at the time of the next inferior conjunction with earth (December 16, 2101). A series of contour maps of the Venusian free-air anomalies is provided.

Mottinger, N. A.↗

NASA NTRS1985

Adaptation of a modern medium helicopter (Sikorsky S-76) to higher harmonic control

Sikorsky Aircraft has performed analytical studies, design analyses, and risk reduction tests have been performed for Higher Harmonic Control (HHC) on the S-76. The S-76 is an 8 to 10,000 lb helicopter which cruises at 145 kts. Flight test hardware has been assembled, main servo frequency response tested and upgraded, aircraft control system shake tested and verified, open loop controllers designed and fabricated, closed loop controllers defined and evaluated, and rotors turning ground and flight tests planned for the near future. Open loop analysis shows that about 2 deg of higher harmonic feathering at the blade 75% radius will be required to eliminate 4P vibration in the cockpit.

Oleary, J. J.↗

NASA NTRS1985

The zonal harmonic and the yearly and half-yearly waves on the Southern Hemisphere in FGGE compared with the mean

The dominant component in the mean annual cycle of sea level pressure in the latitudes north of approx. 40 deg S is the yearly wave, whereas to the south of this latitude a half-yearly wave dominantes. In FGGE, however, the usually small yearly wave was large in the Antarctic, owing to the extraordinarily low pressure there in the winter of 1979. The half-yearly wave's amplitude was appreciably above normal in the center of each ocean in middle latitudes, and below normal at high latitudes. The yearly wave in the zonal geostrophic wind was larger than normal over the Antarctic Ocean. Similar results are obtained at 500 mb. The three month mean stationary waves (zonal harmonic waves) were larger than the mean in FGGE. The intra-annual variation of both amplitude and phase of the monthly mean harmonic waves in FGGE was large, which is not surprising as they contain an appreciable transient component, unlike the long term monthly mean waves.

Vanloon, H.↗

NASA NTRS1985

Finite amplitude gravity waves: Harmonics, advective steepening, breaking and saturation

A simple theory is presented which determines details of the breaking and saturation of a gravity wave as it propagates upward in the atmosphere. Breaking and saturation are here due to nonlinear advection analogous to the breaching of a surface wave and to the breaking of a planetary wave. Much simplification is obtained by the assumption that in a wave packet consisting of a primary wave and its harmonics, the primary wave remains dominant. This assumption, referred to a quasi-monochromatic approximation, is suggested by observations. Determined by this approximate theory are: a detailed picture of the waveform as it steepens and breaks; harmonics of the wave; the turbulence generation; and an underlying relationship between superadiabatic lapse rate and saturation by wave-wave interactions.

Weinstock, J.↗

NASA NTRS1985

Fundamental and harmonic electron cyclotron maser emission

The plasma conditions and features of the energetic electron distribution in electron cyclotron maser emission for which growth in a particular mode is favored when the ratio of the plasma frequency omega(p) to the electron cyclotron frequency Omega(e) is greater than about 0.3 are determined. It is shown that growth at the fundamental is suppressed as omega(p)/Omega(e) increases and emission at harmonics of Omega(e) dominates. Growth at harmonics of Omega(e) is not restricted to the O and X modes, but can also occur for the Z mode. Whether or not growth in a particular mode dominates depends both on omega(p)/Omega(e) and on the form of the distribution. If the density of the energetic electrons is sufficiently large, the dispersion relations of the O and X modes are modified so that the group velocities of the growing O and X mode waves can be comparable to that of the growing Z mode waves.

Winglee, R. M.↗