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NASA NTRS1985

Tidal disruption of dissipative planetesimals

A self-consistent numerical model is developed for the tidal disruption of a solid planetesimal. The planetesimal is treated as a highly viscous, slightly compressible fluid whose disturbed parts are an inviscid, pressureless fluid undergoing distortion and disruption. The distortions were constrained to being symmetrical above and below the equatorial plane. The tidal potential is expanded in terms of Legendre polynomials, which eliminates the center of mass acceleration effects, permitting definition of equations of motion in a noninertial frame. Consideration is given to viscous dissipation and to characteristics of the solid-atmosphere boundary. The model is applied to sample cases in one, two and three dimensions.

Mizuno, H.↗
NASA NTRS2012

Twenty Years of Progress on Global Ocean Tides: The Impact of Satellite Altimetry

At the dawn of the era of high-precision altimetry, before the launch of TOPEX/Poseidon, ocean tides were properly viewed as a source of noise--tidal variations in ocean height would represent a very substantial fraction of what the altimeter measures, and would have to be accurately predicted and subtracted if altimetry were to achieve its potential for ocean and climate studies. But to the extent that the altimetry could be severely contaminated by tides, it also represented an unprecedented global-scale tidal data set. These new data, together with research stimulated by the need for accurate tidal corrections, led to a renaissance in tidal studies in the oceanographic community. In this paper we review contributions of altimetry to tidal science over the past 20 years, emphasizing recent progress. Mapping of tides has now been extended from the early focus on major constituents in the open ocean to include minor constituents, (e.g., long-period tides; non-linear tides in shelf waters, and in the open ocean), and into shallow and coastal waters. Global and spatially local estimates of tidal energy balance have been refined, and the role of internal tide conversion in dissipating barotropic tidal energy is now well established through modeling, altimetry, and in situ observations. However, energy budgets for internal tides, and the role of tidal dissipation in vertical ocean mixing remain controversial topics. Altimetry may contribute to resolving some of these important questions through improved mapping of low-mode internal tides. This area has advanced significantly in recent years, with several global maps now available, and progress on constraining temporally incoherent components. For the future, new applications of altimetry (e.g., in the coastal ocean, where barotropic tidal models remain inadequate), and new mission concepts (studies of the submesoscale with SWOT, which will require correction for internal tides) may bring us full circle, again pushing further development of tidal models as corrections.

Egbert, Gary↗
NASA NTRS2011

Spatial Distribution of Volcanic Hotspots and Paterae on Io: Implications for Tidal Heating Models and Magmatic Pathways

Io, the innermost of Jupiter's Galilean satellites, is the most volcanically active body in the Solar. System. Io's global mean heat flow is approximately 2 W/square m, which is approximately 20 times larger than on Earth. High surface temperatures concentrate within "hotspots" and, to date, 172 Ionian hotspots have been identified by spacecraft and Earth-based telescopes. The Laplace resonance between Io, Europa, and Ganymede maintains these satellites in noncircular orbits and causes displacement of their tidal bulges as the overhead position of Jupiter changes for each moon. Gravitational interactions between Jupiter and Io dominate the orbital evolution of the Laplacian system and generate enormous heat within to as tidal energy is dissipated. If this energy were transferred out of Io at the same rate as it is generated, then the associated surface heat flux would be 2.24 +/- 0.45 W/square m. This estimate is in good agreement with observed global heat flow, but to better constrain tidal dissipation mechanisms and infer how thermal energy is transferred to Io's surface, it is critical to closely examine the spatial distribution of volcanic features. End-member tidal dissipation models either consider that heating occurs completely in the mantle, or completely in the asthenosphere. Mixed models typically favor one-third mantle and two-thirds asthenosphere heating. Recent models also consider the effects of mantle-asthenosphere boundary permeability and asthenospheric instabilities. Deep-mantle heating models predict maximum surface heat flux near the poles, whereas asthenosphere heating models predict maxima near the equator-particularly in the Sub-Jovian and Anti-Jovian hemispheres, with smaller maxima occurring at orbit tangent longitudes. Previous studies have examined the global distribution of Ionian hotspots and patera (i.e., irregular or complex craters with scalloped edges that are generally interpreted to be volcanic calderas), but in this study, we combine a new geospatial analysis technique with an improved hotspot and paterae database .

Hamilton, C. W.↗
NASA NTRS1999

Mantle Q at Period 12.4 Hours: New Estimates from Satellite Tracking and Altimetry

The most convincing estimates of mantle Q at periods of many hours have historically come from extrapolating seismic and free-oscillation estimates via some assumed frequency dependence, sometimes contrained by estimates from the Chandler Wobble. At the semidiurnal tidal period, direct estimates of Q have been difficult to obtain because of the dominating signals of the ocean tides, which account for more than 95!k of the tidal energy dissipation. But knowledge of the ocean tides has been rapidly improving, primarily owing to satellite altimetry, and in 1996 we reported (NATURE, 381, 595-7) an estimate of solid-earth tidal energy dissipation and mantle Q based on combining satellite altimeter measurements with tracking observations of tidally induced satellite orbit perturbations. Tidal estimates from both reveal a small systematic difference in the quadrature component of the degree-2, order-2 spherical harmonic coefficients, which we attribute to a small lag in the earth's body tide. The formalism accounts for this lag in both the altimeter and tracking solutions and also accounts for a very small contribution from the lunar atmospheric tide. Since this original report, both altimeter and tracking estimates have improved. Recent solutions for the body-tide lag at the M2 period are 0.20 +/- 0.09 degrees, implying an energy dissipation of 100 +/- 50 gigawatts and a solid-earth Q of 300. Further new solutions will be discussed, as will the prospects for significantly reducing error bars and for obtaining estimates from other tides in the diurnal band.

Ray, Richard D.↗
NASA NTRS1982

Structure and thermal evolution of the Galilean satellites

Models of the structure and thermal evolution of the Galilean satellites based on the simplest assumptions are described, the most important complications that are likely to exist are identified, and their consequences are explored. Io's internal energy source is modeled based on dissipation of tidal heating due to gravitational interaction with Jupiter and Europa. A proposed contribution due to electrical heating is discussed. Three arguments bearing on Io's dissipation rate are addressed and models for the satellite's interior are discussed. Callisto and Ganymede are treated together in an attempt to model the reasons for their different appearance. It is concluded that the persistence of Ganymede's surface features is due to a prolonged thermal evolution relative to Callisto, probably caused by a higher radioactive content. Theoretical arguments concerning Europa's thermal evolution are more briefly presented.

Cassen, P. M.↗
NASA NTRS2002

Tides and Modern Geodesy

In modem high-precision geodesy, and especially in modem space geodesy, every measurement that one makes contains tidal signals. Generally these signals are considered noise and must somehow be eliminated. The stringent requirements of the latest space geodetic missions place severe demands on tidal models. On the other hand, these missions provide the strongest data for improving tidal models. In particular, TOPEX/POSEIDON altimetry and LAGEOS laser ranging have improved models to such an extent that new geophysical information about the ocean and the solid Earth are coming to light. Presumably GRACE intersatellite ranging data will also add to this information. This paper discusses several of these new geophysical results, with special emphasis given to the dissipation of tidal energy. Strong constraints have recently been placed on the partitioning of energy dissipation among the ocean, atmosphere, and solid earth and between the deep and shallow ocean. The dissipation in deep water is associated with internal tides and has potentially important implications for understanding the ocean's thermohaline circulation.

Ray, Richard D.↗
NASA NTRS1986

Triple interactions involving close binaries in globular clusters

The interaction of a circularized tidal binary system with a third star is studied statistically through the results of about 70,000 numerical scattering experiments. The stars involved are taken to be identical polytropes, with radii equal to one-third of the initial binary separation. The effects of nonzero stellar radii are included by calculating an approximation to the energy dissipated by tidal interactions and estimating the energy lost through physical collisions. These are then used to modify the interaction cross sections obtained in the point-mass approximation. It is found that, for a wide range of assumptions about the details of the triple interactions and the stars' structure, the net rate at which tidal binaries heat the stellar system is substantially reduced (by as much as a factor of 20) below that obtained when the binaries are treated simply as point masses. The most likely outcome of a triple encounter is the coalescence of two, or all three, of the stars involved.

Mcmillan, S. L. W.↗
NASA NTRS1984

Stability of the Galilean Satellites in Deep Orbital Resonance

The Galilean satellites are locked in the Laplace resonance, which is responsible for forcing their significant orbital eccentricites. The eccentricites in turn result in tidal energy dissipation within the satellites that is responsible for the great thermal activity on Io, including both volcanism observed by Voyager and heat flux measured from Earth. The resonance results from the nearly 2:1 commensurability of orbital periods (or equivalently of mean motions) taken by pairs. These geometrical relations enhance the mutual gravitational effects. Yoder pointed out that tides raised on Io tend to drive the system out of resonance (toward greater nu and correspondingly smaller eccentricities) while tides raised on Jupiter tend to drive the system towards deeper resonance (smaller nu and larger forced eccentricities). The possible behavior of the system in deep resonance is investigated.

Richard Greenburg↗
NASA NTRS1986

The lunar nodal tide and the distance to tne Moon during the Precambrian era

The origin and early evolution of life on Earth occurred under physical and chemical conditions distinctly different from those of the present day. The broad goal of this research program is to characterize these conditions. One aspect involves the dynamics of the Earth-Moon system, the distance of the Moon from the Earth, and the length of the day. These have evolved during the course of Earth history as a result of the dissipation of tidal energy. As the moon has receded the amplitude of oceanic tides has decreased while the increasing length of the day should have influenced climate and the circulation of atmosphere and ocean. A 23.3 year periodicity preserved in a 2500 million year old banded iron-formation was interpreted as reflecting the climatic influence of the lunar nodal tide. The corresponding lunar distance would then have been approx. 52 Earth radii. The influence of the lunar nodal tide is also apparent in rocks with an age of 680 million years B.P. The derived value for lunar distance 2500 million years ago is the only datum on the dynamics of the Earth-Moon system during the Precambrian era of Earth history. The implied development of Precambrian tidal friction is in accord with more recent paleontological evidence as well as the long term stability of the lunar orbit.

Walker, J. C. G.↗
NASA NTRS1985

Volcanic hot spots on Io - Correlation with low-albedo calderas

In the present investigation, it is shown that a one-to-one correspondence exists on Io between hot spots and features with normal albedos less than or equal to 0.3 (in the Voyager orange filter). It is pointed out that the hot spots are a manifestation of the prolific volcanic activity and heat flow of Io, widely believed to be due to dissipation of tidal energy resulting from the forward eccentricity of Io's orbit and the tides raised by Jupiter. All of the hot spots identified by the Voyager 1 infrared interferometric spectrometer (IRIS) correspond to relatively low-albedo features. An approach has been developed to identify hot spots not included in the IRIS inventory and to model their temperatures. The spectral reflectivities of the low-albedo features, extracted from the multispectral mosaics produced by Soderblom et al. (1985), are also presented, and compared to the laboratory spectra of a variety of candidate materials.

Mcewen, A. S.↗
NASA NTRS1988

Speculative histories of the Uranian satellite system

The hypothetical histories presently considered for the Uranian satellite system are very sensitive to the assumed satellite masses, within the errors of their determinations, as well as to the Uranian tidal effective dissipation function. A determination is made of the resonances that could have been encountered in these histories, and of the orbital eccentricity constraints that would have led to certain capture. It is found that if the mass of Miranda were of the upper extreme value, it would have been locked in a long-term 2/1 orbital resonance with Ariel to form a Laplacelike, three-body resonance that encompassed Umbriel.

Peale, S. J.↗
NASA NTRS2014

Resonant Tidal Excitation of Internal Waves in the Earth's Fluid Core

It has long been speculated that there is a stably stratified layer below the core-mantle boundary, and two recent studies have improved the constraints on the parameters describing this stratification. Here we consider the dynamical implications of this layer using a simplified model. We first show that the stratification in this surface layer has sensitive control over the rate at which tidal energy is transferred to the core. We then show that when the stratification parameters from the recent studies are used in this model, a resonant configuration arrives whereby tidal forces perform elevated rates of work in exciting core flow. Specifically, the internal wave speed derived from the two independent studies (150 and 155 m/s) are in remarkable agreement with the speed (152 m/s) required for excitation of the primary normal mode of oscillation as calculated from full solutions of the Laplace Tidal Equations applied to a reduced-gravity idealized model representing the stratified layer. In evaluating this agreement it is noteworthy that the idealized model assumed may be regarded as the most reduced representation of the stratified dynamics of the layer, in that there are no non-essential dynamical terms in the governing equations assumed. While it is certainly possible that a more realistic treatment may require additional dynamical terms or coupling, it is also clear that this reduced representation includes no freedom for coercing the correlation described. This suggests that one must accept either (1) that tidal forces resonantly excite core flow and this is predicted by a simple model or (2) that either the independent estimates or the dynamical model does not accurately portray the core surface layer and there has simply been an unlikely coincidence between three estimates of a stratification parameter which would otherwise have a broad plausible range.

Geodynamo↗
NASA NTRS1985

Io: Generation of Silicate Magma by Shear Melting at the Base of a Basaltic Lithosphere

Tidal theory and observational evidence indicates that about 1 w/sq. m. of energy is released at the surface of Io. In order to place limits on how much tidal energy can be dissipated within a rigid lithosphere, depth-temperature profiles were calculated for different lithosphere thickness assuming that the tidal energy was dissipated uniformly throughout the lithosphere. Thus a thick lithosphere implies that a significant fraction of the tidal energy is dissipated below the depth where solidus temperatures are reached. One possibility is that Io has a crust consisting of a low melting temperature fraction such as basalt, overlying a mantle of a high melting temperature fraction such as peridotite. Thus, if the lithosphere of Io is thicker than 30 km, as appears probable, then high rates of silicate volcanism are implied and a significant fraction of the tidal energy must be dissipated by viscous deformation rather than rigid flexure.

Carr, M. H.↗
NASA NTRS1977

A computer software system for the generation of global ocean tides including self-gravitation and crustal loading effects

A computer software system is described which computes global numerical solutions of the integro-differential Laplace tidal equations, including dissipation terms and ocean loading and self-gravitation effects, for arbitrary diurnal and semidiurnal tidal constituents. The integration algorithm features a successive approximation scheme for the integro-differential system, with time stepping forward differences in the time variable and central differences in spatial variables. Solutions for M2, S2, N2, K2, K1, O1, P1 tidal constituents neglecting the effects of ocean loading and self-gravitation and a converged M2, solution including ocean loading and self-gravitation effects are presented in the form of cotidal and corange maps.

Estes, R. H.↗
NASA NTRS1977

A computer software system for the generation of global ocean tides including self-gravitation and crustal loading effects

A computer software system is described which computes global numerical solutions of the integro-differential Laplace tidal equations, including dissipation terms and ocean loading and self-gravitation effects, for arbitrary diurnal and semidiurnal tidal constituents. The integration algorithm features a successive approximation scheme for the integro-differential system, with time stepping forward differences in the time variable and central differences in spatial variables.

Estes, R. H.↗
NASA NTRS1987

Could Ariel have been heated by tidal friction?

For significant dissipation in Ariel, the past existence of an orbital resonance to force and maintain a substantial eccentricity in the orbit is hypothesized. The absolute maximum dissipation rate would be reduced both by the time necessary to establish the resonance in the first place and by a consequently larger minimum Q. The actual heating rate was much less than this maximum so even the existence of a 2/1 resonance may not be sufficient to account for Ariel's smooth sufrace.

Peale, S. J.↗