Tidal gravity and ocean tide loading in Europe

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The results are presented from tidal gravity measurements at five sites in Europe using LaCoste and Romberg ET gravimeters. Improvements that we have made to the accuracies of these gravimeters are discussed. It is shown that the 'standard' calibration of the International Center for Earth Tides, used for worldwide tidal gravity profiles, is 1.2 per cent too high. The M₂ and O₁ observations are compared with model calculations of the Earth's body tide and ocean tide loading and it is shown that there is a very significant improvement in the agreement between observations and models compared to that obtained with previous tidal gravity measurements. For O₁, where the ocean tide loading and attraction in central Europe is only 0.4 per cent of the body tide, our measurements verify that the Dehant-Wahr anelastic body tide model gravimetric factor is accurate to 0.2 per cent. It is also shown that the effects of lateral heterogeneities in Earth structure on tidal gravity are too small to explain the large anomalies in previously published tidal gravity amplitudes. The observations clearly show the importance of conserving tidal mass in the Schwiderski ocean tide model. For sites in central Europe, the M₂ and O₁ observations and the models are in agreement at the 0.1 μgal (10⁻⁹ m s⁻²) level and tidal corrections to this accuracy can now be made to absolute gravity measurements.     

The tidal dynamics of the Irish and Celtic Seas

Summary. Current meter data collected over periods of more than 14 day from the Irish and Celtic Seas are harmonically analysed and presented in maps of tidal stream information. Making use of the analysed current data, and by constructing time series of frictional and inertial stresses which are also harmonically analysed, harmonic constituents of the surface tidal slopes at current meter stations are obtained. Using these with data collected from offshore tide gauges, and in conjunction with coastal tide data, cotidal maps are drawn with some confidence for M ₂, S ₂, O ₁ and K ₁, the M ₂ chart resolving the discrepancy which exists between the different charts of the Celtic Sea already produced. Cotidal maps for M ₃ and M ₄ are also presented.
The mean over a tidal cycle of the energy flux for M ₂, S ₂ and O ₁ is also presented in the form of the total energy flux in these constituents which crosses different sectional lines. A flux of 44 × 10⁶ kW is observed to enter the Celtic Sea from the Atlantic and this is compared with previous estimates. An energy budget is also performed for M ₂, including all the effects of astronomical forcing and Earth tides to enable comparison to be made between the true energy inflow and the estimated frictional dissipation. Finally, comparison is made between the mean of the instantaneous energy flux and the sum of the energy fluxes associated with the major harmonics.     

Tides in the Bristol Channel — an analytical wedge model with friction

Summary. An analytical model of one-dimensional long wave propagation is developed for a wedge-shaped estuary, with bottom friction included in the momentum equation. The model is applied to the M₂ tidal propagation in the Bristol Channel. It appears to represent the amplitude and phase of both elevation and velocity better than the earlier models of Taylor and Bennett. Estimates of energy flux into the estuary are seen to depend critically on the boundary conditions chosen to tune the model. Conditions at Spring and Neap tides are also reproduced by the analytical solution, and the response of the tides to the introduction of a solid barrier at different location is evaluated using the model.     

Tidal gravity in Britain: tidal loading and the spatial distribution of the marine tide

Summary. Tidal gravity measurements have been made at six sites in Britain with two nulled LaCoste and Romberg Earth tide gravitymeters. The M ₂ observations from these and two further sites are compared with calculations of the tidal loading from the seas around the British Isles and the major oceans. Models of the M ₂ marine tides are convolved with Green's functions for appropriate radially stratified Earth models. The differences between the M ₂ observations and the theoretical calculations are less than 0.6 μ gals and it is shown that these differences contain further information concerning the errors in the marine tide models. The M ₂ marine tides on the north-west European continental shelf are reasonably well known and this allows a useful test of the feasibility of using tidal gravity measurements for the inverse ocean tide problem in areas where the ocean tides are less well known. The differential gravity loading signal between pairs of gravity stations is shown to be important for considerations of the uniqueness and accuracy of the inverse problem. M ₂ tidal gravity loading maps for the British Isles and Europe have been produced which are of use in making corrections to various geodetic measurements.     

Observation of Tidal Tilt On Kilauea Volcano, Hawaii

Summary. We have analysed the east-west tilt components, O₁, K₁, N₂, M₂ and S₂ from a continuously recording tiltmeter located in Uwekahuna Vault on Kilauea Volcano, Hawaii, for the period 1971—79. Detailed analysis of the M₂ component gives values of 30.9 ± 2.0 (95 per cent) nrad and 116.0 ± 2.0° for the amplitude and phase, respectively, compared to values of 48.5 nrad and 139.4° for the equilibrium tide. the total theoretical tide, found by summing the equilibrium and load tides, amounts to 37.2 nrad at a phase of 121.7°. the 20 per cent discrepancy with that observed may be due to an inaccurate cotical chart, cavity effects in the vault, strain—tilt coupling or an inappropriate solid earth model. In the vicinity of Hawaii (≤ 3°) two independent cotidal charts give almost identical results for the near field ocean load. At greater distances, we use the Schwiderski (1978) cotidal chart. We estimate that local cavity and strain—tilt coupling effects are less than 12 per cent owing to the agreement between geodetically determined and instrumental tilt but we can not rule out regional effects. Assuming these are small and the cotical charts correct, we find that the M₂ results are brought into satisfactory agreement if, instead of using an average oceanic earth model in the (< 75 km) vicinity of Hawaii, we use an earth model with nearly one-half the oceanic rigidity. Such a low upper mantle and crustal rigidity is consistent with Kilauea's position above the thermal upwelling associated with the Hawaiian hotspot.     

Some features of the seasonal variation of geomagnetic lunar tides

Summary. Mean hourly values have been analysed month by month for a number of stations for which long runs of data are available. It is found that there is an annual variation in the M₂ lunar tide, there being a global enhancement of the tide around January. There is also some evidence from the phase changes with latitude that, near the solstices, the lunar current system consists not of pairs of vortices, one in each hemisphere, but of single vortices with foci in the summer hemisphere.     

The Residual Water Flow through the Solent, South England

Summary. A reduced equation of motion is used to compute the residual velocity and the residual transport through the West Solent from the water levels recorded over an eight-month period at tide gauges at either end of the channel. A coefficient of bottom friction of 5·0 × 10⁻³ is assumed. There was a spring-neap variation and a significant correlation of fluctuations in the residual velocity with meteorological conditions. Westward residual velocities occurred at spring tides with low barometric pressure and south-westerly winds. Eastward residual velocities occurred at neap tides with high pressure and north-easterly winds. Because of the progressive nature of the tidal wave the long term residual transport appeared to be towards the west and the flushing time for the Solent system was long for considerable periods. The maximum velocities experienced during a tidal cycle half way along the channel are towards the west with a probability of values exceeding 160 cm s⁻¹ for 10 min in 5·4 days.     

Fourier contamination of a finite length of data in tidal analysis

Summary. The contamination effect when a discrete Fourier analysis is applied to a short length of data in order to estimate the main diurnal (O₁) and semi-diurnal (M₂) components of the solid body tide is estimated and it is shown that a moderate length of record (30 days) has distinct advantages over larger record lengths of less than 60 days or so.     

The influence of barometric-pressure variations on gravity

Summary. The superconducting gravimeter has been used to measure the influence of barometric pressure on gravity in the frequency range 0.1–10 cycles/day. These measurements show that the incoherent barometric fluctuations are the major cause of random fluctuations in local gravity and account for much of the 'noise' on our gravimeter records. A simple model has been constructed which adequately explains the response of gravity to the local pressure fluctuations. These measurements also show a response to the global atmospheric tides at S ₁ and S ₂ which is much larger than the response to local fluctuations. Although this behaviour is most likely due to the response of the world-wide oceans to the atmospheric tides, no theoretical model has yet been constructed.     

A diurnal resonance in the ocean tide and in the Earth's load response due to the resonant free 'core nutation'

Summary. The luni-solar forced nutations and body tide are believed to be resonant at frequencies near (1 + 1/460) cycle sidereal day⁻¹ as seen from the rotating Earth. This resonance is due to the Earth's rotating, elliptical fluid core. We show here that tides in the open ocean and the Earth's response to those tides must also be resonant at (1 + 1/460) cycle day⁻¹. We examine these resonant oceanic effects on the Earth's nutational motion and on the body tide. Effects on the forced nutations might be as large as 0.002 arcsec at 18.6 yr. The effects on the observed resonance in the body tide are more important. For tidal gravity, for example, the difference between K ₁ and 0 ¹ which is usually used to determine the resonance, can be perturbed by 30 per cent or more due to the oceanic resonance effects.     

The traveltime perturbations for seismic body waves in factorized anisotropic inhomogeneous media

The traveltime perturbation equations for the quasi-compressional and the two quasi-shear waves propagating in a factorized anisotropic inhomogeneous (FAI) media are derived. The concept of FAI media simplifies considerably these equations. In the FAI medium, the density normalized elastic parameters a _ijkl ( X _i ) can be described by the relation a _ijkl ( X _i) = f ²( x _i ) A _ijkl, where A _ijkl are constants, independent of coordinates x _i and f ²( x _i) is a continuous smooth function of x _i . The types of anisotropy ( A _ijkl ) and inhomogeneity [ f ( x _i)] are not restricted. The traveltime perturbations of individual seismic body waves ( q ^P , qS 1 and qS 2) propagating in the FAI medium depend, of course, both on the structural pertubations [δ f ²( x _i)] and on the anisotropy perturbations (δ A _ijkl ), but both these effects are fully separated. The perturbation equations for the time delay between the two qS -waves propagating in the FAI medium are simplified even more. If the unperturbed (background) medium is isotropic, the perturbation of the time delay does not depend on the structural perturbations (δ f ²( x _i) at all. This striking result, valid of course only in the framework of first-order perturbation theory, will simplify considerably the interpretation of the time delay between the two split qS -waves in inhomogeneous anisotropic media. Numerical examples are presented.     

Tides and the evolution of the Earth—Moon system

Summary. A model of the tides in a hemispherical ocean is used to investigate the effect of changes in the Earth's rotation rate on the power dissipated by the ocean tides. The results obtained are then used in an idealized astronomical model to investigate how they affect the history of the Earth—Moon system.
Using the tidal model it is found that at rotation rates higher than that of the present Earth, the power dissipated by the semi-diurnal tides in the ocean drops off rapidly as a result of the increased tidal frequency. Thus if the Earth's rotation rate is doubled from its present value, then the rate of energy dissipation in the ocean is reduced to approximately one-third of its present value and the tidal torque is reduced by a factor of about 6.
The present value for secular acceleration of the Moon, calculated from the results of the tidal model is -30.5 arcsec century^-2. Using this value in the astronomical model, which has the Moon and Sun in circular orbits above the equator, and assuming that the tidal torque is independent of the tidal frequency, the Gerstenkorn event is predicted to have occurred 1.3 × 10⁹ yr ago.
When the astronomical model is run with a torque determined at all times from the tidal model, the reduction in the energy dissipated early in the history of the system, leads to a Gerstenkorn date of 5.3 × 10⁹ yr ago. However, dissipation within the solid earth is found to be important early in the history of the system and when this effect is included it gives a date for the Gerstenkorn event of 3.9 × 10⁹ yr ago.     

Detailed gravity-tide spectrum between one and four cycles per day

Summary. The results of previous work by the authors is used to remove most of the effects of ocean and atmospheric loading from an 18-month Earth gravity-tide record. The remaining signal is examined for additional influence of ocean and atmosphere and for evidence of the frequency-dependence of the response of the solid earth. Variations in time of the measured tides are shown to result from the atmospheric tide at S ₂ and appear to result from variations in ocean tides at other frequencies. The frequency-dependence of the solid earth response near 1 cycle per siderial day is found to be consistent with the nearly diurnal free wobble. However, the influence of the ocean on the small but crucial Ψ₁ tide is uncertain. Anomalous responses are observed at several other frequencies but except for the case of ρ₁ it is argued that anomalous ocean tides are plausible and could therefore explain the observations.     

A comparison of annual and seasonal carbon dioxide effluxes between sub-Arctic Sweden and High-Arctic Svalbard

Recent climate change predictions suggest altered patterns of winter precipitation across the Arctic. It has been suggested that the presence, timing and quantity of snow all affect microbial activity, thus influencing CO₂ production in soil. In this study annual and seasonal emissions of CO₂ were estimated in High-Arctic Adventdalen, Svalbard, and sub-Arctic Latnjajaure, Sweden, using a new trace gas-based method to track real-time diffusion rates through the snow. Summer measurements from snow-free soils were made using a chamber-based method. Measurements were obtained from different snow regimes in order to evaluate the effect of snow depth on winter CO₂ effluxes. Total annual emissions of CO₂ from the sub-Arctic site (0.662–1.487 kg CO₂ m^–2 yr^–1) were found to be more than double the emissions from the High-Arctic site (0.369–0.591 kg CO₂ m^–2 yr^–1). There were no significant differences in winter effluxes between snow regimes or vegetation types, indicating that spatial variability in winter soil CO₂ effluxes are not directly linked to snow cover thickness or soil temperatures. Total winter emissions (0.004–0.248 kg CO₂ m^–2) were found to be in the lower range of those previously described in the literature. Winter emissions varied in their contribution to total annual production between 1 and 18%. Artificial snow drifts shortened the snow-free period by 2 weeks and decreased the annual CO₂ emission by up to 20%. This study suggests that future shifts in vegetation zones may increase soil respiration from Arctic tundra regions.     

On the excitation of resonant double Kelvin waves in the Barents Sea Opening

In the northern Barents Sea Opening (BSO) the K₁ tidal energy is predominant in the diurnal tidal frequency band, suggesting the generation of a topographic wave with the K₁ tidal frequency. Tidal energy of the K₁ component becomes strong where bottom topography undulates in the BSO and the scale of the undulation is close to the wavelength of the K₁ wave. An analytical model is developed to investigate the energy enhancement mechanism of the tidally induced topographic wave due to a resonance between tidal current, a topographic wave and periodic topography. The wave excited by the resonance is identified as a resonant double Kelvin wave (DKW) and the significant K₁ energy in the BSO could be due to the excitation of the resonant DKW.     

Approaches to Modelling the Surface Albedo of a High Arctic Glacier

Broadband surface albedo measurements, made during the summer melt season at three weather stations on John Evans Glacier (79°40 ' N, 74°00 ' W), varied strongly with the solar zenith angle ( θ _z ). Tests were carried out to assess the impact of diurnal variations in surface albedo on seasonal net shortwave radiation ( K ^* ) totals. Removing the diurnal signal from albedo measurements by daily averaging of hourly measurements, or by applying midday measurements to all hours of the day, changed K ^* by up to 16%. Ignoring measurements made at θ _z & 75°, to account for measurement (cosine) error at high θ _z , decreased K ^* by between 5 and 18%. Given the sensitivity of K ^* to diurnal patterns in surface albedo, experiments were carried out with two albedo models. One model accounted for albedo variations with θ _z and one did not. The model driven by θ _z , when implemented within a surface energy balance model for John Evans Glacier, produced better melt estimates. This suggests that diurnal variations in surface albedo should be accounted for in energy balance models of glacier melt.     

The effect of polar wander on the tides of a hemispherical ocean

Summary. A numerical model is constructed of the tides in a hemispherical ocean driven by the forces corresponding to the Y₂^–2 equilibrium tide. The model is used to study how tidal dissipation is affected by changes in the position of the ocean relative to the Earth's rotational axis and to test a hypothesis concerning the Gerstenkorn event.
As the position of the Earth's axis is varied with respect to the ocean, the model shows changes in the dissipation rate due to the changing position and importance of individual resonances of the ocean. However, a cooperative effect is also observed which results, for an ocean of depth 4400 m, in broad frequency bands near 10 rad day⁻¹ and-6 rad day⁻¹ in which the dissipation rate remains high.
The cooperative effect is found to arise from the existence, in an unbounded ocean, of resonances at these frequencies which match the tidal forces. When ocean boundaries are introduced, the new resonances near these frequencies contain a large component of the underlying resonance and as a result are themselves a good match to the driving forces.
For the real ocean, these findings imply that changes in the position of the pole, and also possibly changes in the shape of the ocean, will on average have little effect on the energy dissipated by the tides. However in the past changes in the mean depth and area of the ocean or the increased rotation rate of the Earth may have resulted in a smaller dissipation rate.     

Monthly estimates of carbon dioxide emissions from fossil-fuel consumption in Brazil during the late 1990s and early 2000s

Detailed understanding of global carbon cycling requires estimates of CO ₂ emissions on temporal and spatial scales finer than annual and country. This is the first attempt to derive such estimates for a large, developing, Southern Hemisphere country. Though data on energy use are not complete in terms of time and geography, there are enough data available on the sale or consumption of fuels in Brazil to reasonably approximate the temporal and spatial patterns of fuel use and CO ₂ emissions. Given the available data, a strong annual cycle in emissions from Brazil is not apparent. CO ₂ emissions are unevenly distributed within Brazil as the population density and level of development both vary widely.     

Stacking for the frequencies and Qs of 0S0 and 1S0

Summary. Using nine IDA records for the Indonesian earthquake of 1977 August 19, we have formed an optimal linear combination of the records and have measured the frequency and Q of ₀ S ₀ and ₁ S ₀. The frequency was measured using the moment ratio method. The attenuation was measured by the minimum width method and by the time-lapse method. The frequency and attenuation were measured simultaneously by varying them to obtain a best fit to the data. A 2000-hr stack, the sum of nine individual records, for ₀ S ₀ gave a frequency of 0.814664 mHz±4 ppm. The values for the Q of ₀ S ₀ for the three different methods of measurement were 5600,5833 and 5700, respectively. The error in the estimates of Q ^-1 is about 5 per cent for the minimum power method. For ₁ S ₀ a 300-hr stack yielded a frequency of 1.63151 mHz±30 ppm. The values of Q for this mode were 1960, 1800 and 1850, respectively, with an error in Q ^-1 of about 12 per cent for the minimum power method.     

Coupled normal-mode sensitivity to inner-core shear velocity and attenuation

We have studied the response of normal modes to perturbations in inner-core shear velocity and attenuation, using fully coupled mode synthetics. Our results indicate that (i) mode pairs   _n S _l – _{n ±1} S _l   are strongly coupled by anelasticity, (ii) this coupling causes shear velocity perturbations to strongly affect the Q values of modes through exchange of inner-core characteristics, (iii) there is no evidence for a weakly attenuating inner core in shear, and (iv) the discrepancy between attenuation models returned from normal modes and body waves is small. These results suggest that inversions for inner-core attenuation and shear velocity should be performed simultaneously and should take account of the strong cross-coupling due to attenuation.