Dynamic ocean-tide effects on Earth's rotation

This article develops ‘broad-band’ Liouville equations which are capable of determining the effects on the rotation of the Earth of a periodic excitation even at frequencies as high as semi-diurnal; these equations are then used to predict the rotational effects of altimetric, numerical and 32-constituent spherical harmonic ocean-tide models. The rotational model includes a frequency-dependent decoupled core, the effects of which are especially marked near retrograde diurnal frequencies; and a fully dynamic oceanic response, whose effects appear to be minor despite significant frequency dependence. The model also includes solid-earth effects which are frequency dependent as the result of both anelasticity at long periods and the fluid-core resonance at nearly diurnal periods. The effects of both tidal inertia and relative angular momentum on Earth rotation (polar motion, length of day, ‘nutation’ and Universal Time) are presented for 32 long- and short-period ocean tides determined as solutions to the author’s spherical harmonic tide theory. The lengthening of the Chandler wobble period by the pole tide is also re-computed using the author’s full theory. Additionally, using the spherical harmonic theory, tidal currents and their effects on rotation are determined for available numerical and altimetric tide height models. For all models, we find that the effects of tidal currents are at least as important as those of tide height for diurnal and semi-diurnal constituents.

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title: Effects of dynamic long-period ocean tides on changes in Earth's rotation rate
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title: Tidal variations of earth rotation
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title: Ocean tides for satellite geodesy
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title: The self-consistent dynamic pole tide in non-global oceans
abstract: Fluid dynamic theories of the pole tide are in conflict with pole tide observations. The former predict that the oceanic response to the Chandler wobble should be nearly static; while the latter reveal the tide to possess significant non-equilibrium characteristics. Resolution of the conflict would determine whether mantle anelasticity or non-equilibrium oceans is responsible for a measurable contribution to the Chandler period. 
 
 
 
It is possible that resolution might be achieved through a more exact mathematical investigation of pole tide fluid dynamics than theories have carried out to date. The theories have also neglected to consider a unique property of the pole tide, its ability to lengthen the period of its own forcing function; this property implies that the pole tide characteristics and effects on wobble cannot be correctly determined unless the tide and wobble equations are jointly solved. 
 
 
 
This work constitutes a first attempt to determine, accurately and self-consistently, the nature of the pole tide and its effects on the Chandler wobble. The Laplace tidal equations, modified by the addition of bottom friction but with no simplifying approximations, are employed together with the Liouville equations to elucidate pole tide characteristics. This work is a first attempt because the earth model chosen is a crude one, consisting of global oceans of uniform depth overlying a rigid mantle with a fluid core. Results for this model suggest that departures of the pole tide from equilibrium may indeed be minimal. The relevance of these results to the actual pole tide, however, must await the application of our self-consistent and more exact dynamical theory to the situation of realistic oceans.

title: The self-consistent dynamic pole tide in non-global oceans
abstract: Fluid dynamic theories of the pole tide are in conflict with pole tide observations. The former predict that the oceanic response to the Chandler wobble should be nearly static; while the latter reveal the tide to possess significant non-equilibrium characteristics. Resolution of the conflict would determine whether mantle anelasticity or non-equilibrium oceans is responsible for a measurable contribution to the Chandler period. 
 
 
 
It is possible that resolution might be achieved through a more exact mathematical investigation of pole tide fluid dynamics than theories have carried out to date. The theories have also neglected to consider a unique property of the pole tide, its ability to lengthen the period of its own forcing function; this property implies that the pole tide characteristics and effects on wobble cannot be correctly determined unless the tide and wobble equations are jointly solved. 
 
 
 
This work constitutes a first attempt to determine, accurately and self-consistently, the nature of the pole tide and its effects on the Chandler wobble. The Laplace tidal equations, modified by the addition of bottom friction but with no simplifying approximations, are employed together with the Liouville equations to elucidate pole tide characteristics. This work is a first attempt because the earth model chosen is a crude one, consisting of global oceans of uniform depth overlying a rigid mantle with a fluid core. Results for this model suggest that departures of the pole tide from equilibrium may indeed be minimal. The relevance of these results to the actual pole tide, however, must await the application of our self-consistent and more exact dynamical theory to the situation of realistic oceans.

title: A complete spherical harmonic approach to luni-solar tides
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title: Experiments in tidal mass conservation
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title: Effect of the fluid core on changes in the length of day due to long period tides.
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title: A Search for the Earth's ‘Nearly Diurnal Free Wobble’
abstract: The existence of the Earth's liquid core, besides affecting the period of the Chandler wobble, gives rise to a second free wobble mode, with period short of a sidereal day by a factor of the order of the ellipticity of the coremantle boundary. Seen from space this nearly diurnal geographic motion of the pole should appear as a nutation of the Earth's rotation axis, with a period theoretically calculated to be about 460 days, and (as pointed out by Toomre) an amplitude correspondingly 460 times larger than that of the associated wobble. The curious history of the ‘nearly diurnal wobble’ problem is reviewed, and the relevant dynamics derived and described in terms of Poinsot's scheme. A special method developed for harmonic analysis of non-equispaced data, applied to a 20-year record of observations with Loomis Polar Telescope of Yale University Observatory, sets an upper limit of 0"·12–0"·26 on the amplitude of any free nutation in obliquity in the frequency range 0–4 cpy, i.e. an upper limit of 0"·0003–0"·0006 on the amplitude of the nearly diurnal free wobbble. This supports Toomre's contention that earlier reports of observations of this wobble, with amplitudes in the range 0"·01–0"·02, cannot be correct.

title: The period and Q of the Chandler wobble
abstract: We have extended our calculation of the theoretical period of the Chandler wobble to account for the non-hydrostatic portion of the Earth’s equatorial bulge and the effect of the fluid core upon the lengthening of the period due to the pole tide. We find the theoretical period of a realistic perfectly elastic Earth with an equilibrium pole tide to be 426.7 sidereal days, which is 8.5 day shorter than the observed period of 435.2 day. Using Rayleigh’s principle for a rotating Earth, we exploit this discrepancy together with the observed Chandler Q to place constraints on the frequency dependence of mantle anelasticity. If Q, in the mantle varies with frequency u as u” between 30 s and 14 months and if Q, in the lower mantle is of order 225 at 30 s, we find that 0.04 5 a 5 0.1 1 ; if instead Q, in the lower mantle is of order 350 near 200 s, we find that 0.11 5 a 0.19. In all cases these limits arise from exceeding the 68 per cent confidence limits of ? 2.6 day in the observed period. Since slight departures from an equilibrium pole tide affect the Q much more strongly than the period we believe these limits to be robust.

title: Body tides on an elliptical, rotating, elastic and oceanless earth
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title: THE DAMPING OF THE CHANDLER WOBBLE AND THE POLE TIDE
abstract: The damping of the Chandler wobble is an old yet actively researched problem still awaiting resolution in a number of respects. Due to both observational and theoretical limitations the actual damping rate is uncertain even now; a decay time of 45–70 years or more seems probable. The two most likely candidates for dissipating wobble energy are mantle anelasticity and a dynamic response of the oceans to wobble. Because of the frequency dependence of anelastic mechanisms, study of the oceanic response may be useful for constraining low-frequency anelasticity.

title: The Passive Influence of the Oceans upon the Rotation of the Earth
abstract: A general theory is developed which allows the exact numerical computation of the static equilibrium response of a non-rotating spherically symmetric Earth model covered by thin oceans with geometrically irregular coastlines to the action of an imposed static tidal or centrifugal potential. The theory is self-consistent, and takes into account the gravitational self-attraction of the oceans and the elastic-gravitational response of the Earth model to both the applied potential and the equilibrium oceanic tidal load on the surface. The results are used to determine the influence of an equilibrium pole tide on the free period and the associated rotational eigenfunction of the Chandler wobble. If the pole is globally well represented by this equilibrium approximation, its effect is to increase the Chandler wobble period by 27·6 days. It is shown that a fully self-consistent theory of the rotation of an Earth model with oceans predicts that changes in spin and wobble will be coupled, and that the Chandler wobble should, as a result, be accompanied by an associated periodic change in the length of day. The consequences of spin-wobble coupling are explored quantitatively, and found to be slight.

title: The effects of mantle anelasticity on nutations, earth tides, and tidal variations in rotation rate
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title: Zonal tides and changes in the length of day
abstract: The theory of changes in the length of day produced by the long period zonal tides is examined in an attempt to resolve the discrepancy between theoretical and observational results. It is argued here that as a result of the lack of coupling between core and mantle each conserves angular momentum separately with the result that only the changes in mantle moment of inertia influence the changes in rotation rate. 
 
 
 
The difference between theory and observation is reduced to 2 per cent, well within the error bars of observation. The zonal response coefficient is κ= 0.315.

title: Atlas of ocean tidal charts and maps, Part I: The semidiurnal principal lunar tide M2
abstract: In this paper the author presents the NSWC ocean tide model of the semidiurnal principal lunar (M2) tide in an atlas of ocean tidal charts and maps. The model is the computer result of a unique combination of mathematical and empirical techniques, which was introduced, extensively tested, and evaluated by Schwiderski (1978a, 1980a, b, 1983e). The computed M2 amplitudes and phases are tabulated along with all specially labeled empirical input data on a 1° × 1 ° grid system in 42° × 71° overlapping charts covering the whole oceanic globe. Corresponding global and arctic corange and cotidal maps are included to provide a quick overview of the major tidal phenomena. Significant qualitative and quantitative features are explained and discussed for proper application. In particular, the charted harmonic constants may be used to compute instantaneous M2 ocean tides with an accuracy of better than 5 cm any time and anywhere in the open oceans. Limitations of this accuracy in coastal waters and border sea...

title: An Analysis of Tidal Variations In the Length of Day
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