Earth tides

Earth tides

Cyclic motions of the Earth, sometimes over a foot or so in height, depending on latitude, caused by the same lunar and solar forces which produce tides in the sea. These forces also react on the Moon and Sun, and thus are significant in astronomy in evaluations of the dynamics of the three bodies. For example, the secular spin-down of the Earth due to lunar tidal torques is best computed from the observed acceleration of the Moon's orbital velocity. In oceanography, earth tides and ocean tides are very closely related. See also: Earth, gravity field of; Geodesy; Tide

By far the most widely used earth tide instruments are the tiltmeter and the gravimeter. Both instruments have the merits of portability, high potential precision, and low cost. Thus they are able to advance economically an important mission-the global mapping of earth tides and ocean tides. See also: Gravity meter; Seismographic instrumentation

 

Tide-producing potential

 

Tidal theory begins simply, by evaluating the tidal forces produced on a rigid, unyielding, oceanless, spherical Earth by the Moon and Sun. The Earth, Moon, and Sun can be regarded as perfectly spherical; the results of this idealized assumption will be modified later in this text to relate earth tide observations to deformable Earth models.

In the expansion of the total gravitational potential of the satellite as a sum of solid spherical harmonics, the tide-producing potential consists only of those terms which vary within the Earth and are the source of Earth deformations. The omitted terms are the constant which is arbitrarily chosen to produce null potential at the Earth's center and the first-degree term expressing the uniform acceleration of every part of the Earth toward the satellite (thus also entailing no distortion). Accordingly, the tide-producing potential U is expressible as the sum of solid spherical harmonics of degree 2 and higher in the notation which follows.

If the mass of the satellite is denoted by m, its distance from the Earth's center by R, the distance of the observing point from the Earth's center by r, and the geocentric zenith angle measured from the line of centers by θ, then for points r < R,

 

 

 (1) 

Eq. (1) holds, where G is Newton's universal gravitational constant (6.67 × 10-8 cm3 g-1 s-2).

On an idealized oblate Earth of equatorial radius re and flattening constant f, a point on the surface at geocentric latitude φ has radial distance r = reC(φ), where C(φ) ≡ 1 - f sin2 φ. The vertical (upward) component of tidal gravity ∂U/∂r and the horizontal component r-1∂U/∂θ in the azimuth direction away from the satellite, with P1 n(cos θ) =-∂PN ( θ)/∂θ, are given by Eqs. (2)

 

 

 (2) 

 

 

 

 (3) 

and (3). Here αm ≡ re/Rm and αS ≡ re/RS for the Moon and Sun, respectively.

 

Harmonic constituents

 

Equations (2) and (3) indicate the simple nature of the dependence of tidal gravity on the rigid Earth upon the distance R and the zenith angle θ of the satellite. But the time variations of R and θ are complex, because of the Earth's rotation and the complexity of the orbital motions. The objective in earth tide observations is assistance in determining present relevant properties of the Earth. Table 1 lists the periods and amplitude factors b of the larger tidal harmonic constituents over the broad range of tidal periods. In classifying tidal constituents mnemonically, G. H. Darwin used alphabets in terms of pairs with respect to the Moon (M) and the Sun (S), and A. T. Doodson used argument numbers, which are the coefficients of the six astronomical, independent variables arranged in order of increasing speed. The first three argument numbers are called the constituent number, of which the first two-digit number is the group number. The first digit of the group number designates species 0, 1, and 2 for long-period, diurnal, and semidiurnal, respectively. The purely geometric complexities essential in applied tidal theory are deemphasized here and, when feasible, the simple zenith-centered satellite coordinate system used in Eqs. (1)-(3) is adopted.

 

Tidal torques

 

The orbital acceleration n˙m of the Moon is the critical quantity in Eq. (4),

 

 

 (4) 

giving the lunar torque Nm, which slows the Earth's spin. (Here Me and Mm are masses of the Earth and Moon respectively, and rm is the distance to the Moon.) The value for Nm has been calculated to equal -3.9 × 1023 dyne-cm, to which corresponds the energy loss rate Ė, shown in Eq. (5),

 

 

 (5) 

and the relative spin-down, -ω˙/ω = N/cω = 0.21 eon-1 (1 eon = 109 years). Here ω is the Earth's rotational velocity, and c its principal moment of inertia.

There is an observation equation for the M2 ocean tidal constituent from which an estimate of the tidal acceleration of lunar longitude is made; the value n˙m = -27.4″ ± 3.0″ century-2 is obtained. That means Nm = -4.8 × 1023 dyne-cm, to which corresponds the energy loss rate E = 3.3 × 1019 ergs/s, and the relative spin-down value of 0.26 eon-1.

 

Tidal loss in the solid Earth

 

The gravest free mode of the Earth, period 0.9 h, has the same external form and nearly the same internal geometry as the M2 tide bulge. Its observed Q = 350 ± 100 can be used to estimate the loss rate ĖB in the bodily M2 tide (Q-1 is the loss rate number, and M2 is the principal semidiurnal tide). The result, ĖB = 7 × 1017 ergs/s, is only 3% of the required total in Eq. (5). The rate at which the Earth dissipates the total lunar-solar tidal energy has also been estimated to be 5.7 ± 0.5 × 1019 ergs/s, the share attributed to the oceans is 5.0 ± 0.3 × 1019 ergs/s.

 

Instrumental dimensionless amplitude factors

 

The reading of an earth tide meter is altered by the fact that it is anchored to a yielding Earth. The Earth chosen is assumed to be the radially symmetric standard.

 

Gravimeter

 

Basically, a gravimeter consists of a mass generally supported by a spring. (The superconducting model used a magnetic field.) Variations in gravity are measured by the extensionof the spring or in the null method by the small corrections required to restore the original configuration. This low-drift-rate superconducting gravimeter is designed to increase to several months the intervals between replenishing the helium supply. It is expected to give high precision in gravity measurements. A satellite affects the mean local value of gravity in three ways: by its direct attraction, by the tidal change in elevation of the observing station, and by the redistribution of mass in the deformed Earth.

 

Tiltmeters

 

The tiltmeter measures changes in the angle of tilt between the tiltmeter's foundation and the local vertical, both subject to tidal variations. Several types of tiltmeters are in use. In the horizontal pendulum the rotation axis is fixed at a small angle with the vertical to produce high sensitivity to tilts. In the level tube, which may be several hundred meters or more long, the difference in elevation of a fluid is measured at the two ends. In this way a sample of the tilt of a large region is obtained, but the horizontal pendulum sometimes also samples a large volume.

 

Extensometer

 

The extensometer, or linear strain meter, measures the change in distance between two reference points. In the Benioff design these positions are connected with a quartz tube to bring the points into juxtaposition for precise measurements of relative displacement. By using laser beams, the measurements may be made over distances of kilometers, but the short-baseline strain-mechanical meters still have their use.

 

Indicated and true phase lags

 

The phase indications of a gravimeter or tiltmeter are only a fraction of the true angular lag of the tidal bulge. The tidal bulge is assumed to retain its no-loss equilibrium form, but with the axis carried forward from the Moon's zenith by the small angle ε required to produce the Moon's known orbital acceleration.

 

 

Theories and characteristic numbers

 

Three basic numbers, Love numbers h and k and Shida number l, characterize the elastic behavior of an elastically yielding Earth. Thus h represents the ratio of the tidal height of the yielding Earth to that of the equilibrium tide, k the ratio of the additional potential due to the tidal deformation of the yielding Earth to the deforming potential, and l the ratio of horizontal displacement of theyielding Earth to that of the equilibrium tide.

 

Static theory

 

Static theory of earth tides (the frequency domain) is formulated for a spherical, nonrotating, and isotropic Earth. It is based on the reduction of the problem of elastic deformation of a sphere to a system of six first-order differential equations which are subjected to numerical integration in one formulation. When these equations are integrated with different Earth models by means of the Runge-Kutta method, the second-order Love numbers that are derived are virtually independent of either the presence of the solid inner core or the presence or absence of a low-velocity layer in the upper mantle (Table 2). Also, the continental or oceanic crustal model is only of second order with respect to the effects of the oceans.

 

Dynamic theory

 

Theoretically, a torque-free, nearly diurnal nutation or resonance is indeed possible by the presence of the liquid core inertially coupled to the mantle, and the predicted motion can give a resonance amplification to the nearly diurnal tides. However, theoretical Earth models, while complicated, are a great simplification of the actual Earth.

 

 

Time domain

 

Traditionally, theories of tides for the Earth, the oceans, or the atmosphere were always in the frequency domain. However, tides as observed are typically in the time domain, that is, in time series. Although development of theories of tides in the time domain is more complicated than that in the frequency domain, for practical purposes it would be more realistic to develop theories of tides in the time domain. The Earth, the oceans, and the atmosphere eventually must be treated as a single system, responding to the complete spectrum of the tide-generating forces, wind stresses, and interactions, as well as other periodic and nonperiodic dynamic forcings.

The time-domain earth tide is solved for a spherical, nonrotating, elastic, and isotropic Earth, and also for a rotating Earth. In their formulation, both the spheroidal and toroidal components are simultaneously introduced into the general form of the linearized lagrangian equations of motion, which govern small elasto-gravitational disturbances away from equilibrium of an arbitrary, uniformly rotating, self-gravitating, elastic Earth with an arbitrary initial state of stress field. The resulting equations of motion are those of a set of integro-partial differential equations, the solutions to which satisfy a set of boundary conditions in the Earth. There are strong coupling effects due to the rotation of the Earth.

 

Interaction of earth and ocean tides

 

Neither the problem of earth tides nor that of ocean tides can be solved independently. Characterization of ocean tides in the open oceans is uncertain. Calculations of ocean tides in the open oceans made directly from Laplace's tidal equations without consideration of the elastic yielding of the Earth due to the tidal and load deformation are inadequate. The interdependency of the problems of earth and ocean tides is a problem in geodynamics.

 

Tidal gravity and ocean tides

 

A transcontinental tidal gravity profile, consisting of nine stations across the United States along the 40th parallel, show the consistent results of the dominant effect of the ocean tides in the Pacific and Atlantic oceans (Figs. 1 and 2).

Ocean tides are the primary cause of the observed tidal gravity variations. These perturbations essentially depend only on the amplitude and phase of the ocean tide and the distance between the observation station and the ocean tidal load; land-based and island-based tidal gravity measurements can provide an independent set of observation data. If tidal gravity measurements are made accurately, it is more appropriate to consider the inverse problem of mapping ocean tides in the open oceans with extended earth tidal gravity measurements on the adjacent lands, supplemented by ocean-bottom stations and the ocean tidal information derived from nearshore and island stations.

Open ocean tides can be mapped by solving the inverse problem using land- and island-based tidal gravity measurements, coupled with shore and island ocean tidal measurements and a few ocean-bottom measurements.

 

 

 

Fig. 1  Effect of the ocean tides in the Atlantic and Pacific oceans on the earth tides for the semidiurnal tidal constituent M2. (a) Ocean tides. (b) Comparison of the observed and calculated values. (After J. T. Kuo et al., Transcontinental gravity profile across the United States, Science, 168:968-971, 1970)

 

 

بزرگنمایی تصویر

 

 

Fig. 2  Effect of the ocean tides in the Atlantic and Pacific oceans on the earth tides for the diurnal tidal constituent O1. (a) Ocean tides. (b) Comparison of the observed and calculated values. (After J. T. Kuo et al., Transcontinental gravity profile across the United States, Science, 168:968-971, 1970)

بزرگنمایی تصویر

 

 

 

Modified Laplace's tidal equations

 

Understanding of open ocean tides can be obtained both through numerical integration of Laplace's tidal equations and through direct measurements of tides in the deep oceans. However, the coamplitude and cophase tidal charts, principally of the tidal constituent M2 produced by means of the numerical integration of Laplace's tidal equations, apparently are quite sensitive to the boundaries of the ocean basins and the law of friction, and generally fail to give close agreement with the tidal observations on mid-oceanic islands. Thus, the original formulation of Laplace's tidal equations, which are based on the assumption of a rigid Earth, are inadequate to serve as a theoretical basis for calculating the ocean tides. The contribution of the tidal and loading deformation of the elastic Earth cannot be neglected in the formulation of the tidal equations.

 

Tidal tilt and tidal strain

 

Measurements of tidal tilt and tidal strain are more complicated than those of tidal gravity. Unlike tidal gravity measurements, which are virtually independent of inhomogeneities of the elastic properties in the Earth's crust or even in the upper mantle, tidal tilt and tidal strain measurements, in addition to the influence of ocean tidal loading, are very sensitive to the influences of the inhomogeneities, including the site of instrumental installation, topography, and geological structure.

It is well known that the underground openings in otherwise uniform rock mass strongly distort the elastic strain field. Tiltmeters installed in underground tunnels generally measure cross-tunnel strain-induced tilts, in addition to the regional tilt.

Moreover, the inhomogeneities in the Earth's crust, such as variable surface topography and geological discontinuities and the cavity in which the instruments are installed, are recognized as having significant influence in tidal tilt and tidal strain measurements. Tidal strain was observed from seven strain stations across the continental United States. A comparison of the tidal strain as predicted for a radially symmetrical, stratified, and homogeneous Earth model with corrections for the loading effects of the worldwide ocean tides (derived from the numerical integration of Laplace's tidal equations) indicated that the topographic and geological influences on tidal strain amount to as much as ±25% of the tidal strain for some of the strain stations, and the topographic effect on tidal strain exceeded ±50% of the tidal strain in some extreme cases.

 

Satellite and interferometry measurements

 

It is possible to estimate values for the amplitudes and phases of the M2, K1, and P1 ocean tidal constituents. With improved accuracy of satellite orbital determination, long time spans of accumulated data would permit an improvement in the accuracy of ocean tidal parameters in the near future.

Very long baseline interferometry (VLBI) has also been used to determine vector separations between radio telescopes and positions of radio sources. It has a precision of under 2 in. (5 cm) for baselines from 600 to 2400 mi (1000 to 4000 km) and less than about 4 in. (subdecimeter) for intercontinental lengths of about 3600 mi (6000 km).

 

Nearly diurnal nutation

 

The motion of the nearly diurnal nutation (or resonance) describes the free oscillation of the mantle's inertia axis about the axis of the Earth's rotation. The natural period of such a resonance of the Earth depends on the internal structures and elastic properties of the Earth and is not exactly equal to a sidereal day. However, calculations indicate that only the major K1 and the minor ψ1 tidal constituents with a period close to a sidereal day have amplitudes changed by the resonance effect. The theoretical results of Molodensky model II have been compared with the observed values of the tilt and gravimetric factors obtained by a number of workers at the nearly diurnal resonance of the solar waves ψ1 and K1 (Fig. 3).

 

 

Fig. 3  Observed amplitude spectrum of (a) diurnal and (b) semidiurnal constituents at South Pole. 1 μgal = 1 μm/s2. (After B. V. Jackson and L. B. Slichter, The residual daily earth tides at the South Pole, J. Geophys. Res., 79(11):1711-1715, 1974)

 

 

 

 

 

 

By stacking qravity measurements, the complex eigenfrequency with strength, period, and damping of the nearly diurnal free wobble are determined. The estimate of the real part of the strength amplitude is in close agreement with the first experimental determination of the inner-pressure gravimetric factor.

 

Effects of inertia, ellipticity, and anisotropy

 

In an elliptical and rotating Earth, there is a coupling between the spherical and toroidal displacement fields in the core. In 1981 the Dahlen-Smith formulation of free oscillation of a rotating, elliptical Earth was adopted to calculate the gravimetric factors for M2 and O1 as a function of latitude, which reach a maximum at the Equator and a minimum at the poles. A uniform means of instrumental calibration may bring the observed values of later workers close to that given by the former method.

One intriguing problem in earth tidal studies is the effect of anisotropy in the Earth's mantle on the earth tide and ocean-load tide. Equations of deformation of the Earth with anisotropic regions through lagrangian mechanics are derived. The solution indicates that earth tides are virtually blind, at least to the anisotropy of the upper mantle. However, load tides are affected by as much as 2.5%.

 

Earth tides and earthquakes

 

Volcanic earthquake swarms at Pavlof Volcano (southwestern Alaska Peninsula) correlated significantly with solid earth tidal stress rate for periods just before and just after explosive eruptions. In the 1974 minor eruption sequence, preeruptive earthquake swarms occurred during increasing earth tidal extension. Earth tides may have an effect on certain aspects of volcanic activity at Pavlof, and the polarity of the observed tidal correlation varies systematically during a volcanic eruption. See also: Earthquake

Louis B. Slichter

John T. Kuo

 

Bibliography

 

 

  • O. Francis, and P. Mazzega, What we can learn about ocean tides from tide gauge and gravity loading measurements, Proceedings of the 11th International Symposium on Earth Tides, 1991
  • J. Hinderer, W. Zurn, and H. Legros, Interpretation of the strength of the ''nearly diurnal free wobble'' resonance from stacked gravity tide observations, Proceedings of the 11th International Symposium on Earth Tides, 1991
  • J. T. Kuo, Y. C. Zhang, and Y. H. Chu, Time-domain total Earth tides, Proceedings of the 10th International Symposium on Earth Tides, 1985
  • G. Y. Li and H. T. Hsu, Tidal modeling theory with a lateral inhomogeneous, inelastic mantle, Proceedings of the 11th International Symposium on Earth Tides, 1991
  • S. D. Pagistakis, Effect of anisotropy in the Earth's mantle on body and ocean load tide, Proceedings of the 11th International Symposium on Earth Tides, 1991
  • Y. C. Zhang and J. T. Kuo, Time-domain earth tides for a rotating Earth, Proceedings of the 11th International Symposium on Earth Tides, 1991
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