جریانات کشندی زمین-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
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
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)
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)
Modified
Understanding of open ocean tides can be
obtained both through numerical integration of
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
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
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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
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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
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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
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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
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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
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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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