Earth tide

Earth tide is the sub-meter motion of the Earth of about 12 hours or longer caused by Moon and Sun gravitation, also called body tide which is the largest contribution globally. The largest body tide contribution is from the semidiurnal constituents, but there are also significant diurnal constituents. There also semi-annual and fortnightly contributions due to the axial tilt. The use of the word tide is by analogy, and although the forcing is quite similar, the responses are quite different.

Tidal forcing

A. Lunar tidal forcing: this depicts the Moon directly over 30° N (or 30° S) viewed from above the Northern Hemisphere.

B. This view shows same forcing from 180° from view A. Viewed from above the Northern Hemisphere. Red up, blue down.

The larger of the periodic gravitational forcings is from the Moon but that of the Sun is also important. The images here show lunar tidal forcing when the Moon appears directly over 30° N (or 30° S). This pattern remains fixed with the red area directed toward (or directly away from) the Moon. Red indicates upward pull, blue downward. If, for example the Moon is directly over 90° W (or 90° E), the center of the red areas are centered on the western northern hemisphere, on upper right. Red up, blue down. If for example the Moon is directly over 90° W (90° E), the center of the red area is 30° N, 90° W and 30° S, 90° E, and the center of the bluish band follows the great circle equidistant from those points. At 30° latitude a strong peak occurs once per lunar day, giving significant diurnal forcing at that latitude. Along the equator two equally sized peaks (and depressions) are equally sized, giving semi-diurnal forcing there.

Body tide

Vertical displacements of sectorial movement. Red up, blue down.	File:Sectorialeastwest.jpg East-west displacements of sectorial movement. Red east, blue west.	File:Sectorialnorthsouth.jpg North-south displacements of sectorial movement. Red north, blue south.
File:Tesseralvert.jpg Vertical displacements of tessearal movement. Red up, blue down.	File:Tesseraleastwest.jpg East-West displacements of tessearal movement. Red east, blue west.	File:Tesseralnorthsouth.jpg North-South displacements of tessearal movement. Red north, blue south.

File:Zonalvert.jpg

Vertical displacements of zonal movement. Red up, blue down.

The Earth tide encompasses the entire body of the Earth and is unhindered by the thin crust and land masses of the surface, on scales that make the rigidity of rock irrelevant. Ocean tides are a consequence of the resonance of the same driving forces with water movement periods in ocean basins accumulated over many days, so that their amplitude and timing are quite different and vary over short distances of just a few hundred kilometres. The oscillation periods of the earth as a whole are not near the astronomical periods, so its flexing is due to the forces of the moment.

The tide components with a period near twelve hours have a lunar amplitude (earth bulge/depression distances) that are a little more than twice the height of the solar amplitudes, as tabulated below. At new and full moon, the Sun and the Moon are aligned, and the lunar and the solar tidal maxima and minima (bulges and depressions) add together for the greatest tidal range at particular latitudes. At first- and third-quarter phases of the moon, lunar and solar tides are in opposition, and the tidal range is at a minimum. The semi-diurnal tides go through one full cycle (a high and low tide) about once every 12 hours and one full cycle of maximum height (a spring and neap tide) about once every 14 days.

The classical theory of Earth tides first became established in 1905, primarily to explain nutations, but are also used in Earth rotation predictions. The semi-diurnal tide (one maximum every 12 or so hours) is primarily lunar (only S₂ is purely solar) and gives rise to sectorial deformations which rise and fall at the same time along the same longitude. Sectorial variations of vertical and east-west displacements are maximum at the equator and vanish at the poles. There are two cycles along each latitude, the bulges opposite one another, and the depressions similarly opposed. The diurnal tide is lunisolar, and gives rise to tesseral deformations. The vertical and east-west movement is maximum at 45° latitude and is zero on the equator and at the poles. Tesseral variation have one cycle per latitude, one bulge and one depression; the bulges are opposed (antipodal), that is to say the western part of the northern hemisphere and the eastern part of the southern hemisphere, for example, and similarly the depressions are opposed, the eastern part of the northern hemisphere and the western part of the southern hemisphere, in this case. Finally, fortnightly and semi-annual tides have 'zonal' deformations (constant along a circle of latitude), as the Moon or Sun gravitation is directed alternately away from the northern and southern hemispheres due to tilt. There is zero vertical displacement at 35°16' latitude.

Since these displacements affect the vertical direction east-west and north-south variations are often tabulated in milliarc seconds for astronomical use. The vertical displacement is frequent tabulated in μgal, since the gradient of gravity is location dependent so that the distance conversion is only approximately 3 μgal per cm

Other Earth tide contributors

In coastal areas because the ocean tide is quite out of step with the earth tide, at high ocean tide there is an excess (or at low tide a deficit) of water about what would be the gravitational equilibrium level and the adjacent ground falls (or rises) in response to the resulting differences in weight. Displacements caused by ocean tidal loading can exceed the displacements due to the earth body tide. Sensitive instruments far inland often have to make similar corrections. Atmospheric loading and storm events may also be measurable, though the masses in movement are less weighty.

Tidal constituents

Principal body tide constituents. The amplitudes may vary from those listed within several per cent.

Semi-diurnal
Tidal constituent	Period	Vertical amplitude (mm)	Horizontal amplitude (mm)
M2	12.421 hr	384.83	53.84
S2 (solar semi-diurnal)	12.000 hr	179.05	25.05
N2	12.658 hr	73.69	10.31
K2	11.967 hr	48.72	6.82
Diurnal
Tidal constituent	Period	Vertical amplitude (mm)	Horizontal amplitude (mm)
K1	23.934 hr	191.78	32.01
O1	25.819 hr	158.11	22.05
P1	24.066 hr	70.88	10.36
φ1	23.804 hr	3.44	0.43
ψ1	23.869 hr	2.72	0.21
S1 (solar diurnal)	24.000 hr	1.65	0.25
Long term
Tidal constituent	Period	Vertical amplitude (mm)	Horizontal amplitude (mm)
Mf	13.661 days	40.36	5.59
Mm (moon monthly)	27.555 days	21.33	2.96
Ssa (solar semi-annual)	0.50000 yr	18.79	2.60
Lunar node	18.613 yr	16.92	2.34
Sa (solar annual)	1.0000 yr	2.97	0.41

Earth tide effects

Volcanologists use the regular, predictable Earth tide movements to calibrate and test sensitive volcano deformation monitoring instruments. The tides may also trigger volcanic events. Seismologists have determined that microseismic events are correlated to tidal variations in Central Asia (north of the Himalayas). The semidiurnal amplitude of terrestrial tides can reach about 55 cm at the equator which is important in GPS calibration and VLBI measurements. Also to make precise astronomical angular measurements requires knowledge of the Earth's rate of rotation and nutation, both of which are influenced by earth tides. Terrestrial tides also need to be taken in account in the case of some particle physics experiments. For instance, at the CERN or SLAC, the very large particle accelerators were designed while taking terrestrial tides into account for proper operation. Among the effects that need to be taken into account are circumference deformation for circular accelerators and particle beam energy.

Since tidal forces generate currents of conducting fluids within the interior of the Earth, they affect in turn the Earth's magnetic field itself.

References

• Paul Melchior, Earth Tides, Pergamon Press, Oxford, 1983.

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