Introduction to the Earth Tides

Michel Van CampRoyal Observatory of Belgium

In collaboration with:Olivier Francis (University of Luxembourg)

Simon D.P. Williams (Proudman Oceanographic Laboratory)

Tides – Getijden – Gezeiten – Marées… from old English and German « division of time »and (?) from Greek « to divide »

Tides – Getijden – Gezeiten – Marées

Observing ET has not brought a lot on our knowledge of the Earth interior(e.g. polar motion better constrained by satellites or VLBI…)

But tides affect lot of geodetic measurements (gravity, GPS, Sea level, …)Present sub-cm or µGal accuracy would not be possible without a good knowledge of the Tides

Amazing Tides in the Fundy Bay (Nova Scotia) : 17.5 m

Tidal force = differential force

“Spaghettification”

Newtonian Force ~1/R²Tidal force ~ 1/R3 R

Roche Limit (« extreme tide »)

Within the Roche limit the mass' own gravity can no longer withstand the tidal forces, and the body disintegrates.

The varying orbital speed of the material eventually causes it to form a ring.

http://www.answers.com/topic/roche-limit

Icy fragments of the Icy fragments of the SchoemakerSchoemaker--Levy comet ,1994Levy comet ,1994

A victim of the Roche Limit

Tidal structure in interacting galaxies

NGC4676 (“The mice”)

http://ifa.hawaii.edu/~barnes/saas-fee/mice.mpg

Io volcanic activity :due to the tidal forces of Jupiter, Ganymede and Europa

CERN, Stanford

Stanford Linear Accelerator Center (SLAC): also Pacific ocean loading effect

3 km

http://encyclopedia.laborlawtalk.com/wiki/images/8/8a/Stanford-linear-accelerator-usgs-ortho-kaminski-5900.jpg

Periodic deformations of theStanford and CERN accelerators 4.2 km

Tides on the Earth:

• Periodic movements which are directly related in amplitude and phase to some periodic geophysical force

• The dominant geophysical forcing function is the variation of the gravitational field on the surface of the earth, caused by regular movements of the moon-earth and earth-sun systems.

- Earth tides- Ocean tide loading- Atmospheric tides

In episodic surveys (GPS, gravity), these deformations can be aliased into the longer period deformations being investigated

Imbalance between the centrifugal force due to the Keplerian revolution (same everywhere) and the gravitational force ( 1/R²)

How does it come from?

Inertial reference frame RI :

F = maI

Non-inertial Earth’s reference frame RT :

F + Fcm - 2m[Ω = v ] - 2m[Ω = (Ω= r ) ] = m aE

aE : acceleration in RTFcm= -macm : acceleration of the c.m. of the Earth in RI :

includes the Keplerian revolutionΩ : Earth’s rotation- 2m[Ω = (Ω= r ) ] = macentrifugal

If m at rest in RT : 2m[Ω = v ] = 0aE = 0

Then: F + Fcm + Fcentrifugal + Fcoriolis = m aE

Becomes:F - macm + macentrifugal = 0

Tidal Force

m

F - macm + macentrifugal = 0

In RI :

F = m agt + m agMoon + f= m agt + m agMoon - mg

So:

m agt + m agMoon - mg - macm + macentrifugal = 0

mg = m agt + m (agMoon - acm) + macentrifugal

Tidal force = m (agMoon - acm) [= 0 at the Earth’s c.m.]

Gravity g = Gravitational + Tidal + Centrifugal !!!! Centrifugal: contains Earth rotation only

magtmagMoon

f = - mg : prevent from falling towards the centre of the Earth

m

Tidal Force ?

Tides on the Earth

Center of mass of the system Earth-Moon

Center of mass of the Earth

Tidal force = m (agMoon - acm) More generally:Tidal force = m (ag_Astr - acm)

Differential effect between :

(1) The gravitational attraction from the Moon, function of the position on (in) the Earth and

(2) The acceleration of the centre of mass of the Earth (centripetal) Identical everywhere on the Earth (Keplerian revolution) !!!

Tide and gravity

Gravity g = Gravitational + Tidal + Centrifugal

Tidal effect: 981 000 000 µGal

Usually, in gravimetry :Gravity g = Gravitational + Centrifugal

Centrifugal: 978 Gal (equator) 983 Gal (pole)

Gravitational and Centrifugal forces

Tidal force = m (agMoon - acm)

22 dGM

rGM

FF

mm

lcentrifugagMoon

−=

+=

r

d

O

P

Md

rTidal Force

centripetal force

attractive force

ρ

θ

θ

(θ = lunar zenith angle)

The Potential at P on the Earth’s surface due to the Moon is ρM

MGmPW −=)(

:cos2222 θρ rddr −+=

[ The gravitational force on a particle of unit mass is given by -grad Wp ]

Using ∑∞

=⎟⎠⎞

⎜⎝⎛−=

0

)(cos)(l

l

lM

M Pdr

rGmPW θ

Tidal potential

We have : WM (P) – (Wcentrifug. (P)+ΔWcentrifug.) ∑∞

=⎟⎠⎞

⎜⎝⎛=

2

)(cosl

l

lM P

dr

rGm

θ Tidal potential

Tidal potential

∑∞

=⎟⎠⎞

⎜⎝⎛=

2

)(cosl

l

lM

tid Pdr

rGmW θ

r/d = 1/60.3 (Earth-Moon) r/d = 1/25000 (Earth-Sun)

Rapid convergence : 32 WWWtid +≅

W2 : 98% (Moon); 99% (Sun)Presently available potentials: l = 6 (Moon), l = 3 (Sun), l = 2 (Planets)

Sun effect = 0.46 * Moon effectVenus effect = 0.000054 * Moon effect

Doodson’s development of the tidal potential

( ){ }∑

∑+++++=

⎟⎠⎞

⎜⎝⎛=

∞

=

tpaNapahasaarA

Pdr

rGmW

s

ll

lM

&&&&&& 654321

2

'sin),,(

)(cos

τξϕ

θ

Laplace : development of cos(θ) as a function of the latitude, declination and right ascension

Very complicated time variations due to the complexity of the orbital motions (but diurnal, semi-diurnal and long period tides appear clearly)

Doodson : Harmonic development of the potential as a sum of purely sinusoidal waves, i.e. waves having as argument purely linear functions of the time :

Doodson’s development of the tidal potential

( ){ }∑

∑+++++=

⎟⎠⎞

⎜⎝⎛=

∞

=

tpaNapahasaarA

Pdr

rGmW

s

ll

lM

&&&&&& 654321

2

'sin),,(

)(cos

τξϕ

θ

τ : T ~ 24.8 hours (mean lunar day)s : T ~ 27.3 days (mean Lunar longitude)h : T ~ 365.2 days (tropical year)p : T ~ 8.8 years (Moon’s perigee)N’= -N : T ~ 18.6 years (Regression of the Moon’s node)p : T ~ 20942 years (perihelion)

Today: more than 1200 terms….(e.g. : Tamura 87: 1200, Hartmann-Wenzel 95: 12935)Among them:

Long period (fortnightly [Mf], semi-annual [Ssa], annual [Sa],….) Diurnal [O1, P1, Km

1, Ks1]

Semi-Diurnal [M2, S2]Ter-diurnal [M3]quarter-diurnal [M4]

Tidal waves (Darwin’s notation)

Long periodM0S0SaSsaMSMMmMSFMf 6 µGalMSTMMTMMSQM

DiurnalQ1O1 35 µGalLK1NO1π1P1 16 µGalS1Km

1 33 µGalKS

1 15 µGalψ1φ1J1OO1

Semi-diurnal2N2μ2N2ν2M2 36 µGalλ2T2S2 17 µGalR2Km

2Ks

2

In red : largest amplitudes (at the Membach station)

If:

• The moon’s orbit was exactly circular,

• There was no rotation of the Earth,

then we might only have to deal with Mf (13.7 days)

[and similarly SSa for the Sun (182.6 days)]

But, that’s not the case…….

Resulting periodic deformation

• Taking the Earth’s rotation into account (23h56m),

• And keeping the Moon’s orbital plane aligned with the Earth’s equator,

Then we might only have to deal with M2 (12h25m): relative motion of the Moon as seen from the Earth

[and similarly S2 (12h00m)].

But, that’s not the case…….

The influence of the Earth’s rotation:M2, S2

But

• The Moon’s orbital plane is not aligned with the earth’s equator,

• The Moon’s orbit is elliptic,

• The Earth’s rotational plane is not aligned with the ecliptic,

• The Earth’s orbit about the Sun is elliptic,

Therefore we have to deal with much more waves!

The influence of the Earth’s rotation, the motion of the Moon and the SunMuch more waves !

Why diurnal ?

M1+ M2

δ

Would not exist if the Sun and the Moon were in the Earth’s equatorial plane !

No diurnal if declination δ = 0http://www.astro.oma.be/SEISMO/TSOFT/tsoft.html

EarthSun

Total tidal ellipsoid

Sun’s tidal ellipsoid Moon’s tidal ellipsoid

New moon

Full moon

Spring Tide (from German Springen = to Leap up)

Syzygy

EarthSun

Moon 1st quarter

Moon last quarter

Neap Tide

Lunar quadrature

mvc

NB: you have to observe a signal for at least the beat period to be able to resolve the 2 contributing frequencies.

Beat period TSM⎟⎟⎠

⎞⎜⎜⎝

⎛−=

22

11211

MSSM TTT

M2

S2

Neap Tide and Spring Tide

Equator: no diurnal½ diurnal maximum

Poles: long period only

Mid-latitude: diurnal maximum

Equator – mi-latitude – pole

Other properties… • Semi-diurnal: slows down the Earth rotation. Consequences: the Moon moves away. @ 475 000 km: length of the day ~2 weeks, the Moon and the Earth would present the same face.Slowing down the rotation is a typical tidal effect...even for galaxies!

• Diurnal: the torques producing nutations are those exerted by the diurnal tidal forces. This torque tends to tilt the equatorial plane towards the ecliptic

• Long period: Affect principal moment of inertia C : periodic variations of the length of the day. Its constant part causes the permanent tide and a slight increase of the Earth’s flattening

“Elliptic” waves or “Distance” effect

Δd = 13 % 49% on the tidal force

Modulation of M2 gives N2 and L2

Modulation S of Ks1 gives S1 and ψ1

etc.

d

M2* effect of the distance

effect of the distance

L2N2

M2

“Fine structure”Or “Zeeman effect”

e

+ Perturbations due to the Moon’s perigee, the node, the precession

Node: intercepts Moon’s orbital plane with the ecliptic, rotates in 18.6 years

ecliptic

Perigee: Moon’s orbit rotating in 8.85 years

sdpw

• The period of the solar hour angle is a solar day of 24 hr 0 m.• The period of the lunar hour angle is a lunar day of 24 hr 50.47 m.• Earth’s axis of rotation is inclined 23.45° with respect to the plane of earth’s orbit about the sun. This defines the ecliptic, and the sun’s declination varies between d = ± 23.45°. with a period of one solar year.• The orientation of earth’s rotation axis precesses with respect to the stars with a period of 26 000 years.• The rotation of the ecliptic plane causes d and the vernal equinox to change slowly, and the movement calledthe precession of the equinoxes.• Earth’s orbit about the sun is elliptical, with the sun in one focus. That point in the orbit where the distance between the sun and earth is a minimum is called perigee. The orientation of the ellipse in the ecliptic plane changes slowly with time, causing perigee to rotate with a period of 20 900 years. Therefore Rsun varies with this period.• Moon’s orbit is also elliptical, but a description of moon’s orbit is much more complicated than a description of earth’s orbit. Here are the basics:

• The moon’s orbit lies in a plane inclined at a mean angle of 5.15° relative to the plane of the ecliptic. And lunar declination varies between d = 23.45 ± 5.15° with a period of one tropical month of 27.32 solar days.• The actual inclination of moon’s orbit varies between 4.97°, and 5.32°• The eccentricity of the orbit has a mean value of 0.0549, and it varies between 0.044 and 0.067.• The shape of moon’s orbit also varies.

First, perigee rotates with a period of 8.85 years. Second, the plane of moon’s orbit rotates around earth’s axis of rotation with a period of 18.613 years. Both processes cause variations in Rmoon.

Tidal waves: summary

To calculate Δg induced by Earth tides:

we need a tidal potential, which takes into account the relative position of the Earth, the Moon, the Sun and the planets.

But also a tidal parameter set, which contains:

• The gravimetric factor δ ≈ 1.16 = ΔgObserved / ΔgRigid Earth

= Direct attraction (1.0) + Earth’s deformation (0.6) -Mass redistribution inside the Earth (0.44).

• The phase lag κ = ϕ (observed wave) - ϕ (astronomic wave)

Earth’s transfer function

Solid Earth tides (body tides): deformation of the Earth

The earth’s body tides is the periodic deformation of the earth due to the tidal forces caused by the moon and the sun (Amplitude range 40 cm typically at low latitude).

The body deformation can be computed on the basis of an earth model determined from seismology (“Love’s numbers” : e.g. δ = 1 + h2 - 3/2k2 ~ 1.16).

The gravity body tide can be computed to an accuracy of about 0.1 µGal.

The remaining uncertainty is caused by the effects of the lateral heterogeneities in the earth structure and inelasticity at tidal periods.

Present Earth’s model: 0.1% for δ0.01° for κ

On the other hand, tidal parameter sets can be obtained by performing a tidal analysis

Remark: tidal deformation ~1.3 mm/µGal

Tidal parameter set

Oceanic tidesDynamic process (Coriolis...)Resonance effects

Ocean tides at 5 sites which have very different tidal regimes:

Karumba : diurnal

Musay’id : mixed

Kilindini : semidiurnal

Bermuda : semidiurnal

Courtown : shallow sea distortion

www.physical geography.net/fundamentals/8r.html

Oceanic tides : amphidromic points

M2

Ocean loadingThe ocean loading deformation has a range of more than 10 cm for the vertical displacement in some parts of the world.

2 cm (Brussels)20 cm (Cornwall)

To model the ocean loading deformation at a particular site we need models describing:

1. the ocean tides (main source of error)2. the rheology of the Earth’s interior

Error estimated at about 10-20%

In Membach, loading ~ 1.7 µGal 5 % on M2error ~ 0.25 % on δ and 0.15° (18 s) on κ

Ocean loading

Correcting tidal effects

Using a solid Earth model (e.g. Wahr-Dehant)

...and an ocean loading model

Correcting tidal effects: Ocean tide models

Numerical hydrodynamic models are required to compute the tides in the ocean and in the marginal seas.

The accuracy of the present-day models is mainly determined by - the grid and bathymetry resolution - the approximations used to model the energy dissipation

Data from TOPEX/Poseidon altimetry satellite:- improved the maps of the main tidal harmonics in deep oceans- provide useful constraints in numerical models of shallow waters

Problem for coastal sites (within 100 km of the coasts) due to the resolution of the ocean tide model (1°x1°)

Ground Track of altimetric satellite

Recommended global ocean tides models

Schwiderski: working standard model for 10 years, based on tide gaugesresolution of 1°x1°includes long period tides Mm, Mf, Ssa

± 15 ocean tides models thanks to TOPEX/Poseidon mission

No model is systematically the best for all region amongst the best models:

- CSR3.0 from the University of Texasthe best coverageresolution of 0.5° x 0.5°

- FES95.2 from Grenoble representative of a family of four similar models (includes the Weddell and Ross seas)

(recommended by T/P and Jason Science Working Team)

Ocean loading parameters

(Membach – Schwiderski)Component Amplitude PhasesM2 : 1.7767e-008 57.491sS2 : 5.7559e-009 2.923e+001 sK1 : 2.0613e-009 61.208sO1 : 1.4128e-009 163.723sN2 : 3.6181e-009 73.335sP1 : 6.5538e-010 74.449sK2 : 1.4458e-009 27.716sQ1 : 3.8082e-010 -128.093sMf : 1.4428e-009 4.551sMm : 4.4868e-010 -5.753sSsa : 1.0951e-010 1.178e+001

Examples of tidal effects and corrections(Data from the absolute gravimeter at Membach)

After correction of the solid Earth tide and the ocean loading effect

No correctionAfter correction of the solid Earth tide

Correcting tidal effects using observed tidesAdvantage: take into account all the local effects e.g. ocean loading

Very useful in coastal stations

Disadvantage: a gravimeter must record continuously for 1 month at least

0.000000 0.249951 1.16000 0.0000 MF0.721500 0.906315 1.14660 -0.3219 Q10.9219141 0.940487 1.15028 0.0661 O10.958085 0.974188 1.15776 0.2951 M10.989049 0.998028 1.15100 0.2101 P10.999853 1.011099 1.13791 0.2467 K11.013689 1.044800 1.16053 0.1085 J11.064841 1.216397 1.15964 -0.0457 OO11.719381 1.872142 1.16050 3.6084 2N21.888387 1.906462 1.17730 3.1945 N21.923766 1.942754 1.18889 2.3678 M21.958233 1.976926 1.18465 1.0527 L21.991787 2.002885 1.19403 0.6691 S22.003032 2.182843 1.19451 0.9437 K22.753244 3.081254 1.06239 0.3105 M3

Period (cpd) δ κ

Ocean loading effect

Observed tidal parameter set (Membach):

Tidal analysis (ETERNA, VAV):

provides the “observed” tidal parameter set

Idea: astronomical perturbation well known

fitting the different known waves on the observationsAllows us to resolve more waves than a spectral analysis

…1.719380 1.823400 3N2 .971 1.12590 .01058 2.1258 .60601.825517 1.856953 EPS2 2.552 1.14145 .00444 3.4452 .25461.858777 1.859381 3MJ2 1.639 1.04673 .01183 -1.0228 .67801.859543 1.862429 2N2 8.809 1.14887 .00194 3.5877 .11101.863634 1.893554 MU2 10.763 1.16313 .00105 3.4913 .06021.894921 1.895688 3MK2 6.057 1.06175 .00315 .1165 .18051.895834 1.896748 N2 67.944 1.17253 .00025 3.1479 .01431.897954 1.906462 NU2 12.872 1.16949 .00087 3.2051 .04961.923765 1.942754 M2 359.543 1.18796 .00003 2.4554 .00181.958232 1.963709 LAMB 2.648 1.18656 .00418 2.3112 .23961.965827 1.968566 L2 10.205 1.19297 .00252 1.8996 .14451.968727 1.969169 3MO2 5.641 1.07195 .00678 -.0414 .38831.969184 1.976926 KNO2 2.535 1.18504 .01508 1.7954 .86391.991786 1.998288 T2 9.842 1.19562 .00118 .4525 .06791.999705 2.000767 S2 167.979 1.19293 .00007 .7631 .00412.002590 2.003033 R2 1.383 1.17356 .00668 .1530 .38282.004709 2.013690 K2 45.704 1.19399 .00033 1.0285 .01912.031287 2.047391 ETA2 2.548 1.19032 .00691 .8083 .39562.067579 2.073659 2S2 .408 1.14823 .04493 -2.9513 2.57472.075940 2.182844 2K2 .670 1.19573 .03444 -.7586 1.97312.753243 2.869714 MN3 1.097 1.05723 .00344 .3227 .19732.892640 2.903887 M3 4.005 1.05924 .00094 .4698 .05372.927107 2.940325 ML3 .234 1.09415 .01448 -.0586 .82972.965989 3.081254 MK3 .524 1.06465 .01050 1.0296 .60153.791963 3.833113 N4 .016 .99379 .12679 -86.7406 7.26533.864400 3.901458 M4 .017 .39703 .04408 51.5191 2.5255

Tidal analysis (ETERNA)

adjusted tidal parameters :from to wave ampl. ampl.fac. stdv. ph. lead stdv.[cpd] [cpd] [nm/s**2 ] [deg] [deg].721499 .833113 SIGM 2.650 1.17718 .00988 -.9692 .5661.851182 .859691 2Q1 8.914 1.15445 .00302 -.6510 .1732.860896 .892331 SIGM 10.704 1.14852 .00247 -.5826 .1414.892640 .892950 3MK1 2.632 1.10521 .01542 1.5440 .8834.893096 .896130 Q1 66.963 1.14748 .00057 -.2157 .0325.897806 .906315 RO1 12.706 1.14631 .00202 .0741 .1156.921941 .930449 O1 350.360 1.14950 .00007 .1097 .0041.931964 .940488 TAU1 4.609 1.15939 .00362 .0623 .2073.958085 .965843 LK1 10.002 1.16063 .00568 -.0778 .3258.965989 .966284 M1 8.042 1.07920 .00661 .5365 .3784.966299 .966756 NO1 27.691 1.15522 .00213 .2379 .1222.968565 .974189 CHI1 5.245 1.14413 .00473 .5885 .2712.989048 .995144 PI1 9.543 1.15067 .00214 .2124 .1226.996967 .998029 P1 163.108 1.15011 .00012 .2552 .0072.999852 1.000148 S1 4.021 1.19925 .00744 4.0483 .42681.001824 1.003652 K1 487.579 1.13746 .00005 .2797 .00271.005328 1.005623 PSI1 4.242 1.26511 .00538 1.3458 .30821.007594 1.013690 PHI1 7.167 1.17411 .00290 .4751 .16631.028549 1.034467 TETA 5.272 1.15009 .00462 .2386 .26481.036291 1.039192 J1 27.849 1.16183 .00131 .1711 .07521.039323 1.039649 3MO1 2.994 1.10071 .01413 .2036 .80931.039795 1.071084 SO1 4.604 1.15789 .00587 .5912 .33641.072583 1.080945 OO1 15.154 1.15546 .00248 .0125 .14181.099161 1.216397 NU1 2.891 1.15149 .01258 .4449 .7208

…W4

NDFW

W3

Analysis performed on data from the absolute gravimeter at Membach 1995-1999

g

g

Measuring Earth tides

... Using a gravimeter (but also tiltmeters, strainmeters, long period seismometers)

Spring gravimeter Superconducting gravimeter (magnetic levitation)

GWR Superconducting gravimeter

Advantages :Stability, weak drift (~ 4 µGal / year)Continuously recording

Disadvantages : Not mobileRelativeMaintenance

GWR C021 Superconducting gravimeter at the Membach station

Data from the GWR C021 Superconducting gravimeter

Conclusions

Tidal effects can be corrected at the µGal level (and better) if:

- One uses a good potential (e.g. Tamura 1987)- One uses observed tidal parameter set (esp. along the coast)

Or a tidal parameter set from a solid Earth model AND ocean loading parameters