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Introduction to the Earth Tides

Introduction to the Earth Tides. Michel Van Camp Royal Observatory of Belgium In collaboration with: Olivier Francis (University of Luxembourg) Simon D.P. Williams (Proudman Oceanographic Laboratory).

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Introduction to the Earth Tides

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  1. Introduction to the Earth Tides Michel Van Camp Royal Observatory of Belgium In collaboration with: Olivier Francis (University of Luxembourg) Simon D.P. Williams (Proudman Oceanographic Laboratory)

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

  3. 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

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

  5. Tidal force = differential force Newtonian Force ~1/R² Tidal force ~ 1/R3 R “Spaghettification”

  6. 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. Roche Limit (« extreme tide ») http://www.answers.com/topic/roche-limit

  7. A victim of the Roche Limit Icy fragments of the Schoemaker-Levy comet ,1994

  8. NGC4676 (“The mice”) Tidal structure in interacting galaxies http://ifa.hawaii.edu/~barnes/saas-fee/mice.mpg

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

  10. CERN, Stanford Periodic deformations of the Stanford and CERN accelerators 4.2 km 3 km http://encyclopedia.laborlawtalk.com/wiki/images/8/8a/Stanford-linear-accelerator-usgs-ortho-kaminski-5900.jpg Stanford Linear Accelerator Center (SLAC): also Pacific ocean loading effect

  11. 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

  12. How does it come from? Imbalance between the centrifugal force due to the Keplerian revolution (same everywhere) and the gravitational force (1/R²)

  13. Tidal Force m Inertial reference frame RI : F = maI Non-inertial Earth’s reference frame RT: F+ Fcm- 2m[Wv] - 2m[W(Wr )] = m aE aE : acceleration in RT Fcm= -macm: acceleration of the c.m. of the Earth in RI : includes the Keplerian revolution W:Earth’s rotation - 2m[W(Wr )] = macentrifugal If m at rest in RT: 2m[W v] = 0 aE = 0 Then: F+ Fcm+ Fcentrifugal +Fcoriolis = m aE Becomes: F- macm+ macentrifugal = 0

  14. ? Tidal Force F- macm+ macentrifugal = 0 InRI: 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 m magt magMoon f = - mg : prevent from falling towards the centre of the Earth

  15. Center of mass of the system Earth-Moon Center of mass of the Earth Tides on the Earth • Tidal force = m(agMoon - acm)  More generally: Tidal force = m(ag_Astr - acm) •  Differential effect between : • The gravitational attraction from the Moon, function of the position on (in) the Earth and • The acceleration of the centre of mass of the Earth (centripetal) Identical everywhere on the Earth (Keplerian revolution) !!!

  16. 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)

  17. Gravitational and Centrifugal forces r d Tidal force = m(agMoon - acm)

  18. r Tidal potential centripetal force P  Tidal Force r attractive force q O M d ( = lunar zenith angle) The Potential at P on the Earth’s surface due to the Moon is [ The gravitational force on a particle of unit mass is given by -grad Wp] Using Tidal potential We have :WM (P) – (Wcentrifug. (P)+DWcentrifug.)

  19. Tidal potential • r/d = 1/60.3 (Earth-Moon) • r/d = 1/25000 (Earth-Sun) • Rapid convergence : W2 : 98% (Moon); 99% (Sun) Presently available potentials: l = 6 (Moon), l = 3 (Sun), l = 2 (Planets) Sun effect = 0.46 * Moon effect Venus effect = 0.000054 * Moon effect

  20. Doodson’s development of the tidal potential Laplace : development of cos(q) 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 :

  21. Doodson’s development of the tidal potential t: 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, Km1, Ks1]  Semi-Diurnal [M2, S2]  Ter-diurnal [M3]  quarter-diurnal [M4]

  22. Tidal waves (Darwin’s notation) Diurnal Q1 O1 35 µGal LK1 NO1 p1 P1 16 µGal S1 Km1 33 µGal KS1 15 µGal y1 f1 J1 OO1 Long period M0 S0 Sa Ssa MSM Mm MSF Mf 6 µGal MSTM MTM MSQM Semi-diurnal 2N2 m2 N2 n2 M2 36 µGal l2 T2 S2 17 µGal R2 Km2 Ks2 In red : largest amplitudes (at the Membach station)

  23. Resulting periodic deformation • 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…….

  24. The influence of the Earth’s rotation:M2, S2 • 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…….

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

  26. http://www.astro.oma.be/SEISMO/TSOFT/tsoft.html Why diurnal ? Would not exist if the Sun and the Moon were in the Earth’s equatorial plane ! d M1+ M2 No diurnal if declinationd= 0

  27. Spring Tide (from German Springen = to Leap up) Sun’s tidal ellipsoid Moon’s tidal ellipsoid New moon Earth Sun Full moon Total tidal ellipsoid Syzygy

  28. Neap Tide Moon 1st quarter Earth Sun Moon last quarter Lunar quadrature

  29. Beat period TSM M2 S2 Neap Tide and Spring Tide NB: you have to observe a signal for at least the beat period to be able to resolve the 2 contributing frequencies. mvc

  30. Equator – mi-latitude –pole Equator: no diurnal ½ diurnal maximum Mid-latitude: diurnal maximum Poles: long period only

  31. 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

  32. “Elliptic” waves or “Distance” effect Dd= 13 %  49% on the tidal force  Modulation of M2 gives N2 and L2 Modulation S of Ks1 gives S1 and y1 etc. d M2 effect of the distance N2 L2 M2* effect of the distance “Fine structure” Or “Zeeman effect”

  33. + Perturbations due to the Moon’s perigee, the node, the precession Perigee: Moon’s orbit rotating in 8.85 years ecliptic e Node: intercepts Moon’s orbital plane with the ecliptic, rotates in 18.6 years

  34. Tidal waves: summary • 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 called the 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. sdpw

  35. 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). • To calculate Dg 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 d≈ 1.16 = DgObserved / DgRigid Earth • = Direct attraction (1.0) + Earth’s deformation (0.6) - Mass redistribution inside the Earth (0.44). • The phase lag k = j (observed wave) - j (astronomic wave)

  36. Tidal parameter set The body deformation can be computed on the basis of an earth model determined from seismology (“Love’s numbers” : e.g. d = 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 d 0.01° for k On the other hand, tidal parameter sets can be obtained by performing a tidal analysis Remark: tidal deformation ~1.3 mm/µGal

  37. www.physical geography.net/fundamentals/8r.html Oceanic tides Dynamic 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

  38. Oceanic tides : amphidromic points M2

  39. Ocean loading The 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)

  40. Ocean loading 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 M2  error ~ 0.25 % on d and 0.15° (18 s) on k

  41. Correcting tidal effects Using a solid Earth model (e.g. Wahr-Dehant) ...and an ocean loading model

  42. 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°)

  43. Ground Track of altimetric satellite

  44. Recommended global ocean tides models  Schwiderski: working standard model for 10 years, based on tide gauges resolution 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 Texas the best coverage resolution 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)

  45. Ocean loading parameters (Membach – Schwiderski) Component Amplitude Phase sM2 : 1.7767e-008 57.491 sS2 : 5.7559e-009 2.923e+001 sK1 : 2.0613e-009 61.208 sO1 : 1.4128e-009 163.723 sN2 : 3.6181e-009 73.335 sP1 : 6.5538e-010 74.449 sK2 : 1.4458e-009 27.716 sQ1 : 3.8082e-010 -128.093 sMf : 1.4428e-009 4.551 sMm : 4.4868e-010 -5.753 sSsa : 1.0951e-010 1.178e+001

  46. No correction After correction of the solid Earth tide After correction of the solid Earth tide and the ocean loading effect Examples of tidal effects and corrections (Data from the absolute gravimeter at Membach)

  47. Correcting tidal effects using observed tides Advantage: 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 Observed tidal parameter set (Membach): Period (cpd) d k 0.000000 0.249951 1.16000 0.0000 MF 0.721500 0.906315 1.14660 -0.3219 Q1 0.9219141 0.940487 1.15028 0.0661 O1 0.958085 0.974188 1.15776 0.2951 M1 0.989049 0.998028 1.15100 0.2101 P1 0.999853 1.011099 1.13791 0.2467 K1 1.013689 1.044800 1.16053 0.1085 J1 1.064841 1.216397 1.15964 -0.0457 OO1 1.719381 1.872142 1.16050 3.6084 2N2 1.888387 1.906462 1.17730 3.1945 N2 1.923766 1.942754 1.18889 2.3678 M2 1.958233 1.976926 1.18465 1.0527 L2 1.991787 2.002885 1.19403 0.6691 S2 2.003032 2.182843 1.19451 0.9437 K2 2.753244 3.081254 1.06239 0.3105 M3 Ocean loading effect

  48. Tidal analysis (ETERNA, VAV): provides the “observed” tidal parameter set Idea: astronomical perturbation well known  fitting the different known waves on the observations  Allows us to resolve more waves than a spectral analysis

  49. NDFW W3 W4 Tidal analysis (ETERNA) Analysis performed on data from the absolute gravimeter at Membach 1995-1999   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    .4268      1.001824 1.003652 K1   487.579   1.13746   .00005    .2797    .0027      1.005328 1.005623 PSI1   4.242   1.26511   .00538   1.3458    .3082      1.007594 1.013690 PHI1     7.167   1.17411   .00290    .4751    .1663      1.028549 1.034467 TETA   5.272   1.15009   .00462    .2386    .2648      1.036291 1.039192 J1    27.849   1.16183   .00131    .1711    .07521.039323 1.039649 3MO1   2.994   1.10071   .01413    .2036    .8093      1.039795 1.071084 SO1    4.604   1.15789   .00587    .5912    .3364      1.072583 1.080945 OO1   15.154   1.15546   .00248    .0125    .1418      1.099161 1.216397 NU1    2.891   1.15149   .01258    .4449    .7208    … … 1.719380 1.823400 3N2     .971   1.12590   .01058   2.1258    .6060      1.825517 1.856953 EPS2   2.552   1.14145   .00444   3.4452    .25461.858777 1.859381 3MJ2   1.639   1.04673   .01183  -1.0228    .6780      1.859543 1.862429 2N2    8.809   1.14887   .00194   3.5877    .1110      1.863634 1.893554 MU2   10.763   1.16313   .00105   3.4913    .06021.894921 1.895688 3MK2   6.057   1.06175   .00315    .1165    .1805      1.895834 1.896748 N2    67.944   1.17253   .00025   3.1479    .0143      1.897954 1.906462 NU2   12.872   1.16949   .00087   3.2051    .0496      1.923765 1.942754 M2   359.543   1.18796   .00003   2.4554    .0018      1.958232 1.963709 LAMB   2.648   1.18656   .00418   2.3112    .2396      1.965827 1.968566 L2    10.205   1.19297   .00252   1.8996    .1445  1.968727 1.969169 3MO2   5.641   1.07195   .00678   -.0414    .3883      1.969184 1.976926 KNO2   2.535   1.18504   .01508   1.7954    .8639      1.991786 1.998288 T2     9.842   1.19562   .00118    .4525    .0679      1.999705 2.000767 S2   167.979   1.19293   .00007    .7631    .0041      2.002590 2.003033 R2     1.383   1.17356   .00668    .1530    .3828      2.004709 2.013690 K2    45.704   1.19399   .00033   1.0285    .0191      2.031287 2.047391 ETA2   2.548   1.19032   .00691    .8083    .3956      2.067579 2.073659 2S2     .408   1.14823   .04493  -2.9513   2.5747      2.075940 2.182844 2K2     .670   1.19573   .03444   -.7586   1.97312.753243 2.869714 MN3    1.097   1.05723   .00344    .3227    .1973      2.892640 2.903887 M3     4.005   1.05924   .00094    .4698    .0537      2.927107 2.940325 ML3     .234   1.09415   .01448   -.0586    .8297      2.965989 3.081254 MK3     .524   1.06465   .01050   1.0296    .6015  3.791963 3.833113 N4      .016    .99379   .12679 -86.7406   7.2653      3.864400 3.901458 M4      .017    .39703   .04408  51.5191   2.5255

  50. Measuring Earth tides ... Using a gravimeter (but also tiltmeters, strainmeters, long period seismometers) g g Superconducting gravimeter (magnetic levitation) Spring gravimeter

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