2. 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)

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

4. 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

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

6. Tidal force = differential force “Spaghettification” Newtonian Force ~1/R² Tidal force ~ 1/R 3 R

7. 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

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

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

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

11. 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 the Stanford and CERN accelerators 4.2 km

12. 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

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

14. Inertial reference frame R I : F = ma I Non-inertial Earth’s reference frame R T : F + F cm - 2m[   v ] - 2m[   (   r ) ] = m a E a E : acceleration in R T F cm = -ma cm : acceleration of the c.m. of the Earth in R I : includes the Keplerian revolution  : Earth’s rotation - 2m[   (   r ) ] = ma centrifugal If m at rest in R T : 2m[   v ] = 0 a E = 0 Then: F + F cm + F centrifugal + F coriolis = m a E Becomes: F - ma cm + ma centrifugal = 0 Tidal Force m

15. F - ma cm + ma centrifugal = 0 In R I : F = m a gt + m a gMoon + f = m a gt + m a gMoon - mg So: m a gt + m a gMoon - mg - ma cm + ma centrifugal = 0  mg = m a gt + m (a gMoon - a cm ) + ma centrifugal  Tidal force = m (a gMoon - a cm ) [= 0 at the Earth’s c.m.]  Gravity g = Gravitational + Tidal + Centrifugal !!!! Centrifugal: contains Earth rotation only ma gt ma gMoon f = - mg : prevent from falling towards the centre of the Earth m Tidal Force ?

16. Tides on the Earth Center of mass of the system Earth-Moon Center of mass of the Earth Tidal force = m (a gMoon - a cm )  More generally:Tidal force = m (a g_Astr - a cm )  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) !!!

17. 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)

18. Gravitational and Centrifugal forces Tidal force = m (a gMoon - a cm ) r d

19. O P M d r Tidal Force centripetal force attractive force    (  = 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 W p ] Using Tidal potential We have : W M (P) – (W centrifug. (P)+  W centrifug. ) Tidal potential

20. r/d = 1/60.3 (Earth-Moon) r/d = 1/25000 (Earth-Sun)  Rapid convergence : W 2 : 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

21. Doodson’s development of the tidal potential Laplace : development of cos(  ) 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 :

22. Doodson’s development of the tidal potential  : 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 [M f ], semi-annual [S sa ], annual [S a ],….)  Diurnal [O 1, P 1, K m 1, K s 1 ]  Semi-Diurnal [M 2, S 2 ]  Ter-diurnal [M 3 ]  quarter-diurnal [M 4 ]

23. Tidal waves (Darwin’s notation) Long period M 0 S 0 S a S sa M SM M m M SF M f 6 µGal M STM M TM M SQM Diurnal Q 1 O 1 35 µGal LK 1 NO 1  1 P 1 16 µGal S 1 K m 1 33 µGal K S 1 15 µGal  1  1 J 1 OO 1 Semi-diurnal 2N 2  2 N 2 2 M 2 36 µGal 2 T 2 S 2 17 µGal R 2 K m 2 K s 2 In red : largest amplitudes (at the Membach station)

24. If: The moon’s orbit was exactly circular, There was no rotation of the Earth, then we might only have to deal with M f (13.7 days) [and similarly S Sa for the Sun (182.6 days)] But, that’s not the case……. Resulting periodic deformation

25. 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 M 2 (12h25m): relative motion of the Moon as seen from the Earth [and similarly S 2 (12h00m)]. But, that’s not the case……. The influence of the Earth’s rotation: M 2, S 2

26. 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 Sun Much more waves !

27. Why diurnal ? M 1 + M 2  Would not exist if the Sun and the Moon were in the Earth’s equatorial plane ! No diurnal if declination  = 0 http://www.astro.oma.be/SEISMO/TSOFT/tsoft.html

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

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

30. 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 T SM M2M2 S2S2 Neap Tide and Spring Tide

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

32. 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

33. “Elliptic” waves or “Distance” effect  d = 13 %  49% on the tidal force  Modulation of M 2 gives N 2 and L 2  Modulation S of K s 1 gives S 1 and  1 etc. d M 2 * effect of the distance effect of the distance L2L2 N2N2 M2M2 “Fine structure” Or “Zeeman effect”

34. 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

35. 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 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 R moon. Tidal waves: summary

36. To calculate  g 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  ≈ 1.16 =  g Observed /  g Rigid Earth = Direct attraction (1.0) + Earth’s deformation (0.6) - Mass redistribution inside the Earth (0.44). The phase lag  =  (observed wave) -  (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).

37. The body deformation can be computed on the basis of an earth model determined from seismology (“Love’s numbers” : e.g.  = 1 + h 2 - 3/2k 2 ~ 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  0.01° for  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

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

39. Oceanic tides : amphidromic points M2M2

40. 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)

41. 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 M 2  error ~ 0.25 % on  and 0.15° (18 s) on  Ocean loading

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

43. 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°)

44. Ground Track of altimetric satellite

45. 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)

46. 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

47. 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 correction After correction of the solid Earth tide

48. 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 0.0000000.249951 1.16000 0.0000MF 0.7215000.906315 1.14660 -0.3219Q1 0.92191410.940487 1.15028 0.0661O1 0.9580850.974188 1.15776 0.2951M1 0.9890490.998028 1.15100 0.2101P1 0.9998531.011099 1.13791 0.2467K1 1.0136891.044800 1.16053 0.1085J1 1.0648411.216397 1.15964 -0.0457OO1 1.7193811.872142 1.16050 3.60842N2 1.8883871.906462 1.17730 3.1945N2 1.9237661.942754 1.18889 2.3678M2 1.9582331.976926 1.18465 1.0527L2 1.9917872.002885 1.19403 0.6691S2 2.0030322.182843 1.19451 0.9437K2 2.7532443.081254 1.06239 0.3105M3 Ocean loading effect Observed tidal parameter set (Membach): Period (cpd)  

49. 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

50. … 1.719380 1.823400 3N2.971 1.12590.01058 2.1258.6060 1.825517 1.856953 EPS2 2.552 1.14145.00444 3.4452.2546 1.858777 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 3.4913.0602 1.894921 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 1.9731 2.753243 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 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 4.0483.4268 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 1.16183.00131.1711.0752 1.039323 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 … W4W4 NDFW W3W3 Analysis performed on data from the absolute gravimeter at Membach 1995-1999

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

52. GWR Superconducting gravimeter

53. Advantages : Stability, weak drift (~ 4 µGal / year) Continuously recording Disadvantages : Not mobile Relative Maintenance GWR C021 Superconducting gravimeter at the Membach station

54. Data from the GWR C021 Superconducting gravimeter

55. 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