Introduction to Sea Level and Ocean Tides

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
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
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-kaminsk
i-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

• Inertial reference frame RI
• F maI
• Non-inertial Earths reference frame RT
• F Fcm - 2mW ? v - 2mW ? (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 Earths rotation
• - 2mW ? (W ? r ) macentrifugal
• If m at rest in RT 2mW ? v 0
• aE 0
• Then
• F Fcm Fcentrifugal Fcoriolis m aE

m
14
Tidal Force

• 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
  Earths 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
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 Earths 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
Doodsons 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
Doodsons 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 (Moons
perigee) N -N T 18.6 years (Regression of
the Moons 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 (Darwins notation)
Long period M0 S0 Sa Ssa MSM Mm MSF Mf 6
µGal MSTM MTM MSQM
Diurnal Q1 O1 35 µGal LK1 NO1 p1 P1 16
µGal S1 Km1 33 µGal KS1 15 µGal y1 f1 J1 OO1
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 moons 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, thats not the case.

24
The influence of the Earths rotationM2, S2

• Taking the Earths rotation into account
  (23h56m),
• And keeping the Moons orbital plane aligned with
  the Earths 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, thats not the case.

25
The influence of the Earths rotation, the motion
of the Moon and the SunMuch more waves !

• But
• The Moons orbital plane is not aligned with the
  earths equator,
• The Moons orbit is elliptic,
• The Earths rotational plane is not aligned with
  the ecliptic,
• The Earths orbit about the Sun is elliptic,
• Therefore we have to deal with much more waves!

26
Why diurnal ?
Would not exist if the Sun and the Moon were in
the Earths equatorial plane !
d
M1 M2
No diurnal if declination d 0
27
Spring Tide (from German Springen to Leap up)
Suns tidal ellipsoid
Moons 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
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. _at_ 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 Earths 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
L2
N2
M2 effect of the distance
Fine structure Or Zeeman effect
33
Perturbations due to the Moons perigee, the
node, the precession
Perigee Moons orbit rotating in 8.85 years
ecliptic
e
Node intercepts Moons 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.
• Earths axis of rotation is inclined 23.45 with
  respect to the plane of earths orbit about the
  sun. This defines the ecliptic, and the suns
  declination varies between d 23.45. with a
  period of one solar year.
• The orientation of earths 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.
• Earths 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.
• Moons orbit is also elliptical, but a
  description of moons orbit is much more
  complicated than a description of earths orbit.
  Here are the basics
• The moons 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 moons 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 moons orbit also varies.
• First, perigee rotates with a period of 8.85
  years.
• Second, the plane of moons orbit rotates around
  earths axis of rotation with a period of 18.613
  years. Both processes cause variations in Rmoon.

sdpw
35
Solid Earth tides (body tides) deformation of
the Earth
The earths 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) Earths 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 (Loves 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 Earths 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
Oceanic tides
Dynamic process (Coriolis...) Resonance effects
Ocean tides at 5 sites which have very different
tidal regimes Karumba diurnal Musayid
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 Earths 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 (1x1)

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 1x1
• 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 2.923e001 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 1.178e001
46
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
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/s2                        
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    
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
50
Measuring Earth tides
... Using a gravimeter (but also tiltmeters,
strainmeters, long period seismometers)
g
g
Spring gravimeter
Superconducting gravimeter (magnetic levitation)
51
GWR Superconducting gravimeter
52
GWR C021 Superconducting gravimeter at the
Membach station

• Advantages
• Stability, weak drift ( 4 µGal / year)
• Continuously recording
• Disadvantages
• Not mobile
• Relative
• Maintenance

53
Data from the GWR C021 Superconducting gravimeter
54
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