Transcript Introduction to Sea Level and Ocean 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)
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
Newtonian Force ~1/R²
Tidal force ~ 1/R3
“Spaghettification”
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
A victim of the Roche Limit
Icy fragments of the Schoemaker-Levy comet ,1994
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
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
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
How does it come from?
Imbalance between the centrifugal force due to the Keplerian revolution
(same everywhere) and the gravitational force ( 1/R²)
Tidal Force
Inertial reference frame RI :
m
F = maI
Non-inertial Earth’s reference frame RT :
F + Fcm - 2m[W  v ] - 2m[W  (W  r ) ] = m aE
aE
Fcm= -macm
: acceleration in RT
: 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
Tidal Force
?
F - macm + macentrifugal = 0
m
In RI :
magMoon magt
F = m agt + m agMoon + f
= m agt + m agMoon - mg
f = - mg :
prevent from falling towards
the centre of the Earth
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
Tides on 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) !!!
Center of mass of the
system Earth-Moon
Center of mass of the Earth
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
r
d
Tidal force = m (agMoon - acm)
 FgMoon  Fcentrifugal
GM m GM m
 2 
r
d2
r
Tidal potential
centripetal force
P q
r
O
Tidal Force
q
attractive force
M
d
(q = lunar zenith angle)
The Potential at P on the Earth’s surface due to the Moon is
WM ( P )  
GmM
r
[ The gravitational force on a particle of unit mass is given by -grad Wp ]
GmM
Using r 2  r 2  d 2  2rd cosq : WM ( P)  
r
GmM
We have : WM (P) – (Wcentrifug. (P)+DWcentrifug.)
r

l

r
  Pl (cosq )

l 0  d 
l
Tidal
r
  Pl (cosq ) potential

l 2  d 
Tidal potential
Wtid 
GmM
r

l
r
  Pl (cosq )

l 2  d 
r/d = 1/60.3 (Earth-Moon)
r/d = 1/25000 (Earth-Sun)
 Rapid convergence :
Wtid  W2  W3
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
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 :
GmM
W
r


l
r
  Pl (cos q )
l 2  d 

 A(r , , ) sina   a s  a h  a
1
2
3

4p

 a5 N ' a6 p s t
Doodson’s development of the tidal potential
GmM
W
r


l
r
  Pl (cos q )
l 2  d 

 A(r , , ) sina   a s  a h  a
1
2
3

4p

 a5 N ' a6 p s 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]
Tidal waves (Darwin’s notation)
Long period
M0
S0
Sa
Ssa
MSM
Mm
MSF
Mf
6 µGal
MSTM
MTM
MSQM
Diurnal
Q1
O1
LK1
NO1
p1
P1
S1
Km1
KS1
y1
f1
J1
OO1
35 µGal
16 µGal
33 µGal
15 µGal
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)
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…….
The influence of the Earth’s rotation:
M 2, S 2
• 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,
the motion of the Moon and the Sun
Much 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!
Why diurnal ?
Would not exist if the Sun
and the Moon were in the
Earth’s equatorial plane !
d
M1 + M2
No diurnal if declination d = 0
http://www.astro.oma.be/SEISMO/TSOFT/tsoft.html
Spring Tide (from German Springen = to Leap up)
Sun’s tidal ellipsoid
Moon’s tidal ellipsoid
Earth
New moon
Sun
Full moon
Total tidal ellipsoid
Syzygy
Neap Tide
Moon 1st
quarter
Earth
Moon last
quarter
Lunar quadrature
Sun
Neap Tide and Spring Tide
Beat period TSM
1
1  1
1


TSM 2  TS 2 TM 2




M2
S2
NB: you have to observe a signal for at least
the beat period to be able to resolve the 2
contributing frequencies.
mvc
Equator – mi-latitude – pole
Equator: no diurnal
½ diurnal maximum
Mid-latitude: diurnal maximum
Poles: long period only
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
Dd = 13 %  49% on the tidal force
 Modulation of M2 gives N2 and L2
d
 Modulation S of Ks1 gives S1 and y1
etc.
M2
effect of the distance
M2* effect of the distance
N2
L2
“Fine structure”
Or “Zeeman effect”
+ Perturbations due to the Moon’s perigee, the node,
the precession
Perigee: Moon’s orbit rotating in 8.85 years
e
Node: intercepts Moon’s orbital plane with the ecliptic,
rotates in 18.6 years
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
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 =  (observed wave) -  (astronomic wave)
Earth’s transfer function
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
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
Oceanic tides : amphidromic points
M2
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)
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
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 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)
Ocean loading parameters
(Membach – Schwiderski)
Component
Amplitude
sM2 :
1.7767e-008
sS2 :
5.7559e-009
sK1 :
2.0613e-009
sO1 :
1.4128e-009
sN2 :
3.6181e-009
sP1 :
6.5538e-010
sK2 :
1.4458e-009
sQ1 :
3.8082e-010
sMf :
1.4428e-009
sMm :
4.4868e-010
sSsa :
1.0951e-010
Phase
57.491
2.923e+001
61.208
163.723
73.335
74.449
27.716
-128.093
4.551
-5.753
1.178e+001
Examples of tidal effects and corrections
(Data from the absolute gravimeter at Membach)
No correction
After correction of the solid Earth tide
After correction of the solid Earth tide
and the ocean loading effect
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)
Ocean loading effect
0.000000
0.721500
0.9219141
0.958085
0.989049
0.999853
1.013689
1.064841
1.719381
1.888387
1.923766
1.958233
1.991787
2.003032
2.753244
0.249951
0.906315
0.940487
0.974188
0.998028
1.011099
1.044800
1.216397
1.872142
1.906462
1.942754
1.976926
2.002885
2.182843
3.081254
d
1.16000
1.14660
1.15028
1.15776
1.15100
1.13791
1.16053
1.15964
1.16050
1.17730
1.18889
1.18465
1.19403
1.19451
1.06239
k
0.0000
-0.3219
0.0661
0.2951
0.2101
0.2467
0.1085
-0.0457
3.6084
3.1945
2.3678
1.0527
0.6691
0.9437
0.3105
MF
Q1
O1
M1
P1
K1
J1
OO1
2N2
N2
M2
L2
S2
K2
M3
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
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 NDFW
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
W3
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
W4
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
Measuring Earth tides
... Using a gravimeter (but also tiltmeters, strainmeters, long
period seismometers)
g
g
Spring gravimeter
Superconducting gravimeter
(magnetic levitation)
GWR Superconducting gravimeter
GWR C021 Superconducting gravimeter at the
Membach station
Advantages :
 Stability, weak drift (~ 4 µGal / year)
 Continuously recording
Disadvantages :
 Not mobile
 Relative
 Maintenance
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