Tides

• The tide generating force and the equilibrium tide.

• The origin of tidal constituents.

• Tidal wave amplification across a shelf edge.

• Tides in bays and semi-enclosed seas (co-oscillation).

• Harmonic analysis.

Useful books:

Pugh, D.. Tides, Surges, and Mean Sea Level (Short loan)

Bowden, K.. Physical Oceanography of Coastal Waters. (Short loan).

Tides (1)

Tides (3)

Tides (2)

Space and time scales of processes which affect sea-level.The maximum space and time scales are the dimensions and the age of the earth

One of the most important dynamical processes in shelf seas is the tide.

Ancient civilizations were aware that the rise and fall of sea level was connected somehow to the celestial motions. The earliest evidence of this knowledge probably dates back to 1500 BC. Although quite advanced scientifically, the Greeks and Romans wrote very little about the tide, presumably because the Mediterranean basin responds only slight to tidal forcing.

The English Monk Bede, was aware of tidal behaviour around the Northumbrian coast in 730AD. The earliest tide table for London Bridge dates from 1250AD.

Further steps towards a unified understanding of tides were made by European scientists in the 16th and 17th centuries. The fundamental physical law is the Law of Universal Gravitation which states that the force of gravitational attraction of two

bodies of masses M1 and M2 separated by a distance D is given by:

221

d

MGMF

(Newton, 1687 - the original copy of Newton’s Principia is held in the Trinity College Library at Cambridge)

G is the universal gravitational constant= 6.67 x 10-11 m2/kg2

The Tide Generating Force (TGF).

Remember:

1. Newton’s law of gravitation:

2. The gravitational force exerted by a body on the outside world can be treated as if all of its mass were at its centre of gravity.

221

d

MGMF

C

P

MC of G

d

a

Earth-moon system rotates about common system centre of gravity (like an asymmetric dumbbell)

r

Earth centre of gravity Moon centre of gravity

• All points on the surface of the earth (and inside the earth) rotate about the earth-moon centre of mass with the same radius. Check this movie to convince yourself.

• The centripetal (or centrifugal) force required to keep all points rotating is:

[N kg-1]

• Only at the centre of the earth (C) does the moon’s attractive force exactly balance the required centrifugal force (think what would happen if it did not). Elsewhere there is an imbalance because the distance to the moon changes. i.e. on the previous diagram the distance PM = r, and r < d, which means the moons attractive force at point P is greater than the centrifugal force:

[N kg-1]

2.d

GMF m

cent

2.r

GMF m

attrac

C

P

M

d

aTGF

centrifugal force

attractive force

r

Calculation of the magnitude of the TGF:

Resolve into horizontal and vertical components at P

sin)sin(22 d

GM

r

GMF mm

h

cos)cos(22 d

GM

r

GMF mm

v

( = lunar zenith angle)

Apply cosine rule to triangle CPM: cosadadr 2222

da )cosd

a(dr 2122

Also note: andd

sinasin

1cos

…..to eventually get:

22

33

singd

a

M

MF

e

mh

13 23

cosgd

a

Me

MmFv

2a

GMg e

Knowing that d=60.26a, and Me=81.53Mm we can say:

8106 xg

F

g

F hv

Fv is small compared to gravity, so we ignore it.

Fh is small, but acts perpendicular to gravity and is therefore important…this is the TGF.

i.e. the tidal variation in sea level is not caused by direct attraction under the moon, but the dragging of water from 45 away from the sub-lunar point to pile up under the position of the moon.

The Equilibrium Tide. (After Newton, 1687).

Assumes: the ocean has a uniform depth over the globe, and there is a constant equilibrium between the TGF and the sea surface slope.

d

ds infinitesimal area

a

Fh = Fh() so we expect = ()

adads .sin.2

Slope force = TGF TGFP

a

11

Assume hydrostatic pressure:

agP

a11

2sin23 3

gda

MM

ag

e

m

K

)(coscos32sin23

dKdKa

CKconst

Ka

22 cos23

.cos2

3

Relate to mean sea level by integrating over the entire surface to find C :

surface

ds 0

d.sina.ad.sina.ds 222

dsina.CcosKdsa

22

0

22

3

0

22

2

32 cosdCcosKa

0

32

2

12

cosCcosKa

2

10122 2 CCKa

Therefore:

132

1 23

cosd

a

M

M

a e

m

This describes an ellipsoid of revolution.

HW occurs for = 0, =35 cm

LW occurs for = /2 =-18 cm

Thus, predicted range of equilibrium tide 53 cm.

This is similar to the observed range in the open ocean, but is considerably smaller than the range observed in many coastal seas.

The equilibrium tide analysis can be applied to the earth-sun system in exactly the same way: i.e. the sun also produces a tidal ellipsoid.

Given:

Mass of earth Me = 6 x 1024 kg; Mass of moon Mm = 7 x 1022 kg;

Mass of Sun Msun= 2 x 1030 kg;

Earth - moon distance = 4 x 108 m; Earth - Sun distance = 1.5 x 1011 m.

Calculate the strength of the solar tide relative to that caused of the moon.

Tidal Constituents (1) - the spring-neap cycle.

Imagine the earth rotating about its axis beneath either the lunar or solar tidal ellipsoid. You would see two high tides (HW) and two low tides (LW) per day. These ellipses generate the main two tidal constituents:

M2 : principal lunar semidiurnal period = 12.42 hours

S2 : principal solarsemidiurnal period = 12.00 hours

Now imagine what happens when you have both lunar and solar tides interacting.

EarthSun

Total tidal ellipsoid

Sun’s tidal ellipsoid Moon’s tidal ellipsoid

New moon

Full moon

Spring Tide

EarthSun

Moon 1st quarter

Moon last quarter

Neap Tide

0 5 10 15 20 25T ime (days)

-2

-1

0

1

2

Se

a le

vel

Beat period TSN

22

111

MSSN TTT

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

Investigate the effects of the spring-neap cycle using the tsp6 software (set the port to Liverpool).

Tidal constituents (2) - the other 388 constituents.

If:

• the moon’s orbit was exactly circular,

• the moon’s orbital plane was aligned with the earth’s rotational plane,

• the earth’s rotational plane was aligned with the earth’s orbital plane about the sun,

• the earth’s orbit about the sun was exactly circular,

then we might only have to deal with M2 and S2.

But, that’s not the case…….

For instance, consider the fact that the moon’s orbital plane is at an angle of about 18.5 to the earth’s rotational plane:

To moon

Daily rotation

P

X

At point P you would only see 1 HW per day, or above or below P you might see unequal HWs during the day (the diurnal inequality) the addition of diurnal tidal constituents.

Also, point X precesses around the earth with a period of 18.6 years longer period constituents.

Consider also that the moon’s orbit is an ellipse, not a circle, so there is a difference between successive spring HW.

This manifests as another semi-diurnal constituent (N2)

Earth

Moon

The full expansion of the equilibrium tide was started by Laplace, with further work by Kelvin, Darwin, and eventually completed by Doodson.

This resulted in an equilibrium tide containing 390 tidal constituents.

Symbol Name Period (hrs) Strength (M2=1.0)

M2

S2

N2

K2

K1

O1

P1

Mf

Mm

Principal lunar

Principal solar

Larger lunar elliptic

Luni-solar declinational

Luni-solar declinational

Larger lunar declinational

Larger solar declinational

Lunar fortnightly

Lunar monthly

12.42

12.00

12.66

11.97

23.93

25.82

24.07

13.65 days

27.55 days

1.0000

0.4652

0.1915

0.0402

0.1852

0.4151

0.1932

0.1723

0.0909

Investigate the effects of different tidal constituent using the tsp6 software.

Tidal constituents (3) - shallow water constituents (see Pugh page 110-112).

There is a group of tidal constituents, not predicted by equilibrium theory, that we need to take into account. These are the shallow water constituents, caused by the interaction of the tidal wave with a shoaling seabed.

Remember: ghc

hcresthtrough

htroughhcrest

Deep water, hcrest hmean htrough, so c = constant over wave height.

Shallow water, hcrest > hmean > htrough, so c is greater at crest than at trough. Crest catches up with trough, and wave steepens.

hmeanhmean

Steepen a sinusoidal wave add higher harmonics of the basic wave period.

The M2 tide generates M4 (period 6.21 hours), M6 (period 4.14 hours), M8 (period 3.11 hours)…..etc.

Example: Southampton Water.

Ultimately the tidal wave can reach a situation where the crest overtakes the trough and the wave breaks to form a tidal bore.

Qiantang River, China.

Investigate the effects of shallow water constituents

using the tsp6 software (set the port to Southampton).

Summary of the equilibrium theory:

1. A simple qualitative explanation of the main features of the tide.

2. Defines the fundamental frequencies.

But:

1. It is a gross simplification - based on a simplified equation of motion, without rotation, friction, or acceleration (i.e. misses most of the dynamics).

2. Very simple geometry (no continents, no shelves, constant depth).

The big strength is the determination of the tidal constituents.

Laplace: for any mechanical system, regardless of damping, the output frequencies are the same as the input frequencies.

This is critical to the prediction of tides through harmonic analysis.