Transcript 12. tides

TIDES
Tide - generic term to define alternating rise and fall in sea level with respect to
land and is produced by the balance between the gravitational acceleration (of the
moon and sun mainly) and the centrifugal acceleration.
Tide also occurs in large lakes, in the atmosphere, and within the solid crust
Gravitational Force (Newton’s Law of Gravitation):
F = GmM/R2
G = 6.67×10-11 N m2/kg2
Centrifugal Force
EQUILIBRIUM TIDE
F = GmM/R2
Moon’s Gravitational Force (changes from one side of the earth to the other)
Tide Generating Force (Difference between centrifugal and gravitational)
Center of mass of Earth-Moon system ~1,700 km from Earth’s surface
(because Earth is 81 times heavier than Moon)
How strong is the Tide-Generating Force?
Gravitational Force at A:
B
S
A
 mM 
FA  G 
2
 P  S  
P
Centrifugal Force at A:
m M 
Fc  G  2 
 P 
Imbalance (Tide-generating force at A):
FA  Fc  G
FA  Fc  Gm M
mM
mM

G
P2
P  S 2
2PS  S 2
P 4  2P 3S  P 2S 2
Since P  60 S
Tide-generating Force at A:
FA  Fc  G m M
2S
P3
Tide-generating Force at B: FB  Fc   G m M
2S
P3
How strong is the Tide-Generating Force?
FA  Fc  G m M
2S
P3
Tide-generating Force at B: FB  Fc  G m M
2S
P3
Tide-generating Force at A:
B
S
A
P
The mass of the sun is 2x1027 metric tons while that of the moon is only
7.3x1019 metric tons.
The sun is 390 times farther away from the earth than is the moon.
The relative Tide Generating Force on Earth = [(2x1027/7.3x1019)]/(3903)
or = 2.7x107/5.9x107 = 0.46 or 46%
EQUILIBRIUM TIDE
What alters the range and phase of tides produced by Equilibrium Theory?
Non-astronomical factors:
coastline configuration
bathymetry
atmospheric forcing (wind velocity and barometric pressure)
hydrography
may alter speed, produce resonance effects and seiching, storm surges
In the open ocean, tidally induced variations of sea level are a few cm.
When the tidal wave moves to the continental shelf and into confining channels,
the variations may become greater.
Keep in mind that tidal waves travel as shallow (long) waves
How so?
Typical wavelengths = 4500 km (semidiurnal wave traveling over 1000 m of water)
Ratio of depth / wavelength = 1 / 4500
Then, their phase speed is: C = [ gH ]0.5
The tide observed at any location is the superposition of several constituents
that arise from diverse tidal forcing mechanisms.
Main constituents: Principal Lunar Semidiurnal
M2
12.42 h
Principal Solar Semidiurnal
S2
12.00 h
Larger Lunar Elliptic Semidiurnal
N2
12.66 h
Lunisolar diurnal
Lunar Diurnal
K1
O1
23.93 h
25.82 h
  AM 2 sin( M 2  M 2 )  AS2 sin( S 2  S 2 )  AN 2 sin( N 2  N 2 )  ...
The Form factor F = [ K1 + O1 ] / [ M2 + S2 ] is customarily used to characterize the
tide.
When 0.25 < F < 1.25 the tide is mixed - mainly semidiurnal
When 1.25 < F < 3.00 the tide is mixed - mainly diurnal
F > 3 the tide is diurnal
F < 0.25 the tide is semidiurnal
When 0.25 < F < 1.25 the tide is mixed - mainly semidiurnal
When 1.25 < F < 3.00 the tide is mixed - mainly diurnal
F > 3 the tide is diurnal
F < 0.25 the tide is semidiurnal
Superposition of constituents generates
modulation - e.g. fortnightly, monthly
This applies for both sea level and velocity
Subtidal modulation by two tidal constituents
In Ponce de León Inlet: M2 = 0.41 m; N2 = 0.09 m; O1: 0.06 m; S2: 0.06 m; K1= 0.08 m
GNV
F = [K1 + 01] / [S2 + M2 ] = 0.30
Panama
City
In Panama City, FL: M2 = 0.085 m; N2 = 0.017 m;
O1: 0.442 m; S2: 0.035 m; K1= 0.461 m
F = [K1 + 01] / [S2 + M2 ] = 7.52
From Pinet (1998)
Co-oscillation
Independent tide - caused by gravitational and centrifugal forces directly on
the waters of a basin -- usually negligible effect for typical dimensions of
semienclosed basins
Co-oscillating tide - caused by the ocean tide at the entrance to a basin as
driving force
The wave propagates into the basin and may be subject to RESONANCE and
RECTIFICATION -- alters tidal flows and produces subtidal motions
Resonance of Tidal Wave
At the mouth x = L,   a0 sint
L
Substituting
  a cos x sint
into   a0 sint
a cosL sint  a0 sint
a
 
at x = L
a0
cos L
a0 cos x
sint
cos L
For resonance to exist, the denominator should tend to zero, i.e.,
L 
2

2

L
L

2

2
 2n
2n  1

2
and
L
The natural period of oscillation is then: TN 

4
2n  1 ;   CT
4L  1 
C  2n  1
TN 

4L  1 
C  2n  1

u
a0 cos x
sint
cos L
a C sinx
u 0
cos t
H cos L
L
 4
For an estuary with length < λ /4, u is zero at the head and maximum at the mouth
For longer estuaries u is zero at x = 0, λ / 2, 3 λ / 2,… or where sin κx = 0
and maximum at x = λ /4, 3 λ / 4, 5 λ / 4, …, i.e., where sin κx is max
H (m)
L (km)
C (m/s) TN (h)
Long Island Sound
20
180
14
14
Chesapeake Bay
10
250
10
28
Bay of Fundy
70
250
26
10.7
Merion’s Formula TN 
4L  1 
C  2n  1
Mode 1
(n =1)
Effects of Rotation on a Progressive Tidal Wave in a Semi-enclosed basin
u

 g
t
x

fu  g
y
u
1 

x
H t
Solution:
  ae  y R cosx  t 
U a
C y
e
H
R
cosx  t 
R=C/f
KELVIN WAVE
Effects of Rotation on a Standing Tidal Wave in a Semi-enclosed Basin
Two Kelvin waves of equal amplitude progressing in opposite directions.
  ae  y R cosx  t   ae y R cosx  t 
U a

C y
e
H
R
cosx  t   e y
R

cosx  t 
v 0
Instead of having lines of no motion, we are now reduced to a central region -- amphidromic region-- of no motion at the origin.
The interference of two geostrophically controlled simple harmonic waves
produces a change from a linear standing wave to a rotary wave.
Two Kelvin Waves in Opposite Directions
km
Two Kelvin Waves in Opposite Directions
Effects of Bottom Friction
on an amphydromic
system
(Parker, 1990)
Virtual Amphidromes
(Parker, 1990)
Virtual amphidromes
in Chesapeake Bay