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Ocean Tide Modelling, Part 1
M.S. Bos
Centro Interdisciplinar de Investigacao Marinha e Ambiental (CIIMAR),
University of Porto, Portugal
Introduction to ocean tides – p. 1/57
Who’s your teacher today?
Ph.D. work performed at Proudman Oceanographic
Laboratory, Liverpool, United Kingdom under the
supervision of Prof. Trevor Baker.
Introduction to ocean tides – p. 2/57
Who’s your teacher today?
Ph.D. work performed at Proudman Oceanographic
Laboratory, Liverpool, United Kingdom under the
supervision of Prof. Trevor Baker.
Ph.D. subject: Ocean tide loading.
Introduction to ocean tides – p. 2/57
Who’s your teacher today?
Ph.D. work performed at Proudman Oceanographic
Laboratory, Liverpool, United Kingdom under the
supervision of Prof. Trevor Baker.
Ph.D. subject: Ocean tide loading.
I am more a geodesist than an oceanographer.
Introduction to ocean tides – p. 2/57
Overview of today’s lecture
Short historical overview of tidal research
Introduction to ocean tides – p. 3/57
Overview of today’s lecture
Short historical overview of tidal research
Derivation of the tidal potential
Introduction to ocean tides – p. 3/57
Overview of today’s lecture
Short historical overview of tidal research
Derivation of the tidal potential
Derivation of the Laplace Tidal Equations
Introduction to ocean tides – p. 3/57
Moon & Sun cause ocean tides
Introduction to ocean tides – p. 4/57
Tide gauge at Gibraltar
-0.2
0
0.2
0.4
0.6
0.8
1
12 13 14 15 16 17 18 19 20 21
tide
ga
ug
e (
m)
January 2009
observed
Introduction to ocean tides – p. 5/57
Isaac Newton (1687): Law of Gravity
Introduction to ocean tides – p. 6/57
The gravitational tidal force
Earth
R
r
dMoon
ψ
P
aP = GMr2
Introduction to ocean tides – p. 7/57
The gravitational tidal force
Earth
R
r
dMoon
ψ
P
aP = GMr2
aP = −∇GMr
Introduction to ocean tides – p. 7/57
The gravitational tidal force
Earth
R
r
dMoon
ψ
P
aP = GMr2
aP = −∇GMr
aP = −∇(
GM√d2+R2
−2dR cosψ
)
Introduction to ocean tides – p. 7/57
The gravitational tidal force
Earth
R
r
dMoon
ψ
P
aP = GMr2
aP = −∇GMr
aP = −∇(
GM√d2+R2
−2dR cosψ
)
aP = −GMd∇
∞∑
n=0
(
Rd
)nPn(cosψ)
Introduction to ocean tides – p. 7/57
The gravitational tidal force
Earth
R
r
dMoon
ψ
P
aP = GMr2
aP = −∇GMr
aP = −∇(
GM√d2+R2
−2dR cosψ
)
aP = −GMd∇
∞∑
n=0
(
Rd
)nPn(cosψ)
aP = −∇∞∑
n=0
Un(cosψ)
Introduction to ocean tides – p. 7/57
The gravitational tidal force
Earth
R
r
dMoon
ψ
P
aP = GMr2
aP = −∇GMr
aP = −∇(
GM√d2+R2
−2dR cosψ
)
aP = −GMd∇
∞∑
n=0
(
Rd
)nPn(cosψ)
aP = −∇∞∑
n=0
Un(cosψ)
Only U2 already describes 98% of the tides!
Introduction to ocean tides – p. 7/57
?
Introduction to ocean tides – p. 8/57
Why work with the Tidal potential?
A force has a direction and a magnitude. This is called
a vector.
Introduction to ocean tides – p. 9/57
Why work with the Tidal potential?
A force has a direction and a magnitude. This is called
a vector.
The tidal potential only has a value, no direction.
Introduction to ocean tides – p. 9/57
Why work with the Tidal potential?
A force has a direction and a magnitude. This is called
a vector.
The tidal potential only has a value, no direction.
Its easier to work with a potential than with a vector.
Introduction to ocean tides – p. 9/57
Why work with the Tidal potential?
A force has a direction and a magnitude. This is called
a vector.
The tidal potential only has a value, no direction.
Its easier to work with a potential than with a vector.
A large potential value means it can release a big force
(produce a lot of work)
Low potential
high potential
Introduction to ocean tides – p. 9/57
Shape of U2
2UEarth
Moon
+
− −−
− −
+
Introduction to ocean tides – p. 10/57
Tidal potential, degree 2: U2
ψθ
Λ−λ
δ
Equator
Pole
P
λ Λ
cosψ = sin θ sin δ + cos θ cos δ cos(Λ − λ)
Introduction to ocean tides – p. 11/57
Tidal potential, degree 2: U2
ψθ
Λ−λ
δ
Equator
Pole
P
λ Λ
cosψ = sin θ sin δ + cos θ cos δ cos(Λ − λ)
U2 = 34
GMd
(
Rd
)2 (
cos2 θ cos2 δ cos 2(Λ − λ)+
sin 2θ sin 2δ cos(Λ − λ) +
3(
sin2 θ − 13
) (
sin2 δ − 13
))
Introduction to ocean tides – p. 11/57
Rewrite of U2
D =3
4
GM
d
(
R
d
)2
Introduction to ocean tides – p. 12/57
Rewrite of U2
D =3
4
GM
d
(
R
d
)2
G0 =1
2D(1−3 sin2 θ), G1 = D sin 2θ, G2 = D cos2 θ
Introduction to ocean tides – p. 12/57
Rewrite of U2
D =3
4
GM
d
(
R
d
)2
G0 =1
2D(1−3 sin2 θ), G1 = D sin 2θ, G2 = D cos2 θ
U2 =2
3G0(1 − 3 sin2 δ) +G1 sin 2δ cos(Λ − λ)+
G2 cos2 δ cos 2(Λ − λ)
Introduction to ocean tides – p. 12/57
What are we doing?
We derived the fact that the tidal forcing only depends
on the second degree of the tidal potential: U2.
Introduction to ocean tides – p. 13/57
What are we doing?
We derived the fact that the tidal forcing only depends
on the second degree of the tidal potential: U2.
We rewrote U2 as the sum of three separate functions.
Introduction to ocean tides – p. 13/57
What are we doing?
We derived the fact that the tidal forcing only depends
on the second degree of the tidal potential: U2.
We rewrote U2 as the sum of three separate functions.
A nice property of these three functions is that each
describes the long-period, diurnal and semi-diurnal
variations respectively.
Introduction to ocean tides – p. 13/57
What are we doing?
We derived the fact that the tidal forcing only depends
on the second degree of the tidal potential: U2.
We rewrote U2 as the sum of three separate functions.
A nice property of these three functions is that each
describes the long-period, diurnal and semi-diurnal
variations respectively.
We still need a way to describe the variations in the
inclination and longitude of Sun/Moon.
Introduction to ocean tides – p. 13/57
Geodetic functions
−0.8 −0.4 0.0 0.4 0.8
Long period tides G 0
Introduction to ocean tides – p. 14/57
Geodetic functions
−0.8 −0.4 0.0 0.4 0.8
1G sin(lon)Diurnal tides
Introduction to ocean tides – p. 14/57
Geodetic functions
−0.8 −0.4 0.0 0.4 0.8
Semi−diurnal tides 2G cos(2lon)
Introduction to ocean tides – p. 14/57
Still rewriting U2: account for Λ and δ
U2i =∑
KABC·DEF Gi(θ, R)
cos, for i = 0, 2
sin, for i = 1
(Aτ + Bs + Ch + Dp + EN ′ + Fps)
K0,1,−1·2,−3,1 → 0τ + 1s − 1h + 2p − 3N ′ + 1ps
Introduction to ocean tides – p. 15/57
Still rewriting U2: account for Λ and δ
U2i =∑
KABC·DEF Gi(θ, R)
cos, for i = 0, 2
sin, for i = 1
(Aτ + Bs + Ch + Dp + EN ′ + Fps)
K0,1,−1·2,−3,1 → 0τ + 1s − 1h + 2p − 3N ′ + 1ps
Doodson angles:
τ = local mean lunar time p = perigee of Moon’s orbit
s = mean longitude of Moon N ′ = ascending node of Moon
h = mean longitude of Sun ps = perigee of the Sun
Introduction to ocean tides – p. 15/57
Tidal potential coefficients
Darwin Doodson Frequency
symbol number (cycles/day) K
Ssa 057 · 555 0.00548 0.072732
Mm 065 · 455 0.03629 0.082569
Mf 075 · 555 0.07320 0.156303
Q1 135 · 655 0.89324 0.072136
O1 145 · 555 0.92954 0.376763
P1 163 · 555 0.99726 0.175307
K1 165 · 555 1.00274 -0.529876
N2 245 · 655 1.89598 0.173881
M2 255 · 555 1.93227 0.908184
S2 273 · 555 2.00000 0.422535
K2 275 · 555 2.00548 0.114860
Introduction to ocean tides – p. 16/57
Sir George Howard Darwin
U2 =KM2G2 cos(ωM2
t+ χM2)+
KS2G2 cos(ωS2
t+ χS2)+
KO1G1 sin(ωO1
t+ χO1)+
KMfG0 cos(ωMf t+ χMf)+
. . .
Introduction to ocean tides – p. 17/57
Tamura Tidal potential coefficients (K)
Introduction to ocean tides – p. 18/57
Tidal prediction
Now that we know how to write the tidal potential, we can
also model the tides in the harbour in the same way:
ζ =AM2G2 cos(ωM2
t + χM2+ βM2
)+
AO1G1 sin(ωO1
t + χO1+ βO1
)+
AMfG0 cos(ωMf t + χMf + βMf ) + . . .
Introduction to ocean tides – p. 19/57
Tidal prediction
Now that we know how to write the tidal potential, we can
also model the tides in the harbour in the same way:
ζ =AM2G2 cos(ωM2
t + χM2+ βM2
)+
AO1G1 sin(ωO1
t + χO1+ βO1
)+
AMfG0 cos(ωMf t + χMf + βMf ) + . . .
For each tide gauge, the values of A and β are given for
each harmonic.
Introduction to ocean tides – p. 19/57
Tidal prediction
Now that we know how to write the tidal potential, we can
also model the tides in the harbour in the same way:
ζ =AM2G2 cos(ωM2
t + χM2+ βM2
)+
AO1G1 sin(ωO1
t + χO1+ βO1
)+
AMfG0 cos(ωMf t + χMf + βMf ) + . . .
For each tide gauge, the values of A and β are given for
each harmonic.
The value for ω are known and the value of χ can be
computed.
Introduction to ocean tides – p. 19/57
Tidal values of tide gauge at Gribaltar
http://www.bodc.ac.uk/projects/international/woce/tidal_constants/
Introduction to ocean tides – p. 20/57
Predicted tides at Gibraltar
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
12 13 14 15 16 17 18 19 20 21 22
tid
e g
au
ge
(m
)
January 2009
M2
Introduction to ocean tides – p. 21/57
Predicted tides at Gibraltar
-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
12 13 14 15 16 17 18 19 20 21 22
tid
e g
au
ge
(m
)
January 2009
O1
Introduction to ocean tides – p. 21/57
Predicted tides at Gibraltar
-0.0025
-0.002
-0.0015
-0.001
-0.0005
0
0.0005
0.001
0.0015
12 13 14 15 16 17 18 19 20 21 22
tid
e g
au
ge
(m
)
January 2009
Mf
Introduction to ocean tides – p. 21/57
Predicted tides at Gibraltar
-0.2
0
0.2
0.4
0.6
0.8
1
12 13 14 15 16 17 18 19 20 21 22
tid
e g
au
ge
(m
)
January 2009
observedpredicted
Introduction to ocean tides – p. 21/57
Force due to Tidal potential
dU2
dz
dU2
dxρp+ g
ρ g
p
ρ
Moon
ρ
ρ
Introduction to ocean tides – p. 22/57
Force due to Tidal potential
dU2
dz
dU2
dx
p
ρ
ρ
ρ
dz
dx
Moon
p+ g
g
ρ
ρ
Hydrostatic equilibrium
Introduction to ocean tides – p. 22/57
Remember!
Ocean tides are caused by the horizontal gravitational
force of Moon and Sun, not the vertical force.
Introduction to ocean tides – p. 23/57
Remember!
Ocean tides are caused by the horizontal gravitational
force of Moon and Sun, not the vertical force.
Gravitational force acts on the whole water column.
Introduction to ocean tides – p. 23/57
Modelling the tides
So far, we only discussed the gravitational forcebut there are more forces influencing themodelling of the tides:
The Earth rotates so we have Corriolis forces
Introduction to ocean tides – p. 24/57
Modelling the tides
So far, we only discussed the gravitational forcebut there are more forces influencing themodelling of the tides:
The Earth rotates so we have Corriolis forces
A slope in the sea-surface also causes horizontal force.
Introduction to ocean tides – p. 24/57
Modelling the tides
So far, we only discussed the gravitational forcebut there are more forces influencing themodelling of the tides:
The Earth rotates so we have Corriolis forces
A slope in the sea-surface also causes horizontal force.
Bottom friction and/or lateral eddy dissipation
Introduction to ocean tides – p. 24/57
Force due to slope in sea-level
d ζ
dx
ρ
p
p
xζ
ζ +
p+F
ρp+ g
p
d ζ
dx
Introduction to ocean tides – p. 25/57
Force due to slope in sea-level
d ζ
dx
d ζ
dx
ρ
p
p
xζ
ζ
p+F
x
ρp+ g
p
ρF = g+
d ζ
dx
Introduction to ocean tides – p. 25/57
Force due to slope in sea-level
d ζ
dx
d ζ
dx
xp+F
xp’+F
ρp+ g d ζ
dx
ρ
p
p
ζ
ζx
p
ρF = g+
p’
p’
Introduction to ocean tides – p. 25/57
Conservation of mass
θdy (=Rd )
u
u+du
ζ
v+dv
D
v
λθdx (=R cos d )Introduction to ocean tides – p. 26/57
Conservation of mass
θdy (=Rd )
Fθ
u
u+du
ζ
v+dv
D
v
λθdx (=R cos d )
Fλ
Introduction to ocean tides – p. 26/57
Pierre-Simon Laplace (1776)
Laplace derived the differential
equations for a thin fluid on a
sphere with no vertical motion,
only horizontal motions.
Introduction to ocean tides – p. 27/57
Pierre-Simon Laplace (1776)
Laplace derived the differential
equations for a thin fluid on a
sphere with no vertical motion,
only horizontal motions.
This depth integrated model is
also called a barotropic model.
Introduction to ocean tides – p. 27/57
Laplace Tidal Equations
∂u
∂t+ u · ∇u + f × u = −g∇ζ
U
T
U 2
LfU g
H
L
D
Dt=
∂
∂t+
u
R cos θ
∂
∂λ+v
R
∂
∂θ
Introduction to ocean tides – p. 28/57
Laplace Tidal Equations
Equations of motion in θ and λ direction:
∂u
∂t− (2Ω sin θ)v = − g
R cos θ
∂
∂λ
(
ζ − U2
g
)
+Fλ
ρD
∂v
∂t+ (2Ω sin θ)u = − g
R
∂
∂θ
(
ζ − U2
g
)
+Fθ
ρD
Introduction to ocean tides – p. 29/57
Laplace Tidal Equations
Equations of motion in θ and λ direction:
∂u
∂t− (2Ω sin θ)v = − g
R cos θ
∂
∂λ
(
ζ − U2
g
)
+Fλ
ρD
∂v
∂t+ (2Ω sin θ)u = − g
R
∂
∂θ
(
ζ − U2
g
)
+Fθ
ρD
Conservation of mass:
∂ζ
∂t+
D
R cos θ
(
∂u
∂λ+
∂(v cos θ)
∂θ
)
= 0
Introduction to ocean tides – p. 29/57
What do we have?
A set of ordinary differential equations (ODE’s).
Introduction to ocean tides – p. 30/57
What do we have?
A set of ordinary differential equations (ODE’s).
To solve them, we need boundary conditions.
Introduction to ocean tides – p. 30/57
What do we have?
A set of ordinary differential equations (ODE’s).
To solve them, we need boundary conditions.
Here, it is assumed we have no flow through land,
u = v = 0 at the coast.
Introduction to ocean tides – p. 30/57
What do we have?
A set of ordinary differential equations (ODE’s).
To solve them, we need boundary conditions.
Here, it is assumed we have no flow through land,
u = v = 0 at the coast.
A fact from physics: if a system is influenced by a
periodic force, its reponse will also be periodic.
Introduction to ocean tides – p. 30/57
What do we have?
A set of ordinary differential equations (ODE’s).
To solve them, we need boundary conditions.
Here, it is assumed we have no flow through land,
u = v = 0 at the coast.
A fact from physics: if a system is influenced by a
periodic force, its reponse will also be periodic.
Consequence: We can compute the tides for each
harmonic separately!
Introduction to ocean tides – p. 30/57
LTE in frequency domain
Tides are periodic:
U2(t) = U2eiωt, ζ(t) = ζeiωt, u(t) = ueiωt, v(t) = iveiωt
Introduction to ocean tides – p. 31/57
LTE in frequency domain
Tides are periodic:
U2(t) = U2eiωt, ζ(t) = ζeiωt, u(t) = ueiωt, v(t) = iveiωt
Equations of motion in θ and λ direction:
ωu − (2Ω sin θ)v =gi
R cos θ
∂
∂λ
(
ζ − U2
g
)
ωv − (2Ω sin θ)u = − g
R
∂
∂θ
(
ζ − U2
g
)
Introduction to ocean tides – p. 31/57
LTE in frequency domain
Tides are periodic:
U2(t) = U2eiωt, ζ(t) = ζeiωt, u(t) = ueiωt, v(t) = iveiωt
Equations of motion in θ and λ direction:
ωu − (2Ω sin θ)v =gi
R cos θ
∂
∂λ
(
ζ − U2
g
)
ωv − (2Ω sin θ)u = − g
R
∂
∂θ
(
ζ − U2
g
)
Conservation of mass:
iωζ +D
R cos φ
∂u
∂λ+
D
R
∂iv
∂φ− ivD sinφ
R cos φ= 0
Introduction to ocean tides – p. 31/57
What’s our proges sofar?
The terms with ∂∂t
have disappeared. No more
derivatives with respect to time.
Introduction to ocean tides – p. 32/57
What’s our proges sofar?
The terms with ∂∂t
have disappeared. No more
derivatives with respect to time.
By writing all derivatives of the form ∂yi
∂xas yi+1−yi
∆x, we
transform the ODE’s into a set of linear equations.
Introduction to ocean tides – p. 32/57
What’s our proges sofar?
The terms with ∂∂t
have disappeared. No more
derivatives with respect to time.
By writing all derivatives of the form ∂yi
∂xas yi+1−yi
∆x, we
transform the ODE’s into a set of linear equations.
The set of linear equations can be solved easily.
Introduction to ocean tides – p. 32/57
What’s our proges sofar?
The terms with ∂∂t
have disappeared. No more
derivatives with respect to time.
By writing all derivatives of the form ∂yi
∂xas yi+1−yi
∆x, we
transform the ODE’s into a set of linear equations.
The set of linear equations can be solved easily.
We will call this program BOTM: Basic Ocean Tide
Model.
Introduction to ocean tides – p. 32/57
Solution for a non-rotating Earth
Semidiurnal Tides (G2):
ζ = −KU22
Diurnal Tides (G1):
ζ = −KU21
Long period Tides (G0):
ζ = KU20
K =6gD
ω2R2 − 6gD
Introduction to ocean tides – p. 33/57
Staggered C-grid
v
ζu
Introduction to ocean tides – p. 34/57
Staggered C-grid
Introduction to ocean tides – p. 34/57
Staggered C-grid
0˚ 90˚ 180˚ 270˚
−45˚
0˚
45˚
Introduction to ocean tides – p. 34/57
Why study Earth without topography?
To verify that our BOTM gives reasonable results
Introduction to ocean tides – p. 35/57
Why study Earth without topography?
To verify that our BOTM gives reasonable results
If you program your own ocean tide model, or start
using a model from someone else, you must always,
always, check if it gives good results for cases for
which you know already the answer!
Introduction to ocean tides – p. 35/57
Theoretical versus BOTM
M2
0˚ 90˚ 180˚ 270˚
−45˚
0˚
45˚
−60 −40 −20 0 20 40 60
mm
0˚ 90˚ 180˚ 270˚ 0˚
−45˚
0˚
45˚
−60 −40 −20 0 20 40 60
mm
Theoretical BOTM
Introduction to ocean tides – p. 36/57
Theoretical versus BOTM
O1
0˚ 90˚ 180˚ 270˚
−45˚
0˚
45˚
−2.0 −1.6 −1.2 −0.8 −0.4 0.0 0.4 0.8 1.2 1.6 2.0
m
0˚ 90˚ 180˚ 270˚ 0˚
−45˚
0˚
45˚
−2.0 −1.6 −1.2 −0.8 −0.4 0.0 0.4 0.8 1.2 1.6 2.0
m
Theoretical BOTM
Introduction to ocean tides – p. 36/57
Theoretical versus BOTM
Mf
0˚ 90˚ 180˚ 270˚
−45˚
0˚
45˚
−80 −40 0 40 80
mm
0˚ 90˚ 180˚ 270˚ 0˚
−45˚
0˚
45˚
−80 −40 0 40 80
mm
Theoretical BOTM
Introduction to ocean tides – p. 36/57
Sinning Earth
What happens if we now let the Earth spin on ourocean covered Earth?
Introduction to ocean tides – p. 37/57
M2 tide on an ocean covered Earth
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−1
80
−1
80
−1
80
−1
80
−1
80
−1
80
−1
80
−1
80
−1
80
−1
80
−1
80
−1
80
−1
20
−120
−120
−120
−1
20
−120
−120
−120
−1
20
−120
−120
−120
−1
20
−120
−120
−120
−120
−120
−120
−120−
60
−60
−60
−60 −60
−60
−60
−6
0−
60
−60
−60 −60
−6
0
−60
−60
−60
−6
0−
60
−60
−60
−60
−60
−60
−60−60
00
0
00
00
00
00
0
00
00
60
60
60
606
060
60
60
60
60
60
60
60
60
60
6060 60
60 60
12
0120
120
120
12
0120
120
120
12
0120
120
120
12
0120
120
120120
120
120
120
120
120
120
120
0.0 0.1 0.2 0.3 0.4 0.5 0.6
m
Introduction to ocean tides – p. 38/57
Amplitude and Phase-lag
Colour indicates the size of the amplitude.
amplitude
time
phase−lag
Introduction to ocean tides – p. 39/57
Amplitude and Phase-lag
Colour indicates the size of the amplitude.
contour line indicates how much the tidal signal is
delayed with respect to the phase of the tidal potential.
amplitude
time
phase−lag
Introduction to ocean tides – p. 39/57
tidal ellipses of flow (M2)
0˚ 45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
Introduction to ocean tides – p. 40/57
M2 tide on an ocean covered Earth
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−1
80
−1
80
−1
80
−1
80
−1
80
−1
80
−1
80
−1
80
−1
80
−1
80
−1
80
−1
80
−1
20
−120
−120
−120
−1
20
−120
−120
−120
−1
20
−120
−120
−120
−1
20
−120
−120
−120
−120
−120
−120
−120−
60
−60
−60
−60 −60
−60
−60
−6
0−
60
−60
−60 −60
−6
0
−60
−60
−60
−6
0−
60
−60
−60
−60
−60
−60
−60−60
00
0
00
00
00
00
0
00
00
60
60
60
606
060
60
60
60
60
60
60
60
60
60
6060 60
60 60
12
0120
120
120
12
0120
120
120
12
0120
120
120
12
0120
120
120120
120
120
120
120
120
120
120
0.0 0.1 0.2 0.3 0.4 0.5 0.6
m
Introduction to ocean tides – p. 41/57
Snapshot of sea-level due ove time (M2)t = 0 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 0.7 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 1.4 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 2.1 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 2.8 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 3.4 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 4.1 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 4.8 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 5.5 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 6.2 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 6.9 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 7.6 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 8.3 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 9.0 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 9.6 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 10.3 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 11.0 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 11.7 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)t = 12.4 hours
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−0.4 −0.2 0.0 0.2 0.4
m
Introduction to ocean tides – p. 42/57
O1 tide on an ocean covered Earth
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
−150
−150
−150 −150
−150
−150
−1
50
−1
50
−150
−150 −150
−150
−150
−1
50
−150
−150
−120
−120
−120
−120
−1
20
−120 −120
−120
−120
−120
−120
−1
20
−120 −120−90 −90 −90 −90 −90 −90
−60−
60
−60
−60
−6
0
−60−60 −60 −60 −60 −60 −60 −60
−30−30
−30
−30
−30
−3
0
−3
0
−30 −30
−30
−30 −30 −30
−30
−30 −30 −30 −30 −30 −30
000 0000
00
00
0 0 0
000
0 0 0
0 0 0
00
0 0 0 0 0 0 0
30
3030
30
30
30
30
30
3030 30 30 3030 30 30 30 30 30
30
30
30
30
3030
6060
60
60
60
60 60 60 6060 60 60 60 60
60
60
60
60
60
90 90 9090 90 90
90
90
90
90
90
120
120
120
120
12
0
120120120 120
12
0120
120
120
120150
150
150
150
150
150
150
150
150
150
150 150
0.0 0.1 0.2 0.3 0.4 0.5 0.6
m
Introduction to ocean tides – p. 43/57
tidal ellipses of flow (O2)
0˚ 45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
Introduction to ocean tides – p. 44/57
Mf tide on an ocean covered Earth
45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
0.0 0.1 0.2 0.3 0.4 0.5 0.6
m
180
0
0
Introduction to ocean tides – p. 45/57
tidal ellipses of flow (Mf )
0˚ 45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
Introduction to ocean tides – p. 46/57
Example, Ocean tides around UK
Introduction to ocean tides – p. 47/57
Example, Ocean tides around UK
Introduction to ocean tides – p. 48/57
Example, Ocean tides around UK
Introduction to ocean tides – p. 49/57
Example, Ocean tides around UK
Introduction to ocean tides – p. 49/57
Topography/bathymetry of the Earth
0˚ 45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚ 0˚−90˚
−45˚
0˚
45˚
90˚
−10 −8 −6 −4 −2 0 2 4 6 8 10
km
Introduction to ocean tides – p. 50/57
Staggered C-grid of the Earth
0˚ 90˚ 180˚ 270˚
−45˚
0˚
45˚
Introduction to ocean tides – p. 51/57
Staggered C-grid of the Earth
90˚ 180˚−45˚
0˚
Introduction to ocean tides – p. 51/57
M2 tide of BOTM
90˚ 180˚ 270˚
−45˚
0˚
45˚
−150
−150
−150
−150−150
−120
−120
−120−120
−90
−90
−90−90
−90
−90
−60
−60
−60
−60
−60−60
−60
−30
−30
−30
−30
−30
−30 −30 −30
−30
−30
0
0
0
30
30
30
60
60
60
60
60
60
90
90
90
90
90
9090
90
90
90 90
120
120
120
120
120
120
120120
120
120
120
150
150
150
150
150
150
150
150
150
0.0 0.1 0.2 0.3 0.5 0.8 4.5
m
Introduction to ocean tides – p. 52/57
tidal ellipses of flow (M2)
0˚ 45˚ 90˚ 135˚ 180˚ 225˚ 270˚ 315˚
−45˚
0˚
45˚
Introduction to ocean tides – p. 53/57
M2 BOTM versus FES99
0˚ 90˚ 180˚ 270˚
−45˚
0˚
45˚
0
60
60
60
90
9090
90
120
120
150
150
180210210
240
240
0.0 0.1 0.2 0.3 0.5 0.8 4.5
m
90˚ 180˚ 270˚
−45˚
0˚
45˚
−150
−150
−150
−150−150
−120
−120
−120−120
−90
−90
−90
−90
−90
−90
−60
−60
−60
−60
−60−60
−60
−30
−30
−30
−30
−30
−30 −30 −30
−30
−30
0
0
0
30
30
30
60
60
60
60
60
60
90
90
90
90
90
90
90
90
90
90 90
120
120
120
120
120
120
120
120
120
120
120
150
150
150
150
150
150
150
150
150
0.0 0.1 0.2 0.3 0.5 0.8 4.5
m
BOTM FES99
Introduction to ocean tides – p. 54/57
O1 BOTM versus FES99
0˚ 90˚ 180˚ 270˚
−45˚
0˚
45˚
0
30
3060
606
0
60
90
90120
150
150
180
210
210
210
240
240
240
240
270300
0.0 0.1 0.2 0.3 0.5 0.8 4.5
m
90˚ 180˚ 270˚
−45˚
0˚
45˚
−150
−150
−150
−150
−150
−120
−120
−120−90
−90
−90
−90
−90
−90
−60
−60
−60
−60
−30
−30
−30
−30
−30
00
00
0
30
30
30
60
60
60
60
60
60
60
90
90
90
120
120
120
120
150
150
150
150
150
150
150
0.0 0.1 0.2 0.3 0.5 0.8 4.5
m
BOTM FES99
Introduction to ocean tides – p. 55/57
Mf BOTM versus FES99
0˚ 90˚ 180˚ 270˚
−45˚
0˚
45˚
0
00
00
0
000
0
3030
303030
30
30
30
60
60
90
90
120
120
150
150 180
180
180180
180
180
180
180
180
180180
180 180180180180180
180180180
180 180180
210210
210 210
210
0.00 0.01 0.02 0.03 0.04 0.05 0.06
m
90˚ 180˚ 270˚
−45˚
0˚
45˚
−150
−150 −150
−150
−150−150−
150−
150
−150
−150
−150
−150
−120
−120
−120
−120
−120
−120
−120
−120 −120
−90
−90
−90
−90
−90−90 −
90
−60
−60
−60
−60
−60
−60
−60−60
−60−60
−60
−60−
60
−30
−30
−30 −30
−30
−30
−30
−30
−30 −3
0 −30
−30−30
−30 −30
0
0
0
0
000
0
0
00
0
0
0
0
00
0
0
30
30
30
30
30
303030
30
30
30
30
30 30
303030
3030
30
3030
60
60
6060
60
60
60
6060
60 60
60
60
60
60
60
60
60
60
90
90
90
90
90
90
90 90
90
90
90
90
90
120
120
120 1
20
120
120
120
120
120
120
120 120
150
150
150 150
150
150
150
150
150150
150
150
0.0 0.1 0.2 0.3 0.5 0.8 4.5
m
BOTM FES99
Introduction to ocean tides – p. 56/57
Introduction to ocean tides – p. 57/57