Ocean Tide Modelling, Part 1
M.S. Bos
[email protected]
Centro Interdisciplinar de Investigação 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
1
observed
tide gauge (m)
0.8
0.6
0.4
0.2
0
-0.2
12
13
14
15
16
17
18
19
20
21
January 2009
Introduction to ocean tides – p. 5/57
Isaac Newton (1687): Law of Gravity
Introduction to ocean tides – p. 6/57
The gravitational tidal force
aP =
P
R
ψ
GM
r2
r
d
Moon
Earth
Introduction to ocean tides – p. 7/57
The gravitational tidal force
aP =
P
R
ψ
aP = −∇ GM
r
r
d
GM
r2
Moon
Earth
Introduction to ocean tides – p. 7/57
The gravitational tidal force
aP =
P
R
ψ
r
d
Moon
GM
r2
aP = −∇ GM
r
aP = −∇ √
GM
d2 +R2 −2dR cos ψ
Earth
Introduction to ocean tides – p. 7/57
The gravitational tidal force
aP =
P
R
ψ
r
d
Moon
GM
r2
aP = −∇ GM
r
aP = −∇ √
Earth
aP =
∇
− GM
d
GM
d2 +R2 −2dR cos ψ
∞
P
n=0
R n
d
Pn (cos ψ)
Introduction to ocean tides – p. 7/57
The gravitational tidal force
aP =
P
R
ψ
r
d
Moon
GM
r2
aP = −∇ GM
r
aP = −∇ √
Earth
aP =
∇
− GM
d
aP = −∇
∞
P
GM
d2 +R2 −2dR cos ψ
∞
P
n=0
R n
d
Pn (cos ψ)
Un (cos ψ)
n=0
Introduction to ocean tides – p. 7/57
The gravitational tidal force
aP =
P
R
ψ
r
d
Moon
GM
r2
aP = −∇ GM
r
aP = −∇ √
Earth
aP =
∇
− GM
d
aP = −∇
∞
P
GM
d2 +R2 −2dR cos ψ
∞
P
n=0
R n
d
Pn (cos ψ)
Un (cos ψ)
n=0
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
−
− −
Moon
+
+
Earth
−
U2
−
Introduction to ocean tides – p. 10/57
Tidal potential, degree 2: U2
Pole
P
θ
Λ−λ
ψ
λ Equator
δ
Λ
cos ψ = sin θ sin δ + cos θ cos δ cos(Λ − λ)
Introduction to ocean tides – p. 11/57
Tidal potential, degree 2: U2
Pole
P
θ
Λ−λ
ψ
λ Equator
δ
Λ
cos ψ = sin θ sin δ + cos θ cos δ cos(Λ − λ)
3 GM R 2
U2 = 4 d d
cos2 θ cos2 δ cos 2(Λ − λ)+
sin 2θ sin 2δ cos(Λ − λ) +
2
1
2
1
3 sin θ − 3 sin δ − 3
Introduction to ocean tides – p. 11/57
Rewrite of U2
2
3 GM R
D=
4 d
d
Introduction to ocean tides – p. 12/57
Rewrite of U2
2
3 GM R
D=
4 d
d
1
G0 = D(1−3 sin2 θ),
2
G1 = D sin 2θ, G2 = D cos2 θ
Introduction to ocean tides – p. 12/57
Rewrite of U2
2
3 GM R
D=
4 d
d
1
G0 = D(1−3 sin2 θ),
2
G1 = D sin 2θ, G2 = D cos2 θ
2
U2 = G0 (1 − 3 sin2 δ) + G1 sin 2δ cos(Λ − λ)+
3
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
Long period tides
−0.8
−0.4
0.0
0.4
G0
0.8
Introduction to ocean tides – p. 14/57
Geodetic functions
Diurnal tides
−0.8
−0.4
0.0
0.4
G 1sin(lon)
0.8
Introduction to ocean tides – p. 14/57
Geodetic functions
Semi−diurnal tides
−0.8
−0.4
0.0
0.4
G 2 cos(2lon)
0.8
Introduction to ocean tides – p. 14/57
Still rewriting U2: account for Λ and δ
U2i =
X
KABC·DEF Gi (θ, R)


cos, for i = 0, 2
 (Aτ + Bs + Ch + Dp + EN ′ + F ps )

sin, for i = 1
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 =
X
KABC·DEF Gi (θ, R)


cos, for i = 0, 2
 (Aτ + Bs + Ch + Dp + EN ′ + F ps )

sin, for i = 1
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 =KM2 G2 cos(ωM2 t + χM2 )+
KS2 G2 cos(ωS2 t + χS2 )+
KO1 G1 sin(ωO1 t + χO1 )+
KM f G0 cos(ωM f t + χM f )+
...
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 =AM2 G2 cos(ωM2 t + χM2 + βM2 )+
AO1 G1 sin(ωO1 t + χO1 + βO1 )+
AM f G0 cos(ωM f t + χM f + βM f ) + . . .
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 =AM2 G2 cos(ωM2 t + χM2 + βM2 )+
AO1 G1 sin(ωO1 t + χO1 + βO1 )+
AM f G0 cos(ωM f t + χM f + βM f ) + . . .
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 =AM2 G2 cos(ωM2 t + χM2 + βM2 )+
AO1 G1 sin(ωO1 t + χO1 + βO1 )+
AM f G0 cos(ωM f t + χM f + βM f ) + . . .
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
M2
tide gauge (m)
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
January 2009
Introduction to ocean tides – p. 21/57
Predicted tides at Gibraltar
0.01
0.008
O1
tide gauge (m)
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
January 2009
Introduction to ocean tides – p. 21/57
Predicted tides at Gibraltar
0.0015
tide gauge (m)
0.001
0.0005
Mf
0
-0.0005
-0.001
-0.0015
-0.002
-0.0025
12
13
14
15
16
17
18
19
20
21
22
January 2009
Introduction to ocean tides – p. 21/57
Predicted tides at Gibraltar
1
observed
predicted
tide gauge (m)
0.8
0.6
0.4
0.2
0
-0.2
12
13
14
15
16
17
18
19
20
21
22
January 2009
Introduction to ocean tides – p. 21/57
Force due to Tidal potential
Moon
dU2
ρ
dz
p
ρ
dU2
ρ
dx
p+ ρg
ρg
Introduction to ocean tides – p. 22/57
Force due to Tidal potential
Moon
Hydrostatic equilibrium
p
dz
dU2
ρ
dz
ρ
dx
p+ ρg
dU2
ρ
dx
ρg
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 force
but there are more forces influencing the
modelling 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 force
but there are more forces influencing the
modelling 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 force
but there are more forces influencing the
modelling 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
dζ
ζ + dx
p
p
ρ
ζ
p
p+Fx
p+ ρg d ζ
dx
Introduction to ocean tides – p. 25/57
Force due to slope in sea-level
ζ
d
Fx = ρg
dx
dζ
ζ + dx
p
p
ρ
ζ
p
p+Fx
p+ ρg d ζ
dx
Introduction to ocean tides – p. 25/57
Force due to slope in sea-level
ζ
d
Fx = ρg
dx
dζ
ζ + dx
p
p
ρ
ζ
p
p’
p+Fx
ζ
d
p+ ρg
dx
p’+Fx
p’
Introduction to ocean tides – p. 25/57
Conservation of mass
ζ
v+dv
D
u+du
u
v
dy (=Rdθ)
dx (=R cos θ d λ)
Introduction to ocean tides – p. 26/57
Conservation of mass
ζ
v+dv
D
u+du
u
Fθ
dy (=Rdθ)
v
Fλ
dx (=R cos θ d λ)
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
+ u · ∇u + f × u = −g∇ζ
∂t
U
T
U2
L
fU
H
g
L
D
∂
u
∂
v ∂
=
+
+
Dt ∂t R cos θ ∂λ R ∂θ
Introduction to ocean tides – p. 28/57
Laplace Tidal Equations
Equations of motion in θ and λ direction:
g
∂
∂u
− (2Ω sin θ)v = −
∂t
R cos θ ∂λ
∂v
g ∂
+ (2Ω sin θ)u = −
∂t
R ∂θ
ζ−
U2
g
U2
ζ−
g
+
Fλ
ρD
Fθ
+
ρD
Introduction to ocean tides – p. 29/57
Laplace Tidal Equations
Equations of motion in θ and λ direction:
g
∂
∂u
− (2Ω sin θ)v = −
∂t
R cos θ ∂λ
∂v
g ∂
+ (2Ω sin θ)u = −
∂t
R ∂θ
ζ−
U2
g
U2
ζ−
g
+
Fλ
ρD
Fθ
+
ρD
Conservation of mass:
D
∂ζ
+
∂t
R cos θ
∂u ∂(v cos θ)
+
∂λ
∂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) = Ū2 eiωt , ζ(t) = ζ̄eiωt , u(t) = ūeiωt , v(t) = iv̄eiωt
Introduction to ocean tides – p. 31/57
LTE in frequency domain
Tides are periodic:
U2 (t) = Ū2 eiωt , ζ(t) = ζ̄eiωt , u(t) = ūeiωt , v(t) = iv̄eiωt
Equations of motion in θ and λ direction:
∂
gi
ωū − (2Ω sin θ)v̄ =
R cos θ ∂λ
g ∂
ωv̄ − (2Ω sin θ)ū = −
R ∂θ
Ū2
ζ̄ −
g
Ū2
ζ̄ −
g
Introduction to ocean tides – p. 31/57
LTE in frequency domain
Tides are periodic:
U2 (t) = Ū2 eiωt , ζ(t) = ζ̄eiωt , u(t) = ūeiωt , v(t) = iv̄eiωt
Equations of motion in θ and λ direction:
∂
gi
ωū − (2Ω sin θ)v̄ =
R cos θ ∂λ
g ∂
ωv̄ − (2Ω sin θ)ū = −
R ∂θ
Ū2
ζ̄ −
g
Ū2
ζ̄ −
g
Conservation of mass:
iω ζ̄ +
D ∂ ū D ∂iv̄
iv̄D sin φ
+
−
=0
R cos φ ∂λ
R ∂φ
R cos φ
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.
yi+1 −yi
i
By writing all derivatives of the form ∂y
as
, we
∂x
∆x
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.
yi+1 −yi
i
By writing all derivatives of the form ∂y
as
, we
∂x
∆x
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.
yi+1 −yi
i
By writing all derivatives of the form ∂y
as
, we
∂x
∆x
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 = −K̂U22
Diurnal Tides (G1 ):
ζ = −K̂U21
Long period Tides (G0 ):
ζ = K̂U20
K̂ =
6gD
ω 2 R2 − 6gD
Introduction to ocean tides – p. 33/57
Staggered C-grid
v
u
ζ
Introduction to ocean tides – p. 34/57
Staggered C-grid
Introduction to ocean tides – p. 34/57
Staggered C-grid
45˚
0˚
−45˚
0˚
90˚
180˚
270˚
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
Theoretical
BOTM
45˚
45˚
0˚
0˚
−45˚
−45˚
0˚
90˚
180˚
270˚
0˚ 0˚
90˚
180˚
270˚
mm
mm
−60
−40
−20
0
20
40
60
−60
−40
−20
0
20
40
60
Introduction to ocean tides – p. 36/57
Theoretical versus BOTM
O1
Theoretical
BOTM
45˚
45˚
0˚
0˚
−45˚
−45˚
0˚
90˚
180˚
270˚
0˚ 0˚
90˚
180˚
270˚
m
m
−2.0 −1.6 −1.2 −0.8 −0.4 0.0
0.4
0.8
1.2
1.6
2.0
−2.0 −1.6 −1.2 −0.8 −0.4 0.0
0.4
0.8
1.2
1.6
2.0
Introduction to ocean tides – p. 36/57
Theoretical versus BOTM
Mf
Theoretical
BOTM
45˚
45˚
0˚
0˚
−45˚
−45˚
0˚
90˚
180˚
270˚
0˚ 0˚
90˚
180˚
270˚
mm
mm
−80
−40
0
40
80
−80
−40
0
40
80
Introduction to ocean tides – p. 36/57
Sinning Earth
What happens if we now let the Earth spin on our
ocean covered Earth?
Introduction to ocean tides – p. 37/57
0
−120
−120 −120
−180 −180 −180
60
60
0
0
−180 −180 −180
120
120
120
−120
12
−60
60
0
0
0
0
0
0
−6
0
−6
0
0
0
−60
−60
−60
120
120
120
−180 −180 −180
0
12
−120 −120 −120
−180 −180 −180
60
60
60
0
−60
0
0
0
0
−120
270˚
−60
−120
−120
−60
120
−60
120
225˚
60
2
1 0
60
180˚
120
60
60
−60
120
−120
−60
−120
−60
60
60
135˚
120
60
120
120
60
90˚
−60
0 −1
20
−6
−120 −120 −120
60
45˚
−60
60
2
1 0
60
−45˚
120
60
−60
−120
−60
−60
−120
−120
−60
60
120
0˚
−60
120
60
120
−60
−120
−60
120
−60
120
−60
45˚
120
0
M2 tide on an ocean covered Earth
315˚
m
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Introduction to ocean tides – p. 38/57
Amplitude and Phase-lag
amplitude
Colour indicates the size of the amplitude.
time
phase−lag
Introduction to ocean tides – p. 39/57
Amplitude and Phase-lag
Colour indicates the size of the amplitude.
amplitude
contour line indicates how much the tidal signal is
delayed with respect to the phase of the tidal potential.
time
phase−lag
Introduction to ocean tides – p. 39/57
tidal ellipses of flow (M2)
45˚
0˚
−45˚
0˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
Introduction to ocean tides – p. 40/57
0
−120
−120 −120
−180 −180 −180
60
60
0
0
−180 −180 −180
120
120
120
−120
12
−60
60
0
0
0
0
0
0
−6
0
−6
0
0
0
−60
−60
−60
120
120
120
−180 −180 −180
0
12
−120 −120 −120
−180 −180 −180
60
60
60
0
−60
0
0
0
0
−120
270˚
−60
−120
−120
−60
120
−60
120
225˚
60
2
1 0
60
180˚
120
60
60
−60
120
−120
−60
−120
−60
60
60
135˚
120
60
120
120
60
90˚
−60
0 −1
20
−6
−120 −120 −120
60
45˚
−60
60
2
1 0
60
−45˚
120
60
−60
−120
−60
−60
−120
−120
−60
60
120
0˚
−60
120
60
120
−60
−120
−60
120
−60
120
−60
45˚
120
0
M2 tide on an ocean covered Earth
315˚
m
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Introduction to ocean tides – p. 41/57
Snapshot of sea-level due ove time (M2)
t = 0 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 0.7 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 1.4 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 2.1 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 2.8 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 3.4 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 4.1 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 4.8 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 5.5 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 6.2 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 6.9 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 7.6 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 8.3 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 9.0 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 9.6 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 10.3 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 11.0 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 11.7 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
Snapshot of sea-level due ove time (M2)
t = 12.4 hours
45˚
0˚
−45˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
−0.4
−0.2
0.0
0.2
0.4
Introduction to ocean tides – p. 42/57
O1 tide on an ocean covered Earth
0
12
120
120
120
150
150
−150
−30
0
180˚
225˚
−60
0
0
0
0
30
60
30
60
30
60
30
90
90
90
−150 −150 −150
−120 −120 −120
20
−1
135˚
0
0
270˚
0
0
0
−3
−6
30
0
−60
12
1500 0− 60030
−15 30
−60
150
−60
120
150
−30
120
150
−30
120
150
90˚
−150
−120
120
90
−120
−60
90 −30
50−30
−60
90
60
60
−30
−60
60
0 −30
0−906−30
0−90−30
0−90−30 120
0130
0 30
0 30
30
30
30
60
60
30
90
−60
0−60
−30
120
150
−150
−30
30 0
60
90
120
60
0
30
0
60
−30
30
−120
45˚
1
30
60
90
500
120
120
30
60
50
150 −1
50
−1
−150 −150 −150
30
60
−45˚
−120 −120 −120 −120
60
6030
30
0 −120
0 −120
0
0
−30
−30
−30
−60
−60
−60
−90
−90
−90
−150
−150
030 150 −150
−1
−15
20
0
0
30
60
45˚
0˚
0
0
90
0
90
−30 0 3
0
−30
0
30
315˚
m
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Introduction to ocean tides – p. 43/57
tidal ellipses of flow (O2)
45˚
0˚
−45˚
0˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
Introduction to ocean tides – p. 44/57
Mf tide on an ocean covered Earth
0
45˚
0˚
180
−45˚
0
45˚
90˚
135˚
180˚
225˚
270˚
315˚
m
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Introduction to ocean tides – p. 45/57
tidal ellipses of flow (M f )
45˚
0˚
−45˚
0˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
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
90˚
45˚
0˚
−45˚
−90˚
0˚
45˚
90˚
135˚
180˚
225˚
270˚
315˚
0˚
km
−10
−8
−6
−4
−2
0
2
4
6
8
10
Introduction to ocean tides – p. 50/57
Staggered C-grid of the Earth
45˚
0˚
−45˚
0˚
90˚
180˚
270˚
Introduction to ocean tides – p. 51/57
Staggered C-grid of the Earth
0˚
−45˚
Introduction to ocean tides – p. 51/57
M2 tide of BOTM
9−150
0
0
12
90
0
30
−−91−21
−60 0 050
0
90
12
90
90120
30
150 120
0
0
−30
119526000
−6 −90
0
06
−690
0
−30
0
3
−
−9
0
−6
0
90
120
−30
−90
−3
−45˚
50
−1 120
−
50
−1 20 90
−1 −
2−060
15
10
9
1200
0
−
9−−01125
−6
0
630
0
−30
120
150
0˚
12 90
105
0 12090
15
0
60
60
60
150
0
150
−30
15
45˚
−30
0
−6
−30
−30
90˚
180˚
270˚
m
0.0
0.1
0.2
0.3
0.5
0.8
4.5
Introduction to ocean tides – p. 52/57
tidal ellipses of flow (M2)
45˚
0˚
−45˚
0˚
45˚
90˚
135˚
180˚
225˚
270˚
Introduction to ocean tides – p. 53/57
315˚
M2 BOTM versus FES99
9−150
0
BOTM
0
60
0
90
12
90
60
12 90
105
0 12090
15
0
60
90
90
0
−45˚
24
−30
0
21
90120
0
30
150 120
0
150
119526000
−6 −90
0
06
−690
0
−30
−30
−9
0
−6
0
−30
90
120
−30
0
50
−1 120
−
50
−1 20 90
−1 −
2−060
15
10
−90
−3
−45˚
0˚
0
−6
−9−−01125
0
630
0
−30
120
150
9
1200
120
−−91−21
−60 0 050
60
150
0
0
30
60
0˚
45˚
15
150
−30
60
FES99
90
12
45˚
90
120
11158000
2240
90
0
−6
−30
−30
90˚
180˚
0˚
270˚
90˚
180˚
270˚
m
m
0.0
0.1
0.2
0.3
0.5
0.8
4.5
0.0
0.1
0.2
0.3
0.5
0.8
4.5
Introduction to ocean tides – p. 54/57
O1 BOTM versus FES99
300
270
240 210
18
0
60
BOTM
240
30
0
−3
240
210
0
−6
0
60
0
0
−30
150
90
60
−30
150
0
24
−45˚
0
12 0
0
21
90
60
30
0˚
270˚
90˚
180˚
270˚
m
m
0.0
0.1
0.2
0.3
0.5
60
30
60
120
150
150
180˚
90 60
0
90˚
0 0
−6 −9
0
−9 −120
−1
50
−150
−60
0
−3
60
90
60
15
0˚
60
−60
−90
−90
0
60
030 0
−3
0
−120
15
−45˚
50
−90
0
12
0
120
0˚
−1
150
30−150
115200
FES99
45˚
−9
90
−15020
−1
150
45˚
0.8
4.5
0.0
0.1
0.2
0.3
0.5
0.8
4.5
Introduction to ocean tides – p. 55/57
M f BOTM versus FES99
−150
−1
210
50
−60
0
120
0
180
30
00
0
18
30
−30 6
0
90
18
910
120
−1−150
−
−
9
6
−3 020050 1120
63000 0
50
90
−60
300
60
60
30
150
18
0
21
18
80 0
0 1
180 180
1
180 180 80
180˚
0.1
0.2
0.3
0.5
0.8
4.5
180
180
270˚
m
m
0.0
0 0
120
360
9100
20
60
30
0 0
−3
−−36
00
−60 0
0
60
0
30
−60
−60
180
0
30
0
90˚
0
30
18
−3
0
18
30
30
60
90
120
30
0
18
180
0
18
0
0˚
270˚
21
0
−60
0
0
180
21
18
−030
−12090
−
−150
0
180˚
11880
0
−45˚
60
30
60
90˚
9600
−30
0
0 3
0
0
120
151029000 0
6 −3
0
−30
150 3600
120 120
90
60
−30
30 30
0
90
−90
−90
0
3300 30
−
30
0
90
0
−90
102
−1−5
0
060
0
−6
−6
60
150
300
90
0
−30
60
−30
0
0
0˚
15
300 300
−6
− 0
30
30 −6
306090 6300
12
90
30
150
0˚
0
121050
−150
−1
150
00
1512
150
FES99
45˚
−12
−150
0
360
0
−90
−−6300
0
−−1
1520
0
30
−45˚
−120
20
0
45˚
−15
BOTM
150
−1
−−
151−29
00
900
6
0
9
12010
50
−1
0
20
−15
50
210
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Introduction to ocean tides – p. 56/57
Introduction to ocean tides – p. 57/57