My 102 sheets to discuss the problem of tidal modeling, there references in the slides refer to the notes that I already uploaded to slideshare.

1. Tides
Planetary sciences II
ae4-876 course year 2017-2018
Ernst Schrama
Del? University of Technology
Faculty of Aerospace Engineering
e.j.o.schrama@tudel?.nl

2. Course contents
We use the Lecture notes on Planetary Sciences and Satellite
Orbit DeterminaOon, latest version 11-Jan-2018:
• Fourier frequency analysis (Chapter 4)
• Measurement techniques (Chapter 6)
• Tide generaOng potenOal (Chapter 14)
• ElasOc response of the solid Earth (Chapter 15)
• Ocean Odes, modelling (Chapter 16)
• Data analysis methods (Chapter 17)
• Tidal loading (Chapter 18)
• Satellite AlOmetry and Odes (Chapter 19)
• Tidal energy dissipaOon (Chapter 20)
The lecture notes are on brightspace.

3. OrganisaOon
• We use a flipped classroom method,
effecOvely this means:
• We rely on self-study rather than classical
lectures
• There will be an assignment on Odes
• Details on how to submit the assignment are
discussed during the contact hours.
• The forum on brightspace is your support
offline the contact hours.

4. Contents of this presentaOon
• Coastal observaOons and frequency analysis of
the Odes at a Ode gauge
• Tide generaOng force, relaOon to potenOal
and Odes
• Fluid dynamics, how does it work out?
• Load Odes and satellite alOmetry
• Tidal energy dissipaOon and internal Odes

5. Coastal observaOons
• What do you see?
– Assume that you have a record of water heights
observed by a Ode gauge
– Subject the data record to a frequency analysis,
see chapter 4 in the lecture notes
– We recover amplitudes and the phase of the
signal which are water heights
– It is noOced that we see daily, twice daily and
longer periodic signals.

6. Tides at Pentrez Plage
High Low tide difference of 7
meter (Brittany, France)

7. Saint Malo

8. Cascais, Portugal
England
USA
Phillipines
Vietnam

9. Sea level variaOon in 1995, Aukfield Plaaorm North Sea
1 y
2 w
½ d

10. Spectra sea level variaOons Aukfield plaaorm
Monthly
Once + Twice Daily
M2
S2/K2
N2
See chapter 4

11. Frequency and Ome
Two different domains

12. Frequency vs Ome domain (1)
Normally signals appear in the time domain:
v(t) with t ∈ [0,T]
where T is the length of a record. If we assume that:
v(t +T) = v(t)
then the function is said to be periodic. Furthermore
if v(t) has a finite number of oscillations in [0,T] then
we can develop v(t) in a series:
v(t) = Ai
i=0
N/2
∑ cos(ωit)+ Bi sin(ωit)
which is known as a Fourier series and where Ai and Bi
denote the Euler coefficients.
21/03/18 12

13. Frequency versus Ome domain (2)
• The variable ωi expresses an angular rate as in the
formula: ωi =i.Δω where Δω=2π/T
• The frequency associated with Δω = 2π/T is 1/T Hertz
(Hz) when T is specified in seconds.
• Otherwise we rescale frequencies to “cycles per
period” and period should be defined.
• For a conOnuous funcOon v(t) there are an infinite
number of frequencies, in this case N is unbounded
• Yet all those frequencies are mulOples of the base
frequency 1/T, except for i=0
• This frequency resoluOon Δf=1/T is determined by the
record length T (and not the sampling)
21/03/18 13

14. Frequency versus Ome domain (3)
In order to calculate Ai and Bi from v(t) defined on [0,T] we exploit
the orthogonality properOes
sin(mx)cos(nx)dx = 0
x=0
2π
∫ regardless of m and n
cos(mx)cos(nx)dx =
0
2π
∫
0 : m ≠ n
π : m = n > 0
2π : m = n = 0
#
$
%
&
%
%
sin(mx)sin(nx)dx =
0
2π
∫
π : m = n > 0
0 : m ≠ n, m = n = 0
#
$
%
&%
21/03/18 14

15. Frequency versus Ome domain (4)
v(x)
cos(mx)
sin(mx)
⎧
⎨
⎪
⎩⎪
⎫
⎬
⎪
⎭⎪0
2π
∫ dx ⇒
An cos(nx)+ Bn sin(nx){ }
n=0
N/2
∑
cos(mx)
sin(mx)
⎧
⎨
⎪
⎩⎪
⎫
⎬
⎪
⎭⎪0
2π
∫ dx ⇒
A0 =
1
2π
v(x)dx, B0 = 0
0
2π
∫
Ai =
1
π
v(x)cos(ix)dx, i > 0
0
2π
∫
Bi =
1
π
v(x)sin(ix)dx, i > 0
0
2π
∫
21/03/18 15

16. Frequency versus Ome domain (4)
• Conversion from Ome domain v(t) to frequency
domain (Euler coefficients) via integrals
• When we speak about “the spectrum” we speak
about the existence of Euler coefficients
• There are efficient algorithms that greatly speed
up the computaOon of the integrals, this is what
is called the Fast Fourier Transform, short FFT
• Euler coefficient pairs are o?en wrioen as
complex numbers
• The inverse operator also exists, this is the iFFT
operator. Both FFT and iFFT exist in MATLAB.
21/03/18 16

17. Frequency versus Omeseries (5)
• EssenOally y=FFT(x) carries out a Fourier transform
• FFT algorithm input
– Real vector x(0..N-1) with N datasamples
– The record starts at 0 and is filled to N-1
• FFT algorithm output
– Euler coefficients are stored in the form of complex numbers
– Stored in y(i) are:
y(0) = A0 + I.B0 , y(1) = A1+ I.B1 , …, y(N/2) = AN/2 + I.BN/2
– I is a complex number:
– Be careful with scaling factors, check this always with a test funcOon of
which you now the Euler coefficients in advance
– Suitable test funcOons are for instance linear sin and cos expressions
• FFT algorithms exploit symmetries of sin and cos funcOons, please
use MATLAB because it is thoroughly debugged.
• Be aware that indices in MATLAB vectors start at 1 (and not 0)
I = −1
21/03/18 17

18. We use complex numbers
Real
Imaginary
H
G
Z = H exp( j G ) = H [ cos(G) + j sin(G) ] = A + j B

19. Amplitude and Phase: H cos(x – G)

20. ProperOes of the FFT transform
• The FFT operator implements the discrete version of
the transformaOon
• There are no more than N/2 unique coefficients, this
sets the highest frequency of a series of N samples.
This is what we call the Nyquist limit
• MulOplicaOon of coefficients in the frequency domain
is convoluOon of two funcOon in the Ome domain.
• Parseval’s idenOfy says that the sum of the squares in
the frequency domain is equal to the sum of the
squares in the Ome domain. (proof via auto
convoluOon)
21/03/18 20

21. What is convoluOon
• It means that you shi? two funcOons along one another while you
mulOply the result, input are f and g, the output is h
• With the convoluOon theorem we can build/design/analyze filters
• Most of the filtering in digital radio is implemented by convoluOon
21/03/18 21
h(t) = f (τ )g(t −τ )d
−∞
∞
∫ τ
F(ω) = FFT( f (t))
G(ω) = FFT(g(t))
H(ω) = F(ω)×G(ω) fast
slow

22. f(x)
g(x)
21/03/18 22
h(x) = f (x)⊗ g(x)
This happens when you multiply two block functions

23. Undersampling, frequency resoluOon
• Whenever we discreOze a signal we sample it
at a certain interval Δx (or Δt)
• Undersampling results in aliasing with the
consequence that the spectrum is deformed
• Oversampling is never a problem, the more
the beoer, oversampling counteracts aliasing
• Frequency resoluOon is determined by the
record length, records that are too short
cause a poorly resolved frequencies
21/03/18 TU Delft class ae3-535 23

25. Aliasing = Folding
frequency
power
Watch how a part of the spectrum above the Nyquist frequency folds back
onto the lower part of the spectrum, this is what we call aliasing
Nyquist
frequency

26. Back to Odes

27. Tide generaOng force (ch 14)
Topic was already explained in planetary sciences I

28. Tidal force or potenOal (ch 3+14)
• Force vectors are easy to understand
• Normally we use a potenOal funcOon U
• U is scalar, you can add different U’s
• Gradient of U is an acceleraOon vector
• Gradient of Odal potenOal results in the
Ode generaOng force for a unit mass
• Proof via work integral method (see notes)

29. Tidal potenOal (chapter 14)
€
Ua
=
µm
rem
re
rem
"
#
$
%
&
'
n= 2
3
∑
n
Pn (cosψ)
rem
re
ψ

30. Laplace equaOon and Legendre funcOons (ch 3)
• If you start with ΔU=0 and if you apply the
method of separaOon of variables then
• U(r,λ,θ) = R(r) G(λ,θ)
• R(r) = c1rn + c2r-(n+1)
• G(λ,θ) = [Anmcos(mλ) + Bnmsin(mλ)] Pnm(cosθ)
• See chapter 3 for all properOes of Legendre
funcOons (it is also part of earlier lectures)

31. Equilibrium Ode (secOon 14.2)
The equilibrium Ode assumes that only the Odal potenOal dictates the
shape of the ocean surface.
• In this case the relaOon becomes:
• Idea is equivalent to Epot = m . g . h where:
– m: mass
– g: gravity (just the 9.81 m/s2 on the Earth’s surface)
– h: height
– Epot : potenOal energy
– U : potenOal energy / mass (potenOal := potenOal energy for unit
mass)
Ug 1−
=ζ

32. Tides deforming the Earth (chapter 15)
• The Earth resists forces that cause deformaOon
– Solid earth Odes differ from equilibrium Odes
(geometric effect)
– There is an induced potenOal because of deformaOon
and as a result there is an induced gravity effect
– Both effects can be modeled by scaling factors using
the Love numbers hn and kn
• Consequences
– One unified theory for: solid earth Odes, long periodic
equilibrium Odes in the ocean, acceleraOons on
satellites, gravimeter Odes, reference system
– Details are discussed in chapter 3

33. Love’s theory
€
ζs = g−1
hn
n= 2
∑ Un
UI
= kn
n= 2
∑ Un
ζo = g−1
(1+ kn − hn )
n= 2
∑ Un

34. Ocean tides (chapter 16)
• Purpose of this part is to describe motions of fluids
with a system of differential equations.
• We are only interested in representing currents and
water levels at time and space scales appropriate to
ocean tides. (several km to global, hours to days)
• Where is the input, what is the response, is it linear?

35. Topics in Fluid Dynamics
• ConOnuity equaOons
• Momentum equaOons
• EquaOons of moOon
• Depth averaged transport
• Boundary condiOon and forcing
• FricOon and advecOon
• Turbulence

36. ConOnuity equaOon (1)
A A’
B B’
River Ocean
Flow in an estuary

37. ConOnuity equaOon (2)
A
B
A’
B’
D
C C’
D’
Uin
Uout
w
Uin ABCD - Uout A’B’C’D’ + w DCD’C’ = 0

38. ConOnuity equaOon (3)
• Surfaces ABCD and A’B’C’D’ are known because
the bathymetry is always provided
• Uin and Uout are measured at “secOons”
• The verOcal transport component w automaOcally
follows from the conOnuity condiOon, you can not
easily observe it
• The verOcal velocity component w may be related
to many different effects

39. ConOnuity equaOon (4)
ρ
dx
dy
dz
u u+du
0)(
0)(.....)(.....)(.....
=
∂
∂
+
∂
∂
+
∂
∂
⇔
=+−++−++−
z
w
y
v
x
u
dwwdydxwdydxdvvdzdxvdzdxduudzdyudzdy
ρ
ρρρρρρ

40. ConOnuity equaOon (5)
If we allow variations along x the u component
becomes:
dydzdx
x
u
udx
x
))((
∂
∂
+
∂
∂
+
ρ
ρ
When this is allowed for all components we get:
0
1
=⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
+
∂
∂
+
∂
∂
+
z
w
y
v
x
u
dt
dρ
ρ
For non-compressible fluids the first term will vanish

41. Momentum equaOon (1)
Procedure: Decompose accelerations on a fluid parcel
P∇−
ρ
1
Fu +×− ω2
g
FguP
Dt
uD
++×−∇
−
= ω
ρ
2
1
Gravity vector
Pressure
gradient
Coriolis + forcing +
friction + ….
Plane of equal pressure

42. Momentum equaOon (2)
Why is the pressure gradient like shown? (Pressure = force / area,
Force = pressure times area)
dx
dy
dz
P P+dP
Force effect x direction: zyx
x
P
zydPzydPPzyP δδδδδδδδδ .....).(..
∂
∂
−=−=+−

43. Momentum equaOon (3)
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
∂∂
∂∂
∂∂
−
=∇
−
zP
yP
xP
P
ρρ
11
x
P
∂
∂
−Force per unit volume
Force per unit mass,
(i.e. acceleration) x
P
∂
∂
−
ρ
1
Pressure gradient:

44. Momentum equaOon (4)
How to get the Coriolis term? Answer: consider an inertial
(i) and a rotating coordinate system (a).
a
a
a
a
a
a
i
i
a
a
a
a
i
i
a
a
i
i
xxxx
xxx
xx
eeeex
eeex
eex
!!!!!!!!!!
!!!!
++==
+==
==
2
Unit vectors coordinate basis: e
Point ordinates on a chosen basis: x

45. Momentum equaOon (6)
For a rotating basis we have:
aaa
aa
eωωeωe
eωe
××+×=
×=
!!!
!
And the consequence is:
aaaaI xωxωωxωxx ×+××+×+= !!!!!! 2
“frame”“centrifugal”

46. Momentum equaOon (7)
Du
Dt
=
−1
ρ
∇P − 2ω × u + g + F
The final result is:
Note:
• pressure and density symbols (watch the symbols)
• g does contain a centrifugal term!
• rotation is considered constant
• this equation holds in an Earth fixed frame
• contains Friction and ForcingF

47. Navier Stokes equaOons
€
Du
Dt
=
−1
ρ
∂p
∂x
+ 2Ωsinφv − 2Ωcosφ w + Fx
Dv
Dt
=
−1
ρ
∂p
∂y
− 2Ωsinφ u + Fy
Dw
Dt
=
−1
ρ
∂p
∂z
+ 2Ωcosφ u − g + Fz
Notes:
- w equation: largest terms result in hydrostatic equation
- D./Dt includes local derivative and advection (later)
For a suitable local coordinate frame we find:

48. HydrostaOc equaOon
For the w-equation we get:
€
∂p
∂z
= −gρ
p(z) = − gρdzH
−η
∫ = gρ(H +η)
Example:
• At 1000 m the pressure is approximately 100 bar
• Density changes in the oceans are relatively small (3%)

49. Velocity depth averaged equaOon
f = 2Ωsinϕ
€
Du
Dt
= −g
∂η
∂x
+ fv + Fx
Dv
Dt
= −g
∂η
∂y
− fu + Fy
Dη
Dt
= −(H +η)
∂u
∂x
+
∂v
∂y
%
&
'
(
)
*
In these equations u and v are averaged over the entire water
column. The Coriolis term

50. Laplace Odal equaOons
€
Du
Dt
= −g
∂η
∂x
+ fv + Fx
Dv
Dt
= −g
∂η
∂y
− fu + Fy
Dη
Dt
= −(H +η)
∂u
∂x
+
∂v
∂y
%
&
'
(
)
*
€
Fx
Fy
"
#
$
%
&
' =
∂Γ ∂x
∂Γ ∂y
"
#
$
%
&
' +
rx
ry
"
#
$
%
&
'
Γ = Hω
ω
∑ cos(χω (t) − Gω )
The terms u,v and η represent velocity and water level, f is a
Coriolis term, H is the depth of the ocean, Γ is a forcing
function, r represents dissipative forcing. (Chapter 16)

51. ProperOes Laplace Tidal EquaOons
• Tidal waves behave like any other shallow water wave
• For long gravity waves the propagaOon speed is
approximately (g.H)1/2 with H being the local depth
• Surface speed astronomic Ode is faster than (g.H)1/2
• Oceans and conOnental seas respond like a vibraOng
membrane, actually, the system looks like a Helmholtz
equaOon
• Resonances may occur (think of bays and coastal seas)
• In principle there are three Odal species (long periodic,
diurnal and semi-diurnal) This property is due to the forcing
mechanism AND due to the fact that the problem is nearly
linear

52. Helmholtz equaOon (1)
€
Df
Dt
≈
df
dt
Approach: insert a test function and avoid non-linearity
u(t) = ˆuexp( jωt)
v(t) = ˆvexp( jωt)
η(t) = ˆηexp( jωt)
€
Du
Dt
= −g
∂η
∂x
+ fv + Fx
Dv
Dt
= −g
∂η
∂y
− fu + Fy
Dη
Dt
= −(H +η)
∂u
∂x
+
∂v
∂y
%
&
'
(
)
*
€
Fx
Fy
"
#
$
%
&
' =
∂Γ ∂x
∂Γ ∂y
"
#
$
%
&
' +
C
H
u
v
"
#
$
%
&
'
Γ = ˆΓexp( jωt)
Question: is this a linear problem?

53. Helmholtz equaOon (2)
z = a + jb, j = −1, Re(z) = a, Im(z) = b
αz = αa + jαb
z1 + z2 = (a1 + a2 ) + j(b1 + b2 )
z1z2 = (a1a2 − b1b2) + j(a1b2 + a2b1)
z1 /z2 = (a1 + jb1)/(a2 + jb2 )
z = a − jb
exp( jx) = cos(x) + jsin(x)

54. Helmholtz equaOon (3)
Re
Im
α
R
R.exp( jα) = Rcos(α) + jRsin(α)
αcosR
αsinR

55. Helmholtz equaOon (4)
If you substitute the characteristic solutions for all dependent
variables in the shallow water equations:
(ω2
− f 2
) ˆη + c2 ∂2
ˆη
∂x2
+
∂2
ˆη
∂y2
⎛
⎝⎜
⎞
⎠⎟ ≈ 0
c = gH
In this relation c is the surface speed of the tide. You can
obtain it by substitution of the test function in the Laplace
tidal equations.

56. Examples of Helmholtz soluOons
€
1
r
∂
∂r
r
∂η
∂r
$
%
&
'
(
) +
1
r2
∂2
η
∂ϕ2 + λη = 0 for λ = µ2
⇒
η = AJm (µr) + BYm (µr)[ ]× (Ccosmϕ + Dsinmϕ)

57. Boundary condiOons and forcing
• There are always boundary condiOons.
• No-flux condiOon, inner product (u.n) = 0 at
the shoreline
• IniOal condiOons apply for Ome stepping
models
• Forcing in F is related to FricOon and
GeneraOon

58. Example mixed boundary condiOons
Land
Ocean
η,,vu+
0),( =nv
Closed flux boundary
Open flux boundary

59. Forcing
• Question: what are Fx and Fy in the momentum equation?
• Answer: this depends on the type of problem
• Some examples are:
• Gradients tide generating potential
• Friction
• (Advection is not a forcing term)

60. Linearity
Ocean tides show a linear behavior when sea bottom friction
is linear and when advection is ignored:
rx =
Cu
H
and ry =
Cv
H
€
Γ(t) = ˆΓexp( jωt)
In reality sea bottom friction is quadratic:
r = Cu u

61. Linearity (2)
)exp(ˆ tjωΓ )exp(ˆ tjh ωΓ= *Zh
Z: admittance function

62. Non Linear Response
Some Box)exp(ˆ tjωΓ
)exp(ˆ
11 tjh ω
)exp(ˆ
22 tjh ω

63. AdvecOon
z
u
w
y
u
v
x
u
u
t
u
Dt
Du
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
=
Local
effect
Advection part
Total
derivative
Gradient of u
in x direction is large

64. FricOon
• This is described by
• Fx acts on a molecular level.
• The term ν describes kinemaOc molecular viscosity, ν is
about 10-6 m2/s
• FricOon always introduces dissipaOon
• To apply fricOon in Ode models you have to re-scale fricOon
by means of horizontal and verOcal eddy viscosity terms
• Scale factor depends on numerical consideraOons
Fx =ν
∂2
u
∂x2
+
∂2
u
∂y2
+
∂2
u
∂z2
"
#
$
%
&
'

65. Scale consideraOons
• To assess the forcing terms you have to look at the
scale of certain variables
• Assume that velociOes u and du are scaled like U
• Assume for dx a length scale L.
• Assess the raOo between fricOon and advecOon

66. The Reynolds number
Re
/
/
)(
)(
2
2
22
==≈
∂∂
∂∂
ννν
UL
LU
LU
xu
xuu
Rule of thumb: if Re<1000 then a flow is not
turbulent, turbulence occurs for large Re numbers
Examples: transport in the gulf stream and water
flow through a tube, smoking match example

67. Smoking match example
Task: estimate the
Reynolds number

68. Dispersion relaOon
u(t) = ˆuexp( j(ωt − kx − ly))
v(t) = ˆvexp( j(ωt − kx − ly))
η(t) = ˆηexp( j(ωt − kx − ly))
∇F = 0
Unforced or free waves fulfill the following relation:
Show that this results in the following dispersion relation
)()( 2222
lkgHf +=−ω
Do free waves exist for all frequencies at all latitudes?

69. Simple Ode model
€
h(φ,λ,t) = Hi φ,λ( )
i
∑ cos wi t − t0( )− Gi φ,λ( )( )
Hi φ,λ( ): tidal amplitude
Gi φ,λ( ): Greenwich phase
Deep ocean tides can be predicted by the model:
The tidal frequency is astronomically determined, for a
location at which the tides are predicted it is sufficient to
have a series of H and G values available, t0 is a reference
time, t is the time of observation.

70. M2 tide with phase lines in hours.
North Sea Tides

71. M2 movie

72. Tidal data analysis (chapter 17)
• We already did it in the introducOon!
• Make use of the property that Odal frequencies are
astronomically determined
• Side lines (like 255.545) are a problem, this is why you
need nodal correcOon terms (see secOon 14.3)
• Least squares esOmators do the rest, each Odal frequency
introduces two other parameters to esOmate.
• Response method is a lot easier, but requires you to
define an admioance funcOon

73. Response method
• U is known with astronomical precision
• Z is a smooth funcOon
€
H(ω) = Z(ω) × U(ω)
U =
µm
rem
re
rem
$
%
&
'
(
)
n=2,3
∑
n
Pn cos ψ( )( )

74. Load Odes (chapter 18)
• Tidal moOons cause loading on the Earth
• The Earth shows an elasOc response for Odal
frequencies.
• The surface will flex and this effect can be
seen at a distance from the Odal source
• Load Odes are obtained by convoluOon of the
ocean Odes (see chapter 18)

75. ElasOc deformaOons due to
surface mass layer
nl θ,λ,t( )= G(ψ)dM
Ω
∫ θ'
,λ'
,t( )
G(ψ) =
re
Me
hn
'
n=0
∞
∑ Pn cosψ( )
),(:),(: ''
λθλθ qp
+
+
p
q
qp

76. AlOmetry and Odes (ch. 19)
Source: JPL
Satellite Altimetry is documented in section 6.3.1

77. Topex/Poseidon groundtrack

78. RepeaOng ground tracks
ze
I
ωo
ωe
xe
ye
(xe,ye,ze) : Earth Fixed
ωo = (ω + f )
ωe = (Ω−θ)
!ωo
!ωe
=
Nr
Nd
Nr and Nd do not share factors (primes)
C20 and Earth rotation control !ω !f !Ω and !θ
Study: 6.3.1

79. Mesoscale variability

80. AlOmetry and Aliasing
• Tides occur mostly on a diurnal or semi-diurnal
frequency
• Satellite ground tracks repeat within a few days.
• The consequence is that the Ode signal is
undersampled
• How bad is this for observing ocean Odes with
satellite alOmeter?
• Study chapter 4.2.2 on the Nyquist theorem

81. AlOmetry and Odes
• AlOmetry:
– Topex/Poseidon (and both Jason satellites), provide esOmates
of ocean Odes at one second intervals in the satellite flight
(along track) direcOon.
• Quality Models:
– The quality of these models can be verified by means of an
independent comparison to in-situ Ode gauge observaOons.
– RMS difference for M2: 1.5 cm, S2: 0.94, O1: 0.99, K1: 1.02,
– Other consOtuents are well under the 0.65 cm level,
• Data assimilaOon:
– There are various numerical schemes that assimilate alOmeter
informaOon in barotropic ocean Ode models. (empirical,
representer method, nudging)

82. M2 ocean Ode

83. Tidal Energy DissipaOon (chapter 20)
• Tides are generated by the Sun and the Moon
• ConOnuous forcing shouldn’t result in conOnuous
amplificaOon, somehow a wave has to dissipate its energy
• Conversion of mechanical energy into heat or another form
of energy is called dissipaOon
• Booom fricOon is such a process, and for long it was
assumed this is the only way it works
• Global energy dissipaOon studies: purpose, provide one
number like 2.5 TW for M2 for the enOre planet
• Local energy dissipaOon studies: purpose, provide one
number for e.g. a coastal sea.
• Local dissipaOon is related to internal Odes

84. Tidal energy dissipaOon
3.82 cm/yr
M2 : 2.50 +/- 0.05 TW
Introduced in planetary sciences I, study section 20.1 and 20.2

85. DissipaOon calculaOons
• We combine momentum and conOnuity equaOons in the depth average
LTE, this results in equaOon (20.9)
• In equaOon (20.9) we mulOply currents u Omes a dissipaOve force vector
F, and we average all terms over a Odal cycle
• When you take the average several terms cancel, next for global
dissipaOon calculaOons we integrate over the oceans, furthermore we
impose no flux boundaries this results in a compact equaOon.
• The obtained dissipaOon equaOon is global in nature, thus it is valid for the
enOre system, it does not pin-point where dissipaOon is happening
Section: 20.3

86. Global esOmates Odal energy dissipaOon
Q1 O1 P1 K1 N2 M2 S2 K2
SW80 0.007 0.176 0.033 0.297 0.094 1.896 0.308 0.024
FES99 0.008 0.185 0.033 0.299 0.109 2.438 0.367 0.028
GOT992 0.008 0.181 0.032 0.286 0.110 2.414 0.428 0.029
TPXO51 0.008 0.186 0.032 0.293 0.110 2.409 0.376 0.030
NAO99b 0.007 0.185 0.032 0.294 0.109 2.435 0.414 0.035
Mean 0.008 0.184 0.032 0.294 0.110 2.424 0.396 0.030
Units: TeraWatts
Section 20.3

87. Global Odal energy dissipaOon
• SecOon 20.2 discusses independent astronomic and
geodeOc esOmates:
– Secular trend in Earth Moon distance
– Earth rotaOon slow down
– Phase lags ocean, body or atmospheric Odes
• SecOon 20.3 shows that alOmeter derived ocean Ode
models have roughly the same global dissipaOon
esOmates
• But there are some noOceable differences between
the results

88. Results Global DissipaOon
• High coherence between models, SW80 is an excepOon
because it is before the alOmeter era (TOPEX/Poseidon).
For this reason global dissipaOon esOmates are not a good
quality indicator.
• M2: oceanic 2.42, astronomic 2.51 TW, the difference is
dissipated in the solid Earth Ode (Ray, Eanes and Chao,
1996). Independent body Ode dissipaOon measurements
by gravimeters are not convincing at the moment (only a
0.1 of a degree lag is expected)
• S2: oceanic 0.40, geodeOc 0.20 TW, the difference is
mostly dissipated in the atmosphere (Platzman,1984)
• Challenge: do this for another planet or moon (like Europe
orbiOng around Jupiter)

89. Europa surface

90. Local DissipaOon (1)
∂u
∂t
+ f ×u = −g∇ζ + ∇Γ − F
∂ζ
∂t
= −∇.(uH )
W − P = D
W = ρH < u.∇Γ >
P = gρH∇.< uζ >
D = ρH < u.F >
W: Work
P: Divergence Energy Flux
D: Dissipation
∫
+=
=
=
Ttt
tt
dtFu
T
D
0
0
).(
1

91. Local dissipaOon (2)
• In order to compute local dissipaOons you must
specify the forcings and the velociOes
• AlOmetry only observes Odal heights, it does not
provide us with Odal velociOes (perhaps acousOc
sounding can independent values)
• The computaOon of barotropic velociOes requires
a numerical inversion scheme
• The forcing terms involve self-aoracOon and Odal
loading and the Ode generaOng potenOal.

93. Internal Odes
20 m
5 cm
160 km
ρ1
ρ2
h1
h2

94. Internal Odes seen at the surface
Mitchum and Ray 1996 GRL

96. Internal Odes (1)
• High frequency oscillaOon is imposed on the along track Ode
signal, wavelength typically 160 km for M2, (Mitchum and
Ray, 1996).
• The feature stands above the background noise level.
• The phenomenon is visible for M2 and S2 (hardly for K1).
• There is some contaminaOon in the T/P along track Odes in
regions with increased meso-scale variability.
• “Clean” Along track Ode features are visible around Hawaii,
French Polynesia and East of Mozambique.
• Along track Odes appear near oceanic ridge systems or along
the conOnental shelf.

97. What is the conOnental shelf?

98. Internal Odes (3)
)( 21
212
2
1
hh
hh
gc
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ Δ
=
Δ
−=
ρ
ρ
ρ
ρ
ζ
ζ
m5000
m300
mkg1025
003.0
2
1
3-
=
=
=
=Δ
h
h
ρ
ρρ
c = 3 ms−1
L =12.42 ×3600 ×3 = 140 km
(Apel, 1987)

99. Areas of interest
-30 -20 -10 0 10 20 30 mW/m2
R Ray, GSFC

100. Conclusions (1)
• Global dissipaOon:
– There are consistent values for most models,
• The M2 dissipaOon converges at 2.42 TW to within 2%
– Independent methods to determine the rate of energy
dissipaOon (LLR, satellite geodesy). LLR arrives at 2.5 TW
for M2
– Comparison to astronomic/geodeOc values:
• 0.2 TW at S2 for dissipaOon in the atmosphere
• 0.1 TW at M2 for dissipaOon in the solid earth
• gravimetric confirmaOon of the 0.1 TW is very challenging
– History of Earth rotaOon relies of dissipaOon esOmates from paleo-
oceanographic ocean Ode models.

101. Conclusions (2)
• Local dissipaOon:
– it is the same Odal energeOcs equaOon, the integraOon domain is
however local and you need Odal transport esOmates at the boundary
of the local integraOon domain
– realisOc esOmates are more difficult to obtain and require an
inversion of Odal elevaOons into currents
• Along track Ode signal:
– so far only results for standing waves
– appears as high frequency Odal variaOons in along track alOmetry,
– appear to be related to internal wave features,
– coherence to local dissipaOons,
– visibility: Hawaii, Polynesia, Mozambique, Sulu Celebes region

102. Discussion
• Why relate internal Odes to dissipaOon?
– Mixing in the deep ocean is according to (Egbert and Ray,
2000) parOally caused by internal Odes.
– Their main conclusion is that the deep oceanic esOmate
for M2 is about 0.7 TW.
– Walter Munk stated that 2 TW is required for maintaining
the deep oceanic straOficaOon.
– Approximately 1 TW could come from wind
– The remainder could be caused by internal Odes.