Post on 27-Mar-2015
Slide 1
The Wave Model
ECMWF, Reading, UK
The Wave Model (ECWAM) Slide 2
Lecture notes can be reached at:
http://www.ecmwf.int- News & Events
- Training courses - meteorology and computing
- Lecture Notes: Meteorological Training Course
- (3) Numerical methods and the adiabatic formulation of models
- "The wave model". May 1995 by Peter Janssen
(in both html and pdf format)
Direct link:http://www.ecmwf.int/newsevents/training/rcourse_notes/NUMERICAL_METHODS/
The Wave Model (ECWAM) Slide 3
Description ofECMWF Wave Model (ECWAM):
Available at: http:// www.ecmwf.int
- Research
- Full scientific and technical documentation of the IFS
- VII. ECMWF wave model
(in both html and pdf format)
Direct link:
http://www.ecmwf.int/research/ifsdocs/WAVES/index.html
The Wave Model (ECWAM) Slide 4
Directly Related Books:
“Dynamics and Modelling of Ocean Waves”.
by: G.J. Komen, L. Cavaleri, M. Donelan,
K. Hasselmann, S. Hasselmann,
P.A.E.M. Janssen.
Cambridge University Press, 1996.
“The Interaction of Ocean Waves and Wind”.
By: Peter Janssen
Cambridge University Press, 2004.
Slide 5
INTRODUCTION
The Wave Model (ECWAM) Slide 6
Introduction:
State of the art in wave modelling.
Energy balance equation from ‘first’ principles.
Wave forecasting.
Validation with satellite and buoy data.
Benefits for atmospheric modelling.
The Wave Model (ECWAM) Slide 7
What we are dealing with?
1.010.0 0.03 3x10-3 2x10-5 1x10-5
Frequency (Hz)
Forcing:wind
earthquakemoon/sun
Restoring:gravity
surface tension Coriolis force
12
The Wave Model (ECWAM) Slide 8
What we are dealing with?
Wave Period, T
Wave Length, Wave Height, H
Water surfaceelevation,
The Wave Model (ECWAM) Slide 9
What we are dealing with?
The Wave Model (ECWAM) Slide 10
What we are dealing with?
The Wave Model (ECWAM) Slide 11
What we are dealing with?
The Wave Model (ECWAM) Slide 12
What we are dealing with?
The Wave Model (ECWAM) Slide 13
What we are dealing with?
The Wave Model (ECWAM) Slide 14
What we are dealing with?
The Wave Model (ECWAM) Slide 15
What we are dealing with?
The Wave Model (ECWAM) Slide 16
What we are dealing with?
The Wave Model (ECWAM) Slide 17
What we are dealing with?
The Wave Model (ECWAM) Slide 18
What we are dealing with?
The Wave Model (ECWAM) Slide 19
What we are dealing with?
The Wave Model (ECWAM) Slide 20
What we aredealing with?
The Wave Model (ECWAM) Slide 21
What we are dealing with?
The Wave Model (ECWAM) Slide 22
What we are dealing with?
Fetch, X, or Duration, t
Simplified sketch!
En
erg
y, E
,o
f a
sin
gle
co
mp
on
ent
Slide 23
PROGRAMOF
THE LECTURES
The Wave Model (ECWAM) Slide 24
Program of the lectures:
1. Derivation of energy balance equation
1.1. Preliminaries- Basic Equations- Dispersion relation in deep & shallow water.- Group velocity.- Energy density.- Hamiltonian & Lagrangian for potential flow.- Average Lagrangian.- Wave groups and their evolution.
The Wave Model (ECWAM) Slide 25
Program of the lectures:
1. Derivation of energy balance equation
1.2. Energy balance Eq. - Adiabatic Part - Need of a statistical description of waves: the wave spectrum.- Energy balance equation is obtained from averaged Lagrangian.- Advection and refraction.
The Wave Model (ECWAM) Slide 26
Program of the lectures:
1. Derivation of energy balance equation
1.3. Energy balance Eq. - Physics Diabetic rate of change of the spectrum determined by:- energy transfer from wind (Sin)
- non-linear wave-wave interactions (Snonlin)
- dissipation by white capping (Sdis).
The Wave Model (ECWAM) Slide 27
Program of the lectures:
2. The WAM Model
WAM model solves energy balance Eq.
2.1. Energy balance for wind sea - Wind sea and swell.- Empirical growth curves.- Energy balance for wind sea.- Evolution of wave spectrum.- Comparison with observations (JONSWAP).
The Wave Model (ECWAM) Slide 28
Program of the lectures:
2. The WAM Model
2.2. Wave forecasting - Quality of wind field (SWADE).- Validation of wind and wave analysis using ERS-1/2, ENVISAT and Jason altimeter data and buoy data.- Quality of wave forecast: forecast skill depends on sea state (wind sea or swell).
The Wave Model (ECWAM) Slide 29
Program of the lectures:
3. Benefits for Atmospheric Modelling
3.1. Use as a diagnostic tool Over-activity of atmospheric forecast is studied by comparing monthly mean wave forecast with verifying analysis.
3.2. Coupled wind-wave modelling Energy transfer from atmosphere to ocean is sea state dependent. Coupled wind-wave modelling. Impact on depression and on atmospheric climate.
The Wave Model (ECWAM) Slide 30
Program of the lecture:4. Tsunamis
4.1. Introduction- Tsunami main characteristics.- Differences with respect to wind waves.
4.2. Generation and Propagation- Basic principles.- Propagation characteristics.- Numerical simulation.
4.3. Examples- Boxing-Day (26 Dec. 2004) Tsunami.- 1 April 1946 Tsunami that hit Hawaii
Slide 31
1DERIVATION
OF THEENERGY BALANCE EQUATION
The Wave Model (ECWAM) Slide 32
1. Derivation of the Energy Balance Eqn:
Solve problem with perturbation methods:(i) a / w << 1 (ii) s << 1
Lowest order: free gravity waves.Higher order effects:
wind input, nonlinear transfer & dissipation
The Wave Model (ECWAM) Slide 33
Result:
Application to wave forecasting is a problem:
1. Do not know the phase of waves Spectrum F(k) ~ a*(k) a(k) Statistical description
2. Direct Fourier Analysis gives too many scales:Wavelength: 1 - 250 mOcean basin: 107 m2D 1014 equations
Multiple scale approachshort-scale, O(), solved analyticallylong-scale related to physics.
Result: Energy balance equation that describes large-scale evolution of the wave spectrum.
Deterministic Evolution Equations
Slide 34
1.1PRELIMINARIES
The Wave Model (ECWAM) Slide 35
1.1. Preliminaries:Interface between air and ocean:
Incompressible two-layer fluidNavier-Stokes:
Here
and surface elevation follows from:
Oscillations should vanish for: z ± and z = -D (bottom)
No stresses; a 0 ; irrotational potential flow: (velocity potential )
Then, obeys potential equation.
0 u
gPuut
1
),( tx
z
z
w
a
,
,
wut
u
The Wave Model (ECWAM) Slide 36
Conditions at surface:
Conditions at the bottom:
Conservation of total energy:
with
2
2
2
2
2
2
,0yxz
Laplace’s Equation
][0
;2
212
21 Bernoullig
zt
ztz
0;
z
Dz
xdE
D z
xdg2
2
212
21
The Wave Model (ECWAM) Slide 37
Hamilton Equations
Choose as variables: and
Boundary conditions then follow from Hamilton’s equations:
Homework: Show this!
Advantage of this approach: Solve Laplace equation with boundary conditions:
= (,).
Then evolution in time follows from Hamilton equations.
),,(),( tzxtx
E
t
E
t
;
0),(;),(
Dzxz
zx
The Wave Model (ECWAM) Slide 38
Lagrange Formulation
Variational principle:
with:
gives Laplace’s equation & boundary conditions.
0dx dy dt L
2
212
D
dz g zt z
L
The Wave Model (ECWAM) Slide 39
Intermezzo
Classical mechanics: Particle (p,q) in potential well V(q)
Total energy:
Regard p and q as canonical variables. Hamilton’s equations are:
Eliminate p Newton’s law
= Force
V(q))(2
21 qVE m
p
mp
pE
tqq
qV
qE
tpp
qVqm
The Wave Model (ECWAM) Slide 40
Principle of “least” action. Lagrangian:
Newton’s law Action is extreme, where
Action is extreme if (action) = 0, where
This is applicable for arbitrary q hence (Euler-Lagrange equation)
212 ( ) £ ( , )T V m q V q q q L
2
1
( , )t
t
action dt q q L
( ) ( , ) ( , )action dt q q q q q q L L
q qdt q q L L
q t qdt q
LL
0q qt qm q V L L
The Wave Model (ECWAM) Slide 41
Define momentum, p, as:
and regard now p and q are independent, the Hamiltonian, H , is given as:
Differentiate H with respect to q gives
The other Hamilton equation:
All this is less straightforward to do for a continuum. Nevertheless, Miles obtained the Hamilton equations from the variational principle.
Homework: Derive the governing equations for surface gravity waves from the variational principle.
( )qp q q p L
212( , ) ( )p
q mH p q q V q L L
pHq qq t t
L L q
Htp
qHq q qp p p pq q q
L L L
The Wave Model (ECWAM) Slide 42
Linear TheoryLinearized equations become:
Elementary sines
where a is the wave amplitude, is the wave phase.
Laplace:
02
2
z
gt
ztz ;0
0;
z
Dz
txkezZea ii
;)(;
222 ;0 yx kkkkZkZ
The Wave Model (ECWAM) Slide 43
Constant depth: z = -D Z'(-D) = 0 Z ~ cosh [ k (z+D)]
with
Satisfying Dispersion Relation
Deep water Shallow water
D D 0
dispersion relation:
phase speed:
group speed:
Note: low freq. waves faster! No dispersion.
energy:
gt iea
)(cosh
)(cosh
kD
Dzka
gi
zt
kg2
g
kc
cg
kvg 2
1
2
1
dxaag 1* ;2
DgkDgc
Dgvg
)(tanh2 Dkkg
The Wave Model (ECWAM) Slide 44
Slight generalisation:Slowly varying depth and current, , = intrinsic frequency
Wave GroupsSo far a single wave. However, waves come in groups.
Long-wave groups may be described with geometrical optics approach:
Amplitude and phase vary slowly:
),( txU
Uk
)(tanh Dkkg
(t)
t
..),( ),( ccetxa txi
t
a
aka
a
1,
1
The Wave Model (ECWAM) Slide 45
Local wave number and phase (recall wave phase !)
Consistency: conservation of number of wave crests
Slow time evolution of amplitude is obtained by averaging the Lagrangian over rapid phase, .Average :
For water waves we get
with
txk
kt
;
0 kt
12 d L L
2 2 2 4 23
4
( )1 1 9 10 91 ( )
2 2 8ok U k T T
Og k T g T
L
)(tanh,22
DkTag
The Wave Model (ECWAM) Slide 46
In other words, we have
Evolution equations then follow from the average variational principle
We obtain: a :
:
plus consistency:
Finally, introduce a transport velocity
then
is called the action density.
( , , )k a
L L
( , , ) 0dx dt k a
L
0a
L
0kt x
L L
0 kt
/k
u
L L
0ut x
L L
The Wave Model (ECWAM) Slide 47
Apply our findings to gravity waves. Linear theory, write as
where
Dispersion relation follows from hence, with ,
Equation for action density, N ,
becomes
with
Closed by
12 ( , )D k
L
1)( 2
Tkg
UkD o
0 0a D LTkg
oUk
/N L
0
NvNt g
kvg
/
0
t
k
The Wave Model (ECWAM) Slide 48
Consequencies
Zero flux through boundaries
hence, in case of slowly varying bottom and currents, the wave energy
is not constant.
Of course, the total energy of the system, including currents, ... etc., is constant. However, when waves are considered in isolation (regarded as “the” system), energy is not conserved because of interaction with current (and bottom).
The action density is called an adiabatic invariant.
.constNxdtot
xdE
The Wave Model (ECWAM) Slide 49
Homework: Adiabatic Invariants (study this)
Consider once more the particle in potential well.
Externally imposed change (t). We have
Variational equation is
Calculate average Lagrangian with fixed. If period is = 2/ , then
For periodic motion ( = const.) we have conservation of energy thus also momentum is
The average Lagrangian becomes
( , , ) 0dt q q L
0q q
d
dt L L
02dt
L L
qE q L L),,( Eqqq
qp L ),,( Eqpp
( , , )2
p q E dq E
L
The Wave Model (ECWAM) Slide 50
Allow now slow variations of which give consequent changes in E and . Average variational principle
Define again
Variation with respect to E & gives
The first corresponds to the dispersion relation while the second corresponds to the action density equation. Thus
which is just the classical result of an adiabatic invariant. As the system is modulated, and E vary individually but
remains constant: Analogy waves.
Example: Pendulum with varying length!
( , , ) 0dt E L
0 , 0E
d
dt L L
1const.
2p dq
L
dqpEN
2
1),(
Slide 51
1.2ENERGY BALANCE EQUATION:
THE ADIABATIC PART
The Wave Model (ECWAM) Slide 52
1.2. Energy Balance Eqn (Adiabatic Part):Together with other perturbations
where wave number and frequency of wave packet follow from
Statistical description of waves: The wave spectrum
Random phase Gaussian surface
Correlation homogeneouswith = ensemble average
depends only on
dissipnonlinwindg t
N
t
N
t
NNvN
t
0
t
k
)(tanh DkkgUk o
)()( 21 xxR
12 xx
)()()( xxR
The Wave Model (ECWAM) Slide 53
By definition, wave number spectrum is Fourier transform of correlation R
Connection with Fourier transform of surface elevation.
Two modes
is real
Use this in homogeneous correlation
and correlation becomes
)()2(
1)(
2
RedkF ki
)()( )(ˆ)(ˆ),( txkitxki ekkdekkdtx
)(ˆ)(ˆ * kk
..)(ˆ),( )( ccekkdtx txki
0)(ˆ)(ˆ kk
)()(ˆ)(ˆ)(ˆ2
* kkkkk
..)(ˆ)( )(2
ccekkdR ki
The Wave Model (ECWAM) Slide 54
The wave spectrum becomes
Normalisation: for = 0
For linear, propagating waves:potential and kinetic energy are equal
wave energy E is
Note: Wave height is the distance between crest and trough : given wave height.
Let us now derive the evolution equation for the action density, defined as
2)(ˆ2)( kkF
)()0(2 kFkdR
)(2 kFkdggE ww
24 sH )(kF
)(
)(kF
kN
)(tanh Dkkg
The Wave Model (ECWAM) Slide 55
Starting point is the discrete case.
One pitfall: continuum: , t and independent.discrete: , local wave number.
Connection between discrete and continuous case:
Define
then
Note: !
x
k
),( txkk
otherwise
kkkifk
,0
,1)( 2
121
22;)()(
kkk
kaNkkNkkN
),( txkk
O k
k
The Wave Model (ECWAM) Slide 56
Then, evaluate
using the discrete action balance and the connection.
The result:
This is the action balance equation.
Note that refraction term stems from time and space dependence of local wave number! Furthermore,
and
),,( txkNt
SSSSNNvNt dissipnonlinwindxkgx
)()(
)(tanh, DkkgUk
kvg
/
The Wave Model (ECWAM) Slide 57
Further discussion of adiabatic part of action balance:
Slight generalisation: N(x1, x2, k1 , k2, t)
writing
then the most fundamental form of the transport equation for N in the absence of source terms:
where is the propagation velocity of wave groups in z-space.
This equation holds for any rectangular coordinate system.
For the special case that & are ‘canonical’ the propagation equations (Hamilton’s equations) read
For this special case
),,,( 2121 kkxxz
0)(
Nzz
Nt i
i
z
),( xkx
x
k
i
ii
i xk
kx ;
0/ ii zz
The Wave Model (ECWAM) Slide 58
hence;
thus N is conserved along a path in z-space.
Analogy between particles (H) and wave groups ():
If is independent of t :
(Hamilton equations used for the last equality above)
Hence is conserved following a wave group!
0
Nz
zNt
Ndt
d
ii
0
i
ii
ii
i kk
xx
zz
dt
d
The Wave Model (ECWAM) Slide 59
Lets go back to the general case and apply to spherical geometry (non-canonical)
Choosing(: angular frequency, : direction, : latitude, :longitude)
then
For simplification.Normally, we consider action density in local Cartesian frame (x, y):
R is the radius of the earth.
),,,,(ˆ tN
0)ˆ()ˆ()ˆ()ˆ(ˆ
NNNNNt
0/ t
cosˆ 2RNN
The Wave Model (ECWAM) Slide 60
Result:
With vg the group speed, we have
0)()()()cos(cos
1
NNNNNt
RUv og /)cos(
)cos(/)sin( RVv og
2/)(/tansin kkkRvg
t /
The Wave Model (ECWAM) Slide 61
Properties:
1. Waves propagate along great circle.
2. Shoaling:Piling up of energy when waves, which slow down in shallower water, approach the coast.
3. Refraction:* The Hamilton equations define
a ray light waves* Waves bend towards shallower water!* Sea mountains (shoals) act as lenses.
4. Current effects:Blocking
vg vanishes for k = g / (4 Uo2) !
oUkkg
Slide 62
1.3ENERGY BALANCE EQUATION:
PHYSICS
The Wave Model (ECWAM) Slide 63
1.3. Energy Balance Eqn (Physics):
Discuss wind input and nonlinear transfer in some detail. Dissipation is just given.
Common feature: Resonant Interaction
Wind: Critical layer: c(k) = Uo(zc).
Resonant interaction betweenair at zc and wave
Nonlinear: i = 0 3 and 4 wave interaction
zc
Uo=c
The Wave Model (ECWAM) Slide 64
Transfer from Wind:
Instability of plane parallel shear flow (2D)
Perturb equilibrium: Displacement of streamlinesW = Uo - c (c = /k)
)()(,)(
ˆ,ˆ)(:um”“Equilibri
zdzgzPz
eggezUU
oooo
zxoo
0d
d
1
d
d0
:fluid Adiabatic
t
gPut
u
Ww /
The Wave Model (ECWAM) Slide 65
give Im(c) possible growth of the wave
Simplify by taking no current and constant density in water and air.
Result:
Here, =a /w ~ 10-3 << 1, hence for 0, !
Perturbation expansion
growth rate = Im(k c1)
)()d
d(
d
d 222ooo gWk
zW
z
)(d
d;,0 z
zz oo
0,
)]0(/[)1(,1)0(
0,d
d
d
d
2
222
a
aa
aa
z
kgc
zWkz
Wz
c2 = g/k
)1(,/ 121
11 akooo cckgcccc
The Wave Model (ECWAM) Slide 66
Further ‘simplification’ gives for = w/w(0):
Growth rate:
Wronskian:
0,
)(lim,1)0(
)0at(singular Eq.Rayleigh d
d 22
2
z
cUW
WWkz
W
occ
o
ooo
o
*
0
1( , )
4 zo k
W
* *( )i W
The Wave Model (ECWAM) Slide 67
1. Wronskian is related to wave-induced stress
Indeed, with and the normal mode formulation for u1, w1: (e.g. )
2. Wronskian is a simple function, namely constant except at critical height zc
To see this, calculate d/dz using Rayleigh equation with proper treatment of the singularity at z=zc
where subscript “c” refers to evaluation at critical height zc (Wo = 0)
11 wuw 0 u
..1 cceuu i
)( ** wwwwk
iw
2d2 ( )
d oc c oW Wz
W
zcz
w
The Wave Model (ECWAM) Slide 68
This finally gives for the growth rate (by integrating d/dz to get (z=0) )
Miles (1957): waves grow for which the curvature of wind profile at zc is negative (e.g. log profile).
Consequence: waves grow slowing down wind: Force = dw / dz ~ (z) (step
function)For a single wave, this is singular! Nonlinear theory.
2
2 coc
oc
W
W
k
The Wave Model (ECWAM) Slide 69
Linear stability calculation
Choose a logarithmic wind profile (neutral stability)
= 0.41 (von Karman), u* = friction velocity, = u*2
Roughness length, zo : Charnock (1955) zo = u*
2 / g , 0.015 (for now)
Note: Growth rate, , of the waves ~ = a / w
and depends on
so, short waves have the largest growth.
Action balance equation
oo z
zuzU 1ln)( *
g
u
c
u **
2
*~,2
c
uNSN
t windwind
The Wave Model (ECWAM) Slide 70
Wave growth versus phase speed(comparison of Miles' theory and observations)
The Wave Model (ECWAM) Slide 71
Nonlinear effect: slowing down of wind
Continuum: w is nice function, because of continuum
of critical layers
w is wave induced stress
(u , w): wave induced velocity in air (from Rayleigh Eqn).
. . .
with (sea-state dep. through N!).
Dw > 0 slowing down the wind.
wwaves
o zU
t
wuw
owwaves
o Uz
DUt 2
2
)(233
kNvc
kcD
g
w
The Wave Model (ECWAM) Slide 72
Example:
Young wind sea steep waves. Old wind sea gentle waves.
Charnock parameter depends on sea-state!
(variation of a factor of 5 or so).
Wind speed as function of height for young & old wind sea
old windsea
young windsea
The Wave Model (ECWAM) Slide 73
Non-linear Transfer (finite steepness effects)
Briefly describe procedure how to obtain:
1.
2. Express in terms of canonical variables and = (z =) by solving
iteratively using Fourier transformation.
3. Introduce complex action-variable
nonlint
N
22
212
21
zdzg
0,
,
0
z
zz
z
z
)(ka
)()(2
1)(ˆ *
2/1
kakak
k
)()(2
)(ˆ *2/1
kakak
ik
kgk )(
The Wave Model (ECWAM) Slide 74
gives energy of wave system:
with , … etc.
Hamilton equations: become
Result:
Here, V and W are known functions of
....].[)(ddd
d
d
32*13,2,1321321
*1111
ccaaaVkkkkkk
aak
xE
)( 11 kaa
E
t
E
t*
)(a
Eika
t
43*243214,3,2,1432
321323,2,132111
)(ddd
...)(dd
aaakkkkWkkki
kkkaaVkkiaiat
k
The Wave Model (ECWAM) Slide 75
Three-wave interactions:
Four-wave interactions:
Gravity waves: No three-wave interactions possible.
Sum of two waves does not end up on dispersion curve.
0
0
321
321
kkk
4321
4321
kkkk
dispersion curve
2, k2
1, k1
k
The Wave Model (ECWAM) Slide 76
Phillips (1960) has shown that 4-wave interactions do exist!
Phillips’ figure of 8
Next step is to derive the statistical evolution equation for with N1 is the action density.
Nonlinear Evolution Equation
)( 211*21 kkNaa
hierarchyInfinite
&
&
543214321321
*4
*32132121
aaaaaaaaaaaat
aaaaaaaaat
The Wave Model (ECWAM) Slide 77
Closure is achieved by consistently utilising the assumption of Gaussian probability
Near-Gaussian
Here, R is zero for a Gaussian.
Eventual result:
obtained by Hasselmann (1962).
Raaaaaaaaaaaa *32
*41
*42
*31
*4
*321
)]()([
)(
)(ddd4
32414132
1432
14322
4,3,2,14321
NNNNNNNN
kkkkTkkkNt nonlin
The Wave Model (ECWAM) Slide 78
Properties:
1. N never becomes negative.
2. Conservation laws:
action:
momentum:
energy:
Wave field cannot gain or loose energy through four-wave interactions.
)(d kNk
)(d kNkk
)(d kNk
The Wave Model (ECWAM) Slide 79
Properties: (Cont’d)
3. Energy transfer
Conservation of two scalar quantities has implications
for energy transfer
Two lobe structure is impossible because if action is conserved, energy ~ N cannot be conserved!
)(d
d Nt
The Wave Model (ECWAM) Slide 80
R’ =
(S
nl) f
d/(
Sn
l) dee
p
The Wave Model (ECWAM) Slide 81
Wave Breaking
The Wave Model (ECWAM) Slide 82
Dissipation due to Wave Breaking
Define:
with
Quasi-linear source term: dissipation increases with
increasing integral wave steepness
)(d
)(d
kNk
kNk
)(d
)(d
kNk
kNkkk
NXXmkS odissip2
22)1(
2,/ omkkX
omk2
Slide 83
2THE WAM MODEL
The Wave Model (ECWAM) Slide 84
2. The WAM Model:
Solves energy balance equation, including Snonlin
WAM Group (early 1980’s) State of the art models could not handle rapidly varying
conditions. Super-computers. Satellite observations: Radar Altimeter, Scatterometer,
Synthetic Aperture Radar (SAR) .
Two implementations at ECMWF: Global (0.36 0.36 reduced latitude-longitude).
Coupled with the atmospheric model (2-way).Historically: 3, 1.5, 0.5 then 0.36
Limited Area covering North Atlantic + European Seas (0.25 0.25) Historically: 0.5 covering the Mediterranean only.
Both implementations use a discretisation of 30 frequencies 24 directions (used to be 25 freq.12 dir.)
The Wave Model (ECWAM) Slide 85
The Global Model:
The Wave Model (ECWAM) Slide 86
Global model analysis significant wave height (in metres) for 12 UTC on 1 June 2005.
Wav
e M
od
el D
om
ain
2
2
2
2
2
22
2
3
3
33
4
45
6
80°S80°S
70°S 70°S
60°S60°S
50°S 50°S
40°S40°S
30°S 30°S
20°S20°S
10°S 10°S
0°0°
10°N 10°N
20°N20°N
30°N 30°N
40°N40°N
50°N 50°N
60°N60°N
70°N 70°N
80°N80°N
160°W
160°W 140°W
140°W 120°W
120°W 100°W
100°W 80°W
80°W 60°W
60°W 40°W
40°W 20°W
20°W 0°
0° 20°E
20°E 40°E
40°E 60°E
60°E 80°E
80°E 100°E
100°E 120°E
120°E 140°E
140°E 160°E
160°E
ECMWF Analysis VT:Wednesday 1 June 2005 12UTC Surface: significant wave height (Exp: 0001 )
The Wave Model (ECWAM) Slide 87
Limited Area Model:
The Wave Model (ECWAM) Slide 88
Limited area model mean wave period (in seconds) from the second moment for 12 UTC on 1 June 2005.
4
4
4
5
5
5
5
5
6
6
66
6
7
7
77
18°N18°N
27°N 27°N
36°N36°N
45°N 45°N
54°N54°N
63°N 63°N
72°N72°N
95°W
95°W 85°W
85°W 75°W
75°W 65°W
65°W 55°W
55°W 45°W
45°W 35°W
35°W 25°W
25°W 15°W
15°W 5°W
5°W 5°E
5°E 15°E
15°E 25°E
25°E 35°E
35°E
ECMWF Analysis VT:Wednesday 1 June 2005 12UTC Surface: mean wave period (m2) (Exp: 0001 )
The Wave Model (ECWAM) Slide 89
ECMWF global wave model configurations
Deterministic model40 km grid, 30 frequencies and 24 directions, coupled to the TL799 model, analysis every 6 hrs and 10 day forecasts from 0 and 12Z.
Probabilistic forecasts110 km grid, 30 frequencies and 24 directions, coupled to the TL399 model, (50+1 members) 10 day forecasts from 0 and 12Z.
Monthly forecasts 1.5°x1.5° grid, 25 frequencies and 12 directions, coupled to the TL159 model, deep water physics only.
Seasonal forecasts3.0°x3.0°, 25 frequencies and 12 directions, coupled to the T95 model, deep water physics only.
Slide 90
2.1Energy Balance for Wind Sea
The Wave Model (ECWAM) Slide 91
2.1. Energy Balance for Wind Sea
Summarise knowledge in terms of empirical growth curves.
Idealised situation of duration-limited waves ‘Relevant’ parameters:
, u10(u*) , g , , surface tension, a , w , fo , t
Physics of waves: reduction to
Duration-limited growth not feasible: In practice fully-developed and fetch-limited situations are more relevant.
, u10(u*) , g , t
The Wave Model (ECWAM) Slide 92
Connection between theory and experiments:wave-number spectrum:
Old days H1/3
H1/3 Hs (exact for narrow-band spectrum)
2-D wave number spectrum is hard to observe frequency spectrum
One-dimensional frequency spectrum
Use same symbol, F , for:
)()( kNkF
)(d2 kFkmo
24 sH
dd),(d)(dd),(2 kkkFkkFF
),(),(2 kFv
kF
g
),(d)( 21 FF
)(,),(,)( FFkF
The Wave Model (ECWAM) Slide 93
Let us now return to analysis of wave evolution:Fully-developed:
Fetch-limited:
However, scaling with friction velocity u is to be preferred
over u10 since u10 introduces an additional length scale, z
= 10 m , which is not relevant. In practice, we use u10 (as
u is not available).
)/(/)(~
10510
3 gufuFgF
constant/~ 410
2 umg oconstant/10 gup
)/,/(/)(~ 2
1010510
3 uXggufuFgF
)/(/~ 210
410
2 uXgfumg o
)/(/ 21010 uXgfgup
The Wave Model (ECWAM) Slide 94
JONSWAP fetch relations (1973):
guuXgXumg po /,/~
,/~10
210
410
2
~
X~
Empirical growth relations against measurements
Evolution of spectra with fetch (JONSWAP)
Hz
km
km
km
km
km
The Wave Model (ECWAM) Slide 95
Distinction between wind sea and swell
wind sea: A term used for waves that are under the effect of their
generating wind. Occurs in storm tracks of NH and SH.
Nonlinear.
swell: A term used for wave energy that propagates out of storm
area. Dominant in the Tropics. Nearly linear.
Results with WAM model: fetch-limited. duration-limited [wind-wave interaction].
guuXgXumg po /,/~
,/~10
210
410
2
The Wave Model (ECWAM) Slide 96
Fetch-limited growth
Remarks:
1. WAM model scales with u
Drag coefficient,Using wind profile
CD depends on wind:
Hence, scaling with u gives different results compared
to scaling with u10.
In terms of u10 , WAM model gives a family of growth
curves! [u was not observed during JONSWAP.]
2*
210 ,/ uuCD
guzz
C oo
D /,)/10(ln
2*
2
10u
DC
The Wave Model (ECWAM) Slide 97
Dimensionless energy:
Note: JONSWAP mean wind ~ 8 - 9 m/s
Dimensionless peak frequency:
The Wave Model (ECWAM) Slide 98
2. Phillips constant is a measure of the steepness of high-frequency waves.
Young waves have large steepness.
100
0
p
)(52 Fgp
The Wave Model (ECWAM) Slide 99
Duration-limited growth
Infinite ocean, no advection single grid point.
Wind speed, u10 = 18 m/s u = 0.85 m/s
Two experiments:
1. uncoupled: no slowing-down of air flow Charnock parameter is constant ( = 0.0185)
2. coupled:slowing-down of air flow is taken into account by a parameterisation of Charnock parameter that depends on w:
= { w / } (Komen et al.,
1994)
The Wave Model (ECWAM) Slide 100
Time dependence of wave height for a reference runand a coupled run.
The Wave Model (ECWAM) Slide 101
Time dependence of Phillips parameter, p , for
a reference run and a coupled run.
The Wave Model (ECWAM) Slide 102
Time dependence of wave-induced stress for a reference run and a coupled run.
The Wave Model (ECWAM) Slide 103
Time dependence of drag coefficient, CD , for
a reference run and a coupled run.
The Wave Model (ECWAM) Slide 104
Evolution in time of the one-dimensional frequencyspectrum for the coupled run.
spec
tra
l den
sity
(m
2/H
z)
The Wave Model (ECWAM) Slide 105
The energy balance for young wind sea.
The Wave Model (ECWAM) Slide 106
The energy balance for old wind sea.
0-2
-1
Slide 107
2.2Wave Forecasting
The Wave Model (ECWAM) Slide 108
2.2. Wave Forecasting
Sensitivity to wind-field errors.
For fully developed wind sea:
Hs = 0.22 u102 / g
10% error in u10 20% error in Hs
from observed Hs
Atmospheric state needs reliable wave model.
SWADE case WAM model is a reliable tool.
The Wave Model (ECWAM) Slide 109
Verification of model wind speeds with observations
+ + + observations —— OW/AES winds …… ECMWF winds
(OW/AES: Ocean Weather/Atmospheric Environment Service)
The Wave Model (ECWAM) Slide 110
Verification of WAM wave heights with observations
+ + + observations —— OW/AES winds …… ECMWF winds
(OW/AES: Ocean Weather/Atmospheric Environment Service)
The Wave Model (ECWAM) Slide 111
Validation of wind & wave analysis using satellite & buoy.
Altimeters onboard ERS-1/2, ENVISAT and Jason
Quality is monitored daily.
Monthly collocation plots
SD 0.5 0.3 m for waves (recent SI 12-
15%)
SD 2.0 1.2 m/s for wind (recent SI 16-
18%)
Wave Buoys (and other in-situ instruments)
Monthly collocation plots
SD 0.85 0.45 m for waves (recent SI 16-
20%)
SD 2.6 1.2 m/s for wind (recent SI 16-
21%)
The Wave Model (ECWAM) Slide 112
WAM first-guess wave height againstENVISAT Altimeter measurements
(June 2003 – May 2004)
0 2 4 6 8 10 12 WAM WAVE HEIGHTS (M)
0
2
4
6
8
10
12
EN
VIS
AT
WA
VE
HE
IGH
TS
(M
)
SYMMETRIC SLOPE CORRELATION SCATTER INDEX STANDARD DEVIATION BIAS (ENVISAT - WAM) MEAN ENVISAT MEAN WAM ENTRIES STATISTICS
REGR. CONSTANT REGR. COEFFICIENT
1 . 0 5 0 9 . 9 6 5 7 . 1 3 3 4 . 3 3 8 9 . 1 0 5 5
2 . 6 4 5 3 2 . 5 3 9 8 7 3 2 1 1 0
- . 0 3 8 3 1 . 0 5 6 6
1 5 15 30 50 100 300 500 1000 30000
The Wave Model (ECWAM) Slide 113
Global wave height RMSE between ERS-2 Altimeter and WAM FG (thin navy line is 5-day running mean ….. thick red line is 30-day running mean)
0.25
0.30
0.35
0.40
0.45
0.50
0.55
01/08/97 01/08/98 01/08/99 01/08/00 01/08/01 01/08/02 02/08/03
Wav
e H
eig
ht
RM
SE
(E
RS
2 -
WA
M F
G)
[
m]
4D
VA
R (
25/
11
/97
)
T3
19
(0
1/0
4/9
8)
Co
up
ling
(28
/06
/98
)
Use
of
U1
0 (0
9/0
3/9
9)
NG
co
rre
ctio
n (
13
/07
/99
)
60
Le
vels
(1
2/1
0/9
9)
Pe
aki
ne
ss Q
C (
11
/04
/00
)
12
Hr
4D
VA
R (
11
/09
/00
)
T5
11
(2
0/1
1/0
0)
ZG
M (
01
/06
/01
)
Qu
ikS
CA
T (
21
/01
/02
)
Ya
w c
on
trol
(0
4/0
3/0
2)
Gu
stin
ess
(0
8/0
4/0
2)
SA
R a
ssim
ilatio
n (
13
/01
/03
)
1998
2000
1999
2001
2002
2003
No
ER
S-2
full
cove
rag
e (
22
/06
/03
)
The Wave Model (ECWAM) Slide 114
Analysed wave height and periods against buoy measurements for February to April 2002
2
4
6
8
10
12
14
HS (
m)
mo
de
l
0 2 4 6 8 10 12 14
H S (m) buoy
0
HS ENTRIES:
1 - 3
3 - 8
8 - 22
22 - 62
62 - 173
173 - 482
482 - 1350
0 2 4 6 8 10 12 14 16 18 20
Peak Period Tp (sec.) buoy
0
2
4
6
8
10
12
14
16
18
20
Pea
k P
erio
d T
p
(se
c.)
m
od
el
TP ENTRIES:
1 - 3
3 - 6
6 - 14
14 - 33
33 - 79
79 - 188
188 - 450
SYMMETRIC SLOPE = 0.932CORR COEF = 0.956 SI = 0.175RMSE = 0.438 BIAS = -0.150LSQ FIT: SLOPE = 0.879 INTR = 0.134BUOY MEAN = 2.354 STDEV = 1.389MODEL MEAN = 2.204 STDEV = 1.278ENTRIES = 29470
SYMMETRIC SLOPE = 1.008CORR COEF = 0.801 SI = 0.190RMSE = 1.746 BIAS = 0.072LSQ FIT: SLOPE = 0.813 INTR = 1.785BUOY MEAN = 9.160 STDEV = 2.743MODEL MEAN = 9.232 STDEV = 2.784ENTRIES = 17212
0 2 4 6 8 10 12 14
Mean Period Tz (sec.) buoy
0
2
4
6
8
10
12
14
Me
an
Pe
rio
d T
z (s
ec
.) m
od
el
TZ ENTRIES:
1 - 3
3 - 6
6 - 12
12 - 26
26 - 59
59 - 133
133 - 300
SYMMETRIC SLOPE = 0.985CORR COEF = 0.931 SI = 0.105
RMSE = 0.779 BIAS = -0.149LSQ FIT: SLOPE = 0.989 INTR = -0.069
BUOY MEAN = 7.306 STDEV = 1.979MODEL MEAN = 7.157 STDEV = 2.102
ENTRIES = 6105
Wave height Peak period Mean period
The Wave Model (ECWAM) Slide 115
J93 j J94 j J95 j J96 j J97 j J98 j J99 j J00 j J01 j J02 j J03 j J04 j J05
months (3 month running average)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
RM
SE
(m)
0001 WAVE HEIGHT R.M.S.E. from January 1993 to March 2005
all
MAGICS 6.9.1 muspell - wab Thu May 5 10:02:40 2005 J-R BIDLOT
Global wave height RMSE between buoys and WAM analysis
The Wave Model (ECWAM) Slide 116
Problems with buoys:
1. Atmospheric model assimilates winds from buoys (height ~ 4m) but regards them as 10 m winds [~10% error]
2. Buoy observations are not representative for aerial average.
Problems with satellite Altimeters:
The Wave Model (ECWAM) Slide 117
Quality of wave forecast
Compare forecast with verifying analysis.
Forecast error, standard deviation of error ( ), persistence.
Period: three months (January-March 1995).
Tropics is better predictable because of swell:
Daily errors for July-September 1994Note the start of Autumn.
New: 1. Anomaly correlation
2. Verification of forecast against buoy data.
N.H. Tropics S.H.
tHs1 0.08 0.05 0.08
The Wave Model (ECWAM) Slide 118
Significant wave heightanomaly correlationandst. deviation of errorover 365 daysfor years 1997-2004
Northern Hemisphere (NH)
0 1 2 3 4 5 6 7 8 9 10
Forecast Day
20
30
40
50
60
70
80
90
100%DATE = 20040101 TO 20041231
AREA=N.HEM TIME=12 MEAN OVER 366 CASES
ANOMALY CORRELATION FORECAST
HEIGHT OF WAVES SURFACE LEVEL
FORECAST VERIFICATION 20042003200220012000199919981997
MAGICS 6.7 icarus - dax Tue Jan 11 08:59:01 2005 Verify SCOCOM
0 1 2 3 4 5 6 7 8 9 10
Forecast Day
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1DATE = 20040101 TO 20041231
AREA=N.HEM TIME=12 MEAN OVER 366 CASES
STANDARD DEVIATION OF ERROR FORECAST
HEIGHT OF WAVES SURFACE LEVEL
FORECAST VERIFICATION 20042003200220012000199919981997
MAGICS 6.7 icarus - dax Tue Jan 11 08:59:01 2005 Verify SCOCOM * 1 ERROR(S) FOUND *
The Wave Model (ECWAM) Slide 119
Significant wave heightanomaly correlationandst. deviation of errorover 365 daysfor years 1997-2004
Tropics
0 1 2 3 4 5 6 7 8 9 10
Forecast Day
40
50
60
70
80
90
100%DATE = 20040101 TO 20041231
AREA=TROPICS TIME=12 MEAN OVER 366 CASES
ANOMALY CORRELATION FORECAST
HEIGHT OF WAVES SURFACE LEVEL
FORECAST VERIFICATION 20042003200220012000199919981997
MAGICS 6.7 icarus - dax Tue Jan 11 08:59:02 2005 Verify SCOCOM
0 1 2 3 4 5 6 7 8 9 10
Forecast Day
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5DATE = 20040101 TO 20041231
AREA=TROPICS TIME=12 MEAN OVER 366 CASES
STANDARD DEVIATION OF ERROR FORECAST
HEIGHT OF WAVES SURFACE LEVEL
FORECAST VERIFICATION 20042003200220012000199919981997
MAGICS 6.7 icarus - dax Tue Jan 11 08:59:02 2005 Verify SCOCOM * 1 ERROR(S) FOUND *
The Wave Model (ECWAM) Slide 120
Significant wave heightanomaly correlationandst. deviation of errorover 365 daysfor years 1997-2004
Southern Hemisphere (SH)
0 1 2 3 4 5 6 7 8 9 10
Forecast Day
20
30
40
50
60
70
80
90
100%DATE = 20040101 TO 20041231
AREA=S.HEM TIME=12 MEAN OVER 366 CASES
ANOMALY CORRELATION FORECAST
HEIGHT OF WAVES SURFACE LEVEL
FORECAST VERIFICATION 20042003200220012000199919981997
MAGICS 6.7 icarus - dax Tue Jan 11 08:59:02 2005 Verify SCOCOM
0 1 2 3 4 5 6 7 8 9 10
Forecast Day
0
0.2
0.4
0.6
0.8
1
1.2
1.4DATE = 20040101 TO 20041231
AREA=S.HEM TIME=12 MEAN OVER 366 CASES
STANDARD DEVIATION OF ERROR FORECAST
HEIGHT OF WAVES SURFACE LEVEL
FORECAST VERIFICATION 20042003200220012000199919981997
MAGICS 6.7 icarus - dax Tue Jan 11 08:59:02 2005 Verify SCOCOM * 1 ERROR(S) FOUND *
The Wave Model (ECWAM) Slide 121
RMSE of * significant wave height, * 10m wind speed and * peak wave period of different models as compared to buoy measurements for February to April 2005
0 1 2 3 4 53rd generation fc data only, fc from 0 and 12Z for 0502 to 0504
0
0.10.20.3
0.40.50.6
0.70.80.9
1
RM
SE
(m
)
SIGNIFICANT W AVE HEIGHT ROOT MEAN SQUARE ERROR at all 69 buoys
ECMWF METOF FNMOC MSC NCEP METFR DWD
0 1 2 3 4 53rd generation fc data only, fc from 0 and 12Z for 0502 to 0504
0
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
3.6
4
RM
SE
(m
/s)
10m W IND SPEED ROOT MEAN SQUARE ERROR at all 69 buoys
ECMWF METOF FNMOC MSC NCEP METFR DWD
0 1 2 3 4 53rd generation fc data only, fc from 0 and 12Z for 0502 to 0504
0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
RM
SE
(s
ec
.)
PEAK PERIOD ROOT MEAN SQUARE ERROR at all 63 non UK buoys
ECMWF METOF FNMOC MSC NCEP METFR DWD
Slide 122
3BENEFITS FOR
ATMOSPHERIC MODELLING
Slide 123
3.1Use as Diagnostic Tool
The Wave Model (ECWAM) Slide 124
3.1. Use as Diagnostic Tool
Discovered inconsistency between wind speed and stress and resolved it.
Overactivity of atmospheric model during the forecast: mean forecast error versus time.
Slide 125
3.2Coupled Wind-Wave Modelling
The Wave Model (ECWAM) Slide 126
3.2. Coupled Wind-Wave Modelling
Coupling scheme:
Impact on depression (Doyle).
Impact on climate [extra tropic].
Impact on tropical wind field ocean circulation.
Impact on weather forecasting.
ATM
WAM
t
u10
u10
t
u10
time
time
windww Skk
d,101.0
The Wave Model (ECWAM) Slide 127
WAM – IFS Interface
A t m o s p h e r i c M o d e l
q T P
air
zi / L (U10, V10)
W a v e M o d e l
The Wave Model (ECWAM) Slide 128
Simulated sea-level pressure for uncoupled and coupled simulations for the 60 h time
uncoupled coupled
956.4 mb 963.0 mb
Ulml > 25 m/s1000 km
The Wave Model (ECWAM) Slide 129
Scores of FC 1000 and 500 mb geopotential for SH(28 cases in ~ December 1997)
coupleduncoupled
The Wave Model (ECWAM) Slide 130
Standard deviation of error and systematic error of forecast wave height for Tropics
(74 cases: 16 April until 28 June 1998).
coupled
The Wave Model (ECWAM) Slide 131
Global RMS difference between ECMWF and ERS-2 scatterometer winds
(8 June – 14 July 1998)
coupling
~20 cm/s (~10%) reduction
The Wave Model (ECWAM) Slide 132
Change from 12 to 24 directional bins:Scores of 500 mb geopotential for NH and SH
(last 24 days in August 2000)
Slide 133
END
The Wave Model (ECWAM) Slide 134
J93 j J94 j J95 j J96 j J97 j J98 j J99 j J00 j J01 j J02 j J03 j J04 j J05
months (3 month running average)0.2
0.3
0.4
0.5
0.6
0.7
0.8
RM
SE
(m)
WAVE HEIGHT R.M.S.E. from January 1993 to March 2005
all
MAGICS 6.9.1 muspell - wab Thu May 5 10:42:41 2005 J-R BIDLOT
Global wave height RMSE between buoys and WAM analysis (original list!)