Slide 1 The Wave Model ECMWF, Reading, UK. Slide 2The Wave Model (ECWAM) Resources: Lecture notes available at:

Slide 1

The Wave Model

ECMWF, Reading, UK

Slide 2The Wave Model (ECWAM)

Resources:

Lecture notes available at:http://www.ecmwf.int/newsevents/training/lecture_notes/

pdf_files/NUMERIC/Wave_mod.pdf

Recommended literature can be found at:http://www.ecmwf.int/newsevents/training/2012/nwp/timetable_resources_nwp.html

Model documentation available at:http://www.ecmwf.int/research/ifsdocs/CY36r1/index.html


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 4

INTRODUCTION


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.


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



Wave Period, T

Wave Length, Wave Height, H

Water surfaceelevation,

Wave number: k = 2 π / λ

Wave (angular) frequency: ω = 2 π / T










What we are dealing with?














What we aredealing with


What we are (not) dealing with (this week)



Fetch, X, or Duration, t

Simplified sketch!

En

erg

y, E

,o

f a

sin

gle

co

mp

on

ent

Slide 22

PROGRAMOF

THE LECTURES


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.




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.




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).



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).



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-1/2 altimeter data and buoy data.- Quality of wave forecast: forecast skill depends on sea state (wind sea or swell).



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.


Program of the lecture:4. Error Estimation

4.1. IntroductionDefinition of measurement errors. Types of errors. Accuracy and precision. Data sources. Error estimation when the truth is known.

4.2. Triple Collocation TechniqueError model. Derivation. Assumptions. Practical considerations.

4.3. ExamplesEstimation of wind and wave errors of various measuring systems: buoys, altimeters and model.

Slide 30

1DERIVATION

OF THEENERGY BALANCE EQUATION


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


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 33

1.1PRELIMINARIES


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


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


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


Lagrange Formulation

Variational principle:

with:

gives Laplace’s equation & boundary conditions.

0dx dy dt L

2

212

D

dz g zt z

L


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


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)

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

212 ( ) £ ( , )T V m q V q q q L


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


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


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


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


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

currentmean,)(tanh,22 oUDkTag

2 2 2 4 23

4

( )1 1 9 10 91 ( )

2 2 8ok U k T T

Og k T g T

L


In other words, we have(omitting the brackets )

Evolution equations then follow from the average variational principle

We obtain: a : (dispersion relation)

: (evolution of amplitude)

plus consistency: (conservation of crests)

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


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


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 total action density is Ntot is conserved.

The action density is called an adiabatic invariant.

.constNxdΝ tot

xdE


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


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

dqpEN

2

1),(

1const.

2p dq

L

Slide 50

1.2ENERGY BALANCE EQUATION:

THE ADIABATIC PART


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


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 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


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


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

/


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


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


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


Result:

With vg the group speed, we have

0)()()()cos(cos

1

NNNNNt

RUv og /)cos(

)cos(/)sin( RVv og

2/)(/tansin kkkRvg

t /


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 61

1.3ENERGY BALANCE EQUATION:

PHYSICS


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


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 /


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


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


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


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


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


Wave growth versus phase speed(comparison of Miles' theory and observations)


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


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


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 )(


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


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


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


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


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


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


R’ =

(S

nl) f

d/(

Sn

l) dee

p


Wave Breaking


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 82

2THE WAM MODEL


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 (0.25 0.25 reduced latitude-longitude).

Coupled with the atmospheric model (2-way).Historically: 3, 1.5, 0.5 then 0.36 .

Limited Area = N. Atlantic + European Seas (0.1 0.1) Historically: 0.5 (Mediterranean only), then the current coverage with 0.25.

Both implementations use a discretisation of 36 frequencies 36 directions (used to be 25 freq.12 dir., then 30 freq.24 dir.)


The Global Model:


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 )


Limited Area Model:

10°N

20°N

30°N

40°N

50°N

60°N

70°N

80°N

100°W

100°W

90°W

90°W

80°W

70°W

60°W

50°W 40°W 30°W 20°W 10°W 0° 10°E

20°E

30°E

40°E

50°E

60°E

70°E

70°E

Thursday 25 August 2011 12UTC ECMWF Forecast t+48 VT: Saturday 27 August 2011 12UTC Surface: waves: height_direction (Exp: 0001 )Thursday 25 August 2011 12UTC ECMWF Forecast t+48 VT: Saturday 27 August 2011 12UTC Surface: Signif icant wave height (Exp: 0001 )

10°N

20°N

30°N

40°N

50°N

60°N

70°N

80°N

100°W

100°W

90°W

90°W

80°W

70°W

60°W

50°W 40°W 30°W 20°W 10°W 0° 10°E

20°E

30°E

40°E

50°E

60°E

70°E

70°E m

0.5

1.0

1.5

2.0

3.0

4.0

8.0

14.5


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 )


ECMWF global wave model configurations

Deterministic model28-km grid, 36 frequencies and 36 directions, coupled to the TL1279 atmospheric model, analysis every 6 hrs and 10-day forecasts from 0 and 12Z.

Probabilistic forecasts55-km grid, 30 frequencies and 24 directions, coupled to the T639/T319 (up to 7/15 days) atmospheric model, (50+1 members) 15 (7+8)-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 89

2.1Energy Balance for Wind Sea


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


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


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


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


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


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


Dimensionless energy:

Note: JONSWAP mean wind ~ 8 - 9 m/s

Dimensionless peak frequency:


2. Phillips constant is a measure of the steepness of high-frequency waves.

Young waves have large steepness.

100

0

p

)(52 Fgp


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)


Time dependence of wave height for a reference runand a coupled run.


Time dependence of Phillips parameter, p , for

a reference run and a coupled run.


Time dependence of wave-induced stress for a reference run and a coupled run.


Time dependence of drag coefficient, CD , for

a reference run and a coupled run.


Evolution in time of the one-dimensional frequencyspectrum for the coupled run.

spec

tra

l den

sity

(m

2/H

z)


The energy balance for young wind sea.


The energy balance for old wind sea.

0-2

-1

Slide 106

2.2Wave Forecasting


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.


Verification of model wind speeds with observations

+ + + observations —— OW/AES winds …… ECMWF winds

(OW/AES: Ocean Weather/Atmospheric Environment Service)


Verification of WAM wave heights with observations

+ + + observations —— OW/AES winds …… ECMWF winds

(OW/AES: Ocean Weather/Atmospheric Environment Service)


Validation of wind & wave analysis using satellite & buoy.

Altimeters onboard ERS-1/2, ENVISAT and Jason-1/2

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%)


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


Global comparison of WAM first-guess sig. wave height against ENVISAT Alt. measurements (plot & statistics)

as well as Jason-1/2 (statistics only)(02 February 2010 to 01 February 2011)

Jason-1Jason-2Envisat

0 2 4 6 8 10 12 14WAM Wave Heights (m)

0

2

4

6

8

10

12

14

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. 0026

0. 9786

0. 1051

0. 2733

-0. 0163

2. 5851

2. 6014

1125908

-0. 0587

1. 0163

1510501005001000500010000100000

0 .9983

0 .9791

0 .1044

0 .2826

-0 .0032

2 .7041

2 .7073

1425055

0 .0637

0 .9753

1. 0 397

0. 9 738

0. 1 200

0. 3 232

0. 1 140

2. 8 078

2. 6 939

138 2997

0. 1 072

1. 0 025


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

)

Slide 114

IPF

6.0

2L

04

Jan-

20

04

Jan-

20

05

Jan-

20

06

Jan-

20

07

Jan-

20

08

Jan-

20

09

Jan-

20

10

Jan-

20

11

Jan-

20

120.20

0.25

0.30

0.35

0.40

SW

H S

td. D

ev.

of

Dif

fere

nce

(m

)

Model improvements detected by RA-2 monitoring

Change in RA-2 processing

The Wave Model (ECWAM)

Global wave height SDD between Envisat Alt. (RA-2) and WAM FG(Less along-track averaging of Envisat compared to ERS-2)


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


Analysed wave height and periods against buoy measurements for March to May 2011

0 2 4 6 8 10 12 14 16 18 20 22

Buoy Peak Period (s)

0

2

4

6

8

10

12

14

16

18

20

22

Mo

de

l Pe

ak

Pe

rio

d

(s

) Tp ENTRIES:

1

3

7

19

49

129

341

900

SYMMETRIC SLOPE = 1.021CORR COEF = 0.852 SI = 0.180RMSE = 1.636 BIAS = 0.135LSQ FIT: SLOPE = 0.924 INTR = 0.825BUOY MEAN = 9.05 STDEV = 2.849MODEL MEAN = 9.18 STDEV = 3.089ENTRIES = 35154

0 2 4 6 8 10 12

Buoy SWH (m)

0

2

4

6

8

10

12

Mo

de

l S

WH

(m

)

HsENTRIES:

1

4

11

36

118

388

1276

4200

SYMMETRIC SLOPE = 0.987CORR COEF = 0.969 SI = 0.146RMSE = 0.278 BIAS = -0.017LSQ FIT: SLOPE = 0.944 INTR = 0.089BUOY MEAN = 1.90 STDEV = 1.128MODEL MEAN = 1.88 STDEV = 1.099ENTRIES = 57786

0 2 4 6 8 10 12

Buoy Mean Period (s)

0

2

4

6

8

10

12

Mo

de

l Me

an

Pe

rio

d

(s

) TzENTRIES:

1

3

7

18

44

114

291

750

SYMMETRIC SLOPE = 0.956CORR COEF = 0.933 SI = 0.118RMSE = 0.799 BIAS = -0.362LSQ FIT: SLOPE = 1.066 INTR = -0.761BUOY MEAN = 6.02 STDEV = 1.709MODEL MEAN = 5.65 STDEV = 1.954ENTRIES = 12123

Slide 117

1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

months (median of the 3 month running average)

12

14

16

18

20

22

24

26

28

30

Sc

att

er

In

de

x,S

I

(%

)

0001 WAVE HEIGHT SCATTER INDEX from August 1992 to February 2012

The Wave Model (ECWAM)

Global wave height SI (=SDD/Mean) between buoys and WAM analysis


Problems with buoys:

1. Atmospheric model assimilates winds from buoys (sometimes with unknown height but regards them as 10 m winds [10% maximum error]).

2. Buoy observations are not representative for aerial average.

3. Limited coverage (parts of NH) seasonal cycles.

Problems with satellite Altimeters:

1. They have some difficulties at very low SWH values.

2. Satellite observations may not be available when or where needed.

3. Some Altimeters tend to overestimate SWH values.


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


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



MAGICS 6.7 icarus - dax Tue Jan 11 08:59:01 2005 Verify SCOCOM * 1 ERROR(S) FOUND *



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





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







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





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






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

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



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


Slide 124

3BENEFITS FOR

ATMOSPHERIC MODELLING

Slide 125

3.1Use as Diagnostic Tool


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 127

3.2Coupled Wind-Wave Modelling


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


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


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


Scores of FC 1000 and 500 mb geopotential for SH(28 cases in ~ December 1997)

coupleduncoupled


Standard deviation of error and systematic error of forecast wave height for Tropics

(74 cases: 16 April until 28 June 1998).

coupled


Global RMS difference between ECMWF and ERS-2 scatterometer winds

(8 June – 14 July 1998)

coupling

~20 cm/s (~10%) reduction


Change from 12 to 24 directional bins:Scores of 500 mb geopotential for NH and SH

(last 24 days in August 2000)

Documents

Slide 1 The Wave Model ECMWF, Reading, UK. Slide 2The Wave Model (ECWAM) Resources: Lecture notes available at: