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### The Wave Model

### PROGRAMOF THE LECTURES

### 1DERIVATIONOF THEENERGY BALANCE EQUATION

### 1.1PRELIMINARIES

### 1.2ENERGY BALANCE EQUATION:THE ADIABATIC PART

### 1.3 geometry (ENERGY BALANCE EQUATION:PHYSICS

### 2 assumption of Gaussian probability THE WAM MODEL

### 2.1 assumption of Gaussian probability Energy Balance for Wind Sea

### 2.2 assumption of Gaussian probability Wave Forecasting

### 3 assumption of Gaussian probability BENEFITS FORATMOSPHERIC MODELLING

### 3.1 assumption of Gaussian probability Use as Diagnostic Tool

### 3.2 assumption of Gaussian probability Coupled Wind-Wave Modelling

### END winds

ECMWF, Reading, UK

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)

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)

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.

The Wave Model (ECWAM)

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)

Forcing:

moon/sun

wind

Restoring:

gravity

surface tension

Coriolis force

2

1

10.0

1.0

0.03

3x10-3

2x10-5

1x10-5

Frequency (Hz)

What we are dealing with?The Wave Model (ECWAM)

Water surfaceelevation,

Wave Period, T

Wave Length,

Wave Height, H

What we are dealing with?The Wave Model (ECWAM)

What we are dealing with?

The Wave Model (ECWAM)

What we are dealing with?

The Wave Model (ECWAM)

What we are dealing with?

The Wave Model (ECWAM)

What we are dealing with?

The Wave Model (ECWAM)

What we are dealing with?

The Wave Model (ECWAM)

What we are dealing with?

The Wave Model (ECWAM)

What we are dealing with?

The Wave Model (ECWAM)

What we are dealing with?

The Wave Model (ECWAM)

What we are dealing with?

The Wave Model (ECWAM)

What we are dealing with?

The Wave Model (ECWAM)

What we are dealing with?

The Wave Model (ECWAM)

What we aredealing with?

The Wave Model (ECWAM)

What we are dealing with?

The Wave Model (ECWAM)

Energy, E,of a single component

Simplified sketch!

Fetch, X, or Duration, t

What we are dealing with?The Wave Model (ECWAM)

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)

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)

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)

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)

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)

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)

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

The Wave Model (ECWAM)

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)

Deterministic Evolution Equations

- 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 m Ocean basin: 107 m 2D 1014 equations Multiple scale approach

· short-scale, O(), solved analytically

· long-scale related to physics.

Result: Energy balance equation that describes large-scale evolution of the wave spectrum.

The Wave Model (ECWAM)

1.1. Preliminaries:

- Interface between air and ocean:
- Incompressible two-layer fluid
- Navier-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.

The Wave Model (ECWAM)

Laplace’s Equation

- Conditions at surface:
- Conditions at the bottom:
- Conservation of total energy: with

The Wave Model (ECWAM)

- Hamilton EquationsChoose 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.

The Wave Model (ECWAM)

- Lagrange FormulationVariational principle:with:gives Laplace’s equation & boundary conditions.

The Wave Model (ECWAM)

V(q)

- IntermezzoClassical 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

The Wave Model (ECWAM)

Principle of “least” action. Lagrangian:Newton’s law Action is extreme, whereAction is extreme if (action) = 0, where This is applicable for arbitrary q hence (Euler-Lagrange equation)

The Wave Model (ECWAM)

Define momentum, p, as:and regard now p and q are independent, the Hamiltonian, H , is given as:Differentiate H with respect to q givesThe 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.

The Wave Model (ECWAM)

- Linear TheoryLinearized equations become:Elementary sineswhere a is the wave amplitude, is the wave phase.Laplace:

The Wave Model (ECWAM)

Constant depth: z = -DZ'(-D) = 0Z ~ cosh [ k (z+D)] with

Satisfying Dispersion RelationDeep waterShallow waterDD0

dispersion relation:

phase speed:

group speed:

Note: low freq. waves faster! No dispersion.

energy:

The Wave Model (ECWAM)

(t)

t

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:

The Wave Model (ECWAM)

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 L:

For water waves we getwith

The Wave Model (ECWAM)

Evolution equations then follow from the average variational principle

We obtain:a : : plus consistency:

Finally, introduce a transport velocity

thenL is called the action density.

The Wave Model (ECWAM)

Apply our findings to gravity waves. Linear theory, write L aswhere

Dispersion relation follows fromhence, with ,

Equation for action density, N ,becomeswith

Closed by

The Wave Model (ECWAM)

- 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.

The Wave Model (ECWAM)

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 energythus also momentum is

The average Lagrangian becomes

The Wave Model (ECWAM)

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 LL waves.

Example: Pendulum with varying length!

The Wave Model (ECWAM)

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

The Wave Model (ECWAM)

- By definition, wave number spectrum is Fourier transform of correlation RConnection with Fourier transform of surface elevation.
Two modesis real

Use this in homogeneous correlation and correlation becomes

The Wave Model (ECWAM)

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

The Wave Model (ECWAM)

k

O

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: !

The Wave Model (ECWAM)

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

The Wave Model (ECWAM)

- 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

The Wave Model (ECWAM)

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!

The Wave Model (ECWAM)

- 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.

The Wave Model (ECWAM)

- Properties: geometry (
- Waves propagate along great circle.
- Shoaling:Piling up of energy when waves, which slow down in shallower water, approach the coast.
- Refraction:* The Hamilton equations define a ray light waves* Waves bend towards shallower water!* Sea mountains (shoals) act as lenses.
- Current effects:Blocking vg vanishes for k = g / (4 Uo2) !

The Wave Model (ECWAM)

U geometry (o=c

zc

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

The Wave Model (ECWAM)

- Transfer from Wind geometry (:
Instability of plane parallel shear flow (2D)

Perturb equilibrium: Displacement of streamlinesW = Uo - c (c = /k)

The Wave Model (ECWAM)

give Im(c) geometry ( 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(kc1)

c2 = g/k

The Wave Model (ECWAM)

Further ‘ geometry (simplification’ gives for = w/w(0):

Growth rate:

Wronskian:

The Wave Model (ECWAM)

geometry (

w

zc

z

- Wronskian W is related to wave-inducedstressIndeed, with and the normal mode formulation for u1, w1: (e.g. )
- Wronskian is a simple function, namely constant except at critical height zcTo see this, calculate dW/dzusing Rayleigh equation with proper treatment of the singularity at z=zc where subscript “c” refers to evaluation at critical height zc (Wo = 0)

The Wave Model (ECWAM)

This finally gives for the growth rate (by integrating d geometry (W/dz to get W(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.

The Wave Model (ECWAM)

Linear stability calculation geometry (

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

The Wave Model (ECWAM)

Wave growth versus phase speed geometry (

(comparison of Miles' theory and observations)

The Wave Model (ECWAM)

Nonlinear effect: slowing down of wind geometry (

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.

The Wave Model (ECWAM)

old windsea geometry (

young windsea

Wind speed as function of height for young & old wind sea

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).

The Wave Model (ECWAM)

- Non-linear Transfer geometry ((finite steepness effects)
Briefly describe procedure how to obtain:

- Express in terms of canonical variables and = (z =) by solving iteratively using Fourier transformation.
- Introduce complex action-variable

The Wave Model (ECWAM)

gives energy of wave system: geometry (

with , … etc.

Hamilton equations: become

Result:

Here, V and W are known functions of

The Wave Model (ECWAM)

geometry (

dispersion curve

1, k1

2, k2

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.

The Wave Model (ECWAM)

Phillips (1960) has shown geometry (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

The Wave Model (ECWAM)

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).

The Wave Model (ECWAM)

Properties assumption of Gaussian probability :

- Nnever becomes negative.
- Conservationlaws:action:momentum:energy:Wave field cannot gain or loose energy through four-wave interactions.

The Wave Model (ECWAM)

Properties assumption of Gaussian probability : (Cont’d)

- 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!

The Wave Model (ECWAM)

R’ = (S assumption of Gaussian probability nl)fd/(Snl)deep

The Wave Model (ECWAM)

Wave Breaking assumption of Gaussian probability

The Wave Model (ECWAM)

- Dissipation due to Wave Breaking assumption of Gaussian probability
Define:

with

Quasi-linear source term: dissipation increases with increasing integral wave steepness

The Wave Model (ECWAM)

2. The WAM Model: assumption of Gaussian probability

- 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)

The Global Model: assumption of Gaussian probability

The Wave Model (ECWAM)

Wave Model Domain assumption of Gaussian probability

Global model analysis significant wave height (in metres) for 12 UTC on 1 June 2005.The Wave Model (ECWAM)

Limited Area Model: assumption of Gaussian probability

The Wave Model (ECWAM)

Limited area model assumption of Gaussian probability mean wave period (in seconds) from the second moment for 12 UTC on 1 June 2005.

The Wave Model (ECWAM)

ECMWF global wave model configurations assumption of Gaussian probability

- 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 forecasts1.5°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.

The Wave Model (ECWAM)

2.1. Energy Balance for Wind Sea assumption of Gaussian probability

- Summarise knowledge in terms of empirical growth curves.
- Idealised situation of duration-limited waves‘Relevant’ parameters: , u10(u*) , g , , surface tension, a , w , fo , tPhysics 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)

- Connection between theory and experiments: assumption of Gaussian probability wave-number spectrum:
- Old days H1/3H1/3Hs (exact for narrow-band spectrum)
- 2-D wave number spectrum is hard to observe frequency spectrum
- One-dimensional frequency spectrumUse same symbol, F , for:

The Wave Model (ECWAM)

- Let us now return to analysis of wave evolution: assumption of Gaussian probability 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).

The Wave Model (ECWAM)

km assumption of Gaussian probability

km

km

km

km

Hz

Evolution of spectra with fetch (JONSWAP)

Empirical growth relations against measurements

- JONSWAP fetch relations (1973):

The Wave Model (ECWAM)

- Distinction between assumption of Gaussian probability 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].

The Wave Model (ECWAM)

- Fetch-limited growth assumption of Gaussian probability
Remarks:

- WAM model scales with uDrag coefficient,Using wind profileCD 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.]

The Wave Model (ECWAM)

Dimensionless energy: assumption of Gaussian probability

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

Dimensionless peak

frequency:

The Wave Model (ECWAM)

1000 assumption of Gaussian probability p

- Phillips constant is a measure of the steepness of high-frequency waves.Young waves have large steepness.

The Wave Model (ECWAM)

- Duration-limited growth assumption of Gaussian probability
Infinite ocean, no advection single grid point.

Wind speed, u10 = 18 m/s u = 0.85 m/s

Two experiments:

- uncoupled: no slowing-down of air flow Charnock parameter is constant ( = 0.0185)
- 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)

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

The Wave Model (ECWAM)

Time dependence of assumption of Gaussian probability Phillips parameter, p , for

a reference run and a coupled run.

The Wave Model (ECWAM)

Time dependence of assumption of Gaussian probability wave-induced stress for

a reference run and a coupled run.

The Wave Model (ECWAM)

Time dependence of assumption of Gaussian probability drag coefficient, CD , for

a reference run and a coupled run.

The Wave Model (ECWAM)

spectral density (m assumption of Gaussian probability 2/Hz)

Evolution in time of the one-dimensional frequency

spectrum for the coupled run.

The Wave Model (ECWAM)

The energy balance for young wind sea. assumption of Gaussian probability

The Wave Model (ECWAM)

-1 assumption of Gaussian probability

-2

0

The energy balance for old wind sea.

The Wave Model (ECWAM)

2.2. Wave Forecasting assumption of Gaussian probability

- Sensitivitytowind-fielderrors.
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)

+ + + assumption of Gaussian probability observations

—— OW/AES winds

…… ECMWF winds

Verification of model wind speeds with observations

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

The Wave Model (ECWAM)

+ + + assumption of Gaussian probability observations

—— OW/AES winds

…… ECMWF winds

Verification of WAM wave heights with observations

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

The Wave Model (ECWAM)

- Validation assumption of Gaussian probability ofwind&waveanalysisusingsatellite&buoy.
AltimetersonboardERS-1/2, ENVISATandJason

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%)

WaveBuoys (andotherin-situinstruments)

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)

WAM first-guess wave height against assumption of Gaussian probability ENVISAT Altimeter measurements(June 2003 – May 2004)

The Wave Model (ECWAM)

Global wave height RMSE between ERS-2 Altimeter and WAM FG assumption of Gaussian probability (thin navy line is 5-day running mean ….. thick red line is 30-day running mean)

The Wave Model (ECWAM)

Wave height assumption of Gaussian probability

Peak period

Mean period

14

ENTRIES:

HS

ENTRIES:

TZ

14

12

1 - 3

1 - 3

12

10

3 - 8

3 - 6

10

8

8 - 22

6 - 12

8

(m) model

22 - 62

6

12 - 26

6

Mean Period Tz (sec.) model

62 - 173

4

S

26 - 59

H

4

173 - 482

59 - 133

2

2

482 - 1350

133 - 300

0

ENTRIES = 29470

0

0

2

4

6

8

10

12

14

0

2

4

6

8

10

12

14

H

(m) buoy

MODEL MEAN = 2.204 STDEV = 1.278

Mean Period Tz (sec.) buoy

S

BUOY MEAN = 2.354 STDEV = 1.389

LSQ FIT: SLOPE = 0.879 INTR = 0.134

RMSE = 0.438 BIAS = -0.150

CORR COEF = 0.956 SI = 0.175

ENTRIES = 6105

SYMMETRIC SLOPE = 0.932

MODEL MEAN = 7.157 STDEV = 2.102

20

ENTRIES = 17212

BUOY MEAN = 7.306 STDEV = 1.979

MODEL MEAN = 9.232 STDEV = 2.784

ENTRIES:

TP

LSQ FIT: SLOPE = 0.989 INTR = -0.069

18

BUOY MEAN = 9.160 STDEV = 2.743

RMSE = 0.779 BIAS = -0.149

LSQ FIT: SLOPE = 0.813 INTR = 1.785

1 - 3

CORR COEF = 0.931 SI = 0.105

RMSE = 1.746 BIAS = 0.072

16

SYMMETRIC SLOPE = 0.985

CORR COEF = 0.801 SI = 0.190

SYMMETRIC SLOPE = 1.008

14

3 - 6

12

6 - 14

10

14 - 33

8

Peak Period Tp (sec.) model

33 - 79

6

4

79 - 188

2

188 - 450

0

0

2

4

6

8

10

12

14

16

18

20

Peak Period Tp (sec.) buoy

Analysed wave height and periods against buoy measurements for February to April 2002The Wave Model (ECWAM)

Global wave height RMSE between buoys and WAM analysis assumption of Gaussian probability

The Wave Model (ECWAM)

Problems with buoys: assumption of Gaussian probability

- Atmospheric model assimilates winds from buoys (height ~ 4m) but regards them as 10 m winds [~10% error]
- Buoy observations are not representative for aerial average.
Problems with satellite Altimeters:

The Wave Model (ECWAM)

- Quality assumption of Gaussian probability ofwaveforecast
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.

The Wave Model (ECWAM)

Significant wave height assumption of Gaussian probability anomaly correlationandst. deviation of errorover 365 daysfor years 1997-2004Northern Hemisphere (NH)

The Wave Model (ECWAM)

Significant wave height assumption of Gaussian probability anomaly correlationandst. deviation of errorover 365 daysfor years 1997-2004Tropics

The Wave Model (ECWAM)

Significant wave height assumption of Gaussian probability anomaly correlationandst. deviation of errorover 365 daysfor years 1997-2004Southern Hemisphere (SH)

The Wave Model (ECWAM)

RMSE of assumption of Gaussian probability * significant wave height, * 10m wind speed and * peak wave period of different models as compared to buoy measurements for February to April 2005

The Wave Model (ECWAM)

3.1. Use as Diagnostic Tool assumption of Gaussian probability

- Discovered inconsistency between wind speed and stress and resolved it.
- Overactivity of atmospheric model during the forecast: mean forecast error versus time.

The Wave Model (ECWAM)

assumption of Gaussian probability t

t

ATM

time

u10

u10

u10

WAM

time

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.

The Wave Model (ECWAM)

A t m o s p h e r i c M o d e l assumption of Gaussian probability

q

T

P

(

U

,

V

)

z

/ L

10

10

i

air

W a v e M o d e l

WAM – IFS InterfaceThe Wave Model (ECWAM)

956.4 mb assumption of Gaussian probability

963.0 mb

uncoupled

coupled

1000 km

Ulml > 25 m/s

Simulated sea-level pressure for uncoupled and coupled simulations for the 60 h timeThe Wave Model (ECWAM)

coupled assumption of Gaussian probability

uncoupled

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

coupled assumption of Gaussian probability

Standard deviation of error and systematic error of forecast wave height for Tropics(74 cases: 16 April until 28 June 1998).The Wave Model (ECWAM)

Global RMS difference between ECMWF and ERS-2 scatterometer winds(8 June – 14 July 1998)

~20 cm/s (~10%) reduction

coupling

The Wave Model (ECWAM)

Change from 12 to 24 directional bins: windsScores of 500 mb geopotential for NH and SH(last 24 days in August 2000)

The Wave Model (ECWAM)

Global wave height RMSE between buoys and WAM analysis (original list!)

The Wave Model (ECWAM)

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