SWAN Advanced Course3. Model physics in SWAN

Delft Software Days28 October 2014, Delft

Contents

• SWAN, a third generation wave model

• SWAN, fully spectral

• Physics in SWAN: source terms

2

3

First, second and third generation models

• First generation:> parameters only (Hs, Tp, m)> without nonlinear interactions

• Second generation (Hiswa):> Per discrete direction, Hs and Tp.> crude parametric form of nonlinear interactions

• Third generation (Swan):> Spectral shape as function of frequency and direction> Approximations of Boltzman integral for nonlinear

interactions

Phase-averaged wave models

Gen Sin Snl Sds

1 • based on growthrate meas.

• large inmagnitude

• saturationlimit (on/offlimitspectrum)

2 • based on fluxmeasurements

• smaller than 1stgeneration

• parametricform

• limitedflexibility

• saturationlimit (as in1stgeneration)

3 • based on fluxmeasurements

• stress coupled tosea state

• approximateform ofBolzmannintegral

• explicit form

source term representation: dE/dt = Sin + Snl + Sds

4

Physics in SWAN

Figure courtesy Holthuijsen (TU-Delft)

Generation: wave growth by windPropagation: shoaling, refraction, reflections, diffraction

Transformation: non-linear wave-wave interactionsDissipation: wave breaking, whitecapping, bottom friction

5

In shallow water the Eulerian energy balance equation becomes:

refractionincl. shoaling incl. shoaling

Energy balance equation

x y c Ex y

E c E c E St

6

SWAN: fully spectral E( , )

Based on action balance equation (Action ):

x ySN N c N c N

x yN c ct

refraction (depth, current),diffraction (depth, obstacles)

shoaling (depth) frequency shift (current)

Wave propagation based on linear wave theory

Dispersion relation 2 tanh ,gk kh k U

Action N is conserved in presence of current, energy is NOT !

7

12

2

1 12 1

1

ag

a

g g a

ga

g

C Cm m

C c

cc acc a

Holthuijsen et al. (2003)

Diffraction in SWAN

8

3rd-generation formulations:• Input by wind (Sin)• Wave-wave interactions:

> quadruplets (Snl4)> triads (Snl3)

• Dissipation:> white-capping (Swcap)> depth-induced breaking (Sbr)> bottom friction (Sbot)

Source terms in SWAN

S = Sin + Snl4 + Swcap + Snl3 + Sbr + Sbot

deep shallow

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Physics in SWAN: Wind input

Sin ( , ) = A + B E( , )

•Linear wave growth: Caveleri and Malanotte-Rizzoli (1981):• A = A ( , , w,U*)

•Exponential wave growth:• Komen et al. (1984), Snyder et al. (1981) [WAM-cycle3]

• Janssen (1989, 1991) [WAM-cycle4]

*max 0, 0.25 28 cos 1p

a

w h sw

a e

Uc

B

22* max 0 , cos

phase

aw

w

Uc

B

( : Miles constant) 10

Alternative for exponential wave growth

Yan (1987):

Courtesy:Van der Westhuysen

11

Physics in SWAN: Wind input

2 2* 10DU C UTransformation:

310

310 10

1.2875 10 for 7.5 m/s0.8 0.065 10 for 7.5 m/sD

UC

U U1. Wu (1982):

2. Zijlema et al. (CE 2012):2 30.55 2.97 1.49 10DC U U

10 , 31.5m/sref refU U U U

12

Critical issues:• Effect of gustiness on wind input?• Is wave growth linearly or quadratically proportional to wind

speed?• Is there a limit to momentum transfer from atmosphere to wave

field at extreme wind speeds?• Does wind input depend on wave characteristics in shallow

water (steepness?) ?

Physics in SWAN: Wind input

13

Physics in SWAN: Whitecapping

1p

dsPM

k sCk s

, ,wcapkS Ek

52.36 10 , 0, 4dsC p

Whitecapping is represented by pulse-based model of Hasselmann(1974), reformulated in terms of wave number (for applicability in finite-water depth) by Komen et al. (1984):

with

Tunable coefficients:

• Komen et al. (1984, WAM-cycle3) :

• Janssen (1992, WAM-cycle4):54.10 10 , 0.5, 4dsC p

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Physics in SWAN: Whitecapping

( , ) ( , )qn

wc dsPM

k sS C Esk tots k EKomen et al. (1984):

1. Underprediction of mean wave period (mean and peak)under wind-sea conditions

2. Overprediction of wind-sea when a bit of swell is added

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Saturation-based whitecapping

( , ) ( , )qn

Komen dsPM

k sS C Esk

3( ) ( )gB k c k E

tots k E,

1 12 2

/ 2( )( , ) ( , )

p

Break dsr

B kS C g k EB

,

, ( , ) ( ) 1 ( )wc SB br Break br KomenS f S f S1

21 1 ( )( ) tanh 10 12 2br

r

B kfB

*up fc

Saturation based whitecapping by Van der Westhuysen et al. (2007),related to nonlinear hydrodynamics within wave groups :

Komen et al. (1984):

Adjusted by Van der Westhuysen (2007):

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Saturation-based formulation

Wind-sea part no longer affected byaddition of swell

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Pure wind sea: Lake George, Australia

20 km

Stronger wave growth and betterprediction in spectrum tail bysaturation-based model

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Fetch-limited situations

20 m/s

measured

SWAN default

SWAN saturationbased wcap

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• Deep water, fetches > 5km• position spectral peak improved (used to be at frequencies too

high), low-frequency part better predicted• wave energy in high-frequency tail correctly predicted (used to

be too much)• wave energy better predicted

• Deep water, fetches < 5km• strong overprediction of low-frequency energy (used to be

closer to measurements)

• Shallow water• computed spectral shape deviates from measured spectral

shape (pronounced spectral peak, onset to secondary peak)

Fetch-limited situations

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Physics in SWAN: Quadruplets

Computation of quadruplets is based on Boltzmann integral forsurface gravity waves;

1 2 3 4 1 2 3 4,k k k k

resonance condition:

1 2 3 4 1 2 3 4,

1 2 3 4 1 2 3 4k k k k

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DIA

Xnl

Van der Westhuysen etal. (2005):

• DIA (default) vs. Xnl

• accuracy vs. CPU

Physics in SWAN: Quadruplets

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• Exact methods to solve Boltzmann integral are not suitable foroperational wave models;

• (Initially deep-water) DIA is rather inaccurate, but less time-consuming (Hasselmann et al., 1985);

• Depth effects have been included by WAM scaling.• Quadruplets are of relative importance in relative deep water in

concert with white-capping and wind input.

Physics in SWAN: Quadruplets

Compared to exact method:• DIA provides lower significant wave heights and higher

mean wave periods;• Directional spreading is larger for DIA.

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and proportionality coefficient, fraction of breaking wavesand maximum wave height:

Physics in SWAN: Depth-induced wave breaking

214 2tot BJ b mD Q H

,,br tot

tot

ES D

E

1BJ

Energy dissipation due to depth-induced breaking is modelled by thebore-based model of Battjes and Janssen (1978) :

with

mHbQ

0.73 defaultmH d

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Problem over nearly horizontal beds

Default BJ78( BJ = 0.73)

Physics in SWAN: Depth-induced wave breaking

Apparent upper limit of Hm0/d in SWAN,due to fixed value of

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0.76( ) 0.29BJ pk d

Dependencies of BJ on local variables (vd Westhuysen 2010)

Ruessink et al. (2003):

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Depth breaking based on shallow water nonlinearityBiphase model by Van der Westhuysen, 2010)

3301

04m

tot bfBD H p H dHd

bp H W H p H

From Thornton & Guza (1983):

4,9

n

refref

W H

Introduce a biphase-dependentweighting function on the pdf:

Eldeberky(1996)

33013

16

n

mtot rms

ref

B fD Hd

loc loc44 arctann S S

Boers (1996):

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Calibration and validation of biphase model

Biphase model yields similarimprovement as Ruessink et al.parameterization, but withphysical explanation of modelbehaviour.

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Calibration and validation of biphase model

Amelander Zeegat (18/01/07, 12:20)

Wave growth limit reduced bybiphase model over nearlyhorizontal areas

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Critical issues wrt depth-induced wave breaking

• Does wave breaking depend on local wave characteristics, suchas local wave steepness?

• Is the dissipation rate frequency dependent?

• What is influence of long waves on breaking of shorter waves?

• Knowing that Battjes-Janssen model (BJ) hampers wave growth inshallow water, there is no breaker formulation for the entirespectrum of bottom slopes (ranging from horizontal to reef-type ofslopes) other than the recently implemented but highly empiricalformulation of Salmon et al. (ICCE, 2012).

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Physics in SWAN: Bottom friction

• JONSWAP (Hasselmann et al., 1973):

• Collins (1972):drag-law type

• Madsen et al. (1988):eddy-viscosity type

2bottom w rmsgC f U

2 3

2 3

0.038 m s (swell)0.067 m s (fully-developed sea)bottomC

2

2 2, ,sinhbot bottomS C E

g kd

( 0.015 default)bottom f rms fC C gU C

, 0.05 defaultw w bot N Nf f a K K

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• Triads modelled by Lumped Triad Interaction (LTA) method ofEldeberky (1996).

• In shallow water triads have a significant influence on waveparameters for non-breaking and breaking waves over asubmerged bar or on a sloping beach.

• Present formulation does not include energy transfer to lowerfrequencies. Transfer to higher frequencies oftenoverestimated. Conclusion: Modelling of triads in 2D waveprediction models needs improvement.

Physics in SWAN: Triads

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ENERGY DENSITY SPECTRA (2.61)

0

10

20

30

40

0.0 0.1 0.2 0.3 FREQUENCY (Hz)

EN

ER

GY

DE

NS

ITY

(m2 /H

z)

DEEPMP3MP5MP6TOE

ENERGY DENSITY SPECTRA (2.61)

0

10

20

30

40

0.0 0.1 0.2 0.3

FREQUENCY (Hz)

ENER

GY

DEN

SITY

(m2 /H

z)

DEEPMP3MP5MP6TOE

Measured (flume) Computed (SWAN)

FORESHORE - PETTEN

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-40 -35 -30 -25 -20 -15 -10 -5 0 5

FORESHORE (m)

ELE

VA

TIO

N(m

)

1:30

1:25 1:20 1:1001:25

1:4.5

1:20

1:3

MP3 MP5 MP6DEEP BAR

• No energy transfer to lowfrequencies

• Exaggeration of energy transfer tohigher harmonics

Physics in SWAN: Triads

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Hm0 Tm-1,0

With triads

No triads

Physics in SWAN: Triads

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Depth profile nearPetten Sea defence Tm-1,0 (with triads)Tm-1,0 (no triads)

Physics in SWAN: Triads

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