1.
SWAN Advanced Course
3. Model physics in SWAN
Delft Software Days
28 October 2014, Delft

2.
Contents
• SWAN, a third generation wave model
• SWAN, fully spectral
• Physics in SWAN: source terms
2

3.
First, second and third generation models
3
• First generation:
> parameters only (Hs, Tp, Ĭm)
> 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

4.
Phase-averaged wave models
source term representation: dE/dt = Sin + Snl + Sds
Gen Sin Snl Sds
1 • based on growth
rate meas.
• large in
magnitude
• saturation
limit (on/off
limit
spectrum)
2 • based on flux
measurements
• smaller than 1st
generation
• parametric
form
• limited
flexibility
• saturation
limit (as in
1st
generation)
3 • based on flux
measurements
• stress coupled to
sea state
• approximate
form of
Bolzmann
integral
• explicit form
4

5.
Physics in SWAN
Generation: wave growth by wind
Propagation: shoaling, refraction, reflections, diffraction
Figure courtesy Holthuijsen (TU-Delft)
Transformation: non-linear wave-wave interactions
Dissipation: wave breaking, whitecapping, bottom friction
5

6.
Energy balance equation
In shallow water the Eulerian energy balance equation becomes:
Ec
Tw w

7.
T
w

8.
w
w c E

9.
w x w y w
E cE S
x y
incl. shoaling incl. shoaling refraction
t
6

10.
SWAN: fully spectral E(V,T)
Based on action balance equation (Action ):
N c c t
w
N N c N c N S
w w w w

11.
V V T T V
w x y
x y
w w w w
refraction (depth, current),
diffraction (depth, obstacles)
shoaling (depth) frequency shift (current)
Action N is conserved in presence of current, energy is NOT !
Wave propagation based on linear wave theory
Dispersion relation V 2 gk tanh kh , V Z kU
7

12.
§ w w ·
¨¨ © w

13.
w ¸¸ ¹

14.
1
2
’˜ ’

15.
2
1 1
2 1
1
a
g
a
g g a
g
a
g
C C
m m
C c
cc a
cc a
T
N G
N G
G
G
N
Holthuijsen et al. (2003)
Diffraction in SWAN
8

16.
Source terms in SWAN
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)
S = Sin + Snl4 + Swcap + Snl3 + Sbr + Sbot
deep shallow
9

17.
Physics in SWAN: Wind input
Sin (V,T) = A + B E(V,T)
•Linear wave growth: Caveleri and Malanotte-Rizzoli (1981):
• A = A (V,T, Tw,U*)
•Exponential wave growth:
• Komen et al. (1984), Snyder et al. (1981) [WAM-cycle3]
ª U
­° ½°º « ® ¾»
U
c
max 0, 0.25 28 * cos

18.
1
a
w h s
• Janssen (1989, 1991) [WAM-cycle4]
T T V
U
w
«¬ °¯ p a e
°¿»¼
B

19.
2
* max 0 , cos 2
UE
T T V
U
§ ·
¨¨ ¸¸ ª¬ º¼
© phase ¹
a
w
w
U
c
B
(E : Miles constant) 10

20.
Alternative for exponential wave growth
Yan (1987):
Courtesy:
Van der Westhuysen
11

21.
Physics in SWAN: Wind input
2 2
* D 10 Transformation: U C U
­° ˜
3
U
® °¯
˜ t 1.2875 10 for 7.5m/s

22.
10
D 0.8 0.065 10 3
for 7.5 m/s
10 10
C
U
U
1. Wu (1982):
D 2. Zijlema et al. (CE 2012): 0.55 2.97 U i 1.49 U i 2

23.
˜ 10
3 C
i
10 , 31.5m/s ref ref U U U U
12

24.
Physics in SWAN: Wind input
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?) ?
13

25.
Physics in SWAN: Whitecapping
Whitecapping is represented by pulse-based model of Hasselmann
(1974), reformulated in terms of wave number (for applicability in finite-water
S k E
V ,T

26.
*V V ,T

27.
wcap
k
· * C § ¨ ·§ 1 G

28.
G
k s
¸¨ ¸
© ¹© ¹
p
ds
PM
k s
2.36˜105 ,G 0, 4 dsC p
depth) by Komen et al. (1984):
with
Tunable coefficients:
• Komen et al. (1984, WAM-cycle3) :
• Janssen (1992, WAM-cycle4): 4.10˜105 ,G 0.5, 4 dsC p
14

29.
Physics in SWAN: Whitecapping
1. Underprediction of mean wave period (mean and peak)
n q
§ · § · ¨ ¸ ¨ ¸
S C k s E
Komen et al. (1984): ( V , T ) V ( V , T
)
s k E
wc ds
tot PM
k s
© ¹ © ¹
under wind-sea conditions
2. Overprediction of wind-sea when a bit of swell is added
15

30.
Saturation-based whitecapping
n q
§ · § · ¨ ¸ ¨ ¸
S C k s E
( , ) ( , )
V T V V T
Komen ds
k s
© ¹ © PM
¹
tot , s k E
Saturation based whitecapping by Van der Westhuysen et al. (2007),
related to nonlinear hydrodynamics within wave groups :
( ) 3 ( ) g B k c k E V
ª ( ) º
/ 2 ( , ) ( , )
S C B k g 1 2 k 1
2
E
p
« »
V T V T
Break ds
r
B
¬ ¼
,
*p f u

31.
, ( , ) ( ) 1 ( ) wc SB br Break br Komen S V T f V ˜ S f V ˜ S
§ · ¨ ¸
© ¹
1
c
1 1 ( ) 2 ( ) tanh 10 1
br 2 2
f B k
B
r
V
§ ª º · ¨ « » ¸ ¨ ¬ ¼ ¸ © ¹
Komen et al. (1984):
Adjusted by Van der Westhuysen (2007):
16

32.
Saturation-based formulation
Wind-sea part no longer affected by
addition of swell
17

33.
Pure wind sea: Lake George, Australia
20 km
Stronger wave growth and better
prediction in spectrum tail by
saturation-based model
18

34.
Fetch-limited situations
20 m/s
measured
SWAN default
SWAN saturation
based wcap
19

35.
Fetch-limited situations
• 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)
20

36.
Physics in SWAN: Quadruplets
Computation of quadruplets is based on Boltzmann integral for
surface gravity waves;
k1 r k2 rk3 r k4 , V1 rV 2 rV 3 rV 4
resonance condition:
1 2 3 4 1 2 3 4 k r k rk r k r r r
1 2 3 4 1 2 3 4 r r r , V rV rV rV
21

37.
DIA
Xnl
Van der Westhuysen et
al. (2005):
• DIA (default) vs. Xnl
• accuracy vs. CPU
Physics in SWAN: Quadruplets
22

38.
Physics in SWAN: Quadruplets
• Exact methods to solve Boltzmann integral are not suitable for
operational 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.
Compared to exact method:
• DIA provides lower significant wave heights and higher
mean wave periods;
• Directional spreading is larger for DIA.
23

39.
Physics in SWAN: Depth-induced wave breaking
Energy dissipation due to depth-induced breaking is modelled by the
bore-based model of Battjes and Janssen (1978) :

40.
,

41.
S D
, br tot
V T
E
tot
E
V T
1 2
D V
D
Q § · ¨ ¸
H tot 4 BJ b 2 m S
© ¹
D 1 BJ
with
and proportionality coefficient, fraction of breaking waves
and maximum wave height:
m H
b Q
J J 0.73 default

42.
m H d
24

43.
Physics in SWAN: Depth-induced wave breaking
Problem over nearly horizontal beds
Default BJ78
(JBJ = 0.73)
Apparent upper limit of Hm0/d in SWAN,
due to fixed value of J
25

44.
Dependencies of JBJ on local variables (vd Westhuysen 2010)
J BJ 0.76(kpd) 0.29
Ruessink et al. (2003):
26

45.
Depth breaking based on shallow water nonlinearity
Biphase model by Van der Westhuysen, 2010)
From Thornton Guza (1983):
D B f H p H dH

46.
3
b p
f m
01 ³
3

47.
tot b
d
4 0
H W H p H
Introduce a biphase-dependent
weighting function on the pdf:
§ ·
¨¨ ¸¸
© ¹

48.
, 4
9
n
ref
ref
W H
E S
E
E
Eldeberky
(1996)
n 4 4 arctan Q S S
ª º ¬ ¼
3

49.
loc loc
n
§ ·
D B f H
01 3 3
16
S E
m
¨¨ ¸¸
tot rms
ref
d
E
© ¹
S
Boers (1996):
27

50.
Calibration and validation of biphase model
Biphase model yields similar
improvement as Ruessink et al.
parameterization, but with
physical explanation of model
behaviour.
28

51.
Calibration and validation of biphase model
Amelander Zeegat (18/01/07, 12:20)
Wave growth limit reduced by
biphase model over nearly
horizontal areas
29

52.
Critical issues wrt depth-induced wave breaking
• Does wave breaking depend on local wave characteristics, such
as 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 in
shallow water, there is no breaker formulation for the entire
spectrum of bottom slopes (ranging from horizontal to reef-type of
slopes) other than the recently implemented but highly empirical
formulation of Salmon et al. (ICCE, 2012).
30

53.
Physics in SWAN: Bottom friction
V

54.
V T V T bot bottom S C E
g kd
• JONSWAP (Hasselmann et al., 1973):
• Collins (1972):
drag-law type
• Madsen et al. (1988):
eddy-viscosity type
2
3
2 3
0.038m s (swell)
0.067 m s
(fully-developed sea)
C f g U
bottom w rms
2
­
®¯
bottom C
2
2 2 , ,
sinh
( 0.015 default) bottom f rms f C CgU C
,

55.
0.05 default

56.
w w bot N N f f a K K
31

57.
Physics in SWAN: Triads
• Triads modelled by Lumped Triad Interaction (LTA) method of
Eldeberky (1996).
• In shallow water triads have a significant influence on wave
parameters for non-breaking and breaking waves over a
submerged bar or on a sloping beach.
• Present formulation does not include energy transfer to lower
frequencies. Transfer to higher frequencies often
overestimated. Conclusion: Modelling of triads in 2D wave
prediction models needs improvement.
32

58.
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
Physics in SWAN: Triads
ENERGY DENSITY SPECTRA (2.61)
40
Measured (flume) Computed (SWAN)
30
20
10
0
1:20
0.0 0.1 0.2 0.3
FREQUENCY (Hz)
ENERGY DENSITY (m2/Hz)
DEEP
MP3
MP5
MP6
TOE
ENERGY DENSITY SPECTRA (2.61)
40
Hz)
m2/30
(DENSITY 20
ENERGY 10
0
FREQUENCY (Hz) DEEP
MP3
MP5
MP6
TOE
0.0 0.1 0.2 0.3
FORESHORE - PETTEN
-0.6
-40 -35 -30 -25 -20 -15 -10 -5 0 5
FORESHORE (m)
ELEVATION (m)
1:30
1:25 1:20 1:100
1:25
1:4.5
1:3
DEEP MP3 BAR MP5 MP6
• No energy transfer to low
frequencies
• Exaggeration of energy transfer to
higher harmonics
33

59.
Physics in SWAN: Triads
Hm0 Tm-1,0
No triads
With triads
34

60.
Physics in SWAN: Triads
Depth profile near
Petten Sea defence Tm-1,0 (no triads) Tm-1,0 (with triads)
35