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On the high resolution oastal appli ations with WAVEWATCH III

R©

14

thInternational Workshop on Wave Hind asting and Fore asting, and 5

thCoastal Hazard Symposium

Key West, Florida, USA, Nov 8-13, 2015.

Marion Hu het,

Servi e Hydrographique et O éanographique de la Marine, Brest, Fran e

É ole Centrale Nantes, Nantes, Fran e

Fabien Le kler,


Jean-François Filipot,

Fran e Énergies Marines, Plouzané, Fran e

Aron Roland,

BGS IT&E, Darmstadt, Germany

Fabri e Ardhuin,

Laboratoire de Physique des O éans, UMR 6523 CNRS-IFREMER-IRD-UBO, 29200 Plouzané, Fran e

Mathieu Dutour Sikiri¢,

Institute Ruder Boskovi¢, Zagreb, Croatia

Héloïse Mi haud,


Matthias T. Delpey,

SUEZ Eau Fran e - Rivages Pro Te h, Bidart, Fran e

Guillaume Dodet,

GEOMER - UMR 6554, CNRS-LETG, Institut Universitaire Européen de la Mer, Plouzané, Fran e

1 Introdu tion

Phase-averaged wave models onsider the spe tral de-

omposition of sea surfa e elevation a ross wavenum-

bers k (or frequen ies f) and dire tions θ at point

(x, y) and time t. The evolution of spe tral density

F (k, θ, x, y, t) is resolved using the wave energy equa-

tion [Gel i et al., 1957℄:

dF

dt= Satm + Snl + Swcap + Sbt + Sbr + ... (1)

where the Lagrangian derivative of spe tral density,

on the left-hand side, in ludes the lo al time evolution

and adve tion in both physi al and spe tral spa es

[e.g. WISE Group, 2007℄. The �rst sour e term on

the right-hand side is the atmospheri sour e term

Satm, whi h in ludes the lassi al input of energy Sin

from wind to wave and the energy Sout from waves

to wind

1

, asso iated with fri tion at air-sea interfa e

[Ardhuin et al., 2009℄. The nonlinear sour e term

Snl represents energy transfers in the spe tral domain

due to wave-wave intera tions. Sbt is the sink of en-

ergy due to bottom fri tion. Sbr represents the strong

depth-indu ed wave breaking pro ess on the shore.

Swcap des ribes the wave dissipation due to white-

apping; other e�e ts may also be in luded su h as

Bragg-s attering [see e.g. WISE Group, 2007℄.

With the implementation of unstru tured grids e.g.

[Benoit et al., 1996, Roland et al., 2005, Roland,

2008℄, spe tral wave models be ame appropriate tools

1

The transfer of energy from waves to wind (Sout) is responsible for the swell dissipation over long distan es. A modi� ation

of the Ardhuin et al. [2010℄'s formulation was provided by Le kler et al. [2013℄.

1

for sea state modeling in oastal areas. In parti ular,

re ent developments in WAVEWATCH III

R©[Tolman

et al., 2014, hereinafter WW3℄ in luded the wave

setup pro ess, wave breaking and triad intera tions

for better wave modeling in oastal areas. Never-

theless, the in reasing need of very high resolution

grids also brings up the problem of the limited ompu-

tational resour es for many operational appli ations.

Simulations on su h high resolution grids are not solv-

able at all with expli it methods, due to the small

time step imposed by the Courant�Friedri hs�Lewy

(CFL) riteria. Therefore we have introdu ed into

WW3 the numeri al on ept that was su essfully im-

plemented in WWM-III [Roland et al., 2012, Roland

and Ardhuin, 2014℄.

2 Model developments

We present here the validation of the new numeri-

al s hemes within the WW3 framework. We have

parallelized the urrently available unstru tured grid

(hereinafter UG), using domain de omposition meth-

ods and a new impli it s heme that is 1st order in

time and spa e. The new s heme resembles the fa-

mous SIMPLE algorithm [Patankar, 1980℄ whi h is

widely used for the integration of the Navier-Stokes

equations. The left-hand side of the equation is inte-

grated using 1st order numeri al s hemes in time and

spa e, where in spa e the residual distribution frame-

work in terms of the N-S heme has been introdu ed.

The right hand side of the equation uses the same in-

tegrator as in WW3 but in matrix form. It writes the

fun tional derivate of the sour e terms after dropping

the o�-diagonal ontributions and it linearizes them

on the diagonal of the matrix using Patankar rules.

This way, negative parts are integrated on the new

time level, whi h strengthens the diagonal part of the

matrix and improves onvergen e behavior of the lin-

ear solver. We also linearize the sour e terms within

the integration time step; however, sin e the spe -

tral balan e in deep water needs a limiter to ensure

stable integration, we have splitted the integration of

the sour e terms so as not to limit the solution of

the negative shallow water sour e terms. For these,

fully non-linear integration is also available within the

iterative solver. The latter option is still under devel-

opment. The most important thing in the integration

s heme is that the limiter is not a ting on spatial ad-

ve tion, and therefore it ensures that the transient

modes are well represented a ording to the order of

the s heme itself. This new numeri al methods obey

no splitting error ontrary to the former s hemes on

stru tured and unstru tured grids. They give time-

independent, steady-state solutions and an be used

with reasonable large time steps to integrate WAE

os illation free in time.

3 Model Veri� ation

As the model and the numeri s have just been de-

veloped, we present in this paper the 1st tests on

a ura y and e� ien y of the newly developed meth-

ods. We start with the usual test ase for wave growth

and then look into shoaling/refra tion. Then, in or-

der to test the adve tion term in frequen y-dire tion

spa e, we look at following and opposing urrents.

To ombine propagation e�e ts and evaluate the var-

ious s hemes, we pi ked the Vin ent & Briggs ase

that involves fo using over an ellipti al shoal. We

validate the strong non-linear wave breaking sour e

term on a linear sloping bea h, where we investigate

time step dependen y of the solution and various ef-

fe ts of the limiters in WW3. Last but not least, we

investigate the e�e ts of wave approa hing strongly

under-resolved step-bathymetries and islands, where

e.g. the SWAN model was reported to blow up due

to so alled ina ura ies [see e.g. Dietri h et al., 2013℄

and needed to apply limiters to retain a stable integra-

tion whi h, however, resulted in signi� ant di�eren es

in the solutions [Roland and Ardhuin, 2014℄.

3.1 Case 1 : Wave growing under on-

stant wind

Blowing over the sea, the wind transfers energy to

the waves with a phase speed lower than wind speed,

whereas the waves with a phase speed lower than wind

speed dissipate energy by air-sea fri tion [Ardhuin

et al., 2010℄. In the same time, a part of this en-

ergy dissipates with wave breaking. Another part of

this energy is redistributed in the wave spe trum with

non-linear intera tions. In the spe tral wave model,

these pro esses are respe tively represented by the

sour e terms Satm, Swcap and Snl in the wave energy

balan e equation (1). The sour e term parametriza-

tion here tested is the parameterization TEST451 of

Ardhuin et al. [2010℄ in luding the orre tion for the

swell dissipation of Le kler et al. [2013℄.

Here we onsider sour e terms integration performed

on a small unstru tured grid, orresponding to uni-

form deep o ean onditions. The in�nite o ean is

modeled by dea tivating the wave energy adve tion in

spa e. The wave energy adve tion in spe tral domain

is also disabled for omputational ost onsiderations,

be ause of the uniform depth and the absen e of ur-

rent. The wave energy balan e (1) is then redu ed

to:

∂F

∂t= Satm + Snl + Swcap. (2)

2

0.5

1

1.5h s [m

]

0 5 10 15 20 25 300.2

0.4

0.6

0.8

f p [Hz]

time [h]

0.1 0.2 0.3 0.5 0.9

100

150

200

250

E(f,θ) at T = 5.00 h

θ [d

eg]

freq [Hz]0.1 0.2 0.3 0.5 0.9

100

150

200

250

E(f,θ) at T = 7.00 hθ

[deg

]

freq [Hz]

0.1 0.2 0.3 0.5 0.9

100

150

200

250

E(f,θ) at T = 10.00 h

θ [d

eg]

freq [Hz]0.1 0.2 0.3 0.5 0.9

100

150

200

250

E(f,θ) at T = 70.00 h

θ [d

eg]

freq [Hz]

Semi−Imp. (dt = 10.00s)

Imp. (dt = 10.00s)

Imp. (dt = 60.00s)

Imp. (dt = 300.00s)

Imp. (dt = 600.00s)

Figure 1: Top panels : Time series of the signi� ant wave

height, Hsig (top) and of peak frequen y, fp (bottom). Bot-

tom panels : Shapes of the extra ted spe tra after 5, 7, 10

and 70 hours of integration. The wind speed is onstant

to 10 m/s.

1 2 3 4 510

0

101

102

103

Sem

i−Im

p.dt

=10

.00s

Imp.

dt=

10.0

0s

Imp.

dt=

60.0

0s

Imp.

dt=

300.

00s

Imp.

dt=

600.

00sE

laps

ed ti

me

[s]

U

10 = 5.0 m/s

U10

= 10.0 m/s

U10

= 15.0 m/s

U10

= 20.0 m/s

Figure 2: Comparison of the omputational times with the

di�erent s hemes and with the di�erent time steps. The

shown omputational times are extra ted from the model

log �le and orrespond to a single pro essor run.

The equation is integrated for 72 hours and the wave

evolution is started from rest with onstant winds,

U10, of 5, 10, 15 and 20 m.s

−1. The wave growth

and the non-linear frequen y shifthing obtained with

U10 = 10 m.s

−1are shown in the �gure 1 (top pan-

els), using both the "histori al" semi-impli it s heme

and the newly implemented impli it s heme. The

shape of the obtained spe tra after 5, 7, 10 and 70

hours of integration are plotted on the bottom pan-

els. Both s hemes provide a nearly perfe t �t up to

numeri al errors using the same time step dt = 10 s

(in the �gure, the last plotted urve re overs previ-

ous ones). In reasing the time step with a fa tor

6, the semi-impli it s heme (not plotted here) pro-

vides a learly slower wave growth, whereas the im-

pli it s heme keeps in line. The omputational times

for both s hemes are also investigated and are shown

in �gure 2. The omputational times are extra ted

from the model log �le and orrespond to a single

pro essor run. Typi ally, for the same time step the

semi-impli it s heme is faster by a fa tor 2 to 3 than

the impli it s heme. The advantage of the impli it

s heme omes with the in rease in the time step,

whi h strongly redu es the omputational ost while

keeping appropriate results. The dependen e of the

omputational ost for both s hemes is aused by the

in reasing number of iterations needed to make the

results onvergent.

3.2 Case 2 : Wave- urrent intera -

tions

When propagating in a non-uniform urrent, the wave

spe trum is a�e ted by the energy adve tion o ur-

ing in spe tral spa e. A urrent ollinear to the wave

propagation indu es a frequen y shifting, whereas a

ross urrent implies wave refra tion. For the wave-

urrent intera tion test ases, we onsider a 226-node

unstru tured grid overing an area with longitudes

from 0 degree to 0.072 degree and with latitudes from

0 degree to 0.036 degree.

Moreover, the model was integrated using various

expli it s hemes: a �rst order s heme given by the

swit h PR1 and a third order s heme provided by the

swit hes PR3 and UG (instead of PR1). The expli it

s hemes are here all used with the EXPFSN s heme.

We �rst onsider waves going along an in reasing ur-

rent, to disable the e�e t of the breaking o uring

when waves fa e an in reasing urrent. The o ean

depth is uniform (d = 5000 m) over the onsidered

area. A South-North urrent linearly in reases from

Ucur = 0 m.s

−1at the latitude 0 deg to Ucur = 2m.s

−1

at latitude 0.036 deg and is onstant along longitudes

and in time. The waves are for ed at the southern

boundary, where the urrent is null. The input wave

spe trum is reated using WW3 pre-pro essing tools

ww3_strt and is onstant in time. The for ed bound-

ary wave spe trum is gaussian in frequen y and osi-

3

nus type in dire tion. The peak frequen y is de�ned

to fp = 0.1 Hz with a frequen y spread of 0.01 Hz.

The wave mean dire tion is de�ned to θm = 270 deg

in o eanographi onvention (from South to North,

so that waves propagate with the in reasing urrent)

with a spreading de�ned by a osine power equal

to 20. As a result, the boundary spe trum is very

narrow in frequen ies and dire tions. The �gure 4

ompares the pro�les obtained after 20 hours of inte-

gration for the expli it s hemes [PR3 UG,EXPFSN

and PR1,EXPFSN℄ and the newly oded impli it

s heme. From top to bottom, the pro�les are re-

spe tively the urrent speed pro�le Ucur, the sig-

ni� ant wave height Hsig = 4√E, the mean wave

length Lm = 2πk−1, the mean wave period, Tm 0,2 =

2π/√

σ2, the mean wave period, Tm0,−1 = 2πσ−1

,

the peak frequen y, fp, and the mean wave dire tion

θm = atan

(

ba

)

, with a =∫ 2π

0

∫

∞

0cos(θ)F (σ, θ)dσdθ

and b =∫ 2π

0

∫

∞

0sin(θ)F (σ, θ)dσdθ. As the impli it

s heme must be of �rst order, the �t is obtained

with the �rst order expli it s heme. As expe ted,

the higher order s heme provides slightly di�erent re-

sults, expe ted to be more in line with the physi s.

In reasing the time step, both s hemes diverge from

lower time step resolved results. This is not expe ted

using the impli it s heme. However, the results are

only a bit di�erent and we think that the reason for

this di�eren e is the way we are handling the high

frequen y part of the spe tra, whi h must be further

investigated.

1 2 3 4 510

0

101

102

103

104

Exp

. [P

R3

UG

, EX

PF

SN

]dt

=1.

00s

Exp

. [P

R1,

EX

PF

SN

]dt

=1.

00s

Imp.

dt=

1.00

s

Imp.

dt=

10.0

0s

Imp.

dt=

60.0

0s

Ela

psed

tim

e [s

]

Figure 3: Comparison of the omputational times with the

di�erent s hemes and with the di�erent time steps. The

shown omputational times are extra ted from the model

log �le and orrespond to a MPI 8-pro essor run.

0

1

2

Ucu

r [m/s

]

0.8

0.9

1

Hs [m

]

160

180

200

220

L m [m

]

0.09

0.095

0.1

f p [Hz]

0 0.005 0.01 0.015 0.02 0.025 0.03179

180

181

θ m [d

eg]

latitude [deg]

Exp. [PR3 UG, EXPFSN] (dt = 1.00s)

Exp. [PR1, EXPFSN] (dt = 1.00s)

Imp. (dt = 1.00s)

Imp. (dt = 10.00s)

Imp. (dt = 60.00s)

Figure 4: Comparison of the pro�les obtained after 20

hours of integration for the expli it and impli it s hemes

with di�erent time steps for the waves propagating without

in iden e in the in reasing urrent.

0

1

2

Ucu

r [m/s

]

0.8

0.9

1

Hs [m

]

160

180

200

220

L m [m

]

0.09

0.095

0.1

f p [Hz]

0 0.005 0.01 0.015 0.02 0.025 0.03224

226

228

θ m [d

eg]

latitude [deg]

Exp. [PR3 UG, EXPFSN] (dt = 1.00s)

Exp. [PR1, EXPFSN] (dt = 1.00s)

Imp. (dt = 1.00s)

Imp. (dt = 10.00s)

Imp. (dt = 60.00s)

Figure 5: Comparison of the pro�les obtained after 20

hours of integration for the expli it and impli it s hemes

with di�erent time steps for the waves propagating with

in iden e in the in reasing urrent.

4

On the other hand, we implemented a on�guration

with waves going with an in reasing ross urrent.

Here the South-North urrent linearly in reases from

Ucur = 0 deg on the Eastern and Southern boundaries

to Ucur = 4 m.s

−1at longitude 0.070 deg and latitude

0.035 deg (North-West orner of our area). The waves

are now for ed at the Eastern and Southern bound-

aries, where the urrent is null. The input wave spe -

trum is again reated using WW3 pre-pro essing tools

ww3_strt with the same de�nitions as previously, ex-

ept for the wave mean dire tion whi h is now de�ned

to θm = 235 deg (in o eanographi onvention, from

South-East to North-West) so that waves propagate

in the in reasing South-North urrent with a non-null

in iden e. This non-null in iden e implies the refra -

tion of the waves. Figure 5 shows the pro�les obtained

after 20 hours of integration for both expli it and im-

pli it s hemes, as des ribed above. The on lusions

for this test ase are similar to the previous one: a

�t is obtained with the �rst order expli it s heme,

validating the impli it refra tion s heme, but provid-

ing slightly worse results than higher order s hemes.

Clearly, more investigation is needed for this ase.

3.3 Case 3 : Wave rea hing oast over

a linear bea h

The pro�le of the linear bea h is a onstant slope

of 1:25 on Y -axis. The pro�le length is 300m, from

Y = 0 m with z = 0 m to Y = 300 m with

z = −12 m. The pro�le is onstant along X-axis

and the bea h width is 1000 m from X = −500 m

to X = 500 m. A re tilinear grid with a resolution

de�ned to dX = 10 m (along-shore) and dY = 5 m

( ross-shore) is implemented. Then, by utting the

re tangles of the re tilinear grid in their diagonal, we

reate the triangular grid. The waves are for ed at

the Y = 300 m boundary with a JONSWAP spe -

trum. Two input spe tra are reated with the signi�-

ant wave height hosen to Hsig = 0.5 m and the peak

frequen y to fp = 0.20 Hz. The model is integrated

for 3 minutes.

First, we onsider waves propagating without any in-

iden e angle over the bea h. The pro�les obtained

at the enter of the bea h are plotted in �gure 6

with both expli it and impli it s hemes. The expli it

s heme is run with time steps dt = 0.05 s orrespond-

ing to CFL<1. The impli it s heme is run with the

same time step and with time steps in reased by fa -

tors up to 100 (i.e. dt = 5 s). The expli it time

step for the sour e terms integration is hosen small

enough to well resolve the bathymetri breaking dissi-

pation without any need of the Mi he Limiter (swit h

MLIM in WW3), whi h is then dea tivated for all

on�gurations. Moreover, the model was integrating

using various expli it s hemes, with both a �rst or-

der s heme given by the swit h PR1 and a third or-

der s heme provided by the swit hes PR3 and UG

(instead of PR1). The expli it s hemes are here all

used with the EXPFSN s heme. The results obtained

with the impli it s heme well �t those obtained with

the �rst order expli it s heme [PR1, EXPFSN℄ on

the unstru tred grid; they are in line with the pro-

�les obtained with the third order s heme [PR3 UG,

EXPFSN℄.

0.42

0.44

0.46

0.48

0.5

0.52

h s [m]

−0.5

0

0.5

1

1.5

h s [cm

]

Diff

. to

mea

n pr

ofile

−0.01

0

0.01

0.02

0.03

0.04

θ m [d

eg]

0 50 100 150 200 250 300

0.194

0.196

0.198

0.2

f p [Hz]

Y [m]

Rect. Imp. [PR3 UG, EXPFSN] (dt = 0.05s)

Rect. Imp. [PR1, EXPFSN] (dt = 0.05s)

Unst. Exp. [PR3 UG, EXPFSN] (dt = 0.05s)

Unst. Exp. [PR1, EXPFSN] (dt = 0.05s)

Unst. Imp. (dt = 0.05s)


Figure 6: Cross-shore pro�les obtained after 3 minutes of

integration. The pro�les are extra ted at the enter of the

bea h (X = 0 m) to avoid edge e�e ts. The mean wave

dire tion of propagation at the open boundary is perpendi -

ular to the bea h isobathes. From top to bottom, the �rst

panel shows the signi� ant wave height (Hs) pro�les, the

se ond one is the di�eren e of ea h Hs pro�le to the mean

Hs pro�le (meaning over all s hemes). Then, the third

panel represents the mean wave dire tion, and �nally, the

last one shows the peak frequen y pro�les.

We then onsider waves propagating with a non-null

in iden e angle over the bea h. The input boundary

JONSWAP spe trum is now de�ned to provide an an-

gle of 25�between wave propagation and ross-shore

axis. The depth gradient along wave rest then leads

to wave refra tion. The refra tion of waves is plot-

ted on the bottom pro�les of �gure 7. This ase also

5

shows the same on lusion, with a quite perfe t �t

of the wave refra tion obtained between the impli it

and the expli it �rst order s heme [PR1, EXPFSN℄

on the unstru tured grid.

0.42

0.44

0.46

0.48

h s [m]

−0.5

0

0.5

1

1.5

h s [cm

]

Diff

. to

mea

n pr

ofile

5

10

15

20

25

θ m [d

eg]

0 50 100 150 200 250 300

0.194

0.196

0.198

0.2

0.202

f p [Hz]

Y [m]

Rect. Imp. [PR3 UG, EXPFSN] (dt = 0.05s)

Rect. Imp. [PR1, EXPFSN] (dt = 0.05s)

Unst. Exp. [PR3 UG, EXPFSN] (dt = 0.05s)

Unst. Exp. [PR1, EXPFSN] (dt = 0.05s)



Figure 7: Cross-shore pro�les obtained from the grid af-

ter 3 minutes of integration. The pro�les are extra ted at

the enter of the bea h (X = 0 m) to avoid for the edge

e�e ts. The mean wave propagation at the open boundary

is here for ed with a non-null in iden e angle of 25 deg.

From top to bottom, the �rst panel shows the signi� ant

wave height (Hs) pro�les, the se ond one is the di�eren e

of ea h Hs pro�le to the mean Hs pro�le (meaning over

all s hemes). Then, the third panels represents the mean

wave dire tion, and �nally, the last one shows the peak

frequen y pro�les.

3.4 Case 4 : The Deep Sea Island ase

The next ase is inspired by the paper of Dietri h

et al. [2013℄ to show that we have for su h a ase

monotone and stable results in our numeri al model,

in ontrast to the results shown in the latter work.

The intention of this test is rather to show the model

onvergen e and stability in regions of under-resolved

bathymetry. There is one island de�ned as a hole in

the mesh and the others are submerged, going from

1000m depth to 10m and 15m respe tively (see �gure

8). The results are fully onvergent up to a solver

threshold of 10E-20, stable and monotone (see �g-

ure 9), whi h is expe ted from a 1st order monotone

impli it s heme. However, more tests are needed to

have more eviden e with respe t to the robustness of

the numeri al s heme. The boundary of the island

represented by a hole in the mesh does not take spe -

tra propagation into a ount, whi h in reases onver-

gen e speed. If the island is further resolved refra -

tion e�e ts ome naturally. However, we would like

to stress again that no limiters are used neither on

slope nor on propagation speeds. The s heme devel-

oped here is a building blo k for higher order s hemes,

whi h is the basis and must be onsistent.

Figure 8: Bathymetry of the Deep Sea Island.

Figure 9: Results for the Deep Sea Island ase

6

3.5 Case 5 : Waves over an ellipti

mount

The next ase is inspired by the tank experiment

of Vin ent and Briggs [1989℄ with the motivation to

ompare the refra tion/shoaling hara teristi s of the

various s hemes, as well as to investigate time step de-

penden y of the new developed model. In our simu-

lations, water depth is set at a onstant value outside

the ellipti shoal. The bathymetry is shown in �g-

ure 10 with the dashed lines representing the pro�les

investigated in this paper.

The ellipti shoal is patterned with a major radius

of 4 m along Y , a minor radius of 3 m along X and

a maximum height of hmax = 30.48 m at the en-

ter. Outside the ellipse, the water depth is equal to

dmax = 45.72 m. Therefore, at the top of the el-

lipse the water depth rea hes a minimum equal to

dmin = 15.24 m. The perimeter of the ellipti shoal

is then de�ned with:

(

X

3

)2

+

(

Y

4

)2

= 1. (3)

The depth d is de�ned in the perimeter with:

d(X,Y ) = dmax + hmax ∗

√

1− (X ′

3)2 − (

Y ′

4)2 (4)

and with d(X,Y ) = dmax else.

0 5 7.6 10 150

4

55.6

8

10

15

X [m]

Y [m

]

Depth[m]

0.2

0.25

0.3

0.35

0.4

0.45

Figure 10: Bathymetry inspired of the ellipti mount ex-

periment of Vin ent and Briggs [1989℄.

Three meshes were reated. The �rst mesh is a re -

tilinear grid with resolution dX = dY = 0.2 m. The

se ond mesh is triangular and is formed by utting

squares of the re tilinear grid in their diagonal to re-

ate the triangles. The third mesh is a 2571-node un-

stru tured mesh reated with non-regular triangles.

The results obtained with the two unstru tered grids

are very similar and here we only present the results

obtained with the non-regular, triangular mesh. The

input spe tra are for ed on all boundaries. All the

possible expli it s hemes implemented in WW3 are

tested here, following Roland [2008℄. These s hemes

are implemented in WW3 following the on ept of

the fra tional step method and are either mixed with

1st order upwind s hemes or with 3rd order Ultimate

Qui kest s hemes for spe tral spa e. The impli it

s heme is entirely 1st order in time and spa e.

Four ases are experimented in this study, orre-

sponding to the tests 02, 03, 16 and 17 of Vin ent

and Briggs [1989℄. We keep their test numbers in this

paper. The two �rst ases (TEST02 and TEST03)

orrespond to non-breaking ases, with respe tively

narrow (�gure 11) and broad (�gure 12) input spe -

tra. The two next ases (TEST16 and TEST17) or-

respond to breaking ases, with respe tively broad

(�gure 13) and narrow (�gure 14) input spe tra.

The expli it s heme is run with both �rst and third

order s hemes, with the time step de�ned as dt =0.01 s (CFL<1) on a re tangular grid. The 3rd order

solution an be seen as a referen e solution. The ex-

pli it s hemes up to 2nd order in time and spa e result

in under- and overshooting of the 3rd order results,

but the 1st order results either impli it or expli it are

more or less in line with the expli it results. It seems

that the impli it s heme is a bit more di�usive than

the expli it �u tional splitting s hemes. The impli it

s heme is run with the same time step and aslo with

the time step in reased by a fa tor 100. We �rst no-

ti e that the two impli it runs provide very similar

results, with a omputational time step redu ed by a

fa tor more than 30 for the larger time step and for

all ases. The results for the higher order s hemes of

WW3 are somewhat suspi ious in terms of overshoot-

ing to the 3rd order Ultimate Qui kest, whi h needs

further investigation. Moreover, we need to take into

a ount higher order propagation e�e ts as dis ussed

in e.g. Holthuijsen et al. [2003℄, Toledo et al. [2012℄,

Liau et al. [2011℄. However, this in ludes amplitude

dispersion that makes the left-hand side fully non-

linear, sin e the wave velo ities depend on various

spatial and temporal gradients of the solution itself.

7

0 5 10 15

0.06

0.07

0.08

0.09

Y [m]

Hs [m

]

Test02 : Profile X = 7.6 m

0 5 10 15

−0.4

−0.2

0

Z [m

]

Rect. Exp. [PR3 UG] (dt = 0.01s)

Rect. Exp. [PR1] (dt = 0.01s)

Unst. Exp. [PR3 UG,EXPFSN] (dt = 0.01s)

Unst. Exp. [PR3 UG,EXPFSPSI] (dt = 0.01s)

Unst. Exp. [PR3 UG,EXPFSFCT] (dt = 0.01s)

Unst. Exp. [PR1,EXPFSN] (dt = 0.01s)

Unst. Exp. [PR1,EXPFSPSI] (dt = 0.01s)

Unst. Exp. [PR1,EXPFSFCT] (dt = 0.01s)



0 5 10 15

0.057

0.058

Hs [m

]

Test02 : Profile Y = 4.0 m

0 5 10 15

178

180

182

Dir

[deg

]

X [m]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

0.06

Hs [m

]


0 5 10 15

170

180

190

Dir

[deg

]

X [m]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

0.05

Hs [m

]


0 5 10 15

170

180

190

Dir

[deg

]

X [m]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

−0.4

−0.2

0

Z [m

]

Figure 11: Pro�les obtained after 40 se onds of integration

for TEST02 (non-breaking ase, narrow spe trum)

0 5 10 15

0.06

0.065

0.07

0.075

Y [m]

Hs [m

]


0 5 10 15

−0.4

−0.2

0

Z [m

]











0 5 10 15

0.056

0.057

0.058

Hs [m

]


0 5 10 15

175

180

185

Dir

[deg

]

X [m]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

0.06Hs [m

]


0 5 10 15

170

180

190

Dir

[deg

]

X [m]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

0.06

Hs [m

]


0 5 10 15

175

180

185

Dir

[deg

]

X [m]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

−0.4

−0.2

0

Z [m

]


for TEST03 (non-breaking ase, broad spe trum)

8

0 5 10 15

0.12

0.14

Y [m]

Hs [m

]


0 5 10 15

−0.4

−0.2

0

Z [m

]











0 5 10 15

0.132

0.134

0.136

0.138

0.14

0.142

Hs [m

]


0 5 10 15

175

180

185

Dir

[deg

]

X [m]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

0.12

0.13

0.14

Hs [m

]


0 5 10 15

170

180

190

Dir

[deg

]

X [m]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

0.14

Hs [m

]


0 5 10 15

175

180

185

Dir

[deg

]

X [m]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

−0.4

−0.2

0

Z [m

]


for TEST16 (breaking ase, broad spe trum)

0 5 10 15

0.12

0.14

Y [m]

Hs [m

]


0 5 10 15

−0.4

−0.2

0

Z [m

]











0 5 10 150.132

0.134

0.136

0.138

0.14

0.142

Hs [m

]


0 5 10 15

178

180

182

Dir

[deg

]

X [m]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

0.12

0.14

Hs [m

]


0 5 10 15

170

180

190

Dir

[deg

]

X [m]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

0.12

0.13

0.14

Hs [m

]


0 5 10 15

170

180

190

Dir

[deg

]

X [m]

0 5 10 15

−0.4

−0.2

0

Z [m

]

0 5 10 15

−0.4

−0.2

0

Z [m

]


for TEST17 (breaking ase, narrow spe trum)

9

4 Implementation on the Iroise

Sea

The real ase is implemented on the Iroise Sea, at

the west of Brittany, Fran e. This sea provides both

very strong tide urrents and high tide water level

variations. It is also s attered with many islands and

ro ky shoals. We here implemented an unstru tured

12 518-node mesh represented in �gure 15 [see Ard-

huin et al., 2012℄. This mesh was done using the

POLYMESH tool. The open boundaries are for ed

by 121 for ing boundary nodes linearly spa ed every

5 km. The oast line is resolved with a resolution of

about 200 m.

Figure 15: Unstru tured mesh of the Iroise Sea imple-

mented in WW3, from 5 km resolution o�shore to about

200 m resolution at the oast line.

The boundary spe tra are gotten from the PRE-

VIMER/HOMERE WW3 hind asts [Boudière et al.,

2013℄ whi h provides 3-hour spe tra lose to ea h

boundary point. The wind is obtained from the Eu-

ropean Centre for Medium-Range Weather Fore ast

(ECMWF) hind asts, with 1/4 degree spa e resolu-

tion and 3 hours time resolution. Finally, the wa-

ter levels and the urrents are obtained from PRE-

VIMER MARS-2D hind asts with 250 m spa e reso-

lution and 15 minutes time resolution [see des rip-

tion in Ardhuin et al., 2012℄. These for ing �elds

are then extrapolated on the mesh nodes. The hind-

ast presented here overs January and February,

2014. This period starts with a relative weak sea

state in January, but with many su essive storms

in February, onjugated with high tides [Dodet et al.,

2012℄. Three DATAWELL re orded the wave �eld

for this period at three interesting lo ations. A �rst

DATAWELL (DW2) re orded data far from the is-

lands and in a relatively smooth bathymetry area.

A se ond DATAWELL (DW1) re orded data in the

south of the Sein island. North-West in oming waves

ome to the buoy after rossing a ro ky shoal at the

west of the island. This shoal, named "Chaussée de

Sein", is a shallow water area s attered with numer-

ous ro ks. Finally, a last DATAWELL (DW5) fa es

the West oast of Banne island and is at the boarder

of the "Fromveur" hannel, where the tides provide

strong urrents, up to 4 m.s

−1in the hannel, and

up to 2 m.s

−1at the buoy lo ation. The model is

then integrated for the two months using the newly

implemented full impli it s heme, with the physi al

parametrization TEST451 of Ardhuin et al. [2010℄.

At the DW2 lo ation, over the full time series, the

model slightly underestimates the signi� ant wave

height with a bias of -0.18 m. This bias is due to

the di� ulties of the model to reprodu e the storm

events. Indeed, when looking only at wave �elds with

a signi� ant wave height inferior to 5 m, the model

bias be omes 0.19 m, with a slight overestimation of

the signi� ant wave height. The RMS-Error is 0.43 m

for the global time series, giving a normalized RMS-

Error of 11.5%. This result omes dire tly from the

good propagation of the waves for ed at the boundary

up to the buoy.

The next investigated buoy (DW5, �gure 17) lo-

ated next to the strong tide urrents hannel alled

"Fromveur" learly shows the tidal variations of the

waves, following the urrent and water level varia-

tions. The tidal variations are learly visible on the

signi� ant wave height and the peak wave dire tion

provided by the model. The amplitude of the peak

wave dire tion variation is well in line with the obser-

vations, but the amplitude of the tidal variation of the

wave height provided by the model is slightly under-

stimated ompared to the amplitude of the observed

wave height tidal variations. The peak frequen y time

series provided by the model do not show the variabil-

ity observed at the buoy. We also note that ex ept

10

for the storm event of February 14, 2015, the model

globally overestimates the wave height. The bias is of

0.59 m with a RMSE of 0.73 m (N-RMSE = 23.8%).

This error umulates the slight general wave over-

estimation (ex ept for storm events) at the western

boundary that propagates to the buoy and the un-

derestimation of the e�e ts of the tidal urrents and

water levels on the wave height. These di� ulties for

the wave model to well reprodu e the tidal variation

may be due to an una urate withe apping dissipa-

tion term in presen e of strong urrents.

The DW1 buoy re orded wave parameters on the

southern area of the Sein island. In that on�gu-

ration the in oming prevailing swells ( oming from

West-North-West) must go over a large ro ky shoal

at the western extremity of the island. This shoal

is sprinkled with a large number of small ro ky lus-

ters that blo k the waves. With the mesh used here,

these ro ky lusters are not resolved. As a result, the

in oming waves are not su� iently blo ked by the

shoal and the model provides a strong overestimation

of the signi� ant wave height ompared to the buoy

observations. With this king of strongly unsmoothed

bathymetry, the need of high resolution meshes fastly

resolved with the impli it s heme is highlight. Unfor-

tunately, the results are not yet available.

10−1

f p [Hz]

DW2

0

5

10

h s [m]

150

200

250

300

350

θ p [deg

]

0

10

20

30

U10

[m]

01/05 01/12 01/19 01/26 02/02 02/09 02/16 02/230

0.5

1

time

Cur

[m]

obsevation

model

Interpolated from ECMWF 1/4deg

Interpolated from MARS2D 250m

Figure 16: Comparison of wave buoy observations (DW2)

with the impli it model hind ast results.

10−1

f p [Hz]

DW5

0

5

10

h s [m]

150

200

250

300

350

θ p [deg

]

0

10

20

30

U10

[m]

01/05 01/12 01/19 01/26 02/02 02/09 02/16 02/230

1

2

timeC

ur [m

]

obsevation

model





10−1

f p [Hz]

DW1

0

5

10

h s [m]

150

200

250

300

350

θ p [deg

]

0

10

20

30

U10

[m]

01/05 01/12 01/19 01/26 02/02 02/09 02/16 02/230

0.2

0.4

0.6

time

Cur

[m]

obsevation

model





11

5 Con lusions

We have presented the veri� ation of the numeri al

part and a 1st real ase of a newly developed spe tral

wave model that was in luded in the WW3 frame-

work and that is based on WWM-III. The model re-

sults are promising in terms of a ura y and e� ien y.

The next step will be to validate the full model for

very high resolution bathymetries. Morever, we are

thinking of a full validation test suite for unstru tured

grid models to have a evaluation of numeri s in dif-

ferent environments. Sin e WWM-III was oupled to

SCHISM Roland et al. [2012℄ the presented numeri-

al framework is already well tested within a oupled

wave urrent model in 2d and 3d. The numeri al ba-

sis in the new wave model in WW3 also provides the

basis for future REA (Rapid Environmental Asses-

ment) and other a tivities that need fast and e� ient

downs aling. For expli it models, grids need to be

arefully optimized in order to make the e� ient in-

tegration possible. Fastly generated grids often have

undesirable triangles that strongly redu e the time

step, however in our method this does not pose a hard

onstraint. We are looking forward to extending this

numeri al basis for higher order non-linear methods

and we are at this time developing a fully non-linear

solver that will redu e further the time step depen-

den y of the results and give the basis for the solution

of extended version of the WAE that in lude higher

order propagation e�e ts.

A knowlegments

We thank the te hni al group at the Fren h Navy

Hydrographi and O eanographi Institute (SHOM)

who deployed the buoys. This model development

is part of the PREVIMER proje t o-funded by Eu-

ropean Union, Ifremer, Brittany Region, Finistère

departemental oun il and Brest Métropole O éane.

This work was also supported by the resear h pro-

gram PROTEVS funded by DGA and ondu ted by

SHOM.

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13

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