Impact of waves on the ocean circulation

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To cite this version:

Nicolas Rascle. Impact of waves on the ocean circulation. Ocean, Atmosphere. Université de Bretagne

occidentale - Brest, 2007. English. �tel-00182250�

présentée par

Ni olas Ras le

Impa t des vagues

sur la ir ulation o éanique

soutenue le18 Septembre 2007 devant lejury omposé de :

M. Xavier Carton Président du jury UBO, Brest

M. Vladimir Kudryavtsev Rapporteur NIERSC, Saint-Pétersbourg M. Agustin San hez-Ar illa Rapporteur UPC, Bar elone

M. Philippe Bonneton Examinateur U. Bordeaux 1, Talen e M. Bertrand Chapron Examinateur IFREMER, Plouzané M. Rodrigo Pedreros Examinateur BRGM,Orléans

Ma thèse aété nan ée par une bourse CNRS/DGA.

Etpuisjetiensàremer iertoutd'abordFabri eArdhuinpoursadisponibilité,sa patien e, ses en ouragements etson enthousiasme débordant,et ses autresqualités humainesqui en font un dire teur de thèseex eptionnel.

Mer i à Rudy Magne pour son re ul et ses onseils avisés sur la thèse (et sur la vie aussi aupassage!).

Au CMO, mer i à Stéphanie Louazel, Nadine Paugam, Alain Serpette, Yves Morel de Toulouse.

Mer i au LPOde lafa pour son a ueilen son sein depuis 2006, etles onseils te hniquesglanés àdroiteàgau he auprès de GildasCambon(et bon ouragepour lan), Steven Herbette, BrunoBlanke,Xavier Carton,GuillaumeRoullet.

Mer i à Patri k Mar hesiello età PhillipeEstradepour leur oup de main pour démarrer ave ROMS.

Mer i àRomainBourdallé-Badiede Mer ator àToulouse,pour son tempspassé sur notre projet.

Enn mer i à Bertrand Chapron pour son suivide mathèse ave Fabri e, pour ses onseils, remarques et autresdivagations.

A knowledgements

Thanks to my thesis reviewers, Vladimir Kudryavtsev and Agustin San hez-Ar illa,for the time they spent onmy manus riptthis summer.

Thanks to ea h member of my thesis jury, for their time and for their kind ommentaries.

Introdu tion générale 11

Le ontexte : L'hydrodynamique tièreet de surfa e . . . 11

Problématique : L'impa tdes vagues . . . 12

Plan de l'exposé . . . 13

General introdu tion in english 15 The ontext: The hydrodynami s nearthe oast and near the surfa e . . . 15

The issue: The impa t of waves . . . 16

Contents . . . 17

1 General on epts 19 1.1 Lengths and time s ales . . . 19

1.2 Waves, Stokesdrift, averaging . . . 20

1.3 Wave /Mean owseparation . . . 22

1.4 The Stokes-Coriolisee t. . . 24

I Impa t of waves on the near-surfa e dynami s of the open o ean. One-dimensional study. 27 One-Dimensional study: Introdu tion 29 2 One-Dimensional des ription: Part 1: without strati ation 31 Abstra t . . . 32

2.1 Introdu tion . . . 32

2.2 Wave dynami s . . . 34

2.2.1 Spe tralwave evolution . . . 34

2.2.2 The Stokesdrift . . . 35

2.2.3 Pra ti al al ulation of wave parameters . . . 36

2.3 Wave-averaged mixed layerequations . . . 39

2.3.2 Turbulent losure . . . 41

2.4 Model results and validation . . . 42

2.4.1 Calibrationof the modelwithobserved proles of TKE dissi-pation . . . 42

2.4.2 Eulerianhodographs and shears . . . 46

2.4.3 Lagrangian drift. . . 54

2.5 Generaldis ussion . . . 57

2.6 Con lusion . . . 59

3 One-Dimensional des ription: Part 2: with strati ation 63 Abstra t . . . 64

3.1 Introdu tion . . . 64

3.2 The model . . . 66

3.3 Analysisof the near-surfa e shears - The SMILE data . . . 66

3.3.1 The experiment . . . 66

3.3.2 The model . . . 67

3.3.3 Previous analysis . . . 67

3.3.4 A less onstraining analysis . . . 69

3.4 Analysisof the urrent magnitude - The LOTUS data . . . 73

3.4.1 A simple modelof the diurnal y le . . . 74

3.4.2 A more elaboratemodel: onstraining the strati ation . . . 75

3.4.3 Validatingthe wave-indu ed mixingparameterization.. . . 78

3.4.4 The Stokes-Coriolisee t . . . 78

3.4.5 The wave bias . . . 81

3.5 Surfa e drift . . . 82

3.6 Con lusion . . . 85

4 One-Dimensional study: Epilogue and Perspe tives 89 4.1 Is the surfa e drift due tothe windor due tothe waves ? . . . 89

4.2 Furtherveri ations of the present des ription . . . 90

4.2.1 Thedrifters observationsand themodelofKudryavtsev etal. (2007) . . . 90

4.2.2 Other determinations of the roughness length . . . 93

II Impa t of waves on the o ean mixed layer 95

5.2.2 Wave-indu ed mixingthrough the whole mixed layer . . . 99

5.3 Estimations of the wave-related parameters . . . 100

5.3.1 The Stokesdrift

Us

. . . 100

5.3.2 The Stokestransport

Ts

. . . 102

5.3.3 The roughness length

z

0

. . . 103

5.3.4 The TKE ux

αu

∗

₃

. . . 109

5.4 The spatial and temporal distribution of mixing events. A dire t parameterization from the wind ? . . . 112

5.5 Thedierentkindsofverti almixingmodelsforappli ationsinOGCMs114 5.6 A model toestimatethe impa tof waves onthe mixed layerdepth. . 114

5.7 Preliminary resultson the impa t of waves onthe mixed layer depth. 115 III Impa t of waves on the nearshore and shelf ir ula-tion 119 6 Nearshore and Shelf ir ulation: Introdu tion 121 6.1 Introdu tion . . . 121

6.2 Totalowequations . . . 121

6.3 Mean owequations . . . 123

6.4 On the ouplingof waves and urrent . . . 124

6.5 Models and observations . . . 124

7 Nearshore and Shelf ir ulation: A two dimensional study 129 Abstra t . . . 130

7.1 Introdu tion . . . 130

7.2 GLM des ription of the ow . . . 132

7.2.1 Wave / meanow separation . . . 132

7.2.2 Equationsof motion . . . 133

7.2.3 Verti al boundary onditions. . . 134

7.3 Des ription of the numeri al experiment . . . 135

7.3.1 The wave model. . . 135

7.3.2 The o ean model . . . 136

7.4 Analysis of the solution. . . 136

7.4.1 Des riptionof the dierent termsof the equations . . . 136 7.4.2 Des riptionof thealongshoreand ross-shoremomentum

bal-7.5 Comparisonwith the modelof Newberger and Allen(2007a) . . . 142

7.5.1 Lagrangian mass uxin NA07 . . . 142

7.5.2 Vortex for ein NA07 . . . 146

7.6 Ee t of the urrent shear . . . 147

7.6.1 Ee t of the urrent shearon the Stokes drift . . . 147

7.6.2 Ee t of the urrent shearon the radiationstress . . . 148

7.7 Ee t of the wave non-linearity . . . 149

7.8 Con lusion . . . 151

7.9 Appendix A : modi ation of ROMS tosolve the GLM equations . . 152

7.9.1 Equations in

σ

- oordinates . . . 152

7.9.2 Depth-integrated equations . . . 153

7.10 Complements : Numeri alimplementation of the GLM equations in ROMS . . . 155

7.10.1 Modi ation of the time stepping to in lude the diverging mean ow . . . 155

7.10.2 Dis ussion . . . 156

7.10.3 Momentum for ingterms . . . 157

7.10.4 Volume onservation and boundary onditions . . . 158

General on lusion 159 Brève on lusion générale en français 163 Annexe A : Equations GLM2z 167 8 Annexe A : Equations GLM2z 167 8.1 Introdu tion . . . 171

8.1.1 Air-water separation . . . 173

8.1.2 Separation of wave and urrent momentum uxes . . . 173

8.2 glm2-RANS equations . . . 175

8.2.1 Generalities onGLM and linear wave kinemati s. . . 175

8.2.2 glm2-RANSequations in

z

- oordinates . . . 185

8.3 Relationsbetween the present theory and known equations . . . 190

8.3.1 Depth-integrated GLMfor a onstant density

ρw

. . . 190

8.3.2 Equations of M Williamset al. (2004) . . . 192

8.4 Limitationsof the approximations . . . 194

Le ontexte : L'hydrodynamique tière et de surfa e

Avantdeparlerdevaguesetde ir ulationo éanique,jesouhaiteraisintroduirele ontextegénéralede etteétude.L'étudedela ir ulationo éanique,etdel'o éanen général,est au arrefourde nombreuxenjeux, etilsembleopportund'endresserune listerapide pour avoirune idée des appli ations pratiquesde laprésente re her he. À l'é helle du globe, la ir ulation o éanique intéresse parti ulièrement pour son impa t sur le limat. À l'é helle tière, les intérêts sont plus divers, depuis la onnaissan eetlaprévision desmouvementsdesmasses d'eaupour appli ationàla biologie, à la biogéo himie, l'halieuthique, à la défense sous-marine, jusqu'au suivi des pollutions etau sauvetage en mer. Enn àl'é helle littorale, l'hydrodynamique de l'o éan est étudiéeprin ipalementpour ses appli ations àl'érosion des otes, au transportsédimentaire.

Pour beau oup d'appli ationsparmi elles itées i-dessus, un des pré-requisest de posséder un modèle de l'o éan qui représente orre tement les transports parti- ulairesprès de lasurfa e (suivides pollutions,sauvetage en mer,étude des dérives de larves en halieuthique), qui représente orre tement le mélange des ou hes de surfa e o éaniques , ave les y les diurnes et saisonniers (études limatiques, bio-géo himie),et qui représente orre tement la ir ulation aux abords immédiatsdes plages, dans la zone de déferlement des vagues et au-delà, sur le plateau interne (érosiondes plages,transport des sédiments, suivi des polluantsrejetés de la te). En fait une grande partie des a tivités de re her he o éanographiques né essitent une bonne onnaissan e de l'hydrodynamique près de la surfa e et près de la te, et e n'est pas surprenant puisque l'essentieldes a tivités humaines etanimaless'y on entrent.

Voila e qui onstitue la motivation pour étudier l'hydrodynamique de ette partie de l'o éan, etplus pré isément l'impa tdes vagues sur elle- i.

Problématique : L'impa t des vagues

Les vaguesjouentun rleprépondérantdans ladynamiquede l'o éanauniveau littoral.Parexemple,desvaguesd'in iden eobliquegênèrentdes ourantslelongdes plages,et es ourantssontgénéralementplusimportantsqueles ourants réespar le vent, la marée oules ourants du large. Les modèles hydrodynamiques littoraux intègrent don lesvagues ommeun forçage essentiel.

Au ontraire, dans les modèles tiers d'o éan, à des é helles de grandeur de l'ordred'unebaie,d'unerégionoud'unbassino éanique,l'o éanévolueuniquement en réponse au vent, à la marée et aux diérents forçages sur la température et sur la salinité. Les vagues, et plus généralement l'état de mer, ne sont pas pris en ompte. Hors de la zone de déferlement bathymétrique, les vagues sont ainsi largementignorées.

Pourtant, l'énergie inétique turbulente (TKE) produite par le déferlement des vagues est supérieure, au moins d'un ordre de grandeur, à la produ tion de TKE par le isaillement du ourant d'Ekman (Terray et al., 1996).De même,la pseudo-quantité de mouvement des vagues, intégrée verti alement ( .à.d. le transport de Stokes des vagues) est de l'ordrede grandeur du transport d'Ekman orrespondant au vent qui les a réées : M Williams and Restrepo (1999), ainsi que Polton et al. (2005), ont donné une première estimation de e transport de Stokes à

40%

du transport d'Ekman aux moyennes latitudes. Cependant, une analyse plus réaliste tenant ompte du fait que les vagues sont rarement omplètement développées par vent fort serait ertainement plus pro he de

10%

. En surfa e, la dérive de Stokes des vagues de vent a été estimée par Kenyon (1969) à plus de

3%

de la vitesse du ventà

10

m,une vitesse omparableàladérivedue auventdesparti ulesd'eauàla surfa e. Enn, le transfert de laquantité de mouvement du vent vers l'o éan passe généralement à plus de 80% par les vagues, alors que 20% ou moins sont dus aux frottements visqueux àla surfa e (Donelan,1998; Banner and Peirson, 1998).

Ces diverses observations etanalyses ont amenéà re onsidérer l'importan edes vagues dans lades ription de l'o éan,y omprisloinde la te.En parti ulierpour des problématiques liées aux dérives près de la surfa e ou liées au mélange près de la surfa e, les vagues doivent jouer un rle important, ompte tenu des ordres de grandeurs pré édents.

Également il apparaît un fossé entre la des ription littorale, ave vagues, et la des ription tière,sans vagues,de l'o éan.Pourtant,lazoneintermédiaire,au-delà de la zone de déferlement des vagues et que nous nommerons pré-littorale omme Denamiel (2006), est d'importan e ru ialeen terme de transports de sédiment, de

par les autres forçages tels le vent ou la marée, est ainsi né essaire pour modéliser ette zone pré-littorale.

Plan de l'exposé

Par e que l'impa t général des vagues sur l'hydrodynamique est à heval sur diérents hampsd'investigations,depuisla ltureturbulenteetlemélangeverti al jusqu'à la ir ulation littorale, la bibliographie n'a pas été, omme il est d'usage habituellement, regroupée dans une partie spé ique. Au ontraire, ha une des diérentes partiestraite de sabibliographiespé ique.

Notre exposé s'arti ule en 3parties.

Les aspe ts généraux seront rappelés dans un hapitre préliminaire. J'y ferai notamment une des ription simple des vagues et du transport de masse qui leur est asso ié. La séparation du hamp de vitesse en une partie vagues et une partie ourant moyen y sera présentée, ainsi que la for e de Stokes-Coriolis. Ces deux notionsreviendront de façon ré urrente tout aulong de e travail.

Ensuite une première partie traitera plus pré isément de l'eet des vagues sur l'hydrodynamique loin de la te. L'étude sera alors à une dimension verti ale et s'atta hera àdé rirede façon ohérenteles ourantsd'Ekmanetladérive prèsde la surfa e sous l'eet des vagues. On y abordera également les problèmes de mélange induit par les vagues etde ltureturbulente, etses onséquen es sur lesprols de vitesse près de lasurfa e.

Unedeuxièmepartie reviendrasur lemélangeliéaux vagues,mais ette foissur sonimpa tsur laprofondeurdela ou he demélange.Lesmodèlesetlesparamètres pour prendre en ompte e mélange induit par lesvagues seront dis utés, ainsi que l'impa tsur la formationet l'érosiondes thermo lines.

Ennunetroisièmepartieprésenteraunedes riptiondela ir ulationinduitepar lesvaguesdepuislazonededéferlementjusqu'auplateau ontinental.Lesaspe tsliés àlanon-uniformitédu hampdevagues, onnusparexemplesousleterme"tensions de radiation", seront abordés. L'analyse séparée des vagues et des ourants, ainsi que ses onséquen es sur la ompréhension des ourants de la zone infra-littorale,

The ontext: The hydrodynami s near the oast and

near the surfa e

Before dis ussing the waves and their impa t onthe o ean ir ulation, I would like tointrodu ethe general ontextof this study. The study ofthe o eani ir ulation, and more generally the study of the o ean, might benet to many a tivities and a shortreview mightbehelpfultounderstandthe possibleappli ationsof thepresent thesis.

At global s ale, the o ean ir ulation is under parti ular interest for its impa t on the limate. But at oastal s ale, the motivations are more diverse, from the understanding and fore asting of the water mass transportfor appli ations to biol-ogy,bio hemistry,halieuti ,submarinedefense,topollutantsmonitoringandsear h and res ue. Alsoat nearshore s ale,the o eanhydrodynami sismainlystudied for appli ationsto oastalerosion orsedimentary transport.

Formany oftheappli ations itedabove,onene essary stepistobuildano ean modelwhi h orre tlydes ribesthe transports of parti les lose tothesurfa e (pol-lutantsmonitoring, sear hand res ue, driftof larvae),whi h orre tlydes ribesthe mixinginthe uppero eanwiththe resolutionof thediurnaland seasonal y le ( li-mati studies, bio hemistry), and whi h orre tly represents the ir ulation in the vi inityoftheshore,inthesurf-zoneandbeyondintheinner-shelfzone( oastal ero-sion, sedimentary transport, monitoring of the pollutants reje ted from the oast). A tuallyalargepartoftheo eanresear ha tivitiesneedana urateunderstanding of the near-surfa e and nearshore hydrodynami s, and this is not surprising sin e most animaland humana tivities on entrate in those areas.

This sets up amotivationto study thehydrodynami sof this part ofthe o ean, and more pre isely tostudy the impa tof waves onit.

The issue: The impa t of waves

Waves play adominantrole inthe o ean dynami s lose tothe shore. Forinstan e, obliquelyin identwaves reatealongshore urrents,andthose urrentsaregenerally larger than the urrents reated by the wind, the tides or the o-shore urrents. Therefore nearshore hydrodynami s models use the waves as anessential for ing.

On the ontrary in the oastal models, at the s ale of a bay, of a region or of ano ean,the o eanevolves onlyin response to the wind,the tide and the dierent for ings of the temperature and the salinity. Waves, and more generally the sea state, are then largely ignored outsideof the surf-zone.

Nevertheless,theturbulentkineti energy(TKE)produ edbythewavebreaking isatleast anorderof magnitude largerthan theTKE produ edby the shear ofthe Ekman urrents (Terray et al.,1996). Also the verti ally-integrated waves pseudo-momentum (i.e. the Stokes transport of the waves) is of the order of the Ekman transport orresponding to the wind whi h reated those waves. M Williams and Restrepo(1999)andPoltonetal.(2005)gavearstestimationofthis Stokes trans-port around

40%

of the Ekman transport at mid-latitude. We note however that a more realisti estimation would be lose to

10%

given that the waves are seldom fully-developed under strong winds. At the surfa e, the Stokes drift was estimated by Kenyon (1969) to be more than

3%

of the wind speed at

10

m, a velo ity of the same order asthe drift velo ity of parti lesatthe surfa e. Finally,the momen-tum from the wind transfers to the o ean generally through the wave eld at

80%

, whereas only

20%

or less are due to the vis ous fri tion at the surfa e (Donelan, 1998; Banner and Peirson, 1998).

All those observations have lead us to re onsider the importan e of waves in the des ription of the o ean, even far from the oast. In parti ular for studies of near-surfa e driftor near-surfa emixing, waves mightplay animportantrole given the previously listed orders of magnitude.

Also there is a gap between the nearshore des riptions (with waves) and the oastal des riptions (without waves) of the o ean. Yet the intermediate zone, the inner-shelfzone,isof ru ialimportan eintermsofsedimentarytransport, hemi al orbiologi altransport, sin eallthe materials omingfromthesurf-zone nallyend up there. A oherent des ription of wave-indu ed urrents, from the shore to the open o ean,aswellasthe urrentsindu ed bythe otherfor ings,isane essarystep

As the general impa t of waves on the hydrodynami s deals with many dierent topi s, ranging from the turbulent losure and verti al mixing to the nearshore ir ulation,the bibliography has not been, asit is onventionallydone, gathered in aspe i part. On the ontrary,ea h part deals with itsspe i bibliography.

The thesis is split into threeparts.

The general on epts will be re alled in a preliminary hapter. I will made a simpledes riptionofthewavesandoftheasso iatedmasstransport. Theseparation ofthe velo ityeld intoawavepart and ameanowpart willbepresented, aswell asthe Stokes-Coriolisfor e. Those on epts willappear all alongthe thesis.

Thenarstpartwilldealmorepre iselywiththeee tsofwavesonthedynami s in the open o ean. The study in orporates only one (verti al) dimension and will try todes ribe ina oherent mannerthe Ekman urrents and the drift lose tothe surfa e in the presen e of waves. Also the verti al mixingdue tothe waves will be parameterized with an appropriate turbulen e losure. Its impa t on the velo ity proles lose to the surfa e willbedis ussed.

A se ond part will also fo us on the wave-indu ed mixing, but more pre isely on its impa t on the mixed layer depth. Models and parameters to in lude the wave-indu ed mixing will be dis ussed, as well as the impa t on the thermo line formationand erosion.

Finally,athirdpartbedevotedtothedes riptionofthewave-indu ed ir ulation from the surf-zone to the shelf. Aspe ts linked to horizontally non-uniform wave elds,forinstan eknownasradiationstressee ts,willbedis ussed. Theseparated analysis of waves and urrents, as well as its onsequen e on the understanding of the inner-shelf urrents, willbedis ussed.

This hapter aims to introdu e general on epts used in this study. We rst re all the typi al length and time s ales of the waves, and the impli ations for the wave modelling. Then we introdu e the Stokes drift of the waves and we dis uss on a simple example the two major di ulties whi h appear for the modelling of waves and urrents. Therstdi ultyisthemotionofthefreesurfa e, forwhi haspe ial averagingis needed. The se onddi ulty omes fromthe quitedierent physi sof the mean ow and of the waves Stokes drift, for whi h a separation of waves and meanowisneededtoobtainsuitableparameterizations. Finally,emphasisismade onthe fa t that the mean ow dynami s is dierentthan the total drift dynami s, for instan e with the appearan e of the Stokes-Coriolis for e for an horizontally uniform ase.

1.1 Lengths and time s ales

The present thesis investigates the role of the waves on the 3D dynami s of the upper o ean, and alsoon the 3D dynami s of the nearshore and ostal o eans.

Were allherethatthewaves, i.e. shortgravitywavesatthesurfa e,havetypi al wavelengths of

100

m, heights of

1

m and periods of

10

s. Those s ales are rather small ompared to the typi al length and time s ales of the o ean ir ulation, but it should be noted that the large s ale variations of the wave eld are mu h larger thanthes ale ofasinglewaveand are omparabletothoseo ean ir ulation s ales. The variety of o eani phenomenons inuen ed by waves is large, the typi al horizontal s ales spanning the range from hundreds of kilometers for large s ale variationsof the waveeld toa meterfor the energy ontainingeddies of the wave-stirred turbulent surfa e layer, and even less for the mi ros ale breaking of the smallestwaves.

Be auseofthelimitednumeri alresour esandbe auseonealsoneedstoanalyze thephysi alfeatures,itis ustomaryforthepurposeofo ean ir ulationto onsider horizontallengths alesmallerthanahundred ofmetershorizontallyandone meter verti ally as subgrid phenomenons. The present study will keep in mind these typi al s ales and try to nd adapted des riptions and parameterizations, of small s ale wave-indu ed turbulen e for instan e.

Also,wefo us inthe presentthesis onwave-driven urrents withtimevariations slower thanthe waveperiod. We thereforeuse aspe tral approa h,without resolv-ing the phase of the waves, following the method employed in most of the wave predi tion systems whi h simulate the generation,the propagation and the dissipa-tion of the waves. In this kind of des ription, the sea state is onsidered as a sum of mono hromati waves, spread over a frequen y-dire tional energy spe trum(e.g Komen etal., 1994).

1.2 Waves, Stokes drift, averaging

Let us take a mono hromati wave propagating in the x dire tion, in deep water and without urrent. The equations of motion,validfor

z < η

, are (e.g. Mei, 1989)



















η = a cos(ωt − kx)

u = aω cos(ωt − kx) exp(kz)

w = aω sin(ωt − kx) exp(kz),

(1.1)

where

a

is the amplitude,

ω

the radian frequen y,

k

the wavenumber,

η

the surfa e elevationand

u

,

w

thehorizontalandverti al omponentsofthewavemotion. Note that the previous expressions, aswell as most formulae inthis hapter,are validin the limitof smallwave slope

ka ≪ 1

. For simpli ity,I will not dis uss further the order of ea happroximations.

Nowlet usaverage in time this velo ity,dening the time average

u =

1

T

R

T

0

udt

over awave period

T

. Assuming

u = 0

for

z ≥ η

, weget







u ≃

aω

π

q

1 −

z

a

2

_for

− a < z < a,

u = 0

_{for z < −a.}

(1.2)

InthatEuleriandes ription,the Stokesdrift,i.e. thetime-averagedmasstransport, is on entrate between the rests and the troughs of the waves (g. 1.1, upper

u

L

= u(x(t), z(t), t)

=

∂u

∂x

x(t) +

∂u

∂z

z(t),

(1.3)

where

(x(t), z(t))

are the oordinates of the parti le. The orrelation between the displa ementandthe non-uniform velo ity eldyieldsaresidualmotionoftheuid parti les. In other words, the orbits of the parti les are not losed. The residual driftis the Stokes driftand is equalto

Us

= a

2

ωk exp(2kz),

(1.4)

wherethe verti al oordinates zrepresentsthis time themean positionof aparti le of water.

This simpleexampleillustratesthe ompli ationswhi happear due tothe mov-ing surfa e, even for linear waves, and the ne essity of a areful averaging lose to the surfa e. The hoi e of oordinates to des ribe both wave-indu ed motion and mean urrent isof great importan eand must be dis ussed here.

Most eld measurements are time averages made at almost xed lo ations. Therefore the Euleriandes ription is traditionallyused for o eani ir ulation, and has been hosen in many studies on wave-driven mean ows. In the Eulerian de-s ription,theinterfa eisdistributedbetweenthe rest

z = a

andthetrough

z = −a

. When onsidering the mean elds, it is usually assumed that they an be analyti- ally extended between the trough and the mean surfa e

z = η = 0

. For example, the verti alintegralof the velo ity is dened as

Tm

=

Z

η

−

_h

udz.

(1.5)

In the same time, the wave mass transport of the waves is either assumed to be a surfa e mass transport (Hasselmann, 1971; Stive and Wind, 1986; Newberger and Allen,2007b)equal to

M

w

=

Z

η

η

udz,

(1.6)

or to be distributed a ording to the Lagrangian Stokes drift prole (M Williams et al., 2004). Clearly, the Eulerian averaging pro edure presents some oddities in its surfa e representation. The analyti al extension of the elds is made whereas

the example of the velo ity, it leads to the wave Stokes drift. Although Eulerian averagingmightgive orre trepresentationoftheStokesdriftwitha arefulanalysis (e.g. M Williamsetal.,2004),anot sodetailedanalysismightmissthe fullverti al distribution of the Lagrangian motion (Hasselmann, 1971; Stive and Wind, 1986; Newberger and Allen,2007b).

For this reason, many authors have hosen to use a hange of oordinates to makeaproperaveragingofthewaves(e.g.Jenkins,1986). Themostsimpleonewas re ently proposedby Mellor(2003). It issimplytheuse ofaparti ular

σ

- oordinate system, the one following the uid verti al motion (to lowest order in the wave slope), to bring the parti les ba k to their xed verti al mean lo ation and then to average in time the velo ity (g. 1.1, middle panel). For the ase of the wave motiondis ussed above,the meanvelo ityobtained by this methodisinagreement withtheLagrangiandes riptionoftheStokesdrift. More ompli atedbut alongthe same idea, Andrews and M Intyre (1978a) introdu ed the Generalized Lagrangian Mean (GLM). In that ase, the parti les are also horizontally displa ed ba k to their meanpositionduringa waveperiod(g. 1.1, lowerpanel). The omplexity of the mappingof Andrews andM Intyre (1978a)is ompensated by the simpli ity of exa t phase-averaged equations.

1.3 Wave / Mean ow separation

A key point of the present work is the separation between the wave part and the mean ow. Ifwe goba k tothe mono hromati waves of the previousse tion, on e a proper wave-averaging pro edure is applied (Mellor (2003) or GLM), one gets a mean Lagrangian velo ity equal to the Stokes drift of the waves. Now adding a barotropi urrentbrings nodi ulties. One gets then amean Lagrangian velo ity equal to the Stokes drift of the waves plus the mean urrent. The mean urrent is then similar tothe Eulerian mean urrent belowthe troughs.

Onthatsimpleexampleofmono hromati wavesoverameanbarotropi urrent, we an noti e that the Lagrangian ow is verti ally sheared be ause it in ludes the residual Lagrangian drift due to the wave motion, i.e. the waves Stokes drift. However, applying on the shear of that residual drift a verti al mixing term equal to the verti al mixing we would applied on a omparably sheared mean urrent would be physi ally meaningless. In fa t the turbulen e does not a t similarly on the urrent and on the waves residual Stokes drift. In addition, the Stokes drift propagates with the waves group speed, whereas the mean urrent is adve ted at the mu h smaller urrent velo ity. This leads us a entral idea of this thesis, the

-20

-15

-10

-5

0

-20

-15

-10

-5

0

x (m)

velocity (m/s)

0

40

80

120

160

200 -0.5

0

0.5

-20

-15

-10

-5

z

(elevation,

m)

(a) Eulerian

(c) sigma transform

(b) Generalized Lagrangian

wave-induced

mass

transport

wave

momentum P

quasi-Eulerian

mean current U

L

-P

GLM current U

L

Stokes drift U

s

(known forcing)

sigma-mean

current U

Eulerian mean

current

Figure1.1: Thisgurewastaken fromArdhuinetal.(2007b). Averagingpro edures (left) and examples of resulting velo ity proles (right) in the ase of (a) Eulerian averages(e.g.RiveroandAr illa,1995;M Williamsetal.,2004),(b)theGeneralized LagrangianMean (AndrewsandM Intyre,1978a),and( )sigmatransform(Mellor, 2003;Ardhuin etal.,2007 ). The thi k bla kbars onne t the xed points

x

where theaverage eld isevaluated,tothedispla ed points

x

+ ξ

where the instantaneous eld is evaluated. For averages in moving oordinates the points

x

+ ξ

at a given verti al level

ξ

are along the gray lines. The drift velo ity isthe sum of the (quasi-Eulerian) urrent and the wave-indu ed mass transport. In the present illustration anAirywaveofamplitude3mandwavelength100min30mdepth,issuperimposed onahypotheti al urrentof velo ity

u(z) = −0.5 − 0.01z

m/s forall

z < η(x)

. The urrent prole is not represented in ( ) sin e it is not dire tly given in Mellor's theory, although it an obviously be obtained by taking the dieren e of the other two proles.

parameterizingthemseparatelybe ausetheyarephysi allydierent,one anexpe t signi antimprovement of the modellingof ombined waves and urrent.

as mentioned above. But the dis ussion is strongly ompli ated if we introdu e a verti ally varying urrent. Both averaging of Mellor (2003) and the GLM give the Lagrangian motion

u

L

. That Lagrangian motion an be separated into a residual wave motion

P

and a mean ow

u

b

.

u

L

= P +

u.

b

(1.7)

It shouldbenotedthat the meanowisdes ribed inquasi-Eulerian oordinates. It is dierent fromthe Eulerianmean (at a xed lo ation).

Related tothis, the residualwave part

P =

∂u

l

∂x

x

l

_{(t) +}

∂u

l

∂z

z

l

_(t)

(1.8)

has been alled the wave pseudo-momentum (see M Intyre, 1981, for a full dis us-sion). Here

u

l

is the perturbation of the velo ity eld from the Lagrangian mean, and

x

l

_{, z}

l

isthe displa ement.

P

mightbe dierent fromthe Stokes driftdened as the Lagrangian motion

u

L

minusthe Eulerian mean

u

.

Also,inthe ase ofverti allyvarying urrent,theresidualmotion

P

ofthewaves is dierent than the residual motion without urrent, be ause the verti al shear of the mean urrent an add to the verti al shear of the wave motion, modifying the orrelationbetweenthe velo ityandthe displa ementinformula1.3. Forsimpli ity, the rest of the thesis willignorethis distin tion ex ept inpart III.

A more detaileddes ription ofthe GLMseparation of wavesand mean owhas been made in Ardhuin et al.(2007b).

1.4 The Stokes-Coriolis ee t

Thereisamass transportasso iatedwith thewave motion. A ording tothe linear wavetheory, the verti al integralof the Stokesdriftof amono hromati wave(equ. 1.4) is

M

w

=

Z

0

−

_H

Usdz

=

a

2

_ω

2

.

(1.9)

For large swells (i.e. waves not related to the lo al wind), this transport an be of the order of the Ekman transport of amoderatewind at mid-latitude,as noted by M Williams and Restrepo (1999)and Polton etal. (2005).

introdu ing a for e alled later the Stokes-Coriolis for e or Hasselmann for e. Xu and Bowen (1994) made apparent the physi al meaning of this for e. They made asimple al ulation of the impa t of the Earth rotationonthe wave dynami sand showed that thereis slighttiltingof the orbitsof the parti lesunderpassing waves. As a onsequen e there is an asso iated supplementary ux of momentum to the mean ow, equivalent toa for e equalto

f Us

and oriented tothe right of the wave propagation. Inotherwords,theCoriolisfor ea tsonthewavepseudo-momentum, butthe orrespondinguxofmomentumisreleasedfromthewaveparttothe mean owasathe bodyfor e. Thisfor edrivesaverti ally integratedtransportopposed tothe Stokes transport of the waves.

Now examining the verti al distribution of the wave mass transport and of the Stokes-Coriolisfor e, two typi al length s ales appears, the Stokes s ale

δs

= 1/2k

and the Ekman s ale

δe

(equal to

q

2Kz/f

if the verti al vis osity

Kz

is supposed

uniform). As showed by Polton et al. (2005), if

δs

≫ δ

e

, then

u = −P

b

so that the meanow totally ompensates theStokesdriftof the waves (g. 1.2, upperpanel). This mightbe the ase for a long swell, as studiedin Part III. However the Stokes driftofaspe trumofwindwavesisstronglysurfa etrappedsothat,inthepresen e ofastrongverti almixing,

δ

s

≪ δe

and the meanowdriven by theStokes-Coriolis for e annot ompensate the Stokes driftof the waves lose to the surfa e (g. 1.2, lowerpanel). Thenthe net wave-indu ed drift isapproximatelyequalto the Stokes drift,whi h an besigni ant as shown inPartI.

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

−60

−50

−40

−30

−20

−10

0

Depth [m]

Wave−driven velocities [m/s]

δ

u Stokes−Coriolis

u

st

Stokes drift

u

L

wave−driven

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

−60

−50

−40

−30

−20

−10

0

Depth [m]

Wave−driven velocities [m/s]

δ

u Stokes−Coriolis

u

st

Stokes drift

u

L

wave−driven

Figure 1.2: Verti al proles of the Stokes drift, of the mean ow omponent inthe wavesdire tionand ofthe resultingLagrangiandrift. TheStokes driftis al ulated with the wind waves assumed to be fully-developed with a wind speed of

10

ms

−

1

, themeanowshownisdrivenbytheStokes-Coriolisfor eonly(nowindstress). The upper panel has no verti al mixingwhereas the lower panel in orporates a verti al mixing. In the ase of no verti al mixing, the mean ow ompensates the Stokes drift at ea h depth whereas it does not when the verti al mixing is in luded. The

dynami s of the open o ean.

Westartwiththe studyofthe impa tofwavesonthenearsurfa e o-shore dynam-i s, in itsmost simple des ription : a wind sea withoutany horizontalvariationsof the wind, waves orstrati ation.

First, the Stokes drift of the waves of a wind sea is evaluated. Previous evalu-ations were made using the spe trum of Pierson and Moskowitz (1964) by Kenyon (1969),Lewis andBel her (2004)and Polton etal.(2005),leadingtosurfa e values ofthe Stokesdriftaround

3%

ofthewindspeedat

10

m. Howeverthehighfrequen y rangeofthe spe trum,i.e. thesmallwaves, makesalarge ontributiontotheStokes drift at the surfa e (see gure 4.1). Therefore we used a more realisti spe trum, the one of Kudryavtsev et al. (1999), were the high frequen y range was arefully designed for appli ationsto remotesensing. It leads tovaluesof the surfa e Stokes driftmu h smaller, around

1.2%

of the wind speed at most.

Se ondly, the mean ow driven by the wind, i.e. the Ekman urrent, is evalu-atedusingre entobservations(e.g.Agrawaletal.,1992)andmodels(e.g.Craigand Banner, 1994; Noh,1996) of the strong nearsurfa e mixing, attributedto breaking waves, in moderate and strong winds. Essentially, these models use a TKE al u-lation with a surfa e ux of TKE and use a mixinglength with a pres ribed large value atthe surfa e. The onsequent Ekman urrent isquiteweak atthe surfa e in the presen e of wave breaking.

The impa t of the Stokes-Coriolis for e on the mean urrent is also estimated. Thisimpa tisquiteweak given that thewind stressis alwaysmu hlarger thanthe equivalentStokes-Coriolisstress of the orrespondingwind waves.

The surfa e drift, whi h is the sum of the Stokes drift of the waves and the mean urrent, appears then mainly due to the Stokes drift of the waves, raising thequestion ofdire tparameterizationfromthewindspeed,a ommonengineering pra ti e (see Spaulding, 1999).

Finally,availableobservationsofsurfa e urrentsaredis ussedinthelightofthe presentphysi aldes ription. Essentially,observationsareseparatedintoLagrangian

no available informations onwaves (e.g. Chur hill and Csanady, 1983). Previously used observations, namely, observations of mean urrent shears very lose to the surfa e duringSMILE (Santala, 1991) and of meanEkman spirals during LOTUS3 (Lewisand Bel her, 2004;Poltonetal.,2005)are reanalyzedtond eviden eofthe exposed physi s.

The hapter 2 treats the ase of an uniform o ean without strati ation. The basi softhisphysi aldes riptionare outlinedandbriey omparedtoobservations. The hapter3addsthestrati ation, ommentsonitsee tonthenetwave-indu ed drift,and makesa more rigorous omparison with the observations.

without strati ation

This hapter iswritten asan independent paper :

Drift and mixing under the o ean surfa e. A

oherent one-dimensional des ription with appli ation to unstratied onditions

Ni olas Ras le

(1)

, Fabri e Ardhuin

(1)

, EugeneA. Terray

(2)

Published in Journal of Geophysi al Resear h Mar h 2006

(1)

Centre Militaire d'O éanographie, SHOM, BREST, Fran e

(2)

Dept. of Applied O ean Physi s and Engineering, Woods Hole O eanographi Institution, Woods Hole, Massa husetts, USA

Abstra t

Waveshavemanyee tsonnearsurfa edynami s: breakingwavesenhan emixing, waves are asso iated with a Lagrangian mean drift (the Stokes drift),waves a t on the mean ow by reating Langmuir ir ulations, and also a return ow opposite to the Stokes drift, and, last but not least, waves modify the atmospheri surfa e roughness. Arealisti o eanmodelisproposedtoembra ealltheseaspe ts,fo using on near surfa e mixing and surfa e drift asso iated with the wind and generated waves. The model is based on the Generalized Lagrangian Mean that separates the momentum into a wave pseudo-momentum and a quasi-Eulerian momentum. A wave spe trum with a reasonable high-frequen y range is used to ompute the Stokes drift. A turbulent losure s heme based on a single evolution equation for the turbulent kineti energy in ludes the mixing due to breaking wave ee ts and wave-turbulen eintera tions. Theroughnesslengthofthe losures hemeisadjusted usingobservationsofturbulentkineti energynearthesurfa e. Themodelisapplied to unstratied and horizontally uniform onditions, showing good agreement with observationsofstronglymixed quasi-Eulerian urrentsnearthe surfa e,whenwaves are developed. Model results suggest that a strong surfa e shear persists in the drift urrent, due tothe Stokes drift ontribution. In the present modelthesurfa e drift only rea hes

1.5%

of the wind speed. It is argued that strati ation and the properties of drifting obje ts may lead to a supplementary drift as large as

1%

of the wind speed.

2.1 Introdu tion

The o ean surfa e is where the vast majority of marine a tivities take pla e, and dierentdynami aldes riptionshavebeeninvokedtodes ribethe 100mthat strad-dle both sides of the air-sea interfa e. Dierent solutions have been developed for appli ations su h as wave fore asting for safety at sea [e.g. Komen et al., 1994℄, fore asting of drift for sear h and res ue or pollution mitigation [e.g. Youssef and Spaulding 1993℄, or modelling of the general o ean ir ulation with appli ations to limatestudies [e.g. Semtner, 1995; Ble k, 2002℄.

Unfortunately, these des riptions of the upper o ean are often in oherent, not always based on rst prin iples, and may not give parameters ompatible available measurements that ould onstrain numeri al fore asting models. Work for ea h of the three appli ations listed above have often fo used on one key parameter, the signi ant wave height

Hs

, the surfa e drift urrent

Uς=0

, or the mixed layer temperature

Ts

. The advent of the Global O ean Observing System (GOOS), and

that would involve all the relevant dynami pro esses : geostrophi urrents, o ean waves,tides,internalwaves, andknownturbulentstru turessu haswindrollsinthe atmospheri boundarylayer, andbothbreakingwavesand Langmuir ir ulationsin theo eanmixed layer[Ardhuin etal.,2005℄. Manygoodfundamental ontributions have studied one or two of these pro esses, in luding joint ee ts of wave motion and mean urrents [e.g. Weber, 1981; Jenkins, 1987℄, wave breaking and Langmuir ir ulationsee ts on upper o ean mixing[Agrawal et al.,1992; Craigand Banner, 1994; Thorpe et al., 2003;Mellor and Blumberg, 2004℄.

Are ent onvergen e ofdierentapproa hes totheuppero eandynami sshows a lear in onsisten y. Mellorand Blumberg (2004)demonstrated thata parameter-izationforthe strongmixingduetowavebreaking, previouslyobserved by Agrawal etal.(1992)and others,leadstoimproved hind astsof mixedlayerdepth and tem-perature of the lassi dataset from the Gulf of Alaska station Papa. This strong mixingalsoleadstoaratheruniformEulerian urrentprole,whi hhastobesmall, be ause the depth-integrated transport isthe known Ekman transport. Mellorand Blumberg (2004) nd surfa e urrents less than 0.6% of the wind speed. Su h a value of the Eulerian urrent may be larger than the quasi-Eulerian urrent ob-served by Santalaand Terray(1992),but itisparadoxi ally smallforexperts inthe fore astingof surfa e drift,for whom it is wellestablished that the drift velo ity is often lose to 2 or3% of the wind speed at 10meters,

U10

[Spaulding, 1999℄. Both astrongmixingand astrong velo ityshear atthe surfa e shouldbe obtained when surfa e waves are a ounted for in a onsistent way, in luding both wave breaking and wave-indu ed Stokes drift.

Thegoalofthepresentpaperistoevaluatehowwellasimplebut oherentmodel of the upper o ean performs in terms of drift velo ities, Eulerian velo ities, eddy vis osities and turbulent dissipation. Sin e waves are learly an important part of theo eani mixedlayer, weshall alsoexplore whi hwave parametersareimportant and how the mixed layer is modied. In parti ular the ee t of the Hasselmann for e [Hasselmann, 1970℄ that was reported to be signi ant by Lewis and Bel her (2004)is re-examinedwith arealisti parameterizationof near-surfa emixing. The presentpaperfo useson onditionsthatarestatisti allystationaryandhomogenous in the horizontal dimensions. The wave for ing and resulting wave properties are des ribedinse tion2. ThesedriveamodelforturbulentandmeanEulerian proper-ties,asdes ribed inse tion3. Thatmodelisbasedonthe approximation,tose ond order inthe wave slope, of the Generalized Lagrangian Mean (GLM2, see Andrews

Navier-Stokes(RANS)equations. This GLM2-RANSformalism an beobtainedby subtra ting the wave pseudo-momentum from the total momentum equation given by Mellor (2003), and valid for horizontally-uniform onditions. This step, as well as a derivation from the equations of Andrews and M Intyre (1978a), is des ribed by Ardhuin (2005). Thenumeri al al ulationsuse the omputer ode byCraigand Banner (1994), extended to a ount for wave ee ts spe i to our GML2-RANS equations. In se tion 4, the various ee ts of the waves on the turbulent, Eule-rian and Lagrangian properties are ompared to observations of turbulent kineti energy dissipation, quasi-Eulerian and Lagrangian velo ities. Con lusions followin se tion 5.

2.2 Wave dynami s

2.2.1 Spe tral wave evolution

O eansurfa ewaves, generatedbythe wind,havealargeinuen eonair-seauxes. In parti ular, waves are generally believed to absorb more than 50% the wind-to-o ean momentum ux

τ

a

[Donelan, 1998;Banner and Peirson, 1998℄. This large fra tion of the wind stress

τ

a

is the wave-indu ed stress

τ

in

. However, only a small fra tion of

τ

in

, possibly up to 5%, is radiated in the wave eld momentum ux, the vastmajority is ontinuously lostby waves as they dissipate, essentially due to wave breaking [Donelan, 1998; Janssen et al., 2004; Ardhuin et al., 2004℄. Another ee t ofinterestto oastalo eanographers isthatfor agiven wind speed,

τ

a

an be in reased by as mu h as a fa tor three in oastal areas, due to the dierent nature of the wave eld [e.g. Drennan et al. 2003; Lange et al. 2004℄.

Be ause o ean waves are generated by the wind, many authors have sought a dire tparameterization ofwave ee ts fromthe wind eld. However, wavesare not uniquelydened bythelo alwindspeedanddire tion,inparti ularin oastalareas and marginalseas (likethe Mediterraneansea), where wave developmentis limited by the fet h, but also in the tropi s and mid-latitudes where a large part of the wave energy isdue tolong periodwaves (swell) that have propagated fromdistant storms, sometimes half-way round the Earth [Snodgrass et al. 1966℄. In general, one needs to take into a ount the wave dynami s that are, on these large s ales, statisti allywelldened by the dire tional wave spe trum

E(k, θ)

, that distributes overwavenumbers

k

and dire tions

θ

thewaveenergy

E

w

_{= ρwg}

R

_{E(k, θ)dkdθ}

d

dt

E(k, θ) = S

in

_{(k, θ) + S}

nl

_{(k, θ) + S}

ds

_{(k, θ) + S}

bot

_{(k, θ),}

(2.1)

wherethe Lagrangiantimederivativein ludespropagationee ts, and

S

in

,

S

nl

,

S

ds

,

S

bot

are `sour eterms' (eitherpositivefor true sour esor negative fora tual sinks) thatrepresentthe energygiven tothespe tral omponent

(k, θ)

bythe atmosphere, the other wave omponents,the o ean turbulen e in the water olumn and surfa e boundary layer, and the bottom boundary layer and sediments, respe tively. This equationiseasilyextendedtotakeintoa ountvarying urrents[Komenetal. 1994; White 1999℄. Ea h energy sour e terms an be onverted in a momentum sour e term [e.g. Phillips 1977℄,

τ

i

= ρwg

Z

_S

i

_{(k, θ)}

C

dkdθ,

(2.2)

where C is the wave intrinsi phase speed. Of parti ular interest will be

τ

in

and

−τ

ds

,the momentum uxes, perunit surfa e of the o ean, input towaves fromthe wind,and delivered tothe mean ow by the waves, respe tively.

2.2.2 The Stokes drift

It is also well known that waves possess a pseudo-momentum that is equal to the mass transportvelo ity orStokesdrift

U

s

[e.g. M Intyre 1981℄. This driftarises as the wave-indu ed orbits of parti lesare not exa tly losed. From anEulerian point of viewthis drift is zero everywhere belowthe wavetroughs, and the wave-indu ed masstransporto ursbetween thedeepest troughsandthehighest rests. However, su h an Eulerian view `diuses' the air-sea interfa e over a verti al distan e of the order of the signi ant wave height

Hs

, whi h is not pra ti al for investigatingthe surfa e gradientof any quantity. We shall thusprefer the Lagrangianpointof view [e.g. Andrews and M Intyre, 1976℄, that yields, orre t tose ond order inthe wave slope,the followingexpression [Kenyon,1969℄ for deep-water waves,

U

s

(z) = 2

R

0

2π

R

∞

0

u

θkσe

2kz

E(k, θ)dkdθ

=

2

g

R

2π

0

R

∞

0

u

θσ

3

e

2kz

E(k, θ)dkdθ.

(2.3)

Thatexpressionuses theintrinsi waveradianfrequen y, asgivenbythedeepwater dispersionrelationforlineargravitywaves,

σ =

√

gk

,

g

isthea elerationofgravity,

and

u

θ

= (cos θ, sin θ)

is the unit ve tor in the dire tionof propagation. The origin

U

_s

is learlymu hsmallerthantheorbitalwavevelo ity,byafa tor

ε

thatisthe

waveslope,typi allyless than

0.1

.

U

s

isalsostronglysheared atthesurfa ebe ause the ontributionofea hwave omponentde aysexponentiallyoveritsStokesdepth

1/(2k)

,andthe high-wavenumber omponentsgiveasigni ant ontributionto

U

s

,

butnearthesurfa eonly(gure1). Usingaspe tralshapeproposedbyKudryavtsev et al. (1999), a wind of

U10

= 10

m s

−

₁

yields a surfa e drift of

U

s(z = 0) =

0.11

m s

−

₁

, when only wave omponents with

2π/k > 5

m are in luded , whereas all omponents up to

2π/k = 0.1

m yield up to 0.13 m s

−

₁

. The omparison withamono hromati omponentshows the dieren es between windsea andswell ontributions : the swell-indu ed Stokes drift at the surfa e is typi ally less than

30%

ofthedriftasso iatedwithawindseaofsamepeakperiodandsigni antwave

height. A large swell and a wind sea due to a weak wind an then produ e surfa e Stokes driftsof the same order.

The Stokes transport

M

w

=

Z

0

−

H

U

sdz =

Z

2π

0

Z

∞

0

u

θσE(k, θ)dkdθ

(2.4)

is slightly less inuen ed by the short (and slower) waves. Nevertheless the short waves ontributerelativelymoreto

M

w

thantothewaveenergy,asthe ontribution of ea h spe tral omponent to

M

w

is its surfa e elevation varian e divided by the intrinsi phase speed.

2.2.3 Pra ti al al ulation of wave parameters

Be ause short waves are important, with

U

s

(z = 0)

and

M

w

proportional to the third and rst moments of the frequen y spe trum, respe tively, a numeri al esti-mation of

U

s

based on (2.3) should use a wave spe trum that is well dened in that range. For general appli ations using numeri al wave models su h as WAM [WAMDI Group, 1988℄, the expli itlyresolved spe trum an be arefully extended byahigh-frequen y tail. Inthe presentstudy, weusethefamilyofspe traproposed for remote-sensing appli ations by Kudryavtsev et al. (1999), and governed by the two main parameters that are the wind speed and the stage of wave development. These spe tra have been arefully designed to reprodu e both the long wave spe -trum, with a spe tral shape similar to that of Donelan et al. (1985), and the short wave spe trum with, in parti ular, a se ond moment of the wavenumber spe trum (or fourth moment of the frequen y spe trum) that is well onstrained by the op-ti al measurements of the mean sea surfa e slope by Cox and Munk (1954). One

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

−10

2

−10

1

−10

0

−10

−1

−10

−2

Stokes drift U

s

(m s

−1

)

Elevation (m)

monochromatic

λ

_min

=5m

λ

_min

=1m

λ

_min

=0.1m

Figure 2.1: Stokes drift prole for a wind speed

U10

= 10

ms

−

1

and a fet h larger than 1000 km (fully developed sea), based on the KMC spe trum [Kudryavtsev et al. (1999)℄ , and the integral (2.3). Dierent proles are shown that only in lude wavelengths longer than a minimum value

λmin

. For omparison, the drift due to a single wave omponent is also indi ated. That single omponent has same peak wavelengthand surfa eelevationvarian e(period

T p = 8s

,

Hs

= 2.8m

)asthewave

represented by this model.

Thesespe trayieldvaluesof

U

s

(z = 0)

that anbelargerthantypi almean Eu-lerian urrents, witha transport

M

w

ofthe orderof the Ekman (1905)transportat mid-latitudes,ex eptfor shortfet hes orweakwinds(gure2). Forfullydeveloped waves,

U

s(z = 0) = 0.0125U10

is onsistent with re ent observations of the drift of near-surfa e louds of bubbles by Smith (manus ript submitted to JPO, 2005). In

2

4

6

8

10

12

14

16

18

20

0

5

10

Hs (m)

wind (m/s)

DHH

JONSWAP

ETCV

KMC

PM reviewed by AB

2

4

6

8

10

12

14

16

18

20

1

1.5

2

2.5

3

3.5

wind (m/s)

log

10

(fetch) (km)

0.6

0.7

0.7

0.8

0.8

0.8

0.9

0.9

0.9

1

1

1

1

1.1

1.1

1.1

1.1

1.2

1.2

1.2

1.2

1.3

1

1.5

2

2.5

3

3.5

log

10

(fetch) (km)

5

5

5

10

10

10

10

15

15

15

15

20

20

20

25

30

₂₅

30

25

35

(a) Hs (m)

(b) 100 Us(

z=0

)/U

10

(c) 100 M

w

_/M

Eck

Figure2.2: (a). Signi antwaveheightatfulldevelopmentgivenbyseveral param-eterizations of the wave spe trum

E(k, θ)

: PM is Pierson and Moskowitz (1964), AB is Alves and Banner (2003), DHH is Donelan et al.(1985), JONSWAP is Has-selmann et al.(1973), ETCV is Elfouhaily et al.(1997), and KMC is Kudryavtsev etal.(1999). ForDHH andJONSWAP,fulldevelopmentisobtained by settingthe peak frequen y

fp

to

0.123g/U10

. (b) Surfa e Stokesdriftasa fun tionof fet hand wind speed

U10

for the KMC spe trum, expressed asa per entage of

U10

. ( ) Mag-nitude of the verti ally-integrated Stokes mass transport

M

w

asa fun tion of fet h and

U

10

, expressed as a per entage of the orresponding Ekman (1905) transport

u

2

_∗/f

at mid-latitudes,with

f = 10

−

₄

.

the following al ulations, the wind speed at 10m height

U10

is taken to be inthe dire tion

θ = 0

. The fri tion velo ity

u⋆

is determined from

U10

using Charno k's expression [1955℄,

U

10

=

ua⋆

κ

log



_z

za0



,

(2.5) with

za0

= 0.018 u

2

_a⋆

/g,

(2.6)

are oupled, be ause of the dependan e of the wind prole onthe roughness of the sea [e.g. Janssen, 2004℄. Donelan (1998) gives a parameterization of

za0

that uses

the waveage

cp/U10

(where

cp

isthe phase speed atthe peak ofthe wave frequen y

spe trum) and the signi ant wave height

Hs

,

za0/Hs

_{= 1.67 ∗ 10}

−

₄

(U10/cp)

2.6

.

(2.7)

This ee t willbeevaluated inse tion 2.4.3.

2.3 Wave-averaged mixed layer equations

O eani motions are separated in three omponents, mean ow, waves and tur-bulen e. Turbulen e is separated from other motions by a an average over ow realizations for given wave phases. The mean ow and wave motions are then av-eraged with a Lagrangian mean so that the mean momentum is separated into a mean ow and a wave part. The verti al mean wave momentum is zero while the horizontaltotal mean momentum

ρw

U

issplit in a quasi-Eulerian mean

ρw

u

b

and a Stokes drift,

U

=

u

b

+ Us.

(2.8)

Thisseparation omesnaturallywiththedenitionoftheGeneralizedLagrangian Mean [Andrews and M Intyre, 1978℄. Please note that

U

and

U

s

are

U

L

and

p

in theirnotations and areevaluatedataslightlydispla ed verti alposition[M Intyre, 1988℄. In measurements, this separation may be di ult to a hieve [e.g. Santala and Terray, 1992; Hristov et al., 1998℄. Although the Stokes drift

U

s

orresponds to the wave-indu ed drift that arises from the orrelations of wave-indu ed dis-pla ements and wave-indu ed velo ity gradients, as dened by Phillips (1977), the quasi-Eulerian velo ity

u

b

is more di ult to interpret.

u

b

is the mean velo ity of a water parti le

U

, minus

U

s

, but it is not easily related to Eulerian mean velo -ities. Another interesting velo ity, in parti ular in remote-sensing appli ations, is themeanofthevelo ityatapointthatisxedhorizontallybutmovesup anddown with the surfa e elevation

ζ

. That mean surfa e velo ity is

u(ζ) =

d

u(

b

ζ) + Us(

b

ζ)/2

b

, atse ond order inthe wave slope.

If waves do not intera t with the mean ow,

u

b

is the mean ow velo ity in the limit of vanishingly small wave amplitudes. However, waves do generally intera t

2.3.1 The inuen e of waves on the mean ow

We will use now the equations established in Ardhuin et al. (2004b), whi h are anextensionof Mellor's [2003℄equations,validfor horizontally-uniform onditions. TheseareessentiallyageneralizationinthreedimensionsoftheequationsofGarrett (1976),alsodis ussed inArdhuin etal.(2004a). Theseequationsarealsoequivalent totheGeneralizedLagrangianMeanequationsasgivenbyGroenewegandKlopman (1998), negle ting the modulationsof turbulent properties on the s ale of the wave phase [Ardhuin, 2005℄. Following Ekman (1905) we assume that the wave, velo -ity, and turbulent properties are uniform horizontally. In this ase, the horizontal momentum onservation simpliesas

∂

u

b

∂t

= −fe

z

× (

u

b

+ Us) +

∂

∂z

u

b

′

b

w

′

− T

ds

(z),

(2.9)

with thefollowingboundary onditions,dening our verti al oordinatesothatthe mean sea levelis at

z =

ζ = 0

b

,

b

u

′

b

w

′







z=0

=

τ

a

ρw

−

τ

in

ρw

(2.10) and

u

_|z=−H

= 0.

(2.11) Here

T

ds

isa verti al distribution of

τds

,so that

τds

= ρw

R

0

−

H

T

ds

_dz

.

The inuen e of the wave motion on the quasi-Eulerian ow appears with the Hasselmann for e

−fe

z

× U

s

[Hasselmann, 1970℄, that ombines the Coriolis pa-rameter and the Stokes drift [e.g. Xu and Bowen, 1994℄, and in the momentum transfer from wind to the mean ow. One part of the momentum from the wind goesdire tlytothemeanowviathesurfa eshearstress

ρ

w

u

b

′

b

w

′







z=0

. Itisthedire t mean vis ous drag of air on water. The other part

τ

in

goes to the wave eld, it is the formdrag of wind over water plus the wave-indu ed modulationsofthe vis ous stresses [Longuet-Higgins, 1969℄. Then the wave eld is also dissipated, releasing its momentum to the mean ow. This is the for e

−T

ds

_(z)

. This latter for e is onstitutedby vis ous dissipation(the virtualwave stressis partof it),intera tions of waves with the turbulen e [e.g. Teixeira and Bel her, 2002℄, and wave breaking [Melville et al., 2002℄.

Observations of wave growth with fet h shows that the momentum retained by thewaveeldisaround5%ofthemomentuminput(seese tion2.2.1). Thisleadsto the good approximation

τ

ds

_{≃ −τ}

in

. Furthermore, supposing that the momentum is released by the wave eld at the surfa e (i.e.

T

ds

_{= τ}

ds

_δ(z)/ρw

2.3.2 Turbulent losure

Eq.2.9 involves the divergen e of the Reynolds stresses

u

b

′

b

w

′

that should now be omputed or parameterized. We will use the turbulent losure model of Craig and Banner (1994). It is a "level 2.5" turbulent losure s heme adapted from Mellor and Yamada (1982), with the dissipation of surfa e waves taken into a ount by introdu inga near-surfa e inje tion of turbulentkineti energy (TKE).

The Reynolds stress is assumed to be linearly related to the shear :

u

b

′

b

w

′

₌

Kz∂u/∂z

, with the eddy vis osity

Kz

= lqSm

, where

b = q

2

_/2

is the TKE perunit mass, and

l

the mixinglength. The later is parameterized as

l = κ(z0

_{− z),}

(2.12)

where

κ = 0.4

is the vonKármán's onstantand

z0

isa roughness length.

The bottom has almost no ee t on the near surfa e dynami s, provided that the depth is substantially greater than the Stokes depth(see se tion2.2.2) and the Ekman s ale, whi h is

u

∗

/4f

be ause the turbulent vis osity varies nearly linearly with depth [Craig and Banner, 1994℄. Therefore, the bottom boundary layer of Craigand Banner (1994)is not des ribed here.

The equation for the evolution of TKE is :

∂b

∂t

=

∂

∂z

lqSq

∂b

∂z

!

|

{z

}

a

+ lqSm





∂

u

b

∂z

!

2

+

∂

v

b

∂z

!

2





|

{z

}

b

−

q

3

Bl

|{z}

c

− ϕ

ds

(z)

| {z }

d

,

(2.13)

where

Sm

,

Sq

and

B

are model onstantsfor whi h the appropriatevaluesare 0.39, 0.2 and 16.6.

u

b

and

v

b

are the omponentsof the quasi-Eulerian velo ity

u

b

.

The TKE evolution omes from a transport term(a), a produ tion term by the shear of the mean ow (b), a dissipation term ( ) and a wave-indu ed sour e term (d). The transport termis parameterized by the eddy diusivity

lqSq

.

wave "dissipation"

Φoc

(per unit mass and unit surfa e) of the wave eld,

Φoc

= g

Z

S

ds

(k, θ)dkdθ.

(2.14)

S

ds

isdistributed over depth as

Z

0

−

_H

ϕ

ds

(z)dz = Φoc.

(2.15)

Alternatively[Craig and Banner,1994℄,

Φoc

may bepres ribed asa surfa e ux of TKE and parameterized by

Φoc

= αu

3

∗

with

α ≃ 100

, onsistent with the known

loss of energy from the waves. How the pres ription as a surfa e ux modies the TKE proles will be studied in se tion 2.4.1. The onsequen es of negle ting the variations of

α

with the wave age (from

50

for young waves and fully-developed wavesto

150

otherwise) willbe dealtwith inse tion2.4.3. Theboundary ondition for the TKE is then :

lqSq

∂b

∂z











_z=0

= αu

3

∗,

(2.16)

whi h loses the model.

We will now fo us our attention onthe steady state solutions, when wind- and wave-indu ed inertial os illations are damped. The sea state is again modelled by Kudryavtsev et al.'s [1999℄ spe trum. It is assumed that the wave eld is lo ally uniform even if the sea is not fully developed. In other words, the gradients of the radiation stresses are supposed mu h smaller than the leading terms in the momentumbalan ethataretheCoriolisfor e,theHasselmannfor eandtheverti al mixing (see Ardhuin et al. (2004a) for a dis ussion of the impa t of the radiation stress tensor infet h limited onditions).

2.4 Model results and validation

2.4.1 Calibration of the model with observed proles of TKE dissipation

Two parameters remain unknown in this model : the roughness length

z0

and the s ale

α

ofthe surfa e uxof TKE.

α

may pra ti ally ome fromawave model,and istherefore supposed tobe known [e.g. Janssen et al.,2004℄.

z0

isdeterminedfrom measurements of TKE dissipation near the surfa e.

In termsof TKE, the surfa elayer an be dividedin a"produ tionlayer" and a "diusion layer" [Craig and Banner,1994℄. In the deeper layer, the TKE equation

addition of the Stokes-Coriolis ee t (the Hasselmann for e). This ee t modies the Eulerian velo ities over the whole water olumn (see se tion 2.4.2). But in the diusion layer, the TKE produ tion due to the shear of the mean ow has no importan e. Therefore, the TKE is expe ted toremain un hanged near the surfa e by the addition of the Stokes-Coriolisterm. The Numeri al modelresults onrm this expe tation, with relative hangesin the magnitude of

q

less than 2% near the surfa e.

As a result, we an rely on previous works without the Stokes-Coriolis ee t, providinga parameterizationof

z0

based onmeasurementsof TKE dissipation

ǫ

in the diusion layer. Terray et al. (1996) proposed a s aling of the roughness length withthe signi antwave height

Hs

. It omesfrom thephysi alhypothesisthat the surfa e mixing is proportional to the height of the breaking waves, whi h an be evaluated by

Hs

. Other s alings of

z0

, linked to the wind speed or to the fri tion velo ityarereportedtofail[e.g. Solovievand Lukas,2003℄be auseofnoexpli it de-pendan eonthewavedevelopment. Terrayetal.(2000)usedthemodelofCraigand Banner (1994) to t

z0

using dissipation data from several eld experiments, with various stages of wavedevelopment [Drennanet al.,1996℄. They found

z0

= 1.6Hs

. As was pointed out by the authors, the model does not t very well the data at depths of the order of

Hs

. Therefore they proposed a modied length s ale whi h seems to t better the observations. However, if we attempt a Lagrangian inter-pretation of their Eulerian measurements, there is water between their uppermost datapointsand the surfa e whereTKE dissipationalsoo urs. Even if wesuppose that

ǫ

de ays linearly from

2Φ

oc

/H

s

at

z = −Hs

to

Φ

oc

/H

s

at

z = 0

, the verti ally integrateddissipationrateinthegure1inTerrayetal.(2000),betweenthesurfa e and

−H

s

, isgreaterthanthe waveinputux

Φoc

of TKE.This annotbeexplained bytheprodu tionofTKEbytheshearofthemeanow, whi hisnegligiblenearthe surfa e. Besides, ade rease of

ǫ

between

z = −H

s

andthe surfa eis not supported by the Lagrangian averaged data of Soloviev and Lukas (2003). The data and the modied mixing length of Terray et al. (2000) are not ompatible, unless eviden e isshownof verysmalldissipation ratebetween

z = −H

s

andthe surfa e. Therefore wedonot takethe modiedformofthe mixinglength,asdidMellorand Blumberg (2004), and sti k to (2.12). Soloviev and Lukas (2003) also used measurements of dissipationtoestimate

z

0

,and found

z

0

= 0.6H

s

. However the ontributionof swell tothe signi antwave height wasnot evaluated, whi hmay have leadto an under-estimation of the ratio

z0/Hs

.