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STWAVESteady-State Spectral Wave Model

By

Jane McKee Smith, Donald T. Resio and Alan K. Zundel

US Army Corps o !n"ineers#ater$ays !%periment Station

Coastal &nlets Research 'ro"ram

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PREFACE

The $or( descri)ed herein $as conducted at the U.S. Anny !n"ineer #ater$ays!%periment Station *#!S+, Coastal and ydraulics -a)oratory *C-+ as part o theCoastal &nlets Research 'ro"ram *C&R'+ under the $or( unit Modelin" #aes at&nlets.

/erall pro"ram mana"ement or C&R' is directed )y the ydraulic Desi"n Sectiono the ead0uarters, U.S. Army Corps o !n"ineers *1USAC!+. 'ro"ram Monitors

or the C&R' are Messrs. Barry #. olliday, John Bianco, and Charles B. Chesnutt,1USAC!. The 'ro"ram Mana"er is Mr. !. Clar( Mc2air, C-, and C&R' TechnicalMana"er is Dr. 2icholas C. Kraus, C-.

The purpose o the Modelin" #aes at &nlets $or( unit is to proide ield tools or 0uantiyin" $ae processes at coastal inlets. #ae inormation is re0uired or almostall en"ineerin" studies near inlets to estimate channel shoalin" and mi"ration,shoreline chan"e, scour, orces on structures, and nai"ation saety. This reportdocuments a 3$or(horse3 numerical model or estimatin" nearshore $ae "ro$thand transormation. The purpose o the report is to transer this technolo"y to ieldusers, throu"h "uidance on model application.

Dr. James R. ouston, Director, and Mr. Charles C. Calhoun, Jr. *retired+, Assistant

Director. Direct "uidance $as proided )y Messrs. Thomas #. Richardson, Chie,Coastal Sediments and !n"ineerin" Diision, and Mr. Bruce A. !)ersole, Chie,Coastal 'rocesses Branch *C'B+. This report $as prepared )y Drs. Jane McKeeSmith, C'B, and Donald T. Resio, Senior Research Scientist, C-, and Dr. Alan K.Zundel, !n"ineerin" Computer 4raphics -a)oratory, Bri"ham 5oun" Uniersity. Assistance and reie$ $ere proided )y Mr. S. Jarrell Smith, C'B, Dr. Joon '. Rhee,'rototype Measurement and Analysis Branch, and Ms. -ori adley, Coastalydrodynamics Branch.

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INTRODUCTION

!stimatin" nearshore $ind6$ae "ro$th and transormation is a critical component o most coastal en"ineerin" pro7ects, e."., predictin" )athymetric and shoreline chan"e,estimatin" nai"ation channel shoalin" and mi"ration, desi"nin" or repairin" coastalstructures, assessin" nai"ation conditions, and ealuatin" natural eolution o coastal inlets or )eaches ersus conse0uences o en"ineerin" actions. 2earshore$ae propa"ation is inluenced )y comple% )athymetry *includin" shoals and

nai"ation channels+8 tide6, $ind6, and $ae6"enerated currents8 tide6 and sur"e6induced $ater6leel ariation8 and coastal structures.

Use o numerical $ae models has )ecome $idespread to represent $aetransormation primarily )ecause o their increasin" sophistication and economy o application relatie to the lar"e e%pense o ield measurements or physical modelstudies.

This report descri)es the application o the steady6state spectral $ae model,ST#A9!. The purpose o ST#A9! is to proide an easy6to6apply, le%i)le, andro)ust model or nearshore $ind6$ae "ro$th and propa"ation. Recent up"rades tothe model include $ae6current interaction and steepness6induced $ae )rea(in".

This report descri)es procedures or usin" ST#A9!. An oerie$ o the model

"oernin" e0uation and the numerical discreti:ation is presented.

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MODEL CAPABILITIES

The purpose o applyin" nearshore $ae transormation models is to descri)e0uantitatiely the chan"e in $ae parameters *$ae hei"ht, period, direction, andspectral shape+ )et$een the oshore and the nearshore *typically depths o ;< m or less+.

&n relatiely deep $ater, the $ae ield is airly homo"eneous on the scale o (ilometers8 )ut in the nearshore, $here $aes are stron"ly inluenced )y ariations in)athymetry , $ater leel, and current, $ae parameters may ary si"niicantly on thescale o tens o meters.

/shore $ae inormation is typically aaila)le rom a $ae "au"e or a "lo)al6 or re"ional6scale $ae hindcast or orecast. 2earshore $ae inormation is re0uired or the desi"n o almost all coastal en"ineerin" pro7ects. #aes drie sediment transportand nearshore currents, induce $ae setup and runup, e%cite har)our oscillations, or impact coastal structures. The lon"shore and cross6shore "radients in $ae hei"htand direction can )e as important as the ma"nitude o these parameters or somecoastal desi"n pro)lems.

ST#A9! simulates depth6induced $ae reraction and shoalin", current6 inducedreraction and shoalin", depth6 and steepness6induced $ae )rea(in", diraction,

$ae "ro$th )ecause o $ind input, and $ae6$ae interaction and $hite cappin"that redistri)ute and dissipate ener"y in a "ro$in" $ae ield.

 A $ae spectrum is a statistical representation o a $ae ield. Conceptually, aspectrum is a linear superposition o monochromatic $aes. A spectrum descri)esthe distri)ution o $ae ener"y as a unction o re0uency *one6dimensionalspectrum+ or re0uency and direction *t$o6dimensional spectrum+.

 An e%ample o a one6dimensional $ae spectrum is "ien in =i"ure >. The pea(period o the spectrum is the reciprocal o the re0uency o the pea( o the spectrum.The $ae hei"ht *si"niicant or :ero6moment $ae hei"ht+ is e0ual to our times thearea under the spectrum. =or the e%ample spectrum "ien in =i"ure >, the pea(re0uency is

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=i"ure >. Sample one6dimensional $ae spectrum

Model Assumptions

The assumptions made in ST#A9! ersion ;.< are as ollo$s

Mild bottom slope and negligible wave reflection. ST#A9! is a hal6 plane model,

meanin" that $ae ener"y can propa"ate only rom the o6 shore to$ard thenearshore * .? de" rom the  x a%is o the "rid, $hich is typically theappro%imate shore6normal direction+. #aes relected rom the shoreline or romsteep )ottom eatures trael in directions outside this hal plane and thus arene"lected. =or$ard6scattered $aes, e."., $aes relected o a structure )uttraelin" in the +x direction, are also ne"lected.

Spatially homogeneous offshore wave conditions. The ariation in the $aespectrum alon" the oshore )oundary o a modelin" domain is rarely (no$n, andor domains on the order o tens o (ilometers, is e%pected to )e small. Thus, theinput spectrum in ST#A9! is constant alon" the oshore )oundary. =utureersions o the model may allo$ aria)le input.

Steady-state waves, currents, and winds. ST#A9! is ormulated as a steady6state model. A steady6state ormulation reduces computation time and isappropriate or $ae conditions that ary more slo$ly than the time it ta(es or $aes to transit the computational "rid. =or $ae "eneration, the steady6stateassumption means that the $inds hae remained steady suiciently lon" or the$aes to attain etch6limited or ully deeloped conditions *$aes are not limited)y the duration o the $inds+.

Linear refraction and shoaling. ST#A9! incorporates only linear $ae reractionand shoalin", thus does not represent $ae asymmetry .Model accuracy isthereore reduced at lar"e Ursell num)ers *$ae hei"hts are underestimated+.

Depth-uniform current. The $ae6current interaction in the model is )ased on a

current that is constant throu"h the $ater column. & stron" ertical "radients incurrent occur, their modiication o reraction and shoalin" is not represented in

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the model. =or most applications, three6 dimensional current ields are notaaila)le.

ottom friction is neglected. The si"niicance o )ottom riction on $aedissipation has )een a topic o de)ate in $ae modelin" literature. Bot6 tomriction has oten )een applied as a tunin" coeicient to )rin" model results into

ali"nment $ith measurements. Althou"h )ottom riction is easy to apply in a $aemodel, determinin" the proper riction coeicients is diicult. Also, propa"ationdistances in a nearshore model are relatiely short *tens o (ilometers+, so thatthe cumulatie )ottom riction dissipation is small. =or these reasons, )ottomriction is ne"lected in ST#A9!.

/n"oin" research $ill enhance present model capa)ilities and eliminate somemodel assumptions. The ollo$in" sections descri)e $ae propa"ation andsourceEsin( terms in ST#A9! ersion ;. into !0uation ; and iteratielysolin" or %. The $ae num)er and $aelen"th #L"#&$'%$ are the same in )othreerence rames.Solutions or reraction and shoalin" also re0uire $ae celerities, C, and "roupcelerities, (g, in )oth reerence rames. &n the reerence rame relatie to the current.

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=i"ure ;. Deinition s(etch o $ae and current ectors

k  C r r ω 

=   (3)

   

   +=

2kdsinh

kd2 1C0.5C r gr    (4)

The direction o )oth the relatie celerity and "roup celerity is. , the $ae ortho"onaldirection. &n the a)solute reerence rame,

Ca F Cr  G U cos*δ6α+ *?+

*C"a+i F *C"r +i G *U+i *H+

where i subscript is tensor notation for the x and y components. )he direction of the absolutecelerity is also in the wave orthogonal direction. )he absolute group celerity defines thedirection of the wave ray, so the wave ray direction #*igure &$ is defined as,

   

  

 ++

=   −δ α 

δ α  µ 

coscos

sinsintan

1

U C 

U C 

 gr 

 gr *+

The distinction )et$een the $ae ortho"onal *direction perpendicular to the $aecrest+ and the $ae ray *direction o ener"y propa"ation+ is important in descri)in"$ae6current interaction. #ithout currents, the $ae rays and ortho"onals are thesame, )ut $ith currents, the $ae ener"y moes alon" the rays $hereas the $aedirection is deined )y the ortho"onals.

The $ae ortho"onal direction or steady6state conditions is "ien )y *Mei >@@8Jonsson >@@

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 Dn

 DU 

k 

k 

 Dn

 Dd 

kd 

k C 

 DR

 DC    iir  ga   −−=

2sinh

α *+

where D is a derivative, is a coordinate in the direction of the wave ray, and n is acoordinate normal to the wave orthogonal.

The "oernin" e0uation or steady6state conseration o spectral $ae action alon" a$ae ray is "ien )y *Jonsson >@@

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( ) ( ) ( ) ( )θ θ θ θ    ,,225.0,55.0, 11   a ja ja ja j   w E w E w E w E  −+   ++= *>< proides smoothin" o stron" "radients in $ae hei"ht that occur inshelter re"ions, )ut proides no turnin" o the $aes. This ormulation is "rid6spacin"dependent, $hich is a serious $ea(ness. !orts are on"oin" to implement a moreri"orous diraction representation.

Source/sink terms

Sur6:one $ae )rea(in". The $ae6)rea(in" criterion applied in the irst ersion oST#A9! $as a unction o the ratio o $ae hei"ht to $ater depth,

64.0max0

=d 

 H m*>>+

where / mo is the energy-based 0ero-moment wave height.

 At an inlet, $here $aes steepen )ecause o the $ae6current interaction, $ae)rea(in" is enhanced )ecause o the increased steepenin". Smith, Resio, and9incent *>@@+ perormed la)oratory measurements o irre"ular $ae )rea(in" one)) currents and ound that a )rea(in" relationship in the orm o the Miche criterion*>@?>+ $as simple, ro)ust, and accurate

kd  L H m tanh1.0max0   = *>;+

*see also Batt7es >@; and Batt7es and Janssen >@+.

!0uation >; is applied in ersion ;.< o ST#A9! as a ma%imum limit on the :ero6moment $ae hei"ht.

The ener"y in the spectrum is reduced at each re0uency and direction in proportionto the amount o pre)rea(in" ener"y in re0uency and direction )and. 2on6linear transers o ener"y to hi"h re0uencies that occur durin" )rea(in" are notrepresented in the model.

Wind input.

#aes "ro$ throu"h the transer o momentum rom the $ind ield to the $ae ield.The lu% o ener"y , * in, into the $ae ield in ST#A9! is "ien )y *Resio >@a+,

 g 

uC  F  m

w

ain

2

*85.0 ρ 

 ρ λ = *>I+

1here 2 is a partitioning coefficient that represents percentage of total atmosphere,  ρ a is thedensity of air,  ρ w  is the density of water, ( m is the mean wave celerity and u3  is friction velocity #e!ual to product of wind speed, 4, and s!uare root of drag coefficient, (D".55&+.5555&64$

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&n deep $ater, ST#A9! proides a total ener"y "ro$th rate that is consistent $ithasselmann et al. *>@I+.

The ener"y "ain to the spectrum is calculated )y multiplyin" the ener"y lu% )y thee0uialent time or the $ae to trael across a "rid cell,

m g C 

 xt 

θ β  cos

∆=∆ *>+

where ∆t is the e!uivalent travel time, ∆ x is the grid spacing, β  is a factor e!ual to 5.7 for wind 

seas, ( g  is the average group celerity of spectrum and θ m is the mean wave direction, relative

to grid.

Because ST#A9! is a hal6plane model, only $inds )lo$in" to$ard the shore #+x direction+ are included. #ae dampin" )y oshore $inds and "ro$th o oshore6

traelin" $aes are ne"lected.

#ae6$ae interaction and $hite cappin". As ener"y is ed into the $aes rom the$ind, it is redistri)uted throu"h nonlinear $ae6$ae interaction. !ner"y istranserred rom the pea( o the spectrum to lo$er re0uencies *decreasin" the pea(re0uency or increasin" the pea( period+ and to hi"h re0uencies *$here it isdissipated+.

&n ST#A9!, the re0uency o the spectral pea( is allo$ed to increase $ith etch *or e0uialently propa"ation time across a etch+. The e0uation or this rate o chan"e o f  p is "ien )y,

( ) ( )!33!4

*3!

1 5

"−

+

∆  

 

  

 −=   t 

 g 

u  f    f  

i pi p  ζ  *>?+

where the i and i+ subscripts refer to the grid column indices within S)128 and ξ   is adimensionless constant #esio and 9errie 7:7$.

The ener"y "ained )y the spectrum is distri)uted $ithin re0uencies on the or$ardace o the spectrum *re0uencies lo$er than the pea( re0uency+ in a manner thatretains the sel6 similar shape o the spectrum.

#ae ener"y is dissipated *most nota)ly in an actiely "ro$in" $ae ield+ throu"hener"y transerred to hi"h re0uencies and dissipated throu"h $ae )rea(in" *$hitecappin"+ and tur)ulentEiscous eects. There is a dynamic )alance )et$een ener"yenterin" the $ae ield )ecause o $ind input and ener"y leain" the $ae ield)ecause o nonlinear lu%es to hi"her re0uencies *Resio >@, >@a+. The ener"ylu% to hi"h re0uencies is represented in ST#A9! as

( )d k k  E  g 

 p

 p

 E  4!3

2!"32!1

tanh

∈=Γ  *>H+

where Γ   is the energy flux, ∈ is a coefficient e!ual to ;5, is the total energy in spectrum

and %  p is the wave number associated with the pea% of the spectrum #esio, 7:

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REFERENCES

Batt7es, J.A. *>@;+. 2 case study of wave height variations due to currents in a tidal entrance. (oast. ng. H,6?.

Batt7es, J. A., and &anssen, J.'.=.M. *>@+. nergy loss and set-up due to brea%ing of random waves. 'roc. >Hth Coastal !n". Con., ASC!, ?H@6 ?.

Bri"ham 5oun" Uniersity !n"ineerin" Computer 4raphics -a)oratory. *>@@+.Surface-water modeling system reference manual . Bri"ham 5oun" Uniersity, 'roo,

UT. #http='lhlnet.wes.army.mil'software'sms'docs.htp$

Bou$s, !., 4unther, ., Rosenthal, #., and 9incent, C.-. *>@?+. Similarity of thewind wave spectrum in finite depth waves> . Spectral form. .> 4eophys. Res.@@@;+. S)128 theory and program documentation. Coastal Modelin"System Users Manual. Chapter , &nstruction Report C!RC6@>6l Supplement >, M. A. Cialone, ed., U.S. Army !n"ineer #ater$ays !%periment Station, 5ic(s)ur", MS.

asselmann, K., Barnett, T.'., Bou$s, !., Carlson, ., Cart$ri"ht, D.!., !n(e, K.,

!$in", J.A., 4ienapp, ., asselmann, D.!., Kruseman, '., Meer)ur", A., Muller, '.,/l)ers, D.J., Richter, K., Sell, #., and #alden, . *>@I+. Measurements ofwind-wave growth and swell decay during the ?oint @orth Sea 1ave 9roect #?A@S129$ .Deut. ydro"r. Zo, Suppl. A, *>;+, >6@?.

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-uettich, R.A., #esterin(, J.J., and Schener, 2.#. *>@@;+. 2D(B(= 2n advanced three-dimensional circulation model for shelves, coasts, and estuaries, eport =)heory and methodology of 2D(B(-&DDB and 2D(B(-;DL. Technical Report DR'6@;6H, U.S. Army !n"ineer #ater$ays !%periment Station, 9ic(s)ur", MS.

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action de la houle. Annals des 'ants et Chaussess, >;>e Annee, ;?6I>@ *translated

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