Wave-current Interaction (WEC) in the COAWST Modeling System

Nirnimesh Kumar, SIOwith J.C. Warner, G. Voulgaris, M. Olabarrieta

*see Kumar et al., 2012 (might be in your booklet) Implementation of the vortex force formalism in the coupled ocean-atmosphere-wave-sediment transport (COAWST) modeling system for inner shelf and surf zone applications, Ocean Modelling, Volume 47, 2012, Pages 65-95.*also see Olabarrieta et al., 2012

𝜕𝑼𝜕𝑡

+(𝑼 .𝛻 )𝑼+𝟏𝝆𝟎

𝛻𝑝−ℬ 𝒛+ 𝑓 𝒛×𝑼=𝟎Momentum Balance

Continuity 𝛻 ⋅𝑼=𝟎Eulerian

Governing Equations

;withRadiation Stress

Vortex Force Formalism

(𝑼 ⋅ 𝛻 )𝑼=𝛁|𝑼|𝟐 /𝟐+ (𝜵×𝑼 )×𝑼

This term on phase averaging gives vortex force

Recipes for Wave-Current Interaction

Radiation Stress

𝑺=[𝑆 𝑥𝑥 𝑆 𝑥𝑦𝑆 𝑦𝑥 𝑆 𝑦𝑦 ]=𝐸 [𝑛 ∙ (𝑐𝑜𝑠2𝜃+1 )−0.5

𝑛 ∙𝑠𝑖𝑛 2𝜃2

𝑛 ∙𝑠𝑖𝑛2 𝜃2

𝑛 . (𝑠𝑖𝑛2𝜃+1 )−0.5]2-D Radiation Stress Equations (Longuet-Higgins, 1962, 1964)

Excess flux of momentum due to presence of waves. Explains wave setup, wave setdown, generation of longshore currents, rip currents.

Breaker Zone

Surf Zone

set-upMWLset-down

Swash

Beach Profile

Wave Setup:Balance between quasi-static pressure and radiation stress divergence

Longshore Currents:Generated due to gradient of radiation stress in longshore direction

Adapted from Smith (2006, JPO)

𝑼𝑺𝒕 𝑉 (𝑥)𝜵×𝑼

Cross-shore

Alon

gsho

re

Stokes drift

h-mean ambient flow

𝑽𝑭=𝑉 𝑆𝑡(𝜕𝑉𝜕 𝑥 − 𝜕𝑈𝜕𝑦 ) �̂�+𝑈𝑆𝑡 (𝜕𝑉𝜕 𝑥 − 𝜕𝑈

𝜕𝑦 ) �̂�

Product of Stokes drift and mean flow vorticity (Craik and Leibovich, 1976).

Physically representative of wave refraction due to current shear

𝑼𝑺𝒕

Vortex Force (VF)

Wave Rollers

Stokes-Coriolis Force, Surface and Bottom Streaming

Cross-shore Vel. Alongshore Vel.

From Lentz et al., 2008

Wave-propagation direction

Wave-averaged Eqns.

𝜕𝑢𝑙

𝜕𝑡+{𝜕 (𝑢𝑙𝑢𝑙 )

𝜕𝑥+𝜕 (𝑢𝑙𝑣 𝑙 )𝜕 𝑦

+𝜕 (𝜔𝑠𝑢

𝑙 )𝜕𝑠 }− 𝑓𝑣 𝑙=−𝐻 𝑧 ( 1

𝜌0

𝜕𝑃𝜕 𝑥 |

𝑧)−{𝜕𝑆𝑥𝑥

𝜕 𝑥+𝜕𝑆𝑥𝑦

𝜕 𝑦 }+𝐷𝑥

Local Acc.

Advective accn. Coriolis + Stokes-Coriolis

Pressure Gradient

Radiation Stress

Bottom Stress

Radiation Stress

Accounts for wave breaking

+{𝑣𝑆𝑡 ( 𝜕𝑣𝜕 𝑥 − 𝜕𝑢𝜕 𝑦 )−𝜔𝑠

❑𝑆𝑡 𝜕𝑢𝜕𝑠 }+𝐷𝑥+ℱ𝑤𝑥

Local Acc.

Advective accn. Stokes-Coriolis

Pressure Gradient

Vortex Force Bottom Stress

𝜕𝑢𝜕𝑡

+{𝜕 (𝑢𝑢)𝜕 𝑥

+𝜕 (𝑢𝑣 )𝜕 𝑦

+𝑢 (𝜕𝑢𝑆𝑡

𝜕 𝑥+𝜕𝑣 𝑆𝑡

𝜕 𝑦 )+ 𝜕 (𝜔𝑠𝑢)𝜕 𝑠

+𝑢𝜕 (𝜔𝑠

❑𝑆𝑡 )𝜕 𝑠 }− 𝑓𝑣− 𝑓 𝑣𝑆𝑡=−( 1

𝜌 0

𝜕𝜑𝜕 𝑥 |

𝑧)

Coriolis

Non-conservative forcing

Vortex Force Formalism

Accounts for wave breaking

Wave-averaged Eqns.

+{𝑣𝑆𝑡 ( 𝜕𝑣𝜕 𝑥 − 𝜕𝑢𝜕 𝑦 )−𝜔𝑠

❑𝑆𝑡 𝜕𝑢𝜕𝑠 }+𝐷𝑥+ℱ𝑤𝑥

Local Acc.

Advective accn. Stokes-Coriolis

Pressure Gradient

Vortex Force Bottom Stress

𝜕𝑢𝜕𝑡

+{𝜕 (𝑢𝑢)𝜕 𝑥

+𝜕 (𝑢𝑣 )𝜕 𝑦

+𝑢 (𝜕𝑢𝑆𝑡

𝜕 𝑥+𝜕𝑣 𝑆𝑡

𝜕 𝑦 )+ 𝜕 (𝜔𝑠𝑢)𝜕 𝑠

+𝑢𝜕 (𝜔𝑠

❑𝑆𝑡 )𝜕 𝑠 }− 𝑓𝑣− 𝑓 𝑣𝑆𝑡=−( 1

𝜌 0

𝜕𝜑𝜕 𝑥 |

𝑧)

Coriolis

Non-conservative forcing

Vortex Force Formalism

Accounts for wave breaking

Surface Streaming

Bottom Streaming

Whitecapping

Wave Rollers

Depth-limited breaking

Depth-limited Breaking

𝐁𝑏=(1−α𝑟¿⋅ 𝜖𝑏      

𝜌 0 ⋅𝜎𝐤 ⋅ 𝑓 𝑏 (𝑧 )

= Dissipation due to depth-limited breaking (from empirical formulations or SWAN)

𝑓 𝑏 (𝑧 )= 𝐹𝐵

∫−h

�̂�

𝐹𝐵 .𝑑𝑧

𝐹𝐵  =   cosh ( 2 𝜋𝐻𝑟𝑚𝑠

(𝑧+𝐷 ))

Surface Intensified

Integrate oceanic, atmospheric, wave and morphological processes in the coastal ocean (Warner et al., 2010)

Coupled-Ocean-Atmosphere-Wave-Sediment-Transport (COAWST) modeling system

Methodology

ROMS SWAN

WRF

CSTMShttp://woodshole.er.usgs.gov/operations/modeling/COAWST/index.html

cppdefs.h (COAWST/ROMS/Include)WEC_MELLOR+ Activates the Mellor (2011) method for WEC

WEC_VF(preferred method for 3D)

Activate WEC using the Vortex Force formalism (Uchiyama et al., 2010)

Wave-current Interaction (WEC)

WEC_MELLOR(Mellor, 2011)

WEC_VF(Uchiyama et al., 10)

+ Roller Model+ Streaming

+ Dissipation (depth)+ Roller Model + Wave mixing + Streaming

Implemented in Kumar et al., 2011

Implemented in Kumar et al., 2012

*Processes in italics are optional

Dissipation (Depth-limited wave breaking)WDISS_THORGUZA Wave dissipation based on Thornton and Guza (1983).

See Eqn. (31), pg-71

WDISS_CHURTHOR Wave dissipation based on Church and Thornton (1993). See Eqn. (32), pg-71

WDISS_WAVEMODActivate wave-dissipation from a wave model. If using SWAN wave model, use INRHOG=1 for correct units of wave dissipation

Note: (a) Use WDISS_THORGUZA/CHURTHOR if no information about wave dissipation is

present, and you can’t run the wave model to obtain depth-limited dissipation

(b)If you do not define any of these options, and still define WEC_VF, the model expects a forcing file with information about dissipation

ROLLER MODEL (for Wave Rollers)ROLLER_SVENDSEN Wave roller based on Svendsen (1984). See Warner et al.

(2008), Eqn. 7 and Eqn. 10.

ROLLLER_MONO Wave roller for monochromatic waves from REF-DIF. See Haas and Warner, 2009.

ROLLER_RENIERS Activate wave roller based on Reniers et al. (2004). See Eqn. 34-37 (Advection-Diffusion)

Note: If defining ROLLER_RENIERS, you must specify the parameter

wec_alpha (αr in Eqn. 34, varying from 0-1) in the INPUT file. Here 0 means no percentage of wave dissipation goes into creating wave rollers, while 1 means all the wave dissipation creates wave rollers.

Wave breaking induced mixing

TKE_WAVEDISSZOS_HSIG

• Enhanced vertical mixing from waves within framework of GLS. See Eq. 44, 46 and 47. Based on Feddersen and Trowbridge, 05

• The parameter αw in Eq. 46 can be specified in the INPUT file as ZOS_HSIG_ALPHA (roughness from wave amplitude)

• Parameter Cew in Eqn. 47 is specified in the INPUT file as SZ_ALPHA (roughness from wave dissipation)

Bottom and Surface Streaming

BOTTOM_STREAMINGBottom streaming due to waves using Uchiyama et al. (2010) methodology. See Eqn. 22-26. This method requires dissipation due to bottom friction. If not using a wave model, then uses empirical Eq. 22.

BOTTOM_STREAMING_XU_BOWEN

Bottom streaming due to waves based on methodology of Xu and Bowen, 1994. See Eq. 27.

SURFACE_STREAMING Surface streaming using Xu and Bowen, 1994. See Eq. 28.

Note: (a) BOTTOM_STREAMING_XU_BOWEN was tested in Kumar et al. (2012). It requires

very high resolution close to bottom layer. Suggested Vtransform=2 and Vstretching=3

[0,0]

[1000,-12]

z

x

y

Hsig= 2mTp = 10sθ = 10o

Shoreface Test Case (Obliquely incident waves on a planar beach)

Wave field computed using SWAN One way coupling (only WEC) Application Name: SHOREFACE Header file: COAWST/ROMS/Include/shoreface.h Input file: COAWST/ROMS/External/ocean_shoreface.in

Header File (COAWST/ROMS/Include)

Input File (COAWST/ROMS/External)

Input File (COAWST/ROMS/External)

Requires a wave forcing fileas one way coupling only

Forcing file for one way coupling Data/ROMS/Forcing/swan_shoreface_angle_forc.nc

Should contain the following variables

Wave Height Hwave

Wave Direction Dwave

Wave Length Lwave

Bottom Orbital Vel. Ub_Swan

Depth-limited breaking Dissip_break

Whitecapping induced breaking Dissip_wcap

Bottom friction induced dissip. Dissip_fric

Time Period Pwave_top/Pwave_bot

WEC Related Output

Results (I of III)Significant Wave Height

Sea surface elevation

Results (II of III)Depth-averaged Velocities

Cross-shore Vel. Longshore Vel.

Results (III of III)

Cross-shore

Longshore

Vertical

Eulerian Stokes

WEC related Diagnostics Terms(i.e., contribution to momentum balance)Terms in momentum balance Definition Output

VariableLocal Acc. u_accel/

ubar_accelHorizontal Advection u_hadv/ubar_hadv

Vertical Advection

u_vadv

Coriolis Force u_cor/ubar_cor

Stokes-Coriolis u_stcor/ubar_stcorPressure Gradient

u_prsgrd/ubar_prsgrd

Vortex Force u_hjvf/ubar_hjvf

Vortex Force u_vjvf

Terms in momentum balance Definition Output Var.

(see Eqn. 21)Breaking + Roller Acceleration + Streaming

u_wbrk/ubar_wbrku_wrol/ubar_wrolu_bstm/ubar_bstmu_sstm/ubar_sstm

BREAKING THE PRESSURE GRADIENT TERM

Pressure Gradient u_prsgrd/ubar_prsgrd

Eulerian Contribution ubar_zeta

Quasi-static response, Eqn. 7 ubar_zetw

Bernoulli-head contribution, Eqn. 5 ubar_zbeh

Surface pressure boundary, Eqn. 9 ubar_zqsp

Vertical profile of terms in momentum balance

Breaking Acc.

Hor. Advection

Hor. VF

Pressure Gradient

Vertical Mixing

Vertical Advection

Alongshore Cross-shore

DUCK’ 94, NC-Nearshore Experiment

DUCK’ 94- Nearshore Experiment

Obliquely incident waves on a barred beach (DUCK’ 94 Experiment)

Experiment conducted Oct. 12, 1994 (Elgar et al., 97; Garcez-Faria et al., 98, 00) Wave field from SWAN. One way coupling.

Sea Surface Elevation

Depth averagedcross-shore velocity

Depth averagedlongshore velocity

Wave Height

Wave Parameters, Depth-Averaged Flows

η

u

v

Hrms

εb

Vertical profile of Cross-shore & Longshore Vel.Cr

oss-

shor

e Al

ongs

hore

Eulerian

Kumar et al., 2012

Stokes Drift

Notes:

Vertical Distribution of Wave Dissipation

Comparison to Field Observations

Cross-shore

Alongshore

Obs from Garcez-Faria et al., 1998, 2000 Kumar et al., 2012

Alongshore Cross-shore

Vertical profile of terms in momentum balance

Breaking Acc.

Hor. Advection

Hor. VF

Pressure Gradient

Vertical Mixing

Vertical Advection

Notes:

Over the bar (a) VF balances Breaking

(b) VM balancesAdvection