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

Wave-current Interaction (WEC) in the
COAWST Modeling System
Nirnimesh Kumar, SIO
with 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-atmospherewave-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
Governing Equations
Momentum
Balance
𝜕𝑼
𝟏
+ 𝑼. 𝛻 𝑼 + 𝛻𝑝 − ℬ𝒛 + 𝑓𝒛 × 𝑼 = 𝟎
𝜕𝑡
𝝆𝟎
Continuity
𝛻⋅𝑼=𝟎
Eulerian 𝑼
Mean Flow (𝑼) + Oscillatory Flow (𝑼𝒘 )
Recipes for Wave-Current Interaction
Radiation
Stress
𝑼 ⋅ 𝛻 𝑼 = 𝛁 ⋅ 𝑼𝑼 + 𝑼 𝛁 ⋅ 𝑼 ;with𝛁 ⋅ 𝑼 = 𝟎
Vortex Force
Formalism
𝑼⋅𝛻 𝑼=𝛁 𝑼
𝟐
𝟐+ 𝜵×𝑼 ×𝑼
This term on phase averaging gives vortex force
Radiation Stress
Excess flux of momentum due to presence of
waves. Explains wave setup, wave setdown,
generation of longshore currents, rip currents.
2-D Radiation Stress Equations (Longuet-Higgins, 1962, 1964)
𝑺=
𝑆𝑥𝑥
𝑆𝑦𝑥
𝑆𝑥𝑦
=𝐸
𝑆𝑦𝑦
Swash
𝑛 ∙ 𝑐𝑜𝑠 2 𝜃 + 1 − 0.5
𝑛 ∙ 𝑠𝑖𝑛2𝜃
2
𝑛 ∙ 𝑠𝑖𝑛2𝜃
2
𝑛. 𝑠𝑖𝑛2 𝜃 + 1 − 0.5
Wave Setup:
Balance between
quasi-static pressure
and radiation stress
divergence
Breaker Zone
Surf Zone
set-up
set-down
MWL
Longshore Currents:
Generated due to
gradient of radiation
Beach Profile stress in longshore
direction
Vortex Force (VF)
 Product of Stokes drift and mean
flow vorticity (Craik and Leibovich,
1976).
Alongshore
𝑼𝑺𝒕
 Physically representative of wave
refraction due to current shear
𝑼𝑺𝒕
𝜵×𝑼
= Stokes drift
𝑼 = Depth-mean ambient flow
𝑽𝑭 = 𝑉 𝑆𝑡
𝑼𝑺𝒕
𝑉(𝑥)
𝜕𝑉 𝜕𝑈
𝜕𝑉 𝜕𝑈
−
𝑖 + 𝑈 𝑆𝑡
−
𝑗
𝜕𝑥 𝜕𝑦
𝜕𝑥 𝜕𝑦
Cross-shore
Adapted from Smith (2006, JPO)
Wave Rollers
Stokes-Coriolis Force, Surface and Bottom Streaming
Cross-shore Vel.
Wave-propagation direction
From Lentz et al., 2008
Alongshore Vel.
Wave-averaged Eqns.
Radiation Stress
𝜕 𝑢𝑙 𝑢𝑙
𝜕 𝑢𝑙 𝑣 𝑙
𝜕 𝜔𝑠 𝑢𝑙
𝜕𝑢𝑙
+
+
+
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑠
Local
Acc.
Advective accn.
− 𝑓𝑣𝑙
= −𝐻𝑧
Coriolis +
Stokes-Coriolis
1 𝜕𝑃
𝜌0 𝜕𝑥
𝑧
Pressure
Gradient
𝜕𝑆𝑥𝑥 𝜕𝑆𝑥𝑦
−
+
+ 𝐷𝑥
𝜕𝑥
𝜕𝑦
Radiation
Stress
Bottom
Stress
Accounts for wave breaking
Vortex Force Formalism
𝜕𝑢 𝑆𝑡
𝜕𝑣 𝑆𝑡
𝜕 𝜔𝑠
𝑆𝑡
𝜕𝑢
𝜕 𝑢𝑢
𝜕 𝑢𝑣
𝜕 𝜔𝑠 𝑢
1 𝜕𝜑
+
+
+𝑢
+
+
+𝑢
− 𝑓𝑣 − 𝑓𝑣 𝑆𝑡 = −
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑥
𝜕𝑦
𝜕𝑠
𝜕𝑠
𝜌0 𝜕𝑥 𝑧
Local
Coriolis Stokes
Pressure
Advective accn.
Acc.
-Coriolis Gradient
𝜕𝑣 𝜕𝑢
𝑆𝑡 𝜕𝑢
𝑆𝑡
+ 𝑣
−
− 𝜔𝑠
+ 𝐷𝑥 +
ℱ 𝑤𝑥
Accounts for wave breaking
𝜕𝑥 𝜕𝑦
𝜕𝑠
Vortex Force
Bottom Non-conservative
Stress
forcing
Wave-averaged Eqns.
Vortex Force Formalism
𝜕𝑢 𝑆𝑡
𝜕 𝜔𝑠
𝜕𝑣 𝑆𝑡
𝑆𝑡
𝜕𝑢
𝜕 𝑢𝑢
𝜕 𝑢𝑣
𝜕 𝜔𝑠 𝑢
1 𝜕𝜑
+
+
+𝑢
+
+
+𝑢
− 𝑓𝑣 − 𝑓𝑣 𝑆𝑡 = −
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑥
𝜕𝑦
𝜕𝑠
𝜕𝑠
𝜌0 𝜕𝑥 𝑧
Local
Coriolis Stokes
Pressure
Advective accn.
Acc.
-Coriolis Gradient
𝜕𝑣 𝜕𝑢
𝑆𝑡 𝜕𝑢
+ 𝑣 𝑆𝑡
−
− 𝜔𝑠
+ 𝐷𝑥 +
ℱ 𝑤𝑥
Accounts for wave breaking
𝜕𝑥 𝜕𝑦
𝜕𝑠
Vortex Force
Bottom Non-conservative
Stress
forcing
Depth-limited
breaking
Whitecapping
Wave Rollers
Bottom
Streaming
Surface
Streaming
Depth-limited Breaking
𝑟
𝑏
1
−
α
)
⋅
𝜖
𝐁𝑏 =
𝜌0 ⋅ 𝜎
𝐤
⋅ 𝑓𝑏
𝑧
𝜖 𝑏 = Dissipation due to depth-limited breaking
(from empirical formulations or SWAN)
𝑏
𝑓 𝑧 =
𝐹𝐵
𝜂
𝐹𝐵. 𝑑𝑧
−ℎ
2𝜋
𝐹𝐵 = cosh
𝑧+𝐷
𝐻𝑟𝑚𝑠
Surface
Intensified
Methodology
Coupled-Ocean-Atmosphere-Wave-Sediment-Transport (COAWST) modeling system
Integrate oceanic, atmospheric, wave and morphological processes in the coastal ocean
(Warner et al., 2010)
WRF
SWAN
ROMS
CSTMS
http://woodshole.er.usgs.gov/operations/modeling/COAWST/index.html
Wave-current Interaction (WEC)
WEC_MELLOR
(Mellor, 2011)
+ Roller Model
+ Streaming
Implemented in
Kumar et al., 2011
Implemented in
Kumar et al., 2012
*Processes in italics are optional
WEC_VF
(Uchiyama et al., 10)
+ Dissipation (depth)
+ Roller Model
+ Wave mixing
+ Streaming
cppdefs.h (COAWST/ROMS/Include)
WEC_MELLOR+
WEC_VF
(preferred
method for 3D)
Activates the Mellor (2011) method for WEC
Activate WEC using the Vortex Force formalism (Uchiyama et
al., 2010)
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_WAVEMOD
Activate 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
• 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
TKE_WAVEDISS
INPUT file as ZOS_HSIG_ALPHA (roughness from
ZOS_HSIG
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 streaming due to waves using Uchiyama et al.
(2010) methodology. See Eqn. 22-26. This method
BOTTOM_STREAMING requires dissipation due to bottom friction. If not using
a wave model, then uses empirical Eq. 22.
BOTTOM_STREAMING Bottom streaming due to waves based on methodology
of Xu and Bowen, 1994. See Eq. 27.
_XU_BOWEN
Surface streaming using Xu and Bowen, 1994. See Eq.
SURFACE_STREAMING 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
Shoreface Test Case
(Obliquely incident waves on a planar beach)
Hsig= 2m
Tp = 10s
θ = 10o
[0,0]
z
y
x
[1000,-12]





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 file
as 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)
Eulerian
Cross-shore
Longshore
Vertical
Stokes
WEC related Diagnostics Terms
(i.e., contribution to momentum balance)
Terms in momentum balance
𝜕
𝐻 ⋅𝑢
𝜕𝑡 𝑧
𝜕
𝜕
𝜕
𝜕
𝐻𝑧 𝑢𝑢 +
𝐻𝑧 𝑣𝑢 + 𝑢
𝐻𝑧 𝑢 𝑆𝑡 + 𝑢
𝐻𝑧 𝑣 𝑆𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑥
𝜕𝑦
𝜕
𝜕
𝜔𝑠 𝑢 + 𝑢
𝜔𝑠𝑆𝑡
𝜕𝑠
𝜕𝑠
𝐻𝑧 ⋅ 𝑓 ⋅ 𝑣
𝐻𝑧 ⋅ 𝑓 ⋅ 𝑣 𝑠𝑡
𝜕𝜑𝑐
−𝐻𝑧 ⋅
|
𝜕𝑥 𝑧
𝜕𝑣 𝜕𝑢
𝐻𝑧 𝑣 𝑠𝑡
−
𝜕𝑥 𝜕𝑦
𝜔𝑠𝑆𝑡
𝜕𝑢
𝜕𝑠
Definition
Output
Variable
Local Acc.
u_accel/
ubar_accel
Horizontal
Advection
u_hadv/ubar_hadv
Vertical
Advection
u_vadv
Coriolis Force
u_cor/ubar_cor
Stokes-Coriolis
u_stcor/ubar_stcor
Pressure
Gradient
u_prsgrd/ubar_prs
grd
Vortex Force
u_hjvf/ubar_hjvf
Vortex Force
u_vjvf
Terms in momentum balance
Definition
Breaking + Roller
Acceleration +
Streaming
𝐻𝑧 ℱ 𝑤𝜉 (see Eqn. 21)
Output Var.
u_wbrk/ubar_wbrku_wrol/ub
ar_wrol
u_bstm/ubar_bstmu_sstm/ub
ar_sstm
BREAKING THE PRESSURE GRADIENT TERM
𝜑𝑐
=𝑔
𝜁𝑐
𝑠
− 𝜁 − 𝑔 − 𝒦 |𝜁 𝑐 +
𝜕𝜑 𝑐
−𝐻𝑧 ⋅
|
𝜕𝑥 𝑧
Pressure Gradient
u_prsgrd/ubar_prsgrd
Eulerian Contribution
ubar_zeta
Quasi-static response,
Eqn. 7
ubar_zetw
𝛻⊥ 𝒦|𝜁 𝑐
Bernoulli-head
contribution, Eqn. 5
ubar_zbeh
𝑔𝛻⊥ 𝒫|𝜁 𝑐
Surface pressure
boundary, Eqn. 9
ubar_zqsp
𝜍𝑐
−𝛻⊥
0
𝑔𝜌
− 𝐾 𝐻𝑧 𝑑𝑠 (𝐄𝐪𝐧. 𝟏𝟑, 𝟒𝟗, 𝐓𝐚𝐛𝐥𝐞 𝟐)
𝜌0
𝑔𝜁 𝑐
+
−ℎ
𝑔𝛻⊥ 𝜁
𝑔𝜌
𝑑𝑧
𝜌0
Vertical profile of terms in momentum balance
Cross-shore
Breaking Acc.
𝐵𝑏
Hor. Advection
𝜕 𝑢⋅𝑢
𝜕 𝑢 𝑆𝑡
+𝑢
𝜕𝑥
𝜕𝑥
Hor. VF
𝑢 𝑆𝑡
𝜕 𝑣
𝜕𝑥
Pressure Gradient
−
1 𝜕𝜑
𝜌 𝜕𝑥
Vertical Mixing
𝜕
𝜕𝑢
𝑢′ 𝑤 ′ − 𝑣
𝜕𝑠
𝜕𝑠
Vertical Advection
𝜕 𝑢 ⋅ 𝜔𝑠
𝜕 𝜔 𝑆𝑡
+𝑢
𝜕𝑥
𝜕𝑠
Alongshore
DUCK’ 94- Nearshore Experiment
DUCK’ 94, NC-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.
Wave Parameters, Depth-Averaged Flows
Hrms
Wave Height
Sea Surface Elevation
εb
η
Depth averaged
cross-shore velocity
u
v
Depth averaged
longshore velocity
Vertical profile of Cross-shore & Longshore Vel.
Cross-shore
Eulerian
Stokes Drift
Notes:
𝑢𝑠𝑡 = 𝑢
𝑢𝑠𝑡 (𝑧) ≠ 𝑢(𝑧)
Alongshore
Vertical Distribution
of Wave Dissipation
Kumar et al., 2012
Comparison to Field Observations
Cross-shore
Alongshore
Obs from Garcez-Faria et al., 1998, 2000
Kumar et al., 2012
Vertical profile of terms in momentum balance
Cross-shore
Breaking Acc.
𝐵𝑏
Alongshore
Notes:
Hor. Advection
Over the bar
(a) VF balances
Breaking
Hor. VF
(b) VM balances
Advection
𝜕 𝑢⋅𝑢
𝜕 𝑢 𝑆𝑡
+𝑢
𝜕𝑥
𝜕𝑥
𝜕 𝑣
𝑢 𝑆𝑡
𝜕𝑥
Pressure Gradient
−
1 𝜕𝜑
𝜌 𝜕𝑥
Vertical Mixing
𝜕
𝜕𝑢
𝑢′ 𝑤 ′ − 𝑣
𝜕𝑠
𝜕𝑠
Vertical Advection
𝜕 𝑢 ⋅ 𝜔𝑠
𝜕 𝜔 𝑆𝑡
+𝑢
𝜕𝑥
𝜕𝑠