WAVE-CURRENT INTERACTION Wave-Current Interaction Effects of a Steady Current on Waves • basic properties (H, L, T, C) • wave transformation (shoaling, refraction) • blocking • breaking Example: L shorter for opposing current and longer for following current; gradient in current field cause refraction and shoaling Waves can also affect a mean current Wave Kinematics Longshore Distance y Absolute phase speed: C ga Cgr Ca Cr U cos( ) U Relative phase speed: 1/ g Cr kd k W Cross-Shore Distance x Relative and absolute wave period: Tr L Cr Ta L Ca Dispersion relationship: d kd L gTa2 2 Blocking current E Opposing current Function Value Lo d U cos( )Ta d 1 Lo d L D A No current C B Following current Relative Water Depth (d/L) Wave Orthogonals and Wave Rays Wave energy conserved along wave rays: U sin( ) arctan U cos( ) C gr Absolute group speeed is tangent to wave ray: 1/ C ga Cgr2 U 2C grU cos( ) 2kd Cgr Cr 1 sinh 2kd Wave refraction (Snell’s law): sin 1 sin L1 L2 Wave breaking: Hb 0.88 kd tanh( b ) k 0.88 (steepness- and depth-limited breaking) Wave Blocking (opposing current) Limit of wave penetration (Waves blocked) Calm water Wave direction Ebb current Criterion for wave blocking: gk kd 2 kU cos( ) Ta At blocking: C gr U s cos( ) d d kd L n Lo U s cos( ) n L d kd gTa h Lo Deep water: L Lo U s cos( ) Co Shallow water: 1/ d d L 32 Lo 1/ U s cos( ) d gTa 2 Lo ( Uscos(-)=(gd)1/2 ) Blocking Speed vs Relative Water Depth 1E-1 Blocking Speed (U cos( )/gTa) deepwater asymptot for blocking 1E-2 shallow water asymptot for blocking 1E-3 1E-4 1E-7 1E-6 1E-5 1E-4 1E-3 1E-2 1E-1 1E+0 Relative Water Depth (d/Lo) Longshore Current Model with Wave-Current Interaction • conservation of wave action (E/ r) • roller model • conservation of momentum in xdirection (wave setup/setdown) • conservation of momentum in ydirection (longshore current) Iteration required between the waves and mean current Wave action flux conservation d ECga cos PD Pf dx r r PD ( E Es )Cgr dD (neglect bottom friction) Es gH s2 Hs Hb b (stable wave height) Roller Model From Dally and Osiecki (1994): PD d 1 A A 2 R C cos R g D dx T T Define: mR=rA/T PD d 1 2 mR Cr cos g D mR dx Contributions in momentum equations: M Rx mR Cr cos2 cross-shore M Ry mRCr sin cos longshore Cross-Shore Momentum Equation gd d dS xx CD a W W cos M Rx dx dx C gr 1 S xx gH (cos 1) 2 Cr Alongshore Momentum Equation d dV d dx dx dS xy CD a W W sin M Ry f by dx (bottom friction described as before) S xy C gr gH sin 2 16 Cr Wave Model Validation: CHL Experiment Wave propagating on opposing current (inlet case) Waves on Opposing Current waves Hmo=0.055 m Tp=0.71 s Uc=0 m/s Uc=0.12 m/s current Uc=0.32 m/s Uc=0.24 m/s CHL Data 0.07 Significant Wave Height (m) 0.06 0.05 0.04 calc with current 0.06 0.05 calc with current measured 0.04 calc without current meas ured calc without current 0.03 0.03 12 Cross-Shore Distance (m) 12 Cross-Shore Distance (m) Run 7: Hmo=0.055 m, Tp=1.4 s, Uc=0.24 m/s Run 11: Hmo=0.055 m, Tp=0.7 s, Uc=0.24 m/s Wave Transformation on a Current (opposing and following) 2.4 Current Speed (m/s) 0.0 Root-Mean-Square Wave Height (m) Significant Wave Height (m) 0.07 1.0 2.0 3.0 -1.0 -3.0 1.6 1.2 0.8 0.4 100 200 300 Distance Offshore (m) Current Model Validation: Visser Data Case 0.06 0.5 0.4 meas 0.04 calc with roller calc with roller calc without roller calc without roller Wave Setup (m) Longshore Current (m/s) meas 0.3 0.2 0.02 0.00 0.1 0.0 -0.02 -1 -1 Cross-Shore Distance (m) Cross-Shore Distance (m) Longshore current Wave setup/setdown Current Model Validation: Visser Data Case 0.5 0.020 0.4 meas 0.015 calc with roller calc with roller calc with out roller calc without roller Wave Setup (m) Longshore Current (m/s) meas 0.3 0.2 0.1 0.010 0.005 0.000 0.0 -0.005 -2 Cross-Shore Distance (m) Longshore current 10 -2 10 Cross-Shore Distance (m) Wave setup/setdown Current Model Validation: LSTF Experiment LSTF Data on Longshore Current 0.4 0.5 meas meas calc with roller 0.4 calc with roller calc without roller calc without roller Longshore Current (m/s) Longshore Current (m/s) 0.3 0.3 0.2 0.1 0.2 0.1 0.0 -0.1 0.0 10 15 20 Cross-Shore Distance (m) Monochromatic waves: Href=0.182 m, T=2.5 s, ref=10 deg 25 10 15 20 25 Cross-Shore Distance (m) Random waves: Hmo=0.225 m, T=2.5 s, ref=10 deg ... 1E-4 1E-3 1E-2 1E-1 1E+0 Relative Water Depth (d/Lo) Longshore Current Model with Wave-Current Interaction • conservation of wave action (E/ r) • roller model • conservation of momentum in xdirection