NEARSHORE CURRENTS III (WAVEINDUCED CROSSSHORE CURRENTS)

NEARSHORE CURRENTS III (WAVE-INDUCED CROSS-SHORE CURRENTS) Wave-Induced Cross-Shore Currents Cross-Shore Currents • mass transport • streaming (boundary layer) • undertow (gives vertical structure to the coastal circulation) Important for cross-shore sediment transport Net Flow Velocity in Waves Stokes drift uL u Eulerian Lagrangian u  gH 8Cd uL   A cosh 2kz 8C Boundary Layer Drift (streaming) Real waves (non-breaking): shoreward velocity along the bottom because of boundary layer effects (Longuet-Higgins 1953) Velocity components u and w not 90 deg out of phase in boundary layer due to viscous effects ”Reynold’s stress” term appear  uw Stress Term in Boundary Layer  0 uw at the bed  A   uw ks in the free stream Equation of Motion in Boundary Layer   u  p    u 'w'   uw  z  z   x z   Analytical solution exist for laminar flow and no pressure gradient Streaming is typically not observed in the surf zone (breaking waves) Instead undertow dominates the flow Undertow is related to wave setup Radiation stress gradient is not uniform over the depth, but the opposing pressure gradient almost is Depth-averaged equation: gd d dS   xx dx dx (wave setup/setdown) Cross-Shore Circulation Flow pattern: Onshore mass transport above trough level Offshore flow below the trough (undertow) Undertow current  0.08  0.010 gd (near bottom) Velocity profile determined by: • radiation stress • pressure gradient (sloping water surface) • vertical mixing Undertow (vertical velocity distribution) (vertical distribution of radiation stress and pressure gradient) Measured Cross-Shore Current, Duck, NC longshore bar Undertow Flow Mass conservation => undertow flow: q  ( qdrift  qroller ) CBH d A  T qdrift  qroller T B   / H  dt T 0 (wave shape parameter) Undertow Velocity Distribution Model by Rattanapitikon and Shibayama (2000) Eddy viscosity model:   t dU dz Estimate / t, integrate, and use Um as the boundary condition Shear stress distribution:  k k    2/ D1/B    t z  d (based on Okayasu et al 1988) Integrate velocity: 1/ D    z 1   z   U   B   k4     k5  ln    1   U m     d 2   d   Use bore model for energy dissipation: 1/  gH    z    z   U  b1    b2     0.21 ln  d   1   U m      4Td    d  Coefficient values: b1  0.3  0.7 b2  xb  x xb  xt Transition zone xb  x xb  xt b1  b2  1.0 Inner zone Comparison with Laboratory Data (from Rattanapitikon and Shibayama, 2000) Prediction of Mean Undertow Velocity U m  0.76 BgH BCH  1.12b3 Cd d wave drift roller b3  b3  1/ H  1/ H b 1/ H t 1/ H b b3  1.0 Comparison with Laboratory Data (from Rattanapitikon and Shibayama, 2000)