NEARSHORE CURRENTS III (WAVEINDUCED CROSSSHORE CURRENTS)

WAVE-INDUCED CROSS-SHORE CURRENTSWave-Induced Cross-Shore Currents Cross-Shore Currents • mass transport • streaming boundary layer • undertow Important for cross-shore sediment transpor

(WAVE-INDUCED CROSS-SHORE CURRENTS)

Wave-Induced Cross-Shore Currents

Cross-Shore Currents

• mass transport

• streaming (boundary layer)

• undertow

Important for cross-shore sediment transport.

(gives vertical structure to the

coastal circulation)

Eulerian Lagrangian

2

8

gH

u

Cd

 

u

Stokes drift

cosh 2 8

L

A

C





L

u

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

  

0

4

s

uw

uw

 

 

  

 

 

at the bed

in the free stream

Stress Term in Boundary Layer

  1

' '

        

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.

xx

dS d gd



Depth-averaged equation:

(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

(vertical distribution of

radiation stress and

pressure gradient)

(vertical velocity distribution)

Measured Cross-Shore Current, Duck, NC

longshore bar

Undertow Flow

Mass conservation => undertow flow:

2

(drift roller)

drift

roller

CBH

q

d

A

q

T

  





(wave shape parameter)

0

1

/

T

T

  

Model by Rattanapitikon and Shibayama (2000).

dU dz

  

Estimate / t, integrate, and use U mas

the boundary condition.

2 / 3 1/ 3 4 5

B

t

k

k

D

d z

Shear stress distribution:

(based on Okayasu

et al 1988)

Integrate velocity:

1/ 3

1

2

B

m

         

Use bore model for energy dissipation:

1/ 3

3

1 0.21 ln 1

          

Coefficient values:

1

2

1 2

0.3 0.7

1.0

b

b

x x

b

x x

x x

b

x x

b b











 

Transition zone

Inner zone

(from Rattanapitikon and Shibayama, 2000)

Prediction of Mean Undertow Velocity

2

3 0.76 1.12

m

3

3

3

0

1/ 1/

1.0

b

b

b

b











wave drift roller

Comparison with Laboratory Data

(from Rattanapitikon and Shibayama, 2000)