Water Motion Under Waves vs. Water Depth
DEEP WATER
Wave Direction
Wave Base
Water Particle
Net Movement
SHALLOWER WATER
Nielsen and Nielsen, 1978
Longshore Current
1
Longshore Currents & Oblique Breaking Waves

Longshore component of wave forcing =
where
S xy  En sin  cos  
Frictional drag on flow =
(1) = (2)
vl 
Ry 
1 S xy
d x
(1)
n
gH 2 cos  sin 
8
2
C f um vl

(2)
5 tan *
 b gd sin  cos 
16 C f
vl ,mid  2.7um sin  b cos b
αb
vl
um 
Hb
2
g
hb
(Fig. 2-6)
(Longuet-Higgins 1970) (Komar and Inman 1970)
Nearshore Circulation System – Rip Currents
Cell Circulation
Horisontal Structure
Vertical Structure
2
Nearshore Circulation System – Rip Currents
Rip Currents on Shallow vs. Steep Beach Slopes
Radiation Stress
Radiation stress = ”The excess flow of momentum due to
the presence of the waves” (Longuet-Higgins & Stewart 1960)
Fig. 3-49 alt.

S xx 
 ( p  u
d
2
0
 2kd
1
1

) dz   p0 dz  E 
   E  2n  
2  (3-76 alt.)

 sinh(2kd ) 2 
d
L5-2011
3
Undertow
Undertow is related to wave setup.
Radiation stress gradient is not uniform over the depth,
but the opposing pressure gradient almost is.
Sxx ~ u2
(vertical distribution of radiation
stress and pressure gradient)
Depth-averaged equation:
(wave setup/setdown)
(vertical velocity
distribution)

dS xx
d
 gd
dx
dx
Rip Currents & Cell Circulation
Shallow water
1 3
3

S xx  E  2n    E  gH 2
2 2
16

(3-76 alt.)
Momentum balance in cross-shore direction:
S xx
d
 g (   d )
0
x
x
S xx
x
balanced by
(3-75 alt.)
d
x
Komar, 1976
!
Rip Currents & Cell Circulation
h
H
5
Komar, 1976
4
Edge Waves & Cell Circulation
H
Komar, 1976
H
Water Level Variations
Typical Periods
5
Water Level Variations
• astronomical tides
• tsunamis
• seiches
• wave setup
• wind setup
• storm surge
• climatological variations
Astronomical Tides
Right click to stop animation
Left click to re-start animation
Tide-Induced Water Level Variations
Sun has different mass and distance to Earth = >
effect less than half the moon’s influence
6
Gravitational Force
Attraction force between two bodies:
Fg  f
m1m2
r2
Individual water elements on Earth
attracted by slightly different forces.
Departure from mean net force =>
tides
Newton
Astronomical Tides
Typical Tidal Curves
Semi di-urnal
Di-urnal
7
Global Tidal Variations
Microtides: < 2 m
Mesotides: 2 – 4 m
Macrotide: > 4 m
Low Tide @ Coast of Wales
Tide Gage
8
ADCIRC Model
Simulates tidal
motion + storm surge
Application to Shinnecock,
Long Island
5-10
Wave Shoaling
Wind Waves
deep water
v = 12 m/s
Tsunami
shallow water
v = 4 m/s
d= 2000 m
v = 700 km/h
H=1m
L = 200 km
d= 10 m
v = 36 km/h
H = 13 m
L = 10 km
3 stages:
• Generation
• Propagation
• Inundation
2004 Tsunami in Thailand
9
2011 Tsunami in JAPAN
Seiching
Wave Setdown/Setup
Wave setup = superelevation of mean water level caused by wave action
CEM (in-set after p. 3-98):
b   b2 d b / 16   H b2 / d b / 16
(3-77 alt.)
b  H b / db
(3-80 alt.)
d s ,b  d b  b
1

8 
s  b   1  2  d s ,b
 3 b 
db
Fig. 3-50 alt.
x 
s
d
x
, max  s 
tan   d  / dx
dx
(3-81 alt.)
10
Wind Setup and Storm Surge
Primary factors:
• characteristics of storm
• hydrography of basin
• initial state of system
Other factors:
Astronomical tides, atmospheric pressure
differences, earth’s rotation, rainfall, surface
waves, storm motion effects
surge
Hurricane Isabel
Sept. 18, 2003
wave height
Hurricane Katrina 2005 - The Landfall
11
Katrina – Storm Surge
LA
MS
Main Items
• statistical properties of waves
• determine wind input for wave predictions
• calculate wind-generated waves
• understand various mechanisms for water level variations
5.5-09
Implications of Wave Run-up
Structure Run-up/Overtopping
Beach/Dune Processes
12
Wave Runup & Overtopping
Delft Report by Van Der Meer and Janssen replaces
pages 7.16-7.99.
Ru2%= runup level exceeded by 2% of the waves
Wave Runup & Overtopping
revetment
design water level
Wave Runup Height
Ru 2%  1.6
  h  f    b op
 
3.2
Hs
   h  f 
op 
tan 
Sop
 b op  2
 b op  2
Sop  H s / Lo
13
Wave Runup Height
Ru 2%  1.6
  h  f    b op  1.6  tot op
 
Hs
 3.2
  h  f    3.2  tot /  b
h
f

b
 b op  2
 b op  2
= reduction factor for shallow foreshore
= reduction factor for slope roughness
= reduction factor for wave angle
= reduction factor for berm
 tot   h  f    b , max  0.5!
op 
tan 
Sop
Sop  H s / Lo
Reduction for Slope Roughness  f
for op  3.5, else  f  1.0
Reduction for Shallow Foreshore  h
2

h 
 h  1  0.03  4  m  ,
H
s 

hm
4
Hs
 h  1,
hm
4
Hs
14
Influence of Berm  b
Width factor:
rb  1  tan  eq / tan  ,
Location factor:
dh  2H s
rdh  0.5  d h / H s  , 0  rdh  1
2
 b  1  rb 1  rdh  , 0.6   b  1.0
Reduction factor:
Reduction for Wave Angle  
   1  0.0022
runup
   1  0.0033
overtopping
( in deg)
Total Reduction
 tot   b  h  f   ,
 tot
if  tot  0.5   tot  0.5
Example:
GIVEN: Waves with a height of Hs=1.5 m and T=5 s
approaches the revetment depicted below at a
normal angle.
FIND: The runup height Ru2%.
+1.0
3.0
5.0
SWL
15
General relationship:
Ru 2%
 1.6
  h  f    b op
 max  
3.2
Hs
   h  f 
Reduction for Slope Roughness  f
 b op  2
 b op  2
 1.0
Reduction for Shallow Foreshore
h
+1.0
3.0
5.0
SWL
hm 5.0

 3.33  4
H s 1.5
2

h 
 h  1  0.03  4  m   0.987
Hs 

Influence of Berm  b
+1.0
3.0
5.0
SWL
Width factor:
d h  1.0  2 H s  2.1 
tan 
rb  1  tan  eq / tan 
tan  eq
16
Slopes:
Hs
tan   3 / 8.5
Hs
Width factor: rb
 1  tan  eq / tan   1  8.5 / 10.5  2 / 10.5
Hs
Hs
tan  eq  3 /10.5
Influence of Berm  b
+1.0
3.0
5.0
SWL
Width factor: rb
 1  tan  eq / tan   1  8.5 / 10.5  2 / 10.5
Location factor:
rdh  0.5  d h / H s   0.5 1 / 1.5   0.22, 0  rdh  1
Reduction factor:  b
2
 1  rb 1  rdh   1  2 /10.5 1  0.22   0.85, 0.6   b  1.0
Reduction for Wave Angle
Total Reduction
2
  = 1.0 (normal incidence)
 tot
 tot   b  h  f    0.85  0.987 1.0 1.0  0.84
(if  tot  0.5   tot  0.5 OK)
17
Wave Runup Height
T  5sec  Lo  1.56T 2  39 m 
Sop  H s / Lo  1.5 / 39  0.038 
op 
tan 
0.35

 1.78
Sop
0.038
1.6
  h  f   b op
 
3.2
  h  f 
op 
 b op  0.85 1.78  1.52  2 
 b op  2
 b op  2
tan 
Sop
Ru 2%
 1.6  h  f    b op  1.6  tot op  1.6  0.84 1.78  2.39 
Hs
Ru 2%  2.39 H s  3.59 m
Overtopping Discharge
Breaking waves,
q

op
Qb
Sop / tan 
 2
gH 3 (m3/m/s)
where:
Qb  0.06 exp  4.7 Rb 
Rb 
Rc Sop 1
H s tan   tot
SWL
Rc
h
Overtopping Discharge
Non-breaking waves,

op
 2
q  Qn gH 3
where:
Qn  0.2 exp  2.3Rn 
Rn 
Rc 1
H s  tot
SWL
Rc
h
18
Recommended Overtopping Standards
(backslope conditions)
• 0.1 l/s per m for sandy soils with a poor vegetation
• 1.0 l/s per m for clayey soil with good grass
• 10 l/s per m for a revetment construction
Correction for Wind (SPM)
Qc  Qk '
 h  ds

k '  1.0  W f 
 0.1 sin 
 R

W f  2.0
U on  30m / s
0.5
 15m / s
0.0
 0m / s
(7-12)
h = structure crest height
ds = depth in front of structure
Main Items
• use Delft method to calculate run-up & overtopping
19
THE END
L6-2011
20