cee473_week3notes (1)
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CEE 473 (Thomson): Lecture Notes, Week 3 1
Surf zone wave breaking
Surf similarity: ζ0 = tan(β)(H0
L0
)− 12
= dx
(H0
L0
)− 12
Breaker types:
spilling, ζ0 < 0.5, steep waves, flat beach
plunging, 0.5 < ζ0 < 3.3, steep waves, steep beach
collapsing and surging, ζ0 > 3.3, flat waves, steep beach
Breaker depth: γb = Hbdb
(= 0.8 for solitary KdV waves)... constant shoreward of break (‘saturated’)
Shallow limit of more general Miche criterion H < 0.14L tanh(kd)
Can adjust for beach slope γb = Hbdb
= const+ βkd
Breaker height: Ωb = HbH0
(= 0.56(H0
L0
)− 15
from Komar and Gaughan, assuming γ0 ∼ 1)
Dissipation:dEcgdx = −δ
Battjes and Janssen [1978] predict δ = 14ρgQbfpH
2
Nonlinearity: harmonics, steepness
Infragravity waves
Low frequency waves, 30 < T < 200 s: nonlinear, resonant in shallow water
Types: bound, leaky, edge
Control morphology (e.g., beach cusps) and runup
Setup
Increase in water elevation d = h+ η, where h is depth for still water and η is the wave-averaged adjustment.
The slope of setup or set down η balances gradients of wave momentum flux: dηdx = − 1
ρgddSxxdx
Radiation ‘stress’: Sxx = E(
2kdsinh 2kd + 1
2
), Syy = E kd
sinh 2kd shallow limit: Sxx = 316ρgH
2 = 32E
Offshore (up to) break point, set-down: η = − 18
H22π/Lsinh(4πd/L)
Onshore of break point (saturated surf zone with linear decrease in H), set-up: dηdx = 1
1+ 8
3γ2b
tanβ
At the shoreline (still water level): ηs = ηb + 11+ 8
3γ2b
hb
Runup & Swash
Breaking waves, R/H0 = ζ0 for 0.1 < ζ0 < 2.3
Non-breaking waves, R/H0 = (2π)1/2(π/2β)1/4
Irregular waves, Rmax/H0 = 2.31ζ0.770
CEE 473 (Thomson): Lecture Notes, Week 3 2
Nearshore currents
Depth-averaged momentum balance:
U ∂U∂x + V ∂U
∂y = −g ∂η∂x + Fbx + Lx +Rbx +Rsx, U ∂V∂x + V ∂V
∂y = −g ∂η∂y + Fby + Ly +Rby +Rsy
Continuity: ∂(Ud)∂x + ∂(V d)
∂y = 0
Longshore current, analytic solution [Longuet-Higgins, 1970]: V (x) = 5π16
tan β∗Cf
γb√gd sinα cosα
Derived from Rby = − 1ρd
∂Sxy∂x , Sxy = 1
8cgc ρgH
2 sinα cosα, with optional lateral mixing
where tanβ? = tan β1+3γ2
b/8
to include the effects of set up and Cf 0.005
Longshore current, empirical solution [Komar & Inman, 1970]: V (x = midsurfzone) = 1.17√gHrms,b sinαb cosαb
Cross-shore currents: ‘undertow’ is the mild/diffusive return flow from wave transport and setup,
u ≈ − gd16c
(Hsd
)2cosα [Lentz et al, 2008]
Cross-shore currents: rip currents are the intense/narrow jets caused by variations in setup, ∂η∂y
strength is function of wave energy and water depth, control ratios are H/d and dch/dbar
carry sediment offshore, carve out channels and/or indentations in the coast...
possible mechanism for beach cusps
typically max inside surf zone, but extend offshore few multiples of the surf-zone width
feeder currents → narrow jet → offshore head
unsteady, unstable