cee473_week3notes (1)

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