Mechanisms for Water Exchange in Receiving Waters
Environmental Hydraulics
Mechanisms for Water Exchange
• wind • waves • tide • seiching
1 Wind
Velocity profile:
vz wz()= * ln κ z0
τo v* = ρair
(shear velocity)
Shear Velocity
Rewrite equation: vv wz( )=−** ln z ln z κκ0
Determine v* and zo from lin-log plot
Measured wind velocity profile, Great Lakes
2 Surface Shear Stress
2 τ=010CwD ρair air flow over water/land
2 τ=s CuD ρ water flow over bottom
Force balance:
ρ τ=τ =>uw =air =0.035 w so ρ 10 10
Water surface velocity 3-4% of wind speed
Wind-Induced Flow in Receiving Waters
30% of the water depth
Closed channel
Lake Kösen (11 km2)
Three layers shown
3 Geostrofic Circulation
Effects of Earth’s rotation (Coriolis force). Occurs in large water bodies. A current on the northern hemisphere turns to the right.
Lake Vänern
Water Exchange in a Fjord due to Wind
Byfjorden
Water velocity, water level, and wind velocity
4 Water Exchange in a Coastal Area: Öresund
Modes of flow in Öresund: • two layers; both to north • two layers; both to south • upper layer to north, lower layer to south
Öresund
stratification
Water Exchange in Öresund
Pattern for north- and south- directed flow
Lomma Bay
5 Waves - Induced Flows
• oscillatory flows • mean flows (longshore current, undertow, rip current) • turbulence (bottom boundary, breaking)
Oscillatory flow: not net flow (advection), little mixing
Types of Waves - Generation
* impulse (free wave) - tsunami * constant forcing (forced) - tide, wind
6 Wave Breaking
Duck, NC
Oarai Beach Japan
Breaking Waves
7 Incipient Wave Breaking
Wave Breaking Criteria
H 1 Deep water: o = 0.142 = Lo 7
H Shallow water: b = 0.78 db
⎛⎞Hd ⎛2π ⎞ Intermediate water: ⎜⎟= 0.142tanh ⎜ ⎟ ⎝⎠LLmax ⎝ ⎠
8 Wave-Induced Mean Flows
Wave-Induced Longshore Current
9 Wave-Current Interaction
Shinnecock Inlet
Wave Blocking (opposing current)
Limit of wave penetration (Waves blocked) Calm water Wave direction
Ebb current
10 Cross-Shore Currents
• mass transport • streaming (boundary layer) • undertow ( 0.08 − 0.10 gd )
(gives vertical structure to the coastal circulation)
Important for cross-shore Duck, NC sediment transport.
Tides
Wales
11 Tide-Induced Water Level Variations
Gravitational Force
Attraction force between two bodies: mm Ff= 12 g r 2
Individual water elements on Earth attracted by slightly different forces. Departure from mean net force => Newton tides
12 Schematized Distribution of Forces
fa: local attraction force fa-F yields tidal-generating force
Because of Earth’s rotation: two highs and two lows in a day (approximately)
Moon’s orbital motion => about 12 hr 25 min between highs
13 Types of Tide semi- diurnal
diurnal
mixed
Influence of the Sun
Sun has different mass and distance to Earth = > effect less than half the moon’s influence
14 Typical Tidal Curves
United States coast
Equilibrium Theory
Assume that water responds instantaneously to forces of sun and moon (no inertia and friction).
Tide can be predicted as a sum of harmonic terms:
N ht()=+ hoiiioii∑ fh cos() at + ( V + u ) −κ i=1
Compute coefficients from water level data.
15 Tidal Constituents and Arguments
Time and Phase Shift Effects
16 Dynamic Theory
Tidal motion is viewed as a forced wave.
Basin shape + Coriolis important. => Kelvin Waves Laplace
Global Tidal Variations
Microtides: < 2 m Mesotides: 2 – 4 m Macrotide: > 4 m
17 Tide Gage
ADCIRC Model
Simulates tidal motion + storm surge
Application to Shinnecock, Long Island
18 Seiching
Partial Wave Reflection
19 Partial Wave Reflection
Modes in Closed and Open-Ended Basins
20 Period of Oscillation: Closed Basin
1/2 gT2 ⎛ 2π d ⎞ ⎛⎞2π L L = tanh⎜ ⎟ T = ⎜⎟ 2π ⎝ L ⎠ ⎝⎠gdLtanh(2π / )
1/2 1st harmonic: ⎛⎞4π lB T = ⎜⎟ ⎝⎠gdltanh(π /B ) lLB = /2
1/2 jth harmonic: ⎛⎞4π lB Tj = ⎜⎟ ⎝⎠jgtanh(π jd / lB ) L lj== j1,2,... B 2
Shallow Water Bodies
⎛⎞ππjdjd 21lB tanh⎜⎟≈ Tj = ⎝⎠llBB j gd
Alternative: LgdT=
L lj== j1,2,... B 2
21 Example Page 2-116:
GIVEN: The Baltic Sea has a mean depth of 60 m and its length lB is 1200 km.
FIND: The fundamental frequency of oscillation Tj, if j=1.
Oscillations in the Baltic Sea
1200 km
22 Period of Oscillation: Open-Ended Basin
4l 1st harmonic: T = b gd L l = B 4
41l jth harmonic: T = B j 21j − gd L lj=−()21 B 4
23