Lecture 7a Finish SubTidal Salt Flux Calculations

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Lecture 7a Finish Sub-Tidal Salt Flux
Calculations Lecture 7b Introduction to
Estuarine Boundary Layers

• Outline
• Review Hansen Rattray classification
• Scaling for Length of Salt Intrusion
• Introduction to unstratified boundary layers
• Log-layer scaling
• Eddy-viscosity for a log-layer
• Drag Coefficients

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Cross-sectional and tidal averaged salt balance
Rate of change in mass of salt landward of
cross-section
Down-estuary salt flux due to river discharge.
Up-estuary salt flux (all mechanisms)
Tidal pumping gravitational circulation
oscillatory shear dispersion
Ratio of tidal pumping to total salt flux
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Hudson River Qriver 150 m3/s ltugt 0.01 m/s
Residual Velocity
Residual Salinity
Usurf 0.3 m/s
?S 5
width (km)
width (km)
Chesapeake Bay Qriver 2000 m3/s ltugt 0.01 m/s
Residual Velocity
Residual Salinity
?S 6
Usurf 0.15 m/s
width (km)
width (km)
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Hansen and Rattray Estuarine Classification
Hudson River
Chesapeake Bay
Gravitational circulation
0.99
Tidal Pumping
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For Chesapeake Bay and Hudson River it is
probably reasonable to assume gravitational
circulation dominates up-estuary salt flux.
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Data from San Francisco Bay Monismith et al. JPO
(2002)
LQ-1/7
LQ-1/3
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Estuarine Boundary Layers
For steady, fully developed turbulent flow (with
no rotation)
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For fully developed, stead flow, depth integrated
P.G. is balanced by bed stress.
tb
Viscous sublayer thickness
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From dimensional analysis, the velocity shear
(du/dz) must depend on a velocity scale and a
length scale
The only relevant velocity scale is the shear
velocity (u)
Assumed constant with z, so often referred to as
a constant stress layer
The only relevant length scale is the distance
from the boundary
von Karmans constant 0.41 derived from
numerous empirical experiments.
Integrate in z, to get velocity distribution
Boundary Condition U 0, _at_ z zo
Law of the wall
hrough
Roughness height
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Mixing Length Model
Momentum flux (stress) eddy viscosity
velocity shear
Eddy coefficient is represented as a turbulent
length scale turbulent velocity scale
So, for constant stress layer
Alternatively, if you assume velocity scale is
set by local value of stress
u
L
hbbl
Eddy Viscosity
stress
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Scaling for oscillatory turbulent boundary layer
For unstratified flow Az 0.01 m2/s
For tidal boundary layer T 12.42 hours ?
10-4 s-1
Hbbl 8 m
But what is appropriate value for Az ?
Log-layer scaling
Bed Stress can be represented using a quadratic
drag law
So if
Zo 0.5mm
From log-scaling
CD 0.003
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Homework Problem
Assuming the following

• What is the tidally averaged value of the eddy
  viscosity?
• Plot it as a function of z.
• If dS/dx 410-4 m-1 , what is the predicted
  strength of the residual estuarine circulation
  (Usurf) and stratification (?S) based on your
  estimate of Az.
• How does this compare with the data shown on
  slide 3?
• What value of Az best matches these values in the
  Hudson.
• Using this value, calculate the estimated length
  of the Hudson River salt intrusion if the river
  discharge is 150 m3/s.