Lecture 8 Stratified Boundary Layers

1
Lecture 8 Stratified Boundary Layers

• Outline
• Review Homework Problem
• Introduction to Stratification
• The gradient Richardson number
• Review of the TKE equation
• Potential Energy and Mixing
• Impact of stratification of turbulent length
  scales (Ozmidov Scale and Monin-Obukov Scale)
• Modification of boundary layer scaling to account
  for stratification.

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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.

3
Solutions Assuming Log-Layer Eddy Viscosity
Residual Velocity Profile
Residual Salinity Profile
Log-Layer Eddy Viscosity Profile
Mean Az 0.025 m2/s
Usurf 0.01m/s
?S 0.006
Depth (m)
psu
Az (m2/s)
m/s
Observations from Hudson Usurf 0.30 and ?S
5.5
Over 30 times smaller!!
4
What is the length of the Hudson Salt Intrusion?
W 1.2 km a 1.3 10-5 g 9.8 m/s2 ß
7.810-4 So 25 psu H 15 m Qr 150 m3/s Az
7.210-4 m2/s
5
Must account for the impact of density
stratification on turbulent mixing!
Lighter less dense water
Measure of strength of stratification
Brunt-Vaisala Frequency N
Depth (m)
Heavier more dense water
Density (?) kg/m3
6
A dominant mechanism for mixing in estuaries is
through shear instability
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Growth of a small instability on sheared density
interface
TIME
Kelvin-Helmholtz roll ups or billows
Collapse into turbulence mixing
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Ebb Tide in the Connecticut River
ebb
Clouds over Montana
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Gradient Richardson Number
Stabilizing effect of mean stratification
Mixing by mean shear
0 lt Ri lt 0.25
Ri lt 0
Ri gt 0.25
Flow is unstable, convective mixing likely.
Flow is stable, but mixing due to shear
instability is possible/likely.
Flow is stable, small instabilities will not grow
into turbulence
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Turbulent Kinetic Energy Equation
1) Separate equation of motion into total and
mean component. 2) Subtract mean component from
total to get equation of motion for turbulent
fluctuations. 3) Multiply equation of motion for
turbulent fluctuations by u 4) After a lot of
algebra you get
pressure
turbulent
viscous
Total derivative of TKE
Shear Production
Buoyant Production
Viscous Dissipation
Transport terms
Definitions
In this notation commas denote spatial
derivatives (see Kundu)
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Simplified form of TKE Equation
Gain of turbulence by shear instability
Production/Destruction of TKE by buoyancy
Time rate of change of TKE
dissipation
Energy from the mean flow (i.e mean shear) is
converted into turbulent motion. So there is a
loss of mean kinetic energy and a gain in
turbulent kinetic energy.
Stable conditions
Turbulent kinetic energy is lost in the mixing
process. The resulting buoyancy flux increases
the potential energy of the mean flow at the
expense of the turbulent kinetic energy.
Unstable conditions
Turbulent kinetic energy is gained due to
convection. The potential energy of the mean
flow is reduced and that energy is
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For unstratified, steady, homogenous flow
Shear Production Turbulent Dissipation
What is production for a log-layer
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After mixing
Depth (m)
Before mixing
Density (?) kg/m3
Turbulent Kinetic Energy was used to mix the
water column. The overall potential energy of
was increased ( you raised some mass of salt
upward ).
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Lock Exchange Problem
More dense
Less dense
Ignoring turbulence, mean potential energy is
converted to mean kinetic energy.
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Potential Energy and Mixing PE ?gz
z
Stratified
?h
z h-?h/2
h
??
z (h-?h)/2
?
?o
z
Well-Mixed
z h/2
?
?m
Increase in potential energy (per unit area)
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Shear Production increases the kinetic energy of
turbulence
Some of that TKE is lost to buoyancy flux
?w gt 0
Buoyancy flux
Kinetic energy is transferred from mean flow into
turbulence.
Mixing increases the potential energy of the mean
flow.
Efficiency at which kinetic energy is transferred
to potential energy by turbulent mixing
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How Does Stratification Effect Turbulent Length
Scale?
How much kinetic energy is needed to lift a
parcel of water some distance against the density
gradient?
Alternatively, what is the farthest distance a
given amount of kinetic energy can displace a
parcel of water given some background
stratification
?z ?
More formally, the Ozmidov scale is defined as
the length scale of turbulence where inertial and
buoyant forces are equal.
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Example of Length Scale Limitation from Merrimack
River
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Derivation of log-layer scaling assumes the only
relevant length scale is the distance from the
boundary.
The presence of stratification changes this
From dimensional analysis Monin and Obukov (1954)
defined the following length scale for a
stratified boundary
Log-layer shear is now modified by the function ?M
where a is an empirical constant 5
Integration now results in
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Gradient and Flux Richardson Numbers are Related
Simplified Approaches to Turbulence Closure
Munk and Anderson (1948)