Lecture 4 Estuarine Salinity Structure Vertical

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Lecture 4 Estuarine Salinity Structure
(Vertical)

• Outline
• Review analytical solution for estuarine
  circulation (gravitational circulation), add
  river velocity.
• Derivation of conservation of salinity equation
• Simplified estuarine salt balance
• Derivation of analytical salinity profile.

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Analytic Solution for Residual Circulation
Scale for Residual Circulation
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Scale for Residual Circulation
But what is the appropriate value for Az?
Proposed scaling for Az AoCDUT_rmsH
However, This does not account for impact of
stratification on mixing!!!
See Geyer et al. JPO (2000) and MacCready JPO
(2007)
Ao 0.0325 CD 0.0025
Also see Scully et al. JPO (2008)
Chesapeake Bay ?S 30 Lx 200 km h 15
m UT 0.50 m/s CD 0.0025 ß 7.8 10-4
UE 0.10 m/s
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Derivation of Estuarine Circulation with River
Flow
x-momentum
Integrate twice in z
Apply boundary conditions 1) no stress at
surface 2) no flow at bottom (no slip condition)
This gives
Plug in C1 and C2
Solve for d?/dx, using continuity
Where uriver Q/A
River velocity
Gives
Estuarine velocity
Plug in expression for d?/dx
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Derivation of Estuarine Circulation with River
Flow
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Conservation of Salt Equation
Time rate of change
Advection
Molecular Diffusion
Where ?s 1.510-9 m2s-1
To get equation for mean salinity evolution, use
Reynolds decomposition
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Reynolds Decomposition
Rate of change term
Advection (x-direction)
Advection (y-direction)
Advection (z-direction)
Molecular Diffusion (x-direction)
Molecular Diffusion (y-direction)
Molecular Diffusion (z-direction)
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Reynolds Decomposition
Rate of change term
Advection (x-direction)
Turbulent Flux Terms
Advection (y-direction)
Advection (z-direction)
Molecular Diffusion (x-direction)
Molecular Diffusion (y-direction)
Molecular Diffusion (z-direction)
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Reynolds Decomposition
Rate of change term
Advection (x-direction)
Turbulent Flux Terms
Advection (y-direction)
Advection (z-direction)
Molecular Diffusion (x-direction)
Molecular Diffusion (y-direction)
Molecular Diffusion (z-direction)
Molecular diffusion X-direction
Turbulent diffusion x-direction
If S U
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Reynolds Decomposition
Rate of change term
Advection (x-direction)
Turbulent Flux Terms
Advection (y-direction)
Advection (z-direction)
Molecular Diffusion (x-direction)
Molecular Diffusion (y-direction)
Molecular Diffusion (z-direction)
Molecular Diffusion (z-direction)
11 orders of magnitude bigger!!
Molecular diffusion X-direction
Turbulent diffusion x-direction
If S U
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Turbulent Flux Terms
Continuity
Add Continuity to Flux Terms
Remember
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Reynolds-averaged Conservation of Salinity
Equation
Turbulent fluctuations prevent closure, so we
use eddy diffusivities (analogous to eddy
viscosity)
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We are now going to use the Reynolds-averaging
technique on longer time scales.
Tidally varying, but vertically averaged
Tidally and vertically averaged
Tidally averaged, but vertically varying
What is salt balance at subtidal time scales?
Flux associated with mean river discharge
Flux due to gravitational circulation
Tidal Pumping Flux
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What is the Salt Flux due to Gravitational
Circulation?
Assume lateral homogeneity and ignore vertical
advection
Assume salt balance is steady at sub-tidal time
scales
Scaling of Horizontal Dispersion vs. Vertical
Mixing
Generally KxgtKz but (Lx)-2 ltltlt H-2
1st order balance
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From previous lecture
Assume Kz Az
Integrate once in z
Boundary Condition No flux through surface
C1 0
Integrate again in z
Boundary Condition Vertical Integral of SE0
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Solutions assuming H 10 m and dS/dx 410-4
psu/m