Lecture 12 Lateral Tidal Dynamics

1
Lecture 12 Lateral Tidal Dynamics

• Outline
• Layered Model for Lateral Dynamics
• Differential Advection
• Curvature
• Implications for Along-Channel Momentum Balance

2
Ekman Transport is to the left of the overlying
along-channel flow
z
??
?
?o
Ekman Transport
Looking up-estuary, flow into estuary (flood)
3
Layered (Reduced Gravity) Models
See Cushman-Roisin Intro to GFD for more info.

• For stable stratification, density increase
  monotonically with depth.
• If density is conserved by individual fluid
  parcels, you can transform the equations from a
  coordinate system based on z to a coordinate
  system based on density.
• Define reduced gravity (g) as

?
?z
??
? ??
?x
Baroclinic Pressure Gradient can represented as
the slope of the density interface multiplied by
the reduced gravity
4
Simple Reduced Gravity Model for Tidal, Lateral
Dynamics
What is the 1st order x-momentum balance in
surface layer?
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Simple Reduced Gravity Model for Tidal, Lateral
Dynamics
What is the 1st order x-momentum balance in
surface layer?
Assume that both u1 and surface slope have
periodic solutions
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What is the 1st order x-momentum balance in
bottom layer?
R is linear drag coefficient 110-4
Again, look for periodic solutions
Bottom drag reduces lower layer current and
reduces phase relative to surface layer.
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What is the 1st order lateral momentum balance?
Surface layer
Bottom layer
From previous slide
From previous slide
Plug in expressions for u2 and d?/dy
Integrate in y
Continuity Equation
Plug in expressions for ?o and integrate in y.
Apply BCs that v 0 at y0 and yL gives
solutions for V and ?..
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Simple Reduced Gravity Model for Tidal, Lateral
Dynamics
Governing Equations
Solution
Upper Layer
Lower Layer
x-momentum
y-momentum
continuity
where g g????
assume periodic solutions (u,v,?,? ei?t)
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Observations from the York River
a) Depth-integrated Lateral Momentum Balance
1.0
coriolis
P.G.
0.5
0
Pa
-0.5
-1.0
Dec-7
Dec-14
Dec-21
Dec-28
Jan-04
Jan-11
b) Depth-integ. Momentum Balance
c) Tidal cycle momentum balance
0.4
0.6
r 0.75
coriolis
baroclinic P.G.
0.4
0.2
0.2
0
0
Pa
Pressure Gradient (Pa)
-0.2
-0.2
barotropic P.G.
-0.4
-0.4
-0.6
0
0.4
0.2
-0.2
-0.4
-0.6
0.6
EBB
FLOOD
Coriolis (Pa)
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Comparison of Momentum Balance--Model vs.
Observations
Phase Averaged
Lateral Momentum Balance from Observations
0.4
barotropic P.G.
0.2
coriolis
0
Pa
baroclinic P.G.
-0.2
-0.4
EBB
FLOOD
Lateral Momentum Balance from Lower Layer of
Reduced Gravity Model
0.4
barotropic P.G.
coriolis
0.2
0
Pa
baroclinic P.G.
-0.2
-0.4
EBB
FLOOD
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Phase Averaged Salinity Data from the York River
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Reduced Gravity (Layered) Models Assumes Invisid
Surface Layer
What are Lateral Dynamics Under Well-Mixed
Conditions
Tidal Differential Advection
Lateral Circulation
Fresher
Fresher
Saltier
Flood
(Nunes Simpson, 1985)
Ebb
(from Smith, 1996)
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Differential Advection Leads to an Axial
Convergence Front
(Nunes Simpson, 1985)
Assumed balance
Sideways estuarine circulation
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Channel Curvature
Along-Channel Flow
Outer Bank
Inner Bank
Tilted Surface
Secondary Flow
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Forcing of Secondary Flow by Channel Curvature
Using curvilinear coordinate system
Stream-wise
R radius of curvature
Stream-normal
advection of secondary flow
coriolis
acceleration
friction
Pressure Gradient
centrifugal
(us,un) along-channel, cross-channel
currents (s,n) along-channel, cross-channel
directions R radius of curvature (positive
if outer bank is in positive
y-direction)
s
n
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Simple Steady Stream-wise Momentum
Balance Constant eddy viscosity, Az
Integrate twice and apply BCs us 0 at z -H
and dus/dz 0 at z 0
Depth
us(z)
us(z)
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Simple Stream-normal Momentum Balance
Assume 1) steady (no dun/dt) 2) no
rotation (f 0) 3) no stream-wise
advection of secondary flow. 4)
Homogenous (no baroclinic PG) 5)
Constant Az
Boundary conditions
1) un 0 at z -H, 2) dun/dz 0 at z 0 3)
vertically averaged un 0
Using solution for us from previous slide
Depth
us(z)
un(z)
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Scale for Curvature-Forced Secondary
Circulation Homogeneous Fluid! Geyer, 1993, JGR,
98955-966. Kalkwijk and Booij, 1986, J. Hydraul.
Eng., 2419-37.
U0 Amplitude of tidal current
? Karmans constant (0.41)
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Rotation and Secondary Circulation
Cross-channel momentum equation with rotation
(but still homogenous)
Vertical-average (assume ?vdz 0)
Subtract mean equation from time varying
Curvature forcing
Rotation forcing
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Relative Importance of Curvature and Rotation
in Driving Secondary Flows
Curvature Rossby Number
using
Curvature and rotation both important (R01) when
R5-10 km
(f 1x10-4 s-1 and u0 1 m/s)
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Secondary Flows are Important because they
re-distribute mass and momentum. Act like large
scale turbulence. They play a key role in tidal
dispersion as well as the along-channel momentum
balance
For Next Week we will discuss the following 2
Papers
Lerczak, J. A., and W. R. Geyer, 2004 Modeling
the lateral circulation in straight, stratified
estuaries. J. Phys. Oceanogr., 34,
1410-1428. Geyer, W. R., J. H. Trowbridge, and
M. M. Bowen, 2000 The dynamics of a
partially-mixed estuary. J. Phys. Oceanogr., 30,
2035-2048.
I will email them to you, please be prepared to
discuss them.