typical mean dynamic balances in estuaries along-estuary component 1. barotropic pressure gradient...
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Typical Mean Dynamic Balances in Estuaries
Along-Estuary Component
1. Barotropic pressure gradient vs. friction
2
2
z
uA
xg z
z
zyx zu
Azy
uA
yxu
Ax
dzx
gx
gfvzu
wyu
vxu
utu
Steady state, linear motion, no rotation, homogeneous fluid, friction in the vertical only (Az is a constant)
Sx
SAg
z
u
z
2
2
The balance can then be rewritten as:
z
Sx
x
SAg
z
u
z
2
2Let’s solve this differential equation; integrating once:
1cSzAg
zu
z
Integrating again:
212
2czcSz
Ag
uz
This is the solution, but we need two boundary conditions:
0.,.,0,0@
zu
eizu
Az z This makes c1 = 0
z
b
z
bbz A
ucAz
uzu
AHz2
;,@
z
b
z Auc
SHAg
zu
Hz2
,@
HzcgSH
uSHcg
ubb
@2
Substituting in the solution at z = -H: 2
2
2c
AgSH
cgSH
zb
zb A
gSHcgSH
c2
2
2
z
Sx
x
212
2czcSz
Ag
uz
zb AgSH
cgSH
cc2
02
21
bzz cgSH
AgSH
zAgS
u 22
22
bz c
gSHHz
AgS
u 22
2
General Solution
from boundary conditions
Particular solution, which can be re-arranged:
2. Pressure gradient vs. vertical mixing
2
21
z
uA
xp
z
)( zgp
)( z
zx
gx
gxp
expanding the pressure gradient:
We can write: Gx
Ix
;
The momentum balance then becomes:
zxA
gxA
g
z
u
zz
2
2
zAgG
AgI
z
u
zz
2
2
O.D.E. with general solutionobtained from integrating twice:
12
2cz
AgG
zAgI
zu
zz
2132
62)( czcz
AgG
zAgI
zuzz
2132
62)( czcz
AgG
zAgI
zuzz
General solution:
c1and c2 are determined with boundary conditions:
z
z Acz
zu
A
1gives0)1
2
32
620gives0)2 c
AH
AgGH
AgIH
Hzuzzz
zzz AH
AgGH
AgIH
c
62
32
2
This gives the solution:
Hz
AH
H
zA
gGH
H
zA
gIHzu
zzz11
61
2)( 3
33
2
22
Third degree polynomial proportional to depth and inversely proportional to friction.
Requires knowledge of I, G, and wind stress.
12
2cz
AgG
zAgI
zu
zz
Hz
AH
H
zA
gGH
H
zA
gIHzu
zzz11
61
2)( 3
33
2
22
We can express I in terms of River Discharge R, G ,and wind stress if we restrict thesolution to:
0
)(H
Rzzui.e., the river transport per unit width provides the water added to the system.
Integrating u(z) and making it equal to R, we obtain:
zzz A
H
A
gGH
A
gIHR
2
1
24
3
3
243
Which makes:gHgH
RAGHI z
233
83
3
Note that the effects of G and R are in the same direction, i.e., increase I.The wind stress tends to oppose I.
Hz
AH
H
zA
gGH
H
zA
gIHzu
zzz11
61
2)( 3
33
2
22
gHgH
RAGHI z
233
83
3 Substituting into:
We get:
2
2
2
2
3
3
2
23
131441
123
181948
)(H
zHz
AH
H
zHR
H
z
H
zA
gGHzu
zz
inducedWind
2
2
inducedRiver
2
2
inducedDensity
3
3
2
23
131441
123
181948
)(
H
zHz
AH
H
zHR
H
z
H
zA
gGHzu
zz
Density-induced: sensitive to H and Az; third degree polynomial - two inflection points
River induced: sensitive to H; parabolic profile
Wind-induced: sensitive to H (dubious) and Az; parabolic profile
inducedWind
2
2
inducedRiver
2
2
inducedDensity
3
3
2
23
131441
123
181948
)(
H
zHz
AH
H
zHR
H
z
H
zA
gGHzu
zz
0.0125 Pa
0z
zx
gx
gz
uA
zz
xg
xg z
Momentum balance, steady with friction (tidally averaged)
Geyer et al (2000) approximations: H
UUC
HzETBB
UT is the tidal current amplitude UE is the magnitude of the estuarine circulation.
40 10771
01
.;s
H
UUC
aH
x
sg ETB
o
where ao is an O (1) constant related to the vertical variability of stress.
Also considering that the surface slope scales in proportion to the baroclinic pressure gradient, the momentum eqn becomes
solving for UE
TB
2
oE UC
Hxs
gaU
where ao0.3 based on Hudson data. (Its value depends on whether the layer average or the maximum is considered).
-UE
UE
3. Advection vs. Pressure Gradient
2
Take:
Upper layer homogeneous and mobile
Lower layer immobile
Consider inertial effects
Ignore friction
@ lower layer there is no horizontal pressure gradient
x
h1
1
2
2
1 u1
U2 = 0z
interface
surface
At interface: 111 ghP
At lower layer: zhgghP 112112
dx
zdg
dx
hdg
dx
dP 2
2112 0
112
2 1h
dxd
dxd
The interface slope is of opposite sign to the surface slope
2
The basic balance of forces is
xP
dxdu
xu
u
1
1
211
11
21
02
xP
Over a volume enclosing the upper layer:
dzxP
uhdxd
2
1
12111
But 2111 ,, zPP
x
h1
1
2
2
1 u1
U2 = 0z
interface
surface
Using Leibnitz rule for differentiation under an integral (RHS of last equation; seeOfficer ((1976), p. 103) we get:
0'21
12
111
hguh
dxd
This is the momentum balance integrated over the entire upper layer (i.e., energy balance)The quantity inside the brackets (kinetic and potential energy) must remain constant
2
112
111 '21
hgdxd
uhdxd
2
12
g'g
0'21
12
111
hguh
dxd
Defining transport per unit width:
111 huq
0'21 2
11
21
1
hg
hq
dxd
Total energy (Kinetic plus Potential energy) remains constant along the system
If the density ρ1 and the g’ do not change much along the system, we can estimate the changes in h1 as a function of q1 (i.e. how upper layer depth changes with flow)
x
h1
1
2
2
1 u1
U2 = 0z
interface
surface
constant'21 2
11
21 hghq
121
1
1
1
'
2
hgu
udqdh
0'2
1
11
1
121
21
1
1 dqdh
hgdqdh
h
qhqDifferentiating with
Respect to ‘q1’
21
21
1
12
1
1
1
1 2
'
2
cu
u
hgu
udqdh
Subcritical flow causes 011 dqdh and supercritical flow causes 011 dqdh
This results from a flow slowing down as it moves to deeper regions or accelerating as it moves to shallower waters or through constrictions
x
z
h1 supercritical flow
x
z
h1 subcritical flow
Apparent PARADOX!?
Pressure gradient vs. friction: bz c
gSHHz
AgS
u 22
2i.e., u proportional to H 2
This can also be seen from scaling the balance:
22
2
2
HAgS
UAgS
H
U
AgS
z
u
zz
z
If we include non-linear terms: 02
2
gSz
uA
xu
u z
Which may be scaled as: 02
2
gSH
UALU z 0
22
2
22
gSLH
LA
H
LAU zz
i.e., U proportional to H -2 !!! Ahaa! If L is very large, we go back to u proportional to H 2
Physically, this tells us that when L is small enough the non-linear terms are relevant to the dynamics and the strongest flow will develop over the shallowest areas (fjords). When frictional effects are more important than inertia, then the strongest flow appears over the deepest areas (coastal plain estuaries)!!!
This competition inertia vs. friction to balance the pressure gradient can be exploredwith a non-dimensional number:
ULAHU
H
UALU
zz
22
2
2
LA
UH
z
2 When this ratio > 1, inertia dominates
When the ratio < 1, friction dominates
LCH
HUC
LU
bb
2
2
Alternatively:
bb
CLH
LCH
1 if
When H / L > Cb, inertia dominates
When H / L < Cb, friction dominates
4. Surface Pressure + Advection vs. Interfacial Friction
x
h1
1
2
2
1
u1
u2
z
interface
surface
h2
Similar situation as before (advection vs. presure gradient) but with interfacial friction (fi).
Flow in the lower layer but interface remains.
There is frictional drag between the two layers. The drag slows down the upper layer and drives a weak flow in the lower layer.
In the upper layer, over a volume enclosing the layer: 211
12111 ;
2
1
uCffdzxP
uhdxd
bii
The momentum balance becomes(Officer, 1976; pp. 106-107):
friction
linterfacia
and layer ofthinning by
balanced
2
gradientpressuresfce to dueforce driving
121
31
bCdxd
dxd
qgh
211
211
2111
211 2
1uC
dxd
hghghudxd
b
The solution has a parabolic shape
Boundary conditions:
u1 = u2 at the interfaceu2 =0 at the bottom, z = -H
0
2
2
h
H
u
because interface touches the bottom
In the lower layer, the balance is:2
221
z
uA
xp
z
x
h1
1
2
2
1
u1
u2
z
interface
surface
h2
u
z Salt-wedge