conservation of salt: conservation of heat: equation of state: conservation of mass or continuity:...
TRANSCRIPT
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Conservation of Salt:
zS
zK
zyS
yK
yxS
xK
xzS
wyS
vxS
utS
Conservation of Heat:
zT
zzyT
yyxT
xxzT
wyT
vxT
utT
Equation of State: ],,[ pTS
0
zw
yv
xu
Conservation of Mass or Continuity:
Equations that allow a quantitative look at the OCEAN
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Conservation of Momentum (Equations of Motion)
mF
a
zw
wyw
vxw
utw
zv
wyv
vxv
utv
zu
wyu
vxu
utu
dtdw
dtdv
dtdu
dtVd
a
Fam
Newton’s Second Law:
Conservation of momentum Vm
as they describe changes of momentum in time per unit mass
adtVd
Vmdtd
m
1
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Circulación típica en un fiordo
x
z
mFa
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Aceleraciones
dtdu
zu
wyu
vxu
utu
x
z
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Gradiente de presión
Debido a la pendiente del nivel del mar (barotrópico)
Debido al gradiente de densidad (baroclínico)
dzx
gx
gxp
H
01
x
z
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Fricción
Debida a gradientes verticales de velocidad (divergencia del flujo de momentum)
zu
Az z
x
z
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Coriolis
Debido a la rotación de la Tierra; proporcional a la velocidad
fv
x
z
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Balance de momentum
zp
g
zv
Azy
pfu
dtdv
zu
Azx
pfv
dtdu
z
z
1
1
1x
z
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mF
Pressure gradient + friction + tides+ gravity+ Coriolis
Pressure gradient: Barotropic and Baroclinic
Coriolis: Only in the horizontal
Gravity: Only in the vertical
Friction: Surface, bottom, internal
Tides: Boundary condition
REMEMBER, these are FORCES PER UNIT MASS
Forces per unit mass that produce accelerations in the ocean:
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mF
Pressure gradient + friction+ tides+ gravity+ Coriolis
Pressure gradient: Barotropic and Baroclinic
Coriolis: Only in the horizontal
Gravity: Only in the vertical
Friction: Surface, bottom, internal
Tides: Boundary condition
REMEMBER, these are FORCES PER UNIT MASS
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x
z
y
dy
dz
dx
p dxxp
p
Net Force in ‘x’ = dzdydxxp
Net Force per unit mass in ‘x’ = dzdydxxp
Vol
1xp
1
Total pressure force/unit mass on every face of the fluid element is: pzp
yp
xp
1
,,1
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Illustrate pressure gradient force in the ocean
z
z
1 2
Pressure Gradient?Pressure Gradient
Pressure Gradient Force
Pressure of water column at 1 (hydrostatic pressure) : zgP 1
Hydrostatic pressure at 2 : zzgP 2
x
Pressure gradient force caused by sea level tilt:
xz
gxzg
xPP
xp
1211
BAROTROPIC PRESSURE GRADIENTBAROTROPIC PRESSURE GRADIENT
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Descarga de Agua Dulce
Precipitación pluvial y Ríos
Aporte aproximado por lluvia: 2000 mm por año
area superficial: 350 km por 10 km = 3.5x109 m2
200 m3/s
Dirección Meteorológica de Chile
Aporte aproximado por ríos: 1000 m3/s
Milliman et al. (1995)
dzx
gx
gxp
H
01
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mF
Pressure gradient + friction + tides+ gravity+ Coriolis
Pressure gradient: Barotropic and Baroclinic
Coriolis: Only in the horizontal
Gravity: Only in the vertical
Friction: Surface, bottom, internal
Tides: Boundary condition
REMEMBER, these are FORCES PER UNIT MASS
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Acceleration due to Earth’s Rotation
Remember cross product of two vectors: ),,( 321 aaaA
),,( 321 bbbB
and
321
321
ˆˆˆ
bbb
aaa
kji
BA
)(ˆ 2332 babai )(ˆ 3113 babaj )(ˆ 1221 babak
),,( 122131132332 babababababaC
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Now, let us consider the velocity of a fixed particle on a rotating body at the positionV
The body, for example the earth, rotates at a rate
r
r
V
r
, V
To an observer from space (us):E
Ef rdtrd
dtrd
This gives an operator that relates a fixed frame in space (inertial) to a moving object on a rotating frame on Earth (non-inertial)
EEf
dtd
dtd
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This operator is used to obtain the acceleration of a particle in a reference frame on the rotating earth with respect to a fixed frame in space
EEf r
dtrd
dtrd
EEf
dtd
dtd
EEE
EEf r
dtrd
dtrd
rdtd
dtrd
dtd
dtrd
dtd
r
V
0
EEEf rV
dtVd
dtVd
2
Acceleration of a particle on a rotating Earth with respect to an observer in space
Coriolis Centripetal
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forcesotherprVdtVd
EEE
1
2
The equations of conservation of momentum, up to now look like this:
Coriolis Acceleration
90
Cv
C h
vhvSNWE CC ,,0,,
cos90sin
hC
sin90cos
vC
sin,cos,0
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uuvw
wvu
kji
V cos2,sin2,sin2cos2sincos0
ˆˆˆ
22
h
f
242
sin2
Making:
f is the Coriolis parameter
ufufvwV cos2,,cos22
This can be simplified with two assumptions:
1) Weak vertical velocities in the ocean (w << v, u)
2) Vertical component is ~5 orders of magnitude < acceleration due to gravity
0,,2 fufvV
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0,,2 fufvV
Eastward flow will be deflected to the south
Northward flow will be deflected to the east
f increases with latitude
f is negative in the southern hemisphere
sin2f
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mF
Pressure gradient + friction + tides+ gravity+ Coriolis
Pressure gradient: Barotropic and Baroclinic
Coriolis: Only in the horizontal
Gravity: Only in the vertical
Friction: Surface, bottom, internal
Tides: Boundary condition
forcesother
xp
xp
xp
dtdw
Cfudtdv
Cfvdtdu
y
x
1
1
1
0
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mF
Pressure gradient + friction + tides+ gravity+ Coriolis
Pressure gradient: Barotropic and Baroclinic
Coriolis: Only in the horizontal
Gravity: Only in the vertical
Friction: Surface, bottom, internal
Tides: Boundary condition
REMEMBER, these are FORCES PER UNIT MASS
![Page 23: Conservation of Salt: Conservation of Heat: Equation of State: Conservation of Mass or Continuity: Equations that allow a quantitative look at the OCEAN](https://reader030.vdocuments.us/reader030/viewer/2022032806/56649efd5503460f94c110ce/html5/thumbnails/23.jpg)
Centripetal acceleration and gravity
fg
r
r
forcesotherpgrVdtVd
f 1
2
fg
r
g
),0,0( gg
g has a weak variation with latitude because of the magnitude of the centrifugal acceleration
cos2 rg is maximum at the poles and minimum at the equator (because of both r and lamda)
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Variation in g with latitude is ~ 0.5%, so for practical purposes, g =9.80 m/s2
forcesotherpgVdtVd
1
2
friction
gxp
xp
xp
dtdw
fudtdv
fvdtdu
1
01
01
0
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Friction (wind stress)z
W
u
Vertical Shears (vertical gradients)
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Friction (bottom stress)z
u
bottom
Vertical Shears (vertical gradients)
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Friction (internal stress)z
u1
Vertical Shears (vertical gradients)
u2
Flux of momentum from regions of fast flow to regions of slow flow
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x
z
y
dy
dz
dx
Shear stress has units of kg m-1 s-1 m s-1 m-1 = kg m-1 s-2
Shear stress is proportional to the rate of shear normal to which the stress is exerted zu
zu at molecular scales
µ is the molecular dynamic viscosity = 10-3 kg m-1 s-1 for water is a property of the fluid
or force per unit area or pressure: kg m s-2 m-2 = kg m-1 s-2
xu
dxxu
xxu
y
u
dyyu
yyu
zu
dzzu
zzu
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x
z
y
dy
dz
dx
xu
dxxu
xxu
y
u
dyyu
yyu
zu
dzzu
zzu
Net force per unit mass (by molecular stresses) on u
zu
zyu
yxu
xFx
1
zu
zyu
yxu
x
sm26-10viscositymolecularkinematic
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uzu
yu
xu
Fx2
2
2
2
2
2
2
If viscosity is constant,
zu
zyu
yxu
xFx becomes:
VpgVdtVd
)(1
2 2
And up to now, the equations of motion look like:
These are the Navier-Stokes equations
Presuppose laminar flow!
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Compare non-linear (advective) terms to molecular friction
22
2
2
~
~
LU
xu
LU
xu
u
Inertial to viscous: Re2
2
UL
LULU Reynolds Number
Flow is laminar when Re < 1000
Flow is transition to turbulence when 100 < Re < 105 to 106
Flow is turbulent when Re > 106, unless the fluid is stratified
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Low Re
High Re
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Consider an oceanic flow where U = 0.1 m/s; L = 10 km; kinematic viscosity = 10-6 m2/s
610
100001.0Re 910
Is friction negligible in the ocean?
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Frictional stresses from turbulence are not negligible but molecular friction is negligible at scales > a few m.
'TTT
T 0'' TT
0'
0'
TT
T
TT
- Use these properties of turbulent flows in the Navier Stokes equations-The only terms that have products of fluctuations are the advection terms- All other terms remain the same, e.g., tutututu
0
'
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zu
wyu
vxu
uzu
wyu
vxu
u
'
''
''
'
dtud
z
wu
y
vu
x
uu
''''''
zw
uyv
uxu
uzu
wyu
vxu
u
'
''
''
''
''
''
'
zw
yv
xu
u'''
'
0
'','','' wuvuuu are the Reynolds stressesReynolds stresses
arise from advective (non-linear or inertial) terms
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zu
Awu
yu
Avu
xu
Auu
z
y
x
''
''
''
This relation (fluctuating part of turbulent flow to the mean turbulent flow) is called a
turbulence closureturbulence closure
The proportionality constants (Ax, Ay, Az) are the eddy (or turbulent) viscositieseddy (or turbulent) viscosities and are a property of the flow (vary in space and time)
zu
Azy
uA
yxu
Ax
F zyxx
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Ax, Ay oscillate between 101011 and 101055 mm22/s/s
Az oscillates between 1010-5-5 and 1010-1-1 mm22/s/s
zu
Azy
uA
yxu
Ax
F zyxx
Az << Ax, Ay but frictional forces in vertical are typically stronger
eddy viscosities are up to 1011 times > molecular viscosities
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zw
Azy
wA
yxw
Ax
gzp
dtdw
zv
Azy
vA
yxv
Axy
pfu
dtdv
zu
Azy
uA
yxu
Axx
pfv
dtdu
zyx
zyx
zyx
1
1
1
Equations of motion – conservation of momentum
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Fam
zp
g
zv
Azy
vA
yxv
Axy
pfu
dtdv
zu
Azy
uA
yxu
Axx
pfv
dtdu
zyx
zyx
1
1
1
0
zw
yv
xu
zS
zK
zyS
yK
yxS
xK
xzS
wyS
vxS
utS
],,[ pTS
zT
zzyT
yyxT
xxzT
wyT
vxT
utT
T
S
p
w
v
u