conservation of salt: conservation of heat: equation of state: conservation of mass or continuity:...
TRANSCRIPT
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
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
Circulación típica en un fiordo
x
z
mFa
Aceleraciones
dtdu
zu
wyu
vxu
utu
x
z
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
Fricción
Debida a gradientes verticales de velocidad (divergencia del flujo de momentum)
zu
Az z
x
z
Coriolis
Debido a la rotación de la Tierra; proporcional a la velocidad
fv
x
z
Balance de momentum
zp
g
zv
Azy
pfu
dtdv
zu
Azx
pfv
dtdu
z
z
1
1
1x
z
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:
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
Friction (wind stress)z
W
u
Vertical Shears (vertical gradients)
Friction (bottom stress)z
u
bottom
Vertical Shears (vertical gradients)
Friction (internal stress)z
u1
Vertical Shears (vertical gradients)
u2
Flux of momentum from regions of fast flow to regions of slow flow
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
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
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!
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
Low Re
High Re
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?
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
'
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
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
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
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
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