Conservation of Salt: z S z K z y S y K y x S x K x z S w y S v x S u t S onservation of Heat: z T z z y T y y x T x x z T w y T v x T u t T Equation of State: ] , , [ p T S 0 z w y v x u Conservation of Mass or Continuity: Equations that allow a quantitative look at the OCEAN
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Conservation of Salt: Conservation of Heat: Equation of State: Conservation of Mass or Continuity: Equations that allow a quantitative look at the OCEAN.
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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
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
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