7. conservation of momentum
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Transcript 7. conservation of momentum
Equations that allow a quantitative look at the OCEAN
Equation of State:
[S ,T , p ]
u v w
0
Conservation of Mass or Continuity:
x y z
Conservation of Salt:
S
S
S
S
S
S
S
u
v
w
K
K
K
t
x
y
z x x x y y y z z z
Conservation of Heat:
T
T
T
T
T
u
v
w
t
x
y
z x x x y
T T
y y z z z
Conservation of Momentum (Equations of Motion)
Newton’s Second Law:
a
du
dt
dV dv
a
dt
dt
dw
dt
ma F
F
m
u
u
u
u
u
v
w
t
x
y
z
v
v
v
v
u
v
w
t
x
y
z
w
w
w
w
u
v
w
x
y
z
t
Conservation of momentum mV as they describe changes of momentum in time per unit mass
1 d
dV
mV
a
m dt
dt
Forces per unit mass that produce accelerations in the ocean:
F
Pressure gradient + Coriolis + gravity + friction + tides
m
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
F
Pressure gradient + Coriolis + gravity + friction+ tides
m
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
z
dz
p
p
y
p
dx
x
dy
x
dx
Net Force in ‘x’ =
p
dx dy dz
x
1 p
1 p
dx dy dz
Net Force per unit mass in ‘x’ =
x
Vol x
Total pressure force/unit mass on every face of the fluid element is:
1 p p p
1
,
,
p
x y z
Illustrate pressure gradient force in the ocean
z
Pressure Gradient
Gradient?
z
Pressure Gradient Force
1
2
x
Pressure of water column at 1 (hydrostatic pressure) : P1 g z
Hydrostatic pressure at 2 :
P2 g z z
Pressure gradient force caused by sea level tilt:
1 p
1 P2 P1
gz
z
g
x
x
x
x
BAROTROPIC PRESSURE GRADIENT
F
Pressure gradient + Coriolis + gravity + friction + tides
m
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: A (a1, a2 , a3 ) and
B (b1, b2 , b3 )
iˆ
jˆ
kˆ
A B a1
a2
a3 iˆ(a2b3 a3b2 ) jˆ(a3b1 a1b3 ) kˆ (a1b2 a2 b1 )
b1
b2
b3
C (a2b3 a3b2 , a3b1 a1b3 , a1b2 a2b1)
r
Now, let us consider the velocity V of a fixed particle on a rotating body at the position
The body, for example the earth, rotates at a rate
,r V
V
r
To an observer from space (us):
drf drE
rE
dt
dt
This gives an operator that relates a fixed frame in space (inertial) to a moving object on a
rotating frame on Earth (non-inertial)
d f d E
dt
dt
E
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
V
d f d E
E
dt
dt
drf drE
rE
dt
dt
d drf
dt dt
0
d
d
r
d
d
r
d
r
E
E
E
rE
rE
dt
dt
dt dt dt
dVf dVE
2 VE rE
dt
dt
Acceleration of a particle on a rotating Earth with respect to an observer in space
Coriolis
Centripetal
r
The equations of conservation of momentum, up to now look like this:
dVE
1
2 VE rE p other forces
dt
Coriolis Acceleration
E W , N S , v 0,Ch ,Cv
Ch sin90 cos
Cv cos90 sin
0, cos , sin
90
Cv
90
Cv
E W , N S , v 0,Ch ,Cv
Ch sin90 cos
Cv cos90 sin
0, cos , sin
iˆ
2 V 2 0
u
jˆ
kˆ
cos
v
sin 2 cos w 2 sin v , 2 sin u , 2 cos u
w
Making:
f
f 2 sin
2
24h
is the Coriolis parameter
2 V 2 cos w fv , fu , 2 cos u
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
2 V fv , fu , 0
2 V fv , fu , 0
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
f 2 sin
F
Pressure gradient + Coriolis + gravity + friction + tides
m
Pressure gradient: Barotropic and Baroclinic
Coriolis: Only in the horizontal
Gravity: Only in the vertical
Friction: Surface, bottom, internal
Tides: Boundary condition
du
1 p
fv
C
x
dt
x
dv
1 p
dt fu Cy y
dw
1 p
0
dt
z
other forces
F
Pressure gradient + Coriolis + gravity + friction + tides
m
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
dVE
1
2 VE rE p other forces
dt
dV
1
2 V r gf p other forces
dt
r
gf
g
r
r
gf
g (0, 0, g )
g has a weak variation with latitude because of the magnitude of the centrifugal acceleration
2 r cos
g 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
1
dV
2 V g p other forces
dt
du
1 p
fv
0
dt
x
dv
1 p
dt fu y 0
dw
1 p
dt 0 z g
friction
Friction (wind stress)
z
W
Vertical Shears
(vertical gradients)
u
Friction (bottom stress)
z
u
Vertical Shears
(vertical gradients)
bottom
Friction (internal stress)
z
u1
Vertical Shears
(vertical gradients)
u2
Flux of momentum from regions of fast flow to regions of slow flow
Shear stress is proportional to the rate of shear normal to which the stress is exerted u z
u
z
at molecular scales
µ is the molecular dynamic viscosity = 10-3 kg m-1 s-1 for water;
it is a property of the fluid
Shear stress has units of kg m-1 s-1 m s-1 m-1 = kg m-1 s-2
or force per unit area or pressure: kg m s-2 m-2 = kg m-1 s-2
z
u
z
u
y y
u
y dy
dz
u
x
u
y
y
dy
u u
dz
z z z
x
dx
u u
dx
x x x
z
u
z
u u
dy
y y y
dz
u
x
u
y
u u
dx
x x x
y
dy
u u
dz
z z z
x
dx
Net force per unit mass (by molecular stresses) on u
Fx
1 u
x x y
u u u
u u
y z z x x y y z z
kinematic molecular viscosity 10-6 m 2 s
If viscosity is constant, Fx
u u u
becomes:
x x y y z z
2u
2u
2u
2
Fx
u
2
2
2
x
y
z
And up to now, the equations of motion look like:
1
dV
2
2 V g p ( )V
dt
These are the Navier-Stokes equations
Presuppose laminar flow!
Compare non-linear (advective) terms to molecular friction
u U 2
u
~
x
L
2u U
2~ 2
x
L
U 2 L UL
Inertial to viscous:
Re
2
U L
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
Re
0.110000 109
6
10
Is friction negligible in the ocean?
Frictional stresses from turbulence are not negligible but molecular friction is negligible
at scales > a few m.
T T T'
T T
T
T 'T ' 0
T' 0
T T' 0
- Use these properties of turbulent flows in the Navier Stokes equation
1
dV
2
2 V g p ( )V Navier-Stokes equations
dt
u
u
u
u
1 p
2u
u
v
w
fv
2
t
x
y
z
x
x
x (or E) component
0
Upon applying mean and fluctuating parts to this component of motion:
-The only terms that have products of fluctuations are the advective terms
- All other terms remain the same, e.g.,
u t u t u ' t u t
0
fv fv fv' fv
0
1 p x 1 p x 1 p' x 1 p x
0
What about the advective terms?
u
u
u
u
u '
u '
u '
v
w
u'
v'
w'
x
y
z
x
y
z
du
dt
u' u' u' v ' u' w '
x
y
z
u'
u '
u '
u '
u '
v '
w '
v'
w'
u'
u'
u'
x
y
z
x
y
z
u '
v '
w '
u'
x
y
z
u' u' , u' v ' , u' w '
are the Reynolds
0
stresses
arise from advective (non-linear or inertial) terms
u ' u ' Ax
u
x
u ' v ' Ay
u
y
u ' w ' Az
u
z
This relation (fluctuating part of turbulent flow to the mean turbulent flow) is called a
turbulence closure
The proportionality constants (Ax, Ay, Az) are the eddy
and are a property of the flow (vary in space and time)
Fx
u
Ax
x x y
(or turbulent) viscosities
u u
Ay y z Az z
Fx
u
Ax
x x y
u u
Ay y z Az z
Ax, Ay oscillate between 10-1 and 105 m2/s
Az oscillates between 10-5 and 10-1 m2/s
Az << Ax, Ay
but frictional forces in vertical are typically stronger
eddy viscosities are up to 1011 times > molecular viscosities
Equations of motion – conservation of momentum
du
1 p u
fv
Ax
dt
x x x y
u u
Ay y z Az z
dv
1 p v v v
fu
Ax
Ay
Az
dt
y x x y y z z
dw
1 p
w
g
Ax
dt
z
x
x y
w w
Ay y z Az z
ma F
du
1 p u
fv
Ax
dt
x x x y
u
u u
Ay y z Az z
v
dv
1 p v v v
fu
Ax
Ay
Az
dt
y x x y y z z
p
g
1 p
z
u v w
0
x y z
w
S
T
S
S
S
S
S
S
S
u
v
w
K
K
K
t
x
y
z x x x y y y z z z
T
T
T
T
T
u
v
w
t
x
y
z x x x y
[S ,T , p ]
T T
y y z z z