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




z
Circulación típica en un fiordo
x


a  F m
Aceleraciones
z
x
u
u
u
du u

u
v
w
t
x
y
z
dt
Gradiente de presión
z
x
Debido a la pendiente del nivel del mar (barotrópico)
Debido al gradiente de densidad (baroclínico)
1 p
 g 0 

 g
 
dz
 x
x   H x
Fricción
z
x
Debida a gradientes verticales de velocidad (divergencia del flujo de momentum)
  u 
Az 

z  z 
Coriolis
z
x
Debido a la rotación de la Tierra; proporcional a la velocidad
 fv
Balance de momentum
z
x
du
1 p   u 
 fv  

Az

dt
 x z  z 
dv
1 p   v 
 fu  

Az

dt
 y z  z 
1 p
g
 z
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
gz
z


 g
 x
 x
x
x
BAROTROPIC 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
Aporte aproximado por ríos:
1000 m3/s
1 p
 g 0 

 g
 
dz
 x
x   H x
Milliman et al. (1995)
Dirección Meteorológica de Chile


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   sin90      cos 


Cv   cos90      sin 

  0,  cos  ,  sin  
90  


Cv
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
   x 
 dt


 
 dv
  1 p 


 fu  Cy    
 dt
   x 


 
 dw
  1 p 
 dt  0     x 

 

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

  
  
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

 fu    
 0 

 dt
   x


 

 dw
  1 p
 dt  0     x  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
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 
0.110000   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 '  0
T
T'  0
T T'  0
- 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., u t  u t  u ' t  u t



0
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 101 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 



