EVAT 554
OCEAN-ATMOSPHERE
DYNAMICS
LECTURE 10
FILTERING OF EQUATIONS FOR
OCEAN
(Reference: Peixoto & Oort, Chapter 3,8)
Note that the major
horizontal ocean
circulation
systems mirror
closely the semipermanent high
and low pressure
systems
Scale Analysis
The Ocean
Zonal Momentum Balance:
du / dt  u / t  (V)u  wu / z
 fv  f 'w 1 p /   ( u)   ( u / z)
ˆa cos
z
Meridional Momentum Balance:
H
V
dv / dt  v / t  (V)v  wv / z
  fu  1 p /   ( v)   ( v / z)
ˆa
z
How many
equations?
Vertical Momentum Balance:
dw/ dt  w/ t  (V)w ww/ z
 f 'u  g  1 p / z  ( w)   ( w/ z)
ˆ ˆ
z
How many
variables?
H
H
Continuity:
 1 d  V  w  0
 dt
z
Equation of State:
Heat Equation:
V
V
(incompressible!)
  ( p, S,T )
dT / dt  T / t  (V )T  wT / z
q
q

 ( H T )  (V T / z)  rad  lat  1 dp
z
Cp Cp Cp dt
Zonal Momentum Balance:
du / dt  fv  f 'w
Radius of Earth: a=6.37x
Coriolis parameter: f,f'
H
Depth scale: H103m, h 102m
s-1
Density of Water:  1000 kg
V
Horizontal velocity scale: u,v 10-1 ms-1
m-3
Horizontal Eddy Viscosity: H
105
10-1
V
Length scale: L106m, l105m
106m
10-4
Vertical Eddy Viscosity:
1 p /   ( u)   ( u / z)
ˆa cos
z
m2s-1
m2s-1
Vertical velocity scale: w 10-4 ms-1
Horizontal pressure scale: p 100 mb = 104 Pa
Time Scale: L/u 107s  H/w 107s
u / L  fv f 'wp/ Lˆ  Hu /l Vu / h
2
2
10-8 ms-2 10-5 ms-2 10-8 ms-2
0  fv 
10-5 ms-2
10-6 ms-2
1 p / 
ˆa cos
10-6 ms-2
2
Meridional Momentum Balance:
dv / dt   fu  1 p /   ( v)   ( v / z)
ˆa
z
H
Radius of Earth: a=6.37x
Coriolis parameter: f,f'
Depth scale: H103m, h 102m
s-1
Density of Water:  1000 kg
V
Horizontal velocity scale: u,v 10-1 ms-1
m-3
Horizontal Eddy Viscosity: H
Vertical Eddy Viscosity:
Length scale: L106m, l105m
106m
10-4
V
105
10-1
m2s-1
m2s-1
Vertical velocity scale: w 10-4 ms-1
Horizontal pressure scale: p 100 mb = 104 Pa
Time Scale: L/u 107s  H/w 107s
u / L  fu p / Lˆ  Hu /l Vu / h
2
10-8 ms-2
2
10-5 ms-2
10-5 ms-2
10-6 ms-2
0   fu  1 p / 
ˆa
10-6 ms-2
2
Horizontal Momentum Balance
fv
1 p/ (zonal)
ˆacos
fu  1 p/ (meridional)
ˆa
Geostrophic Balance
Radius of Earth: a=6.37x
Coriolis parameter: f,f'
10-4
Depth scale: H103m, h 102m
s-1
Density of Water:  1000 kg
V
Horizontal velocity scale: u,v 10-1 ms-1
m-3
Horizontal Eddy Viscosity: H
Vertical Eddy Viscosity:
Length scale: L106m, l105m
106m
105
10-1
m2s-1
m2s-1
Vertical velocity scale: w 10-4 ms-1
Horizontal pressure scale: p 100 mb = 104 Pa
Time Scale: L/u 107s  H/w 107s
u / L  fu p / Lˆ  Hu /l Vu / h
2
2
10-8 ms-2
10-5 ms-2
10-5 ms-2
10-6 ms-2
2
10-6 ms-2
1
“Rossby Number”
|
u
/
L
|
|
u
|
10
Ro

 4 6 103
| fu| | f | L 10 10
Geostrophic Balance Holds
2
when Ro << 1
Horizontal Momentum Balance
fv
1 p/ (zonal)
ˆacos
fu  1 p/ (meridional)
ˆa
Geostrophic Balance
Radius of Earth: a=6.37x
Coriolis parameter: f,f'
10-4
Depth scale: H103m, h 102m
s-1
Density of Water:  1000 kg
V
Horizontal velocity scale: u,v 10-1 ms-1
m-3
Horizontal Eddy Viscosity: H
Vertical Eddy Viscosity:
Length scale: L106m, l105m
106m
105
10-1
m2s-1
m2s-1
Vertical velocity scale: w 10-4 ms-1
Horizontal pressure scale: p 100 mb = 104 Pa
Time Scale: L/u 107s  H/w 107s
u / L  fu p / Lˆ  Hu /l Vu / h
2
10-8 ms-2
2
10-5 ms-2
10-5 ms-2
10-6 ms-2
2
10-6 ms-2
“Ekman Number”
5
| H u /l | vH
10
 2  4 10 101
EkH 
Geostrophic Balance Holds
| fu| | f |l 10 10
2
when Ek << 1
Horizontal Momentum Balance
fv
1 p/ (zonal)
ˆacos
fu  1 p/ (meridional)
ˆa
Geostrophic Balance
Radius of Earth: a=6.37x
Coriolis parameter: f,f'
10-4
Depth scale: H103m, h 102m
s-1
Density of Water:  1000 kg
V
Horizontal velocity scale: u,v 10-1 ms-1
m-3
Horizontal Eddy Viscosity: H
Vertical Eddy Viscosity:
Length scale: L106m, l105m
106m
105
10-1
m2s-1
m2s-1
Vertical velocity scale: w 10-4 ms-1
Horizontal pressure scale: p 100 mb = 104 Pa
Time Scale: L/u 107s  H/w 107s
u / L  fu p / Lˆ  Hu /l Vu / h
2
10-8 ms-2
2
10-5 ms-2
10-5 ms-2
10-6 ms-2
2
10-6 ms-2
“Ekman Number”
1
| V u / h | vV
10
EkV 
 4 4 101

2
Geostrophic Balance Holds
| fu| | f |h 10 10
2
when Ek << 1
Horizontal Momentum Balance
fv
1 p/ (zonal)
ˆacos
fu  1 p/ (meridional)
ˆa
Geostrophic Balance
Radius of Earth: a=6.37x
Coriolis parameter: f,f'
10-4
V
Horizontal velocity scale: u,v 10-1 ms-1
m-3
Horizontal Eddy Viscosity: H
3
Depth scale: H103m, h 102m
s-1
Density of Water:  1000 kg
Vertical Eddy Viscosity:
Length scale: L106m, l105m
106m
105
10-1
m2s-1
m2s-1
Ro10 1
Vertical velocity scale: w 10-4 ms-1
Horizontal pressure scale: p 100 mb = 104 Pa
Time Scale: L/u 107s  H/w 107s
1
EkV  EkH 10 1
Note that these approximations are only appropriate for “interior
solutions” and will break down in boundary layers, where
horizontal or vertical shear are large!
Or near the equator!!
Horizontal Momentum Balance
fv
1 p/ (zonal)
ˆacos
fu  1 p/ (meridional)
ˆa
Geostrophic Balance
Horizontal Momentum Balance
fv
1 p/ (zonal)
ˆacos
fu  1 p/ (meridional)
ˆa
Geostrophic Balance
Dynamic Topography
Horizontal Momentum Balance
fv
1 p/ (zonal)
ˆacos
fu  1 p/ (meridional)
ˆa
Geostrophic Balance
Dynamic Topography
Horizontal Momentum Balance
fv
1 p/ (zonal)
ˆacos
fu  1 p/ (meridional)
ˆa
Geostrophic Balance
the dynamic typography is
not a simple consequence of
the overlying sea level
pressure
requires an
understanding of ocean
dynamics and its relation
with atmospheric
windstress
Dynamic Topography
Vertical Momentum Balance:
dw/ dt  f 'u  g  1 p / z  ( w)   ( w/ z)
ˆ ˆ
z
H
Radius of Earth: a=6.37x
Coriolis parameter: f,f'
Depth scale: H103m, h 102m
s-1
Density of Water:  1000 kg
V
Horizontal velocity scale: u,v 10-1 ms-1
m-3
Horizontal Eddy Viscosity: H
Vertical Eddy Viscosity:
Length scale: L106m, l105m
106m
10-4
V
105
10-1
m2s-1
m2s-1
Vertical velocity scale: w 10-4 ms-1
Horizontal pressure scale: p 100 mb = 104 Pa
Time Scale: L/u 107s  H/w 107s
wu / L  f 'u  g  pb / Hˆ  H w/ L V w/ H
2
10-11 ms-2 10-5 ms-2 10 ms-2
10 ms-2
10-11 ms-2
0   g  1 p / z
ˆ ˆ
10-11 ms-2
2
Thermal Wind Balance
As with the atmosphere, we can combine geostrophic and
hydrostatic balance to get
u  g 
z fa 
v   g 
z facos 
But now,  depends on T,S,p
We can’t go proceed until we develop
the equation of state for ocean water…