1. Level of No Motion (LNM)
Pacific:
Deep water is uniform,
current is weak below
1000m.
Atlantic:
A level of no motion at m
“Slope current”: Relative geostrophic current is zero but absolute current is not. May occurs in deep ocean (barotropic?)
Current increases into the deep ocean, unlikely in the real ocean

2. For a barotropic flow, we have is geostrophic current.
 p and ρ surfaces are parallel
For a barotropic flow, we have
is geostrophic current.
Since 
Given a barotropic and hydrostatic conditions,
and
Therefore,
And So
(≈0 in Boussinesq approximation)
The slope of isopycnal is small and undetectable,
for V=0.1 m/s, slope~10-6, i.e., 0.1m height change in 100 km

3. There is no simple relation between the isobars and isopycnals.
Relations between isobaric and isopycnal surfaces and currents
Baroclinic Flow:
and
There is no simple relation between the isobars and isopycnals.
slope of isobar is proportional to velocity
slope of isopycnal is proportional to vertical wind shear.
With a barotropic of mass the water may be stationary but with a baroclinic field, having horizontal density gradients, such as situation is not possible
In the ocean, the barotropic case is most common in deep water while the baroclinic case is most common in the upper 1000 meters where most of the faster currents occur.

4. 1½ layer flow Simplest case of baroclinic flow:
Two layer flow of density ρ1 and ρ2.
The sea surface height is η=η(x,y) (In steady state, η=0). The depth of the upper layer is at z=d(x,y). The lower layer is at rest. 
For z > d,
For z ≤ d,
If we assume 
The slope of the interface between the two layers (isopycnal) =
times the slope of the surface (isobar).
The isopycnal slope is opposite in sign to the isobaric slope.

6. σt A B diff
m --- >50m
m 280m -150m
m 750m -180m
Isopycnals are nearly flat at 100m
Isobars ascend about 0.13m between A and B for upper 150m
Below 100m, isopycnals and isobars slope in opposite directions with 1000 times in size.

9. Example: sea surface height and thermocline depth

10. Comments on the geostrophic equation
If the ocean is in “real” geostrophy, there is no water parcel acceleration. No other forces acting on the parcel. Current should be steady
Present calculation yields only relative currents and the selection of an appropriate level of no motion always presents a problem
One is faced with a problem when the selected level of no motion reaches the ocean bottom as the stations get close to shore
It only yields mean values between stations which are usually tens of kilometers apart
Friction is ignored
Geostrophy breaks down near the equator
The calculated geostrophic currents will include any long-period transient current

11. The β-spiral Determining absolute velocity from density field
Assumptions:
1) Geostrophic
2) incompressible
3) steady state
2) + 3) 

12. Use the thermal wind equation with Boussinesq approximation
Take into
Write u and v to polar format as
In the northern hemisphere,
if w > 0, the current rotates to the right as we go upward (or to the left as we go downward) 

13. Take geostrophic equation
If v≠0, w changes with z and can not be zero everywhere. Thus the β effect makes the rotation of the geostrophic flow with depth likely, hence the term “β spiral”
use

15. Consider an isopycnal surface at
If we go along this surface in the x-direction 
Similarly
Take into  

16. Rewrite the thermal wind relation 
Suppose we have derived u’ and v’ based on some reference level
If the observations should be error free, two levels would be sufficient
Considering the observation errors, particularly noise from time-dependent motions, this equation will not be exactly satisfied. Computationally, a least-square technique is used.