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

2. 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 current 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.

3. 1 and 1/2 layer flow Simplest case of baroclinic flow:
Two layer flow of density 1 and 2.
The sea surface height is =(x,y) (In steady state, =0). The depth of the upper layer is at z=d(x,y)<0. 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.

7. 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.

8. 20oC isotherm anomalies (m)
Relation between fluctuations of SSH () and thermocline depth (h)
20oC isotherm anomalies (m)
COLA ODA
December,1997-February, 1998
SSH anomalies (cm)
TOPEX/Poseidon

11. Example: sea surface height and thermocline depth

12. 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 (within 0.5olatitude, or ±50km)
The calculated geostrophic currents will include any long-period transient current

13. Properties of Sea Water
What is the pressure at the bottom of the ocean relative to sea surface pressure? What unit of pressure is very similar to 1 meter?
What is salinity and why do we use a single chemical constituent (which one?) to determine it? What other physical property of seawater is used to determine salinity? What are the problems with both of these methods?
What properties of seawater determine its density? What is an equation of state?
What happens to the temperature of a parcel of water (or any fluid or gas) when it is compressed adiabatically? What quantity describes the effect of compression on temperature? How does this quantity differ from the measured temperature? (Is it larger or smaller at depth?)
What are the two effects of adiabatic compression on density?
What are t and ? How do they different from the in situ density?
Why do we use different reference pressure levels for potential density?
What are the significant differences between freezing pure water and freezing seawater?

14. Conservation laws
Mass conservation (continuity equation), volume conservation
Salt conservation (evaporation, river run-off, and precipitation)
Heat conservation (short and long-wave radiative fluxes, sensible and evaporative heat fluxes, basic factors controlling the fluxes, parameterizations)
Meridional heat and freshwater transports
Qualitative explanations of the major distributions of the surface heat flux components

15. Basic Dynamics
What are the differences between the centrifugal force and the Coriolis force? Why do we treat them differently in the primitive equation?
How to do a scale analysis for a fluid dynamics problem?
What is the hydrostatic balance? Why does it hold so well in the scales we are interested in?
What is the geostrophic balance?
What is the definition of dynamic height?
In geostrophic flow, what direction is the Coriolis force in relation to the pressure gradient force? What direction is it in relation to the velocity?
Why do we use a method to get current based on temperature and salinity instead of direct current measurements for most of the ocean?
How are temperature and salinity information used to calculate currents? What are the drawbacks to this method?
What is a "level of no motion"? Why do we need a "level of known or no motion" for the calculation of the geostrophic current? (What can we actually compute about the velocity structure given the density distribution and an assumption of geostrophy?)
What are the barotropic and baroclinic flows? Is there a “thermal wind” in a barotropic flow?
What can you expect about the relation between the slopes of the thermocline depth and the sea surface height, based on a 1 and 1/2 layer model?
What factors determine the static stability?