1. Unit 6 – Part I: Geostrophy
Introductory Physical Oceanography (MAR 555) - Fall 2009
Miles A. Sundermeyer
Unit 6 – Part I: Geostrophy
Assigned Reading: OC 3.3 and IPO Chapter 10

2. Key Concepts: Recap rotating (non-inertial) reference frames
Motions on a rotating Earth
Momentum equations
a.) Geostrophic Balance:
b.) Hydrostatic Balance:
c.) Thermal Wind:

3. Recap: Coriolis Force apparent trajectory in rotating ref. frame
actual trajectory in inertial ref frame

4. Recap: Coriolis Force

5. Inertial Circles y y
impulse
impulse x x
velocity
position

6. “Balance” of forces in inertial motion
Centrifugal force
Coriolis Force
In Northern Hemisphere
What’s wrong with this?

7. Steady Forced Inertial Motions
forcing
velocity x y x
position

8. Steady Pressure Driven Motions –
a.k.a. Geostrophy

9. Steady Pressure Driven Motions – a.k.a. Geostrophy

10. Scaling the (u/v)-momentum equations
Basin Scale
We have no easy basis for scaling the horizontal PG
Acc + Adv = PG + Coriolis+ Friction
- Pressure Gradient must be Order 1 to Balance Coriolis 10

11. Scaling the w-momentum equation
Acc + Adv + Adv + Adv = PG + Grav + Diff + Diff + Diff
We can ignore all but the pressure gradient and gravity – AT THESE SCALES
Note:
Advection terms all same order of magnitude
Friction terms all same order of magnitude 11

12. Geostrophy / Thermal Wind
Geostrophic equation:
Hydrostatic equation:
Eliminating pressure implies:
Similarly, for u:
Thermal Wind
Relations

13. Geostrophy / Thermal Wind (cont’d)
Example: Constant density layers
Geostrophy implies: r0 r1 r2 r3
e.g., see OC Fig 3.20 1 2 3 x z y
Dz1
Dz2
Dz3 Dx 
Thus the contributions to DP at each level are:
level DP v 1
(r1-r0)gDz1=DrgDz
pos 2
-(r2-r1)gDz2 =-DrgDz
0/neg 3
(r3-r2)gDz3 =DrgDz

14. Geostrophy / Thermal Wind (cont’d)
Example: Constant density layers
Alternatively, Thermal Wind implies: r0 r1 r2 r3
e.g., see OC Fig 3.20 1 2 3 x z y
Dz1
Dz2
Dz3 Dx r0 1
Dz1 r1
Thus associated with each isopycnal is:
Dz2 2 r2
interface
r/x
v/z v 1
pos
neg
decreases with +z 2
increases with +z 3 3
Dz3 r3 Dx x z y

15. Geostrophy / Thermal Wind (cont’d)
Example: Constant density layers
layer / interface
Geostrophic v
Thermal Wind v 1
pos
decreases with +z 2
0/neg
increases with +z 3 r0 r1 r2 r3 x z y
v(z)

16. Geostrophy / Thermal Wind (cont’d)
Example: Constant density layers r0 r1 r2 r3 x z y
v(z)

17. Geostrophy / Thermal Wind
Geostrophic equation:
Thermal Wind Relations:

18. Dynamic Height (E.g., see OC 3.3.4; Stewart 10.4)
Specific Volume:
Specific Volume Anomaly:
NOTE: Larger d corresponds to lower density
From hydrostatic equation: A B Po rA rB
Implies that lower density requires greater height of water column above

19. Dynamic Height (cont’d)
Change in dynamic height:
Momentum Equation
(geostrophic balance):
Hydrostatic Equation: A B Po rA rB
NOTE: 1 Dynamic meter = 1 geometric m / 9.8

20. Dynamic Height (cont’d)
Example: Global dynamic topography
Introduction to Physical Oceanography, Stuart, Fig 10.2
Ocean Circulation, Open University, Fig 3.21

21. Dynamic Height (cont’d)
Example: Global dynamic topography
Ocean Circulation, Open University, Fig 3.21
Fig Topographic map of the mean sea-surface (i.e., the marine geoid), as determined using a satellite-borne radar altimeter. The mean sea-surface topography reflects the topography of the sea-floor rather than geostrophic current flow, as the effect of the latter is about two orders of magnitude smaller, even in regions of strong current flow.

22. Baroclinic vs. Barotropic
Ocean Circulation, Open University, Fig 3.11
Barotropic: levels of constant pressure are parallel to surfaces of constant density.
Baroclinic: levels of constant pressure are inclined to surfaces of constant density.
Ocean Circulation, Open University, Fig 3.15

23. Computing Currents from Hydrographic Observations
Slopes of sea surface O(1:105 to 1:108)
Isopycnal slopes several hundred times larger
Thermal wind only tells you vertical shear, not absolute velocity
Level of “no” or “known” motion
Small temporal/spatial variations in density confound geostrophic flow estimates
Large-scale measurements allow estimates of average velocities
Can only compute geostrophic velocities normal to hydrographic transects