2. Frictionless Motion
In addition assume no horizontal pressure gradients (no surface slopes) and homogeneous fluid

3. Provided by Coriolis acceleration t = 3π /( 2f )
INERTIAL MOTION N E
radius R (m/s) = U
t = 0
R2 = u2 + v2
Circular path due to centripetal acceleration = U2 / r
t = π /(2f)
Provided by Coriolis acceleration
t = 3π /( 2f )
t = π /f

4. From P. Flament, R. Lumpkin, S. Kennan, E
From P. Flament, R. Lumpkin, S. Kennan, E. Firing School of Ocean and Earth Science and Technology University of Hawai'i (Copyright by Flament, 1996)
“In the rotating frame of reference, the sum of the centrifugal force and gravity are always orthogonal to the surface; thus only the Coriolis force can have a tangential component.
In the fixed frame of reference, the particle oscillates back and forth as a pendulum along the x-axis; the period of the pendulum is exactly the rotation period of the disk. In the moving frame of reference, the particle follows a so-called inertial circle. Note that the particle goes TWICE around this circle in the time of one disk rotation.”

5. N E
Inertial Period -- Minimum at Poles; Maximum at equator

6. r x y u u u u

7. Inertial currents in the North Pacific in October 1987 (days 275–300) measured by holey-sock drifting buoy drogued at a depth of 15m. Positions were observed 10–12 times per day by the Argos system on NOAA polar-orbiting weather satellites and interpolated to positions every three hours. The largest currents were generated by a storm on day 277. Note these are not individual eddies. The entire surface is rotating. A drogue placed anywhere in the region would have the same circular motion. From van Meurs (1998).

8. Frictionless Motion
In addition assume steady state (no local accelerations)

9. GEOSTROPHIC MOTION
light
heavy

10. H L
need to determine this pressure gradient

11. This approach yields one estimate of the horizontal flow? ? ?

12. Diagram of Barotropic Conditions
isobars
isopycnals
increasing depth and density
Diagram of Barotropic Conditions
Geopotential
Surface (surface
where gravity is ┴
everywhere)
If is parallel to isobars then there is no motion

13. z
Level of no motion
isobar parallel to
geopotential 1 2

14. Sea Surface Elevations (slopes can be derived)

15. Strongest currents at location of strongest gradients

16. At Kuroshio, the sea level difference is ~ 1 m in 100 km, i.e.,

17. Diagram of Baroclinic Conditions
geopotential
surface
isobars
isopycnals
increasing depth and density
Diagram of Baroclinic Conditions
How can we express the pressure gradient?
Baroclinic presure gradient (from p = p[ρ, z])

18. isobars
isopycnals
increasing depth and density
geopotential
surface
Diagram of Baroclinic Conditions
How can we express the pressure gradient?
Definitions:
Geopotential distance – distance between two isobaric surfaces located at z1 and z2
Work required to raise a water parcel of mass M by a distance dz against gravity is Mg dz
The change of geopotential is the potential energy/ unit mass gained by the parcel, or

19. geopotential anomaly We need to determine
between z1 and z2 to calculate baroclinic pressure gradients
Integrating
But,
Geopotential distance between z1 and z2 where pressures are p1 and p2
standard geopotential distance
geopotential anomaly
Units of geopotential distance:
m2 / s2 or energy/unit mass

20. “Dynamic Topography” refers to “Geopotential Distances”

21. z3 z1 z4 z2 U1 U2 P1 P2 A A1 A2 B B1 B2 C1 C2

22. Geopotential anomaliesz3 z1 z4 z2 U1 U2 P1 P2 A A1 A2 B B1 B2 C1 C2
Geopotential anomalies

23. T S 1000 m is the LNM; remember that 1 db = 104 Pa A B
Station A: 41°55’N, 50°09’W
m2/s2
Station B: 41°28’N, 50°09’W
m2/s2
Depth
T°C S
(m)
kg/m3
m3/kg
X10-8
104 Pa
5.99
33.71
26.53
148
6.638 25
6.00
33.78
26.61
144
146
0.365
6.273 50
10.30
34.86
26.81
126
135
0.338
5.935 75
34.88
26.83
125
0.315
5.620
100
10.10
34.92
26.89
119
122
0.305
5.315
150
10.25
35.17
27.06
104
112
0.560
4.755
200
8.85
35.03
27.19 93 99
0.455
4.300
300
6.85
34.93
27.41 73 83
0.830
3.470
400
5.55
27.58 57 65
0.650
2.820
600
4.55
34.95
27.71 46 52
1.040
1.780
800
4.25
27.74 45
0.900
0.880
1000
3.90
27.78 43 44
Depth
T°C S
(m)
kg/m3
m3/kg
X10-8
104 Pa
13.04
35.62
26.88
118
7.894 25
13.09
35.63
119
0.298
7.596 50
13.07
7.298 75
13.05
35.64
26.89
7.000
100
121
120
0.300
6.700
150
13.00
35.61
122
0.610
6.090
200
12.65
35.54
26.90
5.480
300
11.30
35.36
27.02
112
117
1.170
4.310
400
8.30
35.09
27.32 83 98
0.980
3.330
600
5.20
34.93
27.61 57 70
1.400
1.930
800
4.20
34.92
27.73 46 52
1.030
0.900
1000
34.97
27.77 44 45
1000 m is the LNM; remember that 1 db = 104 Pa T S A B
Depth (m)
Depth (m)
Depth (m)

24. = 27’ or 27 nautical miles apart
Geopotential anomalies and relative velocities between A and B
Depth
Vrel
(m)
m2/s2
m/s
7.894
6.638
1.256
0.26 25
7.596
6.273
1.323
0.27 50
7.298
5.935
1.363
0.28 75
7.000
5.620
1.380
0.29
100
6.700
5.315
1.385
150
6.090
4.755
1.335
200
5.480
4.300
1.180
0.24
300
4.310
3.470
0.840
0.17
400
3.330
2.820
0.510
0.11
600
1.930
1.780
0.150
0.03
800
0.900
0.880
0.020
0.005
1000
= 27’ or 27 nautical miles apart
27 n.m. x 1852 m/n.m. =50,000 m
f = 9.7x10-5 s-1
Vrel (m/s)
Depth (m)
27.77
27.78
1000 B A
800
600
400
300
200
27.74
27.71
27.58
27.41
27.19
26.56
27.73
27.61
27.32
27.02
26.90
26.88

25. GULF STREAM EXAMPLE

27. Equatorial Current System
SEC
SECC UC
Equatorial
Current
System
NEC
NECC
SEC

29. THERMAL WIND EQUATIONS
Alternate form of the geostrophic equations:
relating horizontal density gradients to flow vertical shears.
Differentiating both components with respect to z:
Using hydrostatics:
THERMAL WIND EQUATIONS

30. Inner shelf data in the South Atlantic Bight
(Blanton et al, 1981, J. Phys. Oceanogr.,11(12), )