1. SCALING AND NON-DIMENSIONAL NUMBERS
Scaling with:
For example: ratio of Inertia to Rotation

2. For example: ratio of Inertia to Rotation
For Ro << 1, e.g., Ro ~ 0.01, inertial accelerations are negligible and the motion is “linear”
Example: U = 0.1 m/s, f = 10-4 s-1, L = 10 km

3. Ratio of Friction to Rotation
For Ev << 1, e.g., Ev ~ 0.01, frictional effects are negligible and the motion is dominated by Coriolis accelerations
Example: Ax = 103 m2/s, f = 10-4 s-1, L = 10 km
Example: Az = 10-3 m2/s, f = 10-4 s-1, H = 10 m

4. H L U 100 105 10-1 1 10 104 U/t U2/L fU AxU/L2 AzU/H2 10-5 10-7 ? 10-8
Scaling is very important to help us diagnose the relevant forces driving the flow in a given area
(horizontal momentum).
Local Inertial Coriolis Pres. Grad Hor. Fric. Ver. Fric. H L U
100
105
10-1 1 10
104
U/t
U2/L fU
AxU/L2
AzU/H2
10-5
10-7 ?
10-8
10-4
10-6
t = 12 h ~ 104 s ; Ax = 103 m2/s; Az = 10-2 m2/s
For vertical momentum the concern is with the stability of the water column (density distribution with depth)

5. STABILITY
< 0
[m-1]

6. Perturbations to the pycnocline (region of maximum stability) cause oscillations.
The frequency of the oscillations (radians / s) is given by:
Buoyancy Frequency or Brunt-Väisälä Frequency
A stable water column does not necessarily represent zero vertical exchange of properties

7. DOUBLE DIFFUSION
S1, T1
S2, T2
S1 > S2
T1 > T2
Salt Fingers

8. Salt Fingers Experiment

9. Example of Salt Fingers (Kuroshio waters interacting with waters from Sea of Japan – through Tsugaru Strait)
AIST Japan
From Miyake et al. (1995, Journal of Oceanogr., 51, )

10. Requirements for Salt Fingers: a) dS/dz > 0 dT/dz > 0
b) Small density ratios
c) Staircase in profiles
From Miyake et al. (1995, Journal of Oceanogr., 51, )

11. From Miyake et al. (1995, Journal of Oceanogr., 51, 99-109)

12. From Miyake et al. (1995, Journal of Oceanogr., 51, 99-109)

13. S1, T1
S2, T2
S2 > S1
T2 > T1
Layering

14. heat flux
from below
Layering Experiment

15. Data from the Arctic
From Kelley et al. (2002, The Diffusive Regime of Double-Diffusive Convection)

16. SHEARED FLOW AND STRATIFICATION
Click on image to see animation
May cause instabilities like the one above (Kelvin-Helmholtz)

17. What will determine whether these waves become unstable?
Richardson
Number

18. Ri < 0.25 necessary condition for instabilities to develop
Overall Richardson Number
Ri < 0.25 necessary condition for instabilities to develop
(0.30 from observations in natural environments)