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SCALING AND NON-DIMENSIONAL NUMBERS

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SCALING AND NON-DIMENSIONAL NUMBERS

Scaling with:

For example: ratio of Inertia to Rotation

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

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

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.

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)

STABILITY

< 0

[m-1]

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

S1 > S2

S1, T1

T1 > T2

S2, T2

DOUBLE DIFFUSION

Salt Fingers

Salt Fingers Experiment

http://www.phys.ocean.dal.ca/programs/doubdiff/labdemos.html

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, 99-109)

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, 99-109)

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

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

S2 > S1

S1, T1

T2 > T1

S2, T2

Layering

heat flux

from below

Layering Experiment

http://www.phys.ocean.dal.ca/programs/doubdiff/labdemos.html

Data from the Arctic

From Kelley et al. (2002, The Diffusive Regime of Double-Diffusive Convection)

SHEARED FLOW AND STRATIFICATION

Click on image to see animation

May cause instabilities like the one above (Kelvin-Helmholtz)

Richardson

Number

What will determine whether these waves become unstable?

Overall Richardson Number

Ri < 0.25 necessary condition for instabilities to develop

(0.30 from observations in natural environments)