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

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)

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

DOUBLE DIFFUSION S1, T1 S2, T2 S1 > S2 T1 > T2 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)

S1, T1 S2, T2 S2 > S1 T2 > T1 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)

What will determine whether these waves become unstable? Richardson Number

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)