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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)**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)