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SCALING AND NON-DIMENSIONAL NUMBERS

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”.

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SCALING AND NON-DIMENSIONAL NUMBERS

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  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. 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)

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

  8. Salt Fingers Experiment http://www.phys.ocean.dal.ca/programs/doubdiff/labdemos.html

  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, 99-109)

  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, 99-109)

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

  14. heat flux from below Layering Experiment http://www.phys.ocean.dal.ca/programs/doubdiff/labdemos.html

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

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

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