Scaling Laws for Residual Flows and Cross-shelf Exchange at an Isolated Submarine Canyon

1. Scaling Laws for Residual Flows and Cross-shelf Exchange at an Isolated Submarine Canyon Dale Haidvogel, IMCS, Rutgers University Don Boyer, Arizona State University With support from: NSF, ONR, Coriolis Lab Gordon Research Conference, 2003

2. Premise and Approach Laboratory datasets complement datasets obtained in the ocean, and are a valuable resource for model testing and validation, and the study of fundamental processes. The approach therefore is the combined application of laboratory and numerical models to idealized, but representative, processes in the coastal ocean.

3. The Physical System The physical system considered is the interaction of an oscillatory, along-slope, barotropic current with an isolated coastal canyon. The flows here are considered to be laminar; however, a subsequent study is underway to consider the effects of boundary layer turbulence. References Perenne, N., D. B. Haidvogel and D. L. Boyer, 2000. JAOT, 18, 235-255. Boyer, D. L., D. B. Haidvogel and N. Perenne, 2003. JPO, submitted. Haidvogel, D. B., 2003. JPO, submitted.

4. The questions What processes and parameters control residual circulation and cross-shelf exchange at an isolated submarine canyon? Do the laboratory and numerical datasets complement each other (e.g., corroborate each other and tell the same dynamical story)?

5. The Laboratory Model

6. Non-dimensional parameters Temporal Rossby Number Rossby Number Burger Number Ekman Number . Geometrical

7. Parameter values (central case)

9. The Numerical Model : Spectral Element Ocean Model • Hydrostatic primitive equations • Unstructured quadrilateral grid • High-order interpolation (7th-order) • (Essentially) zero implicit smoothing • Terrain-following vertical coordinate (but via isoparametric mapping)

10. 0.90 m Elemental partition (Each element contains an 8x8 grid of “points”.) -0.90 m Grid spacing ~ 2 mm Vertical partition: 25 points (8 @ 4) Isobaths (CI = 1 cm) Time step ~ 1 ms

11. Experimental procedure • Spin up for 10 forcing periods (240 seconds) • Run an additional two forcing periods collecting snapshots at 1/20th period • Post-process time series for: - residual circulation (Lab/Numerical) - residual vorticity and divergence (L/N) - on-shelf transport of dense water (N only) - mean and eddy density fluxes (N only) - mean energy budget • Repeat for parameter variations

12. Density at the shelf-break level (first two periods) Colors show the density of water just above the continental shelf break (red: lighter, blue: heavier, grey: unchanged from initial)

13. (a) (b) (c) Figure 2: Vorticity (left) and horizontal divergence (right) fields for the central experiment discussed by PHB (Experiment 01 in the present study) as obtained from (a) the laboratory, (b) the SEOM model using a parameterized shear stress condition along the model floor and (c) the SEOM model using a no-slip condition, including a highly resolved Ekman layer, along the model floor. Time-mean vorticity Time-mean divergence Laboratory Numerical (stress law) Numerical (no-slip)

14. Scaling Laws for Residual Flows • Conservation of Vorticity • Conservation of Energy • Ekman layer dynamics

15. Water parcel passing over canyon rim has a natural vertical length scale set by the depth change it would take to convert KE to PE Conservation of potential vorticity assuming that water column stretches by an amount proportional to this vertical distance Solve for relative vorticity of a parcel, and integrate over the length of the canyon and over a forcing cycle to get total vorticity

16. Equate gain of cyclonic vorticity over a forcing cycle to the loss expected in a laminar Ekman bottom boundary layer Solve for ratio of residual flow strength to magnitude of oscillatory current The equivalent expression in dimensional form

17. N2 L3 L2 N7 N11 N9 N6 N3 N1 L4 N10 L1 N4 Characteristic speed of the normalized time-mean flow at the shelf break level as obtained from the laboratory experiments and the numerical model against the scaling relation . The symbols near the data points correspond to either laboratory (L) or numerical (N) experiments The dashed line is the best fit = (0.9 + 12.7) 10-2.

18. Scaling Laws for Cross-shelf Transport • Linear viscous arguments do not suffice to give a scaling for cross-shelf transport of dense water • - role of advection • - independent roles of mean and eddies • The association of on-shelf pumping with a local increase in potential energy suggests an energy approach

20. What can we do to make progress? • Since we do not know the answer, we try a minimalist dynamical explanation (aka, guesswork) and hope for a lucky break • Let’s assume: • PE gained by cross-shelf transport is proportional to KE in the incident oscillatory current • PE gained is independent of stratification • Conclude: cross-shelf transport is proportional to the square of Ro, inversely proportional to Ro_t, and independent of Bu

21. Lucky Break!!!

24. Summary Scalings are proposed for residual circulation and cross-shelf transport of dense water • The numerical and laboratory models are consistent (e.g., produce the same scaling for the residual flows) • Complicating issues: relationship of laboratory analogue to the “real ocean”; omission of (e.g.) small-scale topographic roughness, multiple canyons, boundary layer turbulence, etc.

28. Rotating tank at the Coriolis Laboratory Grenoble, France Tank is 13m (43 ft) in diameter