Scaling Laws for Residual Flows and Cross-shelf Exchange at an Isolated Submarine Canyon Dale Haidvogel, IMCS, Rutgers University Don Boyer, Arizona State University Terrain Following Ocean Models Meeting, 2003 With support from: NSF, ONR, Coriolis Lab

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.

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

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)?

The Laboratory Model

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

Parameter values (central case)

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

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

Experimental procedure  Spin up for 10 forcing periods (240 seconds)  Run an additional two forcing periods collecting snapshots at 1/20 th 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

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. (a) (b) (c) Time-mean vorticityTime-mean divergence Laboratory Numerical (stress law) Numerical (no-slip)

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)

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

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

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 terms of dimensional parameters

L1 L2 L3 L4 N1 N2 N3 N4 N6 N7 N9 N10 N11 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 = ( ) 

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

Try simplest assumptions 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

Summary and Next Steps  Experiments support simple scaling laws for residual currents and cross-shelf pumping  The numerical and laboratory models are consistent  Companion simulations in the 13 meter tank at the Coriolis Lab are complete (Boyer et al., 2003, in prep)  Numerical experiments in a 13-meter annulus are underway (~50 million gridpoints)  Non-hydrostatic version of SEOM has been developed, and is being tested