Scaling Laws for Residual Scaling Laws for Residual Flows and Cross-shelf Flows and Cross-shelf

Exchange at an Isolated Exchange at an Isolated Submarine CanyonSubmarine Canyon

Dale Haidvogel, IMCS, Rutgers University

Don Boyer, Arizona State University

Gordon Research Conference, 2003

With support from: NSF, ONR, Coriolis Lab

Premise and ApproachPremise 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 SystemThe 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, 2000. JAOT, 18, 235-255.

Boyer, D. L., D. B. Haidvogel and N. Perenne, 2003. JPO, submitted.

Haidvogel, D. B., 2003. 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 questionsThe 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 Numberf

Rot0

Rossby Number

fW

uRo 0

Burger Number 22

22

Wf

hNBu D

Ekman Number Sfh

Ek

GeometricalW

L

h

h

W

h

D

SD ,,.

Non-dimensional parameters

Parameter values (central case)

The Numerical Model : The Numerical Model : Spectral Element Ocean Spectral Element Ocean

ModelModel

Hydrostatic primitive equations

Unstructured quadrilateral grid

High-order interpolation (7th-order)

(Essentially) zero implicit smoothing

Terrain-following vertical coordinate(but via isoparametric mapping)

0.90 m

-0.90 m

Elemental partition

Isobaths (CI = 1 cm)

(Each element contains an 8x8 grid of “points”.)

Vertical partition: 25 points (8 @ 4)

Time step ~ 1 ms

Grid spacing ~ 2 mm

Experimental procedureExperimental 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

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)

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 vorticity Time-mean divergence

Laboratory

Numerical

(stress law)

Numerical

(no-slip)

Scaling Laws for Residual Flows

• Conservation of Vorticity

• Conservation of Energy

• Ekman layer dynamics

*

1/ 2~d

z Roz

h Bu

DSS h

Bu

Roh

f

h

f

2/1

~

LuBuRo

Ro

h

h

tS

D02/1

~

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

WLED B2/1~

2/12/1

0

1

)/(~

EBuRohh

Ro

u

U

tDS

2/1

2/30

0

1

Nfu

u

U

WUB /~ 1

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

0

0.1

0.2

0.3

0 5 10 15

0

1

u

U

L1

L2L3

L4N1

N2

N3

N4

N6

N7N9

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 = (0.9 + 12.7) 10-2.

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

What can we do to make progress?

• 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

• Since we do not know the answer, we try a minimalist dynamical explanation (aka, guesswork) and hope for a

lucky break

Lucky Break!!!

SummarySummary

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.

Rotating tank at the Coriolis Laboratory Grenoble, France

Tank is 13m (43 ft) in diameter