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Benchmark: Ekman motion

Acknowledgments

Problem Statement Away from the equator, a steady wind blowing over an ocean leads to ocean velocities that are not parallel to the direction of the wind. Ekman motion describes the expected behavior, under the following assumptions:

• Steady wind, blowing over an infinitely deep and wide ocean, with constant density.
• Motion based on the balance of friction (wind stress and vertical eddy diffusivity) and Coriolis force.

Distinctive features of Ekman motion, useful to test the implementation of wind stresses in circulation models, include:

• The speed of the current is at the surface approximately 1-3% of the wind magnitude (depending on latitude and vertical eddy viscosity), and decreases exponentially with depth. The surface current flows at 45° to the right (left) of the wind direction in the northern (southern) hemisphere, and the deviation increases over depth in a spiral pattern known as the Ekman spiral.
• At the Ekman depth or depth of frictional influence of the wind, D_E, the direction of the wind-driven current is directly opposite to its direction at the surface, and the magnitude is approximately 4% of the surface magnitude.
• The magnitude of the depth mean current is at a right angle to the wind direction.
• The Ekman transport represents the total volume of water transported at right angles to the wind direction per unit time.

Benchmark set-up Overview: We consider a rectangular (100x100 km² square, centered at [x=0,y=0]), flat-bottom (50m depth) ocean, and apply a steady, uniform wind 30 degrees off the north. The latitude is roughly at 45N, which translates to a Coriolis parameter of 10^-4 rad/s.Domain discretization: Uniform discretization in the horizontal, characterized by x=y=2000m (square elements). Vertical grid resolves the surface Ekman layer more than the deeper depth.Time: Simulation is conducted for 4 days, with dt=5min.Boundary conditions: A uniform water level (MSL) is imposed at all domain boundaries.Forcing: The wind is 30 degrees off the Y axis. Processes and parameters

• Vertical eddy viscosity is uniform over depth (10^-4m²/s).
• The Coriolis factor is uniform (10^-4rad/s).
• Density of seawater (\rho=1025 kg/m³)
• Not included: bottom friction; horizontal diffusion; advection.

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ELCIRC Benchmark Input files

param.in   Input parameters hgrid.gr3   Horizontal grid adv.gr3   Advection flags vgrid.in   Vertical grid wind.th   Wind Time History Elevation results (binary) Horizontal velocity results (binary) Vertical velocity results (binary)

Results

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References
The Open University, 1998. Ocean Circulation, pp.36-38.
Pond,S. and G.G. Pickard, 1989. Introductory Dynamical Oceanography, Butterworth-Heinmann, pp. 106-109.