Boundary Layer under Near-Inertial Internal Waves over a Critically Sloping Bottom

View More View Less
  • 1 Exxon Production Research Company, P.O. Box 2189, Houston, TX 77001
© Get Permissions Rent on DeepDyve
Restricted access

Abstract

Internal waves reflecting off sloping bottoms have been shown to have boundary-layer scales proportional to ν½ (ν is viscosity). As the characteristic slope of the wave approaches the slope of the bottom, the boundary-layer scale increases without limit while the reflected internal wave scale decreases toward zero. When the boundary-layer scale becomes comparable to the internal wave scale, this boundary-layer approximation is no longer valid.

This problem is resolved here by using a different boundary-layer balance in the equations of motion for the critical case. The proper balance becomes apparent after rotating the coordinates of the equations of motion so that the horizontal axis is parallel to the bottom: from this balance arise two boundary layers with thickness of order ν. These solutions may have large amplitudes which cancel at the bottom and decay to zero well above the bottom but which allow strong intensification of motions within a layer thickness of the bottom.

Assuming ν= 1 cm2 s−1, a solution appropriate for the New England continental shelf (where N= cph, slope f = 0.054 cph, slope 103, depth = 85 m) exhibits a 25 m thick boundary layer with amplitudes approximately four times the typical amplitude of inertial motions in the center of the water column. Observations made in this region are consistent with this solution.

Abstract

Internal waves reflecting off sloping bottoms have been shown to have boundary-layer scales proportional to ν½ (ν is viscosity). As the characteristic slope of the wave approaches the slope of the bottom, the boundary-layer scale increases without limit while the reflected internal wave scale decreases toward zero. When the boundary-layer scale becomes comparable to the internal wave scale, this boundary-layer approximation is no longer valid.

This problem is resolved here by using a different boundary-layer balance in the equations of motion for the critical case. The proper balance becomes apparent after rotating the coordinates of the equations of motion so that the horizontal axis is parallel to the bottom: from this balance arise two boundary layers with thickness of order ν. These solutions may have large amplitudes which cancel at the bottom and decay to zero well above the bottom but which allow strong intensification of motions within a layer thickness of the bottom.

Assuming ν= 1 cm2 s−1, a solution appropriate for the New England continental shelf (where N= cph, slope f = 0.054 cph, slope 103, depth = 85 m) exhibits a 25 m thick boundary layer with amplitudes approximately four times the typical amplitude of inertial motions in the center of the water column. Observations made in this region are consistent with this solution.

Save