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Optical altimetry: a new method for observing rotating fluids with applications to Rossby and inertial waves on a polar beta-plane

Published online by Cambridge University Press:  23 January 2007

P. B. RHINES ,
E. G. LINDAHL  and
A. J. MENDEZ
P. B. RHINES
Affiliation:
University of Washington, Seattle, WA 98195, USA
E. G. LINDAHL
Affiliation:
University of Washington, Seattle, WA 98195, USA
A. J. MENDEZ
Affiliation:
University of Washington, Seattle, WA 98195, USA
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Abstract

The entire free-surface elevation field of a rotating fluid in the laboratory can be imaged and analysed, by using it as a parabolic Newtonian telescope mirror. This ‘optical altimetry’ readily achieves a precision of better than 1 μm of surface elevation. The surface topography corresponds to the pressure field just beneath the surface. It is the streamfunction for the geostrophic hydrostatic circulation, which can be resolved to better than 0.1 mm s−1. Still and animated images thus produced, of the entire surface elevation field, are of value in themselves, and using a projected image (a speckle pattern), have the promise of providing quantitative slope and height field data recovered by PIV (particle imaging velocimetry) techniques. With homogeneous fluid, geostrophic flow is the same at all depths. Yet of equal interest are sheared stratified rotating flows where the surface pressure is associated with inertial waves, convection, and other motions, geostrophic or ageostrophic.

Although the technique is designed for experiments in which Coriolis effects are strong, it is possible to use reflective imaging for flows at such high Rossby number that Coriolis effects are negligible, and hence this becomes a tool of more general interest in non-rotating fluid dynamics (for example, illuminating surface gravity waves).

Examples are given, involving (i) the Taylor–Proudman effect with very slow flows over topography; (ii) quasi-geostrophic and inertial-wave flows over a mountain (f-plane); (iii) inertial waves generated by oscillatory forcing; (iv) Kelvin waves (v) free oscillatory Rossby waves on a polar β-plane; and (vi) stationary waves, blocking, jets and wakes with β-plane zonal flow past a mountain. Movies are available with the online version of the paper.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 572 , February 2007 , pp. 389 - 412
Copyright
Copyright © Cambridge University Press 2007

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References

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Rhines et al. supplementary movie

Animation of Figure 6: by peridiocally varying the rotation rate of the system the fluid oscillates back and forth over two circular obstacles, generating inertial waves, Rossby waves, lee vortices and Taylor columns. Time is compressed by a factor of approximately 10 in this and succeeding videos.

Download Rhines et al. supplementary movie(Video)
Video 6.6 MB

Rhines et al. supplementary movie

Animation of Figure 6: by peridiocally varying the rotation rate of the system the fluid oscillates back and forth over two circular obstacles, generating inertial waves, Rossby waves, lee vortices and Taylor columns. Time is compressed by a factor of approximately 10 in this and succeeding videos.

Download Rhines et al. supplementary movie(Video)
Video 3.3 MB

Rhines et al. supplementary movie

Animation of Figure 11: Kelvin waves and inertial waves generated by high-frequemcy oscillation of the table rotation with a shallow fluid layer (H = 5 cm). rate, in the presence of a small (1 cm high) spherical-cap mountain near 6 o'clock. The Kelvin waves propagate with the speed of long gravity waves anticlockwise about the cylinder, trapped along the outer boundary. Note the inertial wave crests moving opposite to the group velocity, hence toward their source. The circular crests are the inertial wave reflected from the circular boundary, whose energy propagates inward toward the center. There are also visible small, cyclonic eddies left of the forcing region which are shed from the oscillating mountain. At larger amplitude these become a dominant part of the response.

Download Rhines et al. supplementary movie(Video)
Video 2.5 MB

Rhines et al. supplementary movie

Animation of Figure 11: Kelvin waves and inertial waves generated by high-frequemcy oscillation of the table rotation with a shallow fluid layer (H = 5 cm). rate, in the presence of a small (1 cm high) spherical-cap mountain near 6 o'clock. The Kelvin waves propagate with the speed of long gravity waves anticlockwise about the cylinder, trapped along the outer boundary. Note the inertial wave crests moving opposite to the group velocity, hence toward their source. The circular crests are the inertial wave reflected from the circular boundary, whose energy propagates inward toward the center. There are also visible small, cyclonic eddies left of the forcing region which are shed from the oscillating mountain. At larger amplitude these become a dominant part of the response.

Download Rhines et al. supplementary movie(Video)
Video 1 MB

Rhines et al. supplementary movie

Animation of Figure 13: Monochrome imaging of surface elevation of a rotating fluid, using optical altimetery (full colour altimetry with quantitative velocity, vorticity and potential vorticity has now been achieved, Afanasyev et al., 2007). Rossby waves on a polar beta-plane generated by periodically oscillating the table rotation rate with a single frequency, in the presence of a spherical cap mountain at 8 o'clock. Note the short Rossby waves to the east, and long Rossby waves to the west (clockwise), which spiral into the North Pole, and fleeting, very short inertial waves to the west, yet not to the east. A dark radial line (the image of a wire near the light source) distorts, showing the azimuthal slope of the free surface (and hence the radial component of geostrophic velocity). Time compression: 12x. Omega = 2.23 -1; oscillation frequency = 0.16 s-1. At smaller amplitude this experiment generates waves of this same frequency; however at the present amplitude we are making time-dependent lee waves to the east of the mountain (the short waves), with a spectrum of intrinsic frequencies, while the long waves west of the mountain are in fact nearly a sharp response at the forcing frequency.

Download Rhines et al. supplementary movie(Video)
Video 5.2 MB

Rhines et al. supplementary movie

Animation of Figure 13: Monochrome imaging of surface elevation of a rotating fluid, using optical altimetery (full colour altimetry with quantitative velocity, vorticity and potential vorticity has now been achieved, Afanasyev et al., 2007). Rossby waves on a polar beta-plane generated by periodically oscillating the table rotation rate with a single frequency, in the presence of a spherical cap mountain at 8 o'clock. Note the short Rossby waves to the east, and long Rossby waves to the west (clockwise), which spiral into the North Pole, and fleeting, very short inertial waves to the west, yet not to the east. A dark radial line (the image of a wire near the light source) distorts, showing the azimuthal slope of the free surface (and hence the radial component of geostrophic velocity). Time compression: 12x. Omega = 2.23 -1; oscillation frequency = 0.16 s-1. At smaller amplitude this experiment generates waves of this same frequency; however at the present amplitude we are making time-dependent lee waves to the east of the mountain (the short waves), with a spectrum of intrinsic frequencies, while the long waves west of the mountain are in fact nearly a sharp response at the forcing frequency.

Download Rhines et al. supplementary movie(Video)
Video 2.5 MB

Rhines et al. supplementary movie

Animation of Figure 14: eastward (anticlockwise) flow over a spherical cap mountain at 2 o'clock. Development of lee Rossby waves, jet-like concentration of flow, spiral flow above the mountain (an arrested topographic Rossby wave), and an upstream (eastward, clockwise) blocking of the circulation. A growing cyclonic wake involving eastward counter-flow forms behind the mountain. The table rotation is steadily ramped downward to maintain the zonal flow, which accounts for the gradually change of the lighting, as the parabolic free surface relaxes. We have left the glass top from the experiment to promote evaporative convection, which produces the mottled pattern of tiny convective cyclones upstream of the mountain. Note how these show the larger scale flow field. In regions of strong shear we see instead convective sheets (tall 'rolls') which thus outline regions of strong shear. The fluid surface deformation is of order 1 micron (10-6 m. in these features, and is of order 1 mm. in the most intense flows.

Download Rhines et al. supplementary movie(Video)
Video 919 KB

Rhines et al. supplementary movie

Animation of Figure 14: eastward (anticlockwise) flow over a spherical cap mountain at 2 o'clock. Development of lee Rossby waves, jet-like concentration of flow, spiral flow above the mountain (an arrested topographic Rossby wave), and an upstream (eastward, clockwise) blocking of the circulation. A growing cyclonic wake involving eastward counter-flow forms behind the mountain. The table rotation is steadily ramped downward to maintain the zonal flow, which accounts for the gradually change of the lighting, as the parabolic free surface relaxes. We have left the glass top from the experiment to promote evaporative convection, which produces the mottled pattern of tiny convective cyclones upstream of the mountain. Note how these show the larger scale flow field. In regions of strong shear we see instead convective sheets (tall 'rolls') which thus outline regions of strong shear. The fluid surface deformation is of order 1 micron (10-6 m. in these features, and is of order 1 mm. in the most intense flows.

Download Rhines et al. supplementary movie(Video)
Video 1 MB